This is a multi-part message in MIME format. ---------------0309050230362 Content-Type: text/plain; name="03-403.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-403.keywords" quantum mechanics, distributions, spectral theorem, Gelfand triplets. ---------------0309050230362 Content-Type: application/postscript; name="damp1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="damp1.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: damp1.dvi %%CreationDate: Fri Sep 05 08:57:21 2003 %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE damp1 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.05:0857 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (damp1.dvi) @start %DVIPSBitmapFont: Fa lasy10 10.95 1 /Fa 1 51 df<007FB812E0B912F0A300F0CAFCB3B3A8B9FCA36C17E0343478B844>50 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy9 9 2 /Fb 2 113 df<007FB712FCB812FEA26C16FC2F047A943C>0 D<1930197819F8A2F001F0 A2F003E0A2F007C0A2F00F80A2F01F00A2183EA260A260A24D5AA24D5AA24D5AA24D5AA2 4DC7FCA2173EA25FA25FA24C5A13C000014B5AEA07E0000F4B5AEA3FF000734B5AEAE3F8 00C14BC8FCEA01FC0000153E7F017E5C137F6D5CA26E485A131F6E485A130F6E485A1307 6E485A13036E48C9FC1301153E14FC01005B14FEEC7EF8147F6E5AA26E5AA26E5AA26E5A 92CAFC3D4C7B8340>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmmi9 9 7 /Fc 7 121 df13 D<123C127EB4FCA21380A2127F123D1201 A412031300A25A1206120E120C121C5A5A126009177A8715>59 D<010FB5D8C03FB5FCA3 9026003FE0C713804B1500A24B5CA2027F14016092C7FCA24A1403605CA201011507605C A20103150F605C91B7FC5B6002F0C7121FA2010F153F605CA2011F157F95C7FC5CA2013F 5D5F5CA2017F14015F91C7FCA24914035F5B00011507B5D8FC03B512F0A340337DB240> 72 D97 D105 D<011F131F90397FC07FE09039E3E1E0F09039C3E380783A01C1 F7007CD981FE133CD983FC133E00035BEB03F0163FEA0707120600025B1200010F147F16 7E5CA2011F14FE16FC5CA2013FEB01F8A291380003F016E0491307ED0FC002801380ED1F 009038FFC03E9038FEE0F89038FC7FE0EC1F80000190C8FCA25BA21203A25BA21207A25B B57EA3283083A027>112 D<90391F801F8090397FE07FE09039E0F0E0703A01C0F9C0F8 3903807D833807007F000E1403000C15F0001C137E0018EC01C002FEC7FC00385B1210C7 FC13015CA31303A25C1640010714E016C0001C5B007E1401010F148000FE1403011FEB07 00011B130E39F839F01C397070F878393FE07FE0390F801F8025227EA02C>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmex9 9 3 /Fd 3 99 df<1430147014E0EB01C0EB03801307EB0F00131E133E133C5B13F85B12015B 1203A2485AA2120F5BA2121F90C7FCA25AA2123EA2127EA5127C12FCB3127C127EA5123E A2123FA27EA27F120FA27F1207A26C7EA212017F12007F13787F133E131E7FEB07801303 EB01C0EB00E0147014301459758223>0 D<12C07E12707E7E121E7E6C7E7F12036C7E7F 12007F1378137CA27FA2133F7FA21480130FA214C0A21307A214E0A5130314F0B314E013 07A514C0A2130FA21480A2131F1400A25B133EA25BA2137813F85B12015B485A12075B48 C7FC121E121C5A5A5A5A14597D8223>I<140C143F4A7E903801FFE0010713F890381FF3 FE90383FC0FF9039FE001FC0D803F8EB07F0D807E0EB01F8D81F80EB007E007CC8EA0F80 00F0ED03C000C015002A0E80B72B>98 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr6 6 2 /Fe 2 51 df<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmti10 10.95 55 /Ff 55 124 df40 D<14031580A2EC01C0EC00E0A21570A215781538 153CA3151EA4151FA2150FA7151FA9153FA2153EA3157EA2157CA215FCA215F8A21401A2 15F0A2140315E0A2140715C0A2EC0F80A2141F15005C143EA25CA25CA2495A5C1303495A 5C130F49C7FC131E5B137C5B5B485A485A485A48C8FC121E5A12705A5A205A7FC325>I< EA01E0EA07F8120FA2EA1FFCA4EA0FF8EA0798EA001813381330A21370136013E013C012 01EA0380EA07001206120E5A5A5A5A5A0E1C7A891C>44 D<387FFFFEA3B5FCA217057995 21>I<120FEA3FC0127FA212FFA31380EA7F00123C0A0A77891C>I<15FE913807FF809138 1F07C091387C01F0ECF000494813F8494813780107147C495A49C7FC167E133E137EA25B A2485AA2000315FEA25B000715FCA2491301120FA34848EB03F8A44848EB07F0A448C7EA 0FE0A316C0007E141F12FE1680153FA2481500A2157EA25DA25D4813015D6C495A127C4A 5A4A5A6C49C7FC143E6C5B380FC1F03803FFC0C648C8FC273F76BC2E>48 D<15031507150F151F151E153E157EEC01FEEC03FC1407141FEB01FF90380FFBF8EB1FC3 EB0E07130015F0A2140FA215E0A2141FA215C0A2143FA21580A2147FA21500A25CA25CA2 1301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CEB7FE0B612F0A215E0203D77 BC2E>I51 D<131EEB3F80137FEBFFC05AA214806C13005B133C90C7FCB3120FEA3FC0127FA2 12FFA35B6CC7FC123C122777A61C>58 D<171C173C177CA217FCA216011603A21607A24C 7EA2161DA216391679167116E1A2ED01C1A2ED038115071601150EA2031C7FA24B7EA25D 15F05D4A5AA24A5AA24AC7FC5C140E5C021FB6FC4A81A20270C7127FA25C13015C495AA2 49C8FCA2130E131E131C133C5B01F882487ED807FEEC01FFB500E0017FEBFF80A25C3941 7BC044>65 D<9339FF8001C0030F13E0033F9038F803809239FF807E07913A03FC001F0F DA0FF0EB071FDA1FC0ECBF00DA7F806DB4FC4AC77E495AD903F86E5A495A130F4948157E 4948157C495A13FF91C9FC4848167812035B1207491670120FA2485A95C7FC485AA3127F 5BA312FF5BA490CCFCA2170FA2170EA2171E171C173C173817786C16706D15F04C5A003F 5E6D1403001F4B5A6D4AC8FC000F151E6C6C5C6C6C14F86C6C495A6C6CEB07C090397FC0 3F8090261FFFFEC9FC010713F0010013803A4272BF41>67 D<49B712C018F818FE903B00 03FE0003FF9438007F804BEC1FC0F00FE0F007F014074BEC03F8F001FCA2140F4BEC00FE A3141F4B15FFA3143F5DA3027F5D5DA219FE14FF92C81203A34917FC4A1507A219F81303 4A150F19F0A20107EE1FE05CF03FC0A2010FEE7F804A16006060011F4B5A4A4A5A4D5AA2 013F4B5A4AEC3FC04DC7FC017F15FEEE03FC4AEB0FF001FFEC7FE0B8128004FCC8FC16E0 403E7BBD45>I<49B812F8A390260003FEC7121F18074B14031801F000F014075DA3140F 5D19E0A2141F4B1338A2EF7801023F027013C04B91C7FCA217F0027F5CED80011603160F 91B65AA3ED001F49EC07805CA3010392C8FC5CF003804C13070107020E14005C93C75A18 0E010F161E4A151C183CA2011F5E5C60A2013F15014A4A5A1707017F150F4D5A4A147F01 FF913807FF80B9FCA295C7FC3D3E7BBD3E>I<49B812F0A390260003FEC7123F180F4B14 03A2F001E014075DA3140F5D19C0A2141F5D1770EFF003023F02E013804B91C7FCA21601 027F5CED8003A2160702FFEB1F8092B5FCA349D9003FC8FC4A7F82A20103140E5CA2161E 0107141C5CA293C9FC130F5CA3131F5CA3133F5CA2137FA25C497EB612E0A33C3E7BBD3B >II<49B648B6FC495DA2D9000390C7000313 004B5D4B5DA2180714074B5DA2180F140F4B5DA2181F141F4B5DA2183F143F4B5DA2187F 147F4B5DA218FF91B8FC96C7FCA292C712015B4A5DA2170313034A5DA2170713074A5DA2 170F130F4A5DA2171F131F4A5DA2173F133F4A5DA2017F157FA24A5D496C4A7EB66CB67E A3483E7BBD44>I<49B6FC5BA2D9000313005D5DA314075DA3140F5DA3141F5DA3143F5D A3147F5DA314FF92C7FCA35B5CA313035CA313075CA3130F5CA3131F5CA3133F5CA2137F A25C497EB67EA3283E7BBD23>I<49B6903807FFFE605ED9000390C7000113E04B6E1300 4B15FC4E5A19E002074B5A4BEC0F804EC7FC183C020F5D4B5C4D5AEF07C0021F4AC8FC4B 131E5F5F023F5C9238C003E0EE07804CC9FC027F5B4B5AEEFF801581ECFF834B7FED0F7F ED1E3F49017C7FECFEF89138FFE01F03C07F491380ED000F4A805C010714074A80A21603 010F815C160183131F4A6D7FA2177F013F825C173F017F82A24A81496C4A7EB6D8800FB5 12C0A261473E7BBD46>75 D<49B612C0A25FD9000390C8FC5D5DA314075DA3140F5DA314 1F5DA3143F5DA3147F5DA314FF92C9FCA35B5CA313035C18C0EF01E0010716C05C170318 80130F4A140718005F131F4A141EA2173E013F5D4A14FC1601017F4A5A16074A131F01FF ECFFF0B8FCA25F333E7BBD39>I<49B5933807FFFC496062D90003F0FC00505ADBBF805E 1A771AEF1407033F923801CFE0A2F1039F020FEE071F020E606F6C140E1A3F021E161C02 1C04385BA2F1707F143C023804E090C7FCF001C0629126780FE0495A02705FF00700F00E 0114F002E0031C5BA2F03803010116704A6C6C5D18E019070103ED01C00280DA03805BA2 943807000F13070200020E5C5FDB03F8141F495D010E4B5CA24D133F131E011CDAF9C05C EEFB80197F013C6DB4C7FC013895C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F049 48B512FE16E01500563E7BBD52>I79 D<49B77E18F018FC903B0003FE0003FEEF00 FF4BEC7F80F03FC00207151F19E05DA2020F16F0A25DA2141FF03FE05DA2023F16C0187F 4B1580A2027FEDFF00604B495A4D5A02FF4A5A4D5A92C7EA3FC04CB4C7FC4990B512FC17 E04ACAFCA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA25C497E B67EA33C3E7BBD3E>II< 49B612FCEFFF8018F0903B0003FE000FF8EF03FE4BEB00FF8419800207ED3FC05DA219E0 140F5DA3021FED7FC05DA2F0FF80143F4B15004D5A60027F4A5A4B495A4D5AEF3F8002FF 02FEC7FC92380007F892B512E01780499038000FE04A6D7E707E707E0103814A130083A2 13075CA25E130F5C5F1603131F5CA3013F020714404A16E05F017F160119C04A01031303 496C1680B6D8800113079438FE0F009338007E1ECAEA3FFCEF07F03B407BBD42>I<9239 0FF001C0ED7FFE4AB5EA0380913907F80FC791390FC003EF91391F8001FF4AC71300027E 805C495A4948143EA2495AA2010F153C5CA3011F1538A38094C7FC80A214FC6DB4FC15F0 15FE6DEBFFC06D14F06D14FC6D80143F020F7F020180EC001F150303007F167F163FA216 1FA212075A5F120EA2001E153F94C7FCA2163E003E157E167C003F15FC4B5A486C5C4B5A 6D495AD87DE0EB1F80D8F8F849C8FC017F13FE39F03FFFF8D8E00F13E048C690C9FC3242 7ABF33>I<48B9FCA25A903AFE001FF00101F89138E0007FD807E0163E49013F141E5B48 C75BA2001E147FA2001C4B131C123C003814FFA2007892C7FC12704A153C00F01738485C C716001403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA25DA214FFA2 92C9FCA25BA25CA21303A25CEB0FFE003FB67E5AA2383D71BC41>I<001FB500F090B512 F0485DA226003FF0C7380FFC004AEC03F04A5D715A017F1503A24A5DA201FF150795C7FC 91C8FCA2485E170E5BA20003161E171C5BA20007163C17385BA2000F167817705BA2001F 16F05F5BA2003F1501A2495DA2007F1503A2495DA2160794C8FC48C8FC5E160E161E6C15 1C163C5E5E5E6C6C13014B5A001F4A5A6C6C011FC9FC6D133E6C6C13F83903FC07F0C6B5 12C0013F90CAFCEB07F83C406FBD44>II<277FFFFE01B500FC90B512E0B5FCA20003902680000790C7380FFC006C 90C701FCEC07F049725A04035EA26350C7FCA20407150EA2040F5D1A3C041F153862163B 6216734F5A6D14E303014B5A6C15C303034BC8FC1683DB0703140E191E030E151C61031C 7F61ED380161157003F04A5A15E002014B5A15C0DA03804AC9FC60DA0700140E60140E60 5C029C5D14B8D97FF85D5C715A5C4A5DA24A92CAFC5F91C7FC705A137E5F137C5F137801 705D53406EBD5B>I<147E49B47E903907C1C38090391F80EFC090383F00FF017E137F49 14804848133F485AA248481400120F5B001F5C157E485AA215FE007F5C90C7FCA2140148 5C5AA21403EDF0385AA21407EDE078020F1370127C021F13F0007E013F13E0003E137FEC F3E1261F01E313C03A0F8781E3803A03FF00FF00D800FC133E252977A72E>97 DIIII<167C 4BB4FC923807C78092380F83C0ED1F87161FED3F3FA2157EA21780EE0E004BC7FCA41401 5DA414035DA30103B512F8A390260007E0C7FCA3140F5DA5141F5DA4143F92C8FCA45C14 7EA414FE5CA413015CA4495AA4495AA4495A121E127F5C12FF49C9FCA2EAFE1EEAF83C12 70EA7878EA3FE0EA0F802A5383BF1C>III<1478EB01FCA21303A314F8EB00E01400AD137C48B4FC38038F 80EA0707000E13C0121E121CEA3C0F1238A2EA781F00701380A2EAF03F140012005B137E 13FE5BA212015BA212035B1438120713E0000F1378EBC070A214F0EB80E0A2EB81C01383 148038078700EA03FEEA00F8163E79BC1C>I<1507ED1FC0A2153FA31680ED0E0092C7FC ADEC07C0EC3FF0EC78F8ECE07CEB01C01303EC807EEB0700A2010E13FE5D131E131CEB3C 01A201005BA21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25CA214 7EA214FEA25CA213015CA2121C387F03F012FF495A5C495A4848C8FCEAF83EEA707CEA3F F0EA0FC0225083BC1C>I108 DIII<903903E001F890390FF807FE90 3A1E7C1E0F80903A1C3E3C07C0013C137801389038E003E0EB783F017001C013F0ED8001 9038F07F0001E015F8147E1603000113FEA2C75AA20101140717F05CA20103140F17E05C A20107EC1FC0A24A1480163F010F15005E167E5E131F4B5A6E485A4B5A90393FB80F80DA 9C1FC7FCEC0FFCEC03E049C9FCA2137EA213FEA25BA21201A25BA21203A2387FFFE0B5FC A22D3A80A72E>I<027E1360903901FF81E0903807C1C390391F80E7C090383F00F7017E 137F5B4848EB3F80485AA2485A000F15005B121F5D4848137EA3007F14FE90C75AA34813 01485CA31403485CA314074A5A127C141F007E133F003E495A14FF381F01EF380F879F39 03FF1F80EA00FC1300143F92C7FCA35C147EA314FE5CA21301130390B512F05AA2233A77 A72A>IIII<137C48B4141C26038F80137EEA07 07000E7F001E15FE121CD83C0F5C12381501EA781F007001805BA2D8F03F130314000000 5D5B017E1307A201FE5C5B150F1201495CA2151F0003EDC1C0491481A2153F1683EE0380 A2ED7F07000102FF13005C01F8EBDF0F00009038079F0E90397C0F0F1C90391FFC07F890 3907F001F02A2979A731>I<017CEB01C048B4EB07F038038F80EA0707000E01C013F812 1E001C1403EA3C0F0038EC01F0A2D8781F130000705BA2EAF03F91C712E012005B017E13 0116C013FE5B1503000115805BA2ED07001203495B150EA25DA25D1578000114706D5B00 00495A6D485AD97E0FC7FCEB1FFEEB03F0252979A72A>I<017C167048B491387001FC3A 038F8001F8EA0707000E01C015FE001E1403001CEDF000EA3C0F0038177C1507D8781F4A 133C00701380A2D8F03F130F020049133812005B017E011F14784C137013FE5B033F14F0 000192C712E05BA2170100034A14C049137E17031880A2EF070015FE170E00010101141E 01F86D131C0000D9039F5BD9FC076D5A903A3E0F07C1E0903A1FFC03FFC0902703F0007F C7FC372979A73C>I<903903F001F890390FFC07FE90393C1E0E0F9026780F1C138001F0 EBB83FD801E013F89039C007F07FEA0380000714E0D9000F140048151C000E4AC7FCA200 1E131FA2C75BA2143F92C8FCA35C147EA314FE4A131CA30101143C001E1538003F491378 D87F811470018314F000FF5D9039077801C039FE0F7C033A7C0E3C078027783C1E1EC7FC 391FF80FFC3907E003F029297CA72A>I<137C48B4143826038F8013FCEA0707000E7F00 1E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F1307020013E012005B 017E130F16C013FE5B151F1201491480A2153F000315005BA25D157EA315FE5D00011301 EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA21407000E5CEA3F80007F495AA2 4A5AD8FF0090C7FC143E007C137E00705B387801F0383803E0381E0FC06CB4C8FCEA03F8 263B79A72C>III E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi6 6 5 /Fg 5 121 df<13F8D803FE1318487E48EB803048EBC060EA3C03397000E0C000601360 48EB71801431C7FCEC3300141B141EA2141CA31418A31438A214301470A45CA35CA21D21 7E9520>13 D 99 D<000F13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E03903C00780A3 EC0F00EA0780A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE0000CEB0780 1F177D9526>110 D<133013785BA4485AA4485AB51280A23803C000485AA448C7FCA412 1EA25B1480383C03001306A25BEA1C38EA0FF0EA07C011217D9F18>116 D<3801F01E3907FC7F80390E1CE1C038180F8100301383007013071260EC0380D8001EC7 FCA45BA21580003014C0397878018012F8EC030038F0FC0638E19C1C387F0FF8381E03E0 1A177D9523>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmex8 8 1 /Fh 1 99 df<143014FCEB03FF010F13C0013F13F090387F03F83901FC00FED807E0EB1F 80D81F80EB07E0007EC7EA01F800F0EC003C00C0150C260C80B027>98 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmr8 8 15 /Fi 15 119 df<013FB5FCA29038007F806EC8FCA6903801FFE0011F13FE90397F3F3F80 D801F8EB07E0D807E0EB01F8D80FC06D7ED81F80147ED83F0080481680A2007E151F00FE 16C0A5007E1680007F153FA26C1600D81F80147ED80FC05CD807E0495AD801F8EB07E0D8 007FEB3F8090261FFFFEC7FC010113E0D9003FC8FCA64A7E013FB5FCA22A2D7CAC33>8 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35AA45A A512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F01370133813 1C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2 EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2EA01 E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>II61 D73 D<15F8141FA214011400ACEB0FE0EB7FF83801F81E3803E007 3807C003380F8001EA1F00481300123E127EA25AA9127C127EA2003E13017EEB8003000F 13073903E00EFC3A01F03CFFC038007FF090391FC0F800222F7EAD27>100 DI<2607C07FEB07F03BFFC3FFC03FFC903AC783 F0783F3C0FCE01F8E01F803B07DC00F9C00F01F8D9FF8013C04990387F000749137EA249 137CB2486C01FEEB0FE03CFFFE0FFFE0FFFEA2371E7E9D3C>109 D<3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F8496C7EA25BA25BB2486C487E3A FFFE1FFFC0A2221E7E9D27>II<3AFFFC01FFC0A23A 0FE0007E000007147C15380003143015706C6C1360A26C6C5BA390387C0180A26D48C7FC A2EB3F07EB1F06A2EB0F8CA214DCEB07D8A2EB03F0A36D5AA26D5A221E7F9C25>118 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmex10 10.95 18 /Fj 18 115 df<140E141E143C147814F01301EB03E0EB07C0A2EB0F80EB1F00A2133E13 7E137C13FC5B1201A2485AA3485AA2120F5BA2121FA25BA2123FA390C7FCA25AA6127E12 FEB3A4127E127FA67EA27FA3121FA27FA2120FA27F1207A26C7EA36C7EA212007F137C13 7E133E7FA2EB0F80EB07C0A2EB03E0EB01F013001478143C141E140E176C72832A>0 D<12E07E12787E7E121F6C7E6C7EA26C7E6C7EA26C7E7F137C137E133E133FA2EB1F80A3 EB0FC0A214E01307A214F0A21303A214F8A31301A214FCA6130014FEB3A414FC1301A614 F8A21303A314F0A21307A214E0A2130F14C0A2EB1F80A3EB3F00A2133E137E137C13FC5B 485AA2485A485AA2485A48C7FC121E5A5A5A5A176C7C832A>II I<12F0B3B3B3A5043B73811E>12 D<17F01601EE03E0EE07C0EE0F80EE1F00163E5E16FC 4B5A4B5A4B5A5E150F4B5A4BC7FCA2157E5D14015D4A5AA24A5A140F5D141F5D143F4AC8 FCA214FEA2495AA2495AA2495AA3495AA2495AA3495AA349C9FCA25B5BA312015BA21203 A25BA21207A25BA2120FA35BA2121FA45BA2123FA65B127FAA48CAFCB3AE6C7EAA123F7F A6121FA27FA4120FA27FA31207A27FA21203A27FA21201A27F1200A37F7FA26D7EA36D7E A36D7EA26D7EA36D7EA26D7EA26D7EA2147FA26E7E141F81140F8114076E7EA26E7E8114 00157E81A26F7E6F7E1507826F7E6F7E6F7E167C8282EE0F80EE07C0EE03E0EE01F01600 2CDA6D8343>18 D<12F07E127C7E7E6C7E6C7E6C7E7F6C7E6C7E137E133E133F6D7E6D7E A26D7E6D7E8013016D7EA2147E147F8081141F816E7EA26E7EA26E7EA26E7EA26E7EA315 7FA26F7EA36F7EA36F7EA2821507A3821503A282A21501A282A21500A282A382A21780A4 163FA217C0A6161F17E0AAEE0FF0B3AEEE1FE0AA17C0163FA61780A2167FA41700A25EA3 5EA21501A25EA21503A25EA215075EA3150F5EA24B5AA34B5AA34BC7FCA215FEA34A5AA2 4A5AA24A5AA24A5AA24A5A5D143F92C8FC5C147E5CA2495A13035C495A495AA2495A49C9 FC133E137E5B485A485A5B485A485A48CAFC123E5A5A5A2CDA7D8343>III<1778EE01F81607161FEE7FE0EEFF8003031300ED07 FC4B5A4B5A4B5A4B5A4B5A93C7FC4A5A14035D14075D140F5DA34A5AB3B3B3A9143F5DA2 147F5DA24AC8FCA2495A13035C495A495A495A495A495A49C9FC485AEA07F8485AEA3FC0 B4CAFC12FCA2B4FCEA3FC0EA0FF06C7EEA01FE6C7E6D7E6D7E6D7E6D7E6D7E6D7E801301 6D7EA26E7EA281143FA281141FB3B3B3A96E7EA38114078114038114016E7E826F7E6F7E 6F7E6F7E6F7E6FB4FC03001380EE7FE0EE1FF816071601EE00782DDA758344>26 D88 D90 D<1560EC01F84A7EEC0FFF023F13C091B512F049EB0FF8903907FC03FE903A1F F000FF80D97FC0EB3FE04848C7EA0FF8D803FCEC03FCD80FF0EC00FFD83F80ED1FC000FE C9EA07F00078EE01E00060EE0060341181C333>98 D104 DI<160F167FED01FF1507ED1FFCED7FE0EDFF804A1300EC03FC4A5A4A5A4A5A4A 5A5D147F92C7FCA25C5CB3B3AA13015CA213035C13075C495A131F495A495A49C8FCEA03 FC485AEA1FE0EA7FC048C9FC12FCB4FCEA7FC0EA1FE0EA07F86C7EC6B4FC6D7E6D7E6D7E 130F6D7E801303801301A2801300B3B3AA8080A281143F816E7E6E7E6E7E6E7E6EB4FC6E 1380ED7FE0ED1FFCED07FF1501ED007F160F28A376833D>110 D<12F012FE6C7E13E0EA 3FF8EA0FFCEA03FEC66C7E6D7E6D7E131F6D7E6D7E130380130180A21300B3B3AA8080A2 81143F816E7E140F6E7E81EC01FC6EB4FCED7F80ED1FE0ED0FF8ED03FEED00FF163F16FF ED03FEED0FF8ED1FE0ED7F80EDFF00EC01FCEC07F85D4A5A141F4A5A5D147F92C7FCA25C 5CB3B3AA1301A25C13035C1307495A495A133F495A495AD803FEC8FCEA0FFCEA3FF8EAFF E0138048C9FC12F028A376833D>I<1CC0F301E0A21B03A21CC0A21B07A21C80A31B0FA2 1C00A263A21B1EA21B3EA21B3CA21B7CA21B78A21BF8A263A31A01A263A21A03A263A21A 07A263A21A0FA298C7FCA262A21A1EA31A3EA21A3CA21A7CA21A78A21AF8A262A21901A2 62A31903A262A21907A262A2190FA297C8FCA261A2191EA2193EA2193CA3197CA21978A2 19F8A261A21801A261A21803A261A21807A201085F1318A20138160F137C96C9FC13FC00 015FA2486C161E1207183E120FD80CFF163C5A0038177C123048177838E07F8012C00000 17F8A2606D7E1701A2606D7E1703A260A26D6C1407A260A2170F6D7E95CAFCA36D6C5CA2 171EA2173E6D7E173CA2177CA26D6C1478A217F8A26E6C5BA31601A2DA3FC05BA21603A2 5FEC1FE01607A25FEC0FF0160FA294CBFCA26E6C5AA2161EA3913803FC3EA2163CA29138 01FE7CA21678A216F8EC00FF5EA46F5AA46F5AA56F5AA493CCFC81150E53DB76835B> 114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmmi8 8 24 /Fk 24 123 df11 D<131FD9FFC013304801F0137000076D13604815 E0D9807C13C0391E003C0148011E13800038EB0E03480107130000605CEC030612E0C713 8EEC018C159C159815B815B0A215F05DA35DA25DA21403A44AC7FCA4140EA45CA3141824 2C7F9D24>13 D15 D<13FC13FFEB1FC0130F6D7EA36D7EA2130180A26D7EA3147EA280A36E7EA2140F81 A24A7E143F147FECF3F0EB01E3EB03C190380781F8130F49C67E133E5B49137E485A4848 7F1207485A4848EB1F8048C7FC127E48EC0FC048EC07E000701403232F7DAD29>21 D<90B612F812035A4815F03A1E0380C000003C130000701301130700E05CEAC00638000E 03A3131CA2133C140713381378A201F07FA21201A2D803E07FA20007130313C0A26C486C 5A251E7E9C29>25 D<0103B512F0131F137F90B612E03A01FC1F80003903F00FC03807C0 0748486C7E121F1300123EA25AA2140700FC5C5AA2140F5D141F92C7FC143E0078133C14 7C007C5B383C01E0381F07C0D807FFC8FCEA01F8241E7D9C28>27 D<15C0140114031580A214071500A25C140EA2141E141CA2143C143814781470A214F05C A213015CA213035C130791C7FCA25B130EA2131E131CA2133C1338A21378137013F05BA2 12015BA212035BA2120790C8FC5A120EA2121E121CA2123C1238A212781270A212F05AA2 1A437CB123>61 D<147F903801FFE090380780F890380E003C497F497F49148001781307 017C14C001FC130316E0A2137090C7FC16F0A314FE903807FF8390381F01C390397C00E7 E049137748481337D807E0133F49131F484814C0121F48C7FCA2481580127EA2ED3F0012 FE48147EA2157C15FC5D4A5A007C495AA26C495A001E49C7FC6C133E3807C0F83803FFE0 38007F8024307DAE25>64 D<1670A216F01501A24B7EA21507150DA2151915391531ED61 FC156015C0EC0180A2EC03005C14064A7F167E5C5CA25C14E05C4948137F91B6FC5B0106 C7123FA25B131C1318491580161F5B5B120112031207000FED3FC0D8FFF8903807FFFEA2 2F2F7DAE35>I<013FB71280A2D900FEC7127F170F4A1407A20101150318005CA21303A2 5C16300107147094C7FC4A136016E0130F15019138C007C091B5FC5BECC0074A6C5AA213 3FA20200EB000CA249151C92C71218017E1538173001FE15705F5B4C5A000115034C5A49 140F161F00034AB4C7FCB8FC5E312D7DAC34>69 D<90273FFFFC0FB5FCA2D900FEC7EA3F 80A24A1500A201015D177E5CA2010315FE5F5CA2010714015F5CA2010F14035F5C91B6FC 5B9139C00007E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7FC5BA20001 5D167E5BA2000315FEB539E03FFFF8A2382D7CAC3A>72 D<90383FFFFEA2010090C8FC5C 5CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7EA0180A2 4914031700017E5C160601FE140EA2495C163C12015E49EB01F84B5A0003141FB7FC5E29 2D7DAC30>76 D97 D99 D101 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8 EA0E3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035BA21207EB C04014C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18> 105 D<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F003FC EC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13F8A2 EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038ECF8 3000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>107 D<137CEA0FFCA21200A213 F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123E A2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E 2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01 263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A4914800081021F 130300015F491400A200034A13076049133E170F0007027EEC8080188149017C131F1801 000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB 03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B801F3870 7F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF80816 18EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E 143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C000 0713F000069038E003E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C013 3EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FF C7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<130E131FA25BA2133EA2137EA2137CA213FCA2B512F8A23801F800A25BA21203A25BA2 1207A25BA2120FA25BA2001F1310143013001470146014E0381E01C0EB0380381F0700EA 0F0EEA07FCEA01F0152B7EA919>116 D<013F137C9038FFC1FF3A01C1E383803A0380F7 03C0390700F60F000E13FE4813FC12180038EC0700003049C7FCA2EA200100005BA31303 5CA301075B5D14C000385CD87C0F130600FC140E011F130C011B131C39F03BE038D87071 13F0393FE0FFC0260F803FC7FC221F7E9D28>120 D<011E1330EB3F809038FFC07048EB E0E0ECF1C03803C0FF9038803F80903800070048130EC75A5C5C5C495A495A49C7FC131E 13385B491340484813C0485A38070001000EEB0380380FE007391FF81F0038387FFF486C 5A38601FFC38E00FF038C003C01C1F7D9D21>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl msbm10 10.95 3 /Fl 3 83 df67 D78 D<007FB612FCB812C06C16F83B03E007C07FFE0000903A0F001F7F80020E9038078FC093 380383E0EFC0F0040113788484EFE00E1600180F84A760180E0401131EEFC01C183C0403 5BEF81F093380787E093381F7FC04BB5C7FC020FB512FC17C004F7C8FC91390E1C078092 381E03C0ED0E01030F7FED078003037FEEC078923801E0380300133C707EEE780EEE380F 93383C0780EE1E03040E7F93380F01E093380780F004031370EFC078706C7E04007F717E 943878078094383803C00003D90F8090383C01E0007FB500FE90381FFFFCB6806C823E3E 7EBD39>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi10 10.95 41 /Fm 41 123 df11 D13 DII<133F14E0EB07F0EB03FC1301 6D7EA3147FA26E7EA36E7EA36E7EA36E7EA36E7EA26E7EA36E7EA3157FA36F7E157F15FF 4A7F5C913807CFE0EC0F8FEC1F0F91383E07F0147C14FC49486C7EEB03F0EB07E049486C 7EEB1F80EB3F00496D7E13FE4848147F485A485A4848EC3F80485A123F4848EC1FC048C8 FC4816E048150F48ED07F0007015032C407BBE35>21 D<011FB612FE017F15FF48B8FC5A 4816FE3B0FC03801C000EA1F00003E1403003C01785B4813705AECF0075AC712E0010191 C7FCA25DEB03C0A313071480A2010F5BA2EB1F0082A2133EA2137E825B150F0001815B12 0315075BC648EB038030287DA634>25 D<020FB512FE027F14FF49B7FC1307011F15FE90 3A3FE03FE00090387F000F01FE6D7E4848130348488048481301485A5B121F5B123F90C7 FC5A127EA2150300FE5D5AA24B5AA2150F5E4B5AA2007C4AC7FC157E157C6C5C001E495A 001FEB07E0390F800F802603E07EC8FC3800FFF8EB3FC030287DA634>27 D<011FB612C090B7FC5A5A481680260FC007C8FC48C65A123E003C130E48131E5A5AA2C7 5AA3147CA2147814F8A4495AA31303A25CA21307A3495AA3131FA25C6DC9FC2A287DA628 >I<1678A21670A216F0A25EA21501A25EA21503A25EA21507A293C7FCA25DA2150EEDFF C0020F13FC91383F9E3F903A01F81C0FC0D903E0EB03E0903A0FC03C01F0D91F00EB00F8 017E0138137C5B48480178133E485A48480170133F120F4901F0131F485A5D48C7FC0201 143F5A007E5CA20203147F00FE167E485C17FE020714FC1601007C020013F8EE03F0007E 49EB07E0A2003E010EEB0FC0003FED1F806C011EEB3F00D80F80147C3A07C01C01F8D803 E0EB03E03A01F03C1F80D8007E01FEC7FC90381FFFF801011380D90078C8FCA21470A214 F0A25CA21301A25CA21303A25CA21307A230527CBE36>30 D32 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919> 58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313 005A120E5A1218123812300B1C798919>I<180E183F18FFEF03FEEF0FF8EF3FE0EFFF80 933803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED1FF8ED7FE0913801FF80DA07FEC8FCEC 1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA 7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FE91 3801FF809138007FE0ED1FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3F E0EF0FF8EF03FEEF00FF183F180E383679B147>II<127012FCB4FCEA7FC0EA1FF0 EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF8EC07FE913801FF8091 38007FE0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF 03FEEF00FFA2EF03FEEF0FF8EF3FE0EFFF80933803FE00EE0FF8EE3FE0EEFF80DB03FEC7 FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1FF8EC7FC04948C9FCEB07FCEB1FF0EB7FC0 4848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270383679B147>I<15FF020713E091381F 00F80278133E4A7F4948EB0F804948EB07C04948EB03E091C7FC496CEB01F002E014F813 1F160017FCA25C0107C812FE90C9FCA7EC03FCEC3FFF9138FE03C1903903F000E1494813 71D91F80133149C7123B017EEC1BFC5B0001151F4848140F484815F8A2485A121F17F048 5A161F17E0127F5BEE3FC0A200FF168090C8127F1700A216FEA2484A5A5E007E1403007F 4A5A5E6C4A5A6C6C495A4BC7FC6C6C13FE6C6C485A3903F80FF06CB512C06C6C90C8FCEB 0FF82F437CC030>64 D<17075F84171FA2173F177FA217FFA25E5EA24C6C7EA2EE0E3F16 1E161C1638A21670A216E0ED01C084ED0380171FED07005D150E5DA25D157815705D844A 5A170F4A5A4AC7FC92B6FC5CA2021CC7120F143C14384A81A24A140713015C495AA249C8 FC5B130E131E4982137C13FED807FFED1FFEB500F00107B512FCA219F83E417DC044>I< DC1FF81307923801FFFE030F9038FF800E923A7FF007E01E4A48C7EAF03EDA03FCEC787E DA0FF0EC3CFCDA3FC0141F4A48140F4AC8FC4948ED07F8EB07F849481503131F4A16F049 481501495A13FF4890C913E05B1203485A19C0485AA2485A95C7FC123F5BA2127F5BA312 FF5BA590CCFC183CA21838A21878187018F06C6C5E17014D5A003F5F6D15074DC7FC001F 161E6C6C5D6D5D6C6C5D00034B5AD801FEEC07C06C6C4AC8FCD97FC0137E90391FF803F8 0107B512E0010114809026001FF8C9FC40427BBF41>67 D<49B912C0A3D9000190C71201 F0003F4B151F190F1A80020316075DA314075D1A00A2140F4BEB0380A205075B021FED00 0E4B92C7FC5FA2023F141E5D173EEE01FE4AB55AA3ED800102FF6D5A92C71278A3491570 5C191C05F0133C01034B13384A167894C71270A2010717F04A5E180161010F16034A4B5A A2180F011F4CC7FC4A5D187E013F16FE4D5A4A140F017F15FFB95AA260423E7DBD43>69 D<49B9FCA3D9000190C7120718004B157F193F191E14035DA314075D191CA2140F5D1707 4D133C021F020E13384B1500A2171E023F141C4B133C177C17FC027FEB03F892B5FCA391 39FF8003F0ED00011600A2495D5CA2160101035D5CA293C9FC13075CA3130F5CA3131F5C A2133FA25C497EB612F8A3403E7DBD3A>II<49B6D8C03FB512F81BF01780D900010180C7383FF000 93C85B4B5EA2197F14034B5EA219FF14074B93C7FCA260140F4B5DA21803141F4B5DA218 07143F4B5DA2180F4AB7FC61A20380C7121F14FF92C85BA2183F5B4A5EA2187F13034A5E A218FF13074A93C8FCA25F130F4A5DA21703131F4A5DA2013F1507A24A5D496C4A7EB6D8 E01FB512FCA2614D3E7DBD4C>I<49B612F0A3D900010180C7FC93C8FC5DA314035DA314 075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9FCA35B5C180C181E0103161C5C 183C183813074A1578187018F0130F4AEC01E0A21703011FED07C04A140F171F013FED3F 8017FF4A1303017F021F1300B9FCA25F373E7DBD3E>76 D<49B712F018FF19C0D9000190 C76C7EF00FF84BEC03FC1801020382727E5DA214071A805DA2140F4E13005DA2021F5E18 034B5D1807023F5E4E5A4B4A5A4E5A027F4B5A06FEC7FC4BEB03FCEF3FF091B712C005FC C8FC92CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C497E B612E0A3413E7DBD3A>80 DI<48B912FCA25A91 3A0003FE000F01F84A1301D807E0EE00F8491307491778000F5D90C7FC001E140FA2001C 4B1470123C0038141FA200785D1270033F15F000F018E0485DC81600157FA25EA215FFA2 93C9FCA25CA25DA21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA2 14FF497F001FB612FCA25E3E3D7FBC35>84 D<007FB500F090387FFFFE19FC5D26007FE0 C7000313804A913800FC004A5D187001FF16F0A291C95AA2481601605BA200031603605B A20007160795C7FC5BA2000F5E170E5BA2001F161E171C5BA2003F163C17385BA2007F16 78A2491570A200FF16F0A290C95AA216015F5A16035F16074CC8FC160E161E5E007F5D5E 6C4A5A6D495A6C6C495A6C6C011FC9FC6C6C137E3903FC03F8C6B512E0013F1380D907FC CAFC3F407ABD3E>I<027FB5D88007B512C091B6FCA2020101F8C7EBF8009126007FE0EC 7F804C92C7FC033F157C701478616F6C495A4E5A6F6C495A4EC8FC180E6F6C5B606F6C5B 6017016F6C485A4D5A6F018FC9FC179E17BCEE7FF85F705AA3707EA283163F167FEEF7FC ED01E7EEC3FEED0383ED070392380E01FF151E4B6C7F5D5D4A486D7E4A5A4A486D7E92C7 FC140E4A6E7E5C4A6E7E14F0495A49486E7E1307D91F806E7ED97FC014072603FFE0EC1F FF007F01FC49B512FEB55CA24A3E7EBD4B>88 D97 D 100 DI<163EEEFFC0923803E1E0923807C0F0ED0F 811687ED1F8F160F153FA217E092387E038093C7FCA45DA514015DA30103B512FCA39026 0003F0C7FCA314075DA4140F5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413015CA4 495AA35CEA1E07127F5C12FF495AA200FE90C9FCEAF81EEA703EEA7878EA1FF0EA07C02C 537CBF2D>I<143C14FEA21301A314FCEB00701400AD137E3801FF803803C7C0EA070300 0F13E0120E121C13071238A2EA780F007013C0A2EAF01F14801200133F14005B137EA213 FE5BA212015B0003130E13F0A20007131EEBE01CA2143CEBC0381478147014E013C13803 E3C03801FF00EA007C173E7EBC1F>105 D107 D110 D112 D<147014FC1301A25CA21303A25CA21307A25C A2130FA25CA2007FB512F0B6FC15E039001F8000133FA291C7FCA25BA2137EA213FEA25B A21201A25BA21203A25BA21207EC01C013E01403000F1480A2EBC0071500140E141E5C00 0713385C3803E1E03801FF80D8003EC7FC1C3A7EB821>116 D<137C48B4EC03802603C7 C0EB0FC0EA0703000F7F000E151F121C010715801238163FEA780F0070491400A2D8F01F 5C5C0000157E133F91C712FEA2495C137E150113FE495CA215030001161C4914F0A21507 173CEEE038150F031F1378000016706D133F017C017313F0017E01E313E0903A3F03C1F1 C0903A0FFF007F80D901FCEB1F002E297EA734>I120 D<137C48B4EC03802603C7C0EB0FC0EA0703000F7F000E151F001C168013071238163FD8 780F150000705BA2D8F01F5C4A137E1200133F91C712FE5E5B137E150113FE495CA21503 00015D5BA215075EA2150F151F00005D6D133F017C137F017E13FF90393F03DF8090380F FF1FEB01FC90C7123F93C7FCA25DD80380137ED80FE013FE001F5C4A5AA24848485A4A5A 6CC6485A001C495A001E49C8FC000E137C380781F03803FFC0C648C9FC2A3B7EA72D>I< 02F8130ED903FE131ED90FFF131C49EB803C49EBC0784914F090397E07F1E09038F800FF 49EB1FC049EB07800001EC0F006C48131E90C75A5D5D4A5A4A5A4A5A4AC7FC143E14785C 495A495A495A49C8FC011E14E05B5B4913014848EB03C0485AD807F8EB078048B4131F3A 1F87E07F00391E03FFFE486C5B00785CD870005B00F0EB7FC048011FC7FC27297DA72A> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmsy10 10.95 26 /Fn 26 113 df<007FB812FEBAFCA26C17FE3804799847>0 D<1506150FB3A9007FB912 E0BA12F0A26C18E0C8000FC9FCB3A6007FB912E0BA12F0A26C18E03C3C7BBB47>6 D<007FB912E0BA12F0A26C18E0C8000FC9FCB3A6007FB912E0BA12F0A26C18E0C8000FC9 FCB3A915063C3C7BAD47>I<180E183F18FFEF03FEEF0FF8EF3FE0EFFF80933803FE00EE 0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1FF8EC7FC049 48C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FE913801FF809138 007FE0ED1FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03 FEEF00FF183F180E1800AE007FB812FEBAFCA26C17FE384879B947>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007F C0EC1FF0EC07FE913801FF809138007FE0ED1FF8ED03FE923800FF80EE3FE0EE0FF8EE03 FE933800FF80EF3FE0EF0FF8EF03FEEF00FFA2EF03FEEF0FF8EF3FE0EFFF80933803FE00 EE0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED7FE0913801FF80DA07FEC8FCEC1FF8EC7FC0 4948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE 007FB812FEBAFCA26C17FE384879B947>I24 D<0203B612FE023F15FF91 B8FC010316FED90FFEC9FCEB1FE0EB7F8001FECAFCEA01F8485A485A485A5B48CBFCA212 3EA25AA21278A212F8A25AA87EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FE EB7F80EB1FE0EB0FFE0103B712FE010016FF143F020315FE383679B147>26 D<19301978A2197C193CA2193E191EA2191F737EA2737E737EA2737E737E1A7C1A7EF21F 80F20FC0F207F0007FBB12FCBDFCA26C1AFCCDEA07F0F20FC0F21F80F27E001A7C624F5A 4F5AA24F5A4F5AA24FC7FC191EA2193E193CA2197C1978A2193050307BAE5B>33 D49 D<0203B512F8023F14FC91B6FC010315F8D90FFEC8FCEB1FE0EB7F 8001FEC9FCEA01F8485A485A485A5B48CAFCA2123EA25AA21278A212F8A25AA2B812F817 FCA217F800F0CAFCA27EA21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80 EB1FE0EB0FFE0103B612F8010015FC143F020314F82E3679B13D>I<007FB5FCB612F015 FC6C14FFC7000113C09138001FE0ED07F8ED01FCED007E82EE1F80EE0FC01607EE03E0A2 EE01F0A2EE00F8A21778A2177CA2173CA2007FB712FCB8FCA27ECA123CA2177CA21778A2 17F8A2EE01F0A2EE03E0A2EE07C0160FEE1F80EE3F00167E4B5AED07F8ED1FE04AB45A00 7FB6C7FCB612FC15F06C91C8FC2E3679B13D>I<1718173C177CA217F8A2EE01F0A2EE03 E0A2EE07C0160F1780EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2 153E157E157C5DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25C13015C495AA2 495AA2495AA249C9FCA2133EA25BA25BA2485AA2485AA2485A120F5B48CAFCA2123EA25A A25AA25A12602E5474C000>54 D<1518153CA2157CA2903803FC7890380FFFF8EB3E0790 387801F0EBF0004848487ED803C07FD807807FA2390F0003EFA248ECCF80001EEB07C700 3E15C01587A2140F007E15E0007C1403A2141FA2141E00FC013E13F0A2143CA2147CA214 78A214F8A214F01301A214E0A21303A214C0A21307A21480D87C0F14E0A21400007E1407 5BA2D83E1E14C0A2133E001FEC0F80133CD80F7C1400A2495B0007141E00035C00015C49 13F83900F801E03901FE07C090B5C7FCEBE3FCD803E0C8FCA25BA26C5A244D7CC52D>59 D<4AB512FC023FECFFE049B712FC0107EEFF80011F8390277FE1FC0114F02601FC01D900 0F7FD803F003017FD807C09238003FFE260F80036F7ED81F001607487113804883007E4A 6E13C012FE48187F00F019E000C00107163FC7FC5D191FA3140F5DA21AC0A24A5AA2F13F 80A24A5A1A0061197E4AC9FC61A2027E4B5A02FE5E18034A4B5A01015F4E5A4A4BC7FC01 03163E604A5D0107ED03F04AEC07C0EF1F80010F037EC8FC4A495A011FEC0FF04AEB7FC0 DB0FFFC9FC49B512FC90B612E04892CAFC4814F84891CBFC433E7EBD46>68 D72 D<4AB6FC023F15F849B712FE0107EEFF80 011F17E090287FE1FC007F13F02601FC01020313F8D803F0030013FC2607C003ED3FFED8 0F80160FD81F00160748EF03FF484A80127E12FE488300F0130712C0C74915FEA319FC02 0F15014B15F8A2F003F0A2021FED07E04B15C0F00F80F01F00183E4A485C4D5AEF03E0EF 0FC04AC7007FC7FCEE0FFE923807FFF8DA7E1F13C0DAFE3F90C8FCED7FF84BC9FC4948CA FCA35C1303A25C1307A25C130F5CA2131F5C133FA291CBFC5B137EA25B13F013C040437E BD3F>80 D83 D90 D<0060EE018000F0EE03C0B3B3A36C1607 A200781780007C160FA26CEE1F00003F5E6C6C157E6C6C5DD807F0EC03F8D803FCEC0FF0 6CB4EC3FE03B007FF003FF80011FB548C7FC010714F8010114E09026001FFEC8FC32397B B63D>II<153FEC03FFEC0FE0EC3F80EC7E00495A 5C495AA2495AB3AA130F5C131F495A91C7FC13FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA 00FE133F806D7E130F801307B3AA6D7EA26D7E80EB007EEC3F80EC0FE0EC03FFEC003F20 5B7AC32D>102 D<12FCEAFFC0EA07F0EA01FCEA007E6D7E131F6D7EA26D7EB3AA801303 806D7E1300147FEC1FC0EC07FEEC00FFEC07FEEC1FC0EC7F0014FC1301495A5C13075CB3 AA495AA2495A133F017EC7FC485AEA07F0EAFFC000FCC8FC205B7AC32D>I<146014F013 01A214E01303A214C01307A2EB0F80A214005BA2131E133EA25BA2137813F8A25B1201A2 5B1203A2485AA25B120FA290C7FC5AA2123EA2123C127CA2127812F8A41278127CA2123C 123EA27EA27E7FA212077FA26C7EA212017FA212007FA21378137CA27FA2131E131FA27F 1480A2EB07C0A2130314E0A2130114F0A213001460145A77C323>I<126012F07EA21278 127CA2123C123EA27EA27E7FA212077FA26C7EA212017FA212007FA21378137CA27FA213 1E131FA27F1480A2EB07C0A2130314E0A2130114F0A414E01303A214C01307A2EB0F80A2 14005BA2131E133EA25BA2137813F8A25B1201A25B1203A2485AA25B120FA290C7FC5AA2 123EA2123C127CA2127812F8A25A1260145A7BC323>I<126012F0B3B3B3B3B11260045B 76C319>I<1A061A0F1A1FA21A3EA21A7CA21AF8A2F101F0A2F103E0A2F107C0A2F10F80 A2F11F00A2193EA261A261A24E5AA24E5AA24E5AA24E5AA24EC7FCA2183EA260A260A24D 5AA24D5A133801F85E486C15071203D80FFE4B5A121D00394CC8FCEAF1FF00C0163EC67F 017F5D80013F5D80011F4A5A80010F4A5A8001074A5AA26E495A13036E49C9FC13016E13 3E7F6F5A147F6F5A143FEDE1F0141FEDE3E015F391380FF7C015FF6E5BA26E90CAFCA26E 5AA26E5AA215781570485B7A834C>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmbx12 14.4 41 /Fo 41 123 df<157815FC14031407141F14FF130F0007B5FCB6FCA2147F13F0EAF800C7 FCB3B3B3A6007FB712FEA52F4E76CD43>49 DI<9138 0FFFC091B512FC0107ECFF80011F15E090263FF8077F9026FF800113FC4848C76C7ED803 F86E7E491680D807FC8048B416C080486D15E0A4805CA36C17C06C5B6C90C75AD801FC16 80C9FC4C13005FA24C5A4B5B4B5B4B13C04B5BDBFFFEC7FC91B512F816E016FCEEFF80DA 000713E0030113F89238007FFE707E7013807013C018E07013F0A218F8A27013FCA218FE A2EA03E0EA0FF8487E487E487EB57EA318FCA25E18F891C7FC6C17F0495C6C4816E001F0 4A13C06C484A1380D80FF84A13006CB44A5A6CD9F0075BC690B612F06D5D011F15800103 02FCC7FCD9001F1380374F7ACD43>I<177C17FEA2160116031607160FA2161F163F167F A216FF5D5DA25D5DED1FBFED3F3F153E157C15FCEC01F815F0EC03E01407EC0FC01580EC 1F005C147E147C5C1301495A495A5C495A131F49C7FC133E5B13FC485A5B485A1207485A 485A90C8FC123E127E5ABA12C0A5C96C48C7FCAF020FB712C0A53A4F7CCE43>III<121F7F7FEBFF8091B81280A45A1900606060A260606048 5F0180C86CC7FC007EC95A4C5A007C4B5A5F4C5A160F4C5A484B5A4C5A94C8FC16FEC812 014B5A5E4B5A150F4B5AA24B5AA24B5A15FFA24A90C9FCA25C5D1407A2140FA25D141FA2 143FA4147F5DA314FFA55BAC6D5BA2EC3FC06E5A395279D043>I<913807FFC0027F13FC 0103B67E010F15E090261FFC0113F8903A3FE0003FFCD97F80EB0FFE49C76C7E48488048 486E1380000717C04980120F18E0177FA2121F7FA27F7F6E14FF02E015C014F802FE4913 806C7FDBC00313009238F007FE6C02F85B9238FE1FF86C9138FFBFF06CEDFFE017806C4B C7FC6D806D81010F15E06D81010115FC010781011F81491680EBFFE748018115C048D900 7F14E04848011F14F048487F48481303030014F8484880161F4848020713FC1601824848 157F173FA2171FA2170FA218F8A27F007F17F06D151FA26C6CED3FE0001F17C06D157F6C 6CEDFF806C6C6C010313006C01E0EB0FFE6C01FCEBFFFC6C6CB612F06D5D010F15800101 02FCC7FCD9000F13C0364F7ACD43>I<171F4D7E4D7EA24D7EA34C7FA24C7FA34C7FA34C 7FA24C7FA34C8083047F80167E8304FE804C7E03018116F8830303814C7E03078116E083 030F814C7E031F81168083033F8293C77E4B82157E8403FE824B800201835D840203834B 800207835D844AB87EA24A83A3DA3F80C88092C97E4A84A2027E8202FE844A82010185A2 4A820103854A82010785A24A82010F855C011F717FEBFFFCB600F8020FB712E0A55B547B D366>65 D<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA 803F0203DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848 C9FC4901F8824949824949824949824949824990CA7E494883A2484983485B1B7F485B48 1A3FA24849181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C7FA21B 0F6C6D1980A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D6D4C5A 6D6D4C5A6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1FF80203 903AFFE001FFF0020091B612C0033F93C8FC030715FCDB007F14E0040101FCC9FC525479 D261>67 D72 DI80 D<93380FFFC00303B6FC031F15E092B712FC0203D9FC0013FF020F01C0010F13C0 023F90C7000313F0DA7FFC02007F902601FFF0ED3FFE49496F7E49496F7F49496F7F4990 C96C7F4948707F4948707F01FF854A177F48864849717EA24849711380A2481BC04A8348 1BE0A24A83481BF0A3481BF8A291CB7EA3B51AFCAF6C1BF8A26E5FA36C1BF0A36C6D4D13 E0A36C1BC06E5F6C1B806E5F6CDB01FE16006C6D902607FF80495A4C13E06C6D013F6D49 5A017F91267F03F85C6D6C90277C00FC015B6D6C49D97E035B6D01806E485B6D6D48D91F 8F5B6D01E0039F90C7FC6D01F06EB45A6DD9FCF85DDA3FFF6E13F0020F6D4913C0020301 FF90B5C8FC020091B512FC031F180C0303181EDB001FEBE3FE93C7EA01FF74133E74137E 7413FEF2F8077290B5FC1CFCA285A21CF8A2851CF07314E0A27314C0731480731400735B 9638007FF8F21FE0576A79D265>II<91260FFF80130791B500F85B010702FF5B011FEDC03F49EDF07F9026FFFC006D 5A4801E0EB0FFD4801800101B5FC4848C87E48488149150F001F824981123F4981007F82 A28412FF84A27FA26D82A27F7F6D93C7FC14C06C13F014FF15F86CECFF8016FC6CEDFFC0 17F06C16FC6C16FF6C17C06C836C836D826D82010F821303010082021F16801400030F15 C0ED007F040714E01600173F050F13F08383A200788200F882A3187FA27EA219E07EA26C EFFFC0A27F6D4B13806D17006D5D01FC4B5A01FF4B5A02C04A5A02F8EC7FF0903B1FFFC0 03FFE0486C90B65AD8FC0393C7FC48C66C14FC48010F14F048D9007F90C8FC3C5479D24B >I97 DI<913801FFF8021FEBFF8091B612F0010315FC010F9038C00FFE903A1FFE0001 FFD97FFC491380D9FFF05B4817C048495B5C5A485BA2486F138091C7FC486F1300705A48 92C8FC5BA312FFAD127F7FA27EA2EF03E06C7F17076C6D15C07E6E140F6CEE1F806C6DEC 3F006C6D147ED97FFE5C6D6CEB03F8010F9038E01FF0010390B55A01001580023F49C7FC 020113E033387CB63C>I<4DB47E0407B5FCA5EE001F1707B3A4913801FFE0021F13FC91 B6FC010315C7010F9038E03FE74990380007F7D97FFC0101B5FC49487F4849143F484980 485B83485B5A91C8FC5AA3485AA412FFAC127FA36C7EA37EA26C7F5F6C6D5C7E6C6D5C6C 6D49B5FC6D6C4914E0D93FFED90FEFEBFF80903A0FFFC07FCF6D90B5128F0101ECFE0FD9 003F13F8020301C049C7FC41547CD24B>I<913803FFC0023F13FC49B6FC010715C04901 817F903A3FFC007FF849486D7E49486D7E4849130F48496D7E48178048497F18C0488191 C7FC4817E0A248815B18F0A212FFA490B8FCA318E049CAFCA6127FA27F7EA218E06CEE01 F06E14037E6C6DEC07E0A26C6DEC0FC06C6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE 903A0FFFC03FF8010390B55A010015C0021F49C7FC020113F034387CB63D>IIII<137F497E 000313E0487FA2487FA76C5BA26C5BC613806DC7FC90C8FCADEB3FF0B5FCA512017EB3B3 A6B612E0A51B547BD325>I 107 DIII<913801FFE0021F13FE91B612C0010315F0010F9038 807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F48496D7F4A147F48834890C8 6C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA2 6C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE011F90C7FC903A0FFF80 7FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I<903A3FF001FFE0B5010F 13FE033FEBFFC092B612F002F301017F913AF7F8007FFE0003D9FFE0EB1FFFC602806D7F 92C76C7F4A824A6E7F4A6E7FA2717FA285187F85A4721380AC1A0060A36118FFA2615F61 6E4A5BA26E4A5B6E4A5B6F495B6F4990C7FC03F0EBFFFC9126FBFE075B02F8B612E06F14 80031F01FCC8FC030313C092CBFCB1B612F8A5414D7BB54B>I<912601FFE0EB0780021F 01F8130F91B500FE131F0103ECFF80010F9039F03FC03F499039800FE07F903A7FFE0003 F04948903801F8FF4849EB00FD4849147F4A805A4849805A4A805AA291C87E5AA35B12FF AC6C7EA37EA2806C5EA26C6D5CA26C6D5C6C6D5C6C93B5FC6C6D5B6D6C5B6DB4EB0FEF01 0F9038C07FCF6D90B5120F010114FED9003F13F80203138091C8FCB1040FB61280A5414D 7CB547>I<90397FE003FEB590380FFF80033F13E04B13F09238FE1FF89139E1F83FFC00 03D9E3E013FEC6ECC07FECE78014EF150014EE02FEEB3FFC5CEE1FF8EE0FF04A90C7FCA5 5CB3AAB612FCA52F367CB537>I<903903FFF00F013FEBFE1F90B7FC120348EB003FD80F F81307D81FE0130148487F4980127F90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C 13FF15F86C14FF16C06C15F06C816C816C81C681013F1580010F15C01300020714E0EC00 3F030713F015010078EC007F00F8153F161F7E160FA27E17E07E6D141F17C07F6DEC3F80 01F8EC7F0001FEEB01FE9039FFC00FFC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C 387CB635>I<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6 FCB8FCA426003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0 FC6DEBFFF86D6C5B021F5B020313802A4D7ECB34>II< B600F00107B5FCA5000101F8C8EA7FE06C6DED3F00A2017F163E6E157E013F167C6E15FC 6D5E6F13016D5E8117036D5E6F13076D5E6F130F6D5E6F131F6D93C7FC815F6E6C133E17 7E023F147C6F13FC6E5C16816E5C16C3A26EEBE3E016E76E5C16FF6E5CA26E91C8FCA26F 5AA36F5AA26F5AA26F5AA26F5A6F5A40367DB447>II<007FB500F090387FFFFEA5C66C48C7000F90C7FC6D6CEC07F86D6D 5C6D6D495A6D4B5A6F495A6D6D91C8FC6D6D137E6D6D5B91387FFE014C5A6E6C485A6EEB 8FE06EEBCFC06EEBFF806E91C9FCA26E5B6E5B6F7E6F7EA26F7F834B7F4B7F92B5FCDA01 FD7F03F87F4A486C7E4A486C7E020F7FDA1FC0804A486C7F4A486C7F02FE6D7F4A6D7F49 5A49486D7F01076F7E49486E7E49486E7FEBFFF0B500FE49B612C0A542357EB447>II<001FB8FC1880A3912680007F130001FCC7B5FC01F0495B495D 49495B495B4B5B48C75C5D4B5B5F003E4A90C7FC92B5FC4A5B5E4A5B5CC7485B5E4A5B5C 4A5B93C8FC91B5FC495B5D4949EB0F805B495B5D495B49151F4949140092C7FC495A485E 485B5C485E485B4A5C48495B4815074849495A91C712FFB8FCA37E31357CB43C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmr9 9 55 /Fp 55 123 df<137813FCA212011203EA07F813E0EA0FC0EA1F801300123C5A5A12400E 0E71B326>19 D24 D<14C01301EB0380EB0F00130E5B133C5B5BA2485A485AA21207 5B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C123EA3121E121FA27E7F12 077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300124A79B71E>40 D<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C133C133EA2131E13 1FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C137813F85BA2485A48 5AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<156015F0B3A4007FB812C0B912 E0A26C17C0C800F0C8FCB3A4156033327CAB3C>43 D<123C127EB4FCA21380A2127F123D 1201A412031300A25A1206120E120C121C5A5A126009177A8715>II<123C127E12FFA4127E123C08087A8715>I<1530157815F8A215F01401A215E01403 A215C01407A21580140FA215005CA2143EA2143C147CA2147814F8A25C1301A25C1303A2 5C1307A2495AA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A248 5AA25B120FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601D4B7CB726>I< EB0FE0EB7FFCEBF83E3903E00F803907C007C0EB8003000F14E0391F0001F0A24814F8A2 003E1300007E14FCA500FE14FEB2007E14FCA56CEB01F8A36C14F0A2390F8003E03907C0 07C0A23903E00F803900F83E00EB7FFCEB0FE01F347DB126>I<13075B5B137FEA07FFB5 FC13BFEAF83F1200B3B3A2497E007FB51280A319327AB126>III<000C14C0380FC00F90B5128015005C5C14F014C0D80C18C7FC 90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07E0A2C713F0 1403A215F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C131F003C14806C EB7F00380F80FE3807FFF8000113E038003F801D347CB126>53 D<1230123C003FB6FCA3 4814FEA215FC0070C7123800601430157015E04814C01401EC0380C7EA07001406140E5C 141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA55BA9131C20347CB126 >55 DI<123C 127E12FFA4127E123C1200B0123C127E12FFA4127E123C08207A9F15>58 D<007FB812C0B912E0A26C17C0CCFCAC007FB812C0B912E0A26C17C033147C9C3C>61 D<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E0 7FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA249800118C77E A24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A33336 7DB53A>65 D67 D71 DII78 DII<007FB712FEA390398007 F001D87C00EC003E0078161E0070160EA20060160600E01607A3481603A6C71500B3AB4A 7E011FB512FCA330337DB237>84 DI91 D93 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>III107 DI<2703F01FE013FF00FF 90267FF80313C0903BF1E07C0F03E0903BF3803E1C01F02807F7003F387FD803FE147049 6D486C7EA2495CA2495CB3486C496C487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F0 1FC000FFEB7FF09038F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F 80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914C049 130F16E0ED07F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF13FC 9038F381F89038F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<903807F0 0390383FFC07EBFC0F3901F8038F3807E001000F14DF48486CB4FC497F123F90C77E5AA2 5A5AA9127FA36C6C5B121F6D5B000F5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE0 90C7FCAAED7F8091380FFFFCA326307DA029>I<3803E07C38FFE1FF9038E38F809038E7 1FC0EA07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A2120312 07381FFFFEB5FCA23803F000AF1403A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0 182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F8000 01EC070015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA290 3807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A12 78D8FC03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2 EB000F003C14E00038EB1FC00030EB3F800070137F1500006013FE495A13035CC6485A49 5AA2495A495A49C7FC153013FE485A12035B48481370485A001F14604913E0485A387F00 0348130F90B5FCA21C207E9F22>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmsy6 6 1 /Fq 1 4 df<136013701360A20040132000E0137038F861F0387E67E0381FFF803807FE 00EA00F0EA07FE381FFF80387E67E038F861F038E060700040132000001300A213701360 14157B9620>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 10.95 84 /Fr 84 128 df0 D<010FB612E0A3D900030180C7FCDA 00FEC8FCA8913807FFC0027F13FC903A03FCFE7F80D90FE0EB0FE0D93F80EB03F8D9FE00 EB00FE4848157F4848ED3F804848ED1FC0000F17E04848ED0FF0003F17F8A24848ED07FC A200FF17FEA8007F17FCA26C6CED0FF8A2001F17F06C6CED1FE0000717C06C6CED3F806C 6CED7F006C6C15FED93F80EB03F8D90FE0EB0FE0D903FCEB7F809027007FFFFCC7FC0207 13C0DA00FEC8FCA8913803FF80010FB612E0A3373E7BBD42>8 D<4AB4EB0FE0021F9038 E03FFC913A7F00F8FC1ED901FC90383FF03FD907F090397FE07F80494801FF13FF494848 5BD93F805C137F0200ED7F00EF003E01FE6D91C7FC82ADB97EA3C648C76CC8FCB3AE486C 4A7E007FD9FC3FEBFF80A339407FBF35>11 D<4AB4FC021F13C091387F01F0903901FC00 78D907F0131C4948133E494813FF49485A137F1400A213FE6F5A163893C7FCAA167FB8FC A33900FE00018182B3AC486CECFF80007FD9FC3F13FEA32F407FBF33>I<4AB47E021F13 F791387F00FFEB01F8903807F001EB0FE0EB1FC0EB3F80137F14008101FE80AEB8FCA3C6 48C77EB3AE486CECFF80007FD9FC3F13FEA32F407FBF33>I<133E133F137F13FFA2EA01 FEEA03FCEA07F813F0EA0FE0EA1FC01380EA3E005A5A1270122010116EBE2D>19 D24 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AC121EAB120CC7FCA812 1EEA7F80A2EAFFC0A4EA7F80A2EA1E000A4179C019>33 D<121EEA7F8012FF13C0A213E0 A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C79BE 19>39 D<1430147014E0EB01C0EB03801307EB0F00131E133E133C5B13F85B12015B1203 A2485AA2120F5BA2121F90C7FCA25AA3123E127EA6127C12FCB2127C127EA6123E123FA3 7EA27F120FA27F1207A26C7EA212017F12007F13787F133E131E7FEB07801303EB01C0EB 00E014701430145A77C323>I<12C07E12707E7E121E7E6C7E7F12036C7E7F12007F1378 137CA27FA2133F7FA21480130FA214C0A3130714E0A6130314F0B214E01307A614C0130F A31480A2131F1400A25B133EA25BA2137813F85B12015B485A12075B48C7FC121E121C5A 5A5A5A145A7BC323>I<1506150FB3A9007FB912E0BA12F0A26C18E0C8000FC9FCB3A915 063C3C7BB447>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312 011380120313005A120E5A1218123812300B1C798919>II<121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919>IIIIII< 150E151E153EA2157EA215FE1401A21403EC077E1406140E141CA214381470A214E0EB01 C0A2EB0380EB0700A2130E5BA25B5BA25B5B1201485A90C7FC5A120E120C121C5AA25A5A B8FCA3C8EAFE00AC4A7E49B6FCA3283E7EBD2D>I<00061403D80780131F01F813FE90B5 FC5D5D5D15C092C7FC14FCEB3FE090C9FCACEB01FE90380FFF8090383E03E090387001F8 496C7E49137E497F90C713800006141FC813C0A216E0150FA316F0A3120C127F7F12FFA4 16E090C7121F12FC007015C012780038EC3F80123C6CEC7F00001F14FE6C6C485A6C6C48 5A3903F80FE0C6B55A013F90C7FCEB07F8243F7CBC2D>II<1238123C123F90B612FCA316F85A16F016E00078C712010070EC 03C0ED078016005D48141E151C153C5DC8127015F04A5A5D14034A5A92C7FC5C141EA25C A2147C147814F8A213015C1303A31307A3130F5CA2131FA6133FAA6D5A0107C8FC26407B BD2D>III<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FC B3121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2779A619>I<121EEA7F80A2EAFFC0A4EA 7F80A2EA1E00C7FCB3121E127FEAFF80A213C0A4127F121E1200A412011380A312031300 5A1206120E120C121C5A1230A20A3979A619>I<007FB912E0BA12F0A26C18E0CDFCAE00 7FB912E0BA12F0A26C18E03C167BA147>61 D<15074B7EA34B7EA34B7EA34B7EA34B7E15 E7A2913801C7FC15C3A291380381FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D7E A34A6D7EA349486D7E91B6FCA249819138800001A249C87EA24982010E157FA2011E8201 1C153FA2013C820138151FA2017882170F13FC00034C7ED80FFF4B7EB500F0010FB512F8 A33D417DC044>65 DIIIIIIII<011FB512FCA3D9000713006E5A 1401B3B3A6123FEA7F80EAFFC0A44A5A1380D87F005B007C130700385C003C495A6C495A 6C495A2603E07EC7FC3800FFF8EB3FC026407CBD2F>IIIII< ED7FE0913807FFFE91391FC03F8091397E0007E04948EB03F8D907F0EB00FE4948147F49 486E7E49486E7E49C86C7E01FE6F7E00018349150300038348486F7EA248486F7EA2001F 188049167F003F18C0A3007F18E049163FA300FF18F0AC007F18E06D167FA4003F18C0A2 6C6CEEFF80A36C6C4B1300A26C6C4B5A00035F6D150700015F6C6C4B5A6D5E6D6C4A5A6D 6C4A5A6D6C4AC7FC6D6C14FED901FCEB03F8D9007FEB0FE091391FC03F80912607FFFEC8 FC9138007FE03C427BBF47>II 82 DI<003FB91280A3903AF0 007FE001018090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00 E0A5C81600B3B14B7E4B7E0107B612FEA33B3D7DBC42>IIII89 D91 D93 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A79BD19>95 D97 DI<49B4FC010F13E090383F 00F8017C131E4848131F4848137F0007ECFF80485A5B121FA24848EB7F00151C007F91C7 FCA290C9FC5AAB6C7EA3003FEC01C07F001F140316806C6C13076C6C14000003140E6C6C 131E6C6C137890383F01F090380FFFC0D901FEC7FC222A7DA828>II< EB01FE90380FFFC090383F03F09038FC01F848486C7E4848137E48487F000F158049131F 001F15C04848130FA2127F16E090C7FCA25AA290B6FCA290C9FCA67EA27F123F16E06C7E 1501000F15C06C6C13036DEB07806C6C1400C66C131E017E5B90381F80F8903807FFE001 0090C7FC232A7EA828>II<167C903903F801FF903A1FFF078F8090397E0FDE1F9038F803F83803F001A23B07 E000FC0600000F6EC7FC49137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F8 03E03903FE0FC026071FFFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E 7E6C15E06C810003813A0FC0001FFC48C7EA01FE003E140048157E825A82A46C5D007C15 3E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313 C0293D7EA82D>III<1478EB01FEA2EB03FFA4EB01FEA2EB0078 1400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14FEA214FCEB81F8EA7F01 387C03F0381E07C0380FFF803801FC00185185BD1C>III<2701F801FE14FF00FF902707FFC00313E0913B1E07E00F03 F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F000101805C01FBD900FF 147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287D A74C>I<3901F801FE00FF903807FFC091381E07E091387803F000079038E001F82603F9 C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733 >I<14FF010713E090381F81F890387E007E01F8131F4848EB0F804848EB07C04848EB03 E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA300 1F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90 380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381FFF8091387C0FE09039FD E003F03A07FFC001FC6C496C7E6C90C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8 AAEE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F89039FDE007F09039FC 780FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>I<02FF131C0107EBC0 3C90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC150748481303121F485A 1501485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F6C7E6C6C133DD8007E 137990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<39 01F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB1401EC00FC01FF133049 1300A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FFF8E03807C03F381F00 0F003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA7FF8EBFFC06C13F86C 13FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800E01303A214017E1400 A27E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FFFC38C01FE01D2A7DA8 24>I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6FCA2D801FCC7FCB3A2 15E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FEEB01F81B397EB723>I< D801FC14FE00FF147FA3000714030003140100011400B3A51501A31503120015076DEB06 FF017E010E13806D4913FC90381FC078903807FFE00100903880FE002E297DA733>IIIII<001FB61280A2EBE000018014004948 5A001E495A121C4A5A003C495A141F00385C4A5A147F5D4AC7FCC6485AA2495A495A130F 5C495A90393FC00380A2EB7F80EBFF005A5B484813071207491400485A48485BA248485B 4848137F00FF495A90B6FCA221277EA628>III<001C130E007FEB3F8039FF807FC0A5397F003F80001CEB0E001A0977BD2D>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx10 10.95 43 /Fs 43 123 df12 D40 D<127012F8127C7EEA3F806C7E6C7E12076C7E7F6C7E6C7EA2137F8013 3F806D7EA280130FA280130780A36D7EA4807FA51580B01500A55B5CA4495AA35C130F5C A2131F5CA2495A5C137F91C7FC13FEA2485A485A5B485A120F485A485A003EC8FC5A5A12 70195A7AC329>I45 D 48 D<140F143F5C495A130F48B5FCB6FCA313F7EAFE071200B3B3A8B712F0A5243C78BB 34>I<903803FF80013F13F890B512FE00036E7E4881260FF80F7F261FC0037F4848C67F 486C6D7E6D6D7E487E6D6D7EA26F1380A46C5A6C5A6C5A0007C7FCC8FC4B1300A25E153F 5E4B5AA24B5A5E4A5B4A5B4A48C7FC5D4A5AEC1FE04A5A4A5A9139FF000F80EB01FC495A 4948EB1F00495AEB1F8049C7FC017E5C5B48B7FC485D5A5A5A5A5AB7FC5EA4293C7BBB34 >I<903801FFE0010F13FE013F6D7E90B612E04801817F3A03FC007FF8D807F06D7E82D8 0FFC131F6D80121F7FA56C5A5E6C48133FD801F05CC8FC4B5A5E4B5A4A5B020F5B902607 FFFEC7FC15F815FEEDFFC0D9000113F06E6C7E6F7E6F7E6F7E1780A26F13C0A217E0EA0F C0487E487E487E487EA317C0A25D491580127F49491300D83FC0495A6C6C495A3A0FFE01 FFF86CB65A6C5DC61580013F49C7FC010313E02B3D7CBB34>II<00071538D80FE0EB01F801FE133F90B6FC5E5E5E 5E93C7FC5D15F85D15C04AC8FC0180C9FCA9ECFFC0018713FC019F13FF90B67E020113E0 9039F8007FF0496D7E01C06D7E5B6CC77FC8120F82A31780A21207EA1FC0487E487E12FF 7FA21700A25B4B5A6C5A01805C6CC7123F6D495AD81FE0495A260FFC075B6CB65A6C92C7 FCC614FC013F13F0010790C8FC293D7BBB34>II<121F7F13F890B712F0A45A17E017C0178017005E5E5A007EC7EA01F8 4B5A007C4A5A4B5A4B5A93C7FC485C157E5DC7485A4A5AA24A5A140F5D141F143F5D147F A214FF92C8FC5BA25BA3495AA3130FA5131FAA6D5A6D5A6D5A2C3F7ABD34>II<903801FFE0010F13FC013F13FF90B612C04801E07F489038003FF048 486D7E000F6E7E485A6F7E123F48488081178012FFA217C0A517E0A4007F5CA4003F5C6C 7E5D6C7E00075C3903FF80FB6C13FF6C6C13F36D13C3010F018313C090380008031400A2 4B1380EA03F0487E486C1500487E4B5AA25E151F4B5A495C6C48EBFFE049485B2607FC0F 5B6CB6C7FC6C14FC6C14F06D13C0D90FFEC8FC2B3D7CBB34>II<16FCA24B7EA24B7EA34B7FA24B7FA34B 7FA24B7FA34B7F157C03FC7FEDF87FA2020180EDF03F0203804B7E02078115C082020F81 4B7E021F811500824A81023E7F027E81027C7FA202FC814A147F49B77EA34982A2D907E0 C7001F7F4A80010F835C83011F8391C87E4983133E83017E83017C81B500FC91B612FCA5 463F7CBE4F>65 D<922607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1 FE01039138803FFB499039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F48 491507485B4A1503481701485B18004890CAFC197E5A5B193E127FA349170012FFAC127F 7F193EA2123FA27F6C187E197C6C7F19FC6C6D16F86C6D150119F06C6D15036C6DED07E0 D97FFEED0FC06D6CED3F80010F01C0ECFF006D01F8EB03FE6D9039FF801FFC010091B55A 023F15E002071580020002FCC7FC030713C03F407ABE4C>67 D75 D77 D80 D<903A03FFC001C0011FEBF803017FEBFE0748B6128F4815DF 48010013FFD80FF8130F48481303497F4848EB007F127F49143F161F12FF160FA27F1607 A27F7F01FC91C7FCEBFF806C13F8ECFFC06C14FCEDFF806C15E016F86C816C816C816C16 806C6C15C07F010715E0EB007F020714F0EC003F1503030013F8167F163F127800F8151F A2160FA27EA217F07E161F6C16E06D143F01E015C001F8EC7F8001FEEB01FF9026FFE007 13004890B55A486C14F8D8F81F5CD8F00314C027E0003FFEC7FC2D407ABE3A>83 D<003FB912FCA5903BFE003FFE003FD87FF0EE0FFE01C0160349160190C71500197E127E A2007C183EA400FC183F48181FA5C81600B3AF010FB712F8A5403D7CBC49>I87 D<903807FFC0013F13F848B6FC48812607FE037F260FF8007F6DEB3FF0486C80 6F7EA36F7EA26C5A6C5AEA01E0C8FC153F91B5FC130F137F3901FFFE0F4813E0000F1380 381FFE00485A5B485A12FF5BA4151F7F007F143F6D90387BFF806C6C01FB13FE391FFF07 F36CEBFFE100031480C6EC003FD91FF890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAF EDFFE0020713FC021FEBFF80027F80DAFF8113F09139FC003FF802F06D7E4A6D7E4A1307 4A80701380A218C082A318E0AA18C0A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0 903AFCFF01FFE0496CB55AD9F01F91C7FCD9E00713FCC7000113C033407DBE3A>IIII<13FFB5FCA512077EAFED1FF8EDFFFE02036D7E4A80DA0FE07F 91381F007F023C805C4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A5333F7CBE3A>104 DII< 13FFB5FCA512077EB3B3AFB512FCA5163F7CBE1D>108 D<01FFD91FF8ECFFC0B590B501 0713F80203DAC01F13FE4A6E487FDA0FE09026F07F077F91261F003FEBF8010007013EDA F9F0806C0178ECFBC04A6DB4486C7FA24A92C7FC4A5CA34A5CB3A4B5D8FE07B5D8F03FEB FF80A551297CA858>I<01FFEB1FF8B5EBFFFE02036D7E4A80DA0FE07F91381F007F0007 013C806C5B4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A533297CA83A>II<01FFEBFFE0B5000713FC021FEBFF80027F80DAFF 8113F09139FC007FF8000701F06D7E6C496D7E4A130F4A6D7E1880A27013C0A38218E0AA 4C13C0A318805E18005E6E5C6E495A6E495A02FCEBFFF0DAFF035B92B55A029F91C7FC02 8713FC028113C00280C9FCACB512FEA5333B7DA83A>I<3901FE01FE00FF903807FF804A 13E04A13F0EC3F1F91387C3FF8000713F8000313F0EBFFE0A29138C01FF0ED0FE0913880 07C092C7FCA391C8FCB3A2B6FCA525297DA82B>114 D<90383FFC1E48B512BE000714FE 5A381FF00F383F800148C7FC007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C13FF 15C06C14F06C806C806C806C80C61580131F1300020713C014000078147F00F8143F151F 7EA27E16806C143F6D140001E013FF9038F803FE90B55A15F0D8F87F13C026E00FFEC7FC 222B7DA929>III119 D121 D<003FB612F8A4D9F80113F001C014E0495A494813C04A138000 7E15005C4A5A007C5C147F4A5A495B5DC65A495B495BA249EB007C495A5C137F494813FC 484913F85C5A48EBC00114804814034813004848130749131F007FECFFF0B7FCA426287D A72E>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr10 10 32 /Ft 32 123 df<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A12 06120E5A5A5A12600A19798817>44 DI<121C127FEAFF80A5EA 7F00121C0909798817>I72 DI<003FB812E0A3D9C003EB001F273E0001FE130348EE 01F00078160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011F B612E0A335397DB83C>84 D87 D97 DIIII<147E903803FF8090380FC1E0EB1F8790 383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8 A31C3B7FBA19>IIII< EB01C0EB07F0EB0FF8A5EB07F0EB01C090C7FCAAEB01F813FFA313071301B3B3A2123C12 7E00FF13F01303A214E038FE07C0127C383C0F00EA0FFEEA03F8154984B719>I108 D<2703F00FF0EB1FE000FFD9 3FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F802603F70013CE01 FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B50083B5FCA34025 7EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC03803F70013FE 496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>II<39 03F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB07E049EB03F0 4914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13076DEB0FE0 01F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB512C0A328357E A42E>II<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03E613EE9038 EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>II<1318A51338A31378A313F812 0112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C03017E13006D5A EB0FFEEB01F81A347FB220>IIIII< B538803FFEA33A0FF8000FF06C48EB07C00003EC03806C7E16007F00001406A2017E5BA2 137F6D5BA26D6C5AA2ECC070010F1360A26D6C5AA214F101035BA2D901FBC7FCA214FF6D 5AA2147CA31438A21430A214701460A25CA2EA7C0100FE5B130391C8FC1306EAFC0EEA70 1C6C5AEA1FF0EA0FC027357EA32C>I<003FB512FCA2EB8003D83E0013F8003CEB07F000 38EB0FE012300070EB1FC0EC3F800060137F150014FE495AA2C6485A495AA2495A495A49 5AA290387F000613FEA2485A485A0007140E5B4848130C4848131CA24848133C48C7127C 48EB03FC90B5FCA21F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmbx10 10 7 /Fu 7 117 df65 D97 D<13FFB5FCA412077EAF4AB47E020F13F0023F13FC9138FE03FFDAF000 13804AEB7FC00280EB3FE091C713F0EE1FF8A217FC160FA217FEAA17FCA3EE1FF8A217F0 6E133F6EEB7FE06E14C0903AFDF001FF80903AF8FC07FE009039F03FFFF8D9E00F13E0D9 C00390C7FC2F3A7EB935>I<903801FFC0010F13FC017F13FFD9FF8013802603FE0013C0 48485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12FFA9127FA27F123FED01 E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB45A010F13F00101138023 257DA42A>I<9038FE03F000FFEB0FFEEC3FFF91387C7F809138F8FFC000075B6C6C5A5C A29138807F80ED3F00150C92C7FC91C8FCB3A2B512FEA422257EA427>114 D<90383FF0383903FFFEF8000F13FF381FC00F383F0003007E1301007C130012FC15787E 7E6D130013FCEBFFE06C13FCECFF806C14C06C14F06C14F81203C614FC131F9038007FFE 140700F0130114007E157E7E157C6C14FC6C14F8EB80019038F007F090B512C000F81400 38E01FF81F257DA426>I<130FA55BA45BA25B5BA25A1207001FEBFFE0B6FCA3000390C7 FCB21578A815F86CEB80F014816CEBC3E090383FFFC06D1380903803FE001D357EB425> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmsy8 8 8 /Fv 8 107 df0 D<130C131EA50060EB01800078130739FC0C0F C0007FEB3F80393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FFE03807 CCF8383F8C7F397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C1A1D7C 9E23>3 D<140381B3A3B812FCA3C7D80380C7FCB3B812FCA32E2F7CAD37>6 D<170EA3170F8384170384170184717E1878187C84180FF007C0BA12F819FC19F8CBEA07 C0F00F00183E601878604D5A60170360170795C7FC5F170EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F1380A3EA3F00A312 3E127E127CA35AA35A0F227EA413>48 DI<027F15189026 07FF801478013F6D14F090B514012601F03F15E02607801F1403380F000F001EEE07C05A 007C5C0078EE0F805A48131FC7ED1F0092C7FCA2173E5CA2023E5CA34A5C010FB6FC5B13 7F5F903900F80001A2494813035FA2495A1607A2495A5F5C130F040F130249C7140E183C 011E167C013EEDE078EFFFF04916C0017016000140EC07F837307EAC3C>72 D<12E0B3B3B3AD034378B114>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr12 12 30 /Fw 30 123 df<131F1480133F137FA2EBFF00485A485A5B485A485A138048C7FC123E12 3C5A12E0124011126CC431>19 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413 E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>44 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30FFA203707FED607FA203E07FEDC03F A2020180ED801FA2DA03007F160FA20206801607A24A6D7EA34A6D7EA34A6D7EA2027081 0260147FA202E08191B7FCA249820280C7121FA249C87F170FA20106821707A2496F7EA3 496F7EA3496F7EA201788313F8486C83D80FFF03037FB500E0027FEBFFC0A342477DC649 >65 D67 DI77 D<49B41303010FEBE007013F13F89039 FE00FE0FD801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E8212 7C00FC81A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C 14FC6C14FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13 C0167F163F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D 147ED8FBC05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C000 3FF02B487BC536>83 D85 D97 DII<167FED3FFFA315018182B3EC7F80903803 FFF090380FC07C90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2 127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C 010E13C0013F011C13FE90380FC0F8903803FFE09026007F0013002F467DC436>IIIIII107 DII<3901FC01FE00FF903807FFC091381E07F0913838 01F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CECFF80 B5D8F83F13FEA32F2C7DAB36>II<3901FC 03FC00FF90380FFF8091383C07E091387001F83A07FDE000FE00030180137FD801FFEC3F 8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217 F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381F FF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I<3903F803F000FFEB1FFCEC3C 3EEC707F0007EBE0FF3803F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB5 12FEA3202C7DAB26>114 D<90383FE0183901FFFC383907E01F78390F0003F8001E1301 481300007C1478127800F81438A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C 13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27E A36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F 2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207001FB512F0B6FCA2C648 C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070903807E0E0903801FF C09038007F001E3E7EBC26>III121 D<003FB612E0A29038C0003F90C713C0 003CEC7F800038ECFF00A20030495A0070495AA24A5A0060495AA24A5A4A5AA2C7485A4A C7FC5B5C495A13075C495A131F4A1360495A495AA249C712C0485AA2485A485A1501485A 48481303A24848EB07804848131F00FF14FF90B6FCA2232B7DAA2B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmbx12 17.28 19 /Fx 19 122 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 401 409 a Fx(Quan)l(tum)53 b(Mec)l(hanics)f(of)i(Damp)t(ed)f (Systems)1427 661 y Fw(Dariusz)32 b(Chru)-5 b(\023)-44 b(sci)s(\023)-51 b(nski)2281 625 y Fv(\003)945 777 y Fw(Departmen)m(t)32 b(of)g(Mathematics)h(and)f(Statistics)1397 894 y(Univ)m(ersit)m(y)i(of)e(Calgary)1307 1010 y(Calgary)-8 b(,)32 b(Alb)s(erta,)g(Canada)1672 1446 y Fu(Abstract)375 1598 y Ft(W)-7 b(e)39 b(sho)n(w)f(that)g(the)h(quan)n(tization)e(of)h (a)g(simple)h(damp)r(ed)f(system)h(leads)e(to)h(a)g(self-adjoin)n(t)251 1698 y(Hamiltonian)d(with)i(a)e(family)h(of)f(complex)g(generalized)g (eigen)n(v)-5 b(alues.)59 b(It)36 b(turns)g(out)g(that)g(they)251 1797 y(corresp)r(ond)e(to)i(the)h(p)r(oles)f(of)g(energy)f(eigen)n(v)n (ectors)f(when)i(con)n(tin)n(ued)g(to)g(the)g(complex)g(energy)251 1897 y(plane.)44 b(Therefore,)30 b(the)h(corresp)r(onding)d (generalized)h(eigen)n(v)n(ectors)f(ma)n(y)h(b)r(e)i(in)n(terpreted)e (as)h(res-)251 1996 y(onan)n(t)e(states.)41 b(W)-7 b(e)30 b(sho)n(w)e(that)h(resonan)n(t)f(states)h(are)f(resp)r(onsible)g(for)g (the)i(irrev)n(ersible)d(quan)n(tum)251 2096 y(dynamics)g(of)h(our)e (simple)i(mo)r(del.)24 2449 y Fs(Mathematical)33 b(Sub)6 b(ject)35 b(Classi\014cations)g(\(2000\):)41 b Fr(46E10,)32 b(46F05,)h(46N50,)g(47A10.)24 2633 y Fs(Key)i(w)m(ords:)40 b Fr(quan)m(tum)30 b(mec)m(hanics,)h(distributions,)26 b(sp)s(ectral)k(theorem,)h(Gelfand)f(triplets.)p 24 4855 1465 4 v 126 4908 a Fq(\003)162 4940 y Fp(On)35 b(the)g(lea)n(v)n(e)g (from)h(Institute)f(of)h(Ph)n(ysics,)i(Nicolaus)f(Cop)r(ernicus)f(Univ) n(ersit)n(y)-6 b(,)37 b(ul.)64 b(Grudzi)r(\030)-36 b(adzk)l(a)35 b(5/7,)k(87-100)24 5031 y(T)-6 b(oru)r(\023)-41 b(n,)26 b(P)n(oland)1832 5281 y Fr(1)p eop %%Page: 2 2 2 1 bop 24 44 a Fo(1)134 b(In)l(tro)t(duction)24 247 y Fr(Standard)32 b(textb)s(o)s(oks)h(on)h(quan)m(tum)e(mec)m(hanics)h (in)m(v)m(estigate)i(mainly)c(the)j(Hamiltonian)d(system,)k(i.e.)24 360 y(b)m(y)25 b(a)g(quan)m(tum)g(system)g(one)h(usually)d(means)i(a)h (Hilb)s(ert)d(space)i Fn(H)h Fr(whic)m(h)e(describ)s(es)g(ph)m(ysical)g (quan)m(tum)24 473 y(states)k(and)f(a)h(self-adjoin)m(t)f(op)s(erator)h (\(Hamiltonian\))f(in)f Fn(H)i Fr(whic)m(h)e(go)m(v)m(erns)j(dynamics)d (of)i(the)g(system.)24 586 y(Ho)m(w)m(ev)m(er,)36 b(most)e(of)g(the)f (classical)g(systems)g(are)h(not)g(Hamiltonian)e(and)h(the)g(quan)m (tum)g(mec)m(hanics)h(of)24 699 y(suc)m(h)f(systems)h(is)f(p)s(o)s (orly)f(understo)s(o)s(d.)49 b(In)33 b(the)h(presen)m(t)f(pap)s(er)g(w) m(e)h(are)h(going)e(to)i(in)m(v)m(estigate)g(one)f(of)24 812 y(the)c(simplest)f(non-Hamiltonian)f(system)j(corresp)s(onding)d (to)j(a)g(damp)s(ed)e(motion)h(in)f(one)i(dimension:)1585 1016 y(_)-40 b Fm(x)25 b Fr(=)g Fn(\000)p Fm(\015)5 b(x)30 b(;)1525 b Fr(\(1.1\))24 1220 y(where)29 b Fm(x)c Fn(2)g Fl(R)s Fr(,)36 b(and)30 b Fm(\015)g(>)25 b Fr(0)30 b(stands)g(for)f (the)i(damping)d(constan)m(t.)42 b(Classically)-8 b(,)28 b(the)j(damping)d(b)s(eha)m(vior)24 1333 y(is)h(describ)s(ed)f(b)m(y)j (the)f(exp)s(onen)m(tial)g(la)m(w)1496 1537 y Fm(x)p Fr(\()p Fm(t)p Fr(\))25 b(=)g Fm(e)1814 1500 y Fv(\000)p Fk(\015)t(t)1939 1537 y Fm(x)31 b(:)1451 b Fr(\(1.2\))24 1742 y(As)27 b(is)g(w)m(ell)g(kno)m(wn)h([1])g(\(see)h(also)f([2)q (]\),)h(within)c(the)j(standard)f(Hilb)s(ert)f(space)j(form)m(ulation)d (of)j(quan)m(tum)24 1854 y(mec)m(hanics)i(there)g(is)f(no)h(ro)s(om)f (for)h(suc)m(h)g(a)g(b)s(eha)m(viour)f(on)h(a)g(quan)m(tum)g(lev)m(el.) 42 b(Therefore,)31 b(in)f(order)g(to)24 1967 y(deal)24 b(with)g(this)g(problem,)h(w)m(e)h(shall)d(use)i(the)g(rigged)g(Hilb)s (ert)e(space)j(approac)m(h)f(to)h(quan)m(tum)f(mec)m(hanics)24 2080 y(whic)m(h)32 b(generalizes)i(the)g(standard)g(Hilb)s(ert)e(space) i(v)m(ersion)f([3)q(,)h(4)q(,)g(5].)52 b(A)34 b(rigged)g(Hilb)s(ert)e (space)i(\(or)h(a)24 2193 y(Gelfand)29 b(triplet\))h(is)f(a)i (collection)f(of)g(spaces)h([6)q(,)f(7)q(]:)1506 2397 y(\010)25 b Fn(\032)g(H)h(\032)f Fr(\010)1958 2360 y Fv(0)2011 2397 y Fm(;)1462 b Fr(\(1.3\))24 2602 y(where)30 b Fn(H)h Fr(is)e(a)i(Hilb)s(ert)e(space,)i(\010)f(is)g(a)h(dense)f(n)m (uclear)g(subspace)g(of)h Fn(H)q Fr(,)f(and)g(\010)2832 2569 y Fv(0)2886 2602 y Fr(denotes)g(its)g(dual,)g(i.e.)24 2715 y(the)g(space)h(of)g(con)m(tin)m(uous)f(functionals)e(on)j(\010)f (\(see)h(section)f(2)h(for)f(a)h(brief)e(review\).)165 2827 y(The)43 b(quan)m(tization)h(of)g(our)g(simple)e(mo)s(del)h (\(1.1\))j(leads)d(to)i(a)g(self-adjoin)m(t)e(Hamiltonian)3504 2804 y Fj(b)3482 2827 y Fm(H)51 b Fr(in)24 2940 y Fn(H)26 b Fr(=)e Fm(L)284 2907 y Fi(2)324 2940 y Fr(\()p Fl(R)s Fr(\).)45 b(In)m(terestingly)-8 b(,)1086 2917 y Fj(b)1064 2940 y Fm(H)31 b Fr(b)s(eing)22 b(self-adjoin)m(t,)j(giv)m(es)f(rise)f (to)h(the)g(family)e(of)i(generalized)f(complex)24 3053 y(eigen)m(v)-5 b(alues.)51 b(Clearly)-8 b(,)35 b(these)f(eigen)m(v)-5 b(alues)34 b(are)g(not)h(elemen)m(ts)f(of)g(the)h(sp)s(ectrum)e Fm(\033)s Fr(\()3066 3030 y Fj(b)3044 3053 y Fm(H)7 b Fr(\))32 b(=)f(\()p Fn(\0001)p Fm(;)15 b Fn(1)p Fr(\).)24 3166 y(The)25 b(corresp)s(onding)f(eigen)m(v)m(ectors)k(do)e(not)g(b)s (elong)f(to)h Fm(L)2023 3133 y Fi(2)2063 3166 y Fr(\()p Fl(R)s Fr(\))32 b(but)25 b(to)i(\010)2560 3133 y Fv(0)2609 3166 y Fr(for)e(an)h(appropriately)e(c)m(hosen)24 3279 y(\010.)48 b(W)-8 b(e)35 b(sho)m(w)e(that)h(these)f(complex)g(eigen)m (v)-5 b(alues)33 b(ha)m(v)m(e)h(man)m(y)g(remark)-5 b(able)32 b(prop)s(erties)g(analogous)h(to)24 3392 y(the)38 b(p)s(oin)m(t)f(sp)s (ectrum)g(of)h(a)g(self-adjoin)m(t)g(op)s(erator.)64 b(In)37 b(particular,)i(they)f(giv)m(e)h(rise)e(to)h(the)h(sp)s(ectral) 24 3505 y(decomp)s(osition)28 b(of)747 3482 y Fj(b)725 3505 y Fm(H)7 b Fr(.)41 b(Moreo)m(v)m(er,)32 b(they)e(are)h(closely)e (related)h(to)h(the)f(con)m(tin)m(uous)f(sp)s(ectrum)g(of)3490 3482 y Fj(b)3468 3505 y Fm(H)7 b Fr(.)41 b(It)24 3618 y(turns)26 b(out)h(that)h(they)g(corresp)s(ond)e(to)i(the)g(p)s(oles)e (of)i(the)f(energy)h(eigen)m(v)m(ectors)h Fm( )2871 3585 y Fk(E)2959 3618 y Fr(when)d(con)m(tin)m(ued)h(to)24 3731 y(the)32 b(complex)g(energy)h(plane)f([8].)47 b(Ph)m(ysicists)31 b(usually)f(called)i(the)h(corresp)s(onding)d(eigen)m(v)m(ectors)k (reso-)24 3844 y(nan)m(t)29 b(states)g([3)q(,)g(9,)g(10)q(])g(\(see)h (also)e([11)q(]\).)41 b(It)29 b(is)f(widely)e(b)s(eliev)m(ed)i(that)h (resonan)m(t)g(states)h(are)f(resp)s(onsible)24 3957 y(for)38 b(the)h(irrev)m(ersible)d(dynamics)h(of)i(ph)m(ysical)e (systems)i(\(see)g(e.g.)66 b(recen)m(t)40 b(collection)e(of)h(pap)s (ers)e([3)q(]\).)24 4069 y(Indeed,)29 b(it)h(is)g(true)g(in)f(our)h (simple)e(mo)s(del.)40 b(T)-8 b(o)30 b(see)h(this)f(w)m(e)h(construct)f (t)m(w)m(o)i(Gelfand)e(triples:)1379 4274 y(\010)1445 4288 y Fv(\006)1529 4274 y Fn(\032)25 b Fm(L)1687 4236 y Fi(2)1726 4274 y Fr(\()p Fl(R)s Fr(\))32 b Fn(\032)25 b Fr(\010)2050 4236 y Fv(0)2050 4296 y(\006)2139 4274 y Fm(;)1334 b Fr(\(1.4\))24 4478 y(suc)m(h)20 b(that)h(\010)472 4492 y Fi(+)530 4478 y Fn(\\)p Fr(\010)657 4492 y Fv(\000)741 4478 y Fr(=)k Fn(f;g)p Fr(.)39 b(Ob)m(viously)-8 b(,)21 b(the)f(time)g(ev)m(olution)g(is)f(p)s(erfectly)h(rev)m(ersible)f(when) g(considered)24 4607 y(on)38 b Fm(L)220 4574 y Fi(2)259 4607 y Fr(\()p Fl(R)s Fr(\).)71 b(It)38 b(is)g(giv)m(en)g(b)m(y)g(the)h (1-parameter)g(group)f(of)g(unitary)f(transformations)h Fm(U)10 b Fr(\()p Fm(t)p Fr(\))39 b(=)f Fm(e)3488 4574 y Fv(\000)p Fk(i)3583 4557 y Fh(b)3567 4574 y Fk(H)5 b(t)3660 4607 y Fr(.)24 4720 y(Ho)m(w)m(ev)m(er,)49 b(when)43 b(restricted)g(to)i(\010)1292 4734 y Fv(\006)1351 4720 y Fr(,)i(it)d(de\014nes)e(only)h(t)m(w)m(o)j(semigroups:)66 b Fm(U)10 b Fr(\()p Fm(t)48 b Fn(\025)f Fr(0\))e(on)e(\010)3407 4734 y Fv(\000)3466 4720 y Fr(,)48 b(and)24 4833 y Fm(U)10 b Fr(\()p Fm(t)25 b Fn(\024)g Fr(0\))30 b(on)g(\010)587 4847 y Fi(+)646 4833 y Fr(.)40 b(Therefore,)30 b(the)g(ev)m(olution)f (on)g(\010)1888 4847 y Fv(\006)1977 4833 y Fr(is)f(irrev)m(ersible.)38 b(This)28 b(irrev)m(ersibilit)m(y)e(is)j(caused)24 4945 y(b)m(y)h(quan)m(tum)g(damping,)f(or,)h(equiv)-5 b(alen)m(tly)d(,)30 b(b)m(y)g(the)h(presence)f(of)h(resonances.)1832 5281 y(2)p eop %%Page: 3 3 3 2 bop 24 44 a Fo(2)134 b(Rigged)46 b(Hilb)t(ert)g(space)24 247 y Fr(Consider)28 b(a)j(rigged)f(Hilb)s(ert)e(space,)j(i.e.)41 b(the)31 b(follo)m(wing)d(collection)i(\(Gelfand)g(triplet\):)1506 451 y(\010)25 b Fn(\032)g(H)h(\032)f Fr(\010)1958 414 y Fv(0)2011 451 y Fm(;)1462 b Fr(\(2.1\))24 656 y(where)35 b Fn(H)h Fr(is)f(a)h(Hilb)s(ert)d(space)j(with)f(the)g(standard)g(norm) g(top)s(ology)h Fm(\034)2570 670 y Fv(H)2634 656 y Fr(,)h(\010)e(is)g (a)h(top)s(ological)f(v)m(ector)24 768 y(space)h(with)f(a)i(top)s (ology)-8 b(,)39 b Fm(\034)1006 782 y Fi(\010)1060 768 y Fr(,)f(stronger)f(than)f Fm(\034)1742 782 y Fv(H)1806 768 y Fr(,)h(and)f(\010)2117 735 y Fv(0)2176 768 y Fr(is)f(the)i(dual)e (space)h(of)h(con)m(tin)m(uous)f(linear)24 881 y(functionals)c(on)i (\010)f([6)q(,)h(7)q(].)51 b(W)-8 b(e)35 b(denote)g(the)f(action)g(of)g (\010)2062 848 y Fv(0)2119 881 y Fr(on)g(\010)g(using)e(Dirac)i (notation,)i(i.e.)51 b(for)34 b(an)m(y)24 994 y Fm(\036)25 b Fn(2)g Fr(\010)30 b(and)g Fm(F)38 b Fn(2)25 b Fr(\010)710 961 y Fv(0)1447 1199 y Fn(h)15 b Fm(\036)p Fn(j)p Fm(F)29 b Fn(i)c Fr(:=)h Fm(F)13 b Fr(\()p Fm(\036)p Fr(\))31 b Fm(:)1402 b Fr(\(2.2\))24 1403 y(An)m(y)30 b(self-adjoin)m(t)g(op)s (erator)1074 1380 y Fj(b)1052 1403 y Fm(A)h Fr(in)e Fn(H)i Fr(ma)m(y)g(b)s(e)f(extended)g(to)h(an)f(op)s(erator)h(on)f(\010)2868 1370 y Fv(0)2891 1403 y Fr(:)1533 1584 y Fj(b)1512 1607 y Fm(A)25 b Fr(:)h(\010)1722 1569 y Fv(0)1770 1607 y Fn(!)f Fr(\010)1952 1569 y Fv(0)2005 1607 y Fm(;)1468 b Fr(\(2.3\))24 1811 y(b)m(y)1351 2015 y Fn(h)15 b Fm(\036)p Fn(j)1502 1992 y Fj(b)1480 2015 y Fm(A)q(F)28 b Fn(i)e Fr(:=)f Fn(h)1889 1992 y Fj(b)1867 2015 y Fm(A\036)p Fn(j)p Fm(F)k Fn(i)i Fm(:)1306 b Fr(\(2.4\))24 2220 y(No)m(w,)31 b(if)e(for)h(an)m(y)h Fm(\036)26 b Fn(2)e Fr(\010)1338 2424 y Fn(h)15 b Fm(\036)p Fn(j)1490 2401 y Fj(b)1467 2424 y Fm(A)q(F)1594 2439 y Fk(\025)1655 2424 y Fn(i)26 b Fr(=)f Fm(\025)p Fn(h)15 b Fm(\036)p Fn(j)p Fm(F)2052 2439 y Fk(\025)2114 2424 y Fn(i)30 b Fm(;)1294 b Fr(\(2.5\))24 2628 y(then)30 b Fm(F)289 2643 y Fk(\025)360 2628 y Fn(2)25 b Fr(\010)512 2595 y Fv(0)565 2628 y Fr(is)k(called)h(a)g(generalized)g (eigen)m(v)m(ector)j(corresp)s(onding)28 b(to)j(a)g(generalized)e (eigen)m(v)-5 b(alue)31 b Fm(\025)p Fr(.)24 2741 y(Omitting)e Fm(\036)h Fr(one)h(simply)d(writes:)1464 2922 y Fj(b)1443 2945 y Fm(A)p Fn(j)p Fm(F)1594 2960 y Fk(\025)1655 2945 y Fn(i)e Fr(=)f Fm(\025)p Fn(j)p Fm(F)1948 2960 y Fk(\025)2009 2945 y Fn(i)31 b Fm(:)1398 b Fr(\(2.6\))24 3150 y(Note,)32 b(that)f(a)f(generalized)g(eigen)m(v)-5 b(alue)31 b Fm(\025)f Fr(ma)m(y)h(b)s(e)f(complex.)40 b(No)m(w,)31 b(if)f(the)g(sp)s(ectrum)f (of)3235 3127 y Fj(b)3214 3150 y Fm(A)1184 3354 y(\033)s Fr(\()1296 3331 y Fj(b)1274 3354 y Fm(A)p Fr(\))d(=)f Fm(\033)1551 3368 y Fk(p)1591 3354 y Fr(\()1647 3331 y Fj(b)1626 3354 y Fm(A)p Fr(\))c Fn([)e Fm(\033)1882 3368 y Fk(c)1917 3354 y Fr(\()1974 3331 y Fj(b)1952 3354 y Fm(A)q Fr(\))56 b Fn(\032)f Fl(R)39 b Fm(;)1139 b Fr(\(2.7\))24 3569 y(with)40 b Fm(\033)294 3583 y Fk(p)334 3569 y Fr(\()391 3546 y Fj(b)369 3569 y Fm(A)q Fr(\))45 b(=)f Fn(f)p Fm(\025)731 3583 y Fi(1)771 3569 y Fm(;)15 b(\025)864 3583 y Fi(2)904 3569 y Fm(;)g(:)g(:)g(:)q Fn(g)p Fr(,)46 b(then)c(the)g(Gelfand-Maurin) e(theorem)i([6)q(,)g(7)q(])g(implies)d(the)k(follo)m(wing)24 3682 y(sp)s(ectral)29 b(decomp)s(ositions:)958 3923 y(1)-20 b(l)1008 3937 y Fi(\010)1089 3923 y Fr(=)1185 3836 y Fj(X)1229 4027 y Fk(n)1331 3923 y Fn(j)p Fm(F)1414 3937 y Fk(n)1477 3923 y Fn(ih)15 b Fm(F)1620 3937 y Fk(n)1668 3923 y Fn(j)21 b Fr(+)1805 3799 y Fj(Z)1855 4005 y Fk(\033)1895 4013 y Fg(c)1928 4005 y Fi(\()1972 3988 y Fh(b)1955 4005 y Fk(A)p Fi(\))2055 3923 y Fm(d\025)15 b Fn(j)p Fm(F)2253 3938 y Fk(\025)2314 3923 y Fn(ih)g Fm(F)2457 3938 y Fk(\025)2504 3923 y Fn(j)30 b Fm(;)914 b Fr(\(2.8\))24 4224 y(and)29 b(of)325 4201 y Fj(b)304 4224 y Fm(A)h Fr(itself:)922 4425 y Fj(b)900 4448 y Fm(A)c Fr(=)1090 4361 y Fj(X)1134 4552 y Fk(n)1236 4448 y Fm(\025)1289 4462 y Fk(n)1336 4448 y Fn(j)p Fm(F)1419 4462 y Fk(n)1482 4448 y Fn(ih)15 b Fm(F)1625 4462 y Fk(n)1673 4448 y Fn(j)21 b Fr(+)1809 4324 y Fj(Z)1860 4530 y Fk(\033)1900 4538 y Fg(c)1932 4530 y Fi(\()1976 4513 y Fh(b)1959 4530 y Fk(A)q Fi(\))2059 4448 y Fm(d\025)15 b(\025)p Fn(j)p Fm(F)2310 4463 y Fk(\025)2372 4448 y Fn(ih)g Fm(F)2515 4463 y Fk(\025)2562 4448 y Fn(j)30 b Fm(:)856 b Fr(\(2.9\))24 4724 y(This)28 b(w)m(a)m(y)k(the)e(rigged)g (Hilb)s(ert)e(space)j(approac)m(h)g(fully)d(justi\014es)h(the)i (standard)e(Dirac)i(notation.)165 4837 y(The)f(c)m(hoice)i(of)f(\010)f (dep)s(ends)f(on)i(the)g(particular)e(problem)g(one)i(deals)f(with.)41 b(In)30 b(the)h(presen)m(t)g(pap)s(er)24 4950 y(w)m(e)38 b(shall)f(consider)g(the)h(follo)m(wing)e(functional)h(spaces:)57 b Fn(D)40 b Fr({)f(the)f(space)h(of)f Fm(C)2860 4917 y Fv(1)2934 4950 y Fr(\()p Fl(R)s Fr(\))45 b(functions)36 b(with)1832 5281 y(3)p eop %%Page: 4 4 4 3 bop 24 44 a Fr(compact)29 b(supp)s(orts)d(equipp)s(ed)f(with)h(the) i(con)m(v)m(ex)i(Sc)m(h)m(w)m(artz)f(top)s(ology)f([12)q(],)h Fn(S)34 b Fr({)28 b(the)g(space)h(of)e Fm(C)3474 11 y Fv(1)3549 44 y Fr(\()p Fl(R)s Fr(\))24 157 y(functions)i(v)-5 b(anishing)28 b(at)j(in\014nit)m(y)d(faster)j(than)f(an)m(y)g(p)s (olynomial)e([12)q(].)41 b(Moreo)m(v)m(er,)33 b(let)d(us)g(de\014ne) 1354 361 y Fn(Z)i Fr(:=)25 b Fn(f)p Fm(F)13 b Fr([)p Fm(\036)p Fr(])i Fn(j)g Fm(\036)27 b Fn(2)e(D)s(g)31 b Fm(;)1264 b Fr(\(2.10\))24 566 y(where)29 b Fm(F)13 b Fr([)p Fm(\036)p Fr(])31 b(denotes)f(the)g(F)-8 b(ourier)29 b(transform)g(of)h Fm(\036)p Fr(.)41 b(It)30 b(turns)f(out)h([13)q(])g (that)h Fn(Z)36 b Fr(is)29 b(isomorphic)f(to)j(the)24 678 y(space)g(of)f(en)m(tire)g(functions)f(of)i(fast)g(decrease)g (along)f Fl(R)s Fr(.)47 b(More)31 b(precisely)-8 b(,)29 b(let)1082 922 y Fm(F)1140 936 y Fk(L)1193 922 y Fr([)p Fm(\036)p Fr(]\()p Fm(z)t Fr(\))e(:=)1636 860 y(1)p 1571 901 177 4 v 1571 919 a Fn(p)p 1647 919 101 4 v 75 x Fr(2)p Fm(\031)1772 798 y Fj(Z)1863 824 y Fv(1)1823 1004 y(\0001)1968 922 y Fm(e)2010 884 y Fk(iz)s(x)2114 922 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)31 b(;)993 b Fr(\(2.11\))24 1179 y(b)s(e)29 b(the)i(F)-8 b(ourier-Laplace)30 b(transform)g(of)h Fm(\036)25 b Fn(2)g(D)s Fr(.)40 b(One)30 b(pro)m(v)m(es)h([12)q(,)g(13) q(])f(the)h(follo)m(wing)24 1367 y Fs(Theorem)j(1)h(\(P)m (aley-Wiener-Sc)m(h)m(w)m(artz\))45 b Ff(L)-5 b(et)34 b Fm(a)26 b(>)f Fr(0)p Ff(.)44 b(A)n(n)32 b(entir)-5 b(e)34 b(function)f Fm(U)10 b Fr(\()p Fm(z)t Fr(\))34 b Ff(is)f(a)h(F)-7 b(ourier-)24 1480 y(L)i(aplac)g(e)34 b(tr)-5 b(ansform)35 b(of)e(a)g(function)g Fm(u)25 b Fn(2)g(D)35 b Ff(with)f(supp)-5 b(ort)1286 1684 y Fr(supp)o(\()p Fm(u)p Fr(\))26 b(=)f Fn(f)p Fm(x)g Fn(2)g Fl(R)18 b Fn(j)d(j)q Fm(x)p Fn(j)31 b(\024)25 b Fm(a)p Fn(g)33 b Fm(;)24 1888 y Ff(if)f(and)h(only)h(if)928 2092 y Fn(j)p Fm(z)t Fn(j)1024 2055 y Fk(n)1072 2092 y Fn(j)p Fm(U)10 b Fr(\()p Fm(z)t Fr(\))p Fn(j)27 b(\024)d Fm(C)1497 2106 y Fk(n)1544 2092 y Fm(e)1586 2055 y Fk(a)p Fv(j)p Fi(Im)p Fk(z)s Fv(j)1820 2092 y Fm(;)243 b(n)25 b Fr(=)g(1)p Fm(;)15 b Fr(2)p Fm(;)g(:)g(:)g(:)50 b(:)24 2296 y Fr(No)m(w,)31 b(for)f Fm(z)g Fr(=)25 b Fm(x)g Fn(2)g Fl(R)r Fr(,)37 b(i.e.)j(Im)15 b Fm(z)30 b Fr(=)25 b(0,)31 b Fm(F)1454 2310 y Fk(L)1506 2296 y Fr([)p Fm(\036)p Fr(])26 b(=)f Fm(F)13 b Fr([)p Fm(\036)p Fr(],)31 b(and)f(the)h(ab)s(o)m(v)m(e)g (theorem)g(implies)1002 2501 y Fn(j)p Fm(x)p Fn(j)1104 2463 y Fk(n)1151 2501 y Fn(j)p Fm(F)13 b Fr([)p Fm(\036)p Fr(]\()p Fm(x)p Fr(\))p Fn(j)28 b(\024)d Fm(C)1687 2515 y Fk(n)1764 2501 y Fm(;)227 b(n)25 b Fr(=)g(1)p Fm(;)15 b Fr(2)p Fm(;)g(:)g(:)g(:)48 b(:)913 b Fr(\(2.12\))24 2705 y(Clearly)-8 b(,)29 b Fn(Z)f(\\)19 b(D)28 b Fr(=)d Fn(f;g)p Fr(.)42 b(Moreo)m(v)m(er,)33 b(one)e(has)1436 2909 y Fn(D)d(\032)d(S)32 b(\032)25 b Fm(L)1875 2872 y Fi(2)1914 2909 y Fr(\()p Fl(R)s Fr(\))37 b Fm(;)1347 b Fr(\(2.13\))24 3113 y(and)1436 3318 y Fn(Z)32 b(\032)25 b(S)32 b(\032)25 b Fm(L)1875 3280 y Fi(2)1915 3318 y Fr(\()p Fl(R)s Fr(\))36 b Fm(;)1347 b Fr(\(2.14\))24 3522 y(and)34 b(b)s(oth)h Fn(D)j Fr(and)d Fn(Z)42 b Fr(are)36 b(dense)f(in)f Fn(S)7 b Fr(.)56 b(One)35 b(pro)m(v)m(es)h([13)q(])g (that)g(the)g(F)-8 b(ourier)34 b(transformation)h(whic)m(h)24 3635 y(de\014nes)29 b(the)i(unitary)e(op)s(erator)1304 3839 y Fm(F)53 b Fr(:)41 b Fm(L)1543 3801 y Fi(2)1582 3839 y Fr(\()p Fl(R)s Fr(\))47 b Fn(\000)-16 b(!)41 b Fm(L)2008 3801 y Fi(2)2047 3839 y Fr(\()p Fl(R)s Fr(\))c Fm(;)1214 b Fr(\(2.15\))24 4043 y(establishes)29 b(an)h(isomorphism)d (b)s(et)m(w)m(een)k Fn(D)i Fr(and)d Fn(Z)7 b Fr(.)24 4330 y Fo(3)134 b(Quan)l(tization)47 b(of)e(damp)t(ed)f(systems)24 4533 y Fr(Let)c(us)f(quan)m(tize)g(a)h(classical)f(damp)s(ed)f(system)i (describ)s(ed)d(b)m(y)j(\(1.1\).)70 b(Clearly)38 b(this)g(system)i(is)f (not)24 4646 y(Hamiltonian.)d(Ho)m(w)m(ev)m(er,)25 b(it)c(is)g(w)m(ell) f(kno)m(wn)h(\(cf.)38 b([14)r(]\))22 b(that)g(an)m(y)f(dynamical)f (system)i(ma)m(y)g(b)s(e)f(rewritten)24 4758 y(in)32 b(a)h(Hamiltonian)f(form.)49 b(Consider)31 b(a)j(dynamical)e(system)h (on)g Fm(n)p Fr(-dimensional)e(con\014guration)h(space)24 4871 y Fm(Q)p Fr(:)1653 4984 y(_)-41 b Fm(x)26 b Fr(=)f Fm(X)7 b Fr(\()p Fm(x)p Fr(\))31 b Fm(;)1427 b Fr(\(3.1\))1832 5281 y(4)p eop %%Page: 5 5 5 4 bop 24 44 a Fr(where)39 b Fm(X)47 b Fr(is)38 b(a)i(v)m(ector)h (\014eld)d(on)i Fm(Q)p Fr(.)68 b(No)m(w,)42 b(de\014ne)d(the)h(follo)m (wing)e(Hamiltonian)g(on)h(the)h(cotangen)m(t)24 157 y(bundle)28 b Fn(P)33 b Fr(=)25 b Fm(T)580 124 y Fv(\003)619 157 y Fm(Q)p Fr(:)1437 270 y Fm(H)7 b Fr(\()p Fm(\013)1613 284 y Fk(x)1657 270 y Fr(\))25 b(:=)h Fm(\013)1897 284 y Fk(x)1941 270 y Fr(\()p Fm(X)7 b Fr(\()p Fm(x)p Fr(\)\))32 b Fm(;)1226 b Fr(\(3.2\))24 426 y(where)30 b Fm(\013)345 440 y Fk(x)414 426 y Fn(2)25 b Fm(T)566 393 y Fv(\003)553 448 y Fk(x)605 426 y Fm(Q)p Fr(.)40 b(Using)30 b(canonical)g(co)s (ordinates)g(\()p Fm(x)1967 393 y Fi(1)2007 426 y Fm(;)15 b(:)g(:)g(:)h(;)f(x)2260 393 y Fk(n)2308 426 y Fm(;)g(p)2394 440 y Fi(1)2433 426 y Fm(;)g(:)g(:)g(:)i(;)e(p)2681 440 y Fk(n)2728 426 y Fr(\))31 b(one)f(obtains:)1379 678 y Fm(H)7 b Fr(\()p Fm(x;)15 b(p)p Fr(\))26 b(=)1836 564 y Fk(n)1792 592 y Fj(X)1793 789 y Fk(k)r Fi(=1)1938 678 y Fm(p)1984 693 y Fk(k)2027 678 y Fm(X)2109 640 y Fk(k)2152 678 y Fr(\()p Fm(x)p Fr(\))31 b Fm(;)1168 b Fr(\(3.3\))24 955 y(where)28 b Fm(X)367 922 y Fk(k)440 955 y Fr(are)h(comp)s(onen)m (ts)h(of)f Fm(X)37 b Fr(in)28 b(the)h(co)s(ordinate)h(basis)e Fm(@)5 b(=@)g(x)2436 922 y Fk(k)2479 955 y Fr(.)41 b(The)29 b(corresp)s(onding)e(Hamilton)24 1068 y(equations)j(tak)m(e)i(the)e (follo)m(wing)f(form:)1161 1251 y(_)-41 b Fm(x)1197 1213 y Fk(k)1322 1251 y Fr(=)83 b Fn(f)p Fm(x)1573 1213 y Fk(k)1616 1251 y Fm(;)15 b(H)7 b Fn(g)26 b Fr(=)f Fm(X)1988 1213 y Fk(k)2031 1251 y Fr(\()p Fm(x)p Fr(\))31 b Fm(;)1289 b Fr(\(3.4\))1169 1458 y(_)-43 b Fm(p)1197 1473 y Fk(k)1322 1458 y Fr(=)83 b Fn(f)p Fm(p)1567 1473 y Fk(k)1610 1458 y Fm(;)15 b(H)7 b Fn(g)26 b Fr(=)f Fn(\000)2030 1344 y Fk(n)1986 1372 y Fj(X)1995 1569 y Fk(l)q Fi(=1)2132 1458 y Fm(p)2178 1473 y Fk(l)2214 1397 y Fm(@)5 b(X)2349 1364 y Fk(l)2376 1397 y Fr(\()p Fm(x)p Fr(\))p 2214 1437 285 4 v 2282 1520 a Fm(@)g(x)2387 1494 y Fk(k)2539 1458 y Fm(;)934 b Fr(\(3.5\))24 1724 y(for)30 b Fm(k)e Fr(=)d(1)p Fm(;)15 b(:)g(:)g(:)i(;)e(n)p Fr(.)41 b(In)29 b(the)h(ab)s(o)m(v)m(e)i (form)m(ulae)e Fn(f)g Fm(;)46 b Fn(g)30 b Fr(denotes)h(the)f(canonical) g(P)m(oisson)g(brac)m(k)m(et)i(on)e Fm(T)3549 1691 y Fv(\003)3588 1724 y Fm(Q)p Fr(:)1101 1976 y Fn(f)p Fm(F)s(;)15 b(G)p Fn(g)26 b Fr(=)1531 1863 y Fk(n)1486 1890 y Fj(X)1488 2088 y Fk(k)r Fi(=1)1633 1848 y Fj(\022)1721 1915 y Fm(@)5 b(F)p 1710 1956 148 4 v 1710 2039 a(@)g(x)1815 2013 y Fk(k)1886 1915 y Fm(@)g(G)p 1877 1956 142 4 v 1877 2039 a(@)g(p)1976 2054 y Fk(k)2049 1976 y Fn(\000)2162 1915 y Fm(@)g(G)p 2150 1956 148 4 v 2150 2039 a(@)g(x)2255 2013 y Fk(k)2327 1915 y Fm(@)g(F)p 2318 1956 142 4 v 2318 2039 a(@)g(p)2417 2054 y Fk(k)2470 1848 y Fj(\023)2582 1976 y Fm(:)891 b Fr(\(3.6\))24 2243 y(Clearly)-8 b(,)29 b(the)i(form)m(ulae)f(\(3.4\))i(repro)s(duce)d(our)h(initial)e (dynamical)h(system)h(\(3.1\))i(on)e Fm(Q)p Fr(.)165 2356 y(Let)44 b(us)f(apply)f(the)i(ab)s(o)m(v)m(e)g(pro)s(cedure)f(to)h (the)g(damp)s(ed)e(system)h(\(1.1\).)83 b(One)43 b(obtains)f(for)i(the) 24 2469 y(Hamiltonian)1510 2582 y Fm(H)7 b Fr(\()p Fm(x;)15 b(p)p Fr(\))26 b(=)f Fn(\000)p Fm(\015)5 b(xp)30 b(;)1299 b Fr(\(3.7\))24 2738 y(and)29 b(hence)i(the)f(corresp)s(onding)f (Hamilton)g(equations)1300 2920 y(_)-41 b Fm(x)25 b Fr(=)g Fn(\000)p Fm(\015)5 b(x)30 b(;)269 b Fr(_)-42 b Fm(p)25 b Fr(=)g Fm(\015)5 b(p)30 b(;)1239 b Fr(\(3.8\))24 3103 y(giv)m(e)30 b(rise)g(to)h(the)g(follo)m(wing)d(Hamiltonian)h(\015o)m (w)i(on)f Fl(R)1927 3070 y Fi(2)1973 3103 y Fr(:)1349 3285 y(\()p Fm(x;)15 b(p)p Fr(\))57 b Fn(\000)-16 b(!)56 b Fr(\()p Fm(e)1893 3248 y Fv(\000)p Fk(\015)t(t)2018 3285 y Fm(x;)15 b(e)2152 3248 y Fk(t\015)2223 3285 y Fm(p)p Fr(\))30 b Fm(:)1139 b Fr(\(3.9\))24 3468 y(No)m(w,)32 b(the)g(quan)m(tization)g(is)e(straigh)m(tforw)m(ard:)43 b(one)32 b(has)g(for)f(the)h(Hilb)s(ert)d(space)k Fn(H)27 b Fr(=)g Fm(L)3165 3435 y Fi(2)3205 3468 y Fr(\()p Fl(R)s Fr(\),)38 b(and)31 b(for)24 3581 y(the)f(Hamiltonian)1403 3740 y Fj(b)1381 3763 y Fm(H)j Fr(=)25 b Fn(\000)1667 3702 y Fm(\015)p 1667 3742 53 4 v 1670 3825 a Fr(2)1728 3763 y(\()s Fj(b)-54 b Fm(x)6 b Fj(b)-56 b Fm(p)20 b Fr(+)25 b Fj(b)-56 b Fm(p)s Fj(b)h Fm(x)p Fr(\))31 b Fm(:)1292 b Fr(\(3.10\))24 3979 y(It)34 b(is)e(eviden)m(t)i(that)g (\(3.10\))i(de\014nes)d(a)i(symmetric)e(op)s(erator)h(on)g Fm(L)2398 3946 y Fi(2)2437 3979 y Fr(\()p Fl(R)s Fr(\).)57 b(In)33 b(section)h(4)g(w)m(e)h(sho)m(w)e(that)45 4069 y Fj(b)24 4092 y Fm(H)k Fr(is)29 b(self-adjoin)m(t)h(and)g(hence)g(it)g (giv)m(es)h(rise)e(to)i(a)g(w)m(ell)f(de\014ned)f(quan)m(tum)h(mec)m (hanical)g(problem.)3492 4059 y Fi(1)p 24 4164 1465 4 v 127 4218 a Fe(1)162 4249 y Fp(Actually)-6 b(,)22 b(this)g (Hamiltonian)f(is)i(w)n(ell)f(kno)n(wn)f(in)h(quan)n(tum)d(optics)k(in) e(connection)h(with)g(the)f(squeezed)h(states)g(of)h(ligh)n(t)24 4341 y([15)q(].)35 b(In)n(tro)r(ducing)24 b Fd(b)-42 b Fc(a)25 b Fp(and)f Fd(b)-42 b Fc(a)871 4309 y Fq(\003)907 4341 y Fp(:)1256 4523 y Fd(b)d Fc(x)21 b Fp(=)1409 4475 y Fd(b)-42 b Fc(a)16 b Fp(+)g Fd(b)-42 b Fc(a)1585 4443 y Fq(\003)p 1410 4506 212 4 v 1464 4522 a Fb(p)p 1528 4522 39 4 v 61 x Fp(2)1656 4523 y Fc(;)197 b Fd(b)-47 b Fc(p)21 b Fp(=)2020 4475 y Fd(b)-42 b Fc(a)16 b Fb(\000)g Fd(b)-42 b Fc(a)2196 4443 y Fq(\003)p 2021 4506 212 4 v 2062 4522 a Fb(p)p 2126 4522 39 4 v 61 x Fp(2)p Fc(i)2267 4523 y(;)24 4709 y Fp(the)25 b(Hamiltonian)g(\(3.10\))i(ma)n(y)e(b)r(e) g(rewritten)h(as)h(follo)n(ws:)1451 4850 y Fd(b)1434 4869 y Fc(H)f Fp(=)1626 4820 y Fc(\015)p 1615 4852 65 4 v 1615 4919 a Fp(2)p Fc(i)1703 4808 y Fd(\000)1737 4869 y(b)-42 b Fc(a)1779 4833 y Fq(\003)p Fe(2)1862 4869 y Fb(\000)16 b Fd(b)-42 b Fc(a)1980 4833 y Fe(2)2014 4808 y Fd(\001)2088 4869 y Fc(;)24 5032 y Fp(whic)n(h)25 b(is)i(exactly)e(a)h(generator)h(of)f(squeezing.)1832 5281 y Fr(5)p eop %%Page: 6 6 6 5 bop 165 44 a Fr(Let)31 b(us)e(observ)m(e)i(that)g(p)s(erforming)d (the)j(canonical)f(transformation)824 282 y Fm(x)25 b Fr(=)1071 220 y(1)p 1007 261 174 4 v 1007 279 a Fn(p)p 1083 279 98 4 v 66 x Fr(2)p Fm(\015)1206 282 y Fr(\()p Fm(\015)5 b(X)28 b Fn(\000)20 b Fm(P)13 b Fr(\))30 b Fm(;)252 b(p)25 b Fr(=)2141 220 y(1)p 2077 261 174 4 v 2077 279 a Fn(p)p 2153 279 98 4 v 66 x Fr(2)p Fm(\015)2275 282 y Fr(\()p Fm(\015)5 b(X)29 b Fr(+)19 b Fm(P)13 b Fr(\))31 b Fm(;)735 b Fr(\(3.11\))24 534 y(the)30 b(classical)g (Hamiltonian)e(\(3.10\))33 b(tak)m(es)f(the)e(follo)m(wing)f(form:)1377 753 y Fj(b)1356 776 y Fm(H)j Fr(=)1570 715 y(1)p 1570 755 46 4 v 1570 839 a(2)1625 776 y(\()1679 753 y Fj(b)1660 776 y Fm(P)1731 739 y Fi(2)1791 776 y Fn(\000)20 b Fm(\015)1934 739 y Fi(2)1997 753 y Fj(b)1974 776 y Fm(X)2056 739 y Fi(2)2096 776 y Fr(\))30 b Fm(;)1267 b Fr(\(3.12\))24 1001 y(that)24 b(is,)g(it)f(corresp)s(onds)g(to)h(the)g(so)g(called)f (rev)m(ersed)h(harmonic)e(oscillator.)38 b(This)22 b(system)i(w)m(as)g (analyzed)24 1114 y(in)29 b([16)q(])h(and)g(recen)m(tly)h(in)e([17)q(,) i(18)q(,)f(19)q(])h(\(see)g(also)f([20)q(,)h(21)q(]\).)24 1401 y Fo(4)134 b(Prop)t(erties)46 b(of)f(the)g(Hamiltonian)24 1604 y Fr(Let)31 b(us)e(in)m(v)m(estigate)j(the)e(basic)g(prop)s (erties)f(of)h(the)h(Hamiltonian)e(de\014ned)g(in)g(\(3.10\).)24 1816 y Fs(Prop)s(osition)36 b(1)46 b Ff(The)32 b(op)-5 b(er)g(ator)1251 1793 y Fj(b)1230 1816 y Fm(H)32 b Fr(=)25 b Fn(\000)1515 1775 y Fk(\015)p 1515 1795 41 4 v 1517 1848 a Fi(2)1565 1816 y Fr(\()s Fj(b)-54 b Fm(x)6 b Fj(b)-56 b Fm(p)20 b Fr(+)25 b Fj(b)-57 b Fm(p)s Fj(b)j Fm(x)p Fr(\))33 b Ff(is)g(self-adjoint)g(in)g Fm(L)2718 1783 y Fi(2)2757 1816 y Fr(\()p Fl(R)s Fr(\))p Ff(.)24 2053 y(Pr)-5 b(o)g(of.)41 b Fr(T)-8 b(o)31 b(pro)m(v)m(e)g(that)901 2030 y Fj(b)879 2053 y Fm(H)38 b Fr(is)29 b(self-adjoin)m(t)h(w)m(e)h (sho)m(w)f(that)h Fm(e)2150 2020 y Fv(\000)p Fk(i)2245 2004 y Fh(b)2229 2020 y Fk(H)2327 2053 y Fr(is)e(unitary)g(in)g Fm(L)2907 2020 y Fi(2)2946 2053 y Fr(\()p Fl(R)s Fr(\))q(.)46 b(One)30 b(has)1045 2285 y Fj(b)1023 2307 y Fm(H)i Fr(=)25 b Fn(\000)1308 2246 y Fm(\015)p 1308 2287 53 4 v 1311 2370 a Fr(2)1370 2307 y(\()s Fj(b)-54 b Fm(x)6 b Fj(b)-57 b Fm(p)20 b Fr(+)26 b Fj(b)-57 b Fm(p)s Fj(b)j Fm(x)p Fr(\))26 b(=)f Fm(i\015)1967 2179 y Fj(\022)2034 2307 y Fm(x)2122 2246 y(d)p 2096 2287 100 4 v 2096 2370 a(dx)2225 2307 y Fr(+)2326 2246 y(1)p 2326 2287 46 4 v 2326 2370 a(2)2382 2179 y Fj(\023)2494 2307 y Fm(:)979 b Fr(\(4.1\))24 2557 y(Let)31 b(us)e(de\014ne)1312 2761 y Fm(U)35 b Fr(=)25 b Fm(e)1547 2723 y Fv(\000)p Fk(i)1642 2707 y Fh(b)1626 2723 y Fk(H)1719 2761 y Fr(=)g Fm(e)1857 2723 y Fk(\015)t(=)p Fi(2)1972 2761 y Fm(e)2014 2723 y Fk(\015)t(x@)2131 2731 y Fg(x)2205 2761 y Fm(:)1268 b Fr(\(4.2\))24 2965 y(Clearly)-8 b(,)1340 3169 y Fm(U)10 b( )s Fr(\()p Fm(x)p Fr(\))26 b(=)f Fm(e)1760 3132 y Fk(\015)t(=)p Fi(2)1875 3169 y Fm( )s Fr(\()p Fm(e)2014 3132 y Fk(\015)2060 3169 y Fm(x)p Fr(\))31 b Fm(;)1295 b Fr(\(4.3\))24 3374 y(for)30 b(an)m(y)g Fm( )f Fn(2)c Fm(L)570 3341 y Fi(2)609 3374 y Fr(\()p Fl(R)s Fr(\))q(.)46 b(The)30 b(op)s(erator)h Fm(U)40 b Fr(de\014nes)30 b(an)g(isometry:)218 3622 y Fn(h)15 b Fm(U)10 b( )s Fn(j)p Fm(U)g(\036)15 b Fn(i)85 b Fr(=)842 3498 y Fj(Z)932 3524 y Fv(1)892 3704 y(\0001)p 1037 3543 258 4 v 1037 3622 a Fm(U)10 b( )s Fr(\()p Fm(x)p Fr(\))q Fm(U)g(\036)p Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)26 b Fr(=)1778 3498 y Fj(Z)1869 3524 y Fv(1)1829 3704 y(\0001)1974 3622 y Fm(e)2016 3584 y Fk(\015)p 2060 3543 272 4 v 2060 3622 a Fm( )s Fr(\()p Fm(e)2199 3596 y Fk(\015)2245 3622 y Fm(x)p Fr(\))p Fm(\036)p Fr(\()p Fm(e)2463 3584 y Fk(\015)2509 3622 y Fm(x)p Fr(\))15 b Fm(dx)26 b Fr(=)2831 3498 y Fj(Z)2922 3524 y Fv(1)2882 3704 y(\0001)p 3027 3543 182 4 v 3027 3622 a Fm( )s Fr(\()p Fm(y)s Fr(\))q Fm(\036)p Fr(\()p Fm(y)s Fr(\))15 b Fm(dy)688 3813 y Fr(=)83 b Fn(h)15 b Fm( )s Fn(j)p Fm(\036)g Fn(i)32 b Fm(:)2358 b Fr(\(4.4\))24 4017 y(Moreo)m(v)m(er,)48 b(due)42 b(to)i(\(4.3\),)k Fm(U)53 b Fr(is)42 b(on)m(to,)47 b(and)42 b(hence)h(it)g(is)f(unitary)f (in)h Fm(L)2670 3984 y Fi(2)2709 4017 y Fr(\()p Fl(R)s Fr(\).)85 b(Therefore,)46 b(Stone's)24 4130 y(theorem)30 b(implies)e(that)904 4107 y Fj(b)883 4130 y Fm(H)37 b Fr(is)30 b(self-adjoin)m(t)f(\(see)j(e.g.)41 b([12)r(]\).)1501 b Fa(2)24 4243 y Fr(Ob)m(viously)-8 b(,)490 4220 y Fj(b)469 4243 y Fm(H)37 b Fr(is)29 b(parit)m(y)h(in)m(v)-5 b(arian)m(t:)1481 4447 y Fs(P)1574 4424 y Fj(b)1553 4447 y Fm(H)7 b Fs(P)1707 4409 y Fv(\000)p Fi(1)1827 4447 y Fr(=)1944 4424 y Fj(b)1923 4447 y Fm(H)37 b(;)1437 b Fr(\(4.5\))24 4651 y(where)30 b(the)g(parit)m(y)g(op)s(erator)h Fm(P)43 b Fr(is)29 b(de\014ned)h(b)m(y:)1063 4855 y Fs(P)s Fj(b)-54 b Fm(x)q Fs(P)1258 4818 y Fv(\000)p Fi(1)1378 4855 y Fr(=)25 b Fn(\000)s Fj(b)-54 b Fm(x)30 b(;)251 b Fs(P)5 b Fj(b)-56 b Fm(p)p Fs(P)2091 4818 y Fv(\000)p Fi(1)2211 4855 y Fr(=)25 b Fn(\000)5 b Fj(b)-56 b Fm(p)30 b(:)1019 b Fr(\(4.6\))1832 5281 y(6)p eop %%Page: 7 7 7 6 bop 24 44 a Fr(No)m(w,)26 b(let)d(us)g(turn)g(to)h(the)g(time)g (rev)m(ersal)g(op)s(erator)g Fs(T)p Fr(.)f(The)h(theory)f(in)m(v)-5 b(arian)m(t)23 b(under)g(the)h(time)f(rev)m(ersal)24 157 y(has)30 b(the)g(follo)m(wing)f(prop)s(ert)m(y:)40 b(if)30 b Fm( )s Fr(\()p Fm(t)p Fr(\))h(is)e(a)i(solution)e(of)i(the)f (Sc)m(hr\177)-45 b(odinger)29 b(equation)h(giv)m(en)h(b)m(y)1481 313 y Fm( )s Fr(\()p Fm(t)p Fr(\))26 b(=)f Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm( )34 b(;)1437 b Fr(\(4.7\))24 485 y(with)29 b Fm(U)10 b Fr(\()p Fm(t)p Fr(\))25 b(=)g Fm(e)569 452 y Fv(\000)p Fk(i)665 436 y Fh(b)648 452 y Fk(H)6 b(t)741 485 y Fr(,)31 b(then)f Fs(T)p Fm( )k Fr(ev)m(olv)m(es)d(in)m(to)1302 642 y(\()p Fs(T)p Fm( )s Fr(\)\()p Fn(\000)p Fm(t)p Fr(\))c(=)e Fm(U)10 b Fr(\()p Fm(t)p Fr(\)\()p Fs(T)p Fm( )s Fr(\))31 b Fm(;)1258 b Fr(\(4.8\))24 798 y(or,)30 b(equiv)-5 b(alen)m(tly)1266 954 y Fs(T)p Fr(\()p Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm( )s Fr(\))27 b(=)e Fm(U)10 b Fr(\()p Fn(\000)p Fm(t)p Fr(\)\()p Fs(T)p Fm( )s Fr(\))31 b Fm(;)1222 b Fr(\(4.9\))24 1110 y(for)40 b(an)m(y)g Fm( )46 b Fn(2)41 b(H)q Fr(.)70 b(No)m(w,)44 b(follo)m(wing)39 b(Wigner)h([22)q(],)j Fs(T)d Fr(is)f(either)h(unitary)f(or)h(an)m(ti-unitary)-8 b(.)70 b(If)40 b Fs(T)g Fr(is)24 1223 y(unitary)-8 b(,)29 b(then)h(\(4.9\))j(implies)1449 1379 y Fs(T)1543 1356 y Fj(b)1522 1379 y Fm(H)27 b Fr(+)1737 1356 y Fj(b)1716 1379 y Fm(H)7 b Fs(T)25 b Fr(=)g(0)30 b Fm(:)1360 b Fr(\(4.10\))24 1536 y(It)30 b(means)g(that)h(if)1504 1669 y Fj(b)1483 1692 y Fm(H)7 b( )1628 1654 y Fk(E)1713 1692 y Fr(=)25 b Fm(E)5 b( )1943 1654 y Fk(E)2034 1692 y Fm(;)1394 b Fr(\(4.11\))24 1848 y(then)1381 1981 y Fj(b)1360 2004 y Fm(H)22 b Fs(T)p Fm( )1593 1967 y Fk(E)1678 2004 y Fr(=)j Fn(\000)p Fm(E)20 b Fs(T)p Fm( )2067 1967 y Fk(E)2157 2004 y Fm(;)1271 b Fr(\(4.12\))24 2160 y(that)40 b(is,)i(an)m(y)e (eigen)m(v)m(ector)i Fm( )1084 2127 y Fk(E)1184 2160 y Fr(with)c(the)i(energy)h Fm(E)k Fr(is)39 b(accompanied)h(b)m(y)f Fs(T)p Fm( )2900 2127 y Fk(E)3000 2160 y Fr(with)g(energy)h Fn(\000)p Fm(E)5 b Fr(.)24 2273 y(Usually)-8 b(,)27 b(this)f(case)i(is) f(excluded)f(since)h(one)g(exp)s(ects)h(that)g(the)f(Hamiltonian)f(is)g (b)s(ounded)g(from)h(b)s(elo)m(w.)24 2386 y(If)j(this)f(is)g(the)i (case,)g(then)g Fs(T)f Fr(is)f(an)m(ti-unitary)g(and)h(\(4.9\))i (implies:)1449 2542 y Fs(T)1543 2519 y Fj(b)1522 2542 y Fm(H)27 b Fn(\000)1737 2519 y Fj(b)1716 2542 y Fm(H)7 b Fs(T)25 b Fr(=)g(0)30 b Fm(:)1360 b Fr(\(4.13\))24 2699 y(Ho)m(w)m(ev)m(er,)35 b(the)e(Hamiltonian)e(de\014ned)h(in)g (\(3.10\))j(is)c(not)i(b)s(ounded)e(from)h(b)s(elo)m(w,)h(and,)g(as)g (w)m(e)h(sho)m(w)e(in)24 2811 y(section)e(6)h(its)f(sp)s(ectrum)f Fm(\033)s Fr(\()1039 2789 y Fj(b)1017 2811 y Fm(H)7 b Fr(\))26 b(=)f(\()p Fn(\0001)p Fm(;)15 b Fn(1)p Fr(\).)41 b(Therefore,)31 b(w)m(e)f(tak)m(e)i Fs(T)e Fr(to)i(b)s(e)d(unitary)g (in)g Fm(L)3281 2778 y Fi(2)3321 2811 y Fr(\()p Fl(R)s Fr(\).)24 2976 y Fs(Prop)s(osition)36 b(2)46 b Ff(The)32 b(time)h(r)-5 b(eversal)34 b(op)-5 b(er)g(ator)36 b Fs(T)c Ff(is)h(r)-5 b(e)g(alize)g(d)34 b(by)f(the)g(F)-7 b(ourier)33 b(tr)-5 b(ansformation:)1510 3132 y Fs(T)p Fm( )28 b Fr(:=)d Fm(F)13 b Fr([)p Fm( )s Fr(])34 b Fm(;)1420 b Fr(\(4.14\))24 3288 y Ff(i.e.)1383 3445 y Fm(F)1454 3407 y Fv(\000)p Fi(1)1569 3422 y Fj(b)1548 3445 y Fm(H)7 b(F)13 b( )29 b Fr(=)c Fn(\000)1978 3422 y Fj(b)1957 3445 y Fm(H)7 b( )36 b(;)1293 b Fr(\(4.15\))24 3601 y Ff(for)33 b(al)5 b(l)33 b Fm( )c Fn(2)c Fm(L)535 3568 y Fi(2)574 3601 y Fr(\()p Fl(R)s Fr(\))p Ff(.)48 b(Mor)-5 b(e)g(over,)1220 3757 y Fs(T)1293 3719 y Fi(2)1333 3757 y Fm( )s Fr(\()p Fm(x)p Fr(\))26 b(=)f Fs(P)p Fm(\036)p Fr(\()p Fm(x)p Fr(\))i(=)e Fm( )s Fr(\()p Fn(\000)p Fm(x)p Fr(\))33 b Fm(:)1131 b Fr(\(4.16\))24 3922 y(Denoting)30 b(b)m(y)h Fs(C)e Fr(the)i(complex)f(conjugation)g Fs(C)p Fm( )f Fr(=)p 1913 3847 63 4 v 25 w Fm( )s Fr(,)i(one)f(immediately)f (\014nds)24 4086 y Fs(Prop)s(osition)36 b(3)46 b Ff(The)32 b(Hamiltonian)j(\(3.10\))f(is)e Fs(CT)g Ff(and)i Fs(PCT)e Ff(invariant,)i(i.e.)1241 4242 y Fr([)1287 4219 y Fj(b)1266 4242 y Fm(H)7 b(;)15 b Fs(CT)o Fr(])26 b(=)f([)1730 4219 y Fj(b)1709 4242 y Fm(H)7 b(;)15 b Fs(PCT)p Fr(])26 b(=)f(0)33 b Fm(:)1151 b Fr(\(4.17\))24 4407 y(Therefore,)30 b(if)1504 4540 y Fj(b)1483 4563 y Fm(H)7 b( )1628 4525 y Fk(E)1713 4563 y Fr(=)25 b Fm(E)5 b( )1943 4525 y Fk(E)2034 4563 y Fm(;)1394 b Fr(\(4.18\))24 4719 y(then)1383 4852 y Fj(b)1362 4875 y Fm(H)7 b(F)13 b Fr([)p 1541 4793 123 4 v Fm( )1603 4849 y Fk(E)1663 4875 y Fr(])25 b(=)g Fm(E)5 b(F)13 b Fr([)p 1977 4793 V Fm( )2039 4849 y Fk(E)2100 4875 y Fr(])31 b Fm(:)1272 b Fr(\(4.19\))24 5032 y(Clearly)-8 b(,)29 b Fs(CT)h Fr(in)m(v)-5 b(ariance)29 b(do)s(es)h(not)h(pro)s (duce)e(an)m(y)i(conserv)m(ed)g(quan)m(tit)m(y)f(since)g Fs(CT)g Fr(is)f(an)m(ti-unitary)-8 b(.)1832 5281 y(7)p eop %%Page: 8 8 8 7 bop 24 44 a Fo(5)134 b(Complex)46 b(eigen)l(v)-7 b(alues)24 247 y Fr(In)m(terestingly)f(,)592 224 y Fj(b)571 247 y Fm(H)38 b Fr(b)s(eing)30 b(self-adjoin)m(t)h(admits)g (generalized)g(eigen)m(v)m(ectors)i(with)d(complex)h(eigen)m(v)-5 b(alues)24 360 y([19)q(,)30 b(23)q(,)h(20)q(,)f(21)q(].)41 b(Let)31 b Fm(f)888 322 y Fv(\006)878 386 y Fi(0)977 360 y Fr(b)s(e)e(distributions)e(satisfying)1249 564 y Fj(b)-54 b Fm(x)15 b(f)1368 526 y Fv(\000)1358 590 y Fi(0)1452 564 y Fr(=)25 b(0)30 b Fm(;)257 b Fj(b)-56 b Fm(p)15 b(f)2016 526 y Fi(+)2006 590 y(0)2100 564 y Fr(=)25 b(0)30 b Fm(:)1202 b Fr(\(5.1\))24 768 y(Clearly)-8 b(,)1127 973 y Fm(f)1182 934 y Fv(\000)1172 999 y Fi(0)1240 973 y Fr(\()p Fm(x)p Fr(\))26 b(=)f Fm(\016)s Fr(\()p Fm(x)p Fr(\))32 b Fm(;)252 b(f)2013 934 y Fi(+)2003 999 y(0)2071 973 y Fr(\()p Fm(x)p Fr(\))26 b(=)f(1)30 b Fm(:)1083 b Fr(\(5.2\))24 1177 y(Its)30 b(easy)h(to)g(see)g(that)1447 1372 y Fj(b)1426 1395 y Fm(H)22 b(f)1579 1357 y Fv(\006)1569 1421 y Fi(0)1663 1395 y Fr(=)j Fn(\006)p Fm(i)1871 1333 y(\015)p 1871 1374 53 4 v 1874 1457 a Fr(2)1948 1395 y Fm(f)2003 1357 y Fv(\006)1993 1421 y Fi(0)2091 1395 y Fm(:)1382 b Fr(\(5.3\))24 1620 y(Let)31 b(us)e(de\014ne)h(t)m(w)m(o)i (families:)883 1849 y Fm(f)938 1812 y Fv(\000)928 1872 y Fk(n)1022 1849 y Fr(:=)1153 1788 y(\()p Fn(\000)p Fm(i)p Fr(\))1325 1755 y Fk(n)p 1153 1828 220 4 v 1185 1847 a Fn(p)p 1261 1847 80 4 v 77 x Fm(n)p Fr(!)1403 1849 y Fj(b)-56 b Fm(p)1444 1812 y Fk(n)1506 1849 y Fm(f)1561 1811 y Fv(\000)1551 1875 y Fi(0)1649 1849 y Fm(;)252 b(f)1981 1812 y Fi(+)1971 1872 y Fk(n)2065 1849 y Fr(:=)2251 1788 y(1)p 2196 1828 156 4 v 2196 1847 a Fn(p)p 2271 1847 80 4 v 2271 1924 a Fm(n)p Fr(!)2380 1849 y Fj(b)-55 b Fm(x)2428 1812 y Fk(n)2490 1849 y Fm(f)2545 1811 y Fi(+)2535 1875 y(0)2634 1849 y Fm(:)839 b Fr(\(5.4\))24 2097 y(One)30 b(\014nds)876 2326 y Fm(f)931 2288 y Fv(\000)921 2348 y Fk(n)990 2326 y Fr(\()p Fm(x)p Fr(\))c(=)1244 2264 y(\()p Fn(\000)p Fr(1\))1430 2231 y Fk(n)p 1244 2305 234 4 v 1283 2323 a Fn(p)p 1359 2323 80 4 v 78 x Fm(n)p Fr(!)1503 2326 y Fm(\016)1546 2288 y Fi(\()p Fk(n)p Fi(\))1648 2326 y Fr(\()p Fm(x)p Fr(\))31 b Fm(;)252 b(f)2133 2288 y Fi(+)2123 2348 y Fk(n)2191 2326 y Fr(\()p Fm(x)p Fr(\))26 b(=)2473 2264 y Fm(x)2525 2231 y Fk(n)p 2445 2305 156 4 v 2445 2323 a Fn(p)p 2521 2323 80 4 v 78 x Fm(n)p Fr(!)2641 2326 y Fm(:)832 b Fr(\(5.5\))24 2572 y(Moreo)m(v)m(er,)1442 2753 y Fj(b)1421 2776 y Fm(H)22 b(f)1574 2739 y Fv(\006)1564 2799 y Fk(n)1657 2776 y Fr(=)j Fn(\006)p Fm(E)1891 2790 y Fk(n)1953 2776 y Fm(f)2008 2739 y Fv(\006)1998 2799 y Fk(n)2097 2776 y Fm(;)1376 b Fr(\(5.6\))24 2981 y(where)1374 3212 y Fm(E)1441 3226 y Fk(n)1513 3212 y Fr(:=)25 b Fm(i\015)1733 3084 y Fj(\022)1800 3212 y Fm(n)20 b Fr(+)1976 3151 y(1)p 1976 3191 46 4 v 1976 3275 a(2)2031 3084 y Fj(\023)2143 3212 y Fm(:)1330 b Fr(\(5.7\))24 3471 y(Clearly)-8 b(,)25 b(b)s(oth)g Fm(f)618 3438 y Fv(\000)608 3494 y Fk(n)702 3471 y Fr(and)g Fm(f)929 3438 y Fv(\000)919 3494 y Fk(n)1013 3471 y Fr(are)h(temp)s (ered)f(distributions,)e(i.e.)39 b Fm(f)2320 3438 y Fv(\006)2310 3494 y Fk(n)2403 3471 y Fn(2)25 b(S)2551 3438 y Fv(0)2575 3471 y Fr(.)39 b(Eviden)m(tly)-8 b(,)25 b(they)h(are)g(related)24 3584 y(b)m(y)k(the)g(F)-8 b(ourier)30 b(transformation:)895 3825 y Fm(F)13 b Fr([)p Fm(f)1046 3787 y Fi(+)1036 3847 y Fk(n)1105 3825 y Fr(])25 b(=)1251 3745 y Fn(p)p 1327 3745 101 4 v 80 x Fr(2)p Fm(\031)t(i)1459 3787 y Fk(n)1506 3825 y Fm(f)1561 3787 y Fv(\000)1551 3847 y Fk(n)1650 3825 y Fm(;)251 b(F)13 b Fr([)p Fm(f)2077 3787 y Fv(\000)2067 3847 y Fk(n)2136 3825 y Fr(])25 b(=)2341 3763 y Fm(i)2372 3730 y Fk(n)p 2292 3804 177 4 v 2292 3822 a Fn(p)p 2368 3822 101 4 v 75 x Fr(2)p Fm(\031)2479 3825 y(f)2534 3787 y Fi(+)2524 3847 y Fk(n)2622 3825 y Fm(:)851 b Fr(\(5.8\))24 4072 y(Let)27 b(us)f(observ)m(e,)j(that)f(these)f(t)m(w)m(o)h(families) d(of)i(generalized)g(eigen)m(v)m(ectors)i(ha)m(v)m(e)f(t)m(w)m(o)g (remark)-5 b(able)27 b(prop-)24 4185 y(erties:)1210 4287 y Fj(Z)1301 4313 y Fv(1)1260 4493 y(\0001)1405 4410 y Fm(f)1460 4373 y Fi(+)1450 4433 y Fk(n)1518 4410 y Fr(\()p Fm(x)p Fr(\))15 b Fm(f)1710 4373 y Fv(\000)1700 4433 y Fk(m)1769 4410 y Fr(\()p Fm(x)p Fr(\))g Fm(dx)27 b Fr(=)e Fm(\016)2168 4424 y Fk(nm)2308 4410 y Fm(;)1165 b Fr(\(5.9\))24 4663 y(and)1201 4800 y Fv(1)1170 4827 y Fj(X)1170 5021 y Fk(n)p Fi(=0)1333 4913 y Fm(f)1388 4876 y Fi(+)1378 4936 y Fk(n)1446 4913 y Fr(\()p Fm(x)p Fr(\))15 b Fm(f)1638 4876 y Fv(\000)1628 4936 y Fk(n)1697 4913 y Fr(\()p Fm(x)1784 4876 y Fv(0)1808 4913 y Fr(\))26 b(=)f Fm(\016)s Fr(\()p Fm(x)c Fn(\000)f Fm(x)2259 4876 y Fv(0)2282 4913 y Fr(\))31 b Fm(:)1080 b Fr(\(5.10\))1832 5281 y(8)p eop %%Page: 9 9 9 8 bop 24 44 a Fr(These)21 b(form)m(ulae)f(remind)g(one)h(of)h(the)f (basic)g(basic)f(prop)s(erties)g(of)h(prop)s(er)f(\(Hilb)s(ert)g (space\))i(eigen)m(v)m(ectors:)24 157 y(if)128 134 y Fj(b)107 157 y Fm(A)30 b Fr(is)g(a)h(self-adjoin)m(t)e(op)s(erator)i (in)e Fn(H)i Fr(and)1521 332 y Fj(b)1499 355 y Fm(A )1626 370 y Fk(k)1694 355 y Fr(=)25 b Fm(\025)1843 370 y Fk(k)1886 355 y Fm( )1945 370 y Fk(k)2018 355 y Fm(;)1410 b Fr(\(5.11\))24 553 y(where)30 b Fm( )346 568 y Fk(k)419 553 y Fr(are)g(normalized)f(v) m(ectors)j(in)d Fn(H)q Fr(,)h(then)1259 664 y Fj(Z)p 1365 713 107 4 v 1365 788 a Fm( )1424 802 y Fk(n)1471 788 y Fr(\()p Fm(x)p Fr(\))p Fm( )1652 802 y Fk(m)1720 788 y Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)26 b Fr(=)f Fm(\016)2118 802 y Fk(nm)2258 788 y Fm(;)1170 b Fr(\(5.12\))24 1022 y(and)1136 1134 y Fj(X)1180 1325 y Fk(n)p 1282 1146 V 1282 1220 a Fm( )1341 1234 y Fk(n)1388 1220 y Fr(\()p Fm(x)p Fr(\))p Fm( )1569 1234 y Fk(n)1617 1220 y Fr(\()p Fm(x)1704 1183 y Fv(0)1728 1220 y Fr(\))15 b Fm(dx)26 b Fr(=)f Fm(\016)s Fr(\()p Fm(x)c Fn(\000)f Fm(x)2293 1183 y Fv(0)2316 1220 y Fr(\))31 b Fm(:)1046 b Fr(\(5.13\))24 1500 y(Ob)m(viously)-8 b(,)29 b(there)h(is)g(no)g(complex)g (conjugation)g(in)f(\(5.9\))k(and)c(\(5.10\))k(since)d Fm(f)2808 1467 y Fv(\006)2798 1523 y Fk(n)2896 1500 y Fr(are)h(real)f(functions.)165 1613 y(No)m(w,)h(for)f(an)m(y)h Fm(\036)25 b Fn(2)g(Z)38 b Fr(one)30 b(has)810 1868 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))c(=)1108 1782 y Fj(X)1152 1973 y Fk(n)1265 1807 y Fm(\036)1319 1774 y Fi(\()p Fk(n)p Fi(\))1421 1807 y Fr(\(0\))p 1265 1847 273 4 v 1361 1931 a Fm(n)p Fr(!)1547 1868 y(\()p Fn(\000)p Fr(1\))1733 1831 y Fk(n)1781 1868 y Fm(x)1833 1831 y Fk(n)1905 1868 y Fr(=)2001 1782 y Fj(X)2045 1973 y Fk(n)2147 1868 y Fm(f)2202 1831 y Fi(+)2192 1891 y Fk(n)2261 1868 y Fr(\()p Fm(x)p Fr(\))p Fn(h)15 b Fm(f)2488 1831 y Fv(\000)2478 1891 y Fk(n)2547 1868 y Fn(j)p Fm(\036)g Fn(i)31 b Fm(:)721 b Fr(\(5.14\))24 2143 y(On)29 b(the)i(other)f(hand,)g(for)g(an)m(y)h Fm(\036)25 b Fn(2)g(D)s Fr(,)30 b(its)g(F)-8 b(ourier)30 b(transform)g Fm(F)13 b Fr([)p Fm(\036)p Fr(])26 b Fn(2)e(Z)7 b Fr(,)31 b(and)414 2404 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))84 b(=)903 2342 y(1)p 838 2383 177 4 v 838 2401 a Fn(p)p 913 2401 101 4 v 913 2476 a Fr(2)p Fm(\031)1039 2280 y Fj(Z)1145 2404 y Fm(e)1187 2366 y Fk(ik)r(x)1294 2404 y Fm(F)13 b Fr([)p Fm(\036)p Fr(]\()p Fm(k)s Fr(\))p Fm(dk)30 b Fr(=)1885 2342 y(1)p 1819 2383 177 4 v 1819 2401 a Fn(p)p 1895 2401 101 4 v 75 x Fr(2)p Fm(\031)2021 2280 y Fj(Z)2127 2404 y Fm(e)2169 2366 y Fk(ik)r(x)2291 2317 y Fj(X)2335 2508 y Fk(n)2447 2342 y Fm(F)13 b Fr([)p Fm(\036)p Fr(])2622 2309 y Fi(\()p Fk(n)p Fi(\))2725 2342 y Fr(\(0\))p 2447 2383 394 4 v 2604 2466 a Fm(n)p Fr(!)2851 2404 y(\()p Fn(\000)p Fr(1\))3037 2366 y Fk(n)3085 2404 y Fm(k)3135 2366 y Fk(n)3197 2404 y Fm(dk)674 2646 y Fr(=)828 2560 y Fj(X)872 2750 y Fk(n)974 2646 y Fm(F)g Fr([)p Fm(f)1125 2608 y Fi(+)1115 2668 y Fk(n)1184 2646 y Fr(]\()p Fm(x)p Fr(\))p Fn(h)i Fm(f)1436 2608 y Fv(\000)1426 2668 y Fk(n)1496 2646 y Fn(j)p Fm(F)e Fr([)p Fm(\036)p Fr(])i Fn(i)26 b Fr(=)1868 2560 y Fj(X)1913 2750 y Fk(n)2015 2646 y Fm(F)13 b Fr([)p Fm(f)2166 2608 y Fi(+)2156 2668 y Fk(n)2225 2646 y Fr(]\()p Fm(x)p Fr(\))p Fn(h)i Fm(F)e Fr([)p Fm(f)2573 2608 y Fv(\000)2563 2668 y Fk(n)2633 2646 y Fr(])p Fn(j)p Fm(\036)i Fn(i)674 2888 y Fr(=)828 2802 y Fj(X)872 2993 y Fk(n)974 2888 y Fm(f)1029 2851 y Fv(\000)1019 2911 y Fk(n)1088 2888 y Fr(\()p Fm(x)p Fr(\))p Fn(h)g Fm(f)1315 2851 y Fi(+)1305 2911 y Fk(n)1374 2888 y Fn(j)p Fm(\036)g Fn(i)31 b Fm(:)1894 b Fr(\(5.15\))24 3158 y(Hence,)31 b(w)m(e)g(ha)m(v)m(e)h(t)m(w)m(o)f(sp)s(ectral)f (decomp)s(ositions:)1065 3469 y Fn(j)p Fm(\036)15 b Fn(i)26 b Fr(=)1316 3383 y Fj(X)1361 3574 y Fk(n)1463 3469 y Fn(j)p Fm(f)1543 3431 y Fi(+)1533 3491 y Fk(n)1617 3469 y Fn(ih)15 b Fm(f)1757 3431 y Fv(\000)1747 3491 y Fk(n)1816 3469 y Fn(j)p Fm(\036)g Fn(i)237 b Fr(in)90 b Fn(Z)37 b Fm(;)976 b Fr(\(5.16\))24 3739 y(and)1057 3937 y Fn(j)p Fm( )19 b Fn(i)26 b Fr(=)1317 3851 y Fj(X)1361 4041 y Fk(n)1463 3937 y Fn(j)p Fm(f)1543 3899 y Fv(\000)1533 3959 y Fk(n)1617 3937 y Fn(ih)15 b Fm(f)1757 3899 y Fi(+)1747 3959 y Fk(n)1816 3937 y Fn(j)p Fm( )k Fn(i)237 b Fr(in)89 b Fn(D)33 b Fm(:)968 b Fr(\(5.17\))24 4232 y(In)29 b(section)i(7)g(w)m (e)f(deriv)m(e)g(\(5.16\))j(and)d(\(5.17\))i(from)e(the)h(sp)s(ectrum)e (of)2514 4209 y Fj(b)2493 4232 y Fm(H)7 b Fr(.)40 b(So)30 b(let)h(us)e(lo)s(ok)h(for)h Fm(\033)s Fr(\()3463 4209 y Fj(b)3442 4232 y Fm(H)7 b Fr(\).)24 4518 y Fo(6)134 b(Sp)t(ectrum)24 4721 y Fr(The)22 b(Hamiltonian)e(\(3.10\))25 b(has)d(a)h(con)m(tin)m(uous)g(sp)s(ectrum)e Fm(\033)s Fr(\()2143 4698 y Fj(b)2121 4721 y Fm(H)7 b Fr(\))26 b(=)f(\()p Fn(\0001)p Fm(;)15 b Fn(1)p Fr(\).)39 b(Since,)23 b(the)g(Hamiltonian)24 4834 y(\(3.10\))32 b(is)e(parit)m(y)g(in)m(v)-5 b(arian)m(t)29 b(eac)m(h)j(generalized)e(eigen)m(v)-5 b(alue)30 b Fm(E)h Fn(2)24 b Fl(R)39 b Fr(is)30 b(doubly)e (degenerated:)1504 5009 y Fj(b)1483 5032 y Fm(H)7 b( )1628 4994 y Fk(E)1625 5054 y Fv(\006)1713 5032 y Fr(=)25 b Fm(E)5 b( )1943 4994 y Fk(E)1940 5054 y Fv(\006)2034 5032 y Fm(:)1439 b Fr(\(6.1\))1832 5281 y(9)p eop %%Page: 10 10 10 9 bop 24 44 a Fr(The)29 b(ab)s(o)m(v)m(e)j(equation)e(ma)m(y)h(b)s (e)f(rewritten)f(as)i(the)g(follo)m(wing)d(di\013eren)m(tial)h (equation)h(for)h Fm( )3234 11 y Fk(E)3231 67 y Fv(\006)3294 44 y Fr(:)1146 305 y Fm(x)1234 244 y(d)p 1208 284 100 4 v 1208 367 a(dx)1318 305 y( )1380 267 y Fk(E)1377 327 y Fv(\006)1440 305 y Fr(\()p Fm(x)p Fr(\))26 b(=)f Fn(\000)1770 177 y Fj(\022)1836 305 y Fm(i)1877 244 y(E)p 1877 284 73 4 v 1887 367 a(\015)1980 305 y Fr(+)2081 244 y(1)p 2081 284 46 4 v 2081 367 a(2)2136 177 y Fj(\023)2219 305 y Fm( )2281 267 y Fk(E)2278 327 y Fv(\006)2371 305 y Fm(:)1102 b Fr(\(6.2\))24 559 y(T)-8 b(o)30 b(solv)m(e)h(\(6.2\))h (let)f(us)e(in)m(tro)s(duce)h(the)g(follo)m(wing)f(distributions)e([13) q(])j(\(see)i(also)e([24)q(]\):)641 817 y Fm(x)693 780 y Fk(\025)693 840 y Fi(+)777 817 y Fr(:=)899 689 y Fj(\032)1008 761 y Fm(x)1060 728 y Fk(\025)1280 761 y Fm(x)25 b Fn(\025)g Fr(0)1008 874 y(0)227 b Fm(x)25 b(<)g Fr(0)1595 817 y Fm(;)252 b(x)1924 780 y Fk(\025)1924 840 y Fv(\000)2008 817 y Fr(:=)2129 689 y Fj(\032)2290 761 y Fr(0)226 b Fm(x)25 b Fn(\025)g Fr(0)2239 874 y Fn(j)p Fm(x)p Fn(j)2341 841 y Fk(\025)2561 874 y Fm(x)g(<)g Fr(0)2876 817 y Fm(;)597 b Fr(\(6.3\))24 1086 y(with)27 b Fm(\025)e Fn(2)g Fl(C)52 b Fr(\(basic)29 b(prop)s(erties)e(of)i Fm(x)1329 1053 y Fk(\025)1329 1109 y Fv(\006)1416 1086 y Fr(are)h(collected)f(in)e (the)i(App)s(endix\).)38 b(It)29 b(is,)f(therefore,)i(clear)e(that)24 1199 y(the)i(generalized)g(eigen)m(v)m(ectors)i Fm( )1220 1166 y Fk(E)1217 1222 y Fv(\006)1311 1199 y Fr(ma)m(y)f(b)s(e)f (written)f(as)i(follo)m(ws:)1175 1448 y Fm( )1237 1411 y Fk(E)1234 1471 y Fv(\006)1297 1448 y Fr(\()p Fm(x)p Fr(\))26 b(:=)1668 1387 y(1)p 1577 1427 229 4 v 1577 1445 a Fn(p)p 1652 1445 153 4 v 1652 1512 a Fr(2)p Fm(\031)s(\015)1830 1448 y(x)1882 1400 y Fv(\000)p Fi(\()p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))1882 1472 y Fv(\006)2342 1448 y Fm(:)1131 b Fr(\(6.4\))24 1711 y(It)29 b(turns)e(out)j(that)f Fm( )775 1678 y Fk(E)772 1733 y Fv(\006)864 1711 y Fr(are)g(w)m(ell)f (de\014ned)g(temp)s(ered)g(distributions)e(for)i(all)g Fm(E)j Fn(2)25 b Fl(R)r Fr(.)46 b(Actually)-8 b(,)30 b(instead)24 1824 y(of)g Fm( )189 1791 y Fk(E)186 1846 y Fv(\006)280 1824 y Fr(one)g(ma)m(y)h(w)m(ork)g(with)e(eigen)m(v)m (ectors)j(of)f(the)f(parit)m(y)g(op)s(erator)h Fs(P)p Fr(:)1295 2073 y Fm( )1357 2035 y Fk(E)1354 2095 y Fi(ev)n(en)1579 2073 y Fr(=)1780 2011 y(1)p 1743 2052 122 4 v 1743 2070 a Fn(p)p 1818 2070 46 4 v 1818 2145 a Fr(2)1889 1999 y Fj(\000)1931 2073 y Fm( )1993 2035 y Fk(E)1990 2095 y Fi(+)2073 2073 y Fr(+)20 b Fm( )2226 2035 y Fk(E)2223 2095 y Fv(\000)2286 1999 y Fj(\001)2373 2073 y Fm(;)1100 b Fr(\(6.5\))1317 2311 y Fm( )1379 2273 y Fk(E)1376 2334 y Fi(o)r(dd)1579 2311 y Fr(=)1780 2249 y(1)p 1743 2290 122 4 v 1743 2308 a Fn(p)p 1818 2308 46 4 v 1818 2383 a Fr(2)1889 2237 y Fj(\000)1931 2311 y Fm( )1993 2273 y Fk(E)1990 2333 y Fi(+)2073 2311 y Fn(\000)20 b Fm( )2226 2273 y Fk(E)2223 2333 y Fv(\000)2286 2237 y Fj(\001)2373 2311 y Fm(:)1100 b Fr(\(6.6\))24 2558 y(Ob)m(viously)967 2762 y Fs(P)15 b Fm( )1115 2725 y Fk(E)1112 2785 y Fi(ev)n(en)1280 2762 y Fr(=)25 b Fm( )1438 2725 y Fk(E)1435 2785 y Fi(ev)n(en)1607 2762 y Fm(;)251 b Fs(P)15 b Fm( )2031 2725 y Fk(E)2028 2785 y Fi(o)r(dd)2174 2762 y Fr(=)25 b Fn(\000)p Fm( )2403 2725 y Fk(E)2400 2785 y Fi(o)r(dd)2550 2762 y Fm(:)923 b Fr(\(6.7\))24 2966 y(These)30 b(distributions)c(of)31 b(de\014nite)e(parit)m(y)h(are)h(giv)m(en)f(b)m(y:)344 3204 y Fm( )406 3166 y Fk(E)403 3226 y Fi(ev)n(en)569 3204 y Fr(=)767 3142 y(1)p 675 3183 229 4 v 675 3266 a(2)720 3209 y Fn(p)p 797 3209 108 4 v 797 3266 a Fm(\031)s(\015)929 3204 y Fn(j)p Fm(x)p Fn(j)1031 3166 y Fv(\000)p Fi(\()p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))1491 3204 y Fm(;)252 b( )1830 3166 y Fk(E)1827 3227 y Fi(o)r(dd)1972 3204 y Fr(=)2170 3142 y(1)p 2078 3183 229 4 v 2078 3266 a(2)2123 3209 y Fn(p)p 2199 3209 108 4 v 57 x Fm(\031)s(\015)2332 3204 y Fr(sign)o(\()p Fm(x)p Fr(\))p Fn(j)p Fm(x)p Fn(j)2713 3166 y Fv(\000)p Fi(\()p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))3174 3204 y Fm(;)299 b Fr(\(6.8\))24 3475 y(\(see)31 b([13)q(])g(and)e([24)r(])h(for)g(the)h(prop)s(erties)e(of)h Fn(j)p Fm(x)p Fn(j)1655 3442 y Fk(\025)1731 3475 y Fr(and)g(sign)o(\()p Fm(x)p Fr(\))p Fn(j)p Fm(x)p Fn(j)2289 3442 y Fk(\025)2336 3475 y Fr(\).)165 3588 y(With)g(the)g(normalization)f(used)h(in)f (\(6.4\))j(one)f(pro)m(v)m(es)g([25)q(])f(orthonormalit)m(y:)1078 3711 y Fj(Z)p 1184 3738 276 4 v 1184 3834 a Fm( )1246 3794 y Fk(E)1298 3803 y Fe(1)1243 3858 y Fv(\006)1337 3834 y Fr(\()p Fm(x)p Fr(\))p Fm( )1521 3794 y Fk(E)1573 3803 y Fe(2)1518 3858 y Fv(\006)1612 3834 y Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)27 b Fr(=)e Fm(\016)s Fr(\()p Fm(E)2116 3848 y Fi(1)2177 3834 y Fn(\000)19 b Fm(E)2334 3848 y Fi(2)2374 3834 y Fr(\))31 b Fm(;)1033 b Fr(\(6.9\))24 4076 y(and)29 b(completeness:)1130 4193 y Fj(Z)p 1236 4229 245 4 v 1236 4317 a Fm( )1298 4285 y Fk(E)1295 4340 y Fv(\006)1358 4317 y Fr(\()p Fm(x)p Fr(\))p Fm( )1542 4279 y Fk(E)1539 4339 y Fv(\006)1603 4317 y Fr(\()p Fm(x)1690 4279 y Fv(0)1713 4317 y Fr(\))15 b Fm(dE)32 b Fr(=)25 b Fm(\016)s Fr(\()p Fm(x)c Fn(\000)f Fm(x)2299 4279 y Fv(0)2322 4317 y Fr(\))31 b Fm(:)1040 b Fr(\(6.10\))24 4558 y(Therefore,)30 b(due)g(to)h(the)f(Gelfand-Maurin)f(sp)s(ectral)h (theorem)g(one)h(has)1127 4799 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))26 b(=)1425 4712 y Fj(X)1463 4904 y Fv(\006)1572 4675 y Fj(Z)1678 4799 y Fm(dE)21 b( )1875 4761 y Fk(E)1872 4821 y Fv(\006)1935 4799 y Fr(\()p Fm(x)p Fr(\))p Fn(h)15 b Fm( )2169 4761 y Fk(E)2166 4821 y Fv(\006)2230 4799 y Fn(j)p Fm(\036)g Fn(i)31 b Fm(;)1038 b Fr(\(6.11\))1809 5281 y(10)p eop %%Page: 11 11 11 10 bop 24 44 a Fr(for)25 b(an)m(y)g Fm(\036)h Fn(2)e(S)7 b Fr(,)27 b(and)d(the)i(corresp)s(onding)d(sp)s(ectral)h(resolution)g (of)h(the)h(Hamiltonian)d(has)i(the)h(follo)m(wing)24 157 y(form:)1235 340 y Fj(b)1214 363 y Fm(H)32 b Fr(=)1418 277 y Fj(X)1456 468 y Fv(\006)1564 239 y Fj(Z)1670 363 y Fm(dE)21 b(E)5 b Fn(j)p Fm( )1964 325 y Fk(E)1961 386 y Fv(\006)2040 363 y Fn(ih)15 b Fm( )2187 325 y Fk(E)2184 386 y Fv(\006)2248 363 y Fn(j)31 b Fm(:)1124 b Fr(\(6.12\))24 645 y(There)28 b(is)f(another)i(family)d(of)j(energy)g(eigen)m(v)m (ectors)h(directly)d(related)i(to)g Fm( )2680 612 y Fk(E)2677 667 y Fv(\006)2740 645 y Fr(.)40 b(Due)28 b(to)i(\(4.19\))g(one)f(has:) 1320 822 y Fj(b)1299 845 y Fm(H)22 b(F)13 b Fr([)p Fm( )1555 807 y Fv(\000)p Fk(E)1552 869 y Fv(\006)1671 845 y Fr(])25 b(=)g Fm(E)5 b(F)13 b Fr([)p Fm( )2047 807 y Fv(\000)p Fk(E)2044 869 y Fv(\006)2163 845 y Fr(])30 b Fm(:)1210 b Fr(\(6.13\))24 1032 y(The)29 b(F)-8 b(ourier)30 b(transform)g(of)h Fm( )1109 999 y Fk(E)1106 1054 y Fv(\006)1199 1032 y Fr(is)e(giv)m(en)i(b)m(y)f(\(cf.)41 b([13)q(])31 b(and)f(the)g(App)s (endix\):)231 1275 y Fm(F)13 b Fr([)p Fm( )389 1237 y Fv(\000)p Fk(E)386 1299 y Fv(\006)504 1275 y Fr(]\()p Fm(k)s Fr(\))26 b(=)f Fn(\006)951 1214 y Fm(i)p 852 1254 229 4 v 852 1338 a Fr(2)p Fm(\031)952 1281 y Fn(p)p 1028 1281 53 4 v 57 x Fm(\015)1106 1275 y Fr(exp)1260 1147 y Fj(\024)1308 1275 y Fn(\006)1389 1214 y Fm(i\031)p 1389 1254 87 4 v 1409 1338 a Fr(2)1500 1147 y Fj(\022)1567 1275 y Fm(i)1608 1214 y(E)p 1608 1254 73 4 v 1618 1338 a(\015)1710 1275 y Fn(\000)1811 1214 y Fr(1)p 1811 1254 46 4 v 1811 1338 a(2)1867 1147 y Fj(\023\025)1997 1275 y Fr(\000)2069 1147 y Fj(\022)2136 1275 y Fm(i)2177 1214 y(E)p 2177 1254 73 4 v 2187 1338 a(\015)2280 1275 y Fr(+)2380 1214 y(1)p 2380 1254 46 4 v 2380 1338 a(2)2436 1147 y Fj(\023)2518 1275 y Fr(\()p Fm(k)f Fn(\006)c Fm(i)p Fr(0\))2826 1238 y Fv(\000)p Fi(\()p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))3287 1275 y Fm(:)141 b Fr(\(6.14\))24 1529 y(One)27 b(sho)m(ws)g([13)q(])g(that)h Fm(F)13 b Fr([)p Fm( )991 1496 y Fk(E)988 1551 y Fv(\006)1052 1529 y Fr(])27 b(are)h(w)m(ell)e (de\014ned)h(temp)s(ered)f(distributions)e(for)j(an)m(y)h Fm(E)i Fn(2)25 b Fl(R)s Fr(.)46 b(Moreo)m(v)m(er,)949 1641 y Fj(Z)p 1055 1667 275 4 v 1055 1764 a Fm(F)13 b Fr([)p Fm( )1213 1724 y Fk(E)1265 1733 y Fe(1)1210 1788 y Fv(\006)1304 1764 y Fr(]\()p Fm(x)p Fr(\))i Fm(F)e Fr([)p Fm( )1624 1724 y Fk(E)1676 1733 y Fe(2)1621 1788 y Fv(\006)1716 1764 y Fr(]\()p Fm(x)p Fr(\))i Fm(dx)27 b Fr(=)e Fm(\016)s Fr(\()p Fm(E)2245 1778 y Fi(1)2306 1764 y Fn(\000)20 b Fm(E)2464 1778 y Fi(2)2503 1764 y Fr(\))31 b Fm(;)859 b Fr(\(6.15\))24 1988 y(and)1001 2070 y Fj(Z)p 1107 2106 244 4 v 1107 2194 a Fm(F)13 b Fr([)p Fm( )1265 2163 y Fk(E)1262 2217 y Fv(\006)1325 2194 y Fr(]\()p Fm(x)p Fr(\))i Fm(F)e Fr([)p Fm( )1645 2156 y Fk(E)1642 2216 y Fv(\006)1706 2194 y Fr(]\()p Fm(x)1818 2156 y Fv(0)1842 2194 y Fr(\))i Fm(dE)32 b Fr(=)25 b Fm(\016)s Fr(\()p Fm(x)c Fn(\000)f Fm(x)2428 2156 y Fv(0)2451 2194 y Fr(\))31 b Fm(:)911 b Fr(\(6.16\))24 2418 y(Hence,)33 b(follo)m(wing)d(the)h(Gelfand-Maurin)f(theorem,)j(w)m (e)f(ha)m(v)m(e)g(further)f(sp)s(ectral)g(decomp)s(ositions:)41 b(for)24 2530 y(an)m(y)30 b Fm( )f Fn(2)c(S)942 2754 y Fm( )s Fr(\()p Fm(x)p Fr(\))i(=)1249 2668 y Fj(X)1287 2859 y Fv(\006)1395 2630 y Fj(Z)1501 2754 y Fm(dE)21 b(F)13 b Fr([)p Fm( )1794 2716 y Fv(\000)p Fk(E)1791 2777 y Fv(\006)1909 2754 y Fr(]\()p Fm(x)p Fr(\))p Fn(h)i Fm(F)e Fr([)p Fm( )2264 2716 y Fv(\000)p Fk(E)2261 2777 y Fv(\006)2381 2754 y Fr(])p Fn(j)p Fm( )19 b Fn(i)31 b Fm(;)853 b Fr(\(6.17\))24 3021 y(and)29 b(for)i(the)f(Hamiltonian)f (itself:)1059 3204 y Fj(b)1037 3227 y Fm(H)k Fr(=)1241 3140 y Fj(X)1280 3332 y Fv(\006)1388 3103 y Fj(Z)1494 3227 y Fm(dE)21 b(E)5 b Fn(j)p Fm(F)13 b Fr([)p Fm( )1884 3188 y Fv(\000)p Fk(E)1881 3250 y Fv(\006)2000 3227 y Fr(])i Fn(ih)g Fm(F)e Fr([)p Fm( )2283 3188 y Fv(\000)p Fk(E)2280 3250 y Fv(\006)2399 3227 y Fr(])p Fn(j)31 b Fm(:)948 b Fr(\(6.18\))24 3574 y Fo(7)134 b(Analyticit)l(y)47 b(of)e(energy)g(eigen)l(v)l(ectors)24 3776 y Fr(Let)23 b(us)g(con)m(tin)m(ue)g(the)h(energy)f(eigen)m(v)m(ectors)j Fm( )1642 3743 y Fk(E)1639 3799 y Fv(\006)1725 3776 y Fr(and)d Fm(F)13 b Fr([)p Fm( )2053 3738 y Fv(\000)p Fk(E)2050 3800 y Fv(\006)2168 3776 y Fr(])23 b(in)m(to)g(the)h(energy)f (complex)g(plane)g Fm(E)30 b Fn(2)25 b Fl(C)18 b Fr(.)24 3889 y(It)37 b(turns)g(out)h([13)q(])f(\(see)i(also)f(the)f(App)s (endix\))f(that)i Fm( )1991 3856 y Fk(E)1988 3912 y Fv(\006)2088 3889 y Fr(has)g(simple)d(p)s(oles)i(at)h Fm(E)43 b Fr(=)37 b Fn(\000)p Fm(E)3260 3903 y Fk(n)3307 3889 y Fr(,)i(whereas)24 4011 y Fm(F)13 b Fr([)p Fm( )182 3972 y Fv(\000)p Fk(E)179 4034 y Fv(\006)297 4011 y Fr(])40 b(has)g(simple)e(p)s(oles)h(at)i Fm(E)46 b Fr(=)41 b(+)p Fm(E)1551 4025 y Fk(n)1598 4011 y Fr(,)i(with)38 b Fm(E)1949 4025 y Fk(n)2036 4011 y Fr(de\014ned)h(in)g(\(5.7\).)71 b(Therefore,)43 b(the)d(p)s(oles)f(of) 24 4124 y(energy)30 b(eigen)m(v)m(ectors)h(considered)e(as)h(functions) f(of)h(the)g(complex)f(energy)h(corresp)s(ond)f(exactly)h(to)h(the)24 4236 y(complex)20 b(eigen)m(v)-5 b(alues)21 b(of)947 4214 y Fj(b)926 4236 y Fm(H)28 b Fr(whic)m(h)19 b(w)m(e)j(found)e(in)g (Section)h(5.)38 b(One)20 b(easily)g(computes)i(the)f(corresp)s(onding) 24 4349 y(residues:)951 4576 y(Res\()p Fm( )1191 4538 y Fk(E)1188 4598 y Fv(\006)1252 4576 y Fr(\()p Fm(x)p Fr(\);)15 b Fn(\000)p Fm(E)1552 4590 y Fk(n)1599 4576 y Fr(\))26 b(=)f Fm(i)15 b Fr(\()p Fn(\007)p Fr(1\))1988 4538 y Fk(n)2036 4443 y Fj(r)p 2127 4443 121 4 v 2161 4514 a Fm(\015)p 2137 4555 101 4 v 2137 4638 a Fr(2)p Fm(\031)2258 4514 y(\016)2301 4481 y Fi(\()p Fk(n)p Fi(\))2403 4514 y Fr(\()p Fm(x)p Fr(\))p 2258 4555 269 4 v 2352 4638 a Fm(n)p Fr(!)2566 4576 y Fm(;)907 b Fr(\(7.1\))24 4803 y(and)812 5015 y(Res\()p Fm(F)13 b Fr([)p Fm( )1148 4977 y Fv(\000)p Fk(E)1145 5038 y Fv(\006)1264 5015 y Fr(\()p Fm(x)p Fr(\)];)i(+)p Fm(E)1589 5029 y Fk(n)1637 5015 y Fr(\))26 b(=)f Fn(\006)1875 4892 y(p)p 1950 4892 53 4 v 1950 4949 a Fm(\015)p 1875 4994 128 4 v 1889 5077 a Fr(2)p Fm(\031)2012 5015 y Fr(\()p Fn(\007)p Fm(i)p Fr(\))2184 4977 y Fk(n)p Fi(+1)2332 4953 y Fr(\()p Fn(\000)p Fr(1\))2518 4920 y Fk(n)p 2332 4994 234 4 v 2409 5077 a Fm(n)p Fr(!)2576 5015 y Fm(x)2628 4977 y Fk(n)2705 5015 y Fm(:)768 b Fr(\(7.2\))1809 5281 y(11)p eop %%Page: 12 12 12 11 bop 24 44 a Fr(Hence,)35 b(residues)d(of)i Fm( )840 11 y Fk(E)837 67 y Fv(\006)934 44 y Fr(and)f Fm(F)13 b Fr([)p Fm( )1272 6 y Fv(\000)p Fk(E)1269 68 y Fv(\006)1387 44 y Fr(])34 b(corresp)s(ond,)f(up)g(to)h(n)m(umerical)e(factors,)k(to) e(the)g(eigen)m(v)m(ectors)24 157 y Fm(f)79 124 y Fv(\006)69 180 y Fk(n)167 157 y Fr(\(5.5\):)1254 345 y(Res\()p Fm( )1494 307 y Fk(E)1491 367 y Fv(\006)1555 345 y Fr(\()p Fm(x)p Fr(\);)15 b Fn(\000)p Fm(E)1855 359 y Fk(n)1903 345 y Fr(\))55 b Fn(\030)h Fm(f)2175 307 y Fv(\000)2165 367 y Fk(n)2263 345 y Fm(;)1210 b Fr(\(7.3\))24 532 y(and)1166 720 y(Res\()p Fm(F)13 b Fr([)p Fm( )1502 682 y Fv(\000)p Fk(E)1499 743 y Fv(\006)1618 720 y Fr(\()p Fm(x)p Fr(\)];)i(+)p Fm(E)1943 734 y Fk(n)1991 720 y Fr(\))56 b Fn(\030)f Fm(f)2263 683 y Fi(+)2253 743 y Fk(n)2352 720 y Fm(:)1121 b Fr(\(7.4\))24 908 y(An)m(y)30 b(function)f Fm(\036)d Fn(2)e(S)33 b(\032)24 b Fm(L)984 875 y Fi(2)1024 908 y Fr(\()p Fl(R)s Fr(\))36 b(giv)m(es)31 b(rise)f(to)h(the)f(follo)m (wing)f(functions)g(of)i(energy:)1334 1095 y Fl(R)i Fn(3)25 b Fm(E)61 b Fn(\000)-15 b(!)55 b(h)15 b Fm( )1952 1058 y Fk(E)1949 1118 y Fv(\006)2013 1095 y Fn(j)p Fm(\036)g Fn(i)26 b(2)f Fl(C)54 b Fm(;)24 1283 y Fr(and)1245 1396 y Fl(R)34 b Fn(3)25 b Fm(E)61 b Fn(\000)-15 b(!)55 b(h)15 b Fm(F)e Fr([)p Fm( )1960 1358 y Fv(\000)p Fk(E)1957 1419 y Fv(\006)2076 1396 y Fr(])p Fn(j)p Fm(\036)i Fn(i)26 b(2)f Fl(C)54 b Fm(:)24 1555 y Fr(Let)34 b(us)f(in)m(tro)s(duce)f(t)m (w)m(o)j(imp)s(ortan)m(t)e(classes)h(of)g(functions)e([26)q(]:)48 b(a)34 b(smo)s(oth)g(function)e Fm(f)40 b Fr(=)31 b Fm(f)10 b Fr(\()p Fm(E)5 b Fr(\))34 b(is)e(in)24 1667 y(the)e(Hardy)g(class)g (from)g(ab)s(o)m(v)m(e)h Fn(H)1221 1634 y Fi(2)1220 1690 y(+)1309 1667 y Fr(\(from)f(b)s(elo)m(w)g Fn(H)1895 1634 y Fi(2)1894 1690 y Fv(\000)1953 1667 y Fr(\))g(if)f Fm(f)10 b Fr(\()p Fm(E)5 b Fr(\))31 b(is)e(a)i(b)s(oundary)d(v)-5 b(alue)30 b(of)g(an)g(analytic)24 1780 y(function)i(in)g(the)i(upp)s (er,)e(i.e.)50 b(Im)15 b Fm(E)36 b Fn(\025)30 b Fr(0)k(\(lo)m(w)m(er,)h (i.e.)50 b(Im)14 b Fm(E)36 b Fn(\024)30 b Fr(0\))35 b(half)d(complex)h Fm(E)5 b Fr(-plane)33 b(v)-5 b(anishing)24 1893 y(faster)30 b(than)h(an)m(y)f(p)s(o)m(w)m(er)h(of)f Fm(E)36 b Fr(at)31 b(the)f(upp)s(er)f(\(lo)m(w)m(er\))i(semi-circle)e Fn(j)p Fm(E)5 b Fn(j)26 b(!)g(1)p Fr(.)40 b(No)m(w,)32 b(de\014ne)1121 2104 y(\010)1187 2118 y Fv(\000)1271 2104 y Fr(:=)1392 2003 y Fj(n)1453 2104 y Fm(\036)25 b Fn(2)g(S)1695 1999 y Fj(\014)1695 2054 y(\014)1695 2108 y(\014)1741 2104 y Fn(h)15 b Fm( )1853 2066 y Fk(E)1850 2126 y Fv(\006)1913 2104 y Fn(j)p Fm(\036)g Fn(i)26 b(2)f(H)2232 2066 y Fi(2)2231 2126 y Fv(\000)2305 2003 y Fj(o)2396 2104 y Fm(;)1077 b Fr(\(7.5\))24 2309 y(and)1033 2497 y(\010)1099 2511 y Fi(+)1183 2497 y Fr(:=)1304 2396 y Fj(n)1365 2497 y Fm(\036)25 b Fn(2)g(S)1607 2392 y Fj(\014)1607 2447 y(\014)1607 2501 y(\014)1652 2497 y Fn(h)15 b Fm(F)e Fr([)p Fm( )1860 2458 y Fv(\000)p Fk(E)1857 2520 y Fv(\006)1976 2497 y Fr(])p Fn(j)p Fm(\036)i Fn(i)27 b(2)e(H)2321 2459 y Fi(2)2320 2519 y(+)2394 2396 y Fj(o)2484 2497 y Fm(:)989 b Fr(\(7.6\))24 2715 y Fs(Prop)s(osition)36 b(4)46 b Fr(\010)753 2729 y Fi(+)831 2715 y Fn(\\)20 b Fr(\010)978 2729 y Fv(\000)1062 2715 y Fr(=)25 b Fn(f;g)p Ff(.)24 2911 y(Pr)-5 b(o)g(of.)39 b Fr(Clearly)-8 b(,)25 b(if)e Fm(\036)i Fn(2)g Fr(\010)938 2925 y Fv(\000)997 2911 y Fr(,)g(then)f Fn(h)15 b Fm( )1360 2878 y Fk(E)1357 2934 y Fv(\006)1420 2911 y Fn(j)p Fm(\036)g Fn(i)25 b Fr(is)e(a)h(smo)s(oth)g(function)f(of)h Fm(E)31 b Fn(2)24 b Fl(R)s Fr(.)45 b(Supp)s(ose,)23 b(that)i Fm(\036)g Fn(2)g Fr(\010)3601 2925 y Fi(+)3660 2911 y Fr(,)24 3024 y(that)31 b(is)1080 3212 y Fn(h)15 b Fm(F)e Fr([)p Fm( )1288 3173 y Fv(\000)p Fk(E)1285 3235 y Fv(\006)1404 3212 y Fr(])p Fn(j)p Fm(\036)i Fn(i)27 b Fr(=)e Fn(h)15 b Fm( )1793 3173 y Fv(\000)p Fk(E)1790 3235 y Fv(\006)1908 3212 y Fn(j)p Fm(F)e Fr([)p Fm(\036)p Fr(])i Fn(i)27 b(2)e(H)2349 3174 y Fi(2)2348 3234 y(+)2437 3212 y Fm(:)1036 b Fr(\(7.7\))24 3399 y(No)m(w,)31 b(due)f(to)h(the)f(P)m(aley-Wiener)h (theorem)g([12)q(])f(the)h(in)m(v)m(erse)f(F)-8 b(ourier)30 b(transform)g(of)g Fm(F)13 b Fr([)p Fm(\036)p Fr(])914 3631 y Fm(F)985 3594 y Fv(\000)p Fi(1)1080 3631 y Fr([)p Fm(F)g Fr([)p Fm(\036)p Fr(]]\()p Fm(E)5 b Fr(\))27 b(=)1646 3570 y(1)p 1580 3610 177 4 v 1580 3628 a Fn(p)p 1656 3628 101 4 v 76 x Fr(2)p Fm(\031)1782 3507 y Fj(Z)1872 3534 y Fv(1)1832 3714 y(\0001)1977 3631 y Fm(F)13 b Fr([)p Fm(\036)p Fr(]\()p Fm(t)p Fr(\))i Fm(e)2312 3594 y Fv(\000)p Fk(itE)2493 3631 y Fm(dt)30 b(;)870 b Fr(\(7.8\))24 3872 y(v)-5 b(anishes)35 b(for)h Fm(E)k(>)35 b Fr(0.)59 b(Therefore,)37 b Fm(\036)p Fr(\()p Fm(E)5 b Fr(\))36 b(=)f(0)i(for)f Fm(E)k(>)35 b Fr(0,)j(and)e(hence)g Fm(f)10 b Fr(\()p Fm(E)5 b Fr(\))37 b(cannot)g(b)s(e)e(a)i(smo)s(oth)24 3985 y(function)29 b(of)h Fm(E)5 b Fr(.)3037 b Fa(2)24 4098 y Fr(Our)29 b(main)g(result)g(consists)h(in)f(the)i(follo)m(wing) 24 4271 y Fs(Theorem)j(2)45 b Ff(F)-7 b(or)34 b(any)f Fm(\036)963 4238 y Fv(\006)1048 4271 y Fn(2)25 b Fr(\010)1200 4285 y Fv(\006)1291 4271 y Ff(one)33 b(has)1196 4470 y Fm(\036)1250 4433 y Fv(\000)1309 4470 y Fr(\()p Fm(x)p Fr(\))26 b(=)1553 4384 y Fj(X)1598 4575 y Fk(n)1700 4470 y Fm(f)1755 4433 y Fv(\000)1745 4493 y Fk(n)1813 4470 y Fr(\()p Fm(x)p Fr(\))p Fn(h)15 b Fm(f)2040 4433 y Fi(+)2030 4493 y Fk(n)2100 4470 y Fn(j)p Fm(\036)2179 4433 y Fv(\000)2253 4470 y Fn(i)33 b Fm(;)1152 b Fr(\(7.9\))24 4730 y Ff(and)1196 4918 y Fm(\036)1250 4880 y Fi(+)1309 4918 y Fr(\()p Fm(x)p Fr(\))26 b(=)1553 4832 y Fj(X)1598 5022 y Fk(n)1700 4918 y Fm(f)1755 4880 y Fi(+)1745 4940 y Fk(n)1813 4918 y Fr(\()p Fm(x)p Fr(\))p Fn(h)15 b Fm(f)2040 4880 y Fv(\000)2030 4940 y Fk(n)2100 4918 y Fn(j)p Fm(\036)2179 4880 y Fi(+)2253 4918 y Fn(i)33 b Fm(:)1107 b Fr(\(7.10\))1809 5281 y(12)p eop %%Page: 13 13 13 12 bop 24 44 a Ff(Pr)-5 b(o)g(of.)41 b Fr(Due)31 b(to)g(the)g(sp)s (ectral)e(form)m(ula)h(\(6.11\))j(one)d(has,)h(for)f Fm(\036)2247 11 y Fv(\000)2331 44 y Fn(2)25 b Fr(\010)2483 58 y Fv(\000)2567 44 y Fn(\032)g(S)7 b Fr(:)1024 293 y Fm(\036)1078 255 y Fv(\000)1137 293 y Fr(\()p Fm(x)p Fr(\))26 b(=)1381 206 y Fj(X)1419 398 y Fv(\006)1527 169 y Fj(Z)1618 195 y Fv(1)1578 375 y(\0001)1722 293 y Fm(dE)21 b( )1919 255 y Fk(E)1916 315 y Fv(\006)1979 293 y Fr(\()p Fm(x)p Fr(\))p Fn(h)15 b Fm( )2213 255 y Fk(E)2210 315 y Fv(\006)2275 293 y Fn(j)p Fm(\036)2354 255 y Fv(\000)2428 293 y Fn(i)31 b Fm(:)934 b Fr(\(7.11\))24 592 y(No)m(w,)30 b(since)f Fn(h)15 b Fm( )589 559 y Fk(E)586 614 y Fv(\006)650 592 y Fn(j)p Fm(\036)729 559 y Fv(\000)804 592 y Fn(i)25 b(2)g(H)1028 559 y Fi(2)1027 614 y Fv(\000)1086 592 y Fr(,)30 b(w)m(e)g(ma)m(y)g(close)g(the)g(in)m(tegration)g(con)m (tour)h(along)e(the)h(lo)m(w)m(er)g(semi-circle)24 705 y Fn(j)p Fm(E)5 b Fn(j)26 b(!)f(1)p Fr(.)40 b(Hence,)32 b(due)e(to)h(the)f(residue)f(theorem)i(one)g(obtains)632 932 y Fm(\036)686 894 y Fv(\000)746 932 y Fr(\()p Fm(x)p Fr(\))26 b(=)f Fn(\000)p Fr(2)p Fm(\031)s(i)1207 845 y Fj(X)1245 1037 y Fv(\006)1354 845 y Fj(X)1398 1036 y Fk(n)1500 932 y Fr(Res\()p Fm( )1740 894 y Fk(E)1737 954 y Fv(\006)1801 932 y Fr(\()p Fm(x)p Fr(\);)15 b Fn(\000)p Fm(E)2101 946 y Fk(n)2149 932 y Fr(\))g Fn(h)g Fm( )2311 894 y Fk(E)2308 954 y Fv(\006)2372 932 y Fn(j)p Fm(\036)2451 894 y Fv(\000)2525 932 y Fn(i)2560 827 y Fj(\014)2560 882 y(\014)2560 936 y(\014)2591 995 y Fk(E)t Fi(=)p Fv(\000)p Fk(E)2809 1003 y Fg(n)2885 932 y Fm(:)543 b Fr(\(7.12\))24 1231 y(Using)29 b(the)i(de\014nition)d(of)i Fm( )1005 1198 y Fk(E)1002 1253 y Fv(\006)205 1484 y Fn(h)15 b Fm( )317 1446 y Fk(E)314 1506 y Fv(\006)378 1484 y Fn(j)p Fm(\036)457 1446 y Fv(\000)531 1484 y Fn(i)26 b Fr(=)789 1422 y(1)p 698 1463 229 4 v 698 1481 a Fn(p)p 774 1481 153 4 v 66 x Fr(2)p Fm(\031)s(\015)951 1360 y Fj(Z)p 1057 1375 482 4 v 1057 1484 a Fm(x)1109 1436 y Fv(\000)p Fi(\()p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))1109 1507 y Fv(\006)1554 1484 y Fm(\036)1608 1446 y Fv(\000)1667 1484 y Fr(\()p Fm(x)p Fr(\))p Fm(dx)h Fr(=)2112 1422 y(1)p 2020 1463 229 4 v 2020 1481 a Fn(p)p 2096 1481 153 4 v 66 x Fr(2)p Fm(\031)s(\015)2274 1360 y Fj(Z)2380 1484 y Fm(x)2432 1436 y Fv(\000)p Fi(\()p Fv(\000)p Fk(iE)t(=\015)t Fi(+1)p Fk(=)p Fi(2\))2432 1507 y Fv(\006)2932 1484 y Fm(\036)2986 1446 y Fv(\000)3045 1484 y Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)31 b(;)116 b Fr(\(7.13\))24 1731 y(one)30 b(\014nds)940 1954 y Fn(h)15 b Fm( )1052 1917 y Fk(E)1049 1977 y Fv(\006)1112 1954 y Fn(j)p Fm(\036)1191 1917 y Fv(\000)1266 1954 y Fn(i)1301 1850 y Fj(\014)1301 1904 y(\014)1301 1959 y(\014)1332 2018 y Fk(E)t Fi(=)p Fv(\000)p Fk(E)1550 2026 y Fg(n)1620 1954 y Fr(=)1818 1893 y(1)p 1726 1933 229 4 v 1726 1952 a Fn(p)p 1802 1952 153 4 v 66 x Fr(2)p Fm(\031)s(\015)1980 1831 y Fj(Z)2086 1954 y Fm(x)2138 1917 y Fk(n)2138 1977 y Fv(\006)2197 1954 y Fm(\036)2251 1917 y Fv(\000)2310 1954 y Fr(\()p Fm(x)p Fr(\))g Fm(dx)32 b(:)850 b Fr(\(7.14\))24 2207 y(Therefore,)30 b(inserting)e(in)m(to)j(\(7.12\))h(the)f(v)-5 b(alue)30 b(of)g(the)h(residue)e(giv)m(en)h(in)f(\(7.1\))j(one)f(gets)g (\014nally)148 2473 y Fm(\036)202 2436 y Fv(\000)261 2473 y Fr(\()p Fm(x)p Fr(\))84 b(=)621 2387 y Fj(X)665 2578 y Fk(n)777 2412 y Fm(\016)820 2379 y Fi(\()p Fk(n)p Fi(\))923 2412 y Fr(\()p Fm(x)p Fr(\))p 777 2452 269 4 v 872 2536 a Fm(n)p Fr(!)1086 2350 y Fj(Z)1192 2400 y(\002)1230 2473 y Fr(\()p Fn(\000)p Fr(1\))1416 2436 y Fk(n)1464 2473 y Fm(x)1516 2436 y Fk(n)1516 2496 y Fi(+)1595 2473 y Fr(+)20 b Fm(x)1738 2436 y Fk(n)1738 2496 y Fv(\000)1797 2400 y Fj(\003)1850 2473 y Fm(\036)1904 2436 y Fv(\000)1963 2473 y Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)26 b Fr(=)2321 2387 y Fj(X)2366 2578 y Fk(n)2453 2473 y Fr(\()p Fn(\000)p Fr(1\))2639 2436 y Fk(n)2696 2412 y Fm(\016)2739 2379 y Fi(\()p Fk(n)p Fi(\))2842 2412 y Fr(\()p Fm(x)p Fr(\))p 2696 2452 V 2791 2536 a Fm(n)p Fr(!)3005 2350 y Fj(Z)3111 2473 y Fm(x)3163 2436 y Fk(n)3210 2473 y Fm(\036)3264 2436 y Fv(\000)3323 2473 y Fr(\()p Fm(x)p Fr(\))15 b Fm(dx)467 2716 y Fr(=)621 2629 y Fj(X)665 2820 y Fk(n)767 2716 y Fm(f)822 2678 y Fv(\000)812 2738 y Fk(n)881 2716 y Fr(\()p Fm(x)p Fr(\))p Fn(h)g Fm(f)1108 2678 y Fi(+)1098 2738 y Fk(n)1167 2716 y Fn(j)p Fm(\036)1246 2678 y Fv(\000)1321 2716 y Fn(i)30 b Fm(:)2042 b Fr(\(7.15\))24 3002 y(T)-8 b(o)30 b(pro)m(v)m(e)i (\(7.10\))g(let)f(us)e(use)h(another)h(sp)s(ectral)f(form)m(ula)f (\(6.17\):)43 b(for)30 b(an)m(y)h Fm(\036)2742 2969 y Fi(+)2826 3002 y Fn(2)25 b Fr(\010)2978 3016 y Fi(+)3062 3002 y Fn(\032)g(S)847 3250 y Fm(\036)901 3213 y Fi(+)960 3250 y Fr(\()p Fm(x)p Fr(\))h(=)1204 3164 y Fj(X)1242 3356 y Fv(\006)1351 3127 y Fj(Z)1442 3153 y Fv(1)1401 3333 y(\0001)1546 3250 y Fm(dE)21 b(F)13 b Fr([)p Fm( )1839 3212 y Fv(\000)p Fk(E)1836 3274 y Fv(\006)1954 3250 y Fr(]\()p Fm(x)p Fr(\))p Fn(h)i Fm(F)e Fr([)p Fm( )2309 3212 y Fv(\000)p Fk(E)2306 3274 y Fv(\006)2426 3250 y Fr(])p Fn(j)p Fm(\036)2530 3213 y Fv(\000)2605 3250 y Fn(i)30 b Fm(:)758 b Fr(\(7.16\))24 3555 y(No)m(w,)31 b(since)f Fn(h)15 b Fm(F)e Fr([)p Fm( )687 3517 y Fv(\000)p Fk(E)684 3578 y Fv(\006)803 3555 y Fr(])p Fn(j)p Fm(\036)907 3522 y Fv(\000)981 3555 y Fn(i)26 b(2)f(H)1206 3522 y Fi(2)1205 3577 y(+)1264 3555 y Fr(,)30 b(w)m(e)h(ma)m(y)g(close)g(the)g (in)m(tegration)f(con)m(tour)h(along)f(the)h(upp)s(er)d(semi-)24 3668 y(circle)h Fn(j)p Fm(E)5 b Fn(j)26 b(!)g(1)p Fr(.)40 b(Hence)31 b(the)g(residue)e(theorem)h(implies)456 3895 y Fm(\036)510 3857 y Fi(+)569 3895 y Fr(\()p Fm(x)p Fr(\))c(=)f(+2)p Fm(\031)s(i)1030 3808 y Fj(X)1069 4000 y Fv(\006)1177 3808 y Fj(X)1222 3999 y Fk(n)1324 3895 y Fr(Res\()p Fm(F)13 b Fr([)p Fm( )1660 3857 y Fv(\000)p Fk(E)1657 3918 y Fv(\006)1776 3895 y Fr(\()p Fm(x)p Fr(\)];)i(+)p Fm(E)2101 3909 y Fk(n)2149 3895 y Fr(\))g Fn(h)g Fm(F)e Fr([)p Fm( )2407 3857 y Fv(\000)p Fk(E)2404 3918 y Fv(\006)2523 3895 y Fr(])p Fn(j)p Fm(\036)2627 3857 y Fi(+)2702 3895 y Fn(i)2737 3790 y Fj(\014)2737 3845 y(\014)2737 3899 y(\014)2767 3958 y Fk(E)t Fi(=+)p Fk(E)2985 3966 y Fg(n)3061 3895 y Fm(:)367 b Fr(\(7.17\))24 4194 y(No)m(w,)31 b(using)e(once)i (more)f(the)h(form)m(ula)f(for)g Fm( )1620 4161 y Fk(E)1617 4217 y Fv(\006)1710 4194 y Fr(one)h(\014nds)907 4443 y Fn(h)15 b Fm(F)e Fr([)p Fm( )1115 4405 y Fv(\000)p Fk(E)1112 4467 y Fv(\006)1231 4443 y Fr(])p Fn(j)p Fm(\036)1335 4406 y Fi(+)1409 4443 y Fn(i)1444 4339 y Fj(\014)1444 4393 y(\014)1444 4448 y(\014)1475 4506 y Fk(E)t Fi(=+)p Fk(E)1693 4514 y Fg(n)1764 4443 y Fr(=)1961 4382 y(1)p 1870 4422 229 4 v 1870 4440 a Fn(p)p 1946 4440 153 4 v 67 x Fr(2)p Fm(\031)s(\015)2108 4443 y Fn(h)i Fm(F)e Fr([)p Fm(x)2306 4406 y Fk(n)2306 4466 y Fv(\006)2366 4443 y Fr(])p Fn(j)p Fm(\036)2470 4406 y Fi(+)2545 4443 y Fn(i)31 b Fm(:)817 b Fr(\(7.18\))1809 5281 y(13)p eop %%Page: 14 14 14 13 bop 24 44 a Fr(Hence,)31 b(inserting)d(the)i(v)-5 b(alues)29 b(of)h(residues)e(\(7.2\))k(in)m(to)e(\(7.17\))i(and)d (using)g(the)h(form)m(ula)f(for)h Fm(F)13 b Fr([)p Fm(x)3419 11 y Fk(n)3419 67 y Fv(\006)3478 44 y Fr(])30 b(\(see)24 157 y(\(A.9\)\))i(one)f(has)330 395 y Fm(\036)384 358 y Fi(+)443 395 y Fr(\()p Fm(x)p Fr(\))84 b(=)885 334 y Fm(i)p 813 374 177 4 v 813 392 a Fn(p)p 888 392 101 4 v 888 468 a Fr(2)p Fm(\031)1014 309 y Fj(X)1058 500 y Fk(n)1145 395 y Fr(\()p Fn(\000)p Fr(1\))1331 358 y Fk(n)1389 334 y Fm(x)1441 301 y Fk(n)p 1389 374 99 4 v 1399 458 a Fm(n)p Fr(!)1513 294 y Fj(h)1556 395 y Fr(\()p Fn(\000)p Fm(i)p Fr(\))1728 358 y Fk(n)p Fi(+1)1866 395 y Fn(h)15 b Fm(F)e Fr([)p Fm(x)2064 358 y Fk(n)2064 418 y Fi(+)2124 395 y Fr(])p Fn(j)p Fm(\036)2228 358 y Fi(+)2303 395 y Fn(i)20 b(\000)g Fm(i)2480 358 y Fk(n)p Fi(+1)2618 395 y Fn(h)15 b Fm(F)e Fr([)p Fm(x)2816 358 y Fk(n)2816 418 y Fv(\000)2875 395 y Fr(])p Fn(j)p Fm(\036)2979 358 y Fi(+)3054 395 y Fn(i)3089 294 y Fj(i)649 666 y Fr(=)885 604 y Fm(i)p 813 645 177 4 v 813 663 a Fn(p)p 888 663 101 4 v 888 738 a Fr(2)p Fm(\031)1014 579 y Fj(X)1058 770 y Fk(n)1145 666 y Fr(\()p Fn(\000)p Fr(1\))1331 628 y Fk(n)1389 604 y Fm(x)1441 571 y Fk(n)p 1389 645 99 4 v 1399 728 a Fm(n)p Fr(!)1528 542 y Fj(Z)1634 565 y(h)1677 666 y Fr(\()p Fn(\000)p Fm(i)p Fr(\))1849 628 y Fk(n)p Fi(+1)p 1987 587 354 4 v 1987 666 a Fm(F)g Fr([)p Fm(x)2135 634 y Fk(n)2135 689 y Fi(+)2195 666 y Fr(]\()p Fm(k)s Fr(\))21 b Fn(\000)f Fm(i)2483 628 y Fk(n)p Fi(+1)p 2620 587 V 2620 666 a Fm(F)13 b Fr([)p Fm(x)2768 634 y Fk(n)2768 689 y Fv(\000)2828 666 y Fr(]\()p Fm(k)s Fr(\))2974 565 y Fj(i)3032 666 y Fm(\036)3086 628 y Fi(+)3145 666 y Fr(\()p Fm(k)s Fr(\))i Fm(dk)649 936 y Fr(=)820 875 y Fm(i)p 813 915 46 4 v 813 999 a Fr(2)883 850 y Fj(X)927 1041 y Fk(n)1015 936 y Fr(\()p Fn(\000)p Fr(1\))1201 899 y Fk(n)1258 875 y Fm(x)1310 842 y Fk(n)p 1258 915 99 4 v 1268 999 a Fm(n)p Fr(!)1382 835 y Fj(h)1425 936 y Fr(\()p Fn(\000)p Fm(i)p Fr(\))1597 899 y Fk(n)p Fi(+1)1735 936 y Fm(i)1766 899 y Fk(n)1834 936 y Fn(\000)20 b Fm(i)1956 899 y Fk(n)p Fi(+1)2093 936 y Fr(\()p Fn(\000)p Fm(i)p Fr(\))2265 899 y Fk(n)2313 835 y Fj(i)2371 813 y(Z)2477 936 y Fm(\016)2520 899 y Fi(\()p Fk(n)p Fi(\))2623 936 y Fr(\()p Fm(k)s Fr(\))p Fm(\036)2797 899 y Fi(+)2857 936 y Fr(\()p Fm(k)s Fr(\))15 b Fm(dk)649 1179 y Fr(=)803 1092 y Fj(X)847 1283 y Fk(n)949 1179 y Fm(f)1004 1141 y Fi(+)994 1201 y Fk(n)1062 1179 y Fr(\()p Fm(x)p Fr(\))p Fn(h)g Fm(f)1289 1141 y Fv(\000)1279 1201 y Fk(n)1349 1179 y Fn(j)p Fm(\036)1428 1141 y Fi(+)1503 1179 y Fn(i)30 b Fm(;)1860 b Fr(\(7.19\))24 1448 y(whic)m(h)29 b(ends)g(the)i(pro)s (of.)2736 b Fa(2)24 1561 y Fr(This)28 b(w)m(a)m(y)k(w)m(e)e(ha)m(v)m(e) i(reco)m(v)m(ered)g(\(5.16\))h(and)c(\(5.17\).)43 b(It)31 b(is)e(not)i(surprising,)c(due)j(to)h(the)f(follo)m(wing)24 1766 y Fs(Prop)s(osition)36 b(5)46 b Fr(\010)753 1780 y Fv(\000)837 1766 y Fr(=)24 b Fn(Z)40 b Ff(and)34 b Fr(\010)1281 1780 y Fi(+)1365 1766 y Fr(=)25 b Fn(D)s Ff(.)24 1971 y Fs(Corollary)35 b(1)45 b Ff(We)33 b(have)g(two)g(sp)-5 b(e)g(ctr)g(al)35 b(de)-5 b(c)g(omp)g(osition)36 b(of)2166 1948 y Fj(b)2144 1971 y Fm(H)7 b Ff(:)1062 2154 y Fj(b)1041 2177 y Fm(H)32 b Fr(=)1245 2090 y Fj(X)1290 2281 y Fk(n)p 1392 2103 73 4 v 1392 2177 a Fm(E)1464 2191 y Fk(n)1511 2177 y Fn(j)p Fm(f)1591 2139 y Fv(\000)1581 2199 y Fk(n)1665 2177 y Fn(ih)15 b Fm(f)1805 2139 y Fi(+)1795 2199 y Fk(n)1864 2177 y Fn(j)236 b Fr(on)98 b(\010)2385 2191 y Fv(\000)2476 2177 y Fm(;)952 b Fr(\(7.20\))24 2446 y Ff(and)1065 2620 y Fj(b)1044 2642 y Fm(H)32 b Fr(=)1248 2556 y Fj(X)1292 2747 y Fk(n)1394 2642 y Fm(E)1461 2656 y Fk(n)1508 2642 y Fn(j)p Fm(f)1588 2605 y Fi(+)1578 2665 y Fk(n)1662 2642 y Fn(ih)15 b Fm(f)1802 2605 y Fv(\000)1792 2665 y Fk(n)1861 2642 y Fn(j)237 b Fr(on)97 b(\010)2382 2656 y Fi(+)2474 2642 y Fm(:)954 b Fr(\(7.21\))24 2983 y Fo(8)134 b(Resonances)46 b(and)f(the)g(quan)l(tum)g(damping)24 3186 y Fr(Finally)-8 b(,)45 b(let)e(us)f(turn)g(to)i(the)g(ev)m (olution)f(generated)h(b)m(y)f(the)g(Hamiltonian)f(\(3.10\).)81 b(Ob)m(viously)-8 b(,)45 b(it)24 3299 y(generates)31 b(a)g(1-parameter)h(unitary)d(group)1488 3511 y Fm(U)10 b Fr(\()p Fm(t)p Fr(\))26 b(=)f Fm(e)1827 3474 y Fv(\000)p Fk(i)1922 3457 y Fh(b)1906 3474 y Fk(H)5 b(t)2029 3511 y Fm(;)1444 b Fr(\(8.1\))24 3708 y(on)30 b Fm(L)212 3675 y Fi(2)251 3708 y Fr(\()p Fl(R)s Fr(\))q(.)46 b(It)31 b(follo)m(ws)e(from)h(\(4.3\))i(that)1096 3905 y Fm( )1155 3919 y Fk(t)1185 3905 y Fr(\()p Fm(x)p Fr(\))26 b(=)f Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm( )s Fr(\()p Fm(x)p Fr(\))27 b(=)e Fm(e)1953 3868 y Fk(\015)t(t=)p Fi(2)2094 3905 y Fm( )s Fr(\()p Fm(e)2233 3868 y Fk(\015)t(t)2304 3905 y Fm(x)p Fr(\))30 b Fm(:)1052 b Fr(\(8.2\))24 4102 y(The)26 b(ab)s(o)m(v)m(e)i(form)m(ula)d(is)h(w)m(ell)g(de\014ned)f (for)h(an)m(y)h Fm(t)e Fn(2)g Fl(R)36 b Fr(and)26 b(clearly)-8 b(,)27 b(as)g(w)m(e)g(already)f(sho)m(w)m(ed,)i(the)f(theory)24 4215 y(is)35 b(time-rev)m(ersal)g(in)m(v)-5 b(arian)m(t:)51 b(if)35 b Fm( )s Fr(\()p Fm(t)p Fr(\))i(solv)m(es)f(the)g(Sc)m(hr\177) -45 b(odinger)34 b(equation)i(so)g(do)s(es)g Fs(T)p Fm( )s Fr(\()p Fm(t)p Fr(\))f(:=)f Fm( )s Fr(\()p Fn(\000)p Fm(t)p Fr(\).)24 4328 y(Therefore,)j(w)m(orking)e(in)f Fm(L)990 4295 y Fi(2)1029 4328 y Fr(\()p Fl(R)s Fr(\))42 b(w)m(e)36 b(do)g(not)g(see)g(an)m(y)g(damping)e(at)j(all.)55 b(No)m(w,)38 b(let)e(us)f(construct)h(t)m(w)m(o)24 4441 y(natural)29 b(Gelfand)h(triplets:)1379 4638 y(\010)1445 4652 y Fv(\006)1529 4638 y Fn(\032)25 b Fm(L)1687 4600 y Fi(2)1726 4638 y Fr(\()p Fl(R)s Fr(\))32 b Fn(\032)25 b Fr(\010)2050 4600 y Fv(0)2050 4660 y(\006)2139 4638 y Fm(:)1334 b Fr(\(8.3\))24 4835 y(If)30 b Fm(\036)169 4802 y Fv(\000)253 4835 y Fn(2)25 b Fr(\010)405 4849 y Fv(\000)464 4835 y Fr(,)30 b(then)849 5032 y Fn(h)15 b Fm( )961 4994 y Fk(E)958 5054 y Fv(\006)1022 5032 y Fn(j)p Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm(\036)1276 4994 y Fv(\000)1351 5032 y Fn(i)26 b Fr(=)f Fn(h)15 b Fm(U)1630 4994 y Fv(\003)1670 5032 y Fr(\()p Fm(t)p Fr(\))g Fm( )1850 4994 y Fk(E)1847 5054 y Fv(\006)1910 5032 y Fn(j)p Fm(\036)1989 4994 y Fv(\000)2064 5032 y Fn(i)26 b Fr(=)f Fm(e)2263 4994 y Fv(\000)p Fk(iE)t(t)2442 5032 y Fn(h)15 b Fm( )2554 4994 y Fk(E)2551 5054 y Fv(\006)2615 5032 y Fn(j)p Fm(\036)2694 4994 y Fv(\000)2768 5032 y Fn(i)31 b Fm(:)639 b Fr(\(8.4\))1809 5281 y(14)p eop %%Page: 15 15 15 14 bop 24 44 a Fr(Hence)36 b Fm(\036)353 11 y Fv(\000)412 44 y Fr(\()p Fm(t)p Fr(\))e Fn(2)g Fr(\010)710 58 y Fv(\000)804 44 y Fr(only)g(for)i Fm(t)d Fn(\025)h Fr(0.)56 b(Similarly)-8 b(,)34 b(if)g Fm(\036)2003 11 y Fi(+)2096 44 y Fn(2)g Fr(\010)2257 58 y Fi(+)2315 44 y Fr(,)j(then)e Fm(\036)2643 11 y Fi(+)2703 44 y Fr(\()p Fm(t)p Fr(\))f Fn(2)f Fr(\010)3000 58 y Fi(+)3094 44 y Fr(only)i(for)g Fm(t)f Fn(\024)f Fr(0.)24 157 y(Therefore,)25 b(the)f(restriction)f(of)i(the)f(unitary)f (group)g Fm(U)10 b Fr(\()p Fm(t)p Fr(\))25 b(on)f Fm(L)2231 124 y Fi(2)2270 157 y Fr(\()p Fl(R)s Fr(\))30 b(to)25 b(\010)2601 171 y Fv(\006)2684 157 y Fr(no)f(longer)g(de\014nes)f(a)h (group.)24 270 y(It)30 b(giv)m(es)h(rise)e(to)i(t)m(w)m(o)h (semigroups:)1067 471 y Fm(U)1129 485 y Fv(\000)1188 471 y Fr(\()p Fm(t)p Fr(\))56 b(:)g(\010)1494 485 y Fv(\000)1608 471 y Fn(\000)-16 b(!)56 b Fr(\010)1876 485 y Fv(\000)1965 471 y Fm(;)197 b Fr(for)91 b Fm(t)25 b Fn(\025)g Fr(0)31 b Fm(;)856 b Fr(\(8.5\))24 673 y(and)1067 786 y Fm(U)1129 800 y Fi(+)1188 786 y Fr(\()p Fm(t)p Fr(\))56 b(:)g(\010)1494 800 y Fi(+)1608 786 y Fn(\000)-16 b(!)56 b Fr(\010)1876 800 y Fi(+)1965 786 y Fm(;)197 b Fr(for)91 b Fm(t)25 b Fn(\024)g Fr(0)31 b Fm(:)856 b Fr(\(8.6\))24 951 y(Due)30 b(to)h(\(7.20\))i(and)d(\(7.21\))i(one)f(has:)813 1166 y Fm(\036)867 1128 y Fv(\000)926 1166 y Fr(\()p Fm(t)p Fr(\))26 b(=)f Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm(\036)1380 1128 y Fv(\000)1465 1166 y Fr(=)1561 1080 y Fj(X)1605 1270 y Fk(n)1707 1166 y Fm(e)1749 1128 y Fv(\000)p Fk(\015)t Fi(\()p Fk(n)p Fi(+1)p Fk(=)p Fi(2\))p Fk(t)2133 1166 y Fn(j)p Fm(f)2213 1128 y Fv(\000)2203 1188 y Fk(n)2286 1166 y Fn(ih)15 b Fm(f)2426 1128 y Fi(+)2416 1188 y Fk(n)2485 1166 y Fn(j)p Fm(\036)2564 1128 y Fv(\000)2639 1166 y Fn(i)31 b Fm(;)768 b Fr(\(8.7\))24 1439 y(for)30 b Fm(t)25 b Fn(\025)g Fr(0,)31 b(and)840 1649 y Fm(\036)894 1612 y Fi(+)953 1649 y Fr(\()p Fm(t)p Fr(\))26 b(=)f Fm(U)10 b Fr(\()p Fm(t)p Fr(\))p Fm(\036)1407 1612 y Fi(+)1492 1649 y Fr(=)1588 1563 y Fj(X)1632 1754 y Fk(n)1734 1649 y Fm(e)1776 1612 y Fk(\015)t Fi(\()p Fk(n)p Fi(+1)p Fk(=)p Fi(2\))p Fk(t)2105 1649 y Fn(j)p Fm(f)2185 1612 y Fi(+)2175 1672 y Fk(n)2259 1649 y Fn(ih)15 b Fm(f)2399 1612 y Fv(\000)2389 1672 y Fk(n)2458 1649 y Fn(j)p Fm(\036)2537 1612 y Fi(+)2612 1649 y Fn(i)30 b Fm(;)796 b Fr(\(8.8\))24 1938 y(for)36 b Fm(t)f Fn(\024)g Fr(0.)59 b(W)-8 b(e)37 b(stress)g(that)g Fm(\036)1147 1900 y Fv(\000)1147 1962 y Fk(t)1242 1938 y Fr(\()p Fm(\036)1331 1900 y Fi(+)1331 1962 y Fk(t)1391 1938 y Fr(\))f(do)s(es)g(b)s(elong)g(to)h Fm(L)2149 1905 y Fi(2)2188 1938 y Fr(\()p Fl(R)s Fr(\))43 b(also)36 b(for)g Fm(t)f(<)g Fr(0)i(\()p Fm(t)e(>)g Fr(0\).)60 b(Ho)m(w)m(ev)m(er,)24 2051 y Fm(\036)78 2013 y Fv(\000)78 2075 y Fk(t)168 2051 y Fn(2)31 b Fr(\010)326 2065 y Fv(\000)419 2051 y Fr(\()p Fm(\036)508 2013 y Fi(+)508 2075 y Fk(t)599 2051 y Fn(2)g Fr(\010)757 2065 y Fi(+)816 2051 y Fr(\))j(only)f(for)h Fm(t)d Fn(\025)g Fr(0)k(\()p Fm(t)c Fn(\024)g Fr(0\).)53 b(This)32 b(w)m(a)m(y)j(the)f(irrev)m(ersibilit)m(y)d(en)m(ters)j(on)g (a)h(purely)24 2164 y(Hamiltonian)28 b(lev)m(el)i(b)m(y)h(restricting)e (dynamics)g(to)i(the)g(dense)f(subspace)g(\010)2679 2178 y Fv(\006)2768 2164 y Fr(of)g Fm(L)2933 2131 y Fi(2)2972 2164 y Fr(\()p Fl(R)s Fr(\))q(.)165 2277 y(Clearly)-8 b(,)29 b(form)m(ulae)h(\(8.7\))j(and)c(\(8.8\))j(are)f(quan)m(tum)f (analogues)h(of)f(the)h(classical)e(damping)g(la)m(ws:)1242 2478 y Fm(x)p Fr(\()p Fm(t)p Fr(\))d(=)f Fm(e)1561 2441 y Fv(\000)p Fk(\015)t(t)1686 2478 y Fm(x)31 b(;)251 b(t)25 b Fn(\025)g Fr(0)31 b Fm(;)1198 b Fr(\(8.9\))24 2679 y(and)1249 2881 y Fm(p)p Fr(\()p Fm(t)p Fr(\))25 b(=)g Fm(e)1561 2843 y Fi(+)p Fk(\015)t(t)1686 2881 y Fm(p)30 b(;)252 b(t)25 b Fn(\024)g Fr(0)31 b Fm(:)1159 b Fr(\(8.10\))24 3082 y(Finally)-8 b(,)33 b(let)h(us)g(recall)f(that)i(the)f(time)g(rev) m(ersal)g(op)s(erator)h Fs(T)e Fr(establishes)g(an)h(isomorphism)d(b)s (et)m(w)m(een)24 3195 y(\010)90 3209 y Fv(\000)179 3195 y Fr(and)e(\010)421 3209 y Fi(+)480 3195 y Fr(.)41 b(Therefore,)30 b(eac)m(h)i(solution)1457 3396 y Fm(\036)1511 3358 y Fv(\000)1511 3420 y Fk(t)1596 3396 y Fr(=)25 b Fm(U)1754 3410 y Fv(\000)1813 3396 y Fr(\()p Fm(t)p Fr(\))p Fm(\036)1970 3359 y Fv(\000)2060 3396 y Fm(;)1368 b Fr(\(8.11\))24 3598 y(with)29 b Fm(\036)285 3565 y Fv(\000)369 3598 y Fn(2)c Fr(\010)521 3612 y Fv(\000)610 3598 y Fr(is)k(mapp)s(ed)g(in)m (to)782 3799 y Fs(T)p Fr(\()p Fm(\036)944 3761 y Fv(\000)944 3823 y Fk(t)1003 3799 y Fr(\))d(=)f Fm(U)1222 3813 y Fv(\000)1281 3799 y Fr(\()p Fn(\000)p Fm(t)p Fr(\))p Fs(T)p Fr(\()p Fm(\036)1617 3762 y Fv(\000)1676 3799 y Fr(\))h(=)f Fm(U)1895 3813 y Fi(+)1954 3799 y Fr(\()p Fm(t)p Fr(\))p Fs(T)p Fr(\()p Fm(\036)2219 3762 y Fv(\000)2279 3799 y Fr(\))30 b Fm(;)137 b(t)25 b Fn(\024)g Fr(0)31 b Fm(:)692 b Fr(\(8.12\))24 4000 y(Con)m(v)m(ersely)-8 b(,)31 b(an)m(y)f(solution)1457 4202 y Fm(\036)1511 4164 y Fi(+)1511 4226 y Fk(t)1596 4202 y Fr(=)25 b Fm(U)1754 4216 y Fi(+)1813 4202 y Fr(\()p Fm(t)p Fr(\))p Fm(\036)1970 4164 y Fi(+)2060 4202 y Fm(;)1368 b Fr(\(8.13\))24 4403 y(with)29 b Fm(\036)285 4370 y Fi(+)369 4403 y Fn(2)c Fr(\010)521 4417 y Fi(+)610 4403 y Fr(is)k(mapp)s(ed)g(in)m(to)782 4604 y Fs(T)p Fr(\()p Fm(\036)944 4566 y Fi(+)944 4629 y Fk(t)1003 4604 y Fr(\))d(=)f Fm(U)1222 4618 y Fi(+)1281 4604 y Fr(\()p Fn(\000)p Fm(t)p Fr(\))p Fs(T)p Fr(\()p Fm(\036)1617 4567 y Fi(+)1676 4604 y Fr(\))h(=)f Fm(U)1895 4618 y Fv(\000)1954 4604 y Fr(\()p Fm(t)p Fr(\))p Fs(T)p Fr(\()p Fm(\036)2219 4567 y Fi(+)2279 4604 y Fr(\))30 b Fm(;)137 b(t)25 b Fn(\025)g Fr(0)31 b Fm(:)692 b Fr(\(8.14\))24 4806 y(Summarizing,)26 b(quan)m(tum)h(dynamics)g(is)g(irrev)m(ersible)f (on)i(\010)2122 4820 y Fv(\000)2209 4806 y Fr(and)f(\010)2449 4820 y Fi(+)2508 4806 y Fr(.)40 b(This)26 b(irrev)m(ersibilit)m(y)f(is) i(caused)24 4919 y(b)m(y)g(quan)m(tum)g(damping,)g(or,)h(equiv)-5 b(alen)m(tly)d(,)28 b(b)m(y)f(the)h(presence)f(of)h(resonan)m(t)g (states)h Fm(f)2994 4886 y Fv(\006)2984 4941 y Fk(n)3080 4919 y Fr(\(5.5\).)41 b(It)28 b(should)24 5032 y(b)s(e)34 b(stressed)h(that)g(it)g(is)f(not)h(an)g(energy)g(that)g(is)f (dissipated.)53 b(Clearly)-8 b(,)35 b(the)g(Hamiltonian)e(\(3.10\))38 b(can)1809 5281 y(15)p eop %%Page: 16 16 16 15 bop 24 44 a Fr(not)26 b(b)s(e)g(in)m(terpreted)f(as)i(a)g(system) f(energy)h(|)f(it)f(w)m(as)i(used)f(to)h(de\014ne)e(a)i(Hamiltonian)d (dynamics)h(of)i(the)24 157 y(enlarged)j(system)h(on)f Fm(L)878 124 y Fi(2)918 157 y Fr(\()p Fl(R)s Fr(\).)48 b(The)30 b(quan)m(tum)g(damp)s(ed)g(system)h(is)f(not)h(de\014ned)e(on) i(the)g(en)m(tire)f Fm(L)3509 124 y Fi(2)3549 157 y Fr(\()p Fl(R)s Fr(\))24 270 y(but)35 b(rather)h(on)g(a)h(dense)f(subset)g(\010) 1294 284 y Fv(\000)1387 270 y Fn(\032)f Fm(L)1555 237 y Fi(2)1594 270 y Fr(\()p Fl(R)t Fr(\).)64 b(As)36 b(w)m(e)h(sa)m(w)g (it)f(imp)s(oses)e(the)j(restriction)e(up)s(on)g(the)24 383 y(time)d(ev)m(olution)h(suc)m(h)f(that)i(it)e(is)g(de\014ned)g (only)g(for)h(p)s(ositiv)m(e)e Fm(t)p Fr(.)48 b(A)33 b(quan)m(tum)g(damping)e(ma)m(y)j(b)s(e)e(seen)24 496 y(as)c(follo)m(ws:)39 b(let)28 b Fm(\036)649 458 y Fv(\000)649 522 y Fi(0)733 496 y Fn(2)d Fr(\010)885 510 y Fv(\000)972 496 y Fr(b)s(e)j(an)g(initial)d(state)30 b(then)d(the)i(probabilit)m(y) d(densit)m(y)h(for)h(a)h(particle)e(p)s(osition)24 609 y(ev)m(olv)m(es)k(in)e(time)h(as)h(follo)m(ws:)892 813 y Fm(p)938 827 y Fk(t)967 813 y Fr(\()p Fm(x)p Fr(\))26 b(=)f Fn(j)p Fm(\036)1290 775 y Fv(\000)1290 837 y Fk(t)1349 813 y Fr(\()p Fm(x)p Fr(\))p Fn(j)1496 775 y Fi(2)1562 813 y Fr(=)g Fm(e)1700 775 y Fk(\015)t(t)1770 813 y Fn(j)p Fm(\036)1849 775 y Fv(\000)1849 839 y Fi(0)1908 813 y Fr(\()p Fm(e)1985 775 y Fk(\015)t(t)2056 813 y Fm(x)p Fr(\))p Fn(j)2168 775 y Fi(2)2233 813 y Fr(=)g Fm(e)2371 775 y Fk(\015)t(t)2442 813 y Fm(p)2488 827 y Fi(0)2527 813 y Fr(\()p Fm(e)2604 775 y Fk(\015)t(t)2674 813 y Fm(x)p Fr(\))31 b Fm(;)636 b Fr(\(8.15\))24 1017 y(and)29 b(hence)i(in)e(the)i(limit)d Fm(t)d Fn(\000)-16 b(!)26 b Fr(+)p Fn(1)p Fr(,)k(one)g(\014nds)f Fm(p)1810 1031 y Fk(t)1839 1017 y Fr(\()p Fm(x)p Fr(\))d Fn(\000)-15 b(!)25 b Fm(\016)s Fr(\()p Fm(x)p Fr(\).)42 b(Indeed,)30 b(for)g(an)m(y)h Fm(\017)25 b(>)g Fr(0)1111 1167 y Fj(Z)1202 1194 y Fk(\017)1161 1374 y Fv(\000)p Fk(\017)1264 1291 y Fm(p)1310 1305 y Fk(t)1339 1291 y Fr(\()p Fm(x)p Fr(\))p Fm(dx)i Fr(=)1683 1167 y Fj(Z)1773 1194 y Fk(\017e)1835 1170 y Fg(\015)s(t)1733 1374 y Fv(\000)p Fk(\017e)1850 1355 y Fg(\015)s(t)1932 1291 y Fm(p)1978 1305 y Fi(0)2017 1291 y Fr(\()p Fm(x)p Fr(\))p Fm(dx)57 b Fn(\000)-16 b(!)56 b Fr(1)31 b Fm(;)855 b Fr(\(8.16\))24 1554 y(for)30 b Fm(t)25 b Fn(\000)-16 b(!)26 b Fr(+)p Fn(1)p Fr(.)40 b(Clearly)-8 b(,)29 b(it)h(corresp)s(onds)f(to)i(the)g(classical)e(b)s (eha)m(vior)h Fm(x)p Fr(\()p Fm(t)p Fr(\))c(=)f Fm(e)2851 1521 y Fv(\000)p Fk(\015)t(t)2976 1554 y Fm(x)3028 1568 y Fi(0)3092 1554 y Fn(\000)-15 b(!)25 b Fr(0.)165 1667 y(In)e(a)i(forthcoming)e(pap)s(er)g(w)m(e)i(are)g(going)f(to)h(sho)m(w) f(that)h(also)f(more)g(complicated)g(damp)s(ed)f(systems,)24 1780 y(e.g.)41 b(the)31 b(damp)s(ed)e(harmonic)g(oscillator,)h(giv)m(e) h(rise)e(to)i(irrev)m(ersible)d(dynamics.)24 2066 y Fo(A)134 b(App)t(endix)24 2269 y Fr(The)29 b(regular)h(temp)s(ered)g (distribution)d Fm(x)1469 2236 y Fk(\025)1469 2292 y Fi(+)1558 2269 y Fr(\(with)i Fm(\025)d Fn(2)e Fl(C)18 b Fr(\))37 b(giv)m(en)30 b(b)m(y)1248 2524 y Fn(h)15 b Fm(\036)p Fn(j)p Fm(x)1429 2487 y Fk(\025)1429 2547 y Fi(+)1504 2524 y Fn(i)25 b Fr(:=)1685 2400 y Fj(Z)1776 2427 y Fv(1)1736 2607 y Fi(0)1866 2524 y Fm(x)1918 2487 y Fk(\025)1963 2524 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)32 b(;)1180 b Fr(\(A.1\))24 2769 y(for)37 b(an)m(y)i Fm(\036)f Fn(2)f(S)7 b Fr(,)40 b(is)d(w)m(ell)g(de\014ned)g(for)h(Re)15 b Fm(\025)38 b(>)g Fn(\000)p Fr(1.)63 b(Ho)m(w)m(ev)m(er,)43 b(it)37 b(ma)m(y)i(b)s(e)e(easily)g(extended)h(to)h(the)24 2882 y(region)30 b(Re)p Fm(\025)25 b(>)g Fn(\000)p Fr(2)31 b(due)e(to)i(the)g(follo)m(wing)e(regularization)g(form)m(ula:)477 3016 y Fj(Z)568 3042 y Fv(1)528 3222 y Fi(0)658 3140 y Fm(x)710 3102 y Fk(\025)756 3140 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)d Fr(=)1153 3016 y Fj(Z)1244 3042 y Fi(1)1203 3222 y(0)1298 3140 y Fm(x)1350 3102 y Fk(\025)1396 3140 y Fr([)p Fm(\036)p Fr(\()p Fm(x)p Fr(\))21 b Fn(\000)f Fm(\036)p Fr(\(0\)])p Fm(dx)i Fr(+)2115 3016 y Fj(Z)2206 3042 y Fv(1)2165 3222 y Fi(1)2296 3140 y Fm(x)2348 3102 y Fk(\025)2393 3140 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)f Fr(+)2810 3078 y Fm(\036)p Fr(\(0\))p 2790 3119 210 4 v 2790 3202 a Fm(\025)f Fr(+)g(1)3040 3140 y Fm(;)410 b Fr(\(A.2\))24 3401 y(whic)m(h)30 b(holds)g(for)h Fm(\025)c Fn(6)p Fr(=)g Fn(\000)p Fr(1.)44 b(In)31 b(the)g(same)h(w)m(a)m(y)g (one)g(ma)m(y)g(extend)g(the)f(distribution)d Fm(x)3082 3368 y Fk(\025)3082 3423 y Fi(+)3173 3401 y Fr(to)k(the)f(region)24 3514 y(Re)15 b Fm(\025)25 b(>)g Fn(\000)p Fm(n)20 b Fn(\000)g Fr(1)30 b(using)f(the)i(form)m(ula)301 3648 y Fj(Z)392 3674 y Fv(1)351 3854 y Fi(0)482 3771 y Fm(x)534 3734 y Fk(\025)579 3771 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)84 b Fr(=)1092 3648 y Fj(Z)1183 3674 y Fi(1)1142 3854 y(0)1237 3771 y Fm(x)1289 3734 y Fk(\025)1350 3643 y Fj(\024)1398 3771 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))21 b Fn(\000)f Fm(\036)p Fr(\(0\))h Fn(\000)f Fm(x\036)2073 3734 y Fv(0)2096 3771 y Fr(\(0\))i Fn(\000)e Fm(:)15 b(:)g(:)21 b Fn(\000)2610 3710 y Fm(x)2662 3677 y Fk(n)p Fv(\000)p Fi(1)p 2551 3751 308 4 v 2551 3834 a Fr(\()p Fm(n)f Fn(\000)g Fr(1\)!)2883 3771 y Fm(\036)2937 3734 y Fi(\()p Fk(n)p Fv(\000)p Fi(1\))3129 3771 y Fr(\(0\))3244 3643 y Fj(\025)3309 3771 y Fm(dx)938 4042 y Fr(+)1092 3919 y Fj(Z)1183 3945 y Fv(1)1142 4125 y Fi(1)1273 4042 y Fm(x)1325 4005 y Fk(\025)1370 4042 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)h Fr(+)1801 3929 y Fk(n)1757 3956 y Fj(X)1758 4154 y Fk(k)r Fi(=1)2028 3981 y Fm(\036)2082 3948 y Fi(\()p Fk(k)r Fv(\000)p Fi(1\))2270 3981 y Fr(\(0\))p 1913 4021 588 4 v 1913 4105 a(\()p Fm(k)j Fn(\000)c Fr(1\)!\()p Fm(\025)i Fr(+)e Fm(k)s Fr(\))2541 4042 y Fm(;)909 b Fr(\(A.3\))24 4341 y(whic)m(h)29 b(holds)h(for)g Fm(\025)d Fn(6)p Fr(=)e Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)j(;)d Fn(\000)p Fm(n)p Fr(.)42 b(The)30 b(ab)s(o)m(v)m(e)i(form)m(ula)e(sho)m(ws)h(that)g Fn(h)15 b Fm(\036)p Fn(j)p Fm(x)2925 4308 y Fk(\025)2925 4364 y Fi(+)3001 4341 y Fn(i)31 b Fr(as)g(a)g(function)f(of)24 4454 y Fm(\025)f Fn(2)g Fl(C)57 b Fr(has)32 b(simple)f(p)s(oles)h(at)i Fm(\025)29 b Fr(=)h Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)r Fr(,)34 b(and)e(the)h(corresp)s(onding)e (residue)g(at)j Fm(\025)c Fr(=)f Fn(\000)p Fm(k)36 b Fr(equals)24 4567 y Fm(\036)78 4534 y Fi(\()p Fk(k)r Fv(\000)p Fi(1\))266 4567 y Fr(\(0\))p Fm(=)p Fr(\()p Fm(k)25 b Fn(\000)20 b Fr(1\)!.)1809 5281 y(16)p eop %%Page: 17 17 17 16 bop 165 44 a Fr(Using)35 b(the)h(same)g(argumen)m(ts)g(one)g(sho) m(ws)f(that)i(the)e(distribution)e Fm(x)2625 11 y Fk(\025)2625 67 y Fv(\000)2719 44 y Fr(ma)m(y)k(b)s(e)e(extended)g(to)i(the)24 157 y(region)30 b(Re)15 b Fm(\025)25 b(>)g Fn(\000)p Fm(n)20 b Fn(\000)g Fr(1)30 b Ff(via)p Fr(:)138 291 y Fj(Z)229 317 y Fi(0)188 497 y Fv(\0001)333 415 y Fm(x)385 377 y Fk(\025)430 415 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)84 b Fr(=)943 291 y Fj(Z)1034 317 y Fv(1)993 497 y Fi(0)1124 415 y Fm(x)1176 377 y Fk(\025)1221 415 y Fm(\036)p Fr(\()p Fn(\000)p Fm(x)p Fr(\))p Fm(dx)789 674 y Fr(=)943 551 y Fj(Z)1034 577 y Fv(1)993 757 y Fi(1)1124 674 y Fm(x)1176 637 y Fk(\025)1236 546 y Fj(\024)1284 674 y Fm(\036)p Fr(\()p Fn(\000)p Fm(x)p Fr(\))21 b Fn(\000)f Fm(\036)p Fr(\(0\))h(+)f Fm(x\036)2030 637 y Fv(0)2054 674 y Fr(\(0\))h Fn(\000)f Fm(:)15 b(:)g(:)21 b Fn(\000)2508 613 y Fr(\()p Fn(\000)p Fr(1\))2694 580 y Fk(n)p Fv(\000)p Fi(1)2832 613 y Fm(x)2884 580 y Fk(n)p Fv(\000)p Fi(1)p 2508 654 514 4 v 2611 737 a Fr(\()p Fm(n)f Fn(\000)g Fr(1\)!)3046 674 y Fm(\036)3100 637 y Fi(\()p Fk(n)p Fv(\000)p Fi(1\))3293 674 y Fr(\(0\))3408 546 y Fj(\025)3472 674 y Fm(dx)789 945 y Fr(+)943 822 y Fj(Z)1034 848 y Fv(1)993 1028 y Fi(1)1124 945 y Fm(x)1176 908 y Fk(\025)1221 945 y Fm(\036)p Fr(\()p Fm(x)p Fr(\))p Fm(dx)h Fr(+)1653 832 y Fk(n)1608 859 y Fj(X)1610 1057 y Fk(k)r Fi(=1)1765 884 y Fr(\()p Fn(\000)p Fr(1\))1951 851 y Fk(k)r Fv(\000)p Fi(1)2084 884 y Fm(\036)2138 851 y Fi(\()p Fk(k)r Fv(\000)p Fi(1\))2326 884 y Fr(\(0\))p 1765 924 678 4 v 1810 1008 a(\()p Fm(k)i Fn(\000)d Fr(1\)!\()p Fm(\025)i Fr(+)e Fm(k)s Fr(\))2483 945 y Fm(;)967 b Fr(\(A.4\))24 1244 y(whic)m(h)27 b(holds)h(for)g Fm(\025)d Fn(6)p Fr(=)g Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)j(;)d Fn(\000)p Fm(n)p Fr(.)40 b(Hence,)30 b Fn(h)15 b Fm(\036)p Fn(j)p Fm(x)1971 1212 y Fk(\025)1971 1267 y Fv(\000)2046 1244 y Fn(i)29 b Fr(has)g(simple)d(p)s(oles)i(at)i Fm(\025)25 b Fr(=)g Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)r Fr(,)30 b(and)24 1357 y(the)g(corresp)s(onding)f (residue)f(at)j Fm(\025)26 b Fr(=)f Fn(\000)p Fm(k)33 b Fr(equals)d(\()p Fn(\000)p Fr(1\))1972 1324 y Fk(k)r Fv(\000)p Fi(1)2105 1357 y Fm(\036)2159 1324 y Fi(\()p Fk(k)r Fv(\000)p Fi(1\))2347 1357 y Fr(\(0\))p Fm(=)p Fr(\()p Fm(k)c Fn(\000)20 b Fr(1\)!.)165 1470 y(The)30 b(F)-8 b(ourier)30 b(transforms)f(of)i Fm(x)1276 1437 y Fk(\025)1276 1493 y Fv(\006)1160 1723 y Fm(F)13 b Fr([)p Fm(x)1308 1686 y Fk(\025)1308 1746 y Fv(\006)1368 1723 y Fr(]\()p Fm(k)s Fr(\))26 b(=)1710 1662 y(1)p 1645 1702 177 4 v 1645 1720 a Fn(p)p 1721 1720 101 4 v 76 x Fr(2)p Fm(\031)1846 1599 y Fj(Z)1953 1723 y Fm(e)1995 1686 y Fk(ik)r(x)2101 1723 y Fm(x)2153 1686 y Fk(\025)2153 1746 y Fv(\006)2227 1723 y Fm(dx)31 b(;)1093 b Fr(\(A.5\))24 1975 y(are)30 b(giv)m(en)h(b)m(y)f(the)h(follo)m(wing)d(form)m(ula)i ([13)q(])815 2219 y Fm(F)13 b Fr([)p Fm(x)963 2182 y Fk(\025)963 2242 y Fv(\006)1022 2219 y Fr(]\()p Fm(k)s Fr(\))26 b(=)f Fn(\006)1442 2158 y Fm(i)p 1370 2198 177 4 v 1370 2217 a Fn(p)p 1446 2217 101 4 v 75 x Fr(2)p Fm(\031)1556 2219 y(e)1598 2182 y Fv(\006)p Fk(i\025\031)r(=)p Fi(2)1836 2219 y Fr(\000\()p Fm(\025)c Fr(+)f(1\)\()p Fm(k)k Fr(+)c Fm(i)p Fr(0\))2481 2182 y Fv(\000)p Fk(\025)p Fv(\000)p Fi(1)2703 2219 y Fm(;)747 b Fr(\(A.6\))24 2472 y(where)30 b(\()p Fm(k)23 b Fn(\006)d Fm(i)p Fr(0\))594 2439 y Fk(\013)675 2472 y Fr(is)29 b(a)i(distribution)c(de\014ned)i(b)m (y:)1235 2676 y(\()p Fm(k)24 b Fn(\006)c Fm(i)p Fr(0\))1543 2638 y Fk(\013)1619 2676 y Fr(=)25 b Fm(k)1765 2638 y Fk(\013)1762 2698 y Fi(+)1841 2676 y Fr(+)20 b Fm(e)1974 2638 y Fv(\006)p Fk(i\013\031)2146 2676 y Fm(k)2196 2638 y Fk(\013)2193 2698 y Fv(\000)2282 2676 y Fm(:)1168 b Fr(\(A.7\))24 2880 y(Due)27 b(to)h(the)g(Euler)d(\000-function)h(the)i (form)m(ula)e(\(A.6\))j(has)e(single)f(p)s(oles)g(at)i Fm(\025)d Fr(=)g Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)r Fr(.)40 b(Note,)29 b(that)24 2993 y(although)36 b(b)s(oth)h Fm(k)681 2960 y Fk(\013)678 3015 y Fi(+)775 2993 y Fr(and)g Fm(k)1009 2960 y Fk(\013)1006 3015 y Fv(\000)1102 2993 y Fr(ha)m(v)m(e)i(p)s(oles)d(at)i Fm(\013)g Fr(=)f Fn(\000)p Fr(1)p Fm(;)15 b Fn(\000)p Fr(2)p Fm(;)g(:)g(:)g(:)r Fr(,)39 b(the)f(distribution)c(\()p Fm(k)28 b Fn(\006)d Fm(i)p Fr(0\))3345 2960 y Fk(\013)3433 2993 y Fr(is)36 b(w)m(ell)24 3106 y(de\014ned)29 b(for)h(all)f Fm(\013)d Fn(2)f Fl(C)17 b Fr(.)47 b(Indeed)1020 3310 y(lim)976 3365 y Fk(\013)p Fv(!\000)p Fk(n)1190 3310 y Fr(\()p Fm(k)23 b Fn(\006)d Fm(i)p Fr(0\))1497 3273 y Fk(\013)1573 3310 y Fr(=)69 b(lim)1669 3365 y Fk(\013)p Fv(!\000)p Fk(n)1883 3310 y Fr(\()p Fm(k)1968 3273 y Fk(\013)1965 3333 y Fi(+)2045 3310 y Fr(+)20 b(\()p Fn(\000)p Fr(1\))2322 3273 y Fk(n)2369 3310 y Fm(k)2419 3273 y Fk(\013)2416 3333 y Fv(\000)2476 3310 y Fr(\))30 b Fm(;)909 b Fr(\(A.8\))24 3549 y(and,)38 b(due)f(to)g(\(A.3\))i(and)e(\(A.4\),)j (the)d(singular)e(parts)i(of)g Fm(k)2148 3516 y Fk(\013)2145 3572 y Fi(+)2242 3549 y Fr(and)f Fm(k)2475 3516 y Fk(\013)2472 3572 y Fv(\000)2532 3549 y Fr(,)j(at)f Fm(\013)f Fr(=)f Fn(\000)p Fm(n)p Fr(,)i(cancel)g(out.)61 b(In)24 3662 y(particular,)29 b(for)h Fm(\025)25 b Fr(=)g Fm(n)g Fn(2)g Fl(N)6 b Fr(,)37 b(one)31 b(obtains)e(\(cf.)41 b([13)r(]\))723 3905 y Fm(F)13 b Fr([)p Fm(x)871 3867 y Fk(n)871 3927 y Fv(\006)930 3905 y Fr(]\()p Fm(k)s Fr(\))27 b(=)1273 3843 y(1)p 1208 3884 177 4 v 1208 3902 a Fn(p)p 1283 3902 101 4 v 1283 3977 a Fr(2)p Fm(\031)1394 3804 y Fj(h)1437 3905 y Fr(\()p Fn(\006)p Fm(i)p Fr(\))1609 3867 y Fk(n)p Fi(+1)1747 3905 y Fm(n)p Fr(!)p Fm(k)1877 3867 y Fv(\000)p Fk(n)p Fv(\000)p Fi(1)2089 3905 y Fr(+)20 b(\()p Fn(\007)p Fm(i)p Fr(\))2352 3867 y Fk(n)2400 3905 y Fm(\031)s(\016)2498 3867 y Fi(\()p Fk(n)p Fi(\))2600 3905 y Fr(\()p Fm(k)s Fr(\))2720 3804 y Fj(i)2794 3905 y Fm(:)656 b Fr(\(A.9\))24 4217 y Fo(Ac)l(kno)l(wledgmen)l(ts)24 4420 y Fr(I)32 b(w)m(ould)g(lik)m(e)g(to)h(thank)g(J\030)-40 b(edrzej)1219 4397 y(\023)1217 4420 y(Sniat)m(yc)m(ki)32 b(for)g(v)m(ery)i(in)m (teresting)e(discussions)e(and)i(his)f(w)m(arm)i(hospi-)24 4533 y(talit)m(y)h(during)f(m)m(y)i(sta)m(y)g(in)f(Calgary)g(and)g (Andrzej)g(Kossak)m(o)m(wski)i(for)e(in)m(tro)s(ducing)e(this)i (problem)f(to)24 4646 y(me)c(and)f(for)h(man)m(y)g(in)m(teresting)f (and)g(stim)m(ulating)g(discussions.)37 b(This)27 b(w)m(ork)i(w)m(as)h (partially)d(supp)s(orted)24 4759 y(b)m(y)j(the)g(P)m(olish)f(State)j (Committee)e(for)h(Scien)m(ti\014c)e(Researc)m(h)i(\(KBN\))h(Gran)m(t)f (no)f(2P03B01619.)1809 5281 y(17)p eop %%Page: 18 18 18 17 bop 24 44 a Fo(References)69 247 y Fr([1])46 b(L.A.)31 b(Khal\014n,)e(JETP)g(Lett.)j Fs(5)e Fr(\(1972\))j(388)69 434 y([2])46 b(C.G)31 b(Hegerfeldt,)g(Ph)m(ys.)f(Rev.)h(Lett.)h Fs(72)e Fr(\(1994\))j(596)69 620 y([3])46 b(A.)29 b(Bohm,)h(H.-D.)f(Do) s(ebner,)g(P)-8 b(.)29 b(Kielano)m(wski,)f Ff(Irr)-5 b(eversibility)31 b(and)h(Causality,)h(Semigr)-5 b(oups)32 b(and)210 733 y(R)n(igge)-5 b(d)33 b(Hilb)-5 b(ert)33 b(Sp)-5 b(ac)g(es)p Fr(,)32 b(Lecture)f(Notes)g(in)e(Ph)m(ysics)h Fs(504)p Fr(,)h(Springer,)e(Berlin,)g(1998.)69 919 y([4])46 b(A.)40 b(Bohm)f(and)g(M.)h(Gadella,)h Ff(Dir)-5 b(ac)41 b(Kets,)i(Gamov)f(V)-7 b(e)i(ctors)41 b(and)h(Gelfand)g(T)-7 b(riplets)p Fr(,)43 b(Lecture)210 1032 y(Notes)32 b(in)d(Ph)m(ysics)h Fs(348)p Fr(,)h(Springer,)d(Berlin,)h(1989)69 1219 y([5])46 b(I.)31 b(An)m(toniou)f(and)g(I.)g(Prigogine,)g(Ph)m(ysica)g Fs(A)35 b(192)c Fr(\(1993\))i(443)69 1405 y([6])46 b(I.M.)33 b(Gelfand)f(and)f(N.Y.)i(Vilenkin,)e Ff(Gener)-5 b(alize)g(d)36 b(F)-7 b(unctions)p Fr(,)34 b(V)-8 b(ol.)32 b(IV,)h(Academic)f(Press,)h (New)210 1518 y(Y)-8 b(ork,)32 b(1964.)69 1705 y([7])46 b(K.)22 b(Maurin,)g Ff(Gener)-5 b(al)26 b(Eigenfunction)e(Exp)-5 b(ansion)26 b(and)f(Unitary)g(R)-5 b(epr)g(esentations)28 b(of)d(T)-7 b(op)i(olo)g(gic)g(al)210 1817 y(Gr)g(oups)p Fr(,)33 b(PWN,)e(W)-8 b(arsza)m(w)m(a,)33 b(1968.)69 2004 y([8])46 b(G.)31 b(P)m(arra)m(vicini,)f(V.)g(Gorini)f(and)h (E.C.G.)h(Sudarshan,)d(J.)i(Math.)i(Ph)m(ys.)e Fs(21)h Fr(\(1980\))i(2208)69 2190 y([9])46 b(S.)41 b(Alb)s(ev)m(erio,)h(L.S.)f (F)-8 b(erreira)40 b(and)g(L.)h(Streit,)i(eds.)d Ff(R)-5 b(esonanc)g(es)44 b({)f(Mo)-5 b(dels)43 b(and)g(Phenomena)p Fr(,)210 2303 y(Lecture)31 b(Notes)h(in)d(Ph)m(ysics)g Fs(211)p Fr(,)j(Springer,)c(Berlin,)h(1984)24 2490 y([10])46 b(E.)35 b(Brandas)g(and)f(N.)h(Elander,)f(eds.)h Ff(R)-5 b(esonanc)g(es)p Fr(,)38 b(Lecture)d(Notes)h(in)d(Ph)m(ysics)h Fs(325)p Fr(,)j(Springer,)210 2603 y(Berlin,)29 b(1989)24 2789 y([11])46 b(M.)34 b(Reed)g(and)f(B.)h(Simon,)e Ff(Metho)-5 b(ds)37 b(of)f(Mo)-5 b(dern)36 b(Mathematic)-5 b(al)38 b(Physics.)d(V)-7 b(ol.)36 b(III.)f(Sc)-5 b(attering)210 2902 y(The)g(ory)p Fr(,)33 b(Academic,)e(New)f(Y)-8 b(ork,)31 b(1979)24 3088 y([12])46 b(K.)31 b(Y)-8 b(osida,)30 b Ff(F)-7 b(unctional)34 b(A)n(nalysis)p Fr(,)d(Springer,)e(Berlin,)g (1978)24 3275 y([13])46 b(I.M.)28 b(Gelfand)e(and)h(G.E.)h(Shilo)m(v,)e Ff(Gener)-5 b(alize)g(d)32 b(functions)p Fr(,)c(V)-8 b(ol.)28 b(I,)f(Academic)g(Press,)h(New)f(Y)-8 b(ork,)210 3388 y(1966)24 3574 y([14])46 b(L.S.)33 b(P)m(on)m(triagin,)g(V.G.)h (Bolta)s(\023)-48 b(nskij,)33 b(R.V.)g(Gamkrelidze,)g(E.F.)h(Miscenk)m (o,)g Ff(The)h(Mathematic)-5 b(al)210 3687 y(The)g(ory)35 b(of)e(Optimal)g(Pr)-5 b(o)g(c)g(esses)p Fr(,)32 b(Wiley)-8 b(,)31 b(New)f(Y)-8 b(ork,)31 b(1962)24 3874 y([15])46 b(D.F.)32 b(W)-8 b(alls,)30 b(Nature,)h Fs(306)h Fr(\(1983\))g(141)24 4060 y([16])46 b(L.D.)32 b(Landau)d(and)h(E.M.)h(Lifshitz,)e Ff(Quantum)k(Me)-5 b(chanics)p Fr(,)31 b(P)m(ergamon,)g(London,)f(1958) 24 4246 y([17])46 b(G.)31 b(Barton,)h(Ann.)d(Ph)m(ys.)i Fs(166)g Fr(\(1986\))i(322)24 4433 y([18])46 b(N.L.)31 b(Balazs)g(and)f(A.)h(V)-8 b(oros,)31 b(Ann.)f(Ph)m(ys.)g Fs(199)i Fr(\(1990\))g(123)24 4619 y([19])46 b(M.)37 b(Castagnino,)g(R.)f(Diener,)h(L.)f(Lara)g(and)f(G.)i(Puccini,)e(In)m (t.)i(Jour.)e(Theor.)g(Ph)m(ys.)h Fs(36)h Fr(\(1997\))210 4732 y(2349)24 4919 y([20])46 b(D.)37 b(Chru)-5 b(\023)-41 b(sci)s(\023)-48 b(nski,)36 b(Op)s(en)f(Sys.)h(Information)f(Dyn.)i Fs(9)f Fr(\(2002\))j(207)f(\(a)m(v)-5 b(ailable)36 b(as)g(LANL)h (e-prin)m(t)210 5032 y(math-ph/0206009\))1809 5281 y(18)p eop %%Page: 19 19 19 18 bop 24 44 a Fr([21])46 b(D.)31 b(Chru)-5 b(\023)-41 b(sci)s(\023)-48 b(nski,)28 b(Wigner)i(function)f(for)i(damp)s(ed)e (systems,)h(LANL)g(e-prin)m(t)g(math-ph/0209008)24 232 y([22])46 b(E.)39 b(P)-8 b(.)39 b(Wigner,)h(Unitary)d(represen)m (tation)i(of)f(the)h(Inhomogeneous)f(Loren)m(tz)h(Group)f(Including)210 345 y(Re\015ections,)k(in)37 b Ff(Gr)-5 b(oup)42 b(The)-5 b(or)g(etic)g(al)43 b(Metho)-5 b(ds)42 b(in)f(Elementary)g(Particle)g (Physics)p Fr(,)g(F.)f(G)s(\023)-48 b(ursa)m(y)210 458 y(\(ed.\),)32 b(Gordon)e(and)g(Breac)m(h,)i(Science)e(Publisher,)d(New) k(Y)-8 b(ork,)31 b(1967)24 645 y([23])46 b(A.)31 b(Kossak)m(o)m(wski,)g (priv)-5 b(ate)30 b(comm)m(unication)24 833 y([24])46 b(R.P)-8 b(.)38 b(Kan)m(w)m(al,)g Ff(Gener)-5 b(alize)g(d)40 b(F)-7 b(unctions:)54 b(The)-5 b(ory)39 b(and)g(T)-7 b(e)i(chniques)p Fr(,)39 b(Mathematics)e(in)e(Science)210 946 y(and)30 b(Engineering)f Fs(177)p Fr(,)i(Academic)g(Press,)f(New)h (Y)-8 b(ork,)31 b(1983)24 1133 y([25])46 b(C.G.)31 b(Bollini)d(and)i (L.E.)g(Oxman,)g(Ph)m(ys.)g(Rev.)h Fs(A)k(47)c Fr(\(1993\))i(2339)24 1321 y([26])46 b(P)-8 b(.L.)31 b(Duren,)f Ff(The)-5 b(ory)35 b(of)d Fn(H)1199 1288 y Fk(p)1271 1321 y Ff(Sp)-5 b(ac)g(es)p Fr(,)32 b(Academic)f(Press,)f(New)h(Y)-8 b(ork,)31 b(1970)1809 5281 y(19)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0309050230362--