Content-Type: multipart/mixed; boundary="-------------0604301644697" This is a multi-part message in MIME format. ---------------0604301644697 Content-Type: text/plain; name="06-140.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-140.keywords" QED, Infrared Problem, Groundstate, Expansion, Algorithm ---------------0604301644697 Content-Type: application/postscript; name="BFP.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BFP.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: BFP.dvi %%Pages: 54 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMSY10 CMBX10 CMSY7 CMSL12 CMBX12 %%+ CMMI12 CMMI8 CMR8 MSBM10 CMTI12 CMSY8 CMEX10 EUFM10 CMR6 CMSY6 CMMI6 %%+ CMBXTI10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o BFP.ps BFP.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.04.30:2310 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: f7b6d320.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: bbad153f.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 % /TeXbbad153fEncoding [ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: aae443f0.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: 74afc74c.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmbxti10 cmff10 cmfi10 cmfib8 cmti10 cmti12 cmti7 cmti8cmti9 cmu10 % /TeX74afc74cEncoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /sterling /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -250 1274 754}readonly def /UniqueID 5000771 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF6BA1ABE5F0F80D96571277EAF86 A3AAFCE3693728E396048C32573DAA11E6D2F036BD9756F1F0A5877DD15DC64A A9C2E20B570464B31E783692E6D10101828C4F6DF30A155808A7697D24C5854D DD1D111B4B4FE343E045B136706A49E8AE009DC1FB981436DAD464861228DE37 013FFDA371B4DC0BFA7E5D606C9716A66221968F9BC406148F4027AEA6A9DF6B 73B120707CD710D9CF211A1350E1BEEA0D934BED2C0B9092C1264EB98D81C625 4391E134B3A487614CE48710F60DBF482372B2F1B2F0ABF06D6E78D7A18074A8 FD18AF17FC89A5B9DB3DA8FEDFDBE78002D31CA796726FF8870BA31625C90F5A EB380DF50BABF49ABC107690566ECD3AFE5A2D47B79D1FA991F1DCA454619CF7 006240AA135984D97F7EE22C346A97AECECC118DFD8E3D97599B0BAF33D949FE CE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 104 /h put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 1458C43CD471029C362871D2FB69E0E5E617BC06F3B8621E3528E4B47E731AE1 0B406A3260424830B3C86A135457A566A1A716189B86883C171DA05829CE2E37 C020AAAD5205DB1A9C1A6A56FFD9008D511B4453C4B33F6B0E977B7C760F8816 4A6F13E41191AA13358515D902E653B81AEF60FE07C4AB673435504C2C376BB2 6BAB038083E5073851178BEB6AA92FE582EFCFA3A22C6795DCAAFAFCA0F3E79D 270BB779E2071AFDF30D3F32082777AEDF90940B6DAA1F34BAC0E9AA08811281 8D4261185EC039EB930024E010615A972BBD0B8EAA09B3931D630F1215D88859 5803A82F87BF0D6B8C64E23358C903C8E022AF457F97845B948D174E063E40F3 FA1F8C2CF0C7C49EC374E22B0A865A90FDC5CE762D643812D9597AC4CA872A23 9CA95A4001607D1E336A8D3C10BBC8BA47B1DD8DAC76B3F7B4E34DA11B11F27E 5CC1B7D0249F7924540CC5E194DD545B39C060A69CFC2145AFED8242A9FEA65F ED03FC2DA7CDD3ADC394D32F22802C8EC818847BA08DE0654277ED11E5C356A7 1E590527E7A231A6AE8B779B88648B69BE95F879361588325BB6F7E01D44C6A1 8422BACAA4C2B6B4802A4452C7A95F232A5B3B9DE2E78780A95F58FF97C276B4 691C2E88AAD4FF004B752120BB3453C12CC6BEDD34165F33D1DACCE965BF81A8 8518AD8EAAD03CCF67CA8BA8A2B78749D66ED2726BACAE757CB7C3BAE0532181 AC95975182CF3EF7B32636EA7C3ABCFD2BCFD2325DE4E377278A34A790804F1B 27FBDEF77921130ED82D188D97D419AC2486AAB33B073D9EDA1ABF2B2339F56C 02B150C69E56B6C84E1C8590BB610F46CFBA0A500F5819E525B9A5670A4A0711 5C03A06ADE01EB3D493BE2C7CA3E99FF5DC64747E1A00B5A9C2EE1DBBEA277C1 8085FAAEAD3E4B1BE6A5CB9C44F313FE57E1A4B160306608DC6B0ADE968D6B77 EEBCBE6EC3B8D4A563A81FE0847717E57BFB42F67F59151A4365AE4289224B40 50DA9AD3DCD89F9AE7142AA168C2DF2F99A867FC716BA89247A93DA37E12C0CE 6EC7A01A08E9F4E831DCF120063FE0E503F838E6C63C79FC6B54D601E358406B 399C33E2B84C23CF22056D818A90D762B6ADA730EE52813D6AF27159BE1BCC87 D0CBF051906A0B7C09103A91CA2899DF158CB01372910C413E7C6EA070110501 61A232902629AEAABF71BDAB08E58E8FA749B9160209349EBB7AA8BF5456D962 719EF38CBBA23EB3D36AACDEC0D9B2D1C4EE1EE1A182B8D9FD4CD78A6D9AA57E FFDD05DBC6F842D0B7C6ACCF5984645FA9F6C2C75276ABE70ED95FBB17DAA776 D50A8D854B5D49852E21244264B544094C4CC3F727212B1B46806235484AE479 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 100 /hatwidest put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 113 /radicalBig put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 1477F5496111E055C7491002AFF272E46ECC46422F0380D093284870022523FB DA1716CC4F2E2CCAD5F173FCBE6EDDB874AD255CD5E5C0F86214393FCB5F5C20 9C3C2BB5886E36FC3CCC21483C3AC193485A46E9D22BD7201894E4D45ADD9BF1 CC5CF6A5010B5654AC0BE0DA903DB563B13840BA3015F72E51E3BC80156388BA F83C7D393392BCBC227771CDCB976E9330253243205FFB3CA5D356218DA3226D F1A87B8617BE385EEAA2D478CD2AEFD997EF6A65BD61F1FE8A3B540866474819 A36E7BCA910A2D81D2697A7DBF5F46EB6E4730EA7A449B0A958741A1ACF81C26 8382D241EC0279F8AFF1075D481166997B553CA96D07D6C195D2E73B36D9B10A 3774EF028C57D2A841B2B1D15CA61638D3BAB6C2CA871B8CA7459263D95EE5F9 6723EE5E2183F218861548AC492250FA114B4717379EB77F0BEC2D308BACB784 7E1AF01DBAFDB1C9913148DF62853A64C887932400EBBF07E4AEF6DCE1FCAFDD 74666FC63867A9C71D0E7DF70B783C5AA23B0E245273B2028252F63A964A3AE4 064A237C35B2AE4922443E8EA49AE003C48698B64A1C87D43E89841D68842F21 46E52564CA02C3E32C907BF1DDED6E79CDD6D8BDB78BD4F04B868C1D5E520F84 BDCC4F4D73DD392A9308DDE0C5DDFE17710B7CC58A18699B62B7E74C1EDE910E C7C462DE0F61C62852D9AA8E6586A16B5FA35F51F61EF9A95D7C499303AA772F 72A4144BC0A8024A80315FDCDFD9299ACE97150B99B42A92A2B7BDB545C13D96 28935D9239A116F2D393052F2A399068C531D3565106C25EDEEDD76B3481F8F3 F1050DA63D12CCAB88426572AEFF17E453BF6CEEA7FA8F3916E994D9D14B2C99 AE456897FFF2AD873EA0266780CAD014DC4C4CCD7CB83F6D0BC80D514858DA5A D6FA2D0D0F429B52DF0FDD9C656B536267E2E97A91ECABBEF2521BB232690392 8348FF7317C68808A31D4162A7D1FF9DE15CAF237383BFC00BFF380920BC09F0 E61F7962232DB9936319DC60A0CE5E3133651035C66A83BD66FC514CA719BB6A 9C566E51965A841FF139667A5CBEBDBB089688BEB17FEA494AB8E34D7F2FF4E1 0FB6EC918345E14E656CF1886423017680BD42D3EFCDA572D6A10CA3DF15EBCA 13B188CBE05E859EE5C61DB3B13B26173160A8EA885E335A3E2F95594C17D6B3 A577DAB8F139B02B70EB14D7A324932418857853060681F5E00181EC43654528 8440068307C4E1A055305A7C301EA57F078FAA8A5CDF6B88F0A9F3D0DC966BEB E1492DC91AE633CAECBD2E78DA93712F4EF84FD4380A8A2532F0916B3E335E3E FBA97DD8112452077FB05333439E89D9BAB34748AAA952BEAFD761C72E6F6978 625A6E1B63D9A1CC05A1CD894278CAB16B77EF0B11322DCB8287834B1015C4E8 13E7F0BA4B7EC2A3EE158000A9669D03B169E9460E7044321269B69C66C0475D 8A7187E0D636DBD68D92487D913ED833CEF0ADDA6772189282EBFACA9A5CF532 890CBB9F892926FFECA19B8D319F6E5E755809F77B5FD595C8B3A5C34CB0E791 4B6C2F6D2B74D9909C95341C3EBA7856F117B37B23CB7C67F4E7DB1859E6916B 59B53B17B9184804DF7330C083E754C1DCAD9836950CCA647650E6FF127BD7F1 197B203D827A4EA4517132F4D639BF9D2E102D9C11B370E30F60E91C02C2B010 A290022C4275B5B9CB644A7D7537952840516D698155302B3D715CBEE5D4A4CB 44AF6DC8E0F58F6CA352CB4472B76B13A09C71DAF095329B6128550189470CC8 006A0D2ACC69E3341631AB4A648D3286CFFC8E740CB55A450BFA0B2C82A2D949 57C629051B2B281D3571375C83EA2267EA310D43F597D600A3473ABADECE64AE 4EE7DD87DC1A11B947D4A6157B27F71483AA5F61D63AC492F3AFC98A609AB1FD B0C1FD4AE9CD1B20C91B1609AF9BD34DF0CE7C9B0AF5F9188C933ADBA1859554 678F0690ACD730F4CDDF6A733FE1C50BDB2442E37C0EA72B030C1DEBE9411A47 911FB099F00A2B48596DCAFCD3E0BD0B36D6ECEADC8A89F01FB03C886D0AEEF5 25F4B28E8298A3921A67DD1A8525A11E8190CCCCE6293EA32C48B502D05D1742 FBE9A24221B8A4F9F36FBB0CF03B2A5DC8445728412200CDC5371E86F36B1588 96B38BB046E14B4F7B64A6779800FEEF4EFB78582023A61BBF9C5AAB41BDAC73 B7B022A6AF4DAD9A950C6028A5598331CCAC2C4931B1957E6270950E623B06D5 D517018C8F1FEC404D95CFE3A9558CABDE812D8520E0FB29702E8C6093A6E039 3C9C33142801B20140469C1F4FA83142CA3CE307B89016322E7AAB7D0B2B946C 1903D3064E6FCE82D76A725631563CA65715C36ABBCF13BA1730035A78CF861B 32CB5BDC6BAF288F28BC9774479C6162588AB35C6F09360064FEB0D39E54F5D5 CBA644041693CF63DBDFB30346C19BB4A8D1B5F28F4C3EA16994C15D23141B02 875BA3DBBEAA7A214C2819E5A9D213D48477B8135309BE1ECA25F405EE40DD59 E1FF103A35B667256DCB7F3C475B63E729E95CB594A205EFB965C7831A3F87C6 C19A8911696D9F3CCD49AC9E33DDD269C31B2D563A3F01F6FBDB774C91A1B370 0E28DE1B1450D15F1ED673937C6838745BF545857047C1D59C49184C2DB8DB16 C9FE767CED61CBAF87E050BE9382D1A4E7EBA084BF80372D19FC62F0FFFB24A1 B63F209CFAF992C8D5BB62701C9CB44D245AE29DB2FFD6775F15A5AEBAD4A27B 3F337000B8EC8C33DC4B169E78E3E0B7B2E1184283D8E95E45B6D14345C7C346 44AA306BC2F3AB44633DD93F148A6395032F892442713D793B4050934DAED828 17EB10144E040E84ECE832A69806AD3038B7015614BB3F14662BC9E46217D325 4964DF4D83BF6C4D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put dup 91 /hatwide put dup 92 /hatwider put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67 3C15C2153C061EB541F66C11E7EE77D5D77C0B11E1AC55101DA976CCACAB6993 EED1406FBB7FF30EAC9E90B90B2AF4EC7C273CA32F11A5C1426FF641B4A2FB2F 4E68635C93DB835737567FAF8471CBC05078DCD4E40E25A2F4E5AF46C234CF59 2A1CE8F39E1BA1B2A594355637E474167EAD4D97D51AF0A899B44387E1FD933A 323AFDA6BA740534A510B4705C0A15647AFBF3E53A82BF320DD96753639BE49C 2F79A1988863EF977B800C9DB5B42039C23EB86953713F730E03EA22FF7BB2C1 D97D33FD77B1BDCC2A60B12CF7805CFC90C5B914C0F30A673DF9587F93E47CEA 5932DD1930560C4F0D97547BCD805D6D854455B13A4D7382A22F562D7C55041F 0FD294BDAA1834820F894265A667E5C97D95FF152531EF97258F56374502865D A1E7C0C5FB7C6FB7D3C43FEB3431095A59FBF6F61CEC6D6DEE09F4EB0FD70D77 2A8B0A4984C6120293F6B947944BE23259F6EB64303D627353163B6505FC8A60 00681F7A3968B6CBB49E0420A691258F5E7B07B417157803FCBE9B9FB1F80FD8 CA0BD3B53F9FD95670A878A64E913C339F38088BB76CF1458CB383805AB5359E BA3A3537C88B7A8CAB42412D8D645B9ACA6D433E26192F237AF00ABC7BC13DA3 0697FB05CC83DBCA72AA53D2025675BA5E2DF9E2C4A2038DCF160C2BB20B02CF 572199F77731E00587F025D73A3C33B647F7BF6664241320667428AD844C9D84 7A7747AB70245A91B895E0183954C8AE181DAA84C3D9C9816A0421224972DF0E E2C5754DA23A64E2F4DEB41B9F817B62180D4D032ACCE3241BB5123386B20B76 62574215FE4AE0B753C93988B80C82CE110ED22B56E4AEBB76C6C9FF6CB9688E 76ED5AA7BAEE635ED9019232C234505B395851637503A6265C869C783F32C1C0 00C073FEC0DC632A7722E0DD2D0D3C24BBC023B87B629A9D554A5CB2A662E847 B4300B4939DF71131FF56BA1F95A84F3D3754A383A46C4882C2FA79DAC48169C DB62C207FBA7E7D0E5C79707C49D2A572DEA8CE4BF3911AA57A9D9BEDD1EC04A 4A220CC0667E5C7E9B201D8F098E83BAC3D0F8A06F2064B32A5C407F990D6A43 94424263E5C129C8B17366DBDA32B15406381269DB448B0B7107537BCAA8FCF3 F1EB7475E396C85116AA83C77C2D7887B361297AE1D1E38DC82512E4336F1034 C5EC2CDDA3AC6C0C12D644B9FB989674111A6031D6D5F626175A38CF69B61233 6DF07330779615B0516AD1B5C4C56944521E76439B5DCA7732A29F37A088B0F3 16B59E7BB99EBC7521DA5BAEF8AED3B848E232336ABF752EE368229466AA66BD 23250F324FA5E2D3B86064CA5DE1745EC472F5BB10A0ED9661C0D75E1E163F64 471D0CEA2C29FE67C00B1B90631809C9ABE15A5805E82C0E104032942570F4FF 1761DAB95CE810FF598F743D418E1D967FAF871CF10BEBBA6599F53AA8DB3E4A BADA43070EF645143F799030E089548F30389BB0DEDDE150317BB0FEAC009E7F D1F08A75DFD726C4D73758C58747199692E982EAAE763CA422D672DA3DC9B6D6 CBAFD65AE92326A2D7B946AF4BFD5CA1F11D9AFE300853D2369005B25EF62046 E2095D9D1DA810E92F821EFEABBF85257C42A6ED10B6C5BC2A8B7790AD6666C7 91D02DA4F0A66A24119201A19266B8B2C570FE3EAFC23CD1FE07580DB0311AF7 E840C409728B918D94642704A029F3D8C48D67EDE02F807FB99015183B246759 C6FB06D680E44A0305C9C9E265E91AE82EDF30F73730DCD4AE70B7D182471A75 A7B62AFD13AFC1300E9C83D776D71142BC19F44C9DAEE69D0403AB22DD0E3339 94DED33814B014BA54C7084308D5D7C254F3BAA0B918B70DFB2C7F858C15B91F 5CE9174545F7F707FF9227C2C6B26B1C5DEA834311914A0E6FB6D76DCF573F38 7DC49288665EDFD8238F4713042605E9099B987A67FB6DBE1314CFCF47DB4642 B4112F13C7E299EB01A7D14A4AD27EFC228C1C290980F5850FF010C638CE28E2 8878D48025DFBE6CC8C2B6966883781C4D8D8AF288962D9D87E0E73F43DD2E91 4EACF7CC7AD6372D0D9144876458F0704F618C9B3157D65B3BECB7B9D139C05D 5B293877AA38564C4CCA58DC69F7F4BECE47886A72FA8379C1478D920CE7B608 BDEB9AB770287656AE30C2EB29187FDE515FA19840D458BF6441E6063AC1E452 D412FD422408D818BC4E0BC11BAA3C460A0DFCC1EFCB29203E5BFCC9B795AB55 B201726C999ACFE0499446027E761E800A4DCEE1D9194035519D6CD736FA5B8E A0B6B2446C6F91F46C8E847DB689C697D51D7145FA3B7DF29B7B8B7F5BA39877 04E1E9F5343B23D2FE9DCCBAC4E79FCEC5DBFC4716296BDCA37F7329A72E4065 EE86B4916613570F0D9B03B72D78AA401B616998A2480A76F0D65DD39873E560 63C4C423766FACC323F85FD8368049ACF479409DE9D725C5A33DAE184A364BE8 D746B8159D8C19BD8B6D6C747D49D00608DEC3E51925ECC9A514B91F0098023C 8CE078CB0680B988D886E0A11B238368263425B66AE9CE6BEE2BA72451052629 8CDFD901F48965CECD5F8FED5052D9C81A3C53CD903CFBB933A884AE05840B6F 71C9FDF6216B0C0ED3E596E9BF53E7BDDFF198147A5D8539A1CA7D4DEB57D8D2 2EDAC86E30F6B0F8A96BE50577BF0D2BEDA035BF7A3116249F016CF70418CD3B 8DC90F559D3945836EC33C342757C2947687A3BAF1556E7AED5440FBF3A8DC37 882983DD53F4B0C6D0B647688699ABDFAD8C02F992C0A9F612EC7C8618E411EA 4C055DC07629BBD7AB37D05DD2551BCEE6C7E5EDB84891AD477CA4C1C62423E1 368A02473B595DB25189654069CC086A0582F2D4447F8A45456FCBA4A465E4D1 61EFD5861E56E0E44C0B50162CB1B13C898833FFECA6834B21BE10BE1D469F8E 709012C0688BCA95D03980A22D36F0109AB75A788F02B7731095AA84F493BAC4 A2E1BCA79D6F0973378B61913280137E02A370E825CD31A45568546BD5FA7E53 C3106175C8221A0B870A9627A031FD2DD144F4E8EFFD39899E5C324EF17A5B1F 7070AE0B7D62F585B5590628EB5DDBBE5F528AC53F54C842E7C410E08DB24175 6A96EFEE8F770E18356068A4D5C52C34A028F55D2BDA969FEB099536C59F6E7A 85245688DA37D85F93DB2E2561ADFBC0853E2F848E48C52C240085E7F03EC3C4 14E4FCB6C8182C2FA364F52A89A31A1DEFA711B5ED82984E61E0646E3DADEFB4 4C05FA865F825A4684F3B0B67D4BAB7B43272BCB1809121D4C3A1C7E925DB7EF 97958651BB7B65C187D7B2BA320D540F4215BD040A3D77AD878E391800013F91 649A0CEF26DEB2CBC730904B9C7C7A6D2E70E687422F4FE67FD4065E7E2309FE C70BFB88ACA8CD65E4C65AB07A9C82B9200B248A0149F3A7A9B46057E0966052 6386C5EC66A88CB411A95D12BF5705F7D8993B0258EC3F38B91ABC60D18B61D6 5BA3DC98D40F660C096DBEF74535B1E337A8A42667FCC46375734BB05FB7B17F 38B2FCFEEA68E1AFC70301D04755CB9F15F3FFBC928D36E020A5A54F681B0013 0137244AE468F0E158BF135BD2A1357AE28239F77ED1C78F6FC6AB2A0A93163D DC0C5A43DE668F09E584D3CADAA74681CA6694DC2C0F44F61B7DAEDAF8B115B9 ACA7416B94877E27489BBB0BAB39D9C58D48ADE58ACF74922F2407E060685013 4A7B712036C552DD07FFC08F6897DDA1F2DA32476D2E19F3B2F467076FE51C8F 63956C1C96EA5644D307C1EFF4E3C4B58961689C1BB62A908A359167C82D2FD1 E1956BADD4C08A56CF886FBB8E38C3CF1BA0A80682A6A141CA47AA84BC6CBC2B 238CC787A063012A94C8D09CC12659A795DF6289BCB7775F3E2F1E2CB03DA947 62F6763C4D1E0099AD1496E07CDD8AA768B4A73C3613A3E6811D2ACA6A7EFD2E 315C4FCDF1C59557671D840B69AA45E70B579F4A96CE030C42085524986D8FE8 BCB61F38E8FF87CDB66929687424C7A9FF942FA1FC3BC8DB9FD7AA85C3E6D838 A8B1F14FA185172144DEDB195C161B42470AF118EDB8908A7657FF4ED93E9BDD FC89BBB3AB8FB47398057032A5834D63F13777E8D07A65EDF55E86B694FDCD35 4B823BD5AB53ED596F81C1A39E75E63543A42C9B12D28E9104F59A40C72AA761 C6620235BD020208C1896A538D607F383D897F0DC7B236106EAB67F2484E28AA FEA5BD635434A8706035A7775A3C94C029E635EECBB4DFFEB7C1AC97A89FBA7D 40E70B743786253D3E1359EF0F1887737F2CE4659A506236E4BE3E2F11407EEB A0898F427DD64C96652AB7D871E8222B9312829E6BF22BB28CB6B3221BB091BD DB1E5D054811E9982B55B0BC1C019AA8B89D621A2557D9249B0714910A3412B5 77ED0F610E3FCFFB386B6D283304063D1271CA18A43662637746E402BB5C9AF6 C2CD431CCE45E1D856598B99F25BE0ED288CC765DF4CC507ACDE207DEE7F98C9 45ACF0C4C267A570F98E4E18E3AB6C0581E05CFEAD1157D6A15730049C24C69D 1BAF3190CF595D7A79190B2EF1697320418FA902ED4AAA21EEE27D65483D1D76 9F7A421891FA19CF04A089D99243C0C04B4411D6130F54CFD104246487CA1FAC 6699E442823A4DB6FFB47740681A28466A8FFF6F59F22950F7D46AE2F6B79A88 D0BEDB7E6A86DAF5207D181B907841A8C3445388F46D72108AEFEA843AADCB63 501AE4479401E88F720AD160676EC12AE3FF01208DEA4912C26B6417AFCC0B1E 3D71E3820F16E7A22B0DBBC962E3E5610A0D483D027A974FF4758A36319E4E8B 58D4AC6C52042C73874321955A6EDFD6C7F89C9D7A1EBF8ED03B38CF16305CF7 FD4BF11B39E19772133D042AD91FBAE8972E8712A35B0D58A88B8CAC9ACDEFDE 2F0C5FC402D34F10DCF93A7DB570AE39CC97CA2204695E5E611283E1CEE47653 FE69C14B570A27AC7718175F20B84138B9538E996B3D379E05F7A589F2414D3C F92D520C7D1F80F940387D910A1B14AD0CC9AF151729985461F0FAB1BB8FCD36 01DEADC9D932947F1D65193EC3C719108127DB88CDDBA5C8E246274A0564DC92 598D23FF7ADCA3279E624075ACBDA5394646BD024C5C61EC9ACAE24DBE96EC59 F1291D2DF3357DEDF85F6FF9177CE9E053912D8B44EB013539BCC26586178CE8 319A6CDDB586772261899F371A8F30DF31C1C6CB5C99F1737A970B91E6A90209 D8E16BEF4009DBBD852BA6B66D5F928444967527C6EF76FBCF51AB9B0B04AE7B 25090888DE6C847BCA4CDE4ECF438F1006E9D86F59297877866B012152D5441C D3DCD66837C6407193FC4CA513DB04901A66A62E2DA8061347567EEF989C3DF5 4F05A35876DC277372BEC77E2304CC14BCD03BCD3252A0E47DEA8C6FE0A8BC1A 51C414C1218A6533956E86C094612D51A48F426D67ECE6E759EC846EDC40ED11 DADEDD38F32FE5087637C9F40CA6086242DE3A925FEB2670C218D348A60A3ACB A799845989E2DA4E57894A8EC94FA989E34E123CC6585FFB4F1C6B5F3B39BD17 83FB4D06A1B7555B541D57002C4737F14710B172D38786FD004CEF1ACA0E2BF3 672DD8BED6B35DB67D1FA0C5EBF715FF4510D6266BAABEBBD6417B9F7A64350E 25DADFFB25FCFCAE66309A40424A43E89794E960C9E98E46F2B3AB24EB44CEC3 0DD315F9D72D509CF1FAC2AE005EA0BD73018B7D3AEBF86B604100F96AB7B44D 5078928DE3C8D9FAB182D4BA456B450CE7A81C97C0EBF87399022F3CF9B2153F 61A13527ABA684E3D271F36BA54EBD6E009E72E91567D73E8F7B188F8F3071B0 B678F544C31C06E33B79059575702371D28B023C7CAAE51A752E720E5C82521F 389F7739CAF13C1EF1EE8DA9A53D468F6D76F0C190B7A2D8716070505D646C6D 867791A50A4D8BE719FCEF742FB8434FC8E90737C70EDD1866F11F34AF33F280 D18D2563657855226BA9F13ADBAB5809577B37EA2E94071870CF40D9F0B2A800 712F0D0081FCBDA40FDB786A4630E73096B3FC2CE8351557F53F7106F85120FC FE4750E907D88D561E4D22CE4311E16917DB115075F7668C7B37DF999E9C4F56 81FCA8DE6BB4B065D25AB19BAEC0602A648911BBFCD9FCF5E95BCF23656DB8AE 991DF334C065A9BDD71C7C3C56412415F483A4434CE395AE7795C5C8C749635F 5A5BE3B3854A9067815BE356C914E8D45D00F876B9F8AEC8615EB8C43D9B0199 86D8D347D175C9D766541BE7AC3C154583FB37999D00758F9370825720828A56 93E410B50051368EE0B5F14A9CBD8480F3E995AE89BB8A4953E3B883D2CC797B 583F9FFF23424D50E280A239BA0B0C12264B9F2E5CE8E455505DDC95B79EBB7B 8A7D0285F84786C8BA32E99FE601FCCB645C91D9C6021BABC2BDDCD1DA4A2122 D90829A774CB64584A91ADB373E85E4A36162AF9587B530C032C56FAC90E3218 EB531817C2E261B386B680A243568F7CD58CCF5920F1AE5D766157D87B544EA7 086FD0CE51AA99A007C7C945B9A7C42A41BD6A42AD1CF4114D9C68D7E7608BE3 208426415868CC61D3959D64E5F2C6D799CD593BE0F4F3957BE46AD0852607C8 84A0AC6D8A6A9E036E49AE6C0A035C9941E076A5BA55DBEF952FB8172152A5ED 0A7FFA714CB552C38E4C1AE38F5A1C1AAFA7C0F4121732B8D939A11EA1CDDCEB 99ABF9A5DC6CFE25FC71A00151968E627547CE2C45A2CDE476A7FB7084C8A350 16B0DA6DF6463C8F3BDDB30479521FFF19CDFD6D2AA8653848D8957A7EAE5CAB DFAE4BEA521EE047DE1533B1CAC2C0A5275BEABC8F0DD3B3AE381DF783F358EC 3D7A9F9A592BAE7F866C9C6BEF69C981C62E0C0F2A7F0CAE736420443CA34F0C FB299816856E7562855E1606E740860AD34AE46494E6191C70CE9925878B19CC 3F9BB502A96800A8F6F22EDD37EF1D02BAB2D2D8AF3BC9B277EB4F9E59247EC0 191D47FF55B101698F97DC481F4A184AAF412202EC7DDD1F80B957E9B3A1C2D2 B35163B390D868983409EEC8315D932118379493A1857E3DFEE38687527ADA76 942EA3B71B0405C9AAFE25E2FA98BB907B73C3A4B624E5B7DD7E2E7A218C0E5F 5BA8A11730DFD6977901C15657E60C534902CFD8249BF61BEBE61D9C3F16AD8B 0D7D5662EB3613440403E74182A379E061BD425CE41B61ECEC285BEAFF4E6E18 68F3D89F1F7502B104329F0E0FAD3097A05877FF7429EC95948F7DD9BC7ECA0D D8E3A7009F908074AE6DF4AF4676C6C72862AB6DFCDE2E12482512C2FBEFEDA7 675D76D9D52378DD6ABDF54941B365F6B9F76663B3757209DDA3DA9E685A2E2A 9DC6DAFFEC2B355D4CF98AAB71398DAE942FFE699064F6DA052B0F0E3F0991A2 0E8C615013A0EBA39DB94B7F0F99FB60007CDE4DEF231BF598737F01BAA017EC 19F1EDB50F6515088F5AA01B593842D0BEB18E9D4FDDDAB5E849FEA17DD6CF01 534139630AC6F062C38E0C2D5F7A5C2295AE436DB527C2A316516F2A143C56B6 928C275C096251E606C799FA866D62D87D71E42743F35E4AD91318610D4ED436 42994003B035593F12EAD928AF0DCB2CCF8E9B692ADAE41151D0CFDD14954A09 5D9EC59FB73A715C14425BCEA590551E5BAA72F527E8E53E2DB281049C6F5008 6F89323859131AF8A66E185143E1E2C7CFC5D4CC960E429939543CC8AABB5453 BBE3BC98A7E0380EC9BB88C1FEF717F0459C285A31DD69B5442E89EE2C986B97 2252D8F1E49847EFE9657833C6CB4D634A75147DB0CCBECE114B36E6B01E7222 20199D601FDACF3299A69A5874276188DF26EF918AA9D9F48448038DC4D25753 A15CD54490FCAFAC29A6FFB1B7330CB445E67EC777E2F2CF3733A884AF3D977C FB301E6B1EFC6333F6C4690BCCE01B172DEC982B757A23AAC52944262C6D4186 58C342FC722E085A266BF7D2B111D2F16F04A14FBBDBC9DA11CF7153A0178579 7DC7CBDED0A8DA7C7505B398AE261D51405745C05885B25A32B64C92729710C4 B24A8B302F12442BB869E75FB57BA1F9F243F52017D234982FDDD154F98F9F25 7679627CCC6AFB225F2F9BDE6DAAA4EDBCEDA07A306D94C6DB8E67F5B5E6AE43 025CBE8E307AEA4537415A5D14BA4AFC913D0708DCC5531847A303F41F815A16 3B8E677B8E82BF842E129BD09377A275C4E627ABAD6E0037B0792E8B6601A32F 9C33C2DFE909E8543D1DD9FE30359E09333249AA3DFC65824DCF5D9A5AD2ABF1 DBD73645B0386BB7DE3D91F94170BAFEEFCED2EAD062AC0DAB06E25215309223 4FDE952128D65CBE9CCFAF729D0F3C88DE68FD48B6D5B8DFE6271537FC4A7F99 94CE7465BBABD089EFF45D1E6AA078E8598B57EFCF76B676B32930DF010C1904 4CF7823CD5E18BB18CB22984EEA212EEEEF458A69A81462DF793D5C98EE2D636 BA509DE6E56C0D1090EACF91586AC38021CFDF424B9F465F0467B0230ECC5718 99888E4A6969EB3981E6726EADFCF2D1E08525EB7B12B15E3D9E19D2186547B0 599627C689A68EE4B4DFCAFC94491243E94DDE1C9BEA557BEDAB4C42FEAAB1CD 443222C79988F3F6DA1BD1C39384A8B4A460D65D43116EF8C5A9C48DCBB18CC7 436FCDCD1E450E3449F7BE838D121B8EBCA2C3DEF7690EF41731B3FEA458FE93 798E64E2A7633A6BDED6B2500C305B7979CCE08CA1D77A15C87C15D739D49404 6041D4621F9D0A30E9D8D677CBAE1767F78EEB6D84D2229C7DC1B409E373DBA3 CFF947319D6D6D00466B2C674E69E1D529B6F5E72D1319448917982E23595412 812DD35B82881D0279DDBD673F79E2D904945ED731571BEE7DDACF3D0F85F8F1 9F39BD6588B7457A9DC35A7CE0387114C7AE16FE5A44643287E373764B9F7D25 C0ECA57DC0424D4592ECD1287C98F89ACEE22685A2094F3C1B53FFB98BA65C43 5F98A6D0B99623E3C4F07AF4B20E52B834ADC03BE2441A311D7A54E89C632E89 345E10126FEAF43F51D9064CF7A70D2D8E4B1CB10A824E08594B57D1F6014002 EE83B778558DF3A48214EBEB2D951B7B24204981DC428093B410377878DC9A17 835F7F30803D880D67E461B78FEE39FDFCE88D81D6A320ED3C174325E7D219DA 5D8FBB8EA1DFE53FFE68CAD46A9D6833C16D6B85135F7E6FC78F0F058A9AD132 C2C11A9666A92CCBB7B2C99C85133F9C008C19AD77ABF5966E545CB1A5723109 F18753080E80680E63701CB23C5FE09D5448B947F3B789740EAACF031CE8A320 B924A81172ED1FDB6E66553D2FA21B87FA891B03010C6A03D6D9B0ECBD86E1EF F41C22B9252F5F9EFF84235DA3ECDE3341F98863110502B566C11D7DBA833FEF 55AB055A680BA33AC55DF3D3452F77F1DC764105A0D118789309A0F1713276D9 AF80E264E2B725632B9439D0DD6A29A4B505B7A32AF17CF52C9A01DB6BA5BAA0 1F7FA8F3B8DE61DEDDB086D025ACD0AD73444462BDEAF0ED3F53199F7D79786C DD388F2466922BB2B0F3752C74CC380725359D36E74880B5CFEA33D698B3E340 555AE6148BC8858ECCD87519435EF6A99EA52A8A0A31B01206A289BCC980C337 00D1940D2BA28F399043C4DDBF486F62C2A00068DEB731B82F1477D3D0440D29 8257DB5516B22B4207DEC5D734793482B0D2EEE5044D8034C6A35E2BFA1E5C6E 10CC29434188969D23AAE6691E09F05D0FA345365E2E1C6324F9E3EECBF14074 999DA395AA5C23FE2E983E1B5D8EA6D71C9A956A1EDAB0608B14443A395124C2 B57460087AE0EE0BDBC673E83A13E1EE966D6B44F2B669CE89DB84EAAA57BD4A 7A2E418930651CB5A9C1A28F25A3EA49B57E0238AEC5D844B70A89A94E1350C7 660E549F13B5CCEEC94EC59DD23CAD838F2AF3AB9D201B0AFF6291B0B0A79B52 5B44F3DCFF87EEE55F0208A75183354A6073A02224811C311A730D56F1445E44 24BB1A44CBD5C14EF6BC0A87AF214E2456BD9FBA4FAB2B01B4F1A836ED9969BA 407DC7A39160976839970569BF90970620C622F43FCE9D42A2CA0FF60761D313 AD8DAB63A231957A451FE57398F1EAA8861ECAF97AEF209CCAD065D0057D2A04 9212CC250BA17C691225512694DBB8CEEDB23C3AD869DEA7DC3BA3395186A940 3C3A896A9299AE8DF7D742FB7AF359054878F235C61CF217E543D6D3145A2077 29A4356A371C8D073BE6C666EBAB34D2C2B39994388AE3591AAE7C73BBA2EB40 9BEA3D67256B7F8DF568992F7F6F34F2E06BDCF2640E51741CC4F5BA4D7AC0D4 3C36C8BDF7FE5ED2AC58D2C5DA23F2469CB2E0E4F8F7BD235303411CFCDBE78B 9AE43E8709FA1EFA1B9C7DE0A153AAB8813BAFA4ED8157F422EB54EE121D0B06 00FB708D2BA5DE96344484007E54DB85C09ED63056D8F72EE23A661B1A170CC7 7DD855B7CB441E64647F4CA5413F9718FB977C89CAB88598425337F494BBB643 05718DC8EDEC6CB3FDD071E4B98445364217C87385E3F225D88919987F883753 012635593A46110546792FFC6C73CC5E16CAC6B1A58E620D0F07805E81BEF5AC C1D402D6F4A08036A8C63CE64C492155A05CEED6D410EDB9CF72E30DA4806E9C A1D728CA92D24348D829B34FE9021A68D048B0CB7BC6259870CD1611F70724E6 E4A8FC870F6F337F50E1BC4608BBD8522D12C8739E28DB4B5164DF90A986E312 462D5DB4DA4E1CC8E4DF111AD1FB7D622910A60BD8852E912E340CB365EAD18C D5B1C98DBA448F5AA9B482100E457FCF00E2CAC54F7BBF74BCCE31BC189C4E0A E9E27D624E5A2DBCEC857DC2FE185F3268BE8FA18C80DE928FB665544ED413E9 877CD4A0AC80E1BC730F0A618CE8FAA19F9B531757413D902896B294116BF218 CD559F659131FE61F19E7EA5DC325B9128D074F4BBC3D09324722208505D389A 3BA4AF22FDC55C7E8694BCED3EAA5CD4977A288A1A173D8620E5FFF7EB1B276B 2AAC42F9F1F28C9DDA72E74DF0DF9450A7EE126977321C0E3E92B719E08637A6 641E93ABA24C26EEC184094D579E88CDB06127C4D0CCB5C1F172194D325771C0 12EE202D91FA3403975CB9A51A94A60C8D71A4C032C709175AD15D961ABC34F3 466D48A34BDAB4AF053D19352D61E98E064B8EA1230AB311FB73AF08477284F3 2BFBB9FDFBB677E2D91C6D1F832578D92572183D40A3E1224D6EF6EE8396A672 C56C21B817C760220B70C1E391D412A94CA2C5011BA9444C676C485A82A98FA5 996ED8140FDA10082738BF991DD11EACC76E7B2C7E098338E0AD3841F4602A26 36F25557DF5A8ADC95A1A31BFCDD6DB4A09C23D66F9BC0DC8B9F61786DCEF405 428BECEC6F15E7805952CE4687014CF14E7ECFF7E8C6B336B8E45E6B751B0ED4 10AC0136EA0DF5CCA134D2A87F305372C210A1244FCFCAF0F93FAD716E732348 D09E95F06C0350E93FC90B719F8D5B048C9632D527FD195B75F0AB862DC59AD0 8FDF43C26D234714CFD643F2E487AEDC298DE4F5A0608C611EB7E3ABCD0208AC BEAA625119D2C918CD0C2768F35F2FDF5E74F252833E0B3469F10B441004842D F1C9BB9950D5572465B2A33084960D31CE640A5E2E46A301A943AFE4913BBC36 51A7F6A25E818E9A9A7276A1A2C43911E77A999885985575B46766F55F36BB00 BCF61D8AB04481CD9B60BEC8D2A49848BFC9C0E87402FA8856C540C77FC7CE67 C40F08035926E5E4BAAE115E99DC3393E8B36230599DD6E5AC14B8B99B80CE36 7EB2B2B69A965849FD8951126FCEBEE339C0F1D27B8E0E8F7251E64771EB5FE3 A8E19804047B8EA3BBABB7247C56F7D4608AC711B858707B6DB8FD0D1C769C80 C4B898966250BB32630AD1D19FE09BD2475AD9D6011FFB75A2EF968D83CA938F FE59B788DF21D5692156809AC568D0E1C1FBC9B04DF223D51034084AC138E9C2 E86D622E0E3CE177E5A6C4771BD5195CC304B86ACE34D55C34DD20103249BABB DE657A2BC4D31E7D6B563B3BBFA9F4E1F25941CBF31DC72E2F3E26F42E6F2898 1006C044AFFC4434E0C15CCA3D38631E0FF731D428B65EEA6298463D5A262134 078AA4EECA9A891D7276B2C3A082C41EFBC567F8937BDE98853A246BD251AE86 CF257ADE2B86CF7A57148EF960F9EC7B460E5F78CA88B8B587FA95DCB056FEFF 34642FD2A9AF3C0957EE19B68367586A1534466E33387ED4DE58A3BBCC59358F C8F194ED0BC9ED9E541990E21029B5C56D5AF73AFD80969B1BB02F25E7EDA20A AA91FBC1304C59462834E2F2E798B6783303BC0353474A8F052AF35FEBA60E15 5276A2B53793B6C6ECA313DD54F20826CAB6BF4A4F38BA08F202CD5B2C3A1939 9A3A586556DC44644A4CEB0DC2E9BBC95C0536208DEC2AB78B06903E8C648E7B A9DD6C28E91A385BEF5D6AC9D8DB0BD179B5C603469DF3BF940FCB1EA713A364 19BC30A8B7C8C5C5F26498AD99792A342C7F073C7CC37E09815BF95BF1DF56C0 21BBF7C851F5E44D0A257C079B1BBF483609B222E01B2A13AD745C8B21370AC3 1F29594848A7A56FF2D42B1A187E430B43DB7772433F87B57B2175FA18F5019E 1AEA0C69753B0CD61642CE9A5A6903958BBCDABC1561FB1305859CC84FEBF0FA 5CD9EDBB97B87428EE91D69CB26A2B397BF71DD8DA50537C777E7F62A2C4F500 680A7DE035BDA52A60F21002D5D1602263CFC07AD0E386078F0527073C43F9AE 1F54DC3BB07899E93032A3CCEE2AEBF5FD196EDD5DD57DCFC5BF3DB59C7CC3F3 AEF643E8F452405E3D72CA060F41C8D458C213B8BEA8812CA709D028B913C4F1 4DB7CCF615F0F5B0A66DA668B72881F6402A95E9F8DBDDE81A8E560ADA09F9D2 39DBFD170AFBB54977C2D60D328F79EE39C9A2CB0409DD5192C551180A975066 0E4076684245EBDD56B67CC09E5361DE4D2998E0B07B7CF949CF4D7006B3A58A 60494DF66E79F306FE75D2B31F5FA7BDA437CD3EDB98D7DC577E2984082BA35B B44ECD123AFE480BCBFDBAB0229BC3BEAB4B20D4B94463D1659BA157AE5FFA8D 4B8E0BCC43CC2D5D1EA82EF712F90AA12ED8D5592AE7DEAF134D9F4B6B730D9F CEF0DDD0753638F2B1F384AE10E98955CCF3998BAA3828323F7A21482DD9E748 A6ADE2E52CFBFED4B5FF3120584D66184541AE934084861C8767777337EFB50D 7EF8B1E1195401DD7690A7BD9720B1A82F3CAAB55C31F34E8618535CC9E4596B 84A30614CD4410F0082A65138B857DBB0F9BECCC7DD3F82D99A5E2495113F021 E451F5C2FD2D4A6A605153645A5CB6A3C09DBAF67F091DEB3DBC442D90E26173 6B36FA568AD745C2EBD2241451CC593174073113CE435E5BB6354503612B9F06 FD2351DF2CCEE6BAFA5BE04B8F20C5F596685441B572FC3F64DC8AF8152F525E 1D22B6E9FB3DAC9471019EF886901826EC39FDDE71992104A965D824279849EE 204A9DF5D44456383B0CDB9B2FF68B2529CED35680FABE8164D78E0BFE7AC38D A75D22697FCF8F323DBFDF1F5DBA27832114D70A294E3142057D48B5E2CC6FA8 75478E9133EC120DF4A57CAC0F6CED62F1BA7E8403BEDDA6896D169439810F66 E326854F2C0A7B79DC85A96A2F5BCC22D62AB572D011A13F1450EDC4A58B6CE8 E81E0DAB93FB7AD51471ACAB9F6C5FC122569659D52857C485060249BA790F3C 3B5377CABFB7251C4E219B7C9A61FBC1472FAE77596CF111F5030899495C8845 DD01B57E31A7D05FCBAB5FDAF2938F0A2DEB98D825CF023A7D746A4FE65347CE F00B8EABCC2FC42F4C33E08D8ADEC22BEA34119A241B4304FCF28F225D012DA9 FD723E368DB619EEE06150DC6F37917F9CD021D0807DAF4EBBE258782406D478 6DE73E70B4D25B439C928FDD20EAF0736C275F297E2F695616C780BBCE49C62A 89B0334FE188A84AA423018A800E91532109970E830E98873D0C4DB9FDC0BA44 70E6097C7AFE46416D3E7B7C4D4E70DBD7D160D03C2F475350E0AC1560FB5215 ED703BB19C1F83FC3CAD24450B54B684A273F900CF9EB28E7C53F5F08386A254 C3818A4594FA4F50F198AD958E17AEE392CC267C857547A3CCFD5C964461BA29 5965DB26D1BED5059CAE674C13446F575841A189E6277C388DC7B40CC9537AEA 7BDD3E7588A669E668F1A71BBC9BCA34E98A6B15F3EFB6D4893173EB65E778D2 1FB0BF5638362474B6E7F5A84697A3B25D6BAB08BF8AD9EAF3F3B47DF22A3B1F 3878620B661593C4D0440E9FCD76914A7A495C5442E3D0A58A0B90E0B0685083 33C2D3586736E2625C3468EE47B4D08B09B4D5E2F182262300889AB3CB931959 4E4C035CD1B33D43728A9A4D8571C2CD009CCB24118420340A8E1D5D28132924 BAAB86C076A8FF88BD584FE82955A210FC978EA63A20646C40D83C5B0EF3CB64 FB55064E56A693AE621FC4B30E98C6D4660D6D355FA2AC0C4184B4D33119567D 9CBA4ABB89B29BDF8A0C3CAF11CA015DFC421F495A182FFC9DC350F4401CFFC2 BFC18A756C15CCF5DA74393CEB390E0C1C177842F6BFF11EC8EECC53779BD95A A65021FC8899B753E57D86DEF84D7B5108D5DFAD818D01AC9F472E35A2F374C8 270D1B1BAC9CC414E29D3D2B421C324F625CD1CCC3B94BE0B1449E1C4302BDE0 A97BEE8E99E45A8602027EB48ED06339A640CAB29DBA9F8788AEC764238E9B66 1D82B164D1CE51765656C25F8E61341955DBAD582CFC3412782E67295650CDFD AAB9AE5128739C52318338A0D7687AF7705EF6F03F6FB8B5D695CCDD79E3A72F A9FDFB506180782D3BF6FF5A0F47519C9D1E3B209A6A733545260F5F0A77BE06 20EB65351DD6F55CA8EF2716273DD4387574D57667B24BB2A9DBD10072FB282B 1DE7B696F343C67B220FE8449C69562ED3EE627E2FD593DA95FCA2D68A69145B 42B604905D80F7E87024BD8C1A2C58FDC81D2FB1420B7B532F8C1167C78AC203 50636255311C8BCCE083C1031112DE534E150D2DBD3A588C320D48117BF711FF 3BDCBB145D0CF90D24AAAE1BAEA5F450E0CEF1964C242817476CB6FF431F9BA8 EB27A0B755437F9C37BE1EEE545F8E8229F8BAB747A6A4E64EE8A4EE1F6C7512 857AF2C5A7C08C3A9AB53C6740EF0926AFD6311A8802AD9808787F305848616C 597C7CA75CB358E51B40A9ADC0C06061F08FB1CD7299DD49082B0F9F4F94C2AC C99AAD6EF8751F97C70D76AD951BAE3BD5C02379306D1E3D7CE0F5F05FCE690C 15D472F465F9001C3D22AABA04642646E7FE24DE47F9D735B4D017E227FB9454 A7ACB2FE1D633A99A10B729AB9DE25BEE925F05734CB87F3FB1FFB6A2BC498C1 1759114FE07124DC997B07749DB6538FE2499744C35C1379A3A7C09EFCE1661A 16B38DD696586BE6831E25F8B0AD43997DF0D46FDB31C01C90CD77B33A186D2D BE3B1732309AB459ECBDEFDAC2FC8E685A64B2BD4503AD210941A0C1267306D1 382CF120D8B3EC279857BC1630787074118223260860FBB28A1EB928DA368F87 443BE260802E465E2745318CCD5B649EA523C8A4A37D3FE46E0B31043E97E62E FA7CFBBB0DDD418600C1327812B302181064C02E9E15116F3F1A9A83B81CF560 CD1D404E7D70310B163FF96DB555AD5A8A667ECD4715B2E880E00343E28DAF55 1BB1477CC69F7D5221BDC9195AA27DB37A95938DB6C6995E8C8E7EF0CE381711 44A005E341B4132772743B9DCD534EAC2BAE80EAD09EDB32ED10AE34E0292FD6 60C0A42A9401C154D5761B10C3BA856515D6437FDC829A3D2262FE246B797778 9B7F0081B5B5606CE5E19282E0091EE0C90B0258E87459EFA48BD7C4B7E8A405 3BF294DA795D5F63BB4C241C93265893CC431C6CBE593AC98D84973E636ECF18 B9E8772BCA74FD9847C71844DFCCC4BB6BC72784D6E11313719CAE499F88C012 639611DFC7E1B2B0E69672E01DE75F52C6EC6BA46FDFD4032ABA6F0774301B69 EB8A235377C58A96B7CAAE3F3A4BB95CF77090AD739ACBEDBF75BB87A700C5CD C97DFB93CE3543377AAE793A0BC601D55CE753CAA40C9956429CB388E071CF47 1F1BD96DFA6CE01923C1BF6F05A7D91BDFFCDE2B17D40D95A0299F00AB47800F B0AF302AFEFE33E9B5742E5E00C9E7EEC2C6143CBE324C537B224755BBC7D914 89736CA305DE90F5C0E2A200BCDD15EBF757A1FC35C05D8CF62319E5332A67EE B89C8B2F21E64D0B0148607BC5E036259E36920E8B5F5EBF92F80DE4DB34E70B E2454E20D22E7913E0868CE2BE29632263ED87CE9E3E1E4F15CD5BDA3E1BF042 66BF4A5392D3A2CF0FF14604CDAAA0F153337D9EF5B05235876A1590D475BCC4 2F5D82A4E190761BE2CFDFA8287A4373B8E568D761F27B07EE8A2B5BA835C004 CDAC46C958CA30C2756BC9280D314E8971E0CD505C46C0DEEEC006A866C0DE07 A54D9663489514C58BCD3418777E7AD818736B2960D5E3A3B8E4FBEC7C20C7F0 0914F556ECA55DDE3C98DEB0F078186B548D4E38C8EC643FD619DF2E32CB18BE F460CD7E591180277841680BA4F31CF6BB64C54568D11B3A7A861BCB52FF97E6 9FA28A4A5A91516240B2AE178D11BE1A5963EA291FA237E9C1FD3408AA3EB1C4 7552CE8C4D715C827B34CDA18938BB6AEBBCADA23462B6DE32B709F2DD00E591 03836262093ABE6A98C49D0F0D175106849C4DDF1958F228C48D97282E479435 49BFEA3F72FE29BDC41ADD154E6AADEE08B5A87AC80B69018030FB9148175394 6F7C7D939D001782BD2B8B7A7B403C7CC908410D27A63F678C9D845B56B96B81 305EA8352C29C4444FA0E4C910A589C35A569B9060C4603AEE7430F4FD301F11 441D5601C59F98DEEFB36E4D9B5F6E18D849128BDD7F7899C81FB067D63FCF24 694CB7B7DC431F554789139B20BFCDD7F0DCC494FDA382ABBF5989843A126CE4 33ECD3E24FD6A6355F30C2BC8A2FCFD152EE32347AC4F0EFD372E5048387D9F5 60CEB7256C67EE46AD70D1FF90ACD3123FBAD035C2E46E2557C3BA0BDE048B56 CD4847CB6829354CC0D106EE7660172F1476D04816205CF1FF0BBDBE929ED6C7 9E56D6ACE108B3AA73BC5CF64472D227729BFF6FA7F2CAFC673558B2102035FE E6F425BAF11EAD57ED091EC08A4F23EB37606ED6C387BD8E024A962873A7EF0A 423A4BA5075C7483458D11FD8CB267D27D6B6045986944050B2198BF60C450F1 250E6CB20CDA4142FFCEB3757032ABE92D07B7DDD1F2F696D5497F69CC40AA8A E4AC921B41D98FEF68DB432366666B4340D893640B2955969D9E9F08B706FF71 A917D433521B4BA5ABCE8C4CAAC8C8BC50E10B2271F1DCBA87BA04CF86026DBA BE3E980B29BA06ABC3BFC399CC3CCEC249844BEC7E0FAD511D2D3DD4DC773E25 B9BBE5D4EEF8427E049D39E83291BC4184B8E8C3AD2C2B2A466A6E35E563CC43 9BCBF4F19E76C08925A5AEBA211A0B9F3C76B315AFB95DBE3C492C2A3D2C0B6F B6C3891F012BA15E4382312ECAA5C7C84B767C47D5DDEF79A606ABCC24ECF8B7 D0AD55EE735A36984832B447398A2A8C77E5A3079936354A25D994744E1B3616 7C72A0F98B88A8B91EF9FD6918C64AFD65934FA3E7355DCC96127232FC9DED72 9845865C292EC0171B849E937EBEC8DE31AC7D263AF9079D6D0A9496318A1885 29B9318A7EE334D141F461DAF4FB0DE4BA3CCCC7B132236987CBEAC3191F23D1 FF9EF42D9FBECF56CF15193AC27E011640AD71DA154769F0D25BAF89C5045089 9DFAF285CF46B8E920F4B9C713655D4EAF6C7A1916E2F0B2483DCDD91C2E3D9F D1A8141823B3C6E0129DE48BA9FE5F33BE496FD8903C155231F28EF6F64889C7 6BB21603342E7FA99A13CD434287110A9A35818666019DDB1889A4421D641AB6 5ABFDF58F8D8486906788F86EA41084B9F0FF7F3E5B414DB4BED1E76BB7DD571 9FBB7D7B602BBC98323BDC8C2DD0C100F29634C71CFC1786F8E5FC6D10FF7B93 E3D7B85DCF1929E831ED6393B62DE97CDB715E82D0CCA22E621C0B9F26B77444 70E4279AAFBC6F376DE829AF87BDA092434AE59C74F5E068F5A9A2556450F04D 2AAD5D18226F180C906903ABFC9961A0C80CD32DA533D252195076EF1865A339 66C2CC7322CE85C2217316459EA3D120135CFF8C053F3AC5F4E164ADD9057AF7 935C07FCDAB5023E5F66418CDA9E6A49F56096F235F5EB65FD1C62975629825F 52C147EB9FD2EF1E4184703736DCF2279252D4BDA635A0CF3DBF01652EFD0063 19D41EA7DDF578C606D00405C999719B44AF146AA44216618003C3209521A5D6 0A37B82A76D8403B26526DDA9785D727D53CF0622B5A46101B2BDAB2AACBB0C7 54A4554451A04EFE67845E7F87829E8115ACD692185E2E507B2470FD132ABF3F F749A004CAA2D8CEEFA51A18CA6B06CDC3C6A1EC0C6D571775690BB160519EEC 73FEAEB3A1E6870A5510706A564F9B8D13D540EA809C088663566CAA3633E268 01B8A011CB1E0C48A4DCC42478DAE54D3B05F74E079519C0E45E4F3109711A6A 397A9F73E4CCA1A56215A04353D75041F51304F5E8C1737DC22161CB07731E20 A10F670AD515103B81F0ABCF54EF3873414D57C895461B9ED44EF9AC861D613F 8CE5BC1ECEA92485B3808538F5BF07ECC3BE8457D5E18FE0B5C8594EE12ABB06 11EEBD4A759C21D584667BB6CCA559B18B79A9E1B855DBCF5BDA69AF3BC36DED C592BA25B2368B4130029FCB2038FBE3C221076679F66674D0D133A6F3B1DD4B 112F7A4FF29507F062163C1FC9BF38FE7C1680933CF8C2BE7B87CF8A8C292879 FD9F2748D6124B5539971DFDCB3230A671B810BD8079CEBFC8169E1E4CBC0DDB 3A50A21BA0AF5B049CAE56D6F2761D992B666213C1AB21B40140F7DFE47AC388 FCF367982B92A2E1A1BB50F542F25950E85CAA59FEC70F52A1B400F618150151 BE547F73C81973AB2EB41EBAA363BC225CDB759A7303D3741C5C642948A60A13 817FACD029B6D81203E944930DF76DD8AA774D43FC4E5ABE2773BDC9E981121E 6FD65DF07AEC11EB4DED803687399CC167B3ABEA1065878E10D85A9F110AEFF0 A312399FDB374A8CAD9EB70AD2926C9AA1D0B4D26EEC63BD77CA333E2351B197 FFF980806D12D5FB88F9F932A7A6A88A2C091B73EC09C22E449EB26C2B188E92 84AC84658BCA1E7460DDC7B8CEF96A9DB4B39A1FB65379A005F55B8FED6EF456 5AF85AE7E8A51CCC6C626DB681DA5C8B00A05F261F882BB9F12DE23ED952C700 6959AD346EF7E4EBED0DD603BDC89979923185A74AFA3E99E06E8C349FDB43CE 3F4DE9CB8ECEB41DBC32FB96D216F59C00971D0A2D5FFE70BAF96259CC95EA5C 748FDC8CD7CFA1AE6FA2E69A92917ABF373840481CE6D3CA71433138A500B776 C607F2FC2F86A2140F53B2473FFE7A1ACDC12D164103BDB647529A7E24677588 DC6994549D06DDD2C90513D6443B4458AB14C809001552718B6F60DE17FAD425 E2893BD0CC2C5DC962AC2EE8B8DA7D2AC094F7B421FB71B35ED16582E57D0FB8 341CB6D58C6D1E30705A796A23FCD3FBB51C4E5C2EE60FAD202E07708F1DE999 9C7C203C9C72177B4A317E96A8D286D49A4E8EDC08A031AFD5EDF83441FC5757 961DE08DE5FECA646DA5B8CD499580C155F7AE8A76B5B6E471122AC07E3F7D20 5E91346B7C86C3B3CB28607F78D567FC0BE3F7162AC51A0E794C4EBA9D4D96AF 26BA9B2BA1460753166882DEDACEDCACA7F4584D1B1DC523A702E524914CE1D3 9C1C217DBC89DF41D59EE03132D51F91D782A054D40BC8529EAE752FB704AD01 85DBE24445EB9DC6BB84DBD37DFD7C55D6A3239594C7BFC0F168865A59E8E2D8 AB96682DA05662FAFE1BE79253F85F789645FB75E3E6E5771D0C8EB2BF305A12 8F21D5E1CBF5D8C97ED13C2B1056F2A74A52A1FE243E9D8C50193228D662E86D 54668522B15C388C0C9FCF40C7755822EEF952C394D4E60041E3719A5B3F8FC2 80C37B452CDA6E7BDD80D55CBB0711FB5C363E81D943D061B496884B509D7FFF C6B76C4CAB62E249BE26A3196DEEC41EB7F58C5879178FEC0E35BF9A92B27BA5 643EA41A7129CB66CAE7BAE5DDCDD77B1AFD0C4E1E6DC74CDEDF7A85831580A9 B1EA0692F9A6228FDB6C5796F74583C28340C95353534E2372A97A2CF9F47960 A917197B8C1CC2085F22BC593ED82DD7B4FAE6546CD68FF9575C72FF8F2BEC3C 832E329CA0DE29FEBDA415D5B841EBABED6B15F6FC0CC9F5BFE9388311D5D3A0 890D12A5A26EA946D18C96DDC91393E03C292B3FFCD3F495503C075C61A94668 6E90B27858E5D94EB933006BFA3721383B24549EFFF49F85B43ECD9A14110055 0C7E0A0E01862B24651149DBDDD208BA94EC7BA70B387640BFC398383373900F 46F8C7FB5163ED5788D5CA5AA88225ACF7B4CAD2A852B905A43F4B40117899F3 1C07AB5BC8CDE290C2191A651ECAB29E23D1FB0ABF60606B110327DA8E292B0A 39253D4A766AB1AADCFDAEC16FF9650F35824FB0912A1A40A77CA1D6443F4ECA A63641CA62706676D4C26D55698DFC6ED0DC1D1DD2149267D52FF710039C2D17 03AD4B8DCED753428A4E3C0C1929171D8A019837CB37520EB8B5852D54BF8C3F 35D74AEFB862941DA6D6DAEA809FAF83CE32F087CE9A3FAD29656ADA14C3E03E F79AEDAE3648E0A3F04F72ECF9B64B857338658618280B0DC51FB8E724A42747 8DE62D933844637838617DDD5DFE096329BB15DE4B4B85FF3CDB864C922484F1 04C2BC4488888D3630A5C364117F6A37E70CA73F4DBF2AC6CCA22AB2C765ACC3 7699728CBB5DCC962278CE1184FD7B588B34AE444F928E74407F8D08BE3FB67C FD2FB3C0CE32BA3A00EED59378DB3B1A98DD1B8A77B28B48F19DD119AEB49D39 BB598CA7E6A59E5B287A372927B4A9814B64B0C8D2E71A826E63C7C707EFDA9A EDFCE89D43C2429957A86C596EEF7E1AC0F6075BD3D390D9F7A94100E0FC034B A8C58B97E4B56A006254A5722D6E822C74BD107007EBF35B77A372C91BD81E6B D2E28A3B7659F062BF9B871FC668DCFE6A0B26E0E38667F047C5C740CB36AEFB A1A08985232B6EC26CFAFBBD576E36AE9C50C84CBD0FE9A156E3EF7A6863D82F 9CAB2143E715D10D66D18C4A97701E297DB873298A5AE30811DF960E0C83CA31 F283348F935C93A8D1AB1F37E0329810DA2A788C9037DA8F5C36322F956F8B39 A2EF679E301D554A8450028ACD48887D0591C5918F2941D3A18136EA69C329E5 F5D4161662B742612ED8747A59A8CAC4436D43F5FE75C8025B0E1D078358C87E C8C6D0CA63DAF16976EEC02711A99971A0535A57718BD92BED7391A162A5622F 4E94A95CD29F4A853CF4ED3218A9D8A849205C393E7DE01B05F085E0B39F5BA3 529DF4B596B9C0064313966379F2F491566EA75C33A214421EC6C321FAFA7F4E 6103B5894251D2DEF88D5D594094FD6880A96DFA37CC154C37F8925ADA52974D 489FFBD04C14311826E60304FC55D875C00E9593E22A409037368FE8FCC56FE4 CB3618D5CD261E96B56F72C29406987B809144B4A74EA2E6E7A93C87107BDCFD 52B04AF6182EEF281BF755AF1F67EAD4E8B484B5AC1AE8E8368262AB03F8FAB7 064496C75CCDBE07D2A6D8185E7057D51CDC2EE7B2786DC5B93B9CCF103C2331 8BBCD4E0D7B0DE83A7F54A701AA59A63FFDD81A7664BF58F966D4601B86960F9 CC200C3D89200A9A2DF625C3A74B9A9792254917B3573CEEA3A40FA69A7B5733 6921B869D866EC4F4BDBFC2CA15F5DC6302B2C0C1C98445D1846E7E698A8C440 77CCEC243B8DB50CAAEA923B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-56 -251 1102 750}readonly def /UniqueID 5000799 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 BDB00E9B9B086BDBE6EDCF841A3EAFC6F5284FED3C634085BA4EE0FC6A026E96 96D55575481B007BF93CA452EE3F71D83FAAB3D9DEDD2A8F96C5840EAE5BE5DC 9322E81DFF5E250DEB386E12A49FC9FBF9B4C25C3283F3CEA74B8278A1B09DA7 E9AE4FBAAF23EDF5A3E07D39385D521547C3AAAB8EB70549756EBA8EF445AF4A 497CA924ACCC3DD5456F8E2C7E36946A5BF14E2E959895F7C94F49137256BE46 4A238684D52792234869EAE1A6D8ADF4E138B79472D2A90A6CA99E2394CC20CD 3841733046175B20CEBE372327BF13428EED6A3E2FDF84C2DBA4B0AD584EE9DF B51828D3B8F385846158C29C9AC3496CB9692DD10219697B2ED4D425C3957FD8 C4600D76E045C561216EF05D38177243C314877A69A1C22E3BEC611A2EE5A216 9B7C264CF6D1839DBBD78A40610F2C0D7C2FE09FFA9822FF55035AD52546970F 83EED2D30EABB1F303091EBC11A5379B12BB3F405E371519A53EA9D66174ED25 A2E55463EC71A97BE4C04B39E68112956117C8252DB6FB14AB64534B4BCD568B 246DB833982B38CDE7268BBF74B6B0C18091E1B1F87D32D66F4DD023D1F10D2A 7736A960F72AC01F733A11023832CD68FB6288A5977743F6F3F23E0C1657CF5D E8374835BDBD2DED3690C84A1EBB8E2383A5E49E610B6F5F0F5F5EC43CFD16FF 24FEEFB92425CCB577E17FDE4EA6C50E1448DC5726A21888E25B6D6B52DA3D3C E4C4C6A73C176DFEB60B6B6191B336AC4F5BAA123E1B3B6FAE4B3FA9DC8F7E39 335277EF2294315BE95F64EBDB1F393B293FD0FDB9DEE6C89082232013130D28 9234FF12DF47D454558A1EE8603B2832772E5CA07D18B34A4763D5B890F7173F B8FC0C5BFE5294C37FEFDB97B53EE19AAB54A02407EF851FF973498D3EAB44FB AFC73B5DC3D247D126CD3DF3A511BA48F57CB0247C1E845AB3986271177755A5 552C1023367A6E6F2D364DD413095331FEEA764D8D52A1759C8778457B084F41 F4D97B39B42E7E1699B4126104999BC25C20D14D4320A35774160AF25DA44D45 C0040C534090929D6D37531E2B1A9675E87B5734BFC387DF1727A73292B39508 C0D1B53F325DD21517E55083F3502C2F44979E699C907C4B8038453BB57ED859 41ED42D0F77ADBBE4C53C351F919DB682EE14C2007760DFC0382A99CED3AA3BB 0CC16C1384738E4F0B785E2D17C3D8FEC9E9D62A6E596585A724544B32A9675F BD417327D71F0B355693A87F8D47C17151342923B6EFFD2D90BF05250A3F7D4F 3AE37633FE89C9DF6560592580A2695D465F5556186C469B332DC37F86A5466A 1491BD7E0D87D87DC2A47FB0975B20401C6863D25E0355DD041DC01B7A690AE3 6B62A459CA07F86A8830D3B15628B0F48A961F2BA9CE1C47B1B1E6A18407836D AA6625D73A9B02F75666F60079E1C83FF1131C699C0E438377D9E8838AAB39BE F4DCE13F34FA2FA58D2CE63D6F9CA675452BE80B0CDE4AE8DFA8F432535F6707 54B9776CE0E04EF2922C7404289A624D88F1003190FF86E6D0965665C04A34AB BC982B134BB9B30C2D959D64103FD456D9A7B0CFB19CEB6486A3F433FF77D6D7 DDE2921AE1B97D291F95A8960A53198FDA9A649537CA4651CAB7822F5CEBB41E 676723E376B614E288CD1EB2C2FEC49AA50844A1B9C10408A6EAE0E0C094F1DC F04062527B500AD1FDAF6BB10C33B73BEF5A25493B07C09798747F6FD6D1BB65 997171B8C1449C230A398FC252A1A6CDB17A2A81B39E242B363CC1798C83BB3F D788E9547FAE4A3C688AC99B39BE77066D6446500DFC6E6441CD9399F2441BD6 E39D0307EC8A32EBA2EF519BB4ED7AB5103F626EF132130764CAC6D891EAE431 463C35DEE8360CC6E8E5C26B279DE9DC159B31296064731E685BC1463E62092D CE87163B1AAC63B66953234A4B4D6024183489A8B257E8C2EE738733286BFD4C 294712C4EC130F9999220CED953B2A85491733CFEDE0CF24EB6F1F9BC2470FDD 0B45D1F71A616CA82A33887A7916CD767E4E93D34762FFB37CB1CE016058DD54 AF54A6E5A6BC8F9AE2768C14210395C47CCD27CA487B7AA6F61EEB68D90B5E84 9D0526B153C66C364B8AD618CD6F9D73CCD7DAB6B5085008212BFBE16B270BDA 55CBEBC6D57E8FB226211956DC224ED036BF55E74685E061143E3E9133359317 14AD902DC5BE4C7B04544F2449602DE0421276A2723DB70EF1AE1D21329D29F2 3E6DD3754EA97A470FE6B5ADE6C063225FCFA4CFF74A61D7843BBE3B6C6BD497 C6102FD9D4EC83E62723D55B6D4C67908A15BA75E1EE1B87F97845676F85AF78 A86998AA3EDAA4886329A0554CC91F881437A4A7C29B06FF9AA6609E9F9219E0 D626DFBC2CCCD8AFC606F714318FE4ACEDE4B4A3533CCEB56E82ECA6A72CE28A 0848895655939EB6C8B0C915CFFB85742BFC0B02F465297C6AC6CABCAA256B48 8E0DBD94572EE5DF00F75FCB1E273C3B1A7496C3A1E20F40FFED918F5AB56913 FD86680482DEDBD85BFF4CFE940036D0D3A8711D8447FB7C1A2218EAE7B5E250 C8F7DD88E2D1419A9F96020BB00FE35F9FFDC5C7A0FFB197743B0BF3F51C1FF1 49EE6A81E012AB597D0E3C0C3BA55CF3D0C3E4958ED3F2E84273C0F68325E850 3B5606DE9168805398A3661298C0CDCAF7A2A37D71BBF04084A99287C84A734B 83B667E85FFB7039983DDAEEB9DCB1E5CB428E446CF505F847319371F0C402FD 90E4C0F0519CBC8BA8A7B9A45E883376954BC44D16E064BC466DB5C7541D7107 4AE1563ACA3860CC6FD96CE7596F46AA03A56EA18B3F7DE55F6BA694E7ABA9D7 3EA6954163536AEE1D4F42AD2E858A5F16DC51C96718BF8E932B7F4980730A66 A48D2B3800A3DC50ED618A2AA720577AE2DF1A349BCCA7DEC3BB49355A4A0AF8 42F7B40464AF0394CC9C838D942BDFD32B2FD92059DE940B21E4BBBE5897CF37 E19A8E74A6D99A028F057EA2E42F8144FB253C4C083FEE5F871D75FD8027C254 BB7B04D9D3E48D836A87D1C818A2013683EF731AE387D70E97FA05869B5CF64C 6ADAB261C2FC28139FA2B374B81DE82997FA09D78E889FD50D7601C0E3E11C0A 7A6E85C9C4D63B5099EC9932D10B8DA3854AE559E08ACCF20A4D8785AC75CCFB 407CF3D9BD4E8EAE9F0E9814BB1EB4DF9C3A2386C18A5061C1C50C1053CB4D0F 552E10B28D4A355FE08700286C0A08E6C461056A4C0398A4FFB9FF89A78F1BE8 254933A2DFC4D339AF0C495268DA0817589A6B1520B4F12EADB04A53CD5CA71F D1BFFEB3EF6A3F5B853F9B49E956F0485D32061C6AA2191156F9658C68951AD3 7F93934AC9E97AE6A768C731C62057C5109B6A0B1C9E0D23C4CE950A9696F040 B5919E5564F6A35547980675FE9563EAF796F81975DCC13852E55A3065A17330 9A5C9FCEB72C2462D2D8ECE90021CF72DF9D9AA36A14912C118E7304BC797094 09A2AC421ED8D869B5A2042C1D3684861599B7E862D36B253745D9D1B0EA8EB4 B0E23547594F1926F6778F29B3F7C4B926A9A89334660AEA794CABC99FCF1BD3 BBAB2151D5216B7FD571FB4B51BFE524BFB87DA4DF7778EBEC8288E5435D9D95 820D02A5D424775087A2EE5A81A0BAE67A674089F24E7468648B64EF8CA8A111 328B4BD3971C4CC295030409EAF92FD10F983E47A8E1D59FFFBEE7FAA7C4E245 1E98C742957EBE289A419BF5BA49E01B02F0DAF2ADD38A7C88564205B06BAF51 0670FA63928BAC7F7A251C36E2FC5402CDC4A97500909A35015CECBC23BE4297 CD636B393221FF5547FEC937AC42E79151232AC82EF37C0DD3035C9654B1C052 C02280ADF801564482878D4AA1FB8F6321B58953AD85FE826FB3D4CAD0A31F01 C74D17D004787C761FF1AB7B6789D074B9C1DF01DAEC26A323D97931C2C504E7 574D4992E61EEB69D46D58D4E7EAF55D3178C08A54B0AB136E3EB9B2B21602C8 3B498CBB598DA129D8A2B6C3C41D95D634FC1D5C17FD720FA679504413058F67 93F0F13D0240C1C9DF1D2D80F08E1ABE1025978C0626FBD873F9954F0901AC05 CF8B7011E3092424F0065EF5614A790D47A995125EB26399B5F0784244877BCA 7859A52DF50BA15A045C894E35FC0A57DD129D7B8F57EAD42DBB0D8876C3802D 16A0FC9BC4ED392647931F89244C2C3717A137B897949D400116B3C3E7550A7E 06CF4D4EA8BFA1E9AFD5018C81EEACC7A4E8D4E9052D03DBED16558C9E8E71F4 2CE7066CCC1141EF875F99FEDFB9A1E95CA35C309607A9750D10B5905079709B 34EB3E53F9085CEE4D775D656E76BDDF42AAC4B38657CEEC12B54FBBA96C3F7C 13B3DBEDDEF57C1DB3A19D56BFCFB4324E63355B3729A0EAD83FB85042A0BEF9 DDC7FEBA636BA8BF04CA0F743755043A6675ED1CB4BBAA9C6CC20B9B42EEC46B A692F1C9B7566D6927EBF0C8D7B6539E721EA252715553FE6F183C7A7A0F5662 A0ABBB7847CECBDB15D9CDCB47ADC18876D9466D72E4B2A18A1AEC66AFA65CAA 350E0B70D818A9DB67606919E60C7F42DAB31CBAA46128AB6FA87E66C7BEF366 0538683217E57FB7B3D8BFDB283A353BA7739B0F3650CD127D98CBAB9140633C 2930D09AE6BD90DC3EA4E6590E98FDBA948169754E954DC3AF4EE7211D68E0FA 647F9759B1183C5A643A8C218562F194416F0FDB1DA5E697582CB717F2A7DFE9 EA2535BBEDFBA73A96C40A8986C693435A15839C0C06CD8D394067CF1C2B09C6 03FA5AE30C8E35CF2A9F5C4C4272F32D0837B76CF0AC3321965C8AB711A8E16E FA8B9E55403B91827A9F119F5F0B113FEDA9A7BFEED2630DD9A80831DBE6C07B 72DF198BF18BB6DD98955D8279A156D57B0F3A5F76CEBCAA2AEB523AC0A41CE2 44500C9E29ECA57BDBAC40F47AF929FE2D152C00E57A0404E4D1DCAE5FA48F26 E7E1BA0FB65FE604CCC29234A83964D368ECCC0DA6C548E2BF05B77A8E3986DB ABC9917D9E849178252F8C1DF6156229E71DE865AB04E562BF139623013A3043 45EEF70355210BE919A574E7AD021DC70988FE3EECCC2FEF9F00FFB15D20C5C2 71C2D2D5F4CF47F77BC763B9A8E11210B667FBD053E62B8D023860A10B912CBC 81DE6476D251EC5341F09CB9E42A615162D7239360030108B45AFFCAE298D5D8 3FCB85A8DDDE3C1FA1F2321951A1DC8654931616FDB80D4DAD23B791694ABBD8 8F6284203B388EDD1AFDC9F139AD84079AA22D42FEFFA4AC9582CB63597D8869 ABDFA9175A4DCE57AADFFFB90F4A1CDF4A0F83696FA1B6862578BB747011B51D C751766745E4663C1BB85278339EB820CA18372CC6EE7202A627205CD60C6FB2 960CAD61D38D363DAB0DDEDA6545BD39B2BC6A998BB71E1B0960F418CE23AD8F 2C841C0F8105FDE9F8DCA6D15369F208E2DD0D93E87EBA28FCD32714ACDDC751 EC5AB99F256DA504A3A4201A5A20F5FBE4B0AC18235A15DCF19ABC26411BEC76 F11E0D1F89C3A5A01298A2C2F0B6775D4BFCA55BEFB7EABB1BF7A866801EE716 A03837EA85DF134CBBEA6AEEC365B60C287C4773F2725838A63E16E9A68DC1A0 B39AC051DBA51EB1D97CEDEDD1D73C694EBFC3ADC3DF39916BE7062BCF10E35B 7625EC0C8A68FDA4B07F62FE87C0CC20FC4E38FFFB114FC277407503D0E9F765 AC029004F11B55E5553337469EAF95B22EDA1EB533EA74040804294ADB3DADD4 10493569BEF594B2A96E5863E622983B47B6EA3DC283DAB3891F0D537D8BC132 9CFA40F09D1EED5522D29922ED35BAD1B6114355435641BDDDEC1EAF4C657D6C D5CAEED39D810F5CE9BBDD7B14D1B6A2BA34C86BD4ED594B5B16A3088AB9DBC5 B5B88F2CEDF7D4A11B68FE333BBF17C1ACC994130A91CC82D6CC32CF1FF737A3 5B7C9C55231A4C7127BFF4A73CE49AA71E3B73FAB49423AD71AEF1CD0376AAB1 44BFDD6221D18D5EE8CF90BB962E36E80FE82E680741CF215C43CBB9E8FF8EDF D037CB6B20B5850F77AA67125F773406CB688337D0443281B8EF1024C99E1605 AB36345B58A6A99B596B1FC7A75271DFBE256D66723AF71530AB623617277CFB F1CFF7D83FAEF67926CD3C3A8C122B51EC04A79B829389124A0B57F933D6CEF7 1101334A567A01345CAB351276822F508C76143034179789E76CE922DA90B742 8679DDA1452070D626AE8AD2C3D7B342D09AECCDE6D71FF04FB24D1B4B681EBB FF3C3F60F4364E67408666CAEFC5E4F7D0EFDEADA5BB822DDC23B47108670807 F0978D7FE486B832F8F975E73E83C729372F983D28D5DAFC68ADBBBDAC6ABDD5 6B9B2C6234B203A1EBDC954859FCA9CAAC2EAF5956DF607BA0A260F0B71E6A6F 98641B7F21D66F3B597660C9CD8A29450CB74C057EF25D431BF6E7F06E284C6F 554BA592F81EC413C8D29C0B179E0D73C4574A46382FFDF76DE91D4C47DDC95E BEDF6782D98D19370DA2CFAE43AC89BE4F0F0A5C7592BAA4441BD1835C9F2F88 E7503D0B892B0854FCC595AEA8F33D5C58B310CDA1A43F384ADEE3AD67CF9395 E123279AEDD1FA0E5FB6CC449F2D90F9956D969EA07B92D8F0CC5FC1FC130B02 2F6A7FF78D1098714302FA4546EB1B55B663531128734C7BC35E6B4749AD399B E032CAC7FE79E26326925C4C6E2301A893BFCD52CDA5E7E707E996D4828F3DD8 4AE2341F0E4A5B16496CAF85BCC9875E74B4F00F99310927D82DC8EBE5DC6A0F ADEA7B0E5FC2F42B47D35048400655479F2BDD42CB0800EEB2921BCB25FBB42C D2C9CD9573798277EFE9DB85B3D47D893B113A38CC3316FC678B15240E5CBB2A 3C9A05311047FACF72FCFF94E400385A42DD8C436C62ED9F4C83F13345407286 8FF942D6BF35000420EC9132060A0831561C8E3B07C85B0D078259739E455F9B 72047927B06E5E77EB961956E8588B4520A19501A712F90207638DF7E32CF96A 6043697BFC997D0DED458D1789ADE6E6C50EB08D1A83CA7B3D125B1BCD524BD5 AFCF63AD048B3C2D6F76D84AD7F7EC28BF1A2757A69593C53A320AC15C1FD530 80D73F39AE3872B72B7F56DD4FB6E82B7531CDF83B432EEF889284E046383969 598973C450BE7F925713364249DE9804B35D68054511F9AC2DF9B73CAB18008B 4C106794C363E2E36BE74DA075C9D539CB63F6A2FD8A4E750E55CA1F08B339F9 D7BB18719AAF5736144D949CFF67FF06C921388EB1191043CBF58E3E12E7AC07 7D3D8FC4A74C27ED123F010A3AFAFB4262294547526ADA3975402A0BAA6642E5 EEE99C0FF7B8F46F52A07258E52B29FCFB87DD0D38385B3A7E271496D16B9295 E898502C58373C387FAF464D2C7EE43F1A3A1A4512404BA73CA991CFF88B1FF8 EC8B94F87A8EA8736A770A2F906F47FE7DEBF3E1D77D08E8C51DE90BB16E4A7B A7C55E17E395F4F2C2771E4AE628FF8DD016B645BA02B1D4F708F8A0A5420C96 59B98C5D0CC3A6D3E68E1FF5BB968A34B1F1820773760B4D45A8B5873804C31C 9D9B4190EF0CB0CFC0E5A709DBEAFD3158D4E591AC64A528A603DA46F615821E BE48122412F0C712564D26F4E6EF3D469B6ED96DFDD8C278CB9ACA417DDECD57 C29227C135EF11CBA0EC76D9F6EFE35657BA9880282BC02923265EA39E984323 6887F83C4660ABCAF3050C88EC6557C2B6A4FF20CCB4EC8FD6646C3D8C236D62 4E87E21D24BC1379CFD9F08A39D1185AAE5D91F622903DE7584E046704C9CE06 8061D76A86C6B7C82FD4D03791D66482071AF2615F83FB462689A4E66686047B 76E4D866A9B7652321A78985DC89A9B08E84787B9BA7BF3FE7D0D6DE4F0A7450 04FA16CD33915E78CFFE0CAC3D813E1C80AB6AEAFCA393A44DA717C4D208439E CC5F37F6780B34D8FD2AC554C2363B307E7E8577E5E533288D153B5EA1604100 6520248C1C0C9A07D81495E66D3645A50CFBF498 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (BFP.dvi) @start /Fa 172[65 10[89 19[59 59 59 59 7[47 41[{ TeX74afc74cEncoding ReEncodeFont }7 99.6264 /CMBXTI10 rf /Fb 140[29 1[28 32 1[38 54 2[25 22 1[30 9[25 8[36 5[48 58 44[41 32[{ TeXaae443f0Encoding ReEncodeFont }13 49.8132 /CMMI6 rf /Fc 134[71 71 1[71 75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 1[65 12[94 1[100 1[92 101 105 4[50 2[85 88 3[102 13[67 67 67 2[37 45 32[75 12[{ TeXf7b6d320Encoding ReEncodeFont }35 119.552 /CMBX12 rf /Fd 207[18 47[48{ TeXbbad153fEncoding ReEncodeFont }2 49.8132 /CMSY6 rf /Fe 134[59 59 2[62 44 44 46 1[62 56 62 93 31 59 1[31 62 1[34 51 62 50 1[54 12[78 1[84 1[77 2[106 67 2[42 88 2[74 86 81 11[56 56 56 56 56 56 56 2[31 33[62 12[{ TeXf7b6d320Encoding ReEncodeFont }38 99.6264 /CMBX12 rf /Ff 140[24 11[30 41[47 9[30 30 30 30 4[47 1[24 24 40[{ TeXf7b6d320Encoding ReEncodeFont }10 49.8132 /CMR6 rf /Fg 151[52 104[{}1 99.6264 /EUFM10 rf /Fh 130[37 37 37 37 8[83 1[55 55 4[39 39 120 83 46 120 9[46 106 120 5[39 1[88 125 92 125 43[66 66 10[44 44 61 61 50 50 2[46 28 12[{}31 83.022 /CMEX10 rf /Fi 148[35 20 30[54 11[55 12[47 71 19 14[71 11[55 14[55 2[35 1[20 55{ TeXbbad153fEncoding ReEncodeFont }13 66.4176 /CMSY8 rf /Fj 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 9[97 1[72 70 55 2[66 2[87 2[51 38 72 75 64 66 74 70 1[72 5[30 30 1[50 50 50 50 50 50 50 50 50 1[30 35 30 2[40 40 20[50 6[55 60 11[{ TeX74afc74cEncoding ReEncodeFont }60 99.6264 /CMTI12 rf /Fk 163[232 188 66 7[72 3[72 10[72 67[{}6 99.6264 /MSBM10 rf /Fl 129[35 10[28 4[39 1[20 4[35 41[55 6[35 1[35 35 35 35 35 4[55 1[27 27 17[35 11[51 9[44{ TeXf7b6d320Encoding ReEncodeFont }18 66.4176 /CMR8 rf /Fm 129[35 3[33 1[40 3[25 33 32 32 36 1[43 62 21 37 29 24 41 34 1[33 3[37 29 2[27 27 7[41 3[45 1[56 68 48 3[58 4[50 5[35 1[20 20 22[42 33 1[46 1[42 2[40 1[41 1[35 42 41 2[33 2[29 31 1[40 45 11[{ TeXaae443f0Encoding ReEncodeFont }45 66.4176 /CMMI8 rf /Fn 141[83 3[50 2[50 28 39 39 50 50 9[66 66 3[98 3[60 2[69 79 3[76 2[84 59 72 53 1[52 65 80 8[55 0 4[66 100 15[100 6[77 1[77 2[77 77 2[77 1[50 4[77 1[77 1[77 3[77 28 77{ TeXbbad153fEncoding ReEncodeFont }39 99.6264 /CMSY10 rf /Fo 129[49 3[45 1[55 1[47 56 35 46 1[43 49 47 58 85 29 51 40 33 56 47 48 45 51 42 41 51 41 6[57 1[92 57 2[60 74 1[63 1[78 94 66 1[54 43 81 77 63 72 81 70 74 73 2[76 49 76 27 27 18[64 3[57 46 61 63 1[58 1[42 55 50 55 43 1[59 57 56 3[43 39 43 1[55 62 11[{ TeXaae443f0Encoding ReEncodeFont }68 99.6264 /CMMI12 rf /Fp 134[85 85 1[85 90 63 64 66 1[90 81 90 134 45 2[45 90 81 49 74 90 72 90 78 12[112 90 120 1[110 2[153 97 2[60 4[124 2[122 10[81 81 6[45 54 45[{ TeXf7b6d320Encoding ReEncodeFont }34 143.462 /CMBX12 rf /Fq 128[49 49 2[49 43 51 51 70 51 54 38 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54 49 2[49 27 49 27 60 2[100 73 73 70 54 72 76 66 76 73 89 61 76 50 35 73 77 64 66 75 70 69 73 3[76 1[27 27 49 49 49 49 49 49 49 49 49 49 1[27 33 27 76 49 38 38 27 3[81 49 27 10[49 1[49 5[81 54 54 57 70 2[76 3[68 1[81 61{ TeXf7b6d320Encoding ReEncodeFont }92 99.6264 /CMR12 rf /Fr 135[51 3[38 1[38 1[54 3[27 2[27 21[54 2[66 8[77 64 66 2[69 73 10[49 3[49 1[49 2[33 27 44[{ TeXf7b6d320Encoding ReEncodeFont }18 99.6264 /CMSL12 rf /Fs 137[44 46 32 33 33 2[42 2[23 6[37 3[42 7[62 5[46 9[30 62 2[57 2[59 20[28 23 44[{ TeXf7b6d320Encoding ReEncodeFont }17 83.022 /CMR10 rf /Ft 252[34 3[{ TeXbbad153fEncoding ReEncodeFont }1 58.1154 /CMSY7 rf /Fu 139[41 41 43 14[46 58 51 31[79 65[{ TeXf7b6d320Encoding ReEncodeFont }7 90.9091 /CMBX10 rf /Fv 252[42 3[{ TeXbbad153fEncoding ReEncodeFont }1 83.022 /CMSY10 rf /Fw 128[45 4[40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 6[56 4[68 66 51 2[62 2[83 3[33 68 71 59 62 69 66 64 1[71 4[25 1[45 45 2[45 1[45 2[45 1[25 30 25 2[35 35 27[51 53 11[{ TeXf7b6d320Encoding ReEncodeFont }55 90.9091 /CMR10 rf /Fx 128[59 4[52 4[65 1[46 46 1[65 59 65 1[33 62 1[33 65 59 1[52 65 52 1[59 10[88 5[80 5[60 3[76 3[83 88 10[59 3[59 1[59 2[39 45[{ TeXf7b6d320Encoding ReEncodeFont }27 119.552 /CMR12 rf /Fy 133[70 1[83 115 1[88 61 62 61 1[88 79 88 133 43 2[43 88 79 47 70 88 70 1[79 12[115 1[117 124 7[56 1[125 104 108 122 2[119 18[43 52 45[{ TeXf7b6d320Encoding ReEncodeFont }31 172.188 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 0 1 TeXDict begin 0 0 bop 666 976 a Fy(Infrared-Finite)53 b(Algorithms)f(in)f(QED)368 1183 y(I)t(I.)j(The)e(Expansion)g(of)g(the) g(Groundstate)g(of)g(an)587 1391 y(A)l(tom)f(In)l(teracting)h(with)g (the)g(Quan)l(tized)1338 1598 y(Radiation)g(Field)1567 1894 y Fx(V)-10 b(olk)m(er)37 b(Bac)m(h)829 2043 y Fw(FB)31 b(Mathematik;)h(Univ)m(ersit\177)-45 b(at)32 b(Mainz;)g(D-55099)h (Mainz;)986 2141 y(German)m(y)e(\(vbac)m(h@mathematik.uni-mainz.de\)) 1535 2349 y Fx(J)s(\177)-62 b(urg)38 b(F)-10 b(r\177)-59 b(ohlic)m(h)2219 2306 y Fv(\003)923 2499 y Fw(Inst.)30 b(f.)h(Theoretisc)m(he)g(Ph)m(ysik;)g(ETH)f(H\177)-45 b(onggerb)s(erg;)802 2597 y(CH-8093)33 b(Z)s(\177)-48 b(uric)m(h;)30 b(Switzerland)g(\(juerg@itp.ph)m(ys.ethz.c)m(h\))1441 2805 y Fx(Alessandro)37 b(Pizzo)923 2954 y Fw(Inst.)30 b(f.)h(Theoretisc)m(he)g(Ph)m(ysik;)g(ETH)f(H\177)-45 b(onggerb)s(erg;)801 3052 y(CH-8093)33 b(Z)s(\177)-48 b(uric)m(h;)29 b(Switzerland)i(\(pizzo@itp.ph)m(ys.ethz.c)m(h\))1530 3286 y Fx(20-April-2006)1676 3761 y Fu(Abstract)638 3929 y Fw(In)37 b(this)h(pap)s(er,)i(w)m(e)f(presen)m(t)f(an)g(explicit)i (and)d(constructiv)m(e)j(algorithm)501 4042 y(enabling)34 b(us)f(to)h(calculate)i(the)e(groundstate)g(and)e(the)i(groundstate)g (energy)501 4155 y(of)39 b(a)f(nonrelativistic)i(atom)f(minimally)g (coupled)f(to)h(the)f(quan)m(tized)h(radia-)501 4268 y(tion)c(\014eld)f(up)f(to)i(an)g(error)f(of)g(arbitrary)g(\014nite)g (order)g(in)g(the)h(feinstructure)501 4381 y(constan)m(t.)41 b(Because)26 b(of)f(infrared)g(div)m(ergences,)i(whic)m(h)e(in)m(v)-5 b(alidate)27 b(a)f(straigh)m(t-)501 4494 y(forw)m(ard)g(T)-8 b(a)m(ylor)28 b(expansion,)g(an)e(iterativ)m(e)j(construction)e(is)g (emplo)m(y)m(ed)h(to)f(re-)501 4607 y(mo)m(v)m(e)35 b(the)d(infrared)g (cuto\013)i(in)e(photon)h(momen)m(tum)f(space)i(and)e(to)h(pro)s(duce) 501 4720 y(a)e(con)m(v)m(ergen)m(t)i(algorithm.)p 257 5297 1296 4 v 369 5358 a Ft(\003)407 5388 y Fs(also)26 b(at)i(IHES,)f(Bures-sur-Yv)n(ette)p eop end %%Page: 1 2 TeXDict begin 1 1 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(1)257 573 y Fp(I)162 b(Description)60 b(of)j(the)e(Problem)i(and)f (Summary)479 756 y(of)54 b(Main)g(Results)257 975 y Fq(In)43 b(this)g(pap)s(er)g(w)m(e)g(pursue)h(our)e(program)g(to)g(con)m(trol)h (the)g(infrared)f(b)s(eha)m(viour)i(of)257 1095 y(nonrelativistic)26 b(QED.)d(W)-8 b(e)24 b(construct)h(an)e(explicit)i(and)f(constructiv)m (e)i(infrared-\014nite)257 1215 y(algorithm)34 b(for)f(the)h (groundstate)h(and)e(the)i(groundstate)f(energy)h(of)e(a)g (nonrelativis-)257 1336 y(tic)45 b(atom)f(in)m(teracting)h(with)g(the)f (quan)m(tized)i(radiation)e(\014eld,)k(determining)e(these)257 1456 y(quan)m(tities)41 b(up)e(to)f(error)g(terms)i(of)e(arbitrary)g (\014nite)h(order)g(in)g(the)g(\014nite)g(structure)257 1576 y(constan)m(t.)j(In)23 b(a)g(previous)h(pap)s(er,)h(w)m(e)f(ha)m (v)m(e)g(presen)m(ted,)k(using)23 b(an)g(iterativ)m(e)h(sc)m(heme,)j(a) 257 1697 y(con)m(v)m(ergen)m(t)f(expansion)g(for)d(the)i(groundstate)f (and)g(the)g(groundstate)h(energy)-8 b(,)26 b(assum-)257 1817 y(ing)33 b(that)g(the)h(feinstructure)h(constan)m(t)f Fo(\013)29 b Fn(\030)g Fq(1)p Fo(=)p Fq(137)j(is)h(a)g(su\016cien)m (tly)j(small)e(p)s(ositiv)m(e)257 1938 y(n)m(um)m(b)s(er.)57 b(Starting)36 b(from)g(this)h(result,)i(w)m(e)e(will)g(deriv)m(e)h(an)e (explicit)i(represen)m(tation)257 2058 y(for)i(the)h(groundstate)g(and) f(the)h(groundstate)f(energy)i(in)e(the)h(form)f(of)g(\014nite)h(sums) 257 2178 y(of)30 b(expressions)k(computable)d(in)g(terms)g(of)f (\014nitely)i(man)m(y)f(con)m(v)m(ergen)m(t)i(explicit)f(in)m(te-)257 2299 y(grals)i(and)g(an)g(error)g(term)h(b)s(ounded)f(ab)s(o)m(v)m(e)h (b)m(y)g Fo(const)23 b Fn(\001)g Fo(\013)2474 2263 y Fm(N)2541 2299 y Fq(,)35 b(for)e(an)m(y)i(giv)m(en)g Fo(N)40 b(>)30 b Fq(0.)257 2419 y(In)39 b(a)g(forthcoming)g(pap)s(er,)h (w)m(e)g(will)f(presen)m(t)i(a)d(mathematically)i(precise)h (justi\014ca-)257 2540 y(tion)f(of)e(Bohr's)i(frequency)h(condition)f (in)f(atomic)h(sp)s(ectroscop)m(y)-8 b(,)43 b(b)m(y)d(constructing)257 2660 y(an)27 b(asymptotic)i(expansion)f(of)e(the)i(scattering)g (amplitudes)g(for)f(Ra)m(yleigh)g(scattering)257 2780 y(of)32 b(ligh)m(t)h(at)g(an)f(atom)g([2].)404 2901 y(In)27 b(this)h(pap)s(er,)h(an)e(atom)g(is)h(describ)s(ed)h(as)e(a)g(quan)m (tum-mec)m(hanical)j(b)s(ound)e(state)257 3021 y(consisting)46 b(of)f(a)f(static,)49 b(p)s(ositiv)m(ely)d(c)m(harged,)j(p)s(oin)m (tlik)m(e)d(n)m(ucleus)h(surrounded)f(b)m(y)257 3141 y(electrons.)f(The)33 b(electrons)g(are)f(describ)s(ed)h(as)f (nonrelativistic,)i(p)s(oin)m(tlik)m(e)f(quan)m(tum-)257 3262 y(mec)m(hanical)43 b(particles)f(with)f(electric)h(c)m(harge)f Fn(\000)p Fo(e)h Fq(and)e(spin)2640 3223 y Fl(1)p 2640 3239 36 4 v 2640 3296 a(2)2685 3262 y Fq(,)j(as)e(originally)g(pro-)257 3382 y(p)s(osed)30 b(b)m(y)f(P)m(auli.)43 b(They)30 b(are)f(b)s(ound)g (to)f(the)i(n)m(ucleus)g(b)m(y)g(the)f(electrostatic)h(Coulom)m(b)257 3503 y(\014eld)h(and)f(in)m(teract)g(with)h(the)f(soft)g(mo)s(des)g(of) f(the)h(quan)m(tized)i(electromagnetic)f(\014eld.)404 3623 y(T)-8 b(o)46 b(k)m(eep)i(our)e(exp)s(osition)h(as)f(simple)i(as)e (p)s(ossible,)51 b(w)m(e)d(consider)f(a)f(h)m(ydrogen)257 3743 y(atom)e(consisting)i(of)e(a)g(single,)k(static)d(proton)f(of)g(c) m(harge)h Fo(e)g Fq(surrounded)g(b)m(y)h(only)257 3864 y(one)41 b(electron.)69 b(The)42 b(spin)g(of)e(the)h(electron)h(then)f (turns)g(out)g(to)f(b)s(e)h(an)g(inessen)m(tial)257 3984 y(complication,)53 b(and,)f(for)47 b(simplicit)m(y)-8 b(,)54 b(w)m(e)49 b(neglect)g(the)f(Zeeman)h(coupling)f(of)g(the)257 4104 y(magnetic)35 b(momen)m(t)f(of)f(the)i(electron)f(to)f(the)h(quan) m(tized)i(magnetic)e(\014eld.)47 b(It)34 b(w)m(ould,)257 4225 y(ho)m(w)m(ev)m(er,)28 b(not)23 b(b)s(e)g(di\016cult)h(to)e (include)j(the)e(Zeeman)h(term)f(in)g(our)g(analysis.)42 b(Through)257 4345 y(this)32 b(pap)s(er)g(w)m(e)g(follo)m(w)f(the)h (notation)f(and)g(con)m(v)m(en)m(tion)i(of)e([1].)43 b(F)-8 b(or)30 b(the)i(con)m(v)m(enience)257 4466 y(of)g(the)h(reader,) h(w)m(e)f(recall)g(the)g(main)g(de\014nitions.)404 4586 y(The)41 b(Hilb)s(ert)h(space)f(of)g(pure)g(state)g(v)m(ectors)i(of)d (the)h(system)i(just)e(describ)s(ed)i(is)257 4706 y(giv)m(en)34 b(b)m(y)1524 4827 y Fn(H)62 b Fq(:=)e Fn(H)1917 4842 y Fm(el)1998 4827 y Fn(\012)22 b(F)32 b Fo(;)1080 b Fq(\(I.1\))257 5001 y(where)33 b Fn(H)622 5016 y Fm(el)709 5001 y Fq(=)28 b Fo(L)879 4965 y Fl(2)918 5001 y Fq(\()p Fk(R)1028 4965 y Fl(3)1068 5001 y Fq(\))j(is)i(the)f(Hilb)s(ert)g(space)h(appropriate) f(for)f(the)i(description)g(of)e(a)257 5121 y(single)26 b(electron,)h(disregarding)e(its)g(spin,)j(and)c Fn(F)34 b Fq(is)25 b(the)g(F)-8 b(o)s(c)m(k)25 b(space)h(used)f(to)g(describ)s (e)257 5242 y(the)j(states)f(of)g(the)g(transv)m(erse)i(mo)s(des)e(of)g (the)g(quan)m(tized)h(electromagnetic)h(\014eld,)f(i.e.,)p eop end %%Page: 2 3 TeXDict begin 2 2 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(2)257 573 y(the)33 b Fj(photons)p Fq(.)43 b(More)33 b(explicitly)-8 b(,)1176 835 y Fn(F)69 b Fq(:=)1522 727 y Fi(1)1494 752 y Fh(M)1481 936 y Fm(N)7 b Fl(=0)1650 835 y Fn(F)1732 794 y Fl(\()p Fm(N)g Fl(\))1876 835 y Fo(;)159 b Fn(F)2144 794 y Fl(\(0\))2265 835 y Fq(=)28 b Fk(C)17 b Fq(\012)22 b Fo(;)731 b Fq(\(I.2\))257 1112 y(where)47 b(\012)f(is)f(the)h(v)-5 b(acuum)46 b(v)m(ector,)k(i.e.,)f (the)c(state)h(of)f(the)h(electromagnetic)g(\014eld)257 1232 y(without)33 b(an)m(y)h(excited)g(mo)s(des,)f(and)1222 1511 y Fn(F)1304 1470 y Fl(\()p Fm(N)7 b Fl(\))1486 1511 y Fq(:=)60 b Fn(S)1709 1526 y Fm(N)1841 1403 y(N)1810 1428 y Fh(O)1811 1611 y Fm(j)t Fl(=1)1952 1511 y Fg(h)22 b Fo(;)158 b(N)38 b Fn(\025)29 b Fq(1)22 b Fo(;)777 b Fq(\(I.3\))257 1799 y(where)34 b(the)f(Hilb)s(ert)g(space)h Fg(h)e Fq(of)g(state)h(v)m(ectors)h(of)f(a)f(single)h(photon)g(is)1465 2009 y Fg(h)60 b Fq(:=)g Fo(L)1806 1968 y Fl(2)1846 2009 y Fq([)p Fk(R)1945 1968 y Fl(3)2006 2009 y Fn(\002)23 b Fk(Z)2172 2024 y Fl(2)2212 2009 y Fq(])f Fo(;)1020 b Fq(\(I.4\))257 2220 y(with)31 b(photon)f(momen)m(tum)h(space)g(giv)m (en)g(b)m(y)g Fk(R)2029 2184 y Fl(3)2068 2220 y Fq(,)g(and)f Fk(Z)2379 2235 y Fl(2)2449 2220 y Fq(is)g(accoun)m(ting)h(for)e(the)h (t)m(w)m(o)257 2340 y(indep)s(enden)m(t)35 b(transv)m(erse)f(p)s (olarizations,)f(or)e(helicities,)k(of)c(a)h(photon.)44 b(In)32 b(Eq.)h(\(I.2\),)257 2460 y Fn(S)317 2475 y Fm(N)417 2460 y Fq(denotes)f(the)g(orthogonal)f(pro)5 b(jection)33 b(on)m(to)e(the)h(subspace)i(of)2787 2394 y Fh(N)2879 2420 y Fm(N)2879 2485 y(j)t Fl(=1)3022 2460 y Fg(h)e Fq(of)f(totally)257 2581 y(symmetric)38 b Fo(N)10 b Fq(-photon)35 b(w)m(a)m(v)m(e)j(functions,)f(in)f(accordance)g(with)g(the)g(fact)f (that)h(pho-)257 2701 y(tons)27 b(satisfy)h(Bose-Einstein)h (statistics.)43 b(Th)m(us,)29 b Fn(F)2165 2665 y Fl(\()p Fm(N)7 b Fl(\))2314 2701 y Fq(is)27 b(the)g(subspace)i(of)d Fn(F)36 b Fq(of)27 b(state)257 2822 y(v)m(ectors)k(for)d (con\014gurations)h(of)f(exactly)i Fo(N)40 b Fq(photons.)i(It)29 b(is)g(con)m(v)m(enien)m(t)j(to)c(represen)m(t)257 2942 y(the)34 b(Hilb)s(ert)f(space)i Fn(H)f Fq(as)f(the)h(space)g(of)f (square-in)m(tegrable)i(w)m(a)m(v)m(e)g(functions)f(on)f(the)257 3062 y(electron)h(p)s(osition)f(space)g Fk(R)1330 3026 y Fl(3)1402 3062 y Fq(with)g(v)-5 b(alues)34 b(in)f(the)g(photon)f(F)-8 b(o)s(c)m(k)33 b(space)h Fn(F)10 b Fq(,)32 b(i.e.,)1480 3273 y Fn(H)1626 3245 y(\030)1627 3277 y Fq(=)1763 3273 y Fo(L)1829 3232 y Fl(2)1869 3176 y Fh(h)1908 3273 y Fk(R)1980 3232 y Fl(3)2036 3273 y Fq(;)39 b Fn(F)2184 3176 y Fh(i)2245 3273 y Fo(:)1036 b Fq(\(I.5\))257 3483 y(The)34 b(dynamics)g(of)e(the)h(system)i(is)e(generated)g(b)m(y)h(the) f(Hamiltonian)783 3717 y Fo(H)8 b Fq(\()p Fo(\013)983 3648 y Ff(1)p 983 3660 31 4 v 983 3702 a(2)1027 3717 y Fq(\))60 b(:=)1288 3620 y Fh(\020)1360 3717 y Fn(\000)23 b Fo(i)1502 3691 y(~)1493 3717 y Fn(r)1575 3732 y Fm(~)-34 b(x)1642 3717 y Fq(+)22 b Fo(\013)1803 3676 y Fl(3)p Fm(=)p Fl(2)1931 3691 y Fo(~)1913 3717 y(A)p Fq(\()p Fo(\013)o(~)-47 b(x)p Fq(\))2180 3620 y Fh(\021)2229 3643 y Fl(2)2319 3717 y Fn(\000)50 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))50 b(+)2857 3691 y(\024)2831 3717 y Fo(H)30 b(:)339 b Fq(\(I.6\))257 3961 y(In)40 b(Eq.)g(\(I.6\),)834 3936 y Fo(~)824 3961 y Fn(r)906 3976 y Fm(~)-34 b(x)990 3961 y Fq(denotes)40 b(the)g(gradien)m(t)f(with)h(resp)s(ect)g(to)f (the)g(electron)h(p)s(osition)256 4090 y Fo(~)-48 b(x)43 b Fn(2)g Fk(R)536 4054 y Fl(3)575 4090 y Fq(,)h Fo(\013)751 4063 y Fn(\030)752 4094 y Fq(=)871 4090 y(1)p Fo(=)p Fq(137)c(is)h(the)h(feinstructure)h(constan)m(t,)2469 4065 y Fo(~)2451 4090 y(A)q Fq(\()n Fo(~)-47 b(x)p Fq(\))41 b(denotes)h(the)g(v)m(ector)257 4211 y(p)s(oten)m(tial)e(of)f(the)h (transv)m(erse)i(mo)s(des)e(of)f(the)h(quan)m(tized)h(electromagnetic)g (\014eld)f(in)257 4331 y(the)33 b Fj(Coulomb)h(gauge)p Fq(,)1537 4426 y Fo(~)1528 4451 y Fn(r)1610 4466 y Fm(~)-34 b(x)1677 4451 y Fn(\001)1744 4426 y Fo(~)1726 4451 y(A)q Fq(\()n Fo(~)-47 b(x)p Fq(\))60 b(=)g(0)22 b Fo(;)1083 b Fq(\(I.7\))257 4622 y Fo(V)56 b Fq(is)34 b(the)h(Coulom)m(b)g(p)s (oten)m(tial)f(of)g(electrostatic)h(attraction)f(of)g(the)g(electron)h (to)f(the)257 4742 y(n)m(ucleus.)63 b(The)39 b(general)f(assumption)i (on)e Fo(V)59 b Fq(for)38 b(whic)m(h)h(our)f(analysis)i(w)m(orks)f(is)g (the)257 4862 y(follo)m(wing:)257 5057 y Fe(Hyp)s(othesis)d(1.)41 b Fj(The)32 b(form)h(domain)f(of)h Fo(V)54 b Fj(includes)33 b(the)g(form)g(domain)f Fq(H)3156 5021 y Fl(1)3195 5057 y Fq(\()p Fk(R)3305 5021 y Fl(3)3344 5057 y Fq(\))i Fj(of)257 5178 y Fn(\000)p Fq(\001)i Fj(and,)e(for)h(any)g Fo(")27 b(>)h Fq(0)p Fj(,)34 b(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(onstant)35 b Fo(b)2327 5193 y Fm(")2392 5178 y Fo(<)27 b Fn(1)p Fj(,)34 b(such)h(that)1363 5388 y Fn(\006)p Fo(V)85 b Fn(\024)63 b Fo(")17 b Fq(\()p Fn(\000)p Fq(\001\))44 b(+)g Fo(b)2224 5403 y Fm(")2284 5388 y Fn(\001)21 b Fe(1)919 b Fq(\(I.8\))p eop end %%Page: 3 4 TeXDict begin 3 3 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(3)257 573 y Fj(on)39 b Fq(H)474 537 y Fl(1)514 573 y Fq(\()p Fk(R)624 537 y Fl(3)663 573 y Fq(\))p Fj(.)58 b(Mor)-5 b(e)g(over,)40 b Fq(lim)1389 588 y Fi(j)p Fm(x)p Fi(j!1)1630 573 y Fo(V)21 b Fq(\()p Fo(x)p Fq(\))36 b(=)g(0)p Fj(,)k(and)f Fo(e)2345 588 y Fm(el)2440 573 y Fq(:=)d(inf)24 b Fo(\033)t Fq(\()p Fo(H)2892 588 y Fm(el)2950 573 y Fq(\))36 b Fo(<)g Fq(0)p Fj(,)k(wher)-5 b(e)257 693 y Fo(H)338 708 y Fm(el)425 693 y Fq(:=)27 b Fn(\000)p Fq(\001)12 b(+)g Fo(V)23 b Fj(,)32 b(is)e(an)g(isolate)-5 b(d)30 b(eigenvalue)f(of)h(multiplicity)h(one)f(with)h(c)-5 b(orr)g(esp)g(ond-)257 814 y(ing)35 b(normalize)-5 b(d)33 b(eigenve)-5 b(ctor)34 b Fo(')1487 829 y Fm(el)1573 814 y Fn(2)28 b(H)1751 829 y Fm(el)1810 814 y Fj(,)35 b(i.e.,)1075 1003 y Fq(\001)p Fo(e)1201 1018 y Fm(el)1323 1003 y Fq(:=)63 b(inf)7 b Fn(f)p Fo(\033)t Fq(\()p Fo(H)1835 1018 y Fm(el)1915 1003 y Fn(\000)23 b Fo(e)2060 1018 y Fm(el)2119 1003 y Fq(\))f Fn(\\)g Fk(R)2339 961 y Fl(+)2398 1003 y Fn(g)28 b Fo(>)f Fq(0)22 b Fo(:)631 b Fq(\(I.9\))283 1166 y(\024)257 1191 y Fo(H)44 b Fq(is)36 b(the)h(Hamiltonian)f(of)g(the)g(quan)m (tized,)j(free)d(electromagnetic)i(\014eld.)55 b(This)37 b(op-)257 1312 y(erator)c(is)g(giv)m(en)g(b)m(y)1129 1500 y(\024)1103 1525 y Fo(H)68 b Fq(:=)1431 1442 y Fh(X)1415 1627 y Fm(\025)p Fl(=)p Fi(\006)1583 1408 y Fh(Z)1682 1525 y Fo(d)1733 1484 y Fl(3)1773 1525 y Fo(k)30 b(a)1905 1484 y Fi(\003)1945 1525 y Fq(\()1978 1499 y Fo(~)1983 1525 y(k)s(;)17 b(\025)p Fq(\))g Fn(j)2216 1499 y Fo(~)2221 1525 y(k)r Fn(j)g Fo(a)p Fq(\()2403 1499 y Fo(~)2408 1525 y(k)r(;)g(\025)p Fq(\))22 b Fo(;)611 b Fq(\(I.10\))257 1816 y(where)42 b Fo(a)598 1780 y Fi(\003)637 1816 y Fq(\()670 1790 y Fo(~)675 1816 y(k)s(;)17 b(\025)p Fq(\))40 b(and)g Fo(a)p Fq(\()1189 1790 y Fo(~)1194 1816 y(k)t(;)17 b(\025)p Fq(\))39 b(are)i(the)f(usual)h(photon)f(creation-)g(and)h (annihilation)257 1936 y(op)s(erators,)33 b(whic)m(h)h(satisfy)g(the)f (canonical)g(comm)m(utation)g(relations)860 2125 y([)p Fo(a)938 2089 y Fi(\003)978 2125 y Fq(\()1011 2099 y Fo(~)1016 2125 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)1336 2089 y Fi(\003)1376 2125 y Fq(\()1409 2099 y Fo(~)1414 2125 y(k)1468 2089 y Fi(0)1491 2125 y Fo(;)17 b(\025)1592 2089 y Fi(0)1615 2125 y Fq(\)])56 b(=)f([)p Fo(a)p Fq(\()1978 2099 y Fo(~)1983 2125 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)p Fq(\()2336 2099 y Fo(~)2341 2125 y(k)2395 2089 y Fi(0)2419 2125 y Fo(;)17 b(\025)2520 2089 y Fi(0)2543 2125 y Fq(\)])55 b(=)h(0)21 b Fo(;)368 b Fq(\(I.11\))1096 2281 y([)p Fo(a)p Fq(\()1207 2254 y Fo(~)1212 2281 y(k)s(;)17 b(\025)p Fq(\))g Fo(;)32 b(a)1532 2245 y Fi(\003)1572 2281 y Fq(\()1605 2254 y Fo(~)1610 2281 y(k)1664 2245 y Fi(0)1687 2281 y Fo(;)17 b(\025)1788 2245 y Fi(0)1811 2281 y Fq(\)])56 b(=)f Fo(\016)2106 2296 y Fm(\025\025)2188 2277 y Fd(0)2232 2281 y Fo(\016)t Fq(\()2312 2254 y Fo(~)2317 2281 y(k)25 b Fn(\000)2487 2254 y Fo(~)2492 2281 y(k)2546 2245 y Fi(0)2570 2281 y Fq(\))d Fo(;)603 b Fq(\(I.12\))1550 2436 y Fo(a)p Fq(\()1634 2410 y Fo(~)1639 2436 y(k)s(;)17 b(\025)p Fq(\))g(\012)55 b(=)g(0)22 b Fo(;)1057 b Fq(\(I.13\))257 2635 y(for)32 b(all)537 2609 y Fo(~)542 2635 y(k)s(;)635 2609 y(~)640 2635 y(k)694 2599 y Fi(0)745 2635 y Fn(2)c Fk(R)911 2599 y Fl(3)983 2635 y Fq(and)k Fo(\025;)17 b(\025)1330 2599 y Fi(0)1381 2635 y Fn(2)28 b Fk(Z)1541 2650 y Fl(2)1608 2635 y Fn(\021)h(f\006g)p Fq(.)404 2765 y(The)k(v)m(ector)h(p)s(oten)m(tial,)1354 2739 y Fo(~)1336 2765 y(A)p Fq(\()n Fo(~)-47 b(x)p Fq(\),)33 b(in)g(the)g(Coulom)m(b)g (gauge)g(is)g(giv)m(en)g(b)m(y)297 2937 y Fo(~)279 2962 y(A)p Fq(\()n Fo(~)-47 b(x)q Fq(\))60 b(:=)2613 b(\(I.14\))578 3099 y(1)p 455 3144 294 4 v 455 3235 a(\(2)p Fo(\031)t Fq(\))639 3206 y Fl(3)p Fm(=)p Fl(2)791 3084 y Fh(X)775 3268 y Fm(\025)p Fl(=)p Fi(\006)943 3050 y Fh(Z)1118 3099 y Fo(d)1169 3063 y Fl(3)1208 3099 y Fo(k)p 1053 3144 275 4 v 1053 3160 a Fh(q)p 1136 3160 192 4 v 112 x Fq(2)17 b Fn(j)1225 3245 y Fo(~)1230 3272 y(k)r Fn(j)1354 3167 y Fq(\003\()1455 3141 y Fo(~)1460 3167 y(k)r Fq(\))1568 3071 y Fh(n)1622 3167 y Fo(~)-48 b(")p Fq(\()1702 3141 y Fo(~)1707 3167 y(k)s(;)17 b(\025)p Fq(\))p Fo(e)1945 3126 y Fi(\000)p Fm(i)2021 3108 y(~)2024 3126 y(k)q Fi(\001)o Fm(~)-34 b(x)2126 3167 y Fo(a)2177 3126 y Fi(\003)2216 3167 y Fq(\()2249 3141 y Fo(~)2254 3167 y(k)s(;)17 b(\025)p Fq(\))38 b(+)g Fo(~)-48 b(")p Fq(\()2679 3141 y Fo(~)2684 3167 y(k)s(;)17 b(\025)p Fq(\))2877 3126 y Fi(\003)2916 3167 y Fo(e)2961 3126 y Fm(i)2982 3108 y(~)2985 3126 y(k)q Fi(\001)o Fm(~)-34 b(x)3087 3167 y Fo(a)p Fq(\()3171 3141 y Fo(~)3176 3167 y(k)s(;)17 b(\025)p Fq(\))3369 3071 y Fh(o)3446 3167 y Fo(;)257 3480 y Fq(where)35 b(\003\()641 3454 y Fo(~)646 3480 y(k)s Fq(\))e(is)h(the)f(c)m(haracteristic)i (function)f(of)f(the)h(ball)f Fn(f)2537 3454 y Fo(~)2542 3480 y(k)f Fn(2)d Fk(R)2792 3444 y Fl(3)2848 3480 y Fn(j)g(j)2928 3454 y Fo(~)2933 3480 y(k)r Fn(j)g(\024)g Fo(\024)p Fn(g)34 b Fq(\(or)e(a)257 3611 y(nonnegativ)m(e,)c(smo)s(oth)c(appro)m (ximation)i(thereof)7 b(\),)26 b(and)d Fo(~)-48 b(")p Fq(\()2452 3585 y Fo(~)2457 3611 y(k)s(;)17 b Fq(+\),)25 b Fo(~)-48 b(")o Fq(\()2800 3585 y Fo(~)2805 3611 y(k)s(;)17 b Fn(\000)p Fq(\))25 b(are)f(photon)257 3731 y(p)s(olarization)33 b(v)m(ectors,)h(i.e.,)f(t)m(w)m(o)g(unit)g(v)m(ectors)h(in)f Fk(C)22 b Fn(\012)h Fk(R)2440 3695 y Fl(3)2512 3731 y Fq(satisfying)913 3920 y Fo(~)-48 b(")p Fq(\()993 3894 y Fo(~)998 3920 y(k)s(;)17 b(\025)p Fq(\))1191 3879 y Fi(\003)1252 3920 y Fn(\001)j Fo(~)-47 b(")o Fq(\()1380 3894 y Fo(~)1385 3920 y(k)s(;)17 b(\026)p Fq(\))55 b(=)g Fo(\016)1809 3935 y Fm(\025\026)1919 3920 y Fo(;)2147 3894 y(~)2152 3920 y(k)25 b Fn(\001)c Fo(~)-48 b(")p Fq(\()2357 3894 y Fo(~)2362 3920 y(k)s(;)17 b(\025)p Fq(\))54 b(=)i(0)22 b Fo(;)421 b Fq(\(I.15\))257 4122 y(for)43 b Fo(\025;)17 b(\026)44 b Fq(=)i Fn(\006)p Fq(.)75 b(The)44 b(equation)1536 4096 y Fo(~)1541 4122 y(k)33 b Fn(\001)27 b Fo(~)-47 b(")o Fq(\()1760 4096 y Fo(~)1765 4122 y(k)s(;)17 b(\025)p Fq(\))45 b(=)g(0)e(expresses)j(the)d(Coulom)m (b)h(gauge)257 4243 y(condition.)404 4363 y(The)34 b(function)g(\003\() 1089 4337 y Fo(~)1094 4363 y(k)r Fq(\))f(insures)i(that)e(mo)s(des)h (of)f(the)h(electromagnetic)g(\014eld)g(corre-)257 4494 y(sp)s(onding)k(to)f(w)m(a)m(v)m(e)i(v)m(ectors)1375 4467 y Fo(~)1380 4494 y(k)h Fq(with)e Fn(j)1721 4467 y Fo(~)1726 4494 y(k)s Fn(j)d(\025)h Fo(\024)h Fq(do)h(not)f(in)m (teract)h(with)f(the)h(electron;)257 4614 y(i.e.,)51 b(\003)45 b(represen)m(ts)k(an)d Fj(ultr)-5 b(aviolet)47 b(cuto\013)68 b Fq(that)46 b(will)g(b)s(e)h(k)m(ept)g(\014xed)g (throughout)257 4734 y(our)d(analysis.)79 b(The)44 b(v)m(ector)h(p)s (oten)m(tial)f(de\014ned)h(in)f(\(I.14\))g(is)g(th)m(us)h(cut)f(o\013)g (in)g(the)257 4855 y(ultra)m(violet.)404 4975 y(In)33 b(the)g(follo)m(wing,)g(w)m(e)g(simplify)h(our)f(notation)f(b)m(y)h (setting)257 5188 y Fo(k)52 b Fq(:=)c(\()544 5162 y Fo(~)549 5188 y(k)s(;)17 b(\025)p Fq(\))k Fo(;)111 b(!)t Fq(\()p Fo(k)s Fq(\))48 b Fn(\021)h(j)p Fo(k)s Fn(j)f Fq(:=)g Fn(j)1602 5162 y Fo(~)1607 5188 y(k)s Fn(j)21 b Fo(;)111 b Fq(and)2116 5071 y Fh(Z)2216 5188 y Fo(f)11 b Fq(\()p Fo(k)s Fq(\))20 b Fo(dk)52 b Fq(:=)2746 5105 y Fh(X)2730 5290 y Fm(\025)p Fl(=)p Fi(\006)2898 5071 y Fh(Z)2997 5188 y Fo(f)11 b Fq(\()3089 5162 y Fo(~)3094 5188 y(k)s(;)17 b(\025)p Fq(\))g Fo(d)3355 5147 y Fl(3)3393 5188 y Fo(k)24 b(;)3260 5388 y Fq(\(I.16\))p eop end %%Page: 4 5 TeXDict begin 4 4 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(4)257 573 y(for)32 b(an)m(y)i(in)m(tegrable)f(\(p)s(ossibly)h(v)m (ector-v)-5 b(alued\))34 b(function)f Fo(f)11 b Fq(.)404 693 y(Next,)33 b(w)m(e)h(summarize)g(the)f(plan)g(and)g(the)g(main)f (results)i(of)f(this)g(pap)s(er.)404 814 y(In)i(the)h(remainder)g(of)e (this)i(section,)h(w)m(e)f(recall)f(the)h(main)f(ideas)h(of)f(the)g Fj(iter)-5 b(ate)g(d)257 934 y(analytic)29 b(p)-5 b(erturb)g(ation)29 b(metho)-5 b(d)p Fq(,)26 b(in)m(tro)s(duced)h(in)g([7],)g(and)f(some)h (basic)f(notation)g(used)257 1063 y(for)35 b(the)g(construction)h(in)f ([1])g(of)f(the)h(groundstate)h(of)e(the)h(Hamiltonian)g Fo(H)8 b Fq(\()p Fo(\013)3263 999 y Ff(1)p 3262 1011 31 4 v 3262 1053 a(2)3307 1063 y Fq(\).)50 b(It)257 1184 y(is)30 b(pro)m(v)m(en)g(that)e(the)h(groundstate)g(is)h(unique,)h(pro) m(vided)f(the)f(feinstructure)h(constan)m(t)257 1304 y Fo(\013)j Fq(is)f(small)h(enough,)f(dep)s(ending)h(on)f(the)g(v)-5 b(alue)32 b(of)f(the)i(ultra)m(violet)f(cuto\013)g Fo(\024)p Fq(.)44 b(\(F)-8 b(or)30 b(a)257 1424 y(discussion)c(of)d(related)h (results)g(in)g(the)g(literature)g(w)m(e)g(refer)g(to)f([1]\).)40 b(The)24 b(main)g(feature)257 1545 y(of)30 b(the)g(analysis)i(in)e([1]) g(is)h(the)f(ac)m(hiev)m(emen)m(t)j(of)d(an)g Fj(explicit,)i(inductive) g(c)-5 b(onstruction)257 1674 y Fq(of)44 b(\()p Fo(E)490 1689 y Fl(gs)604 1674 y Fn(\021)k Fo(E)801 1689 y Fl(gs)869 1674 y Fq(\()p Fo(\013)q Fq(\))p Fo(;)17 b(\036)1110 1689 y Fl(gs)1223 1674 y Fn(\021)48 b Fo(\036)1406 1689 y Fl(gs)1473 1674 y Fq(\()p Fo(\013)1584 1610 y Ff(1)p 1583 1622 V 1583 1663 a(2)1628 1674 y Fq(\)\),)e(based)f(on)f(a)g (decomp)s(osition)h(of)f(the)g(terms)257 1806 y(in)38 b(the)g(Hamiltonian)g Fo(H)44 b Fn(\021)36 b Fo(H)8 b Fq(\()p Fo(\013)1553 1742 y Ff(1)p 1553 1754 V 1553 1795 a(2)1597 1806 y Fq(\))38 b(that)f(describ)s(e)i(the)f(in)m(teractions)h (b)s(et)m(w)m(een)h(the)257 1927 y(electron)31 b(and)f(the)g(photons)g (in)m(to)g(a)f(sum)i(of)e(terms)i(with)f(the)g(prop)s(ert)m(y)h(that)e (at)h(least)257 2047 y(one)j(photon)g(creation-)f(or)g(annihilation)h (op)s(erator)f(is)h(lo)s(calized)h(in)f(a)f(shell)287 2171 y Fh(n)342 2267 y Fo(k)424 2167 y Fh(\014)424 2217 y(\014)424 2267 y(\014)g Fo(\024\014)601 2226 y Fm(n)p Fl(+1)766 2267 y Fn(\024)c(j)p Fo(k)s Fn(j)f(\024)h Fo(\024\014)1230 2226 y Fm(n)1277 2171 y Fh(o)1354 2267 y Fo(;)88 b(\024)28 b Fn(\025)g Fq(1)22 b Fo(;)87 b Fq(0)28 b Fo(<)f(\014)34 b(<)27 b(\024)2271 2226 y Fi(\000)p Fl(1)2388 2267 y Fo(;)87 b(n)28 b Fn(2)g Fk(N)2754 2282 y Fl(0)2821 2267 y Fq(=)g Fn(f)p Fq(0)p Fo(;)17 b Fq(1)p Fo(;)g Fq(2)p Fo(;)g(:)g(:)g(:)n Fn(g)22 b Fo(;)3260 2400 y Fq(\(I.17\))257 2520 y(of)i(photon)f(momen)m(tum)i(space.)42 b(In)24 b(the)g(presen)m(t)h(pap)s(er)f(w)m(e)h(set)f Fo(\014)33 b Fq(:=)28 b Fo(\013)c Fq(whic)m(h)h(is)f(com-)257 2641 y(patible)32 b(with)f([1].)43 b(Our)31 b(inductiv)m(e)i(construction)f (yields)g(a)f(sequence)i(\()p Fo(E)3015 2656 y Fm(n)3062 2641 y Fo(;)17 b(\036)3164 2656 y Fm(n)3211 2641 y Fq(\))3249 2604 y Fi(1)3249 2665 y Fm(n)p Fl(=0)3417 2641 y Fq(of)257 2761 y(appro)m(ximate)28 b(groundstate)f(energies)g(and)g(groundstates) g(constructed)h(from)e(Hamil-)257 2881 y(tonians)34 b Fo(H)681 2896 y Fm(n)760 2881 y Fq(with)f(an)g(infrared)g(cuto\013)g (at)f Fn(j)p Fo(k)s Fn(j)c Fq(=)g Fo(\024\013)2238 2845 y Fm(n)2284 2881 y Fq(.)44 b(More)34 b(precisely)-8 b(,)35 b(the)e(expres-)257 3002 y(sion)47 b(for)f Fo(H)716 3017 y Fm(n)809 3002 y Fq(is)h(obtained)f(from)g(the)h(one)f(for)g Fo(H)54 b Fq(b)m(y)47 b(replacing)g(the)g(cuto\013)f(\003)g(in)257 3122 y(De\014nition)38 b(\(I.14\))e(of)g(the)i(v)m(ector)g(p)s(oten)m (tial)2007 3097 y Fo(~)1989 3122 y(A)p Fq(\()n Fo(~)-47 b(x)p Fq(\))37 b(b)m(y)h(the)f(c)m(haracteristic)i(function)257 3253 y(of)32 b Fn(f)413 3226 y Fo(~)418 3253 y(k)f Fn(2)d Fk(R)666 3216 y Fl(3)722 3253 y Fn(j)f Fo(\024\013)896 3216 y Fm(n)971 3253 y Fo(<)g Fn(j)1097 3226 y Fo(~)1102 3253 y(k)s Fn(j)h(\024)g Fo(\024)p Fn(g)p Fq(.)43 b(It)33 b(has)g(b)s(een)g(sho)m(wn)h(in)f([1])f(that)1208 3473 y(lim)1184 3522 y Fm(n)p Fi(!1)1384 3473 y Fo(E)1456 3488 y Fm(n)1531 3473 y Fq(=)c Fo(E)1707 3488 y Fl(gs)1796 3473 y Fo(;)159 b Fq(lim)1958 3522 y Fm(n)p Fi(!1)2159 3473 y Fo(\036)2217 3488 y Fm(n)2291 3473 y Fq(=)28 b Fo(\036)2453 3488 y Fl(gs)2542 3473 y Fo(:)691 b Fq(\(I.18\))257 3708 y(These)40 b(results)f(and)e(some)h(underlying)h(tec)m(hnical)g (to)s(ols)f(are)f(collected)i(in)f(Sect.)g(I.2)257 3828 y(and)31 b(will)g(b)s(e)f(widely)i(used)g(in)e(the)h(re-expansion)h(of) e Fo(\036)2291 3843 y Fl(gs)2388 3828 y Fq(in)g Fo(\013)2572 3764 y Ff(1)p 2572 3776 V 2572 3817 a(2)2617 3828 y Fq(.)42 b(T)-8 b(o)31 b(outline)g(the)f(gen-)257 3949 y(eral)k(ideas)h (underlying)g(our)f(re-expansion)h(algorithm)f(is)g(the)g(purp)s(ose)h (of)e(Sect.)i(I.3.)257 4069 y(W)-8 b(e)37 b(include)h(this)f(section)h (for)e(the)h(con)m(v)m(enience)j(of)c(the)h(reader,)h(to)e(ease)h(the)g (read-)257 4189 y(ing)d(of)f(Sects.)i(I)s(I)e(and)h(I)s(I)s(I,)g(in)f (whic)m(h)j(the)d(actual)h(construction)h(of)e(the)h(re-expanded)257 4310 y(groundstate)f(v)m(ector)h(is)f(carried)g(out.)404 4430 y(In)e(Sect.)h(I)s(I,)f(w)m(e)h(start)f(b)m(y)h(exploiting)f(the)h (infrared)f(asymptotic)h(freedom)f(of)g(the)257 4551 y(mo)s(del:)44 b(As)33 b(follo)m(ws)h(from)e([1],)h(w)m(e)g(ha)m(v)m(e) 1367 4771 y Fn(j)p Fo(E)1467 4786 y Fm(n)1536 4771 y Fn(\000)23 b Fo(E)1708 4786 y Fl(gs)1776 4771 y Fn(j)82 b(\024)i(O)s Fq(\()p Fo(\013)2230 4729 y Fm(N)2297 4771 y Fq(\))22 b Fo(;)876 b Fq(\(I.19\))1352 4916 y Fn(k)p Fo(\036)1460 4931 y Fm(n)1529 4916 y Fn(\000)22 b Fo(\036)1686 4931 y Fl(gs)1753 4916 y Fn(k)83 b(\024)h(O)s Fq(\()p Fo(\013)2230 4875 y Fm(N)2297 4916 y Fq(\))22 b Fo(;)876 b Fq(\(I.20\))257 5136 y(for)35 b(an)m(y)h(\014nite)g Fo(N)43 b(>)33 b Fq(0,)j(pro)m(vided)h Fo(n)32 b Fn(\025)h Fo(const)25 b Fn(\001)40 b Fo(N)10 b Fq(,)37 b(b)s(ecause)g(the)e(in)m (teraction)i(term)257 5256 y(in)47 b(the)g(Hamiltonian)g Fo(H)54 b Fq(b)s(eha)m(v)m(es)48 b(as)f(it)f(w)m(as)i Fj(irr)-5 b(elevant)55 b Fq(in)47 b(the)g Fj(infr)-5 b(ar)g(e)g(d)p Fq(,)49 b(when)257 5377 y(restricted)34 b(to)f(a)f(sp)s(ectral)h(region)g(close)h(to)e(the)h(groundstate)g (energy)-8 b(.)p eop end %%Page: 5 6 TeXDict begin 5 5 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(5)257 573 y(W)-8 b(e)34 b(then)f(in)m(tro)s(duce)h(some)f(in)m (termediate)i(ob)5 b(jects)34 b(at)e(all)h(scales)h Fo(m)p Fq(,)g(with)f(0)28 b Fn(\024)g Fo(m)h Fn(\024)257 693 y Fo(const)g Fn(\001)f Fo(N)10 b Fq(;)46 b(namely)d(truncated)f(ground) f(states,)k(\()p Fo(\036)2283 708 y Fm(m)2349 693 y Fq(\))2387 657 y Fm(T)2442 693 y Fq(,)f(and)d(truncated)i(ground-)257 814 y(state)e(energy)g(shifts,)i(\()p Fo(E)1234 777 y Fm(m)1228 838 y(m)p Fl(+1)1384 814 y Fq(\))1422 777 y Fm(T)1477 814 y Fq(.)66 b(They)41 b(allo)m(w)g(us)f(to)g(compute)h(the) f(quan)m(tities)i(w)m(e)257 934 y(are)31 b(in)m(terested)i(in)e(up)h (to)e(an)h(arbitrary)g(c)m(hosen)h(accuracy)g(in)g Fo(\013)f Fq(b)m(y)h(p)s(erforming)f(a)g(\014-)257 1054 y(nite)e(\(b)s(ecause)g (of)f(the)g(truncation\))h(n)m(um)m(b)s(er)g(of)f(op)s(erations)g(in)g (the)h(scale-dep)s(enden)m(t)257 1175 y(resolv)m(en)m(ts)36 b(of)c(the)h(Hamiltonians)g Fo(H)1665 1190 y Fm(n)1744 1175 y Fq(and)g(on)f(in)m(teractions)i(terms.)404 1295 y(In)39 b(Sect.)i(I)s(I)s(I,)g(w)m(e)f(presen)m(t)h(the)f(asymptotic)h (expansion)g(of)e Fo(\036)2773 1310 y Fm(g)r(s)2885 1295 y Fq(and)g Fo(E)3153 1310 y Fm(g)r(s)3226 1295 y Fq(.)64 b(It)40 b(is)257 1416 y(deriv)m(ed)34 b(from)e(a)g(re-expansion)h(of)f (the)g(\014nitely)i(man)m(y)f(scale-dep)s(enden)m(t)h(quan)m(tities,) 257 1536 y(de\014ning)j(the)g(truncated)g(groundstates)g(and)f (groundstate)h(energy)g(shifts)g(in)f(terms)257 1656 y(of)27 b(\014nitely)h(man)m(y)f(bare)g(quan)m(tities,)j(\(con)m(v)m (ergen)m(t)f(in)m(tegrals)f(in)f(resolv)m(en)m(ts)i(of)d Fo(H)3272 1671 y Fl(0)3338 1656 y Fq(and)257 1791 y(com)m(binations)40 b(of)d Fo(~)-49 b(p)36 b Fq(=)h Fn(\000)p Fo(i)1290 1766 y(~)1281 1791 y Fn(r)1363 1806 y Fm(~)-34 b(x)1446 1791 y Fq(and)38 b(of)f Fo(e)1802 1755 y Fi(\006)p Fm(i)1878 1737 y(~)1881 1755 y(k)r Fi(\001)p Fm(\013)o(~)-34 b(x)2028 1791 y Fq(\).)59 b(In)38 b(Corollary)g(I)s(I)s(I.7)g(w)m(e)h(form)m (ulate)257 1911 y(our)33 b(\014nal)g(result,)g(whic)m(h)h(w)m(e)g (restate)f(b)s(elo)m(w.)257 2032 y Fe(Main)39 b(Result)404 2152 y Fj(F)-7 b(or)32 b(any)i Fo(N)43 b Fj(and)33 b(for)h Fo(\013)g Fj(in)f(a)g(\014xe)-5 b(d)33 b(interval)g Fq(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\(2)p Fo(N)29 b Fn(\000)19 b Fq(1\)\))p Fj(,)34 b(wher)-5 b(e)42 b Fq(\026)-59 b Fo(\013)q Fq(\(2)p Fo(N)29 b Fn(\000)20 b Fq(1\))257 2281 y Fj(is)29 b Fo(N)10 b Fj(-dep)-5 b(endent,)30 b Fo(E)1033 2296 y Fl(gs)1128 2281 y Fn(\021)e Fo(E)1305 2296 y Fl(gs)1373 2281 y Fq(\()p Fo(\013)q Fq(\))g Fj(and)h Fo(\036)1782 2296 y Fl(gs)1876 2281 y Fn(\021)g Fo(\036)2040 2296 y Fl(gs)2107 2281 y Fq(\()p Fo(\013)2218 2217 y Ff(1)p 2217 2229 31 4 v 2217 2270 a(2)2262 2281 y Fq(\))g Fj(have)f(exp)-5 b(ansions)28 b(of)h(the)g(form)1036 2576 y Fo(E)1108 2591 y Fl(gs)1176 2576 y Fq(\()p Fo(\013)q Fq(\))82 b(=)h Fo(E)1628 2591 y Fl(0)1690 2576 y Fq(+)1816 2468 y Fm(N)1788 2493 y Fh(X)1788 2677 y Fm(`)p Fl(=3)1924 2576 y Fo(")1970 2591 y Fm(`)2020 2576 y Fq(\()p Fo(\013)q Fq(\))17 b Fo(\013)2239 2535 y Fm(`)2320 2576 y Fq(+)50 b Fo(o)p Fq(\()p Fo(\013)2594 2535 y Fm(N)2661 2576 y Fq(\))561 b(\(I.21\))996 2880 y Fo(\036)1054 2895 y Fl(gs)1121 2880 y Fq(\()p Fo(\013)1232 2812 y Ff(1)p 1232 2824 V 1232 2865 a(2)1276 2880 y Fq(\))83 b(=)g Fo(\036)1614 2895 y Fl(0)1675 2880 y Fq(+)1784 2772 y Fl(2)p Fm(N)1773 2797 y Fh(X)1774 2982 y Fm(`)p Fl(=3)1910 2880 y Fo(')1974 2895 y Fm(`)2007 2880 y Fq(\()p Fo(\013)q Fq(\))17 b Fo(\013)2226 2839 y Fm(`=)p Fl(2)2378 2880 y Fq(+)50 b Fo(o)p Fq(\()p Fo(\013)2652 2839 y Fm(N)2719 2880 y Fq(\))503 b(\(I.22\))257 3164 y Fj(with)1024 3381 y Fq(lim)1016 3438 y Fm(\013)p Fi(!)p Fl(0)1184 3381 y Fo(\013)1247 3340 y Fm(\016)1284 3381 y Fn(j)p Fo(")1358 3396 y Fm(`)1407 3381 y Fq(\()p Fo(\013)q Fq(\))p Fn(j)27 b Fq(=)g(0)598 b Fn(8)p Fo(\016)32 b(>)c Fq(0)626 b(\(I.23\))961 3567 y(lim)953 3623 y Fm(\013)p Fi(!)p Fl(0)1121 3567 y Fo(\013)1184 3526 y Fm(\016)1222 3567 y Fn(k)p Fo(')1336 3582 y Fm(`)1385 3567 y Fq(\()p Fo(\013)q Fq(\))p Fn(k)27 b Fq(=)g(0)598 b Fn(8)p Fo(\016)32 b(>)c Fq(0)626 b(\(I.24\))257 3811 y Fj(for)47 b(arbitr)-5 b(ary)48 b Fo(N)61 b Fq(=)50 b(1)p Fo(;)17 b Fq(2)p Fo(;)g Fq(3)p Fo(;)g(:)g(:)g(:)n Fj(.)82 b(In)46 b(exp)-5 b(ansions)46 b(\(I.21\))g(and)h(\(I.22\),)h Fo(")3122 3826 y Fm(`)3155 3811 y Fq(\()p Fo(\013)q Fq(\))f Fj(and)257 3932 y Fo(')321 3947 y Fm(`)354 3932 y Fq(\()p Fo(\013)q Fq(\))34 b Fj(ar)-5 b(e)34 b(c)-5 b(omputable)34 b(in)g(terms)g(of)g(\014nitely)h(many)f(c)-5 b(onver)g(gent)33 b(inte)-5 b(gr)g(als,)34 b(for)g(any)257 4052 y Fq(3)28 b Fn(\024)g Fo(`)g(<)f Fn(1)p Fj(.)257 4173 y Fq(Eqs.)40 b(\(I.23\),)f(\(I.24\))e(accoun)m(t)i(for)f(the)g(p)s(ossible)h(app)s (earance)g(of)f(p)s(o)m(w)m(ers)h(of)f(ln[1)p Fo(=\013)q Fq(])257 4293 y(\(\\infrared)29 b(logarithms"\).)43 b(W)-8 b(e)29 b(exp)s(ect)h(that)f(suc)m(h)h(infrared)f(logarithms)g(are)g (not)g(ar-)257 4413 y(tifact)f(of)g(our)g(algorithm,)h(but)g(are)f(an)h (expression)h(of)e(infrared)g(div)m(ergences)k(in)c(naiv)m(e)257 4534 y(p)s(erturbation)i(theory:)43 b(The)30 b(quan)m(tities)i Fo(E)1886 4549 y Fl(gs)1953 4534 y Fq(\()p Fo(\013)q Fq(\))d(and)h Fo(\036)2366 4549 y Fl(gs)2433 4534 y Fq(\()p Fo(\013)2544 4470 y Ff(1)p 2544 4482 V 2544 4523 a(2)2588 4534 y Fq(\))g(are)g(not)f(analytic)h(nor)257 4666 y(ev)m(en)42 b(smo)s(oth)e(at)g Fo(\013)1039 4602 y Ff(1)p 1039 4614 V 1039 4655 a(2)1124 4666 y Fq(=)g(0,)i(but)e(deriv)-5 b(ativ)m(es)42 b(in)f Fo(\013)2232 4602 y Ff(1)p 2231 4614 V 2231 4655 a(2)2316 4666 y Fq(of)e(su\016cien)m(tly)k(high)d (order)h(of)257 4786 y(these)34 b(quan)m(tities)h(div)m(erge)f(as)e Fo(\013)d Fn(!)e Fq(0.)404 4907 y(T)-8 b(o)31 b(our)g(kno)m(wledge,)j (a)d(rigorous)g(asymptotic)i(expansion)f Fj(up)i(to)g(any)g(or)-5 b(der)33 b(in)h Fo(\013)257 5027 y Fq(of)40 b(the)g(groundstate)h(and)f (the)g(groundstate)g(energy)h(in)g(nonrelativistic)g(QED)f(has)257 5147 y(nev)m(er)c(b)s(een)e(pro)m(vided)h(in)f(the)h(literature)f(b)s (efore.)47 b(An)34 b(expansion)h(to)f(leading)g(order)257 5268 y(follo)m(ws)24 b(from)e(the)h(pro)s(of)f(of)g(the)h(existence)i (of)d(the)h(groundstate)g(con)m(tained)g(in)g([4];)j(\(see)257 5388 y(also)35 b([3])g(for)f(an)h(approac)m(h)g(to)g(the)g(problem)h(b) m(y)g(renormalization)f(group)g(analysis\).)p eop end %%Page: 6 7 TeXDict begin 6 6 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(6)257 573 y(W)-8 b(e)43 b(stress)g(that)f(the)g(expansion)h(of)e (the)i(groundstate)f(is)g(crucial)h(in)f(constructing)257 693 y(similar)i(expansions)h(of)d(ph)m(ysical)j(quan)m(tities,)i(suc)m (h)e(as)e(scattering)g(amplitudes)i(of)257 814 y(Ra)m(yleigh)28 b(scattering.)43 b(The)28 b(expansion)h(of)e(the)g(groundstate)h (energy)g(is)g(a)f(b)m(ypro)s(duct)257 934 y(that)40 b(is)h(less)h(signi\014can)m(t.)67 b(T)-8 b(ec)m(hnical)42 b(impro)m(v)m(emen)m(ts)i(in)c(the)h(calculation)g(of)e(some)257 1054 y(high)k(order)g(con)m(tributions)h(to)e(the)h(groundstate)f (energy)i(ha)m(v)m(e)g(b)s(een)f(pro)m(vided)h(in)257 1175 y([5],[6].)257 1464 y Fc(I.1)136 b(Notation)257 1648 y Fq(W)-8 b(e)40 b(in)m(tro)s(duce)g(some)h(more)e(notation)g (useful)i(throughout)e(this)h(pap)s(er.)64 b(Giv)m(en)40 b(an)257 1769 y(op)s(erator-v)-5 b(alued)32 b(function)i Fo(F)41 b Fq(:)28 b Fk(R)1570 1733 y Fl(3)1631 1769 y Fn(\002)23 b Fk(Z)1797 1784 y Fl(2)1864 1769 y Fn(!)28 b(B)s Fq(\()p Fn(H)2182 1784 y Fm(el)2241 1769 y Fq(\),)k(w)m(e)i (write)1230 2019 y Fo(a)1281 1978 y Fi(\003)1321 2019 y Fq(\()p Fo(F)14 b Fq(\))82 b(:=)1742 1902 y Fh(Z)1842 2019 y Fo(F)14 b Fq(\()p Fo(k)s Fq(\))22 b Fn(\012)g Fo(a)2221 1978 y Fi(\003)2261 2019 y Fq(\()p Fo(k)s Fq(\))p Fo(dk)767 b Fq(\(I.25\))1269 2236 y Fo(a)p Fq(\()p Fo(F)14 b Fq(\))83 b(:=)1742 2119 y Fh(Z)1842 2236 y Fo(F)14 b Fq(\()p Fo(k)s Fq(\))2049 2195 y Fi(\003)2110 2236 y Fn(\012)23 b Fo(a)p Fq(\()p Fo(k)s Fq(\))p Fo(dk)s(:)737 b Fq(\(I.26\))257 2483 y(Thereb)m(y)35 b(the)e(v)m(elo)s(cit)m(y)i(op)s (erator)27 b Fo(~)-45 b(v)37 b Fq(can)c(b)s(e)f(written)i(as)1215 2703 y Fo(~)-45 b(v)64 b Fq(:=)c Fn(\000)p Fo(i)1612 2678 y(~)1603 2703 y Fn(r)1686 2718 y Fm(x)1769 2703 y Fq(+)39 b Fo(a)1935 2662 y Fi(\003)1974 2703 y Fq(\()2027 2678 y Fo(~)2012 2703 y(G)p Fq(\))g(+)g Fo(a)p Fq(\()2385 2678 y Fo(~)2370 2703 y(G)p Fq(\))22 b Fo(;)726 b Fq(\(I.27\))257 2932 y(where)554 2907 y Fo(~)539 2932 y(G)28 b Fq(:)g Fk(R)771 2896 y Fl(3)832 2932 y Fn(\002)23 b Fk(Z)998 2947 y Fl(2)1065 2932 y Fn(!)28 b(B)s Fq(\()p Fn(H)1383 2947 y Fm(el)1442 2932 y Fq(\))1480 2896 y Fl(3)1552 2932 y Fq(are)k(the)h(m)m(ultiplication)h(op)s(erators)f(de\014ned)h(b) m(y)1093 3193 y Fo(~)1079 3218 y(G)p Fq(\()p Fo(k)s Fq(\))60 b(:=)1579 3151 y Fo(\013)1642 3114 y Fl(3)p Fm(=)p Fl(2)p 1519 3195 294 4 v 1519 3286 a Fq(\(2)p Fo(\031)t Fq(\))1703 3257 y Fl(3)p Fm(=)p Fl(2)1898 3151 y Fq(\003\()p Fo(k)s Fq(\))p 1860 3195 275 4 v 1860 3211 a Fh(q)p 1943 3211 192 4 v 100 x Fq(2)17 b Fn(j)p Fo(k)s Fn(j)2172 3218 y Fo(e)2217 3177 y Fi(\000)p Fm(i\013)2338 3159 y(~)2341 3177 y(k)r Fi(\001)o Fm(~)-34 b(x)2470 3218 y Fo(~)-48 b(")p Fq(\()p Fo(k)s Fq(\);)586 b(\(I.28\))257 3535 y(moreo)m(v)m(er,) 47 b(in)c(terms)h(of)e(the)h(v)m(elo)s(cit)m(y)h(op)s(erator,)h(the)e (Hamiltonian)g(assumes)i(the)257 3655 y(simple)34 b(form)1270 3775 y Fo(H)8 b Fq(\()p Fo(\013)1470 3707 y Ff(1)p 1469 3719 31 4 v 1469 3760 a(2)1513 3775 y Fq(\))60 b(=)c Fo(~)-45 b(v)1809 3734 y Fl(2)1888 3775 y Fn(\000)39 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))39 b(+)2393 3750 y(\024)2367 3775 y Fo(H)8 b(:)777 b Fq(\(I.29\))257 3950 y(W)-8 b(e)38 b(kno)m(w)g(recall)g(the)g(m)m(ultiscale)h(structure)g(underlying)f (our)g(construction)g(of)f(the)257 4070 y(groundstate)c(of)g Fo(H)i Fn(\021)28 b Fo(H)8 b Fq(\()p Fo(\013)1327 4006 y Ff(1)p 1326 4018 V 1326 4059 a(2)1371 4070 y Fq(\).)257 4190 y(F)-8 b(or)42 b Fo(\013)j(<)f(\024)725 4154 y Fi(\000)p Fl(1)819 4190 y Fq(,)h(where)f Fo(\024)1239 4154 y Fi(\000)p Fl(1)1375 4190 y Fq(is)f(the)g(ultra)m(violet)g(cuto\013)g(in)f (Eq.\(I.14\),)k(w)m(e)d(de\014ne)g(a)257 4311 y(decreasing)34 b(sequence)i(\()p Fo(\033)1227 4326 y Fm(n)1274 4311 y Fq(\))1312 4275 y Fi(1)1312 4335 y Fm(n)p Fl(=0)1482 4311 y Fq(of)c(energy)h(scales)h(b)m(y)g(setting)1595 4531 y Fo(\033)1650 4546 y Fm(n)1757 4531 y Fq(:=)61 b Fo(\024)22 b(\013)2062 4490 y Fm(n)2131 4531 y Fo(:)1102 b Fq(\(I.30\))257 4751 y(Note,)33 b(in)g(passing,)h(that,)e(for)g(all)h Fo(n)27 b Fn(2)i Fk(N)1779 4766 y Fl(0)1818 4751 y Fq(,)k(w)m(e)g(ha)m (v)m(e)h(that)e Fo(\033)2512 4766 y Fm(n)p Fl(+1)2678 4751 y Fn(\024)c Fo(\024\013)g(<)g Fq(1.)257 4871 y(W)-8 b(e)32 b(factorize)g(F)-8 b(o)s(c)m(k)32 b(space)g Fn(F)37 b Fq(=)28 b Fn(F)10 b Fq(\()p Fg(h)p Fq(\))31 b(in)m(to)g(tensor)h(pro) s(ducts)g(b)m(y)h(in)m(tro)s(ducing)f(suit-)257 4992 y(able)e(subspaces)h(corresp)s(onding)f(to)f(the)h(one-photon)f(Hilb)s (ert)g(space)h(corresp)s(onding)257 5112 y(to)j(di\013eren)m(t)g (energy)h(scales:)1029 5332 y Fg(h)1081 5347 y Fm(n)1188 5332 y Fq(:=)60 b Fo(L)1417 5291 y Fl(2)1457 5236 y Fh(h)1496 5332 y Fn(K)1572 5347 y Fm(n)1619 5236 y Fh(i)1753 5332 y Fq(and)94 b Fg(h)2056 5291 y Fm(m)2056 5357 y(n)2183 5332 y Fq(:=)60 b Fo(L)2412 5291 y Fl(2)2452 5236 y Fh(h)2491 5332 y Fn(K)2568 5291 y Fm(m)2567 5357 y(n)2635 5236 y Fh(i)2697 5332 y Fo(;)536 b Fq(\(I.31\))p eop end %%Page: 7 8 TeXDict begin 7 7 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(7)257 573 y(where)512 790 y(for)32 b(0)27 b Fn(\024)h Fo(n)g Fn(\024)g(1)p Fq(:)94 b Fn(K)1330 805 y Fm(n)1470 790 y Fq(:=)1650 694 y Fh(n)1705 790 y Fq(\()1738 764 y Fo(~)1743 790 y(k)s(;)17 b(\034)11 b Fq(\))28 b Fn(2)g Fk(R)2126 749 y Fl(3)2188 790 y Fn(\002)22 b Fk(Z)2353 805 y Fl(2)2421 691 y Fh(\014)2421 741 y(\014)2421 790 y(\014)32 b Fo(\033)2536 805 y Fm(n)2611 790 y Fn(\024)c Fo(!)t Fq(\()2814 764 y Fo(~)2819 790 y(k)s Fq(\))2911 694 y Fh(o)2988 790 y Fo(;)245 b Fq(\(I.32\))295 973 y(for)32 b(0)c Fn(\024)g Fo(m)g(<)f(n)h Fn(\024)g(1)p Fq(:)83 b Fn(K)1320 937 y Fm(m)1319 998 y(n)1470 973 y Fq(:=)1650 877 y Fh(n)1705 973 y Fq(\()1738 947 y Fo(~)1743 973 y(k)s(;)17 b(\034)11 b Fq(\))28 b Fn(2)g Fk(R)2126 932 y Fl(3)2188 973 y Fn(\002)22 b Fk(Z)2353 988 y Fl(2)2421 874 y Fh(\014)2421 923 y(\014)2421 973 y(\014)32 b Fo(\033)2536 988 y Fm(n)2611 973 y Fn(\024)c Fo(!)t Fq(\()2814 947 y Fo(~)2819 973 y(k)s Fq(\))f Fo(<)h(\033)3097 988 y Fm(m)3164 877 y Fh(o)3241 973 y Fo(:)-8 b Fq(\(I.33\))257 1207 y(Note)31 b(that)g Fn(K)778 1171 y Fl(0)777 1232 y Fm(n)852 1207 y Fn(\032)d(K)1033 1222 y Fm(n)1110 1207 y Fq(is)k(a)e(prop)s(er)h(subset.)45 b(F)-8 b(or)30 b(in)m(tegers)i(1) 27 b Fn(\024)i Fo(m)f(<)f(n)h(<)g(`)f Fn(\024)h(1)p Fq(,)j(w)m(e)257 1328 y(ha)m(v)m(e)40 b(the)e(disjoin)m(t)h(decomp)s(osition)g Fn(K)1738 1343 y Fm(`)1808 1328 y Fq(=)e Fn(K)1997 1343 y Fm(m)2090 1328 y Fn([)26 b(K)2259 1292 y Fm(m)2258 1352 y(n)2352 1328 y Fn([)g(K)2521 1292 y Fm(n)2520 1352 y(`)2606 1328 y Fq(and)38 b(hence)i(the)e(direct)257 1448 y(sum)1421 1569 y Fg(h)1473 1584 y Fm(`)1566 1541 y Fn(\030)1566 1573 y Fq(=)1703 1569 y Fg(h)1755 1584 y Fm(m)1844 1569 y Fn(\010)23 b Fg(h)1996 1527 y Fm(m)1996 1593 y(n)2084 1569 y Fn(\010)g Fg(h)2236 1527 y Fm(n)2236 1593 y(`)2305 1569 y Fo(;)928 b Fq(\(I.34\))257 1742 y(whic)m(h)34 b(giv)m(es)g(rise)g(to)e(the)h(isomorphism)1371 1960 y Fn(F)1443 1975 y Fm(`)1536 1932 y Fn(\030)1537 1964 y Fq(=)1674 1960 y Fn(F)1746 1975 y Fm(m)1834 1960 y Fn(\012)23 b(F)2016 1918 y Fm(m)2006 1984 y(n)2104 1960 y Fn(\012)g(F)2286 1918 y Fm(n)2276 1984 y(`)2354 1960 y Fo(;)879 b Fq(\(I.35\))257 2177 y(with)34 b Fn(F)552 2192 y Fm(n)626 2177 y Fq(:=)27 b Fn(F)10 b Fq(\()p Fg(h)928 2192 y Fm(n)975 2177 y Fq(\))32 b(and)h Fn(F)1317 2141 y Fm(m)1307 2202 y(n)1411 2177 y Fq(:=)27 b Fn(F)10 b Fq(\()p Fg(h)1713 2141 y Fm(m)1713 2202 y(n)1779 2177 y Fq(\).)44 b(In)33 b(particular,)f(for)g(an)m(y)i Fo(n)28 b Fn(2)g Fk(N)p Fq(,)1146 2395 y Fn(F)70 b Fq(=)60 b Fn(F)1496 2410 y Fi(1)1630 2367 y Fn(\030)1631 2399 y Fq(=)1768 2395 y Fn(F)1840 2410 y Fm(n)1909 2395 y Fn(\012)22 b(F)2090 2354 y Fm(n)2080 2419 y(n)p Fl(+1)2239 2395 y Fn(\012)h(F)2421 2354 y Fm(n)p Fl(+1)2411 2419 y Fi(1)2580 2395 y Fo(:)653 b Fq(\(I.36\))257 2612 y(The)38 b(in)m(teraction)g(is)g (cut)f(in)m(to)g(slices)i(corresp)s(onding)f(to)e(ev)m(er)j(lo)m(w)m (er)f(energy)g(scales,)257 2733 y(w)m(e)c(mak)m(e)g(use)f(of)f(the)h (op)s(erators)318 2925 y Fo(~)303 2950 y(G)380 2965 y Fm(n)427 2950 y Fq(\()p Fo(k)s Fq(\))60 b(:=)g Fe(1)836 2854 y Fh(\020)886 2950 y Fo(\033)941 2965 y Fm(n)1016 2950 y Fn(\024)28 b(j)p Fo(k)s Fn(j)1231 2854 y Fh(\021)1317 2925 y Fo(~)1302 2950 y(G)p Fq(\()p Fo(k)s Fq(\))22 b Fo(;)112 b Fq(and)1936 2925 y Fo(~)1921 2950 y(G)1998 2909 y Fm(m)1998 2975 y(n)2065 2950 y Fq(\()p Fo(k)s Fq(\))60 b(:=)g Fe(1)2474 2854 y Fh(\020)2524 2950 y Fo(\033)2579 2965 y Fm(n)2654 2950 y Fn(\024)28 b(j)p Fo(k)s Fn(j)f Fo(<)h(\033)3055 2965 y Fm(m)3122 2854 y Fh(\021)3208 2925 y Fo(~)3193 2950 y(G)p Fq(\()p Fo(k)s Fq(\))22 b Fo(;)3260 3083 y Fq(\(I.37\))257 3213 y(for)35 b(all)f Fo(k)h Fn(2)d Fk(R)802 3177 y Fl(3)865 3213 y Fn(\002)24 b Fk(Z)1032 3228 y Fl(2)1106 3213 y Fq(and)35 b Fo(m;)17 b(n)32 b Fn(2)g Fk(N)1687 3228 y Fl(0)1726 3213 y Fq(,)k(with)f Fo(m)d(<)f(n)p Fq(.)51 b(Note)34 b(that)2839 3188 y Fo(~)2824 3213 y(G)d Fq(=)3040 3146 y Fh(P)3127 3173 y Fi(1)3127 3237 y Fm(n)p Fl(=0)3296 3188 y Fo(~)3281 3213 y(G)3358 3177 y Fm(n)3358 3237 y(n)p Fl(+1)257 3349 y Fq(and)43 b(that)694 3324 y Fo(~)679 3349 y(G)756 3364 y Fm(n)846 3349 y Fq(is)g(the)h(coupling)f(function)h (of)e(the)i(in)m(teraction)g(regularized)g(in)f(the)257 3469 y(infrared)33 b(region)g(\()p Fn(j)p Fo(k)s Fn(j)27 b(\025)h Fo(\033)1251 3484 y Fm(n)1298 3469 y Fq(\).)44 b(W)-8 b(e)33 b(set)916 3687 y Fn(H)1000 3702 y Fm(n)1107 3687 y Fq(:=)60 b Fn(H)1354 3702 y Fm(el)1435 3687 y Fn(\012)23 b(F)1607 3702 y Fm(n)1748 3687 y Fq(and)95 b Fn(H)2085 3646 y Fm(m)2084 3711 y(n)2212 3687 y Fq(:=)60 b Fn(H)2459 3702 y Fm(el)2540 3687 y Fn(\012)22 b(F)2721 3646 y Fm(m)2711 3711 y(n)2810 3687 y Fo(;)423 b Fq(\(I.38\))257 3904 y(and)23 b(for)f(energy-scale)j(indices)f Fo(m;)17 b(n)28 b Fn(2)g Fk(N)1803 3919 y Fl(0)1842 3904 y Fq(,)d Fo(m)j(<)g(n)p Fq(,)d(w)m(e)e(de\014ne)h(the)f(v)m(elo)s(cit)m(y)i(op)s (erator)253 4025 y Fo(~)-45 b(v)304 4040 y Fm(n)351 4025 y Fq(,)33 b(the)g(\014eld)g(energy)h(op)s(erators)1559 4000 y(\024)1533 4025 y Fo(H)1614 4040 y Fm(n)1661 4025 y Fq(,)1746 4000 y(\024)1721 4025 y Fo(H)1810 3989 y Fm(m)1802 4049 y(n)1876 4025 y Fq(,)f(and)f(the)h(Hamiltonian)g Fo(H)2935 4040 y Fm(n)3014 4025 y Fq(b)m(y)890 4251 y Fo(~)-45 b(v)941 4266 y Fm(n)1071 4251 y Fq(:=)83 b Fn(\000)p Fo(i)1376 4226 y(~)1367 4251 y Fn(r)1450 4266 y Fm(x)1534 4251 y Fq(+)38 b Fo(a)1699 4210 y Fi(\003)1739 4251 y Fq(\()1792 4226 y Fo(~)1777 4251 y(G)1854 4266 y Fm(n)1901 4251 y Fq(\))g(+)h Fo(a)p Fq(\()2196 4226 y Fo(~)2181 4251 y(G)2258 4266 y Fm(n)2305 4251 y Fq(\))22 b Fo(;)868 b Fq(\(I.39\))886 4401 y(\024)860 4426 y Fo(H)941 4441 y Fm(n)1071 4426 y Fq(:=)1257 4309 y Fh(Z)1357 4426 y Fe(1)p Fq(\()p Fo(\033)1506 4441 y Fm(n)1581 4426 y Fn(\024)28 b(j)p Fo(k)s Fn(j)p Fq(\))21 b Fo(!)t Fq(\()p Fo(k)s Fq(\))h Fo(a)2123 4385 y Fi(\003)2163 4426 y Fq(\()p Fo(k)s Fq(\))p Fo(a)p Fq(\()p Fo(k)s Fq(\))g Fo(dk)j(;)610 b Fq(\(I.40\))858 4619 y(\024)833 4644 y Fo(H)922 4603 y Fm(m)914 4669 y(n)1071 4644 y Fq(:=)1257 4527 y Fh(Z)1357 4644 y Fe(1)p Fq(\()p Fo(\033)1506 4659 y Fm(n)1581 4644 y Fn(\024)28 b(j)p Fo(k)s Fn(j)f Fo(<)h(\033)1982 4659 y Fm(m)2049 4644 y Fq(\))22 b Fo(!)t Fq(\()p Fo(k)s Fq(\))f Fo(a)2376 4603 y Fi(\003)2416 4644 y Fq(\()p Fo(k)s Fq(\))p Fo(a)p Fq(\()p Fo(k)s Fq(\))h Fo(dk)j(;)357 b Fq(\(I.41\))860 4832 y Fo(H)941 4847 y Fm(n)1071 4832 y Fq(:=)79 b Fo(~)-45 b(v)1319 4791 y Fl(2)1304 4857 y Fm(n)1398 4832 y Fn(\000)39 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))39 b(+)1903 4807 y(\024)1877 4832 y Fo(H)1958 4847 y Fm(n)2027 4832 y Fo(;)1206 b Fq(\(I.42\))257 5050 y(as)45 b(op)s(erators)g(on)g Fn(H)1065 5065 y Fm(n)1157 5050 y Fq(and)g Fn(H)1444 5014 y Fm(m)1443 5075 y(n)1511 5050 y Fq(,)j(resp)s(ectiv)m(ely)-8 b(.)83 b(W)-8 b(e)45 b(in)m(tro)s(duce)h(the)g(groundstate)257 5170 y(energy)34 b(on)e(scale)i Fo(n)f Fq(and)f(groundstate)i(energy)f (di\013erences)935 5388 y Fo(E)1007 5403 y Fm(n)1114 5388 y Fq(:=)60 b(inf)24 b Fo(\033)t Fq(\()p Fo(H)1590 5403 y Fm(n)1637 5388 y Fq(\))94 b(and)h Fo(E)2099 5347 y Fm(m)2093 5413 y(n)2226 5388 y Fq(:=)60 b Fo(E)2461 5403 y Fm(m)2550 5388 y Fn(\000)22 b Fo(E)2721 5403 y Fm(n)2791 5388 y Fo(:)442 b Fq(\(I.43\))p eop end %%Page: 8 9 TeXDict begin 8 8 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(8)257 573 y(W)-8 b(e)33 b(de\014ne)f(the)h(p)s(ositiv)m(e)g(op)s (erators)e Fo(H)1752 537 y Fl(+)1744 597 y Fm(n)1843 573 y Fq(and)2040 541 y Fh(f)2031 573 y Fo(H)2120 537 y Fl(+)2112 597 y Fm(n)2211 573 y Fq(on)g Fn(H)2429 588 y Fm(n)2508 573 y Fq(and)h Fn(H)2781 588 y Fm(n)p Fl(+1)2918 573 y Fq(,)g(resp)s(ectiv)m(ely)-8 b(,)257 693 y(b)m(y)1109 913 y Fo(H)1198 872 y Fl(+)1190 938 y Fm(n)1339 913 y Fq(:=)83 b Fo(H)1606 928 y Fm(n)1692 913 y Fn(\000)39 b Fo(E)1880 928 y Fm(n)1949 913 y Fo(;)1284 b Fq(\(I.44\))1117 1026 y Fh(f)1109 1059 y Fo(H)1198 1017 y Fl(+)1190 1083 y Fm(n)1339 1059 y Fq(:=)83 b Fo(H)1614 1017 y Fl(+)1606 1083 y Fm(n)1695 1059 y Fn(\012)22 b Fe(1)1850 1017 y Fm(n)1850 1083 y(n)p Fl(+1)2032 1059 y Fq(+)44 b Fe(1)2208 1074 y Fm(n)2277 1059 y Fn(\012)2402 1033 y Fq(\024)2377 1059 y Fo(H)2466 1017 y Fm(n)2458 1083 y(n)p Fl(+1)2617 1059 y Fo(;)616 b Fq(\(I.45\))257 1279 y(where)46 b(w)m(e)f(denote)g (the)f(iden)m(tit)m(y)i(op)s(erator)e(on)g Fn(H)2219 1294 y Fm(n)2310 1279 y Fq(and)g(on)g Fn(F)2740 1242 y Fm(m)2730 1303 y(n)2850 1279 y Fq(b)m(y)h Fe(1)3053 1294 y Fm(n)3144 1279 y Fq(and)g Fe(1)3402 1242 y Fm(m)3402 1303 y(n)3468 1279 y Fq(,)257 1399 y(resp)s(ectiv)m(ely)-8 b(.)61 b(De\014nitions)38 b(\(I.44\))f(and)h(\(I.45\))f(are)g(mean)m(t) h(to)f(exploit)h(regular)g(p)s(er-)257 1519 y(turbation)i(theory)g(in)f (the)h(step)g(from)f(the)h(Hamiltonian)g Fo(H)2580 1534 y Fm(n)2666 1519 y Fq(to)f(the)h(Hamiltonian)257 1640 y Fo(H)338 1655 y Fm(n)p Fl(+1)475 1640 y Fq(,)h(at)d(successiv)m(e)43 b(energy)d(scales,)h(in)e(next)h(equation)g(\(I.48\),)g(where)3100 1607 y Fh(f)3092 1640 y Fo(H)3181 1604 y Fl(+)3173 1664 y Fm(n)3278 1640 y Fq(pla)m(ys)257 1760 y(the)c(role)f(of)f(the)h(unp)s (erturb)s(ed)h(op)s(erator.)50 b(W)-8 b(e)35 b(iden)m(tify)e Fo(~)-45 b(v)2482 1775 y Fm(n)2564 1760 y Fq(with)31 b Fo(~)-45 b(v)2835 1775 y Fm(n)2906 1760 y Fn(\012)24 b Fe(1)3063 1724 y Fm(n)3063 1785 y(n)p Fl(+1)3235 1760 y Fq(acting)257 1880 y(on)33 b Fn(H)477 1895 y Fm(n)p Fl(+1)614 1880 y Fq(.)44 b(Note)32 b(that,)h(for)f Fo(n)c Fn(2)g Fk(N)1560 1895 y Fl(0)1599 1880 y Fq(,)1096 2100 y Fo(~)-45 b(v)1147 2115 y Fm(n)p Fl(+1)1345 2100 y Fq(=)56 b Fo(~)-45 b(v)1528 2115 y Fm(n)1614 2100 y Fq(+)38 b Fo(a)1779 2059 y Fi(\003)1819 2100 y Fq(\()1872 2075 y Fo(~)1857 2100 y(G)1934 2059 y Fm(n)1934 2125 y(n)p Fl(+1)2071 2100 y Fq(\))g(+)h Fo(a)p Fq(\()2366 2075 y Fo(~)2351 2100 y(G)2428 2059 y Fm(n)2428 2125 y(n)p Fl(+1)2565 2100 y Fq(\))22 b Fo(:)608 b Fq(\(I.46\))257 2320 y(Similarly)-8 b(,)34 b(giv)m(en)g Fo( )1009 2335 y Fm(n)1084 2320 y Fn(2)28 b(H)1262 2335 y Fm(n)1309 2320 y Fq(,)33 b(w)m(e)g(de\014ne)h(a)e(v)m(ector)1286 2514 y(~)1266 2540 y Fo( )1329 2555 y Fm(n)1437 2540 y Fq(:=)60 b Fo( )1663 2555 y Fm(n)1732 2540 y Fn(\012)23 b Fq(\012)1902 2499 y Fm(n)1902 2565 y(n)p Fl(+1)2100 2540 y Fn(2)50 b(H)2300 2555 y Fm(n)p Fl(+1)2459 2540 y Fo(;)774 b Fq(\(I.47\))257 2760 y(where)42 b(\012)617 2775 y Fm(n)704 2760 y Fq(and)e(\012)971 2724 y Fm(m)971 2785 y(n)1079 2760 y Fq(denote)g(the)h(v)-5 b(acuum)41 b(v)m(ectors)h(in)e Fn(F)2476 2775 y Fm(n)2563 2760 y Fq(and)g Fn(F)2842 2724 y Fm(m)2832 2785 y(n)2908 2760 y Fq(,)i(resp)s(ectiv)m(ely)-8 b(.)257 2881 y(With)33 b(these)h(notations,)f(w)m(e)h(ha)m(v)m(e)f(that)1231 3101 y Fo(H)1320 3060 y Fl(+)1312 3126 y Fm(n)p Fl(+1)1476 3101 y Fq(=)1588 3069 y Fh(f)1580 3101 y Fo(H)1669 3060 y Fl(+)1661 3126 y Fm(n)1766 3101 y Fq(+)39 b Fo(W)1987 3060 y Fm(n)1973 3126 y(n)p Fl(+1)2149 3101 y Fq(+)f Fo(E)2341 3060 y Fm(n)2335 3126 y(n)p Fl(+1)2495 3101 y Fo(;)738 b Fq(\(I.48\))257 3321 y(where)369 3541 y Fo(W)475 3500 y Fm(n)461 3566 y(n)p Fl(+1)681 3541 y Fq(:=)83 b(\()l Fo(~)-45 b(v)952 3556 y Fm(n)p Fl(+1)1090 3541 y Fq(\))1128 3500 y Fl(2)1206 3541 y Fn(\000)39 b Fq(\()l Fo(~)-45 b(v)1407 3556 y Fm(n)1454 3541 y Fq(\))1492 3500 y Fl(2)3260 3541 y Fq(\(I.49\))695 3722 y(=)96 b(2)17 b Fo(a)984 3681 y Fi(\003)1023 3722 y Fq(\()1076 3697 y Fo(~)1061 3722 y(G)1138 3681 y Fm(n)1138 3746 y(n)p Fl(+1)1275 3722 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)1432 3737 y Fm(n)1529 3722 y Fq(+)50 b(2)13 b Fo(~)-45 b(v)1768 3737 y Fm(n)1837 3722 y Fn(\001)21 b Fo(a)p Fq(\()1990 3697 y Fo(~)1975 3722 y(G)2052 3681 y Fm(n)2052 3746 y(n)p Fl(+1)2190 3722 y Fq(\))49 b(+)2403 3625 y Fh(\020)2453 3722 y Fo(a)2504 3681 y Fi(\003)2543 3722 y Fq(\()2596 3697 y Fo(~)2581 3722 y(G)2658 3681 y Fm(n)2658 3746 y(n)p Fl(+1)2795 3722 y Fq(\))22 b(+)g Fo(a)p Fq(\()3057 3697 y Fo(~)3042 3722 y(G)3119 3681 y Fm(n)3119 3746 y(n)p Fl(+1)3257 3722 y Fq(\))3295 3625 y Fh(\021)3344 3648 y Fl(2)695 3901 y Fq(=)96 b(2)17 b Fo(a)984 3860 y Fi(\003)1023 3901 y Fq(\()1076 3876 y Fo(~)1061 3901 y(G)1138 3860 y Fm(n)1138 3926 y(n)p Fl(+1)1275 3901 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)1432 3916 y Fm(n)1529 3901 y Fq(+)50 b(2)13 b Fo(~)-45 b(v)1768 3916 y Fm(n)1837 3901 y Fn(\001)21 b Fo(a)p Fq(\()1990 3876 y Fo(~)1975 3901 y(G)2052 3860 y Fm(n)2052 3926 y(n)p Fl(+1)2190 3901 y Fq(\))49 b(+)h Fo(a)2454 3860 y Fi(\003)2494 3901 y Fq(\()2547 3876 y Fo(~)2532 3901 y(G)2609 3860 y Fm(n)2609 3926 y(n)p Fl(+1)2746 3901 y Fq(\))22 b Fn(\001)g Fo(a)2907 3860 y Fi(\003)2946 3901 y Fq(\()2999 3876 y Fo(~)2984 3901 y(G)3061 3860 y Fm(n)3061 3926 y(n)p Fl(+1)3198 3901 y Fq(\))895 4082 y(+)28 b Fo(a)p Fq(\()1103 4057 y Fo(~)1088 4082 y(G)1165 4041 y Fm(n)1165 4107 y(n)p Fl(+1)1302 4082 y Fq(\))22 b Fn(\001)g Fo(a)p Fq(\()1516 4057 y Fo(~)1501 4082 y(G)1578 4041 y Fm(n)1578 4107 y(n)p Fl(+1)1715 4082 y Fq(\))50 b(+)f(2)17 b Fo(a)2045 4041 y Fi(\003)2084 4082 y Fq(\()2137 4057 y Fo(~)2122 4082 y(G)2199 4041 y Fm(n)2199 4107 y(n)p Fl(+1)2336 4082 y Fq(\))22 b Fn(\001)g Fo(a)p Fq(\()2550 4057 y Fo(~)2535 4082 y(G)2612 4041 y Fm(n)2612 4107 y(n)p Fl(+1)2749 4082 y Fq(\))50 b(+)2963 3982 y Fh(\015)2963 4032 y(\015)2963 4082 y(\015)3024 4057 y Fo(~)3009 4082 y(G)3086 4041 y Fm(n)3086 4107 y(n)p Fl(+1)3223 3982 y Fh(\015)3223 4032 y(\015)3223 4082 y(\015)3269 4009 y Fl(2)3331 4082 y Fo(;)257 4356 y Fq(with)478 4256 y Fh(\015)478 4306 y(\015)478 4356 y(\015)539 4331 y Fo(~)524 4356 y(G)601 4320 y Fm(n)601 4380 y(n)p Fl(+1)738 4256 y Fh(\015)738 4306 y(\015)738 4356 y(\015)784 4282 y Fl(2)852 4356 y Fq(:=)982 4285 y Fh(R)1054 4356 y Fn(j)1097 4331 y Fo(~)1082 4356 y(G)1159 4320 y Fm(n)1159 4380 y(n)p Fl(+1)1296 4356 y Fq(\()p Fo(k)s Fq(\))p Fn(j)1454 4320 y Fl(2)1510 4356 y Fo(dk)s Fq(.)42 b(In)32 b(Eq.)g(\(I.48\),)f(w)m(e)h(mak)m(e)g (use)g(of)f(the)g(Coulom)m(b)257 4510 y(gauge)i(condition)971 4485 y Fo(~)962 4510 y Fn(r)22 b(\001)1135 4485 y Fo(~)1117 4510 y(A)p Fq(\()p Fo(x)p Fq(\))28 b(=)g(0,)k(whic)m(h)i(implies)g (that)311 4739 y Fo(a)362 4698 y Fi(\003)401 4739 y Fq(\()454 4714 y Fo(~)439 4739 y(G)516 4698 y Fm(n)516 4764 y(n)p Fl(+1)653 4739 y Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)810 4754 y Fm(n)917 4739 y Fq(=)57 b Fo(~)-45 b(v)1101 4754 y Fm(n)1170 4739 y Fn(\001)22 b Fo(a)1271 4698 y Fi(\003)1310 4739 y Fq(\()1363 4714 y Fo(~)1348 4739 y(G)1425 4698 y Fm(n)1425 4764 y(n)p Fl(+1)1562 4739 y Fq(\))95 b(and)f Fo(a)p Fq(\()2050 4714 y Fo(~)2035 4739 y(G)2112 4698 y Fm(n)2112 4764 y(n)p Fl(+1)2250 4739 y Fq(\))22 b Fn(\001)17 b Fo(~)-45 b(v)2406 4754 y Fm(n)2514 4739 y Fq(=)56 b Fo(~)-45 b(v)2697 4754 y Fm(n)2766 4739 y Fn(\001)22 b Fo(a)p Fq(\()2920 4714 y Fo(~)2905 4739 y(G)2982 4698 y Fm(n)2982 4764 y(n)p Fl(+1)3119 4739 y Fq(\))g Fo(:)54 b Fq(\(I.50\))257 5028 y Fc(I.2)136 b(Preliminary)46 b(results)f(and)g(outline)h(of)f(strategy)257 5213 y Fq(Here,)33 b(w)m(e)g(describ)s(e)h(some)f(results)g(deriv)m(ed)h(in)f ([1)o(],)g(concerning)g(the)f(construction)i(of)257 5342 y(the)43 b(groundstate)f(of)g Fo(H)51 b Fn(\021)45 b Fo(H)8 b Fq(\()p Fo(\013)1556 5278 y Ff(1)p 1555 5290 31 4 v 1555 5331 a(2)1599 5342 y Fq(\),)45 b(that)d(will)g(b)s(e)h (used)g(again)f(and)g(again)f(in)i(the)p eop end %%Page: 9 10 TeXDict begin 9 9 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1940 b Fq(9)257 573 y(re-expansion)45 b(pro)s(cedure.)76 b(A)43 b(k)m(ey)h(ingredien)m(t)h(used)f(in)f(our)g(construction)h(of)f(the) 257 693 y(groundstate)33 b(of)g Fo(H)40 b Fq(is)33 b(the)g(simple)h (iden)m(tit)m(y)1544 860 y(2)13 b Fo(~)-45 b(v)63 b Fq(=)d Fo(i)17 b Fq([)p Fo(H)25 b(;)31 b(~)-47 b(x)p Fq(])p Fo(;)1052 b Fq(\(I.51\))257 1027 y(whic)m(h)24 b(implies)f(that)f(the)h (in)m(teraction)g(term)f(in)g Fo(H)8 b Fq(,)24 b(whic)m(h)g(is)e Fj(mar)-5 b(ginal)31 b Fq(in)23 b(the)f Fj(infr)-5 b(ar)g(e)g(d)257 1147 y Fq(region)30 b(\(in)g(the)g(sense)i(of)d(p)s(o)m(w)m(er)i(coun)m (ting\),)g(is)f(actually)g Fj(infr)-5 b(ar)g(e)g(d-irr)g(elevant)38 b Fq(on)30 b(the)257 1267 y(subspace)35 b(of)d(all)g(those)i(states)f (where)h(the)f(electron)h(is)f(b)s(ound)f(to)h(the)g(n)m(ucleus.)404 1388 y(In)43 b(the)g(next)g(prop)s(osition,)j(the)d(main)g(results)h (are)f(the)g(gap)f(estimates)i(\(I.52\),)257 1508 y(\(I.53\))c(and)g (the)g(Neumann)h(series)h(expansion)f(\(I.57\))f(of)f(the)h(sp)s (ectral)h(pro)5 b(jections)257 1628 y(on)33 b(the)g(groundstates)g(of)f (the)i(Hamiltonians)f Fo(H)2096 1643 y Fm(n)2143 1628 y Fq(.)257 1786 y Fe(Prop)s(osition)53 b(I.1.)48 b Fj(Assume)f(Hyp)-5 b(othesis)47 b(1.)81 b(Then)46 b(ther)-5 b(e)47 b(exist)f(c)-5 b(onstants)47 b Fq(0)i Fo(<)257 1907 y(C)334 1870 y Fi(0)408 1907 y Fn(\024)i Fo(C)7 b Fj(,)50 b(with)d Fo(C)58 b Fn(\025)51 b Fq(4)c Fj(such)g(that,)j(for)d(al)5 b(l)47 b Fo(\013)k(<)2344 1867 y Fl(1)p 2317 1883 91 4 v 2317 1941 a(2)p Fm(C)2417 1907 y Fj(,)f Fo(E)2569 1922 y Fm(n)2664 1907 y Fj(is)d(an)f(eigenvalue)h(of)257 2027 y(multiplicity)35 b(one.)45 b(Mor)-5 b(e)g(over)836 2233 y Fq(inf)970 2136 y Fh(h)1009 2233 y Fo(\033)t Fq(\()p Fo(H)1195 2191 y Fl(+)1187 2257 y Fm(n)1254 2233 y Fq(\))22 b Fn(n)g(f)p Fq(0)p Fn(g)1535 2136 y Fh(i)1657 2233 y Fq(=:)83 b(gap)1994 2256 y Fm(n)2131 2233 y Fn(\025)91 b Fq([1)22 b Fn(\000)2507 2165 y Fq(3)p 2507 2209 49 4 v 2507 2301 a(4)2566 2233 y Fo(C)7 b(\013)q Fq(])17 b Fo(\033)2805 2248 y Fm(n)2873 2233 y Fo(;)360 b Fq(\(I.52\))836 2434 y(inf)970 2337 y Fh(h)1009 2434 y Fo(\033)t Fq(\()1114 2401 y Fh(f)1106 2434 y Fo(H)1195 2393 y Fl(+)1187 2458 y Fm(n)1254 2434 y Fq(\))22 b Fn(n)g(f)p Fq(0)p Fn(g)1535 2337 y Fh(i)1657 2434 y Fq(=:)1858 2427 y Fh(g)1843 2434 y Fq(gap)1994 2457 y Fm(n)2131 2434 y Fq(=)91 b Fo(\033)2353 2449 y Fm(n)p Fl(+1)2512 2434 y Fo(;)721 b Fq(\(I.53\))257 2600 y Fj(and)700 2803 y Fq(sup)650 2882 y Fm(z)s Fi(2)p Fl(\000)777 2891 y Fb(n)p Ff(+1)913 2678 y Fh(\015)913 2728 y(\015)913 2778 y(\015)913 2828 y(\015)987 2682 y(\022)1193 2736 y Fq(1)p 1058 2780 320 4 v 1066 2852 a Fh(f)1058 2884 y Fo(H)1147 2856 y Fl(+)1139 2909 y Fm(n)1228 2884 y Fn(\000)22 b Fo(z)1387 2682 y Fh(\023)1448 2705 y Fl(1)p Fm(=)p Fl(2)1558 2707 y Fh(\020)1630 2803 y Fn(\000)g Fo(W)1835 2762 y Fm(n)1821 2828 y(n)p Fl(+1)1959 2707 y Fh(\021)2008 2682 y(\022)2215 2736 y Fq(1)p 2079 2780 V 2087 2852 a Fh(f)2079 2884 y Fo(H)2168 2856 y Fl(+)2160 2909 y Fm(n)2249 2884 y Fn(\000)h Fo(z)2408 2682 y Fh(\023)2470 2705 y Fl(1)p Fm(=)p Fl(2)2607 2678 y Fh(\015)2607 2728 y(\015)2607 2778 y(\015)2607 2828 y(\015)2736 2803 y Fn(\024)84 b Fo(C)23 b(\013)g(;)158 b Fq(\(I.54\))257 3056 y Fj(wher)-5 b(e)41 b Fq(\000)600 3071 y Fm(n)p Fl(+1)776 3056 y Fq(:=)e Fn(f)978 3017 y Fl(1)p 978 3033 36 4 v 978 3090 a(4)1023 3056 y Fo(\033)1078 3071 y Fm(n)p Fl(+1)1216 3056 y Fo(e)1261 3020 y Fm(i#)1370 3056 y Fn(2)h Fk(C)p Fn(j)p Fo(#)f Fn(2)h Fq([0)p Fo(;)17 b Fq(2)p Fo(\031)t Fq(\))p Fn(g)p Fj(.)62 b(The)40 b(sp)-5 b(e)g(ctr)g(al)41 b(pr)-5 b(oje)g(ctions)3267 3024 y Fh(e)3243 3056 y Fo(P)3306 3071 y Fm(n)3392 3056 y Fq(:=)257 3203 y Fn(j)340 3143 y Fl(~)330 3161 y Fm(\036)372 3169 y Fb(n)p 295 3180 156 4 v 295 3246 a Fi(k)339 3228 y Fl(~)330 3246 y Fm(\036)372 3254 y Fb(n)415 3246 y Fi(k)460 3203 y Fn(ih)592 3143 y Fl(~)583 3161 y Fm(\036)625 3169 y Fb(n)p 547 3180 V 547 3246 a Fi(k)591 3228 y Fl(~)582 3246 y Fm(\036)624 3254 y Fb(n)667 3246 y Fi(k)712 3203 y Fn(j)33 b Fj(and)f Fo(P)1023 3218 y Fm(n)p Fl(+1)1188 3203 y Fq(:=)c Fn(j)1392 3157 y Fm(\036)1434 3166 y Fb(n)p Ff(+1)p 1356 3180 233 4 v 1356 3238 a Fi(k)p Fm(\036)1433 3247 y Fb(n)p Ff(+1)1553 3238 y Fi(k)1598 3203 y Fn(ih)1721 3157 y Fm(\036)1763 3166 y Fb(n)p Ff(+1)p 1686 3180 V 1686 3238 a Fi(k)p Fm(\036)1763 3247 y Fb(n)p Ff(+1)1882 3238 y Fi(k)1928 3203 y Fn(j)p Fj(,)33 b(wher)-5 b(e)2304 3177 y Fq(~)2292 3203 y Fo(\036)2350 3218 y Fm(n)2429 3203 y Fj(and)33 b Fo(\036)2675 3218 y Fm(n)p Fl(+1)2845 3203 y Fj(ar)-5 b(e)32 b(gr)-5 b(oundstate)257 3365 y(ve)g(ctors)35 b(of)708 3333 y Fh(f)699 3365 y Fo(H)780 3380 y Fm(n)862 3365 y Fj(and)f Fo(H)1132 3380 y Fm(n)p Fl(+1)1269 3365 y Fj(,)h(r)-5 b(esp)g(e)g(ctively,)34 b(c)-5 b(orr)g(esp)g(ond)34 b(to)1280 3546 y Fh(e)1256 3578 y Fo(P)1319 3593 y Fm(n)1429 3578 y Fq(=)1585 3511 y Fn(\000)p Fq(1)p 1577 3555 141 4 v 1577 3646 a(2)p Fo(\031)t(i)1745 3461 y Fh(Z)1791 3650 y Fl(\000)1835 3659 y Fb(n)p Ff(+1)2094 3511 y Fo(dz)2190 3526 y Fm(n)p Fl(+1)p 1985 3555 452 4 v 1993 3627 a Fh(f)1985 3659 y Fo(H)2074 3631 y Fl(+)2066 3684 y Fm(n)2155 3659 y Fn(\000)22 b Fo(z)2299 3674 y Fm(n)p Fl(+1)2469 3578 y Fo(;)764 b Fq(\(I.55\))257 3808 y Fj(and)1011 3953 y Fo(P)1074 3968 y Fm(n)p Fl(+1)1273 3953 y Fq(=)1429 3885 y Fn(\000)p Fq(1)p 1422 3929 141 4 v 1422 4021 a(2)p Fo(\031)t(i)1589 3836 y Fh(Z)1635 4024 y Fl(\000)1679 4033 y Fb(n)p Ff(+1)2140 3885 y Fo(dz)2236 3900 y Fm(n)p Fl(+1)p 1829 3929 854 4 v 1829 4021 a Fo(H)1910 4036 y Fm(n)p Fl(+1)2070 4021 y Fn(\000)22 b Fo(E)2241 4036 y Fm(n)p Fl(+1)2401 4021 y Fn(\000)g Fo(z)2545 4036 y Fm(n)p Fl(+1)2715 3953 y Fo(;)518 b Fq(\(I.56\))257 4165 y Fj(r)-5 b(esp)g(e)g(ctively,)46 b(with)d Fq(\000)1089 4180 y Fm(n)1181 4165 y Fq(:=)h Fn(f)1388 4125 y Fl(1)p 1388 4141 36 4 v 1388 4199 a(4)1433 4165 y Fo(\033)1488 4180 y Fm(n)1535 4165 y Fo(e)1580 4128 y Fm(i#)1695 4165 y Fn(2)h Fk(C)p Fn(j)p Fo(#)f Fn(2)h Fq([0)p Fo(;)17 b Fq(2)p Fo(\031)t Fq(\))p Fn(g)p Fj(.)70 b(Their)44 b(di\013er)-5 b(enc)g(e)42 b(has)h(a)257 4285 y(norm)35 b(c)-5 b(onver)g(gent)34 b(series)g(exp)-5 b(ansion)779 4504 y Fo(P)842 4519 y Fm(n)p Fl(+1)1002 4504 y Fn(\000)1124 4471 y Fh(e)1101 4504 y Fo(P)1164 4519 y Fm(n)1274 4504 y Fq(=)1429 4436 y Fn(\000)p Fq(1)p 1422 4480 141 4 v 1422 4572 a(2)p Fo(\031)t(i)1619 4396 y Fi(1)1594 4421 y Fh(X)1589 4602 y Fm(\027)t Fl(=1)1735 4387 y Fh(Z)1781 4575 y Fl(\000)1825 4584 y Fb(n)p Ff(+1)1949 4504 y Fq(\()p Fn(\000)p Fq(1\))2151 4463 y Fm(\027)2217 4504 y Fo(Y)2295 4453 y Fl(\()p Fm(\027)t Fl(\))2274 4525 y Fm(n)p Fl(+1)2410 4504 y Fq(\()p Fo(z)2493 4519 y Fm(n)p Fl(+1)2631 4504 y Fq(\))22 b Fo(dz)2787 4519 y Fm(n)p Fl(+1)2946 4504 y Fo(;)287 b Fq(\(I.57\))257 4734 y Fj(wher)-5 b(e)1068 4874 y Fo(Y)1146 4823 y Fl(\()p Fm(\027)t Fl(\))1125 4896 y Fm(n)p Fl(+1)1262 4874 y Fq(\()p Fo(z)t Fq(\))28 b(:=)1691 4806 y(1)p 1556 4851 320 4 v 1564 4923 a Fh(f)1556 4955 y Fo(H)1645 4926 y Fl(+)1637 4980 y Fm(n)1726 4955 y Fn(\000)22 b Fo(z)1885 4753 y Fh(\022)1946 4874 y Fo(W)2052 4833 y Fm(n)2038 4898 y(n)p Fl(+1)2337 4806 y Fq(1)p 2202 4851 V 2210 4923 a Fh(f)2202 4955 y Fo(H)2291 4926 y Fl(+)2283 4980 y Fm(n)2372 4955 y Fn(\000)g Fo(z)2531 4753 y Fh(\023)2592 4776 y Fm(\027)2657 4874 y Fo(;)576 b Fq(\(I.58\))257 5088 y Fj(and)1264 5209 y Fn(k)p Fo(P)1377 5224 y Fm(n)p Fl(+1)1536 5209 y Fn(\000)1659 5176 y Fh(e)1635 5209 y Fo(P)1698 5224 y Fm(n)1745 5209 y Fn(k)63 b(\024)g Fo(C)2075 5168 y Fi(0)2114 5209 y Fo(\013)2177 5168 y Fl(\()p Fm(n)p Fl(+2\))p Fm(=)p Fl(2)2462 5209 y Fo(:)771 b Fq(\(I.59\))257 5360 y Fj(The)35 b(eigenvalue)e Fo(E)999 5375 y Fi(1)1102 5360 y Fn(\021)28 b Fo(E)1279 5375 y Fm(g)r(s)1387 5360 y Fj(is)35 b(non)f(de)-5 b(gener)g(ate.)p eop end %%Page: 10 11 TeXDict begin 10 10 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(10)257 573 y Fc(I.3)136 b(Outline)45 b(of)g(the)h(Re-Expansion)f(Algorithm)257 758 y Fq(Our)30 b(goal)g(is)g(to)g(deriv)m(e)h(an)f(explicit)i(appro)m(ximation)e(of) 2431 710 y Fm(\036)2473 718 y Fb(g)r(s)p 2396 734 178 4 v 2396 792 a Fi(k)p Fm(\036)2473 800 y Fb(g)r(s)2538 792 y Fi(k)2611 758 y Fq(:=)e(lim)2877 773 y Fm(n)p Fi(!1)3111 715 y Fm(\036)3153 723 y Fb(n)p 3075 734 156 4 v 3075 792 a Fi(k)p Fm(\036)3152 800 y Fb(n)3195 792 y Fi(k)3270 758 y Fq(up)j(to)257 900 y(an)i(error)f(of)g(order)h Fo(o)p Fq(\()p Fo(\013)1145 864 y Fm(N)1212 900 y Fq(\).)43 b(W)-8 b(e)33 b(ma)m(y)g(as)g(w)m(ell)h(consider)f(the)g(v)m(ector)h Fo(\036)2915 915 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)3107 900 y Fq(,)33 b(b)s(ecause)1139 1029 y Fh(\015)1139 1078 y(\015)1139 1128 y(\015)1139 1178 y(\015)1245 1086 y Fo(\036)1303 1101 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)p 1195 1130 351 4 v 1195 1221 a Fn(k)p Fo(\036)1303 1236 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)1495 1221 y Fn(k)1577 1153 y(\000)1737 1086 y Fo(\036)1795 1101 y Fm(g)r(s)p 1687 1130 231 4 v 1687 1221 a Fn(k)p Fo(\036)1795 1236 y Fm(g)r(s)1867 1221 y Fn(k)1927 1029 y Fh(\015)1927 1078 y(\015)1927 1128 y(\015)1927 1178 y(\015)2033 1153 y Fn(\024)61 b(O)s Fq(\()p Fo(\013)2354 1112 y Fm(N)7 b Fl(+)2482 1085 y Ff(1)p 2482 1097 31 4 v 2482 1138 a(2)2526 1153 y Fq(\))22 b Fo(:)647 b Fq(\(I.60\))257 1416 y(The)29 b(b)s(ound)f(\(I.60\))f(follo)m(ws)i(from)f(Prop)s (osition)g(I.1.)41 b(Up)29 b(to)e(normalization,)i(the)f(v)m(ec-)257 1536 y(tor)449 1488 y Fm(\036)491 1499 y Ff(2)p Fb(N)5 b Fd(\000)p Ff(1)p 414 1513 280 4 v 414 1571 a Fi(k)p Fm(\036)491 1582 y Ff(2)p Fb(N)g Fd(\000)p Ff(1)657 1571 y Fi(k)724 1536 y Fq(is)23 b(giv)m(en)f(b)m(y)h(the)f(pro)s(duct)g Fo(P)1755 1551 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1964 1536 y Fn(\001)17 b(\001)g(\001)e Fo(P)2160 1551 y Fl(1)2221 1536 y Fq(of)22 b(the)g(pro)5 b(jections)23 b Fo(P)3033 1551 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)3225 1536 y Fo(;)17 b(:)g(:)g(:)f(;)h(P)3507 1551 y Fl(1)257 1678 y Fq(applied)43 b(to)f(the)h(groundstate,)i Fo(\036)1549 1693 y Fl(0)1588 1678 y Fq(,)g(of)d(the)g(bare)h(Hamiltonian)f Fo(H)2844 1637 y Fl(+)2836 1700 y(0)2903 1678 y Fq(,)i(i.e.,)i(there)d(is)257 1799 y Fo(\025)314 1814 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)535 1799 y Fn(2)28 b Fk(C)22 b Fn(n)g(f)p Fq(0)p Fn(g)32 b Fq(suc)m(h)i(that)1168 1985 y Fo(\036)1226 2000 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)p 1118 2029 351 4 v 1118 2120 a Fn(k)p Fo(\036)1226 2135 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)1419 2120 y Fn(k)1538 2052 y Fq(=)61 b Fo(\025)1732 2067 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1946 2052 y Fn(\001)22 b Fo(P)2059 2067 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2268 2052 y Fn(\001)17 b(\001)g(\001)e Fo(P)2464 2067 y Fl(1)2520 2052 y Fo(\036)2578 2067 y Fl(0)2617 2052 y Fo(:)616 b Fq(\(I.61\))257 2312 y(A)34 b(more)h(precise)g(v)m(ersion)h(of)d(F)-8 b(orm)m(ula)34 b(\(I.61\))g(is)g(giv)m(en)h(in)f(the)h(next)f(section,) i(where)257 2432 y(it)g(is)h(sho)m(wn)g(that)f(the)g(v)m(ector)h Fo(P)1504 2447 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1713 2432 y Fn(\001)17 b(\001)g(\001)e Fo(P)1909 2447 y Fl(1)1965 2432 y Fo(\036)2023 2447 y Fl(0)2098 2432 y Fq(is)37 b(an)e(appro)m(ximate)i(expression)i(for)257 2553 y Fo(\036)315 2568 y Fm(g)r(s)388 2553 y Fq(,)46 b(up)e(to)g(a)f(remainder)i(term)f (of)f(order)h Fo(o)p Fq(\()p Fo(\013)2088 2517 y Fm(N)2155 2553 y Fq(\).)76 b(F)-8 b(rom)43 b(Prop)s(osition)h(I.1,)j(more)257 2673 y(precisely)35 b(from)c(Estimate)i(\(I.54\),)f(w)m(e)h(infer)g (that)e(the)i(expansion)g(in)f(Eq.)h(\(I.57\))f(can)257 2793 y(b)s(e)26 b(truncated)g(at)f(a)g(\014nite)h(order)g(in)g(suc)m(h) g(a)g(w)m(a)m(y)g(that)f(the)h(remainder)h(term)f(is)f Fo(o)p Fq(\()p Fo(\013)3363 2757 y Fm(N)3430 2793 y Fq(\).)257 2914 y(Th)m(us,)35 b(the)e(v)m(ector)1298 3034 y Fo(\036)1356 2993 y Fm(T)1356 3059 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1609 3034 y Fq(:=)60 b Fo(P)1849 2993 y Fm(T)1835 3059 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2044 3034 y Fn(\001)17 b(\001)g(\001)e Fo(P)2254 2993 y Fm(T)2240 3059 y Fl(1)2308 3034 y Fo(\036)2366 3049 y Fl(0)2428 3034 y Fo(;)805 b Fq(\(I.62\))257 3202 y(where)440 3455 y Fo(P)517 3414 y Fm(T)503 3480 y(m)632 3455 y Fq(:=)813 3388 y Fn(\000)p Fq(1)p 805 3432 141 4 v 805 3524 a(2)p Fo(\031)t(i)1003 3347 y Fm(N)974 3372 y Fh(X)973 3554 y Fm(j)t Fl(=0)1112 3338 y Fh(Z)1158 3527 y Fl(\000)1202 3535 y Fb(m)1503 3388 y Fq(1)p 1291 3432 472 4 v 1299 3504 a Fh(f)1291 3537 y Fo(H)1380 3495 y Fl(+)1372 3558 y Fm(m)p Fi(\000)p Fl(1)1551 3537 y Fn(\000)23 b Fo(z)1696 3552 y Fm(m)1795 3334 y Fh(\024)1839 3455 y Fq(\()p Fn(\000)p Fo(W)2060 3414 y Fm(m)p Fi(\000)p Fl(1)2046 3480 y Fm(m)2216 3455 y Fq(\))2492 3388 y(1)p 2281 3432 V 2289 3504 a Fh(f)2281 3537 y Fo(H)2370 3495 y Fl(+)2362 3558 y Fm(m)p Fi(\000)p Fl(1)2541 3537 y Fn(\000)f Fo(z)2685 3552 y Fm(m)2762 3334 y Fh(\025)2806 3357 y Fm(j)2865 3455 y Fo(dz)2961 3470 y Fm(m)3049 3455 y Fo(;)184 b Fq(\(I.63\))257 3749 y(is)30 b(an)e(appro)m(ximate)i (expression)h(for)d Fo(\036)1712 3764 y Fm(g)r(s)1785 3749 y Fq(,)h(up)g(to)g(a)f(remainder)i(term)f(of)g(order)f Fo(o)p Fq(\()p Fo(\013)3363 3713 y Fm(N)3430 3749 y Fq(\).)257 3869 y(Note)33 b(that)f(the)h(truncation)g(in)g(the)g(de\014nition)h (of)e Fo(P)2253 3833 y Fm(T)2239 3894 y(m)2340 3869 y Fq(only)h(dep)s(ends)h(on)f Fo(N)10 b Fq(.)404 3990 y(F)-8 b(rom)30 b(no)m(w)h(on,)g(w)m(e)g(fo)s(cus)g(our)f(atten)m(tion)h(on)f (the)h(analysis)h(of)e(the)g(\014nitely)i(man)m(y)257 4110 y(resolv)m(en)m(ts)k(app)s(earing)c(on)h(the)g(righ)m(t)f(side)i (of)e(\(I.63\).)404 4231 y(Our)g(deriv)-5 b(ation)33 b(of)f(explicit)i(\014nite)g(expressions)h(for)d(the)h(v)m(ector)1439 4435 y Fo(\036)1497 4393 y Fm(T)1497 4459 y(m)1624 4435 y Fq(:=)60 b Fo(P)1864 4393 y Fm(T)1850 4459 y(m)1935 4435 y Fn(\001)17 b(\001)g(\001)e Fo(P)2145 4393 y Fm(T)2131 4459 y Fl(1)2216 4435 y Fo(\036)2274 4450 y Fl(0)3260 4435 y Fq(\(I.64\))257 4639 y(relies)33 b(on)f(three)h(op)s(erations)f (\(i\),)g(\(ii\),)g(and)g(\(iii\),)g(to)g(b)s(e)g(iterated)g(a)g (\014nite)g(n)m(um)m(b)s(er)i(of)257 4759 y(times:)349 4949 y(\(i\))49 b(The)k(photon)e(creation-)g(and)h(annihilation)g(op)s (erators)f(are)g(Wic)m(k-ordered,)501 5069 y(shell)34 b(b)m(y)f(shell.)322 5268 y(\(ii\))49 b(The)30 b(iden)m(tit)m(y)h(op)s (erator,)f Fe(1)1529 5283 y Fm(n)1576 5268 y Fq(,)g(in)f(the)h(space)g Fn(H)2249 5283 y Fm(n)2326 5268 y Fq(is)f(decomp)s(osed)i(in)m(to)f (the)f(sum)501 5388 y Fo(P)564 5403 y Fm(n)633 5388 y Fq(+)22 b Fo(P)808 5352 y Fi(?)794 5413 y Fm(n)867 5388 y Fq(.)p eop end %%Page: 11 12 TeXDict begin 11 11 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(11)295 573 y(\(iii\))49 b(Tw)m(o)29 b(sligh)m(tly)g(di\013eren)m(t,)i(truncated)d(Neumann)h(series)h (expansions,)h(whic)m(h)e(w)m(e)501 693 y(call)g Fn(A)e Fq(and)h Fn(B)s Fq(,)i(are)e(used)h(to)f(re-expand)h(the)g(con)m (tributions)g(asso)s(ciated)g(to)f Fo(P)3437 657 y Fi(?)3423 718 y Fm(n)501 814 y Fq(coming)33 b(from)g(op)s(eration)f(ii\).)257 987 y(As)h(a)g(\014rst)g(step,)g(w)m(e)h(expand)g(the)f(resolv)m(en)m (ts)1852 1143 y(1)p 1641 1187 472 4 v 1649 1259 a Fh(f)1641 1292 y Fo(H)1730 1251 y Fl(+)1722 1314 y Fm(m)p Fi(\000)p Fl(1)1900 1292 y Fn(\000)23 b Fo(z)2045 1307 y Fm(m)3260 1210 y Fq(\(I.65\))257 1475 y(app)s(earing)40 b(in)g(the)g(truncated)h (pro)5 b(jection)40 b Fo(P)2012 1439 y Fm(T)1998 1499 y(m)2106 1475 y Fq(in)g(\(I.64\))g(un)m(til)g(only)g(resolv)m(en)m(ts)j (of)257 1595 y(the)38 b(Hamiltonian)f Fo(H)1084 1554 y Fl(+)1076 1617 y Fm(m)p Fi(\000)p Fl(2)1269 1595 y Fq(are)g(left.)56 b(W)-8 b(e)37 b(then)g(put)g(the)h(expressions)h (obtained)e(in)g(a)257 1715 y(form)26 b(that)g(enables)h(us)f(to)g (iterate)g(the)g(op)s(eration,)h(un)m(til)g(only)f(resolv)m(en)m(ts)i (of)e(the)g(bare)257 1836 y(Hamiltonian)33 b Fo(H)907 1795 y Fl(+)899 1858 y(0)998 1836 y Fq(are)g(left.)404 1956 y(This)53 b(can)g(b)s(e)g(accomplished)h(if)e(w)m(e)i(tak)m(e)f (in)m(to)g(accoun)m(t)g(that)f(the)h(op)s(erator)257 2077 y(\(I.65\))40 b(is)h(applied)g(to)f(a)h(v)m(ector)g(con)m(taining) g(only)g(a)f(\014nite,)j Fo(N)10 b Fq(-dep)s(enden)m(t,)44 b(but)d Fo(\013)q Fq(-)257 2197 y(indep)s(enden)m(t,)k(n)m(um)m(b)s(er) c(of)f(photons)h(with)g(momen)m(ta)g(in)f(the)h(shell)g Fn(K)2972 2161 y Fm(m)p Fi(\000)p Fl(1)2971 2222 y Fm(m)3129 2197 y Fq(;)j(in)d(par-)257 2317 y(ticular,)f(they)f(are)g(at)e(most)i (2)p Fo(N)48 b Fq(b)s(ecause)40 b(of)e(the)g(truncation)h(in)f Fo(P)2874 2281 y Fm(T)2860 2342 y(m)2967 2317 y Fq(and)g(the)g(fact)257 2438 y(that)30 b(the)h(slice)g(in)m(teraction)g Fo(W)1435 2402 y Fm(m)p Fi(\000)p Fl(1)1421 2462 y Fm(m)1621 2438 y Fq(is)g(quadratic)g(in)f(the)g(creation)h(and)f(annihilation)257 2558 y(op)s(erators.)68 b(After)40 b(Wic)m(k-ordering)i(of)e(these)i (photon)f(creation-)f(and)h(annihilation)257 2679 y(op)s(erators)j(in)h (the)g(shell)g Fn(K)1317 2642 y Fm(m)p Fi(\000)p Fl(1)1316 2703 y Fm(m)1474 2679 y Fq(,)i(the)e(original)f(resolv)m(en)m(t)i (\(I.65\))e(is)h(replaced)g(b)m(y)g(a)257 2799 y(\014nite,)31 b Fo(N)10 b Fq(-dep)s(enden)m(t)31 b(n)m(um)m(b)s(er)g({)e(at)g(most)h Fo(N)10 b Fq(!)30 b({)f(of)g(terms)h(con)m(taining)g(resolv)m(en)m(ts)i (of)257 2919 y(the)h(form)1852 2992 y(1)p 1439 3036 875 4 v 1439 3128 a Fo(H)1528 3086 y Fl(+)1520 3149 y Fm(m)p Fi(\000)p Fl(1)1699 3128 y Fn(\000)22 b Fo(z)1843 3143 y Fm(m)1932 3128 y Fq(+)2030 3061 y Fh(P)2118 3148 y Fm(j)2171 3128 y Fn(j)p Fo(k)2250 3143 y Fm(j)2286 3128 y Fn(j)3260 3059 y Fq(\(I.66\))257 3283 y(applied)32 b(to)f(a)g(v)m(ector)h(in)f Fn(H)1282 3298 y Fm(m)p Fi(\000)p Fl(1)1439 3283 y Fq(,)g(where)1777 3217 y Fh(P)1865 3304 y Fm(j)1918 3283 y Fn(j)p Fo(k)1997 3298 y Fm(j)2033 3283 y Fn(j)f Fq(is)i(a)f(\014nite)g(sum)h(of)f(photon)g(energies)257 3403 y Fn(j)p Fo(k)336 3418 y Fm(j)372 3403 y Fn(j)g Fq(,)h(with)g(a)f(n)m(um)m(b)s(er)i(of)e(terms)h(dep)s(ending)h(only)f (on)f Fo(N)10 b Fq(;)32 b(in)g(fact)f(they)h(are)g(at)f(most)257 3524 y(2)p Fo(N)10 b Fq(.)44 b(Note)33 b(that)930 3642 y Fh(P)1018 3729 y Fm(j)1071 3709 y Fn(j)p Fo(k)1150 3724 y Fm(j)1186 3709 y Fn(j)60 b(2)g Fo(J)1454 3724 y Fm(m)1576 3709 y Fq(:=)c Fn(f)p Fq(0)p Fn(g)43 b([)i Fq([)p Fo(\033)2120 3724 y Fm(m)2215 3709 y Fo(;)72 b Fq(2)p Fo(N)32 b Fn(\001)22 b Fo(\033)2578 3724 y Fm(m)p Fi(\000)p Fl(1)2735 3709 y Fq(\))g Fo(;)438 b Fq(\(I.67\))984 3836 y Fl(1)p 984 3852 36 4 v 984 3909 a(4)1029 3875 y Fo(\033)1084 3890 y Fm(m)1211 3875 y Fn(\024)1349 3775 y Fh(\014)1349 3825 y(\014)1349 3875 y(\014)1393 3809 y(P)1481 3896 y Fm(j)1534 3875 y Fn(j)p Fo(k)1613 3890 y Fm(j)1649 3875 y Fn(j)21 b(\000)i Fo(z)1843 3890 y Fm(m)1910 3775 y Fh(\014)1910 3825 y(\014)1910 3875 y(\014)60 b Fn(\024)h Fq(\(2)p Fo(N)32 b Fq(+)22 b(1\))p Fo(\033)2573 3890 y Fm(m)p Fi(\000)p Fl(1)2752 3875 y Fo(:)481 b Fq(\(I.68\))257 4072 y(The)29 b(k)m(ey)f(idea)g(underlying)g(the)g(re-expansion)h(of)d Fo(\036)2191 4087 y Fm(g)r(s)2291 4072 y Fq(is)i(to)e(split)i(the)g(op) s(erator)f(\(I.66\))257 4193 y(in)m(to)33 b(t)m(w)m(o)g(pieces,)i (namely)-8 b(,)1387 4338 y Fl(1)p 1086 4354 637 4 v 1086 4419 a Fm(H)1149 4389 y Ff(+)1144 4442 y Fb(m)p Fd(\000)p Ff(1)1281 4419 y Fi(\000)p Fm(z)1369 4427 y Fb(m)1428 4419 y Fl(+)1483 4361 y Fh(P)1570 4448 y Fb(j)1615 4419 y Fi(j)p Fm(k)1672 4429 y Fb(j)1703 4419 y Fi(j)1750 4377 y Fo(P)1813 4392 y Fm(m)p Fi(\000)p Fl(1)2029 4377 y Fq(=)2305 4331 y Fm(P)2350 4340 y Fb(m)p Fd(\000)p Ff(1)p 2175 4354 442 4 v 2175 4416 a Fi(\000)p Fm(z)2263 4424 y Fb(m)2322 4416 y Fl(+)2377 4358 y Fh(P)2464 4445 y Fb(j)2509 4416 y Fi(j)p Fm(k)2566 4426 y Fb(j)2597 4416 y Fi(j)2649 4377 y Fo(;)584 b Fq(\(I.69\))1716 4534 y Fl(1)p 1415 4550 637 4 v 1415 4615 a Fm(H)1478 4585 y Ff(+)1473 4638 y Fb(m)p Fd(\000)p Ff(1)1610 4615 y Fi(\000)p Fm(z)1698 4623 y Fb(m)1757 4615 y Fl(+)1812 4557 y Fh(P)1899 4644 y Fb(j)1943 4615 y Fi(j)p Fm(k)2000 4625 y Fb(j)2032 4615 y Fi(j)2078 4573 y Fo(P)2155 4537 y Fi(?)2141 4598 y Fm(m)p Fi(\000)p Fl(1)2320 4573 y Fo(:)913 b Fq(\(I.70\))257 4801 y(The)34 b(term)g(in)f(\(I.69\))g(is)h (a)f(m)m(ultiple)h(of)f Fo(P)1813 4816 y Fm(m)p Fi(\000)p Fl(1)1969 4801 y Fq(,)h(the)f(factor)g(of)g(prop)s(ortionalit)m(y)g(b)s (eing)257 4922 y(an)g(explicit)h(n)m(um)m(b)s(er.)45 b(Since)367 5106 y Fo(P)430 5121 y Fm(m)p Fi(\000)p Fl(1)609 5106 y Fn(\000)23 b Fo(P)786 5065 y Fm(T)772 5131 y(m)p Fi(\000)p Fl(1)989 5106 y Fq(=)2195 b(\(I.71\))534 5316 y(=)654 5249 y Fn(\000)p Fq(1)p 647 5293 141 4 v 647 5384 a(2)p Fo(\031)t(i)900 5208 y Fi(1)875 5233 y Fh(X)815 5417 y Fm(j)t Fl(=)p Fm(N)7 b Fl(+1)1072 5199 y Fh(Z)1118 5387 y Fl(\000)1162 5396 y Fb(m)p Fd(\000)p Ff(1)1586 5249 y Fq(1)p 1330 5293 562 4 v 1338 5365 a Fh(f)1330 5397 y Fo(H)1419 5356 y Fl(+)1411 5419 y Fm(m)p Fi(\000)p Fl(2)1589 5397 y Fn(\000)23 b Fo(z)1734 5412 y Fm(m)p Fi(\000)p Fl(1)1923 5195 y Fh(\024)1967 5316 y Fq(\()p Fn(\000)p Fo(W)2188 5275 y Fm(m)p Fi(\000)p Fl(2)2174 5341 y Fm(m)p Fi(\000)p Fl(1)2345 5316 y Fq(\))2666 5249 y(1)p 2409 5293 V 2417 5365 a Fh(f)2409 5397 y Fo(H)2498 5356 y Fl(+)2490 5419 y Fm(m)p Fi(\000)p Fl(2)2669 5397 y Fn(\000)g Fo(z)2814 5412 y Fm(m)p Fi(\000)p Fl(1)2981 5195 y Fh(\025)3024 5218 y Fm(j)3083 5316 y Fo(dz)3179 5331 y Fm(m)p Fi(\000)p Fl(1)3358 5316 y Fo(;)p eop end %%Page: 12 13 TeXDict begin 12 12 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(12)257 573 y(w)m(e)35 b(ma)m(y)g(appro)m (ximate)g Fo(P)1243 588 y Fm(m)p Fi(\000)p Fl(1)1434 573 y Fq(b)m(y)f Fo(P)1647 537 y Fm(T)1633 597 y(m)p Fi(\000)p Fl(1)1824 573 y Fq(up)g(to)g(a)f(remainder)i(of)f(order)g Fo(o)p Fq(\()p Fo(\013)3148 537 y Fm(N)3215 573 y Fq(\).)47 b(The)257 693 y(term)f(in)f(\(I.70\))f(is)i(analyzed)g(b)m(y)g(using)f (the)g(Neumann)h(series)h(expansion)f(b)s(elo)m(w,)257 814 y(whic)m(h)34 b(w)m(e)g(call)f(of)f(t)m(yp)s(e)i Fn(A)p Fq(:)740 983 y(1)p 327 1027 875 4 v 327 1118 a Fo(H)416 1077 y Fl(+)408 1140 y Fm(m)p Fi(\000)p Fl(1)587 1118 y Fn(\000)22 b Fo(z)731 1133 y Fm(m)820 1118 y Fq(+)918 1052 y Fh(P)1006 1139 y Fm(j)1059 1118 y Fn(j)p Fo(k)1138 1133 y Fm(j)1174 1118 y Fn(j)1228 1050 y Fo(P)1305 1009 y Fi(?)1291 1075 y Fm(m)p Fi(\000)p Fl(1)3260 1050 y Fq(\(I.72\))400 1317 y(=)83 b Fo(P)636 1276 y Fi(?)622 1341 y Fm(m)p Fi(\000)p Fl(1)1202 1249 y Fq(1)p 789 1294 V 797 1366 a Fh(f)789 1398 y Fo(H)878 1357 y Fl(+)870 1420 y Fm(m)p Fi(\000)p Fl(2)1048 1398 y Fn(\000)23 b Fo(z)1193 1413 y Fm(m)1282 1398 y Fq(+)1380 1332 y Fh(P)1468 1419 y Fm(j)1521 1398 y Fn(j)p Fo(k)1600 1413 y Fm(j)1636 1398 y Fn(j)1690 1317 y Fo(P)1767 1276 y Fi(?)1753 1341 y Fm(m)p Fi(\000)p Fl(1)1959 1317 y Fq(+)50 b Fo(P)2162 1276 y Fi(?)2148 1341 y Fm(m)p Fi(\000)p Fl(1)2728 1249 y Fq(1)p 2315 1294 V 2323 1366 a Fh(f)2315 1398 y Fo(H)2404 1357 y Fl(+)2396 1420 y Fm(m)p Fi(\000)p Fl(2)2574 1398 y Fn(\000)23 b Fo(z)2719 1413 y Fm(m)2808 1398 y Fq(+)2906 1332 y Fh(P)2994 1419 y Fm(j)3047 1398 y Fn(j)p Fo(k)3126 1413 y Fm(j)3162 1398 y Fn(j)3216 1317 y Fo(P)3293 1276 y Fi(?)3279 1341 y Fm(m)p Fi(\000)p Fl(1)614 1610 y Fn(\002)734 1502 y Fi(1)710 1527 y Fh(X)708 1709 y Fm(j)t Fl(=1)848 1489 y Fh(\024)891 1514 y(\020)963 1610 y Fn(\000)g Fo(W)1169 1569 y Fm(m)p Fi(\000)p Fl(2)1155 1634 y Fm(m)p Fi(\000)p Fl(1)1347 1610 y Fn(\000)g Fo(E)1525 1569 y Fm(m)p Fi(\000)p Fl(2)1519 1634 y Fm(m)p Fi(\000)p Fl(1)1682 1514 y Fh(\021)2154 1542 y Fq(1)p 1741 1587 V 1749 1659 a Fh(f)1741 1691 y Fo(H)1830 1650 y Fl(+)1822 1713 y Fm(m)p Fi(\000)p Fl(2)2001 1691 y Fn(\000)g Fo(z)2146 1706 y Fm(m)2235 1691 y Fq(+)2333 1625 y Fh(P)2420 1712 y Fm(j)2473 1691 y Fn(j)p Fo(k)2552 1706 y Fm(j)2588 1691 y Fn(j)2643 1610 y Fo(P)2720 1569 y Fi(?)2706 1634 y Fm(m)p Fi(\000)p Fl(1)2862 1489 y Fh(\025)2906 1512 y Fm(j)2965 1610 y Fo(:)257 1900 y Fq(This)j(expansion)g(con)m(v)m(erges)h(b)s(ecause)f (the)g(absolute)f(v)-5 b(alue)25 b(of)f(the)i(energy)f(shift)g Fo(E)3338 1858 y Fm(m)p Fi(\000)p Fl(2)3332 1921 y Fm(m)p Fi(\000)p Fl(1)257 2020 y Fq(is)33 b(v)m(ery)i(small,)e(namely)h(of)e (order)g Fn(O)s Fq(\()p Fo(\013)1739 1984 y Fm(m)p Fl(+1)1896 2020 y Fq(\))g(and)h(b)s(ecause)956 2137 y Fh(\015)956 2187 y(\015)956 2237 y(\015)956 2287 y(\015)1425 2195 y Fq(1)p 1012 2239 V 1020 2311 a Fh(f)1012 2343 y Fo(H)1101 2302 y Fl(+)1093 2365 y Fm(m)p Fi(\000)p Fl(2)1272 2343 y Fn(\000)23 b Fo(z)1417 2358 y Fm(m)1506 2343 y Fq(+)1604 2277 y Fh(P)1691 2364 y Fm(j)1744 2343 y Fn(j)p Fo(k)1823 2358 y Fm(j)1859 2343 y Fn(j)1913 2262 y Fo(P)1990 2221 y Fi(?)1976 2287 y Fm(m)p Fi(\000)p Fl(1)2133 2137 y Fh(\015)2133 2187 y(\015)2133 2237 y(\015)2133 2287 y(\015)2239 2262 y Fn(\024)61 b(O)s Fq(\()p Fo(\033)2556 2221 y Fi(\000)p Fl(1)2552 2287 y Fm(m)p Fi(\000)p Fl(1)2709 2262 y Fq(\))22 b Fo(;)464 b Fq(\(I.73\))257 2545 y(thanks)38 b(to)f(the)g(presence)i (of)e(the)g(orthogonal)f(pro)5 b(jection)38 b Fo(P)2595 2508 y Fi(?)2581 2569 y Fm(m)p Fi(\000)p Fl(1)2738 2545 y Fq(.)56 b(These)39 b(t)m(w)m(o)f(facts)257 2665 y(and)32 b(Estimate)h(\(I.54\))e(imply)h(that)g(the)f Fo(j)1821 2629 y Fm(th)1923 2665 y Fq(term)h(in)g(the)g(series)h(expansion)g(in)f (\(I.72\))257 2785 y(is)g(b)s(ounded)f(b)m(y)h(\()p Fo(C)7 b(\013)q Fq(\))1100 2749 y Fm(j)1136 2785 y Fq(,)32 b(for)e(some)i (constan)m(t)f Fo(C)k Fn(2)28 b Fk(R)2247 2749 y Fl(+)2306 2785 y Fq(.)43 b(Th)m(us,)33 b(the)e(series)i(expansion)257 2906 y(\(I.72\))e(is)g(con)m(v)m(ergen)m(t)j(and)d(can)g(b)s(e)g(appro) m(ximated)h(up)g(to)e(terms)i(of)f(order)g Fo(o)p Fq(\()p Fo(\013)3257 2870 y Fm(N)3323 2906 y Fq(\))g(b)m(y)257 3026 y(its)i(truncation)g(at)g(order)f Fo(j)i Fq(=)27 b Fo(N)33 b Fq(+)22 b(1,)32 b(where)i(w)m(e)g(replace)g Fo(P)2572 2990 y Fi(?)2558 3051 y Fm(m)p Fi(\000)p Fl(1)2747 3026 y Fq(b)m(y)f Fe(1)22 b Fn(\000)h Fo(P)3137 2990 y Fm(T)3123 3051 y(m)p Fi(\000)p Fl(1)3280 3026 y Fq(.)404 3147 y(As)41 b(a)f(result)i(of)e(the)h(previous)h(op)s(erations)f (\(the)g(decomp)s(osition)h(\(I.69\)-\(I.70\),)257 3267 y(the)26 b(truncation)g(\(I.71\))e(of)h(the)h(series)h(expansion)f(of)f Fo(P)2293 3231 y Fm(T)2279 3292 y(m)p Fi(\000)p Fl(1)2436 3267 y Fq(,)h(and)g(the)f(Neumann)i(series)257 3387 y(expansion)k Fn(A)e Fq(in)h(\(I.72\)\),)f(the)h(resolv)m(en)m(t)h(\(I.65\))e(is)h (represen)m(ted)i(b)m(y)e(a)f(p)s(olynomial)h(in)257 3508 y(the)j(follo)m(wing)g(op)s(erators:)403 3692 y Fn(\017)48 b Fq(the)33 b(resolv)m(en)m(ts)1528 3841 y(1)p 1115 3886 V 1123 3958 a Fh(f)1115 3990 y Fo(H)1204 3949 y Fl(+)1196 4012 y Fm(m)p Fi(\000)p Fl(2)1375 3990 y Fn(\000)22 b Fo(z)1519 4005 y Fm(m)1608 3990 y Fq(+)1706 3924 y Fh(P)1794 4011 y Fm(j)1847 3990 y Fn(j)p Fo(k)1926 4005 y Fm(j)1962 3990 y Fn(j)2022 3909 y Fo(;)2527 3841 y Fq(1)p 2271 3886 562 4 v 2279 3958 a Fh(f)2271 3990 y Fo(H)2360 3949 y Fl(+)2352 4012 y Fm(m)p Fi(\000)p Fl(2)2530 3990 y Fn(\000)h Fo(z)2675 4005 y Fm(m)p Fi(\000)p Fl(1)2864 3909 y Fo(;)369 b Fq(\(I.74\))403 4231 y Fn(\017)48 b Fq(the)33 b(slice)h(in)m(teraction)g Fo(W)1478 4190 y Fm(m)p Fi(\000)p Fl(2)1464 4253 y Fm(m)p Fi(\000)p Fl(1)1634 4231 y Fq(,)f(and)403 4428 y Fn(\017)48 b Fq(the)33 b(energy)h(shift)f Fo(E)1279 4387 y Fm(m)p Fi(\000)p Fl(2)1273 4450 y Fm(m)p Fi(\000)p Fl(1)1436 4428 y Fq(.)257 4613 y(Returning)42 b(to)e(the)h(expression)i(\(I.64\),)g(w)m(e)f(then) f(Wic)m(k-order)h(the)f(photon)g(op)s(era-)257 4733 y(tors)34 b(corresp)s(onding)g(to)e(photon)h(momen)m(ta)h(in)g(the)f(shell)h Fn(K)2540 4692 y Fm(m)p Fi(\000)p Fl(2)2539 4755 y Fm(m)p Fi(\000)p Fl(1)2698 4733 y Fq(.)45 b(This)34 b(yields)h(again)257 4853 y(\014nitely)27 b(man)m(y)f(terms)g(and)f(the)g(resolv)m(en)m(ts)j (in)d(Eq.)h(\(I.74\))e(are)h(replaced)i(b)m(y)f(resolv)m(en)m(ts)257 4974 y(of)32 b(the)h(form)1834 5132 y Fl(1)p 1503 5148 698 4 v 1503 5213 a Fm(H)1566 5183 y Ff(+)1561 5235 y Fb(m)p Fd(\000)p Ff(2)1698 5213 y Fi(\000)p Fm(z)1786 5222 y Fb(m)p Fd(\000)p Ff(1)1923 5213 y Fl(+)1978 5155 y Fh(P)2065 5242 y Fb(i)2103 5213 y Fi(j)p Fm(q)2155 5223 y Fb(i)2181 5213 y Fi(j)2233 5171 y Fo(;)1000 b Fq(\(I.75\))1834 5318 y Fl(1)p 1395 5334 915 4 v 1395 5399 a Fm(H)1458 5369 y Ff(+)1453 5422 y Fb(m)p Fd(\000)p Ff(2)1590 5399 y Fi(\000)p Fm(z)1678 5407 y Fb(m)1736 5399 y Fl(+)1791 5341 y Fh(P)1878 5428 y Fb(j)1923 5399 y Fi(j)p Fm(k)1980 5409 y Fb(j)2012 5399 y Fi(j)o Fl(+)2086 5341 y Fh(P)2174 5428 y Fb(i)2212 5399 y Fi(j)p Fm(q)2264 5409 y Fb(i)2289 5399 y Fi(j)2341 5358 y Fo(;)892 b Fq(\(I.76\))p eop end %%Page: 13 14 TeXDict begin 13 13 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(13)257 573 y(where)36 b(the)f(photon)f(momen) m(ta)h Fo(q)1513 588 y Fm(i)1575 573 y Fq(all)g(b)s(elong)f(to)g(the)g (shell)i Fn(K)2618 532 y Fm(m)p Fi(\000)p Fl(2)2617 595 y Fm(m)p Fi(\000)p Fl(1)2809 573 y Fq(and)e(of)g(terms,)i(in)257 627 y Fh(P)345 714 y Fm(i)390 693 y Fn(j)p Fo(q)461 708 y Fm(i)489 693 y Fn(j)p Fq(,)42 b(is)f(b)s(ounded)g(b)m(y)h(an)e Fo(N)10 b Fn(\000)p Fq(dep)s(enden)m(t)43 b(n)m(um)m(b)s(er,)h(thanks)e (to)e(the)h(truncation)257 814 y(in)f Fo(P)455 777 y Fm(T)441 838 y(m)p Fi(\000)p Fl(1)598 814 y Fq(.)64 b(The)40 b(resolv)m(en)m(t)i(in)e(\(I.75\))f(has)h(already)g(the)g(desired)h (form)e(that)g(can)h(b)s(e)257 934 y(traced)29 b(bac)m(k)g(to)f (\(I.66\),)h(but)f(the)h(one)f(in)g(\(I.76\))g(do)s(esn't.)43 b(As)28 b(b)s(efore,)i(w)m(e)f(decomp)s(ose)257 1054 y(the)36 b(resolv)m(en)m(ts)i(\(I.75\),)e(\(I.76\))f(in)m(to)h(t)m(w)m (o)g(pieces,)i(using)e(the)f(pro)5 b(jections)37 b Fo(P)3146 1069 y Fm(m)p Fi(\000)p Fl(2)3338 1054 y Fq(and)257 1175 y Fo(P)334 1139 y Fi(?)320 1199 y Fm(m)p Fi(\000)p Fl(2)477 1175 y Fq(.)k(This)26 b(yields)g(terms)f(prop)s(ortional)f(to)h(the)g (pro)5 b(jection)25 b Fo(P)2632 1190 y Fm(m)p Fi(\000)p Fl(2)2789 1175 y Fq(,)h(with)f(an)g(explicit)257 1295 y(factor)32 b(of)h(prop)s(ortionalit)m(y)-8 b(,)32 b(and)h(terms)g(of)g (the)g(form)824 1487 y(1)p 378 1531 941 4 v 378 1622 a Fo(H)467 1581 y Fl(+)459 1644 y Fm(m)p Fi(\000)p Fl(2)637 1622 y Fn(\000)23 b Fo(z)782 1637 y Fm(m)p Fi(\000)p Fl(1)961 1622 y Fq(+)1059 1556 y Fh(P)1147 1643 y Fm(i)1192 1622 y Fn(j)p Fo(q)1263 1637 y Fm(i)1291 1622 y Fn(j)1345 1554 y Fo(P)1422 1513 y Fi(?)1408 1579 y Fm(m)p Fi(\000)p Fl(2)1587 1554 y Fo(;)2438 1487 y Fq(1)p 1835 1531 1255 4 v 1835 1622 a Fo(H)1924 1581 y Fl(+)1916 1644 y Fm(m)p Fi(\000)p Fl(2)2095 1622 y Fn(\000)g Fo(z)2240 1637 y Fm(m)2329 1622 y Fq(+)2427 1556 y Fh(P)2514 1643 y Fm(j)2568 1622 y Fn(j)p Fo(k)2647 1637 y Fm(j)2683 1622 y Fn(j)e Fq(+)2830 1556 y Fh(P)2918 1643 y Fm(i)2963 1622 y Fn(j)p Fo(q)3034 1637 y Fm(i)3062 1622 y Fn(j)3116 1554 y Fo(P)3193 1513 y Fi(?)3179 1579 y Fm(m)p Fi(\000)p Fl(2)3358 1554 y Fo(:)3260 1739 y Fq(\(I.77\))257 1859 y(The)k(\014rst)f(term)g(in)g (Eq.\(I.77\))g(is)g(analyzed)h(b)m(y)f(using)h(the)f(Neumann)g(series)i (expansion)257 1980 y(of)40 b(t)m(yp)s(e)g Fn(A)p Fq(,)h(see)g (\(I.72\),)g(with)f Fo(m)27 b Fn(\000)h Fq(1)39 b(replaced)i(b)m(y)f Fo(m)28 b Fn(\000)f Fq(2.)64 b(The)41 b(second)g(term)f(in)257 2100 y(Eq.)h(\(I.77\))e(is)h(treated)g(b)m(y)g(applying)g(the)g(follo)m (wing)g(expansion,)i(whic)m(h)f(w)m(e)g(call)f(of)257 2221 y(t)m(yp)s(e)34 b Fn(B)s Fq(:)1033 2412 y(1)p 431 2456 V 431 2548 a Fo(H)520 2507 y Fl(+)512 2570 y Fm(m)p Fi(\000)p Fl(2)690 2548 y Fn(\000)23 b Fo(z)835 2563 y Fm(m)924 2548 y Fq(+)1022 2481 y Fh(P)1110 2569 y Fm(j)1163 2548 y Fn(j)p Fo(k)1242 2563 y Fm(j)1278 2548 y Fn(j)f Fq(+)1426 2481 y Fh(P)1513 2569 y Fm(i)1558 2548 y Fn(j)p Fo(q)1629 2563 y Fm(i)1657 2548 y Fn(j)1711 2480 y Fo(P)1788 2438 y Fi(?)1774 2504 y Fm(m)p Fi(\000)p Fl(2)1991 2480 y Fq(=)504 2746 y(=)83 b Fo(P)740 2705 y Fi(?)726 2771 y Fm(m)p Fi(\000)p Fl(2)1338 2679 y Fq(1)p 892 2723 941 4 v 900 2795 a Fh(f)892 2828 y Fo(H)981 2786 y Fl(+)973 2849 y Fm(m)p Fi(\000)p Fl(3)1152 2828 y Fn(\000)23 b Fo(z)1297 2843 y Fm(m)p Fi(\000)p Fl(1)1476 2828 y Fq(+)1574 2761 y Fh(P)1661 2848 y Fm(i)1706 2828 y Fn(j)p Fo(q)1777 2843 y Fm(i)1805 2828 y Fn(j)1860 2746 y Fo(P)1937 2705 y Fi(?)1923 2771 y Fm(m)p Fi(\000)p Fl(2)663 3021 y Fq(+)17 b Fo(P)833 2980 y Fi(?)819 3046 y Fm(m)p Fi(\000)p Fl(2)1431 2954 y Fq(1)p 985 2998 V 993 3070 a Fh(f)985 3102 y Fo(H)1074 3061 y Fl(+)1066 3124 y Fm(m)p Fi(\000)p Fl(3)1244 3102 y Fn(\000)23 b Fo(z)1389 3117 y Fm(m)p Fi(\000)p Fl(1)1568 3102 y Fq(+)1666 3036 y Fh(P)1754 3123 y Fm(i)1799 3102 y Fn(j)p Fo(q)1870 3117 y Fm(i)1898 3102 y Fn(j)1952 3021 y Fo(P)2029 2980 y Fi(?)2015 3046 y Fm(m)p Fi(\000)p Fl(2)3260 3021 y Fq(\(I.78\))695 3326 y Fn(\002)819 3218 y Fm(N)791 3243 y Fh(X)789 3425 y Fm(j)t Fl(=1)928 3205 y Fh(\024)972 3230 y(\020)1044 3326 y Fn(\000)g Fo(W)1250 3285 y Fm(m)p Fi(\000)p Fl(3)1236 3351 y Fm(m)p Fi(\000)p Fl(2)1428 3326 y Fn(\000)g Fo(E)1606 3285 y Fm(m)p Fi(\000)p Fl(3)1600 3351 y Fm(m)p Fi(\000)p Fl(2)1763 3230 y Fh(\021)2285 3259 y Fq(1)p 1839 3303 V 1847 3375 a Fh(f)1839 3407 y Fo(H)1928 3366 y Fl(+)1920 3429 y Fm(m)p Fi(\000)p Fl(3)2098 3407 y Fn(\000)g Fo(z)2243 3422 y Fm(m)p Fi(\000)p Fl(1)2422 3407 y Fq(+)2520 3341 y Fh(P)2608 3428 y Fm(i)2653 3407 y Fn(j)p Fo(q)2724 3422 y Fm(i)2752 3407 y Fn(j)2806 3326 y Fo(P)2883 3285 y Fi(?)2869 3351 y Fm(m)p Fi(\000)p Fl(2)858 3622 y Fq(+)934 3525 y Fh(\020)983 3622 y Fo(z)1028 3637 y Fm(m)1117 3622 y Fn(\000)g Fo(z)1262 3637 y Fm(m)p Fi(\000)p Fl(1)1441 3622 y Fq(+)1539 3539 y Fh(X)1583 3721 y Fm(j)1676 3622 y Fn(j)p Fo(k)1755 3637 y Fm(j)1791 3622 y Fn(j)1818 3525 y Fh(\021)2340 3554 y Fq(1)p 1894 3599 V 1902 3671 a Fh(f)1894 3703 y Fo(H)1983 3662 y Fl(+)1975 3725 y Fm(m)p Fi(\000)p Fl(3)2154 3703 y Fn(\000)g Fo(z)2299 3718 y Fm(m)p Fi(\000)p Fl(1)2478 3703 y Fq(+)2576 3637 y Fh(P)2664 3724 y Fm(i)2708 3703 y Fn(j)p Fo(q)2779 3718 y Fm(i)2807 3703 y Fn(j)2862 3622 y Fo(P)2939 3581 y Fi(?)2925 3646 y Fm(m)p Fi(\000)p Fl(2)3081 3501 y Fh(\025)3125 3524 y Fm(j)663 3907 y Fq(+)1413 3840 y(1)p 765 3884 1345 4 v 765 3975 a Fo(H)854 3934 y Fl(+)846 3997 y Fm(m)p Fi(\000)p Fl(2)1025 3975 y Fn(\000)f Fo(z)1169 3990 y Fm(m)p Fi(\000)p Fl(1)1349 3975 y Fq(+)1447 3909 y Fh(P)1534 3996 y Fm(j)1587 3975 y Fn(j)p Fo(k)1666 3990 y Fm(j)1702 3975 y Fn(j)g Fq(+)1850 3909 y Fh(P)1938 3996 y Fm(i)1983 3975 y Fn(j)p Fo(q)2054 3990 y Fm(i)2082 3975 y Fn(j)2136 3907 y Fo(P)2213 3866 y Fi(?)2199 3932 y Fm(m)p Fi(\000)p Fl(2)3260 3907 y Fq(\(I.79\))695 4174 y Fn(\002)772 4053 y Fh(\024)817 4078 y(\020)888 4174 y Fn(\000)h Fo(W)1094 4133 y Fm(m)p Fi(\000)p Fl(3)1080 4199 y Fm(m)p Fi(\000)p Fl(2)1272 4174 y Fn(\000)g Fo(E)1450 4133 y Fm(m)p Fi(\000)p Fl(3)1444 4199 y Fm(m)p Fi(\000)p Fl(2)1607 4078 y Fh(\021)2129 4107 y Fq(1)p 1683 4151 941 4 v 1691 4223 a Fh(f)1683 4255 y Fo(H)1772 4214 y Fl(+)1764 4277 y Fm(m)p Fi(\000)p Fl(3)1943 4255 y Fn(\000)f Fo(z)2087 4270 y Fm(m)p Fi(\000)p Fl(1)2267 4255 y Fq(+)2365 4189 y Fh(P)2452 4276 y Fm(i)2497 4255 y Fn(j)p Fo(q)2568 4270 y Fm(i)2596 4255 y Fn(j)2650 4174 y Fo(P)2727 4133 y Fi(?)2713 4199 y Fm(m)p Fi(\000)p Fl(2)858 4461 y Fq(+)934 4364 y Fh(\020)983 4461 y Fo(z)1028 4476 y Fm(m)1117 4461 y Fn(\000)h Fo(z)1262 4476 y Fm(m)p Fi(\000)p Fl(1)1441 4461 y Fq(+)1539 4378 y Fh(X)1583 4560 y Fm(j)1676 4461 y Fn(j)p Fo(k)1755 4476 y Fm(j)1791 4461 y Fn(j)1818 4364 y Fh(\021)2340 4393 y Fq(1)p 1894 4437 V 1902 4510 a Fh(f)1894 4542 y Fo(H)1983 4501 y Fl(+)1975 4564 y Fm(m)p Fi(\000)p Fl(3)2154 4542 y Fn(\000)g Fo(z)2299 4557 y Fm(m)p Fi(\000)p Fl(1)2478 4542 y Fq(+)2576 4476 y Fh(P)2664 4563 y Fm(i)2708 4542 y Fn(j)p Fo(q)2779 4557 y Fm(i)2807 4542 y Fn(j)2862 4461 y Fo(P)2939 4420 y Fi(?)2925 4485 y Fm(m)p Fi(\000)p Fl(2)3081 4340 y Fh(\025)3125 4363 y Fm(N)7 b Fl(+1)3305 4461 y Fo(:)257 4764 y Fq(In)44 b(this)f(truncated)h(expansion,)j(the)c(remainder)h (term)f(\(I.79\))g(is)g(pro)m(v)m(en)h(to)f(b)s(e)g(of)257 4884 y(order)32 b Fo(o)p Fq(\()p Fo(\013)659 4848 y Fm(N)725 4884 y Fq(\))f(with)h(resp)s(ect)h(to)d(the)i(original)f(expression)i (in)f(\(I.77\).)42 b(T)-8 b(o)32 b(see)g(this,)g(w)m(e)257 5005 y(use)40 b(the)f(b)s(ounds)g(on)g(the)f(shift)h(of)g(the)g(in)m (tegration)f(v)-5 b(ariable,)41 b Fo(z)2731 5020 y Fm(m)2824 5005 y Fn(\000)26 b Fo(z)2972 5020 y Fm(m)p Fi(\000)p Fl(1)3130 5005 y Fq(,)40 b(and)e(on)257 5125 y(the)30 b(sum)625 5059 y Fh(P)712 5146 y Fm(j)765 5125 y Fn(j)p Fo(k)844 5140 y Fm(j)880 5125 y Fn(j)p Fq(:)42 b(Both)28 b(quan)m(tities)j(are)e(of)g(order,)h(at)e(most,)i Fo(\033)2641 5140 y Fm(m)p Fi(\000)p Fl(1)2799 5125 y Fq(,)f(the)h(\014rst)f(one)g (b)m(y)257 5245 y(de\014nition)39 b(of)d Fo(z)856 5260 y Fm(m)960 5245 y Fq(and)i Fo(z)1200 5260 y Fm(m)p Fi(\000)p Fl(1)1357 5245 y Fq(,)g(and)g(the)f(second)i(one)f(b)s(ecause)g(of)f (the)h(fact)f(that)g(the)257 5366 y(n)m(um)m(b)s(er)f(of)e(terms)h(in) 1116 5299 y Fh(P)1203 5387 y Fm(j)1256 5366 y Fn(j)p Fo(k)1335 5381 y Fm(j)1371 5366 y Fn(j)f Fq(is)h(\014nite)g(and)f (indep)s(enden)m(t)j(of)d Fo(\013)q Fq(.)48 b(The)35 b(replacemen)m(t)p eop end %%Page: 14 15 TeXDict begin 14 14 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(14)257 573 y(of)44 b(the)i(in)m(tegration)e (v)-5 b(ariable)45 b Fo(z)1491 588 y Fm(m)1603 573 y Fq(b)m(y)g(the)g(v)-5 b(ariable)45 b Fo(z)2353 588 y Fm(m)p Fi(\000)p Fl(1)2555 573 y Fq(is)g(p)s(ossible)h(b)s(ecause)g(of) 257 693 y(Estimate)37 b(\(I.53\))e(on)g(the)h(sp)s(ectral)g(gap.)51 b(One)36 b(then)g(expands)h(in)e(the)h(energies)h Fn(j)p Fo(k)3405 708 y Fm(j)3440 693 y Fn(j)p Fq(,)257 814 y(whic)m(h)g(are)e (b)s(ounded)h(b)m(y)h Fo(const)24 b Fn(\001)f Fo(\033)1602 829 y Fm(m)p Fi(\000)p Fl(1)1760 814 y Fq(.)51 b(The)36 b(norms)g(of)f(the)h(remaining)g(resolv)m(en)m(ts)257 934 y(in)41 b(\(I.79\))f(are)h(b)s(ounded)g(b)m(y)g Fo(const)28 b Fn(\001)g Fo(\033)1746 893 y Fi(\000)p Fl(1)1742 956 y Fm(m)p Fi(\000)p Fl(2)1939 934 y Fq(thanks)42 b(to)e Fo(P)2469 898 y Fi(?)2455 959 y Fm(m)p Fi(\000)p Fl(2)2611 934 y Fq(.)68 b(Finally)41 b(w)m(e)g(replace)257 1054 y Fo(P)334 1018 y Fi(?)320 1079 y Fm(m)p Fi(\000)p Fl(2)510 1054 y Fq(b)m(y)33 b Fe(1)701 1069 y Fm(m)p Fi(\000)p Fl(2)880 1054 y Fn(\000)23 b Fo(P)1057 1018 y Fm(T)1043 1079 y(m)p Fi(\000)p Fl(2)1232 1054 y Fq(in)32 b(the)h(truncated)h (expansion)g(\(I.78\).)404 1175 y(Note)40 b(a)g(new)h(feature)g(of)e (the)i(expansion)h(of)e(t)m(yp)s(e)h Fn(B)s Fq(,)h(as)f(compared)g(to)f (an)g(ex-)257 1295 y(pansion)j(of)e(t)m(yp)s(e)i Fn(A)p Fq(:)62 b(W)-8 b(e)43 b(do)e(not)h(con)m(trol)h(the)f(gro)m(wth)g(of)g (the)g(co)s(e\016cien)m(t)i Fo(c)3305 1310 y Fm(N)3414 1295 y Fq(in)257 1416 y Fn(k)p Fq(\()p Fo(I)8 b(:)p Fq(79\))p Fn(k)27 b(\024)h Fo(c)783 1431 y Fm(N)867 1416 y Fo(\013)929 1431 y Fm(N)7 b Fl(+1)1087 1416 y Fq(,)32 b(as)h Fo(N)39 b Fn(!)27 b(1)p Fq(,)32 b(and)h(so)g(w)m(e)h(do)e(not)h(assert)g Fj(c)-5 b(onver)g(genc)g(e)39 b Fq(of)32 b(the)257 1536 y(series)j(of)d(t)m(yp)s(e)h Fn(B)j Fq(-)c(it)h(is)g(only)g (asymptotic.)404 1656 y(Hence,)f(starting)e(with)g(a)g(resolv)m(en)m(t) i(of)e(the)g(form)g(giv)m(en)h(in)f(Eq.)h(\(I.66\),)f(and)h(then)257 1777 y(p)s(erforming)e(the)f(op)s(erations)g(just)h(describ)s(ed)h(w)m (e)f(end)g(up)f(with)h(a)f(p)s(olynomial)g(in)h(the)257 1897 y(follo)m(wing)k(op)s(erators:)403 2100 y Fn(\017)48 b Fq(the)33 b(resolv)m(en)m(ts)1528 2273 y(1)p 1082 2317 941 4 v 1090 2389 a Fh(f)1082 2421 y Fo(H)1171 2380 y Fl(+)1163 2443 y Fm(m)p Fi(\000)p Fl(3)1342 2421 y Fn(\000)22 b Fo(z)1486 2436 y Fm(m)p Fi(\000)p Fl(1)1666 2421 y Fq(+)1764 2355 y Fh(P)1851 2442 y Fm(i)1896 2421 y Fn(j)p Fo(q)1967 2436 y Fm(i)1995 2421 y Fn(j)2055 2340 y Fo(;)2560 2273 y Fq(1)p 2304 2317 562 4 v 2312 2389 a Fh(f)2304 2421 y Fo(H)2393 2380 y Fl(+)2385 2443 y Fm(m)p Fi(\000)p Fl(3)2563 2421 y Fn(\000)h Fo(z)2708 2436 y Fm(m)p Fi(\000)p Fl(2)2897 2340 y Fo(;)336 b Fq(\(I.80\))403 2683 y Fn(\017)48 b Fq(the)33 b(slice)h(in)m(teraction)g Fo(W)1478 2642 y Fm(m)p Fi(\000)p Fl(3)1464 2705 y Fm(m)p Fi(\000)p Fl(2)1634 2683 y Fq(,)403 2887 y Fn(\017)48 b Fq(the)33 b(energy)h(shift)f Fo(E)1279 2846 y Fm(m)p Fi(\000)p Fl(3)1273 2909 y Fm(m)p Fi(\000)p Fl(2)1436 2887 y Fq(,)403 3090 y Fn(\017)48 b Fq(the)33 b(di\013erences)i Fo(z)1189 3105 y Fm(m)1278 3090 y Fn(\000)23 b Fo(z)1423 3105 y Fm(m)p Fi(\000)p Fl(1)1612 3090 y Fq(of)32 b(in)m(tegration)h(v)-5 b(ariables,)403 3294 y Fn(\017)48 b Fq(sums)34 b(of)e(single-photon)h (energies)1828 3227 y Fh(P)1916 3314 y Fm(j)1969 3294 y Fn(j)p Fo(k)2048 3309 y Fm(j)2084 3294 y Fn(j)p Fq(.)257 3497 y(W)-8 b(e)37 b(pro)s(ceed)g(b)m(y)g(Wic)m(k-ordering)g(the)g (photon)f(creation-)g(and)g(annihilation)h(op)s(era-)257 3617 y(tors)29 b(in)g(the)g(shell)h Fn(K)1019 3576 y Fm(m)p Fi(\000)p Fl(3)1018 3639 y Fm(m)p Fi(\000)p Fl(2)1205 3617 y Fq(in)f(ev)m(ery)i(term)e(obtained)g(so)g(far,)g(starting)g (from)g(the)g(v)m(ector)257 3738 y(in)41 b(Eq.)f(\(I.64\),)i(using)f (\(I.63\),)g(and)f(then)h(expanding)g(the)g(resolv)m(en)m(ts)h(as)e (describ)s(ed)257 3858 y(ab)s(o)m(v)m(e.)45 b(Tw)m(o)33 b(t)m(yp)s(es)h(of)e(resolv)m(en)m(ts)j(result:)1161 4050 y(1)p 503 4094 1365 4 v 503 4185 a Fo(H)592 4144 y Fl(+)584 4207 y Fm(m)p Fi(\000)p Fl(3)762 4185 y Fn(\000)23 b Fo(z)907 4200 y Fm(m)p Fi(\000)p Fl(1)1086 4185 y Fq(+)1184 4119 y Fh(P)1272 4206 y Fm(i)1317 4185 y Fn(j)p Fo(q)1388 4200 y Fm(i)1416 4185 y Fn(j)f Fq(+)1564 4119 y Fh(P)1651 4206 y Fm(i)1675 4187 y Fd(0)1718 4185 y Fn(j)p Fo(q)1793 4151 y Fi(0)1789 4211 y Fm(i)1813 4192 y Fd(0)1840 4185 y Fn(j)1975 4117 y Fo(;)2497 4050 y Fq(1)p 2029 4094 986 4 v 2029 4185 a Fo(H)2118 4144 y Fl(+)2110 4207 y Fm(m)p Fi(\000)p Fl(3)2288 4185 y Fn(\000)h Fo(z)2433 4200 y Fm(m)p Fi(\000)p Fl(2)2612 4185 y Fq(+)2710 4119 y Fh(P)2798 4206 y Fm(i)2822 4187 y Fd(0)2865 4185 y Fn(j)p Fo(q)2940 4151 y Fi(0)2936 4211 y Fm(i)2960 4192 y Fd(0)2987 4185 y Fn(j)3260 4117 y Fq(\(I.81\))257 4397 y(where)47 b(the)e(new)h(sum,)1195 4330 y Fh(P)1282 4418 y Fm(i)1306 4399 y Fd(0)1349 4397 y Fn(j)p Fo(q)1424 4361 y Fi(0)1420 4421 y Fm(i)1444 4403 y Fd(0)1471 4397 y Fn(j)o Fq(,)j(corresp)s(onds)d(to)f(photon)g(momen)m(ta,)k Fo(q)3111 4361 y Fi(0)3107 4421 y Fm(i)3131 4403 y Fd(0)3158 4397 y Fq(,)f(in)e(the)257 4517 y(shell)c Fn(K)565 4476 y Fm(m)p Fi(\000)p Fl(3)564 4539 y Fm(m)p Fi(\000)p Fl(2)722 4517 y Fq(,)g(and)e(the)g(n)m(um)m(b)s(er)h(of)f(terms)h(is)f(b)s (ounded)h(b)m(y)g(an)f Fo(N)10 b Fq(-dep)s(enden)m(t,)44 b(but)257 4638 y Fo(\013)q Fq(-indep)s(enden)m(t)34 b(in)m(teger.)404 4758 y(After)42 b(inserting)h(the)f(partition)g(of)g(unit)m(y)-8 b(,)45 b Fe(1)2134 4773 y Fm(m)p Fi(\000)p Fl(3)2335 4758 y Fq(=)f Fo(P)2518 4773 y Fm(m)p Fi(\000)p Fl(3)2703 4758 y Fq(+)28 b Fo(P)2884 4722 y Fi(?)2870 4783 y Fm(m)p Fi(\000)p Fl(3)3027 4758 y Fq(,)44 b(w)m(e)f(arriv)m(e)257 4878 y(at)37 b(the)h(expressions)i(in)e(\(I.77\),)g(but)g(with)g Fo(m)25 b Fn(\000)h Fq(2)37 b(replaced)i(b)m(y)f Fo(m)26 b Fn(\000)g Fq(3.)57 b(Th)m(us,)40 b(our)257 4999 y(re-expansion)34 b(pro)s(cedure,)f(based)g(on)f(truncated)h(Neumann)h(expansions)g(of)e (t)m(yp)s(e)h Fn(A)257 5119 y Fq(and)g Fn(B)j Fq(and)d(on)f(Wic)m (k-ordering,)i(can)f(b)s(e)g(iterated.)257 5239 y(By)h(applying)f(the)h (re-expansion)g(pro)s(cedure,)g(scale)g(b)m(y)g(scale,)g(to)e(all)h (the)h(resolv)m(en)m(ts)257 5360 y(app)s(earing)22 b(in)h(the)f (truncated)h(pro)5 b(jections)23 b(of)f(Expression)i(\(I.62\),)g(w)m(e) f(ev)m(en)m(tually)h(end)p eop end %%Page: 15 16 TeXDict begin 15 15 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(15)257 573 y(up)29 b(with)f(an)g(expansion)h (of)f(the)g(v)m(ector)h(in)f(\(I.62\))g(in)m(v)m(olving)h(only)g (\\bare")e(resolv)m(en)m(ts,)257 693 y(i.e.,)34 b(resolv)m(en)m(ts)h (of)d(the)h(form)1369 886 y(1)p 814 930 1158 4 v 814 1022 a Fo(H)903 981 y Fl(+)895 1044 y(0)984 1022 y Fn(\000)23 b Fo(z)1129 1037 y Fl(2)1191 1022 y Fq(+)1289 955 y Fh(P)1376 1043 y Fm(i)1421 1022 y Fn(j)p Fo(q)1492 1037 y Fm(i)1520 1022 y Fn(j)f Fq(+)1668 955 y Fh(P)1756 1043 y Fm(i)1780 1024 y Fd(0)1823 1022 y Fn(j)p Fo(q)1898 988 y Fi(0)1894 1047 y Fm(i)1918 1028 y Fd(0)1944 1022 y Fn(j)2079 954 y Fo(;)2498 886 y Fq(1)p 2133 930 779 4 v 2133 1022 a Fo(H)2222 981 y Fl(+)2214 1044 y(0)2303 1022 y Fn(\000)g Fo(z)2447 1037 y Fl(1)2509 1022 y Fq(+)2607 955 y Fh(P)2695 1043 y Fm(i)2719 1024 y Fd(0)2762 1022 y Fn(j)p Fo(q)2837 988 y Fi(0)2833 1047 y Fm(i)2857 1028 y Fd(0)2883 1022 y Fn(j)2921 954 y Fo(;)312 b Fq(\(I.82\))257 1171 y(with)34 b(momen)m(ta)f Fn(f)p Fo(q)998 1186 y Fm(i)1026 1171 y Fn(g)p Fq(,)g Fn(f)p Fo(q)1233 1134 y Fi(0)1229 1195 y Fm(i)1253 1176 y Fd(0)1279 1171 y Fn(g)g Fq(b)s(elonging)f(to)h(the)g (shells)h Fn(K)2429 1134 y Fl(1)2428 1195 y(2)2501 1171 y Fq(and)f Fn(K)2768 1134 y Fl(0)2767 1195 y(1)2808 1171 y Fq(,)f(resp)s(ectiv)m(ely)-8 b(.)404 1291 y(As)34 b(w)m(e)g(will)g (see)h(in)e(Sections)i(I)s(I)f(and)f(I)s(I)s(I,)h(the)f(main)h (di\016cult)m(y)i(in)d(the)h(inductiv)m(e)257 1411 y(pro)s(of)c(of)h (con)m(v)m(ergence)i(of)d(the)i(re-expansion)g(pro)s(cedure)g(is)f (related)g(to)g(determining)257 1532 y(the)36 b(energy)g(shifts)g Fo(E)1081 1496 y Fm(n)1075 1556 y(m)1177 1532 y Fq(more)g(explicitly)-8 b(.)53 b(In)36 b(re-expanding)g Fo(\036)2652 1547 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2844 1532 y Fq(,)36 b(all)f(the)h(energy)257 1652 y(shifts)h(up)f(to)f(scale)i(2)p Fo(N)e Fn(\000)25 b Fq(1)35 b(app)s(ear)h(in)g(our)f(form)m(ulas.)54 b(The)37 b(energy)f(shifts)h(can)f(b)s(e)257 1773 y(expressed)28 b(in)e(terms)g(of)f(the)h(groundstate)f(v)m(ectors)i(on)e(scales)i(up)f (to)f(2)p Fo(N)17 b Fn(\000)7 b Fq(2.)42 b(Explicit)257 1893 y(expressions)c(for)c(the)i(energy)g(shifts)f Fo(E)1742 1857 y Fm(n)1736 1918 y(m)1838 1893 y Fq(are)g(obtained)g(from)g(the)g (re-expansions)i(of)257 2013 y Fn(f)p Fo(\036)365 2028 y Fm(i)393 2013 y Fn(j)p Fo(i)28 b Fq(=)f(1)p Fo(;)17 b(:::;)g Fq(2)p Fo(N)32 b Fn(\000)23 b Fq(2)p Fn(g)p Fq(,)32 b(up)h(to)f(remainder)i(terms)f(of)f Fo(o)p Fq(\()p Fo(\013)2471 1977 y Fm(N)2538 2013 y Fq(\).)404 2134 y(The)42 b(groundstate)g(v)m(ector,)i(fully)e(re-expanded)h(up)e(to)g (order)h Fo(\013)2883 2098 y Fm(N)2950 2134 y Fq(,)h(is)f(obtained)257 2254 y(in)d(Corollary)g(I)s(I)s(I.7)f(for)g Fo(\013)h Fq(in)g(an)f(in)m(terv)-5 b(al)39 b(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\))39 b(where)49 b(\026)-59 b Fo(\013)39 b Fq(is)g Fo(N)10 b Fq(-indep)s(enden)m(t.)64 b(In)257 2374 y(the)35 b(in)m(terv)-5 b(al)35 b(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\(2)p Fo(N)33 b Fn(\000)24 b Fq(1\)\),)34 b(where)45 b(\026)-59 b Fo(\013)q Fq(\(2)p Fo(N)34 b Fn(\000)23 b Fq(1\))34 b(is)h Fo(N)10 b Fq(-dep)s(enden)m(t) 36 b(\(see)f(Theorem)257 2495 y(I)s(I)s(I.4\),)43 b(the)d(re-expanded)i (groundstate)f(v)m(ector)h(is)f(of)f(the)h(follo)m(wing)f(form:)59 b(It)41 b(can)257 2615 y(b)s(e)29 b(written)h(as)f(an)g Fo(\013)q Fq(-indep)s(enden)m(t)h(and)f(only)h Fo(N)10 b Fq(-dep)s(enden)m(t)31 b(sum)f(of)e(expressions)k(of)257 2736 y(the)h(t)m(yp)s(e)257 2883 y Fh(Z)303 3071 y Fl(\000)347 3082 y Ff(2)p Fb(N)5 b Fd(\000)p Ff(1)535 3000 y Fo(dz)631 3015 y Fl(2)p Fm(N)i Fi(\000)p Fl(1)843 2867 y(\026)845 2883 y Fm(j)s Fl(\(2)p Fm(N)g Fi(\000)p Fl(2\))930 2917 y Fh(Y)840 3099 y Fm(j)869 3110 y Ff(2)p Fb(N)e Fd(\000)p Ff(2)1035 3099 y Fl(=1)1142 2883 y Fh(Z)1188 3071 y Fl(\000)1232 3082 y Ff(2)p Fb(N)g Fd(\000)p Ff(2)1420 3000 y Fo(dz)1520 2947 y Fm(j)1549 2958 y Ff(2)p Fb(N)g Fd(\000)p Ff(2)1516 3024 y Fl(2)p Fm(N)i Fi(\000)p Fl(2)1719 3000 y Fn(\001\001\001\001) 1860 2867 y Fl(\026)1862 2883 y Fm(j)s Fl(\(3\))1870 2917 y Fh(Y)1847 3099 y Fm(j)1876 3108 y Ff(3)1910 3099 y Fl(=1)2016 2883 y Fh(Z)2063 3071 y Fl(\000)2107 3080 y Ff(3)2162 3000 y Fo(dz)2262 2953 y Fm(j)2291 2962 y Ff(3)2258 3022 y Fl(3)2360 2867 y(\026)2362 2883 y Fm(j)s Fl(\(2\))2370 2917 y Fh(Y)2346 3099 y Fm(j)2375 3108 y Ff(2)2409 3099 y Fl(=1)2516 2883 y Fh(Z)2562 3071 y Fl(\000)2606 3080 y Ff(2)2662 3000 y Fo(dz)2762 2953 y Fm(j)2791 2962 y Ff(2)2758 3022 y Fl(2)2860 2867 y(\026)2861 2883 y Fm(j)t Fl(\(1\))2870 2917 y Fh(Y)2846 3099 y Fm(j)2875 3108 y Ff(1)2909 3099 y Fl(=1)3016 2883 y Fh(Z)3062 3071 y Fl(\000)3106 3080 y Ff(1)3162 3000 y Fo(dz)3262 2953 y Fm(j)3291 2962 y Ff(1)3258 3022 y Fl(1)3329 3000 y Fo(A')3466 3015 y Fm(el)3525 3000 y Fn(\012)p Fq(\012)3260 3208 y(\(I.83\))257 3329 y(where)31 b Fo(')600 3344 y Fm(el)689 3329 y Fq(is)f(the)g(groundstate)g(of)g(the)g(Coulom)m(b)h (Hamiltonian)f Fo(H)2808 3344 y Fm(el)2866 3329 y Fq(,)h(and)f(the)g(n) m(um-)257 3449 y(b)s(ers,)492 3427 y(\026)494 3449 y Fo(j)6 b Fq(\()p Fo(m)p Fq(\),)32 b(1)c Fn(\024)g Fo(m)g Fn(\024)g Fq(2)p Fo(N)33 b Fn(\000)22 b Fq(1,)33 b(of)f(con)m(tour)h (in)m(tegrations)g(are)g Fo(\013)q Fq(-indep)s(enden)m(t;)257 3570 y(The)f(sym)m(b)s(ol)g Fo(A)f Fq(denotes)h(an)e(in)m(tegral)i(in)e (all)h(the)g(photon)g(momen)m(ta)h(shells)g(do)m(wn)g(to)257 3690 y Fn(K)334 3649 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(2)333 3714 y(2)p Fm(N)g Fi(\000)p Fl(1)445 3968 y Fo(A)28 b Fq(=)649 3851 y Fh(Z)695 4040 y Fi(K)749 4010 y Ff(2)p Fb(N)5 b Fd(\000)p Ff(2)749 4064 y(2)p Fb(N)g Fd(\000)p Ff(1)942 3834 y Fl(\026)943 3851 y Fm(l)q Fl(\(2)p Fm(N)i Fi(\000)p Fl(1\))1022 3885 y Fh(Y)937 4070 y Fm(l)958 4081 y Ff(2)p Fb(N)e Fd(\000)p Ff(1)1124 4070 y Fl(=1)1231 3968 y Fo(dk)1333 3983 y Fm(l)1354 3994 y Ff(2)p Fb(N)g Fd(\000)p Ff(1)1541 3851 y Fh(Z)1587 4040 y Fi(K)1641 4010 y Ff(2)p Fb(N)g Fd(\000)p Ff(3)1641 4064 y(2)p Fb(N)g Fd(\000)p Ff(2)1834 3834 y Fl(\026)1835 3851 y Fm(l)p Fl(\(2)p Fm(N)i Fi(\000)p Fl(2\))1914 3885 y Fh(Y)1828 4070 y Fm(l)1849 4081 y Ff(2)p Fb(N)e Fd(\000)p Ff(2)2016 4070 y Fl(=1)2123 3968 y Fo(dk)2225 3983 y Fm(l)2246 3994 y Ff(2)p Fb(N)g Fd(\000)p Ff(2)2416 3968 y Fo(:::)2514 3851 y Fh(Z)2560 4040 y Fi(K)2614 4017 y Ff(0)2614 4062 y(1)2685 3834 y Fl(\026)2686 3851 y Fm(l)q Fl(\(1\))2689 3885 y Fh(Y)2669 4070 y Fm(l)2690 4079 y Ff(1)2725 4070 y Fl(=1)2832 3968 y Fo(dk)2934 3983 y Fm(l)2955 3992 y Ff(1)2993 3968 y Fo(B)193 b Fq(\(I.84\))257 4245 y(with)478 4218 y(\026)478 4245 y Fo(l)r Fq(\(2)p Fo(N)30 b Fn(\000)20 b Fq(1\))p Fo(;)d(::;)1029 4218 y Fq(\026)1030 4245 y Fo(l)q Fq(\(1\))31 b(\014nite)h(and)f Fo(\013)q Fq(-indep)s(enden)m(t,) 2326 4174 y Fh(R)2366 4274 y Fi(K)2420 4244 y Ff(2)p Fb(N)5 b Fd(\000)p Ff(2)2420 4299 y(2)p Fb(N)g Fd(\000)p Ff(1)2590 4245 y Fo(;)17 b(:::;)2759 4174 y Fh(R)2798 4270 y Fi(K)2852 4247 y Ff(0)2852 4293 y(1)2922 4245 y Fq(m)m(ultiple)33 b(in)m(te-)257 4383 y(grals;)257 4503 y(The)h(op)s(erator)e Fo(B)37 b Fq(is)d(a)e(tensor)h(pro)s(duct)g (of)f(op)s(erators)1453 4682 y Fo(B)h Fq(=)28 b Fo(D)s Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))22 b Fn(\012)g Fo(C)7 b Fq(\()p Fo(k)s Fq(\))961 b(\(I.85\))257 4860 y(where:)403 5027 y Fn(\017)48 b Fo(C)7 b Fq(\()p Fo(k)s Fq(\))28 b(is)h(applied)g(to)f(the)h(F)-8 b(o)s(c)m(k)29 b(space)g Fn(F)2012 4991 y Fl(0)2002 5052 y Fi(1)2105 5027 y Fq(and)f(is)h(the)g (pro)s(duct)f(of)g(a)g(\014nite)h(and)501 5147 y Fo(\013)q Fq(-indep)s(enden)m(t)37 b(n)m(um)m(b)s(er)g(of)f(photon)f(creation)i (op)s(erators,)f(con)m(tained)h(in)f(the)501 5268 y(range)d Fn(K)844 5232 y Fl(0)843 5292 y(2)p Fm(N)7 b Fi(\000)p Fl(1)1036 5268 y Fq(;)33 b(here)h Fo(k)i Fq(stands)e(for)f(all)g(the)g (momen)m(ta)h(\(and)f(helicities\))i(of)e(the)501 5388 y(photon)g(creation)g(op)s(erators;)p eop end %%Page: 16 17 TeXDict begin 16 16 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(16)403 573 y Fn(\017)48 b Fo(D)s Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))43 b(is)f(applied)h(to)f Fn(H)1522 588 y Fm(el)1610 573 y Fn(\012)29 b(F)1788 588 y Fl(0)1870 573 y Fq(and)42 b(is)h(the)f(pro)s(duct)h(of)f(a)g (\014nite)h(and)f Fo(\013)q Fq(-)501 693 y(indep)s(enden)m(t)d(n)m(um)m (b)s(er)e(of)f(factors)g(written)h(b)s(elo)m(w)g(and)g(prop)s(erly)f (com)m(bined;)501 814 y(for)c(details)h(see)h(De\014nition)f(I)s(I)s (I.3)f(and)g(Theorem)i(I)s(I)s(I.4.)43 b(In)33 b Fo(D)s Fq(\()p Fo(k)s(;)17 b(z)t Fq(\),)33 b Fo(k)i Fq(stands)501 934 y(for)d(all)h(the)g(momen)m(ta)g(\(and)g(helicities\))h(in)m (tegration)f(v)-5 b(ariables.)501 1093 y(The)34 b(factors)e(include:) 618 1292 y(*)49 b(The)33 b(bare)g(resolv)m(en)m(ts)i(and)e(the)g(pro)5 b(jection)33 b(on)g(the)g(v)m(ector)h Fo(')3063 1307 y Fm(el)3143 1292 y Fn(\012)23 b Fq(\012)1532 1495 y Fo(P)1595 1510 y Fl(0)3260 1495 y Fq(\(I.86\))1884 1617 y(1)p 1542 1661 734 4 v 1542 1753 a Fo(H)1631 1712 y Fl(+)1623 1775 y(0)1712 1753 y Fn(\000)f Fo(z)1856 1768 y Fl(1)1918 1753 y Fq(+)2016 1687 y Fh(P)2104 1774 y Fm(i)2149 1753 y Fn(j)p Fo(q)2220 1768 y Fm(i)2248 1753 y Fn(j)2285 1685 y Fo(P)2362 1643 y Fi(?)2348 1709 y Fl(0)3260 1685 y Fq(\(I.87\))2096 1875 y(1)p 1542 1919 1158 4 v 1542 2011 a Fo(H)1631 1970 y Fl(+)1623 2033 y(0)1712 2011 y Fn(\000)g Fo(z)1856 2026 y Fl(2)1918 2011 y Fq(+)2016 1945 y Fh(P)2104 2032 y Fm(i)2149 2011 y Fn(j)p Fo(q)2220 2026 y Fm(i)2248 2011 y Fn(j)g Fq(+)2396 1945 y Fh(P)2483 2032 y Fm(i)2507 2013 y Fd(0)2550 2011 y Fn(j)p Fo(q)2625 1977 y Fi(0)2621 2037 y Fm(i)2645 2018 y Fd(0)2672 2011 y Fn(j)2710 1943 y Fo(P)2787 1902 y Fi(?)2773 1967 y Fl(0)3260 1943 y Fq(\(I.88\))716 2206 y(where)993 2140 y Fh(P)1081 2227 y Fm(i)1126 2206 y Fo(q)1169 2221 y Fm(i)1225 2206 y Fq(and)1410 2140 y Fh(P)1498 2227 y Fm(i)1522 2208 y Fd(0)1565 2206 y Fo(q)1612 2170 y Fi(0)1608 2231 y Fm(i)1632 2212 y Fd(0)1687 2206 y Fq(are)28 b(sums)h(of)f(photon)g(momen)m(ta)h(in)f Fn(K)3125 2170 y Fl(0)3124 2231 y(1)3193 2206 y Fq(and)g Fn(K)3455 2170 y Fl(1)3454 2231 y(2)716 2327 y Fq(resp)s(ectiv)m(ely)-8 b(,)35 b(the)e(n)m(um)m(b)s(er)h(of)f(terms)g(b)s(eing)g Fo(\013)q Fq(-indep)s(enden)m(t;)618 2484 y(*)49 b(the)33 b(bare)g(v)m(ertices)1472 2717 y Fo(~)-50 b(p)22 b Fn(\001)1634 2649 y Fo(~)-48 b(")p Fq(\()p Fo(k)1770 2664 y Fm(i)1798 2649 y Fq(\))p 1602 2693 267 4 v 1602 2710 a Fh(q)p 1685 2710 184 4 v 100 x Fq(2)p Fn(j)p Fo(k)1813 2825 y Fm(i)1841 2810 y Fn(j)1878 2717 y Fo(e)1923 2675 y Fi(\000)p Fm(i\013)2044 2658 y(~)2047 2675 y(k)2084 2685 y Fb(i)2111 2675 y Fi(\001)o Fm(~)-34 b(x)3260 2717 y Fq(\(I.89\))1513 2964 y Fo(~)-48 b(")p Fq(\()p Fo(k)1649 2979 y Fm(i)1677 2964 y Fq(\))p 1481 3008 267 4 v 1481 3025 a Fh(q)p 1564 3025 184 4 v 100 x Fq(2)p Fn(j)p Fo(k)1692 3140 y Fm(i)1720 3125 y Fn(j)1779 3031 y(\001)1871 2964 y Fo(~)g(")p Fq(\()p Fo(k)2007 2979 y Fm(j)2043 2964 y Fq(\))p 1839 3008 275 4 v 1839 3025 a Fh(q)p 1922 3025 192 4 v 98 x Fq(2)p Fn(j)p Fo(k)2050 3138 y Fm(j)2086 3123 y Fn(j)2124 3031 y Fo(e)2169 2990 y Fi(\000)p Fm(i\013)2290 2973 y(~)2293 2990 y(k)2330 3000 y Fb(i)2356 2990 y Fi(\001)o Fm(~)-34 b(x)2420 3031 y Fo(e)2465 2990 y Fi(\000)p Fm(i\013)2586 2973 y(~)2589 2990 y(k)2626 3000 y Fb(j)2658 2990 y Fi(\001)o Fm(~)g(x)2819 3031 y Fo(;)3260 3279 y Fq(\(I.90\))1530 3412 y Fo(~)-48 b(")o Fq(\()p Fo(k)1665 3427 y Fm(i)1693 3412 y Fq(\))p 1498 3456 267 4 v 1498 3472 a Fh(q)p 1581 3472 184 4 v 100 x Fq(2)p Fn(j)p Fo(k)1709 3587 y Fm(i)1736 3572 y Fn(j)1796 3479 y(\001)1868 3412 y Fo(~)g(")p Fq(\()p Fo(k)2004 3427 y Fm(j)2040 3412 y Fq(\))2078 3376 y Fi(\003)p 1856 3456 275 4 v 1856 3472 a Fh(q)p 1939 3472 192 4 v 99 x Fq(2)p Fn(j)p Fo(k)2067 3586 y Fm(j)2103 3571 y Fn(j)2140 3479 y Fo(e)2185 3438 y Fi(\000)p Fm(i\013)2306 3420 y(~)2309 3438 y(k)2346 3448 y Fb(i)2373 3438 y Fi(\001)o Fm(~)-34 b(x)2436 3479 y Fo(e)2481 3438 y Fm(i\013)2547 3420 y(~)2550 3438 y(k)2587 3448 y Fb(j)2620 3438 y Fi(\001)o Fm(~)g(x)716 3788 y Fq(and)27 b(their)g(adjoin)m(ts,)i Fo(k)1570 3803 y Fm(i)1598 3788 y Fq(,)f Fo(k)1704 3803 y Fm(j)1767 3788 y Fq(are)f(photon)g(momen)m(ta)h(in)f(the)h(range)f Fn(K)3277 3752 y Fl(0)3276 3813 y(2)p Fm(N)7 b Fi(\000)p Fl(1)3468 3788 y Fq(;)618 3945 y(*)49 b(the)33 b(c-n)m(um)m(b)s(ers)h (op)s(erators)1100 4120 y(1)p 892 4164 466 4 v 892 4189 a Fh(P)979 4276 y Fm(i)1024 4255 y Fn(j)p Fo(q)1095 4270 y Fm(i)1123 4255 y Fn(j)22 b(\000)h Fo(z)1318 4270 y Fl(1)1368 4187 y Fe(1)1424 4202 y Fl(0)3315 4187 y Fq(\(I.91\))1499 4378 y(1)p 892 4422 1264 4 v 892 4447 a Fh(P)979 4534 y Fm(i)1003 4542 y Fb(m)1083 4514 y Fn(j)p Fo(q)1154 4529 y Fm(i)1178 4537 y Fb(m)1241 4514 y Fn(j)e Fq(+)1388 4447 y Fh(P)1476 4534 y Fm(i)1500 4543 y Fb(m)p Fd(\000)p Ff(1)1658 4514 y Fn(j)p Fo(q)1729 4529 y Fm(i)1753 4538 y Fb(m)p Fd(\000)p Ff(1)1894 4514 y Fn(j)h(\000)h Fo(z)2089 4529 y Fm(m)2165 4445 y Fe(1)2221 4460 y Fl(0)2277 4445 y Fq(;)2628 4378 y(1)p 2348 4422 611 4 v 2348 4447 a Fh(P)2435 4534 y Fm(i)2459 4542 y Fb(m)2539 4514 y Fn(j)p Fo(q)2610 4529 y Fm(i)2634 4537 y Fb(m)2696 4514 y Fn(j)f(\000)h Fo(z)2891 4529 y Fm(m)2968 4445 y Fe(1)3024 4460 y Fl(0)3315 4445 y Fq(\(I.92\))917 4592 y Fh(X)882 4774 y Fm(i)906 4785 y Ff(2)p Fb(N)5 b Fd(\000)p Ff(1)1089 4675 y Fn(j)p Fo(k)1168 4690 y Fm(i)1192 4701 y Ff(2)p Fb(N)g Fd(\000)p Ff(1)1362 4675 y Fn(j)p Fe(1)1446 4690 y Fl(0)1485 4675 y Fo(;)17 b(:::;)1654 4592 y Fh(X)1685 4774 y Fm(i)1709 4783 y Ff(3)1791 4675 y Fn(j)p Fo(k)1870 4690 y Fm(i)1894 4699 y Ff(3)1932 4675 y Fn(j)p Fe(1)2016 4690 y Fl(0)2072 4675 y Fq(;)g(\()g Fo(z)2216 4690 y Fm(m)p Fi(\000)p Fl(1)2394 4675 y Fn(\000)23 b Fo(z)2539 4690 y Fm(m)2605 4675 y Fq(\))p Fe(1)2699 4690 y Fl(0)899 4926 y Fn(h)p Fo(W)1044 4885 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(2)p Fi(\000)p Fl(1)1030 4951 y(2)p Fm(N)g Fi(\000)p Fl(1)1326 4926 y Fn(i)1365 4941 y Fl(\012)1416 4952 y Ff(2)p Fb(N)e Fd(\000)p Ff(1)1586 4926 y Fe(1)1642 4941 y Fl(0)1682 4926 y Fo(;)17 b(:::;)g Fn(h)p Fo(W)1996 4885 y Fl(0)1982 4951 y(1)2034 4926 y Fn(i)2073 4941 y Fl(\012)2124 4950 y Ff(1)2163 4926 y Fe(1)2219 4941 y Fl(0)2258 4926 y Fq(;)33 b(\()p Fo(E)2434 4885 y Fm(m)2496 4861 y Fd(0)2519 4885 y Fi(\000)p Fl(1)2428 4952 y Fm(m)2490 4933 y Fd(0)2613 4926 y Fq(\))2651 4885 y Fm(T)2651 4951 y(])2706 4926 y Fe(1)2762 4941 y Fl(0)3315 4926 y Fq(\(I.93\))716 5129 y(where:)716 5268 y(The)41 b(n)m(um)m(b)s(er)h(of)e(terms)h(in)g(all)f (the)h(sums)h(is)f Fo(\013)q Fq(-indep)s(enden)m(t,)j Fo(q)3213 5283 y Fm(i)3237 5291 y Fb(m)3340 5268 y Fq(is)d(a)716 5388 y(photon)32 b(momen)m(tum)j(in)d(the)h(shell)h Fn(K)2140 5352 y Fm(m)p Fi(\000)p Fl(1)2139 5413 y Fm(m)2330 5388 y Fq(with)f(2)27 b Fn(\024)i Fo(m)f Fn(\024)g Fq(2)p Fo(N)k Fn(\000)23 b Fq(1;)p eop end %%Page: 17 18 TeXDict begin 17 17 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(17)716 573 y(\014nally)-8 b(,)35 b Fn(h)p Fo(W)1177 532 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(2)1163 597 y(2)p Fm(N)g Fi(\000)p Fl(1)1369 573 y Fn(i)1408 588 y Fl(\012)1459 599 y Ff(2)p Fb(N)e Fd(\000)p Ff(1)1664 573 y Fq(is)35 b(the)g(v)-5 b(acuum)35 b(exp)s(ectation)g(v)-5 b(alue)35 b(of)f(the)h(op-)716 707 y(erator)26 b Fo(W)1103 665 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(2)1089 731 y(2)p Fm(N)g Fi(\000)p Fl(1)1295 707 y Fq(,)27 b(and)g(the)f(sym)m(b)s(ol)i(\()p Fo(E)2138 665 y Fm(m)2200 642 y Fd(0)2222 665 y Fi(\000)p Fl(1)2132 732 y Fm(m)2194 713 y Fd(0)2317 707 y Fq(\))2355 670 y Fm(T)2355 731 y(])2410 707 y Fq(,)f(1)h Fn(\024)g Fo(m)2731 670 y Fi(0)2782 707 y Fn(\024)g Fq(2)p Fo(N)19 b Fn(\000)9 b Fq(2,)28 b(stands)716 845 y(for)37 b(an)h(explicit)i (appro)m(ximate)f(expression)h(of)e(the)g(energy)i(shift)e Fo(E)3316 803 y Fm(m)3378 780 y Fd(0)3401 803 y Fi(\000)p Fl(1)3310 870 y Fm(m)3372 851 y Fd(0)716 965 y Fq(whic)m(h)33 b(will)g(b)s(e)g(de\014ned)g(iterativ)m(ely)h(in)e(De\014nition)h(I)s (I)s(I.3,)f(where)h(a)f(more)716 1085 y(precise)i(de\014nition)g(of)e (a)g(fully)h(re-expanded)h(v)m(ector)g(is)f(giv)m(en.)257 1289 y(W)-8 b(e)30 b(no)m(w)g(brie\015y)h(summarize)g(the)f(mix)g(of)f (ingredien)m(ts)j(in)e(our)f(construction)i(and)e(w)m(e)257 1409 y(commen)m(t)39 b(on)e(the)h(\014nal)g(result)g(concerning)g(the)g (structure)h(of)e(the)g(infrared)h(singu-)257 1530 y(larit)m(y)-8 b(.)257 1650 y(The)29 b(main)g(source)g(of)f(ingredien)m(ts)h(for)f (our)g(expansion)i(is)e(the)h(scaling)g(analysis)g(in)f([1])257 1770 y(inspired)c(b)m(y)g(the)f(analogous)f(one)h(in)g([7].)40 b(The)23 b(estimate)h(in)f(\(I.60\))f(implicitly)i(pro)m(vides)257 1891 y(a)31 b(h)m(uge)g(resummation)h(of)e(terms)i(in)f(a)f(formal)g(T) -8 b(a)m(ylor)32 b(expansion.)44 b(A)m(t)31 b(\014xed)h Fo(N)10 b Fq(,)32 b(this)257 2011 y(result)37 b(enables)h(us)f(to)f (basically)h(w)m(ork)g(with)g(\014nitely)g(man)m(y)h(photons,)f(the)g (n)m(um)m(b)s(er)257 2132 y(of)48 b(them)g(b)s(eing)g(only)h Fo(N)10 b Fq(-dep)s(enden)m(t)49 b(and)f(uniform)g(in)g Fo(\013)q Fq(,)k(for)47 b Fo(\013)h Fq(small)h(enough.)257 2252 y(This)40 b(manifestly)h(app)s(ears)e(in)g(the)g(truncation)g(of)g (the)g(groundstate)g(pro)5 b(jection)40 b(in)257 2372 y(Eq.)k(\(I.63\),)h(as)d(w)m(ell)i(as)f(in)g(the)g(fact)f(that,)j (shell)f(b)m(y)f(shell,)k(w)m(e)c(m)m(ust)h(Wic)m(k-order)257 2493 y(only)c(\014nitely)g(man)m(y)g(photon)f(creation-)f(and)h (annihilation)g(op)s(erators,)i(and)e(in)g(the)257 2613 y(truncated)34 b(expansions)g(of)e(t)m(yp)s(e)i Fn(A)e Fq(and)h Fn(B)s Fq(.)257 2733 y(Our)41 b(expansion)h(sc)m(heme)h(also)e (relies)h(on)f(the)g(gap)g(estimates)h(scale)g(b)m(y)g(scale,)i(still) 257 2854 y(related)c(to)f(the)h Fj(asymptotic)h(fr)-5 b(e)g(e)g(dom)45 b Fq(of)39 b(the)h(mo)s(del)g(displa)m(y)m(ed)h(b)m(y) g(the)e(results)i(in)257 2974 y([1].)53 b(This)37 b(information)f(pla)m (ys)h(a)e(crucial)i(role)f(in)g(all)f(the)h(expansions)i(as)e(w)m(ell)h (as)f(in)257 3095 y(the)f(decomp)s(osition)h(of)f(the)g(resolv)m(en)m (ts,)j(scale)d(b)m(y)h(scale,)g(in)m(to)f(a)f(part)h(prop)s(ortional) 257 3215 y(to)41 b(the)h(groundstate)g(pro)5 b(jection)42 b(and)f(an)h(orthogonal)e(one,)k(whic)m(h)f(enables)g(us)f(to)257 3335 y(separate)h(an)e(explicit)j(and)d(main)h(con)m(tribution)h(-)e (corresp)s(onding)i(to)e(the)h(parallel)257 3456 y(part)c(-)f(from)h(a) g(smaller)h(one)f(that)f(can)h(b)s(e)h(further)f(expanded)h(-)f (corresp)s(onding)h(to)257 3576 y(the)33 b(orthogonal)f(one.)257 3697 y(The)37 b(relev)-5 b(ance)37 b(of)f(the)g(structural)h(prop)s (erties)f(con)m(trolled)h(in)f([1])g(is)g(eviden)m(t)i(in)e(the)257 3817 y(details)28 b(of)f(the)g(expansions)i(\(I.21\))e(and)g(\(I.22\),) g(con)m(tained)h(in)g(Corollary)f(I)s(I)s(I.7.)41 b(F)-8 b(or)26 b(a)257 3937 y(\014xed)31 b(order)e Fo(N)10 b Fq(,)30 b(w)m(e)h(split)e(an)h Fo(N)10 b Fq(-indep)s(enden)m(t)31 b(in)m(terv)-5 b(al)30 b(\(0)p Fo(;)c Fq(\026)-58 b Fo(\013)p Fq(\))29 b(in)m(to)g(a)g(lo)m(w)m(er)h(in)m(terv)-5 b(al)257 4058 y(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(M)39 b Fq(=)27 b(2)p Fo(N)f Fn(\000)15 b Fq(1\)\))29 b(and)g(in)m(to)g(an)g(upp)s(er)h(one)f([)10 b(\026)-59 b Fo(\013)q Fq(\()p Fo(M)39 b Fq(=)27 b(2)p Fo(N)f Fn(\000)15 b Fq(1\))p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\))29 b(\(discussed)j(in)257 4178 y(the)f(Remark)f(after)g(Corollary)g(I)s(I) s(I.7\),)h(where)41 b(\026)-59 b Fo(\013)q Fq(\()p Fo(M)38 b Fq(=)28 b(2)p Fo(N)f Fn(\000)17 b Fq(1\))30 b(is)g(an)g Fo(N)10 b Fq(-dep)s(enden)m(t)257 4298 y(p)s(ositiv)m(e)43 b(n)m(um)m(b)s(er.)72 b(In)41 b(the)h(lo)m(w)m(er)h(in)m(terv)-5 b(al,)44 b(the)e(expansion)h(is)f(con)m(trolled)g(b)m(y)h(the)257 4419 y(algorithm.)j(In)34 b(the)f(upp)s(er)h(one)g(w)m(e)g(can)g (basically)g(use)g(plain)g(p)s(erturbation)g(theory)-8 b(,)257 4539 y(since)33 b(w)m(e)g(ha)m(v)m(e)g(an)e(infrared)h (cut-o\013)f(that)g(only)h(dep)s(ends)h(on)e Fo(N)42 b Fq(but)32 b(the)g(gap)f(in)h(the)257 4660 y(sp)s(ectrum)39 b(of)d(the)h(Hamiltonian)g(is)g Fo(\013)q Fq(-indep)s(enden)m(t.)58 b(No)m(w)37 b(-)f(also)h(for)f(the)h(purp)s(ose)257 4780 y(of)e(comparison)h(with)g([5],)g([6],)g(where)h(mo)s(derate)e(high)h (order)f(con)m(tributions)i(to)e(the)257 4900 y(groundstate)k(energy)g (ha)m(v)m(e)g(b)s(een)g(calculated)g(-)f(it)g(is)h(clear)f(that)g(the)g (di\016cult)i(task)257 5021 y(is)h(to)e(extend)i(the)f(upp)s(er)g (region)g(do)m(wn)g(to)g Fo(\013)g Fq(=)g(0,)h(for)e(large)h(v)-5 b(alues)40 b(of)f Fo(N)10 b Fq(.)65 b(Here)257 5141 y(w)m(e)38 b(encoun)m(ter)g(the)f(true)g(nature)f(of)g(the)h(infrared)g (singularit)m(y)-8 b(.)56 b(F)-8 b(or)36 b(large)h Fo(N)10 b Fq(,)38 b(ev)m(en)257 5261 y(in)m(tro)s(ducing)29 b(an)e Fo(\013)q Fq(-dep)s(enden)m(t)i(infrared)e(cut-o\013,)i(plain)e(p)s (erturbation)h(theory)g(is)g(not)257 5382 y(a-priori)g(w)m(ell)h (de\014ned.)44 b(Nev)m(ertheless,)32 b(w)m(e)d(can)g(co)m(v)m(er)g(the) g(lo)m(w)m(er)g(in)m(terv)-5 b(al)29 b(and)f(study)p eop end %%Page: 18 19 TeXDict begin 18 18 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(18)257 573 y(the)40 b(asymptotic)h(b)s(eha)m (vior)f(of)f(the)h(groundstate)g(and)f(of)g(the)h(groundstate)g(energy) 257 693 y(as)g Fo(\013)f Fn(!)g Fq(0,)i Fj(up)g(to)g(any)g(or)-5 b(der)41 b Fo(N)10 b Fq(,)41 b(thanks)f(to)f(the)h(a-priori)e(results)j (deriving)f(from)257 814 y(the)35 b(scaling)g(analysis)h(and)f(thanks)g (to)f(the)h(shell)h(b)m(y)f(shell)g(re-expansion)h(pro)s(cedure)257 934 y(whic)m(h)44 b(is)e(someho)m(w)i(guided)f(b)m(y)g(the)f(non-p)s (erturbativ)m(e)h(metho)s(d)f(of)g(constructing)257 1054 y(the)33 b(groundstate.)404 1175 y(In)g(the)g(remaining)h(part)e(of)h (this)g(Section,)h(w)m(e)g(explain)g(the)g(reason)f(that)g(ln[1)p Fo(=\013)q Fq(])257 1295 y(terms)42 b(ma)m(y)-8 b(,)43 b(in)e(principle,)j(o)s(ccur)c(in)h(the)g(asymptotic)h(expansions)g(in) f(Eqs.)h(\(I.21\))257 1416 y(and)33 b(\(I.22\).)257 1536 y(This)g(is)f(p)s(ossible)h(b)m(y)f(isolating)g(some)g(particular)f (string)h(of)f(op)s(erators)g(con)m(tained)i(in)257 1656 y(Expression)i(\(I.83\),)e(for)f(instance)i(the)f(string:)546 1921 y Fn(\001)22 b(\001)f(\001)q Fo(~)-50 b(p)22 b Fn(\001)g Fo(a)p Fq(\()898 1896 y Fo(~)883 1921 y(G)960 1880 y Fm(m)p Fi(\000)p Fl(1)960 1945 y Fm(m)1117 1921 y Fq(\))1331 1853 y(1)p 1165 1897 382 4 v 1165 1989 a Fo(H)1254 1948 y Fl(+)1246 2011 y(0)1334 1989 y Fn(\000)h Fo(z)1479 2004 y Fm(m)1556 1921 y Fq(\()p Fo(E)1672 1880 y Fm(m)p Fi(\000)p Fl(1)1666 1945 y Fm(m)1829 1921 y Fq(\))1867 1880 y Fm(T)1867 1945 y(])2098 1853 y Fq(1)p 1932 1897 V 1932 1989 a Fo(H)2021 1948 y Fl(+)2013 2011 y(0)2101 1989 y Fn(\000)g Fo(z)2246 2004 y Fm(m)2323 1921 y Fo(~)-49 b(p)22 b Fn(\001)g Fo(a)2495 1880 y Fi(\003)2534 1921 y Fq(\()2587 1896 y Fo(~)2572 1921 y(G)2649 1880 y Fm(m)p Fi(\000)p Fl(1)2649 1945 y Fm(m)2806 1921 y Fq(\))g Fn(\001)g(\001\001) 288 b Fq(\(I.94\))257 2188 y(F)-8 b(or)34 b(the)h(expression)i(ab)s(o)m (v)m(e)f(w)m(e)f(assume)i(-)d(it)g(is)i(actually)f(the)g(case)h(for)e (some)h(terms)257 2308 y(when)f Fo(m)f Fq(is)g(large)f(enough)h(-)g (that:)257 2429 y(When)49 b(the)g(v)-5 b(ariable)48 b Fo(z)1167 2444 y Fm(m)1282 2429 y Fq(is)h(in)m(tegrated)g(out,)j(the)c (in)m(tegration)g(around)h(the)f(p)s(ole)257 2549 y Fo(z)302 2564 y Fm(m)397 2549 y Fq(=)27 b(0)33 b(is)g(not)f(v)-5 b(anishing;)257 2669 y(Except)44 b(the)f(con)m(traction)g(with)g (initial)g(and)f(\014nal)h(p)s(oin)m(t)f(in)h(the)g(string,)i(no)d (other)257 2790 y(con)m(traction)34 b(of)e(photon)g(op)s(erators)h (crosses)h(the)f(resolv)m(en)m(ts.)257 2910 y(Then,)41 b(inserting)f(the)e(complete)i(set)f(of)f(states)h(corresp)s(onding)g (to)f(the)g(eigenstates)257 3031 y(of)31 b(the)g(unp)s(erturb)s(ed)h (Hamiltonian,)f Fo(H)1770 2989 y Fl(+)1762 3052 y(0)1829 3031 y Fq(,)g(b)s(et)m(w)m(een)i Fo(~)-49 b(p)18 b Fn(\001)g Fo(a)p Fq(\()2479 3005 y Fo(~)2464 3031 y(G)2541 2994 y Fm(m)p Fi(\000)p Fl(1)2541 3055 y Fm(m)2698 3031 y Fq(\))31 b(and)g Fo(~)-50 b(p)19 b Fn(\001)f Fo(a)3119 2994 y Fi(\003)3159 3031 y Fq(\()3212 3005 y Fo(~)3197 3031 y(G)3274 2994 y Fm(m)p Fi(\000)p Fl(1)3274 3055 y Fm(m)3430 3031 y Fq(\),)257 3151 y(w)m(e)35 b(consider)g(the)f(con)m (tribution)g(of)f(the)h(pro)5 b(jector)34 b(asso)s(ciated)g(to)f(the)h (unp)s(erturb)s(ed)257 3271 y(groundstate.)42 b(The)26 b(computation)f(of)g(the)g(con)m(traction)g(of)g(the)g(creation-)g(and) g(annihi-)257 3392 y(lation)31 b(op)s(erators)g(giv)m(es)i(the)e(follo) m(wing)g(result)h(\(up)g(to)e(a)h(m)m(ultiplicativ)m(e)j(co)s (e\016cien)m(t\))1406 3656 y(\()p Fo(E)1522 3615 y Fm(m)p Fi(\000)p Fl(1)1516 3681 y Fm(m)1679 3656 y Fq(\))1717 3615 y Fm(T)1717 3681 y(])1788 3539 y Fh(Z)1834 3728 y Fi(K)1888 3698 y Fb(m)p Fd(\000)p Ff(1)1888 3740 y Fb(m)2046 3656 y Fo(dk)2211 3589 y Fq(1)p 2161 3633 149 4 v 2161 3725 a Fn(j)p Fo(k)s Fn(j)2271 3696 y Fl(3)2320 3656 y Fo(;)913 b Fq(\(I.95\))257 3932 y(where)36 b(the)e(in)m(tegral)h (in)f Fo(k)k Fq(yields)d(a)f(ln)q([1)p Fo(=\013)q Fq(])f(term)i(but)f (some)h(p)s(o)m(w)m(er)g(of)f Fo(\013)h Fq(is)g(alw)m(a)m(ys)257 4052 y(con)m(tained)26 b(in)f(\()p Fo(E)913 4016 y Fm(m)p Fi(\000)p Fl(1)907 4077 y Fm(m)1069 4052 y Fq(\))1107 4016 y Fm(T)1107 4077 y(])1162 4052 y Fq(,)i(b)m(y)e(construction.)42 b(A)m(t)25 b(this)g(p)s(oin)m(t,)h(t)m(w)m(o)f(imp)s(ortan)m(t)g (remarks)257 4173 y(are)33 b(necessary:)257 4293 y(1\))28 b(Because)i(of)e(the)g(shell-slicing)i(in)e(the)h(photon)f(momen)m(ta)h (space)g(no)f(singularit)m(y)i(in)257 4414 y Fo(\013)f Fq(=)e(0)g(can)h(app)s(ear)f(as)g(output)h(of)f(the)g(algorithm.)42 b(This)28 b(is)g(the)g(con)m(ten)m(t)g(of)f(Theorem)257 4534 y(I)s(I)s(I.4)34 b(in)g(Section)h(I)s(I)s(I.)f(More)g(explicitly) -8 b(,)36 b(concerning)f(the)f(string)g(in)g(Eq.)h(\(I.94\),)f(it)g(is) 257 4654 y(clear)f(that)g(more)g(blo)s(c)m(ks)h(of)e(the)h(t)m(yp)s(e) 1493 4913 y(\()p Fo(E)1609 4872 y Fm(m)p Fi(\000)p Fl(1)1603 4938 y Fm(m)1766 4913 y Fq(\))1804 4872 y Fm(T)1804 4938 y(])2035 4846 y Fq(1)p 1869 4890 382 4 v 1869 4982 a Fo(H)1958 4940 y Fl(+)1950 5003 y(0)2038 4982 y Fn(\000)23 b Fo(z)2183 4997 y Fm(m)3260 4913 y Fq(\(I.96\))257 5181 y(inside)29 b(the)f(string)g(pro)s(duce)h(more)f(p)s(o)m(w)m(ers)h(of)e Fo(\013)q Fq(,)h(b)s(ecause)h(the)f(in)m(v)m(erse)j(p)s(o)m(w)m(er)d (due)h(to)257 5301 y(the)h(in)m(tegral)f(in)h(the)f(photons)h(momen)m (ta)g(shell)g(is)g(comp)s(ensated)h(b)m(y)f(the)f(n)m(umerator;)p eop end %%Page: 19 20 TeXDict begin 19 19 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(19)257 573 y(2\))22 b(The)h(logarithmic)f (terms)g(cannot)g(sum)h(up)f(to)f(yield)i(some)g(p)s(o)m(w)m(er)g(of)e Fo(\013)q Fq(,)j(for)d(instance)257 693 y(through)33 b(the)g(iden)m(tit)m(y)1626 814 y Fo(e)1671 772 y Fi(\000)p Fl(ln1)p Fm(=\013)1933 814 y Fq(=)27 b Fo(\013)q(;)1134 b Fq(\(I.97\))257 988 y(b)s(ecause,)32 b(if)d(w)m(e)h(\014x)g(a)f(p)s (o)m(w)m(er)i(of)e Fo(\013)h Fq(whic)m(h)g(m)m(ultiplies)i(logarithmic) d(terms,)i(the)f(terms)257 1108 y(of)e(this)h(t)m(yp)s(e)g(cannot)g(b)s (e)f(in\014nitely)i(man)m(y)-8 b(.)43 b(In)28 b(fact,)h(starting)g (from)f(a)g(scale)h Fo(n)f Fq(with)h Fo(n)257 1229 y Fq(large)j(enough,)h(the)f(new)h(terms)g(are)f(of)f(higher)h(order)h (in)f Fo(\013)q Fq(,)f(due)i(to)f(the)g(estimate)h(in)257 1349 y(Eq.\(I.60\).)44 b(On)31 b(the)h(other)f(hand,)h(for)f(a)g (\014xed)i(scale,)f(the)g(n)m(um)m(b)s(er)h(of)d(terms)j(is)e(\014nite) 257 1469 y(b)s(ecause)j(of)e(the)h(truncation)g(pro)s(cedure.)404 1590 y(The)d(dep)s(endence)h(on)e Fo(\013)g Fq(of)f(the)i(co)s (e\016cien)m(ts)h Fo(")2151 1605 y Fm(l)2176 1590 y Fq(\()p Fo(\013)q Fq(\))e(and)g(of)f(the)h(v)m(ectors)i Fo(')3193 1605 y Fm(l)3219 1590 y Fq(\()p Fo(\013)q Fq(\),)e(in)257 1710 y(Eqs.)39 b(\(I.21\))e(and)g(\(I.22\),)h(resp)s(ectiv)m(ely)-8 b(,)42 b(is)37 b(also)h(due)f(to)g(the)h(Coulom)m(b)g(in)m(teraction,) 257 1831 y(through)49 b(the)g(rescaling)g(of)f(the)h(unit)g(of)f(the)h (electron)g(p)s(osition)g(underlying)h(the)257 1951 y(de\014nition)g (of)e(Hamiltonian)h(\(I.6\),)j(see)e([1].)92 b(This)49 b(dep)s(endence)i(is)e(ho)m(w)m(ev)m(er)i(not)257 2071 y(singular)33 b(as)g Fo(\013)c Fn(!)e Fq(0.)p eop end %%Page: 20 21 TeXDict begin 20 20 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(20)257 573 y Fp(I)t(I)162 b(T)-13 b(runcated)68 b(groundstate)e(v)l(ectors)h(and)h(trun-)543 756 y(cated)54 b(energy)e(shifts)257 975 y Fq(Starting)38 b(from)g(Prop)s(osition)h(I.1)f(a)g(sequence)j(of)d(unnormalized)h (groundstate)g(v)m(ec-)257 1095 y(tors,)33 b(denoted)h(b)m(y)f Fn(f)p Fo(\036)1092 1110 y Fm(n)1139 1095 y Fn(g)p Fq(,)f Fn(f)1311 1069 y Fq(~)1298 1095 y Fo(\036)1356 1110 y Fm(n)1403 1095 y Fn(g)p Fq(,)g(is)h(recursiv)m(ely)j(de\014ned)e(b)m(y) 1539 1270 y Fo(\036)1597 1285 y Fm(n)p Fl(+1)1762 1270 y Fq(:=)27 b Fo(P)1955 1285 y Fm(n)p Fl(+1)2105 1244 y Fq(~)2092 1270 y Fo(\036)2150 1285 y Fm(n)3270 1270 y Fq(\(I)s(I.1\))257 1445 y(with)492 1419 y(~)480 1445 y Fo(\036)538 1460 y Fm(n)612 1445 y Fq(:=)h Fo(\036)801 1460 y Fm(n)869 1445 y Fn(\012)23 b Fq(\012)1039 1409 y Fm(n)1039 1470 y(n)p Fl(+1)1177 1445 y Fq(,)1249 1419 y(~)1236 1445 y Fo(\036)1294 1460 y Fl(0)1361 1445 y Fq(:=)28 b Fo(\036)1550 1460 y Fl(0)1611 1445 y Fn(\012)23 b Fq(\012)1781 1409 y Fl(0)1781 1470 y(1)1853 1445 y Fq(and)33 b Fo(\036)2101 1460 y Fl(0)2168 1445 y Fq(:=)27 b Fo(')2362 1460 y Fm(el)2443 1445 y Fn(\012)c Fq(\012)2613 1460 y Fl(0)2653 1445 y Fq(,)32 b(with)h Fn(k)p Fo(\036)3042 1460 y Fl(0)3081 1445 y Fn(k)28 b Fq(=)f(1.)257 1566 y(By)j(Prop)s(osition)g(I.1,)f(there)h(exist)h(p)s(ositiv)m(e)f (constan)m(ts)h Fo(C)2433 1530 y Fi(0)2485 1566 y Fq(and)f Fo(\016)t Fq(,)f(indep)s(enden)m(t)j(of)d Fo(\013)257 1686 y Fq(and)k(of)f Fo(n;)17 b(m)28 b Fn(2)g Fk(N)p Fq(,)33 b(suc)m(h)h(that)775 1871 y Fn(k)p Fo(\036)883 1886 y Fm(m)971 1871 y Fn(\000)23 b Fo(\036)1129 1886 y Fm(n)p Fi(\000)p Fl(1)1288 1871 y Fn(\012)g Fq(\012)1458 1829 y Fm(n)p Fi(\000)p Fl(1)1458 1895 y Fm(m)1595 1871 y Fn(k)28 b(\024)g Fo(C)1855 1829 y Fi(0)1878 1871 y Fo(\013)1951 1801 y Fb(n)p Ff(+1)p 1951 1815 116 4 v 1993 1855 a(2)2470 1871 y Fq(for)97 b Fo(m)28 b Fn(\025)h Fo(n)309 b Fq(\(I)s(I.2\))257 2046 y(and)1673 2166 y Fn(k)p Fo(\036)1781 2181 y Fm(n)1828 2166 y Fn(k)27 b(\025)i Fo(\016)n(:)1191 b Fq(\(I)s(I.3\))257 2321 y(Estimates)33 b(\(I)s(I.2\),)f(\(I)s(I.3\))f(enable)h(us)g(to)g(carry)f(out)h(the)f (\014rst)h(imp)s(ortan)m(t)g(step)g(in)g(the)257 2442 y(re-expansion)i(pro)s(cedure)g(of)e(the)h(\(unnormalized\))h (groundstate)f(v)m(ector)g Fo(\036)3126 2457 y Fl(gs)1435 2617 y Fo(\036)1493 2632 y Fl(gs)1588 2617 y Fq(:=)52 b(lim)1718 2667 y Fm(n)p Fi(!1)1919 2617 y Fo(\036)1977 2632 y Fm(n)2046 2617 y Fn(\012)22 b Fq(\012)2215 2576 y Fm(n)2215 2642 y Fi(1)2291 2617 y Fo(:)952 b Fq(\(I)s(I.4\))257 2820 y(If)27 b(an)f(error)h(term)g(of)f(order)g Fo(o)p Fq(\()p Fo(\013)1439 2784 y Fm(N)1506 2820 y Fq(\))g(is)h(tolerated,)h (it)f(su\016ces)i(to)d(re-expand)h(the)g(v)m(ector)257 2940 y Fo(\036)315 2955 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)508 2940 y Fq(.)43 b(By)33 b(telescoping)734 3116 y Fo(\036)792 3131 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1068 3116 y Fq(=)83 b Fo(P)1290 3131 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1495 3089 y Fq(~)1482 3116 y Fo(\036)1540 3131 y Fl(2)p Fm(N)g Fi(\000)p Fl(2)1755 3116 y Fn(\000)1867 3089 y Fq(~)1854 3116 y Fo(\036)1912 3131 y Fl(2)p Fm(N)g Fi(\000)p Fl(2)3270 3116 y Fq(\(I)s(I.5\))1227 3335 y(+)1320 3227 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)1353 3252 y Fh(X)1352 3435 y Fm(j)t Fl(=2)1508 3335 y Fq(\()p Fo(P)1609 3350 y Fl(2)p Fm(N)g Fi(\000)p Fm(j)1811 3309 y Fq(~)1798 3335 y Fo(\036)1856 3350 y Fl(2)p Fm(N)g Fi(\000)p Fm(j)t Fi(\000)p Fl(1)2158 3335 y Fn(\000)2270 3309 y Fq(~)2258 3335 y Fo(\036)2316 3350 y Fl(2)p Fm(N)g Fi(\000)p Fm(j)t Fi(\000)p Fl(1)2595 3335 y Fq(\))22 b Fn(\012)h Fq(\012)2825 3288 y Fl(2)p Fm(N)7 b Fi(\000)p Fm(j)2825 3360 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)3270 3335 y Fq(\(I)s(I.6\))1227 3581 y(+)1316 3555 y(~)1303 3581 y Fo(\036)1361 3596 y Fl(0)1422 3581 y Fn(\012)22 b Fq(\012)1591 3540 y Fl(1)1591 3606 y(2)p Fm(N)7 b Fi(\000)p Fl(1)3270 3581 y Fq(\(I)s(I.7\))257 3757 y(and)33 b(similarly)1266 3946 y Fo(E)1338 3961 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1559 3946 y Fq(=)28 b Fo(E)1735 3961 y Fl(0)1796 3946 y Fn(\000)1896 3838 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1930 3863 y Fh(X)1929 4045 y Fm(j)t Fl(=1)2101 3946 y Fo(E)2179 3899 y Fl(2)p Fm(N)g Fi(\000)p Fm(j)t Fi(\000)p Fl(1)2173 3970 y(2)p Fm(N)g Fi(\000)p Fm(j)2459 3946 y Fo(:)784 b Fq(\(I)s(I.8\))257 4184 y(Eac)m(h)34 b(di\013erence)h(in)e(\(I)s(I.5\))g(and)g(\(I)s (I.6\))f(will)i(b)s(e)f(giv)m(en)h(as)f(a)g(series)h(expansion)h(in)e (the)257 4305 y(spirit)j(of)d(Prop)s(osition)i(I.1.)49 b(The)36 b(next)f(step)h(will)f(consist)g(in)g(k)m(eeping)h(only)f(a)f (\014nite,)257 4425 y Fo(N)10 b Fq(-dep)s(enden)m(t,)50 b(n)m(um)m(b)s(er)45 b(of)f(terms)i(in)e(eac)m(h)i(of)d(these)j(series) g(expansions,)j(with)c(a)257 4545 y(remainder)34 b(term)f(of)f(order)h Fo(o)p Fq(\()p Fo(\013)1465 4509 y Fm(N)1532 4545 y Fq(\).)257 4666 y Fe(Remark)257 4786 y Fq(Throughout)51 b(our)f(pap)s(er,)55 b(the)c(sym)m(b)s(ol)g Fo(o)p Fq(\()p Fo(\013)2016 4750 y Fm(N)2083 4786 y Fq(\))f(stands)h(for)f(a)g(p)s(ositiv)m(e)i(quan)m (tit)m(y)257 4907 y(b)s(ounded)33 b(b)m(y)g Fo(const)22 b Fn(\001)f Fo(\013)1152 4870 y Fm(N)7 b Fl(+)p Fm(\017)1302 4907 y Fq(,)33 b(for)e(an)m(y)i Fo(\017)28 b(>)g Fq(0,)k(where)h(the)g (m)m(ultiplicativ)m(e)i(constan)m(t)e(is)257 5027 y(only)h(uniform)g (in)g Fo(\013)q Fq(,)f(unless)i(otherwise)g(sp)s(eci\014ed.)48 b(The)35 b(sym)m(b)s(ol)f Fn(O)s Fq(\()p Fo(\013)2954 4991 y Fm(N)3021 5027 y Fq(\))g(is)g(de\014ned)257 5147 y(similarly)-8 b(.)404 5268 y(Prop)s(osition)35 b(I.1)g(enables)h(us)f (to)g(prop)s(ose)g(a)g(meaningful)g(and)g(useful)h(de\014nition)257 5388 y(of)c(truncated)i(v)m(ectors:)p eop end %%Page: 21 22 TeXDict begin 21 21 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(21)257 573 y Fe(De\014nition)35 b(I)s(I.1.)k Fj(F)-7 b(or)30 b Fo(\013)j Fj(which)e(ful\014l)5 b(ls)31 b(the)h(assumptions)f(of)h(Pr)-5 b(op)g(osition)31 b(I.1,)g(the)257 693 y(trunc)-5 b(ate)g(d)36 b(ve)-5 b(ctors)877 892 y Fq(\()p Fo(\036)973 907 y Fm(m)1039 892 y Fq(\))1077 851 y Fm(T)1167 892 y Fo(;)51 b Fq(\()1296 865 y(~)1283 892 y Fo(\036)1341 907 y Fm(m)1408 892 y Fq(\))1446 851 y Fm(T)1916 892 y Fq(0)27 b Fn(\024)h Fo(m)g Fn(\024)g Fo(M)39 b Fq(=)27 b(2)p Fo(N)33 b Fn(\000)23 b Fq(1)410 b(\(I)s(I.9\))257 1090 y Fj(ar)-5 b(e)35 b(de\014ne)-5 b(d)34 b(r)-5 b(e)g(cursively)34 b(starting)h(fr)-5 b(om)1328 1289 y Fq(\()p Fo(\036)1424 1304 y Fl(0)1463 1289 y Fq(\))1501 1247 y Fm(T)1583 1289 y Fq(:=)28 b Fo(\036)1772 1304 y Fl(0)1846 1289 y Fo(;)52 b Fq(\()1976 1262 y(~)1963 1289 y Fo(\036)2021 1304 y Fl(0)2060 1289 y Fq(\))2098 1247 y Fm(T)2180 1289 y Fq(:=)2324 1262 y(~)2311 1289 y Fo(\036)2369 1304 y Fl(0)3222 1289 y Fq(\(I)s(I.10\))257 1487 y Fj(ac)-5 b(c)g(or)g(ding)34 b(to)h(the)g(formulas)1311 1685 y Fq(\()p Fo(\036)1407 1700 y Fm(m)1474 1685 y Fq(\))1512 1644 y Fm(T)1650 1685 y Fq(:=)83 b(\()p Fo(P)1937 1700 y Fm(m)2003 1685 y Fq(\))2041 1644 y Fm(T)2096 1685 y Fq(\()2147 1659 y(~)2134 1685 y Fo(\036)2192 1700 y Fm(m)p Fi(\000)p Fl(1)2348 1685 y Fq(\))2386 1644 y Fm(T)3222 1685 y Fq(\(I)s(I.11\))1311 1831 y(\()1362 1804 y(~)1349 1831 y Fo(\036)1407 1846 y Fm(m)1474 1831 y Fq(\))1512 1790 y Fm(T)1650 1831 y Fq(:=)g(\()p Fo(\036)1932 1846 y Fm(m)1998 1831 y Fq(\))2036 1790 y Fm(T)2113 1831 y Fn(\012)23 b Fq(\012)2283 1790 y Fm(m)2283 1855 y(m)p Fl(+1)3222 1831 y Fq(\(I)s(I.12\))257 2029 y Fj(wher)-5 b(e)389 2277 y Fq(\()p Fo(P)490 2292 y Fm(m)556 2277 y Fq(\))594 2236 y Fm(T)677 2277 y Fq(:=)27 b Fn(\000)941 2210 y Fq(1)p 894 2254 141 4 v 894 2346 a(2)p Fo(\031)t(i)1092 2169 y Fm(N)1064 2194 y Fh(X)1062 2376 y Fm(j)t Fl(=0)1201 2160 y Fh(Z)1248 2349 y Fl(\000)1292 2357 y Fb(m)1371 2277 y Fo(dz)1467 2292 y Fm(m)1755 2210 y Fq(1)p 1544 2254 472 4 v 1552 2326 a Fh(f)1544 2359 y Fo(H)1633 2317 y Fl(+)1625 2380 y Fm(m)p Fi(\000)p Fl(1)1803 2359 y Fn(\000)c Fo(z)1948 2374 y Fm(m)2025 2181 y Fh(h)2064 2277 y Fq(\()p Fn(\000)p Fo(W)2285 2236 y Fm(m)p Fi(\000)p Fl(1)2271 2302 y Fm(m)2442 2277 y Fq(\))2718 2210 y(1)p 2506 2254 V 2514 2326 a Fh(f)2506 2359 y Fo(H)2595 2317 y Fl(+)2587 2380 y Fm(m)p Fi(\000)p Fl(1)2766 2359 y Fn(\000)g Fo(z)2911 2374 y Fm(m)2987 2181 y Fh(i)3027 2204 y Fm(j)3063 2277 y Fo(:)132 b Fq(\(I)s(I.13\))257 2553 y Fj(The)35 b(de\014nition)e(is)i(me)-5 b(aningful)34 b(b)-5 b(e)g(c)g(ause)1272 2794 y Fn(k)p Fq(\()p Fo(\036)1418 2809 y Fm(m)1484 2794 y Fq(\))1522 2753 y Fm(T)1577 2794 y Fn(k)28 b(\025)1760 2698 y Fh(h)1809 2727 y Fq(1)22 b Fn(\000)h Fq(2)p Fo(C)7 b(\013)p 1809 2771 359 4 v 1833 2863 a Fq(1)22 b Fn(\000)h Fo(C)7 b(\013)2178 2698 y Fh(i)2217 2721 y Fm(m)2284 2794 y Fn(k)p Fo(\036)2392 2809 y Fl(0)2431 2794 y Fn(k)741 b Fq(\(I)s(I.14\))257 3032 y Fj(thanks)35 b(to)g(the)g(estimate)f(in)h(Eq.)f(\(I.54\).)404 3217 y Fq(Under)28 b(the)h(assumptions)h(of)d(De\014nition)i(I)s(I.1,)g (the)f(follo)m(wing)h(k)m(ey)g(prop)s(erties)g(are)257 3338 y(trivially)34 b(ful\014lled)1383 3458 y Fn(k)p Fq(\()p Fo(\036)1529 3473 y Fm(m)1595 3458 y Fq(\))1633 3417 y Fm(T)1710 3458 y Fn(\000)23 b Fo(\036)1868 3473 y Fm(m)1934 3458 y Fn(k)28 b(\024)g Fo(o)p Fq(\()p Fo(\013)2265 3417 y Fm(N)2332 3458 y Fq(\))852 b(\(I)s(I.15\))257 3623 y(and)968 3767 y Fn(j)p Fq(\()1047 3740 y(~)1034 3767 y Fo(\036)1092 3782 y Fm(m)1158 3767 y Fq(\))1196 3726 y Fm(T)1268 3767 y Fo(;)33 b Fq(\()p Fo(\036)1424 3782 y Fm(m)p Fl(+1)1580 3767 y Fq(\))1618 3726 y Fm(T)1673 3767 y Fq(\))p Fn(j)28 b(\025)1872 3670 y Fh(h)1921 3699 y Fq(1)22 b Fn(\000)h Fq(2)p Fo(C)7 b(\013)p 1921 3744 V 1945 3835 a Fq(1)22 b Fn(\000)h Fo(C)7 b(\013)2290 3670 y Fh(i)2329 3693 y Fl(2)p Fm(m)p Fl(+1)2521 3767 y Fn(k)p Fo(\036)2629 3782 y Fl(0)2668 3767 y Fn(k)2718 3726 y Fl(2)2757 3767 y Fo(:)438 b Fq(\(I)s(I.16\))257 3966 y(Next,)34 b(taking)f(in)m(to)f(accoun)m(t)i(the)f(iden)m(tit)m(y) 960 4231 y Fo(E)1038 4190 y Fm(m)p Fi(\000)p Fl(1)1032 4256 y Fm(m)1223 4231 y Fq(=)27 b Fo(E)1398 4246 y Fm(m)p Fi(\000)p Fl(1)1577 4231 y Fn(\000)c Fo(E)1749 4246 y Fm(m)1843 4231 y Fq(=)28 b Fn(\000)2034 4164 y Fq(\()p Fo(W)2178 4128 y Fm(m)p Fi(\000)p Fl(1)2164 4188 y Fm(m)2347 4137 y Fq(~)2335 4164 y Fo(\036)2393 4179 y Fm(m)p Fi(\000)p Fl(1)2549 4164 y Fo(;)17 b(\036)2651 4179 y Fm(m)2717 4164 y Fq(\))p 2034 4208 721 4 v 2166 4310 a(\()2217 4284 y(~)2204 4310 y Fo(\036)2262 4325 y Fm(m)p Fi(\000)p Fl(1)2418 4310 y Fo(;)g(\036)2520 4325 y Fm(m)2586 4310 y Fq(\))2765 4231 y Fo(;)430 b Fq(\(I)s(I.17\))257 4521 y(truncated)53 b(energy)f(shifts)h(are)f(de\014ned)h(b)m(y)f(\()p Fo(E)2136 4485 y Fm(m)p Fi(\000)p Fl(1)2130 4546 y Fm(m)2293 4521 y Fq(\))2331 4485 y Fm(T)2446 4521 y Fq(:=)60 b(\001)p Fn(E)2752 4425 y Fh(\020)2801 4521 y Fq(\()p Fo(\036)2897 4536 y Fm(m)2964 4521 y Fq(\))3002 4485 y Fm(T)3057 4521 y Fo(;)17 b Fq(\()3152 4495 y(~)3139 4521 y Fo(\036)3197 4536 y Fm(m)p Fi(\000)p Fl(1)3353 4521 y Fq(\))3391 4485 y Fm(T)3446 4425 y Fh(\021)257 4642 y Fq(where)647 4888 y(\001)p Fn(E)790 4791 y Fh(\020)840 4888 y Fq(\()p Fo(\036)936 4903 y Fm(m)1002 4888 y Fq(\))1040 4847 y Fm(T)1095 4888 y Fo(;)g Fq(\()1190 4861 y(~)1177 4888 y Fo(\036)1235 4903 y Fm(m)p Fi(\000)p Fl(1)1391 4888 y Fq(\))1429 4847 y Fm(T)1484 4791 y Fh(\021)1561 4888 y Fq(:=)28 b Fn(\000)1779 4820 y Fq(\()p Fo(W)1923 4784 y Fm(m)p Fi(\000)p Fl(1)1909 4845 y Fm(m)2080 4820 y Fq(\()2131 4794 y(~)2118 4820 y Fo(\036)2176 4835 y Fm(m)p Fi(\000)p Fl(1)2332 4820 y Fq(\))2370 4784 y Fm(T)2458 4820 y Fo(;)17 b Fq(\()p Fo(\036)2598 4835 y Fm(m)2664 4820 y Fq(\))2702 4784 y Fm(T)2757 4820 y Fq(\))p 1779 4865 1016 4 v 1911 4967 a(\(\()2000 4941 y(~)1987 4967 y Fo(\036)2045 4982 y Fm(m)p Fi(\000)p Fl(1)2201 4967 y Fq(\))2239 4938 y Fm(T)2326 4967 y Fo(;)g Fq(\()p Fo(\036)2466 4982 y Fm(m)2532 4967 y Fq(\))2570 4938 y Fm(T)2625 4967 y Fq(\))2804 4888 y Fo(:)391 b Fq(\(I)s(I.18\))257 5147 y(The)33 b(truncated)g(energy)g (shifts)g(will)f(pla)m(y)h(an)f(imp)s(ortan)m(t)g(role)g(in)g(our)g (re-expansion.)257 5268 y(This)45 b(hinges)f(on)f(the)h(estimates)g(in) g(next)g(lemma)g(whic)m(h)g(con)m(trol)g(their)g(v)m(ery)g(fast)257 5388 y(decreasing)34 b(rate)f(in)g Fo(\013)g Fq(for)f(large)g Fo(m)p Fq(.)p eop end %%Page: 22 23 TeXDict begin 22 22 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(22)257 573 y Fe(Lemma)36 b(I)s(I.2.)j Fj(Under)32 b(the)h(assumptions)e(of)h(De\014nition)f(II.1,)g(the)h (estimates)g(b)-5 b(elow)257 693 y(hold)35 b(true:)1468 814 y Fn(j)p Fo(E)1574 772 y Fm(m)p Fi(\000)p Fl(1)1568 838 y Fm(m)1730 814 y Fn(j)28 b(\024)g(O)s Fq(\()p Fo(\013)2074 772 y Fl(2+)p Fm(m)2230 814 y Fq(\))954 b(\(I)s(I.19\))257 988 y Fj(and)1175 1108 y Fn(j)p Fo(E)1281 1067 y Fm(m)p Fi(\000)p Fl(1)1275 1133 y Fm(m)1460 1108 y Fn(\000)22 b Fq(\()p Fo(E)1675 1067 y Fm(m)p Fi(\000)p Fl(1)1669 1133 y Fm(m)1832 1108 y Fq(\))1870 1067 y Fm(T)1925 1108 y Fn(j)27 b(\024)i Fo(o)p Fq(\()p Fo(\013)2244 1040 y Ff(1)p 2243 1052 31 4 v 2243 1093 a(2)2283 1067 y Fl(+)p Fm(m)p Fl(+)p Fm(N)2523 1108 y Fq(\))661 b(\(I)s(I.20\))257 1283 y Fj(with)35 b Fq(1)28 b Fn(\024)g Fo(m)g Fn(\024)g Fo(M)38 b Fq(=)28 b(2)p Fo(N)k Fn(\000)23 b Fq(1)p Fj(.)257 1486 y(Pr)-5 b(o)g(of.)257 1606 y Fq(Pro)s(of)32 b(of)g(Inequalit)m(y)j (\(I)s(I.19\))257 1727 y(W)-8 b(e)33 b(start)g(from)f(the)h (denominator)g(in)g(expression)i(\(I)s(I.17\):)914 1947 y Fn(j)p Fq(\()993 1921 y(~)980 1947 y Fo(\036)1038 1962 y Fm(m)p Fi(\000)p Fl(1)1227 1947 y Fo(\036)1285 1962 y Fm(m)1351 1947 y Fq(\))p Fn(j)28 b Fq(=)f Fn(j)p Fq(\()1627 1921 y(~)1614 1947 y Fo(\036)1672 1962 y Fm(m)p Fi(\000)p Fl(1)1828 1947 y Fo(;)17 b(P)1935 1962 y Fm(m)2014 1921 y Fq(~)2001 1947 y Fo(\036)2059 1962 y Fm(m)p Fi(\000)p Fl(1)2216 1947 y Fq(\))p Fn(j)27 b(\025)h Fo(\016)2461 1906 y Fi(0)2485 1947 y Fn(k)2548 1921 y Fq(~)2535 1947 y Fo(\036)2593 1962 y Fm(m)p Fi(\000)p Fl(1)2749 1947 y Fn(k)2799 1906 y Fl(2)3222 1947 y Fq(\(I)s(I.21\))257 2167 y(for)41 b(some)h Fo(\016)715 2131 y Fi(0)780 2167 y Fo(>)h Fq(0,)g(indep)s(enden)m(t)g(of)e Fo(\013)g Fq(and)h Fo(m)g Fn(2)h Fk(N)p Fq(.)69 b(This)42 b(follo)m(ws)g(from)f(Prop)s(o-) 257 2287 y(sition)47 b(I.1.)85 b(T)-8 b(o)47 b(estimate)h(the)e(n)m (umerator)i(in)e(\(I)s(I.17\),)k(w)m(e)d(again)f(use)i(the)f(series)257 2408 y(expansion)34 b(of)f Fo(P)885 2423 y Fm(m)951 2408 y Fq(:)662 2628 y(\()p Fo(W)806 2587 y Fm(m)p Fi(\000)p Fl(1)792 2652 y Fm(m)975 2601 y Fq(~)963 2628 y Fo(\036)1021 2643 y Fm(m)p Fi(\000)p Fl(1)1210 2628 y Fo(;)17 b(\036)1312 2643 y Fm(m)1378 2628 y Fq(\))27 b(=)503 2773 y(=)83 b(\()p Fo(W)806 2732 y Fm(m)p Fi(\000)p Fl(1)792 2798 y Fm(m)975 2747 y Fq(~)963 2773 y Fo(\036)1021 2788 y Fm(m)p Fi(\000)p Fl(1)1210 2773 y Fo(;)1266 2747 y Fq(~)1254 2773 y Fo(\036)1312 2788 y Fm(m)p Fi(\000)p Fl(1)1468 2773 y Fq(\))1716 b(\(I)s(I.22\))662 2976 y Fn(\000)796 2908 y Fq(1)p 749 2953 141 4 v 749 3044 a(2)p Fo(\031)t(i)943 2868 y Fi(1)919 2893 y Fh(X)917 3075 y Fm(j)t Fl(=1)1056 2859 y Fh(Z)1103 3047 y Fl(\000)1147 3055 y Fb(m)1226 2976 y Fo(dz)1322 2991 y Fm(m)1389 2880 y Fh(\020)1438 2976 y Fo(W)1544 2935 y Fm(m)p Fi(\000)p Fl(1)1530 3001 y Fm(m)1713 2950 y Fq(~)1701 2976 y Fo(\036)1759 2991 y Fm(m)p Fi(\000)p Fl(1)1948 2976 y Fo(;)1247 b Fq(\(I)s(I.23\))1443 3272 y Fo(;)1708 3204 y Fq(1)p 1496 3248 472 4 v 1504 3321 a Fh(f)1496 3353 y Fo(H)1585 3312 y Fl(+)1577 3375 y Fm(m)p Fi(\000)p Fl(1)1756 3353 y Fn(\000)23 b Fo(z)1901 3368 y Fm(m)1978 3150 y Fh(\024)2021 3272 y Fq(\()p Fn(\000)p Fo(W)2242 3231 y Fm(m)p Fi(\000)p Fl(1)2228 3296 y Fm(m)2399 3272 y Fq(\))2691 3204 y(1)p 2480 3248 V 2488 3321 a Fh(f)2480 3353 y Fo(H)2569 3312 y Fl(+)2561 3375 y Fm(m)p Fi(\000)p Fl(1)2739 3353 y Fn(\000)g Fo(z)2884 3368 y Fm(m)2961 3150 y Fh(\025)3005 3173 y Fm(j)3054 3245 y Fq(~)3041 3272 y Fo(\036)3099 3287 y Fm(m)p Fi(\000)p Fl(1)3256 3175 y Fh(\021)3305 3272 y Fo(:)257 3564 y Fq(Next)34 b(w)m(e)f(b)s(ound)g(the)g(t)m(w)m(o)g(terms,)h(\(I)s (I.22\))e(and)h(\(I)s(I.23\):)257 3685 y(T)-8 b(erm)34 b(\(I)s(I.22\):)650 3964 y(\()p Fo(W)794 3923 y Fm(m)p Fi(\000)p Fl(1)780 3989 y Fm(m)963 3938 y Fq(~)950 3964 y Fo(\036)1008 3979 y Fm(m)p Fi(\000)p Fl(1)1197 3964 y Fo(;)1254 3938 y Fq(~)1241 3964 y Fo(\036)1299 3979 y Fm(m)p Fi(\000)p Fl(1)1456 3964 y Fq(\))27 b(=)1720 3897 y Fo(\013)1783 3860 y Fl(3)p 1635 3941 272 4 v 1635 4032 a Fq(3\(2)p Fo(\031)t Fq(\))1868 4003 y Fl(2)1916 3964 y Fq(\()p Fo(\033)2013 3923 y Fl(2)2009 3989 y Fm(m)p Fi(\000)p Fl(1)2189 3964 y Fn(\000)22 b Fo(\033)2347 3923 y Fl(2)2343 3989 y Fm(m)2410 3964 y Fq(\))p Fn(k)2511 3938 y Fq(~)2498 3964 y Fo(\036)2556 3979 y Fm(m)p Fi(\000)p Fl(1)2713 3964 y Fn(k)2763 3923 y Fl(2)2802 3964 y Fq(;)393 b(\(I)s(I.24\))257 4240 y(T)-8 b(erm)54 b(\(I)s(I.23\):)40 b(W)-8 b(e)26 b(m)m(ust)i(b)s(ound)e(the)h(absolute)g(v)-5 b(alue)27 b(of)f(the)h(scalar)f(pro)s(duct)h(inside)257 4360 y(the)33 b(con)m(tour)g(in)m(tegral)g(as)g(follo)m(ws)738 4530 y Fh(\014)738 4580 y(\014)738 4630 y(\014)766 4533 y(\020)816 4630 y Fo(W)922 4589 y Fm(m)908 4654 y(m)1000 4603 y Fq(~)988 4630 y Fo(\036)1046 4645 y Fm(m)p Fi(\000)p Fl(1)1235 4630 y Fo(;)1500 4562 y Fq(1)p 1288 4606 472 4 v 1296 4679 a Fh(f)1288 4711 y Fo(H)1377 4670 y Fl(+)1369 4733 y Fm(m)p Fi(\000)p Fl(1)1548 4711 y Fn(\000)23 b Fo(z)1693 4726 y Fm(m)1770 4508 y Fh(\024)1813 4630 y Fq(\()p Fn(\000)p Fo(W)2034 4589 y Fm(m)p Fi(\000)p Fl(1)2020 4654 y Fm(m)2191 4630 y Fq(\))2483 4562 y(1)p 2272 4606 V 2280 4679 a Fh(f)2272 4711 y Fo(H)2361 4670 y Fl(+)2353 4733 y Fm(m)p Fi(\000)p Fl(1)2531 4711 y Fn(\000)g Fo(z)2676 4726 y Fm(m)2753 4508 y Fh(\025)2797 4531 y Fm(j)2846 4603 y Fq(~)2833 4630 y Fo(\036)2891 4645 y Fm(m)p Fi(\000)p Fl(1)3048 4533 y Fh(\021)3097 4530 y(\014)3097 4580 y(\014)3097 4630 y(\014)28 b Fn(\024)578 4905 y(\024)738 4806 y Fh(\015)738 4856 y(\015)738 4905 y(\015)784 4809 y(\020)1055 4838 y Fq(1)p 844 4882 V 852 4954 a Fh(f)844 4987 y Fo(H)933 4946 y Fl(+)925 5009 y Fm(m)p Fi(\000)p Fl(1)1104 4987 y Fn(\000)22 b Fo(z)1248 5002 y Fm(m)1325 4809 y Fh(\021)1385 4805 y Ff(1)p 1385 4817 31 4 v 1385 4858 a(2)1429 4905 y Fo(W)1535 4864 y Fm(m)p Fi(\000)p Fl(1)1521 4930 y Fm(m)1704 4879 y Fq(~)1692 4905 y Fo(\036)1750 4920 y Fm(m)p Fi(\000)p Fl(1)1906 4806 y Fh(\015)1906 4856 y(\015)1906 4905 y(\015)1952 4832 y Fl(2)2014 4905 y Fn(\002)1131 b Fq(\(I)s(I.25\))933 5181 y Fn(\002)1010 5082 y Fh(\015)1010 5131 y(\015)1010 5181 y(\015)1057 5085 y(\020)1328 5114 y Fq(1)p 1117 5158 472 4 v 1125 5230 a Fh(f)1117 5263 y Fo(H)1206 5221 y Fl(+)1198 5284 y Fm(m)p Fi(\000)p Fl(1)1376 5263 y Fn(\000)23 b Fo(z)1521 5278 y Fm(m)1598 5085 y Fh(\021)1657 5081 y Ff(1)p 1657 5093 31 4 v 1657 5134 a(2)1702 5181 y Fo(W)1808 5140 y Fm(m)p Fi(\000)p Fl(1)1794 5206 y Fm(m)1964 5085 y Fh(\020)2235 5114 y Fq(1)p 2024 5158 472 4 v 2032 5230 a Fh(f)2024 5263 y Fo(H)2113 5221 y Fl(+)2105 5284 y Fm(m)p Fi(\000)p Fl(1)2284 5263 y Fn(\000)f Fo(z)2428 5278 y Fm(m)2505 5085 y Fh(\021)2565 5081 y Ff(1)p 2565 5093 31 4 v 2565 5134 a(2)2609 5082 y Fh(\015)2609 5131 y(\015)2609 5181 y(\015)2655 5108 y Fm(j)t Fi(\000)p Fl(1)2782 5082 y Fh(\014)2782 5131 y(\014)2782 5181 y(\014)2851 5114 y Fq(1)p 2820 5158 112 4 v 2820 5250 a Fo(z)2865 5265 y Fm(m)2941 5082 y Fh(\014)2941 5131 y(\014)2941 5181 y(\014)p Fo(:)p eop end %%Page: 23 24 TeXDict begin 23 23 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(23)257 573 y(Using)42 b(Estimate)g(\(I.54\))e (in)h(Prop)s(osition)g(I.1,)i(the)e(absolute)h(v)-5 b(alue)41 b(of)f(the)h(sum)h(in)257 693 y(Eq.)34 b(\(I)s(I.23\))e(is)h(seen)h(to) e(b)s(e)h(b)s(ounded)g(b)m(y)1065 885 y(1)p 934 929 311 4 v 934 1020 a(1)22 b Fn(\000)h Fo(C)7 b(\013)1318 952 y Fq(sup)1271 1031 y Fm(z)1304 1039 y Fb(m)1362 1031 y Fi(2)p Fl(\000)1453 1039 y Fb(m)1529 952 y Fn(k)p Fq(\()1625 920 y Fh(f)1617 952 y Fo(H)1706 911 y Fl(+)1698 977 y Fm(m)p Fi(\000)p Fl(1)1876 952 y Fn(\000)23 b Fo(z)2021 967 y Fm(m)2088 952 y Fq(\))2126 911 y Fi(\000)2191 884 y Ff(1)p 2191 896 31 4 v 2191 937 a(2)2235 952 y Fo(W)2341 911 y Fm(m)p Fi(\000)p Fl(1)2327 977 y Fm(m)2510 926 y Fq(~)2497 952 y Fo(\036)2555 967 y Fm(m)p Fi(\000)p Fl(1)2712 952 y Fn(k)2762 911 y Fl(2)2801 952 y Fo(:)394 b Fq(\(I)s(I.26\))257 1219 y(Our)33 b(next)g(step)h(consists)g(in)f (pro)m(ving)g(that)903 1433 y(sup)856 1512 y Fm(z)889 1520 y Fb(m)948 1512 y Fi(2)p Fl(\000)1039 1520 y Fb(m)1114 1433 y Fn(k)p Fq(\()1210 1401 y Fh(f)1202 1433 y Fo(H)1291 1392 y Fl(+)1283 1458 y Fm(m)p Fi(\000)p Fl(1)1461 1433 y Fn(\000)23 b Fo(z)1606 1448 y Fm(m)1673 1433 y Fq(\))1711 1392 y Fi(\000)1776 1365 y Ff(1)p 1776 1377 V 1776 1418 a(2)1820 1433 y Fo(W)1926 1392 y Fm(m)p Fi(\000)p Fl(1)1912 1458 y Fm(m)2095 1407 y Fq(~)2083 1433 y Fo(\036)2141 1448 y Fm(m)p Fi(\000)p Fl(1)2297 1433 y Fn(k)2347 1392 y Fl(2)2414 1433 y Fn(\024)28 b(O)s Fq(\()p Fo(\013)2702 1392 y Fl(2+)p Fm(m)2859 1433 y Fq(\))325 b(\(I)s(I.27\))257 1705 y(whic)m(h)32 b(pro)m(v)m(es)g(inequalit)m(y)f(\(I)s(I.19\))f (claimed)h(in)f(the)h(statemen)m(t)g(of)f(the)g(Lemma.)43 b(The)257 1826 y(v)m(ector)31 b(inside)g(the)f(norm)f(on)h(the)g(left)g (side)g(of)f(\(I)s(I.27\))g(can)h(b)s(e)g(split)g(as)g Fo(')3007 1841 y Fl(1)3062 1826 y Fq(+)16 b Fo(')3218 1841 y Fl(2)3273 1826 y Fq(+)g Fo(')3429 1841 y Fl(3)3468 1826 y Fq(,)257 1946 y(where)370 2160 y Fo(')434 2175 y Fl(1)557 2160 y Fq(:=)83 b(\()789 2128 y Fh(f)781 2160 y Fo(H)870 2119 y Fl(+)862 2185 y Fm(m)p Fi(\000)p Fl(1)1040 2160 y Fn(\000)23 b Fo(z)1185 2175 y Fm(m)1252 2160 y Fq(\))1290 2119 y Fi(\000)1355 2092 y Ff(1)p 1355 2104 V 1355 2145 a(2)1395 2160 y Fo(~)-45 b(v)1446 2175 y Fm(m)p Fi(\000)p Fl(1)1625 2160 y Fn(\001)22 b Fo(a)1726 2119 y Fi(\003)1766 2160 y Fq(\()1819 2135 y Fo(~)1804 2160 y(G)1881 2119 y Fm(m)p Fi(\000)p Fl(1)1881 2185 y Fm(m)2037 2160 y Fq(\))2088 2134 y(~)2075 2160 y Fo(\036)2133 2175 y Fm(m)p Fi(\000)p Fl(1)3222 2160 y Fq(\(I)s(I.28\))570 2362 y(=)97 b(\()813 2294 y Fo(\013)p 791 2339 108 4 v 791 2430 a Fq(2)p Fo(\031)908 2362 y Fq(\))956 2294 y Ff(3)p 956 2306 31 4 v 956 2347 a(2)1017 2245 y Fh(Z)1063 2433 y Fi(K)1117 2403 y Fb(m)p Fd(\000)p Ff(1)1117 2445 y Fb(m)1370 2294 y Fo(dk)p 1285 2339 275 4 v 1285 2355 a Fh(q)p 1368 2355 192 4 v 100 x Fq(2)17 b Fn(j)p Fo(k)s Fn(j)1570 2362 y Fo(a)1621 2321 y Fi(\003)1660 2362 y Fq(\()p Fo(k)s Fq(\)\()1836 2329 y Fh(f)1828 2362 y Fo(H)1917 2321 y Fl(+)1909 2386 y Fm(m)p Fi(\000)p Fl(1)2088 2362 y Fn(\000)22 b Fo(z)2232 2377 y Fm(m)2322 2362 y Fq(+)g Fn(j)p Fo(k)s Fn(j)p Fq(\))2568 2321 y Fi(\000)2633 2294 y Ff(1)p 2632 2306 31 4 v 2632 2347 a(2)2698 2362 y Fn(\002)447 b Fq(\(I)s(I.29\))1035 2641 y Fn(\002)o Fo(~)-48 b(")p Fq(\()p Fo(k)s Fq(\))23 b Fn(\001)17 b Fo(~)-45 b(v)1407 2656 y Fm(m)p Fi(\000)p Fl(1)1564 2641 y Fo(e)1609 2600 y Fi(\000)p Fm(i\013)1730 2582 y(~)1733 2600 y(k)r Fi(\001)o Fm(~)-34 b(x)1848 2614 y Fq(~)1836 2641 y Fo(\036)1894 2656 y Fm(m)p Fi(\000)p Fl(1)370 2799 y Fo(')434 2814 y Fl(2)557 2799 y Fq(:=)83 b(\()789 2767 y Fh(f)781 2799 y Fo(H)870 2758 y Fl(+)862 2824 y Fm(m)p Fi(\000)p Fl(1)1040 2799 y Fn(\000)23 b Fo(z)1185 2814 y Fm(m)1252 2799 y Fq(\))1290 2758 y Fi(\000)1355 2731 y Ff(1)p 1355 2743 V 1355 2784 a(2)1399 2799 y Fo(a)1450 2758 y Fi(\003)1490 2799 y Fq(\()1543 2774 y Fo(~)1528 2799 y(G)1605 2758 y Fm(m)p Fi(\000)p Fl(1)1605 2824 y Fm(m)1761 2799 y Fq(\))g Fn(\001)e Fo(a)1922 2758 y Fi(\003)1962 2799 y Fq(\()2015 2774 y Fo(~)2000 2799 y(G)2077 2758 y Fm(m)p Fi(\000)p Fl(1)2077 2824 y Fm(m)2234 2799 y Fq(\))2285 2773 y(~)2272 2799 y Fo(\036)2330 2814 y Fm(m)p Fi(\000)p Fl(1)3222 2799 y Fq(\(I)s(I.30\))570 3001 y(=)97 b(\()813 2933 y Fo(\013)p 791 2977 108 4 v 791 3069 a Fq(2)p Fo(\031)908 3001 y Fq(\))946 2959 y Fl(3)1002 2883 y Fh(Z)1048 3072 y Fi(K)1102 3042 y Fb(m)p Fd(\000)p Ff(1)1102 3084 y Fb(m)1260 2883 y Fh(Z)1306 3072 y Fi(K)1360 3042 y Fb(m)p Fd(\000)p Ff(1)1360 3084 y Fb(m)1607 2933 y Fo(dk)19 b(dq)p 1528 2977 377 4 v 1528 2994 a Fh(q)p 1611 2994 294 4 v 100 x Fq(2)e Fn(j)p Fo(k)s Fn(j)g(j)p Fo(q)t Fn(j)1914 3001 y Fo(~)-48 b(")o Fq(\()p Fo(k)s Fq(\))22 b Fn(\001)f Fo(~)-48 b(")p Fq(\()p Fo(q)t Fq(\))22 b Fn(\002)792 b Fq(\(I)s(I.31\))1133 3280 y Fn(\002)p Fo(a)1261 3238 y Fi(\003)1301 3280 y Fq(\()p Fo(k)s Fq(\))p Fo(a)1482 3238 y Fi(\003)1522 3280 y Fq(\()p Fo(q)t Fq(\)\()1691 3247 y Fh(f)1683 3280 y Fo(H)1772 3238 y Fl(+)1764 3304 y Fm(m)p Fi(\000)p Fl(1)1942 3280 y Fn(\000)23 b Fo(z)2087 3295 y Fm(m)2176 3280 y Fq(+)f Fn(j)p Fo(k)s Fn(j)f Fq(+)h Fn(j)p Fo(q)t Fn(j)p Fq(\))2644 3238 y Fi(\000)2709 3211 y Ff(1)p 2708 3223 31 4 v 2708 3264 a(2)2753 3280 y Fo(e)2798 3238 y Fi(\000)p Fm(i\013)p Fl(\()2946 3221 y Fm(~)2949 3238 y(k)r Fl(+)p Fm(~)-35 b(q)r Fl(\))p Fi(\001)o Fm(~)h(x)3180 3253 y Fq(~)3168 3280 y Fo(\036)3226 3295 y Fm(m)p Fi(\000)p Fl(1)370 3491 y Fo(')434 3506 y Fl(3)557 3491 y Fq(:=)837 3423 y Fo(\013)900 3387 y Fl(3)p 753 3467 272 4 v 753 3559 a Fq(3\(2)p Fo(\031)t Fq(\))986 3530 y Fl(2)1034 3491 y Fq(\()p Fo(\033)1131 3450 y Fl(2)1127 3515 y Fm(m)p Fi(\000)p Fl(1)1307 3491 y Fn(\000)22 b Fo(\033)1465 3450 y Fl(2)1461 3515 y Fm(m)1528 3491 y Fq(\)\()1612 3458 y Fh(f)1604 3491 y Fo(H)1693 3450 y Fl(+)1685 3515 y Fm(m)p Fi(\000)p Fl(1)1864 3491 y Fn(\000)g Fo(z)2008 3506 y Fm(m)2075 3491 y Fq(\))2113 3450 y Fi(\000)2178 3422 y Ff(1)p 2178 3434 31 4 v 2178 3476 a(2)2235 3464 y Fq(~)2223 3491 y Fo(\036)2281 3506 y Fm(m)p Fi(\000)p Fl(1)3222 3491 y Fq(\(I)s(I.32\))257 3755 y(The)34 b(corresp)s(onding)f (norms)h(are:)829 4014 y Fn(k)p Fo(')943 4029 y Fl(1)982 4014 y Fn(k)1032 3973 y Fl(2)1099 4014 y Fq(=)27 b(\()1273 3946 y Fo(\013)p 1250 3990 108 4 v 1250 4082 a Fq(2)p Fo(\031)1368 4014 y Fq(\))1406 3973 y Fl(3)1462 3897 y Fh(Z)1508 4085 y Fi(K)1562 4055 y Fb(m)p Fd(\000)p Ff(1)1562 4097 y Fb(m)1776 3946 y Fo(dk)p 1730 3990 198 4 v 1730 4098 a Fq(2)p Fn(j)1802 4071 y Fo(~)1807 4098 y(k)s Fn(j)1889 4069 y Fl(2)1938 4014 y Fq(\()o Fo(~)-48 b(")o Fq(\()p Fo(k)s Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)2270 4029 y Fm(m)p Fi(\000)p Fl(1)2427 4014 y Fo(e)2472 3973 y Fi(\000)p Fm(i\013)p Fi(\001)o Fm(~)-34 b(x)2673 3987 y Fq(~)2660 4014 y Fo(\036)2718 4029 y Fm(m)p Fi(\000)p Fl(1)2891 4014 y Fo(;)304 b Fq(\(I)s(I.33\))1140 4298 y Fo(;)1517 4231 y Fn(j)p Fo(k)s Fn(j)p 1193 4275 757 4 v 1193 4379 a(j)1229 4347 y Fh(f)1221 4379 y Fo(H)1310 4338 y Fl(+)1302 4401 y Fm(m)p Fi(\000)p Fl(1)1481 4379 y Fn(\000)22 b Fo(z)1625 4394 y Fm(m)1714 4379 y Fq(+)g Fn(j)p Fo(k)s Fn(jj)1958 4298 y Fo(~)-48 b(")p Fq(\()p Fo(k)s Fq(\))22 b Fn(\001)c Fo(~)-45 b(v)2254 4313 y Fm(m)p Fi(\000)p Fl(1)2411 4298 y Fo(e)2456 4257 y Fi(\000)p Fm(i\013)2577 4239 y(~)2580 4257 y(k)r Fi(\001)o Fm(~)-34 b(x)2695 4272 y Fq(~)2682 4298 y Fo(\036)2740 4313 y Fm(m)p Fi(\000)p Fl(1)2897 4298 y Fq(\))257 4589 y(whic)m(h)33 b(is)e(b)s(ounded)h(b)m(y)g Fo(C)1232 4604 y Fl(1)1271 4589 y Fo(\013)1334 4553 y Fm(m)p Fl(+2)1490 4589 y Fq(,)f Fo(C)1618 4604 y Fl(1)1689 4589 y Fq(a)f(n)m(umerical)i(co)s(e\016cien) m(t,)h(as)f(follo)m(ws)f(from)g(the)257 4710 y(inequalities)544 4994 y(\()586 4886 y Fm(l)q Fl(=3)582 4911 y Fh(X)586 5095 y Fm(l)q Fl(=1)719 4994 y Fn(k)p Fo(v)820 4952 y Fm(l)816 5018 y(m)p Fi(\000)p Fl(1)973 4994 y Fo(e)1018 4952 y Fi(\000)p Fm(i\013)1139 4935 y(~)1142 4952 y(k)r Fi(\001)o Fm(~)-34 b(x)1244 4994 y Fo(\036)1302 5009 y Fm(m)p Fi(\000)p Fl(1)1459 4994 y Fn(k)1509 4952 y Fl(2)1548 4994 y Fq(\))1596 4925 y Ff(1)p 1596 4937 31 4 v 1596 4978 a(2)1723 4994 y Fq(=:)83 b Fn(k)l Fo(~)-45 b(v)2006 5009 y Fm(m)p Fi(\000)p Fl(1)2163 4994 y Fo(e)2208 4952 y Fi(\000)p Fm(i\013)2329 4935 y(~)2332 4952 y(k)r Fi(\001)o Fm(~)-34 b(x)2434 4994 y Fo(\036)2492 5009 y Fm(m)p Fi(\000)p Fl(1)2649 4994 y Fn(k)1737 5243 y Fq(=)96 b Fn(k)p Fo(e)2004 5202 y Fi(\000)p Fm(i\013)2125 5184 y(~)2128 5202 y(k)r Fi(\001)o Fm(~)-34 b(x)2230 5243 y Fq(\()l Fo(~)-45 b(v)2315 5258 y Fm(m)p Fi(\000)p Fl(1)2494 5243 y Fn(\000)23 b Fo(\013)2652 5217 y(~)2657 5243 y(k)s Fq(\))p Fo(\036)2807 5258 y Fm(m)p Fi(\000)p Fl(1)2963 5243 y Fn(k)209 b Fq(\(I)s(I.34\))1736 5388 y Fn(\024)96 b Fo(\013)q(\024)p Fn(k)p Fo(\036)2136 5403 y Fm(m)p Fi(\000)p Fl(1)2292 5388 y Fn(k)22 b Fq(+)g Fn(k)l Fo(~)-45 b(v)2559 5403 y Fm(m)p Fi(\000)p Fl(1)2716 5388 y Fo(\036)2774 5403 y Fm(m)p Fi(\000)p Fl(1)2931 5388 y Fn(k)241 b Fq(\(I)s(I.35\))p eop end %%Page: 24 25 TeXDict begin 24 24 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(24)257 573 y(and)1369 624 y Fh(\015)1369 673 y(\015)1369 723 y(\015)1721 656 y Fn(j)p Fo(k)s Fn(j)p 1425 700 701 4 v 1433 772 a Fh(f)1425 805 y Fo(H)1514 763 y Fl(+)1506 826 y Fm(m)p Fi(\000)p Fl(1)1685 805 y Fn(\000)22 b Fo(z)1829 820 y Fm(m)1918 805 y Fq(+)h Fn(j)p Fo(k)s Fn(j)2136 624 y Fh(\015)2136 673 y(\015)2136 723 y(\015)28 b Fn(\024)2325 656 y Fq(4)p 2325 700 49 4 v 2325 792 a(3)3222 723 y(\(I)s(I.36\))257 964 y(for)k Fo(\024)c(>)g Fn(j)p Fo(k)s Fn(j)f(\025)h Fo(\033)891 979 y Fm(m)958 964 y Fq(,)33 b Fo(z)1063 979 y Fm(m)1157 964 y Fn(2)28 b Fq(\000)1312 979 y Fm(m)1379 964 y Fq(.)257 1085 y(Next,)688 1335 y Fn(k)p Fo(')802 1350 y Fl(2)841 1335 y Fn(k)891 1293 y Fl(2)1014 1335 y Fq(=)83 b(2\()1292 1267 y Fo(\013)p 1270 1311 108 4 v 1270 1403 a Fq(2)p Fo(\031)1387 1335 y Fq(\))1425 1293 y Fl(6)1481 1217 y Fh(Z)1527 1406 y Fi(K)1581 1376 y Fb(m)p Fd(\000)p Ff(1)1581 1418 y Fb(m)1739 1217 y Fh(Z)1785 1406 y Fi(K)1839 1376 y Fb(m)p Fd(\000)p Ff(1)1839 1418 y Fb(m)2064 1267 y Fo(dk)19 b(dq)p 2007 1311 334 4 v 2007 1403 a Fq(2)e Fn(j)p Fo(k)s Fn(j)2183 1374 y Fl(2)2238 1403 y Fn(j)p Fo(q)t Fn(j)2350 1335 y Fq(\()o Fo(~)-48 b(")o Fq(\()p Fo(k)s Fq(\))23 b Fn(\001)d Fo(~)-48 b(")p Fq(\()p Fo(q)t Fq(\)\))2842 1293 y Fl(2)2903 1335 y Fn(\002)242 b Fq(\(I)s(I.37\))1173 1624 y Fn(\002)1250 1525 y Fh(\015)1250 1575 y(\015)1250 1624 y(\015)1741 1557 y Fn(j)p Fo(k)s Fn(j)1861 1493 y Ff(1)p 1860 1505 31 4 v 1860 1546 a(2)p 1306 1601 1033 4 v 1306 1707 a Fn(j)1342 1675 y Fh(f)1334 1707 y Fo(H)1423 1666 y Fl(+)1415 1729 y Fm(m)p Fi(\000)p Fl(1)1594 1707 y Fn(\000)22 b Fo(z)1738 1722 y Fm(m)1827 1707 y Fq(+)g Fn(j)p Fo(k)s Fn(j)g Fq(+)g Fn(j)p Fo(q)t Fn(jj)2296 1643 y Ff(1)p 2295 1655 31 4 v 2295 1697 a(2)2349 1624 y Fo(e)2394 1583 y Fi(\000)p Fm(i\013)p Fl(\()2542 1566 y Fm(~)2545 1583 y(k)s Fl(+)p Fm(~)-35 b(q)q Fl(\))p Fi(\001)o Fm(~)h(x)2764 1624 y Fo(\036)2822 1639 y Fm(m)p Fi(\000)p Fl(1)2979 1525 y Fh(\015)2979 1575 y(\015)2979 1624 y(\015)3025 1551 y Fl(2)257 1899 y Fq(whic)m(h)34 b(is)g(b)s(ounded)f(b)m(y)897 2082 y(2)p 897 2126 49 4 v 897 2217 a(3)956 2149 y(\()1026 2082 y Fo(\013)p 1004 2126 108 4 v 1004 2217 a Fq(2)p Fo(\031)1121 2149 y Fq(\))1159 2108 y Fl(6)1199 2053 y Fh(\020)1265 2032 y(Z)1311 2221 y Fi(K)1365 2191 y Fb(m)p Fd(\000)p Ff(1)1365 2232 y Fb(m)1523 2032 y Fh(Z)1569 2221 y Fi(K)1623 2191 y Fb(m)p Fd(\000)p Ff(1)1623 2232 y Fb(m)1848 2082 y Fo(dk)19 b(dq)p 1791 2126 334 4 v 1791 2217 a Fq(2)e Fn(j)p Fo(k)s Fn(j)1967 2189 y Fl(2)2022 2217 y Fn(j)p Fo(q)t Fn(j)2134 2149 y Fq(\()o Fo(~)-48 b(")o Fq(\()p Fo(k)s Fq(\))23 b Fn(\001)d Fo(~)-48 b(")p Fq(\()p Fo(q)t Fq(\)\))2626 2108 y Fl(2)2665 2053 y Fh(\021)2737 2149 y Fn(\002)985 2418 y(\002)99 b Fq(sup)1079 2501 y Fi(j)p Fm(k)r Fi(j\025)p Fm(\033)1253 2510 y Fb(n)p Ff(+1)1371 2501 y Fi(j)1407 2418 y Fn(k)1763 2350 y(j)p Fo(k)s Fn(j)p 1467 2394 701 4 v 1475 2467 a Fh(f)1467 2499 y Fo(H)1556 2458 y Fl(+)1548 2521 y Fm(m)p Fi(\000)p Fl(1)1727 2499 y Fn(\000)23 b Fo(z)1872 2514 y Fm(m)1961 2499 y Fq(+)f Fn(j)p Fo(k)s Fn(j)2178 2418 y(k)27 b(\024)i Fo(C)2431 2433 y Fl(2)2470 2418 y Fo(\013)2533 2376 y Fl(3+3)p Fm(m)3222 2418 y Fq(\(I)s(I.38\))257 2691 y(and)717 2931 y Fn(k)p Fo(')831 2946 y Fl(3)870 2931 y Fn(k)920 2890 y Fl(2)1043 2931 y Fq(=)83 b Fn(k)1347 2864 y Fo(\013)1410 2828 y Fl(3)p 1262 2908 272 4 v 1262 2999 a Fq(3\(2)p Fo(\031)t Fq(\))1495 2971 y Fl(2)1544 2931 y Fq(\()p Fo(\033)1641 2890 y Fl(2)1637 2956 y Fm(m)p Fi(\000)p Fl(1)1816 2931 y Fn(\000)23 b Fo(\033)1975 2890 y Fl(2)1971 2956 y Fm(m)2038 2931 y Fq(\)\()2122 2899 y Fh(f)2114 2931 y Fo(H)2203 2890 y Fl(+)2195 2956 y Fm(m)p Fi(\000)p Fl(1)2373 2931 y Fn(\000)g Fo(z)2518 2946 y Fm(m)2585 2931 y Fq(\))2623 2890 y Fi(\000)2688 2863 y Ff(1)p 2688 2875 31 4 v 2688 2916 a(2)2732 2931 y Fo(\036)2790 2946 y Fm(m)p Fi(\000)p Fl(1)2947 2931 y Fn(k)2997 2890 y Fl(2)1042 3142 y Fn(\024)83 b Fo(C)1272 3157 y Fl(3)1312 3142 y Fo(\013)1375 3100 y Fl(2+3)p Fm(m)1566 3142 y Fo(:)1629 b Fq(\(I)s(I.39\))257 3348 y(with)34 b Fo(C)550 3363 y Fl(2)621 3348 y Fq(and)f Fo(C)881 3363 y Fl(3)953 3348 y Fq(n)m(umerical)h(co)s(e\016cien)m(ts.)257 3468 y(Th)m(us)g(the)f(b)s(ound)g(\(I)s(I.27\))f(is)h(pro)m(v)m(ed.)257 3588 y(Pro)s(of)f(of)g(Inequalit)m(y)j(\(I)s(I.20\))257 3709 y(According)f(to)e(\(I)s(I.17\),)g(\(I)s(I.18\),)g(the)h (di\013erence)1515 3915 y Fo(E)1593 3874 y Fm(m)p Fi(\000)p Fl(1)1587 3939 y Fm(m)1772 3915 y Fn(\000)23 b Fq(\()p Fo(E)1988 3874 y Fm(m)p Fi(\000)p Fl(1)1982 3939 y Fm(m)2144 3915 y Fq(\))2182 3874 y Fm(T)3222 3915 y Fq(\(I)s(I.40\))257 4121 y(corresp)s(onds)34 b(to)862 4394 y Fn(\000)949 4327 y Fq(\()p Fo(W)1093 4291 y Fm(m)p Fi(\000)p Fl(1)1079 4351 y Fm(m)1263 4300 y Fq(~)1250 4327 y Fo(\036)1308 4342 y Fm(m)p Fi(\000)p Fl(1)1497 4327 y Fo(;)17 b(\036)1599 4342 y Fm(m)1665 4327 y Fq(\))p 949 4371 754 4 v 1081 4473 a(\()1132 4447 y(~)1119 4473 y Fo(\036)1177 4488 y Fm(m)p Fi(\000)p Fl(1)1366 4473 y Fo(;)g(\036)1468 4488 y Fm(m)1534 4473 y Fq(\))1735 4394 y(+)1843 4327 y(\()p Fo(W)1987 4291 y Fm(m)p Fi(\000)p Fl(1)1973 4351 y Fm(m)2143 4327 y Fq(\()2194 4300 y(~)2181 4327 y Fo(\036)2239 4342 y Fm(m)p Fi(\000)p Fl(1)2396 4327 y Fq(\))2434 4291 y Fm(T)2521 4327 y Fo(;)g Fq(\()p Fo(\036)2661 4342 y Fm(m)2727 4327 y Fq(\))2765 4291 y Fm(T)2820 4327 y Fq(\))p 1843 4371 1016 4 v 2040 4473 a(\(\()2129 4447 y(~)2116 4473 y Fo(\036)2174 4488 y Fm(m)p Fi(\000)p Fl(1)2330 4473 y Fq(\))2368 4444 y Fm(T)2455 4473 y Fo(;)g(\036)2557 4444 y Fm(T)2557 4498 y(m)2623 4473 y Fq(\))3222 4394 y(\(I)s(I.41\))703 4691 y(=)83 b Fn(\000)949 4624 y Fq(\()p Fo(W)1093 4588 y Fm(m)p Fi(\000)p Fl(1)1079 4648 y Fm(m)1263 4597 y Fq(~)1250 4624 y Fo(\036)1308 4639 y Fm(m)p Fi(\000)p Fl(1)1497 4624 y Fo(;)17 b(\036)1599 4639 y Fm(m)1665 4624 y Fq(\)[\(\()1819 4597 y(~)1806 4624 y Fo(\036)1864 4639 y Fm(m)p Fi(\000)p Fl(1)2020 4624 y Fq(\))2058 4588 y Fm(T)2146 4624 y Fo(;)g Fq(\()p Fo(\036)2286 4639 y Fm(m)2352 4624 y Fq(\))2390 4588 y Fm(T)2445 4624 y Fq(\))22 b Fn(\000)g Fq(\()2655 4597 y(~)2642 4624 y Fo(\036)2700 4639 y Fm(m)p Fi(\000)p Fl(1)2889 4624 y Fo(;)17 b(\036)2991 4639 y Fm(m)3057 4624 y Fq(\)])p 949 4668 2173 4 v 1414 4770 a(\(\()1503 4744 y(~)1490 4770 y Fo(\036)1548 4785 y Fm(m)p Fi(\000)p Fl(1)1705 4770 y Fq(\))1743 4741 y Fm(T)1830 4770 y Fo(;)g Fq(\()p Fo(\036)1970 4785 y Fm(m)2036 4770 y Fq(\))2074 4741 y Fm(T)2129 4770 y Fq(\)\()2218 4744 y(~)2205 4770 y Fo(\036)2263 4785 y Fm(m)p Fi(\000)p Fl(1)2452 4770 y Fo(;)g(\036)2554 4785 y Fm(m)2620 4770 y Fq(\))3222 4691 y(\(I)s(I.42\))862 4988 y Fn(\000)949 4921 y Fq(\()p Fo(W)1093 4885 y Fm(m)p Fi(\000)p Fl(1)1079 4945 y Fm(m)1263 4894 y Fq(~)1250 4921 y Fo(\036)1308 4936 y Fm(m)p Fi(\000)p Fl(1)1497 4921 y Fo(;)g(\036)1599 4936 y Fm(m)1665 4921 y Fq(\))22 b Fn(\000)h Fq(\()p Fo(W)1969 4885 y Fm(m)p Fi(\000)p Fl(1)1955 4945 y Fm(m)2125 4921 y Fq(\()2176 4894 y(~)2163 4921 y Fo(\036)2221 4936 y Fm(m)p Fi(\000)p Fl(1)2377 4921 y Fq(\))2415 4885 y Fm(T)2503 4921 y Fo(;)17 b Fq(\()p Fo(\036)2643 4936 y Fm(m)2709 4921 y Fq(\))2747 4885 y Fm(T)2802 4921 y Fq(\))p 949 4965 1891 4 v 1518 5067 a(\(\()1607 5041 y(~)1594 5067 y Fo(\036)1652 5082 y Fm(m)p Fi(\000)p Fl(1)1809 5067 y Fq(\))1847 5039 y Fm(T)1934 5067 y Fo(;)g Fq(\()p Fo(\036)2074 5082 y Fm(m)2140 5067 y Fq(\))2178 5039 y Fm(T)2233 5067 y Fq(\))3222 4988 y(\(I)s(I.43\))257 5268 y(The)31 b(term)e(in)h(Eq.)g(\(I)s(I.42\))e(is)i(easily)h(con)m (trolled)f(b)m(y)g(a)f(quan)m(tit)m(y)i Fo(o)p Fq(\()p Fo(\013)2826 5232 y Fl(2+)p Fm(m)p Fl(+)p Fm(N)3100 5268 y Fq(\))e(b)s(ecause)257 5388 y(of)g(Estimate)h(\(I)s(I.15\),)g(b)s (ecause)g(of)f(the)h(\014rst)f(part)g(of)g(the)g(lemma)h(and)g(the)f (inequalit)m(y)p eop end %%Page: 25 26 TeXDict begin 25 25 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(25)257 573 y(in)38 b(Eq.)h(\(I)s(I.16\).)58 b(In)37 b(order)h(to)f(b)s(ound)h(the)g(term)g(in)g(Eq.)g(\(I)s (I.43\),)h(the)f(denominator)257 693 y(is)g(con)m(trolled)g(as)g(in)f (\(I)s(I.42\).)57 b(Concerning)38 b(the)g(n)m(umerator,)h(w)m(e)f (\014rst)g(rewrite)g(it)f(as)257 814 y(follo)m(ws)1349 1034 y(\()p Fo(W)1493 992 y Fm(m)p Fi(\000)p Fl(1)1479 1058 y Fm(m)1649 1034 y Fq(\()1700 1007 y(~)1687 1034 y Fo(\036)1745 1049 y Fm(m)p Fi(\000)p Fl(1)1924 1034 y Fn(\000)23 b Fq(\()2075 1007 y(~)2062 1034 y Fo(\036)2120 1049 y Fm(m)p Fi(\000)p Fl(1)2276 1034 y Fq(\))2314 992 y Fm(T)2369 1034 y Fq(\))32 b Fo(;)17 b(\036)2541 1049 y Fm(m)2607 1034 y Fq(\))577 b(\(I)s(I.44\))1190 1179 y(+)83 b(\()p Fo(W)1493 1138 y Fm(m)p Fi(\000)p Fl(1)1479 1204 y Fm(m)1649 1179 y Fq(\()1700 1153 y(~)1687 1179 y Fo(\036)1745 1194 y Fm(m)p Fi(\000)p Fl(1)1902 1179 y Fq(\))1940 1138 y Fm(T)2027 1179 y Fo(;)17 b(\036)2129 1194 y Fm(m)2217 1179 y Fn(\000)23 b Fq(\()p Fo(\036)2413 1194 y Fm(m)2479 1179 y Fq(\))2517 1138 y Fm(T)2572 1179 y Fq(\))p Fo(:)585 b Fq(\(I)s(I.45\))257 1399 y(W)-8 b(e)31 b(then)g(notice)g(that,)f(b)m(y)h(steps)h(similar)f(to)e(those)i (in)g(the)f(\014rst)h(part)f(of)g(the)g(lemma,)257 1519 y(the)42 b(absolute)f(v)-5 b(alue)42 b(of)e(the)h(\014rst)h(scalar)f (pro)s(duct,)i(\(I)s(I.44\),)g(can)e(b)s(e)g(con)m(trolled)h(in)257 1640 y(terms)34 b(of)1349 1812 y(1)p 1218 1856 311 4 v 1218 1948 a(1)22 b Fn(\000)h Fo(C)7 b(\013)1602 1879 y Fq(sup)1555 1958 y Fm(z)1588 1966 y Fb(m)1646 1958 y Fi(2)p Fl(\000)1737 1966 y Fb(m)1813 1879 y Fn(k)p Fq(\()1909 1847 y Fh(f)1901 1879 y Fo(H)1990 1838 y Fl(+)1982 1904 y Fm(m)p Fi(\000)p Fl(1)2160 1879 y Fn(\000)23 b Fo(z)2305 1894 y Fm(m)2372 1879 y Fq(\))2410 1838 y Fi(\000)2475 1811 y Ff(1)p 2474 1823 31 4 v 2474 1864 a(2)2519 1879 y Fo(W)2625 1838 y Fm(m)p Fi(\000)p Fl(1)2611 1904 y Fm(m)2794 1853 y Fq(~)2781 1879 y Fo(\036)2839 1894 y Fm(m)p Fi(\000)p Fl(1)2996 1879 y Fn(k)f(\002)419 2108 y(\002)65 b Fq(sup)513 2187 y Fm(z)546 2195 y Fb(m)605 2187 y Fi(2)p Fl(\000)696 2195 y Fb(m)771 2108 y Fn(k)p Fq(\()867 2076 y Fh(f)859 2108 y Fo(H)948 2067 y Fl(+)940 2133 y Fm(m)p Fi(\000)p Fl(1)1119 2108 y Fn(\000)22 b Fo(z)1263 2123 y Fm(m)1330 2108 y Fq(\))1368 2067 y Fi(\000)1433 2040 y Ff(1)p 1433 2052 V 1433 2093 a(2)1477 2108 y Fo(W)1583 2067 y Fm(m)p Fi(\000)p Fl(1)1569 2133 y Fm(m)1740 2108 y Fq(\()1791 2082 y(~)1778 2108 y Fo(\036)1836 2123 y Fm(m)p Fi(\000)p Fl(1)2014 2108 y Fn(\000)h Fq(\()2165 2082 y(~)2152 2108 y Fo(\036)2210 2123 y Fm(m)p Fi(\000)p Fl(1)2366 2108 y Fq(\))2404 2067 y Fm(T)2459 2108 y Fq(\))p Fn(k)28 b(\024)g Fo(o)p Fq(\()p Fo(\013)2828 2067 y Fl(2+)p Fm(m)p Fl(+)p Fm(N)3102 2108 y Fq(\))p Fo(:)55 b Fq(\(I)s(I.46\))257 2386 y(This)34 b(is)f(p)s(ossible)h(b)s(ecause)g(of)e(\(I)s(I.15\))g (and)h(pro)m(vided)1139 2606 y Fn(k)l Fo(~)-45 b(v)1236 2621 y Fm(m)p Fi(\000)p Fl(1)1393 2606 y Fq(\()p Fo(\036)1489 2621 y Fm(m)p Fi(\000)p Fl(1)1668 2606 y Fn(\000)22 b Fq(\()p Fo(\036)1863 2621 y Fm(m)p Fi(\000)p Fl(1)2020 2606 y Fq(\))2058 2565 y Fm(T)2113 2606 y Fq(\))p Fn(k)27 b(\024)h Fo(o)p Fq(\()p Fo(\013)2481 2565 y Fm(N)2548 2606 y Fq(\))p Fo(;)609 b Fq(\(I)s(I.47\))257 2826 y(whic)m(h)34 b(is)g(easy)f(to)f(pro)m(v)m(e,)i(b)m(y)g(construction)g(of)e(\()p Fo(\036)2160 2841 y Fm(m)p Fi(\000)p Fl(1)2316 2826 y Fq(\))2354 2790 y Fm(T)2409 2826 y Fq(.)257 2946 y(F)-8 b(or)32 b(the)h(scalar)g(pro)s(duct)g(\(I)s(I.45\),)f(the)h(b)s(ound) 1382 3166 y Fn(k)l Fo(~)-45 b(v)1479 3181 y Fm(m)p Fi(\000)p Fl(1)1636 3166 y Fq(\()p Fo(\036)1732 3181 y Fm(m)p Fi(\000)p Fl(1)1888 3166 y Fq(\))1926 3125 y Fm(T)1981 3166 y Fn(k)28 b(\024)g(O)s Fq(\(1\))851 b(\(I)s(I.48\))257 3386 y(ensures)51 b(the)e(desired)h(result,)k(\(I)s(I.20\),)e(after)c(applying)i(the)f (Sc)m(h)m(w)m(arz)h(inequalit)m(y)-8 b(.)p 3425 3507 4 66 v 3429 3444 59 4 v 3429 3507 V 3487 3507 4 66 v 404 3747 a(F)g(rom)32 b(Lemma)h(I)s(I.2,)g(w)m(e)g(easily)h(deriv)m(e)g (the)f(follo)m(wing)g(corollary)-8 b(.)257 3951 y Fe(Corollary)28 b(I)s(I.3.)34 b Fj(Assume)26 b(that)g Fq(\001)p Fn(E)1672 3855 y Fh(\020)1721 3951 y Fo( )1788 3915 y Fm(appr)n(:)1784 3975 y(m)1950 3951 y Fo(;)2014 3925 y Fq(~)1994 3951 y Fo( )2061 3904 y Fm(appr)n(:)2057 3973 y(m)p Fi(\000)p Fl(1)2223 3855 y Fh(\021)2298 3951 y Fj(is)g(wel)5 b(l)25 b(de\014ne)-5 b(d)24 b(for)i(the)g(ve)-5 b(ctors)257 4101 y(b)g(elonging)26 b(to)h(the)g(se)-5 b(quenc)g(es)26 b Fn(f)p Fo( )1481 4065 y Fm(appr)n(:)1477 4126 y(m)1643 4101 y Fn(g)h Fj(and)g Fn(f)1972 4075 y Fq(~)1952 4101 y Fo( )2019 4065 y Fm(appr)n(:)2015 4126 y(m)2208 4101 y Fq(:=)h Fo( )2406 4054 y Fm(appr)n(:)2402 4123 y(m)p Fi(\000)p Fl(1)2573 4101 y Fn(\012)5 b Fq(\012)2725 4065 y Fm(m)p Fi(\000)p Fl(1)2725 4126 y Fm(m)2883 4101 y Fn(g)p Fj(,)28 b(r)-5 b(esp)g(e)g(ctively,)257 4231 y(with)45 b Fo( )546 4195 y Fm(appr)n(:)542 4255 y(m)754 4231 y Fn(2)i(H)951 4246 y Fm(m)1062 4231 y Fj(and)d Fn(j)p Fq(\()1347 4204 y(~)1327 4231 y Fo( )1394 4184 y Fm(appr)n(:)1390 4253 y(m)p Fi(\000)p Fl(1)1556 4231 y Fo(;)17 b( )1667 4195 y Fm(appr)n(:)1663 4255 y(m)1829 4231 y Fq(\))p Fn(j)46 b Fo(>)f(\016)50 b(>)c Fq(0)f Fj(wher)-5 b(e)44 b Fo(\016)49 b Fj(is)44 b Fo(\013)q Fj(-indep)-5 b(endent.)257 4351 y(Assume)45 b(that)f(ine)-5 b(qualities)44 b(as)f(in)h(Eqs.)g (\(II.15\),\(II.46\))d(and)i(\(II.48\))g(ar)-5 b(e)44 b(ful\014l)5 b(le)-5 b(d)257 4472 y(when)34 b Fq(\()p Fo(\036)602 4487 y Fm(m)669 4472 y Fq(\))707 4435 y Fm(T)797 4472 y Fj(is)g(r)-5 b(eplac)g(e)g(d)34 b(by)h Fo( )1465 4435 y Fm(appr)n(:)1461 4496 y(m)1627 4472 y Fj(,)g(then)981 4705 y Fn(j)p Fo(E)1087 4664 y Fm(m)p Fi(\000)p Fl(1)1081 4729 y Fm(m)1265 4705 y Fn(\000)23 b Fq(\001)p Fn(E)1508 4608 y Fh(\020)1557 4705 y Fo( )1624 4664 y Fm(appr)n(:)1620 4729 y(m)1787 4705 y Fo(;)1850 4678 y Fq(~)1831 4705 y Fo( )1898 4658 y Fm(appr)n(:)1894 4727 y(m)p Fi(\000)p Fl(1)2059 4608 y Fh(\021)2109 4705 y Fn(j)k(\024)h Fo(o)p Fq(\()p Fo(\013)2427 4636 y Ff(1)p 2427 4648 31 4 v 2427 4690 a(2)2467 4664 y Fl(+)p Fm(m)p Fl(+)p Fm(N)2707 4705 y Fq(\))p Fo(:)450 b Fq(\(I)s(I.49\))p eop end %%Page: 26 27 TeXDict begin 26 26 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(26)257 573 y Fp(I)t(I)t(I)163 b(Re-expansion)257 792 y Fq(The)36 b(truncated)f(v)m(ectors)h Fn(f)p Fq(\()p Fo(\036)1385 807 y Fm(m)1451 792 y Fq(\))1489 756 y Fm(T)1574 792 y Fq(:)31 b(1)f Fn(\024)h Fo(m)g Fn(\024)g Fq(2)p Fo(N)j Fn(\000)24 b Fq(1)p Fn(g)33 b Fq(are)i(written)g(in)f(terms)i(of)d(a)257 912 y(\014nite)40 b(n)m(um)m(b)s(er)g(of)e(resolv)m(en)m(ts)k(of)c(the)h(op)s(erators)f Fn(f)p Fo(H)2313 876 y Fl(+)2305 937 y Fm(m)2410 912 y Fq(:)55 b(0)38 b Fn(\024)g Fo(m)h Fn(\024)f Fq(2)p Fo(N)f Fn(\000)27 b Fq(2)p Fn(g)38 b Fq(and)257 1033 y(in)m(teraction)31 b(terms)g Fn(f)p Fo(W)1170 991 y Fm(m)1232 968 y Fd(0)1254 991 y Fi(\000)p Fl(1)1156 1058 y Fm(m)1218 1039 y Fd(0)1376 1033 y Fq(:)44 b(1)28 b Fn(\024)g Fo(m)1714 997 y Fi(0)1765 1033 y Fn(\024)g Fq(2)p Fo(N)f Fn(\000)16 b Fq(1)p Fn(g)p Fq(,)31 b(where)g Fo(N)40 b Fq(is)30 b(the)h(order)e(in)h Fo(\013)h Fq(up)257 1153 y(to)c(whic)m(h)h(w)m(e)g(in)m(tend)g(to)e(re-expand.)43 b(The)27 b(re-expansion)h(consists)h(in)e(expressing)i(the)257 1273 y(resolv)m(en)m(ts)j(of)d(the)g(op)s(erators)g Fn(f)p Fo(H)1541 1237 y Fl(+)1533 1298 y Fm(m)1627 1273 y Fq(:)44 b(0)28 b Fn(\024)g Fo(m)g Fn(\024)g Fq(2)p Fo(N)e Fn(\000)15 b Fq(2)p Fn(g)29 b Fq(in)g(the)g(original)g(de\014nition)257 1394 y(\(De\014nition)45 b(I)s(I.1\))f(of)f Fn(f)p Fq(\()p Fo(\036)1260 1409 y Fm(m)1326 1394 y Fq(\))1364 1358 y Fm(T)1419 1394 y Fn(g)h Fq(in)g(terms)h(of)f(sums)h(of)f(pro)s(duct)g (of)g(\014nitely)h(man)m(y)257 1528 y(resolv)m(en)m(ts)36 b(of)c Fo(H)906 1487 y Fl(+)898 1550 y(0)964 1528 y Fq(,)h(b)s(oson)g (creation)g(op)s(erators,)f(the)h(op)s(erators)g Fo(~)-49 b(p)32 b Fq(and)h Fo(e)3052 1492 y Fi(\006)p Fm(i)3128 1475 y(~)3131 1492 y(k)r Fi(\001)p Fm(\013)o(~)-34 b(x)3278 1528 y Fq(,)33 b(and)257 1649 y(explicitly)27 b(computable)f(quan)m (tities,)i(up)d(to)g(an)f(error)h(of)f(order)h Fo(\013)2677 1613 y Fm(N)2744 1649 y Fq(.)41 b(The)26 b(dep)s(endence)257 1769 y(of)i(eac)m(h)g(v)m(ector)h(\()p Fo(\036)962 1784 y Fm(m)1028 1769 y Fq(\))1066 1733 y Fm(T)1149 1769 y Fq(on)e(the)i(previous)g(one,)g(\()p Fo(\036)2124 1784 y Fm(m)p Fi(\000)p Fl(1)2280 1769 y Fq(\))2318 1733 y Fm(T)2373 1769 y Fq(,)g(as)f(displa)m(y)m(ed)i(in)e(De\014nition)257 1889 y(I)s(I.1,)h(links)f(the)g(re-expansion)g(of)f(\()p Fo(\036)1613 1904 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1805 1889 y Fq(\))1843 1853 y Fm(T)1925 1889 y Fq(to)27 b(all)g(the)g (re-expanded)i(expressions)h(for)257 2010 y Fn(f)p Fq(\()p Fo(\036)403 2025 y Fm(m)469 2010 y Fq(\))507 1974 y Fm(T)599 2010 y Fq(:)54 b(1)36 b Fn(\024)i Fo(m)f Fn(\024)g Fq(2)p Fo(N)f Fn(\000)26 b Fq(2)p Fn(g)p Fq(.)59 b(The)39 b(strategy)g(to)e (cop)s(e)i(with)f(all)g(these)h(features,)257 2130 y(whic)m(h)30 b(has)f(b)s(een)g(previously)h(outlined,)g(is)f(dev)m(elop)s(ed)h(in)e (detail)h(in)f(this)h(Section.)43 b(T)-8 b(o)257 2251 y(b)s(egin)32 b(with,)g(w)m(e)g(pro)m(vide)g(some)g(further)g (necessary)h(notations.)43 b(W)-8 b(e)32 b(assume)h(that)e Fo(\013)257 2371 y Fq(is)i(small)h(enough)f(to)f(ensure)i(all)e(the)h (results)h(stated)g(in)e(the)h(previous)h(sections.)257 2651 y Fc(I)t(I)t(I.1)135 b(Notations)47 b(and)d(de\014nitions)257 2836 y Fq(W)-8 b(e)29 b(use)h(planar)e(p)s(olar)h(co)s(ordinates)g(to)f (parameterizes)i(our)f(in)m(tegrations.)43 b(Eac)m(h)29 b Fo(z)3401 2851 y Fm(m)3468 2836 y Fq(,)257 2956 y Fo(m)f Fn(\025)h Fq(1,)j(is)h(written)g(as)f Fo(\032)1192 2971 y Fm(m)1259 2956 y Fo(e)1304 2920 y Fm(i\022)1400 2956 y Fq(with)h Fo(\032)1672 2971 y Fm(m)1766 2956 y Fq(=)28 b Fn(j)p Fo(z)1943 2971 y Fm(m)2009 2956 y Fn(j)k Fq(o)m(v)m(er)i(the)f (con)m(tour)f(in)m(tegral)h(\000)3216 2971 y Fm(m)3283 2956 y Fq(.)43 b(F)-8 b(or)257 3077 y Fo(m)28 b Fn(\025)h Fq(2,)j(w)m(e)i(also)e(de\014ne)1219 3239 y Fo(z)1268 3188 y Fl(\()p Fm(m)p Fl(\))1264 3261 y Fm(m)p Fi(\000)p Fl(1)1449 3239 y Fq(:=)c Fo(z)1629 3198 y Fl(\()p Fm(m)p Fl(\))1625 3264 y Fm(m)1773 3239 y Fq(+)22 b Fo(\032)1921 3254 y Fm(m)p Fi(\000)p Fl(1)2078 3239 y Fo(e)2123 3198 y Fm(i\022)2209 3239 y Fn(\000)g Fo(\032)2358 3254 y Fm(m)2425 3239 y Fo(e)2470 3198 y Fm(i\022)3232 3239 y Fq(\(I)s(I)s(I.1\))257 3419 y(where)35 b Fo(z)589 3383 y Fl(\()p Fm(m)p Fl(\))585 3443 y Fm(m)740 3419 y Fn(\021)30 b Fo(z)892 3434 y Fm(m)959 3419 y Fq(;)k(the)g(sup)s(erscript)h(\()p Fo(m)p Fq(\))f(in)f Fo(z)2050 3368 y Fl(\()p Fm(m)p Fl(\))2046 3440 y Fm(m)p Fi(\000)p Fl(1)2237 3419 y Fq(is)h(a)f(lab)s(el)g(to)h(k) m(eep)h(trac)m(k)f(of)f(the)257 3539 y(scale)44 b(of)f(the)g(original)g (in)m(tegration)g(v)-5 b(ariable,)46 b Fo(z)2121 3554 y Fm(m)2188 3539 y Fq(,)g(from)d(whic)m(h)h(it)f(deriv)m(es)i(in)f(the) 257 3659 y(re-expansion)34 b(to)e(b)s(e)h(describ)s(ed.)46 b(The)33 b(general)g(de\014nition)h(is:)1061 3822 y Fo(z)1110 3771 y Fl(\()p Fm(m)p Fl(\))1106 3845 y Fm(m)p Fi(\000)p Fm(j)t Fi(\000)p Fl(1)1378 3822 y Fq(:=)28 b Fo(z)1558 3771 y Fl(\()p Fm(m)p Fl(\))1554 3845 y Fm(m)p Fi(\000)p Fm(j)1730 3822 y Fq(+)22 b Fo(\032)1878 3837 y Fm(m)p Fi(\000)p Fm(j)t Fi(\000)p Fl(1)2122 3822 y Fo(e)2167 3781 y Fm(i\022)2253 3822 y Fn(\000)g Fo(\032)2402 3837 y Fm(m)p Fi(\000)p Fm(j)2556 3822 y Fo(e)2601 3781 y Fm(i\022)2665 3822 y Fo(:)540 b Fq(\(I)s(I)s(I.2\))257 3984 y(for)35 b(0)d Fn(\024)h Fo(j)39 b Fn(\024)33 b Fo(m)24 b Fn(\000)h Fq(2.)51 b(The)37 b(basic)f(ob)5 b(jects)36 b(featured)g(in)g(our)f(re-expansion)i(come)f(in)257 4104 y(t)m(w)m(o)e(v)-5 b(arieties:)349 4258 y(\(i\))49 b(Resolv)m(en)m(ts)35 b(of)d(the)h(Hamiltonians)h Fo(H)1951 4217 y Fl(+)1943 4280 y Fm(m)p Fi(\000)p Fl(2)2132 4258 y Fq(and)2330 4226 y Fh(f)2322 4258 y Fo(H)2411 4217 y Fl(+)2403 4280 y Fm(m)p Fi(\000)p Fl(3)774 4469 y Fo(R)849 4428 y Fl(+)848 4493 y Fm(m)p Fi(\000)p Fl(2)1005 4469 y Fq(\()p Fo(z)1092 4418 y Fl(\()p Fm(m)1181 4394 y Fd(0)1205 4418 y Fl(\))1088 4490 y Fm(m)p Fi(\000)p Fl(1)1267 4469 y Fn(\000)23 b Fo(S)1427 4484 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1660 4469 y Fq(\()p Fo(k)s Fq(\)\))83 b(:=)2635 4401 y(1)p 2107 4445 1105 4 v 2107 4556 a Fo(H)2196 4515 y Fl(+)2188 4578 y Fm(m)p Fi(\000)p Fl(2)2367 4556 y Fn(\000)23 b Fo(z)2516 4505 y Fl(\()p Fm(m)2605 4486 y Fd(0)2629 4505 y Fl(\))2512 4578 y Fm(m)p Fi(\000)p Fl(1)2691 4556 y Fq(+)f Fo(S)2849 4572 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))3082 4556 y Fq(\()p Fo(k)s Fq(\))3232 4469 y(\(I)s(I)s(I.3\))624 4753 y Fo(R)699 4712 y Fl(+)698 4778 y Fm(m)p Fi(\000)p Fl(2)855 4753 y Fq(\()p Fo(z)942 4712 y Fl(\()p Fm(m)1031 4689 y Fd(0)1055 4712 y Fl(\))938 4778 y Fm(m)1109 4753 y Fn(\000)g Fo(S)1268 4769 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1660 4753 y Fq(\()p Fo(k)s Fq(\)\))83 b(:=)2711 4686 y(1)p 2107 4730 1256 4 v 2107 4841 a Fo(H)2196 4799 y Fl(+)2188 4862 y Fm(m)p Fi(\000)p Fl(2)2367 4841 y Fn(\000)23 b Fo(z)2516 4790 y Fl(\()p Fm(m)2605 4771 y Fd(0)2629 4790 y Fl(\))2512 4848 y Fm(m)2682 4841 y Fq(+)f Fo(S)2840 4856 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))3233 4841 y Fq(\()p Fo(k)s Fq(\))3232 4753 y(\(I)s(I)s(I.4\))1342 5005 y Fh(e)1319 5037 y Fo(R)1394 4996 y Fl(+)1393 5062 y Fm(m)p Fi(\000)p Fl(3)1550 5037 y Fq(\()p Fo(z)1637 4987 y Fl(\()p Fm(m)1726 4963 y Fd(0)1750 4987 y Fl(\))1633 5059 y Fm(m)p Fi(\000)p Fl(2)1790 5037 y Fq(\))83 b(:=)2364 4970 y(1)p 2107 5014 562 4 v 2115 5093 a Fh(f)2107 5125 y Fo(H)2196 5084 y Fl(+)2188 5147 y Fm(m)p Fi(\000)p Fl(3)2367 5125 y Fn(\000)23 b Fo(z)2516 5074 y Fl(\()p Fm(m)2605 5055 y Fd(0)2629 5074 y Fl(\))2512 5147 y Fm(m)p Fi(\000)p Fl(2)3232 5037 y Fq(\(I)s(I)s(I.5\))797 5286 y Fh(e)774 5318 y Fo(R)849 5277 y Fl(+)848 5343 y Fm(m)p Fi(\000)p Fl(3)1005 5318 y Fq(\()p Fo(z)1092 5267 y Fl(\()p Fm(m)1181 5244 y Fd(0)1205 5267 y Fl(\))1088 5340 y Fm(m)p Fi(\000)p Fl(1)1267 5318 y Fn(\000)g Fo(S)1427 5334 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1660 5318 y Fq(\()p Fo(k)s Fq(\)\))83 b(:=)2635 5251 y(1)p 2107 5295 1105 4 v 2115 5374 a Fh(f)2107 5406 y Fo(H)2196 5365 y Fl(+)2188 5428 y Fm(m)p Fi(\000)p Fl(3)2367 5406 y Fn(\000)23 b Fo(z)2516 5355 y Fl(\()p Fm(m)2605 5336 y Fd(0)2629 5355 y Fl(\))2512 5428 y Fm(m)p Fi(\000)p Fl(1)2691 5406 y Fq(+)f Fo(S)2849 5421 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))3082 5406 y Fq(\()p Fo(k)s Fq(\))3232 5318 y(\(I)s(I)s(I.6\))p eop end %%Page: 27 28 TeXDict begin 27 27 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(27)501 573 y(with)36 b Fo(m)811 537 y Fi(0)867 573 y Fn(\025)c Fo(m)p Fq(,)37 b Fo(m)32 b Fn(\025)h Fq(2)i(in)g(Eqs.)i(\(I)s(I)s(I.3\))e(and)g(\(I)s(I)s (I.4\),)g Fo(m)e Fn(\025)g Fq(3)h(in)i(Eqs.)g(\(I)s(I)s(I.5\))501 693 y(and)d(\(I)s(I)s(I.6\),)f(where)714 1001 y Fo(S)774 1017 y Fm(l)q Fl(\()p Fm(m)p Fl(\))918 1001 y Fq(\()p Fo(k)s Fq(\)\))27 b(:=)1244 884 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1253 918 y Fh(X)1256 1101 y Fm(i)p Fl(=1)1400 1001 y Fn(j)p Fo(k)1479 1016 y Fm(i)1506 1001 y Fq(\()p Fo(m)p Fq(\))p Fn(j)293 b Fo(k)30 b Fq(=)e(\()p Fo(k)2262 1016 y Fl(1)2301 1001 y Fq(\()p Fo(m)p Fq(\))p Fo(;)17 b Fn(\001)22 b(\001)g(\001)p Fo(;)17 b(k)2729 1016 y Fm(l)t(m)2820 1001 y Fq(\()p Fo(m)p Fq(\)\))213 b(\(I)s(I)s(I.7\))501 1492 y Fo(S)561 1507 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))953 1492 y Fq(\()p Fo(k)s Fq(\))83 b(:=)1352 1375 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1362 1409 y Fh(X)1365 1591 y Fm(i)p Fl(=1)1508 1492 y Fn(j)p Fo(k)1587 1507 y Fm(i)1615 1492 y Fq(\()p Fo(m)p Fq(\))p Fn(j)22 b Fq(+)1924 1375 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1979 1409 y Fh(X)2026 1591 y Fm(i)2170 1492 y Fn(j)p Fo(k)2249 1507 y Fm(i)2276 1492 y Fq(\()p Fo(m)h Fn(\000)f Fq(1\))p Fn(j)731 b Fq(\(I)s(I)s(I.8\))1352 1707 y Fo(k)31 b Fq(=)d(\()p Fo(k)1627 1722 y Fl(1)1666 1707 y Fq(\()p Fo(m)p Fq(\))p Fo(;)17 b Fn(\001)22 b(\001)f(\001)p Fo(;)c(k)2093 1722 y Fm(l)t(m)2185 1707 y Fq(\()p Fo(m)p Fq(\))p Fo(;)g(k)2441 1722 y Fl(1)2480 1707 y Fq(\()p Fo(m)23 b Fn(\000)f Fq(1\))p Fo(;)17 b Fn(\001)k(\001)h(\001)p Fo(;)17 b(k)3078 1722 y Fm(l)t(m)p Fi(\000)p Fl(1)3260 1707 y Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\)\))501 1927 y(and)h(our)g(notation)f(for)g(the)h(photon)g(momen)m(ta,)i Fo(k)2345 1942 y Fm(i)2373 1927 y Fq(\()p Fo(m)p Fq(\),)g Fo(k)2638 1942 y Fm(i)2666 1927 y Fq(\()p Fo(m)t Fn(\000)t Fq(1\),)g(are)d(referred)501 2048 y(to)32 b(the)h(momen)m(tum-space)i (shells)g Fn(K)1900 2011 y Fm(m)p Fi(\000)p Fl(1)1899 2072 y Fm(m)2057 2048 y Fq(,)d Fn(K)2193 2006 y Fm(m)p Fi(\000)p Fl(2)2192 2069 y Fm(m)p Fi(\000)p Fl(1)2350 2048 y Fq(,)h(resp)s(ectiv)m(ely:)708 2268 y Fo(\033)763 2283 y Fm(m)858 2268 y Fn(\024)28 b(j)p Fo(k)1042 2283 y Fm(i)1070 2268 y Fq(\()p Fo(m)p Fq(\))p Fn(j)f(\024)h Fo(\033)1446 2283 y Fm(m)p Fi(\000)p Fl(1)1701 2268 y Fo(;)114 b(\033)1897 2283 y Fm(m)p Fi(\000)p Fl(1)2082 2268 y Fn(\024)28 b(j)p Fo(k)2266 2283 y Fm(i)2294 2268 y Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\))p Fn(j)k(\024)h Fo(\033)2841 2283 y Fm(m)p Fi(\000)p Fl(2)2998 2268 y Fo(:)207 b Fq(\(I)s(I)s(I.9\))501 2488 y(The)32 b(non-negativ)m(e)g(n)m (um)m(b)s(ers)h Fo(l)r Fq(\()p Fo(m)p Fq(\),)f Fo(l)r Fq(\()p Fo(m)19 b Fn(\000)h Fq(1\))31 b(of)f(terms)i(in)f(the)h(sums)g (\(I)s(I)s(I.7\),)501 2608 y(\(I)s(I)s(I.8\))j(are)g(b)s(ounded)h(and)f Fo(\013)q Fq(-indep)s(enden)m(t;)i Fo(l)r Fq(\()p Fo(m)p Fq(\))32 b(=)g(0,)j Fo(l)r Fq(\()p Fo(m)25 b Fn(\000)f Fq(1\))31 b(=)h(0)j(mean)501 2728 y(no)49 b(sums.)92 b(The)49 b(lo)m(w)m(er)h(b)s(ounds)f(in)f(Eq.)i(\(I)s(I)s(I.9\))e(and)g (our)g(b)s(ounds)h(on)g(the)501 2849 y(energy)33 b(gaps)f(in)h Fo(\033)t Fq(\()p Fo(H)1334 2808 y Fl(+)1326 2871 y Fm(m)p Fi(\000)p Fl(2)1482 2849 y Fq(\),)f Fo(\033)t Fq(\()1684 2816 y Fh(f)1676 2849 y Fo(H)1765 2808 y Fl(+)1757 2871 y Fm(m)p Fi(\000)p Fl(3)1914 2849 y Fq(\))g(ab)s(o)m(v)m(e)h(the)f (groundstate)h(energies)g(\(see)501 2969 y(Eqs.)24 b(\(I.52\),)g (\(I.53\)\))e(ensure)h(that)f(the)h(op)s(erators)f(\(I)s(I)s(I.3\),\(I) s(I)s(I.4\),\(I)s(I)s(I.5\),\(I)s(I)s(I.6\))501 3090 y(are)33 b(b)s(ounded.)322 3293 y(\(ii\))49 b(The)34 b(scale)f(dep)s(enden)m(t)i(in)m(teraction)e(v)m(ertexes)1077 3568 y(\007)1153 3527 y Fl(+)1153 3593 y Fm(m)1219 3568 y Fq(\()p Fo(k)1308 3583 y Fm(i)1336 3568 y Fq(\()p Fo(m)1459 3527 y Fi(0)1483 3568 y Fq(\)\))83 b(:=)c Fo(~)-45 b(v)1875 3583 y Fm(m)1964 3568 y Fn(\001)2055 3500 y Fo(~)d(")p Fq(\()p Fo(k)2191 3515 y Fm(i)2219 3500 y Fq(\()p Fo(m)2342 3464 y Fi(0)2365 3500 y Fq(\)\))p 2023 3545 451 4 v 2023 3561 a Fh(q)p 2106 3561 368 4 v 100 x Fq(2)p Fn(j)p Fo(k)2234 3676 y Fm(i)2262 3661 y Fq(\()p Fo(m)2385 3632 y Fi(0)2408 3661 y Fq(\))p Fn(j)2484 3568 y Fo(e)2529 3527 y Fi(\000)p Fm(i\013)2650 3509 y(~)2653 3527 y(k)2690 3537 y Fb(i)2716 3527 y Fl(\()p Fm(m)2805 3503 y Fd(0)2829 3527 y Fl(\))p Fi(\001)o Fm(~)-34 b(x)3184 3568 y Fq(\(I)s(I)s(I.10\))1077 3822 y(\007)1153 3781 y Fi(\000)1153 3846 y Fm(m)1219 3822 y Fq(\()p Fo(k)1308 3837 y Fm(i)1336 3822 y Fq(\()p Fo(m)1459 3781 y Fi(0)1483 3822 y Fq(\)\))83 b(:=)g(\(\007)1942 3781 y Fl(+)1942 3846 y Fm(m)2008 3822 y Fq(\()p Fo(k)2097 3837 y Fm(i)2125 3822 y Fq(\()p Fo(m)2248 3781 y Fi(0)2271 3822 y Fq(\)\))2347 3781 y Fi(\003)3184 3822 y Fq(\(I)s(I)s(I.11\))501 4141 y(\000)562 4100 y Fl(+)p Fm(;)p Fl(+)696 4141 y Fq(\()p Fo(k)785 4156 y Fm(i)813 4141 y Fq(\()p Fo(m)936 4100 y Fi(0)959 4141 y Fq(\))p Fo(;)17 b(k)1092 4156 y Fm(j)1128 4141 y Fq(\()p Fo(m)1251 4100 y Fi(00)1294 4141 y Fq(\)\))96 b(=)h(\(\000)1738 4100 y Fi(\000)p Fm(;)p Fi(\000)1871 4141 y Fq(\()p Fo(k)1960 4156 y Fm(i)1988 4141 y Fq(\()p Fo(m)2111 4100 y Fi(0)2134 4141 y Fq(\))p Fo(;)17 b(k)2267 4156 y Fm(j)2303 4141 y Fq(\()p Fo(m)2426 4100 y Fi(00)2469 4141 y Fq(\)\)\))2583 4100 y Fi(\003)3263 4141 y Fq(\(I)s(I)s(I.12\))1453 4345 y(:=)1680 4277 y Fo(~)-48 b(")p Fq(\()p Fo(k)1816 4292 y Fm(i)1844 4277 y Fq(\()p Fo(m)1967 4241 y Fi(0)1990 4277 y Fq(\)\))p 1649 4321 451 4 v 1649 4338 a Fh(q)p 1732 4338 368 4 v 100 x Fq(2)p Fn(j)p Fo(k)1860 4453 y Fm(i)1887 4438 y Fq(\()p Fo(m)2010 4409 y Fi(0)2034 4438 y Fq(\))p Fn(j)2131 4345 y(\001)2223 4277 y Fo(~)g(")o Fq(\()p Fo(k)2358 4292 y Fm(j)2395 4277 y Fq(\()p Fo(m)2518 4241 y Fi(00)2560 4277 y Fq(\)\))p 2191 4321 479 4 v 2191 4338 a Fh(q)p 2274 4338 396 4 v 99 x Fq(2)p Fn(j)p Fo(k)2402 4452 y Fm(j)2438 4437 y Fq(\()p Fo(m)2561 4408 y Fi(00)2603 4437 y Fq(\))p Fn(j)2679 4345 y Fo(e)2724 4304 y Fi(\000)p Fm(i\013)2845 4286 y(~)2848 4304 y(k)2885 4314 y Fb(i)2912 4304 y Fl(\()p Fm(m)3001 4280 y Fd(0)3024 4304 y Fl(\))p Fi(\001)o Fm(~)-34 b(x)3115 4345 y Fo(e)3160 4304 y Fi(\000)p Fm(i\013)3281 4286 y(~)3284 4304 y(k)3321 4314 y Fb(j)3353 4304 y Fl(\()p Fm(m)3442 4280 y Fd(00)3484 4304 y Fl(\))p Fi(\001)o Fm(~)g(x)501 4773 y Fq(\000)562 4732 y Fl(+)p Fm(;)p Fi(\000)696 4773 y Fq(\()p Fo(k)785 4788 y Fm(i)813 4773 y Fq(\()p Fo(m)936 4732 y Fi(0)959 4773 y Fq(\))p Fo(;)17 b(k)1092 4788 y Fm(j)1128 4773 y Fq(\()p Fo(m)1251 4732 y Fi(00)1294 4773 y Fq(\)\))96 b(=)h(\(\000)1738 4732 y Fi(\000)p Fm(;)p Fl(+)1871 4773 y Fq(\()p Fo(k)1960 4788 y Fm(i)1988 4773 y Fq(\()p Fo(m)2111 4732 y Fi(0)2134 4773 y Fq(\))p Fo(;)17 b(k)2267 4788 y Fm(j)2303 4773 y Fq(\()p Fo(m)2426 4732 y Fi(00)2469 4773 y Fq(\)\)\))2583 4732 y Fi(\003)3313 4773 y Fq(\(I)s(I)s(I.13\))1453 4976 y(:=)1680 4909 y Fo(~)-48 b(")p Fq(\()p Fo(k)1816 4924 y Fm(i)1844 4909 y Fq(\()p Fo(m)1967 4873 y Fi(0)1990 4909 y Fq(\)\))p 1649 4953 451 4 v 1649 4969 a Fh(q)p 1732 4969 368 4 v 101 x Fq(2)p Fn(j)p Fo(k)1860 5085 y Fm(i)1887 5070 y Fq(\()p Fo(m)2010 5041 y Fi(0)2034 5070 y Fq(\))p Fn(j)2131 4976 y(\001)2191 4909 y Fq(\()o Fo(~)g(")p Fq(\()p Fo(k)2364 4924 y Fm(j)2400 4909 y Fq(\()p Fo(m)2523 4873 y Fi(00)2565 4909 y Fq(\)\)\))2679 4873 y Fi(\003)p 2191 4953 528 4 v 2257 5044 a Fq(2)p Fn(j)p Fo(k)2385 5059 y Fm(j)2421 5044 y Fq(\()p Fo(m)2544 5016 y Fi(00)2587 5044 y Fq(\))p Fn(j)2729 4976 y Fo(e)2774 4935 y Fi(\000)p Fm(i\013)2895 4917 y(~)2898 4935 y(k)2935 4945 y Fb(i)2961 4935 y Fl(\()p Fm(m)3050 4912 y Fd(0)3073 4935 y Fl(\))p Fi(\001)o Fm(~)-34 b(x)3164 4976 y Fo(e)3209 4935 y Fl(+)p Fm(i\013)3330 4917 y(~)3333 4935 y(k)3370 4945 y Fb(j)3403 4935 y Fl(\()p Fm(m)3492 4912 y Fd(00)3534 4935 y Fl(\))p Fi(\001)o Fm(~)g(x)501 5297 y Fq(with)33 b Fo(m)28 b Fn(\025)h Fq(0,)j Fo(m)1135 5260 y Fi(0)1159 5297 y Fo(;)17 b(m)1288 5260 y Fi(00)1358 5297 y Fn(\025)28 b Fq(0.)p eop end %%Page: 28 29 TeXDict begin 28 28 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(28)257 573 y Fc(I)t(I)t(I.2)135 b(T)-11 b(runcation)45 b(map)257 758 y Fq(As)k(outlined)g(in)g(Section) g(I,)f(the)h(strategy)f(of)g(the)h(re-expansion)g(is)g(to)f(split)h (the)257 878 y(resolv)m(en)m(ts)40 b(\(I)s(I)s(I.3\))e(and)f(\(I)s(I)s (I.4\))g(in)m(to)h(t)m(w)m(o)g(parts,)h(resp)s(ectiv)m(ely:)57 b(a)37 b(parallel)h(and)f(an)257 998 y(orthogonal)g(one)i(to)e(the)h (sp)s(ectral)h(pro)5 b(jection)39 b Fo(P)2136 1013 y Fm(m)p Fi(\000)p Fl(2)2292 998 y Fq(.)60 b(Since,)40 b(as)e(an)g(op)s(erator,)h(the)257 1119 y(parallel)33 b(part)g(is)g(just)g(the)g(pro)5 b(jection)33 b Fo(P)1801 1134 y Fm(m)p Fi(\000)p Fl(2)1990 1119 y Fq(times)h(an)e(explicit)i(n)m (um)m(b)s(er)578 1342 y Fo(R)653 1291 y Fl(+)12 b Fi(k)652 1364 y Fm(m)p Fi(\000)p Fl(2)809 1342 y Fq(\()p Fo(z)896 1291 y Fl(\()p Fm(m)985 1268 y Fd(0)1009 1291 y Fl(\))892 1364 y Fm(m)p Fi(\000)p Fl(1)1071 1342 y Fn(\000)23 b Fo(S)1231 1358 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1464 1342 y Fq(\()p Fo(k)s Fq(\)\))83 b(:=)g Fo(R)1976 1301 y Fl(+)1975 1367 y Fm(m)p Fi(\000)p Fl(2)2132 1342 y Fq(\()p Fo(z)2219 1291 y Fl(\()p Fm(m)2308 1268 y Fd(0)2332 1291 y Fl(\))2215 1364 y Fm(m)p Fi(\000)p Fl(1)2394 1342 y Fn(\000)23 b Fo(S)2554 1358 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2787 1342 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)3018 1357 y Fm(m)p Fi(\000)p Fl(2)3184 1342 y Fq(\(I)s(I)s(I.14\)) 1729 1509 y(=)96 b(\()p Fo(S)1999 1525 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2232 1509 y Fq(\()p Fo(k)s Fq(\))22 b Fn(\000)h Fo(z)2533 1459 y Fl(\()p Fm(m)2622 1435 y Fd(0)2646 1459 y Fl(\))2529 1531 y Fm(m)p Fi(\000)p Fl(1)2686 1509 y Fq(\))2724 1468 y Fi(\000)p Fl(1)2818 1509 y Fo(P)2881 1524 y Fm(m)p Fi(\000)p Fl(2)281 1833 y Fo(R)356 1782 y Fl(+)12 b Fi(k)355 1855 y Fm(m)p Fi(\000)p Fl(2)512 1833 y Fq(\()p Fo(z)599 1792 y Fl(\()p Fm(m)688 1769 y Fd(0)712 1792 y Fl(\))595 1858 y Fm(m)766 1833 y Fn(\000)23 b Fo(S)926 1849 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1318 1833 y Fq(\()p Fo(k)s Fq(\)\))83 b(:=)g Fo(R)1830 1792 y Fl(+)1829 1858 y Fm(m)p Fi(\000)p Fl(2)1986 1833 y Fq(\()p Fo(z)2073 1792 y Fl(\()p Fm(m)2162 1769 y Fd(0)2186 1792 y Fl(\))2069 1858 y Fm(m)2239 1833 y Fn(\000)23 b Fo(S)2399 1849 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2791 1833 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)3022 1848 y Fm(m)p Fi(\000)p Fl(2)3184 1833 y Fq(\(I)s(I)s(I.15\))1755 1991 y(=)k(\()p Fo(S)1956 2006 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2348 1991 y Fq(\()p Fo(k)s Fq(\))22 b Fn(\000)h Fo(z)2649 1950 y Fl(\()p Fm(m)2738 1926 y Fd(0)2762 1950 y Fl(\))2645 2015 y Fm(m)2794 1991 y Fq(\))2832 1950 y Fi(\000)p Fl(1)2926 1991 y Fo(P)2989 2006 y Fm(m)p Fi(\000)p Fl(2)257 2202 y Fq(our)j(expansion)g(for)f(the) h(ground)f(state)h(v)m(ector)h(\()p Fo(\036)2109 2217 y Fm(m)2171 2198 y Fd(0)2197 2202 y Fq(\))2235 2166 y Fm(T)2290 2202 y Fq(,)g(re-expanded)g(on)e(scale)i Fo(m)7 b Fn(\000)g Fq(2,)257 2323 y(2)29 b Fn(\024)g Fo(m)h Fn(\024)f Fo(m)747 2286 y Fi(0)800 2323 y Fn(\024)g Fq(2)p Fo(N)k Fn(\000)23 b Fq(1,)33 b(will)h(con)m(tain,)h(as)e(far)g(as)g (resolv)m(en)m(ts)j(and)e(pro)5 b(jections)34 b(are)257 2443 y(concerned,)h(pro)s(ducts)e(of)f(the)h(follo)m(wing)g(elemen)m (tary)i(op)s(erators:)986 2775 y Fo(P)1049 2790 y Fm(m)p Fi(\000)p Fl(2)3184 2775 y Fq(\(I)s(I)s(I.16\))986 2935 y Fo(R)1061 2894 y Fl(+)1060 2960 y Fm(m)p Fi(\000)p Fl(2)1217 2935 y Fq(\()p Fo(z)1304 2884 y Fl(\()p Fm(m)1393 2861 y Fd(0)1417 2884 y Fl(\))1300 2957 y Fm(m)p Fi(\000)p Fl(1)1479 2935 y Fn(\000)22 b Fo(S)1638 2951 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1872 2935 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)2117 2894 y Fi(?)2103 2960 y Fm(m)p Fi(\000)p Fl(2)2287 2935 y Fq(=)1571 3103 y(=:)28 b Fo(R)1777 3061 y Fl(+)11 b Fi(?)1776 3127 y Fm(m)p Fi(\000)p Fl(2)1933 3103 y Fq(\()p Fo(z)2020 3052 y Fl(\()p Fm(m)2109 3028 y Fd(0)2132 3052 y Fl(\))2016 3124 y Fm(m)p Fi(\000)p Fl(1)2195 3103 y Fn(\000)22 b Fo(S)2354 3118 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2588 3103 y Fq(\()p Fo(k)s Fq(\)\))428 b(\(I)s(I)s(I.17\))986 3260 y Fo(R)1061 3219 y Fl(+)1060 3285 y Fm(m)p Fi(\000)p Fl(2)1217 3260 y Fq(\()p Fo(z)1304 3219 y Fl(\()p Fm(m)1393 3196 y Fd(0)1417 3219 y Fl(\))1300 3285 y Fm(m)1470 3260 y Fn(\000)23 b Fo(S)1630 3276 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2022 3260 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)2267 3219 y Fi(?)2253 3285 y Fm(m)p Fi(\000)p Fl(2)2437 3260 y Fq(=)1571 3418 y(=:)28 b Fo(R)1777 3376 y Fl(+)11 b Fi(?)1776 3442 y Fm(m)p Fi(\000)p Fl(2)1933 3418 y Fq(\()p Fo(z)2020 3376 y Fl(\()p Fm(m)2109 3353 y Fd(0)2132 3376 y Fl(\))2016 3442 y Fm(m)2186 3418 y Fn(\000)23 b Fo(S)2346 3433 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2738 3418 y Fq(\()p Fo(k)s Fq(\)\))p Fo(:)251 b Fq(\(I)s(I)s(I.18\))257 3629 y(W)-8 b(e)43 b(no)m(w)h(sp)s(ecify)g(the)f(expansions)i Fn(A)d Fq(and)h Fn(B)j Fq(and)d(to)f(this)h(purp)s(ose)h(w)m(e)g (de\014ne)f(a)257 3750 y(map)31 b Fo(\034)525 3713 y Fm(N)593 3750 y Fq(,)g(the)h(\\truncation)f(map")f(acting)h(on)g(the)g (op)s(erators)g(\(I)s(I)s(I.16\),)g(\(I)s(I)s(I.17\))f(and)257 3870 y(\(I)s(I)s(I.18\))g(according)h(to)f(the)h(rules)g(b)s(elo)m(w.) 44 b(After)31 b(an)f(application)h(of)f Fo(\034)2914 3834 y Fm(N)2982 3870 y Fq(,)g(the)h(Hamil-)257 3990 y(tonian)23 b(app)s(earing)g(in)g(the)g(new)h(expressions)h(is)2042 3958 y Fh(f)2034 3990 y Fo(H)2123 3949 y Fl(+)2115 4012 y Fm(m)p Fi(\000)p Fl(3)2271 3990 y Fq(.)40 b(As)24 b(in)m(termediate)g (elemen)m(tary)257 4111 y(op)s(erators)33 b(app)s(earing)f(in)h(the)g (re-expansion)h(w)m(e)g(encoun)m(ter)g(the)f(follo)m(wing)g(ones:)1084 4290 y Fh(e)1061 4322 y Fo(P)1124 4337 y Fm(m)p Fi(\000)p Fl(3)3184 4322 y Fq(\(I)s(I)s(I.19\))1083 4449 y Fh(e)1061 4481 y Fo(R)1136 4440 y Fl(+)1135 4506 y Fm(m)p Fi(\000)p Fl(3)1292 4481 y Fq(\()p Fo(z)1379 4430 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1375 4503 y Fm(m)p Fi(\000)p Fl(2)1591 4481 y Fq(\))1652 4449 y Fh(e)1629 4481 y Fo(P)1706 4440 y Fi(?)1692 4506 y Fm(m)p Fi(\000)p Fl(3)1876 4481 y Fq(=:)2030 4449 y Fh(e)2007 4481 y Fo(R)2082 4440 y Fl(+)12 b Fi(?)2081 4506 y Fm(m)p Fi(\000)p Fl(3)2238 4481 y Fq(\()p Fo(z)2325 4430 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2321 4503 y Fm(m)p Fi(\000)p Fl(2)2537 4481 y Fq(\))609 b(\(I)s(I)s(I.20\)) 1083 4616 y Fh(e)1061 4648 y Fo(R)1136 4607 y Fl(+)1135 4673 y Fm(m)p Fi(\000)p Fl(3)1292 4648 y Fq(\()p Fo(z)1379 4597 y Fl(\()p Fm(m)1468 4574 y Fd(0)1492 4597 y Fl(\))1375 4670 y Fm(m)p Fi(\000)p Fl(1)1554 4648 y Fn(\000)22 b Fo(S)1713 4664 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1947 4648 y Fq(\()p Fo(k)s Fq(\)\))2138 4616 y Fh(e)2115 4648 y Fo(P)2192 4607 y Fi(?)2178 4673 y Fm(m)p Fi(\000)p Fl(3)2362 4648 y Fq(=)1646 4815 y(=:)1799 4783 y Fh(e)1777 4815 y Fo(R)1852 4774 y Fl(+)11 b Fi(?)1851 4840 y Fm(m)p Fi(\000)p Fl(3)2008 4815 y Fq(\()p Fo(z)2095 4764 y Fl(\()p Fm(m)2184 4741 y Fd(0)2208 4764 y Fl(\))2091 4837 y Fm(m)p Fi(\000)p Fl(1)2270 4815 y Fn(\000)22 b Fo(S)2429 4831 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2663 4815 y Fq(\()p Fo(k)s Fq(\)\))p Fo(:)326 b Fq(\(I)s(I)s(I.21\))257 5027 y Fj(De\014nition)34 b(of)h(the)f(map)h Fo(\034)1261 4991 y Fm(N)257 5147 y Fq(In)41 b(the)f(next)h(de\014nition)g(\(i\),)h (w)m(e)g(just)e(rephrase)h(the)g(expansion)g(in)g(Eq.)g(\(I.63\))f(ac-) 257 5268 y(cording)32 b(to)e(the)i(new)g(notations.)43 b(The)32 b(de\014nition)g(\(ii\))f(follo)m(ws)h(from)e(\(i\))h(b)m(y)h (de\014ning)257 5388 y(the)h(truncation)g(of)f(the)h(orthogonal)f(pro)5 b(jection,)34 b Fo(P)2237 5352 y Fi(?)2223 5413 y Fm(m)p Fi(\000)p Fl(2)2379 5388 y Fq(,)f(as)g Fe(1)2615 5403 y Fm(m)p Fi(\000)p Fl(2)2794 5388 y Fn(\000)22 b Fo(\034)2946 5352 y Fm(N)3014 5388 y Fq(\()p Fo(P)3115 5403 y Fm(m)p Fi(\000)p Fl(2)3271 5388 y Fq(\).)p eop end %%Page: 29 30 TeXDict begin 29 29 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(29)257 573 y(\(i\))629 793 y Fo(\034)682 752 y Fm(N)749 793 y Fq(\()p Fo(P)850 808 y Fm(m)p Fi(\000)p Fl(2)1007 793 y Fq(\))27 b(:=)2009 b(\(I)s(I)s(I.22\))470 1013 y(=)652 980 y Fh(e)629 1013 y Fo(P)692 1028 y Fm(m)p Fi(\000)p Fl(3)870 1013 y Fn(\000)1026 945 y Fq(1)p 980 989 141 4 v 980 1081 a(2)p Fo(\031)t(i)1177 905 y Fm(N)1149 930 y Fh(X)1147 1112 y Fm(j)t Fl(=1)1287 895 y Fh(Z)1333 1084 y Fl(\000)1377 1093 y Fb(m)p Fd(\000)p Ff(2)1535 1013 y Fo(dz)1635 962 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1631 1034 y Fm(m)p Fi(\000)p Fl(2)1847 916 y Fh(\020)1896 1013 y Fq(\()p Fn(\000)p Fo(z)2056 1028 y Fm(m)p Fi(\000)p Fl(2)2214 1013 y Fq(\))2252 971 y Fi(\000)p Fl(1)2369 980 y Fh(e)2346 1013 y Fo(P)2409 1028 y Fm(m)p Fi(\000)p Fl(3)2588 1013 y Fq(+)2708 980 y Fh(e)2686 1013 y Fo(R)2761 971 y Fl(+)11 b Fi(?)2760 1037 y Fm(m)p Fi(\000)p Fl(3)2917 1013 y Fq(\()p Fo(z)3000 1028 y Fm(m)p Fi(\000)p Fl(2)3157 1013 y Fq(\))3195 916 y Fh(\021)3266 1013 y Fn(\002)824 1284 y(\002)901 1187 y Fh(h)962 1284 y Fn(\000)23 b Fo(W)1168 1242 y Fm(m)p Fi(\000)p Fl(3)1154 1308 y Fm(m)p Fi(\000)p Fl(2)1324 1187 y Fh(\020)1374 1284 y Fq(\()p Fn(\000)p Fo(z)1534 1299 y Fm(m)p Fi(\000)p Fl(2)1692 1284 y Fq(\))1730 1242 y Fi(\000)p Fl(1)1847 1251 y Fh(e)1824 1284 y Fo(P)1887 1299 y Fm(m)p Fi(\000)p Fl(3)2066 1284 y Fq(+)2186 1251 y Fh(e)2164 1284 y Fo(R)2239 1242 y Fl(+)11 b Fi(?)2238 1308 y Fm(m)p Fi(\000)p Fl(3)2395 1284 y Fq(\()p Fo(z)2478 1299 y Fm(m)p Fi(\000)p Fl(2)2635 1284 y Fq(\))2673 1187 y Fh(\021)o(i)2761 1210 y Fm(j)257 1516 y Fq(\(ii\))499 1736 y Fo(\034)552 1695 y Fm(N)620 1736 y Fq(\()p Fo(P)735 1695 y Fi(?)721 1761 y Fm(m)p Fi(\000)p Fl(2)878 1736 y Fq(\))27 b(:=)h Fe(1)1130 1751 y Fm(m)p Fi(\000)p Fl(2)1309 1736 y Fn(\000)22 b Fo(\034)1461 1695 y Fm(N)1529 1736 y Fq(\()p Fo(P)1630 1751 y Fm(m)p Fi(\000)p Fl(2)1787 1736 y Fq(\))1433 b(\(I)s(I)s(I.23\))340 1962 y(=)83 b Fe(1)555 1977 y Fm(m)p Fi(\000)p Fl(2)734 1962 y Fn(\000)857 1930 y Fh(e)834 1962 y Fo(P)897 1977 y Fm(m)p Fi(\000)p Fl(3)1076 1962 y Fq(+)1230 1895 y(1)p 1184 1939 V 1184 2031 a(2)p Fo(\031)t(i)1381 1854 y Fm(N)1352 1879 y Fh(X)1351 2062 y Fm(j)t Fl(=1)1490 1845 y Fh(Z)1536 2034 y Fl(\000)1580 2043 y Fb(m)p Fd(\000)p Ff(2)1738 1962 y Fo(dz)1838 1911 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1834 1984 y Fm(m)p Fi(\000)p Fl(2)2050 1866 y Fh(\020)2100 1962 y Fq(\()p Fn(\000)p Fo(z)2260 1977 y Fm(m)p Fi(\000)p Fl(2)2417 1962 y Fq(\))2455 1921 y Fi(\000)p Fl(1)2573 1930 y Fh(e)2550 1962 y Fo(P)2613 1977 y Fm(m)p Fi(\000)p Fl(3)2791 1962 y Fq(+)2912 1930 y Fh(e)2889 1962 y Fo(R)2964 1921 y Fl(+)12 b Fi(?)2963 1987 y Fm(m)p Fi(\000)p Fl(3)3120 1962 y Fq(\()p Fo(z)3203 1977 y Fm(m)p Fi(\000)p Fl(2)3360 1962 y Fq(\))3398 1866 y Fh(\021)3470 1962 y Fn(\002)694 2233 y(\002)771 2137 y Fh(h)833 2233 y Fn(\000)23 b Fo(W)1039 2192 y Fm(m)p Fi(\000)p Fl(3)1025 2258 y Fm(m)p Fi(\000)p Fl(2)1195 2137 y Fh(\020)1245 2233 y Fq(\()p Fn(\000)p Fo(z)1405 2248 y Fm(m)p Fi(\000)p Fl(2)1562 2233 y Fq(\))1600 2192 y Fi(\000)p Fl(1)1718 2201 y Fh(e)1695 2233 y Fo(P)1758 2248 y Fm(m)p Fi(\000)p Fl(3)1936 2233 y Fq(+)2057 2201 y Fh(e)2034 2233 y Fo(R)2109 2192 y Fl(+)12 b Fi(?)2108 2258 y Fm(m)p Fi(\000)p Fl(3)2265 2233 y Fq(\()p Fo(z)2348 2248 y Fm(m)p Fi(\000)p Fl(2)2505 2233 y Fq(\))2543 2137 y Fh(\021i)2632 2160 y Fm(j)404 2466 y Fq(No)m(w,)33 b(at)f(p)s(oin)m(t)h(iii\))f(and)h(iv\))g(b)s(elo)m(w,)g(w)m(e)h(sp)s (ecify)g(the)f(truncation)g(of)f(the)h(expan-)257 2586 y(sion)g(of)g(t)m(yp)s(e)g Fn(A)f Fq(and)h Fn(B)s Fq(,)g(resp)s(ectiv)m (ely)-8 b(.)257 2707 y(\(iii\))33 b(By)g(iterativ)m(ely)h(using)g(the)f (resolv)m(en)m(t)h(equation)762 2927 y Fo(R)837 2885 y Fl(+)836 2951 y Fm(m)p Fi(\000)p Fl(2)993 2927 y Fq(\()p Fo(\020)8 b Fq(\))82 b(=)1383 2894 y Fh(e)1361 2927 y Fo(R)1436 2885 y Fl(+)1435 2951 y Fm(m)p Fi(\000)p Fl(3)1592 2927 y Fq(\()p Fo(\020)8 b Fq(\))1465 b(\(I)s(I)s(I.24\))1361 3091 y(+)p Fo(R)1512 3050 y Fl(+)1511 3116 y Fm(m)p Fi(\000)p Fl(2)1668 3091 y Fq(\()p Fo(\020)8 b Fq(\))1795 2995 y Fh(\020)1865 3091 y Fn(\000)23 b Fo(W)2071 3050 y Fm(m)p Fi(\000)p Fl(3)2057 3116 y Fm(m)p Fi(\000)p Fl(2)2250 3091 y Fn(\000)f Fo(E)2427 3050 y Fm(m)p Fi(\000)p Fl(3)2421 3116 y Fm(m)p Fi(\000)p Fl(2)2584 2995 y Fh(\021)2656 3059 y(e)2634 3091 y Fo(R)2709 3050 y Fl(+)2708 3116 y Fm(m)p Fi(\000)p Fl(3)2865 3091 y Fq(\()p Fo(\020)8 b Fq(\))257 3351 y(where)34 b Fo(\020)h Fq(:=)28 b Fo(z)797 3300 y Fl(\()p Fm(m)886 3276 y Fd(0)910 3300 y Fl(\))793 3372 y Fm(m)p Fi(\000)p Fl(1)972 3351 y Fn(\000)22 b Fo(S)1131 3366 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1365 3351 y Fq(\()p Fo(k)s Fq(\),)32 b(w)m(e)i(can)f(easily)h(write)f(the)g (expansion)536 3571 y Fo(R)611 3530 y Fl(+)610 3595 y Fm(m)p Fi(\000)p Fl(2)767 3571 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)971 3530 y Fi(?)957 3595 y Fm(m)p Fi(\000)p Fl(2)1196 3571 y Fq(=)82 b Fo(P)1431 3530 y Fi(?)1417 3595 y Fm(m)p Fi(\000)p Fl(2)1574 3571 y Fo(R)1649 3530 y Fl(+)1648 3595 y Fm(m)p Fi(\000)p Fl(2)1805 3571 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2009 3530 y Fi(?)1995 3595 y Fm(m)p Fi(\000)p Fl(2)3184 3571 y Fq(\(I)s(I)s(I.25\))1196 3728 y(=)82 b Fo(P)1431 3687 y Fi(?)1417 3752 y Fm(m)p Fi(\000)p Fl(2)1597 3695 y Fh(e)1574 3728 y Fo(R)1649 3687 y Fl(+)1648 3752 y Fm(m)p Fi(\000)p Fl(3)1805 3728 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2009 3687 y Fi(?)1995 3752 y Fm(m)p Fi(\000)p Fl(2)2173 3728 y Fq(+)935 b(\(I)s(I)s(I.26\)) 1354 3892 y(+)1430 3796 y Fh(\020)1480 3892 y Fo(P)1557 3851 y Fi(?)1543 3917 y Fm(m)p Fi(\000)p Fl(2)1722 3860 y Fh(e)1700 3892 y Fo(R)1775 3851 y Fl(+)1774 3917 y Fm(m)p Fi(\000)p Fl(3)1931 3892 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2135 3851 y Fi(?)2121 3917 y Fm(m)p Fi(\000)p Fl(2)2276 3796 y Fh(\021)2348 3892 y Fn(\002)759 b Fq(\(I)s(I)s(I.27\)) 1452 4137 y Fn(\002)1576 4029 y Fm(N)1547 4054 y Fh(X)1546 4236 y Fm(j)t Fl(=1)1685 4040 y Fh(h\020)1796 4137 y Fn(\000)23 b Fo(W)2002 4096 y Fm(m)p Fi(\000)p Fl(3)1988 4161 y Fm(m)p Fi(\000)p Fl(2)2180 4137 y Fn(\000)g Fo(E)2358 4096 y Fm(m)p Fi(\000)p Fl(3)2352 4161 y Fm(m)p Fi(\000)p Fl(2)2515 4040 y Fh(\021)2587 4104 y(e)2564 4137 y Fo(R)2639 4096 y Fl(+)2638 4161 y Fm(m)p Fi(\000)p Fl(3)2795 4137 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2999 4096 y Fi(?)2985 4161 y Fm(m)p Fi(\000)p Fl(2)3141 4040 y Fh(i)3180 4063 y Fm(j)1354 4378 y Fq(+)p Fo(R)1505 4337 y Fl(+)1504 4402 y Fm(m)p Fi(\000)p Fl(2)1661 4378 y Fq(\()p Fo(\020)g Fq(\))p Fo(P)1865 4337 y Fi(?)1851 4402 y Fm(m)p Fi(\000)p Fl(2)2029 4378 y Fn(\002)1078 b Fq(\(I)s(I)s(I.28\))1452 4561 y Fn(\002)p Fq([)1556 4465 y Fh(\020)1628 4561 y Fn(\000)23 b Fo(W)1834 4520 y Fm(m)p Fi(\000)p Fl(3)1820 4586 y Fm(m)p Fi(\000)p Fl(2)2012 4561 y Fn(\000)g Fo(E)2190 4520 y Fm(m)p Fi(\000)p Fl(3)2184 4586 y Fm(m)p Fi(\000)p Fl(2)2347 4465 y Fh(\021)2419 4529 y(e)2396 4561 y Fo(R)2471 4520 y Fl(+)2470 4586 y Fm(m)p Fi(\000)p Fl(3)2627 4561 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2831 4520 y Fi(?)2817 4586 y Fm(m)p Fi(\000)p Fl(2)2973 4465 y Fh(i)3012 4488 y Fm(N)f Fl(+1)257 4794 y Fq(Finally)-8 b(,)37 b(starting)e(from)g(the) h(expressions)i(in)e(\(I)s(I)s(I.26\))e(and)i(\(I)s(I)s(I.27\),)f(w)m (e)i(de\014ne)f(our)257 4914 y(truncation)g(of)f(the)g(orthogonal)f (part)h(of)g(the)h(resolv)m(en)m(t)h(in)e(Eq.)h(\(I)s(I)s(I.25\).)51 b(The)36 b(de\014-)p eop end %%Page: 30 31 TeXDict begin 30 30 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(30)257 573 y(nition)33 b(is)847 749 y Fo(\034)900 708 y Fm(N)968 749 y Fq(\()p Fo(R)1081 708 y Fl(+)12 b Fi(?)1080 773 y Fm(m)p Fi(\000)p Fl(2)1237 749 y Fq(\()p Fo(\020)c Fq(\))27 b(:=)1690 b(\(I)s(I)s(I.29\))688 913 y(=)847 817 y Fh(h)887 913 y Fo(\034)940 872 y Fm(N)1007 913 y Fq(\()p Fo(P)1122 872 y Fi(?)1108 938 y Fm(m)p Fi(\000)p Fl(2)1265 913 y Fq(\))1303 817 y Fh(\020)1352 913 y Fq(\()p Fn(\000)p Fo(\020)8 b Fq(\))1556 872 y Fi(\000)p Fl(1)1673 881 y Fh(e)1650 913 y Fo(P)1713 928 y Fm(m)p Fi(\000)p Fl(3)1892 913 y Fq(+)2013 881 y Fh(e)1990 913 y Fo(R)2065 872 y Fl(+)k Fi(?)2064 938 y Fm(m)p Fi(\000)p Fl(3)2221 913 y Fq(\()p Fo(\020)c Fq(\))2348 817 y Fh(\021)2397 913 y Fo(\034)2450 872 y Fm(N)2517 913 y Fq(\()p Fo(P)2632 872 y Fi(?)2618 938 y Fm(m)p Fi(\000)p Fl(2)2775 913 y Fq(\))2813 817 y Fh(i)847 1096 y Fq(+)923 1000 y Fh(h)962 1096 y Fo(\034)1015 1055 y Fm(N)1083 1096 y Fq(\()p Fo(P)1198 1055 y Fi(?)1184 1120 y Fm(m)p Fi(\000)p Fl(2)1341 1096 y Fq(\))1379 1000 y Fh(\020)1428 1096 y Fq(\()p Fn(\000)p Fo(\020)g Fq(\))1632 1055 y Fi(\000)p Fl(1)1749 1063 y Fh(e)1726 1096 y Fo(P)1789 1111 y Fm(m)p Fi(\000)p Fl(3)1968 1096 y Fq(+)2088 1063 y Fh(e)2066 1096 y Fo(R)2141 1055 y Fl(+)k Fi(?)2140 1120 y Fm(m)p Fi(\000)p Fl(3)2297 1096 y Fq(\()p Fo(\020)c Fq(\))2424 1000 y Fh(\021)2472 1096 y Fo(\034)2525 1055 y Fm(N)2593 1096 y Fq(\()p Fo(P)2708 1055 y Fi(?)2694 1120 y Fm(m)p Fi(\000)p Fl(2)2851 1096 y Fq(\))2889 1000 y Fh(i)2950 1096 y Fn(\002)945 1340 y(\002)1069 1233 y Fm(N)1040 1257 y Fh(X)1039 1440 y Fm(j)t Fl(=1)1178 1244 y Fh(h\020)1289 1340 y Fn(\000)23 b Fo(W)1495 1299 y Fm(m)p Fi(\000)p Fl(3)1481 1365 y Fm(m)p Fi(\000)p Fl(2)1673 1340 y Fn(\000)g Fq(\()p Fo(E)1889 1299 y Fm(m)p Fi(\000)p Fl(3)1883 1365 y Fm(m)p Fi(\000)p Fl(2)2046 1340 y Fq(\))2084 1299 y Fm(T)2139 1244 y Fh(\021)2210 1340 y Fn(\002)1042 1611 y(\002)1119 1515 y Fh(\020)1170 1611 y Fq(\()p Fn(\000)p Fo(\020)8 b Fq(\))1374 1570 y Fi(\000)p Fl(1)1491 1579 y Fh(e)1467 1611 y Fo(P)1530 1626 y Fm(m)p Fi(\000)p Fl(3)1709 1611 y Fq(+)1830 1579 y Fh(e)1807 1611 y Fo(R)1882 1570 y Fl(+)k Fi(?)1881 1636 y Fm(m)p Fi(\000)p Fl(3)2038 1611 y Fq(\()p Fo(\020)c Fq(\))2165 1515 y Fh(\021)2214 1611 y Fo(\034)2267 1570 y Fm(N)2334 1611 y Fq(\()p Fo(P)2449 1570 y Fi(?)2435 1636 y Fm(m)p Fi(\000)p Fl(2)2592 1611 y Fq(\))2630 1515 y Fh(i)2669 1538 y Fm(j)2706 1611 y Fo(:)257 1787 y Fq(Notice)32 b(that)f(the)h(expression)h(ab)s(o)m(v)m(e)g(is)e(written)h(in)g(terms) g(of)f(the)g(elemen)m(tary)j(op)s(er-)257 1908 y(ators)f(\(I)s(I)s (I.19\),)g(\(I)s(I)s(I.20\))f(and)h(\(I)s(I)s(I.21\);)g(in)g (particular,)g(the)h(op)s(erator)2956 1875 y Fh(e)2933 1908 y Fo(P)3010 1872 y Fi(?)2996 1932 y Fm(m)p Fi(\000)p Fl(3)3153 1908 y Fq(,)f(arising)257 2028 y(from)38 b(the)g(truncated)h (expansion)g Fo(\034)1627 1992 y Fm(N)1695 2028 y Fq(\()p Fo(P)1810 1992 y Fi(?)1796 2053 y Fm(m)p Fi(\000)p Fl(2)1952 2028 y Fq(\),)g(is)f(alw)m(a)m(ys)i(attac)m(hed)e(to)f(a)h(resolv)m(en) m(t;)257 2170 y(hence)44 b(either)f(it)g(yields)h(a)e(term)h(of)f(t)m (yp)s(e)1923 2138 y Fh(e)1900 2170 y Fo(R)1975 2129 y Fl(+)12 b Fi(?)1974 2192 y Fm(m)p Fi(\000)p Fl(3)2131 2170 y Fq(\()p Fo(z)2218 2119 y Fl(\()p Fm(m)2307 2096 y Fd(0)2331 2119 y Fl(\))2214 2192 y Fm(m)p Fi(\000)p Fl(1)2400 2170 y Fn(\000)29 b Fo(S)2566 2185 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2800 2170 y Fq(\()p Fo(k)s Fq(\)\))42 b(or)g(it)g(is)h(not)257 2290 y(presen)m(t)35 b(at)d(all.)257 2411 y(\(iv\))h(Analogously)-8 b(,)34 b(w)m(e)f(start)g(from)f(the)h(equalit)m(y)h(b)s(elo)m(w)625 2587 y Fo(R)700 2546 y Fl(+)699 2611 y Fm(m)p Fi(\000)p Fl(2)856 2587 y Fq(\()p Fo(\020)945 2546 y Fi(0)967 2587 y Fq(\))p Fo(P)1082 2546 y Fi(?)1068 2611 y Fm(m)p Fi(\000)p Fl(2)1252 2587 y Fq(=)1856 b(\(I)s(I)s(I.30\))708 2744 y(=)83 b Fo(P)944 2703 y Fi(?)930 2768 y Fm(m)p Fi(\000)p Fl(2)1109 2711 y Fh(e)1086 2744 y Fo(R)1161 2703 y Fl(+)1160 2768 y Fm(m)p Fi(\000)p Fl(3)1317 2744 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)1521 2703 y Fi(?)1507 2768 y Fm(m)p Fi(\000)p Fl(2)3184 2744 y Fq(\(I)s(I)s(I.31\))867 2908 y(+)943 2812 y Fh(\020)992 2908 y Fo(P)1069 2867 y Fi(?)1055 2933 y Fm(m)p Fi(\000)p Fl(2)1234 2876 y Fh(e)1212 2908 y Fo(R)1287 2867 y Fl(+)1286 2933 y Fm(m)p Fi(\000)p Fl(3)1443 2908 y Fq(\()p Fo(\020)g Fq(\))p Fo(P)1647 2867 y Fi(?)1633 2933 y Fm(m)p Fi(\000)p Fl(2)1789 2812 y Fh(\021)1860 2908 y Fn(\002)1247 b Fq(\(I)s(I)s(I.32\))964 3153 y Fn(\002)1088 3045 y Fm(N)1060 3070 y Fh(X)1058 3252 y Fm(j)t Fl(=1)1198 3007 y Fh( )1438 3083 y Fn(\000)p Fo(W)1621 3042 y Fm(m)p Fi(\000)p Fl(3)1607 3105 y Fm(m)p Fi(\000)p Fl(2)1800 3083 y Fn(\000)23 b Fo(E)1978 3042 y Fm(m)p Fi(\000)p Fl(3)1972 3105 y Fm(m)p Fi(\000)p Fl(2)1263 3221 y Fq(+)p Fo(z)1388 3185 y Fl(\()p Fm(m)1477 3161 y Fd(0)1501 3185 y Fl(\))1384 3246 y Fm(m)1555 3221 y Fn(\000)g Fo(z)1704 3170 y Fl(\()p Fm(m)1793 3147 y Fd(0)1817 3170 y Fl(\))1700 3243 y Fm(m)p Fi(\000)p Fl(1)1879 3221 y Fq(+)f Fo(S)2037 3236 y Fm(l)q Fl(\()p Fm(m)p Fl(\))2180 3221 y Fq(\()p Fo(k)s Fq(\))2393 3007 y Fh(!)2498 3120 y(e)2475 3153 y Fo(R)2550 3112 y Fl(+)2549 3177 y Fm(m)p Fi(\000)p Fl(3)2706 3153 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2910 3112 y Fi(?)2896 3177 y Fm(m)p Fi(\000)p Fl(2)3052 3056 y Fh(i)3091 3079 y Fm(j)867 3394 y Fq(+)p Fo(R)1018 3353 y Fl(+)1017 3418 y Fm(m)p Fi(\000)p Fl(2)1174 3394 y Fq(\()p Fo(\020)1263 3353 y Fi(0)1285 3394 y Fq(\))p Fo(P)1400 3353 y Fi(?)1386 3418 y Fm(m)p Fi(\000)p Fl(2)1565 3394 y Fn(\002)1542 b Fq(\(I)s(I)s(I.33\))964 3613 y Fn(\002)1058 3466 y Fh( )1299 3543 y Fn(\000)p Fo(W)1482 3501 y Fm(m)p Fi(\000)p Fl(3)1468 3564 y Fm(m)p Fi(\000)p Fl(2)1661 3543 y Fn(\000)23 b Fo(E)1839 3501 y Fm(m)p Fi(\000)p Fl(3)1833 3564 y Fm(m)p Fi(\000)p Fl(2)1124 3681 y Fq(+)p Fo(z)1249 3645 y Fl(\()p Fm(m)1338 3621 y Fd(0)1362 3645 y Fl(\))1245 3705 y Fm(m)1416 3681 y Fn(\000)f Fo(z)1564 3630 y Fl(\()p Fm(m)1653 3606 y Fd(0)1677 3630 y Fl(\))1560 3703 y Fm(m)p Fi(\000)p Fl(1)1740 3681 y Fq(+)g Fo(S)1898 3696 y Fm(l)q Fl(\()p Fm(m)p Fl(\))2041 3681 y Fq(\()p Fo(k)s Fq(\))2254 3466 y Fh(!)2359 3580 y(e)2336 3613 y Fo(R)2411 3571 y Fl(+)2410 3637 y Fm(m)p Fi(\000)p Fl(3)2567 3613 y Fq(\()p Fo(\020)8 b Fq(\))p Fo(P)2771 3571 y Fi(?)2757 3637 y Fm(m)p Fi(\000)p Fl(2)2913 3516 y Fh(i)2952 3539 y Fm(N)f Fl(+1)257 3868 y Fq(where)49 b Fo(\020)605 3832 y Fi(0)679 3868 y Fq(:=)k Fo(z)884 3832 y Fl(\()p Fm(m)973 3808 y Fd(0)997 3832 y Fl(\))880 3893 y Fm(m)1060 3868 y Fn(\000)33 b Fo(S)1230 3884 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1622 3868 y Fq(\()p Fo(k)s Fq(\))47 b(and)g(where)h(w)m(e)g(recall)g(that)f Fo(S)3018 3884 y Fm(l)q Fl(\()p Fm(m)p Fl(\))3161 3868 y Fq(\()p Fo(k)s Fq(\))g(and)257 3988 y Fo(S)317 4004 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))710 3988 y Fq(\()p Fo(k)s Fq(\))32 b(are)h(de\014ned)h(in)e(Eqs.)i(\(I)s(I) s(I.7\))f(and)f(\(I)s(I)s(I.8\),)h(resp)s(ectiv)m(ely)-8 b(.)257 4109 y(W)g(e)35 b(de\014ne)g(our)f(truncation)h(of)e(the)i (orthogonal)e(part)h(of)g(the)g(resolv)m(en)m(t)i(in)f(\(I)s(I)s (I.30\),)257 4229 y(b)m(y)40 b(inserting)h(\()p Fo(E)924 4188 y Fm(m)p Fi(\000)p Fl(3)918 4251 y Fm(m)p Fi(\000)p Fl(2)1080 4229 y Fq(\))1118 4193 y Fm(T)1213 4229 y Fq(in)e(place)h(of) f Fo(E)1785 4188 y Fm(m)p Fi(\000)p Fl(3)1779 4251 y Fm(m)p Fi(\000)p Fl(2)1981 4229 y Fq(and)g Fo(\034)2230 4193 y Fm(N)2298 4229 y Fq(\()p Fo(P)2413 4193 y Fi(?)2399 4254 y Fm(m)p Fi(\000)p Fl(2)2555 4229 y Fq(\))g(in)h(place)g(of)e Fo(P)3203 4193 y Fi(?)3189 4254 y Fm(m)p Fi(\000)p Fl(2)3346 4229 y Fq(,)j(in)257 4350 y(expressions)36 b(\(I)s(I)s(I.31\))c(and)g (\(I)s(I)s(I.32\):)479 4526 y Fo(\034)532 4484 y Fm(N)599 4526 y Fq(\()p Fo(R)712 4484 y Fl(+)12 b Fi(?)711 4550 y Fm(m)p Fi(\000)p Fl(2)868 4526 y Fq(\()p Fo(\020)957 4484 y Fi(0)980 4526 y Fq(\)\))27 b(:=)1998 b(\(I)s(I)s(I.34\))562 4690 y(=)720 4594 y Fh(h)760 4690 y Fo(\034)813 4649 y Fm(N)880 4690 y Fq(\()p Fo(P)995 4649 y Fi(?)981 4715 y Fm(m)p Fi(\000)p Fl(2)1138 4690 y Fq(\))1176 4594 y Fh(\020)1226 4690 y Fq(\()p Fn(\000)p Fo(\020)8 b Fq(\))1430 4649 y Fi(\000)p Fl(1)1547 4658 y Fh(e)1523 4690 y Fo(P)1586 4705 y Fm(m)p Fi(\000)p Fl(3)1765 4690 y Fq(+)1886 4658 y Fh(e)1863 4690 y Fo(R)1938 4649 y Fl(+)k Fi(?)1937 4715 y Fm(m)p Fi(\000)p Fl(3)2094 4690 y Fq(\()p Fo(\020)c Fq(\))2221 4594 y Fh(\021)2270 4690 y Fo(\034)2323 4649 y Fm(N)2390 4690 y Fq(\()p Fo(P)2505 4649 y Fi(?)2491 4715 y Fm(m)p Fi(\000)p Fl(2)2648 4690 y Fq(\))2686 4594 y Fh(i)720 4873 y Fq(+)796 4776 y Fh(h)836 4873 y Fo(\034)889 4831 y Fm(N)956 4873 y Fq(\()p Fo(P)1071 4831 y Fi(?)1057 4897 y Fm(m)p Fi(\000)p Fl(2)1214 4873 y Fq(\))1252 4776 y Fh(\020)1301 4873 y Fq(\()p Fn(\000)p Fo(\020)g Fq(\))1505 4831 y Fi(\000)p Fl(1)1622 4840 y Fh(e)1599 4873 y Fo(P)1662 4888 y Fm(m)p Fi(\000)p Fl(3)1841 4873 y Fq(+)1962 4840 y Fh(e)1939 4873 y Fo(R)2014 4831 y Fl(+)k Fi(?)2013 4897 y Fm(m)p Fi(\000)p Fl(3)2170 4873 y Fq(\()p Fo(\020)c Fq(\))2297 4776 y Fh(\021)2346 4873 y Fo(\034)2399 4831 y Fm(N)2466 4873 y Fq(\()p Fo(P)2581 4831 y Fi(?)2567 4897 y Fm(m)p Fi(\000)p Fl(2)2724 4873 y Fq(\))2762 4776 y Fh(i)2823 4873 y Fn(\002)818 5117 y(\002)942 5009 y Fm(N)913 5034 y Fh(X)912 5216 y Fm(j)t Fl(=1)1051 5021 y Fh(h\020)1162 5117 y Fn(\000)23 b Fo(W)1368 5076 y Fm(m)p Fi(\000)p Fl(3)1354 5142 y Fm(m)p Fi(\000)p Fl(2)1546 5117 y Fn(\000)g Fq(\()p Fo(E)1762 5076 y Fm(m)p Fi(\000)p Fl(3)1756 5142 y Fm(m)p Fi(\000)p Fl(2)1919 5117 y Fq(\))1957 5076 y Fm(T)2034 5117 y Fq(+)f Fo(z)2181 5076 y Fl(\()p Fm(m)2270 5053 y Fd(0)2294 5076 y Fl(\))2177 5142 y Fm(m)2348 5117 y Fn(\000)g Fo(z)2496 5066 y Fl(\()p Fm(m)2585 5043 y Fd(0)2609 5066 y Fl(\))2492 5139 y Fm(m)p Fi(\000)p Fl(1)2671 5117 y Fq(+)g Fo(S)2829 5133 y Fm(l)q Fl(\()p Fm(m)p Fl(\))2973 5117 y Fq(\()p Fo(k)s Fq(\))3103 5021 y Fh(\021)3174 5117 y Fn(\002)818 5388 y(\002)895 5292 y Fh(\020)945 5388 y Fq(\()p Fn(\000)p Fo(\020)8 b Fq(\))1149 5347 y Fi(\000)p Fl(1)1266 5356 y Fh(e)1243 5388 y Fo(P)1306 5403 y Fm(m)p Fi(\000)p Fl(3)1485 5388 y Fq(+)1605 5356 y Fh(e)1583 5388 y Fo(R)1658 5347 y Fl(+)j Fi(?)1657 5413 y Fm(m)p Fi(\000)p Fl(3)1814 5388 y Fq(\()p Fo(\020)d Fq(\))1941 5292 y Fh(\021)1989 5388 y Fo(\034)2042 5347 y Fm(N)2110 5388 y Fq(\()p Fo(P)2225 5347 y Fi(?)2211 5413 y Fm(m)p Fi(\000)p Fl(2)2368 5388 y Fq(\))2406 5292 y Fh(i)2445 5315 y Fm(j)p eop end %%Page: 31 32 TeXDict begin 31 31 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(31)257 573 y(In)27 b(Theorem)g(I)s(I)s(I.4,)g (b)m(y)g(basically)g(com)m(bining)g(the)f(map)g Fo(\034)2426 537 y Fm(N)2520 573 y Fq(and)g(the)g(Wic)m(k-ordering,)257 693 y(w)m(e)43 b(reco)m(v)m(er)g(the)f(elemen)m(tary)i(op)s(erators)d (\(I)s(I)s(I.16\),\(I)s(I)s(I.17\))g(and)g(\(I)s(I)s(I.18\),)j(on)d (scale)257 814 y Fo(m)25 b Fn(\000)f Fq(3.)52 b(Before)36 b(implemen)m(ting)h(this)f(program,)g(it)f(is)h(ho)m(w)m(ev)m(er)i (necessary)g(to)d(c)m(hec)m(k)257 934 y(that,)52 b(b)m(y)d(applying)g (our)f(map)g Fo(\034)1554 898 y Fm(N)1622 934 y Fq(,)j(only)e(terms)g (of,)i(at)d(most,)k(order)d Fo(o)p Fq(\()p Fo(\013)3213 898 y Fm(N)3279 934 y Fq(\))f(are)257 1054 y(neglected;)35 b(i.e.,)1410 1264 y Fn(k)p Fq(\()p Fo(I)8 b(d)22 b Fn(\000)g Fo(\034)1774 1223 y Fm(N)1842 1264 y Fq(\)\()p Fo(P)1981 1279 y Fm(m)p Fi(\000)p Fl(2)2138 1264 y Fq(\))p Fn(k)958 b Fq(\(I)s(I)s(I.35\))1493 1410 y Fn(\024)84 b Fo(C)7 b(\013)1794 1369 y Fm(N)g Fl(+1)1973 1410 y Fn(\001)22 b(k)p Fo(P)2136 1425 y Fm(m)p Fi(\000)p Fl(2)2292 1410 y Fn(k)986 1724 y(k)p Fq(\()p Fo(I)8 b(d)21 b Fn(\000)i Fo(\034)1350 1683 y Fm(N)1417 1724 y Fq(\)\()p Fo(R)1568 1677 y Fl(+)p Fm(;)p Fi(?)1567 1746 y Fm(m)p Fi(\000)p Fl(2)1724 1724 y Fq(\()p Fo(z)1811 1673 y Fl(\()p Fm(m)1900 1650 y Fd(0)1924 1673 y Fl(\))1807 1746 y Fm(m)p Fi(\000)p Fl(1)1986 1724 y Fn(\000)g Fo(S)2146 1740 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2380 1724 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)586 b Fq(\(I)s(I)s(I.36\))1069 1892 y Fn(\024)83 b Fo(C)7 b(\013)1369 1851 y Fm(N)g Fl(+)1497 1823 y Ff(1)p 1497 1835 31 4 v 1497 1877 a(2)1563 1892 y Fn(\001)22 b(k)p Fo(R)1738 1845 y Fl(+)p Fm(;)p Fi(?)1737 1914 y Fm(m)p Fi(\000)p Fl(2)1894 1892 y Fq(\()p Fo(z)1981 1841 y Fl(\()p Fm(m)2070 1817 y Fd(0)2094 1841 y Fl(\))1977 1914 y Fm(m)p Fi(\000)p Fl(1)2156 1892 y Fn(\000)h Fo(S)2316 1907 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2549 1892 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)906 2214 y(k)p Fq(\()p Fo(I)8 b(d)22 b Fn(\000)g Fo(\034)1270 2172 y Fm(N)1338 2214 y Fq(\)\()p Fo(R)1489 2166 y Fl(+)p Fm(;)p Fi(?)1488 2235 y Fm(m)p Fi(\000)p Fl(2)1645 2214 y Fq(\()p Fo(z)1732 2163 y Fl(\()p Fm(m)1821 2139 y Fd(0)1845 2163 y Fl(\))1728 2235 y Fm(m)p Fi(\000)p Fl(1)1907 2214 y Fn(\000)h Fo(S)2067 2229 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2459 2214 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)507 b Fq(\(I)s(I)s(I.37\))989 2381 y Fn(\024)84 b Fo(C)7 b(\013)1290 2340 y Fm(N)g Fl(+)1418 2313 y Ff(1)p 1417 2325 V 1417 2366 a(2)1484 2381 y Fn(\001)22 b(k)p Fo(R)1659 2334 y Fl(+)p Fm(;)p Fi(?)1658 2403 y Fm(m)p Fi(\000)p Fl(2)1815 2381 y Fq(\()p Fo(z)1902 2330 y Fl(\()p Fm(m)1991 2307 y Fd(0)2015 2330 y Fl(\))1898 2403 y Fm(m)p Fi(\000)p Fl(1)2077 2381 y Fn(\000)g Fo(S)2236 2396 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2629 2381 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)257 2591 y Fq(where)32 b Fo(I)8 b(d)31 b Fq(is)g(the)h(iden)m(tit) m(y)g(map)f(and)g Fo(C)38 b Fq(an)31 b Fo(\013)q Fq(-indep)s(enden)m(t) h(constan)m(t.)44 b(Concerning)257 2711 y(the)e(pro)5 b(jection)42 b Fo(P)969 2726 y Fm(m)p Fi(\000)p Fl(2)1126 2711 y Fq(,)h(Prop)s(ert)m(y)f(\(I)s(I)s(I.35\))f(has)g(b)s(een)h(pro)m (v)m(en)h(in)e(Prop)s(osition)h(I.I.)257 2832 y(F)-8 b(or)29 b(the)i(orthogonal)e(part)g(of)h(the)g(resolv)m(en)m(t,)i (expanded)g(at)d(p)s(oin)m(t)h(\(iii\))g(and)g(\(iv\),)h(w)m(e)257 2952 y(need)j(the)f(follo)m(wing)g(t)m(w)m(o)g(lemmata.)257 3147 y Fe(Lemma)39 b(I)s(I)s(I.1.)j Fj(F)-7 b(or)34 b Fo(m)1213 3111 y Fi(0)1264 3147 y Fn(\025)28 b Fo(m)23 b Fn(\000)f Fq(1)28 b Fn(\025)g Fq(2)1192 3357 y Fn(k)p Fq(\()p Fo(I)8 b(d)22 b Fn(\000)g Fo(\034)1556 3316 y Fm(N)1624 3357 y Fq(\)\()p Fo(R)1775 3310 y Fl(+)p Fm(;)p Fi(?)1774 3379 y Fm(m)p Fi(\000)p Fl(2)1931 3357 y Fq(\()p Fo(z)2018 3306 y Fl(\()p Fm(m)2107 3283 y Fd(0)2131 3306 y Fl(\))2014 3379 y Fm(m)p Fi(\000)p Fl(1)2193 3357 y Fn(\000)h Fo(S)2353 3373 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2586 3357 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)380 b Fq(\(I)s(I)s(I.38\))1032 3524 y Fn(\024)83 b Fo(C)7 b(\013)1332 3483 y Fm(N)g Fl(+)1460 3456 y Ff(1)p 1460 3468 V 1460 3509 a(2)1527 3524 y Fn(\001)21 b(k)p Fo(R)1701 3477 y Fl(+)p Fm(;)p Fi(?)1700 3546 y Fm(m)p Fi(\000)p Fl(2)1857 3524 y Fq(\()p Fo(z)1944 3474 y Fl(\()p Fm(m)2033 3450 y Fd(0)2057 3474 y Fl(\))1940 3546 y Fm(m)p Fi(\000)p Fl(1)2119 3524 y Fn(\000)i Fo(S)2279 3540 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2512 3524 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)p Fo(:)257 3735 y Fj(Pr)-5 b(o)g(of.)40 b Fq(By)22 b(the)g(Neumann)h(series)g(expansion)h(in)e(Eqs.)h(\(I)s(I)s(I.26\),)g (\(I)s(I)s(I.27\))e(and)h(\(I)s(I)s(I.28\),)257 3855 y(Expression)i(\(I)s(I)s(I.29\))e(corresp)s(onds)h(to)e(expression)j (\(I)s(I)s(I.25\),)g(with)e Fo(\034)2791 3819 y Fm(N)2859 3855 y Fq(\()p Fo(P)2974 3819 y Fi(?)2960 3880 y Fm(m)p Fi(\000)p Fl(2)3116 3855 y Fq(\),)i Fo(\034)3258 3819 y Fm(N)3326 3855 y Fq(\()p Fo(P)3427 3870 y Fm(m)p Fi(\000)p Fl(2)3583 3855 y Fq(\))257 3975 y(and)35 b(\()p Fo(E)565 3934 y Fm(m)p Fi(\000)p Fl(3)559 3997 y Fm(m)p Fi(\000)p Fl(2)722 3975 y Fq(\))760 3939 y Fm(T)849 3975 y Fq(replaced)g(b)m(y)g Fo(P)1449 3939 y Fi(?)1435 4000 y Fm(m)p Fi(\000)p Fl(2)1626 3975 y Fq(,)p Fo(P)1716 3990 y Fm(m)p Fi(\000)p Fl(2)1907 3975 y Fq(and)g Fo(E)2177 3934 y Fm(m)p Fi(\000)p Fl(3)2171 3997 y Fm(m)p Fi(\000)p Fl(2)2334 3975 y Fq(,)f(resp)s(ectiv)m(ely)-8 b(,)38 b(up)d(to)f(the)g(re-)257 4096 y(mainder)g(term)732 4306 y Fo(R)807 4265 y Fl(+)806 4330 y Fm(m)p Fi(\000)p Fl(2)963 4306 y Fq(\()p Fo(z)1050 4255 y Fl(\()p Fm(m)1139 4231 y Fd(0)1163 4255 y Fl(\))1046 4328 y Fm(m)p Fi(\000)p Fl(1)1225 4306 y Fn(\000)23 b Fo(S)1385 4321 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1619 4306 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)1864 4265 y Fi(?)1850 4330 y Fm(m)p Fi(\000)p Fl(2)2028 4306 y Fn(\002)1079 b Fq(\(I)s(I)s(I.39\))732 4490 y Fn(\002)809 4393 y Fh(h)q(\020)921 4490 y Fn(\000)22 b Fo(W)1126 4449 y Fm(m)p Fi(\000)p Fl(3)1112 4514 y Fm(m)p Fi(\000)p Fl(2)1305 4490 y Fn(\000)g Fo(E)1482 4449 y Fm(m)p Fi(\000)p Fl(3)1476 4514 y Fm(m)p Fi(\000)p Fl(2)1639 4393 y Fh(\021)1711 4457 y(e)1689 4490 y Fo(R)1764 4449 y Fl(+)1763 4514 y Fm(m)p Fi(\000)p Fl(3)1920 4490 y Fq(\()p Fo(z)2007 4439 y Fl(\()p Fm(m)2096 4415 y Fd(0)2120 4439 y Fl(\))2003 4512 y Fm(m)p Fi(\000)p Fl(1)2182 4490 y Fn(\000)h Fo(S)2342 4505 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2575 4490 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)2820 4449 y Fi(?)2806 4514 y Fm(m)p Fi(\000)p Fl(2)2963 4393 y Fh(i)3002 4416 y Fm(N)7 b Fl(+1)3159 4490 y Fo(:)257 4712 y Fq(W)-8 b(e)33 b(\014rst)g(pro)m(v)m(e)h(that)f(the)g(norm)f(of) g(\(I)s(I)s(I.39\))g(is)i(less)f(than)1107 4937 y Fo(C)7 b(\013)1247 4896 y Fm(N)g Fl(+)1375 4869 y Ff(1)p 1375 4881 V 1375 4922 a(2)1442 4937 y Fn(\001)21 b(k)p Fo(R)1616 4890 y Fl(+)p Fm(;)p Fi(?)1615 4959 y Fm(m)p Fi(\000)p Fl(2)1772 4937 y Fq(\()p Fo(z)1859 4886 y Fl(\()p Fm(m)1948 4863 y Fd(0)1972 4886 y Fl(\))1855 4959 y Fm(m)p Fi(\000)p Fl(1)2034 4937 y Fn(\000)i Fo(S)2194 4953 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2428 4937 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)538 b Fq(\(I)s(I)s(I.40\))257 5147 y(for)24 b(an)g Fo(\013)q Fq(-indep)s(enden)m(t)i(constan)m(t)f Fo(C)7 b Fq(.)41 b(Later)24 b(on,)i(w)m(e)g(will)f(sho)m(w)g(that)g(the)f (truncations)257 5268 y Fo(\034)310 5232 y Fm(N)378 5268 y Fq(\()p Fo(P)493 5232 y Fi(?)479 5292 y Fm(m)p Fi(\000)p Fl(2)636 5268 y Fq(\),)p Fo(\034)754 5232 y Fm(N)821 5268 y Fq(\()p Fo(P)922 5283 y Fm(m)p Fi(\000)p Fl(2)1079 5268 y Fq(\))44 b(and)g(\()p Fo(E)1478 5227 y Fm(m)p Fi(\000)p Fl(3)1472 5290 y Fm(m)p Fi(\000)p Fl(2)1635 5268 y Fq(\))1673 5232 y Fm(T)1772 5268 y Fq(yield)h(a)f(remainder)h (of)e(the)i(same)g(order)f(as)257 5388 y(\(I)s(I)s(I.40\).)p eop end %%Page: 32 33 TeXDict begin 32 32 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(32)257 573 y(T)-8 b(o)24 b(con)m(trol)g (Expression)h(\(I)s(I)s(I.39\),)g(w)m(e)g(re-write)f(it)f(in)h(terms)g (of)f(a)g(pro)s(duct)h(of)f(op)s(erator)257 693 y(blo)s(c)m(ks:)598 808 y Fh(\020)648 904 y Fq(\()p Fo(R)761 863 y Fl(+)760 929 y Fm(m)p Fi(\000)p Fl(2)917 904 y Fq(\()p Fo(u)p Fq(\))1049 808 y Fh(\021)1108 804 y Ff(1)p 1108 816 31 4 v 1108 857 a(2)1152 904 y Fo(P)1229 863 y Fi(?)1215 929 y Fm(m)p Fi(\000)p Fl(2)1372 904 y Fn(\002)1735 b Fq(\(I)s(I)s(I.41\))764 1088 y Fn(\002)841 992 y Fh(h)881 1088 y Fq(\()p Fo(R)994 1047 y Fl(+)993 1113 y Fm(m)p Fi(\000)p Fl(2)1150 1088 y Fq(\()p Fo(u)p Fq(\)\))1330 1020 y Ff(1)p 1329 1032 V 1329 1073 a(2)1374 1088 y Fo(P)1451 1047 y Fi(?)1437 1113 y Fm(m)p Fi(\000)p Fl(2)1593 992 y Fh(\020)1665 1088 y Fn(\000)23 b Fo(W)1871 1047 y Fm(m)p Fi(\000)p Fl(3)1857 1113 y Fm(m)p Fi(\000)p Fl(2)2049 1088 y Fn(\000)g Fo(E)2227 1047 y Fm(m)p Fi(\000)p Fl(3)2221 1113 y Fm(m)p Fi(\000)p Fl(2)2384 992 y Fh(\021)2433 1088 y Fq(\()2494 1056 y Fh(e)2471 1088 y Fo(R)2546 1047 y Fl(+)2545 1113 y Fm(m)p Fi(\000)p Fl(3)2702 1088 y Fq(\()p Fo(u)p Fq(\)\))2882 1020 y Ff(1)p 2882 1032 V 2882 1073 a(2)2926 992 y Fh(i)2987 1088 y Fn(\002)120 b Fq(\(I)s(I)s(I.42\))764 1290 y Fn(\002)841 1193 y Fh(h)881 1290 y Fq(\()942 1257 y Fh(e)919 1290 y Fo(R)994 1249 y Fl(+)993 1314 y Fm(m)p Fi(\000)p Fl(3)1150 1290 y Fq(\()p Fo(u)p Fq(\)\))1330 1221 y Ff(1)p 1329 1233 V 1329 1275 a(2)1374 1290 y Fo(P)1451 1249 y Fi(?)1437 1314 y Fm(m)p Fi(\000)p Fl(2)1593 1193 y Fh(\020)1665 1290 y Fn(\000)23 b Fo(W)1871 1249 y Fm(m)p Fi(\000)p Fl(3)1857 1314 y Fm(m)p Fi(\000)p Fl(2)2049 1290 y Fn(\000)g Fo(E)2227 1249 y Fm(m)p Fi(\000)p Fl(3)2221 1314 y Fm(m)p Fi(\000)p Fl(2)2384 1193 y Fh(\021)2433 1290 y Fq(\()2494 1257 y Fh(e)2471 1290 y Fo(R)2546 1249 y Fl(+)2545 1314 y Fm(m)p Fi(\000)p Fl(3)2702 1290 y Fq(\()p Fo(u)p Fq(\)\))2882 1221 y Ff(1)p 2882 1233 V 2882 1275 a(2)2926 1193 y Fh(i)2965 1216 y Fm(N)3055 1290 y Fn(\002)52 b Fq(\(I)s(I)s(I.43\))764 1473 y Fn(\002)p Fq(\()902 1441 y Fh(e)879 1473 y Fo(R)954 1432 y Fl(+)953 1498 y Fm(m)p Fi(\000)p Fl(3)1111 1473 y Fq(\()p Fo(u)p Fq(\)\))1291 1405 y Ff(1)p 1290 1417 V 1290 1458 a(2)1335 1473 y Fo(P)1412 1432 y Fi(?)1398 1498 y Fm(m)p Fi(\000)p Fl(2)3184 1473 y Fq(\(I)s(I)s(I.44\))257 1685 y(with)34 b Fo(u)27 b Fq(=)g Fo(z)715 1634 y Fl(\()p Fm(m)804 1611 y Fd(0)828 1634 y Fl(\))711 1707 y Fm(m)p Fi(\000)p Fl(1)891 1685 y Fn(\000)22 b Fo(S)1050 1701 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1284 1685 y Fq(\()p Fo(k)s Fq(\).)43 b(No)m(w)33 b(b)s(ecause)h(of)e(the)h (expansion)644 1941 y Fo(P)721 1900 y Fi(?)707 1966 y Fm(m)p Fi(\000)p Fl(2)946 1941 y Fq(=)1128 1909 y Fh(e)1105 1941 y Fo(P)1182 1900 y Fi(?)1168 1966 y Fm(m)p Fi(\000)p Fl(3)1347 1941 y Fq(+)1501 1874 y(1)p 1455 1918 141 4 v 1455 2009 a(2)p Fo(\031)t(i)1648 1833 y Fi(1)1624 1858 y Fh(X)1622 2040 y Fm(j)t Fl(=1)1761 1824 y Fh(Z)1808 2012 y Fl(\000)1852 2021 y Fb(m)p Fd(\000)p Ff(2)2009 1941 y Fo(dz)2109 1890 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2105 1963 y Fm(m)p Fi(\000)p Fl(2)2344 1909 y Fh(e)2322 1941 y Fo(R)2397 1900 y Fl(+)2396 1966 y Fm(m)p Fi(\000)p Fl(3)2553 1941 y Fq(\()p Fo(z)2640 1890 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2636 1963 y Fm(m)p Fi(\000)p Fl(2)2852 1941 y Fq(\))22 b Fn(\002)195 b Fq(\(I)s(I)s(I.45\))1885 2212 y Fn(\002)1962 2116 y Fh(h)2024 2212 y Fn(\000)23 b Fo(W)2230 2171 y Fm(m)p Fi(\000)p Fl(3)2216 2236 y Fm(m)p Fi(\000)p Fl(2)2409 2180 y Fh(e)2386 2212 y Fo(R)2461 2171 y Fl(+)2460 2236 y Fm(m)p Fi(\000)p Fl(3)2617 2212 y Fq(\()p Fo(z)2704 2161 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2700 2234 y Fm(m)p Fi(\000)p Fl(2)2917 2212 y Fq(\))2955 2116 y Fh(i)2994 2139 y Fm(j)257 2403 y Fq(w)m(e)29 b(can)g(pro)m(vide,)h(b) m(y)f(usual)g(metho)s(ds,)h(a)d(b)s(ound)i(of)e(order,)i(at)f(most,)h Fo(\013)g Fq(for)f(the)g(norm)257 2523 y(of)j(the)h(op)s(erator)f(blo)s (c)m(k)h(in)g(\(I)s(I)s(I.42\))f(and)g(for)g(the)h(norm)f(of)g(eac)m(h) i(of)e(the)g Fo(N)42 b Fq(blo)s(c)m(ks)33 b(in)257 2643 y(\(I)s(I)s(I.43\);)41 b(in)d(this)h(p)s(o)m(w)m(er)g(coun)m(ting)g(it) f(is)g(imp)s(ortan)m(t)h(to)e(tak)m(e)i(in)m(to)g(accoun)m(t)g(that)e (a)257 2773 y(factor)e Fo(\013)612 2709 y Ff(1)p 612 2721 31 4 v 612 2762 a(2)691 2773 y Fq(is)h(gained)f(b)s(ecause)i(of)d (the)i(pro)5 b(jection)36 b Fo(P)2294 2736 y Fi(?)2280 2797 y Fm(m)p Fi(\000)p Fl(2)2472 2773 y Fq(in)f(the)h(op)s(erator)e (blo)s(c)m(k)i(in)257 2893 y(fron)m(t)d(of)f(eac)m(h)h(of)g(them.)44 b(In)33 b(the)g(end,)g(w)m(e)h(easily)f(get)g(the)g(b)s(ound)g(in)g (Eq.)g(\(I)s(I)s(I.40\).)257 3013 y(T)-8 b(o)33 b(complete)h(our)e(pro) s(of,)g(w)m(e)i(m)m(ust)f(sho)m(w)h(that)e(the)h(expression)669 3213 y Fo(P)746 3172 y Fi(?)732 3238 y Fm(m)p Fi(\000)p Fl(2)911 3181 y Fh(e)888 3213 y Fo(R)963 3172 y Fl(+)962 3238 y Fm(m)p Fi(\000)p Fl(3)1119 3213 y Fq(\()p Fo(z)1206 3163 y Fl(\()p Fm(m)1295 3139 y Fd(0)1319 3163 y Fl(\))1202 3235 y Fm(m)p Fi(\000)p Fl(1)1382 3213 y Fn(\000)22 b Fo(S)1541 3229 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1775 3213 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)2020 3172 y Fi(?)2006 3238 y Fm(m)p Fi(\000)p Fl(2)2184 3213 y Fq(+)924 b(\(I)s(I)s(I.46\)) 510 3381 y(+)669 3284 y Fh(h)708 3381 y Fo(P)785 3340 y Fi(?)771 3405 y Fm(m)p Fi(\000)p Fl(2)950 3348 y Fh(e)928 3381 y Fo(R)1003 3340 y Fl(+)1002 3405 y Fm(m)p Fi(\000)p Fl(3)1159 3381 y Fq(\()p Fo(z)1246 3330 y Fl(\()p Fm(m)1335 3306 y Fd(0)1359 3330 y Fl(\))1242 3403 y Fm(m)p Fi(\000)p Fl(1)1421 3381 y Fn(\000)22 b Fo(S)1580 3396 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1814 3381 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)2059 3340 y Fi(?)2045 3405 y Fm(m)p Fi(\000)p Fl(2)2201 3284 y Fh(i)2263 3381 y Fn(\002)864 3625 y(\002)988 3518 y Fm(N)959 3542 y Fh(X)958 3725 y Fm(j)t Fl(=1)1097 3529 y Fh(h\020)1208 3625 y Fn(\000)h Fo(W)1414 3584 y Fm(m)p Fi(\000)p Fl(3)1400 3650 y Fm(m)p Fi(\000)p Fl(2)1592 3625 y Fn(\000)g Fo(E)1770 3584 y Fm(m)p Fi(\000)p Fl(3)1764 3650 y Fm(m)p Fi(\000)p Fl(2)1927 3529 y Fh(\021)1999 3593 y(e)1976 3625 y Fo(R)2051 3584 y Fl(+)2050 3650 y Fm(m)p Fi(\000)p Fl(3)2207 3625 y Fq(\()p Fo(z)2294 3575 y Fl(\()p Fm(m)2383 3551 y Fd(0)2407 3575 y Fl(\))2290 3647 y Fm(m)p Fi(\000)p Fl(1)2469 3625 y Fn(\000)g Fo(S)2629 3641 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2862 3625 y Fq(\()p Fo(k)s Fq(\)\))p Fo(P)3107 3584 y Fi(?)3093 3650 y Fm(m)p Fi(\000)p Fl(2)3250 3529 y Fh(i)3289 3552 y Fm(j)257 3899 y Fq(di\013ers)42 b(from)f(\(I)s(I)s(I.29\))f(b)m(y)i (a)e(remainder)i(term)f(whose)h(norm)f(is)g(con)m(trolled)h(b)m(y)g (the)257 4020 y(b)s(ound)32 b(in)f(Eq.)h(\(I)s(I)s(I.40\).)43 b(Concerning)32 b(the)g(replacemen)m(t)h(of)e Fo(E)2638 3978 y Fm(m)p Fi(\000)p Fl(3)2632 4041 y Fm(m)p Fi(\000)p Fl(2)2826 4020 y Fq(b)m(y)h(\()p Fo(E)3076 3978 y Fm(m)p Fi(\000)p Fl(3)3070 4041 y Fm(m)p Fi(\000)p Fl(2)3233 4020 y Fq(\))3271 3984 y Fm(T)3326 4020 y Fq(,)f(w)m(e)257 4140 y(use)i(the)f(estimate)h(\(I)s(I.20\))e(pro)m(v)m(ed)i(in)f(Lemma) h(I)s(I.2.)43 b(F)-8 b(or)31 b(the)h(truncations)g(of)g Fo(P)3326 4104 y Fi(?)3312 4165 y Fm(m)p Fi(\000)p Fl(2)3468 4140 y Fq(,)257 4260 y(the)h(result)h(follo)m(ws)f(b)s(ecause)h(the)f (truncation)g Fo(\034)2076 4224 y Fm(N)2176 4260 y Fq(amoun)m(ts)h(to)e (neglecting)613 4395 y Fi(1)589 4420 y Fh(X)528 4604 y Fm(j)t Fl(=)p Fm(N)7 b Fl(+1)786 4386 y Fh(Z)832 4575 y Fl(\000)876 4584 y Fb(m)p Fd(\000)p Ff(2)1034 4503 y Fo(dz)1134 4452 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1130 4525 y Fm(m)p Fi(\000)p Fl(2)1368 4471 y Fh(e)1346 4503 y Fo(R)1421 4462 y Fl(+)1420 4528 y Fm(m)p Fi(\000)p Fl(3)1577 4503 y Fq(\()p Fo(z)1664 4452 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1660 4525 y Fm(m)p Fi(\000)p Fl(2)1876 4503 y Fq(\)[)p Fn(\000)p Fo(W)2124 4462 y Fm(m)p Fi(\000)p Fl(2)2110 4528 y Fm(m)p Fi(\000)p Fl(3)2303 4471 y Fh(e)2281 4503 y Fo(R)2356 4462 y Fl(+)2355 4528 y Fm(m)p Fi(\000)p Fl(3)2512 4503 y Fq(\()p Fo(z)2599 4452 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2595 4525 y Fm(m)p Fi(\000)p Fl(2)2811 4503 y Fq(\)])2876 4462 y Fm(j)3184 4503 y Fq(\(I)s(I)s(I.47\))257 4786 y(in)35 b(the)h(expansion)g(of)e Fo(P)1189 4750 y Fi(?)1175 4811 y Fm(m)p Fi(\000)p Fl(2)1367 4786 y Fq(carried)h(out)g(in)g(Eqs.)h(\(I)s(I)s(I.45\).)50 b(When)36 b(estimating)g(the)257 4907 y(norms)42 b(of)e(all)g(the)h(factors)g(in) g(the)g(remainder,)j(w)m(e)e(\014nd)f(that)f(either)i(the)f(norm)g(of) 257 5027 y(at)e(least)g(one)g(factor)f(in)h(\(I)s(I)s(I.42\),)h(\(I)s (I)s(I.43\))e(is,)j(at)d(most,)j Fn(O)s Fq(\()p Fo(\013)2651 4990 y Fm(N)7 b Fl(+)2779 4963 y Ff(1)p 2779 4975 V 2779 5016 a(2)2823 5027 y Fq(\))39 b(or)f(w)m(e)i(gain)e(an)257 5147 y(extra-factor)32 b Fo(\013)851 5111 y Fm(N)7 b Fl(+1)1041 5147 y Fq(in)33 b(blo)s(c)m(ks)h(of)e(t)m(yp)s(e)h(\(I)s(I)s (I.44\).)p 3425 5147 4 66 v 3429 5085 59 4 v 3429 5147 V 3487 5147 4 66 v 404 5388 a(In)g(the)g(next)g(lemma,)h(w)m(e)f(pro)m (vide)h(the)f(norm)g(b)s(ound)f(in)h(Eq.)h(\(I)s(I)s(I.37\).)p eop end %%Page: 33 34 TeXDict begin 33 33 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(33)257 573 y Fe(Lemma)39 b(I)s(I)s(I.2.)j Fj(F)-7 b(or)34 b Fo(m)1213 537 y Fi(0)1264 573 y Fn(\025)28 b Fo(m)g Fn(\025)g Fq(3)1110 793 y Fn(k)p Fq(\()p Fo(I)8 b(d)22 b Fn(\000)g Fo(\034)1474 752 y Fm(N)1542 793 y Fq(\)\()p Fo(R)1693 746 y Fl(+)p Fm(;)p Fi(?)1692 815 y Fm(m)p Fi(\000)p Fl(2)1849 793 y Fq(\()p Fo(z)1936 742 y Fl(\()p Fm(m)2025 719 y Fd(0)2049 742 y Fl(\))1932 815 y Fm(m)p Fi(\000)p Fl(1)2111 793 y Fn(\000)h Fo(S)2271 808 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2663 793 y Fq(\()p Fo(k)s Fq(\)\))p Fn(jj)297 b Fq(\(I)s(I)s(I.48\))950 960 y Fn(\024)83 b Fo(C)7 b(\013)1250 919 y Fm(N)g Fl(+)1378 892 y Ff(1)p 1378 904 31 4 v 1378 945 a(2)1444 960 y Fn(\001)22 b(k)p Fo(R)1619 913 y Fl(+)p Fm(;)p Fi(?)1618 982 y Fm(m)p Fi(\000)p Fl(2)1775 960 y Fq(\()p Fo(z)1862 909 y Fl(\()p Fm(m)1951 886 y Fd(0)1975 909 y Fl(\))1858 982 y Fm(m)p Fi(\000)p Fl(1)2037 960 y Fn(\000)h Fo(S)2197 976 y Fm(l)q Fl(\()p Fm(m)p Fl(\))p Fm(;l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2589 960 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)p Fo(:)257 1180 y Fj(Pr)-5 b(o)g(of.)49 b Fq(The)35 b(pro)s(of)e(is)i(analogous)f (to)g(Lemma)h(I)s(I)s(I.2,)g(since)h(the)f(t)m(w)m(o)g(more)f(terms)i (to)257 1301 y(b)s(e)c(added)h(to)f(the)g(n)m(umerator,)h Fn(\000)p Fo(W)1646 1259 y Fm(m)p Fi(\000)p Fl(2)1632 1322 y Fm(m)p Fi(\000)p Fl(3)1823 1301 y Fn(\000)22 b Fo(E)2000 1259 y Fm(m)p Fi(\000)p Fl(2)1994 1322 y Fm(m)p Fi(\000)p Fl(3)2157 1301 y Fq(,)32 b(in)g(the)g(op)s(erator)f(blo)s(c)m (ks)j(\(I)s(I)s(I.42\))257 1421 y(and)f(\(I)s(I)s(I.43\))f(are)h(small) g(enough,)g(namely)1335 1662 y Fn(j)p Fo(z)1412 1621 y Fl(\()p Fm(m)1501 1597 y Fd(0)1524 1621 y Fl(\))1408 1686 y Fm(m)1578 1662 y Fn(\000)23 b Fo(z)1727 1611 y Fl(\()p Fm(m)1816 1587 y Fd(0)1840 1611 y Fl(\))1723 1684 y Fm(m)p Fi(\000)p Fl(1)1880 1662 y Fn(j)k(\024)h(O)s Fq(\()p Fo(\013)2223 1621 y Fm(m)p Fi(\000)p Fl(1)2380 1662 y Fq(\))766 b(\(I)s(I)s(I.49\))257 1836 y(b)m(y)34 b(construction,)g(and)1413 1956 y Fn(j)p Fo(S)1501 1972 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1644 1956 y Fq(\()p Fo(k)s Fq(\))p Fn(j)28 b(\024)g(O)s Fq(\()p Fo(\013)2118 1915 y Fm(m)p Fi(\000)p Fl(1)2274 1956 y Fq(\))p Fo(;)845 b Fq(\(I)s(I)s(I.50\))257 2152 y(b)s(ecause,)33 b(b)m(y)f (de\014nition,)g Fo(S)1297 2167 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1440 2152 y Fq(\()p Fo(k)s Fq(\))c(=)1701 2085 y Fh(P)1789 2101 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1789 2174 y Fm(i)p Fl(=1)1949 2152 y Fn(j)p Fo(k)2028 2167 y Fm(i)2055 2152 y Fq(\()p Fo(m)p Fq(\))p Fn(j)p Fq(,)k(with)f Fo(l)r Fq(\()p Fo(m)p Fq(\))g(an)g Fo(\013)q Fq(-indep)s(enden)m(t)257 2272 y(n)m(um)m(b)s(er.)p 3425 2272 4 66 v 3429 2209 59 4 v 3429 2272 V 3487 2272 4 66 v 257 2513 a Fe(Remark)257 2633 y Fq(Notice)k(that)f(the)h(results)h(in)e(Lemmata)h(I)s(I)s(I.1)f (and)g(I)s(I)s(I.2)g(are)g(still)h(v)-5 b(alid)34 b(if)h(\()p Fo(E)3246 2592 y Fm(m)p Fi(\000)p Fl(3)3240 2655 y Fm(m)p Fi(\000)p Fl(2)3402 2633 y Fq(\))3440 2597 y Fm(T)257 2753 y Fq(is)h(replaced)f(b)m(y)h(an)m(y)f(other)g(appro)m(ximation,)h (\()p Fo(E)2121 2712 y Fm(m)p Fi(\000)p Fl(3)2115 2775 y Fm(m)p Fi(\000)p Fl(2)2278 2753 y Fq(\))2316 2717 y Fm(appr)n(:)2478 2753 y Fq(,)f(of)g Fo(E)2732 2712 y Fm(m)p Fi(\000)p Fl(3)2726 2775 y Fm(m)p Fi(\000)p Fl(2)2923 2753 y Fq(satisfying)h(the)257 2886 y(prop)s(ert)m(y)e Fn(j)p Fo(E)762 2845 y Fm(m)p Fi(\000)p Fl(3)756 2908 y Fm(m)p Fi(\000)p Fl(2)940 2886 y Fn(\000)23 b Fq(\()p Fo(E)1156 2845 y Fm(m)p Fi(\000)p Fl(3)1150 2908 y Fm(m)p Fi(\000)p Fl(2)1313 2886 y Fq(\))1351 2850 y Fm(appr)n(:)1513 2886 y Fn(j)k(\024)h Fo(o)p Fq(\()p Fo(\013)1821 2850 y Fm(m)p Fl(+)p Fm(N)7 b Fi(\000)2066 2823 y Ff(3)p 2066 2835 31 4 v 2066 2876 a(2)2111 2886 y Fq(\).)257 3175 y Fc(I)t(I)t(I.3)135 b(F)-11 b(ully)45 b(re-expanded)g(v)l(ectors)257 3360 y Fq(In)22 b(this)h(paragraph)e(w)m(e)i(presen)m(t)g(the)f(full)g (re-expansion)h(of)f(the)g(appro)m(ximated)h(ground)257 3480 y(state)33 b(v)m(ector)h(\()p Fo(\036)885 3495 y Fm(m)951 3480 y Fq(\))989 3444 y Fm(T)1044 3480 y Fq(.)257 3684 y Fe(De\014nition)54 b(I)s(I)s(I.3.)48 b Fj(We)f(say)h(that)f(the) g Fo(\013)q Fj(-dep)-5 b(endent)46 b(ve)-5 b(ctor)47 b Fq(\()p Fo(\036)2815 3699 y Fm(m)2881 3684 y Fq(\))2919 3648 y Fm(T)2974 3684 y Fj(,)k Fq(1)f Fn(\024)h Fo(m)g Fn(\024)257 3813 y Fo(M)58 b Fq(=)46 b(2)p Fo(N)40 b Fn(\000)30 b Fq(1)p Fj(,)48 b(c)-5 b(an)44 b(b)-5 b(e)45 b(r)-5 b(e-exp)g(ande)g(d)43 b(up)j(to)f(or)-5 b(der)45 b Fq(2)p Fo(N)10 b Fj(,)47 b(in)e Fo(\013)2749 3749 y Ff(1)p 2749 3761 V 2749 3802 a(2)2793 3813 y Fj(,)j(in)d(the)g (interval)257 3933 y Fo(\013)29 b Fn(2)f Fq(\(0)p Fo(;)e Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)p Fq(\)\))p Fj(,)35 b(wher)-5 b(e)44 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)p Fq(\))28 b Fo(>)f Fq(0)p Fj(,)35 b(if)f(it)h(c)-5 b(an)35 b(b)-5 b(e)34 b(split)h(as)1438 4153 y Fq(\()p Fo(\036)1534 4168 y Fm(m)1600 4153 y Fq(\))1638 4112 y Fm(T)1721 4153 y Fq(=)28 b(\()p Fo(\036)1921 4168 y Fm(m)1987 4153 y Fq(\))2025 4112 y Fm(T)2025 4178 y(])2102 4153 y Fq(+)22 b Fo(\030)2248 4112 y Fm(N)3184 4153 y Fq(\(I)s(I)s(I.51\))257 4373 y Fj(wher)-5 b(e:)257 4494 y(i\))35 b Fn(k)p Fq(\()p Fo(\036)508 4509 y Fm(m)574 4494 y Fq(\))612 4458 y Fm(T)612 4518 y(])667 4494 y Fn(k)27 b(\025)i Fo(\016)i(>)d Fq(0)p Fj(,)34 b(with)h Fo(\016)k Fj(an)34 b Fo(\013)q Fj(-indep)-5 b(endent)33 b(c)-5 b(onstant;)257 4614 y(ii\))35 b Fn(k)p Fo(\030)490 4578 y Fm(N)556 4614 y Fn(k)28 b(\024)g Fo(o)p Fq(\()p Fo(\013)887 4578 y Fm(N)954 4614 y Fq(\))p Fj(;)257 4735 y(iii\))35 b Fq(\()p Fo(\036)518 4750 y Fm(m)584 4735 y Fq(\))622 4698 y Fm(T)622 4759 y(])712 4735 y Fj(is)f(of)h(the)g(form)f(given)g(b)-5 b(elow.)257 4855 y(The)39 b(ve)-5 b(ctor)38 b Fq(\()p Fo(\036)848 4870 y Fm(m)915 4855 y Fq(\))953 4819 y Fm(T)953 4880 y(])1047 4855 y Fj(c)-5 b(an)38 b(b)-5 b(e)39 b(explicitly)f(c)-5 b(ompute)g(d)39 b(in)f(terms)h(of)g(b)-5 b(ar)g(e)38 b(quantities.)57 b(It)257 4975 y(c)-5 b(onsists)40 b(of)f(a)h (\014nite,)h Fo(N)10 b Fj(-dep)-5 b(endent,)40 b(but)h Fo(\013)q Fj(-indep)-5 b(endent)38 b(numb)-5 b(er)39 b(of)h(expr)-5 b(essions)p eop end %%Page: 34 35 TeXDict begin 34 34 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(34)257 573 y Fj(whose)34 b(norm)h(is,)f(at)h (most,)g(of)f(or)-5 b(der)35 b Fn(O)s Fq(\(1\))f Fj(in)h Fo(\013)q Fj(.)44 b(They)35 b(ar)-5 b(e)35 b(of)f(the)h(typ)-5 b(e)819 767 y Fh(Z)865 956 y Fl(\000)909 965 y Ff(2)964 885 y Fo(\013)1027 843 y Fm(m)p Fi(\000)p Fl(2)1184 885 y Fo(dz)1284 834 y Fl(\()p Fm(m)p Fl(\))1280 906 y(2)1421 752 y(\026)1422 767 y Fm(j)t Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1489 802 y Fh(Y)1481 984 y Fm(j)t Fl(=1)1679 767 y Fh(Z)1725 956 y Fl(\000)1769 965 y Ff(2)1824 885 y Fo(\013)1887 843 y Fm(m)p Fi(\000)p Fl(3)2044 885 y Fo(dz)2144 834 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)2140 906 y Fl(2)2430 885 y Fn(\002)22 b Fo(:::)574 b Fq(\(I)s(I)s(I.52\))819 1217 y Fo(:::)22 b Fn(\002)1020 1085 y Fl(\026)1022 1100 y Fm(j)s Fl(\(3\))1030 1134 y Fh(Y)1022 1316 y Fm(j)t Fl(=1)1161 1100 y Fh(Z)1207 1289 y Fl(\000)1251 1298 y Ff(2)1306 1217 y Fo(\013)q(dz)1469 1166 y Fl(\(3\))p Fm(;j)1465 1239 y Fl(2)1631 1085 y(\026)1632 1100 y Fm(j)t Fl(\(2\))1641 1134 y Fh(Y)1632 1316 y Fm(j)t Fl(=1)1772 1100 y Fh(Z)1818 1289 y Fl(\000)1862 1298 y Ff(2)1917 1217 y Fo(dz)2017 1166 y Fl(\(2\))p Fm(;j)2013 1239 y Fl(2)2179 1085 y(\026)2181 1100 y Fm(j)s Fl(\(1\))2189 1134 y Fh(Y)2181 1316 y Fm(j)t Fl(=1)2320 1100 y Fh(Z)2366 1289 y Fl(\000)2410 1298 y Ff(1)2465 1217 y Fo(dz)2565 1166 y Fl(\(1\))p Fm(;j)2561 1239 y Fl(1)2712 1217 y Fo(A')2849 1232 y Fm(el)2930 1217 y Fn(\012)g Fq(\012)257 1515 y Fj(wher)-5 b(e:)257 1564 y Fh(R)297 1660 y Fl(\000)341 1669 y Ff(2)396 1635 y Fo(\013)459 1599 y Fm(m)p Fi(\000)p Fl(3)615 1635 y Fo(dz)715 1584 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)711 1657 y Fl(2)979 1635 y Fj(\()p Fq(=)1129 1564 y Fh(R)1168 1660 y Fl(\000)1212 1669 y Fb(m)p Fd(\000)p Ff(1)1370 1635 y Fo(dz)1470 1584 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)1466 1657 y(m)p Fi(\000)p Fl(1)1734 1635 y Fj(\))39 b(denotes)e(a)h(c)-5 b(ontour)39 b(inte)-5 b(gr)g(al)38 b(with)g(p)-5 b(ath)38 b Fq(\000)3456 1650 y Fl(2)257 1786 y Fj(in)i(the)g(inte)-5 b(gr)g(ation)39 b(variable)g Fo(z)1462 1735 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1458 1807 y(2)1714 1786 y Fj(\(original)g(variable)g Fo(z)2534 1801 y Fm(m)p Fi(\000)p Fl(1)2691 1786 y Fj(\).We)g(note)h (that)g(the)257 1906 y(numb)-5 b(ers,)673 1884 y Fq(\026)675 1906 y Fo(j)6 b Fq(\()p Fo(l)r Fq(\))p Fj(,)34 b(of)h(c)-5 b(ontour)35 b(inte)-5 b(gr)g(ations)34 b(ar)-5 b(e)35 b Fo(\013)q Fj(-indep)-5 b(endent;)257 2026 y(The)35 b(symb)-5 b(ol)34 b Fo(A)h Fj(denotes)f(the)h(inte)-5 b(gr)g(al)404 2335 y Fo(A)28 b Fq(=)608 2218 y Fh(Z)655 2406 y Fi(K)709 2376 y Fb(m)p Fd(\000)p Ff(1)709 2418 y Fb(m)866 2200 y Fl(\026)866 2218 y Fm(l)q Fl(\()p Fm(m)p Fl(\))883 2252 y Fh(Y)880 2436 y Fm(l)q Fl(=1)1022 2335 y Fo(dk)1124 2350 y Fm(l)1150 2335 y Fq(\()p Fo(m)p Fq(\))1328 2218 y Fh(Z)1374 2406 y Fi(K)1428 2376 y Fb(m)p Fd(\000)p Ff(2)1428 2429 y Fb(m)p Fd(\000)p Ff(1)1585 2200 y Fl(\026)1585 2218 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1647 2252 y Fh(Y)1644 2436 y Fm(l)q Fl(=1)1831 2335 y Fo(dk)1933 2350 y Fm(l)1959 2335 y Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\))p Fo(:::)2389 2218 y Fh(Z)2435 2406 y Fi(K)2489 2383 y Ff(0)2489 2429 y(1)2543 2200 y Fl(\026)2544 2218 y Fm(l)q Fl(\(1\))2547 2252 y Fh(Y)2544 2436 y Fm(l)q Fl(=1)2673 2335 y Fo(dk)2775 2350 y Fm(l)2800 2335 y Fq(\(1\))33 b Fo(B)152 b Fq(\(I)s(I)s(I.53\))257 2635 y Fj(with)468 2609 y Fq(\026)469 2635 y Fo(l)r Fq(\()p Fo(m)p Fq(\))p Fo(;)17 b(::;)802 2609 y Fq(\026)803 2635 y Fo(l)r Fq(\(1\))35 b Fj(\014nite)f(and)h Fo(\013)q Fj(-indep)-5 b(endent,)2103 2564 y Fh(R)2142 2663 y Fi(K)2196 2633 y Fb(m)p Fd(\000)p Ff(1)2196 2674 y Fb(m)2337 2635 y Fo(;)17 b(:::;)2506 2564 y Fh(R)2545 2660 y Fi(K)2599 2638 y Ff(0)2599 2683 y(1)2673 2635 y Fj(multiple)34 b(inte)-5 b(gr)g(als;)257 2755 y(The)35 b(op)-5 b(er)g(ator)34 b Fo(B)40 b Fj(c)-5 b(an)35 b(b)-5 b(e)34 b(written)h(as)1453 2975 y Fo(B)e Fq(=)28 b Fo(D)s Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))22 b Fn(\012)g Fo(C)7 b Fq(\()p Fo(k)s Fq(\))885 b(\(I)s(I)s(I.54\))257 3195 y Fj(wher)-5 b(e:)403 3399 y Fn(\017)48 b Fo(C)7 b Fq(\()p Fo(k)s Fq(\))45 b Fj(is)g(the)h(pr)-5 b(o)g(duct)45 b(of)g(a)g(\014nite)g(and)g Fo(\013)q Fj(-indep)-5 b(endent)43 b(numb)-5 b(er)45 b(of)g(photon)501 3519 y(cr)-5 b(e)g(ation)31 b(op)-5 b(er)g(ators,)31 b(c)-5 b(ontaine)g(d)30 b(in)h(the)g(r)-5 b(ange)30 b Fn(K)2364 3483 y Fl(0)2363 3544 y Fm(m)2429 3519 y Fj(;)j(her)-5 b(e)30 b Fo(k)k Fj(stands)d(for)g(al)5 b(l)30 b(the)501 3640 y(momenta)k(\(and)g(helicities\))g(of)h(the)g (photon)f(cr)-5 b(e)g(ation)34 b(op)-5 b(er)g(ators;)403 3843 y Fn(\017)48 b Fo(D)s Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))48 b Fj(is)f(the)h(pr)-5 b(o)g(duct)48 b(of)f(a)h(\014nite)f(and) g Fo(\013)q Fj(-indep)-5 b(endent)46 b(numb)-5 b(er)47 b(of)h(fac-)501 3963 y(tors)36 b(b)-5 b(elonging)34 b(to)h(thr)-5 b(e)g(e)36 b(gr)-5 b(oups,)35 b Fn(G)1883 3927 y Fl(1)1877 3988 y(0)1923 3963 y Fj(,)g Fn(G)2053 3927 y Fl(2)2047 3988 y(0)2128 3963 y Fj(and)g Fn(G)2383 3927 y Fl(3)2377 3988 y(0)2423 3963 y Fj(,)g(sp)-5 b(e)g(ci\014e)g(d)34 b(b)-5 b(elow;)35 b(her)-5 b(e)35 b(the)501 4084 y(ar)-5 b(gument)38 b Fo(k)k Fj(denotes)c(al)5 b(l)38 b(the)g(momenta)g(\(and)f (helicities\))h(inte)-5 b(gr)g(ation)38 b(vari-)501 4204 y(ables)f(and)f(the)h(ar)-5 b(gument)37 b Fo(z)42 b Fj(denotes)37 b(al)5 b(l)36 b(the)h(c)-5 b(ontour)38 b(inte)-5 b(gr)g(ation)36 b(variables)501 4325 y(in)f(Expr)-5 b(ession)34 b(\(III.52\).)562 4528 y Fn(G)627 4492 y Fl(1)621 4553 y(0)1674 4748 y Fo(P)1737 4763 y Fl(0)3184 4748 y Fq(\(I)s(I)s(I.55\))1674 4907 y Fo(R)1749 4866 y Fl(+)11 b Fi(?)1748 4931 y Fl(0)1874 4907 y Fq(\()p Fo(z)1961 4856 y Fl(\(1\))1957 4929 y(1)2079 4907 y Fn(\000)22 b Fo(S)2238 4922 y Fm(l)q Fl(\(1\))2354 4907 y Fq(\()p Fo(k)s Fq(\)\))662 b(\(I)s(I)s(I.56\))1674 5074 y Fo(R)1749 5033 y Fl(+)11 b Fi(?)1748 5099 y Fl(0)1874 5074 y Fq(\()p Fo(z)1961 5023 y Fl(\()p Fm(m)2050 5000 y Fd(0)2074 5023 y Fl(\))1957 5096 y(2)2128 5074 y Fn(\000)23 b Fo(S)2288 5090 y Fm(l)q Fl(\(1\))p Fm(;l)q Fl(\(2\))2536 5074 y Fq(\()p Fo(k)s Fq(\)\))480 b(\(I)s(I)s(I.57\))716 5294 y Fj(wher)-5 b(e)34 b Fq(0)28 b Fn(\024)g Fo(l)r Fq(\(1\))f Fn(\024)1461 5268 y Fq(\026)1461 5294 y Fo(l)s Fq(\(1\))p Fj(,)34 b Fq(0)28 b Fn(\024)g Fo(l)r Fq(\(2\))f Fn(\024)2152 5268 y Fq(\026)2152 5294 y Fo(l)s Fq(\(2\))p Fj(,)34 b Fq(2)28 b Fn(\024)g Fo(m)2640 5258 y Fi(0)2691 5294 y Fn(\024)g Fo(m)p Fj(;)p eop end %%Page: 35 36 TeXDict begin 35 35 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(35)562 573 y Fn(G)627 537 y Fl(2)621 597 y(0)858 793 y Fq(\007)934 752 y Fi(\006)934 817 y Fl(0)993 793 y Fq(\()p Fo(k)1082 808 y Fm(i)1110 793 y Fq(\()p Fo(m)1233 752 y Fi(0)1256 793 y Fq(\)\))399 b Fo(;)415 b Fq(\000)2234 752 y Fi(\006)p Fm(;)p Fi(\006)2368 793 y Fq(\()p Fo(k)2457 808 y Fm(i)2484 793 y Fq(\()p Fo(m)2607 752 y Fi(0)2631 793 y Fq(\))p Fo(;)17 b(k)2764 808 y Fm(j)2800 793 y Fq(\()p Fo(m)2923 752 y Fi(00)2966 793 y Fq(\)\))142 b(\(I)s(I)s(I.58\))716 1013 y Fj(with)35 b Fq(1)27 b Fn(\024)h Fo(m)1194 977 y Fi(0)1218 1013 y Fo(;)17 b(m)1347 977 y Fi(00)1417 1013 y Fn(\024)28 b Fo(m)p Fj(,)35 b Fq(1)28 b Fn(\024)g Fo(i)g Fn(\024)2019 987 y Fq(\026)2020 1013 y Fo(l)r Fq(\()p Fo(m)2174 977 y Fi(0)2198 1013 y Fq(\))p Fj(,)34 b Fq(1)28 b Fn(\024)g Fo(j)34 b Fn(\024)2660 987 y Fq(\026)2661 1013 y Fo(l)r Fq(\()p Fo(m)2815 977 y Fi(00)2858 1013 y Fq(\))p Fj(;)562 1175 y Fn(G)627 1139 y Fl(3)621 1199 y(0)1153 1372 y Fq(1)p 892 1416 572 4 v 892 1526 a Fo(S)952 1542 y Fm(l)q Fl(\(1\))1068 1526 y Fq(\()p Fo(k)s Fq(\))22 b Fn(\000)h Fo(z)1369 1475 y Fl(\(1\))1365 1548 y(1)1474 1439 y Fe(1)1530 1454 y Fl(0)3601 1439 y Fq(\(I)s(I)s(I.59\))1496 1656 y(1)p 892 1700 1257 4 v 892 1811 a Fo(S)952 1832 y Fm(l)q Fl(\()p Fm(m)1063 1813 y Fd(00)1104 1832 y Fl(\))p Fm(;)1150 1815 y Fl(\026)1151 1832 y Fm(l)q Fl(\()p Fm(m)1262 1813 y Fd(00)1304 1832 y Fi(\000)p Fl(1\))1425 1811 y Fq(\()p Fo(k)s Fq(\))f Fn(\000)h Fo(\013)1740 1782 y Fm(m)1802 1763 y Fd(00)1843 1782 y Fi(\000)p Fl(2)1937 1811 y Fo(z)1986 1760 y Fl(\()p Fm(m)2075 1741 y Fd(00)2117 1760 y Fl(\))1982 1833 y(2)2159 1723 y Fe(1)2215 1738 y Fl(0)2271 1723 y Fq(;)2800 1656 y(1)p 2341 1700 968 4 v 2341 1811 a Fo(S)2401 1826 y Fm(l)q Fl(\()p Fm(m)2512 1807 y Fd(00)2553 1826 y Fl(\))2585 1811 y Fq(\()p Fo(k)s Fq(\))f Fn(\000)h Fo(\013)2900 1782 y Fm(m)2962 1763 y Fd(00)3002 1782 y Fi(\000)p Fl(2)3097 1811 y Fo(z)3146 1760 y Fl(\()p Fm(m)3235 1741 y Fd(00)3277 1760 y Fl(\))3142 1833 y(2)3319 1723 y Fe(1)3375 1738 y Fl(0)3601 1723 y Fq(\(I)s(I)s(I.60\))882 1984 y Fo(S)942 1999 y Fm(l)q Fl(\()p Fm(m)p Fl(\))1085 1984 y Fq(\()p Fo(k)s Fq(\))p Fe(1)1271 1999 y Fl(0)1311 1984 y Fo(;)17 b(:::;)g(S)1540 1999 y Fm(l)q Fl(\(3\))1655 1984 y Fq(\()p Fo(k)s Fq(\))p Fe(1)1841 1999 y Fl(0)1898 1984 y Fq(;)49 b(;)34 b(\()p Fo(z)2122 1933 y Fl(\()p Fm(m)2211 1910 y Fd(0)2235 1933 y Fl(\))2118 2009 y Fm(m)2180 1991 y Fd(00)2221 2009 y Fi(\000)p Fl(1)2337 1984 y Fn(\000)23 b Fo(z)2486 1933 y Fl(\()p Fm(m)2575 1910 y Fd(0)2599 1933 y Fl(\))2482 2009 y Fm(m)2544 1991 y Fd(00)2631 1984 y Fq(\))p Fe(1)2725 1999 y Fl(0)3601 1984 y Fq(\(I)s(I)s(I.61\)) 882 2142 y Fn(h)p Fo(W)1027 2101 y Fm(m)p Fi(\000)p Fl(1)1013 2167 y Fm(m)1183 2142 y Fn(i)1222 2157 y Fl(\012)1273 2165 y Fb(m)1336 2142 y Fe(1)1392 2157 y Fl(0)1431 2142 y Fo(;)17 b(::::;)g Fn(h)p Fo(W)1772 2101 y Fl(0)1758 2167 y(1)1811 2142 y Fn(i)1850 2157 y Fl(\012)1901 2166 y Ff(1)1939 2142 y Fe(1)1995 2157 y Fl(0)2051 2142 y Fq(\()p Fo(E)2167 2101 y Fm(m)2229 2077 y Fd(000)2288 2101 y Fi(\000)p Fl(1)2161 2167 y Fm(m)2223 2149 y Fd(000)2383 2142 y Fq(\))2421 2101 y Fm(T)2421 2167 y(])2476 2142 y Fe(1)2532 2157 y Fl(0)716 2362 y Fj(wher)-5 b(e)39 b Fq(2)c Fn(\024)i Fo(m)1279 2326 y Fi(00)1357 2362 y Fn(\024)g Fo(m)1556 2326 y Fi(0)1615 2362 y Fn(\024)g Fo(m)p Fj(,)j Fq(0)c Fn(\024)g Fo(l)r Fq(\()p Fo(m)2236 2326 y Fi(00)2279 2362 y Fq(\))g Fn(\024)2466 2336 y Fq(\026)2467 2362 y Fo(l)r Fq(\()p Fo(m)2621 2326 y Fi(00)2663 2362 y Fq(\))p Fj(,)41 b Fq(0)35 b Fn(\024)i Fo(l)r Fq(\()p Fo(m)3124 2326 y Fi(00)3192 2362 y Fn(\000)26 b Fq(1\))36 b Fn(\024)715 2456 y Fq(\026)716 2482 y Fo(l)r Fq(\()p Fo(m)870 2446 y Fi(00)927 2482 y Fn(\000)15 b Fq(1\))p Fj(,)32 b(and)f(the)h(symb)-5 b(ol)31 b Fq(\()p Fo(E)1942 2441 y Fm(m)2004 2418 y Fd(000)2063 2441 y Fi(\000)p Fl(1)1936 2508 y Fm(m)1998 2489 y Fd(000)2157 2482 y Fq(\))2195 2446 y Fm(T)2195 2507 y(])2250 2482 y Fj(,)h Fq(1)c Fn(\024)g Fo(m)2579 2446 y Fi(000)2668 2482 y Fn(\024)h Fo(m)15 b Fn(\000)g Fq(1)p Fj(,)31 b(stands)g(for)716 2603 y(an)e(explicit)h(expr)-5 b(ession)28 b(which)h(c)-5 b(onsists)30 b(of)f(a)h(\014nite)g(and)f Fo(\013)q Fj(-indep)-5 b(endent)716 2723 y(numb)g(er)32 b(of)g(inte)-5 b(gr)g(als)32 b(involving)g(only)g(b)-5 b(ar)g(e)32 b(quantities)h(ac)-5 b(c)g(or)g(ding)31 b(to)i(the)716 2843 y(formula)1233 3137 y Fq(\()p Fo(E)1349 3096 y Fm(m)1411 3073 y Fd(0000)1489 3096 y Fi(\000)p Fl(1)1343 3163 y Fm(m)1405 3144 y Fd(000)1583 3137 y Fq(\))1621 3096 y Fm(T)1621 3162 y(])1704 3137 y Fq(:=)27 b Fn(\000)1921 3063 y Fq(\()p Fo(W)2065 3022 y Fm(m)2127 2998 y Fd(0000)2205 3022 y Fi(\000)p Fl(1)2051 3089 y Fm(m)2113 3070 y Fd(000)2299 3063 y Fq(\()2350 3037 y(~)2337 3063 y Fo(\036)2395 3078 y Fm(m)2457 3059 y Fd(000)2516 3078 y Fi(\000)p Fl(1)2610 3063 y Fq(\))2648 3027 y Fm(T)2648 3088 y(])2738 3063 y Fo(;)17 b Fq(\()p Fo(\036)2878 3078 y Fm(m)2940 3059 y Fd(000)3003 3063 y Fq(\))3041 3027 y Fm(T)3041 3088 y(])3096 3063 y Fq(\))p 1921 3114 1213 4 v 2091 3216 a(\(\()2180 3190 y(~)2167 3216 y Fo(\036)2225 3231 y Fm(m)2287 3212 y Fd(000)2346 3231 y Fi(\000)p Fl(1)2441 3216 y Fq(\))2479 3182 y Fm(T)2479 3241 y(])2568 3216 y Fo(;)g Fq(\()p Fo(\036)2708 3231 y Fm(m)2770 3212 y Fd(000)2833 3216 y Fq(\))2871 3182 y Fm(T)2871 3241 y(])2926 3216 y Fq(\))3184 3137 y(\(I)s(I)s(I.62\)) 1233 3380 y(\()1284 3354 y(~)1271 3380 y Fo(\036)1329 3395 y Fm(m)1391 3376 y Fd(000)1450 3395 y Fi(\000)p Fl(1)1545 3380 y Fq(\))1583 3339 y Fm(T)1583 3405 y(])1665 3380 y Fq(:=)28 b(\()p Fo(\036)1892 3395 y Fm(m)1954 3376 y Fd(000)2013 3395 y Fi(\000)p Fl(1)2107 3380 y Fq(\))2145 3339 y Fm(T)2145 3405 y(])2222 3380 y Fn(\012)23 b Fq(\012)2392 3339 y Fm(m)2454 3315 y Fd(000)2513 3339 y Fi(\000)p Fl(1)2392 3406 y Fm(m)2454 3387 y Fd(000)3184 3380 y Fq(\(I)s(I)s(I.63\))716 3600 y Fj(wher)-5 b(e)29 b(it)g(is)h(assume)-5 b(d)28 b(that)i(al)5 b(l)29 b(the)h(ve)-5 b(ctors)29 b(in)g Fn(f)p Fq(\()p Fo(\036)2624 3615 y Fm(j)2660 3600 y Fq(\))2698 3564 y Fm(T)2753 3600 y Fq(\))p Fn(j)p Fo(j)k Fq(=)28 b(1)p Fo(;)17 b(:::;)g(m)10 b Fn(\000)g Fq(1)p Fn(g)716 3720 y Fj(c)-5 b(an)30 b(b)-5 b(e)30 b(ful)5 b(ly)30 b(r)-5 b(e-exp)g(ande)g(d)29 b(in)h Fq(\(0)p Fo(;)c Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)p Fq(\)\))31 b Fj(and)f Fn(j)p Fq(\(\()2585 3694 y(~)2572 3720 y Fo(\036)2630 3735 y Fm(j)t Fi(\000)p Fl(1)2756 3720 y Fq(\))2794 3684 y Fm(T)2794 3745 y(])2879 3720 y Fo(;)17 b Fq(\()p Fo(\036)3019 3735 y Fm(j)3055 3720 y Fq(\))3093 3684 y Fm(T)3093 3745 y(])3148 3720 y Fq(\))p Fn(j)27 b Fo(>)h(\016)j(>)716 3841 y Fq(0)p Fj(,)j(wher)-5 b(e)35 b Fo(\016)j Fj(is)d(an)g Fo(\013)q Fj(-indep)-5 b(endent)33 b(c)-5 b(onstant.)257 4044 y Fe(Remark)257 4165 y Fq(The)26 b(de\014nition)g(of)e(\()p Fo(\036)1075 4180 y Fm(m)1141 4165 y Fq(\))1179 4128 y Fm(T)1179 4189 y(])1259 4165 y Fq(dep)s(ends)i(only)f(on)g(the)g(v)m (ectors)h(\()p Fo(\036)2541 4180 y Fm(m)2603 4161 y Fd(0)2630 4165 y Fq(\))2668 4128 y Fm(T)2668 4189 y(])2747 4165 y Fq(with)f(1)j Fn(\024)g Fo(m)3228 4128 y Fi(0)3279 4165 y Fo(<)g(m)p Fq(.)257 4424 y Fe(I)s(I)s(I.3.1)112 b(Explicit)37 b(truncation)h(map)g Fo(\034)1943 4388 y Fm(N)1932 4449 y(])2049 4424 y Fe(and)g(main)h(theorem)257 4609 y Fq(In)49 b(the)g(next)h(theorem)f(w)m(e)h(pro)m(v)m(e)g(that)f (the)g(v)m(ector)g(\()p Fo(\036)2465 4624 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2658 4609 y Fq(\))2696 4573 y Fm(T)2799 4609 y Fq(can)49 b(b)s(e)g(fully)g(re-)257 4729 y(expanded)24 b(and)e(the)g(related)g(energy)h(shift)g(has)f(an)g(explicit)h (expression)h(up)e(to)g(order)g Fo(N)257 4850 y Fq(in)j Fo(\013)q Fq(.)40 b(These)25 b(results,)j(com)m(bined)d(with)g(\(I)s (I.15\),)g(\(I)s(I.20\),)g(yield)g(the)g(expansions)h(\(I.21\))257 4970 y(and)40 b(\(I.22\))f(announced)i(in)f(Section)g(I,)g(for)f Fo(\013)h Fq(in)f(an)h Fo(N)10 b Fq(-dep)s(enden)m(t)41 b(neigh)m(b)s(orho)s(o)s(d)257 5091 y(of)31 b(0.)42 b(Before)31 b(stating)g(the)g(theorem,)h(w)m(e)g(m)m(ust)g(de\014ne)g(the)f Fj(explicit)i(trunc)-5 b(ation)33 b(map)257 5211 y Fo(\034)310 5175 y Fm(N)299 5236 y(])411 5211 y Fq(whic)m(h)h(is)f(used)h(in)e(the) h(pro)s(of.)257 5331 y Fj(De\014nition)h(of)h(the)f(map)h Fo(\034)1261 5295 y Fm(N)1250 5356 y(])p eop end %%Page: 36 37 TeXDict begin 36 36 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(36)257 573 y(The)26 b(map)e Fo(\034)711 537 y Fm(N)700 597 y(])803 573 y Fq(deriv)m(es)j(from)d Fo(\034)1396 537 y Fm(N)1488 573 y Fq(de\014ned)i(in)e(Section)i(I)s(I) s(I.2)e(b)m(y)h(replacing)g(the)g(v)-5 b(ariable)257 693 y(\()p Fo(E)373 657 y Fm(m)p Fi(\000)p Fl(1)367 718 y Fm(m)530 693 y Fq(\))568 657 y Fm(T)662 693 y Fq(b)m(y)41 b(the)e(explicit)i(quan)m(tit)m(y)g(\()p Fo(E)1843 657 y Fm(m)p Fi(\000)p Fl(1)1837 718 y Fm(m)2000 693 y Fq(\))2038 657 y Fm(T)2038 718 y(])2132 693 y Fq(\(see)g(De\014nition)f(I)s(I)s (I.3\),)g(1)f Fn(\024)h Fo(m)g Fn(\024)257 814 y Fo(M)f Fq(=)27 b(2)p Fo(N)19 b Fn(\000)8 b Fq(1,)27 b(whic)m(h)g(will)f(turn)g (out)g(to)f(b)s(e)h(w)m(ell)g(de\014ned)i(and)d(ful\014lling)h(the)g (prop)s(ert)m(y)257 934 y(\(see)34 b(Theorem)g(I)s(I)s(I.4\))1091 1167 y Fn(j)27 b Fq(\()p Fo(E)1262 1126 y Fm(m)p Fi(\000)p Fl(1)1256 1192 y Fm(m)1419 1167 y Fq(\))1457 1126 y Fm(T)1457 1192 y(])1534 1167 y Fn(\000)22 b Fq(\()p Fo(E)1749 1126 y Fm(m)p Fi(\000)p Fl(1)1743 1192 y Fm(m)1906 1167 y Fq(\))1944 1126 y Fm(T)2027 1167 y Fn(j\024)28 b Fo(o)p Fq(\()p Fo(\013)2308 1126 y Fm(m)p Fl(+)2435 1099 y Ff(1)p 2434 1111 31 4 v 2434 1152 a(2)2475 1126 y Fl(+)p Fm(N)2597 1167 y Fq(\))p Fo(:)522 b Fq(\(I)s(I)s(I.64\))257 1387 y(Therefore)32 b(Lemmata)g(I)s(I)s(I.1)e(and)h(I)s(I)s(I.2)g(hold)g (for)f Fo(\034)2169 1351 y Fm(N)2158 1412 y(])2268 1387 y Fq(as)h(w)m(ell;)i(see)f(the)f(Remark)g(after)257 1508 y(Lemma)k(I)s(I)s(I.2.)49 b(In)35 b(the)g(case)g(of)f(the)h(op)s (erators)f Fn(f)p Fo(P)2201 1523 y Fm(m)2267 1508 y Fn(j)17 b Fo(m)31 b Fq(=)f(1)p Fo(;)17 b(:::;)g Fq(2)p Fo(N)33 b Fn(\000)24 b Fq(1)p Fn(g)p Fq(,)35 b(the)f(t)m(w)m(o)257 1628 y(maps,)g Fo(\034)593 1592 y Fm(N)693 1628 y Fq(and)f Fo(\034)936 1592 y Fm(N)925 1653 y(])1003 1628 y Fq(,)g(coincide.)257 1831 y Fe(Theorem)49 b(I)s(I)s(I.4.)d Fj(F)-7 b(or)43 b(any)g Fo(N)54 b Fn(2)44 b Fk(N)p Fj(,)h(the)f(ve)-5 b(ctor)43 b Fq(\()p Fo(\036)2377 1846 y Fm(M)7 b Fl(=2)p Fm(N)g Fi(\000)p Fl(1)2699 1831 y Fq(\))2737 1795 y Fm(T)2836 1831 y Fj(c)-5 b(an)43 b(b)-5 b(e)43 b(ful)5 b(ly)43 b(r)-5 b(e-)257 1961 y(exp)g(ande)g(d,)35 b(up)g(to)h(the)f(or)-5 b(der)36 b Fq(2)p Fo(N)10 b Fj(,)35 b(in)h Fo(\013)1773 1897 y Ff(1)p 1772 1909 V 1772 1950 a(2)1817 1961 y Fj(,)f(for)h Fo(\013)g Fj(in)f(the)h(interval)f Fq(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(M)10 b Fq(\)\))p Fj(,)36 b(wher)-5 b(e)267 2081 y Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(M)10 b Fq(\))46 b Fo(>)f Fq(0)p Fj(.)73 b(In)44 b(the)g(same)g (interval,)i Fq(\()p Fo(E)1901 2040 y Fm(M)7 b Fi(\000)p Fl(1)1895 2105 y Fm(M)2070 2081 y Fq(\))2108 2045 y Fm(T)2108 2106 y(])2207 2081 y Fj(is)44 b(wel)5 b(l)44 b(de\014ne)-5 b(d)44 b(and)f(ful\014l)5 b(ls)44 b(the)257 2201 y(c)-5 b(onstr)g(aint)1133 2322 y Fn(j)p Fq(\()p Fo(E)1277 2281 y Fm(M)7 b Fi(\000)p Fl(1)1271 2346 y Fm(M)1446 2322 y Fq(\))1484 2281 y Fm(T)1561 2322 y Fn(\000)23 b Fq(\()p Fo(E)1777 2281 y Fm(M)7 b Fi(\000)p Fl(1)1771 2346 y Fm(M)1946 2322 y Fq(\))1984 2281 y Fm(T)1984 2346 y(])2039 2322 y Fn(j)27 b(\024)h Fo(o)p Fq(\()p Fo(\013)2347 2281 y Fl(3)p Fm(N)7 b Fi(\000)2510 2253 y Ff(1)p 2510 2265 V 2510 2307 a(2)2555 2322 y Fq(\))p Fo(:)564 b Fq(\(I)s(I)s(I.65\))257 2525 y Fj(Pr)-5 b(o)g(of.)50 b Fq(The)36 b(pro)s(of)e(is)i(b)m(y)g (induction)g(on)e(a)h(\014nite)h(set)f Fn(f)p Fo(m)17 b Fn(j)p Fq(1)31 b Fn(\024)h Fo(m)g Fn(\024)g Fq(2)p Fo(N)i Fn(\000)25 b Fq(1)p Fn(g)p Fq(.)50 b(W)-8 b(e)257 2646 y(assume)36 b(that)e(certain)g(prop)s(erties)h(are)f(ful\014lled)h (for)e(1)d Fn(\024)h Fo(q)i Fn(\024)e Fo(m)23 b Fn(\000)h Fq(1,)34 b(and)g(w)m(e)h(sho)m(w)257 2766 y(that)e(they)g(hold)g(true)g (for)f Fo(q)f Fq(=)d Fo(m)p Fq(.)257 2886 y Fj(Inductive)34 b(hyp)-5 b(otheses)257 3007 y Fq(F)d(or)32 b(1)c Fn(\024)g Fo(q)j Fn(\024)d Fo(m)23 b Fn(\000)f Fq(1:)257 3127 y Fa(H1\))35 b Fq(The)c(v)m(ector)h(\()p Fo(\036)1072 3142 y Fm(q)1110 3127 y Fq(\))1148 3091 y Fm(T)1234 3127 y Fq(can)f(b)s(e)g(fully)g(re-expanded)h(up)f(to)g(the)g(order)g(2)p Fo(N)41 b Fq(in)31 b Fo(\013)3339 3063 y Ff(1)p 3339 3075 V 3339 3116 a(2)3414 3127 y Fq(in)257 3247 y(an)i(in)m(terv)-5 b(al)33 b(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)21 b Fn(\000)i Fq(1\)\))31 b(where)44 b(\026)-59 b Fo(\013)q Fq(\()p Fo(m)21 b Fn(\000)i Fq(1\))k Fo(>)g Fq(0,)33 b(i.e.,)g(a)f(v)m(ector)h(\()p Fo(\036)2906 3262 y Fm(q)2944 3247 y Fq(\))2982 3211 y Fm(T)2982 3272 y(])3069 3247 y Fq(exists)h(and)257 3368 y(has)f(all)g(the)g(prop)s (erties)g(sp)s(eci\014ed)i(in)e(De\014nition)f(I)s(I)s(I.3;)257 3488 y Fa(H2\))k Fq(F)-8 b(or)31 b Fo(\013)d Fn(2)g Fq(\(0)p Fo(;)e Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)21 b Fn(\000)g Fq(1\)\),)32 b(the)g(truncated)g(energy)h(shifts)g(are)f(w)m(ell)h (de\014ned)g(and)257 3609 y(ful\014ll)g(the)g(prop)s(ert)m(y)1177 3829 y(\()p Fo(E)1293 3787 y Fm(q)r Fi(\000)p Fl(1)1287 3853 y Fm(q)1421 3829 y Fq(\))1459 3787 y Fm(T)1541 3829 y Fq(=)28 b(\()p Fo(E)1761 3787 y Fm(q)r Fi(\000)p Fl(1)1755 3853 y Fm(q)1889 3829 y Fq(\))1927 3787 y Fm(T)1927 3853 y(])2004 3829 y Fq(+)22 b Fo(o)p Fq(\()p Fo(\013)2250 3787 y Fm(q)r Fl(+)2349 3760 y Ff(1)p 2349 3772 V 2349 3814 a(2)2389 3787 y Fl(+)p Fm(N)2511 3829 y Fq(\);)608 b(\(I)s(I)s(I.66\))257 4049 y Fa(H3\))46 b Fq(Giv)m(en)e(the)f(natural) f(n)m(um)m(b)s(er)i Fo(a)e Fq(suc)m(h)i(that)e(2)j Fn(\024)g Fo(q)j Fn(\024)d Fo(a)f Fn(\024)h Fq(2)p Fo(N)40 b Fn(\000)29 b Fq(1,)45 b(in)d(an)257 4169 y(in)m(terv)-5 b(al)41 b Fo(\013)f Fn(2)h Fq(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)1019 4184 y Fm( )1065 4192 y Fb(q)1103 4169 y Fq(\),)41 b(where)51 b(\026)-59 b Fo(\013)q Fq(\()p Fo(q)31 b Fn(\000)d Fq(1\))39 b Fn(\025)50 b Fq(\026)-58 b Fo(\013)2084 4184 y Fm( )2130 4192 y Fb(q)2208 4169 y Fo(>)40 b Fq(0,)h(it)f(is)g(p)s(ossible)i(to)d (fully)h(re-)257 4306 y(expand)25 b(a)e(v)m(ector)i Fo( )1006 4321 y Fm(q)1044 4306 y Fq(\()p Fo(\013)q Fq(\),)g(de\014ned)g(b)s(elo) m(w,)h(up)e(to)f(the)h(order)g(2)p Fo(N)34 b Fq(in)23 b Fo(\013)2843 4242 y Ff(1)p 2843 4254 V 2843 4296 a(2)2887 4306 y Fq(,)j(i.e.,)g(w)m(e)e(adapt)257 4427 y(De\014nition)38 b(I)s(I)s(I.3)e(to)h(the)g(v)m(ector)h Fo( )1595 4442 y Fm(q)1670 4427 y Fq(in)f(a)g(natural)g(w)m(a)m(y)-8 b(,)39 b(b)m(y)f(dropping)f(condition)g(i\))257 4547 y(\(in)f(De\014nition)f(I)s(I)s(I.3\))g(to)g(include)h(v)m(ectors)h Fo( )1985 4562 y Fm(q)2059 4547 y Fq(whose)f(norm)f(is)h(not)f(b)s (ounded)h(a)m(w)m(a)m(y)257 4667 y(from)d(0)f(uniformly)h(in)g Fo(\013)q Fq(.)43 b(The)34 b(de\014nition)f(of)f(the)h(v)m(ector)h Fo( )2529 4682 y Fm(q)2600 4667 y Fq(is:)505 4982 y Fo( )568 4997 y Fm(q)690 4982 y Fq(:=)875 4874 y Fm(a)841 4899 y Fh(Y)820 5084 y Fm(p)856 5065 y Fd(0)878 5084 y Fl(=)p Fm(q)984 4886 y Fh(\020)1048 4850 y Fl(\026)1050 4865 y Fm(j)s Fl(\()p Fm(p)1145 4841 y Fd(0)1167 4865 y Fl(\))1069 4899 y Fh(Y)1061 5081 y Fm(j)t Fl(=1)1211 4865 y Fh(Z)1258 5054 y Fl(\000)1302 5062 y Fb(q)1357 4982 y Fo(\013)1420 4941 y Fm(p)1456 4918 y Fd(0)1477 4941 y Fi(\000)p Fm(q)1570 4982 y Fo(dz)1670 4941 y Fl(\()p Fm(p)1733 4918 y Fd(0)1756 4941 y Fl(\))p Fm(;j)1666 5007 y(q)1840 4886 y Fh(\021)1904 4850 y Fl(\026)1906 4865 y Fm(j)s Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1958 4899 y Fh(Y)1950 5081 y Fm(j)t Fl(=1)2134 4865 y Fh(Z)2180 5054 y Fl(\000)2224 5063 y Fb(q)r Fd(\000)p Ff(1)2357 4982 y Fo(dz)2457 4931 y Fl(\()p Fm(q)r Fi(\000)p Fl(1\))p Fm(;j)2453 5004 y(q)r Fi(\000)p Fl(1)2715 4982 y Fn(\002)392 b Fq(\(I)s(I)s(I.67\))690 5317 y Fn(\002)784 5209 y Fm(a)p Fl(+1)795 5234 y Fh(Y)785 5409 y Fm(p)p Fl(=)p Fm(q)928 5221 y Fh(\020)994 5200 y(Z)1040 5389 y Fi(K)1094 5356 y Fb(p)p Fd(\000)p Ff(2)1094 5411 y Fb(p)p Fd(\000)p Ff(1)1224 5185 y Fl(\026)1230 5200 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1279 5234 y Fh(Y)1275 5417 y Fm(i)p Fl(=1)1451 5317 y Fo(dk)1553 5332 y Fm(i)1581 5317 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))1876 5221 y Fh(\021)1925 5317 y Fo(D)2006 5332 y Fm(q)r Fi(\000)p Fl(2)2135 5317 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2454 5332 y Fm(q)r Fi(\000)p Fl(2)2582 5317 y Fq(\))2620 5276 y Fm(T)2697 5317 y Fn(\012)23 b Fo(C)2874 5276 y Fm(q)r Fi(\000)p Fl(2)2867 5342 y Fm(a)3002 5317 y Fq(\()p Fo(k)s Fq(\)\012)3202 5276 y Fm(q)r Fi(\000)p Fl(2)3202 5342 y Fm(a)p eop end %%Page: 37 38 TeXDict begin 37 37 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(37)257 573 y(where)257 693 y(i\))354 671 y(\026)355 693 y Fo(j)6 b Fq(\()p Fo(p)488 657 y Fi(0)511 693 y Fq(\))p Fo(;)591 671 y Fq(\026)593 693 y Fo(j)g Fq(\()p Fo(q)26 b Fn(\000)c Fq(1\))p Fo(;)968 671 y Fq(\026)976 693 y Fo(i)p Fq(\()p Fo(p)g Fn(\000)h Fq(1\))32 b(are)g Fo(\013)q Fq(-indep)s(enden)m(t)i(natural)f(n)m(um)m (b)s(ers;)257 814 y(ii\))26 b Fo(C)452 777 y Fm(q)r Fi(\000)p Fl(2)445 838 y Fm(a)580 814 y Fq(\()p Fo(k)s Fq(\))g(is)g(a)g(\014nite) g(pro)s(duct)g(of)g(an)f Fo(\013)q Fq(-indep)s(enden)m(t)j(n)m(um)m(b)s (er)f(of)e(photon)h(creation)257 934 y(op)s(erators)33 b(with)g(momen)m(ta)g(con)m(tained)h(in)f(the)g(range)f Fn(K)2402 898 y Fm(q)r Fi(\000)p Fl(2)2401 959 y Fm(a)2531 934 y Fq(;)257 1054 y(iii\))50 b Fo(D)507 1069 y Fm(q)r Fi(\000)p Fl(2)635 1054 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))49 b(is)h(a)e(\014nite)i(pro)s(duct)f(of)g(an)g Fo(\013)q Fq(-indep)s(enden)m(t)h(n)m(um)m(b)s(er)h(of)d(factors)257 1175 y(among)36 b(the)g(three)g(groups)g(b)s(elo)m(w)g(and)g(with)g (the)g(constrain)m(ts)h(giv)m(en)g(b)m(y)f(t)m(w)m(o)h(prop-)257 1295 y(erties,)d Fn(P)8 b Fq(1)33 b(and)g Fn(P)8 b Fq(2,)32 b(discussed)k(later.)257 1416 y Fj(Gr)-5 b(oups)403 1616 y Fn(\017)48 b(G)566 1580 y Fl(1)560 1640 y Fm(q)r Fi(\000)p Fl(2)1535 1832 y Fo(P)1598 1847 y Fm(q)r Fi(\000)p Fl(2)3184 1832 y Fq(\(I)s(I)s(I.68\))1535 1994 y Fo(R)1610 1953 y Fl(+)12 b Fi(?)1609 2019 y Fm(q)r Fi(\000)p Fl(2)1738 1994 y Fq(\()p Fo(z)1825 1943 y Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1821 2016 y Fm(q)r Fi(\000)p Fl(1)2031 1994 y Fn(\000)22 b Fo(S)2190 2010 y Fm(l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2395 1994 y Fq(\()p Fo(k)s Fq(\)\))621 b(\(I)s(I)s(I.69\))1535 2157 y Fo(R)1610 2116 y Fl(+)12 b Fi(?)1609 2182 y Fm(q)r Fi(\000)p Fl(2)1738 2157 y Fq(\()p Fo(z)1825 2116 y Fl(\()p Fm(p)1888 2092 y Fd(0)1911 2116 y Fl(\))1821 2182 y Fm(q)1965 2157 y Fn(\000)22 b Fo(S)2124 2178 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;)2253 2161 y Fl(\026)2254 2178 y Fm(l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2459 2157 y Fq(\()p Fo(k)s Fq(\)\))557 b(\(I)s(I)s(I.70\))501 2382 y(where)32 b Fo(q)g Fn(\024)c Fo(p)1010 2346 y Fi(0)1061 2382 y Fn(\024)g Fo(a)p Fq(,)j(and)g(0)c Fn(\024)h Fo(l)r Fq(\()p Fo(q)t Fq(\))g Fn(\024)1931 2355 y Fq(\026)1931 2382 y Fo(l)r Fq(\()p Fo(q)t Fq(\),)j(0)d Fn(\024)g Fo(l)r Fq(\()p Fo(q)22 b Fn(\000)d Fq(1\))27 b Fn(\024)2773 2355 y Fq(\026)2774 2382 y Fo(l)r Fq(\()p Fo(q)22 b Fn(\000)d Fq(1\))30 b(with)3341 2355 y(\026)3341 2382 y Fo(l)s Fq(\()p Fo(q)t Fq(\))501 2502 y(and)690 2476 y(\026)691 2502 y Fo(l)r Fq(\()p Fo(q)c Fn(\000)d Fq(1\))32 b Fo(\013)q Fq(-indep)s(enden)m(t)i(n)m(um)m(b)s(ers.)403 2705 y Fn(\017)48 b(G)566 2668 y Fl(2)560 2729 y Fm(q)r Fi(\000)p Fl(2)782 2921 y Fq(\007)858 2880 y Fi(\006)858 2946 y Fm(q)r Fi(\000)p Fl(2)986 2921 y Fq(\()p Fo(k)1075 2936 y Fm(i)1103 2921 y Fq(\()p Fo(q)1188 2880 y Fi(0)1211 2921 y Fq(\)\))390 b Fo(;)407 b Fq(\000)2172 2880 y Fi(\006)p Fm(;)p Fi(\006)2306 2921 y Fq(\()p Fo(k)2395 2936 y Fm(i)2423 2921 y Fq(\()p Fo(q)2508 2880 y Fi(0)2531 2921 y Fq(\))p Fo(;)17 b(k)2664 2936 y Fm(j)2700 2921 y Fq(\()p Fo(q)2785 2880 y Fi(00)2827 2921 y Fq(\)\))281 b(\(I)s(I)s(I.71\))501 3137 y(where)37 b Fo(q)28 b Fn(\000)d Fq(1)32 b Fn(\024)i Fo(q)1198 3101 y Fi(0)1221 3137 y Fo(;)17 b(q)1312 3101 y Fi(00)1386 3137 y Fn(\024)34 b Fo(a)p Fq(,)i(1)d Fn(\024)g Fo(i)g Fn(\024)1979 3111 y Fq(\026)1980 3137 y Fo(l)r Fq(\()p Fo(q)2096 3101 y Fi(0)2119 3137 y Fq(\),)j(1)d Fn(\024)g Fo(j)39 b Fn(\024)2600 3111 y Fq(\026)2601 3137 y Fo(l)r Fq(\()p Fo(q)2717 3101 y Fi(00)2760 3137 y Fq(\))c(with)3058 3111 y(\026)3058 3137 y Fo(l)r Fq(\()p Fo(q)3174 3101 y Fi(0)3198 3137 y Fq(\),)3298 3111 y(\026)3299 3137 y Fo(l)r Fq(\()p Fo(q)3415 3101 y Fi(00)3457 3137 y Fq(\))501 3258 y Fo(\013)q Fq(-indep)s(enden)m(t)f(n)m(um)m(b)s(ers.) 403 3460 y Fn(\017)48 b(G)566 3424 y Fl(3)560 3485 y Fm(q)r Fi(\000)p Fl(2)1149 3666 y Fq(1)p 798 3710 750 4 v 798 3821 a Fo(S)858 3836 y Fm(l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1063 3821 y Fq(\()p Fo(k)s Fq(\))22 b Fn(\000)h Fo(z)1364 3770 y Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1360 3843 y Fm(q)r Fi(\000)p Fl(1)1558 3734 y Fe(1)1614 3749 y Fm(q)r Fi(\000)p Fl(2)3184 3734 y Fq(\(I)s(I)s(I.72\)) 1311 3953 y(1)p 798 3997 1076 4 v 798 4108 a Fo(S)858 4123 y Fm(l)q Fl(\()p Fm(p)943 4104 y Fd(0)965 4123 y Fl(\))p Fm(;l)q Fl(\()p Fm(p)1097 4104 y Fd(0)1120 4123 y Fi(\000)p Fl(1\))1241 4108 y Fq(\()p Fo(k)s Fq(\))f Fn(\000)h Fo(\013)1556 4079 y Fm(p)1592 4060 y Fd(0)1614 4079 y Fi(\000)p Fm(q)1707 4108 y Fo(z)1756 4057 y Fl(\()p Fm(p)1819 4038 y Fd(0)1842 4057 y Fl(\))1752 4115 y Fm(q)1883 4020 y Fe(1)1939 4035 y Fm(q)r Fi(\000)p Fl(2)2084 4020 y Fq(;)2545 3953 y(1)p 2154 3997 831 4 v 2154 4108 a Fo(S)2214 4123 y Fm(l)q Fl(\()p Fm(p)2299 4104 y Fd(0)2322 4123 y Fl(\))2353 4108 y Fq(\()p Fo(k)s Fq(\))f Fn(\000)h Fo(\013)2668 4079 y Fm(p)2704 4060 y Fd(0)2725 4079 y Fi(\000)p Fm(q)2818 4108 y Fo(z)2867 4057 y Fl(\()p Fm(p)2930 4038 y Fd(0)2953 4057 y Fl(\))2863 4115 y Fm(q)2995 4020 y Fe(1)3051 4035 y Fm(q)r Fi(\000)p Fl(2)3184 4020 y Fq(\(I)s(I)s(I.73\))788 4263 y Fn(f)p Fo(\030)886 4216 y Fm(q)r Fi(\000)p Fl(2)881 4285 y Fm(q)r Fi(\000)p Fl(1)1014 4263 y Fe(1)1070 4278 y Fm(q)r Fi(\000)p Fl(2)1198 4263 y Fn(g)p Fo(;)17 b(::::;)g Fn(f)p Fo(\030)1542 4222 y Fm(a)p Fi(\000)p Fl(1)1537 4288 y Fm(a)1673 4263 y Fe(1)1729 4278 y Fm(q)r Fi(\000)p Fl(2)1857 4263 y Fn(g)1277 b Fq(\(I)s(I)s(I.74\))501 4492 y(where)40 b Fo(q)i Fn(\024)c Fo(p)1038 4456 y Fi(0)1099 4492 y Fn(\024)g Fo(a)h Fq(and)f(eac)m(h)h (of)f(the)h(families)h Fn(f)p Fo(\030)2475 4445 y Fm(q)r Fi(\000)p Fl(2)2470 4514 y Fm(q)r Fi(\000)p Fl(1)2602 4492 y Fe(1)2658 4507 y Fm(q)r Fi(\000)p Fl(2)2787 4492 y Fn(g)p Fo(;)17 b(::::;)g Fn(f)p Fo(\030)3131 4456 y Fm(a)p Fi(\000)p Fl(1)3126 4517 y Fm(a)3261 4492 y Fe(1)3317 4507 y Fm(q)r Fi(\000)p Fl(2)3445 4492 y Fn(g)501 4626 y Fq(ob)s(ey)35 b Fn(j)p Fo(\030)812 4579 y Fm(j)t Fi(\000)p Fl(2)807 4649 y Fm(j)t Fi(\000)p Fl(1)937 4626 y Fn(j)30 b(\024)g(O)s Fq(\()p Fo(\013)1285 4590 y Fm(j)t Fi(\000)p Fl(1)1412 4626 y Fq(\))k(and)g(#)p Fn(f)p Fo(\030)1854 4579 y Fm(j)t Fi(\000)p Fl(2)1849 4649 y Fm(j)t Fi(\000)p Fl(1)1980 4626 y Fe(1)2036 4641 y Fm(q)r Fi(\000)p Fl(2)2164 4626 y Fn(g)c(\024)g Fo(L)p Fq(\()p Fo(N)10 b Fq(\))35 b(where)g Fo(L)p Fq(\()p Fo(N)10 b Fq(\))35 b(is)f(an)g Fo(\013)q Fq(-)501 4747 y(indep)s(enden)m(t)39 b(n)m(um)m(b)s(er.)58 b(The)37 b(c-n)m(um)m(b)s(ers)i(are)e(supp)s(osed)h(to)e(b)s(e)h (explicit,)j(i.e.)501 4867 y(they)34 b(can)f(b)s(e)f(computed)i(in)f (terms)g(of)g(bare)f(quan)m(tities.)257 5067 y Fj(Pr)-5 b(op)g(erties)35 b Fn(P)8 b Fq(1)p Fj(,)p Fn(P)g Fq(2)326 5268 y Fn(P)g Fq(1)49 b(Next)32 b(to)f(eac)m(h)h(op)s(erator)e Fo(g)1516 5232 y Fl(2)1512 5292 y Fm(q)r Fi(\000)p Fl(2)1668 5268 y Fn(2)e(G)1827 5232 y Fl(2)1821 5292 y Fm(q)r Fi(\000)p Fl(2)1949 5268 y Fq(,)k(there)f(is)h(an)f(op)s(erator)f Fo(g)2928 5232 y Fl(1)2924 5292 y Fm(q)r Fi(\000)p Fl(2)3080 5268 y Fn(2)e(G)3239 5232 y Fl(1)3233 5292 y Fm(q)r Fi(\000)p Fl(2)3392 5268 y Fq(on)501 5388 y(the)33 b(left)g(and)g(on)f(the)h (righ)m(t)g(side.)p eop end %%Page: 38 39 TeXDict begin 38 38 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(38)326 573 y Fn(P)8 b Fq(2)49 b(The)39 b(follo)m(wing)f(estimate)h(of)e(the)h(norm)g(of)f(the)i(v)m (ector)f(\(I)s(I)s(I.67\))g(is,)h(at)f(most,)501 693 y Fn(O)s Fq(\(1\))32 b(in)h Fo(\013)777 875 y Fh(\015)777 925 y(\015)777 975 y(\015)777 1024 y(\015)894 891 y Fm(a)860 916 y Fh(Y)839 1101 y Fm(p)875 1082 y Fd(0)897 1101 y Fl(=)p Fm(q)1003 903 y Fh(\020)1067 867 y Fl(\026)1069 882 y Fm(j)s Fl(\()p Fm(p)1164 859 y Fd(0)1186 882 y Fl(\))1088 916 y Fh(Y)1080 1099 y Fm(j)t Fl(=1)1230 882 y Fh(Z)1277 1071 y Fl(\000)1321 1079 y Fb(q)1376 999 y Fo(\013)1439 958 y Fm(p)1475 935 y Fd(0)1496 958 y Fi(\000)p Fm(q)1589 999 y Fo(dz)1689 958 y Fl(\()p Fm(p)1752 935 y Fd(0)1775 958 y Fl(\))p Fm(;j)1685 1024 y(q)1859 903 y Fh(\021)1923 867 y Fl(\026)1925 882 y Fm(j)s Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1977 916 y Fh(Y)1969 1099 y Fm(j)t Fl(=1)2153 882 y Fh(Z)2199 1071 y Fl(\000)2243 1080 y Fb(q)r Fd(\000)p Ff(1)2376 999 y Fo(dz)2476 949 y Fl(\()p Fm(q)r Fi(\000)p Fl(1\))p Fm(;j)2472 1021 y(q)r Fi(\000)p Fl(1)2734 999 y Fn(\002)373 b Fq(\(I)s(I)s(I.75\))777 1335 y Fn(\002)871 1227 y Fm(a)p Fl(+1)881 1252 y Fh(Y)872 1426 y Fm(p)p Fl(=)p Fm(q)1015 1238 y Fh(\020)1081 1217 y(Z)1127 1406 y Fi(K)1181 1374 y Fb(p)p Fd(\000)p Ff(2)1181 1429 y Fb(p)p Fd(\000)p Ff(1)1311 1202 y Fl(\026)1317 1217 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1366 1252 y Fh(Y)1362 1434 y Fm(i)p Fl(=1)1538 1335 y Fo(dk)1640 1350 y Fm(i)1667 1335 y Fq(\()p Fo(p)22 b Fn(\000)h Fq(1\))1963 1238 y Fh(\021)2012 1335 y Fo(D)2093 1350 y Fm(q)r Fi(\000)p Fl(2)2222 1335 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2541 1350 y Fm(q)r Fi(\000)p Fl(2)2669 1335 y Fq(\))2707 1294 y Fm(T)2784 1335 y Fn(\012)23 b Fo(C)2961 1294 y Fm(q)r Fi(\000)p Fl(2)2954 1359 y Fm(a)3089 1335 y Fq(\()p Fo(k)s Fq(\)\012)3289 1294 y Fm(q)r Fi(\000)p Fl(2)3289 1359 y Fm(a)3417 1210 y Fh(\015)3417 1260 y(\015)3417 1310 y(\015)3417 1360 y(\015)616 1659 y Fn(\024)831 1551 y Fm(a)797 1576 y Fh(Y)777 1761 y Fm(p)813 1742 y Fd(0)834 1761 y Fl(=)p Fm(q)940 1563 y Fh(\020)1005 1527 y Fl(\026)1006 1542 y Fm(j)s Fl(\()p Fm(p)1101 1519 y Fd(0)1124 1542 y Fl(\))1026 1576 y Fh(Y)1017 1759 y Fm(j)t Fl(=1)1168 1659 y Fo(\013)1231 1618 y Fm(p)1267 1595 y Fd(0)1288 1618 y Fi(\000)p Fm(q)1381 1659 y Fn(j)p Fq(\000)1470 1618 y Fm(j)1470 1684 y(q)1508 1659 y Fn(j)1536 1563 y Fh(\021)1600 1527 y Fl(\026)1602 1542 y Fm(j)s Fl(\()p Fm(q)r Fi(\000)p Fl(1\))1654 1576 y Fh(Y)1646 1759 y Fm(j)t Fl(=1)1830 1659 y Fn(j)p Fq(\000)1919 1612 y Fm(j)1919 1681 y(q)r Fi(\000)p Fl(1)2047 1659 y Fn(j)e(\002)1011 b Fq(\(I)s(I)s(I.76\))777 1994 y(sup)923 2018 y Fm(z)963 1898 y Fh(n)1035 1887 y Fm(a)p Fl(+1)1046 1911 y Fh(Y)1037 2086 y Fm(p)p Fl(=)p Fm(q)1179 1898 y Fh(\020)1246 1877 y(Z)1292 2066 y Fi(K)1346 2033 y Fb(p)p Fd(\000)p Ff(2)1346 2088 y Fb(p)p Fd(\000)p Ff(1)1475 1862 y Fl(\026)1481 1877 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1530 1911 y Fh(Y)1526 2094 y Fm(i)p Fl(=1)1702 1994 y Fo(dk)1804 2009 y Fm(i)1832 1994 y Fq(\()p Fo(p)22 b Fn(\000)h Fq(1\))2128 1898 y Fh(\021)2193 1887 y Fm(a)p Fl(+1)2204 1911 y Fh(Y)2195 2086 y Fm(p)p Fl(=)p Fm(q)2338 1898 y Fh(\020)2404 1877 y(Z)2450 2066 y Fi(K)2504 2033 y Fb(p)p Fd(\000)p Ff(2)2504 2088 y Fb(p)p Fd(\000)p Ff(1)2634 1862 y Fl(\026)2639 1877 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))2689 1911 y Fh(Y)2685 2094 y Fm(i)p Fl(=1)2861 1994 y Fo(dk)2966 1953 y Fi(0)2963 2019 y Fm(i)2990 1994 y Fq(\()p Fo(p)f Fn(\000)h Fq(1\))3286 1898 y Fh(\021)3357 1994 y Fn(\002)777 2273 y(\002)871 2190 y Fh(X)909 2364 y Fm(\031)1007 2273 y Fq(\001)1088 2288 y Fm(\031)1135 2273 y Fq(\()p Fo(k)1227 2232 y Fi(0)1251 2273 y Fo(;)17 b(k)s Fq(\))p Fn(k)p Fo(D)1521 2232 y Fi(\003)1518 2297 y Fm(q)r Fi(\000)p Fl(2)1646 2273 y Fq(\()p Fo(k)1738 2232 y Fi(0)1761 2273 y Fo(;)g(z)t Fq(\))p Fo(D)1973 2288 y Fm(q)r Fi(\000)p Fl(2)2101 2273 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))p Fn(k)23 b(\001)f(k)p Fq(\()p Fo(\036)2593 2288 y Fm(q)r Fi(\000)p Fl(2)2720 2273 y Fq(\))2758 2232 y Fm(T)2813 2273 y Fn(k)2863 2232 y Fl(2)2903 2177 y Fh(o)2968 2172 y Ff(1)p 2968 2184 31 4 v 2968 2226 a(2)501 2544 y Fq(where:)618 2746 y(*)49 b Fo(\031)33 b Fq(stands)e(for)e(eac)m(h)h(pairing)f(of)g(the)h (photon)g(creation-)f(and)h(annihilation)716 2866 y(op)s(erators)i(in)g ([)p Fo(C)1364 2830 y Fm(q)r Fi(\000)p Fl(2)1357 2891 y Fm(a)1492 2866 y Fq(])1519 2830 y Fi(\003)1559 2866 y Fq(\()p Fo(k)1651 2830 y Fi(0)1674 2866 y Fq(\)[)p Fo(C)1816 2830 y Fm(q)r Fi(\000)p Fl(2)1809 2891 y Fm(a)1944 2866 y Fq(]\()p Fo(k)s Fq(\))g(and)g(\001)2403 2881 y Fm(\031)2451 2866 y Fq(\()p Fo(k)2543 2830 y Fi(0)2566 2866 y Fo(;)17 b(k)s Fq(\))32 b(is)g(the)h(corresp)s(ond-)716 2987 y(ing)f(pro)s(duct)h(of)f(delta-functions)i(and)e(Kronec)m(k)m (er's)j(sym)m(b)s(ols;)618 3147 y(*)49 b Fn(j)p Fq(\000)805 3111 y Fm(j)805 3172 y(q)843 3147 y Fn(j)27 b Fq(=)g(2)p Fo(\031)t(\013)1172 3111 y Fm(q)1242 3147 y Fq(is)33 b(the)g(length)g(of)g(the)g(j-th)f(con)m(tour)h(path;)618 3307 y(*)49 b(sup)863 3331 y Fm(z)948 3307 y Fq(is)d(the)g(sup)h(in)e (the)h(in)m(tegration)g(v)-5 b(ariables)47 b Fo(z)j Fq(running)c(o)m(v) m(er)h(the)716 3428 y(con)m(tours)33 b Fn(f)p Fq(\000)1220 3392 y Fm(j)1220 3452 y(q)1258 3428 y Fn(g)p Fq(,)g Fn(f)p Fq(\000)1479 3380 y Fm(j)1479 3449 y(q)r Fi(\000)p Fl(1)1607 3428 y Fn(g)p Fq(.)716 3568 y(W)-8 b(e)32 b(w)m(arn)h(the)f(reader)h (that,)f(ev)m(en)h(if)f(w)m(e)h(use)g(the)f(same)h(sym)m(b)s(ol,)g(not) f(all)716 3688 y(the)27 b Fo(k)s(;)17 b(k)1030 3652 y Fi(0)1079 3688 y Fq(in)m(tegration)26 b(v)-5 b(ariables)27 b(are)g(necessarily)h(paired,)h(in)d(general)h(the)716 3809 y(pairings)i Fn(f)p Fo(\031)t Fn(g)g Fq(are)g(related)h(to)f(a)f (subset)j(of)e(them,)h(those)g(ones)g(con)m(tained)716 3929 y(in)j Fo(C)907 3893 y Fm(q)r Fi(\000)p Fl(2)900 3954 y Fm(a)1035 3893 y Fi(\003)1074 3929 y Fq(\()p Fo(k)1166 3893 y Fi(0)1190 3929 y Fq(\))p Fo(C)1305 3893 y Fm(q)r Fi(\000)p Fl(2)1298 3954 y Fm(a)1433 3929 y Fq(\()p Fo(k)s Fq(\))f(\(see)i(the)f(commen)m(t)h(b)s(elo)m(w)f(Eq.)h(\(I)s(I)s (I.54\)\).)257 4131 y(No)m(w)43 b(w)m(e)h(sp)s(ecify)g(the)f(estimate)h (of)e Fn(k)p Fo(D)1807 4146 y Fm(q)r Fi(\000)p Fl(2)1935 4131 y Fq(\()p Fo(k)2027 4095 y Fi(0)2050 4131 y Fo(;)17 b(z)t Fq(\))2181 4095 y Fi(\003)2221 4131 y Fo(D)2302 4146 y Fm(q)r Fi(\000)p Fl(2)2430 4131 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))p Fn(k)43 b Fq(starting)f(from)h(the)257 4251 y(constrain)m(ts)35 b(on)f(the)g(structure)h(of)e Fo(D)1673 4266 y Fm(q)r Fi(\000)p Fl(2)1802 4251 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\).)47 b(Let)34 b Fo(D)2356 4266 y Fm(q)r Fi(\000)p Fl(2)2484 4251 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))34 b(b)s(e)g(a)g (generic)g(com-)257 4372 y(bination)f(of)f(op)s(erators)h(whic)m(h)h (ful\014lls)f(Prop)s(erties)h Fn(P)8 b Fq(1)33 b(and)f Fn(P)8 b Fq(2,)33 b(for)f(instance)1103 4586 y Fo(g)1154 4545 y Fl(1)1150 4611 y Fm(m)p Fi(\000)p Fl(2)1306 4586 y Fo(g)1357 4545 y Fl(2)1353 4611 y Fm(m)p Fi(\000)p Fl(2)1510 4586 y Fo(g)1561 4545 y Fl(1)1557 4611 y Fm(m)p Fi(\000)p Fl(2)1730 4586 y Fn(\001)17 b(\001)g(\001)e Fo(g)1914 4545 y Fl(1)1910 4611 y Fm(m)p Fi(\000)p Fl(2)2066 4586 y Fo(g)2117 4545 y Fl(3)2113 4611 y Fm(m)p Fi(\000)p Fl(2)2286 4586 y Fn(\001)i(\001)g(\001)e Fo(g)2470 4545 y Fl(3)2466 4611 y Fm(m)p Fi(\000)p Fl(2)2623 4586 y Fo(:)534 b Fq(\(I)s(I)s(I.77\))257 4801 y(By)42 b(con)m(v)m(en)m(tion,) k(w)m(e)c(push)g(all)f(the)h Fo(g)1711 4765 y Fl(3)1707 4825 y Fm(m)p Fi(\000)p Fl(2)1905 4801 y Fq(op)s(erators)f(to)g(the)g (v)m(ery)i(righ)m(t;)j(b)s(eing)41 b(c-)257 4921 y(n)m(um)m(b)s(ers)24 b(this)f(op)s(eration)e(is)i(neutral.)40 b(In)22 b(\(I)s(I)s(I.77\))g (w)m(e)h(ha)m(v)m(e)g(omitted)f(the)h(dep)s(endence)257 5042 y(on)37 b Fo(k)j Fq(and)c(on)h Fo(z)t Fq(;)i(w)m(e)f(will)f(tak)m (e)h(it)e(in)m(to)h(accoun)m(t)h(when)f(necessary)-8 b(.)58 b(Eac)m(h)38 b(op)s(erator)257 5162 y Fo(g)308 5126 y Fl(2)304 5187 y Fm(m)p Fi(\000)p Fl(2)491 5162 y Fq(is)30 b(estimated)i(in)e(companion)g(with)h(its)g(neigh)m(b)s(or)f Fo(g)2433 5126 y Fl(1)2429 5187 y Fm(m)p Fi(\000)p Fl(2)2615 5162 y Fq(on)g(the)h(righ)m(t;)g(e.g.,)g(for)411 5388 y Fo(g)462 5347 y Fl(2)458 5413 y Fm(m)p Fi(\000)p Fl(2)642 5388 y Fq(=)d(\007)822 5347 y Fi(\006)822 5413 y Fm(q)r Fi(\000)p Fl(2)950 5388 y Fq(\()p Fo(k)1039 5403 y Fm(i)1067 5388 y Fq(\()p Fo(q)1152 5347 y Fi(0)1175 5388 y Fq(\)\))97 b(and)h Fo(g)1654 5347 y Fl(1)1650 5413 y Fm(m)p Fi(\000)p Fl(2)1834 5388 y Fq(=)28 b Fo(R)2013 5347 y Fl(+)12 b Fi(?)2012 5413 y Fm(q)r Fi(\000)p Fl(2)2140 5388 y Fq(\()p Fo(z)2227 5347 y Fl(\()p Fm(p)2290 5324 y Fd(0)2313 5347 y Fl(\))2223 5413 y Fm(q)2367 5388 y Fn(\000)23 b Fo(S)2527 5404 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2862 5388 y Fq(\()p Fo(k)s Fq(\)\))154 b(\(I)s(I)s(I.78\))p eop end %%Page: 39 40 TeXDict begin 39 39 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(39)257 573 y(w)m(e)34 b(can)f(easily)h (estimate)751 773 y Fn(k)p Fq(\007)877 732 y Fi(\006)877 798 y Fm(q)r Fi(\000)p Fl(2)1005 773 y Fq(\()p Fo(k)1094 788 y Fm(i)1122 773 y Fq(\()p Fo(q)1207 732 y Fi(0)1230 773 y Fq(\)\))p Fo(R)1381 732 y Fl(+)12 b Fi(?)1380 798 y Fm(q)r Fi(\000)p Fl(2)1509 773 y Fq(\()p Fo(z)1596 732 y Fl(\()p Fm(p)1659 709 y Fd(0)1682 732 y Fl(\))1592 798 y Fm(q)1735 773 y Fn(\000)23 b Fo(S)1894 777 y Fl(\026)1895 794 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2230 773 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)k(\024)632 b Fq(\(I)s(I)s(I.79\))1113 976 y Fn(\024)28 b Fo(C)h Fn(\001)1558 909 y Fq(1)p 1377 953 413 4 v 1377 969 a Fh(q)p 1460 969 330 4 v 100 x Fq(2)p Fn(j)p Fo(k)1588 1084 y Fm(i)1615 1069 y Fq(\()p Fo(q)1700 1041 y Fi(0)1723 1069 y Fq(\))p Fn(j)1799 976 y(jj)p Fo(R)1930 935 y Fl(+)11 b Fi(?)1929 1001 y Fm(q)r Fi(\000)p Fl(2)2056 976 y Fq(\()p Fo(z)2143 935 y Fl(\()p Fm(p)2206 911 y Fd(0)2229 935 y Fl(\))2139 1001 y Fm(q)2283 976 y Fn(\000)23 b Fo(S)2442 980 y Fl(\026)2443 997 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2778 976 y Fq(\()p Fo(k)s Fq(\)\))p Fn(jj)257 1273 y Fq(where)34 b Fo(C)40 b Fq(is)33 b Fo(\013)q Fq(-indep)s(enden)m(t;)h(analogously)-8 b(,)526 1473 y Fn(k)p Fq(\000)637 1432 y Fi(\006)p Fm(;)p Fi(\006)771 1473 y Fq(\()p Fo(k)860 1488 y Fm(i)888 1473 y Fq(\()p Fo(q)973 1432 y Fi(0)996 1473 y Fq(\))p Fo(;)17 b(k)1129 1488 y Fm(j)1164 1473 y Fq(\()p Fo(q)1249 1432 y Fi(00)1292 1473 y Fq(\)\))p Fo(R)1443 1432 y Fl(+)11 b Fi(?)1442 1498 y Fm(q)r Fi(\000)p Fl(2)1570 1473 y Fq(\()p Fo(z)1657 1432 y Fl(\()p Fm(p)1720 1409 y Fd(0)1743 1432 y Fl(\))1653 1498 y Fm(q)1797 1473 y Fn(\000)22 b Fo(S)1956 1489 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2292 1473 y Fq(\()p Fo(k)s Fq(\)\))p Fn(jj)k(\024)565 b Fq(\(I)s(I)s(I.80\))888 1676 y Fn(\024)28 b Fo(C)h Fn(\001)1547 1608 y Fq(1)p 1152 1652 841 4 v 1152 1669 a Fh(q)p 1235 1669 758 4 v 98 x Fq(2)p Fn(j)p Fo(k)1363 1782 y Fm(i)1390 1767 y Fq(\()p Fo(q)1475 1739 y Fi(0)1498 1767 y Fq(\))p Fn(j)22 b(\001)g Fq(2)p Fn(j)p Fo(k)1764 1782 y Fm(j)1799 1767 y Fq(\()p Fo(q)1884 1739 y Fi(00)1926 1767 y Fq(\))p Fn(j)2002 1676 y(jj)p Fo(R)2133 1634 y Fl(+)11 b Fi(?)2132 1700 y Fm(q)r Fi(\000)p Fl(2)2260 1676 y Fq(\()p Fo(z)2347 1634 y Fl(\()p Fm(p)2410 1611 y Fd(0)2433 1634 y Fl(\))2343 1700 y Fm(q)2487 1676 y Fn(\000)22 b Fo(S)2646 1691 y Fm(l)q Fl(\()p Fm(q)r Fl(\))p Fm(;l)q Fl(\()p Fm(q)r Fi(\000)p Fl(1\))2981 1676 y Fq(\()p Fo(k)s Fq(\)\))p Fn(k)p Fo(:)257 1978 y Fq(F)-8 b(ollo)m(wing)39 b(the)g(rules)h(in)e (Eqs.)i(\(I)s(I)s(I.79\))e(and)h(\(I)s(I)s(I.80\),)h(and)e(using)i(the) f(usual)g(norm)257 2098 y(estimates)31 b(for)e(the)h(remaining)g Fn(f)p Fo(g)1547 2062 y Fl(1)1543 2123 y Fm(m)p Fi(\000)p Fl(2)1699 2098 y Fn(g)f Fq(op)s(erators)h(and)f(the)h(c-n)m(um)m(b)s (ers)h Fn(f)p Fo(g)3125 2062 y Fl(3)3121 2123 y Fm(m)p Fi(\000)p Fl(2)3277 2098 y Fn(g)p Fq(,)f(w)m(e)257 2219 y(de\014ne)681 2419 y Fn(E)743 2323 y Fh(\020)792 2419 y Fo(D)873 2434 y Fm(q)r Fi(\000)p Fl(2)1002 2419 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))1225 2323 y Fh(\021)1302 2419 y Fq(:=)1779 b(\(I)s(I)s(I.81\))847 2586 y(=)28 b Fn(k)p Fo(g)1052 2545 y Fl(1)1048 2611 y Fm(m)p Fi(\000)p Fl(2)1204 2586 y Fn(kk)p Fo(g)1355 2545 y Fl(2)1351 2611 y Fm(m)p Fi(\000)p Fl(2)1507 2586 y Fo(g)1558 2545 y Fl(1)1554 2611 y Fm(m)p Fi(\000)p Fl(2)1711 2586 y Fn(k)1761 2545 y Fi(0)1800 2586 y Fn(\001)17 b(\001)g(\001)e(k)p Fo(g)2034 2545 y Fl(1)2030 2611 y Fm(m)p Fi(\000)p Fl(2)2186 2586 y Fn(kj)p Fo(g)2315 2545 y Fl(3)2311 2611 y Fm(m)p Fi(\000)p Fl(2)2484 2586 y Fn(\001)i(\001)g(\001)e Fo(g)2668 2545 y Fl(3)2664 2611 y Fm(m)p Fi(\000)p Fl(2)2820 2586 y Fn(j)p Fq(\()p Fo(k)s(;)i(z)t Fq(\))257 2786 y(where,)56 b(dep)s(ending)51 b(on)f Fo(g)1278 2750 y Fl(2)1274 2811 y Fm(m)p Fi(\000)p Fl(2)1431 2786 y Fq(,)k Fn(k)p Fo(g)1613 2750 y Fl(2)1609 2811 y Fm(m)p Fi(\000)p Fl(2)1765 2786 y Fo(g)1816 2750 y Fl(1)1812 2811 y Fm(m)p Fi(\000)p Fl(2)1969 2786 y Fn(k)2019 2750 y Fi(0)2092 2786 y Fq(is)c(either)h (the)f(righ)m(t)h(hand)f(side)h(in)257 2907 y(Eq.)34 b(\(I)s(I)s(I.79\))e(or)g(in)h(Eq.)g(\(I)s(I)s(I.80\).)43 b(W)-8 b(e)33 b(can)g(state)g(the)g(upp)s(er)g(b)s(ound)516 3120 y Fn(jj)p Fo(D)653 3135 y Fm(q)r Fi(\000)p Fl(2)781 3120 y Fq(\()p Fo(k)873 3078 y Fi(0)896 3120 y Fo(;)17 b(z)t Fq(\))1027 3078 y Fi(\003)1067 3120 y Fo(D)1148 3135 y Fm(q)r Fi(\000)p Fl(2)1276 3120 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))p Fn(jj)28 b(\024)g(E)1750 3023 y Fh(\020)1799 3120 y Fo(D)1880 3135 y Fm(q)r Fi(\000)p Fl(2)2008 3120 y Fq(\()p Fo(k)2100 3078 y Fi(0)2123 3120 y Fo(;)17 b(z)t Fq(\))2254 3023 y Fh(\021)2304 3120 y Fn(E)2366 3023 y Fh(\020)2415 3120 y Fo(D)2496 3135 y Fm(q)r Fi(\000)p Fl(2)2625 3120 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))2848 3023 y Fh(\021)2898 3120 y Fo(:)259 b Fq(\(I)s(I)s(I.82\))257 3332 y(Finally)-8 b(,)25 b(b)m(y)d(de\014nition,)j(Prop)s(ert)m(y)e Fn(P)8 b Fq(2)22 b(ensures)h(that,)h(b)m(y)f(inserting)g(estimate)g (\(I)s(I)s(I.82\),)257 3453 y(the)g(m)m(ultiple)g(in)m(tegral)f(\(I)s (I)s(I.76\))f(turns)i(out)e(to)g(b)s(e)h(con)m(v)m(ergen)m(t)i(and)e (uniformly)h(b)s(ounded)257 3573 y(in)33 b Fo(\013)q Fq(.)43 b(Therefore)34 b(the)f(same)g(holds)h(for)e(the)h(m)m(ultiple)h (in)m(tegral)f(\(I)s(I)s(I.75\).)257 3693 y Fe(Remark)257 3814 y Fq(It)46 b(is)f(imp)s(ortan)m(t)h(not)f(to)f(confuse)j(the)e (induction)h(index)h Fo(q)i Fq(and)c(the)h(scale)g(of)e(re-)257 3934 y(expansion,)35 b Fo(q)25 b Fn(\000)e Fq(2,)33 b(of)f(the)h (expression)h(in)f(Eq.\(I)s(I)s(I.67\).)257 4055 y Fa(H4\))46 b Fq(The)c(remainder)h(obtained)f(from)g(the)g(re-expansion)h(of)e(\(I) s(I)s(I.67\))h(is)g(a)f(\014nite,)257 4175 y Fo(\013)q Fq(-indep)s(enden)m(t,)34 b(sum)g(of)e(expressions)j(of)d(the)h(t)m(yp) s(e)471 4354 y Fm(a)437 4378 y Fh(Y)405 4563 y Fm(p)441 4545 y Fd(0)463 4563 y Fl(=)p Fm(q)552 4545 y Fd(0)591 4365 y Fh(\020)655 4329 y Fl(\026)657 4344 y Fm(j)s Fl(\()p Fm(p)752 4321 y Fd(0)775 4344 y Fl(\))676 4378 y Fh(Y)668 4561 y Fm(j)t Fl(=1)819 4344 y Fh(Z)865 4533 y Fl(\000)909 4552 y Fb(q)939 4538 y Fd(0)986 4462 y Fo(\013)1049 4420 y Fm(p)1085 4397 y Fd(0)1107 4420 y Fi(\000)p Fm(q)1196 4397 y Fd(0)1222 4462 y Fo(dz)1322 4411 y Fl(\()p Fm(p)1385 4387 y Fd(0)1408 4411 y Fl(\))p Fm(;j)1318 4487 y(q)1352 4468 y Fd(0)1492 4365 y Fh(\021)1556 4329 y Fl(\026)1558 4344 y Fm(j)s Fl(\()p Fm(q)1651 4321 y Fd(0)1674 4344 y Fi(\000)p Fl(1\))1622 4378 y Fh(Y)1613 4561 y Fm(j)t Fl(=1)1808 4344 y Fh(Z)1854 4533 y Fl(\000)1898 4552 y Fb(q)1928 4538 y Fd(0)1951 4552 y(\000)p Ff(1)2054 4462 y Fo(dz)2154 4411 y Fl(\()p Fm(q)2215 4387 y Fd(0)2238 4411 y Fi(\000)p Fl(1\))p Fm(;j)2150 4487 y(q)2184 4468 y Fd(0)2206 4487 y Fi(\000)p Fl(1)2412 4462 y Fn(\002)695 b Fq(\(I)s(I)s(I.83\))571 4797 y Fn(\002)675 4689 y Fm(a)p Fl(+1)685 4714 y Fh(Y)665 4899 y Fm(p)p Fl(=)p Fm(q)790 4880 y Fd(0)828 4700 y Fh(\020)895 4680 y(Z)941 4868 y Fi(K)995 4836 y Fb(p)p Fd(\000)p Ff(2)995 4891 y Fb(p)p Fd(\000)p Ff(1)1125 4664 y Fl(\026)1130 4680 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1179 4714 y Fh(Y)1175 4896 y Fm(i)p Fl(=1)1351 4797 y Fo(dk)1453 4812 y Fm(i)1481 4797 y Fq(\()p Fo(p)22 b Fn(\000)h Fq(1\))1777 4700 y Fh(\021)1848 4775 y Fk([)1826 4797 y Fo(D)1907 4812 y Fm(q)1941 4793 y Fd(0)1963 4812 y Fi(\000)p Fl(2)2058 4797 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2377 4812 y Fm(q)2411 4793 y Fd(0)2433 4812 y Fi(\000)p Fl(2)2527 4797 y Fq(\))2565 4756 y Fm(T)2642 4797 y Fn(\012)23 b Fo(C)2819 4756 y Fm(q)2853 4732 y Fd(0)2875 4756 y Fi(\000)p Fl(2)2812 4821 y Fm(a)2969 4797 y Fq(\()p Fo(k)s Fq(\)\012)3169 4756 y Fm(q)3203 4732 y Fd(0)3226 4756 y Fi(\000)p Fl(2)3169 4821 y Fm(a)3320 4797 y Fo(;)257 5081 y Fq(with)34 b(2)27 b Fn(\024)h Fo(q)708 5045 y Fi(0)759 5081 y Fn(\024)g Fo(q)k(<)27 b(a)h Fn(\024)g Fq(2)p Fo(N)33 b Fn(\000)22 b Fq(1,)33 b(where)349 5268 y(\(i\))49 b Fo(C)578 5232 y Fm(q)612 5208 y Fd(0)634 5232 y Fi(\000)p Fl(2)571 5292 y Fm(a)729 5268 y Fq(\()p Fo(k)s Fq(\))e(is)h(a)g(\014nite)g(pro)s(duct)g(of)f(an)h Fo(\013)q Fq(-indep)s(enden)m(t)h(n)m(um)m(b)s(er)g(of)e(photon)501 5388 y(creation)33 b(op)s(erators)g(with)g(momen)m(ta)g(con)m(tained)h (in)f(the)g(shell)g Fn(K)2977 5352 y Fm(q)3011 5329 y Fd(0)3034 5352 y Fi(\000)p Fl(2)2976 5413 y Fm(a)3128 5388 y Fq(;)p eop end %%Page: 40 41 TeXDict begin 40 40 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(40)322 577 y(\(ii\))523 555 y Fk([)501 577 y Fo(D)582 592 y Fm(q)616 573 y Fd(0)638 592 y Fi(\000)p Fl(2)733 577 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))40 b(corresp)s(onds)i(to)d Fo(D)1746 592 y Fm(q)1780 573 y Fd(0)1802 592 y Fi(\000)p Fl(2)1897 577 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))40 b(except)i(that)e(at)f(least)i(one)f(of)g (the)501 698 y(op)s(erators)c Fo(g)987 662 y Fl(1)983 722 y Fm(q)1017 703 y Fd(0)1039 722 y Fi(\000)p Fl(2)1167 698 y Fn(2)e(G)1332 662 y Fl(1)1326 722 y Fm(q)1360 703 y Fd(0)1383 722 y Fi(\000)p Fl(2)1513 698 y Fq(con)m(tained)j(in)g Fo(D)2157 713 y Fm(q)2191 694 y Fd(0)2213 713 y Fi(\000)p Fl(2)2307 698 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))37 b(is)g(replaced)g(b) m(y)g Fo(g)3247 662 y Fl(1)3243 722 y Fm(q)3277 703 y Fd(0)3299 722 y Fi(\000)p Fl(2)3418 698 y Fn(\000)501 818 y Fo(\034)554 782 y Fm(N)543 843 y(])622 818 y Fq(\()p Fo(g)711 782 y Fl(1)707 843 y Fm(q)741 824 y Fd(0)763 843 y Fi(\000)p Fl(2)857 818 y Fq(\).)404 1017 y Fj(Structur)-5 b(e)36 b(of)e(the)h(pr)-5 b(o)g(of.)257 1138 y Fq(The)34 b(pro)s(of)e(consists)i(of)e(three)h(main)g(steps.)257 1258 y Fa(S1\))40 b Fq(Assuming)f Fj(H2\))p Fq(,)f Fj(H3\))j Fq(and)c Fj(H4\))k Fq(for)c(v)m(ectors)h(of)f(t)m(yp)s(e)h Fo( )2640 1273 y Fm(q)2678 1258 y Fq(,)h(up)e(to)g Fo(q)i Fq(=)c Fo(m)26 b Fn(\000)f Fq(1,)257 1378 y(w)m(e)35 b(pro)m(v)m(e)g Fj(H3\))j Fq(and)c Fj(H4\))j Fq(for)c(a)h(v)m(ector)g (of)g(t)m(yp)s(e)g Fo( )2178 1393 y Fm(m)2245 1378 y Fq(,)g(b)m(y)h(pic)m(king)g(an)44 b(\026)-59 b Fo(\013)2979 1393 y Fm( )3025 1401 y Fb(m)3118 1378 y Fo(>)29 b Fq(0)34 b(with)267 1499 y(\026)-59 b Fo(\013)319 1514 y Fm( )365 1522 y Fb(m)456 1499 y Fn(\024)38 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\).)257 1619 y Fa(S2\))33 b Fq(F)-8 b(rom)29 b Fj(H3\))34 b Fq(for)29 b Fo(q)i Fq(=)d Fo(m)i Fq(and)g Fj(H2\))j Fq(up)d(to)g Fo(q)h Fq(=)d Fo(m)16 b Fn(\000)g Fq(1,)31 b(w)m(e)g(deriv)m(e)g Fj(H1\))j Fq(for)29 b Fo(q)i Fq(=)d Fo(m)257 1740 y Fq(b)m(y)34 b(pic)m(king)g(an)42 b(\026)-59 b Fo(\013)q Fq(\()p Fo(m)p Fq(\))28 b Fo(>)f Fq(0)33 b(small)g(enough)g(and)g(suc)m(h)h(that)42 b(\026)-59 b Fo(\013)q Fq(\()p Fo(m)p Fq(\))28 b Fn(\024)37 b Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)23 b Fn(\000)f Fq(1\).)257 1860 y Fa(S3\))52 b Fq(F)-8 b(rom)48 b Fj(H1\))k Fq(and)c Fj(H4\))k Fq(for)c Fo(q)59 b Fq(=)54 b Fo(m)33 b Fn(\000)h Fq(1,)52 b(w)m(e)d(get)f Fj(H2\))53 b Fq(up)48 b(to)g Fo(q)59 b Fq(=)54 b Fo(m)p Fq(,)f(for)257 1980 y(0)28 b Fo(<)f(\013)i(<)37 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)p Fq(\).)257 2101 y(In)33 b(the)g(end)h(w)m(e)f(c)m(hec)m (k)i(the)e(initial)g(step)g(of)f(the)h(induction.)257 2221 y Fe(Remark)257 2341 y Fq(Concerning)28 b(terminology)-8 b(,)28 b(in)e(order)g(to)g(a)m(v)m(oid)g(am)m(biguities,)j(w)m(e)e (will)g(alw)m(a)m(ys)h(use)f(the)257 2462 y(w)m(ord)d Fj(expr)-5 b(ession)29 b Fq(for)23 b(a)f(v)m(ector)j(with)e(the)h (structure)g(displa)m(y)m(ed)i(either)d(in)h(Eq.)g(\(I)s(I)s(I.52\))257 2582 y(or)33 b(in)h(Eq.)g(\(I)s(I)s(I.67\),)e(or)h(in)h(Eq.)g(\(I)s(I)s (I.83\).)44 b(As)34 b(w)m(e)g(will)g(see,)g(the)g(map)f Fo(\034)2935 2546 y Fm(N)2924 2607 y(])3036 2582 y Fq(pro)s(duces)h(a) 257 2703 y(sum)c(of)f Fj(expr)-5 b(essions)36 b Fq(starting)29 b(from)g(an)g(original)f(one;)j(in)e(other)h(w)m(ords,)h(it)e(is)g (helpful)257 2823 y(to)j(think)h(of)f(it)g(as)g(a)g(map)g(that)g(pro)s (duces)h(p)s(olynomials)g(starting)g(from)e(monomials.)257 2943 y(F)-8 b(or)29 b(the)h(norm)g(estimate)h(of)e(a)h(v)m(ector)g (corresp)s(onding)h(to)e(an)h Fj(expr)-5 b(ession)p Fq(,)29 b(w)m(e)i(alw)m(a)m(ys)257 3064 y(assume)j(the)f(estimate)h(pro)s (cedure)g(presen)m(ted)h(in)d(Prop)s(ert)m(y)i Fn(P)8 b Fq(2,)33 b(Hyp)s(othesis)h Fj(H3\))p Fq(.)257 3184 y Fa(S1\))i Fq(W)-8 b(e)33 b(start)g(from)f(the)h(follo)m(wing)g (expression)i(re-expanded)f(on)e(scale)i Fo(m)22 b Fn(\000)h Fq(2)390 3491 y Fo( )453 3506 y Fm(m)547 3491 y Fq(:=)747 3383 y Fm(a)713 3408 y Fh(Y)678 3593 y Fm(p)714 3574 y Fd(0)736 3593 y Fl(=)p Fm(m)870 3395 y Fh(\020)934 3358 y Fl(\026)936 3374 y Fm(j)s Fl(\()p Fm(p)1031 3350 y Fd(0)1054 3374 y Fl(\))955 3408 y Fh(Y)947 3590 y Fm(j)t Fl(=1)1098 3374 y Fh(Z)1144 3562 y Fl(\000)1188 3570 y Fb(m)1267 3491 y Fo(\013)1330 3450 y Fm(p)1366 3426 y Fd(0)1388 3450 y Fi(\000)p Fm(m)1509 3491 y Fo(dz)1609 3450 y Fl(\()p Fm(p)1672 3426 y Fd(0)1695 3450 y Fl(\))p Fm(;j)1605 3515 y(m)1779 3395 y Fh(\021)1843 3358 y Fl(\026)1845 3374 y Fm(j)s Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1912 3408 y Fh(Y)1904 3590 y Fm(j)t Fl(=1)2101 3374 y Fh(Z)2147 3562 y Fl(\000)2191 3571 y Fb(m)p Fd(\000)p Ff(1)2349 3491 y Fo(dz)2449 3440 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)2445 3513 y(m)p Fi(\000)p Fl(1)2713 3491 y Fn(\002)394 b Fq(\(I)s(I)s(I.84\))556 3826 y Fn(\002)662 3718 y Fm(a)p Fl(+1)673 3743 y Fh(Y)650 3917 y Fm(p)p Fl(=)p Fm(m)819 3730 y Fh(\020)885 3709 y(Z)932 3898 y Fi(K)986 3865 y Fb(p)p Fd(\000)p Ff(2)986 3920 y Fb(p)p Fd(\000)p Ff(1)1115 3694 y Fl(\026)1121 3709 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1170 3743 y Fh(Y)1166 3925 y Fm(i)p Fl(=1)1342 3826 y Fo(dk)1444 3841 y Fm(i)1472 3826 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))1767 3730 y Fh(\021)1817 3826 y Fo(D)1898 3841 y Fm(m)p Fi(\000)p Fl(2)2054 3826 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2373 3841 y Fm(m)p Fi(\000)p Fl(2)2530 3826 y Fq(\))2568 3785 y Fm(T)2645 3826 y Fn(\012)23 b Fo(C)2822 3785 y Fm(m)p Fi(\000)p Fl(2)2815 3851 y Fm(a)2979 3826 y Fq(\()p Fo(k)s Fq(\)\012)3179 3785 y Fm(m)p Fi(\000)p Fl(2)3179 3851 y Fm(a)3336 3826 y Fo(:)257 4116 y Fq(Let)48 b Fo(D)528 4131 y Fm(m)p Fi(\000)p Fl(2)685 4116 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))47 b(b)s(e,)52 b(for)47 b(example,)52 b(the)c(com)m(bination)g(of)f(op)s(erators)g (considered)257 4237 y(b)s(efore)33 b(in)g(Hyp)s(othesis)h Fj(H3)14 b Fq(:)1116 4452 y Fo(g)1167 4411 y Fl(1)1163 4476 y Fm(m)p Fi(\000)p Fl(2)1320 4452 y Fo(g)1371 4411 y Fl(2)1367 4476 y Fm(m)p Fi(\000)p Fl(2)1523 4452 y Fo(g)1574 4411 y Fl(1)1570 4476 y Fm(m)p Fi(\000)p Fl(2)1743 4452 y Fn(\001)j(\001)g(\001)e Fo(g)1927 4411 y Fl(1)1923 4476 y Fm(m)p Fi(\000)p Fl(2)2080 4452 y Fo(g)2131 4411 y Fl(3)2127 4476 y Fm(m)p Fi(\000)p Fl(2)2300 4452 y Fn(\001)i(\001)g(\001)e Fo(g)2484 4411 y Fl(3)2480 4476 y Fm(m)p Fi(\000)p Fl(2)3184 4452 y Fq(\(I)s(I)s(I.85\))257 4667 y(whic)m(h)34 b(ful\014lls)g(Prop)s(erties)g Fn(P)8 b Fq(1,)32 b Fn(P)8 b Fq(2,)33 b(b)m(y)h(h)m(yp)s(othesis.)257 4787 y(W)-8 b(e)31 b(replace)g(eac)m(h)h(op)s(erator)d(in)i Fo(D)1552 4802 y Fm(m)p Fi(\000)p Fl(2)1709 4787 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))30 b(and)h(b)s(elonging)f(to)g Fn(G)2771 4751 y Fl(1)2765 4812 y Fm(m)p Fi(\000)p Fl(2)2952 4787 y Fq(b)m(y)i(the)e(corre-)257 4907 y(sp)s(onding)k(one)g(under)h (the)f(map)f Fo(\034)1570 4871 y Fm(N)1559 4932 y(])1672 4907 y Fq(whic)m(h)i(is)f(w)m(ell)h(de\014ned)g(thanks)f(to)f(Hyp)s (othesis)257 5028 y Fj(H2\))p Fq(,)g(assuming)h(that)e(0)c Fo(<)f(\013)i(<)37 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\):)775 5243 y Fo(g)826 5202 y Fl(1)822 5267 y Fm(m)p Fi(\000)p Fl(2)978 5243 y Fo(g)1029 5202 y Fl(2)1025 5267 y Fm(m)p Fi(\000)p Fl(2)1182 5243 y Fo(g)1233 5202 y Fl(1)1229 5267 y Fm(m)p Fi(\000)p Fl(2)1402 5243 y Fn(\001)17 b(\001)g(\001)e Fo(g)1586 5202 y Fl(1)1582 5267 y Fm(m)p Fi(\000)p Fl(2)1738 5243 y Fo(g)1789 5202 y Fl(3)1785 5267 y Fm(m)p Fi(\000)p Fl(2)1964 5243 y Fn(\001)22 b(\001)p Fo(g)2093 5202 y Fl(3)2089 5267 y Fm(m)p Fi(\000)p Fl(2)2272 5243 y Fn(7!)812 b Fq(\(I)s(I)s(I.86\))941 5388 y Fo(\034)994 5347 y Fm(N)983 5413 y(])1062 5388 y Fq(\()p Fo(g)1151 5347 y Fl(1)1147 5413 y Fm(m)p Fi(\000)p Fl(2)1303 5388 y Fq(\))p Fo(g)1392 5347 y Fl(2)1388 5413 y Fm(m)p Fi(\000)p Fl(2)1544 5388 y Fo(\034)1597 5347 y Fm(N)1586 5413 y(])1665 5388 y Fq(\()p Fo(g)1754 5347 y Fl(1)1750 5413 y Fm(m)p Fi(\000)p Fl(2)1907 5388 y Fq(\))22 b Fn(\001)f(\001)p Fo(\034)2097 5347 y Fm(N)2086 5413 y(])2165 5388 y Fq(\()p Fo(g)2254 5347 y Fl(1)2250 5413 y Fm(m)p Fi(\000)p Fl(2)2406 5388 y Fq(\))p Fo(g)2495 5347 y Fl(3)2491 5413 y Fm(m)p Fi(\000)p Fl(2)2670 5388 y Fn(\001)h(\001)p Fo(g)2799 5347 y Fl(3)2795 5413 y Fm(m)p Fi(\000)p Fl(2)2951 5388 y Fo(:)206 b Fq(\(I)s(I)s (I.87\))p eop end %%Page: 41 42 TeXDict begin 41 41 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(41)257 573 y(Exploiting)39 b(Lemmata)f(I)s(I)s(I.1,)h(I)s(I)s(I.2)e(and)h(Hyp)s(othesis)i Fj(H2\))p Fq(,)f(the)f(op)s(erator)f(norm)h(of)257 693 y(the)33 b(di\013erence)i(b)s(et)m(w)m(een)f(\(I)s(I)s(I.86\))e(and)h (\(I)s(I)s(I.87\))f(is)h(con)m(trolled)h(as)e(follo)m(ws)521 913 y Fn(k)p Fo(g)622 872 y Fl(1)618 938 y Fm(m)p Fi(\000)p Fl(2)774 913 y Fo(g)825 872 y Fl(2)821 938 y Fm(m)p Fi(\000)p Fl(2)978 913 y Fo(g)1029 872 y Fl(1)1025 938 y Fm(m)p Fi(\000)p Fl(2)1198 913 y Fn(\001)17 b(\001)g(\001)e Fo(g)1382 872 y Fl(1)1378 938 y Fm(m)p Fi(\000)p Fl(2)1534 913 y Fo(g)1585 872 y Fl(3)1581 938 y Fm(m)p Fi(\000)p Fl(2)1754 913 y Fn(\001)i(\001)g(\001)e Fo(g)1938 872 y Fl(3)1934 938 y Fm(m)p Fi(\000)p Fl(2)862 1059 y Fn(\000)p Fo(\034)992 1017 y Fm(N)981 1083 y(])1060 1059 y Fq(\()p Fo(g)1149 1017 y Fl(1)1145 1083 y Fm(m)p Fi(\000)p Fl(2)1302 1059 y Fq(\))p Fo(g)1391 1017 y Fl(2)1387 1083 y Fm(m)p Fi(\000)p Fl(2)1543 1059 y Fo(\034)1596 1017 y Fm(N)1585 1083 y(])1664 1059 y Fq(\()p Fo(g)1753 1017 y Fl(1)1749 1083 y Fm(m)p Fi(\000)p Fl(2)1905 1059 y Fq(\))i Fn(\001)g(\001)g(\001) e Fo(\034)2146 1017 y Fm(N)2135 1083 y(])2213 1059 y Fq(\()p Fo(g)2302 1017 y Fl(1)2298 1083 y Fm(m)p Fi(\000)p Fl(2)2455 1059 y Fq(\))p Fo(g)2544 1017 y Fl(3)2540 1083 y Fm(m)p Fi(\000)p Fl(2)2713 1059 y Fn(\001)i(\001)g(\001)e Fo(g)2897 1017 y Fl(3)2893 1083 y Fm(m)p Fi(\000)p Fl(2)3049 1059 y Fn(k)27 b(\024)604 1230 y(\024)84 b Fo(C)7 b(\013)905 1189 y Fm(N)g Fl(+)1033 1161 y Ff(1)p 1032 1173 31 4 v 1032 1215 a(2)1099 1230 y Fn(\001)22 b(E)1211 1133 y Fh(\020)1260 1230 y Fo(g)1311 1189 y Fl(1)1307 1254 y Fm(m)p Fi(\000)p Fl(2)1463 1230 y Fo(g)1514 1189 y Fl(2)1510 1254 y Fm(m)p Fi(\000)p Fl(2)1667 1230 y Fo(g)1718 1189 y Fl(1)1714 1254 y Fm(m)p Fi(\000)p Fl(2)1892 1230 y Fn(\001)g(\001)p Fo(g)2021 1189 y Fl(1)2017 1254 y Fm(m)p Fi(\000)p Fl(2)2173 1230 y Fo(g)2224 1189 y Fl(3)2220 1254 y Fm(m)p Fi(\000)p Fl(2)2393 1230 y Fn(\001)17 b(\001)g(\001)e Fo(g)2577 1189 y Fl(3)2573 1254 y Fm(m)p Fi(\000)p Fl(2)2729 1133 y Fh(\021)3184 1230 y Fq(\(I)s(I)s(I.88\))257 1450 y(where)34 b Fo(C)40 b Fq(is)33 b Fo(\013)q Fq(-indep)s(enden)m(t.)257 1570 y(Then)40 b(w)m(e)g(split)g(the)f(v)m(ertex)i(op)s(erators)e Fn(f)p Fq(\007)1932 1529 y Fi(\006)1932 1592 y Fm(m)p Fi(\000)p Fl(2)2088 1570 y Fq(\()p Fo(k)2177 1585 y Fm(i)2205 1570 y Fq(\()p Fo(m)2328 1534 y Fi(0)2351 1570 y Fq(\)\))p Fn(g)p Fq(,)i Fo(m)2630 1534 y Fi(0)2692 1570 y Fn(\025)d Fo(m)27 b Fn(\000)g Fq(2,)40 b(in)m(to)f(t)m(w)m(o)257 1690 y(pieces:)417 1986 y(\007)493 1945 y Fl(+)493 2011 y Fm(m)p Fi(\000)p Fl(3)650 1986 y Fq(\()p Fo(k)739 2001 y Fm(i)767 1986 y Fq(\()p Fo(m)890 1945 y Fi(0)913 1986 y Fq(\)\))83 b(;)g([)p Fo(a)p Fq(\()1313 1961 y Fo(~)1298 1986 y(G)1375 1945 y Fm(m)p Fi(\000)p Fl(1)1375 2011 y Fm(m)1532 1986 y Fq(\))22 b(+)g Fo(a)1741 1945 y Fi(\003)1781 1986 y Fq(\()1834 1961 y Fo(~)1819 1986 y(G)1896 1945 y Fm(m)p Fi(\000)p Fl(3)1896 2011 y Fm(m)p Fi(\000)p Fl(2)2053 1986 y Fq(\)])g Fn(\001)f Fo(~)-48 b(")o Fq(\()p Fo(k)2324 2001 y Fm(i)2352 1986 y Fq(\()p Fo(m)2475 1945 y Fi(0)2499 1986 y Fq(\)\))2585 1919 y Fo(e)2630 1883 y Fi(\000)p Fm(i\013)2751 1865 y(~)2754 1883 y(k)2791 1893 y Fb(i)2817 1883 y Fl(\()p Fm(m)2906 1859 y Fd(0)2929 1883 y Fl(\)\))p Fi(\001)o Fm(~)-34 b(x)p 2585 1963 464 4 v 2591 1980 a Fh(q)p 2674 1980 368 4 v 100 x Fq(2)p Fn(j)p Fo(k)2802 2095 y Fm(i)2829 2080 y Fq(\()p Fo(m)2952 2051 y Fi(0)2976 2080 y Fq(\))p Fn(j)3184 1986 y Fq(\(I)s(I)s(I.89\)) 417 2328 y(\007)493 2287 y Fi(\000)493 2352 y Fm(m)p Fi(\000)p Fl(3)650 2328 y Fq(\()p Fo(k)739 2343 y Fm(i)767 2328 y Fq(\()p Fo(m)890 2287 y Fi(0)913 2328 y Fq(\)\))83 b(;)g([)p Fo(a)p Fq(\()1313 2303 y Fo(~)1298 2328 y(G)1375 2287 y Fm(m)p Fi(\000)p Fl(3)1375 2352 y Fm(m)p Fi(\000)p Fl(2)1532 2328 y Fq(\))22 b(+)g Fo(a)1741 2287 y Fi(\003)1781 2328 y Fq(\()1834 2303 y Fo(~)1819 2328 y(G)1896 2287 y Fm(m)p Fi(\000)p Fl(3)1896 2352 y Fm(m)p Fi(\000)p Fl(2)2053 2328 y Fq(\)])g Fn(\001)f Fo(~)-48 b(")o Fq(\()p Fo(k)2324 2343 y Fm(i)2352 2328 y Fq(\()p Fo(m)2475 2287 y Fi(0)2499 2328 y Fq(\)\))2575 2287 y Fi(\003)2624 2260 y Fo(e)2669 2224 y Fl(+)p Fm(i\013)2790 2207 y(~)2793 2224 y(k)2830 2234 y Fb(i)2856 2224 y Fl(\()p Fm(m)2945 2201 y Fd(0)2969 2224 y Fl(\)\))p Fi(\001)o Fm(~)-34 b(x)p 2624 2305 464 4 v 2630 2321 a Fh(q)p 2713 2321 368 4 v 100 x Fq(2)p Fn(j)p Fo(k)2841 2436 y Fm(i)2869 2421 y Fq(\()p Fo(m)2992 2392 y Fi(0)3015 2421 y Fq(\))p Fn(j)3119 2328 y Fo(;)38 b Fq(\(I)s(I)s(I.90\))257 2657 y(while)f(the)f(other)g(v)m(ertex)h(op)s(erators,)g(\000)1766 2615 y Fi(\006)1766 2678 y Fm(m)p Fi(\000)p Fl(2)1922 2657 y Fq(,)g(remain)f(unc)m(hanged.)54 b(W)-8 b(e)36 b(use)h(the)f(no-)257 2777 y(tation)g Fo(\016)t(g)646 2741 y Fl(2)642 2802 y Fm(m)p Fi(\000)p Fl(2)834 2777 y Fq(for)g(the)g(general)h(elemen)m(t)h(in)e(the)g(new)h(family)g(of)e (v)m(ertex)j(op)s(erators.)257 2897 y(Hence)28 b(w)m(e)f(can)f(rewrite) h(\(I)s(I)s(I.86\))e(as)h(an)g Fo(\013)q Fq(-indep)s(enden)m(t)h(sum)g (of)e(the)i(follo)m(wing)f(pro)s(d-)257 3018 y(ucts)654 3138 y Fo(\034)707 3097 y Fm(N)696 3163 y(])774 3138 y Fq(\()p Fo(g)863 3097 y Fl(1)859 3163 y Fm(m)p Fi(\000)p Fl(2)1016 3138 y Fq(\))p Fo(\016)t(g)1152 3097 y Fl(2)1148 3163 y Fm(m)p Fi(\000)p Fl(2)1304 3138 y Fo(\034)1357 3097 y Fm(N)1346 3163 y(])1425 3138 y Fq(\()p Fo(g)1514 3097 y Fl(1)1510 3163 y Fm(m)p Fi(\000)p Fl(2)1666 3138 y Fq(\))c Fn(\001)g(\001)p Fo(\034)1857 3097 y Fm(N)1846 3163 y(])1925 3138 y Fq(\()p Fo(g)2014 3097 y Fl(1)2010 3163 y Fm(m)p Fi(\000)p Fl(2)2166 3138 y Fq(\))p Fo(g)2255 3097 y Fl(3)2251 3163 y Fm(m)p Fi(\000)p Fl(2)2424 3138 y Fn(\001)17 b(\001)g(\001)e Fo(g)2608 3097 y Fl(3)2604 3163 y Fm(m)p Fi(\000)p Fl(2)2760 3138 y Fo(:)397 b Fq(\(I)s(I)s (I.91\))257 3330 y(Next,)43 b(in)e(Expression)h(\(I)s(I)s(I.84\),)g(w)m (e)f(rewrite)g(\()p Fo(\036)2120 3345 y Fm(m)p Fi(\000)p Fl(2)2277 3330 y Fq(\))2315 3294 y Fm(T)2410 3330 y Fq(as)f Fo(\034)2590 3294 y Fm(N)2658 3330 y Fq(\()p Fo(P)2759 3345 y Fm(m)p Fi(\000)p Fl(2)2916 3330 y Fq(\)\()3005 3304 y(~)2992 3330 y Fo(\036)3050 3345 y Fm(m)p Fi(\000)p Fl(3)3206 3330 y Fq(\))3244 3294 y Fm(T)3299 3330 y Fq(;)k(see)257 3451 y(the)33 b(de\014nitions)h(in)f(Eqs.)h(\(I)s(I.11\),)e(\(I)s (I.13\))h(and)f(\(I)s(I)s(I.22\).)257 3571 y(W)-8 b(e)26 b(denote)h(b)m(y)860 3539 y Fh(f)854 3571 y Fo(D)935 3586 y Fm(m)p Fi(\000)p Fl(3)1092 3571 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))26 b(the)g(general)g(term,)h(i.e.,)h(pro)s(duct)e(of)f(op)s (erators,)i(that)e(w)m(e)257 3691 y(obtain)h(from)f Fo(D)858 3706 y Fm(m)p Fi(\000)p Fl(2)1015 3691 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))p Fo(P)1301 3706 y Fm(m)p Fi(\000)p Fl(2)1484 3691 y Fq(under)26 b(the)g(action)g(of)f(the)h(map)g Fo(\034)2728 3655 y Fm(N)2717 3716 y(])2821 3691 y Fq(and)g(after)f (break-)257 3832 y(ing)31 b(the)g(v)m(ertex)i(op)s(erators)d(according) h(to)f(\(I)s(I)s(I.89\))g(and)h(\(I)s(I)s(I.90\).)42 b(In)2898 3800 y Fh(f)2892 3832 y Fo(D)2973 3847 y Fm(m)p Fi(\000)p Fl(3)3130 3832 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))31 b(w)m(e)257 3952 y(distinguish)k(three)e(di\013eren)m(t)h (groups)f(of)f(elemen)m(tary)i(ob)5 b(jects:)257 4148 y Fh(e)236 4181 y Fn(G)301 4144 y Fl(1)295 4205 y Fm(m)p Fi(\000)p Fl(3)1468 4380 y Fh(e)1444 4412 y Fo(P)1507 4427 y Fm(m)p Fi(\000)p Fl(3)3184 4412 y Fq(\(I)s(I)s(I.92\))1467 4540 y Fh(e)1444 4573 y Fo(R)1519 4531 y Fl(+)1518 4597 y Fm(m)p Fi(\000)p Fl(3)1675 4573 y Fq(\()p Fo(z)1762 4522 y Fl(\()p Fm(m)1851 4498 y Fd(0)1875 4522 y Fl(\))1758 4594 y Fm(m)p Fi(\000)p Fl(2)1915 4573 y Fq(\))1977 4540 y Fh(e)1953 4573 y Fo(P)2030 4531 y Fi(?)2016 4597 y Fm(m)p Fi(\000)p Fl(3)3184 4573 y Fq(\(I)s(I)s(I.93\))1467 4707 y Fh(e)1444 4739 y Fo(R)1519 4698 y Fl(+)1518 4764 y Fm(m)p Fi(\000)p Fl(3)1675 4739 y Fq(\()p Fo(z)1762 4688 y Fl(\()p Fm(m)1851 4665 y Fd(0)1875 4688 y Fl(\))1758 4761 y Fm(m)p Fi(\000)p Fl(1)1938 4739 y Fn(\000)22 b Fo(S)2097 4755 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2331 4739 y Fq(\()p Fo(k)s Fq(\)\))2522 4707 y Fh(e)2499 4739 y Fo(P)2576 4698 y Fi(?)2562 4764 y Fm(m)p Fi(\000)p Fl(3)3184 4739 y Fq(\(I)s(I)s(I.94\))501 4959 y(where)34 b Fo(m)868 4923 y Fi(0)919 4959 y Fn(\025)29 b Fo(m)22 b Fn(\000)h Fq(1,)32 b(0)27 b Fn(\024)i Fo(l)r Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\))k Fn(\024)2017 4933 y Fq(\026)2017 4959 y Fo(l)s Fq(\()p Fo(m)22 b Fn(\000)h Fq(1\).)p eop end %%Page: 42 43 TeXDict begin 42 42 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(42)257 541 y Fh(e)236 573 y Fn(G)301 537 y Fl(2)295 597 y Fm(m)p Fi(\000)p Fl(3)752 760 y Fq(\007)828 719 y Fi(\006)828 785 y Fm(m)p Fi(\000)p Fl(3)984 760 y Fq(\()p Fo(k)1073 775 y Fm(i)1101 760 y Fq(\()p Fo(q)1186 719 y Fi(0)1209 760 y Fq(\)\))98 b(;)114 b(\000)1585 719 y Fi(\006)p Fm(;)p Fi(\006)1718 760 y Fq(\()p Fo(k)1807 775 y Fm(i)1835 760 y Fq(\()p Fo(q)1920 719 y Fi(0)1943 760 y Fq(\))p Fo(;)17 b(k)2076 775 y Fm(j)2112 760 y Fq(\()p Fo(q)2197 719 y Fi(00)2239 760 y Fq(\)\))869 b(\(I)s(I)s(I.95\))752 994 y(\001\007)909 953 y Fl(+)909 1018 y Fm(m)p Fi(\000)p Fl(3)1066 994 y Fq(\()p Fo(k)1155 1009 y Fm(i)1182 994 y Fq(\()p Fo(q)1267 953 y Fi(0)1290 994 y Fq(\)\))28 b(:=)g([)p Fo(a)p Fq(\()1656 969 y Fo(~)1641 994 y(G)1718 953 y Fm(m)p Fi(\000)p Fl(3)1718 1018 y Fm(m)p Fi(\000)p Fl(2)1875 994 y Fq(\))22 b(+)g Fo(a)2084 953 y Fi(\003)2123 994 y Fq(\()2176 969 y Fo(~)2161 994 y(G)2238 953 y Fm(m)p Fi(\000)p Fl(3)2238 1018 y Fm(m)p Fi(\000)p Fl(2)2395 994 y Fq(\)])g Fn(\001)f Fo(~)-48 b(")p Fq(\()p Fo(k)2667 1009 y Fm(i)2694 994 y Fq(\()p Fo(m)2817 953 y Fi(0)2841 994 y Fq(\)\))2927 926 y Fo(e)2972 890 y Fi(\000)p Fm(i\013)3093 873 y(~)3096 890 y(k)3133 900 y Fb(i)3159 890 y Fl(\()p Fm(q)3220 867 y Fd(0)3243 890 y Fl(\)\))p Fi(\001)o Fm(~)-34 b(x)p 2927 970 435 4 v 2938 987 a Fh(q)p 3021 987 330 4 v 100 x Fq(2)p Fn(j)p Fo(k)3149 1102 y Fm(i)3176 1087 y Fq(\()p Fo(q)3261 1058 y Fi(0)3285 1087 y Fq(\))p Fn(j)3184 1241 y Fq(\(I)s(I)s(I.96\))752 1468 y(\001\007)909 1427 y Fi(\000)909 1493 y Fm(m)p Fi(\000)p Fl(3)1066 1468 y Fq(\()p Fo(k)1155 1483 y Fm(i)1182 1468 y Fq(\()p Fo(q)1267 1427 y Fi(0)1290 1468 y Fq(\)\))28 b(:=)g([)p Fo(a)p Fq(\()1656 1443 y Fo(~)1641 1468 y(G)1718 1427 y Fm(m)p Fi(\000)p Fl(3)1718 1493 y Fm(m)p Fi(\000)p Fl(2)1875 1468 y Fq(\))22 b(+)g Fo(a)2084 1427 y Fi(\003)2123 1468 y Fq(\()2176 1443 y Fo(~)2161 1468 y(G)2238 1427 y Fm(m)p Fi(\000)p Fl(3)2238 1493 y Fm(m)p Fi(\000)p Fl(2)2395 1468 y Fq(\)])g Fn(\001)f Fo(~)-48 b(")p Fq(\()p Fo(k)2667 1483 y Fm(i)2694 1468 y Fq(\()p Fo(m)2817 1427 y Fi(0)2841 1468 y Fq(\)\))2917 1427 y Fi(\003)2966 1401 y Fo(e)3011 1364 y Fl(+)p Fm(i\013)3132 1347 y(~)3135 1364 y(k)3172 1374 y Fb(i)3199 1364 y Fl(\()p Fm(q)3260 1341 y Fd(0)3283 1364 y Fl(\)\))p Fi(\001)o Fm(~)-34 b(x)p 2966 1445 435 4 v 2978 1461 a Fh(q)p 3061 1461 330 4 v 100 x Fq(2)p Fn(j)p Fo(k)3189 1576 y Fm(i)3216 1561 y Fq(\()p Fo(q)3301 1533 y Fi(0)3324 1561 y Fq(\))p Fn(j)752 1725 y Fo(W)858 1683 y Fm(m)p Fi(\000)p Fl(3)844 1749 y Fm(m)p Fi(\000)p Fl(2)1036 1725 y Fn(\000)23 b(h)p Fo(W)1281 1683 y Fm(m)p Fi(\000)p Fl(3)1267 1749 y Fm(m)p Fi(\000)p Fl(2)1437 1725 y Fn(i)1476 1740 y Fl(\012)1527 1749 y Fb(m)p Fd(\000)p Ff(2)3184 1725 y Fq(\(I)s(I)s(I.97\))501 1912 y(where)34 b Fo(m)23 b Fn(\000)f Fq(2)28 b Fn(\024)g Fo(q)1219 1876 y Fi(0)1242 1912 y Fo(;)17 b(q)1333 1876 y Fi(00)1403 1912 y Fn(\024)28 b Fo(a)p Fq(,)k(1)c Fn(\024)g Fo(i;)17 b(j)34 b Fn(\024)2048 1890 y Fq(\026)2056 1912 y Fo(i)p Fq(\()p Fo(p)22 b Fn(\000)h Fq(1\))32 b(with)h Fo(m)28 b Fn(\024)g Fo(p)g Fn(\024)g Fo(a)23 b Fq(+)f(1.)501 2069 y(In)g(group)897 2044 y(~)878 2069 y Fn(G)943 2033 y Fl(2)937 2094 y Fm(m)p Fi(\000)p Fl(3)1094 2069 y Fq(,)i(w)m(e)f (distinguish)g(the)f(op)s(erators)g(in)g(Eqs.)h(\(I)s(I)s(I.95\))e(and) h(\(I)s(I)s(I.96\),)501 2190 y(whic)m(h)36 b(come)f(from)g(a)f(v)m (ertex)i(op)s(erator)e(in)g Fn(G)2221 2153 y Fl(2)2215 2214 y Fm(m)p Fi(\000)p Fl(2)2407 2190 y Fq(and)g(that)g(w)m(e)i(ha)m (v)m(e)g(already)501 2310 y(denoted)f(b)m(y)f Fo(\016)t(g)1105 2274 y Fl(2)1101 2335 y Fm(m)p Fi(\000)p Fl(2)1257 2310 y Fq(,)g(from)g(the)f(op)s(erators)h(in)f(Eq.)i(\(I)s(I)s(I.97\))e (whic)m(h)i(w)m(e)f(denote)501 2430 y(b)m(y)g Fn(W)743 2389 y Fm(m)p Fi(\000)p Fl(3)735 2452 y Fm(m)p Fi(\000)p Fl(2)900 2430 y Fq(.)44 b(This)33 b(notation)f(is)i(useful)f(in)g(the)g (pro)s(of)f(of)g(next)h(Lemma)g(I)s(I)s(I.5.)257 2601 y Fh(e)236 2633 y Fn(G)301 2597 y Fl(3)295 2658 y Fm(m)p Fi(\000)p Fl(3)1056 2804 y Fq(1)p 677 2849 807 4 v 677 2959 a Fo(S)737 2975 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(2\))971 2959 y Fq(\()p Fo(k)s Fq(\))22 b Fn(\000)g Fo(z)1271 2908 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1267 2981 y Fm(m)p Fi(\000)p Fl(2)1494 2872 y Fe(1)1550 2887 y Fm(m)p Fi(\000)p Fl(2)3464 2872 y Fq(\(I)s(I)s(I.98\))1267 3089 y(1)p 677 3133 1230 4 v 677 3244 a Fo(S)737 3259 y Fm(l)q Fl(\()p Fm(p)822 3240 y Fd(0)844 3259 y Fl(\))p Fm(;l)q Fl(\()p Fm(p)976 3240 y Fd(0)999 3259 y Fi(\000)p Fl(1\))1120 3244 y Fq(\()p Fo(k)s Fq(\))h Fn(\000)f Fo(\013)1435 3215 y Fm(p)1471 3196 y Fd(0)1493 3215 y Fi(\000)p Fm(m)p Fi(\000)p Fl(1)1704 3244 y Fo(z)1753 3193 y Fl(\()p Fm(p)1816 3174 y Fd(0)1839 3193 y Fl(\))1749 3265 y Fm(m)p Fi(\000)p Fl(1)1916 3156 y Fe(1)1972 3171 y Fm(m)p Fi(\000)p Fl(2)2146 3156 y Fq(;)2684 3089 y(1)p 2216 3133 985 4 v 2216 3244 a Fo(S)2276 3259 y Fm(l)q Fl(\()p Fm(p)2361 3240 y Fd(0)2383 3259 y Fl(\))2415 3244 y Fq(\()p Fo(k)s Fq(\))g Fn(\000)h Fo(\013)2730 3215 y Fm(p)2766 3196 y Fd(0)2787 3215 y Fi(\000)p Fm(m)p Fi(\000)p Fl(1)2999 3244 y Fo(z)3048 3193 y Fl(\()p Fm(p)3111 3174 y Fd(0)3134 3193 y Fl(\))3044 3265 y Fm(m)p Fi(\000)p Fl(1)3211 3156 y Fe(1)3267 3171 y Fm(m)p Fi(\000)p Fl(2)3543 3156 y Fo(;)-106 b Fq(\(I)s(I)s(I.99\))667 3411 y Fo(S)727 3426 y Fm(l)q Fl(\()p Fm(m)p Fl(\))871 3411 y Fq(\()p Fo(k)s Fq(\))p Fe(1)1057 3426 y Fm(m)p Fi(\000)p Fl(2)1213 3411 y Fo(;)17 b Fq(\()p Fo(z)1344 3370 y Fl(\()p Fm(p)1407 3346 y Fd(00)1448 3370 y Fl(\))1340 3436 y Fm(m)1502 3411 y Fn(\000)23 b Fo(z)1651 3360 y Fl(\()p Fm(p)1714 3337 y Fd(00)1755 3360 y Fl(\))1647 3433 y Fm(m)p Fi(\000)p Fl(1)1804 3411 y Fq(\))p Fe(1)1898 3426 y Fm(m)p Fi(\000)p Fl(2)2054 3411 y Fo(;)1335 b Fq(\(I)s(I)s(I.100\))667 3556 y Fn(h)p Fo(W)812 3515 y Fm(m)p Fi(\000)p Fl(3)798 3581 y Fm(m)p Fi(\000)p Fl(2)968 3556 y Fn(i)1007 3571 y Fl(\012)1058 3580 y Fb(m)p Fd(\000)p Ff(2)1199 3556 y Fe(1)1255 3571 y Fm(m)p Fi(\000)p Fl(2)1412 3556 y Fq(;)17 b(\()p Fo(E)1572 3515 y Fm(m)p Fi(\000)p Fl(3)1566 3581 y Fm(m)p Fi(\000)p Fl(2)1729 3556 y Fq(\))1767 3515 y Fm(T)1767 3581 y(])1822 3556 y Fe(1)1878 3571 y Fm(m)p Fi(\000)p Fl(2)2034 3556 y Fq(;)g Fn(f)p Fo(\030)2176 3515 y Fm(m)p Fi(\000)p Fl(2)2171 3581 y Fm(m)p Fi(\000)p Fl(1)2332 3556 y Fe(1)2388 3571 y Fm(m)p Fi(\000)p Fl(2)2545 3556 y Fn(g)p Fo(:)g(::::)g Fn(f)p Fo(\030)2862 3515 y Fm(a)p Fi(\000)p Fl(1)2857 3581 y Fm(a)2992 3556 y Fe(1)3048 3571 y Fm(m)p Fi(\000)p Fl(2)3205 3556 y Fn(g)p Fo(:)501 3744 y Fq(with)33 b Fo(m)23 b Fn(\000)g Fq(1)k Fn(\024)h Fo(p)1161 3707 y Fi(0)1212 3744 y Fn(\024)g Fo(a)p Fq(,)33 b Fo(m)28 b Fn(\024)g Fo(p)1695 3707 y Fi(00)1765 3744 y Fn(\024)g Fo(a)p Fq(.)257 3935 y(With)33 b(the)g(de\014nitions)h(ab)s(o)m(v)m(e,)1459 3902 y Fh(f)1453 3935 y Fo(D)1534 3950 y Fm(m)p Fi(\000)p Fl(3)1691 3935 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))33 b(has)f(the)h(follo)m(wing)g(form) 667 4081 y Fl(\026)669 4097 y Fm(j)632 4127 y Fh(Y)624 4309 y Fm(j)t Fl(=1)763 4093 y Fh(Z)809 4282 y Fl(\000)853 4291 y Fb(m)p Fd(\000)p Ff(2)1011 4210 y Fo(dz)1111 4159 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))p Fm(;j)1107 4232 y(m)p Fi(\000)p Fl(2)1395 4210 y Fq(~)-53 b Fo(g)1442 4169 y Fl(1)1438 4235 y Fm(m)p Fi(\000)p Fl(3)1598 4210 y Fq(~)h Fo(g)1646 4169 y Fl(2)1642 4235 y Fm(m)p Fi(\000)p Fl(3)1802 4210 y Fq(~)f Fo(g)1849 4169 y Fl(1)1845 4235 y Fm(m)p Fi(\000)p Fl(3)2024 4210 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2153 4169 y Fl(1)2149 4235 y Fm(m)p Fi(\000)p Fl(3)1391 4276 y Fh(|)p 1428 4276 383 10 v 383 w({z)p 1885 4276 V 383 w(})2321 4210 y Fn(\002)737 b Fq(\(I)s(I)s(I.101\))916 4464 y Fn(\002)21 b Fq(~)-53 b Fo(g)1061 4423 y Fl(2)1057 4489 y Fm(m)p Fi(\000)p Fl(3)1010 4531 y Fh(|)p 1047 4531 28 10 v 28 w({z)p 1149 4531 V 28 w(})1230 4464 y Fn(\001)22 b(\001)j Fq(~)-52 b Fo(g)1381 4423 y Fl(1)1377 4489 y Fm(m)p Fi(\000)p Fl(3)1537 4464 y Fq(~)f Fo(g)1584 4423 y Fl(2)1580 4489 y Fm(m)p Fi(\000)p Fl(3)1740 4464 y Fq(~)h Fo(g)1788 4423 y Fl(1)1784 4489 y Fm(m)p Fi(\000)p Fl(3)1962 4464 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2091 4423 y Fl(1)2087 4489 y Fm(m)p Fi(\000)p Fl(3)1330 4531 y Fh(|)p 1367 4531 383 10 v 383 w({z)p 1824 4531 V 383 w(})2264 4464 y Fq(~)f Fo(g)2311 4423 y Fl(3)2307 4489 y Fm(m)p Fi(\000)p Fl(3)2486 4464 y Fn(\001)21 b(\001)s Fq(~)-52 b Fo(g)2614 4423 y Fl(3)2610 4489 y Fm(m)p Fi(\000)p Fl(3)2260 4531 y Fh(|)p 2297 4531 179 10 v 179 w({z)p 2550 4531 V 179 w(})2767 4464 y Fq(\()p Fo(k)s(;)17 b(z)t(;)g(z)3045 4414 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))3041 4486 y Fm(m)p Fi(\000)p Fl(2)3257 4464 y Fq(\))257 4700 y(where:)349 4876 y(\(i\))49 b(The)41 b(argumen)m(ts)g Fo(k)i Fq(and)d Fo(z)45 b Fq(are)40 b(the)g(momen)m(ta)h(\(and)f(helicities\))i(in)m (tegration)501 4996 y(v)-5 b(ariables)40 b(and)g(the)g(con)m(tour)f(in) m(tegration)h(v)-5 b(ariables)40 b(in)g(expression)i(\(I)s(I)s(I.84\),) 501 5130 y(resp)s(ectiv)m(ely;)29 b(the)23 b(argumen)m(t)g Fo(z)1686 5079 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1682 5152 y Fm(m)p Fi(\000)p Fl(2)1921 5130 y Fq(stands)h(for)d(all)i(the)g (in)m(tegration)f(v)-5 b(ariables)501 5268 y({)42 b Fn(f)p Fo(z)691 5217 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))p Fm(;j)687 5290 y(m)p Fi(\000)p Fl(2)999 5268 y Fq(:)i Fo(j)49 b Fq(=)44 b(1)p Fo(;)17 b Fn(\001\001)p Fo(;)1470 5245 y Fq(\026)1472 5268 y Fo(j)t Fn(g)p Fq(,)44 b(where)1927 5245 y(\026)1929 5268 y Fo(j)k Fq(is)42 b Fo(\013)q Fq(-indep)s(enden)m (t)h({)f(in)g(the)h(con)m(tour)501 5388 y(in)m(tegrals)34 b(whic)m(h)g(are)e(related)i(to)e(the)h(expansion)h(of)e Fo(P)2576 5403 y Fm(m)p Fi(\000)p Fl(2)2765 5388 y Fq(and)h Fo(P)3032 5352 y Fi(?)3018 5413 y Fm(m)p Fi(\000)p Fl(2)3174 5388 y Fq(;)p eop end %%Page: 43 44 TeXDict begin 43 43 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(43)322 573 y(\(ii\))49 b(The)c(\014rst)f (collection)h(of)e(op)s(erators)h(corresp)s(onds,)k(up)c(to)f(m)m (ultiplicativ)m(e)j(c-)501 693 y(n)m(um)m(b)s(ers)35 b(\(see)f(b)s(elo)m(w,)h(p)s(oin)m(t)e(iv\)\),)h(to)e(one)i(of)e(the)i (terms)g(in)f Fo(\034)2884 657 y Fm(N)2873 718 y(])2952 693 y Fq(\()p Fo(g)3041 657 y Fl(1)3037 718 y Fm(m)p Fi(\000)p Fl(2)3194 693 y Fq(\),)g Fo(g)3343 657 y Fl(1)3339 718 y Fm(m)p Fi(\000)p Fl(2)501 814 y Fq(b)s(eing)g(the)g(\014rst)g(op) s(erator)f(in)h(\(I)s(I)s(I.86\);)295 1002 y(\(iii\))49 b(The)28 b(second)f(collection)g(of)f(op)s(erators)g(consists)i(of)e (one)h(op)s(erator)f(whic)m(h)h(corre-)501 1122 y(sp)s(onds)f(to)e(one) h(of)f(t)m(yp)s(e)h Fo(\016)t(g)1511 1086 y Fl(2)1507 1147 y Fm(m)p Fi(\000)p Fl(2)1663 1122 y Fq(,)i Fo(g)1768 1086 y Fl(2)1764 1147 y Fm(m)p Fi(\000)p Fl(2)1945 1122 y Fq(b)s(eing)d(the)h(second)h(op)s(erator)e(in)h(\(I)s(I)s(I.86\);)298 1310 y(\(iv\))49 b(The)41 b(other)f(collections)h(of)e(op)s(erators)h (ha)m(v)m(e)h(an)e(analogous)h(origin)f(from)h(the)501 1431 y(com)m(bination)f(of)f(op)s(erators)f(in)i(\(I)s(I)s(I.86\),)f (except)i(the)e(last)h(one)f(whic)m(h)h(is)g(the)501 1551 y(collection)c(of)e(all)g(c-n)m(um)m(b)s(ers,)j(the)e(original)g (c-n)m(um)m(b)s(er)g(op)s(erators)g(in)g(\(I)s(I)s(I.85\))501 1672 y(and)40 b(the)f(ones)h(whic)m(h)h(come)f(from)f(the)h(terms)g(in) g Fo(\034)2502 1635 y Fm(N)2491 1696 y(])2570 1672 y Fq(\()p Fo(g)2659 1635 y Fl(1)2655 1696 y Fm(m)p Fi(\000)p Fl(2)2811 1672 y Fq(\))f(and)h(that)f(ha)m(v)m(e)501 1792 y(b)s(een)34 b(pushed)g(to)e(the)h(v)m(ery)h(righ)m(t.)257 1950 y(Hyp)s(othesis)24 b Fj(H2\))h Fq(together)d(with)h(the)f(results) h(in)f(Lemmata)g(I)s(I)s(I.1,)i(I)s(I)s(I.2)d(and)h(Eq.)h(\(I.54\))257 2070 y(imply)430 2237 y Fn(k)486 2205 y Fh(f)480 2237 y Fo(D)561 2252 y Fm(m)p Fi(\000)p Fl(3)718 2237 y Fq(\()p Fo(k)810 2196 y Fi(0)833 2237 y Fo(;)17 b(z)t Fq(\))964 2196 y Fi(\003)1010 2205 y Fh(f)1004 2237 y Fo(D)1085 2252 y Fm(m)p Fi(\000)p Fl(3)1241 2237 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))p Fn(k)84 b(\024)f(k)1814 2205 y Fh(f)1808 2237 y Fo(D)1889 2252 y Fm(m)p Fi(\000)p Fl(3)2046 2237 y Fq(\()p Fo(k)2138 2196 y Fi(0)2161 2237 y Fo(;)17 b(z)t Fq(\))2292 2196 y Fi(\003)2332 2237 y Fn(kk)2438 2205 y Fh(f)2432 2237 y Fo(D)2513 2252 y Fm(m)p Fi(\000)p Fl(3)2669 2237 y Fq(\()p Fo(k)s(;)g(z)t Fq(\))p Fn(k)1598 2382 y(\024)83 b Fo(C)29 b Fn(\001)22 b(E)9 b Fq(\()p Fo(D)2088 2397 y Fm(m)p Fi(\000)p Fl(2)2244 2382 y Fq(\()p Fo(k)2336 2341 y Fi(0)2360 2382 y Fo(;)17 b(z)t Fq(\)\))p Fn(E)9 b Fq(\()p Fo(D)2710 2397 y Fm(m)p Fi(\000)p Fl(2)2866 2382 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\))8 b(\(I)s(I)s(I.102\))257 2549 y(where)34 b Fo(C)40 b Fq(is)33 b(an)f Fo(\013)q Fq(-indep)s(enden)m(t)i(constan)m(t.)257 2669 y(Then,)k(starting)d (from)g(the)h(v)m(ector)h(in)e(Eq.)i(\(I)s(I)s(I.84\),)f(under)g(the)g (action)f(of)g(the)h(map)257 2790 y Fo(\034)310 2754 y Fm(N)299 2814 y(])378 2790 y Fq(,)i(and)g(after)f(the)g(partial)g (re-expansion)i(of)d(\()p Fo(\036)2147 2805 y Fm(m)p Fi(\000)p Fl(2)2304 2790 y Fq(\))2342 2754 y Fm(T)2434 2790 y Fq(\(=)p Fo(\034)2601 2754 y Fm(N)2668 2790 y Fq(\()p Fo(P)2769 2805 y Fm(m)p Fi(\000)p Fl(2)2926 2790 y Fq(\)\()3015 2763 y(~)3002 2790 y Fo(\036)3060 2805 y Fm(m)p Fi(\000)p Fl(3)3216 2790 y Fq(\))3254 2754 y Fm(T)3309 2790 y Fq(\))h(w)m(e)257 2910 y(get)c(an)f Fo(\013)q Fq(-indep)s(enden)m(t)i(sum)g(of)e(expressions)j(of)d(the)h (t)m(yp)s(e)459 3055 y Fm(a)424 3080 y Fh(Y)390 3265 y Fm(p)426 3246 y Fd(0)448 3265 y Fl(=)p Fm(m)581 3067 y Fh(\020)648 3046 y(Z)694 3235 y Fl(\000)738 3243 y Fb(m)817 3163 y Fo(\013)880 3122 y Fm(p)916 3099 y Fd(0)938 3122 y Fi(\000)p Fm(m)1074 3031 y Fl(\026)1076 3046 y Fm(j)s Fl(\()p Fm(p)1171 3023 y Fd(0)1194 3046 y Fl(\))1095 3080 y Fh(Y)1087 3262 y Fm(j)t Fl(=1)1238 3163 y Fo(dz)1338 3122 y Fl(\()p Fm(p)1401 3099 y Fd(0)1423 3122 y Fl(\))p Fm(;j)1334 3188 y(m)1507 3067 y Fh(\021)1573 3046 y(Z)1619 3235 y Fl(\000)1663 3244 y Fb(m)p Fd(\000)p Ff(1)1820 3031 y Fl(\026)1821 3046 y Fm(j)t Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1888 3080 y Fh(Y)1880 3262 y Fm(j)t Fl(=1)2078 3163 y Fo(dz)2178 3112 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)2174 3185 y(m)p Fi(\000)p Fl(1)2442 3163 y Fn(\002)616 b Fq(\(I)s(I)s(I.103\))556 3498 y Fn(\002)662 3390 y Fm(a)p Fl(+1)673 3415 y Fh(Y)650 3590 y Fm(p)p Fl(=)p Fm(m)819 3402 y Fh(\020)885 3381 y(Z)932 3570 y Fi(K)986 3537 y Fb(p)p Fd(\000)p Ff(2)986 3592 y Fb(p)p Fd(\000)p Ff(1)1115 3366 y Fl(\026)1121 3381 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1170 3415 y Fh(Y)1166 3598 y Fm(i)p Fl(=1)1342 3498 y Fo(dk)1444 3513 y Fm(i)1472 3498 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))1767 3402 y Fh(\021)1822 3466 y(f)1817 3498 y Fo(D)1898 3513 y Fm(m)p Fi(\000)p Fl(3)2054 3498 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()2328 3472 y(~)2315 3498 y Fo(\036)2373 3513 y Fm(m)p Fi(\000)p Fl(3)2530 3498 y Fq(\))2568 3457 y Fm(T)2645 3498 y Fn(\012)23 b Fo(C)2822 3457 y Fm(m)p Fi(\000)p Fl(2)2815 3523 y Fm(a)2979 3498 y Fq(\()p Fo(k)s Fq(\)\012)3179 3457 y Fm(m)p Fi(\000)p Fl(2)3179 3523 y Fm(a)3336 3498 y Fo(:)257 3740 y Fq(Because)36 b(of)d(Prop)s(ert)m(y)i Fn(P)8 b Fq(2)34 b(whic)m(h)h(holds)g(for)e Fo(D)2084 3755 y Fm(m)p Fi(\000)p Fl(2)2241 3740 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\),)34 b(b)m(y)h(h)m(yp)s(othesis,)i(and)c(b)s(e-)257 3861 y(cause)46 b(of)f(inequalit)m(y)i(\(I)s(I)s(I.102\),)g(the)f(norm) f(of)f(eac)m(h)i(Expression)i(\(I)s(I)s(I.103\))c(is,)49 b(at)257 3981 y(most,)29 b(of)e(order)h Fn(O)s Fq(\(1\))f(in)h Fo(\013)q Fq(.)41 b(In)28 b(addition,)h(b)s(ecause)g(of)e(Inequalit)m (y)i(\(I)s(I)s(I.88\),)f(the)g(sum)257 4101 y(of)k(the)h(new)g (expressions)i(of)c(t)m(yp)s(e)i(\(I)s(I)s(I.103\))f(di\013ers)h(from)f (the)g(original)g(one)h(\(I)s(I)s(I.84\))257 4222 y(b)m(y)h(a)e (remainder)i(of)e(order,)h(at)f(most,)h Fo(o)p Fq(\()p Fo(\013)1861 4186 y Fm(N)1928 4222 y Fq(\))f(in)h(norm.)257 4342 y(Under)g(the)f(Wic)m(k-ordering)g(of)g(the)g(photon)f(creation-)h (and)f(annihilation)h(op)s(erators)257 4463 y(in)i(the)f(shell)h Fn(K)840 4421 y Fm(m)p Fi(\000)p Fl(3)839 4484 y Fm(m)p Fi(\000)p Fl(2)997 4463 y Fq(,)g(b)m(y)g(decomp)s(osing)g(the)f(iden)m (tit)m(y)i Fe(1)2357 4478 y Fm(m)p Fi(\000)p Fl(3)2547 4463 y Fq(in)m(to)e Fo(P)2808 4478 y Fm(m)p Fi(\000)p Fl(3)2987 4463 y Fq(+)22 b Fo(P)3162 4426 y Fi(?)3148 4487 y Fm(m)p Fi(\000)p Fl(3)3338 4463 y Fq(and)257 4604 y(using)27 b(the)f(c)m(hange)g(of)f(v)-5 b(ariables)27 b Fo(z)1533 4568 y Fl(\()p Fm(p)1596 4545 y Fd(0)1619 4568 y Fl(\))p Fm(;j)1529 4629 y(m)1730 4604 y Fq(=)h Fo(\013)q(z)1946 4554 y Fl(\()p Fm(p)2009 4530 y Fd(0)2031 4554 y Fl(\))p Fm(;j)1942 4626 y(m)p Fi(\000)p Fl(1)2115 4604 y Fq(,)f(eac)m(h)g(Expression)h(\(I)s(I)s(I.103\))d(yields)257 4725 y(an)g Fo(\013)q Fq(-indep)s(enden)m(t)i(sum)e(of)g(expressions)j (of)c(the)h(t)m(yp)s(e)h(in)f(Eq.)h(\(I)s(I)s(I.84\))f(on)g(scale)h Fo(m)7 b Fn(\000)g Fq(3)459 4875 y Fm(a)424 4900 y Fh(Y)345 5085 y Fm(p)381 5066 y Fd(0)402 5085 y Fl(=)p Fm(m)p Fi(\000)p Fl(1)627 4887 y Fh(\020)691 4851 y Fl(\026)693 4866 y Fm(j)s Fl(\()p Fm(p)788 4843 y Fd(0)810 4866 y Fl(\))712 4900 y Fh(Y)704 5083 y Fm(j)t Fl(=1)854 4866 y Fh(Z)901 5055 y Fl(\000)945 5064 y Fb(m)p Fd(\000)p Ff(1)1102 4983 y Fo(\013)1165 4942 y Fm(p)1201 4919 y Fd(0)1223 4942 y Fi(\000)p Fm(m)p Fi(\000)p Fl(1)1435 4983 y Fo(dz)1535 4932 y Fl(\()p Fm(p)1598 4909 y Fd(0)1620 4932 y Fl(\))p Fm(;j)1531 5005 y(m)p Fi(\000)p Fl(1)1704 4887 y Fh(\021)1769 4851 y Fl(\026)1770 4866 y Fm(j)t Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1837 4900 y Fh(Y)1829 5083 y Fm(j)t Fl(=1)2027 4866 y Fh(Z)2073 5055 y Fl(\000)2117 5064 y Fb(m)p Fd(\000)p Ff(2)2275 4983 y Fo(dz)2375 4932 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))p Fm(;j)2371 5005 y(m)p Fi(\000)p Fl(2)2639 4983 y Fn(\002)419 b Fq(\(I)s(I)s(I.104\))511 5319 y Fn(\002)662 5211 y Fm(a)p Fl(+1)673 5236 y Fh(Y)605 5417 y Fm(p)p Fl(=)p Fm(m)p Fi(\000)p Fl(1)864 5222 y Fh(\020)930 5201 y(Z)977 5390 y Fi(K)1031 5358 y Fb(p)p Fd(\000)p Ff(2)1031 5413 y Fb(p)p Fd(\000)p Ff(1)1160 5186 y Fl(\026)1166 5201 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1215 5236 y Fh(Y)1211 5418 y Fm(i)p Fl(=1)1387 5319 y Fo(dk)1489 5334 y Fm(i)1517 5319 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))1812 5222 y Fh(\021)1862 5319 y Fo(D)1943 5334 y Fm(m)p Fi(\000)p Fl(3)2100 5319 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2419 5334 y Fm(m)p Fi(\000)p Fl(3)2575 5319 y Fq(\))2613 5277 y Fm(T)2690 5319 y Fn(\012)23 b Fo(C)2867 5277 y Fm(m)p Fi(\000)p Fl(3)2860 5343 y Fm(a)3024 5319 y Fq(\()p Fo(k)s Fq(\)\012)3224 5277 y Fm(m)p Fi(\000)p Fl(3)3224 5343 y Fm(a)3381 5319 y Fo(:)p eop end %%Page: 44 45 TeXDict begin 44 44 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(44)257 573 y(F)-8 b(or)26 b(the)h (expressions)j(of)c(t)m(yp)s(e)i(\(I)s(I)s(I.104\),)f(Prop)s(erties)h Fn(P)8 b Fq(1,)28 b Fn(P)8 b Fq(2)27 b(are)f(correctly)i(ful\014lled) 257 693 y(b)m(y)38 b(the)f(op)s(erators)g(in)g(the)g(term)h Fo(D)1614 708 y Fm(m)p Fi(\000)p Fl(3)1770 693 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\).)57 b(Prop)s(ert)m(y)38 b Fn(P)8 b Fq(1)37 b(is)g(trivial)g(b)s(ecause)i(of)257 814 y(the)44 b(structure)g(of)e(the)h(map)g Fo(\034)1443 777 y Fm(N)1432 838 y(])1511 814 y Fq(.)74 b(Prop)s(ert)m(y)44 b Fn(P)8 b Fq(2)43 b(is)h(a)e(corollary)h(of)g(the)g(follo)m(wing)257 934 y(lemma:)257 1137 y Fe(Lemma)36 b(I)s(I)s(I.5.)j Fj(In)32 b(an)f(expr)-5 b(ession)31 b(of)h(the)g(typ)-5 b(e)33 b(given)e(in)h(Eq.\(III.103\),)e(the)i(Wick-)257 1258 y(or)-5 b(dering)30 b(of)g(the)g(photon)g(cr)-5 b(e)g(ation-)29 b(and)h(annihilation)f(op)-5 b(er)g(ators)29 b(in)h(the)h(shel)5 b(l)29 b Fn(K)3338 1217 y Fm(m)p Fi(\000)p Fl(3)3337 1280 y Fm(m)p Fi(\000)p Fl(2)257 1378 y Fj(pr)-5 b(o)g(duc)g(es)25 b(a)f(\014nite,)j Fo(\013)q Fj(-indep)-5 b(endent,)25 b(numb)-5 b(er)24 b(of)h(expr)-5 b(essions)23 b(of)i(the)g(typ)-5 b(e)25 b(in)g(Eq.)f(\(III.104\).)257 1499 y(The)31 b(or)-5 b(der)31 b(in)g Fo(\013)h Fj(of)e(the)i(norm)e (estimate)h(of)g(e)-5 b(ach)30 b(of)h(these)g(expr)-5 b(essions)30 b(is)h(higher)f(or)257 1619 y(e)-5 b(qual)35 b(to)g(the)g(or)-5 b(der)34 b(of)h(the)g(norm)f(estimate)h(of)f(the)h (original)f(expr)-5 b(ession)34 b(\(III.103\).)257 1822 y(Pr)-5 b(o)g(of.)43 b Fq(In)33 b(the)g(App)s(endix.)257 1943 y(W)-8 b(e)41 b(can)g(no)m(w)f(pic)m(k)52 b(\026)-59 b Fo(\013)1106 1958 y Fm( )1152 1966 y Fb(m)1256 1943 y Fo(>)40 b Fq(0,)i Fo(N)10 b Fq(-dep)s(enden)m(t)43 b(only)-8 b(,)42 b(with)51 b(\026)-59 b Fo(\013)2625 1958 y Fm( )2671 1966 y Fb(m)2775 1943 y Fn(\024)51 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)27 b Fn(\000)h Fq(1\))40 b(and)267 2063 y(\026)-59 b Fo(\013)319 2078 y Fm( )365 2086 y Fb(m)462 2063 y Fn(\024)43 b Fq(\026)-58 b Fo(\013)635 2085 y Fm( )683 2062 y Fb(i)681 2107 y(m)p Fd(\000)p Ff(1)858 2063 y Fq(for)35 b(eac)m(h)i Fo(i)p Fq(,)g(where)g(the)f(v)m (ectors)i(in)e Fn(f)p Fo( )2355 2027 y Fm(i)2351 2088 y(m)p Fi(\000)p Fl(1)2508 2063 y Fn(g)g Fq(are)g(the)g(\014nitely)h (man)m(y)257 2197 y({)c(the)h(n)m(um)m(b)s(er)g(b)s(eing)g Fo(\013)q Fq(-indep)s(enden)m(t)g({)f(expressions)j(of)d(t)m(yp)s(e)h (\(I)s(I)s(I.104\))e(that)h(ha)m(v)m(e)257 2317 y(b)s(een)i(obtained.) 47 b(In)34 b(other)g(w)m(ords,)h(w)m(e)f(pic)m(k)45 b(\026)-59 b Fo(\013)2031 2332 y Fm( )2077 2340 y Fb(m)2170 2317 y Fo(>)29 b Fq(0)k(small)i(enough)f(suc)m(h)h(that)e(w)m(e)257 2438 y(can)c(apply)g(Hyp)s(othesis)i Fj(H3\))h Fq(up)d(to)f Fo(q)j Fq(=)d Fo(m)14 b Fn(\000)g Fq(1,)30 b(and)e(w)m(e)i(can)f(get)f (the)h(claim)g(for)f Fo( )3401 2453 y Fm(m)3468 2438 y Fq(.)257 2558 y(Concerning)35 b(Hyp)s(othesis)f Fj(H4\))p Fq(,)f(w)m(e)h(observ)m(e)h(that)e(the)g(structure)h(of)f(the)g (remainder)257 2679 y(-)38 b(of)f(order,)i(at)f(most,)h Fo(o)p Fq(\()p Fo(\013)1277 2642 y Fm(N)1344 2679 y Fq(\))f(-)f(that)h (has)g(b)s(een)g(neglected)i(in)e(the)g(step)h(from)e(scale)257 2799 y Fo(m)23 b Fn(\000)f Fq(2)33 b(to)f Fo(m)22 b Fn(\000)h Fq(3)32 b(is)h(clearly)h(an)f Fo(\013)q Fq(-indep)s(enden)m(t)h(sum)f (of)f(expressions)j(of)e(the)g(t)m(yp)s(e)447 2997 y Fm(a)413 3022 y Fh(Y)378 3207 y Fm(p)414 3188 y Fd(0)436 3207 y Fl(=)p Fm(m)570 3009 y Fh(\020)635 2973 y Fl(\026)636 2988 y Fm(j)s Fl(\()p Fm(p)731 2965 y Fd(0)754 2988 y Fl(\))656 3022 y Fh(Y)647 3204 y Fm(j)t Fl(=1)798 2988 y Fh(Z)844 3177 y Fl(\000)888 3185 y Fb(m)967 3105 y Fo(\013)1030 3064 y Fm(p)1066 3041 y Fd(0)1088 3064 y Fi(\000)p Fm(m)1209 3105 y Fo(dz)1309 3064 y Fl(\()p Fm(p)1372 3041 y Fd(0)1395 3064 y Fl(\))p Fm(;j)1305 3130 y(m)1479 3009 y Fh(\021)1544 2973 y Fl(\026)1545 2988 y Fm(j)s Fl(\()p Fm(m)p Fi(\000)p Fl(1\))1612 3022 y Fh(Y)1604 3204 y Fm(j)t Fl(=1)1801 2988 y Fh(Z)1848 3177 y Fl(\000)1892 3186 y Fb(m)p Fd(\000)p Ff(1)2049 3105 y Fo(dz)2149 3054 y Fl(\()p Fm(m)p Fi(\000)p Fl(1\))p Fm(;j)2145 3127 y(m)p Fi(\000)p Fl(1)2413 3105 y Fn(\002)645 b Fq(\(I)s(I)s(I.105\))544 3440 y Fn(\002)651 3332 y Fm(a)p Fl(+1)662 3357 y Fh(Y)638 3532 y Fm(p)p Fl(=)p Fm(m)808 3344 y Fh(\020)874 3323 y(Z)957 3350 y Fm(\033)997 3359 y Fb(p)p Fd(\000)p Ff(2)920 3512 y Fm(\033)960 3521 y Fb(p)p Fd(\000)p Ff(1)1127 3308 y Fl(\026)1133 3323 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1182 3357 y Fh(Y)1178 3540 y Fm(i)p Fl(=1)1354 3440 y Fo(dk)1456 3455 y Fm(i)1483 3440 y Fq(\()p Fo(p)22 b Fn(\000)h Fq(1\))1779 3344 y Fh(\021)1831 3418 y Fk(\\)1828 3440 y Fo(D)1909 3455 y Fm(m)p Fi(\000)p Fl(2)2066 3440 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2385 3455 y Fm(m)p Fi(\000)p Fl(2)2542 3440 y Fq(\))2580 3399 y Fm(T)2657 3440 y Fn(\012)23 b Fo(C)2834 3399 y Fm(m)p Fi(\000)p Fl(2)2827 3465 y Fm(a)2990 3440 y Fq(\()p Fo(k)s Fq(\)\012)3190 3399 y Fm(m)p Fi(\000)p Fl(2)3190 3465 y Fm(a)3347 3440 y Fo(:)257 3736 y Fq(W)-8 b(e)32 b(collect)g(the)g(remainders)h(that)e(follo)m(w)h (from)f(Hyp)s(othesis)i Fj(H4\))i Fq(up)c(to)g Fo(q)h Fq(=)27 b Fo(m)20 b Fn(\000)g Fq(1)257 3856 y(and)39 b(w)m(e)g(add)f(the)h(remainder)g(\(I)s(I)s(I.105\))e(obtained)i(ab)s (o)m(v)m(e.)61 b(Th)m(us)39 b(Hyp)s(othesis)h Fj(H4\))257 3976 y Fq(holds)34 b(true)f(also)f(for)g Fo(q)g Fq(=)27 b Fo(m)p Fq(.)257 4097 y Fa(S2\))36 b Fq(W)-8 b(e)33 b(start)g(from)f(the)h(expression)1347 4317 y(\()p Fo(\036)1443 4332 y Fm(m)1510 4317 y Fq(\))1548 4276 y Fm(T)1630 4317 y Fq(=)28 b Fo(\034)1787 4276 y Fm(N)1855 4317 y Fq(\()p Fo(P)1956 4332 y Fm(m)2022 4317 y Fq(\)\()2111 4290 y(~)2098 4317 y Fo(\036)2156 4332 y Fm(m)p Fi(\000)p Fl(1)2312 4317 y Fq(\))2350 4276 y Fm(T)3135 4317 y Fq(\(I)s(I)s(I.106\))257 4537 y(and)40 b(w)m(e)h(assume)g(Hyp)s(othesis)g Fj(H2\))j Fq(up)c(to)f Fo(q)44 b Fq(=)39 b Fo(m)27 b Fn(\000)h Fq(1)39 b(and)h(0)g Fo(<)f(\013)h(<)50 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)27 b Fn(\000)g Fq(1\).)257 4657 y(Then)34 b(w)m(e)g(successiv)m(ely)j(apply)c(the)g(follo)m(wing)g(op)s (erations:)420 4861 y(-)48 b(The)34 b(Wic)m(k-ordering)g(of)e(the)h (photon)f(op)s(erators)h(in)f(the)h(shell)h Fn(K)2961 4824 y Fm(m)p Fi(\000)p Fl(1)2960 4885 y Fm(m)3118 4861 y Fq(;)420 5064 y(-)48 b(The)33 b(map)f Fo(\034)970 5028 y Fm(N)959 5089 y(])1069 5064 y Fq(applied)h(to)e(the)h(op)s(erators)g Fo(R)2201 5023 y Fl(+)11 b Fi(?)2200 5086 y Fm(m)p Fi(\000)p Fl(1)2357 5064 y Fq(\()p Fo(z)2444 5028 y Fl(\()p Fm(m)p Fl(\))2440 5089 y Fm(m)2586 5064 y Fn(\000)21 b Fo(S)2743 5068 y Fl(\026)2744 5085 y Fm(l)q Fl(\()p Fm(m)p Fl(\))2887 5064 y Fq(\()p Fo(k)s Fq(\)\))32 b(and)g Fo(P)3339 5079 y Fm(m)p Fi(\000)p Fl(1)501 5184 y Fq(whic)m(h)g(are)f(pro)s(duced)h (after)e(the)h(Wic)m(k-ordering)h(and)f(under)g(the)g(decomp)s(osi-)501 5305 y(tion)i(of)f Fe(1)869 5320 y Fm(m)p Fi(\000)p Fl(1)1058 5305 y Fq(in)m(to)h Fo(P)1319 5320 y Fm(m)p Fi(\000)p Fl(1)1498 5305 y Fq(+)22 b Fo(P)1673 5269 y Fi(?)1659 5329 y Fm(m)p Fi(\000)p Fl(1)1815 5305 y Fq(.)p eop end %%Page: 45 46 TeXDict begin 45 45 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(45)257 573 y(W)-8 b(e)44 b(k)m(eep)g(the)f (only)h(expressions)h(of)e(t)m(yp)s(e)h(\(I)s(I)s(I.84\))e(on)h(scale)h Fo(m)29 b Fn(\000)h Fq(2)42 b(and)h(neglect)257 693 y(a)48 b(remainder)h Fo(o)p Fq(\()p Fo(\013)978 657 y Fm(N)1045 693 y Fq(\);)55 b(here)49 b(w)m(e)f(exploit)h(Hyp)s(othesis)h Fj(H2\))h Fq(for)d Fo(q)57 b Fq(=)d Fo(m)33 b Fn(\000)g Fq(1)47 b(and)257 814 y(Lemma)34 b(I)s(I)s(I.1.)43 b(Then)34 b(w)m(e)f(can)g(pic)m(k)44 b(\026)-59 b Fo(\013)q Fq(\()p Fo(m)p Fq(\))28 b Fo(>)f Fq(0,)33 b(and)g(w)m(e)g(can)g(single)h(out)e (the)h(v)m(ector)257 934 y(\()p Fo(\036)353 949 y Fm(m)420 934 y Fq(\))458 898 y Fm(T)458 959 y(])549 934 y Fq(b)m(y)k(the)f(pro)s (cedure)h(in)f(step)h Fj(S1\))p Fq(.)53 b(In)37 b(particular,)g(b)m(y)g (c)m(ho)s(osing)46 b(\026)-58 b Fo(\013)p Fq(\()p Fo(m)p Fq(\))36 b(small)257 1054 y(enough,)29 b(w)m(e)e(require)h(that,)g(in)f (the)g(in)m(terv)-5 b(al)27 b(\(0)p Fo(;)f Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)p Fq(\)\),)28 b(the)f(so)g(de\014ned)h (re-expanded)257 1175 y(v)m(ector)36 b(is)g(not)f(zero)g(and)g(that)g (the)h(o)m(v)m(erlap)g(with)g(\()p Fo(\036)2276 1190 y Fm(m)p Fi(\000)p Fl(1)2432 1175 y Fq(\))2470 1139 y Fm(T)2470 1199 y(])2560 1175 y Fq(do)s(es)f(not)g(v)-5 b(anish.)52 b(This)257 1295 y(is)38 b(p)s(ossible)g(b)s(ecause)g Fn(k)p Fq(\()p Fo(\036)1242 1310 y Fm(m)p Fi(\000)p Fl(1)1398 1295 y Fq(\))1436 1259 y Fm(T)1436 1320 y(])1516 1295 y Fn(\000)26 b Fq(\()p Fo(\036)1715 1310 y Fm(m)p Fi(\000)p Fl(1)1871 1295 y Fq(\))1909 1259 y Fm(T)1964 1295 y Fn(k)35 b(\024)h Fo(o)p Fq(\()p Fo(\013)2310 1259 y Fm(N)2376 1295 y Fq(\))h(and)g Fn(k)p Fq(\()p Fo(\036)2791 1310 y Fm(m)2857 1295 y Fq(\))2895 1259 y Fm(T)2895 1320 y(])2975 1295 y Fn(\000)26 b Fq(\()p Fo(\036)3174 1310 y Fm(m)3240 1295 y Fq(\))3278 1259 y Fm(T)3333 1295 y Fn(k)35 b(\024)257 1416 y Fo(o)p Fq(\()p Fo(\013)405 1379 y Fm(N)472 1416 y Fq(\).)257 1536 y Fa(S3\))h Fq(W)-8 b(e)33 b(\014rst)g(pro)m(v)m(e)h (that)1098 1729 y Fn(k)l Fo(~)-45 b(v)1195 1744 y Fm(m)p Fi(\000)p Fl(1)1352 1729 y Fq([\()p Fo(\036)1475 1744 y Fm(m)p Fi(\000)p Fl(1)1632 1729 y Fq(\))1670 1688 y Fm(T)1670 1754 y(])1747 1729 y Fn(\000)22 b Fq(\()p Fo(\036)1942 1744 y Fm(m)p Fi(\000)p Fl(1)2099 1729 y Fq(\))2137 1688 y Fm(T)2192 1729 y Fq(])p Fn(k)27 b(\024)i Fo(o)p Fq(\()p Fo(\013)2550 1688 y Fm(N)2616 1729 y Fq(\))481 b(\(I)s(I)s(I.107\))257 1923 y(b)m(y)31 b(taking)f(in)m(to)g(accoun)m(t)g(the)g(structure)h(of) f(eac)m(h)g(expression)i(\(I)s(I)s(I.83\),)e(with)g Fo(a)e Fq(=)g Fo(m)p Fq(,)257 2043 y(con)m(tained)40 b(in)e(the)h(remainder)g (\()p Fo(\036)1560 2058 y Fm(m)p Fi(\000)p Fl(1)1717 2043 y Fq(\))1755 2007 y Fm(T)1755 2068 y(])1836 2043 y Fn(\000)26 b Fq(\()p Fo(\036)2035 2058 y Fm(m)p Fi(\000)p Fl(1)2192 2043 y Fq(\))2230 2007 y Fm(T)2285 2043 y Fq(.)61 b(In)38 b(details,)j(for)d(eac)m(h)h Fo(q)3271 2007 y Fi(0)3332 2043 y Fn(\025)e Fq(2)257 2164 y(with)d Fo(m)28 b(>)f(q)743 2128 y Fi(0)788 2164 y Fn(\000)c Fq(1,)32 b(w)m(e)i(split)29 b Fo(~)-45 b(v)1404 2179 y Fm(m)p Fi(\000)p Fl(1)1594 2164 y Fq(in)m(to)1092 2357 y Fo(~)g(v)1143 2372 y Fm(m)p Fi(\000)p Fl(1)1327 2357 y Fq(=)24 b Fo(~)-45 b(v)1478 2372 y Fm(q)1512 2353 y Fd(0)1534 2372 y Fi(\000)p Fl(2)1651 2357 y Fq(+)22 b Fo(a)p Fq(\()1853 2332 y Fo(~)1838 2357 y(G)1915 2310 y Fm(q)1949 2286 y Fd(0)1971 2310 y Fi(\000)p Fl(2)1915 2379 y Fm(m)p Fi(\000)p Fl(1)2072 2357 y Fq(\))g(+)g Fo(a)2281 2316 y Fi(\003)2320 2357 y Fq(\()2373 2332 y Fo(~)2358 2357 y(G)2435 2310 y Fm(q)2469 2286 y Fd(0)2491 2310 y Fi(\000)p Fl(2)2435 2379 y Fm(m)p Fi(\000)p Fl(1)2592 2357 y Fq(\))p Fo(:)478 b Fq(\(I)s(I)s(I.108\))257 2551 y(and)44 b(w)m(e)g(notice)g(that)f(the)h(ratio)f Fn(k)p Fo(v)1656 2515 y Fm(l)1652 2575 y(q)1686 2556 y Fd(0)1708 2575 y Fi(\000)p Fl(2)1803 2551 y Fo(g)1854 2515 y Fl(1)1850 2575 y Fm(q)1884 2556 y Fd(0)1906 2575 y Fi(\000)p Fl(2)2000 2551 y Fn(k)p Fo(=)p Fn(k)p Fo(g)2200 2515 y Fl(1)2196 2575 y Fm(q)2230 2556 y Fd(0)2251 2575 y Fi(\000)p Fl(2)2346 2551 y Fn(k)p Fq(,)i Fo(l)k Fq(=)d(1)p Fo(;)17 b Fq(2)p Fo(;)g Fq(3,)44 b(is)g(uniformly)257 2671 y(b)s(ounded)36 b(in)f Fo(q)821 2635 y Fi(0)879 2671 y Fq(and)g(in)g Fo(\013)h Fq(for)e(an)m(y)i(op)s(erator)e Fo(g)2069 2635 y Fl(1)2065 2696 y Fm(q)2099 2677 y Fd(0)2121 2696 y Fi(\000)p Fl(2)2215 2671 y Fq(.)51 b(As)35 b(far)f(as)d Fo(~)-45 b(v)2759 2686 y Fm(q)2793 2667 y Fd(0)2816 2686 y Fi(\000)p Fl(2)2945 2671 y Fq(is)35 b(concerned,)257 2791 y(this)e(is)f(enough)g(to)g(get)g(the)g(inequalit)m(y)i(in)e(\(I)s (I)s(I.107\).)42 b(Next)33 b(w)m(e)g(pass)f(to)g(con)m(trol)g(the)257 2912 y(t)m(w)m(o)i(op)s(erators)1157 3032 y([)p Fo(a)p Fq(\()1288 3007 y Fo(~)1273 3032 y(G)1350 2985 y Fm(q)1384 2962 y Fd(0)1406 2985 y Fi(\000)p Fl(2)1350 3054 y Fm(m)p Fi(\000)p Fl(1)1507 3032 y Fq(\))p Fo(;)17 b(C)1666 2991 y Fm(q)1700 2968 y Fd(0)1722 2991 y Fi(\000)p Fl(2)1659 3057 y Fm(a)1816 3032 y Fq(])98 b(and)f Fo(a)2246 2991 y Fi(\003)2286 3032 y Fq(\()2339 3007 y Fo(~)2324 3032 y(G)2401 2985 y Fm(q)2435 2962 y Fd(0)2457 2985 y Fi(\000)p Fl(2)2401 3054 y Fm(m)p Fi(\000)p Fl(1)2558 3032 y Fq(\))539 b(\(I)s(I)s(I.109\))257 3195 y(when)35 b(they)g(are)e(applied)i(to)e (an)h(expression)i(lik)m(e)f(in)e(Eq.)i(\(I)s(I)s(I.83\).)46 b(F)-8 b(or)33 b(that)h(w)m(e)g(use)257 3316 y(the)f(result)h(in)257 3496 y Fe(Lemma)53 b(I)s(I)s(I.6.)47 b Fj(The)e(norm)g(of)g(the)h(ve)-5 b(ctor)46 b(obtaine)-5 b(d)44 b(by)i(applying)f(the)h(op)-5 b(er)g(ator)257 3617 y Fo(a)p Fq(\()361 3591 y Fo(~)346 3617 y(G)423 3569 y Fm(q)457 3546 y Fd(0)480 3569 y Fi(\000)p Fl(2)423 3638 y Fm(m)p Fi(\000)p Fl(1)580 3617 y Fq(\))38 b Fj(to)f(a)g(ve)-5 b(ctor)38 b(of)f(the)g(typ)-5 b(e)38 b(in)f(Eq.\(III.83\))e(c)-5 b(an)37 b(b)-5 b(e)37 b(estimate)-5 b(d)37 b(fr)-5 b(om)37 b(ab)-5 b(ove.)257 3737 y(Its)44 b(or)-5 b(der)44 b(in)f Fo(\013)i Fj(is)f(higher)f(or)h(e)-5 b(qual)43 b(to)h(the)g(or)-5 b(der)44 b(of)g(the)g(norm)f(estimate)h (of)f(the)257 3857 y(original)34 b(ve)-5 b(ctor.)257 4038 y(Pr)g(o)g(of.)43 b Fq(In)33 b(the)g(App)s(endix.)257 4158 y(Moreo)m(v)m(er,)i(from)d(Eq.)h(\(I)s(I)s(I.107\))f(w)m(e)i(also) e(get)h(that)1427 4352 y Fn(k)l Fo(~)-45 b(v)1524 4367 y Fm(m)p Fi(\000)p Fl(1)1681 4352 y Fq(\()p Fo(\036)1777 4367 y Fm(m)1843 4352 y Fq(\))1881 4311 y Fm(T)1881 4377 y(])1936 4352 y Fn(k)28 b(\024)g(O)s Fq(\(1\))809 b(\(I)s(I)s(I.110\)) 257 4545 y(b)s(ecause)39 b(of)d(the)h(inequalit)m(y)i Fn(k)l Fo(~)-45 b(v)1464 4560 y Fm(m)p Fi(\000)p Fl(1)1621 4545 y Fq(\()p Fo(\036)1717 4560 y Fm(m)1783 4545 y Fq(\))1821 4509 y Fm(T)1876 4545 y Fn(k)35 b(\024)g(O)s Fq(\(1\))i(whic)m(h)h (trivially)g(follo)m(ws)f(from)257 4666 y(the)e(de\014nitions)h(in)e (Eqs.)i(\(I)s(I.11\),)f(\(I)s(I.12\))e(and)i(\(I)s(I.13\).)48 b(Then)36 b(Hyp)s(othesis)g Fj(H2\))i Fq(for)257 4786 y Fo(q)32 b Fq(=)27 b Fo(m)d Fq(follo)m(ws)g(from)f(the)h(de\014nition) g(in)g(Eq.)g(\(I)s(I)s(I.62\))f(and)g(b)s(ecause)i(of)e(Eqs.)i(\(I)s(I) s(I.107\),)257 4907 y(\(I)s(I)s(I.110\))34 b(and)g(due)h(to)f(the)h (fact)f(that)g Fn(j)p Fq(\(\()p Fo(\036)1889 4922 y Fm(m)p Fi(\000)p Fl(1)2045 4907 y Fq(\))2083 4870 y Fm(T)2083 4931 y(])2138 4907 y Fo(;)17 b Fq(\()p Fo(\036)2278 4922 y Fm(m)2344 4907 y Fq(\))2382 4870 y Fm(T)2382 4931 y(])2437 4907 y Fq(\))p Fn(j)30 b(\025)h Fo(\016)k(>)30 b Fq(0,)35 b(where)g Fo(\016)j Fq(is)d Fo(\013)q Fq(-)257 5027 y(indep)s(enden)m (t,)j(in)d(the)h(in)m(terv)-5 b(al)35 b(0)c Fo(<)h(\013)g(<)41 b Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(m)p Fq(\))32 b Fo(<)41 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(m)24 b Fn(\000)g Fq(1\),)35 b(whic)m(h)i(follo)m(ws)e(from)257 5147 y(step)f Fj(S2\))p Fq(.)43 b(This)33 b(can)g(b)s(e)g(deriv)m(ed)i(from)d (Corollary)h(I)s(I.3.)404 5268 y(W)-8 b(e)33 b(no)m(w)g(c)m(hec)m(k)i (the)e(\014rst)g(step)g(of)f(the)h(inductiv)m(e)i(pro)s(cedure.)257 5388 y(F)-8 b(or)32 b Fo(\013)h Fq(in)g(the)g(in)m(terv)-5 b(al)33 b(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\(3\)\))33 b(where)43 b(\026)-59 b Fo(\013)q Fq(\(3\))27 b Fo(>)h Fq(0:)p eop end %%Page: 46 47 TeXDict begin 46 46 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(46)366 573 y(1\))48 b(The)34 b(follo)m(wing)f(truncated)g(v)m(ectors)h(can)f(b)s(e)g(fully)g (re-expanded)708 783 y(\()p Fo(\036)804 798 y Fl(1)843 783 y Fq(\))881 742 y Fm(T)1019 783 y Fq(:=)1217 757 y(~)1205 783 y Fo(\036)1263 798 y Fl(0)3135 783 y Fq(\(I)s(I)s(I.111\)) 1205 1003 y Fn(\000)1338 935 y Fq(1)p 1292 980 141 4 v 1292 1071 a(2)p Fo(\031)t(i)1489 895 y Fm(N)1461 920 y Fh(X)1460 1102 y Fm(j)t Fl(=1)1599 886 y Fh(Z)1645 1074 y Fl(\000)1689 1083 y Ff(1)1744 1003 y Fo(dz)1840 1018 y Fl(1)2042 935 y Fq(1)p 1890 980 355 4 v 1898 1052 a Fh(f)1890 1084 y Fo(H)1979 1043 y Fl(+)1971 1106 y(0)2060 1084 y Fn(\000)22 b Fo(z)2204 1099 y Fl(1)2254 1003 y Fq([\()p Fn(\000)p Fo(W)2502 962 y Fl(0)2488 1027 y(1)2541 1003 y Fq(\))2759 935 y(1)p 2606 980 V 2614 1052 a Fh(f)2606 1084 y Fo(H)2695 1043 y Fl(+)2687 1106 y(0)2776 1084 y Fn(\000)g Fo(z)2920 1099 y Fl(1)2970 1003 y Fq(])2997 962 y Fm(j)3046 977 y Fq(~)3034 1003 y Fo(\036)3092 1018 y Fl(0)708 1249 y Fq(\()p Fo(\036)804 1264 y Fl(2)843 1249 y Fq(\))881 1208 y Fm(T)1019 1249 y Fq(:=)83 b(\()1256 1223 y(~)1243 1249 y Fo(\036)1301 1264 y Fl(1)1340 1249 y Fq(\))1378 1208 y Fm(T)3135 1249 y Fq(\(I)s(I)s(I.112\))1205 1469 y Fn(\000)1338 1401 y Fq(1)p 1292 1445 141 4 v 1292 1537 a(2)p Fo(\031)t(i)1489 1361 y Fm(N)1461 1386 y Fh(X)1460 1568 y Fm(j)t Fl(=1)1599 1352 y Fh(Z)1645 1540 y Fl(\000)1689 1549 y Ff(2)1744 1469 y Fo(dz)1840 1484 y Fl(2)2042 1401 y Fq(1)p 1890 1445 355 4 v 1898 1518 a Fh(f)1890 1550 y Fo(H)1979 1509 y Fl(+)1971 1572 y(1)2060 1550 y Fn(\000)22 b Fo(z)2204 1565 y Fl(2)2254 1469 y Fq([\()p Fn(\000)p Fo(W)2502 1428 y Fl(1)2488 1493 y(2)2541 1469 y Fq(\))2759 1401 y(1)p 2606 1445 V 2614 1518 a Fh(f)2606 1550 y Fo(H)2695 1509 y Fl(+)2687 1572 y(1)2776 1550 y Fn(\000)g Fo(z)2920 1565 y Fl(2)2970 1469 y Fq(])2997 1428 y Fm(j)3034 1469 y Fq(\()3085 1442 y(~)3072 1469 y Fo(\036)3130 1484 y Fl(1)3169 1469 y Fq(\))3207 1428 y Fm(T)708 1715 y Fq(\()p Fo(\036)804 1730 y Fl(3)843 1715 y Fq(\))881 1674 y Fm(T)1019 1715 y Fq(:=)83 b(\()1256 1688 y(~)1243 1715 y Fo(\036)1301 1730 y Fl(2)1340 1715 y Fq(\))1378 1674 y Fm(T)3135 1715 y Fq(\(I)s(I)s(I.113\))1205 1934 y Fn(\000)1338 1867 y Fq(1)p 1292 1911 141 4 v 1292 2003 a(2)p Fo(\031)t(i)1489 1827 y Fm(N)1461 1851 y Fh(X)1460 2034 y Fm(j)t Fl(=1)1599 1817 y Fh(Z)1645 2006 y Fl(\000)1689 2015 y Ff(3)1744 1934 y Fo(dz)1840 1949 y Fl(3)2042 1867 y Fq(1)p 1890 1911 355 4 v 1898 1983 a Fh(f)1890 2016 y Fo(H)1979 1975 y Fl(+)1971 2038 y(2)2060 2016 y Fn(\000)22 b Fo(z)2204 2031 y Fl(3)2254 1934 y Fq([\()p Fn(\000)p Fo(W)2502 1893 y Fl(2)2488 1959 y(3)2541 1934 y Fq(\))2759 1867 y(1)p 2606 1911 V 2614 1983 a Fh(f)2606 2016 y Fo(H)2695 1975 y Fl(+)2687 2038 y(2)2776 2016 y Fn(\000)g Fo(z)2920 2031 y Fl(3)2970 1934 y Fq(])2997 1893 y Fm(j)3034 1934 y Fq(\()3085 1908 y(~)3072 1934 y Fo(\036)3130 1949 y Fl(2)3169 1934 y Fq(\))3207 1893 y Fm(T)3262 1934 y Fq(;)366 2268 y(2\))48 b(The)34 b(truncated)f(energy)h(shifts)f(can)g(b)s(e)g (split)g(as)1312 2491 y(\()p Fo(E)1428 2450 y Fm(q)r Fi(\000)p Fl(1)1422 2516 y Fm(q)1556 2491 y Fq(\))1594 2450 y Fm(T)1677 2491 y Fq(=)27 b(\()p Fo(E)1896 2450 y Fm(q)r Fi(\000)p Fl(1)1890 2516 y Fm(q)2025 2491 y Fq(\))2063 2450 y Fm(T)2063 2516 y(])2140 2491 y Fq(+)22 b Fo(o)p Fq(\()p Fo(\013)2386 2450 y Fm(q)r Fl(+)2485 2423 y Ff(1)p 2484 2435 31 4 v 2484 2476 a(2)2524 2450 y Fl(+)p Fm(N)2647 2491 y Fq(\))450 b(\(I)s(I)s(I.114\))501 2702 y(for)32 b(1)c Fn(\024)g Fo(q)j Fn(\024)e Fq(3;)366 2902 y(3\))48 b(It)25 b(is)f(p)s(ossible)i(to)e(fully)g(re-expand,)j (up)e(to)f(the)g(order)h(2)p Fo(N)34 b Fq(in)25 b Fo(\013)2834 2838 y Ff(1)p 2833 2850 V 2833 2892 a(2)2878 2902 y Fq(,)h(an)e (expression)501 3023 y(of)32 b(the)h(t)m(yp)s(e)805 3319 y Fo( )868 3334 y Fl(1)991 3319 y Fq(:=)1121 3223 y Fh(\020)1186 3187 y Fl(\026)1188 3202 y Fm(j)s Fl(\(3\))1196 3236 y Fh(Y)1188 3418 y Fm(j)t Fl(=1)1327 3202 y Fh(Z)1373 3391 y Fl(\000)1417 3400 y Ff(3)1472 3319 y Fo(dz)1572 3268 y Fl(\(3\))p Fm(;j)1568 3341 y Fl(3)1734 3187 y(\026)1736 3202 y Fm(j)s Fl(\(2\))1744 3236 y Fh(Y)1736 3418 y Fm(j)t Fl(=1)1875 3202 y Fh(Z)1921 3391 y Fl(\000)1965 3400 y Ff(2)2020 3319 y Fo(dz)2120 3268 y Fl(\(2\))p Fm(;j)2116 3341 y Fl(2)2267 3223 y Fh(\021)2339 3319 y Fn(\002)719 b Fq(\(I)s(I)s(I.115\))991 3652 y Fn(\002)1130 3544 y Fl(4)1095 3569 y Fh(Y)1085 3750 y Fm(p)p Fl(=3)1227 3555 y Fh(\020)1293 3535 y(Z)1339 3723 y Fi(K)1393 3691 y Fb(p)p Fd(\000)p Ff(2)1393 3746 y Fb(p)p Fd(\000)p Ff(1)1523 3519 y Fl(\026)1529 3535 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1578 3569 y Fh(Y)1574 3751 y Fm(i)p Fl(=1)1750 3652 y Fo(dk)1852 3667 y Fm(i)1880 3652 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))2175 3555 y Fh(\021)2225 3652 y Fo(D)2306 3667 y Fl(1)2345 3652 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2664 3667 y Fl(1)2704 3652 y Fq(\))2742 3611 y Fm(T)2819 3652 y Fn(\012)22 b Fo(C)2995 3611 y Fl(1)2988 3676 y(3)3035 3652 y Fq(\()p Fo(k)s Fq(\)\012)3235 3611 y Fl(1)3235 3676 y(3)501 3938 y Fq(and)31 b(the)f(remainder)i(is)f(giv) m(en)g(as)g(an)f Fo(\013)q Fq(-indep)s(enden)m(t)i(n)m(um)m(b)s(er)g (of)d(expressions)501 4059 y(of)j(the)h(t)m(yp)s(e)926 4259 y Fh(\020)991 4223 y Fl(\026)992 4238 y Fm(j)s Fl(\(3\))1000 4272 y Fh(Y)992 4454 y Fm(j)t Fl(=1)1131 4238 y Fh(Z)1177 4427 y Fl(\000)1221 4436 y Ff(3)1277 4355 y Fo(dz)1377 4304 y Fl(\(3\))p Fm(;j)1373 4377 y Fl(3)1539 4223 y(\026)1540 4238 y Fm(j)s Fl(\(2\))1548 4272 y Fh(Y)1540 4454 y Fm(j)t Fl(=1)1679 4238 y Fh(Z)1725 4427 y Fl(\000)1769 4436 y Ff(2)1825 4355 y Fo(dz)1925 4304 y Fl(\(2\))p Fm(;j)1921 4377 y Fl(2)2071 4259 y Fh(\021)2143 4355 y Fn(\002)915 b Fq(\(I)s(I)s(I.116\))926 4688 y Fn(\002)1065 4580 y Fl(4)1030 4605 y Fh(Y)1020 4786 y Fm(p)p Fl(=3)1162 4591 y Fh(\020)1228 4571 y(Z)1275 4759 y Fi(K)1329 4727 y Fb(p)p Fd(\000)p Ff(2)1329 4782 y Fb(p)p Fd(\000)p Ff(1)1458 4555 y Fl(\026)1464 4571 y Fm(i)p Fl(\()p Fm(p)p Fi(\000)p Fl(1\))1513 4605 y Fh(Y)1509 4787 y Fm(i)p Fl(=1)1685 4688 y Fo(dk)1787 4703 y Fm(i)1815 4688 y Fq(\()p Fo(p)22 b Fn(\000)g Fq(1\))2110 4591 y Fh(\021)2160 4655 y(d)2160 4688 y Fo(D)2241 4703 y Fl(1)2280 4688 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()p Fo(\036)2599 4703 y Fl(1)2639 4688 y Fq(\))2677 4647 y Fm(T)2754 4688 y Fn(\012)22 b Fo(C)2930 4647 y Fl(2)2923 4712 y(3)2970 4688 y Fq(\()p Fo(k)s Fq(\)\012)3170 4647 y Fl(2)3170 4712 y(3)3210 4688 y Fo(:)257 5027 y Fq(The)30 b(claim)e(on)h(the)f(truncated)h(energy)h (shift)f(\()p Fo(E)2084 4991 y Fl(0)2078 5052 y(1)2123 5027 y Fq(\))2161 4991 y Fm(T)2244 5027 y Fq(and)g(on)f(the)h (truncated)g(ground-)257 5147 y(state)j(\()p Fo(\036)591 5162 y Fl(1)631 5147 y Fq(\))669 5111 y Fm(T)755 5147 y Fq(is)g(ob)m(vious,)i(b)m(y)e(de\014nition,)h(in)f(a)f(su\016cien)m (tly)k(small)d(in)m(terv)-5 b(al)32 b(\(0)p Fo(;)27 b Fq(\026)-59 b Fo(\013)q Fq(\(1\)\).)257 5268 y(Then)33 b(w)m(e)g(observ)m(e)h(that,)e(after)g(Wic)m(k-ordering)h(the)f(photon) g(creation-)g(and)g(annihi-)257 5388 y(lation)f(op)s(erators)g(in)h (the)f(shell)i Fn(K)1540 5352 y Fl(1)1539 5413 y(2)1580 5388 y Fq(,)e(w)m(e)h(can)g(pic)m(k)42 b(\026)-59 b Fo(\013)q Fq(\(2\))31 b(suc)m(h)h(that)f(the)h(v)m(ector)g(\()p Fo(\036)3363 5403 y Fl(2)3402 5388 y Fq(\))3440 5352 y Fm(T)p eop end %%Page: 47 48 TeXDict begin 47 47 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(47)257 573 y(can)43 b(b)s(e)g(fully)g (re-expanded)h(in)f(the)g(in)m(terv)-5 b(al)43 b(0)h Fo(<)h(\013)g(<)54 b Fq(\026)-59 b Fo(\013)q Fq(\(2\))44 b Fn(\024)55 b Fq(\026)-59 b Fo(\013)q Fq(\(1\))42 b(b)m(y)i(simply)257 693 y(applying)31 b(Lemma)f(I)s(I)s(I.1)f(and)g(using)i(\()p Fo(E)1769 657 y Fl(0)1763 718 y(1)1808 693 y Fq(\))1846 657 y Fm(T)1929 693 y Fn(\021)d Fq(\()p Fo(E)2150 657 y Fl(0)2144 718 y(1)2189 693 y Fq(\))2227 657 y Fm(T)2227 718 y(])2282 693 y Fq(.)43 b(By)30 b(follo)m(wing)f(the)h(pro)s(cedure) 257 814 y(in)g Fj(S3\))p Fq(,)f(w)m(e)i(can)e(pro)m(v)m(e)i(the)f (claim)g(concerning)g(\()p Fo(E)2167 777 y Fl(1)2161 838 y(2)2206 814 y Fq(\))2244 777 y Fm(T)2299 814 y Fq(;)h(i.e.,)g(w)m (e)f(can)f(compute)i(\()p Fo(E)3363 777 y Fl(1)3357 838 y(2)3402 814 y Fq(\))3440 777 y Fm(T)3440 838 y(])257 934 y Fq(for)h(0)c Fo(<)f(\013)i(<)37 b Fq(\026)-59 b Fo(\013)q Fq(\(2\).)43 b(.)257 1054 y(Expression)g(\(I)s(I)s(I.115\))c (can)h(b)s(e)h(fully)f(re-expanded)i(in)e(an)g(in)m(terv)-5 b(al)41 b(0)f Fo(<)h(\013)g(<)50 b Fq(\026)-59 b Fo(\013)3383 1069 y Fm( )3429 1078 y Ff(1)3468 1054 y Fq(,)267 1175 y(\026)g Fo(\013)319 1190 y Fm( )365 1199 y Ff(1)432 1175 y Fn(\024)38 b Fq(\026)-59 b Fo(\013)q Fq(\(1\),)27 b(b)m(y)g(the)f(pro)s(cedure)h(describ)s(ed)h(in)e Fj(S1\))j Fq(since)e(\()p Fo(E)2569 1139 y Fl(0)2563 1199 y(1)2609 1175 y Fq(\))2647 1139 y Fm(T)2728 1175 y Fq(is)f(already)h(explicit.) 257 1295 y(The)34 b(remainder)g(is)f(trivially)g(of)f(the)h(form)g(in)f (Eq.)i(\(I)s(I)s(I.116\).)257 1416 y(By)40 b(the)g(pro)s(cedure)h(in)f (step)g Fj(S2\))p Fq(,)h(w)m(e)f(can)g(set)g(the)g(in)m(terv)-5 b(al)41 b(0)e Fo(<)g(\013)i(<)49 b Fq(\026)-59 b Fo(\013)q Fq(\(3\),)41 b(with)257 1536 y(0)34 b Fo(<)44 b Fq(\026)-59 b Fo(\013)q Fq(\(3\))33 b Fn(\024)44 b Fq(\026)-58 b Fo(\013)p Fq(\(2\))34 b Fn(\024)44 b Fq(\026)-59 b Fo(\013)q Fq(\(1\),)37 b(where)g(the)g(v)m(ector)g(\()p Fo(\036)2216 1551 y Fl(3)2256 1536 y Fq(\))2294 1500 y Fm(T)2385 1536 y Fq(can)f(b)s(e)h(reduced)h(to)d(a)h(\014nite)257 1656 y(and)41 b Fo(\013)q Fq(-indep)s(enden)m(t)h(n)m(um)m(b)s(er)f(of)f (expressions)k(of)39 b(the)i(t)m(yp)s(e)h(in)e(Eq.)h(\(I)s(I)s(I.115\)) f(and)257 1777 y(then)34 b(fully)f(re-expanded;)h(here)f(w)m(e)h(use)f (\()p Fo(E)1918 1741 y Fl(1)1912 1801 y(2)1958 1777 y Fq(\))1996 1741 y Fm(T)1996 1801 y(])2083 1777 y Fq(previously)i (determined.)257 1897 y(Finally)-8 b(,)37 b(follo)m(wing)f(the)g(pro)s (cedure)g(in)g Fj(S3\))p Fq(,)f(\()p Fo(E)2093 1861 y Fl(2)2087 1922 y(3)2133 1897 y Fq(\))2171 1861 y Fm(T)2171 1922 y(])2261 1897 y Fq(is)h(obtained)g(for)f Fo(\013)e Fn(2)g Fq(\(0)p Fo(;)27 b Fq(\026)-59 b Fo(\013)q Fq(\(3\)\).)p 3425 2017 4 66 v 3429 1955 59 4 v 3429 2017 V 3487 2017 4 66 v 257 2138 a(In)33 b(the)g(follo)m(wing)g(corollary)g(the)g (\014nal)g(result)g(is)g(ac)m(hiev)m(ed.)257 2322 y Fe(Corollary)40 b(I)s(I)s(I.7.)h Fj(F)-7 b(or)33 b(any)i Fo(N)10 b Fj(,)35 b(for)g Fo(\013)g Fj(in)g(a)f(\014xe)-5 b(d)34 b(interval)g Fq(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\))35 b Fj(wher)-5 b(e)44 b Fq(\026)-59 b Fo(\013)28 b(>)g Fq(0)34 b Fj(is)257 2451 y Fo(N)10 b Fj(-dep)-5 b(endent,)34 b Fo(E)938 2466 y Fl(gs)1034 2451 y Fn(\021)28 b Fo(E)1211 2466 y Fl(gs)1278 2451 y Fq(\()p Fo(\013)q Fq(\))35 b Fj(and)f Fo(\036)1699 2466 y Fl(gs)1794 2451 y Fn(\021)28 b Fo(\036)1957 2466 y Fl(gs)2024 2451 y Fq(\()p Fo(\013)2135 2387 y Ff(1)p 2134 2399 31 4 v 2134 2441 a(2)2179 2451 y Fq(\))35 b Fj(have)f(exp)-5 b(ansions)33 b(of)i(the)g(form)1036 2726 y Fo(E)1108 2741 y Fl(gs)1176 2726 y Fq(\()p Fo(\013)q Fq(\))82 b(=)h Fo(E)1628 2741 y Fl(0)1690 2726 y Fq(+)1816 2619 y Fm(N)1788 2643 y Fh(X)1788 2828 y Fm(`)p Fl(=3)1924 2726 y Fo(")1970 2741 y Fm(`)2020 2726 y Fq(\()p Fo(\013)q Fq(\))17 b Fo(\013)2239 2685 y Fm(`)2320 2726 y Fq(+)50 b Fo(o)p Fq(\()p Fo(\013)2594 2685 y Fm(N)2661 2726 y Fq(\))436 b(\(I)s(I)s(I.117\))996 3031 y Fo(\036)1054 3046 y Fl(gs)1121 3031 y Fq(\()p Fo(\013)1232 2963 y Ff(1)p 1232 2975 V 1232 3016 a(2)1276 3031 y Fq(\))83 b(=)g Fo(\036)1614 3046 y Fl(0)1675 3031 y Fq(+)1784 2923 y Fl(2)p Fm(N)1773 2948 y Fh(X)1774 3133 y Fm(`)p Fl(=3)1910 3031 y Fo(')1974 3046 y Fm(`)2007 3031 y Fq(\()p Fo(\013)q Fq(\))17 b Fo(\013)2226 2990 y Fm(`=)p Fl(2)2378 3031 y Fq(+)50 b Fo(o)p Fq(\()p Fo(\013)2652 2990 y Fm(N)2719 3031 y Fq(\))378 b(\(I)s(I)s(I.118\))257 3296 y Fj(with)1024 3493 y Fq(lim)1016 3550 y Fm(\013)p Fi(!)p Fl(0)1184 3493 y Fo(\013)1247 3452 y Fm(\016)1284 3493 y Fn(j)p Fo(")1358 3508 y Fm(`)1407 3493 y Fq(\()p Fo(\013)q Fq(\))p Fn(j)27 b Fq(=)g(0)598 b Fn(8)p Fo(\016)32 b(>)c Fq(0)501 b(\(I)s(I)s(I.119\))961 3679 y(lim)953 3735 y Fm(\013)p Fi(!)p Fl(0)1121 3679 y Fo(\013)1184 3638 y Fm(\016)1222 3679 y Fn(k)p Fo(')1336 3694 y Fm(`)1385 3679 y Fq(\()p Fo(\013)q Fq(\))p Fn(k)27 b Fq(=)g(0)598 b Fn(8)p Fo(\016)32 b(>)c Fq(0)501 b(\(I)s(I)s(I.120\))257 3904 y Fj(for)44 b(arbitr)-5 b(ary)44 b Fo(N)55 b Fq(=)44 b(1)p Fo(;)17 b Fq(2)p Fo(;)g Fq(3)p Fo(;)g(:)g(:)g(:)n Fj(.)71 b(In)43 b(exp)-5 b(ansions)42 b(\(III.117\))f(and)j(\(III.118\),)f Fo(")3324 3919 y Fm(`)3357 3904 y Fq(\()p Fo(\013)q Fq(\))257 4025 y Fj(and)34 b Fo(')510 4040 y Fm(`)543 4025 y Fq(\()p Fo(\013)q Fq(\))g Fj(ar)-5 b(e)33 b(c)-5 b(omputable)34 b(in)g(terms)g(of)g(\014nitely)g(many)g(c)-5 b(onver)g(gent)33 b(inte)-5 b(gr)g(als,)33 b(for)257 4145 y(any)i Fq(3)28 b Fn(\024)g Fo(`)f(<)h Fn(1)p Fj(.)257 4329 y(Pr)-5 b(o)g(of.)44 b Fq(F)-8 b(or)33 b(an)m(y)g(N,)g(w)m(e)h(can)f(select)i(the)e(v)-5 b(alue)43 b(\026)-58 b Fo(\013)p Fq(\()p Fo(M)39 b Fq(=)28 b(2)p Fo(N)33 b Fn(\000)23 b Fq(1\))33 b(suc)m(h)h(that)f(expan-)257 4450 y(sions)c(of)e(the)h(t)m(yp)s(e)g(in)g(Eqs.)h(\(I)s(I)s(I.117\),)e (\(I)s(I)s(I.118\))g(hold)h(for)f Fo(\013)h Fn(2)g Fq(\(0)p Fo(;)e Fq(\026)-58 b Fo(\013)p Fq(\()p Fo(M)39 b Fq(=)27 b(2)p Fo(N)22 b Fn(\000)12 b Fq(1\)\).)257 4570 y(In)41 b(fact,)g(b)s(ecause)h(of)d(Theorem)j(I)s(I)s(I.4)d(and)h(b)s(ecause)i (of)d(the)i(results)g(in)f(Section)h(I)s(I,)257 4745 y(the)36 b(quan)m(tities)926 4681 y Fl(\()p Fm(\036)995 4692 y Fb(M)5 b Ff(=2)p Fb(N)g Fd(\000)p Ff(1)1271 4681 y Fl(\))1298 4658 y Fb(T)1298 4705 y(])1347 4681 y Fl(\()p Fm(\013)1429 4630 y Ff(1)p 1430 4639 V 1430 4674 a(2)1475 4681 y Fl(\))p 890 4722 648 4 v 890 4809 a Fi(k)p Fl(\()p Fm(\036)994 4820 y Fb(M)g Ff(=2)p Fb(N)g Fd(\000)p Ff(1)1271 4809 y Fl(\))1298 4786 y Fb(T)1298 4834 y(])1347 4809 y Fl(\()p Fm(\013)1429 4757 y Ff(1)p 1430 4766 31 4 v 1430 4801 a(2)1475 4809 y Fl(\))p Fi(k)1582 4745 y Fq(and)35 b Fo(E)1846 4760 y Fl(0)1910 4745 y Fn(\000)2011 4679 y Fh(P)2098 4705 y Fm(m)p Fl(=)p Fm(M)2098 4770 y(m)p Fl(=1)2295 4745 y Fq(\()p Fo(E)2411 4709 y Fm(m)p Fi(\000)p Fl(1)2405 4770 y Fm(m)2568 4745 y Fq(\))2606 4709 y Fm(T)2606 4770 y(])2661 4745 y Fq(\()p Fo(\013)q Fq(\))f(are)h(asymptotic,)257 4983 y(up)i(to)e(order)h(2)p Fo(N)46 b Fq(in)36 b Fo(\013)1145 4919 y Ff(1)p 1145 4931 V 1145 4972 a(2)1189 4983 y Fq(,)h(as)f Fo(\013)e Fn(!)f Fq(0,)j(to)1885 4935 y Fm(\036)1927 4943 y Ff(gs)1986 4935 y Fl(\()p Fm(\013)2068 4884 y Ff(1)p 2068 4893 V 2068 4928 a(2)2113 4935 y Fl(\))p 1850 4960 327 4 v 1850 5046 a Fi(k)p Fm(\036)1927 5054 y Ff(gs)1986 5046 y Fl(\()p Fm(\013)2068 4995 y Ff(1)p 2068 5004 31 4 v 2068 5039 a(2)2113 5046 y Fl(\))p Fi(k)2221 4983 y Fq(and)g Fo(E)2486 4998 y Fl(gs)2554 4983 y Fq(\()p Fo(\013)q Fq(\),)g(resp)s(ectiv)m(ely)-8 b(.)56 b(The)257 5147 y(co)s(e\016cien)m(ts)34 b Fo(")796 5162 y Fm(`)828 5147 y Fq(\()p Fo(\013)q Fq(\))d(and)g(the)h(v)m(ectors)h Fo(')1747 5162 y Fm(`)1780 5147 y Fq(\()p Fo(\013)q Fq(\))d(are)h (expressed)j(in)e(terms)g(of)f(bare)g(quan-)257 5268 y(tities)j(according)f(to)f(De\014nition)h(I)s(I)s(I.3.)p 3425 5268 4 66 v 3429 5205 59 4 v 3429 5268 V 3487 5268 4 66 v eop end %%Page: 48 49 TeXDict begin 48 48 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(48)257 573 y Fe(Remark)257 693 y Fq(In)32 b(order)g(to)f(\014nd)h(appro)m(ximate)g(expressions)j (for)c Fo(\013)h Fq(in)f(an)h Fo(N)10 b Fq(-indep)s(enden)m(t)33 b(in)m(terv)-5 b(al)257 814 y(of)26 b(the)g(t)m(yp)s(e)h(\(0)p Fo(;)f Fq(\026)-58 b Fo(\013)p Fq(\),)28 b(up)e(to)g(corrections)h(of)e (the)i(t)m(yp)s(e)g Fo(const)9 b Fn(\001)g Fo(\013)2572 777 y Fm(N)e Fl(+)p Fm(\017)2722 814 y Fq(,)27 b Fo(\017)h(>)g Fq(0,)f(w)m(e)g(can)f(use)257 934 y(regular)36 b(p)s(erturbation)f (theory)-8 b(.)52 b(The)36 b(argumen)m(t)g(is)f(the)h(follo)m(wing.)52 b(F)-8 b(rom)34 b(Prop)s(osi-)257 1054 y(tion)f(I.I,)g(w)m(e)h(kno)m(w) f(that,)g(for)f(an)m(y)h(v)-5 b(alue)33 b(of)f Fo(\013)h Fq(ful\014lling)g(the)g(assumptions,)1025 1204 y Fh(\015)1025 1254 y(\015)1025 1304 y(\015)1025 1354 y(\015)1131 1262 y Fo(\036)1189 1277 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1382 1262 y Fq(\()p Fo(\013)1493 1198 y Ff(1)p 1492 1210 31 4 v 1492 1251 a(2)1537 1262 y Fq(\))p 1081 1306 544 4 v 1081 1412 a Fn(k)p Fo(\036)1189 1427 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)1382 1412 y Fq(\()p Fo(\013)1493 1348 y Ff(1)p 1492 1360 31 4 v 1492 1401 a(2)1537 1412 y Fq(\))p Fn(k)1657 1329 y(\000)1816 1262 y Fo(\036)1874 1277 y Fl(gs)1941 1262 y Fq(\()p Fo(\013)2052 1198 y Ff(1)p 2052 1210 V 2052 1251 a(2)2096 1262 y Fq(\))p 1766 1306 418 4 v 1766 1412 a Fn(k)p Fo(\036)1874 1427 y Fl(gs)1941 1412 y Fq(\()p Fo(\013)2052 1348 y Ff(1)p 2052 1360 31 4 v 2052 1401 a(2)2096 1412 y Fq(\))p Fn(k)2194 1204 y Fh(\015)2194 1254 y(\015)2194 1304 y(\015)2194 1354 y(\015)2268 1329 y Fn(\024)28 b Fo(C)2450 1288 y Fi(00)2492 1329 y Fo(\013)2555 1288 y Fm(N)7 b Fl(+)2683 1261 y Ff(1)p 2683 1273 V 2683 1314 a(2)3135 1329 y Fq(\(I)s(I)s(I.121\))257 1604 y(where)47 b Fo(C)629 1568 y Fi(00)717 1604 y Fq(is)f Fo(\013)q Fq(-)f(and)h Fo(N)10 b Fq(-indep)s(enden)m(t.)85 b(No)m(w)46 b(consider)h(the)f(Hamiltonian)g(with)257 1725 y(infrared)39 b(cut-o\013)e Fo(\033)k Fq(=)c Fo(\013)1212 1689 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1442 1725 y Fq(in)38 b(the)h(in)m(teraction)f(term,)i(with)f Fj(r)-5 b(adiative)45 b Fq(coupling)257 1845 y(constan)m(t)34 b Fo(\014)k Fq(and)33 b(acting)f(on)h(the)g(space)g Fn(H)1874 1860 y Fm(el)1955 1845 y Fn(\012)23 b(F)2127 1860 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2319 1845 y Fq(.)44 b(It)32 b(is)i(de\014ned)g(as)e(follo)m(ws)472 2024 y(^)446 2049 y Fo(H)527 2064 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)720 2049 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\))26 b(:=)2042 b(\(I)s(I)s(I.122\))529 2203 y(=)83 b(\()p Fn(\000)p Fo(i)845 2178 y(~)836 2203 y Fn(r)919 2218 y Fm(x)1002 2203 y Fq(+)39 b Fo(a)1168 2162 y Fi(\003)1208 2203 y Fq(\()1261 2178 y Fo(~)1246 2203 y(G)1323 2218 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1515 2203 y Fq(\()p Fo(\014)f Fq(\)\))38 b(+)h Fo(a)p Fq(\()1947 2178 y Fo(~)1932 2203 y(G)2009 2218 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2202 2203 y Fq(\()p Fo(\014)f Fq(\)\)\))2427 2162 y Fl(2)2504 2203 y Fn(\000)39 b Fo(V)22 b Fq(\()p Fo(x)p Fq(\))39 b(+)3009 2178 y(\024)2984 2203 y Fo(H)3065 2218 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)3279 2203 y Fo(;)257 2407 y Fq(where)1039 2537 y Fo(~)1024 2562 y(G)p Fq(\()p Fo(k)s Fq(\)\()p Fo(\014)f Fq(\))60 b(:=)1697 2494 y Fo(\014)1758 2458 y Fl(3)p 1601 2538 294 4 v 1601 2630 a Fq(\(2)p Fo(\031)t Fq(\))1785 2601 y Fl(3)p Fm(=)p Fl(2)1980 2494 y Fq(\003\()p Fo(k)s Fq(\))p 1942 2538 275 4 v 1942 2555 a Fh(q)p 2025 2555 192 4 v 100 x Fq(2)17 b Fn(j)p Fo(k)s Fn(j)2254 2562 y Fo(e)2299 2521 y Fi(\000)p Fm(i\013)2420 2503 y(~)2423 2521 y(k)r Fi(\001)o Fm(~)-34 b(x)2552 2562 y Fo(~)-48 b(")p Fq(\()p Fo(k)s Fq(\))406 b(\(I)s(I)s(I.124\))257 2831 y(F)-8 b(rom)34 b(the)h(analysis)h(in)f([1])f(and)h(underlying)h (the)f(results)h(in)f(Prop)s(osition)g(I.1,)g(for)f Fo(\014)257 2960 y Fq(in)41 b(an)f(in)m(terv)-5 b(al)41 b(\()p Fn(\000)10 b Fq(\026)-59 b Fo(\013)1070 2897 y Ff(1)p 1070 2909 31 4 v 1070 2950 a(2)1115 2960 y Fo(;)26 b Fq(\026)-58 b Fo(\013)1232 2897 y Ff(1)p 1231 2909 V 1231 2950 a(2)1276 2960 y Fq(\))40 b(where)51 b(\026)-58 b Fo(\013)41 b Fq(is)g(a)f(p)s(ositiv)m(e)i Fo(N)10 b Fq(-indep)s(enden)m(t)42 b(n)m(um)m(b)s(er,)i(w)m(e)257 3081 y(can)29 b(construct)h(the)g(v)m (ector)1323 3055 y(^)1310 3081 y Fo(\036)1368 3096 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1561 3081 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\),)28 b(that)h(is)g(groundstate)g(of)g(the)g(Hamiltonian) 283 3176 y(^)257 3201 y Fo(H)338 3216 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)531 3201 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\),)34 b(b)m(y)i(de\014nition.)51 b(F)-8 b(urthermore,)36 b(w)m(e)g(kno)m(w)g(that)f(gap[)2868 3176 y(^)2843 3201 y Fo(H)2924 3216 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)3116 3201 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\)])31 b Fn(\025)257 3322 y Fo(c)22 b Fn(\001)g Fo(\024\013)490 3285 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)715 3322 y Fq(where)34 b(0)28 b Fo(<)f(c)h(<)f Fq(1.)257 3442 y(Consider)38 b(\026)-59 b Fo(\013)29 b Fn(\025)38 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(M)10 b Fq(\).)42 b(W)-8 b(e)27 b(are)f(ultimately)i(in)m(terested)h (in)2509 3416 y(^)2497 3442 y Fo(\036)2555 3457 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2747 3442 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\))p Fn(j)3019 3491 y Fm(\014)s Fl(=)p Fm(\013)3172 3440 y Ff(1)p 3171 3449 V 3171 3484 a(2)3246 3442 y Fq(where)257 3587 y Fo(\013)29 b Fn(\025)38 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(M)38 b Fq(=)28 b(2)p Fo(N)g Fn(\000)20 b Fq(1\);)31 b(then)g(w)m(e)h(can)f(apply)h(regular)f(p)s(erturbation)g(theory)g (and)g(w)m(e)257 3716 y(get)i(that,)f(in)h(particular,)g(the)g(v)m (ector)1722 3690 y(^)1709 3716 y Fo(\036)1767 3731 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1960 3716 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\))31 b(is)i Fn(C)2391 3680 y Fi(1)2499 3716 y Fq(in)f Fo(\014)38 b Fq(for)32 b Fo(\014)i Fn(2)28 b Fq(\()p Fn(\000)10 b Fq(\026)-59 b Fo(\013)3225 3652 y Ff(1)p 3225 3664 V 3225 3706 a(2)3269 3716 y Fo(;)27 b Fq(\026)-59 b Fo(\013)3386 3652 y Ff(1)p 3386 3664 V 3386 3706 a(2)3430 3716 y Fq(\).)257 3837 y(Therefore)36 b(w)m(e)f(can)g(use)g(T)-8 b(a)m(ylor)35 b(form)m(ula)f(and)g(w)m(e)i (can)e(write)h(for)43 b(\026)-58 b Fo(\013)31 b(>)f(\013)h Fn(\025)41 b Fq(\026)-59 b Fo(\013)q Fq(\()p Fo(M)41 b Fq(=)257 3957 y(2)p Fo(N)33 b Fn(\000)22 b Fq(1\):)874 4174 y Fo(\036)932 4189 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1124 4174 y Fq(\()p Fo(\013)1235 4106 y Ff(1)p 1235 4118 V 1235 4159 a(2)1279 4174 y Fq(\))28 b Fn(\021)1463 4148 y Fq(^)1450 4174 y Fo(\036)1508 4189 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1700 4174 y Fq(\()p Fo(\013)1811 4106 y Ff(1)p 1811 4118 V 1811 4159 a(2)1856 4174 y Fo(;)17 b(\013)q Fq(\))27 b(=)1031 b(\(I)s(I)s(I.125\))957 4401 y(=)1128 4374 y(^)1116 4401 y Fo(\036)1174 4416 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)1366 4401 y Fq(\(0)p Fo(;)17 b(\013)q Fq(\))k(+)1729 4293 y Fl(2)p Fm(N)1718 4318 y Fh(X)1717 4500 y Fm(j)t Fl(=1)1866 4333 y Fo(\013)1940 4265 y Fb(j)p 1939 4282 V 1939 4323 a Ff(2)p 1866 4377 118 4 v 1888 4469 a Fo(j)6 b Fq(!)2003 4333 y Fo(d)2054 4297 y Fm(j)2103 4307 y Fq(^)2091 4333 y Fo(\036)2149 4348 y Fl(2)p Fm(N)h Fi(\000)p Fl(1)2341 4333 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\))p 2003 4377 581 4 v 2220 4469 a Fo(d\014)2332 4440 y Fm(j)2594 4301 y Fh(\014)2594 4351 y(\014)2594 4401 y(\014)2621 4455 y Fm(\014)s Fl(=0)2781 4401 y Fq(+)278 b(\(I)s(I)s(I.126\))1116 4721 y(+)1282 4654 y Fo(\013)1355 4588 y Ff(2)p Fb(N)5 b Ff(+1)p 1355 4602 162 4 v 1420 4643 a(2)p 1201 4698 410 4 v 1201 4789 a Fq(\(2)p Fo(N)33 b Fq(+)22 b(1\)!)1630 4654 y Fo(d)1681 4618 y Fl(2)p Fm(N)7 b Fl(+1)1886 4627 y Fq(^)1874 4654 y Fo(\036)1932 4669 y Fl(2)p Fm(N)g Fi(\000)p Fl(1)2124 4654 y Fq(\()p Fo(\014)f(;)17 b(\013)q Fq(\))p 1630 4698 737 4 v 1847 4789 a Fo(d\014)1959 4761 y Fl(2)p Fm(N)7 b Fl(+1)2377 4621 y Fh(\014)2377 4671 y(\014)2377 4721 y(\014)2405 4775 y Fm(\014)s Fl(=)p Fm(\013)2548 4743 y Fd(0)2576 4724 y Ff(1)p 2576 4733 31 4 v 2576 4768 a(2)3135 4721 y Fq(\(I)s(I)s(I.127\))257 5002 y(where)34 b Fo(\013)602 4965 y Fi(0)631 4938 y Ff(1)p 631 4950 V 631 4991 a(2)708 5002 y Fq(is)f(a)f(p)s(oin)m(t)h(in)g(\(0)p Fo(;)17 b(\013)1460 4938 y Ff(1)p 1459 4950 V 1459 4991 a(2)1504 5002 y Fq(\).)257 5122 y(A)39 b(similar)h(expansion)g(can)f(b) s(e)g(pro)m(vided)h(for)f Fo(E)2114 5137 y Fl(2)p Fm(N)7 b Fi(\000)p Fl(1)2307 5122 y Fq(\()p Fo(\013)q Fq(\).)61 b(Finally)-8 b(,)41 b(in)e(Expansions)257 5243 y(\(I)s(I)s(I.117\),)j (\(I)s(I)s(I.118\),)f(w)m(e)g(can)f(\014x)h(the)f(constan)m(ts)h(in)f (the)h(error)f(terms)h Fo(o)p Fq(\()p Fo(\013)3183 5207 y Fm(N)3249 5243 y Fq(\))f(uni-)257 5363 y(formly)33 b(in)g(\(0)p Fo(;)26 b Fq(\026)-58 b Fo(\013)p Fq(\).)p eop end %%Page: 49 50 TeXDict begin 49 49 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(49)257 573 y Fp(A)162 b(Pro)t(ofs)53 b(of)h(Lemma)h(I)t(I)t(I.5)g(and)e(Lemma)i(I)t(I)t(I.6)257 792 y Fe(Pro)s(of)38 b(of)f(Lemma)i(I)s(I)s(I.5.)257 912 y Fq(W)-8 b(e)30 b(m)m(ust)h(study)f(the)g(norm)g(of)f(eac)m(h)h (distinct)h(v)m(ector)f(in)g(the)g(sum)g(corresp)s(onding)h(to)257 1033 y(Wic)m(k-ordering)j(in)f(the)g(shell)h Fn(K)1476 991 y Fm(m)p Fi(\000)p Fl(3)1475 1054 y Fm(m)p Fi(\000)p Fl(2)1665 1033 y Fq(of)1465 1232 y Fh(f)1459 1264 y Fo(D)1540 1279 y Fm(m)p Fi(\000)p Fl(3)1697 1264 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\)\()1971 1238 y(~)1958 1264 y Fo(\036)2016 1279 y Fm(m)p Fi(\000)p Fl(3)2173 1264 y Fq(\))2211 1223 y Fm(T)2266 1264 y Fo(;)977 b Fq(\(A.1\))257 1484 y(where)545 1452 y Fh(f)539 1484 y Fo(D)620 1499 y Fm(m)p Fi(\000)p Fl(3)777 1484 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))33 b(is)g(giv)m(en)h (in)e(Eq.)i(\(I)s(I)s(I.101\).)257 1604 y(F)-8 b(or)28 b(eac)m(h)h(of)f(these)i(v)m(ectors,)h(w)m(e)f(ev)m(en)m(tually)h(pro)m (v)m(e)e(that)g(the)g(order)f(in)h Fo(\013)g Fq(of)f(its)h(norm)257 1725 y(is,)34 b(at)e(most,)h(equal)g(to)g(the)g(order)f(of)g(the)h (norm)g(estimate)h(of)1120 1945 y Fn(k)1176 1912 y Fh(f)1170 1945 y Fo(D)1251 1960 y Fm(m)p Fi(\000)p Fl(3)1408 1945 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))p Fn(k)28 b(\024)g(E)9 b Fq(\()p Fo(D)1995 1960 y Fm(m)p Fi(\000)p Fl(2)2151 1945 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))p Fo(P)2451 1904 y Fm(T)2437 1969 y(m)p Fi(\000)p Fl(2)2594 1945 y Fq(\))638 b(\(A.2\))257 2165 y(that)42 b(w)m(e)g(get)g(in)f(Eq.)i(\(I)s(I)s (I.102\))d(as)i(a)f(b)m(ypro)s(duct)i(of)e(the)h(previous)h(con)m(trol) f(of)f(the)257 2285 y(truncation)33 b(map)g(\(see)h(Lemmata)f(I)s(I)s (I.1,)f(I)s(I)s(I.2\).)257 2405 y(Without)f(loss)h(of)e(generalit)m(y) -8 b(,)33 b(w)m(e)f(study)g(the)f(term)2221 2373 y Fh(f)2215 2405 y Fo(D)2296 2420 y Fm(m)p Fi(\000)p Fl(3)2453 2405 y Fq(\()p Fo(k)s(;)17 b(z)t Fq(\))31 b(considered)i(in)e(The-)257 2526 y(orem)i(I)s(I)s(I.4)488 2680 y Fl(\026)489 2695 y Fm(j)452 2726 y Fh(Y)444 2908 y Fm(j)t Fl(=1)583 2692 y Fh(Z)629 2880 y Fl(\000)673 2889 y Fb(m)p Fd(\000)p Ff(2)831 2809 y Fo(dz)931 2758 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))p Fm(;j)927 2831 y(m)p Fi(\000)p Fl(2)1215 2809 y Fq(~)-52 b Fo(g)1263 2768 y Fl(1)1259 2834 y Fm(m)p Fi(\000)p Fl(3)1419 2809 y Fq(~)f Fo(g)1466 2768 y Fl(2)1462 2834 y Fm(m)p Fi(\000)p Fl(3)1622 2809 y Fq(~)h Fo(g)1670 2768 y Fl(1)1666 2834 y Fm(m)p Fi(\000)p Fl(3)1844 2809 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)1973 2768 y Fl(1)1969 2834 y Fm(m)p Fi(\000)p Fl(3)1212 2875 y Fh(|)p 1249 2875 383 10 v 383 w({z)p 1706 2875 V 383 w(})2142 2809 y Fn(\002)1051 b Fq(\(A.3\))903 3063 y Fn(\002)20 b Fq(~)-52 b Fo(g)1048 3022 y Fl(2)1044 3088 y Fm(m)p Fi(\000)p Fl(3)997 3130 y Fh(|)p 1034 3130 28 10 v 28 w({z)p 1136 3130 V 28 w(})1217 3063 y Fn(\001)22 b(\001)j Fq(~)-53 b Fo(g)1367 3022 y Fl(1)1363 3088 y Fm(m)p Fi(\000)p Fl(3)1523 3063 y Fq(~)h Fo(g)1571 3022 y Fl(2)1567 3088 y Fm(m)p Fi(\000)p Fl(3)1727 3063 y Fq(~)f Fo(g)1774 3022 y Fl(1)1770 3088 y Fm(m)p Fi(\000)p Fl(3)1949 3063 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2078 3022 y Fl(1)2074 3088 y Fm(m)p Fi(\000)p Fl(3)1316 3130 y Fh(|)p 1353 3130 383 10 v 383 w({z)p 1810 3130 V 383 w(})2250 3063 y Fq(~)f Fo(g)2297 3022 y Fl(3)2293 3088 y Fm(m)p Fi(\000)p Fl(3)2472 3063 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2601 3022 y Fl(3)2597 3088 y Fm(m)p Fi(\000)p Fl(3)2246 3130 y Fh(|)p 2283 3130 179 10 v 179 w({z)p 2536 3130 V 179 w(})2753 3063 y Fq(\()p Fo(k)s(;)17 b(z)t(;)g(z)3031 3012 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))3027 3085 y Fm(m)p Fi(\000)p Fl(2)3244 3063 y Fq(\))p Fo(:)257 3342 y Fq(As)27 b(explained)h(in)f(the)f(theorem,)j(the)d(op)s(erators)h(of)e(t)m(yp)s (e)31 b(~)-52 b Fo(g)2452 3305 y Fl(2)2448 3366 y Fm(m)p Fi(\000)p Fl(3)2630 3342 y Fq(in)27 b(the)f(\014rst)h(collection)257 3462 y(are)33 b(in)g(fact)f(of)g(t)m(yp)s(e)i Fn(W)1163 3421 y Fm(m)p Fi(\000)p Fl(3)1155 3484 y Fm(m)p Fi(\000)p Fl(2)1352 3462 y Fq(\(see)g(Eq.)f(\(I)s(I)s(I.97\)\);)f(the)h(op)s (erator)j(~)-53 b Fo(g)2746 3426 y Fl(2)2742 3487 y Fm(m)p Fi(\000)p Fl(3)2931 3462 y Fq(in)33 b(the)g(second)257 3582 y(collection)46 b(is)e(of)g(t)m(yp)s(e)h Fo(\016)t(g)1264 3546 y Fl(2)1260 3607 y Fm(m)p Fi(\000)p Fl(2)1417 3582 y Fq(,)i(since)e(it)g(deriv)m(es)h(from)e(a)g(v)m(ertex)i(op)s(erator)d (in)i(the)257 3703 y(group)32 b Fn(G)598 3667 y Fl(2)592 3727 y Fm(m)p Fi(\000)p Fl(2)749 3703 y Fq(.)43 b(In)32 b(the)g(same)h(w)m(a)m(y)g(it)e(is)h(easy)h(to)e(sp)s(ecify)i(the)g(op) s(erators)e(in)h(the)g(other)257 3823 y(collections.)404 3943 y(F)-8 b(or)44 b(deriving)h(the)g(estimate)h(in)f(Eq.)h(\(A.2\),)i (among)c(all)h(the)g(p)s(ossible)h(com)m(bi-)257 4064 y(nations)d(of)f(op)s(erators)h Fn(W)1278 4023 y Fm(m)p Fi(\000)p Fl(3)1270 4086 y Fm(m)p Fi(\000)p Fl(2)1478 4064 y Fq(and)f(op)s(erators)h Fo(\016)t(g)2217 4028 y Fl(2)2213 4088 y Fm(m)p Fi(\000)p Fl(2)2412 4064 y Fq(whic)m(h)h(ma)m(y)f(b)s(e)g(presen)m(t)h(in)257 4184 y(\(A.3\),)39 b(w)m(e)f(select)g(the)g(leading)g(order)f(ones;)j(i.e,)f (in)e(applying)h(the)g(results)g(of)f(Lem-)257 4305 y(mata)31 b(I)s(I)s(I.1,)g(I)s(I)s(I.2,)g(w)m(e)h(implicitly)g(assume)h(the)e(op) s(erators)f Fo(\016)t(g)2600 4268 y Fl(2)2596 4329 y Fm(m)p Fi(\000)p Fl(2)2783 4305 y Fq(b)s(e)h(di\013eren)m(t)h(from)257 4441 y(\001\007)414 4399 y Fi(\006)414 4462 y Fm(m)p Fi(\000)p Fl(3)571 4441 y Fq(\()p Fo(k)660 4456 y Fm(i)688 4441 y Fq(\()p Fo(q)773 4404 y Fi(0)796 4441 y Fq(\)\))e(and)g(the)h (op)s(erators)e Fn(W)1789 4399 y Fm(m)p Fi(\000)p Fl(3)1781 4462 y Fm(m)p Fi(\000)p Fl(2)1977 4441 y Fq(b)s(e)h(either)h(equal)g (to)25 b Fo(~)-45 b(v)2799 4456 y Fm(m)p Fi(\000)p Fl(3)2973 4441 y Fn(\001)17 b Fo(a)p Fq(\()3122 4415 y Fo(~)3107 4441 y(G)3184 4399 y Fm(m)p Fi(\000)p Fl(3)3184 4462 y Fm(m)p Fi(\000)p Fl(2)3341 4441 y Fq(\))30 b(or)257 4574 y(to)f Fo(~)-45 b(v)424 4589 y Fm(m)p Fi(\000)p Fl(3)603 4574 y Fn(\001)21 b Fo(a)703 4538 y Fi(\003)743 4574 y Fq(\()796 4549 y Fo(~)781 4574 y(G)858 4533 y Fm(m)p Fi(\000)p Fl(3)858 4596 y Fm(m)p Fi(\000)p Fl(2)1015 4574 y Fq(\).)257 4694 y(W)-8 b(e)40 b(also)f(note)h(that,)h(for)d (determining)j(the)e(estimates)i(stated)f(in)g(Lemmata)f(I)s(I)s(I.1,) 257 4815 y(I)s(I)s(I.2)30 b(and)f(Prop)s(osition)h(I.1)g(and)f (underlying)j(Inequalit)m(y)f(\(A.2\),)f(w)m(e)h(can)f(as)f(w)m(ell)i (use)257 4935 y(the)i(follo)m(wing)g(p)s(o)m(w)m(er)h(coun)m(ting)f (rules.)257 5055 y Fj(Power-c)-5 b(ounting)34 b(rules)366 5259 y Fq(1\))48 b(F)-8 b(or)32 b(eac)m(h)h Fn(W)1001 5218 y Fm(m)p Fi(\000)p Fl(3)993 5281 y Fm(m)p Fi(\000)p Fl(2)1191 5259 y Fq(a)g(factor)f Fo(\013)1615 5223 y Fl(2)1676 5259 y Fn(\001)22 b Fo(\013)1789 5223 y Fm(m)p Fi(\000)p Fl(3)1978 5259 y Fq(is)33 b(gained.)p eop end %%Page: 50 51 TeXDict begin 50 50 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(50)366 573 y(2\))48 b(F)-8 b(or)32 b(eac)m(h)h(op)s(erator)f(of)g(the)h(t)m(yp)s(e)1577 734 y Fh(e)1554 767 y Fo(R)1629 726 y Fl(+)p Fi(?)1628 791 y Fm(m)p Fi(\000)p Fl(3)1785 767 y Fq(\()p Fo(z)1872 726 y Fm(m)p Fi(\000)p Fl(2)1868 791 y Fm(m)p Fi(\000)p Fl(2)2030 767 y Fq(\))1202 b(\(A.4\))1577 901 y Fh(e)1554 933 y Fo(R)1629 892 y Fl(+)p Fi(?)1628 958 y Fm(m)p Fi(\000)p Fl(3)1785 933 y Fq(\()p Fo(z)1872 883 y Fl(\()p Fm(m)1961 859 y Fd(0)1985 883 y Fl(\))1868 955 y Fm(m)p Fi(\000)p Fl(1)2047 933 y Fn(\000)23 b Fo(S)2207 949 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2440 933 y Fq(\()p Fo(k)s Fq(\)\))662 b(\(A.5\))501 1127 y(a)33 b(factor)f Fo(\013)925 1091 y Fm(m)p Fi(\000)p Fl(2)1114 1127 y Fq(is)h(lost.)366 1323 y(3\))48 b(The)28 b(op)s(erators)f Fo(\016)t(g)1220 1287 y Fl(2)1216 1348 y Fm(m)p Fi(\000)p Fl(2)1399 1323 y Fq(and)1606 1291 y Fh(e)1582 1323 y Fo(P)1645 1338 y Fm(m)p Fi(\000)p Fl(3)1829 1323 y Fq(are)g(neutral,)h(i.e.,)h(no)d(p) s(o)m(w)m(er)i(of)e Fo(\013)i Fq(is)f(gained)501 1444 y(or)32 b(lost.)366 1640 y(4\))48 b(The)34 b(c-n)m(um)m(b)s(er)g(op)s (erators)e(giv)m(e)i(either)f(p)s(ositiv)m(e)h(or)e(negativ)m(e)i(p)s (o)m(w)m(ers)g(of)e Fo(\013)q Fq(.)366 1835 y(5\))48 b(Eac)m(h)34 b(con)m(tour)f(in)m(tegration)g(in)f(Eq.)i(\(A.3\))e (yields)i(a)f(p)s(o)m(w)m(er)g Fo(\013)2846 1799 y Fm(m)p Fi(\000)p Fl(2)3003 1835 y Fq(.)257 2016 y(No)m(w)41 b(w)m(e)g(return)f(to)g(the)g(v)m(ectors)h(in)f(Eq.)h(\(A.1\).)65 b(Let)40 b Fo( )k Fq(b)s(e)c(an)m(y)h(v)m(ector)g(obtained)257 2137 y(from)33 b(the)g(Wic)m(k-ordering)g(in)g(Eq.)h(\(A.1\).)43 b(The)34 b(norm)e(of)g Fo( )37 b Fq(is)c(b)s(ounded)g(b)m(y)545 2290 y Fl(\026)546 2305 y Fm(j)509 2336 y Fh(Y)501 2518 y Fm(j)t Fl(=1)640 2419 y Fn(j)p Fq(\000)729 2371 y Fm(j)729 2440 y(m)p Fi(\000)p Fl(2)885 2419 y Fn(j)p Fq(sup)1060 2457 y Fm(z)1096 2422 y Ff(\()p Fb(m)p Fd(\000)p Ff(2\))1093 2480 y Fb(m)p Fd(\000)p Ff(2)1281 2457 y Fm(;z)s(;k)1398 2322 y Fh(h)1437 2419 y Fn(j)p Fq([)s(~)-52 b Fo(g)1543 2378 y Fl(3)1539 2443 y Fm(m)p Fi(\000)p Fl(3)1696 2419 y Fq(]\()p Fo(k)s(;)17 b(z)t(;)g(z)2001 2368 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1997 2440 y Fm(m)p Fi(\000)p Fl(2)2213 2419 y Fq(\))p Fn(j)22 b(\002)1176 b Fq(\(A.6\))340 2672 y Fn(\002)501 2576 y Fh(\020)571 2672 y Fq(~)-53 b Fo(g)618 2631 y Fl(1)614 2697 y Fm(m)p Fi(\000)p Fl(3)774 2672 y Fq(~)h Fo(g)822 2631 y Fl(2)818 2697 y Fm(m)p Fi(\000)p Fl(3)977 2672 y Fq(~)g Fo(g)1025 2631 y Fl(1)1021 2697 y Fm(m)p Fi(\000)p Fl(3)1200 2672 y Fn(\001)21 b(\001)s Fq(~)-52 b Fo(g)1328 2631 y Fl(1)1324 2697 y Fm(m)p Fi(\000)p Fl(3)567 2738 y Fh(|)p 604 2738 383 10 v 383 w({z)p 1061 2738 V 383 w(})1501 2672 y Fq(~)f Fo(g)1548 2631 y Fl(2)1544 2697 y Fm(m)p Fi(\000)p Fl(3)1497 2738 y Fh(|)p 1534 2738 28 10 v 28 w({z)p 1636 2738 V 28 w(})1717 2672 y Fn(\001)22 b(\001)j Fq(~)-52 b Fo(g)1868 2631 y Fl(1)1864 2697 y Fm(m)p Fi(\000)p Fl(3)2024 2672 y Fq(~)f Fo(g)2071 2631 y Fl(2)2067 2697 y Fm(m)p Fi(\000)p Fl(3)2227 2672 y Fq(~)h Fo(g)2275 2631 y Fl(1)2271 2697 y Fm(m)p Fi(\000)p Fl(3)2449 2672 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2578 2631 y Fl(1)2574 2697 y Fm(m)p Fi(\000)p Fl(3)1817 2738 y Fh(|)p 1854 2738 383 10 v 383 w({z)p 2311 2738 V 383 w(})2730 2672 y Fq(\()2781 2646 y(~)2768 2672 y Fo(\036)2826 2687 y Fm(m)p Fi(\000)p Fl(3)2983 2672 y Fq(\))3021 2631 y Fm(T)3076 2672 y Fo(;)518 2929 y(;)20 b Fq(~)-52 b Fo(g)613 2888 y Fl(1)609 2954 y Fm(m)p Fi(\000)p Fl(3)768 2929 y Fq(~)g Fo(g)816 2888 y Fl(2)812 2954 y Fm(m)p Fi(\000)p Fl(3)972 2929 y Fq(~)f Fo(g)1019 2888 y Fl(1)1015 2954 y Fm(m)p Fi(\000)p Fl(3)1194 2929 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)1323 2888 y Fl(1)1319 2954 y Fm(m)p Fi(\000)p Fl(3)561 2996 y Fh(|)p 598 2996 V 383 w({z)p 1055 2996 V 383 w(})1495 2929 y Fq(~)f Fo(g)1542 2888 y Fl(2)1538 2954 y Fm(m)p Fi(\000)p Fl(3)1491 2996 y Fh(|)p 1528 2996 28 10 v 28 w({z)p 1630 2996 V 28 w(})1711 2929 y Fn(\001)22 b(\001)j Fq(~)-52 b Fo(g)1862 2888 y Fl(1)1858 2954 y Fm(m)p Fi(\000)p Fl(3)2018 2929 y Fq(~)f Fo(g)2065 2888 y Fl(2)2061 2954 y Fm(m)p Fi(\000)p Fl(3)2221 2929 y Fq(~)h Fo(g)2269 2888 y Fl(1)2265 2954 y Fm(m)p Fi(\000)p Fl(3)2443 2929 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2572 2888 y Fl(1)2568 2954 y Fm(m)p Fi(\000)p Fl(3)1811 2996 y Fh(|)p 1848 2996 383 10 v 383 w({z)p 2305 2996 V 383 w(})2724 2929 y Fq(\()2775 2903 y(~)2762 2929 y Fo(\036)2820 2944 y Fm(m)p Fi(\000)p Fl(3)2977 2929 y Fq(\))3015 2888 y Fm(T)3070 2833 y Fh(\021)3129 2829 y Ff(1)p 3129 2841 31 4 v 3129 2882 a(2)3119 2983 y Fm(\031)3160 2995 y Fb( )3211 2929 y Fq(\()p Fo(k)s(;)17 b(z)t(;)g(z)3489 2878 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))3485 2951 y Fm(m)p Fi(\000)p Fl(2)3701 2929 y Fq(\))3739 2833 y Fh(i)257 3171 y Fq(where)33 b(the)e(sym)m(b)s(ol)i(of)d(scalar)i(pro)s(duct)f (lab)s(eled)h(b)m(y)g Fo(\031)2307 3186 y Fm( )2391 3171 y Fq(means)g(the)g(con)m(tribution)g(to)257 3292 y(the)h(square)h(norm) 432 3486 y Fn(k)20 b Fq(~)-52 b Fo(g)550 3444 y Fl(1)546 3510 y Fm(m)p Fi(\000)p Fl(3)705 3486 y Fq(~)g Fo(g)753 3444 y Fl(2)749 3510 y Fm(m)p Fi(\000)p Fl(3)909 3486 y Fq(~)f Fo(g)956 3444 y Fl(1)952 3510 y Fm(m)p Fi(\000)p Fl(3)1131 3486 y Fn(\001)21 b(\001)s Fq(~)-52 b Fo(g)1259 3444 y Fl(1)1255 3510 y Fm(m)p Fi(\000)p Fl(3)498 3552 y Fh(|)p 535 3552 383 10 v 383 w({z)p 992 3552 V 383 w(})1432 3486 y Fq(~)f Fo(g)1479 3444 y Fl(2)1475 3510 y Fm(m)p Fi(\000)p Fl(3)1428 3552 y Fh(|)p 1465 3552 28 10 v 28 w({z)p 1567 3552 V 28 w(})1648 3486 y Fn(\001)22 b(\001)j Fq(~)-52 b Fo(g)1799 3444 y Fl(1)1795 3510 y Fm(m)p Fi(\000)p Fl(3)1955 3486 y Fq(~)f Fo(g)2002 3444 y Fl(2)1998 3510 y Fm(m)p Fi(\000)p Fl(3)2158 3486 y Fq(~)h Fo(g)2206 3444 y Fl(1)2202 3510 y Fm(m)p Fi(\000)p Fl(3)2380 3486 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)2509 3444 y Fl(1)2505 3510 y Fm(m)p Fi(\000)p Fl(3)1748 3552 y Fh(|)p 1785 3552 383 10 v 383 w({z)p 2242 3552 V 383 w(})2661 3486 y Fq(\()2712 3459 y(~)2699 3486 y Fo(\036)2757 3501 y Fm(m)p Fi(\000)p Fl(3)2914 3486 y Fq(\))2952 3444 y Fm(T)3007 3486 y Fn(k)3057 3444 y Fl(2)3270 3486 y Fq(\(A.7\))257 3728 y(corresp)s(onding)23 b(to)e(the)i(sp)s(eci\014c)g (pairings-com)m(bination)g Fo(\031)2436 3743 y Fm( )2510 3728 y Fq(-)f(of)f(the)h(photon)g(creation-)257 3848 y(and)33 b(annihilation)g(op)s(erators)f(con)m(tained)i(in)f(the)g (terms)g Fn(W)2521 3807 y Fm(m)p Fi(\000)p Fl(3)2513 3870 y Fm(m)p Fi(\000)p Fl(2)2711 3848 y Fq(-)f(asso)s(ciated)i(to)e (the)257 3986 y(square)i(norm)f(of)f(the)i(giv)m(en)f(v)m(ector)h Fo( )t Fq(;)f([)s(~)-52 b Fo(g)1851 3950 y Fl(3)1847 4011 y Fm(m)p Fi(\000)p Fl(3)2004 3986 y Fq(]\()p Fo(k)s(;)17 b(z)t(;)g(z)2309 3935 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2305 4008 y Fm(m)p Fi(\000)p Fl(2)2521 3986 y Fq(\))33 b(is)g(an)g (abbreviated)h(no-)257 4127 y(tation)f(for)i(~)-52 b Fo(g)745 4085 y Fl(3)741 4151 y Fm(m)p Fi(\000)p Fl(3)919 4127 y Fn(\001)22 b(\001)s Fq(~)-52 b Fo(g)1048 4085 y Fl(3)1044 4151 y Fm(m)p Fi(\000)p Fl(3)694 4193 y Fh(|)p 731 4193 179 10 v 179 w({z)p 984 4193 V 179 w(})1200 4127 y Fq(\()p Fo(k)s(;)17 b(z)t(;)g(z)1478 4076 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1474 4148 y Fm(m)p Fi(\000)p Fl(2)1691 4127 y Fq(\).)257 4301 y(W)-8 b(e)33 b(can)g(pro)m(vide)h(an) f(upp)s(er)g(b)s(ound)f(to)h(the)g(quan)m(tit)m(y)h(in)f(Eq.)g(\(A.6\)) f(as)h(follo)m(ws.)257 4421 y(F)-8 b(or)40 b(an)m(y)h(distinct)g (couple)g(of)f(creation-)g(and)h(annihilation)f(op)s(erator)g(that)g (will)h(b)s(e)257 4542 y(con)m(tracted)c(in)f(the)f(pairings)h(com)m (bination)h Fo(\031)2013 4557 y Fm( )2066 4542 y Fq(,)f(there)g(is)g(a) f(distinct)i(resolv)m(en)m(t,)h(b)s(e-)257 4662 y(t)m(w)m(een)d(the)e (t)m(w)m(o)g(photon)g(op)s(erators,)f(of)g(the)h(t)m(w)m(o)g(p)s (ossible)h(t)m(yp)s(es)1004 4833 y Fh(e)981 4865 y Fo(R)1056 4818 y Fl(+)p Fm(;)p Fi(?)1055 4887 y Fm(m)p Fi(\000)p Fl(3)1212 4865 y Fq(\()p Fo(z)1299 4814 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1295 4887 y Fm(m)p Fi(\000)p Fl(2)1511 4865 y Fq(\))98 b Fo(;)1713 4833 y Fh(e)1690 4865 y Fo(R)1765 4818 y Fl(+)p Fm(;)p Fi(?)1764 4887 y Fm(m)p Fi(\000)p Fl(3)1921 4865 y Fq(\()p Fo(z)2008 4814 y Fl(\()p Fm(m)2097 4791 y Fd(0)2121 4814 y Fl(\))2004 4887 y Fm(m)p Fi(\000)p Fl(1)2184 4865 y Fn(\000)22 b Fo(S)2343 4881 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2577 4865 y Fq(\()p Fo(k)s Fq(\)\))p Fo(:)498 b Fq(\(A.8\))257 5059 y(Because)43 b(of)e(the)g(pull-through)g(form)m(ula,)i(after)e(a)g(con)m(traction,)j (the)d(resolv)m(en)m(ts)j(in)257 5179 y(Eq.)34 b(\(A.8\))e(assume)i (the)f(form)787 5388 y Fo(R)862 5347 y Fl(+)861 5413 y Fm(m)p Fi(\000)p Fl(3)1018 5388 y Fq(\()p Fo(z)1105 5337 y Fl(\()p Fm(m)1194 5314 y Fd(0)1218 5337 y Fl(\))1101 5410 y Fm(m)p Fi(\000)p Fl(2)1280 5388 y Fn(\000)22 b(j)p Fo(q)t Fn(j)p Fq(\))97 b Fo(;)17 b(R)1736 5347 y Fl(+)1735 5413 y Fm(m)p Fi(\000)p Fl(3)1892 5388 y Fq(\()p Fo(z)1979 5337 y Fl(\()p Fm(m)2068 5314 y Fd(0)2092 5337 y Fl(\))1975 5410 y Fm(m)p Fi(\000)p Fl(1)2154 5388 y Fn(\000)23 b Fo(S)2314 5404 y Fm(l)q Fl(\()p Fm(m)p Fi(\000)p Fl(1\))2547 5388 y Fq(\()p Fo(k)s Fq(\))f Fn(\000)h(j)p Fo(q)t Fn(j)p Fq(\))p Fo(;)303 b Fq(\(A.9\))p eop end %%Page: 51 52 TeXDict begin 51 51 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(51)257 573 y(with)38 b Fo(q)g Fn(2)e(K)744 532 y Fm(m)p Fi(\000)p Fl(3)743 595 y Fm(m)p Fi(\000)p Fl(2)901 573 y Fq(.)56 b(In)37 b(the)g(case)g(of)g(a)f(con)m (traction)h(in)m(v)m(olving)i(the)e(op)s(erators)c Fo(~)-45 b(v)3286 588 y Fm(m)p Fi(\000)p Fl(3)3468 573 y Fn(\001)257 706 y Fo(a)p Fq(\()369 681 y(\024)346 706 y Fo(G)423 665 y Fm(m)p Fi(\000)p Fl(3)423 728 y Fm(m)p Fi(\000)p Fl(2)580 706 y Fq(\))33 b(and)28 b Fo(~)-45 b(v)887 721 y Fm(m)p Fi(\000)p Fl(3)1066 706 y Fn(\001)22 b Fo(a)1167 670 y Fi(\003)1207 706 y Fq(\()1260 681 y Fo(~)1245 706 y(G)1322 665 y Fm(m)p Fi(\000)p Fl(3)1322 728 y Fm(m)p Fi(\000)p Fl(2)1478 706 y Fq(\),)33 b(an)g(elemen)m(tary)h(calculation) f(yields)560 852 y Fh(\015)560 902 y(\015)560 952 y(\015)560 1002 y(\015)606 977 y Fo(\013)669 936 y Fl(3)725 860 y Fh(Z)771 1049 y Fi(K)825 1058 y Fb(m)p Fd(\000)p Ff(2)982 977 y Fo(dq)1086 910 y(~)-45 b(v)1137 925 y Fm(m)p Fi(\000)p Fl(3)1316 910 y Fn(\001)20 b Fo(~)-48 b(")p Fq(\()p Fo(q)t Fq(\))1534 873 y Fi(\003)1573 910 y Fo(e)1618 873 y Fm(i\013~)-35 b(q)s Fl(\))p Fi(\001)o Fm(~)h(x)p 1090 954 724 4 v 1358 970 a Fh(q)p 1441 970 103 4 v 100 x Fn(j)p Fo(q)t Fn(j)2217 910 y Fq(1)p 1849 954 784 4 v 1849 1064 a Fo(H)1938 1023 y Fl(+)1930 1086 y Fm(m)p Fi(\000)p Fl(3)2109 1064 y Fn(\000)22 b Fo(z)2257 1014 y Fl(\()p Fm(m)2346 995 y Fd(0)2370 1014 y Fl(\))2253 1086 y Fm(m)p Fi(\000)p Fl(2)2433 1064 y Fq(+)g Fn(j)p Fo(q)t Fn(j)2665 910 y Fo(~)-45 b(v)2716 925 y Fm(m)p Fi(\000)p Fl(3)2895 910 y Fn(\001)21 b Fo(~)-48 b(")p Fq(\()p Fo(q)t Fq(\))p Fo(e)3159 873 y Fi(\000)p Fm(i\013~)-35 b(q)r Fi(\001)o Fm(~)h(x)p 2669 954 712 4 v 2932 970 a Fh(q)p 3015 970 103 4 v 100 x Fn(j)p Fo(q)t Fn(j)3390 852 y Fh(\015)3390 902 y(\015)3390 952 y(\015)3390 1002 y(\015)399 1281 y Fn(\024)84 b Fo(C)7 b(\013)700 1240 y Fl(3)755 1164 y Fh(Z)801 1353 y Fi(K)855 1362 y Fb(m)p Fd(\000)p Ff(2)1013 1281 y Fo(dq)1147 1214 y Fq(1)p 1121 1258 V 1121 1350 a Fn(j)p Fo(q)t Fn(j)1535 1214 y Fq(1)p 1259 1258 602 4 v 1259 1369 a Fn(j)22 b(\000)h Fo(z)1458 1318 y Fl(\()p Fm(m)1547 1299 y Fd(0)1571 1318 y Fl(\))1454 1391 y Fm(m)p Fi(\000)p Fl(2)1633 1369 y Fq(+)f Fn(j)p Fo(q)t Fn(jj)1898 1281 y(\024)28 b Fo(C)2080 1240 y Fi(0)2104 1281 y Fo(\013)2167 1240 y Fm(m)2233 1281 y Fo(:)962 b Fq(\(A.10\))257 1559 y(where)33 b Fo(C)7 b Fq(,)31 b Fo(C)750 1523 y Fi(0)804 1559 y Fq(are)g Fo(\013)q Fq(-indep)s(enden)m(t)i(constan)m(ts.)44 b(An)32 b(analogous)e(result)i(holds)g(for)f(the)257 1680 y(second)41 b(resolv)m(en)m(t)h(in)e(Eq.)g(\(A.9\).)65 b(F)-8 b(rom)39 b(the)h(calculation)g(ab)s(o)m(v)m(e,)j(w)m(e)d(reco)m(v)m(er)i(the)257 1800 y(same)34 b(estimate)g(deriv)m(ed)g(b)m(y)f(the)g Fj(p)-5 b(ower)35 b(c)-5 b(ounting)34 b(rules)8 b Fq(:)44 b(In)33 b(fact)1303 2001 y Fo(\013)1366 1960 y Fm(m)1460 2001 y Fq(=)27 b(\()p Fo(\013)1664 1960 y Fl(2)1726 2001 y Fn(\001)21 b Fo(\013)1838 1960 y Fm(m)p Fi(\000)p Fl(3)1995 2001 y Fq(\))2033 1960 y Fl(2)2072 2001 y Fq(\()p Fo(\013)2173 1960 y Fi(\000)p Fm(m)p Fl(+2)2385 2001 y Fq(\))p Fo(;)772 b Fq(\(A.11\))257 2202 y(and)34 b(the)h(righ)m(t)f(hand)g(side)h(in)f (\(A.11\))g(corresp)s(onds)h(exactly)h(to)d(the)i(factors)f(carried)257 2323 y(b)m(y)k(t)m(w)m(o)e(op)s(erator)g Fn(W)1087 2281 y Fm(m)p Fi(\000)p Fl(3)1079 2344 y Fm(m)p Fi(\000)p Fl(2)1280 2323 y Fq(and)h(a)f(resolv)m(en)m(t)i(in)f(\(A.3\),)g (according)f(to)g(Rules)h(1\))f(and)257 2443 y(2\),)d(resp)s(ectiv)m (ely)-8 b(.)257 2563 y(Therefore,)51 b(in)46 b(the)g(in)m(tegral)g(in)g (Eq.)h(\(A.6\),)i(w)m(e)d(can)g(asso)s(ciate)h(to)e(eac)m(h)i(distinct) 257 2684 y(con)m(traction)36 b(a)e(distinct)i(resolv)m(en)m(t)h(in)e (the)h(middle)g(-)e(this)i(is)f(b)s(ecause)h(of)f(the)g(c)m(hoice)257 2804 y(of)26 b(a)g(leading)h(order)g(com)m(bination)g(of)f(op)s (erators)g Fn(W)2201 2763 y Fm(m)p Fi(\000)p Fl(3)2193 2826 y Fm(m)p Fi(\000)p Fl(2)2385 2804 y Fq(and)g Fo(\016)t(g)2666 2768 y Fl(2)2662 2829 y Fm(m)p Fi(\000)p Fl(2)2845 2804 y Fq(in)h(\(A.3\).)41 b(Then,)257 2924 y(w)m(e)29 b(use)g(estimates)g (analogous)e(to)h(\(A.10\))f(for)g(the)h(con)m(tractions;)i(w)m(e)f (estimate)g(in)f(the)257 3045 y(usual)39 b(w)m(a)m(y)g(all)f(the)g(pro) 5 b(jections,)41 b(all)d(the)g(remaining)h(resolv)m(en)m(ts)h(and)e (the)g(v)m(elo)s(cit)m(y)257 3165 y(op)s(erators)30 b(next)i(to)d(them) i(\(see)h(Eq.)f(\(I)s(I)s(I.79\).)42 b(In)31 b(the)f(end,)i(b)s(ecause) f(of)f(the)h(result)g(in)257 3286 y(Eq.)g(\(A.10\),)e(w)m(e)i(get)e (the)h(same)g(order)g(in)g Fo(\013)g Fq(obtained)f(b)m(y)i(the)f(p)s(o) m(w)m(er)g(coun)m(ting)g(rules)257 3406 y(for)i(the)h(estimate)h(of)e (the)h(en)m(tire)h(expression)h(\(A.3\).)257 3526 y(After)53 b(the)g(Wic)m(k-ordering,)59 b(w)m(e)53 b(decomp)s(ose)h(the)f(iden)m (tit)m(y)i(op)s(erator)d Fe(1)3121 3541 y Fm(m)p Fi(\000)p Fl(3)3330 3526 y Fq(in)m(to)257 3647 y Fo(P)320 3662 y Fm(m)p Fi(\000)p Fl(3)508 3647 y Fq(+)31 b Fo(P)692 3611 y Fi(?)678 3671 y Fm(m)p Fi(\000)p Fl(3)881 3647 y Fq(in)46 b(fron)m(t)g(of)f(all)h(the)h(resolv)m(en)m(ts.)86 b(Then,)50 b(according)d(to)e(our)h(de\014-)257 3767 y(nitions,)39 b(the)f(v)m(ector)g Fo( )j Fq(corresp)s(onds)d(to)f(the)h (sum)f(of)g(a)g(\014nite)h(and)f Fo(\013)q Fq(-indep)s(enden)m(t)257 3888 y(n)m(um)m(b)s(er)d(of)e(expressions)k(of)c(the)h(t)m(yp)s(e)519 4046 y Fl(\026)521 4061 y Fm(j)484 4092 y Fh(Y)476 4274 y Fm(j)t Fl(=1)615 4058 y Fh(Z)661 4246 y Fl(\000)705 4255 y Fb(m)p Fd(\000)p Ff(2)863 4175 y Fo(dz)963 4124 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))p Fm(;j)959 4196 y(m)p Fi(\000)p Fl(2)1243 4058 y Fh(Z)1290 4246 y Fi(K)1344 4255 y Fb(m)p Fd(\000)p Ff(2)1495 4042 y Fl(\026)1501 4058 y Fm(i)p Fl(\()p Fm(m)p Fi(\000)p Fl(2\))1564 4092 y Fh(Y)1560 4274 y Fm(i)p Fl(=1)1749 4175 y Fo(dk)1851 4190 y Fm(i)1879 4175 y Fq(\()p Fo(m)22 b Fn(\000)h Fq(2\))17 b Fo(g)2279 4134 y Fl(1)2275 4199 y Fm(m)p Fi(\000)p Fl(3)2430 4175 y Fo(g)2481 4134 y Fl(2)2477 4199 y Fm(m)p Fi(\000)p Fl(3)2634 4175 y Fo(g)2685 4134 y Fl(1)2681 4199 y Fm(m)p Fi(\000)p Fl(3)2854 4175 y Fn(\001)g(\001)g(\001)e Fo(g)3038 4134 y Fl(1)3034 4199 y Fm(m)p Fi(\000)p Fl(3)2227 4241 y Fh(|)p 2264 4241 407 10 v 407 w({z)p 2745 4241 V 407 w(})3207 4175 y Fn(\002)-62 b Fq(\(A.12\))837 4429 y Fn(\002)17 b Fo(g)982 4388 y Fl(2)978 4454 y Fm(m)p Fi(\000)p Fl(3)931 4496 y Fh(|)p 968 4496 28 10 v 28 w({z)p 1070 4496 V 28 w(})1151 4429 y Fn(\001)g(\001)g(\001)e Fo(g)1335 4388 y Fl(1)1331 4454 y Fm(m)p Fi(\000)p Fl(3)1487 4429 y Fo(g)1538 4388 y Fl(2)1534 4454 y Fm(m)p Fi(\000)p Fl(3)1691 4429 y Fo(g)1742 4388 y Fl(1)1738 4454 y Fm(m)p Fi(\000)p Fl(3)1916 4429 y Fn(\001)22 b(\001)p Fo(g)2045 4388 y Fl(1)2041 4454 y Fm(m)p Fi(\000)p Fl(3)1284 4496 y Fh(|)p 1321 4496 383 10 v 383 w({z)p 1778 4496 V 383 w(})2197 4429 y Fq(\()p Fo(k)s(;)17 b(k)s Fq(\()p Fo(m)22 b Fn(\000)h Fq(2\))p Fo(;)17 b(z)t(;)g(z)2905 4378 y Fl(\()p Fm(m)p Fi(\000)p Fl(2\))2901 4451 y Fm(m)p Fi(\000)p Fl(2)3117 4429 y Fq(\))22 b Fn(\002)1032 4640 y(\002)p Fq(\()p Fo(\036)1205 4655 y Fm(m)p Fi(\000)p Fl(3)1362 4640 y Fq(\))1400 4599 y Fm(T)1477 4640 y Fn(\012)h Fo(C)1654 4599 y Fm(m)p Fi(\000)p Fl(3)1647 4665 y Fm(m)p Fi(\000)p Fl(2)1810 4640 y Fq(\()p Fo(k)s Fq(\()p Fo(m)g Fn(\000)f Fq(2\)\)\012)2342 4599 y Fm(m)p Fi(\000)p Fl(2)2342 4665 y Fm(m)p Fi(\000)p Fl(3)257 4841 y Fq(where)43 b(w)m(e)e(distinguish)i (the)e(new)h(momen)m(ta)g Fo(k)s Fq(\()p Fo(m)28 b Fn(\000)g Fq(2\))41 b(in)g(our)g(presen)m(t)h(notation.)257 4962 y(F)-8 b(urthermore,)40 b(notice)e(that)g(the)g(b)s(ound)g(deriv)m(ed)h (in)f(Eq.)h(\(A.10\))e(holds)i(separately)257 5082 y(when)34 b(the)f(resolv)m(en)m(t)i(is)e(decomp)s(osed)h(in)m(to)f(the)g(t)m(w)m (o)g(orthogonal)f(pieces)1098 5254 y(1)p 871 5299 502 4 v 871 5409 a Fn(\000)p Fo(z)997 5358 y Fl(\()p Fm(m)1086 5339 y Fd(0)1110 5358 y Fl(\))993 5431 y Fm(m)p Fi(\000)p Fl(2)1173 5409 y Fq(+)22 b Fn(j)p Fo(q)t Fn(j)1383 5322 y Fo(P)1446 5337 y Fm(m)p Fi(\000)p Fl(3)1700 5322 y Fo(;)2219 5254 y Fq(1)p 1851 5299 784 4 v 1851 5409 a Fo(H)1940 5368 y Fl(+)1932 5431 y Fm(m)p Fi(\000)p Fl(3)2111 5409 y Fn(\000)h Fo(z)2260 5358 y Fl(\()p Fm(m)2349 5339 y Fd(0)2373 5358 y Fl(\))2256 5431 y Fm(m)p Fi(\000)p Fl(2)2435 5409 y Fq(+)f Fn(j)p Fo(q)t Fn(j)2645 5322 y Fo(P)2722 5281 y Fi(?)2708 5346 y Fm(m)p Fi(\000)p Fl(3)2864 5322 y Fo(:)331 b Fq(\(A.13\))p eop end %%Page: 52 53 TeXDict begin 52 52 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(52)257 573 y(Then,)26 b(b)s(ecause)d(of)f (this)g(last)g(remark)h(concerning)g(the)g(op)s(erators)e(in)i(Eq.)f (\(A.13\),)i(eac)m(h)257 693 y(Expression)33 b(\(A.12\))d(has,)h(at)f (most,)h(the)g(same)g(order)f(in)h Fo(\013)g Fq(of)e(the)i(quan)m(tit)m (y)h(in)e(\(A.6\))257 814 y(and)j(Prop)s(ert)m(y)h Fn(P)8 b Fq(2)32 b(is)h(therefore)h(ful\014lled)f(for)f(the)h(expressions)j (in)c(Eq.)i(\(I)s(I)s(I.104\).)257 934 y(If)e(w)m(e)i(consider)f (\\subleading)g(com)m(binations")h(of)d(the)i(op)s(erators)f Fn(W)2866 893 y Fm(m)p Fi(\000)p Fl(3)2858 956 y Fm(m)p Fi(\000)p Fl(2)3056 934 y Fq(and)g Fo(\016)t(g)3343 898 y Fl(2)3339 959 y Fm(m)p Fi(\000)p Fl(2)257 1054 y Fq(in)d(\(A.3\),)g (similar)g(argumen)m(ts)g(apply)-8 b(.)42 b(All)29 b(the)f(other)h(c)m (hoices)h(for)d Fo(\016)t(g)2866 1018 y Fl(2)2862 1079 y Fm(m)p Fi(\000)p Fl(2)3047 1054 y Fq(and)h Fn(W)3338 1013 y Fm(m)p Fi(\000)p Fl(3)3330 1076 y Fm(m)p Fi(\000)p Fl(2)257 1175 y Fq(\(di\013eren)m(t)48 b(from)d(the)i(ones)f (underlying)i(the)e(op)s(erator)g(norm)g(estimate)h(of)e(\(A.2\)\))257 1295 y(turn)i(out)f(to)g(b)s(e)h(subleading)h(b)s(ecause)g(the)e(op)s (erators)h Fo(a)2505 1259 y Fm(\\)2536 1295 y Fq(\(\()p Fo(G)2689 1254 y Fm(m)p Fi(\000)p Fl(3)2689 1317 y Fm(m)p Fi(\000)p Fl(2)2846 1295 y Fq(\))2884 1259 y Fm(l)2910 1295 y Fq(\),)j Fo(l)j Fq(=)e(1)p Fo(;)17 b Fq(2)p Fo(;)g Fq(3,)257 1416 y(carry)33 b(further)f(p)s(ositiv)m(e)h(p)s(o)m(w)m(ers) g(of)e Fo(\013)q Fq(,)h(b)s(oth)f(directly)j(and)d(as)h(output)g(of)g (a)f(con)m(trac-)257 1536 y(tion.)p 3425 1536 4 66 v 3429 1473 59 4 v 3429 1536 V 3487 1536 4 66 v 257 1777 a Fe(Pro)s(of)38 b(of)f(Lemma)i(I)s(I)s(I.6.)257 1897 y Fq(In)f(order)g(to)f(pro)m(v)m(e)h(the)g(desired)h(result,)h(it)d(is) h(su\016cien)m(t)i(to)d(observ)m(e)i(that,)g(for)d(an)m(y)257 2017 y Fo(n)p Fq(-photon)d(v)m(ector)g(and)g(an)m(y)g Fo(h)28 b Fn(2)g Fo(L)1589 1981 y Fl(2)1629 2017 y Fq(\()p Fk(R)1739 1981 y Fl(3)1800 2017 y Fn(\002)23 b Fk(Z)1966 2032 y Fl(2)2006 2017 y Fq(\),)33 b(w)m(e)g(ha)m(v)m(e)885 2267 y Fn(k)p Fo(a)p Fq(\()p Fo(h)p Fq(\))p Fo(\036)p Fn(k)83 b Fq(=)h Fn(k)1536 2150 y Fh(Z)1635 2267 y Fo(dq)p 1733 2183 179 4 v 4 w(h)p Fq(\()p Fo(q)t Fq(\))o Fo(a)p Fq(\()p Fo(q)t Fq(\))p Fo(dq)t(\036)p Fn(k)931 b Fq(\(A.14\))1308 2485 y Fn(\024)1469 2368 y Fh(Z)1568 2485 y Fn(j)p 1596 2400 V Fo(h)p Fq(\()p Fo(q)t Fq(\))p Fn(jk)p Fo(a)p Fq(\()p Fo(q)t Fq(\))p Fo(\036)p Fn(k)p Fo(dq)993 b Fq(\(A.15\))1308 2703 y Fn(\024)84 b Fq(\()1507 2586 y Fh(Z)1606 2703 y Fn(j)p 1634 2618 V Fo(h)p Fq(\()p Fo(q)t Fq(\))p Fn(j)1841 2662 y Fl(2)1880 2703 y Fn(j)p Fo(dq)t Fq(\))2054 2635 y Ff(1)p 2053 2647 31 4 v 2053 2688 a(2)2120 2703 y Fn(\001)22 b Fq(\()2208 2586 y Fh(Z)2307 2703 y Fn(k)p Fo(a)p Fq(\()p Fo(q)t Fq(\))p Fo(\036)p Fn(k)2639 2662 y Fl(2)2678 2703 y Fo(dq)t Fq(\))2824 2635 y Ff(1)p 2823 2647 V 2823 2688 a(2)3222 2703 y Fq(\(A.16\))1309 2904 y(=)84 b Fn(k)p Fo(h)p Fn(k)1625 2920 y Fm(L)1673 2902 y Ff(2)1734 2904 y Fn(\001)21 b Fo(n)1851 2835 y Ff(1)p 1852 2847 V 1852 2889 a(2)1896 2904 y Fn(k)p Fo(\036)p Fn(k)1168 b Fq(\(A.17\))p 3425 3124 4 66 v 3429 3061 59 4 v 3429 3124 V 3487 3124 4 66 v 257 3577 a Fp(References)306 3796 y Fq([1])49 b(V.)24 b(Bac)m(h,)i(J.)f(F)-8 b(r\177)-49 b(ohlic)m(h,)26 b(and)e(A.)g(Pizzo.)30 b(Infrared-Finite)24 b(Algorithms)h(in)f(QED)g (I.)458 3916 y(The)31 b(Groundstate)f(of)g(an)g(A)m(tom)h(In)m (teracting)g(with)g(the)f(Quan)m(tized)i(Radiation)458 4037 y(Field.)44 b Fj(Comm.)33 b(Math.)i(Phys.)e Fq(2006,)f(264\(1\))f (145-165)306 4240 y([2])49 b(V.)40 b(Bac)m(h,)i(J.)e(F)-8 b(r\177)-49 b(ohlic)m(h,)43 b(and)c(A.)i(Pizzo.)66 b(An)40 b(Infrared-Finite)g(Algorithm)g(for)458 4361 y(Ra)m(yleigh)h (Scattering)h(Amplitudes,)i(and)d(Bohr's)h(F)-8 b(requency)42 b(Condition.)69 b Fj(in)458 4481 y(pr)-5 b(ep)g(ar)g(ation.)306 4684 y Fq([3])49 b(V.)28 b(Bac)m(h,)i(J.)f(F)-8 b(r\177)-49 b(ohlic)m(h,)30 b(and)e(I.)h(M.)g(Sigal.)36 b(Renormalization)29 b(group)f(analysis)i(of)458 4805 y(sp)s(ectral)e(problems)h(in)f(quan)m (tum)g(\014eld)h(theory)-8 b(.)35 b Fj(A)-5 b(dv.)30 b(in)g(Math.)h Fq(,)d(137:205{298,)458 4925 y(1998.)306 5128 y([4])49 b(V.)37 b(Bac)m(h,)j(J.)d(F)-8 b(r\177)-49 b(ohlic)m(h,)39 b(and)f(I.)f(M.)h(Sigal.)58 b(Sp)s(ectral)37 b(analysis)i(for)e(systems)j(of)458 5249 y(atoms)45 b(and)f(molecules)i (coupled)g(to)e(the)h(quan)m(tized)h(radiation)f(\014eld.)79 b Fj(Com-)458 5369 y(mun.)34 b(Math.)i(Phys.)p Fq(,)c (207\(2\):249{290,)e(1999.)p eop end %%Page: 53 54 TeXDict begin 53 53 bop 257 266 a Fr(BFP-ExpGrSt,)33 b(20-April-2006)1892 b Fq(53)306 573 y([5])49 b(C.)36 b(Hainzl,)h(R.)e(Seiringer.)54 b(Mass)37 b(Renormalization)f(and)f (Energy)i(Lev)m(el)g(Shift)458 693 y(in)c(Non-Relativistic)g(QED)43 b Fj(A)-5 b(dv.)35 b(The)-5 b(or.)34 b(Math.)h(Phys.)g Fq(,)e(6,)f(847)g(\(2002\))306 897 y([6])49 b(C.)40 b(Hainzl,)i(M.)e (Hirok)-5 b(a)m(w)m(a)40 b(and)g(H.)f(Sp)s(ohn.)65 b(Binding)40 b(energy)g(for)g(h)m(ydrogen-)458 1017 y(lik)m(e)30 b(atoms)g(in)f(the) g(Nelson)i(mo)s(del)e(without)h(cuto\013s)38 b Fj(Journal)31 b(of)h(F)-7 b(unct.)31 b(A)n(nal.)p Fq(,)458 1137 y(220\(2\):424-459)f (\(2005\))306 1341 y([7])49 b(A.)43 b(Pizzo.)76 b(One-particle)44 b(\(improp)s(er\))f(states)h(in)g(Nelson's)g(massless)i(mo)s(del.)458 1461 y Fj(A)n(nn.)34 b(H.)h(Poinc)-5 b(ar)n(\023)-47 b(e)p Fq(,)31 b(4\(3\):439{486,)g(2003.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0604301644697--