Content-Type: multipart/mixed; boundary="-------------0812290351289" This is a multi-part message in MIME format. ---------------0812290351289 Content-Type: text/plain; name="08-243.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-243.keywords" Bogolubov-Hartree-Fock Theory, Chandrasekar Limit,Neutron Stars, Attractive Interactions, BCS Theory ---------------0812290351289 Content-Type: application/postscript; name="BFJ-08-12-19.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BFJ-08-12-19.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: BFJ-08-12-19.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMSY10 CMBX10 CMR10 CMSY7 CMTT10 CMSL12 %%+ CMBX12 CMMI12 CMMI8 CMTI12 CMR8 CMEX10 EUFM10 CMSY8 MSBM10 EUFM7 %%+ CMR6 CMMI6 rsfs10 MSAM10 MSBM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o BFJ-08-12-19.ps BFJ-08-12-19 %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.12.21:1823 %%BeginProcSet: tex.pro 0 0 %! /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/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]{Ci}imagemask restore}B/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: texps.pro 0 0 %! 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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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 33 /omega put dup 35 /theta1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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%%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % 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 (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 76 /L put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly 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 26 /rho put dup 78 /N put dup 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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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 40 /parenleft put dup 41 /parenright put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % 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 (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 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 72 /H put readonly def /FontBBox{0 -250 1193 750}readonly 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 72 /H put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly 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 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 78 /N put dup 80 /P put dup 121 /dagger put readonly def /FontBBox{-30 -955 1185 779}readonly 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 65 /A put dup 70 /F put dup 72 /H put dup 77 /M put dup 81 /Q put dup 83 /S put dup 100 /d put dup 104 /h put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 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%%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. 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dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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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 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put 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 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 86 /logicalandtext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 8 /Phi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 59 /semicolon put dup 61 /equal put dup 99 /c put dup 110 /n put dup 111 /o put dup 112 /p put dup 116 /t put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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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 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /beta put dup 13 /gamma put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 33 /omega put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 66 /B put dup 70 /F put dup 72 /H put dup 75 /K put dup 77 /M put dup 78 /N put dup 81 /Q put dup 84 /T put dup 86 /V put dup 96 /lscript put dup 100 /d put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 116 /t put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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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 /Gamma put dup 5 /Pi put dup 6 /Sigma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 22 /macron put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 56 /eight put dup 57 /nine put dup 66 /B put dup 68 /D put dup 70 /F put dup 74 /J put dup 99 /c put dup 101 /e put readonly def /FontBBox{-56 -251 1102 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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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 (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 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 45 /hyphen put dup 46 /period put dup 64 /at put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{-4 -235 731 800}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 51 /three put dup 52 /four put dup 53 /five put dup 56 /eight put dup 57 /nine put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /asteriskmath put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-15 -951 1252 782}readonly 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 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 77B9F65C23E3BAB41EFAE344DDC9AB1B3CCBC0618290D83DC756F9D5BEFECB18 2DB0E39997F264D408BD076F65A50E7E94C9C88D849AB2E92005CFA316ACCD91 FF524AAD7262B10351C50EBAD08FB4CD55D2E369F6E836C82C591606E1E5C73F DE3FA3CAD272C67C6CBF43B66FE4B8677DAFEEA19288428D07FEB1F4001BAA68 7AAD6DDBE432714E799CFA49D8A1A128F32E8B280524BC8041F1E64ECE4053C4 9F0AEC699A75B827002E9F95826DB3F643338F858011008E338A899020962176 CF66A62E3AEF046D91C88C87DEB03CE6CCDF4FB651990F0E86D17409F121773D 6877DF0085DFB269A3C07AA6660419BD0F0EF3C53DA2318BA1860AB34E28BAC6 E82DDB1C43E5203AC9DF9277098F2E42C0F7BD03C6D90B629DE97730245B8E8E 8903B9225098079C55A37E4E59AE2A9E36B6349FA2C09BB1F5F4433E4EEFC75E 3F9830EB085E7E6FBE2666AC5A398C2DF228062ACF9FCA5656390A15837C4A99 EC3740D873CFEF2E248B44CA134693A782594DD0692B4DBF1F16C4CDECA692C4 0E44FDBEF704101118BC53575BF22731E7F7717934AD715AC33B5D3679B784C9 4046E6CD3C0AD80ED1F65626B14E33CFDA6EB2825DC444FA6209608D3976637A 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%!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 45 /hyphen put dup 65 /A put dup 66 /B put dup 70 /F put dup 72 /H put dup 73 /I put dup 77 /M put dup 78 /N put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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b(ably)d(,)42 b(for)c(the)i(mo)s(del)257 4524 y(of)29 b(main)g(in)m(terest)i(-)d (pseudorelativistic)k(fermions)e(in)m(teracting)g(b)m(y)g(Newtonain)f (gra)m(v-)257 4644 y(itational)37 b(forces)h(-)e(the)h(existence)j(of)c (minimizers)j(and)e(their)h(c)m(haracterization)g(has)257 4764 y(b)s(een)30 b(ac)m(hiev)m(ed)h(in)d([11].)42 b(Indeed,)31 b([11])d(pro)m(vides)j(a)d(series)i(of)e(results)i(complemen)m(ting)257 4885 y(our)j(presen)m(t)h(pap)s(er.)404 5005 y(Bogolub)s(o)m (v-Hartree-F)-8 b(o)s(c)m(k)31 b(theory)h(is)h(not)e(the)h(only)g(p)s (ossible)h(generalization)g(of)257 5126 y(Hartree-F)-8 b(o)s(c)m(k)32 b(theory)-8 b(.)44 b(The)33 b(M)s(\177)-51 b(uller)31 b(functional,)i(in)m(tro)s(duced)f(and)g(analyzed)h(in)f ([6],)257 5246 y(constitutes)38 b(an)f(in)m(teresting)h(alternativ)m(e) f(generalization)g(of)f(Hartree-F)-8 b(o)s(c)m(k)37 b(theory)-8 b(,)257 5366 y(whic)m(h)34 b(has)f(also)g(b)s(een)g(inspiring)h(for)e (the)h(presen)m(t)h(pap)s(er.)p eop end %%Page: 2 3 TeXDict begin 2 2 bop 257 266 a Fv(BFJ-19-Dec-2008)2452 b Fu(2)257 573 y Fl(I.1)136 b(The)44 b(Hartree-F)-11 b(o)t(c)l(k)45 b(Appro)l(ximation)257 758 y Fu(One)39 b(of)f(the)h(most)g(imp)s(ortan)m(t)f(appro)m(ximation)h(metho)s(ds)h (in)e(quan)m(tum)i(mec)m(hanics)257 878 y(of)29 b(man)m(y)h(particles)g (is)g(the)f(Hartree-F)-8 b(o)s(c)m(k)29 b(\(HF\))g(appro)m(ximation.)43 b(It)29 b(de\014nes)h(a)f(qual-)257 998 y(itativ)m(ely)44 b(accurate)f(theory)f(for)g(molecular)h(binding,)i(and)d(m)m(uc)m(h)i (of)d(our)h(in)m(tuitiv)m(e)257 1119 y(understanding)h(of)e(atomic)g (and)h(molecular)g(ph)m(ysics)i(is)d(based)i(on)e(it.)69 b(Its)42 b(ground)257 1239 y(state,)36 b(the)g(HF)e(ground)h(state,)h (is)f(de\014ned)i(b)m(y)e(restricting)h(the)f(Raleigh-Ritz)g(v)-5 b(aria-)257 1359 y(tional)22 b(principle)i(for)e(the)g(lo)m(w)m(est)i (eigen)m(v)-5 b(alue)24 b(of)d(the)i(Hamiltonian)f(to)g(a)g (minimization)257 1480 y(o)m(v)m(er)34 b(Slater)f(determinan)m(ts)1084 1659 y(\010)61 b(=)f Fs(')1415 1674 y 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Fj(\024)p Fr(i)g(D)s Fu(\()p Fs(\015)5 b(;)17 b(\015)5 b Fu(\))38 b(+)h Fs(D)s Fu(\()t(~)-53 b Fs(\015)t(;)21 b Fu(~)-53 b Fs(\015)5 b Fu(\))p Fs(;)647 b Fu(\(I)s(I.59\))257 1387 y(for)40 b(all)g Fs(\025)g Fq(2)g Fi(R)p Fu(,)i(whic)m(h)f(sho)m (ws)h(that)d(minimization)j(tends)f(to)e(lo)s(calize)i(the)f(spatial) 257 1507 y(supp)s(ort)j(of)f(the)h(1-p)s(dm.)73 b(By)43 b(a)g(v)-5 b(arian)m(t)42 b(of)g(Lion's)h(concen)m(tration)h (compactness,)257 1628 y(this)34 b(idea)f(is)g(implemen)m(ted)i(in)e ([11)o(])g(and)g(yields)h(the)f(existence)i(of)d(a)g(minimizer.)404 1748 y(Minimizers)27 b(of)e(energy)i(functionals)f(are)g(necessarily)i (stationary)e(p)s(oin)m(ts)g(of)f(these)257 1869 y(and)j(ful\014ll)h (the)f(corresp)s(onding)h(Euler-Lagrange)f(equations.)43 b(In)28 b(the)h(con)m(text)g(of)e(HF)257 1989 y(or)49 b(BHF)f(theory)-8 b(,)54 b(these)c(are)f(also)f(often)h(called)h Fp(self-c)-5 b(onsistent)48 b(e)-5 b(quations)p Fu(.)92 b(W)-8 b(e)257 2109 y(form)m(ulate)43 b(these)h(only)f(for)f(HF)g (theory)h(b)s(ecause)g(its)g(later)g(use)g(in)g(\(I)s(I)s(I.159\))o(.) 73 b(W)-8 b(e)257 2230 y(assume)32 b(that)d Fs(V)49 b Fq(\025)29 b Fu(0)g(is)h(relativ)m(e)h Fs(h)p Fu(-form)e(b)s(ounded)i (with)f(relativ)m(e)h(b)s(ound)f Fs(a)e(<)g Fq(1)h Fu(in)257 2350 y(the)34 b(repulsiv)m(e)i(case.)48 b(In)34 b(the)g(attractiv)m(e)h (case,)g(w)m(e)f(require)h Fs(a)30 b(<)f Fu(\()p Fs(N)34 b Fu(+)22 b(1\))3049 2314 y Fj(\000)p Fo(1)3143 2350 y Fu(,)34 b(but)g(w)m(e)257 2471 y(k)m(eep)h(in)e(mind)h(that)f(w)m(e)g (ma)m(y)h(w)m(eak)m(en)h(this)f(assumption)g(to)f Fs(a)28 b(<)g(cN)2890 2434 y Fj(\000)p Fo(2)p Fr(=)p Fo(3)3089 2471 y Fu(according)257 2591 y(to)e(the)g(remark)g(made)h(after)e (Lemma)h(1.)41 b(Then,)28 b(for)e(a)f(giv)m(en)i(1-p)s(dm)e(0)j Fq(\024)g Fs(\015)33 b Fq(\024)28 b Fu(1,)f(with)257 2711 y(T)-8 b(r)17 b Fs(\015)33 b Fu(=)27 b Fs(N)41 b Fu(and)31 b(T)-8 b(r)p Fq(f)p Fs(h\015)5 b Fq(g)28 b Fs(<)f Fq(1)p 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b(elative)27 b(b)-5 b(ound)257 3666 y Fs(a)28 b(<)g Fq(1)f Fp(in)g(the)h(r)-5 b(epulsive)27 b(c)-5 b(ase)27 b(\(+\))g(and)g Fs(a)h(<)f(N)2065 3629 y Fj(\000)p Fo(1)2188 3666 y Fp(in)g(the)h(attr) -5 b(active)27 b(c)-5 b(ase)27 b(\(-\),)i(wher)-5 b(e)257 3786 y Fs(N)39 b(>)27 b Fu(0)p Fp(.)42 b(Then,)29 b(for)e(a)h(given)f (1-p)-5 b(dm)27 b Fu(0)g Fq(\024)h Fs(\015)33 b Fq(\024)28 b Fu(1)p Fp(,)h(with)f Fu(T)-8 b(r)17 b Fs(\015)32 b Fu(=)c Fs(N)38 b Fp(and)27 b Fu(T)-8 b(r)q Fq(f)p Fs(h\015)5 b Fq(g)27 b Fs(<)h Fq(1)p Fp(,)257 3906 y(the)35 b(e\013e)-5 b(ctive)34 b(Hamiltonian)g Fs(H)8 b Fu(\()p Fs(\015)d Fu(\))35 b Fp(de\014nes)f(a)g(semib)-5 b(ounde)g(d)34 b(quadr)-5 b(atic)34 b(form.)257 4116 y(Pr)-5 b(o)g(of:)80 b Fu(W)-8 b(e)49 b(in)m(tro)s(duce)h Fs(#)55 b Fu(:=)1499 4054 y Fq(p)p 1582 4054 57 4 v 62 x Fs(\015)e Fu(and)c Fs(#)1949 4131 y Fr(z)1989 4116 y Fu(\()p Fs(x)p Fu(\))56 b(:=)f Fs(#)p Fu(\()p Fs(z)t(;)17 b(x)p Fu(\),)54 b(as)49 b(in)g(the)g(pro)s(of)f(of)257 4236 y(Lemma)34 b(1.)43 b(Then)33 b Fq(h)p Fs(')p Fq(j)17 b Fs(H)8 b Fu(\()p Fs(\015)d Fu(\))p Fs(')p Fq(i)31 b Fu(can)i(b)s(e)g(written)g(as)257 4488 y Fq(h)p Fs(')p Fq(j)27 b Fs(H)8 b Fu(\()p Fs(\015)d Fu(\))p Fs(')p Fq(i)53 b Fu(=)h Fq(h)p Fs(')p Fq(j)9 b Fs(h')p Fq(i)18 b(\006)1335 4353 y Fm(Z)1452 4378 y(n)1518 4488 y Fq(h)p Fs(')9 b Fq(\012)g Fs(#)1773 4503 y Fr(z)1814 4488 y Fq(j)28 b Fs(V)38 b Fu(\()p Fs(')9 b Fq(\012)g Fs(#)2219 4503 y Fr(z)2260 4488 y Fu(\))p Fq(i)26 b(\000)g(h)p Fs(#)2562 4503 y Fr(z)2611 4488 y Fq(\012)9 b Fs(')p Fq(j)29 b Fs(V)38 b Fu(\()p Fs(')9 b Fq(\012)g Fs(#)3167 4503 y Fr(z)3208 4488 y Fu(\))p Fq(i)3285 4378 y Fm(o)3368 4488 y Fs(dz)t(:)3222 4657 y Fu(\(I)s(I.61\))257 4777 y(By)33 b(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)k(Inequalit)m(y)-8 b(,)894 4981 y Fq(h)p Fs(')22 b Fq(\012)g Fs(#)1175 4996 y Fr(z)1216 4981 y Fq(j)27 b Fs(V)38 b Fu(\()p Fs(')22 b Fq(\012)h Fs(#)1647 4996 y Fr(z)1687 4981 y Fu(\))p Fq(i)60 b(\025)h(h)p Fs(#)2058 4996 y Fr(z)2120 4981 y Fq(\012)22 b Fs(')p Fq(j)28 b Fs(V)38 b Fu(\()p Fs(')22 b Fq(\012)g Fs(#)2714 4996 y Fr(z)2755 4981 y Fu(\))p Fq(i)p Fs(;)363 b Fu(\(I)s(I.62\))257 5185 y(and)33 b(w)m(e)h (therefore)f(ha)m(v)m(e)1235 5388 y Fq(h)p Fs(')p Fq(j)17 b Fs(H)8 b Fu(\()p Fs(\015)d Fu(\))p Fs(')p Fq(i)58 b(\025)j(h)p Fs(')p Fq(j)17 b Fs(h')p Fq(i)58 b(\025)j Fs(e)2451 5403 y Fo(0)2491 5388 y Fs(;)704 b Fu(\(I)s(I.63\))p eop end %%Page: 14 15 TeXDict begin 14 14 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(14)257 573 y(in)33 b(the)g(repulsiv)m(e)i(case)f(\(+\).)404 693 y(In)f(the)g(attractiv)m(e)g(case)g(\(-\),)f(w)m(e)i(observ)m(e)g (that)298 903 y Fq(h)p Fs(')p Fq(j)27 b Fs(H)8 b Fu(\()p Fs(\015)d Fu(\))p Fs(')p Fq(i)2442 b Fu(\(I)s(I.64\))381 1097 y Fq(\025)83 b(h)p Fs(')p Fq(j)22 b Fs(h')p Fq(i)44 b(\000)1019 962 y Fm(Z)1118 1097 y Fq(h)p Fs(')22 b Fq(\012)h Fs(#)1400 1112 y Fr(z)1440 1097 y Fq(j)k Fs(V)39 b Fu(\()p Fs(')21 b Fq(\012)i Fs(#)1871 1112 y Fr(z)1911 1097 y Fu(\))p Fq(i)17 b Fs(dz)381 1352 y Fq(\025)83 b(h)p Fs(')p Fq(j)22 b Fs(h')p Fq(i)44 b(\000)h Fs(a)1103 1216 y Fm(Z)1219 1241 y(D)1280 1352 y Fs(')22 b Fq(\012)h Fs(#)1523 1367 y Fr(z)1563 1237 y Fm(\014)1563 1297 y(\014)1563 1357 y(\014)1624 1352 y Fq(f)p Fu(\()p Fs(h)f Fu(+)g Fs(b)p Fu(\))h Fq(\012)f Fn(1)g Fu(+)g Fn(1)h Fq(\012)f Fu(\()p Fs(h)h Fu(+)f Fs(b)p Fu(\))p Fq(g)17 b Fu(\()p Fs(')k Fq(\012)i Fs(#)3084 1367 y Fr(z)3124 1352 y Fu(\))3162 1241 y Fm(E)3240 1352 y Fs(dz)381 1545 y Fq(\025)83 b(\000)p Fs(b)46 b Fu(+)e(\(1)22 b Fq(\000)g Fs(aN)10 b Fu(\))17 b Fq(h)p Fs(')p Fq(j)22 b Fu(\()p Fs(h)g Fu(+)g Fs(b)p Fu(\))p Fs(')p Fq(i)45 b(\000)f Fs(a)34 b Fu(T)-8 b(r)p Fq(f)p Fu(\()p Fs(h)22 b Fu(+)g Fs(b)p Fu(\))p Fs(\015)5 b Fq(g)381 1691 y(\025)83 b(\000)p Fs(b)46 b Fq(\000)e Fs(a)34 b Fu(T)-8 b(r)p Fq(f)p Fu(\()p Fs(h)22 b Fu(+)g Fs(b)p Fu(\))p Fs(\015)5 b Fq(g)p Fs(:)1890 b Fd(\003)404 1901 y Fu(W)-8 b(e)24 b(remark)g(that)f(m)m(uc)m(h)i(of)e(the)h (conclusion)h(of)e(Lieb's)h(v)-5 b(ariational)23 b(principle)i(\(I.8\)) 257 2021 y(holds)47 b(true)e(for)g(general)h(p)s(oten)m(tials)h (\(repulsiv)m(e,)k(attractiv)m(e,)e(or)d(ev)m(en)h(inde\014nite\))257 2142 y(under)41 b(the)g(additional)f(assumption)h(of)e(existence)k(of)c (a)h(minimizer,)k(i.e.,)e(that)e(the)257 2262 y(in\014m)m(um)46 b(on)e(the)g(righ)m(t)g(side)h(of)50 b(\(I.8\))44 b(is)h(actually)f(a)g (minim)m(um.)79 b(W)-8 b(e)44 b(pro)m(v)m(e)i(this)257 2382 y(assertion)34 b(in)f(Lemma)g(3)f(b)s(elo)m(w.)257 2577 y Fn(Lemma)39 b(3)49 b Fp(L)-5 b(et)48 b Fs(N)61 b Fq(2)51 b Fi(N)d Fp(and)e Fs(\015)1568 2592 y Fo(0)1655 2577 y Fp(b)-5 b(e)47 b(a)g(minimizer)f(of)h Fq(E)2541 2592 y Fr(H)5 b(F)2709 2577 y Fp(with)47 b Fu(T)-8 b(r)q(\()p Fs(\015)3123 2592 y Fo(0)3162 2577 y Fu(\))50 b(=)h Fs(N)10 b Fp(.)257 2698 y(Then)39 b(the)h(e\013e)-5 b(ctive)39 b(Hamiltonian)g Fs(H)8 b Fu(\()p Fs(\015)1810 2713 y Fo(0)1848 2698 y Fu(\))40 b Fp(has)f(at)h(le)-5 b(ast)39 b Fs(N)50 b Fp(eigenvalues)39 b(\(c)-5 b(ounting)257 2818 y(multiplicities\))42 b(b)-5 b(elow)42 b(its)h(c)-5 b(ontinuous)42 b(sp)-5 b(e)g(ctrum,)45 b(and)d Fs(\015)2493 2833 y Fo(0)2574 2818 y Fp(ne)-5 b(c)g(essarily)42 b(ful\014l)5 b(ls)42 b(the)257 2939 y(self-c)-5 b(onsistent)34 b(e)-5 b(quation)1109 3149 y Fn(1)1165 3164 y Fr(N)7 b Fj(\000)p Fo(1)1323 3068 y Fm(\002)1364 3149 y Fs(H)h Fu(\()p Fs(\015)1542 3164 y Fo(0)1581 3149 y Fu(\))1619 3068 y Fm(\003)1723 3149 y Fq(\024)63 b Fs(\015)1914 3164 y Fo(0)2015 3149 y Fq(\024)g Fn(1)2211 3164 y Fr(N)2279 3068 y Fm(\002)2320 3149 y Fs(H)8 b Fu(\()p Fs(\015)2498 3164 y Fo(0)2537 3149 y Fu(\))2575 3068 y Fm(\003)2616 3149 y Fs(;)579 b Fu(\(I)s(I.65\))257 3359 y Fp(wher)-5 b(e)34 b Fn(1)588 3374 y Fr(k)631 3359 y Fu([)p Fs(H)8 b Fu(\()p Fs(\015)836 3374 y Fo(0)875 3359 y Fu(\)])34 b Fp(denotes)g(the)g(pr)-5 b(oje)g(ction)34 b(onto)g(the)h Fs(k)j Fp(lowest)c(eigenvalues)f Fs(H)8 b Fu(\()p Fs(\015)3389 3374 y Fo(0)3427 3359 y Fu(\))p Fp(.)257 3554 y(Pr)-5 b(o)g(of:)52 b Fu(Let)33 b(0)c Fq(\024)h Fs(\015)977 3569 y Fo(0)1045 3554 y Fq(\024)f Fu(1)k(b)s(e)h(a)f(minimizer)i(and)e(0)c Fq(\024)h Fs(\015)2331 3569 y Fo(1)2399 3554 y Fq(\024)f Fu(1)k(b)s(e)h(some)g(other)g(1-p)s (dm)257 3674 y(of)c(trace)h(T)-8 b(r)q Fq(f)p Fs(\015)810 3689 y Fo(1)848 3674 y Fq(g)28 b Fu(=)f(T)-8 b(r)q Fq(f)p Fs(\015)1231 3689 y Fo(0)1270 3674 y Fq(g)27 b Fu(=)h Fs(N)10 b Fu(.)43 b(Set)31 b Fs(\015)1826 3689 y Fr(\025)1898 3674 y Fu(:=)d(\(1)18 b Fq(\000)g Fs(\025)p Fu(\))p Fs(\015)2375 3689 y Fo(0)2432 3674 y Fu(+)g Fs(\025\015)2634 3689 y Fo(1)2673 3674 y Fu(,)31 b(so)f(0)e Fq(\024)g Fs(\015)3081 3689 y Fr(\025)3154 3674 y Fq(\024)g Fu(1)i(and)257 3795 y(T)-8 b(r)q Fq(f)p Fs(\015)459 3810 y Fr(\025)503 3795 y Fq(g)28 b Fu(=)f Fs(N)10 b Fu(,)34 b(for)e(all)g(0)27 b Fq(\024)i Fs(\025)e Fq(\024)h Fu(1.)44 b(An)32 b(easy)i(computation)f (yields)307 4005 y Fq(E)360 4020 y Fr(H)5 b(F)481 4005 y Fu(\()p Fs(\015)570 4020 y Fr(\025)615 4005 y Fu(\))15 b Fq(\000)g(E)813 4020 y Fr(H)5 b(F)936 4005 y Fu(\()p Fs(\015)1025 4020 y Fo(0)1064 4005 y Fu(\))56 b(=)h Fs(\025)33 b Fu(T)-8 b(r)1498 3924 y Fm(\010)1556 4005 y Fs(H)1637 4020 y Fo(0)1693 4005 y Fu(\()p Fs(\015)1782 4020 y Fo(1)1836 4005 y Fq(\000)15 b Fs(\015)1979 4020 y Fo(0)2019 4005 y Fu(\))2057 3924 y Fm(\011)2146 4005 y Fu(+)31 b Fs(\025)2310 3964 y Fo(2)2365 4005 y Fs(Q)p Fu([)p Fs(\015)2520 4020 y Fo(1)2575 4005 y Fq(\000)15 b Fs(\015)2718 4020 y Fo(0)2758 4005 y Fs(;)44 b(\015)2880 4020 y Fo(1)2935 4005 y Fq(\000)15 b Fs(\015)3078 4020 y Fo(0)3118 4005 y Fu(])p Fs(;)50 b Fu(\(I)s(I.66\))257 4215 y(where)34 b(w)m(e)g(abbreviate)f Fs(H)1243 4230 y Fo(0)1310 4215 y Fu(:=)28 b Fs(H)8 b Fu(\()p Fs(\015)1619 4230 y Fo(0)1657 4215 y Fu(\))33 b(and)g(use)410 4474 y Fs(Q)p Fu([)p Fs(\021)t(;)17 b(\024)p Fu(])60 b(:=)917 4338 y Fm(Z)1033 4393 y(\010)1091 4474 y Fs(\021)t Fu(\()p Fs(x;)17 b(x)p Fu(\))g Fs(\024)p Fu(\()p Fs(y)t(;)g(y)t Fu(\))42 b Fq(\000)j Fs(\021)t Fu(\()p Fs(x;)17 b(y)t Fu(\))g Fs(\024)p Fu(\()p Fs(y)t(;)g(x)p Fu(\))2413 4393 y Fm(\011)2469 4474 y Fs(W)d Fu(\()p Fs(x;)j(y)t Fu(\))g Fs(dx)g(dy)t(:)150 b Fu(\(I)s(I.67\))257 4727 y(Since)34 b Fs(\015)563 4742 y Fo(0)635 4727 y Fu(is)f(a)f(minimizer,)i(this)g(yields)1312 4937 y(T)-8 b(r)1429 4856 y Fm(\010)1487 4937 y Fs(H)1568 4952 y Fo(0)1629 4937 y Fs(\015)1680 4952 y Fo(0)1719 4856 y Fm(\011)1838 4937 y Fq(\024)60 b Fu(T)-8 b(r)2092 4856 y Fm(\010)2150 4937 y Fs(H)2231 4952 y Fo(0)2293 4937 y Fs(\015)2344 4952 y Fo(1)2383 4856 y Fm(\011)3222 4937 y Fu(\(I)s(I.68\))257 5147 y(b)m(y)38 b(taking)e(the)h(limit)g Fs(\025)d Fq(!)f Fu(0.)55 b(No)m(w)37 b(let)g(\006)d(:=)g(inf)24 b Fs(\033)2267 5162 y Fo(con)n(t)2403 5147 y Fu([)p Fs(H)2511 5162 y Fo(0)2550 5147 y Fu(])36 b(and)h Fs(e)2852 5162 y Fo(1)2926 5147 y Fq(\024)d Fs(e)3082 5162 y Fo(2)3156 5147 y Fq(\024)g(\001)17 b(\001)g(\001)33 b(\024)257 5268 y Fs(e)302 5283 y Fr(M)418 5268 y Fs(<)k Fu(\006)i(b)s(e)f(the)g (eigen)m(v)-5 b(alues)40 b(of)d Fs(H)1658 5283 y Fo(0)1736 5268 y Fu(\(coun)m(ting)h(m)m(ultiplicities\))j(strictly)e(b)s(elo)m(w) g(\006)257 5388 y(\(p)s(ossibly)-8 b(,)34 b Fs(M)39 b Fu(=)27 b(0)33 b(or)f Fs(M)38 b Fu(=)28 b Fq(1)p Fu(\).)p eop end %%Page: 15 16 TeXDict begin 15 15 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(15)404 573 y(First)28 b(assume)i(that)e Fs(M)39 b(<)27 b(N)10 b Fu(.)43 b(Since)30 b Fs(e)1867 588 y Fr(M)1973 573 y Fs(<)e Fu(\006)h(and)f(T)-8 b(r)q Fq(f)p Fs(\015)2563 588 y Fo(0)2601 573 y Fq(g)28 b Fu(=)g Fs(N)38 b(>)27 b(M)10 b Fu(,)30 b(w)m(e)g(ha)m(v)m(e)892 886 y Fs(\016)64 b Fu(:=)c(T)-8 b(r)1279 805 y Fm(\010)1337 886 y Fu(\()p Fs(H)1456 901 y Fo(0)1517 886 y Fq(\000)23 b Fu(\006\))17 b Fs(\015)1793 901 y Fo(0)1832 805 y Fm(\011)1934 886 y Fq(\000)2095 761 y Fr(M)2061 791 y Fm(X)2056 1000 y Fr(m)p Fo(=1)2209 886 y Fu(\()p Fs(e)2292 901 y Fr(m)2381 886 y Fq(\000)22 b Fu(\006\))61 b Fs(>)f Fu(0)p Fs(:)361 b Fu(\(I)s(I.69\))257 1193 y(By)24 b(the)g(de\014nition)g(of)e(the)i (con)m(tin)m(uous)h(sp)s(ectrum,)h Fs(\015)2219 1208 y Fo(1)2281 1193 y Fu(can)e(b)s(e)f(c)m(hosen)i(as)e Fn(1)3043 1208 y Fr(M)3122 1193 y Fu([)p Fs(H)3230 1208 y Fo(0)3269 1193 y Fu(])s(+)s Fs(\015)3434 1157 y Fj(0)3429 1217 y Fo(1)3468 1193 y Fu(,)257 1313 y(where)36 b Fs(\015)597 1277 y Fj(0)592 1338 y Fo(1)666 1313 y Fu(is)f(a)f(rank)h Fs(N)f Fq(\000)24 b Fs(M)45 b Fu(pro)5 b(jection)36 b(with)f Fs(\015)2174 1277 y Fj(0)2169 1338 y Fo(1)2239 1313 y Fn(1)2295 1328 y Fr(M)2374 1313 y Fu([)p Fs(H)2482 1328 y Fo(0)2521 1313 y Fu(])c(=)g Fn(1)2742 1328 y Fr(M)2821 1313 y Fu([)p Fs(H)2929 1328 y Fo(0)2969 1313 y Fu(])g Fs(\015)3083 1277 y Fj(0)3078 1338 y Fo(1)3148 1313 y Fu(=)g(0)j(and)257 1434 y(0)28 b Fq(\024)g Fu(T)-8 b(r)p Fq(f)p Fu(\()p Fs(H)708 1449 y Fo(0)769 1434 y Fq(\000)23 b Fu(\006\))17 b Fs(\015)1050 1397 y Fj(0)1045 1458 y Fo(1)1084 1434 y Fq(g)28 b(\024)g Fs(\016)t(=)p Fu(2.)43 b(Th)m(us)603 1654 y(T)-8 b(r)720 1573 y Fm(\010)778 1654 y Fu(\()p Fs(H)897 1669 y Fo(0)958 1654 y Fq(\000)23 b Fu(\006\))f Fs(\015)1239 1669 y Fo(0)1278 1573 y Fm(\011)1381 1654 y Fq(\000)44 b Fu(T)-8 b(r)1619 1573 y Fm(\010)1677 1654 y Fu(\()p Fs(H)1796 1669 y Fo(0)1858 1654 y Fq(\000)22 b Fu(\006\))h Fs(\015)2139 1669 y Fo(1)2178 1573 y Fm(\011)687 1901 y Fu(=)83 b(T)-8 b(r)963 1821 y Fm(\010)1021 1901 y Fu(\()p Fs(H)1140 1916 y Fo(0)1202 1901 y Fq(\000)22 b Fu(\006\))h Fs(\015)1483 1916 y Fo(0)1522 1821 y Fm(\011)1624 1901 y Fq(\000)1781 1777 y Fr(M)1746 1807 y Fm(X)1751 2016 y Fr(n)p Fo(=1)1890 1901 y Fu(\()p Fs(e)1973 1916 y Fr(n)2042 1901 y Fq(\000)g Fu(\006\))44 b Fq(\000)h Fu(T)-8 b(r)2533 1821 y Fm(\010)2591 1901 y Fu(\()p Fs(H)2710 1916 y Fo(0)2772 1901 y Fq(\000)22 b Fu(\006\))h Fs(\015)3058 1860 y Fj(0)3053 1926 y Fo(1)3092 1821 y Fm(\011)686 2196 y Fq(\025)857 2128 y Fs(\016)p 856 2173 49 4 v 856 2264 a Fu(2)975 2196 y Fs(>)60 b Fu(0)p Fs(;)2035 b Fu(\(I)s(I.70\))257 2441 y(whic)m(h)34 b(implies)h(that)663 2661 y(T)-8 b(r)780 2581 y Fm(\010)838 2661 y Fs(H)919 2676 y Fo(0)981 2661 y Fs(\015)1032 2676 y Fo(0)1071 2581 y Fm(\011)1212 2661 y Fu(=)83 b(T)-8 b(r)1488 2581 y Fm(\010)1546 2661 y Fu(\()p Fs(H)1665 2676 y Fo(0)1726 2661 y Fq(\000)23 b Fu(\006\))f Fs(\015)2007 2676 y Fo(0)2046 2581 y Fm(\011)2127 2661 y Fu(+)g(\006)17 b Fs(N)832 b Fu(\(I)s(I.71\))1212 2807 y Fs(>)83 b Fu(T)-8 b(r)1488 2726 y Fm(\010)1546 2807 y Fu(\()p Fs(H)1665 2822 y Fo(0)1726 2807 y Fq(\000)23 b Fu(\006\))f Fs(\015)2007 2822 y Fo(1)2046 2726 y Fm(\011)2127 2807 y Fu(+)g(\006)17 b Fs(N)70 b Fu(=)61 b(T)-8 b(r)2713 2726 y Fm(\010)2771 2807 y Fs(H)2852 2822 y Fo(0)2914 2807 y Fs(\015)2965 2822 y Fo(1)3004 2726 y Fm(\011)3062 2807 y Fs(;)257 3027 y Fu(con)m(tradicting)38 b(\(I)s(I.68\))o(.)51 b(Therefore,)37 b Fs(M)43 b Fq(\025)32 b Fs(N)46 b Fu(and)35 b Fs(e)2285 3042 y Fr(N)2384 3027 y Fs(<)d Fu(\006,)k(whic)m(h,)i(in)d (particular,)257 3147 y(implies)1405 3355 y(T)-8 b(r)1522 3274 y Fm(\010)1580 3355 y Fs(H)1661 3370 y Fo(0)1722 3355 y Fs(\015)1773 3370 y Fo(0)1812 3274 y Fm(\011)1930 3355 y Fq(\025)2109 3230 y Fr(N)2068 3260 y Fm(X)2074 3469 y Fr(n)p Fo(=1)2229 3355 y Fs(e)2274 3370 y Fr(n)2321 3355 y Fs(:)874 b Fu(\(I)s(I.72\))257 3616 y(Note)33 b(that)f(Inequalit)m(y)k(\(I)s(I.72\))d(is)g(strict)g(unless)549 3836 y Fs(\015)600 3851 y Fo(0)699 3836 y Fu(=)60 b Fn(1)p Fu([)p Fs(H)999 3851 y Fo(0)1066 3836 y Fs(<)28 b(e)1215 3851 y Fr(N)1282 3836 y Fu(])22 b(+)k(~)-53 b Fs(\015)1480 3851 y Fo(0)1519 3836 y Fs(;)212 b Fu(Ran)20 b(~)-52 b Fs(\015)2001 3851 y Fo(0)2100 3836 y Fq(\022)61 b Fu(Ran)16 b Fn(1)p Fu([)p Fs(H)2593 3851 y Fo(0)2660 3836 y Fu(=)28 b Fs(e)2809 3851 y Fr(N)2876 3836 y Fu(])p Fs(;)292 b Fu(\(I)s(I.73\))257 4056 y(for)32 b(whic)m(h)j(it)d(b)s(ecomes)i(an)f (equalit)m(y)-8 b(.)1755 b Fd(\003)257 4345 y Fl(I)t(I.1)135 b(The)45 b(Bogolub)t(o)l(v-Hartree-F)-11 b(o)t(c)l(k)46 b(Appro)l(ximation)600 4494 y(for)f(Repulsiv)l(e)i(P)l(oten)l(tials)257 4679 y Fu(W)-8 b(e)36 b(b)s(egin)f(this)g(section)h(b)m(y)g(de\014ning) g(the)f(Bogolub)s(o)m(v-Hartree-F)-8 b(o)s(c)m(k)35 b(\(BHF\))f(func-) 257 4799 y(tional)382 5019 y Fq(E)435 5034 y Fr(B)s(H)5 b(F)612 5019 y Fu(\(\000\))60 b(:=)2310 b(\(I)s(I.74\))477 5213 y(T)-8 b(r)p Fq(f)p Fs(h)17 b(\015)5 b Fq(g)44 b Fu(+)980 5146 y(1)p 980 5190 V 980 5281 a(2)1055 5078 y Fm(Z)1171 5103 y(n)1238 5213 y Fs(\032)1288 5228 y Fr(\015)1332 5213 y Fu(\()p Fs(x)p Fu(\))p Fs(\032)1513 5228 y Fr(\015)1559 5213 y Fu(\()p Fs(y)t Fu(\))21 b Fq(\000)h(j)p Fs(\015)5 b Fu(\()p Fs(x;)17 b(y)t Fu(\))p Fq(j)2146 5172 y Fo(2)2206 5213 y Fu(+)22 b Fq(j)p Fs(\013)q Fu(\()p Fs(x;)17 b(y)t Fu(\))p Fq(j)2650 5172 y Fo(2)2688 5103 y Fm(o)2771 5213 y Fs(W)d Fu(\()p Fs(x;)j(y)t Fu(\))g Fs(dx)g(dy)t(;)p eop end %%Page: 16 17 TeXDict begin 16 16 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(16)257 573 y(with)37 b(\000)c(=)g(\000\()p Fs(\015)5 b(;)17 b(\013)q Fu(\))32 b Fq(2)h Fk(A)p Fu(,)k(suc)m(h)h(that)d(all)h (terms)g(on)g(the)g(righ)m(t)g(side)h(exist,)h(and)e(then)257 693 y(the)d(BHF)g(energy)g(b)m(y)1025 913 y Fs(E)1097 928 y Fr(B)s(H)5 b(F)1275 913 y Fu(\()p Fs(N)10 b Fu(\))61 b(:=)f(inf)1797 833 y Fm(\010)1856 913 y Fq(E)1909 928 y Fr(B)s(H)5 b(F)2086 913 y Fu(\(\000\))2256 829 y Fm(\014)2256 888 y(\014)2321 913 y Fu(\000)28 b Fq(2)g Fk(A)2575 928 y Fr(N)2643 833 y Fm(\011)2701 913 y Fs(:)494 b Fu(\(I)s(I.75\))257 1133 y(Note)33 b(that,)g(due)g(to)f(\(I.36\),)g(w)m(e)i(ha)m(v)m(e)g (that)1447 1353 y Fs(!)t Fu(\()p Fi(H)p Fu(\))60 b(=)g Fq(E)1914 1368 y Fr(B)s(H)5 b(F)2091 1353 y Fu(\(\000)2190 1368 y Fr(!)2241 1353 y Fu(\))p Fs(;)916 b Fu(\(I)s(I.76\))257 1573 y(for)32 b(an)m(y)i(quasifree)g(state)f Fs(!)e Fq(2)d Fk(Q)p Fu(,)33 b(and)f(hence)1106 1793 y Fs(E)1178 1808 y Fr(B)s(H)5 b(F)1356 1793 y Fu(\()p Fs(N)10 b Fu(\))61 b(=)f(inf)1852 1713 y Fm(\010)1910 1793 y Fs(!)t Fu(\()p Fi(H)p Fu(\))2160 1709 y Fm(\014)2160 1768 y(\014)2226 1793 y Fs(!)31 b Fq(2)d Fk(Q)2494 1808 y Fr(N)2561 1713 y Fm(\011)2620 1793 y Fs(:)575 b Fu(\(I)s(I.77\))257 2013 y(If)33 b Fs(W)46 b Fu(is)33 b(repulsiv)m(e,)i(i.e.,)e Fs(W)14 b Fu(\()p Fs(x;)j(y)t Fu(\))27 b Fq(\025)h Fu(0,)33 b(for)f(all)g Fs(x;)17 b(y)31 b Fq(2)d Fs(M)10 b Fu(,)33 b(then)h(w)m(e)f(observ)m(e)i(that)471 2282 y Fq(E)524 2297 y Fr(B)s(H)5 b(F)701 2201 y Fm(\000)747 2282 y Fu(\000\()p Fs(\015)g(;)17 b(\013)q Fu(\))1047 2201 y Fm(\001)1176 2282 y Fu(=)83 b Fq(E)1388 2297 y Fr(B)s(H)5 b(F)1566 2201 y Fm(\000)1612 2282 y Fu(\000\()p Fs(\015)g(;)17 b Fu(0\))1898 2201 y Fm(\001)1987 2282 y Fu(+)2117 2214 y(1)p 2117 2259 49 4 v 2117 2350 a(2)2193 2146 y Fm(Z)2309 2282 y Fq(j)p Fs(\013)q Fu(\()p Fs(x;)g(y)t Fu(\))p Fq(j)2655 2241 y Fo(2)2709 2282 y Fs(W)d Fu(\()p Fs(x;)j(y)t Fu(\))g Fs(dx)g(dy)1175 2485 y Fq(\025)83 b(E)1388 2500 y Fr(B)s(H)5 b(F)1566 2405 y Fm(\000)1612 2485 y Fu(\000\()p Fs(\015)g(;)17 b Fu(0\))1898 2405 y Fm(\001)1176 2631 y Fu(=)83 b Fq(E)1388 2646 y Fr(H)5 b(F)1510 2631 y Fu(\()p Fs(\015)g Fu(\))p Fs(;)1553 b Fu(\(I)s(I.78\))257 2851 y(whic)m(h)34 b(implies)h(that)611 3071 y Fs(E)683 3086 y Fr(B)s(H)5 b(F)861 3071 y Fu(\()p Fs(N)10 b Fu(\))61 b(=)f(inf)1356 2990 y Fm(\010)1414 3071 y Fq(E)1467 3086 y Fr(B)s(H)5 b(F)1645 3071 y Fu(\(\000\))1814 2986 y Fm(\014)1814 3046 y(\014)1880 3071 y Fu(\000)28 b Fq(2)g Fk(A)2134 3020 y Fo(\(0\))2134 3098 y Fr(N)2229 2990 y Fm(\011)2347 3071 y Fu(=)60 b Fs(E)2555 3086 y Fr(H)5 b(F)2677 3071 y Fu(\()p Fs(N)10 b Fu(\))p Fs(:)354 b Fu(\(I)s(I.79\))257 3291 y(In)43 b(other)f(w)m(ords,)k Fp(BHF)d(ener)-5 b(gy)43 b(and)g(HF)h(ener)-5 b(gy)43 b(agr)-5 b(e)g(e)43 b(for)h(r)-5 b(epulsive)43 b(p)-5 b(otentials)p Fu(,)257 3411 y(and)28 b(the)h(v)-5 b(ariation)27 b(o)m(v)m(er)i(more)g(general)f(quasifree)i(states)e(than)g(the)h (gauge)f(in)m(v)-5 b(arian)m(t)257 3531 y(ones)34 b(is)f(obsolete.)p eop end %%Page: 17 18 TeXDict begin 17 17 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(17)257 573 y Ft(I)t(I)t(I)163 b(The)53 b(BHF)i(Appro)l(ximation) 608 756 y(for)f(A)l(ttractiv)l(e)e(P)l(oten)l(tials)257 975 y Fu(The)c(main)f(purp)s(ose)h(of)e(this)h(pap)s(er)g(is)g(to)g (discuss)i(the)e(BHF)f(appro)m(ximation)i(in)257 1095 y(case)36 b(of)e Fp(attr)-5 b(active)42 b Fu(pair)34 b(p)s(oten)m(tials.)50 b(It)34 b(turns)h(out)f(that)h(our)f(result)h (requires)h(some)257 1215 y(additional)d(assumptions.)404 1336 y(Besides)38 b(the)f(attractiv)m(eness)i(of)c(the)i(p)s(oten)m (tial,)h(a)e(second)i(imp)s(ortan)m(t)e(assump-)257 1456 y(tion)22 b(is)h(that)f(the)g(fermions)h(in)f(the)h(system)g(under)g (consideration)g(are)f(spin-)3069 1417 y Fo(1)p 3069 1433 36 4 v 3069 1490 a(2)3137 1456 y Fu(particles)257 1594 y(\(e.g.,)27 b(neutrons\).)42 b(That)26 b(is,)h(w)m(e)f(assume)h (the)e(one-particle)h(Hilb)s(ert)f(space)h Fk(h)i Fu(=)3242 1568 y(^)3241 1594 y Fk(h)6 b Fq(\012)h Fi(C)3455 1558 y Fo(2)257 1714 y Fu(of)29 b(a)f(spin-)658 1675 y Fo(1)p 658 1691 V 658 1749 a(2)704 1714 y Fu(,)h(where)1039 1688 y(^)1038 1714 y Fk(h)f Fu(is)i(again)e(assumed)i(to)e(b)s(e)h(a)g (Hilb)s(ert)g(space)g Fs(L)2832 1678 y Fo(2)2872 1714 y Fu(\()p Fs(M)10 b Fu(\))29 b(of)f(square-)257 1835 y(in)m(tegrable)35 b(functions)f(on)f(a)g(con\014guration)h(space)g Fs(M)10 b Fu(.)46 b(Under)35 b(this)e(assumption)i(w)m(e)257 1955 y(essen)m(tially)j(pro)m(v)m(e)e(that)f(the)h(minimizing)g(state)g (is)f(a)g(spin)h(singlet)g(state,)g(i.e.,)h(it)e(al-)257 2075 y(lo)m(ws)29 b(us)g(to)e(mo)m(v)m(e)j(the)e(an)m(tisymmetry)j (requiremen)m(t)f(of)d(the)i(v)-5 b(ariational)27 b(state)h(under)257 2196 y(consideration)34 b(en)m(tirely)g(to)f(its)g(spin)g(part;)g(see)g (Eqs.)h(\(I)s(I)s(I.88\))e(and)h(\(I)s(I)s(I.97\))o(.)404 2316 y(Ev)m(en)i(though)f(w)m(e)h(assume)h(the)e(fermions)h(to)f(ha)m (v)m(e)h(spin)2595 2277 y Fo(1)p 2595 2293 V 2595 2351 a(2)2640 2316 y Fu(,)f(b)s(oth)g(the)g(one-b)s(o)s(dy)257 2437 y(Hamiltonian)25 b Fs(h)j Fu(:)g Fk(h)f Fq(!)h Fk(h)c Fu(and)h(the)g(t)m(w)m(o-b)s(o)s(dy)g(in)m(teraction)g Fs(W)41 b Fu(=)28 b Fq(\000)p Fs(V)50 b Fu(:)27 b Fk(h)6 b Fq(\012)g Fk(h)28 b Fq(!)f Fk(h)6 b Fq(\012)g Fk(h)257 2557 y Fu(\(note)34 b(the)f(min)m(us)i(sign,)f Fs(V)50 b(>)28 b Fu(0\))33 b(are)g(supp)s(osed)i(to)d(b)s(e)i(diagonal)e(in)i (spin)g(space)g(and)257 2677 y(real.)46 b(More)33 b(precisely)-8 b(,)36 b(w)m(e)e(require)g(that)f Fs(h)c Fu(:=)2060 2651 y(^)2059 2677 y Fs(h)23 b Fq(\012)g Fn(1)2294 2694 y Fc(C)2342 2675 y Fg(2)2380 2677 y Fu(,)34 b(where)2724 2651 y(^)2723 2677 y Fs(h)29 b Fu(:)2865 2651 y(^)2864 2677 y Fk(h)f Fq(!)3074 2651 y Fu(^)3072 2677 y Fk(h)34 b Fu(is)f(a)g(real)257 2798 y(quadratic)f(form)e(de\014ned)i(on)e(a)g (dense)i(domain)2075 2779 y(^)2075 2798 y Fk(d)17 b Fq(\002)2238 2779 y Fu(^)2238 2798 y Fk(d)27 b Fq(\022)2422 2772 y Fu(^)2420 2798 y Fk(h)18 b Fq(\002)2587 2772 y Fu(^)2585 2798 y Fk(h)31 b Fu(and)f(b)s(ounded)h(b)s(elo)m(w)257 2918 y(b)m(y)689 3038 y Fs(e)734 3053 y Fo(0)834 3038 y Fu(:=)60 b(inf)1132 2958 y Fm(\010)1190 3038 y Fq(h)p Fs(')p Fq(j)1338 3012 y Fu(^)1338 3038 y Fs(h)16 b(')p Fq(i)1540 2954 y Fm(\014)1540 3014 y(\014)1606 3038 y Fs(')27 b Fq(2)1792 3019 y Fu(^)1791 3038 y Fk(d)p Fs(;)82 b Fq(k)p Fs(')p Fq(k)27 b Fu(=)h(1)2294 2958 y Fm(\011)2412 3038 y Fs(>)60 b Fq(\0001)p Fs(:)432 b Fu(\(I)s(I)s(I.80\))404 3227 y(Similarly)-8 b(,)52 b(w)m(e)c(assume)h(that)f(there)g(is)g(a)f (family)h Fq(f)2453 3202 y Fm(b)2429 3227 y Fs(X)2510 3242 y Fr(\021)2551 3227 y Fq(g)2601 3242 y Fr(\021)r Fj(2N)2811 3227 y Fq(\022)53 b(B)s Fu(\()3049 3201 y(^)3047 3227 y Fk(h)q Fu(\))47 b(of)g(real,)257 3347 y(orthogonal)27 b(pro)5 b(jections)30 b(and)d(a)h(p)s(ositiv)m(e)h(measure)g Fs(\026)f Fu(on)g(a)f(measure)i(space)g Fq(N)15 b Fu(,)29 b(suc)m(h)257 3468 y(that)1329 3612 y Fs(V)82 b Fu(=)1604 3476 y Fm(Z)1659 3702 y Fj(N)1770 3612 y Fs(X)1851 3627 y Fr(\021)1915 3612 y Fq(\012)22 b Fs(X)2095 3627 y Fr(\021)2159 3612 y Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(:)760 b Fu(\(I)s(I)s(I.81\))257 3855 y(where)26 b Fs(X)612 3870 y Fr(\021)681 3855 y Fu(:=)837 3830 y Fm(b)812 3855 y Fs(X)893 3870 y Fr(\021)939 3855 y Fq(\012)5 b Fn(1)1077 3872 y Fc(C)1125 3853 y Fg(2)1164 3855 y Fu(.)41 b(The)25 b(class)g(of)f(in)m(teraction)h(p)s(oten)m(tials)f(sp)s(eci\014ed)i(b)m (y)h(\(I)s(I)s(I.81\))257 3976 y(con)m(tains)34 b(relev)-5 b(an)m(t)34 b(examples,)g(suc)m(h)g(as)f(the)g(Newtonian)g(p)s(oten)m (tial)812 4160 y(1)p 695 4205 284 4 v 695 4296 a Fq(j)p Fs(x)22 b Fq(\000)h Fs(y)t Fq(j)1049 4228 y Fu(=)1185 4092 y Fm(Z)1285 4119 y Fj(1)1240 4318 y Fo(0)1418 4160 y Fs(dr)p 1386 4205 162 4 v 1386 4296 a(\031)d(r)1508 4267 y Fo(5)1591 4092 y Fm(Z)1646 4318 y Fc(R)1694 4299 y Fg(3)1749 4228 y Fs(d)1800 4187 y Fo(3)1839 4228 y Fs(z)38 b Fn(1)1978 4243 y Fr(B)s Fo(\()p Fr(z)s(;r)r Fo(\))2182 4228 y Fu(\()p Fs(x)p Fu(\))28 b Fn(1)2397 4243 y Fr(B)s Fo(\()p Fr(z)s(;r)r Fo(\))2602 4228 y Fu(\()p Fs(y)t Fu(\))p Fs(;)427 b Fu(\(I)s(I)s(I.82\))257 4495 y(for)36 b(neutrons)h(in)m(teracting)g(b)m(y)g(gra)m(vitational)f (forces,)i(where)g Fs(B)5 b Fu(\()p Fs(z)t(;)17 b(r)s Fu(\))34 b(:=)g Fq(f)p Fs(x)g Fq(2)g Fi(R)3395 4459 y Fo(3)3468 4495 y Fu(:)257 4615 y Fq(j)p Fs(x)23 b Fq(\000)f Fs(z)33 b(<)27 b(r)s Fq(g)32 b Fu(denotes)i(the)f(ball)g(or)f(radius)h Fs(r)i Fu(ab)s(out)d Fs(z)t Fu(,)i(or)1451 4839 y Fs(\016)1494 4854 y Fr(x;y)1655 4839 y Fu(=)1798 4745 y Fm(X)1791 4956 y Fr(z)s Fj(2)p Fr(M)1965 4839 y Fs(\016)2008 4854 y Fr(x;z)2135 4839 y Fs(\016)2178 4854 y Fr(y)r(;z)2275 4839 y Fs(;)882 b Fu(\(I)s(I)s(I.83\))257 5147 y(for)27 b(the)g(case)h(of)e(the)h(attractiv)m(e)h(Hubbard)f(mo)s(del,)i(with)e Fs(M)39 b Fq(\022)28 b Fi(Z)2706 5111 y Fr(d)2774 5147 y Fu(and)f Fs(\016)3001 5162 y Fr(x;z)3127 5147 y Fu(denoting)257 5268 y(the)g(usual)g(Kronec)m(k)m(er)i(delta.)41 b(It)27 b(is)g(easy)g(to)g(see)g(that)f(if)h Fs(V)48 b Fu(is)27 b(a)f(translation)h(in)m(v)-5 b(arian)m(t)257 5388 y(p)s(oten)m(tial)47 b(on)f Fi(R)904 5352 y Fr(d)990 5388 y Fu(\(with)h(trivial)g(action)f (on)g(the)g(spin)h(v)-5 b(ariables\))47 b(then)g(the)f(form)p eop end %%Page: 18 19 TeXDict begin 18 18 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(18)257 573 y(\(I)s(I)s(I.81\))33 b(implies)i(that)e(b)s(oth)g Fs(V)55 b Fu(and)33 b(its)h(F)-8 b(ourier)33 b(transform)g(are)h(p)s (ositiv)m(e.)47 b(A)33 b(fairly)257 693 y(general)27 b(class)g(of)f(p)s(oten)m(tials)h(admitting)f(a)g(decomp)s(osition)h (of)f(the)h(form)f(\(I)s(I)s(I.81\))f(w)m(as)257 814 y(iden)m(ti\014ed)35 b(in)e([10)o(],)g(the)g(form)f(\(I)s(I)s(I.82\))g (originates)h(in)g([5].)257 1099 y Fl(I)t(I)t(I.1)135 b(Elimination)47 b(of)e Fb(\013)h Fl(as)g(a)f(Degree)h(of)f(F)-11 b(reedom)257 1284 y Fu(Giv)m(en)34 b(a)e(g1-p)s(dm)g(\000)c(=)f(\000\() p Fs(\015)5 b(;)17 b(\013)q Fu(\))27 b Fq(2)h Fk(A)33 b Fu(with)g Fs(\015)g Fu(=)28 b Fs(\015)2159 1247 y Fj(\003)2231 1284 y Fu(and)k Fs(\013)d Fu(=)e Fq(\000)p Fs(\013)2754 1247 y Fr(T)2809 1284 y Fu(,)33 b(w)m(e)h(de\014ne)1048 1537 y Fs(\015)60 b Fu(=:)1318 1397 y Fm(\022)1397 1476 y Fu(^)-52 b Fs(\015)1445 1491 y Fj(")1574 1476 y Fu(^)f Fs(\015)1621 1491 y Fr(o)1395 1597 y Fu(^)g Fs(\015)1447 1561 y Fj(\003)1442 1621 y Fr(o)1573 1597 y Fu(^)g Fs(\015)1620 1612 y Fj(#)1660 1397 y Fm(\023)1749 1537 y Fs(;)82 b(\013)56 b Fu(=:)2135 1397 y Fm(\022)2264 1476 y Fu(^)-58 b Fs(\013)2317 1491 y Fj(")2496 1476 y Fu(^)g Fs(\013)2549 1491 y Fr(o)2208 1597 y Fq(\000)10 b Fu(^)-59 b Fs(\013)2348 1561 y Fr(T)2347 1621 y(o)2496 1597 y Fu(^)g Fs(\013)2548 1612 y Fj(#)2588 1397 y Fm(\023)2678 1537 y Fs(;)479 b Fu(\(I)s(I)s(I.84\))257 1786 y(with)340 1981 y Fq(h)p Fs(f)11 b Fq(j)31 b Fu(^)-53 b Fs(\015)544 1996 y Fj(")600 1981 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)22 b Fs(\016)1109 1996 y Fj(")1149 1981 y Fq(j)27 b Fs(\015)22 b Fu(\()p Fs(g)j Fq(\012)e Fs(\016)1530 1996 y Fj(")1570 1981 y Fu(\))p Fq(i)p Fs(;)134 b Fq(h)p Fs(f)11 b Fq(j)37 b Fu(^)-59 b Fs(\013)2023 1996 y Fj(")2079 1981 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)23 b Fs(\016)2589 1996 y Fj(")2628 1981 y Fq(j)37 b Fu(^)-58 b Fs(\013)17 b Fu(\()p Fs(g)25 b Fq(\012)e Fs(\016)3016 1996 y Fj(")3056 1981 y Fu(\))p Fq(i)p Fs(;)31 b Fu(\(I)s(I)s(I.85\))340 2127 y Fq(h)p Fs(f)11 b Fq(j)31 b Fu(^)-53 b Fs(\015)544 2142 y Fj(#)600 2127 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)22 b Fs(\016)1109 2142 y Fj(#)1149 2127 y Fq(j)27 b Fs(\015)22 b Fu(\()p Fs(g)j Fq(\012)e Fs(\016)1530 2142 y Fj(#)1570 2127 y Fu(\))p Fq(i)p Fs(;)134 b Fq(h)p Fs(f)11 b Fq(j)37 b Fu(^)-59 b Fs(\013)2023 2142 y Fj(#)2079 2127 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)23 b Fs(\016)2589 2142 y Fj(#)2628 2127 y Fq(j)37 b Fu(^)-58 b Fs(\013)17 b Fu(\()p Fs(g)25 b Fq(\012)e Fs(\016)3016 2142 y Fj(#)3056 2127 y Fu(\))p Fq(i)p Fs(;)31 b Fu(\(I)s(I)s(I.86\))342 2272 y Fq(h)p Fs(f)11 b Fq(j)30 b Fu(^)-53 b Fs(\015)545 2287 y Fr(o)600 2272 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)22 b Fs(\016)1109 2287 y Fj(#)1149 2272 y Fq(j)27 b Fs(\015)22 b Fu(\()p Fs(g)j Fq(\012)e Fs(\016)1530 2287 y Fj(")1570 2272 y Fu(\))p Fq(i)p Fs(;)134 b Fq(h)p Fs(f)11 b Fq(j)37 b Fu(^)-59 b Fs(\013)2023 2287 y Fr(o)2078 2272 y Fs(g)t Fq(i)27 b Fu(:=)g Fq(h)p Fs(f)33 b Fq(\012)22 b Fs(\016)2587 2287 y Fj(#)2627 2272 y Fq(j)37 b Fu(^)-59 b Fs(\013)18 b Fu(\()p Fs(g)25 b Fq(\012)e Fs(\016)3015 2287 y Fj(")3054 2272 y Fu(\))p Fq(i)p Fs(;)33 b Fu(\(I)s(I)s(I.87\))257 2479 y(where)i Fs(f)5 b(;)17 b(g)32 b Fq(2)812 2453 y Fu(^)811 2479 y Fk(h)h Fu(and)g Fs(A)1159 2443 y Fr(T)1214 2479 y Fu(\()p Fs(x;)17 b(y)t Fu(\))28 b(:=)h Fs(A)p Fu(\()p Fs(y)t(;)17 b(x)p Fu(\),)32 b(for)h Fs(x;)17 b(y)32 b Fq(2)d Fs(M)10 b Fu(.)46 b(Note)33 b(that,)h(apart)e(from)257 2599 y(the)h(an)m(tisymmetry)45 b(^)-59 b Fs(\013)1108 2614 y Fj(")1176 2599 y Fu(=)27 b Fq(\000)10 b Fu(^)-59 b Fs(\013)1419 2563 y Fr(T)1418 2625 y Fj(")1507 2599 y Fu(and)42 b(^)-58 b Fs(\013)1759 2614 y Fj(#)1826 2599 y Fu(=)27 b Fq(\000)10 b Fu(^)-59 b Fs(\013)2069 2563 y Fr(T)2068 2625 y Fj(#)2124 2599 y Fu(,)33 b(i.e.,)615 2795 y(^)-59 b Fs(\013)667 2810 y Fj(")706 2795 y Fu(\()p Fs(x;)17 b(y)t Fu(\))60 b(=)g Fq(\000)10 b Fu(^)-59 b Fs(\013)1268 2810 y Fj(")1308 2795 y Fu(\()p Fs(y)t(;)17 b(x)p Fu(\))93 b(and)105 b(^)-59 b Fs(\013)1942 2810 y Fj(#)1982 2795 y Fu(\()p Fs(x;)17 b(y)t Fu(\))59 b(=)h Fq(\000)10 b Fu(^)-59 b Fs(\013)2543 2810 y Fj(#)2583 2795 y Fu(\()p Fs(y)t(;)17 b(x)p Fu(\))p Fs(;)347 b Fu(\(I)s(I)s (I.88\))257 2991 y(and)26 b(the)h(selfadjoin)m(tness)h(of)h(^)-53 b Fs(\015)1394 3006 y Fj(")1459 2991 y Fu(and)30 b(^)-53 b Fs(\015)1693 3006 y Fj(#)1732 2991 y Fu(,)27 b(no)f(other)g (constrain)m(t)h(on)f(the)g(matrix)h(en)m(tries)257 3111 y(of)32 b Fs(\015)38 b Fu(and)33 b Fs(\013)g Fu(arises)h(from)e(the)h (CAR.)404 3231 y(Inserting)i(\(I)s(I)s(I.81\))d(in)m(to)h(\(I)s(I.74\)) o(,)g(w)m(e)g(obtain)273 3427 y Fq(E)326 3442 y Fr(B)s(H)5 b(F)504 3346 y Fm(\000)550 3427 y Fu(\000\()p Fs(\015)g(;)17 b(\013)q Fu(\))850 3346 y Fm(\001)955 3427 y Fu(:=)60 b Fs(!)t Fu(\()p Fs(H)8 b Fu(\))1836 b(\(I)s(I)s(I.89\))297 3631 y(=)61 b(T)-8 b(r)p Fq(f)p Fs(h\015)5 b Fq(g)22 b(\000)877 3563 y Fu(1)p 877 3608 49 4 v 877 3699 a(2)953 3495 y Fm(Z)1008 3721 y Fj(N)1096 3520 y Fm(h)1143 3550 y(\000)1206 3631 y Fu(T)-8 b(r)p Fq(f)p Fs(X)1437 3646 y Fr(\021)1479 3631 y Fs(\015)5 b Fq(g)1585 3550 y Fm(\001)1630 3572 y Fo(2)1714 3631 y Fq(\000)45 b Fu(T)-8 b(r)p Fq(f)p Fs(X)2067 3646 y Fr(\021)2109 3631 y Fs(\015)5 b(X)2246 3646 y Fr(\021)2288 3631 y Fs(\015)g Fq(g)44 b Fu(+)g(T)-8 b(r)p Fq(f)p Fs(X)2789 3646 y Fr(\021)2831 3631 y Fs(\013)q(X)2975 3646 y Fr(\021)3016 3631 y Fs(\013)3079 3590 y Fj(\003)3119 3631 y Fq(g)3169 3520 y Fm(i)3215 3631 y Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(;)257 3873 y Fu(and)33 b(w)m(e)h(recall)f(the)g (de\014nition)g(of)f(the)h(BHF)g(energy)g(with)h(particle)f(n)m(um)m(b) s(er)h Fs(N)10 b Fu(,)347 4069 y Fs(E)419 4084 y Fr(B)s(H)5 b(F)597 4069 y Fu(\()p Fs(N)10 b Fu(\))61 b(=)f(inf)1092 3989 y Fm(\010)1150 4069 y Fq(E)1203 4084 y Fr(B)s(H)5 b(F)1381 4069 y Fu(\(\000\))1540 3985 y Fm(\014)1540 4044 y(\014)1638 4069 y Fu(\000)28 b(=)g(\000\()p Fs(\015)5 b(;)17 b(\013)q Fu(\))26 b Fq(2)i Fk(A)2322 4084 y Fr(N)2390 4069 y Fs(;)49 b Fu(T)-8 b(r)q Fq(f)p Fs(h\015)5 b Fq(g)27 b Fs(<)h Fq(1)3010 3989 y Fm(\011)3067 4069 y Fs(:)90 b Fu(\(I)s(I)s(I.90\))257 4265 y(Our)25 b(\014rst)f(main)h(result)g(is) g(the)f(follo)m(wing)h(simpli\014cation)h(of)d(the)i(v)-5 b(ariational)24 b(principle)257 4385 y(\(I)s(I)s(I.90\).)257 4568 y Fn(Theorem)38 b(4)49 b Fp(L)-5 b(et)1012 4542 y Fu(^)1011 4568 y Fs(h)38 b Fu(=)1220 4542 y(^)1219 4568 y Fs(h)1275 4532 y Fj(\003)1352 4568 y Fu(=)1467 4542 y(^)1466 4568 y Fs(h)1522 4532 y Fr(T)1617 4568 y Fp(b)-5 b(e)40 b(a)g(self-adjoint,)h(semib)-5 b(ounde)g(d)38 b(op)-5 b(er)g(ator,)42 b(de-)257 4688 y(\014ne)-5 b(d)35 b(on)f(a)h(dense)f(domain)1334 4669 y Fu(^)1334 4688 y Fk(d)27 b Fq(\022)1518 4662 y Fu(^)1516 4688 y Fk(h)p Fp(.)45 b(Assume)d Fu(\(I)s(I)s(I.81\))35 b Fp(and)f(that)2774 4663 y Fm(b)2749 4688 y Fs(X)2830 4703 y Fr(\021)2900 4688 y Fu(=)3029 4663 y Fm(b)3004 4688 y Fs(X)3093 4652 y Fo(2)3085 4713 y Fr(\021)3160 4688 y Fu(=)3288 4663 y Fm(b)3263 4688 y Fs(X)3352 4652 y Fj(\003)3344 4713 y Fr(\021)3419 4688 y Fu(=)282 4806 y Fm(b)257 4831 y Fs(X)346 4795 y Fr(T)338 4856 y(\021)439 4831 y Fp(ar)-5 b(e)38 b(selfadjoint)f(and)g(r)-5 b(e)g(al,)38 b(for)g(al)5 b(l)37 b Fs(\021)g Fq(2)d(N)15 b Fp(,)38 b(such)g(that)g Fs(V)60 b Fp(is)37 b(de\014ne)-5 b(d)37 b(on)h Fk(d)24 b Fq(\012)h Fk(d)257 4952 y Fp(and)44 b(is)f(an)h(r)-5 b(elatively)43 b(b)-5 b(ounde)g(d)43 b(p)-5 b(erturb)g(ation)44 b(of)g Fs(h)29 b Fq(\012)g Fn(1)g Fu(+)f Fn(1)h Fq(\012)g Fs(h)p Fp(.)72 b(Then,)45 b(for)f(al)5 b(l)257 5072 y Fs(N)39 b Fq(2)28 b Fi(N)p Fp(,)278 5268 y Fs(E)350 5283 y Fr(B)s(H)5 b(F)529 5268 y Fu(\()p Fs(N)10 b Fu(\))63 b(=)f(2)979 5243 y Fm(b)960 5268 y Fs(E)1032 5283 y Fr(B)s(H)5 b(F)1210 5268 y Fu(\()p Fs(N)r(=)p Fu(2\))62 b(:=)h(inf)1826 5187 y Fm(\010)1885 5268 y Fu(2)1948 5243 y Fm(b)1934 5268 y Fq(E)1987 5283 y Fr(B)s(H)5 b(F)2164 5268 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))2318 5183 y Fm(\014)2318 5243 y(\014)2390 5268 y Fu(^)-53 b Fs(\015)33 b Fq(2)2572 5242 y Fm(b)2564 5268 y Fk(A)2635 5283 y Fr(N)q(=)p Fo(2)2767 5268 y Fs(;)52 b Fu(T)-8 b(r)p Fq(f)2997 5241 y Fu(^)2996 5268 y Fs(h)t Fu(^)-53 b Fs(\015)5 b Fq(g)28 b Fs(<)f Fq(1)3389 5187 y Fm(\011)3447 5268 y Fs(;)3184 5388 y Fu(\(I)s(I)s(I.91\))p eop end %%Page: 19 20 TeXDict begin 19 19 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(19)257 573 y Fp(\(p)-5 b(ossibly,)34 b Fs(E)755 588 y Fr(B)s(H)5 b(F)934 573 y Fu(\()p Fs(N)10 b Fu(\))28 b(=)1249 548 y Fm(b)1229 573 y Fs(E)1301 588 y Fr(B)s(H)5 b(F)1480 573 y Fu(\()p Fs(N)r(=)p Fu(2\))27 b(=)h Fq(\0001)p Fp(\))35 b(wher)-5 b(e)431 778 y Fm(b)423 804 y Fk(A)494 819 y Fr(N)q(=)p Fo(2)689 804 y Fu(:=)854 723 y Fm(\010)916 804 y Fu(^)-53 b Fs(\015)33 b Fq(2)28 b(B)s Fu(\()1198 778 y(^)1196 804 y Fk(h)q Fu(\))1314 719 y Fm(\014)1314 779 y(\014)1417 804 y Fu(0)g Fq(\024)j Fu(^)-52 b Fs(\015)32 b Fu(=)g(^)-53 b Fs(\015)1842 762 y Fj(\003)1909 804 y Fu(=)31 b(^)-52 b Fs(\015)2069 762 y Fr(T)2151 804 y Fq(\024)29 b Fn(1)p Fs(;)51 b Fu(T)-8 b(r\()t(^)-53 b Fs(\015)5 b Fu(\))28 b(=)g Fs(N)r(=)p Fu(2)2960 723 y Fm(\011)3184 804 y Fu(\(I)s(I)s(I.92\))257 1024 y Fp(and)303 1238 y Fm(b)289 1263 y Fq(E)342 1278 y Fr(B)s(H)5 b(F)519 1263 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))63 b(:=)f(T)-8 b(r)p Fq(f)p Fs(h)t Fu(^)-53 b Fs(\015)5 b Fq(g)50 b(\000)g(D)s Fu(\()t(^)-53 b Fs(\015)5 b(;)20 b Fu(^)-52 b Fs(\015)5 b Fu(\))49 b(+)1865 1196 y(1)p 1865 1240 49 4 v 1865 1332 a(2)1924 1263 y Fq(P)8 b Fu(\()t(^)-53 b Fs(\015)5 b(;)21 b Fu(^)-53 b Fs(\015)5 b Fu(\))50 b Fq(\000)2420 1196 y Fu(1)p 2420 1240 V 2420 1332 a(2)2479 1263 y Fq(P)2556 1182 y Fm(\000)2602 1172 y(p)p 2702 1172 274 4 v 2706 1263 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-53 b Fs(\015)2935 1234 y Fo(2)2975 1263 y Fs(;)3019 1172 y Fm(p)p 3118 1172 V 3122 1263 a Fu(^)g Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)3352 1234 y Fo(2)3391 1182 y Fm(\001)3437 1263 y Fs(;)3184 1415 y Fu(\(I)s(I)s(I.93\))257 1535 y Fp(and)35 b(with)967 1778 y Fq(D)s Fu(\()p Fs(a;)17 b(b)p Fu(\))83 b(:=)1528 1643 y Fm(Z)1584 1868 y Fj(N)1672 1778 y Fu(T)-8 b(r)p Fq(f)1847 1753 y Fm(b)1822 1778 y Fs(X)1903 1793 y Fr(\021)1962 1778 y Fs(a)p Fq(g)38 b Fu(T)-8 b(r)q Fq(f)2276 1753 y Fm(b)2252 1778 y Fs(X)2333 1793 y Fr(\021)2391 1778 y Fs(b)p Fq(g)22 b Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(;)415 b Fu(\(I)s(I)s(I.94\))970 2037 y Fq(P)8 b Fu(\()p Fs(a;)17 b(b)p Fu(\))83 b(:=)1528 1902 y Fm(Z)1584 2127 y Fj(N)1672 2037 y Fu(T)-8 b(r)p Fq(f)1847 2012 y Fm(b)1822 2037 y Fs(X)1903 2052 y Fr(\021)1962 2037 y Fs(a)2054 2012 y Fm(b)2030 2037 y Fs(X)2111 2052 y Fr(\021)2169 2037 y Fs(b)p Fq(g)22 b Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(:)637 b Fu(\(I)s(I)s(I.95\))257 2333 y Fp(Pr)-5 b(o)g(of:)50 b Fu(Giv)m(en)38 b(^)-53 b Fs(\015)33 b Fq(2)1033 2307 y Fm(b)1025 2333 y Fk(A)1096 2349 y Fr(N)q(=)p Fo(2)1228 2333 y Fu(,)g(w)m(e)g(de\014ne)h Fs(\015)5 b(;)17 b(\013)28 b Fq(2)g(B)s Fu(\()p Fk(h)p Fu(\))33 b(b)m(y)808 2618 y Fs(\015)88 b Fu(:=)1133 2478 y Fm(\022)1210 2557 y Fu(^)-53 b Fs(\015)92 b Fu(0)1210 2678 y(0)e(^)-53 b Fs(\015)1401 2478 y Fm(\023)1535 2618 y Fu(=)63 b(^)-52 b Fs(\015)27 b Fq(\012)22 b Fn(1)1904 2635 y Fc(C)1952 2616 y Fg(2)1991 2618 y Fs(;)1166 b Fu(\(I)s(I)s(I.96\))801 2902 y Fs(\013)84 b Fu(:=)1133 2762 y Fm(\022)1407 2841 y Fu(0)1739 2755 y Fm(p)p 1839 2755 V 1843 2841 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)2073 2813 y Fo(2)1206 2972 y Fq(\000)1283 2885 y Fm(p)p 1383 2885 V 1387 2972 a Fu(^)f Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)1617 2943 y Fo(2)1901 2972 y Fu(0)2112 2762 y Fm(\023)2246 2902 y Fu(=)2382 2811 y Fm(p)p 2481 2811 V 2485 2902 a Fu(^)g Fs(\015)28 b Fq(\000)e Fu(^)-53 b Fs(\015)2715 2874 y Fo(2)2777 2902 y Fq(\012)22 b Fs(\034)6 b(;)233 b Fu(\(I)s(I)s(I.97\))257 3211 y(where)39 b Fs(\034)48 b Fu(:=)36 b(\()855 3176 y Fo(0)55 b(1)828 3233 y Fj(\000)p Fo(1)27 b(0)997 3211 y Fu(\).)58 b(T)-8 b(rivially)g(,)40 b(T)-8 b(r)q Fq(f)p Fs(h\015)5 b Fq(g)36 b Fs(<)g Fq(1)h Fu(i\013)g(T)-8 b(r)q Fq(f)2409 3185 y Fu(^)2408 3211 y Fs(h)s Fu(^)-52 b Fs(\015)5 b Fq(g)36 b Fs(<)g Fq(1)p Fu(.)58 b(Since)42 b(^)-52 b Fs(\015)42 b Fu(ob)s(eys)257 3332 y(the)i(admissibilit)m(y)i (conditions)e(0)i Fq(\024)k Fu(^)-53 b Fs(\015)51 b Fu(=)f(^)-53 b Fs(\015)1992 3296 y Fj(\003)2077 3332 y Fu(=)50 b(^)-53 b Fs(\015)2255 3296 y Fr(T)2356 3332 y Fq(\024)47 b Fn(1)c Fu(and)g(T)-8 b(r)q(\()t(^)-53 b Fs(\015)5 b Fu(\))45 b(=)h Fs(N)r(=)p Fu(2,)g(it)257 3452 y(follo)m(ws)34 b(that)e Fs(\015)h Fu(=)27 b Fs(\015)1032 3416 y Fj(\003)1099 3452 y Fu(=)h Fs(\015)1259 3416 y Fr(T)1314 3452 y Fu(,)33 b Fs(\013)1437 3416 y Fr(T)1519 3452 y Fu(=)28 b Fq(\000)p Fs(\013)q Fu(,)k(T)-8 b(r)q(\()p Fs(\015)5 b Fu(\))27 b(=)h(2)17 b(T)-8 b(r)o(\()t(^)-53 b Fs(\015)5 b Fu(\))28 b(=)f Fs(N)10 b Fu(,)34 b(and)427 3879 y(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])60 b(=)901 3589 y Fm(0)901 3764 y(B)901 3824 y(B)901 3884 y(B)901 3948 y(@)1150 3687 y Fu(^)-53 b Fs(\015)448 b Fu(0)484 b(0)2510 3601 y Fm(p)p 2610 3601 V 2614 3687 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2844 3659 y Fo(2)1150 3818 y Fu(0)446 b(^)-53 b Fs(\015)285 b Fq(\000)2054 3731 y Fm(p)p 2154 3731 V 2158 3818 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2388 3789 y Fo(2)2673 3818 y Fu(0)1150 3948 y(0)245 b Fq(\000)1521 3862 y Fm(p)p 1621 3862 V 1625 3948 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)1855 3920 y Fo(2)2085 3948 y Fn(1)23 b Fq(\000)j Fu(^)-53 b Fs(\015)359 b Fu(0)988 3992 y Fm(p)p 1087 3992 V 1091 4079 a Fu(^)-53 b Fs(\015)28 b Fq(\000)e Fu(^)-53 b Fs(\015)1321 4050 y Fo(2)1645 4079 y Fu(0)484 b(0)353 b Fn(1)22 b Fq(\000)k Fu(^)-52 b Fs(\015)2883 3589 y Fm(1)2883 3764 y(C)2883 3824 y(C)2883 3884 y(C)2883 3948 y(A)2987 3879 y Fs(:)170 b Fu(\(I)s(I)s(I.98\))257 4306 y(It)36 b(is)f(easily)i(c)m(hec)m(k)m(ed)h(that)d(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])35 b(is)g(a)g(real,)h(orthogonal)e(pro)5 b(jection)36 b(and,)g(in)g(par-)257 4426 y(ticular,)d(a)g(g1-p)s(dm,)f (0)27 b Fq(\024)h Fu(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])28 b Fq(\024)g Fn(1)p Fu(.)43 b(Moreo)m(v)m(er,)34 b(T)-8 b(r)q(\()p Fs(\015)5 b Fu(\))27 b(=)h Fs(N)10 b Fu(,)33 b(so)1561 4646 y(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])60 b Fq(2)h Fk(A)2097 4661 y Fr(N)2164 4646 y Fs(:)993 b Fu(\(I)s(I)s(I.99\))p eop end %%Page: 20 21 TeXDict begin 20 20 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(20)257 573 y(Moreo)m(v)m(er,)35 b(with)e Fs(\015)5 b(;)17 b(\013)33 b Fu(according)g(to)f(\(I)s(I)s(I.96\))o(,)463 793 y Fq(E)516 808 y Fr(B)s(H)5 b(F)694 712 y Fm(\000)739 793 y Fu(\000[)p Fs(\015)g(;)17 b(\013)q Fu(])1017 712 y Fm(\001)546 938 y Fu(=)83 b(T)-8 b(r)p Fq(f)p Fs(h\015)5 b Fq(g)50 b(\000)715 1064 y Fu(1)p 715 1109 49 4 v 715 1200 a(2)790 996 y Fm(Z)845 1222 y Fj(N)934 1021 y Fm(h)981 1051 y(\000)1043 1132 y Fu(T)-8 b(r)p Fq(f)p Fs(X)1274 1147 y Fr(\021)1316 1132 y Fs(\015)5 b Fq(g)1422 1051 y Fm(\001)1468 1073 y Fo(2)1551 1132 y Fq(\000)45 b Fu(T)-8 b(r)p Fq(f)p Fs(X)1904 1147 y Fr(\021)1946 1132 y Fs(\015)5 b(X)2083 1147 y Fr(\021)2125 1132 y Fs(\015)g Fq(g)44 b Fu(+)g(T)-8 b(r)p Fq(f)p Fs(X)2626 1147 y Fr(\021)2668 1132 y Fs(\013)q(X)2812 1147 y Fr(\021)2854 1132 y Fs(\013)2917 1091 y Fj(\003)2956 1132 y Fq(g)3006 1021 y Fm(i)3053 1132 y Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))546 1391 y(=)83 b(2)17 b(T)-8 b(r)o Fq(f)p Fs(h)t Fu(^)-53 b Fs(\015)5 b Fq(g)50 b(\000)1259 1255 y Fm(Z)1315 1481 y Fj(N)1403 1280 y Fm(h)1450 1391 y Fu(2)1499 1310 y Fm(\000)1561 1391 y Fu(T)-8 b(r)p Fq(f)1736 1365 y Fm(b)1711 1391 y Fs(X)1792 1406 y Fr(\021)1838 1391 y Fu(^)-53 b Fs(\015)5 b Fq(g)1940 1310 y Fm(\001)1986 1332 y Fo(2)2069 1391 y Fq(\000)45 b Fu(T)-8 b(r)p Fq(f)2366 1365 y Fm(b)2341 1391 y Fs(X)2422 1406 y Fr(\021)2468 1391 y Fu(^)-53 b Fs(\015)2545 1365 y Fm(b)2520 1391 y Fs(X)2601 1406 y Fr(\021)2647 1391 y Fu(^)g Fs(\015)5 b Fq(g)1436 1628 y Fu(+)38 b(T)-8 b(r)1667 1548 y Fm(\010)1750 1603 y(b)1725 1628 y Fs(X)1806 1643 y Fr(\021)1848 1537 y Fm(p)p 1948 1537 274 4 v 1952 1628 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-53 b Fs(\015)2181 1600 y Fo(2)2246 1603 y Fm(b)2221 1628 y Fs(X)2302 1643 y Fr(\021)2344 1537 y Fm(p)p 2443 1537 V 2447 1628 a Fu(^)g Fs(\015)28 b Fq(\000)e Fu(^)-53 b Fs(\015)2677 1600 y Fo(2)2717 1548 y Fm(\011)2775 1518 y(i)2822 1628 y Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))546 1823 y(=)83 b(2)768 1798 y Fm(b)754 1823 y Fq(E)807 1838 y Fr(B)s(H)5 b(F)984 1823 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))p Fs(:)1992 b Fu(\(I)s(I)s(I.100\))257 2043 y(Therefore,)470 2263 y(2)555 2238 y Fm(b)536 2263 y Fs(E)608 2278 y Fr(B)s(H)5 b(F)786 2263 y Fu(\()p Fs(N)r(=)p Fu(2\))84 b(=)f(inf)1418 2183 y Fm(\010)1476 2263 y Fu(2)1539 2238 y Fm(b)1525 2263 y Fq(E)1578 2278 y Fr(B)s(H)5 b(F)1756 2263 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))1910 2179 y Fm(\014)1910 2239 y(\014)2012 2263 y Fu(^)-53 b Fs(\015)33 b Fq(2)2194 2237 y Fm(b)2186 2263 y Fk(A)2257 2279 y Fr(N)q(=)p Fo(2)2417 2183 y Fm(\011)1124 2431 y Fu(=)83 b(inf)1418 2350 y Fm(\010)1476 2431 y Fq(E)1529 2446 y Fr(B)s(H)5 b(F)1707 2431 y Fu(\()t(^)-53 b Fs(\015)27 b Fq(\012)c Fn(1)1979 2447 y Fc(C)2027 2429 y Fg(2)2065 2431 y Fs(;)2109 2340 y Fm(p)p 2209 2340 V 2213 2431 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-53 b Fs(\015)2442 2402 y Fo(2)2504 2431 y Fq(\012)23 b Fs(\034)11 b Fu(\))2717 2346 y Fm(\014)2717 2406 y(\014)2819 2431 y Fu(^)-53 b Fs(\015)33 b Fq(2)3001 2405 y Fm(b)2993 2431 y Fk(A)3064 2446 y Fr(N)q(=)p Fo(2)3224 2350 y Fm(\011)1123 2576 y Fq(\025)83 b Fs(E)1355 2591 y Fr(B)s(H)5 b(F)1534 2576 y Fu(\()p Fs(N)10 b Fu(\))p Fs(:)1410 b Fu(\(I)s(I)s(I.101\))404 2843 y(Con)m(v)m(ersely)-8 b(,)35 b(supp)s(ose)f(that)f(w)m(e)g(are)g (giv)m(en)h(\()p Fs(\015)5 b(;)17 b(\013)q Fu(\))26 b Fq(2)j Fk(A)2487 2858 y Fr(N)2554 2843 y Fu(,)k(with)1075 3121 y Fs(\015)60 b Fu(=)1318 2981 y Fm(\022)1397 3060 y Fu(^)-52 b Fs(\015)1445 3075 y Fj(")1574 3060 y Fu(^)f Fs(\015)1621 3075 y Fr(o)1395 3181 y Fu(^)g Fs(\015)1447 3145 y Fj(\003)1442 3205 y Fr(o)1573 3181 y Fu(^)g Fs(\015)1620 3196 y Fj(#)1660 2981 y Fm(\023)1749 3121 y Fs(;)82 b(\013)56 b Fu(=)2107 2981 y Fm(\022)2237 3060 y Fu(^)-58 b Fs(\013)2290 3075 y Fj(")2469 3060 y Fu(^)g Fs(\013)2522 3075 y Fr(o)2181 3181 y Fq(\000)10 b Fu(^)-59 b Fs(\013)2321 3145 y Fr(T)2320 3205 y(o)2469 3181 y Fu(^)g Fs(\013)2521 3196 y Fj(#)2560 2981 y Fm(\023)2650 3121 y Fs(:)458 b Fu(\(I)s(I)s(I.102\))257 3406 y(As)33 b(\()p Fs(\015)5 b(;)17 b(\013)q Fu(\))27 b Fq(2)h Fk(A)832 3421 y Fr(N)900 3406 y Fu(,)33 b(w)m(e)g(ha)m(v)m(e)h Fs(\015)f Fu(=)27 b Fs(\015)1571 3370 y Fj(\003)1611 3406 y Fu(,)32 b Fs(\013)d Fu(=)e Fq(\000)p Fs(\013)2004 3370 y Fr(T)2059 3406 y Fu(,)33 b(T)-8 b(r\()p Fs(\015)5 b Fu(\))28 b(=)f Fs(N)10 b Fu(,)33 b(and)g(0)28 b Fq(\024)g Fu(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])27 b Fq(\024)h Fn(1)p Fu(.)257 3527 y(W)-8 b(e)33 b(then)h(de\014ne)1265 3667 y(^)-52 b Fs(\015)65 b Fu(:=)1551 3599 y(1)p 1551 3644 49 4 v 1551 3735 a(4)1610 3586 y Fm(\000)1659 3667 y Fu(^)-53 b Fs(\015)1706 3682 y Fj(")1768 3667 y Fu(+)25 b(^)-52 b Fs(\015)1917 3682 y Fj(#)1978 3667 y Fu(+)25 b(^)-52 b Fs(\015)2132 3626 y Fr(T)2127 3691 y Fj(")2209 3667 y Fu(+)26 b(^)-53 b Fs(\015)2363 3626 y Fr(T)2358 3691 y Fj(#)2418 3586 y Fm(\001)2464 3667 y Fs(;)644 b Fu(\(I)s(I)s(I.103\))257 3872 y(Since)34 b(0)28 b Fq(\024)g Fu(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])27 b Fq(\024)h Fn(1)p Fu(,)33 b(w)m(e)g(also)g(ha)m(v)m(e)h(0)27 b Fq(\024)32 b Fu(^)-53 b Fs(\015)2016 3887 y Fj(")2056 3872 y Fs(;)20 b Fu(^)-52 b Fs(\015)2151 3887 y Fj(#)2217 3872 y Fq(\024)28 b Fn(1)33 b Fu(and)f(th)m(us)1317 4092 y(0)60 b Fq(\024)65 b Fu(^)-53 b Fs(\015)65 b Fu(=)f(^)-53 b Fs(\015)1872 4051 y Fj(\003)1972 4092 y Fu(=)64 b(^)-53 b Fs(\015)2164 4051 y Fr(T)2247 4092 y Fq(\024)28 b Fn(1)p Fs(:)700 b Fu(\(I)s(I)s(I.104\))257 4312 y(F)-8 b(urthermore,)743 4575 y(T)g(r)p Fq(f)t Fu(^)-53 b Fs(\015)5 b Fq(g)60 b Fu(=)1205 4507 y(1)p 1205 4552 V 1205 4643 a(2)1280 4575 y(T)-8 b(r)p Fq(f)t Fu(^)-53 b Fs(\015)1481 4590 y Fj(")1542 4575 y Fu(+)26 b(^)-53 b Fs(\015)1691 4590 y Fj(#)1730 4575 y Fq(g)60 b Fu(=)1987 4507 y(1)p 1987 4552 V 1987 4643 a(2)2062 4575 y(T)-8 b(r)p Fq(f)p Fs(\015)5 b Fq(g)60 b Fu(=)2524 4507 y Fs(N)p 2524 4552 89 4 v 2544 4643 a Fu(2)2622 4575 y Fs(;)486 b Fu(\(I)s(I)s(I.105\))257 4820 y(and)33 b(th)m(us)1643 4941 y(^)-52 b Fs(\015)65 b Fq(2)1891 4915 y Fm(b)1882 4941 y Fk(A)1953 4956 y Fr(N)q(=)p Fo(2)2086 4941 y Fs(:)1022 b Fu(\(I)s(I)s(I.106\))257 5115 y(Our)33 b(goal)f(is)h(to)f(pro)m(v)m(e)i(that)1337 5335 y Fq(E)1390 5350 y Fr(B)s(H)5 b(F)1568 5335 y Fu(\()p Fs(\015)g(;)17 b(\013)q Fu(\))59 b Fq(\025)i Fu(2)2067 5310 y Fm(b)2053 5335 y Fq(E)2106 5350 y Fr(B)s(H)5 b(F)2284 5335 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))719 b(\(I)s(I)s(I.107\))p eop end %%Page: 21 22 TeXDict begin 21 21 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(21)257 573 y(whic)m(h)34 b(then)f(implies)h(that)e Fs(E)1372 588 y Fr(B)s(H)5 b(F)1550 573 y Fu(\()p Fs(N)10 b Fu(\))29 b Fq(\025)f Fu(2)1933 548 y Fm(b)1914 573 y Fs(E)1986 588 y Fr(B)s(H)5 b(F)2163 573 y Fu(\()p Fs(N)r(=)p Fu(2\))33 b(and)f(concludes)i(the)f(pro)s(of.)257 703 y(T)-8 b(o)29 b(this)g(end,)h(w)m(e)f(observ)m(e)i(that)d(the)h(realit) m(y)g(of)f Fs(h)g Fu(=)f Fs(h)2294 667 y Fr(T)2378 703 y Fu(and)2588 678 y Fm(b)2563 703 y Fs(X)2644 718 y Fr(\021)2714 703 y Fu(=)2842 678 y Fm(b)2817 703 y Fs(X)2906 667 y Fr(T)2898 728 y(\021)2989 703 y Fu(implies)j(that)535 969 y(T)-8 b(r)p Fq(f)p Fs(h\015)5 b Fq(g)83 b Fu(=)g(T)-8 b(r)p Fq(f)1240 943 y Fu(^)1239 969 y Fs(h)p Fu(\()t(^)-53 b Fs(\015)1384 984 y Fj(")1445 969 y Fu(+)26 b(^)-53 b Fs(\015)1594 984 y Fj(#)1633 969 y Fu(\))p Fq(g)60 b Fu(=)1927 902 y(1)p 1927 946 49 4 v 1927 1037 a(2)2003 969 y(T)-8 b(r)p Fq(f)p Fs(h)p Fu(\()t(^)-53 b Fs(\015)2298 984 y Fj(")2359 969 y Fu(+)26 b(^)-53 b Fs(\015)2508 984 y Fj(#)2547 969 y Fu(\))22 b(+)g Fs(h)2761 928 y Fr(T)2816 969 y Fu(\()t(^)-53 b Fs(\015)2910 928 y Fr(T)2905 994 y Fj(")2987 969 y Fu(+)26 b(^)-52 b Fs(\015)3142 928 y Fr(T)3137 994 y Fj(#)3197 969 y Fu(\))p Fq(g)930 1166 y Fu(=)83 b(2)17 b(T)-8 b(r)p Fq(f)1306 1140 y Fu(^)1305 1166 y Fs(h)s Fu(^)-52 b Fs(\015)5 b Fq(g)p Fs(;)1641 b Fu(\(I)s(I)s(I.108\))468 1321 y(T)-8 b(r)q Fq(f)p Fs(X)700 1336 y Fr(\021)741 1321 y Fs(\015)5 b Fq(g)83 b Fu(=)g(T)-8 b(r)p Fq(f)1264 1296 y Fm(b)1239 1321 y Fs(X)1320 1336 y Fr(\021)1362 1321 y Fu(\()t(^)-53 b Fs(\015)1451 1336 y Fj(")1512 1321 y Fu(+)26 b(^)-53 b Fs(\015)1661 1336 y Fj(#)1700 1321 y Fu(\))p Fq(g)60 b Fu(=)g(2)17 b(T)-8 b(r)p Fq(f)2224 1296 y Fm(b)2200 1321 y Fs(X)2281 1336 y Fr(\021)2326 1321 y Fu(^)-52 b Fs(\015)5 b Fq(g)p Fs(:)679 b Fu(\(I)s(I)s(I.109\))257 1541 y(\(In)32 b(particular,)h(T)-8 b(r)p Fq(f)1044 1515 y Fu(^)1043 1541 y Fs(h)s Fu(^)-52 b Fs(\015)5 b Fq(g)27 b Fs(<)h Fq(1)j Fu(i\013)h(T)-8 b(r)p Fq(f)p Fs(h\015)5 b Fq(g)28 b Fs(<)f Fq(1)p Fu(.\))43 b(Hence,)33 b(comparing)f Fq(E)3079 1556 y Fr(B)s(H)5 b(F)3257 1541 y Fu(\()p Fs(\015)g(;)17 b(\013)q Fu(\))257 1672 y(to)33 b(2)440 1647 y Fm(b)426 1672 y Fq(E)479 1687 y Fr(B)s(H)5 b(F)656 1672 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\),)33 b(with)j(^)-52 b Fs(\015)37 b Fu(giv)m(en)d(in)f(\(I)s(I)s (I.103\))o(,)g(w)m(e)g(obtain)405 1891 y Fq(E)458 1906 y Fr(B)s(H)5 b(F)636 1891 y Fu(\()p Fs(\015)g(;)17 b(\013)q Fu(\))49 b Fq(\000)h Fu(2)1114 1865 y Fm(b)1100 1891 y Fq(E)1153 1906 y Fr(B)s(H)5 b(F)1331 1891 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))488 2084 y(=)647 1949 y Fm(Z)702 2174 y Fj(N)791 1974 y Fm(h)854 2084 y Fu(T)-8 b(r)p Fq(f)p Fs(X)1085 2099 y Fr(\021)1127 2084 y Fs(\015)5 b(X)1264 2099 y Fr(\021)1306 2084 y Fs(\015)g Fq(g)44 b(\000)h Fu(2)17 b(T)-8 b(r)p Fq(f)1818 2059 y Fm(b)1794 2084 y Fs(X)1875 2099 y Fr(\021)1920 2084 y Fu(^)-53 b Fs(\015)1997 2059 y Fm(b)1972 2084 y Fs(X)2053 2099 y Fr(\021)2099 2084 y Fu(^)g Fs(\015)5 b Fq(g)934 b Fu(\(I)s(I)s (I.110\))905 2322 y(+)22 b(2)17 b(T)-8 b(r)p Fq(f)1244 2297 y Fm(b)1219 2322 y Fs(X)1300 2337 y Fr(\021)1342 2231 y Fm(p)p 1441 2231 274 4 v 1445 2322 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)1675 2293 y Fo(2)1739 2297 y Fm(b)1714 2322 y Fs(X)1795 2337 y Fr(\021)1837 2231 y Fm(p)p 1937 2231 V 1941 2322 a Fu(^)g Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)2171 2293 y Fo(2)2210 2322 y Fq(g)44 b(\000)h Fu(T)-8 b(r)p Fq(f)p Fs(X)2657 2337 y Fr(\021)2699 2322 y Fs(\013)q(X)2843 2337 y Fr(\021)2885 2322 y Fs(\013)2948 2281 y Fj(\003)2987 2322 y Fq(g)3037 2211 y Fm(i)3084 2322 y Fs(d\026)p Fu(\()p Fs(\021)t Fu(\))p Fs(:)257 2558 y Fu(Th)m(us)34 b(\(I)s(I)s(I.107\))e (follo)m(ws)i(from)e(the)h(t)m(w)m(o)g(inequalities)869 2777 y(2)17 b(T)-8 b(r)p Fq(f)1110 2752 y Fm(b)1085 2777 y Fs(X)11 b Fu(^)-53 b Fs(\015)1254 2752 y Fm(b)1230 2777 y Fs(X)11 b Fu(^)-53 b Fs(\015)5 b Fq(g)83 b(\024)h Fu(T)-8 b(r)p Fq(f)p Fs(X)8 b(\015)d(X)j(\015)d Fq(g)p Fs(;)950 b Fu(\(I)s(I)s(I.111\))882 2934 y(T)-8 b(r)p Fq(f)p Fs(X)8 b(\013)q(X)g(\013)1336 2893 y Fj(\003)1374 2934 y Fq(g)83 b(\024)h Fu(2)17 b(T)-8 b(r)o Fq(f)1908 2909 y Fm(b)1883 2934 y Fs(X)1972 2843 y Fm(p)p 2072 2843 V 2076 2934 a Fu(^)-53 b Fs(\015)27 b Fq(\000)f Fu(^)-53 b Fs(\015)2305 2906 y Fo(2)2370 2909 y Fm(b)2345 2934 y Fs(X)2434 2843 y Fm(p)p 2533 2843 V 2537 2934 a Fu(^)g Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2767 2906 y Fo(2)2806 2934 y Fq(g)p Fs(;)252 b Fu(\(I)s(I)s(I.112\))257 3153 y(whic)m(h)39 b(w)m(e)f(assert)g(to)e(hold)h(true)h(for)e(an)m(y)h (real)h(orthogonal)e(pro)5 b(jection)3046 3128 y Fm(b)3021 3153 y Fs(X)43 b Fu(=)3281 3128 y Fm(b)3256 3153 y Fs(X)3345 3117 y Fo(2)3419 3153 y Fu(=)282 3248 y Fm(b)257 3273 y Fs(X)346 3237 y Fj(\003)413 3273 y Fu(=)542 3248 y Fm(b)517 3273 y Fs(X)606 3237 y Fr(T)661 3273 y Fu(,)32 b(with)h Fs(X)j Fu(:=)1214 3248 y Fm(b)1190 3273 y Fs(X)29 b Fq(\012)23 b Fn(1)1456 3290 y Fc(C)1504 3271 y Fg(2)1543 3273 y Fu(.)404 3394 y(T)-8 b(o)25 b(establish)i(\(I)s(I)s(I.111\))o(,) g(w)m(e)f(in)m(tro)s(duce)k(~)-53 b Fs(\015)33 b Fu(:=)2136 3355 y Fo(1)p 2136 3371 36 4 v 2136 3428 a(2)2181 3394 y Fu(\()t(^)-53 b Fs(\015)2270 3409 y Fj(")2316 3394 y Fu(+)11 b(^)-53 b Fs(\015)2450 3409 y Fj(#)2489 3394 y Fu(\))28 b(=)j(~)-53 b Fs(\015)2714 3358 y Fj(\003)2754 3394 y Fu(,)26 b(so)j(^)-52 b Fs(\015)32 b Fu(=)3117 3355 y Fo(1)p 3117 3371 V 3117 3428 a(2)3162 3394 y Fu(\()t(~)-53 b Fs(\015)12 b Fu(+)f(~)-53 b Fs(\015)3402 3358 y Fr(T)3457 3394 y Fu(\))257 3514 y(and)33 b(th)m(us)872 3743 y(2)17 b(T)-8 b(r)p Fq(f)1113 3717 y Fm(b)1088 3743 y Fs(X)11 b Fu(^)-52 b Fs(\015)1257 3717 y Fm(b)1233 3743 y Fs(X)11 b Fu(^)-53 b Fs(\015)5 b Fq(g)84 b Fu(=)1681 3675 y(1)p 1681 3720 49 4 v 1681 3811 a(2)1756 3743 y(T)-8 b(r)p Fq(f)1931 3717 y Fm(b)1906 3743 y Fs(X)8 b Fu(\()t(~)-53 b Fs(\015)27 b Fu(+)f(~)-53 b Fs(\015)2265 3701 y Fr(T)2320 3743 y Fu(\))2383 3717 y Fm(b)2358 3743 y Fs(X)8 b Fu(\()t(~)-53 b Fs(\015)27 b Fu(+)f(~)-53 b Fs(\015)2717 3701 y Fr(T)2772 3743 y Fu(\))p Fq(g)1511 3941 y Fu(=)84 b(T)-8 b(r)p Fq(f)1846 3916 y Fm(b)1821 3941 y Fs(X)11 b Fu(~)-52 b Fs(\015)1990 3916 y Fm(b)1966 3941 y Fs(X)11 b Fu(~)-53 b Fs(\015)5 b Fq(g)50 b Fu(+)g(T)-8 b(r)p Fq(f)2511 3916 y Fm(b)2486 3941 y Fs(X)11 b Fu(~)-52 b Fs(\015)2655 3916 y Fm(b)2631 3941 y Fs(X)11 b Fu(~)-53 b Fs(\015)2775 3900 y Fr(T)2830 3941 y Fq(g)1510 4097 y(\024)84 b Fu(2)17 b(T)-8 b(r)o Fq(f)1911 4071 y Fm(b)1886 4097 y Fs(X)12 b Fu(~)-53 b Fs(\015)2056 4071 y Fm(b)2031 4097 y Fs(X)11 b Fu(~)-52 b Fs(\015)5 b Fq(g)p Fs(;)882 b Fu(\(I)s(I)s(I.113\))257 4305 y(using)34 b(the)f(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)j(inequalit)m(y) -8 b(.)45 b(Similarly)-8 b(,)818 4553 y(2)17 b(T)-8 b(r)p Fq(f)1059 4528 y Fm(b)1034 4553 y Fs(X)11 b Fu(~)-53 b Fs(\015)1203 4528 y Fm(b)1179 4553 y Fs(X)11 b Fu(~)-53 b Fs(\015)5 b Fq(g)84 b Fu(=)1627 4486 y(1)p 1627 4530 V 1627 4621 a(2)1702 4553 y(T)-8 b(r)p Fq(f)1877 4528 y Fm(b)1852 4553 y Fs(X)8 b Fu(\()t(^)-53 b Fs(\015)2030 4568 y Fj(")2091 4553 y Fu(+)26 b(^)-53 b Fs(\015)2240 4568 y Fj(#)2279 4553 y Fu(\))2342 4528 y Fm(b)2317 4553 y Fs(X)8 b Fu(\()t(^)-53 b Fs(\015)2495 4568 y Fj(")2556 4553 y Fu(+)25 b(^)-52 b Fs(\015)2705 4568 y Fj(#)2744 4553 y Fu(\))p Fq(g)1456 4752 y(\024)84 b Fu(T)-8 b(r)p Fq(f)1792 4727 y Fm(b)1767 4752 y Fs(X)11 b Fu(^)-52 b Fs(\015)1907 4767 y Fj(")1970 4727 y Fm(b)1946 4752 y Fs(X)11 b Fu(^)-53 b Fs(\015)2085 4767 y Fj(")2124 4752 y Fq(g)50 b Fu(+)g(T)-8 b(r)p Fq(f)2525 4727 y Fm(b)2500 4752 y Fs(X)11 b Fu(^)-52 b Fs(\015)2640 4767 y Fj(#)2703 4727 y Fm(b)2679 4752 y Fs(X)11 b Fu(^)-53 b Fs(\015)2818 4767 y Fj(#)2857 4752 y Fq(g)p Fs(:)201 b Fu(\(I)s(I)s(I.114\))257 4961 y(Eqs.)34 b(\(I)s(I)s(I.113\))e(and)h(\(I)s(I)s(I.114\))f(imply)j (\(I)s(I)s(I.111\))c(b)s(ecause)j(of)e(the)h(iden)m(tit)m(y)307 5179 y(T)-8 b(r)p Fq(f)p Fs(X)8 b(\015)d(X)j(\015)d Fq(g)60 b Fu(=)g(T)-8 b(r)p Fq(f)1168 5154 y Fm(b)1143 5179 y Fs(X)11 b Fu(^)-52 b Fs(\015)1283 5194 y Fj(")1346 5154 y Fm(b)1322 5179 y Fs(X)11 b Fu(^)-53 b Fs(\015)1461 5194 y Fj(")1500 5179 y Fq(g)50 b Fu(+)f(T)-8 b(r)q Fq(f)1900 5154 y Fm(b)1876 5179 y Fs(X)11 b Fu(^)-53 b Fs(\015)2015 5194 y Fj(#)2079 5154 y Fm(b)2054 5179 y Fs(X)12 b Fu(^)-53 b Fs(\015)2194 5194 y Fj(#)2233 5179 y Fq(g)49 b Fu(+)h(2)17 b(T)-8 b(r)p Fq(f)2698 5154 y Fm(b)2674 5179 y Fs(X)11 b Fu(^)-53 b Fs(\015)2813 5194 y Fr(o)2876 5154 y Fm(b)2851 5179 y Fs(X)11 b Fu(^)-52 b Fs(\015)2996 5138 y Fj(\003)2991 5204 y Fr(o)3035 5179 y Fq(g)50 b Fu(\(I)s(I)s(I.115\))257 5388 y(and)33 b(the)g(fact)f(that)h(the)g(last)g(term)g(on)f(the)h (righ)m(t)g(side)h(of)39 b(\(I)s(I)s(I.115\))31 b(is)i(negativ)m(e.)p eop end %%Page: 22 23 TeXDict begin 22 22 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(22)404 573 y(The)33 b(k)m(ey)h(elemen)m(t)h(of)d(the)h(pro)s(of)f (is)h(Estimate)g(\(I)s(I)s(I.112\).)43 b(W)-8 b(e)33 b(\014rst)g(write)1368 763 y Fs(G)60 b Fu(:=)g(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])22 b Fq(\000)h Fu(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])2346 722 y Fo(2)3135 763 y Fu(\(I)s(I)s(I.116\))257 953 y(as)33 b(a)f(4)22 b Fq(\002)h Fu(4)32 b(matrix)h(with)g Fq(B)s Fu(\()1357 927 y(^)1355 953 y Fk(h)q Fu(\)-v)-5 b(alued)32 b(en)m(tries,)j(i.e.,)689 1321 y(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])22 b Fq(\000)h Fu(\000[)p Fs(\015)5 b(;)17 b(\013)q Fu(])1367 1280 y Fo(2)1466 1321 y Fu(=:)1629 1061 y Fm(0)1629 1236 y(B)1629 1296 y(B)1629 1360 y(@)1716 1140 y Fs(G)1793 1155 y Fo(11)1951 1140 y Fs(G)2028 1155 y Fo(12)2186 1140 y Fs(G)2263 1155 y Fo(13)2420 1140 y Fs(G)2497 1155 y Fo(14)1716 1260 y Fs(G)1793 1275 y Fo(21)1951 1260 y Fs(G)2028 1275 y Fo(22)2186 1260 y Fs(G)2263 1275 y Fo(23)2420 1260 y Fs(G)2497 1275 y Fo(24)1716 1380 y Fs(G)1793 1395 y Fo(31)1951 1380 y Fs(G)2028 1395 y Fo(32)2186 1380 y Fs(G)2263 1395 y Fo(33)2420 1380 y Fs(G)2497 1395 y Fo(34)1716 1501 y Fs(G)1793 1516 y Fo(41)1951 1501 y Fs(G)2028 1516 y Fo(42)2186 1501 y Fs(G)2263 1516 y Fo(43)2420 1501 y Fs(G)2497 1516 y Fo(44)2572 1061 y Fm(1)2572 1236 y(C)2572 1296 y(C)2572 1360 y(A)2676 1321 y Fs(;)432 b Fu(\(I)s(I)s(I.117\))257 1684 y(and)33 b(note)g(that)f Fs(G)c Fq(\025)g Fu(0)k(particularly)i (implies)953 1874 y(^)-53 b Fs(\015)1000 1889 y Fj(")1062 1874 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1217 1833 y Fo(2)1212 1899 y Fj(")1279 1874 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1429 1889 y Fr(o)1471 1874 y Fu(^)g Fs(\015)1523 1833 y Fj(\003)1518 1899 y Fr(o)1585 1874 y Fq(\000)32 b Fu(^)-59 b Fs(\013)1746 1889 y Fj(")1796 1874 y Fu(^)g Fs(\013)1849 1833 y Fj(\003)1848 1899 y(")1910 1874 y Fq(\000)33 b Fu(^)-59 b Fs(\013)2072 1889 y Fr(o)2120 1874 y Fu(^)g Fs(\013)2173 1833 y Fj(\003)2172 1899 y Fr(o)2300 1874 y Fu(=)88 b Fs(G)2541 1889 y Fo(11)2699 1874 y Fq(\025)83 b Fu(0)p Fs(;)200 b Fu(\(I)s(I)s(I.118\))805 2031 y(^)-53 b Fs(\015)852 2046 y Fj(#)913 2031 y Fq(\000)26 b Fu(^)-52 b Fs(\015)1069 1990 y Fo(2)1064 2056 y Fj(#)1130 2031 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1286 1990 y Fj(\003)1281 2056 y Fr(o)1329 2031 y Fu(^)h Fs(\015)1377 2046 y Fr(o)1437 2031 y Fq(\000)32 b Fu(^)-59 b Fs(\013)1598 2046 y Fj(#)1647 2031 y Fu(^)h Fs(\013)1701 1990 y Fj(\003)1700 2056 y(#)1762 2031 y Fq(\000)32 b Fu(^)-58 b Fs(\013)1925 1990 y Fr(T)1924 2056 y(o)1979 2031 y Fu(\()10 b(^)-59 b Fs(\013)2080 1990 y Fj(\003)2079 2056 y Fr(o)2119 2031 y Fu(\))2157 1990 y Fr(T)2300 2031 y Fu(=)88 b Fs(G)2541 2046 y Fo(22)2699 2031 y Fq(\025)83 b Fu(0)p Fs(:)200 b Fu(\(I)s(I)s(I.119\))3135 2176 y(\(I)s(I)s(I.120\))257 2366 y(W)-8 b(e)33 b(use)h(again)i(~)-53 b Fs(\015)33 b Fu(:=)1078 2327 y Fo(1)p 1078 2344 36 4 v 1078 2401 a(2)1124 2366 y Fu(\()t(^)-53 b Fs(\015)1213 2381 y Fj(")1274 2366 y Fu(+)25 b(^)-52 b Fs(\015)1423 2381 y Fj(#)1462 2366 y Fu(\))27 b(=)k(~)-52 b Fs(\015)1687 2330 y Fj(\003)1759 2366 y Fu(and)36 b(^)-52 b Fs(\015)32 b Fu(=)2146 2327 y Fo(1)p 2146 2344 V 2146 2401 a(2)2191 2366 y Fu(\()t(~)-53 b Fs(\015)27 b Fu(+)f(~)-53 b Fs(\015)2461 2330 y Fr(T)2516 2366 y Fu(\))33 b(and)f(observ)m(e)j(that)1054 2610 y(^)-52 b Fs(\015)1107 2569 y Fo(2)1229 2610 y Fu(=)1398 2543 y(1)p 1398 2587 49 4 v 1398 2679 a(4)1457 2530 y Fm(\002)1502 2610 y Fu(~)f Fs(\015)28 b Fu(+)d(~)-52 b Fs(\015)1731 2569 y Fr(T)1786 2530 y Fm(\003)1827 2552 y Fo(2)1927 2610 y Fq(\024)2074 2543 y Fu(1)p 2074 2587 V 2074 2679 a(2)2133 2530 y Fm(\002)2178 2610 y Fu(~)g Fs(\015)2231 2569 y Fo(2)2292 2610 y Fu(+)22 b(\()t(~)-53 b Fs(\015)2484 2569 y Fo(2)2524 2610 y Fu(\))2562 2569 y Fr(T)2617 2530 y Fm(\003)2658 2610 y Fs(;)450 b Fu(\(I)s(I)s(I.121\)) 1054 2843 y(~)-52 b Fs(\015)1107 2802 y Fo(2)1229 2843 y Fu(=)1398 2776 y(1)p 1398 2820 V 1398 2912 a(4)1457 2763 y Fm(\002)1502 2843 y Fu(^)f Fs(\015)1549 2858 y Fj(")1611 2843 y Fu(+)25 b(^)-52 b Fs(\015)1760 2858 y Fj(#)1799 2763 y Fm(\003)1840 2785 y Fo(2)1940 2843 y Fq(\024)2087 2776 y Fu(1)p 2087 2820 V 2087 2912 a(2)2146 2763 y Fm(\002)2191 2843 y Fu(^)g Fs(\015)2244 2802 y Fo(2)2239 2868 y Fj(")2305 2843 y Fu(+)26 b(^)-53 b Fs(\015)2459 2802 y Fo(2)2454 2868 y Fj(#)2499 2763 y Fm(\003)2540 2843 y Fs(;)568 b Fu(\(I)s(I)s(I.122\))257 3033 y(so)1142 3173 y(^)-53 b Fs(\015)1194 3132 y Fo(2)1294 3173 y Fq(\024)1441 3106 y Fu(1)p 1441 3150 V 1441 3242 a(4)1500 3092 y Fm(\002)1545 3173 y Fu(^)h Fs(\015)1598 3132 y Fo(2)1593 3198 y Fj(")1659 3173 y Fu(+)26 b(^)-53 b Fs(\015)1813 3132 y Fo(2)1808 3198 y Fj(#)1875 3173 y Fu(+)22 b(\()t(^)-53 b Fs(\015)2067 3132 y Fo(2)2062 3198 y Fj(")2106 3173 y Fu(\))2144 3132 y Fr(T)2221 3173 y Fu(+)22 b(\()t(^)-53 b Fs(\015)2413 3132 y Fo(2)2408 3198 y Fj(#)2453 3173 y Fu(\))2491 3132 y Fr(T)2546 3092 y Fm(\003)2587 3173 y Fs(:)521 b Fu(\(I)s(I)s(I.123\)) 257 3366 y(Th)m(us,)32 b(adding)d(up)g(\(I)s(I)s(I.118\))g(and)g(\(I)s (I)s(I.119\))f(and)h(using)i(\(I)s(I)s(I.123\))e(and)g(the)g(an)m (tisym-)257 3486 y(metry)34 b(of)42 b(^)-59 b Fs(\013)715 3501 y Fj(")p Fr(;)p Fj(#)809 3486 y Fu(,)33 b(w)m(e)h(obtain)721 3723 y(^)-52 b Fs(\015)27 b Fq(\000)f Fu(^)-53 b Fs(\015)951 3682 y Fo(2)1074 3723 y Fq(\025)1244 3656 y Fu(1)p 1244 3700 V 1244 3792 a(4)1303 3643 y Fm(\002)1354 3723 y Fu(^)-58 b Fs(\013)1407 3738 y Fj(")1456 3723 y Fu(^)f Fs(\013)1509 3682 y Fj(\003)1508 3748 y(")1570 3723 y Fu(+)32 b(^)-59 b Fs(\013)1730 3738 y Fj(#)1779 3723 y Fu(^)h Fs(\013)1833 3682 y Fj(\003)1832 3748 y(#)1894 3723 y Fu(+)32 b(^)-59 b Fs(\013)2054 3738 y Fr(o)2102 3723 y Fu(^)g Fs(\013)2155 3682 y Fj(\003)2154 3748 y Fr(o)2217 3723 y Fu(+)22 b(\()10 b(^)-59 b Fs(\013)2416 3682 y Fj(\003)2415 3748 y Fr(o)2464 3723 y Fu(^)h Fs(\013)2517 3738 y Fr(o)2555 3723 y Fu(\))2593 3682 y Fr(T)1352 3911 y Fu(+\()10 b(^)-59 b Fs(\013)1528 3926 y Fj(")1577 3911 y Fu(^)h Fs(\013)1631 3870 y Fj(\003)1630 3936 y(")1670 3911 y Fu(\))1708 3870 y Fr(T)1785 3911 y Fu(+)22 b(\()10 b(^)-59 b Fs(\013)1983 3926 y Fj(#)2032 3911 y Fu(^)g Fs(\013)2085 3870 y Fj(\003)2084 3936 y(#)2125 3911 y Fu(\))2163 3870 y Fr(T)2240 3911 y Fu(+)22 b(\()10 b(^)-59 b Fs(\013)2438 3926 y Fr(o)2486 3911 y Fu(^)g Fs(\013)2539 3870 y Fj(\003)2538 3936 y Fr(o)2578 3911 y Fu(\))2616 3870 y Fr(T)2693 3911 y Fu(+)32 b(^)-59 b Fs(\013)2854 3870 y Fj(\003)2853 3936 y Fr(o)2903 3911 y Fu(^)g Fs(\013)2955 3926 y Fr(o)2994 3831 y Fm(\003)1075 4114 y Fu(=)1244 4046 y(1)p 1244 4091 V 1244 4182 a(4)1303 4033 y Fm(\002)1354 4114 y Fu(^)h Fs(\013)1407 4129 y Fj(")1456 4114 y Fu(^)f Fs(\013)1509 4073 y Fj(\003)1508 4138 y(")1570 4114 y Fu(+)32 b(^)-59 b Fs(\013)1731 4073 y Fj(\003)1730 4138 y(")1780 4114 y Fu(^)g Fs(\013)1832 4129 y Fj(")1894 4114 y Fu(+)32 b(^)-59 b Fs(\013)2054 4129 y Fj(#)2103 4114 y Fu(^)h Fs(\013)2157 4073 y Fj(\003)2156 4138 y(#)2218 4114 y Fu(+)31 b(^)-58 b Fs(\013)2379 4073 y Fj(\003)2378 4138 y(#)2428 4114 y Fu(^)f Fs(\013)2480 4129 y Fj(#)1352 4302 y Fu(+)10 b(^)-59 b Fs(\013)1490 4317 y Fr(o)1538 4302 y Fu(^)h Fs(\013)1592 4261 y Fj(\003)1591 4326 y Fr(o)1653 4302 y Fu(+)32 b(^)-59 b Fs(\013)1814 4261 y Fj(\003)1813 4326 y Fr(o)1863 4302 y Fu(^)g Fs(\013)1915 4317 y Fr(o)1975 4302 y Fu(+)22 b(\()10 b(^)-59 b Fs(\013)2173 4317 y Fr(o)2221 4302 y Fu(^)h Fs(\013)2275 4261 y Fj(\003)2274 4326 y Fr(o)2314 4302 y Fu(\))2352 4261 y Fr(T)2429 4302 y Fu(+)22 b(\()10 b(^)-59 b Fs(\013)2628 4261 y Fj(\003)2627 4326 y Fr(o)2677 4302 y Fu(^)g Fs(\013)2729 4317 y Fr(o)2767 4302 y Fu(\))2805 4261 y Fr(T)2860 4221 y Fm(\003)2902 4302 y Fs(:)206 b Fu(\(I)s(I)s(I.124\))257 4507 y(Since)34 b Fs(A)28 b Fq(7!)740 4422 y(p)p 823 4422 74 4 v 85 x Fs(A)33 b Fu(is)g(op)s(erator)f(monotone,)h(this)g(implies)h(that)1283 4671 y Fm(\020)1414 4657 y Fo(8)1360 4687 y Fm(X)1367 4896 y Fr(\027)t Fo(=1)1520 4781 y Fs(A)1593 4796 y Fr(\027)1636 4671 y Fm(\021)1696 4693 y Fo(1)p Fr(=)p Fo(2)1866 4781 y Fq(\024)61 b Fu(2)2053 4690 y Fm(p)p 2152 4690 290 4 v 2156 4781 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2386 4753 y Fo(2)2442 4781 y Fs(;)666 b Fu(\(I)s(I)s(I.125\)) 257 5053 y(where)34 b(the)f(p)s(ositiv)m(e)h(selfadjoin)m(t)f(op)s (erators)g Fs(A)2042 5068 y Fo(1)2081 5053 y Fs(;)17 b(:)g(:)g(:)f(;)h(A)2373 5068 y Fo(8)2440 5053 y Fq(\025)28 b Fu(0)33 b(are)f(de\014ned)i(b)m(y)751 5243 y Fs(A)824 5258 y Fo(1)891 5243 y Fu(:=)j(^)-58 b Fs(\013)1084 5258 y Fj(")1133 5243 y Fu(^)f Fs(\013)1186 5207 y Fj(\003)1185 5269 y(")1225 5243 y Fs(;)82 b(A)1407 5258 y Fo(2)1474 5243 y Fu(:=)37 b(^)-58 b Fs(\013)1668 5207 y Fj(\003)1667 5269 y(")1717 5243 y Fu(^)f Fs(\013)1769 5258 y Fj(")1808 5243 y Fs(;)82 b(A)1990 5258 y Fo(3)2057 5243 y Fu(:=)38 b(^)-59 b Fs(\013)2250 5258 y Fj(#)2299 5243 y Fu(^)g Fs(\013)2352 5207 y Fj(\003)2351 5269 y(#)2392 5243 y Fs(;)81 b(A)2573 5258 y Fo(4)2641 5243 y Fu(:=)37 b(^)-59 b Fs(\013)2834 5207 y Fj(\003)2833 5269 y(#)2883 5243 y Fu(^)g Fs(\013)2935 5258 y Fj(#)2975 5243 y Fs(;)133 b Fu(\(I)s(I)s(I.126\))605 5388 y Fs(A)678 5403 y Fo(5)745 5388 y Fu(:=)37 b(^)-58 b Fs(\013)938 5403 y Fr(o)986 5388 y Fu(^)f Fs(\013)1039 5352 y Fj(\003)1038 5413 y Fr(o)1078 5388 y Fs(;)82 b(A)1260 5403 y Fo(6)1327 5388 y Fu(:=)37 b(^)-58 b Fs(\013)1521 5352 y Fj(\003)1520 5413 y Fr(o)1569 5388 y Fu(^)g Fs(\013)1622 5403 y Fr(o)1660 5388 y Fs(;)82 b(A)1842 5403 y Fo(7)1909 5388 y Fu(:=)37 b(^)-58 b Fs(\013)2103 5352 y Fr(T)2102 5413 y(o)2157 5388 y Fu(\()10 b(^)-59 b Fs(\013)2258 5352 y Fj(\003)2257 5413 y Fr(o)2297 5388 y Fu(\))2335 5352 y Fr(T)2390 5388 y Fs(;)82 b(A)2572 5403 y Fo(8)2639 5388 y Fu(:=)28 b(\()10 b(^)-59 b Fs(\013)2871 5352 y Fj(\003)2870 5413 y Fr(o)2910 5388 y Fu(\))2948 5352 y Fr(T)3013 5388 y Fu(^)g Fs(\013)3066 5352 y Fr(T)3065 5413 y(o)3121 5388 y Fs(:)-13 b Fu(\(I)s(I)s(I.127\))p eop end %%Page: 23 24 TeXDict begin 23 23 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(23)404 573 y(If)35 b Fs(M)47 b Fu(is)36 b(a)g(densely)i (de\014ned,)g(closed)f(op)s(erator)e(then)i(the)f(p)s(olar)f(decomp)s (osition)257 693 y(yields)50 b(a)d(unitary)h Fs(U)59 b Fu(suc)m(h)49 b(that)e Fs(M)64 b Fu(=)54 b Fs(U)10 b Fq(j)p Fs(M)g Fq(j)48 b Fu(and)g Fs(M)2469 657 y Fj(\003)2562 693 y Fu(=)54 b Fq(j)p Fs(M)10 b Fq(j)p Fs(U)2928 657 y Fj(\003)2968 693 y Fu(.)88 b(Here,)53 b(the)257 814 y(absolute)36 b(v)-5 b(alue)35 b(of)f(an)h(op)s(erator)f Fs(M)45 b Fu(is)35 b(de\014ned)h(as)f(usual:)48 b Fq(j)p Fs(M)10 b Fq(j)32 b Fu(:=)2867 729 y Fq(p)p 2950 729 266 4 v 85 x Fs(M)3054 785 y Fj(\003)3094 814 y Fs(M)27 b Fu(.)50 b(Note)257 934 y(that)38 b Fq(j)p Fs(M)606 898 y Fj(\003)645 934 y Fq(j)e Fu(=)821 850 y Fq(p)p 904 850 249 4 v 84 x Fs(M)10 b(M)1112 905 y Fj(\003)1189 934 y Fu(=)36 b Fs(U)10 b Fq(j)p Fs(M)g Fq(j)p Fs(U)1613 898 y Fj(\003)1653 934 y Fu(.)58 b(Using)38 b(the)g(Cauc)m(h)m(y-Sc)m (h)m(w)m(arz)j(inequalit)m(y)f(for)257 1054 y(traces,)34 b(w)m(e)f(then)h(observ)m(e)g(that)595 1228 y(T)-8 b(r)p Fq(f)770 1203 y Fm(b)745 1228 y Fs(X)30 b(M)1007 1203 y Fm(b)982 1228 y Fs(X)g(M)1197 1187 y Fj(\003)1238 1228 y Fq(g)87 b Fu(=)h(T)-8 b(r)p Fq(f)1714 1203 y Fm(b)1689 1228 y Fs(X)30 b(U)j Fq(j)p Fs(M)10 b Fq(j)2106 1203 y Fm(b)2081 1228 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(j)22 b Fs(U)2450 1187 y Fj(\003)2490 1228 y Fq(g)678 1383 y Fu(=)84 b(T)-8 b(r)955 1303 y Fm(\010\000)1059 1383 y Fq(j)p Fs(M)10 b Fq(j)1219 1342 y Fo(1)p Fr(=)p Fo(2)1351 1383 y Fs(U)1427 1342 y Fj(\003)1514 1358 y Fm(b)1489 1383 y Fs(X)30 b(U)j Fq(j)p Fs(M)10 b Fq(j)1859 1342 y Fo(1)p Fr(=)p Fo(2)1969 1303 y Fm(\001\000)2060 1383 y Fq(j)p Fs(M)g Fq(j)2220 1342 y Fo(1)p Fr(=)p Fo(2)2378 1358 y Fm(b)2353 1383 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(j)2624 1342 y Fo(1)p Fr(=)p Fo(2)2734 1303 y Fm(\001)o(\011)678 1590 y Fq(\024)838 1479 y Fm(\020)914 1590 y Fu(T)-8 b(r)p Fq(f)p Fs(U)1140 1548 y Fj(\003)1227 1564 y Fm(b)1203 1590 y Fs(X)29 b(U)k Fq(j)p Fs(M)10 b Fq(j)23 b Fs(U)1671 1548 y Fj(\003)1758 1564 y Fm(b)1733 1590 y Fs(X)30 b(U)i Fq(j)p Fs(M)10 b Fq(jg)23 b(\001)e Fu(T)-8 b(r)q Fq(f)2399 1564 y Fm(b)2375 1590 y Fs(X)29 b Fq(j)p Fs(M)10 b Fq(j)2692 1564 y Fm(b)2668 1590 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(jg)2989 1479 y Fm(\021)3048 1501 y Fo(1)p Fr(=)p Fo(2)678 1826 y Fu(=)838 1715 y Fm(\020)914 1826 y Fu(T)-8 b(r)p Fq(f)1089 1800 y Fm(b)1064 1826 y Fs(X)30 b Fq(j)p Fs(M)1307 1785 y Fj(\003)1347 1826 y Fq(j)1422 1800 y Fm(b)1397 1826 y Fs(X)g Fq(j)p Fs(M)1640 1785 y Fj(\003)1680 1826 y Fq(jg)21 b(\001)h Fu(T)-8 b(r)p Fq(f)2004 1800 y Fm(b)1979 1826 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(j)2297 1800 y Fm(b)2272 1826 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(jg)2593 1715 y Fm(\021)2653 1737 y Fo(1)p Fr(=)p Fo(2)678 2055 y Fq(\024)848 1988 y Fu(1)p 848 2032 49 4 v 848 2124 a(2)907 1975 y Fm(\000)969 2055 y Fu(T)-8 b(r)p Fq(f)1144 2030 y Fm(b)1119 2055 y Fs(X)30 b Fq(j)p Fs(M)1362 2014 y Fj(\003)1402 2055 y Fq(j)1477 2030 y Fm(b)1452 2055 y Fs(X)g Fq(j)p Fs(M)1695 2014 y Fj(\003)1735 2055 y Fq(jg)21 b Fu(+)h(T)-8 b(r)p Fq(f)2107 2030 y Fm(b)2082 2055 y Fs(X)30 b Fq(j)p Fs(M)10 b Fq(j)2400 2030 y Fm(b)2376 2055 y Fs(X)29 b Fq(j)p Fs(M)10 b Fq(jg)2696 1975 y Fm(\001)2742 2055 y Fs(:)366 b Fu(\(I)s(I)s(I.128\))257 2260 y(W)-8 b(e)33 b(deriv)m(e)i(the)e (follo)m(wing)f(upp)s(er)i(b)s(ound)e(on)h(T)-8 b(r)p Fq(f)p Fs(X)2239 2275 y Fr(\021)2281 2260 y Fs(\013)q(X)2425 2275 y Fr(\021)2466 2260 y Fs(\013)2529 2224 y Fj(\003)2568 2260 y Fq(g)p Fu(,)339 2434 y(T)g(r)p Fq(f)p Fs(X)8 b(\013)q(X)g(\013) 793 2393 y Fj(\003)832 2434 y Fq(g)423 2589 y Fu(=)84 b(T)-8 b(r)p Fq(f)758 2564 y Fm(b)733 2589 y Fs(X)17 b Fu(^)-58 b Fs(\013)884 2604 y Fj(")948 2564 y Fm(b)923 2589 y Fs(X)18 b Fu(^)-59 b Fs(\013)1075 2548 y Fj(\003)1074 2614 y(")1114 2589 y Fq(g)22 b Fu(+)g(T)-8 b(r)p Fq(f)1459 2564 y Fm(b)1434 2589 y Fs(X)17 b Fu(^)-58 b Fs(\013)1585 2604 y Fj(#)1649 2564 y Fm(b)1624 2589 y Fs(X)18 b Fu(^)-59 b Fs(\013)1776 2548 y Fj(\003)1775 2614 y(#)1815 2589 y Fq(g)22 b Fu(+)g(2)17 b(T)-8 b(r)p Fq(f)2225 2564 y Fm(b)2201 2589 y Fs(X)17 b Fu(^)-59 b Fs(\013)2351 2604 y Fr(o)2415 2564 y Fm(b)2390 2589 y Fs(X)17 b Fu(^)-59 b Fs(\013)2541 2548 y Fj(\003)2540 2614 y Fr(o)2581 2589 y Fq(g)423 2757 y Fu(=)84 b(T)-8 b(r)p Fq(f)758 2732 y Fm(b)733 2757 y Fs(X)17 b Fu(^)-58 b Fs(\013)884 2772 y Fj(")948 2732 y Fm(b)923 2757 y Fs(X)18 b Fu(^)-59 b Fs(\013)1075 2716 y Fj(\003)1074 2782 y(")1114 2757 y Fq(g)22 b Fu(+)g(T)-8 b(r)p Fq(f)1459 2732 y Fm(b)1434 2757 y Fs(X)17 b Fu(^)-58 b Fs(\013)1585 2772 y Fj(#)1649 2732 y Fm(b)1624 2757 y Fs(X)18 b Fu(^)-59 b Fs(\013)1776 2716 y Fj(\003)1775 2782 y(#)1815 2757 y Fq(g)22 b Fu(+)g(T)-8 b(r)p Fq(f)2160 2732 y Fm(b)2135 2757 y Fs(X)18 b Fu(^)-59 b Fs(\013)2286 2772 y Fr(o)2349 2732 y Fm(b)2324 2757 y Fs(X)18 b Fu(^)-59 b Fs(\013)2476 2716 y Fj(\003)2475 2782 y Fr(o)2515 2757 y Fq(g)22 b Fu(+)g(T)-8 b(r)p Fq(f)2860 2732 y Fm(b)2835 2757 y Fs(X)18 b Fu(^)-59 b Fs(\013)2987 2716 y Fr(T)2986 2782 y(o)3067 2732 y Fm(b)3042 2757 y Fs(X)8 b Fu(\()i(^)-59 b Fs(\013)3232 2716 y Fj(\003)3231 2782 y Fr(o)3271 2757 y Fu(\))3309 2716 y Fr(T)3364 2757 y Fq(g)422 3005 y(\024)593 2938 y Fu(1)p 593 2982 V 593 3073 a(2)722 2881 y Fo(8)668 2910 y Fm(X)675 3120 y Fr(\027)t Fo(=1)829 3005 y Fu(T)-8 b(r)p Fq(f)1003 2980 y Fm(b)979 3005 y Fs(X)1090 2913 y Fm(p)p 1189 2913 133 4 v 1189 3005 a Fs(A)1262 3020 y Fr(\027)1375 2980 y Fm(b)1350 3005 y Fs(X)1439 2913 y Fm(p)p 1538 2913 V 1538 3005 a Fs(A)1611 3020 y Fr(\027)1671 3005 y Fq(g)p Fs(;)1387 b Fu(\(I)s(I)s(I.129\))257 3266 y(with)34 b Fs(A)553 3281 y Fo(1)592 3266 y Fs(;)17 b(:)g(:)g(:)f(;)h(A)884 3281 y Fo(8)956 3266 y Fu(as)32 b(in)h(\(I)s(I)s(I.126\){\(I)s(I)s (I.127\))o(,)f(b)m(y)i(applying)g(\(I)s(I)s(I.128\).)404 3386 y(No)m(w)j(w)m(e)h(use)f(a)g(\(sp)s(ecial)g(case)h(of)e(a)h(non)m (trivial\))g(trace)g(inequalit)m(y)i(due)e(to)g(Lieb)257 3507 y([14]:)54 b(The)39 b(map)g(from)e(the)i(p)s(ositiv)m(e)g(op)s (erators)f(in)m(to)g(the)h(p)s(ositiv)m(e)g(n)m(um)m(b)s(ers)h(giv)m (en)257 3637 y(b)m(y)53 b(\()p Fs(A;)17 b(B)5 b Fu(\))61 b Fq(7!)g Fu(T)-8 b(r)p Fq(f)1081 3612 y Fm(b)1056 3637 y Fs(X)8 b(A)1218 3601 y Fo(1)p Fr(=)p Fo(2)1353 3612 y Fm(b)1328 3637 y Fs(X)f(B)1495 3601 y Fo(1)p Fr(=)p Fo(2)1606 3637 y Fq(g)52 b Fu(is)g Fp(c)-5 b(onc)g(ave)p Fu(,)56 b(i.e.,)i(giv)m(en)53 b(p)s(ositiv)m(e)g(op)s(erators)257 3758 y Fs(A)330 3773 y Fo(1)370 3758 y Fs(;)17 b(A)487 3773 y Fo(2)526 3758 y Fs(;)g(B)644 3773 y Fo(1)683 3758 y Fs(;)g(B)801 3773 y Fo(2)868 3758 y Fq(\025)29 b Fu(0)j(and)g(a)h(n)m (um)m(b)s(er)h Fs(\025)27 b Fq(2)h Fu([0)p Fs(;)17 b Fu(1],)32 b(the)h(estimate)654 3945 y Fs(\025)g Fu(T)-8 b(r)861 3864 y Fm(\010)944 3920 y(b)919 3945 y Fs(X)24 b(A)1097 3894 y Fo(1)p Fr(=)p Fo(2)1097 3969 y(1)1249 3920 y Fm(b)1224 3945 y Fs(X)g(B)1408 3894 y Fo(1)p Fr(=)p Fo(2)1403 3969 y(1)1518 3864 y Fm(\011)1626 3945 y Fu(+)50 b(\(1)22 b Fq(\000)g Fs(\025)p Fu(\))34 b(T)-8 b(r)2205 3864 y Fm(\010)2288 3920 y(b)2263 3945 y Fs(X)25 b(A)2442 3894 y Fo(1)p Fr(=)p Fo(2)2442 3969 y(2)2593 3920 y Fm(b)2569 3945 y Fs(X)f(B)2753 3894 y Fo(1)p Fr(=)p Fo(2)2748 3969 y(2)2863 3864 y Fm(\011)3135 3945 y Fu(\(I)s(I)s(I.130\))737 4110 y Fq(\024)83 b Fu(T)-8 b(r)1014 4029 y Fm(\010)1097 4085 y(b)1072 4110 y Fs(X)25 b Fu([)p Fs(\025A)1335 4125 y Fo(1)1396 4110 y Fu(+)d(\(1)g Fq(\000)h Fs(\025)p Fu(\))p Fs(A)1871 4125 y Fo(2)1910 4110 y Fu(])1937 4069 y Fo(1)p Fr(=)p Fo(2)2089 4085 y Fm(b)2064 4110 y Fs(X)h Fu([)p Fs(\025B)2327 4125 y Fo(1)2389 4110 y Fu(+)e(\(1)g Fq(\000)g Fs(\025)p Fu(\))p Fs(B)2864 4125 y Fo(2)2904 4110 y Fu(])2931 4069 y Fo(1)p Fr(=)p Fo(2)3041 4029 y Fm(\011)257 4284 y Fu(holds)45 b(true.)80 b(This)45 b(of)f(course)h(extends)i(to)d(a)m (v)m(erages)h(of)f(p)s(ositiv)m(e)i(op)s(erators,)h(and)257 4404 y(hence)34 b(w)m(e)g(obtain)588 4643 y(T)-8 b(r)705 4563 y Fm(\010)763 4643 y Fs(X)8 b(\013)q(X)g(\013)1067 4602 y Fj(\003)1105 4563 y Fm(\011)1246 4643 y Fq(\024)84 b Fu(4)22 b Fq(\001)1538 4576 y Fu(1)p 1538 4620 49 4 v 1538 4712 a(8)1667 4519 y Fo(8)1613 4549 y Fm(X)1620 4758 y Fr(\027)t Fo(=1)1773 4643 y Fu(T)-8 b(r)1890 4563 y Fm(\010)1973 4618 y(b)1948 4643 y Fs(X)36 b(A)2138 4602 y Fo(1)p Fr(=)p Fo(2)2138 4668 y Fr(\027)2300 4618 y Fm(b)2276 4643 y Fs(X)f(A)2465 4602 y Fo(1)p Fr(=)p Fo(2)2465 4668 y Fr(\027)2575 4563 y Fm(\011)1246 4978 y Fq(\024)84 b Fu(4)17 b(T)-8 b(r)1589 4868 y Fm(n)1680 4953 y(b)1655 4978 y Fs(X)1772 4868 y Fm(\020)1841 4911 y Fu(1)p 1841 4955 V 1841 5047 a(8)1971 4854 y Fo(8)1917 4884 y Fm(X)1924 5093 y Fr(\027)t Fo(=1)2077 4978 y Fs(A)2150 4993 y Fr(\027)2194 4868 y Fm(\021)2253 4890 y Fo(1)p Fr(=)p Fo(2)2416 4953 y Fm(b)2391 4978 y Fs(X)2507 4868 y Fm(\020)2577 4911 y Fu(1)p 2577 4955 V 2577 5047 a(8)2706 4854 y Fo(8)2652 4884 y Fm(X)2659 5093 y Fr(\027)t Fo(=1)2813 4978 y Fs(A)2886 4993 y Fr(\027)2929 4868 y Fm(\021)2988 4890 y Fo(1)p Fr(=)p Fo(2)3098 4868 y Fm(o)1247 5313 y Fu(=)1417 5246 y(1)p 1417 5290 V 1417 5382 a(2)1492 5313 y(T)g(r)1609 5203 y Fm(n)1700 5288 y(b)1675 5313 y Fs(X)1792 5203 y Fm(\020)1922 5189 y Fo(8)1868 5219 y Fm(X)1875 5428 y Fr(\027)t Fo(=1)2028 5313 y Fs(A)2101 5328 y Fr(\027)2145 5203 y Fm(\021)2204 5225 y Fo(1)p Fr(=)p Fo(2)2367 5288 y Fm(b)2342 5313 y Fs(X)2458 5203 y Fm(\020)2589 5189 y Fo(8)2535 5219 y Fm(X)2542 5428 y Fr(\027)t Fo(=1)2695 5313 y Fs(A)2768 5328 y Fr(\027)2811 5203 y Fm(\021)2871 5225 y Fo(1)p Fr(=)p Fo(2)2981 5203 y Fm(o)3047 5313 y Fs(:)61 b Fu(\(I)s(I)s(I.131\))p eop end %%Page: 24 25 TeXDict begin 24 24 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(24)257 573 y(Finally)-8 b(,)33 b(if)g Fs(A)27 b Fq(\025)i Fs(B)j Fq(\025)d Fu(0)j(are)g(p)s(ositiv)m(e)i(op)s(erators,) f(then)566 762 y(T)-8 b(r)682 682 y Fm(\010)765 737 y(b)741 762 y Fs(X)24 b(B)967 737 y Fm(b)942 762 y Fs(X)g(B)1126 682 y Fm(\011)1245 762 y Fq(\024)60 b Fu(T)-8 b(r)1499 682 y Fm(\010)1582 737 y(b)1557 762 y Fs(X)25 b(B)1783 737 y Fm(b)1758 762 y Fs(X)g(A)1937 682 y Fm(\011)2055 762 y Fq(\024)61 b Fu(T)-8 b(r)2310 682 y Fm(\010)2393 737 y(b)2368 762 y Fs(X)24 b(A)2588 737 y Fm(b)2563 762 y Fs(X)g(A)2741 682 y Fm(\011)2800 762 y Fs(;)308 b Fu(\(I)s(I)s (I.132\))257 952 y(whic)m(h,)56 b(together)49 b(with)h(\(I)s(I)s (I.125\))f(and)h(\(I)s(I)s(I.131\))e(yields)j(the)f(desired)h (inequalit)m(y)257 1072 y(\(I)s(I)s(I.112\):)526 1352 y(T)-8 b(r)p Fq(f)p Fs(X)8 b(\013)q(X)g(\013)980 1311 y Fj(\003)1018 1352 y Fq(g)83 b(\024)1322 1284 y Fu(1)p 1322 1329 49 4 v 1322 1420 a(2)1397 1352 y(T)-8 b(r)1514 1241 y Fm(n)1605 1327 y(b)1580 1352 y Fs(X)1697 1241 y Fm(\020)1827 1227 y Fo(8)1773 1257 y Fm(X)1780 1466 y Fr(\027)t Fo(=1)1933 1352 y Fs(A)2006 1367 y Fr(\027)2050 1241 y Fm(\021)2109 1263 y Fo(1)p Fr(=)p Fo(2)2272 1327 y Fm(b)2247 1352 y Fs(X)2363 1241 y Fm(\020)2494 1227 y Fo(8)2439 1257 y Fm(X)2447 1466 y Fr(\027)t Fo(=1)2600 1352 y Fs(A)2673 1367 y Fr(\027)2716 1241 y Fm(\021)2776 1263 y Fo(1)p Fr(=)p Fo(2)2886 1241 y Fm(o)3135 1352 y Fu(\(I)s(I)s(I.133\))1151 1609 y Fq(\024)84 b Fu(2)17 b(T)-8 b(r)1494 1528 y Fm(\010)1577 1584 y(b)1552 1609 y Fs(X)1669 1517 y Fm(p)p 1768 1517 290 4 v 1772 1609 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2002 1580 y Fo(2)2110 1584 y Fm(b)2086 1609 y Fs(X)2202 1517 y Fm(p)p 2302 1517 V 2306 1609 a Fu(^)g Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)2536 1580 y Fo(2)2592 1528 y Fm(\011)2650 1609 y Fs(:)472 b Fd(\003)257 1893 y Fl(I)t(I)t(I.2)135 b(Occurence)44 b(of)i(P)l(airing,)g Fb(\013)34 b Fa(6)p FC(=)g(0)257 2078 y Fu(Our)k(next)g(goal)f(is)i(to)e(deriv)m(e)i (su\016cien)m(t)h(conditions)f(for)e(the)h(o)s(ccurence)h(of)e Fp(p)-5 b(airing)257 2198 y Fs(\013)319 2213 y Fo(0)406 2198 y Fq(6)p Fu(=)46 b(0)e(for)f(minimizers)j(\000[)p Fs(\015)1425 2213 y Fo(0)1464 2198 y Fs(;)17 b(\013)1570 2213 y Fo(0)1609 2198 y Fu(])90 b Fq(2)h Fk(A)1954 2213 y Fr(N)2065 2198 y Fu(of)44 b Fq(E)2241 2213 y Fr(B)s(H)5 b(F)2419 2198 y Fu(.)76 b(In)45 b(view)g(of)50 b(\(I)s(I.75\))43 b(and)257 2319 y(\(I)s(I.79\),)33 b(this)g(is)g(equiv)-5 b(alen)m(t)34 b(to)1378 2508 y Fs(E)1450 2523 y Fr(H)5 b(F)1572 2508 y Fu(\()p Fs(N)10 b Fu(\))61 b Fs(>)f(E)2005 2523 y Fr(B)s(H)5 b(F)2183 2508 y Fu(\()p Fs(N)10 b Fu(\))p Fs(;)761 b Fu(\(I)s(I)s(I.134\))257 2698 y(Thanks)40 b(to)e(Theorem)i(4,)f(Eq.)h(\(I)s(I)s(I.134\))d(is)i(equiv)-5 b(alen)m(t)40 b(to)e(pro)m(ving)h(that)f(the)h(min-)257 2828 y(imizers)44 b(^)-52 b Fs(\015)646 2843 y Fo(0)723 2828 y Fu(of)855 2803 y Fm(e)840 2828 y Fq(E)893 2843 y Fr(B)s(H)5 b(F)1110 2828 y Fu(are)39 b(not)f(orthogonal)g(pro)5 b(jections.)63 b(Put)39 b(di\013eren)m(tly)-8 b(,)42 b(pairing)257 2948 y(o)s(ccurs)34 b(i\013)1308 3043 y Fm(b)1289 3069 y Fs(E)1361 3084 y Fr(H)5 b(F)1483 3069 y Fu(\()p Fs(N)r(=)p Fu(2\))60 b Fs(>)1952 3043 y Fm(b)1933 3069 y Fs(E)2005 3084 y Fr(B)s(H)5 b(F)2183 3069 y Fu(\()p Fs(N)r(=)p Fu(2\))p Fs(;)671 b Fu(\(I)s(I)s(I.135\))257 3230 y(where)636 3394 y Fm(b)617 3419 y Fs(E)689 3434 y Fr(B)s(H)5 b(F)867 3419 y Fu(\()p Fs(N)r(=)p Fu(2\))83 b(:=)g(inf)1525 3339 y Fm(\010)1597 3394 y(b)1583 3419 y Fq(E)1636 3434 y Fr(B)s(H)5 b(F)1814 3419 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))1968 3335 y Fm(\014)1968 3394 y(\014)2037 3419 y Fu(^)-53 b Fs(\015)33 b Fq(2)2219 3393 y Fm(b)2211 3419 y Fk(A)2282 3435 y Fr(N)q(=)p Fo(2)2415 3419 y Fs(;)49 b Fu(T)-8 b(r)p Fq(f)2642 3393 y Fu(^)2641 3419 y Fs(h)t Fu(^)-53 b Fs(\015)5 b Fq(g)28 b Fs(<)f Fq(1)3034 3339 y Fm(\011)3092 3419 y Fs(;)16 b Fu(\(I)s(I)s(I.136\))693 3560 y Fm(b)673 3585 y Fs(E)745 3600 y Fr(H)5 b(F)867 3585 y Fu(\()p Fs(N)r(=)p Fu(2\))83 b(:=)g(inf)1525 3505 y Fm(\010)1597 3560 y(b)1583 3585 y Fq(E)1636 3600 y Fr(H)5 b(F)1757 3585 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))1911 3501 y Fm(\014)1911 3560 y(\014)1981 3585 y Fu(^)-53 b Fs(\015)33 b Fq(2)2163 3559 y Fm(b)2155 3585 y Fk(A)2226 3601 y Fr(N)q(=)p Fo(2)2358 3585 y Fs(;)49 b Fu(T)-8 b(r)q Fq(f)2586 3559 y Fu(^)2585 3585 y Fs(h)s Fu(^)-52 b Fs(\015)5 b Fq(g)27 b Fs(<)h Fq(1)2978 3505 y Fm(\011)3035 3585 y Fs(;)73 b Fu(\(I)s(I)s(I.137\))257 3775 y(and)862 3959 y Fm(b)848 3984 y Fq(E)901 3999 y Fr(H)5 b(F)1023 3984 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))82 b(:=)h(T)-8 b(r)q Fq(f)p Fs(h)t Fu(^)-53 b Fs(\015)5 b Fq(g)49 b(\000)i(D)s Fu(\()t(^)-53 b Fs(\015)t(;)21 b Fu(^)-53 b Fs(\015)5 b Fu(\))50 b(+)2409 3916 y(1)p 2409 3961 49 4 v 2409 4052 a(2)2468 3984 y Fq(P)8 b Fu(\()t(^)-53 b Fs(\015)6 b(;)20 b Fu(^)-52 b Fs(\015)t Fu(\))p Fs(;)331 b Fu(\(I)s(I)s(I.138\)) 806 4192 y Fm(b)792 4217 y Fq(E)845 4232 y Fr(B)s(H)5 b(F)1023 4217 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))96 b(=)1438 4192 y Fm(b)1423 4217 y Fq(E)1476 4232 y Fr(H)5 b(F)1598 4217 y Fu(\()t(^)-53 b Fs(\015)5 b Fu(\))49 b Fq(\000)1917 4150 y Fu(1)p 1917 4194 V 1917 4285 a(2)1975 4217 y Fq(P)2052 4136 y Fm(\000)2099 4126 y(p)p 2198 4126 274 4 v 2202 4217 a Fu(^)-53 b Fs(\015)27 b Fq(\000)g Fu(^)-53 b Fs(\015)2432 4188 y Fo(2)2471 4217 y Fs(;)2515 4126 y Fm(p)p 2615 4126 V 2619 4217 a Fu(^)g Fs(\015)27 b Fq(\000)f Fu(^)-52 b Fs(\015)2849 4188 y Fo(2)2888 4136 y Fm(\001)2934 4217 y Fs(:)174 b Fu(\(I)s(I)s(I.139\))257 4432 y(W)-8 b(e)30 b(\014rst)h(remark)f(that)g(pairing)f(do)s(es)h(not) g(alw)m(a)m(ys)h(o)s(ccur.)43 b(W)-8 b(e)30 b(demonstrate)h(this)f(b)m (y)257 4552 y(a)24 b(simple)i(example.)43 b(F)-8 b(or)23 b Fs(N)5 b(;)17 b(K)35 b Fq(2)28 b Fi(N)p Fu(,)e(with)f Fs(N)38 b(<)28 b(K)7 b Fu(,)26 b(and)e(a)g(small)h(coupling)g(constan)m (t)257 4673 y Fs(g)42 b(>)d Fu(0,)h(let)f Fk(h)g Fu(:=)f Fi(C)1028 4637 y Fr(K)1097 4673 y Fu(,)i Fs(V)60 b Fu(:=)39 b Fs(g)t Fn(1)p Fu(,)h Fs(h)e Fu(:=)1833 4598 y Fm(P)1938 4624 y Fr(K)1938 4702 y(k)r Fo(=1)2088 4673 y Fs(h)2144 4688 y Fr(k)2186 4673 y Fq(j)p Fs(k)s Fq(ih)p Fs(k)s Fq(j)g Fu(and)h Fs(P)2725 4688 y Fr(N)2831 4673 y Fu(:=)2973 4598 y Fm(P)3078 4624 y Fr(N)3078 4702 y(k)r Fo(=1)3227 4673 y Fq(j)p Fs(k)s Fq(ih)p Fs(k)s Fq(j)p Fu(,)257 4793 y(where)c Fs(h)596 4808 y Fo(1)666 4793 y Fs(<)30 b(h)828 4808 y Fo(2)897 4793 y Fs(<)g Fq(\001)17 b(\001)g(\001)28 b Fs(<)h(h)1310 4808 y Fr(K)1379 4793 y Fu(.)47 b(Our)34 b(goal)f(is)i(to)e(sho)m(w)i(a)f(certain)g(rigidit)m(y)h(of)e(the)h(HF) 257 4913 y(and)29 b(BHF)g(minimizer)h(in)f(this)h(case:)42 b(F)-8 b(or)28 b Fs(g)j(>)c Fu(0)i(su\016cien)m(tly)i(small,)g Fs(P)2925 4928 y Fr(N)3020 4913 y Fu(or)e(\000\()p Fs(P)3298 4928 y Fr(N)3365 4913 y Fs(;)17 b Fu(0\))257 5034 y(p)s(ersists)35 b(to)d(b)s(e)h(the)g(minimizer)h(of)e Fq(E)1648 5049 y Fr(H)5 b(F)1802 5034 y Fu(or)32 b Fq(E)1974 5049 y Fr(B)s(H)5 b(F)2152 5034 y Fu(,)32 b(resp)s(ectiv)m(ely)-8 b(.)47 b(Then)745 5310 y Fq(E)798 5325 y Fr(B)s(H)5 b(F)975 5310 y Fu([\000\()p Fs(\015)g(;)17 b(\013)q Fu(\)])60 b(=)1565 5186 y Fr(K)1525 5216 y Fm(X)1533 5428 y Fr(k)r Fo(=1)1686 5200 y Fm(n)1752 5310 y Fs(h)1808 5325 y Fr(k)1867 5310 y Fs(\032)1917 5325 y Fr(k)1983 5310 y Fq(\000)2092 5243 y Fs(g)p 2092 5288 51 4 v 2093 5379 a Fu(2)2169 5310 y Fq(j)p Fs(\013)q Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))p Fq(j)2516 5269 y Fo(2)2554 5200 y Fm(o)2621 5310 y Fs(;)487 b Fu(\(I)s(I)s(I.140\))p eop end %%Page: 25 26 TeXDict begin 25 25 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(25)257 573 y(using)40 b Fs(\032)569 588 y Fr(k)651 573 y Fu(:=)e Fs(\015)5 b Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))39 b(and)g(that)g Fq(D)s Fu(\()p Fs(\015)5 b(;)17 b(\015)5 b Fu(\))38 b(=)h Fq(P)8 b Fu(\()p Fs(\015)d(;)17 b(\015)5 b Fu(\))39 b(in)g(this)h(case.)63 b(In)40 b(particular,)257 693 y Fq(E)310 708 y Fr(B)s(H)5 b(F)488 693 y Fu([\000\()p Fs(P)677 708 y Fr(N)744 693 y Fs(;)17 b Fu(0\)])27 b(=)1033 619 y Fm(P)1138 645 y Fr(N)1138 722 y(k)r Fo(=1)1288 693 y Fs(h)1344 708 y Fr(k)1386 693 y Fu(.)44 b(Since)463 995 y Fq(j)p Fs(\013)q Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))p Fq(j)810 954 y Fo(2)908 995 y Fq(\024)61 b(j)1074 904 y Fm(p)p 1173 904 274 4 v 1173 995 a Fs(\015)27 b Fq(\000)c Fs(\015)1407 966 y Fo(2)1446 995 y Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))p Fq(j)1702 954 y Fo(2)1801 995 y Fq(\024)1939 825 y Fm(\()2146 828 y(\014)2146 887 y(\014)2179 850 y Fq(p)p 2262 850 57 4 v 62 x Fs(\015)5 b Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))2546 828 y Fm(\014)2546 887 y(\014)2579 854 y Fo(2)2787 912 y Fq(\024)169 b Fs(\032)3083 927 y Fr(k)3126 912 y Fs(;)2061 1008 y Fm(\014)2061 1068 y(\014)2094 1020 y Fq(p)p 2177 1020 227 4 v 73 x Fu(1)22 b Fq(\000)h Fs(\015)5 b Fu(\()p Fs(k)s(;)17 b(k)s Fu(\))2632 1008 y Fm(\014)2632 1068 y(\014)2665 1034 y Fo(2)2787 1093 y Fq(\024)84 b Fu(1)22 b Fq(\000)g Fs(\032)3168 1108 y Fr(k)3211 1093 y Fs(;)3135 1212 y Fu(\(I)s(I)s(I.141\))257 1332 y(w)m(e)34 b(ha)m(v)m(e)533 1533 y Fq(E)586 1548 y Fr(B)s(H)5 b(F)764 1533 y Fu([\000\()p Fs(\015)g(;)17 b(\013)q Fu(\)])k Fq(\000)i(E)1292 1548 y Fr(B)s(H)5 b(F)1470 1533 y Fu([\000\()p Fs(P)1659 1548 y Fr(N)1726 1533 y Fs(;)17 b Fu(0\)])1251 b(\(I)s(I)s(I.142\))616 1771 y Fq(\025)84 b(\000)911 1647 y Fr(N)871 1676 y Fm(X)878 1889 y Fr(k)r Fo(=1)1031 1661 y Fm(n)1098 1690 y(\000)1143 1771 y Fs(h)1199 1786 y Fr(k)1264 1771 y Fu(+)1372 1704 y Fs(g)p 1372 1748 51 4 v 1373 1839 a Fu(2)1432 1690 y Fm(\001)17 b(\000)1540 1771 y Fu(1)22 b Fq(\000)h Fs(\032)1761 1786 y Fr(k)1804 1690 y Fm(\001)1850 1661 y(o)1938 1771 y Fu(+)2127 1647 y Fr(K)2088 1676 y Fm(X)2036 1889 y Fr(k)r Fo(=)p Fr(N)7 b Fo(+1)2299 1661 y Fm(n)2366 1690 y(\000)2412 1771 y Fs(h)2468 1786 y Fr(k)2532 1771 y Fq(\000)2642 1704 y Fs(g)p 2642 1748 V 2643 1839 a Fu(2)2702 1690 y Fm(\001)2765 1771 y Fs(\032)2815 1786 y Fr(k)2858 1661 y Fm(o)2924 1771 y Fs(;)617 2119 y Fu(=)817 1994 y Fr(N)777 2024 y Fm(X)784 2236 y Fr(k)r Fo(=1)937 2008 y Fm(n)1004 2038 y(\000)1049 2119 y Fs(\026)22 b Fq(\000)h Fs(h)1286 2134 y Fr(k)1351 2119 y Fq(\000)1460 2051 y Fs(g)p 1460 2096 V 1461 2187 a Fu(2)1520 2038 y Fm(\001)17 b(\000)1628 2119 y Fu(1)22 b Fq(\000)h Fs(\032)1849 2134 y Fr(k)1892 2038 y Fm(\001)1938 2008 y(o)2026 2119 y Fu(+)2215 1994 y Fr(K)2176 2024 y Fm(X)2124 2236 y Fr(k)r Fo(=)p Fr(N)7 b Fo(+1)2387 2008 y Fm(n)2454 2038 y(\000)2499 2119 y Fs(h)2555 2134 y Fr(k)2620 2119 y Fq(\000)23 b Fs(\026)f Fq(\000)2910 2051 y Fs(g)p 2910 2096 V 2911 2187 a Fu(2)2971 2038 y Fm(\001)3033 2119 y Fs(\032)3083 2134 y Fr(k)3126 2008 y Fm(o)3192 2119 y Fs(;)257 2424 y Fu(for)32 b(an)m(y)i Fs(\026)27 b Fq(2)h Fi(R)p Fu(.)44 b(Cho)s(osing)32 b Fs(\026)c Fu(:=)1566 2385 y Fo(1)p 1566 2401 36 4 v 1566 2458 a(2)1612 2424 y Fu(\()p Fs(h)1706 2439 y Fr(N)1795 2424 y Fu(+)22 b Fs(h)1949 2439 y Fr(N)7 b Fo(+1)2107 2424 y Fu(\),)32 b(w)m(e)i(obtain)398 2625 y Fq(8)22 b Fs(g)53 b(<)d(h)757 2640 y Fr(N)7 b Fo(+1)937 2625 y Fq(\000)22 b Fs(h)1092 2640 y Fr(N)1187 2625 y Fu(:)223 b Fq(E)1490 2640 y Fr(B)s(H)5 b(F)1668 2625 y Fu([\000\()p Fs(\015)g(;)17 b(\013)q Fu(\)])22 b Fq(\000)g(E)2196 2640 y Fr(B)s(H)5 b(F)2374 2625 y Fu([\000\()p Fs(P)2563 2640 y Fr(N)2630 2625 y Fs(;)17 b Fu(0\)])27 b Fs(>)h Fu(0)p Fs(;)140 b Fu(\(I)s(I)s(I.143\))257 2826 y(and)33 b(hence)h(no)f(pairing)f(o)s(ccurs.)257 3033 y Fn(Theorem)38 b(5)g(\(Su\016cien)m(t)g(Condition)g(for)f(P)m(airing\))49 b Fp(Assume)32 b(the)f(same)h(hyp)-5 b(o-)257 3164 y(thesis)31 b(as)g(in)g(The)-5 b(or)g(em)31 b(4)g(and)f(supp)-5 b(ose)31 b(that)k Fu(^)-52 b Fs(\015)2030 3179 y Fo(0)2097 3164 y Fq(2)2199 3138 y Fm(b)2191 3164 y Fk(A)2262 3179 y Fr(n)2340 3164 y Fp(is)31 b(a)h(minimizer)e(of)3109 3139 y Fm(b)3095 3164 y Fq(E)3148 3179 y Fr(H)5 b(F)3269 3164 y Fp(,)32 b(i.e.,)272 3259 y Fm(b)257 3284 y Fq(E)310 3299 y Fr(H)5 b(F)432 3284 y Fu(\()t(^)-53 b Fs(\015)521 3299 y Fo(0)560 3284 y Fu(\))30 b(=)754 3259 y Fm(b)734 3284 y Fs(E)806 3299 y Fr(H)5 b(F)928 3284 y Fu(\()p Fs(n)p Fu(\))p Fp(,)37 b(wher)-5 b(e)36 b Fs(n)31 b Fu(:=)f Fs(N)r(=)p Fu(2)p Fp(.)49 b(L)-5 b(et)2077 3259 y Fm(b)2055 3284 y Fs(H)2136 3299 y Fo(0)2206 3284 y Fu(:=)2362 3259 y Fm(b)2340 3284 y Fs(H)7 b Fu(\()t(^)-53 b Fs(\015)2517 3299 y Fo(0)2556 3284 y Fu(\))36 b Fp(b)-5 b(e)36 b(the)h(c)-5 b(orr)g(esp)g(onding)257 3404 y(e\013e)g(ctive)34 b(Hamiltonian)h (de\014ne)-5 b(d)33 b(by)307 3568 y Fm(\002)371 3623 y(b)349 3648 y Fs(H)430 3663 y Fo(0)469 3648 y Fs(')533 3568 y Fm(\003)574 3648 y Fu(\()p Fs(x)p Fu(\))54 b(=)f([)916 3622 y(^)915 3648 y Fs(h')p Fu(]\()p Fs(x)p Fu(\))r Fq(\000)1273 3513 y Fm(Z)1390 3568 y(\010)1452 3648 y Fu(^)-53 b Fs(\015)1499 3663 y Fo(0)1538 3648 y Fu(\()p Fs(y)t(;)17 b(y)t Fu(\))g Fs(')p Fu(\()p Fs(x)p Fu(\))r Fq(\000)6 b Fu(^)-53 b Fs(\015)2106 3663 y Fo(0)2143 3648 y Fu(\()p Fs(x;)17 b(y)t Fu(\))g Fs(')p Fu(\()p Fs(y)t Fu(\))2579 3568 y Fm(\011)2635 3648 y Fs(V)k Fu(\()p Fs(x;)c(y)t Fu(\))g Fs(dy)t(;)48 b Fu(\(I)s(I)s(I.144\))257 3905 y Fp(and)32 b(denote)f(by)h Fs(e)921 3920 y Fr(k)996 3905 y Fp(its)g Fs(k)1184 3869 y Fr(th)1286 3905 y Fp(eigenvalue)e(\(c)-5 b(ounting)32 b(multiplicities\))f(with)h(c)-5 b(orr)g(esp)g(ond-)257 4026 y(ing)35 b(eigenve)-5 b(ctor)34 b Fs(')993 4041 y Fr(k)1063 4026 y Fq(2)1159 4000 y Fu(^)1157 4026 y Fk(h)p Fp(.)45 b(If)329 4270 y Fs(e)374 4285 y Fr(n)p Fo(+1)533 4270 y Fq(\000)23 b Fs(e)678 4285 y Fr(n)787 4270 y Fs(<)926 4134 y Fm(Z)1042 4159 y(\020)1101 4270 y Fq(j)p Fs(')1193 4285 y Fr(n)1240 4270 y Fu(\()p Fs(x)p Fu(\))p Fq(j)1399 4228 y Fo(2)1455 4270 y Fq(j)p Fs(')1547 4285 y Fr(n)1594 4270 y Fu(\()p Fs(y)t Fu(\))p Fq(j)1750 4228 y Fo(2)1832 4270 y Fu(+)44 b Fq(j)p Fs(')2044 4285 y Fr(n)p Fo(+1)2181 4270 y Fu(\()p Fs(x)p Fu(\))p Fq(j)2340 4228 y Fo(2)2396 4270 y Fq(j)p Fs(')2488 4285 y Fr(n)p Fo(+1)2625 4270 y Fu(\()p Fs(y)t Fu(\))p Fq(j)2781 4228 y Fo(2)2819 4159 y Fm(\021)2878 4270 y Fs(V)22 b Fu(\()p Fs(x;)17 b(y)t Fu(\))g Fs(dx)g(dy)3135 4438 y Fu(\(I)s(I)s(I.145\))257 4559 y Fp(then)35 b(p)-5 b(airing)34 b(o)-5 b(c)g(curs,)1425 4654 y Fm(b)1406 4679 y Fs(E)1478 4694 y Fr(H)5 b(F)1600 4679 y Fu(\()p Fs(n)p Fu(\))63 b Fs(>)1954 4654 y Fm(b)1935 4679 y Fs(E)2007 4694 y Fr(B)s(H)5 b(F)2185 4679 y Fu(\()p Fs(n)p Fu(\))p Fs(:)789 b Fu(\(I)s(I)s(I.146\))257 4897 y Fp(Pr)-5 b(o)g(of:)50 b Fu(F)-8 b(or)36 b(^)-53 b Fs(\015)790 4912 y Fo(0)829 4897 y Fs(;)21 b Fu(^)-53 b Fs(\015)924 4912 y Fo(1)991 4897 y Fq(2)1093 4871 y Fm(b)1085 4897 y Fk(A)1156 4912 y Fr(n)1236 4897 y Fu(w)m(e)33 b(set)817 5098 y(^)-53 b Fs(\015)864 5113 y Fr(\025)992 5098 y Fu(:=)83 b(\(1)22 b Fq(\000)h Fs(\025)p Fu(\))d(^)-52 b Fs(\015)1550 5113 y Fo(0)1633 5098 y Fu(+)44 b Fs(\025)20 b Fu(^)-52 b Fs(\015)1878 5113 y Fo(1)1976 5098 y Fq(2)2111 5072 y Fm(b)2103 5098 y Fk(A)2174 5113 y Fr(N)q(=)p Fo(2)2306 5098 y Fs(;)802 b Fu(\(I)s(I)s(I.147\))813 5243 y Fs(a)864 5258 y Fr(\025)992 5243 y Fu(:=)87 b(^)-53 b Fs(\015)1229 5258 y Fr(\025)1296 5243 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1452 5202 y Fo(2)1447 5267 y Fr(\025)1552 5243 y Fq(\025)61 b Fu(0)p Fs(;)1369 b Fu(\(I)s(I)s(I.148\))868 5388 y Fs(b)83 b Fu(:=)k(^)-53 b Fs(\015)1229 5403 y Fo(0)1285 5388 y Fu(\()p Fn(1)22 b Fq(\000)k Fu(^)-52 b Fs(\015)1552 5403 y Fo(1)1591 5388 y Fu(\))20 b(^)-52 b Fs(\015)1697 5403 y Fo(0)1757 5388 y Fu(+)22 b(\()p Fn(1)h Fq(\000)j Fu(^)-53 b Fs(\015)2122 5403 y Fo(0)2161 5388 y Fu(\))20 b(^)-52 b Fs(\015)2267 5403 y Fo(1)2322 5388 y Fu(\()p Fn(1)23 b Fq(\000)j Fu(^)-53 b Fs(\015)2589 5403 y Fo(0)2628 5388 y Fu(\))60 b Fq(\025)h Fu(0)p Fs(;)195 b Fu(\(I)s(I)s(I.149\))p eop end %%Page: 26 27 TeXDict begin 26 26 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(26)257 573 y(for)32 b(all)h(0)27 b Fq(\024)h Fs(\025)g Fq(\024)g Fu(1.)43 b(Straigh)m(tforw)m(ard)33 b(computation)g(yields) 448 801 y Fs(a)499 816 y Fr(\025)627 801 y Fu(=)87 b(^)-53 b Fs(\015)837 816 y Fo(0)920 801 y Fu(+)45 b Fs(\025)p Fu(\()t(^)-53 b Fs(\015)1187 816 y Fo(1)1248 801 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1398 816 y Fo(0)1437 801 y Fu(\))44 b Fq(\000)1641 720 y Fm(\002)1686 801 y Fu(^)-52 b Fs(\015)1734 816 y Fo(0)1817 801 y Fu(+)44 b Fs(\025)p Fu(\()t(^)-53 b Fs(\015)2083 816 y Fo(1)2144 801 y Fq(\000)26 b Fu(^)-52 b Fs(\015)2295 816 y Fo(0)2334 801 y Fu(\))2372 720 y Fm(\003)2413 742 y Fo(2)627 946 y Fu(=)87 b(^)-53 b Fs(\015)837 961 y Fo(0)898 946 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1054 905 y Fo(2)1049 971 y(0)1138 946 y Fu(+)44 b Fs(\025)1315 865 y Fm(\002)1360 946 y Fu(^)-53 b Fs(\015)1407 961 y Fo(1)1468 946 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1619 961 y Fo(0)1680 946 y Fu(+)22 b(2)t(^)-53 b Fs(\015)1883 905 y Fo(2)1878 971 y(0)1945 946 y Fq(\000)26 b Fu(^)-53 b Fs(\015)2095 961 y Fo(0)2138 946 y Fu(^)g Fs(\015)2185 961 y Fo(1)2246 946 y Fq(\000)27 b Fu(^)-53 b Fs(\015)2397 961 y Fo(1)2440 946 y Fu(^)g Fs(\015)2487 961 y Fo(0)2526 865 y Fm(\003)2590 946 y Fq(\000)22 b Fs(\025)2746 905 y Fo(2)2786 946 y Fu(\()t(^)-53 b Fs(\015)2875 961 y Fo(1)2936 946 y Fq(\000)26 b Fu(^)-53 b Fs(\015)3086 961 y Fo(0)3125 946 y Fu(\))3163 905 y Fo(2)627 1091 y Fu(=)87 b(^)-53 b Fs(\015)837 1106 y Fo(0)898 1091 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1054 1050 y Fo(2)1049 1116 y(0)1138 1091 y Fu(+)44 b Fs(\025)1315 1010 y Fm(\002)1360 1091 y Fu(^)-53 b Fs(\015)1407 1106 y Fo(1)1468 1091 y Fq(\000)27 b Fu(^)-53 b Fs(\015)1624 1050 y Fo(2)1619 1116 y(1)1686 1091 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1836 1106 y Fo(0)1897 1091 y Fu(+)26 b(^)-53 b Fs(\015)2051 1050 y Fo(2)2046 1116 y(0)2113 1091 y Fu(+)22 b(\()t(^)-53 b Fs(\015)2300 1106 y Fo(1)2361 1091 y Fq(\000)26 b Fu(^)-52 b Fs(\015)2512 1106 y Fo(0)2551 1091 y Fu(\))2589 1050 y Fo(2)2628 1010 y Fm(\003)2692 1091 y Fq(\000)22 b Fs(\025)2848 1050 y Fo(2)2888 1091 y Fu(\()t(^)-53 b Fs(\015)2977 1106 y Fo(1)3038 1091 y Fq(\000)26 b Fu(^)-52 b Fs(\015)3189 1106 y Fo(0)3228 1091 y Fu(\))3266 1050 y Fo(2)627 1236 y Fu(=)83 b(\(1)22 b Fq(\000)h Fs(\025)p Fu(\))17 b Fs(a)1158 1251 y Fo(0)1241 1236 y Fu(+)44 b Fs(\025)17 b(a)1486 1251 y Fo(1)1570 1236 y Fu(+)44 b(\()p Fs(\025)22 b Fq(\000)g Fs(\025)1963 1195 y Fo(2)2003 1236 y Fu(\))17 b(\()t(^)-53 b Fs(\015)2147 1251 y Fo(1)2207 1236 y Fq(\000)27 b Fu(^)-53 b Fs(\015)2358 1251 y Fo(0)2397 1236 y Fu(\))2435 1195 y Fo(2)3135 1236 y Fu(\(I)s(I)s(I.150\))257 1456 y(and)579 1675 y(\()t(^)g Fs(\015)668 1690 y Fo(0)729 1675 y Fq(\000)26 b Fu(^)-52 b Fs(\015)880 1690 y Fo(1)919 1675 y Fu(\))957 1634 y Fo(2)1079 1675 y Fu(=)87 b(^)-53 b Fs(\015)1294 1634 y Fo(2)1289 1700 y(0)1356 1675 y Fu(+)25 b(^)-52 b Fs(\015)1510 1634 y Fo(2)1505 1700 y(1)1571 1675 y Fq(\000)26 b Fu(^)-52 b Fs(\015)1722 1690 y Fo(0)1764 1675 y Fu(^)g Fs(\015)1812 1690 y Fo(1)1873 1675 y Fq(\000)26 b Fu(^)-52 b Fs(\015)2024 1690 y Fo(1)2066 1675 y Fu(^)g Fs(\015)2114 1690 y Fo(0)1079 1820 y Fu(=)87 b(^)-53 b Fs(\015)1294 1779 y Fo(2)1289 1845 y(0)1356 1820 y Fu(+)25 b(^)-52 b Fs(\015)1505 1835 y Fo(1)1566 1820 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1716 1835 y Fo(0)1759 1820 y Fu(^)g Fs(\015)1806 1835 y Fo(1)1868 1820 y Fq(\000)26 b Fu(^)-53 b Fs(\015)2018 1835 y Fo(1)2061 1820 y Fu(^)g Fs(\015)2108 1835 y Fo(0)2169 1820 y Fq(\000)23 b Fs(a)2320 1835 y Fo(1)1079 1966 y Fu(=)87 b(^)-53 b Fs(\015)1289 1981 y Fo(0)1328 1966 y Fu(\()p Fn(1)22 b Fq(\000)k Fu(^)-52 b Fs(\015)1595 1981 y Fo(1)1634 1966 y Fu(\))t(^)f Fs(\015)1723 1981 y Fo(0)1784 1966 y Fq(\000)26 b Fu(^)-52 b Fs(\015)1935 1981 y Fo(0)1977 1966 y Fu(^)g Fs(\015)2025 1981 y Fo(1)2064 1966 y Fu(\()p Fn(1)22 b Fq(\000)k Fu(^)-53 b Fs(\015)2330 1981 y Fo(0)2369 1966 y Fu(\))22 b(+)k(^)-53 b Fs(\015)2578 1981 y Fo(1)2618 1966 y Fu(\()p Fn(1)22 b Fq(\000)k Fu(^)-53 b Fs(\015)2884 1981 y Fo(0)2923 1966 y Fu(\))22 b Fq(\000)h Fs(a)3134 1981 y Fo(1)1079 2111 y Fu(=)83 b Fs(b)23 b Fq(\000)f Fs(a)1452 2126 y Fo(1)1492 2111 y Fs(:)1616 b Fu(\(I)s(I)s(I.151\))257 2330 y(Since)34 b(\()t(^)-53 b Fs(\015)601 2345 y Fo(0)662 2330 y Fq(\000)26 b Fu(^)-52 b Fs(\015)813 2345 y Fo(1)852 2330 y Fu(\))890 2294 y Fo(2)957 2330 y Fq(\025)28 b Fu(0,)k(this)i(implies)g(that)1447 2549 y(0)60 b Fq(\024)h Fs(a)1745 2564 y Fo(1)1845 2549 y Fq(\024)g Fs(b)f Fq(\024)h Fn(1)p Fs(;)830 b Fu(\(I)s(I)s(I.152\))257 2769 y(and)33 b(b)m(y)h(inserting)g(\(I)s(I)s(I.151\))e(in)m(to)h(\(I)s (I)s(I.150\))o(,)g(w)m(e)h(obtain)1040 2988 y Fs(a)1091 3003 y Fr(\025)1196 2988 y Fu(=)60 b(\(1)22 b Fq(\000)h Fs(\025)p Fu(\))17 b Fs(a)1704 3003 y Fo(0)1787 2988 y Fu(+)44 b(\()p Fs(\025)22 b Fq(\000)h Fs(\025)2181 2947 y Fo(2)2220 2988 y Fu(\))17 b Fs(b)45 b Fq(\000)f Fs(\025)2539 2947 y Fo(2)2595 2988 y Fs(a)2646 3003 y Fo(1)2686 2988 y Fs(:)422 b Fu(\(I)s(I)s(I.153\))257 3218 y(Recall)41 b(that)j(^)-53 b Fs(\015)829 3233 y Fo(0)909 3218 y Fq(2)1025 3192 y Fm(b)1017 3218 y Fk(A)1088 3233 y Fr(n)1176 3218 y Fu(is)41 b(a)f(minimizer)i(of)1967 3193 y Fm(b)1953 3218 y Fq(E)2006 3233 y Fr(H)5 b(F)2127 3218 y Fu(,)42 b(i.e.,)2405 3193 y Fm(b)2390 3218 y Fq(E)2443 3233 y Fr(H)5 b(F)2565 3218 y Fu(\()t(^)-53 b Fs(\015)2654 3233 y Fo(0)2693 3218 y Fu(\))41 b(=)2908 3193 y Fm(b)2889 3218 y Fs(E)2961 3233 y Fr(H)5 b(F)3082 3218 y Fu(\()p Fs(N)r(=)p Fu(2\).)67 b(If)257 3338 y Fs(n)54 b(=)-60 b Fq(2)43 b Fi(N)e Fu(then)k(^)-52 b Fs(\015)862 3353 y Fo(0)942 3338 y Fu(cannot)41 b(p)s(ossibly)i(b)s(e)e(an)g(orthogonal) f(pro)5 b(jection,)45 b(and)c(\(I)s(I)s(I.135\))257 3459 y(holds)34 b(true)g(trivially)-8 b(.)45 b(So,)34 b(w)m(e)g(ma)m(y)g (henceforth)g(assume)g(that)f Fs(n)c Fq(2)g Fi(N)p Fu(,)k(i.e.,)h Fs(N)39 b Fu(=)28 b(2)p Fs(n)257 3579 y Fu(is)k(an)e(ev)m(en)j(in)m (teger)e(and)k(^)-53 b Fs(\015)1267 3594 y Fo(0)1337 3579 y Fu(an)31 b(orthogonal)f(pro)5 b(jection)31 b(of)g(rank)g Fs(n)d Fu(=)f Fs(N)r(=)p Fu(2.)43 b(In)31 b(fact,)257 3700 y(according)i(to)g(Lemma)g(3,)1503 3820 y(^)-53 b Fs(\015)1550 3835 y Fo(0)1650 3820 y Fu(=)60 b Fn(1)1842 3835 y Fr(n)1889 3739 y Fm(\002)1952 3795 y(b)1930 3820 y Fs(H)8 b Fu(\()t(^)-53 b Fs(\015)2108 3835 y Fo(0)2147 3820 y Fu(\))2185 3739 y Fm(\003)2226 3820 y Fs(;)882 b Fu(\(I)s(I)s(I.154\))257 3994 y(is)47 b(a)e(rank-)p Fs(n)h Fu(pro)5 b(jection)46 b(on)m(to)g(the)g Fs(n)g Fu(lo)m(w)m(est)h(eigen)m(v)-5 b(alues)48 b(\(coun)m(ting)e(m)m (ultiplici-)257 4124 y(ties\))40 b(of)e(the)h(e\013ectiv)m(e)i (Hamiltonian)1744 4099 y Fm(b)1722 4124 y Fs(H)1803 4139 y Fo(0)1881 4124 y Fu(de\014ned)f(in)f(\(I)s(I)s(I.144\).)61 b(In)40 b(particular,)k(^)-53 b Fs(\015)3456 4139 y Fo(0)257 4255 y Fu(comm)m(utes)35 b(with)960 4229 y Fm(b)938 4255 y Fs(H)1019 4270 y Fo(0)1086 4255 y Fu(:=)1239 4229 y Fm(b)1217 4255 y Fs(H)7 b Fu(\()t(^)-53 b Fs(\015)1394 4270 y Fo(0)1433 4255 y Fu(\).)44 b(Therefore,)480 4484 y(T)-8 b(r)q Fq(f)652 4459 y Fm(b)631 4484 y Fs(H)712 4499 y Fo(0)751 4484 y Fu(\()t(^)-53 b Fs(\015)840 4499 y Fo(1)901 4484 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1051 4499 y Fo(0)1090 4484 y Fu(\))p Fq(g)60 b Fu(=)g(T)-8 b(r)q Fq(f)p Fu(\()1584 4459 y Fm(b)1563 4484 y Fs(H)1644 4499 y Fo(0)1705 4484 y Fq(\000)23 b Fs(e)1850 4499 y Fr(n)1897 4484 y Fu(\)\()t(^)-53 b Fs(\015)2024 4499 y Fo(1)2085 4484 y Fq(\000)26 b Fu(^)-53 b Fs(\015)2235 4499 y Fo(0)2274 4484 y Fu(\))p Fq(g)773 b Fu(\(I)s(I)s(I.155\))563 4639 y(=)83 b(T)-8 b(r)q Fq(f)p Fu(\()932 4614 y Fm(b)911 4639 y Fs(H)992 4654 y Fo(0)1053 4639 y Fq(\000)22 b Fs(e)1197 4654 y Fr(n)1244 4639 y Fu(\))t(^)-53 b Fs(\015)1333 4654 y Fo(0)1372 4639 y Fu(\()t(^)g Fs(\015)1461 4654 y Fo(1)1523 4639 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1673 4654 y Fo(0)1712 4639 y Fu(\))t(^)g Fs(\015)1801 4654 y Fo(0)1840 4639 y Fq(g)44 b Fu(+)g(T)-8 b(r)q Fq(f)p Fu(\()2264 4614 y Fm(b)2243 4639 y Fs(H)2324 4654 y Fo(0)2385 4639 y Fq(\000)22 b Fs(e)2529 4654 y Fr(n)2577 4639 y Fu(\))t(^)-53 b Fs(\015)2671 4598 y Fj(?)2666 4664 y Fo(0)2730 4639 y Fu(\()t(^)g Fs(\015)2819 4654 y Fo(1)2880 4639 y Fq(\000)26 b Fu(^)-53 b Fs(\015)3030 4654 y Fo(0)3069 4639 y Fu(\))t(^)g Fs(\015)3163 4598 y Fj(?)3158 4664 y Fo(0)3222 4639 y Fq(g)563 4794 y Fu(=)83 b(T)-8 b(r)q Fq(f\000)p Fu(\()1010 4769 y Fm(b)988 4794 y Fs(H)1069 4809 y Fo(0)1130 4794 y Fq(\000)23 b Fs(e)1275 4809 y Fr(n)1322 4794 y Fu(\))t(^)-53 b Fs(\015)1411 4809 y Fo(0)1450 4794 y Fu(\()p Fn(1)22 b Fq(\000)k Fu(^)-52 b Fs(\015)1717 4809 y Fo(1)1756 4794 y Fu(\))t(^)f Fs(\015)1845 4809 y Fo(0)1884 4794 y Fq(g)44 b Fu(+)g(T)-8 b(r)p Fq(f)p Fu(\()2308 4769 y Fm(b)2286 4794 y Fs(H)2367 4809 y Fo(0)2428 4794 y Fq(\000)23 b Fs(e)2573 4809 y Fr(n)2620 4794 y Fu(\))t(^)-53 b Fs(\015)2714 4753 y Fj(?)2709 4819 y Fo(0)2793 4794 y Fu(^)h Fs(\015)2841 4809 y Fo(1)2900 4794 y Fu(^)f Fs(\015)2952 4753 y Fj(?)2947 4819 y Fo(0)3011 4794 y Fq(g)563 4950 y Fu(=)83 b(T)-8 b(r)q Fq(fj)922 4924 y Fm(b)901 4950 y Fs(H)982 4965 y Fo(0)1042 4950 y Fq(\000)23 b Fs(e)1187 4965 y Fr(n)1234 4950 y Fq(j)f Fs(b)p Fq(g)p Fs(:)257 5169 y Fu(Moreo)m(v)m(er,)35 b Fs(a)764 5184 y Fo(0)831 5169 y Fu(=)c(^)-53 b Fs(\015)985 5184 y Fo(0)1047 5169 y Fq(\000)26 b Fu(^)-53 b Fs(\015)1202 5133 y Fo(2)1197 5193 y(0)1269 5169 y Fu(=)28 b(0)k(and)h(therefore) 1328 5388 y Fs(a)1379 5403 y Fr(\025)1485 5388 y Fu(=)60 b(\()p Fs(\025)22 b Fq(\000)g Fs(\025)1894 5347 y Fo(2)1934 5388 y Fu(\))17 b Fs(b)44 b Fu(+)g Fs(\025)2251 5347 y Fo(2)2307 5388 y Fs(a)2358 5403 y Fo(1)2398 5388 y Fs(:)710 b Fu(\(I)s(I)s(I.156\))p eop end %%Page: 27 28 TeXDict begin 27 27 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(27)257 573 y(In)33 b(particular,)1355 693 y(\()p Fs(\025)22 b Fq(\000)g Fs(\025)1628 652 y Fo(2)1668 693 y Fu(\))17 b Fs(b)60 b Fq(\024)g Fs(a)2012 708 y Fr(\025)2118 693 y Fq(\024)h Fs(\025)17 b(b;)737 b Fu(\(I)s(I)s(I.157\))257 847 y(b)s(ecause)49 b(0)k Fq(\024)h Fs(a)917 862 y Fo(1)1009 847 y Fq(\024)g Fs(b)p Fu(.)89 b(By)48 b(the)g(op)s(erator)f (monotonicit)m(y)h(of)f(the)h(square)h(ro)s(ot,)257 967 y(\(I)s(I)s(I.157\))32 b(and)h(\(I)s(I)s(I.155\))f(yield)309 1123 y Fm(b)295 1148 y Fq(E)348 1163 y Fr(B)s(H)5 b(F)525 1148 y Fu(\()t(^)-53 b Fs(\015)614 1163 y Fr(\025)659 1148 y Fu(\))22 b Fq(\000)833 1123 y Fm(b)819 1148 y Fq(E)872 1163 y Fr(B)s(H)5 b(F)1050 1148 y Fu(\()t(^)-53 b Fs(\015)1139 1163 y Fo(0)1178 1148 y Fu(\))83 b(=)g Fs(\025)33 b Fu(T)-8 b(r)p Fq(f)1720 1123 y Fm(b)1698 1148 y Fs(H)1779 1163 y Fo(0)1818 1148 y Fu(\()t(^)-53 b Fs(\015)1907 1163 y Fo(1)1968 1148 y Fq(\000)26 b Fu(^)-52 b Fs(\015)2119 1163 y Fo(0)2158 1148 y Fu(\))p Fq(g)49 b(\000)i Fs(\025)2480 1107 y Fo(2)2519 1148 y Fq(D)s Fu(\()t(^)-53 b Fs(\015)2688 1163 y Fo(1)2749 1148 y Fq(\000)26 b Fu(^)-53 b Fs(\015)2899 1163 y Fo(0)2938 1148 y Fs(;)21 b Fu(^)-53 b Fs(\015)3033 1163 y Fo(1)3094 1148 y Fq(\000)27 b Fu(^)-53 b Fs(\015)3245 1163 y Fo(0)3284 1148 y Fu(\))1485 1352 y(+)1599 1285 y Fs(\025)1656 1249 y Fo(2)p 1599 1329 97 4 v 1623 1421 a Fu(2)1705 1352 y Fq(P)8 b Fu(\()t(^)-53 b Fs(\015)1871 1367 y Fo(1)1933 1352 y Fq(\000)26 b Fu(^)-53 b Fs(\015)2083 1367 y Fo(0)2122 1352 y Fs(;)21 b Fu(^)-53 b Fs(\015)2217 1367 y Fo(1)2278 1352 y Fq(\000)26 b Fu(^)-52 b Fs(\015)2429 1367 y Fo(0)2468 1352 y Fu(\))50 b Fq(\000)2693 1285 y Fu(1)p 2693 1329 49 4 v 2693 1421 a(2)2752 1352 y Fq(P)2829 1272 y Fm(\000)2875 1283 y Fq(p)p 2958 1283 97 4 v 69 x Fs(a)3009 1367 y Fr(\025)3054 1352 y Fs(;)3098 1283 y Fq(p)p 3181 1283 V 69 x Fs(a)3232 1367 y Fr(\025)3278 1272 y Fm(\001)1299 1586 y Fu(=)83 b Fs(\025)1515 1475 y Fm(h)1578 1586 y Fu(T)-8 b(r)p Fq(fj)1778 1560 y Fm(b)1756 1586 y Fs(H)1837 1601 y Fo(0)1898 1586 y Fq(\000)23 b Fs(e)2043 1601 y Fr(n)2090 1586 y Fq(j)f Fs(b)p Fq(g)50 b(\000)2418 1518 y Fu(1)p 2418 1563 49 4 v 2418 1654 a(2)2477 1586 y Fq(P)2554 1505 y Fm(\000)2600 1496 y Fq(p)p 2683 1496 42 4 v 90 x Fs(b)q(;)2769 1496 y Fq(p)p 2851 1496 V 2851 1586 a Fs(b)2893 1505 y Fm(\001)2939 1475 y(i)3035 1586 y Fu(+)g Fq(O)3243 1505 y Fm(\000)3289 1586 y Fs(\025)3346 1544 y Fo(2)3385 1505 y Fm(\001)3431 1586 y Fs(:)3135 1762 y Fu(\(I)s(I)s(I.158\))257 1933 y(As)33 b(a)g(result,)570 2143 y Fs(@)621 2158 y Fr(\025)681 2117 y Fm(b)667 2143 y Fq(E)720 2158 y Fr(B)s(H)5 b(F)898 2143 y Fu(\()t(^)-53 b Fs(\015)987 2158 y Fr(\025)1032 2143 y Fu(\))1070 2058 y Fm(\014)1070 2118 y(\014)1103 2182 y Fr(\025)p Fo(=0)1299 2143 y Fu(=)60 b(T)-8 b(r)p Fq(fj)1634 2117 y Fm(b)1613 2143 y Fs(H)1694 2158 y Fo(0)1755 2143 y Fq(\000)22 b Fs(e)1899 2158 y Fr(n)1947 2143 y Fq(j)f Fs(b)p Fq(g)50 b(\000)2275 2075 y Fu(1)p 2275 2120 49 4 v 2275 2211 a(2)2333 2143 y Fq(P)2410 2062 y Fm(\000)2457 2053 y Fq(p)p 2540 2053 42 4 v 90 x Fs(b;)2625 2053 y Fq(p)p 2708 2053 V 90 x Fs(b)2749 2062 y Fm(\001)2795 2143 y Fs(:)313 b Fu(\(I)s(I)s(I.159\))257 2361 y(Next)43 b(w)m(e)g(c)m(ho)s (ose)f Fs(\015)1027 2376 y Fo(1)1110 2361 y Fu(:=)1256 2286 y Fm(P)1361 2313 y Fr(n)p Fj(\000)p Fo(1)1361 2390 y Fr(k)r Fo(=1)1515 2361 y Fq(j)p Fs(')1607 2376 y Fr(k)1649 2361 y Fq(ih)p Fs(')1791 2376 y Fr(k)1833 2361 y Fq(j)28 b Fu(+)h Fq(j)p Fs(')2086 2376 y Fr(n)p Fo(+1)2222 2361 y Fq(ih)p Fs(')2364 2376 y Fr(n)p Fo(+1)2501 2361 y Fq(j)41 b Fu(and)h(recall)g(that)g Fs(\015)3310 2376 y Fo(0)3392 2361 y Fu(:=)257 2407 y Fm(P)363 2433 y Fr(n)363 2511 y(k)r Fo(=1)512 2482 y Fq(j)p Fs(')604 2497 y Fr(k)646 2482 y Fq(ih)p Fs(')788 2497 y Fr(k)830 2482 y Fq(j)p Fu(.)h(With)33 b(this)h(c)m(hoice,)1217 2653 y Fs(b)61 b Fu(=)f Fq(j)p Fs(')1547 2668 y Fr(n)1593 2653 y Fq(ih)p Fs(')1735 2668 y Fr(n)1782 2653 y Fq(j)44 b Fu(+)g Fq(j)p Fs(')2066 2668 y Fr(n)p Fo(+1)2202 2653 y Fq(ih)p Fs(')2344 2668 y Fr(n)p Fo(+1)2481 2653 y Fq(j)p Fs(;)599 b Fu(\(I)s(I)s(I.160\)) 257 2824 y(whose)34 b(insertion)g(in)m(to)f(\(I)s(I)s(I.159\))e (immediately)k(giv)m(es)269 3004 y Fs(@)320 3019 y Fr(\025)380 2979 y Fm(b)366 3004 y Fq(E)419 3019 y Fr(B)s(H)5 b(F)597 3004 y Fu(\()t(^)-53 b Fs(\015)686 3019 y Fr(\025)731 3004 y Fu(\))769 2920 y Fm(\014)769 2980 y(\014)802 3044 y Fr(\025)p Fo(=0)352 3212 y Fu(=)83 b Fs(e)556 3227 y Fr(n)p Fo(+1)716 3212 y Fq(\000)22 b Fs(e)860 3227 y Fr(n)930 3212 y Fq(\000)1029 3077 y Fm(Z)1145 3102 y(\020)1205 3212 y Fq(j)p Fs(')1297 3227 y Fr(n)1344 3212 y Fu(\()p Fs(x)p Fu(\))p Fq(j)1503 3171 y Fo(2)1559 3212 y Fq(j)p Fs(')1651 3227 y Fr(n)1697 3212 y Fu(\()p Fs(y)t Fu(\))p Fq(j)1853 3171 y Fo(2)1913 3212 y Fu(+)g Fq(j)p Fs(')2103 3227 y Fr(n)p Fo(+1)2240 3212 y Fu(\()p Fs(x)p Fu(\))p Fq(j)2399 3171 y Fo(2)2455 3212 y Fq(j)p Fs(')2547 3227 y Fr(n)p Fo(+1)2684 3212 y Fu(\()p Fs(y)t Fu(\))p Fq(j)2840 3171 y Fo(2)2878 3102 y Fm(\021)2937 3212 y Fs(V)g Fu(\()p Fs(x;)17 b(y)t Fu(\))g Fs(dx)g(dy)352 3406 y(<)83 b Fu(0)p Fs(;)2548 b Fu(\(I)s(I)s(I.161\))257 3577 y(thanks)34 b(to)e(\(I)s(I)s(I.145\))o(.)44 b(This)34 b(implies)g(the)f(asserted)h(claim.)942 b Fd(\003)404 3697 y Fu(W)-8 b(e)37 b(close)h(with)g(the)f(remark)h(that)f(if)g (pairing)g(o)s(ccurs)h(then)g(the)f(corresp)s(onding)257 3818 y(1-p)s(dm)c(has)g(full)f(rank.)257 3979 y Fn(Lemma)39 b(6)49 b Fp(Assume)44 b(the)f(same)g(hyp)-5 b(othesis)43 b(as)g(in)g(The)-5 b(or)g(em)42 b(4)h(and)g(additional)5 b(ly)257 4100 y(assume)37 b(that)i Fq(h)p Fs( )t Fq(j)p Fs(V)20 b( )t Fq(i)33 b Fs(>)g Fu(0)p Fp(,)38 b(whenever)f Fs( )f Fq(6)p Fu(=)d(0)p Fp(.)53 b(F)-7 b(urthermor)i(e,)38 b(supp)-5 b(ose)37 b(that)k Fu(^)-52 b Fs(\015)3357 4115 y Fo(0)3429 4100 y Fq(2)265 4205 y Fm(b)257 4231 y Fk(A)328 4246 y Fr(n)414 4231 y Fp(is)39 b(a)f(minimizer)f(of)1211 4206 y Fm(b)1197 4231 y Fq(E)1250 4246 y Fr(H)5 b(F)1372 4231 y Fp(,)39 b(i.e.,)1659 4206 y Fm(b)1645 4231 y Fq(E)1698 4246 y Fr(H)5 b(F)1819 4231 y Fu(\()t(^)-53 b Fs(\015)1908 4246 y Fo(0)1947 4231 y Fu(\))34 b(=)2149 4206 y Fm(b)2130 4231 y Fs(E)2202 4246 y Fr(H)5 b(F)2323 4231 y Fu(\()p Fs(n)p Fu(\))p Fp(,)40 b(wher)-5 b(e)38 b Fs(n)c Fu(:=)h Fs(N)r(=)p Fu(2)p Fp(.)55 b(If)42 b Fu(^)-53 b Fs(\015)3456 4246 y Fo(0)257 4351 y Fp(is)35 b(not)g(a)g(pr)-5 b(oje)g(ction)34 b(then)1554 4471 y Fq(h)p Fs(')p Fq(j)d Fu(^)-53 b Fs(\015)1763 4486 y Fo(0)1819 4471 y Fs(')p Fq(i)62 b Fs(>)g Fu(0)p Fs(;)937 b Fu(\(I)s(I)s(I.162\))257 4625 y Fp(for)35 b(al)5 b(l)35 b Fs(')27 b Fq(2)740 4599 y Fu(^)738 4625 y Fk(h)22 b Fq(n)g(f)p Fu(0)p Fq(g)p Fp(.)257 4786 y Fu(The)38 b(pro)s(of)d(of)h(Lemma)h(6)g(is)g(similar)g(to)f(the)h (corresp)s(onding)g(statemen)m(t)h(in)f([6])f(and)257 4907 y(w)m(as)e(also)f(giv)m(en)g(in)g([11].)257 5027 y Fn(Ac)m(kno)m(wledgmen)m(ts:)69 b Fu(W)-8 b(e)44 b(are)h(grateful)f (to)h(E.)g(Lenzmann)g(and)g(M.)g(Lewin)h(for)257 5147 y(v)-5 b(arious)33 b(useful)g(discussions)i(and)d(for)f(making)i(their) f(man)m(uscript)i([11])e(accessible)j(to)257 5268 y(us)k(prior)e(to)g (publication.)60 b(V.)38 b(B.)g(thanks)g(E.)g(Lieb)g(and)g(H.)g(Sieden) m(top)h(for)e(helpful)257 5388 y(discussions)f(and)c(explanations)i(of) e([6].)p eop end %%Page: 28 29 TeXDict begin 28 28 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(28)257 573 y Ft(References)306 792 y Fu([1])49 b(V.)56 b(Bac)m(h.)114 b(Error)55 b(b)s(ound)i(for)e(the)h(Hartree-Fo)s(c)m(k)g (energy)h(of)f(atoms)g(and)458 912 y(molecules.)45 b Fp(Commun.)34 b(Math.)h(Phys.)g Fu(,)e(147:527{548,)d(1992.)306 1116 y([2])49 b(V.)28 b(Bac)m(h,)h(E.)f(H.)f(Lieb,)i(and)f(J.)f(P)-8 b(.)28 b(Solo)m(v)m(ej.)36 b(Generalized)29 b(Hartree-Fo)s(c)m(k)f (theory)458 1236 y(and)33 b(the)g(Hubbard)g(mo)s(del.)43 b Fp(J.)35 b(Stat.)h(Phys.)e Fu(,)f(76:3{90,)e(1994.)306 1439 y([3])49 b(O.)23 b(Bratteli)h(and)f(D.)g(Robinson.)29 b Fp(Op)-5 b(er)g(ator)26 b(A)n(lgebr)-5 b(as)25 b(and)h(Quantum)g (Statistic)-5 b(al)458 1560 y(Me)g(chanics)31 b(1)p Fu(.)37 b(T)-8 b(ext)30 b(and)g(Monographs)f(in)g(Ph)m(ysics.)j(Springer-V)-8 b(erlag,)30 b(Berlin,)458 1680 y(2)i(edition,)h(1987.)306 1884 y([4])49 b(O.)23 b(Bratteli)h(and)f(D.)g(Robinson.)29 b Fp(Op)-5 b(er)g(ator)26 b(A)n(lgebr)-5 b(as)25 b(and)h(Quantum)g (Statistic)-5 b(al)458 2004 y(Me)g(chanics)31 b(2)p Fu(.)37 b(T)-8 b(ext)30 b(and)g(Monographs)f(in)g(Ph)m(ysics.)j(Springer-V)-8 b(erlag,)30 b(Berlin,)458 2124 y(2)i(edition,)h(1996.)306 2328 y([5])49 b(C.)42 b(F)-8 b(e\013erman)42 b(and)g(R.)g(de)g(la)g (Lla)m(v)m(e.)72 b(Relativistic)43 b(stabilit)m(y)g(of)e(matter)h({)g (I.)458 2448 y Fp(R)-5 b(evista)34 b(Matematic)-5 b(a)35 b(Ib)-5 b(er)g(o)g(americ)g(ana)p Fu(,)31 b(2\(1,)h(2\):119{161,)e (1986.)306 2652 y([6])49 b(R.)28 b(F)-8 b(rank,)30 b(E.H.)f(Lieb,)h(R.) e(Seiringer,)j(and)d(H.)h(Sieden)m(top.)39 b(M)s(\177)-51 b(uller's)29 b(exc)m(hange-)458 2772 y(correlation)43 b(energy)g(in)g(densit)m(y-matrix-functional)i(theory)-8 b(.)74 b Fp(Phys.)44 b(R)-5 b(ev.)44 b(A)p Fu(,)458 2892 y(page)33 b(052517,)e(2007.)306 3096 y([7])49 b(J.)25 b(F)-8 b(r\177)-49 b(ohlic)m(h,)27 b(L.)e(Jonsson,)j(and)d(E.)h (Lenzmann.)32 b(Boson)25 b(stars)h(as)f(solitary)h(w)m(a)m(v)m(es.)458 3216 y Fp(Commun.)34 b(Math.)h(Phys.)p Fu(,)d(274\(1\):1{30,)f(2007.) 306 3420 y([8])49 b(J.)28 b(F)-8 b(r\177)-49 b(ohlic)m(h,)30 b(L.)e(Jonsson,)i(and)f(E.)f(Lenzmann.)38 b(E\013ectiv)m(e)30 b(dynamics)g(for)e(b)s(oson)458 3540 y(stars.)44 b Fp(Nonline)-5 b(arity)p Fu(,)32 b(20:1031{1075,)f(2007.)306 3743 y([9])49 b(J.)41 b(F)-8 b(r\177)-49 b(ohlic)m(h)42 b(and)f(E.)h(Lenzmann.)70 b(Dynamical)42 b(collapse)g(of)f(white)h(dw)m(arfs)g(in)458 3864 y(Hartree-)33 b(and)h(Hartree-Fo)s(c)m(k)g(theory)-8 b(.)47 b Fp(Commun.)34 b(Math.)i(Phys.)p Fu(,)e(274\(3\):737{)458 3984 y(750,)e(2007.)257 4187 y([10])49 b(Ch.)33 b(Hainzl)g(and)g(R.)f (Seiringer.)44 b(General)33 b(decomp)s(osition)h(of)e(radial)g (functions)458 4308 y(on)h Fs(r)641 4272 y Fr(n)720 4308 y Fu(and)g(applications)h(to)f(n-b)s(o)s(dy)g(quan)m(tum)h(systems.)47 b Fp(L)-5 b(ett.)36 b(Math.)f(Phys.)p Fu(,)458 4428 y(61:75{84,)c (2002.)257 4632 y([11])49 b(E.)66 b(Lenzmann)h(and)f(M.)h(Lewin.)144 b(Minimiziers)68 b(for)d(the)h(Hartree-Fo)s(c)m(k-)458 4752 y(Bogoliub)s(o)m(v)33 b(theory)g(of)f(neutron)h(stars.)44 b Fp(arXiv:0809.2560)p Fu(,)31 b(2008.)257 4955 y([12])49 b(E.)35 b(Lieb)g(and)f(H.-T.)h(Y)-8 b(au.)49 b(The)36 b(Chandrasek)-5 b(ar)35 b(theory)g(of)f(stellar)h(collapse)h(as)458 5076 y(the)23 b(limit)g(of)f(quan)m(tum)i(mec)m(hanics.)29 b Fp(Commun.)c(Math.)h(Phys.)p Fu(,)f(112\(1\):147{174,)458 5196 y(1987.)p eop end %%Page: 29 30 TeXDict begin 29 29 bop 257 266 a Fv(BFJ-19-Dec-2008)2404 b Fu(29)257 573 y([13])49 b(E.)88 b(H.)f(Lieb.)207 b(V)-8 b(ariational)87 b(principle)i(for)d(man)m(y-fermion)j(systems.)458 693 y Fp(Phys.)35 b(R)-5 b(ev.)34 b(L)-5 b(ett.)36 b Fu(,)c(46\(7\):457{459,)f(1981.)257 897 y([14])49 b(E.H.)36 b(Lieb.)51 b(Con)m(v)m(ex)37 b(trace)e(functions)h(and)f(the)h (Wigner-Yanase-Dyson)f(con-)458 1017 y(jecture.)44 b Fp(A)-5 b(dv.)35 b(Math.)p Fu(,)e(11:267{288,)e(1973.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0812290351289--