Content-Type: multipart/mixed; boundary="-------------0206171109504" This is a multi-part message in MIME format. ---------------0206171109504 Content-Type: text/plain; name="02-269.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-269.comments" 18 pages ---------------0206171109504 Content-Type: text/plain; name="02-269.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-269.keywords" exponential decay, ionization threshold, quantum electrodynamics, localization, IMS formula ---------------0206171109504 Content-Type: application/postscript; name="decay.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="decay.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: decay.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips decay %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.06.17:1801 %%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 TeXDict begin 39158280 55380996 1000 600 600 (decay.dvi) @start %DVIPSBitmapFont: Fa cmsy12 12 3 /Fa 3 71 df10 D33 D<040FB712C0047F16F00303B812F8151F5D9226F801FCC7121FDA01C0 EE07F0912607800316C0DA0F004915004A94C7FC143E147E4A13075F13015CEB03E04A49 5A0102C7FC90C8FC5F161FA34C5AA394CAFC5EA2167E16FEA293B612FE4B5D61614B15C0 04F0CAFCA215075E150F5EA24B5AA2153F93CBFC5D157EA25DA24A5AA24A5AA24A5A1207 001F495AEA3F80007F495AD8FFC090CCFCEBE03E387FFC3CEBFFF86C13E06C5B6C90CDFC EA01F84D477EC346>70 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmmi12 12 1 /Fb 1 74 df<91B612E0A30200EBE000ED7F80A24BC7FCA44A5AA44A5AA44A5AA44A5AA4 4A5AA44A5AA44A5AA44AC8FCA4495AA4495AA4495AA4495AA4495AA4495AA2EBFFE0B612 E0A32B447CC32B>73 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmbx12 12 29 /Fc 29 123 df12 D45 DI 49 DII65 DII73 D<923807FFC092B512FE0207ECFFC091261F FE0013F0DA7FF0EB1FFC902601FFC0EB07FF010790C7000113C049486E7F49486F7E4948 6F7E49486F7E49486F7E48496F7E4819804A814819C091C97E4819E0A248487013F0A200 3F19F8A3007F19FC49177FA400FF19FEAD007F19FC6D17FFA3003F19F8A36C6C4C13F0A3 6C6D4B13E0A26C6D4B13C06C19806E5D6C19006C6D4B5A6D6C4B5A6D6C4B5A6D6C4B5A6D 6C4A5B6D01C001075B010101F0011F90C7FC6D01FEEBFFFE023FB612F8020715C002004A C8FC030713C047467AC454>79 D<923807FFC092B512FE0207ECFFC091261FFE0013F0DA 7FF0EB1FFC902601FFC0EB07FF010790C7000113C049486E7F49486F7E49486F7E49486F 7E49486F7E488448496F1380A248496F13C0A24890C96C13E0A24819F04982003F19F8A3 007F19FCA249177FA300FF19FEAD007F19FCA36D17FF003F19F8A3001F19F06D5EA26C19 E06E5D6C03FE15C06C912603FF8014806E486D5A6C9028E00F01E00F13006C91261E00F0 5B90267FF01C9038781FFC6D6C486D485AD91FFC6E485AD90FFE4B5AD907FF6E5B010101 DC92C7FC6D01FE495ADA3FFFEBFFF8020790B500E01302020017070307EBC7F092C7EA03 F872130F72131F9538FF80FF71EBFFFEA47114FCA27213F8A27213F0A27213E07213C072 1300F000FC48587AC454>81 D83 D<003FBA12E0A49026FE000FEB800301F0EE007FD87FC0EF1FF049170F90C716 07007E1803007C1801A300781800A400F819F8481978A5C81700B3B3A40107B8FCA44543 7CC24E>I<903807FFF0017F13FF48B612C03A03FC007FF0486CEB1FF86F7E486C6D7E6F 7E83A26C487F836C5AEA00F090C7FCA4150F023FB5FC0103B6FC011F1301EB7FF03801FF C000071300485A485A485A5B127F5B12FFA35DA26C6C5B15066C6C5B6C6C013813F03C0F FF80F07FFFC00003EBFFE0C6EC801F90390FFE0007322C7DAB36>97 D99 DII104 D<1378EA01FE487E481380A24813C0A46C1380A26C13006C5AEA007890C7FCACEB7FC0EA 7FFFA412037EB3B0B6FCA418467CC520>I108 D<90397F8007FEB590383FFFC04B7F913981F03FF0913983C00FF80003 D987007F6C018E80029C1307149802B080A214E0A35CB3A8B60083B512FEA4372C7CAB3E >110 DI<90397FC01FF8B500C1B5FC02C7 14E09139CFC03FF09139FF000FFC000301FC6D7E6C01F06D7E4A6D13804A15C08218E0EF 7FF0A218F8A2173FA218FCAA18F8177FA218F0A2EFFFE0A24C13C018806E5B6E15006EEB 0FFE02FEEB1FF89139CFC07FF002C7B512C002C191C7FC9138C03FF092C9FCAFB67EA436 3F7DAB3E>I<90387F807FB53881FFC0028313F091388787F891388E0FFC0003138C6C90 389C1FFE149814B0A29138F00FFC14E0ED07F8ED01E092C7FC5CB3A7B612E0A4272C7DAB 2E>114 D<90391FFE038090B512CF000314FF380FF003391F80007F90C7123F48141F00 7E140FA200FE1407A27EA201C090C7FC13F0EBFF806C13FCECFF806C14E06C14F86C806C 806C80C61580013F14C01301EB000F020013E00070147F00F0141FA26C140FA36C15C07E 16806C141F6D140001E0137E9038F801FC00F8B55AD8F03F13E026E007FEC7FC232C7CAB 2C>III<001FB71280A39026FC000F130001E0495A49133F495C90C748 5A4B5A003E5B5E4A5B003C5B5E4A90C7FC4A5A143FC75B4A5A4A5A5B5D495B5B5D499038 000780495A133F5C4948130F494814005A5C48495B5A5C4890C75A48485C003F5D49EB03 FE4848131FB7FCA3292C7DAB32>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd eufm10 10.95 1 /Fd 1 105 df<1307EAC01E5BEA4078EA61F0EA63E01267EA7FC0A25BA390C8FCA7140C 141E147FECFFC04913F8010713FE5B013F13FFEB7E1FEBF007EBE001EB800090C7FCA515 7FAA5AA67FA26D137E7FA26C7E6C4813FE5B6C5A90C712FC120E120CC8FCEC01F815F0A2 15E0EC03C01580EC0700140E140C14385C14F0495AEB07C06D5A6DC7FC205179BD2F> 104 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr6 6 1 /Fe 1 49 df<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131F AD0078131EA2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA0 1E>48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmbx11 10.95 31 /Ff 31 120 df46 D<140F5C147F495A130F48B5FCB6FCA213F7EAFE071200B3B3AA007FB612 F8A4253C79BB34>49 D<903803FF80013F13F890B512FE00036E7E2607F80F7F260FC001 13F048486C7F48C76C7ED87FC06D7E7F486C6D7E7FA26F1380A46C5AA2EA1FC00007C7FC C814005DA25E153F5E4B5A5E4B5A4A5B5E4A90C7FC4A5AEC0FF84A5A5DEC3F804AC8FC14 FED901F8EB0780495A495A4948EB0F00495A013EC7FC5B495C485A90B7FC485D5A5A5A5A 5AB7FC5EA3293C7BBB34>II<16F81501A215031507150F151FA2153F157F15FF5CA2EC03DFEC079F140F 151F141E143C147814F814F0EB01E0EB03C0EB0780130F1400131E5B137C5B5B485A485A 1207485A90C7FC121E5A127C5AB812F8A4C8383FF800AB91B612F8A42D3C7DBB34>I<00 0E1518D80FC014F801FC131F90B65AA25E5E5E93C7FC15FC5D15E092C8FC14F80180C9FC A9903881FFC0018F13F801BF13FE9039FF01FF809039F8007FE001E06D7E4980496D7E6C C7FCC87F150F82A31780A2120FEA3FC0487E487EA41700A25B6C48495A5B007CC75B6C14 3F003F5D6C6C495AD80FE0495A2607FC075B0001B6C7FC6C14FC013F13F0010790C8FC29 3D7BBB34>II<121E121F13F0 90B712F0A35A17E017C0178017005EA2485D007CC7EA01F84B5A00784A5A5E150F4B5A48 4AC7FC157E5DC85A14014A5A4A5AA24A5A141F5D143FA24AC8FCA25CA2495AA21303A313 075CA2130FA5131FAA6D5A6D5A6D5A2C3F7ABD34>III<16F84B7EA24B 7EA34B7EA24B7FA34B7FA24B7FA34B7F157D03FD7F15F8A2020180EDF07F020380EDE03F A20207804B7E020F814B7EA2021F814B7E4A81023E7FA2027E81027C7F02FC814A7FA249 488191B7FC4982A2498202C0C7123F83010F834A80011F8391C8FC834983013E81017E83 137C8348B483B500FC49B612F8A4453F7CBE4E>65 DI76 D<903A03FF8001C0011FEBF003017FEBFE0748B6120F4848C613DFD807F8EB0FFF484813 0348487F48487F167F007F153F49141F160F12FF1607A27FA26D1403A27F01FC91C7FC13 FF6C13F0ECFF8015F86C14FF16C06C15F06C816C816C816C816C6C1580131F010715C0EB 007F020714E0EC003F1507030113F081167F12F0163FA2161FA27EA217E07EA26CED3FC0 7E6DEC7F807F01F0ECFF0001FE495A3AFEFFE00FFCD8FC3FB55AD8F80F14E0D8F0031480 27E0003FFCC7FC2C407ABE39>83 D<003FB912F8A4903BFC007FFC007F01E0160FD87F80 EE03FC90C71501007E1700A2007C187CA20078183CA548181EA5C81600B3B1011FB712F0 A43F3D7CBC48>I<903807FF80013F13F048B512FC3903FC03FF2607E00013C0D80FF86D 7E001F6E7E7F6F7EA26F7EA26C5A6C5AEA01C0C8FCA3EC07FF49B5FC130F017F130F3801 FFF000071380481300EA1FFC485A127F5B12FF5BA3151FA26D133F127F6D49B4FC273FFC 01F713FC390FFE07E36CB5128100019038FE007F26001FF890C7FC2E2B7DA932>97 D99 DII<903A03FF8003F0013F9038F81FFC90B538FE7FFE00 03903801FFFC3A07FC007FE1000F15E04848EB3FF0003FEDF87C49011F1300A2007F81A7 003F5DA26D133F001F5D6C6C495A00075D9039FF01FF80DAFFFEC7FCD80F3F13F8010313 80001ECAFCA2121FA37F7F90B6FC6C15F016FC6C15FF17806C16C017E0120F271FC00001 13F04848EB001F48C8EA0FF8160712FE1603A46C15076C16F06D140F6C6CEC1FE06C6CEC 3FC0D80FF8ECFF803B03FF800FFE00C690B512F8011F14C0010101FCC7FC2F3D7DA834> 103 D<13FFB5FCA412077EB0ED3FE0913801FFFC020713FFDA0F817FDA3E007F4A6D7E14 784A133F4A805CA25CA391C7FCB3A5B5D8FC0FB512C0A4323F7CBE39>I<13FFB5FCA412 077EB1923803FFFEA4030013804CC7FC4B5AED03F04B5AED1FC04B5A037EC8FC5DEC03F8 EC07E04A7E4A7EEC7FFC14FF818102E77F02837F1401496C7F826F7E6F7E151F6F7E826F 7F6F7F816F7F83B5D8F807EBFFC0A4323F7DBE37>107 D<13FFB5FCA412077EB3B3B1B5 12FCA4163F7CBE1D>IIII<01FFEBFFC0B5000713F8021F13FFDA7F017FDAF80013 E00007496D7E6C01C06D7E4A6D7E91C77F160F83821880A38218C0AA18805EA318005E5F 6E495AA26E495A6E495A02F8EBFFC0DA7E035B91261FFFFEC7FC020713F80201138091CA FCADB512FCA4323B7DA839>I<3901FE01F800FFEB0FFF4A13C091383E3FE01478000790 38F07FF0000313E013FF14C0A2ED3FE09138801FC0ED070092C7FCA291C8FCB3A4B6FCA4 24297DA82A>114 D<90381FF80E48B5123E000714FE380FE007381F800148C7FC007E14 7E007C143E12FC151EA27E7E6D90C7FC13E013FF6C13FCECFF806C14E06C14F86C800003 80C680133F01031480EB000F020113C000F0EB007F153F6C141FA2150F7EA26C1580151F 6C15006D5B01E0137E9038F803FC00FCB512F0D8F03F13C026E007FEC7FC222B7DA929> II119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmti11 10.95 54 /Fg 54 128 df11 D<933807FF80043F13F09338FE00FCDB01F0131EDB07E0130F4B48131F4C133F031F147F 4BC7FCA2187E037E14381800A215FE5DA31401A25DA414030103B712F0A218E0903A0003 F000070207140F4B14C0A3171F020F15805DA2173F1800141F5D5F177EA2143F92C712FE 5FA34A1301027EECF81CA3160302FEECF03C4A1538A21878187013014A010113F018E093 3800F1C0EF7F804948EC1F0094C7FCA35C1307A25C121CEA7E0F00FE5BA249CAFC12FCEA F81E485AEA7878EA3FF0EA07C0385383BF33>I19 D39 DI<14031580A2EC01C0EC00 E0A21570A215781538153CA3151EA4151FA2150FA7151FA9153FA2153EA3157EA2157CA2 15FCA215F8A21401A215F0A2140315E0A2140715C0A2EC0F80A2141F15005C143EA25CA2 5CA2495A5C1303495A5C130F49C7FC131E5B137C5B5B485A485A485A48C8FC121E5A1270 5A5A205A7FC325>I44 D<387FFFFCA3B5FCA21605799520>I<120EEA3F80127F12FFA31300127E123C09097688 1C>I<1506150E151EA2153C157C15FC1401EC07F8140F143FEB01FF90380FF3F0EB1FC3 EB0E07130015E0A2140FA215C0A2141FA21580A2143FA21500A25CA2147EA214FEA25CA2 1301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CEB7FC0B612E0A215C01F3D76 BC2E>49 D<15FE913803FFC091380F01F091383C00F84A137C4A7F4948133F49487F4A14 8049C7FC5BEB0E0C011E15C0EB1C0EEB3C06133813781370020E133FD9F00C148013E014 1C0218137F00011600EBC0384A13FEEC600102E05B3A00E3C003F89039FF0007F0013C49 5A90C7485A5E037FC7FC15FC4A5A4A5AEC0FC04AC8FC147E14F8EB03E0495A011FC9FC13 3E49141801F0143C48481438485A1678485A48C85A120E001E4A5AD83FE0130301FF495A 397C3FF01FD8780FB55AD8700391C7FCD8F0015B486C6C5A6E5AEC07C02A3F79BC2E>I< ED7F80913803FFE091380F80F891383C007C02F87FD901E07F494814804948130F49C7FC 010E15C0131EEB1C18EB3C1CEB380C0178141F17801370A2021C133F6D4814004A5BD91F E0137ED90F805B90C8FC4B5A4B5A4B5AED1F8003FFC7FCECFFFC15F0A2EC00FC153E153F 8182150F82A4151FA3123E007E143F00FE5DA3484AC7FC12E015FE5D14016C495A007049 5A0078EB0FC00038495A6C017EC8FC380F01F83803FFE0C690C9FC2A3F78BC2E>I<131E 137FEBFF80A31400A25B133890C7FCB3A3120EEA3F80127F12FFA390C7FC127E123C1127 76A61C>58 D<17381778A217FCA21601A216031607A2160FA2161DA21639A2167116F116 E1ED01C183ED0380A2ED07005D150E5DA25DA25DA25D14015D4A5AA24AC77E83140EA202 1FB6FC5CA20270C77EA25CA2495A13035C130791C8FC010E1680A249153F133C13381378 13F80001167FD807FCEDFFC0B500C0013F13FFA338417BC043>65 D67 D<49B812F8A390260003FCC7123F18074B 14031801F000F014075DA3140F5D19E0A2141F4B1338A2EF7801023F027013C04B91C7FC A217F0027F5CED00011603160F91B65AA39138FE001F0101EC07805CA3010392C8FC5C18 074C5B0107020E130E5C93C7121E181C010F163C4A15381878A2011F5E5C4D5AA2013F15 034A4A5A170F017F151F4D5A91C812FF49020F90C7FCB9FCA25F3D3E7BBD3D>69 D<49B812F0A390260003FCC7123F180F4B14071803F001E014075DA3140F5D19C0A2141F 5D1770EFF003023F02E013804B91C7FCA21601027F5CED0003A216074AEB1F8092B5FCA3 902701FE003FC8FC4A7F82A20103140E5CA2161E0107141C5CA293C9FC130F5CA3131F5C A3133F5CA2137FA291CBFC497EB612C0A25D3C3E7BBD3B>I<49B5D8FC01B512FCA39026 0003FEC73803FE004B5D4B5DA2180714074B5DA2180F140F4B5DA2181F141F4B5DA2183F 143F4B5DA2187F147F92C890C7FCA26091B8FC60A24AC7120113014A5DA2170313034A5D A2170713074A5DA2170F130F4A5DA2171F131F4A5DA2173F133F4A5DA2017F157FA291C8 90C8FC496C4A7EB690B6FCA24A5D463E7BBD43>72 D<49B512FE5BA290390003FE005D5D A314075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C5CA313015CA313035CA31307 5CA3130F5CA3131F5CA3133F5CA2137FA291C8FC497EB6FCA3273E7BBD23>I<4AB512FC A391C71300A25EA215015EA315035EA315075EA3150F5EA3151F5EA3153F5EA3157F93C7 FCA35DA25DA21401A25DA21403A25DA21407121FD87F805BA2140FD8FF005B141F485C00 F8495A12E0007049C8FC14FE387801FC383C03F06C485A3807FF80D801FCC9FC2E4078BD 2F>I<49B6FCA390260003FEC8FC5D5DA314075DA3140F5DA3141F5DA3143F5DA3147F92 C9FCA35C5CA313015CA313035CEF0180EF03C0010716805C17071800130F4A5C170E171E 131F4A5CA2177C013F5D4A1301A2017FEC07F0160F91C7123F4949B45AB8FCA25F323E7B BD38>76 D<902601FFFE93381FFFC062A2D9000394387FE000505A6303BFED01DF1AFF02 07EE03BF033FDB073FC7FCA2F10E7F140F020EEE1C7E19381AFE021E167091261C1F805D 19E019E1023CED01C1023892380381F8A2F0070314780270030E5B181C19079126F00FC0 133802E05F1870F0E00F130102C0DA01C05BA2943803801F13030280DA07005B170EDB07 E0143F01075D020094C8FC5F4D5B5B010E4B137EA2DCE1C013FE011EECE380011C5F04E7 C7FCDB03F71301013C14FE01385F5E01784A130313F8486C4A5CD807FE4C7EB500F0D9E0 07B512F016C0DAE0015E523E7ABD51>I<49B77E18F018FC903B0003FC0003FEEF00FF4B EC3F80F01FC01407F00FE05DA2020F16F0A25DA2141FF01FE05DA2023F16C0183F4B1580 187F027F160018FE92C7485A604AEC07F04D5A4AEC3F804CB4C7FC49B612F817E002FCCA FCA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CBFC497EB6 FCA25C3C3E7BBD3D>80 D<49B612FCEFFF8018F0903B0003FC000FF8EF01FE4B6D7E84F0 3F800207ED1FC05DA219E0140F5DA3021FED3FC05DA2F07F80143F4B150018FE4D5A027F 5D92C7485AEF0FE0EF1F804A027EC7FC4AEB07F891B612E01780903A01FE000FE04AEB03 F0707E707E0103814A147E177FA213075CA25F130F5C5F1601131F5CA3013F020314404A 16E05F017F160119C091C700011303496C1680B61507933900FE0F004AEC7E1ECAEA1FFC EF07F03B407BBD42>82 D<92390FF001C0ED7FFE4AB5EA0380913907F00FC791390FC003 EF91381F0001023E903800FF004A805C495A4948143EA2495AA2010F153C5CA3011F1538 A46E91C7FCA2806D7E14FCECFFC06D13F8EDFF806D14E06D14F86D6C7F021F7F02037FEC 003F03077F1500167F707E161FA2160FA212075A5F120EA2001E151F94C7FCA2163E123E 5E123F5E486C495A4B5A6D1307D87DE0EB0F80D8F8F849C8FCD8F07F13FE90383FFFF8D8 E00F13E048C690C9FC32427ABF33>I<48B812FEA3489039001FE00301F81500D807E04A 137E49163E49163C000F143F90C749131C183C121E157F001C92C7FC003C173812385D00 785C1270187800F001011570C7491400A314035DA314075DA3140F5DA3141F5DA3143F5D A3147F92C9FCA35C5CA313015CA21303A25CEB0FFC003FB6FCA3373E71BD40>I<001FB5 D8C001B512C0A326003FE0C7380FF8004AEC07E04A5D715A017F1507A291C890C7FCA249 5D170E5BA20001161E171C5BA20003163C17385BA20007167817705BA2000F16F05F5BA2 001F15015F5BA2003F1503A2495DA2007F1507A290C890C8FCA25E160E12FE161E161C16 3C007E153816785E5E15016C4A5A4B5A6C6C49C9FC153E6C6C5B3907E001F03903F80FE0 C6B51280D93FFECAFCEB0FF03A406FBD43>I<147E49B47E903907C1C38090391F80EFC0 90383F00FF017E137F4914804848133F485AA248481400120F5B001F5C157E485AA215FE 007F5C90C7FCA21401485C5AA21403EDF0385AA21407EDE078020F1370127C021F13F000 7E013F13E0003E137FECF3E1261F01E313C03A0F8781E3803A03FF00FF00D800FC133E25 2977A72E>97 DIII<143F90 3801FFE0903807C0F090381F0078137E49133C485A485A12074848137C491378121F4848 13F8EC01F0007FEB07E09038001FC0903801FF00B512F8148048C8FCA45AA6150C151C15 3C007C147815F0007EEB01E0003EEB03C06CEB0F806CEB1E00380780FC3803FFE0C690C7 FC1E2976A729>I<167C4BB4FC923807C78092380F83C0151F160FED3F1FA2157E1780EE 0F0093C7FC5DA414015DA414035DA30103B512F8A390260007E0C7FCA3140F5DA5141F5D A4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495AA4495AA2121C007E5B12FE49 C9FCA2EAFC1E485A12F0EA7878EA3FE0EA0F802A5383BF1C>III<1470EB01FCA314F8A2EB00E01400AD137C48B4FC 38038F80EA0707000E13C0121E121CEA3C0F1238A2EA781F00701380A2EAF03F14001200 5B137E13FE5BA212015BA212035B1438120713E0000F1378EBC070A214F0EB80E0A2EB81 C01383148038078700EA03FEEA00F8163D79BB1C>IIIIIII<903903E001F890390FF807FE903A1E7C1E0F80903A1C3E3C 07C0013C137801389038E003E0EB783F017001C013F0ED80019038F07F0001E015F8147E 1603000113FEA2C75AA20101140717F05CA20103140F17E05CA20107EC1FC0A24A148016 3F010F15005E167E5E131F4B5A6E485A4B5A90393FB80F80DA9C1FC7FCEC0FFCEC03E049 C9FCA2137EA213FEA25BA21201A25BA21203A2387FFFE0B5FCA22D3A80A72E>I<027C13 C0903801FF03903807C38790391F01CF8090383E00EF4913FF01FC137F48481400485AA2 485A000F147E5B121F15FE48485BA3007F130101005BA3481303485CA31407485CA3140F 5D007C131F143F147F6C495A5B381F03DF380F07BF2607FE3FC7FCEA01F8C7FC5C147EA3 14FE5CA313015CA21303130748B512E0A3223A77A729>II II<137C48B4143826038F8013FCEA0707000E7F001E1401121CD83C0F5C123815 03EA781F007001805BA2D8F03F1307140000005D5B017E130FA201FE5C5B151F1201495C A2153F0003ED8380491403A2157F1607037E1300A2EDFE0F160E000113019039F803BE1C 0000EB073E90397C1E1E3890393FF80FF0903907E003E0292979A730>I<017CEB038048 B4EB07E039038F800FEA0707000E01C013F0121E001C1407EA3C0F0038140316E0D8781F 130100701380A2EAF03F020013C012005B017E1303168013FE5B1507000115005BA2150E 12035B5DA25DA25DA200015C4A5AEBF8030000495AD97E0FC7FCEB1FFCEB07F0242979A7 29>I<017C167048B491387001FC3A038F8001F8EA0707000E01C015FE001E1403001CED F000EA3C0F0038177C1507D8781F4A133C00701380A2D8F03F130F020049133812005B01 7E011F14784C137013FE5B033F14F0000192C712E05BA2170100034A14C049137E170318 80A2EF070015FE170E00010101141E01F86D131C0000D9039F5BD9FC076D5A903A3E0F07 C1E0903A1FFC03FFC0902703F0007FC7FC372979A73C>I<903903E003F090390FF80FFC 90393C3C1C1F9039701E380F9039E01F703F000102F013803B03C00FE07F001380000714 C0D9001F131C4892C7FC000E5CA2001E133FA2C790C8FCA25C147EA314FE5CA313014A13 38A3010314781670001C4913F0007E5D0107130100FE5D010F495A90380EF80727781C78 0FC7FC9038383C3C393FF01FF83907C007E029297CA729>I<137C48B4143826038F8013 FCEA0707000E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F13 07020013E012005B017E130F16C013FE5B151F1201491480A2153F000315005BA25D157E A315FE5D00011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA214075D121F 486C485AA24A5AD87F0090C7FC007E133E0038137E5CEB01F06C485A381E0FC0D807FFC8 FCEA01F8263B79A72C>II<000E131E383F807F007FEBFF8012FFA215005B007E5B003C133819 0968BD2E>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmsl8 8 26 /Fh 26 125 df<120E121FEA3F80A4121D1201A2EA0300A31206A25A5A5A5A5A5A09147B 8614>44 D<16181638A2167016F016E0ED01C0A2ED0380A2ED0700A2150EA25DA25DA25D A25DA24A5A14035D4AC7FCA2140EA25CA25CA25CA25CA2495AA2495AA249C8FC5B130E5B A25BA25BA25BA2485AA2485AA248C9FCA2120EA25A123C12385AA25A126025437FB123> 47 DI<1430 14F01301130FEA03FDEBF3E0EA0003A5EB07C0A6EB0F80A6EB1F00A6133EA65BA513FC48 7EB512FCA2162C79AB23>I<147F903803FFE090380F81F890381C007C01307F49133F49 7F48481480487E7F7F1207A212036C48133FC81300A2157EA25D4A5A5D4A5A4A5A4A5A4A C7FC143E14785C495AEB038049C8FC130E49130C01705B5B485A48485B0006C71270000F B512F0485C5A5AB65A212C7DAB23>I<120C121E001FB512FCA24814F815F015E00070C7 12C00060EB0180EC03004813065C5CC7FC5C5C5C495A49C7FC5B13065B131C5BA2137813 7013F0485AA21203A2485AA3120F5BA2121FA448C8FCA2121E120C1E2E76AC23>55 D<48B612F016FE3B000FF0007F806D48EB0FC0EE03E0707E49486D7E83177C177E173E17 3F495AA2EF1F80A449C8123FA6017E16005FA2177EA217FE495DA24C5A5F16034C5A4848 4A5A5F043FC7FC167E16F80003EC03F00007EC1FC0B7C8FC15F0312D7CAC36>68 D<48B712F8A23A000FF000076D48130017781738495A1718A5495AED0180A21700A21503 D93F0090C7FC5D151F91B5FCA2EC003F017E130EA21506A217181730495B92C7FC1760A2 17E017C048481401A2EE03801607160F0003ED3F0000074AB4FCB8FC5E2D2D7DAC30>I< 9238FF8004020FEBE00C91393F80781C9139FC001C3CD903E0EB067C4948EB03FC011FC7 1201013EEC00F85B4915781201485A49153800071630485AA2121F491520003F1600A248 CAFCA5127E12FE92381FFFFEA2007E9138003FC0161F1780A2127F7EA27E6DEC3F00120F 6C7E6C6C5C1201D800F8EB01DF017EEB078E90391F803E06903907FFF802010001C0C7FC 2F2F79AD37>71 D<49B512C0A290390003FC006E5AA34A5AA64A5AA64A5AA64A5AA64AC7 FCA4121C127E147E12FEA2485B485B386001F06C485A381C0FC0D80FFFC8FCEA03F8222E 7CAC24>74 D97 D99 D101 D<130E131FEB3F80A3EB1F00130E90C7FCAA133EEA07FE5B 1200137CA45BA6485AA6485AA51207A2EAFFFEA2112E7EAD14>105 D108 D<90263E07F8EB7F803C07FE1FFE01FFE03C0FFC781F0781F0 0000903AE00F8E00F890277D8007981378D97F0013B0017E02E0137C017C5C01FC010F14 F8495CA5484890391F0001F0A64848013EEB03E0A50007027E1307A23CFFFE0FFFE0FFFE A2371D7E9C3C>I<90383E07F03907FE3FFC390FFC781E0000EBC01F90387D800FEB7F00 017E1480137C01FCEB1F005BA54848133EA648485BA5000714FCA23AFFFE1FFFC0A2221D 7E9C27>II<90380F83F83901FF8FFF 4890383C0F803A001F7003C09138C001E0028013F091C712F816FC133E167CA2167EA349 14FCA4ED01F8A249EB03F016E06DEB07C0ED0F806DEB1F006D133E3901F381F89038F0FF E06EC7FC91C8FCA3485AA61207A2EAFFFEA2272A809C27>I<90387C0F803907FC3FE039 0FF871F0000013C3EB7983EB7B0390387E01E091C7FC5BA25BA4485AA6485AA51207A2B5 FCA21C1D7E9C1C>114 D<90381FF08090387FFFC03901F00F803803C003EA0700481301 A34814006DC7FCEA0FF0EBFF806C13F06C7FC67FEB3FFEEB01FFEB003F0020EB0F801230 00701400A30078130E5C007C5B00F713F038E3FFE0004090C7FC1A1D7E9C1C>I<1318A2 5BA21370A213F0A2485A12031207121FB512F0A23807C000A6485AA648C7FCA314C0A338 3E0180A4381E03001306EA0F0EEA07FCEA01F014297AA81B>II<3A07FFC03FF8A23A007F00 1FC06D14006D131C6E5A010F5B6D6C5AECC1C0903803E18002F3C7FCEB01F6EB00FCA214 7C147E14FEEB019F9038030F801306496C7E01187FEB3003496C7E13E000036D7E000F49 7E3AFFF007FFC0A2251D7F9C25>120 D<3A03FFF007FEA23A003F8003F091380001E06D 14C01680ED03001480010F1306150E150C6E5A13075D1570ECE06001035BA2ECE18014F1 D901F3C7FC14F714F614FC13005CA214701460A25C13015C49C8FC1278EAFC06130E5B48 5AEAF0F0EA7FC0001FC9FC272A809C25>I124 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi6 6 5 /Fi 5 107 df<127812FCA212FEA2127E1206A3120CA2121C121812301260124007107A 8513>59 D78 D<90B512FCEDFF809039 07C007C0ED01F090380F800016F8A349C7FCA3ED01F0013E14E0ED03C0ED0F80ED7E0090 387FFFF85D90387C00FC153E5B81A34848133EA448485B1606A20007EC3C0CD8FFFEEB3E 18ED1FF0C8EA07E027237CA12E>82 D<1338137CA2137813701300A8EA0780EA1FC0EA31 E0126012E112C1A2EA03C0A3EA0780A2EA0F0013041306EA1E0CA213181330EA0FE0EA07 C00F227DA116>105 D<1418143C147CA214381400A8EB0F80EB1FE0EB70F0136013C0EA 0180A2380001E0A4EB03C0A4EB0780A4EB0F00A4131EA21238EA783CEAF8785BEA7FE0EA 3F80162C81A119>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmex10 10.95 25 /Fj 25 126 df<140E141E143C147814F01301EB03E0EB07C0A2EB0F80EB1F00A2133E13 7E137C13FC5B1201A2485AA3485AA2120F5BA2121FA25BA2123FA390C7FCA25AA6127E12 FEB3A4127E127FA67EA27FA3121FA27FA2120FA27F1207A26C7EA36C7EA212007F137C13 7E133E7FA2EB0F80EB07C0A2EB03E0EB01F013001478143C141E140E176C72832A>0 D<12E07E12787E7E121F6C7E6C7EA26C7E6C7EA26C7E7F137C137E133E133FA2EB1F80A3 EB0FC0A214E01307A214F0A21303A214F8A31301A214FCA6130014FEB3A414FC1301A614 F8A21303A314F0A21307A214E0A2130F14C0A2EB1F80A3EB3F00A2133E137E137C13FC5B 485AA2485A485AA2485A48C7FC121E5A5A5A5A176C7C832A>I10 D<126012F07EA21278127CA27EA27EA27E7FA26C7EA26C7E A212017FA26C7EA2137CA2133C133EA27FA27F80A26D7EA26D7EA2130180A26D7EA2147C A2143C143EA280A2EC0F80A2140715C0A41580140FA2EC1F00A2143EA2143C147CA25CA2 495AA25C1303A2495AA2495AA291C7FC5BA2133EA2133C137CA25BA2485AA25B1203A248 5AA2485AA290C8FC5AA2123EA25AA2127812F8A25A12601A6C79832B>I<12F0B3B3B3A5 043B73811E>I<00F0130FB3B3B3A5183B738133>I[<173E177E17FCEE01F8160317F0EE 07E0EE0FC0EE1F80163F1700167E16FE4B5A5E15034B5A5E150F4B5AA24B5A4BC7FCA215 FEA24A5AA24A5AA24A5A140F5D141F5DA2143F5D147F92C8FC5CA2495AA25C1303A2495A A3495AA3495AA3133F5CA3495AA313FF91C9FCA35A5BA31203A25BA31207A25BA3120FA3 5BA3121FA55BA2123FA75B127FAD5B12FFB3B3A4127F7FAD123F7FA7121FA27FA5120FA3 7FA31207A37FA21203A37FA21201A37F7EA380137FA36D7EA380131FA36D7EA36D7EA36D 7EA2130180A26D7EA28081143F81141FA281140F8114076E7EA26E7EA26E7EA2157FA26F 7E6F7EA26F7E1507826F7E1501826F7E167E821780161FEE0FC0EE07E0EE03F017F81601 EE00FC177E173E>47 272 107 131 72 32 D[<12F87E127E7E7F121F6C7E6C7E6C7E7F 12016C7E7F137F7F806D7E130F806D7EA26D7E6D7EA26D7EA2147FA26E7EA26E7E81140F 811407A281140381140181A26E7EA28182A26F7EA36F7EA36F7EA3821507A36F7EA38215 01A38281A31780A2167FA317C0A2163FA317E0A3161FA317F0A5160FA217F8A7160717FC AD160317FEB3B3A417FC1607AD17F8160FA717F0A2161FA517E0A3163FA317C0A3167FA2 1780A316FFA21700A35D5EA315035EA34B5AA3150F5EA34B5AA34B5AA34B5AA293C7FC5D A24A5AA25D14035D14075DA2140F5D141F5D4A5AA24AC8FCA214FEA2495AA2495A495AA2 495A5C131F495A91C9FC5B13FE5B485A12035B485A485A485A123F90CAFC127E5A5A>47 272 125 131 72 I[<177CEE01FC1607160F163FEE7FF0EEFFE04B1380030713004B5A4B 5A5E4B5A4B5A4B5A5E5C4A90C7FC5D14075D140F5DA2141F5DA3143F5DB3B3B3B3A6147F 5DA44A5AA34990C8FCA2495AA2495AA2495AA2495A495A5C137F495A4890C9FC485A485A EA0FF0EA3FE0485A48CAFC5AA27EEA7FC06C7EEA0FF0EA07FC6C7E6C7E6C7F6D7E133F80 6D7E6D7EA26D7EA26D7EA26D7EA26D7FA36E7EA481143FB3B3B3B3A681141FA381140FA2 811407811403816E7F80826F7E6F7E6F7E826F7E6F7E030113806F13E0EE7FF0EE3FFC16 0F16071601EE007C>46 272 115 131 73 40 D[<12F812FE6C7E7F7FEA3FF86C7EEA07 FE6C7E6C7F6C7F6D7E6D7E131F806D7E1307801303807F817FA281147FA381143FB3B3B3 B3A681141FA381140FA2811407A26E7EA26E7EA26E7F6F7EA26F7E6F7E6F7E6F7E6F7E6F 7E6F1380EE7FC0EE3FF0EE0FF8EE07FC1601A21607EE0FF8EE3FF0EE7FC0EEFF804B1300 4B5A4B5A4B5A4B5A4B5A4B5AA24B5A4A90C7FCA24A5AA24A5AA2140F5DA2141F5DA3143F 5DB3B3B3B3A6147F5DA314FF5DA25B92C8FC5B5C13075C130F495A5C133F495A495A485B 4890C9FC485AEA1FFC485AEAFFE05B5B48CAFC12F8>46 272 115 131 73 I<1638167CA216FC16F8A2150116F0150316E0150716C0A2150F1680151F1600 A25D153E157E157C15FC5DA214015D14035DA214075D140F5D141F92C7FCA25C143E147E 147C14FC5CA213015C13035CA213075C130F5C131F91C8FCA25B133E137E137CA213FC5B 12015B12035BA212075B120F5BA2121F90C9FC5A123E127E127CA212FC5A7E127CA2127E 123E123F7E7F120FA27F12077F1203A27F12017F12007F137CA2137E133E133F7FA28013 0F801307801303A2801301801300A280147C147E143E143F80A281140F811407811403A2 811401811400A281157C157E153E153F81A21680150F16C01507A216E0150316F0150116 F81500A216FC167CA2163826A3768338>68 D<127012F8A27E127CA2127E123E123F7E7F 120FA27F12077F1203A27F12017F12007F137CA2137E133E133F7FA280130F8013078013 03A280130180130080147CA2147E143E143F80A281140F811407811403A2811401811400 A281157C157E153E153F81A21680150F16C01507A216E0150316F0150116F81500A216FC 167C16FC16F8A2150116F0150316E0150716C0A2150F1680151F1600A25D153E157E157C 15FC5DA214015D14035DA214075D140F5D141F92C7FCA25C143E147E147CA214FC5C1301 5C13035CA213075C130F5C131F91C8FCA25B133E137E137CA213FC5B12015B12035BA212 075B120F5BA2121F90C9FC5A123E127E127CA212FC5AA2127026A3798338>I80 D82 D88 D90 D104 DI<160F167FED01FF1507ED1FFCED7FE0EDFF804A1300EC03FC4A5A4A5A4A5A4A 5A5D147F92C7FCA25C5CB3B3AA13015CA213035C13075C495A131F495A495A49C8FCEA03 FC485AEA1FE0EA7FC048C9FC12FCB4FCEA7FC0EA1FE0EA07F86C7EC6B4FC6D7E6D7E6D7E 130F6D7E801303801301A2801300B3B3AA8080A281143F816E7E6E7E6E7E6E7E6EB4FC6E 1380ED7FE0ED1FFCED07FF1501ED007F160F28A376833D>110 D<12F012FE6C7E13E0EA 3FF8EA0FFCEA03FEC66C7E6D7E6D7E131F6D7E6D7E130380130180A21300B3B3AA8080A2 81143F816E7E140F6E7E81EC01FC6EB4FCED7F80ED1FE0ED0FF8ED03FEED00FF163F16FF ED03FEED0FF8ED1FE0ED7F80EDFF00EC01FCEC07F85D4A5A141F4A5A5D147F92C7FCA25C 5CB3B3AA1301A25C13035C1307495A495A133F495A495AD803FEC8FCEA0FFCEA3FF8EAFF E0138048C9FC12F028A376833D>I<1CC0F301E01B03A2F307C0A2F30F80A2F31F00A21B 3EA263A263A2505AA2631A03A2505AA2505AA250C7FCA21A3EA262A262A24F5AA24F5AA2 4F5AA24F5AA24FC8FCA2193EA261A261A24E5AA24E5AA2611807131801384C5A137CD801 FC4CC9FC487E0007173EEA1EFF00385F5A486C6C5D12006D6C4A5AA26D6C4A5AA24D5A6D 7E4D5A6D7E4DCAFC6D7E173EA26D6C5CA26D6C5CA26E6C485AA24C5AEC3FC04C5AEC1FE0 5F91380FF00FA24CCBFCEC07F8163EEC03FC5EEC01FE5EA26E6C5AA26F5AA26F5AA25E15 1F93CCFC150E536D76835B>I122 DI<12F87E7E7EA26C7E6C7E 7F6C7E6C7E13FE6C7E6C13C06C7F6C13F86DB4FC6D13E06D13FC6DEBFFF00103ECFFF87F 7F143F140F14031400153F1507150016072D1E839D29>I<17F8160116031607A2EE0FF0 EE1FE0163FEE7FC0EEFF8015034B1300ED1FFE4B5AEDFFF802075B023F5B49B55A017F5C B648C7FC5D5D15E015804AC8FC14F814E091C9FC13F890CAFC2D1E819D29>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk msbm10 10.95 2 /Fk 2 83 df67 D<007FB612FCB812C06C16F83B03E007C07FFE0000903A0F001F7F80020E9038078FC093 380383E0EFC0F0040113788484EFE00E1600180F84A760180E0401131EEFC01C183C0403 5BEF81F093380787E093381F7FC04BB5C7FC020FB512FC17C004F7C8FC91390E1C078092 381E03C0ED0E01030F7FED078003037FEEC078923801E0380300133C707EEE780EEE380F 93383C0780EE1E03040E7F93380F01E093380780F004031370EFC078706C7E04007F717E 943878078094383803C00003D90F8090383C01E0007FB500FE90381FFFFCB6806C823E3E 7EBD39>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr8 8 22 /Fl 22 127 df0 D<156015F0A24A7E4A7EA24A7E1406EC0E7F14 0C4A6C7E151F02307F150F02607F9138E007F014C001016D7E148001036D7E140001066D 7E167E49147F82498101386E7E133001706E7E136001E06E7E5B00016F7E5B48C86C7E16 00000682177E48167F001CEE3F801218003FB812C0A24817E0A2B912F0342E7DAD3B>I< 1406140FA34A7EA34A7EA3EC6FE01467ECE7F014C3A201017F1481A290380301FC1400A2 01067F157EA249137F81011C800118131FA20138800130130FA249801507A249801503A2 4848801501000381150000071401D81FE0497ED8FFF890383FFFF0A22C2E7EAD31>3 D<14FF010713E090381F80F090383E00380178137C4913FC1201485A157892C7FCA8B612 FCA23803E000157CB3A5486C13FE3A7FFF0FFFE0A2232E7FAD27>12 D<13031307130C131C1338137013E0A2EA01C0EA0380A2EA0700A25A120E121EA2121C12 3CA35AA512F8A25AAB7EA21278A57EA3121C121EA2120E120F7EA2EA0380A2EA01C0EA00 E0A213701338131C130C1307130310437AB11B>40 D<7E7E126012707E7E7EA27EEA0380 A2EA01C0A213E0120013F0A213701378A3133CA5133EA2131EAB133EA2133CA51378A313 7013F0A213E0120113C0A2EA0380A2EA0700120EA25A5A5A12605A5A0F437BB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>I II61 D99 D<15F8141FA214011400ADEB0FE0EB7FF83801F81C3803E00738 078003380F0001481300123EA2127E127C12FCA7127CA2127E123E6C13017E3807800338 03C00E3901F03CFC3A007FF0FFC0EB1FC0222E7EAD27>II108 D<3807C0FFD8FFC313C09038C703E0390FCC01F0EA07D89038F000F85BA25BB2486C487E 3AFFFE1FFFC0A2221D7E9C27>110 DI<3807 C0FE39FFC7FF809038CF03E0390FD800F0D807F0137849137C497F811680A2150F16C0A7 1680151F16005D153E6D5B6D5B9038D801F09038CE07E09038C7FF80D9C1FCC7FC01C0C8 FCAA487EEAFFFEA2222A7E9C27>I<3801FE18380FFFB8381E01F8383800784813381418 12F0A27E6C1300EA7F8013FC383FFF806C13E0000713F0C613F8EB07FC13000040133E12 C0141E7EA26C131C6C133C6C133838FF01F038E3FFC000C01300171D7E9C1C>115 D117 D<38078008380FE01C381FF838383FFFF03870 7FE038E01FC03840078016077AAC23>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi8 8 30 /Fm 30 123 df<157E913803FF8091380F81C091381C00E0023013F04A13F85C495A49C7 FCA213065BED01F05BED03E016C049EB078091381FEF00EC3FFEA29038601FEF91380007 8016C0A25BED03E01507A2485AA448C7EA0FC0A3ED1F80481500A26D133E5DD80CC01378 5D90387003E090383C0F8026180FFFC7FCEB03F890C9FCA25AA45AA45AA3253B7EAD28> 12 D<13F813FE133F6D7E130F80130780A21303801301A280130080A2147C147E143EA2 143F8081A34A7E14E7903801C7E01303EB078390380F03F0131EEB3C0101787FEBF00048 487F485A0007147C4848137E48C7123E127E48143F48804815800060140F212E7CAD29> 21 D<011CEB0380013EEB07C0A249EB0F80A449EB1F00A44848133EA448485BA3160848 48EBF818A31401260FE00313301407020E13609039F83878C03A1F7FF03F8090391FC00F 0090C9FCA2123EA45AA45AA31270252B7E9C2A>I34 D<0002EC0F800006EC3FE0EDFFF0484913F848903803E078913807801C48EB06004A130C 48131C14185C5A5CA2144002C01318A24A13300101147000E015E00060EC01C0D87003EB 0380003C9038000F00001F143E390FF703FC6CB512F000015C6C6C1380D90FF8C7FC010E C8FC131EA35BA3137CA2137813F8A31370262B7C9C2E>39 D<1238127C12FEA212FF127F 123B1203A41206A3120CA2121812301260124008147A8614>59 D<17C01603EE0F80EE3F 0016FCED03F0ED0FC0033FC7FC15FCEC03F0EC0FC0023FC8FC14FCEB03F0EB0FC0013FC9 FC13FCEA03F0EA0FC0003FCAFC12FC12F012FC123FEA0FC0EA03F0EA00FC133FEB0FC0EB 03F0EB00FC143FEC0FC0EC03F0EC00FC153FED0FC0ED03F0ED00FC163FEE0FC016031601 2A2B7AA537>I<15C01401A2EC0380A3EC0700A2140EA35CA35CA35CA35CA2495AA3495A A349C7FCA3130EA25BA35BA35BA35BA3485AA2485AA348C8FCA3120EA35AA35AA25AA35A A25A1A437CB123>I<12C012F012FC123FEA0FC0EA03F0EA00FC133FEB0FC0EB03F0EB00 FC143FEC0FC0EC03F0EC00FC153FED0FC0ED03F0ED00FC163FEE0FC01603160FEE3F0016 FCED03F0ED0FC0033FC7FC15FCEC03F0EC0FC0023FC8FC14FCEB03F0EB0FC0013FC9FC13 FCEA03F0EA0FC0003FCAFC12FC12F012C02A2B7AA537>I<011FB512FEEEFFC0903A00FE 0007F04AEB01F8EE007E838349481580170F18C01707494815E0A4495AA4495A170FA349 4815C0171FA2188049C8123F1800A2177E137E5F5F1601494A5A4C5A4C5A4CC7FC484814 3E16FCED03F00003EC1FC0B7C8FC15F8332D7CAC3A>68 D78 D<011FB6FC17E0903A00FE0007F04AEB01FCEE007EA283495AA21880180049485CA3 17FE495A5F4C5A4C5A4948495AEE1F8004FFC7FC91B512F84914C00280C9FCA349CAFCA4 137EA45BA4485AA31203B512C0A2312D7DAC2D>80 D<011FB512F816FF903A00FE001FC0 4AEB07E0707E707E707E495A83A34948495AA34C5A49485C4C5A4C5A4C5A4948017EC7FC ED03F891B512E093C8FC90391F8007C06F7E6F7E8249C7FCA282A2017E495AA4491303A3 EF01804848ED0300A21706000302015BB539C000FC38EE7FF0C9EA0FC0312E7CAC35>82 D<90261FFFF0EBFFFEA201000180EB1FE06EC71300171E6E6C13185F021F5C6F5B020F49 5A6F48C7FC020713066F5A5E6E6C5A5E6E6C5AEDFD806EB4C8FC5D157FA26F7E157F15DF 9138019FC0EC030F02067FEC0E07021C7F14384A6C7E14E049486C7EEB038049C77E130E 49147F5B496E7E5B1203000F4B7ED8FFFC903803FFFEA2372D7DAC3A>88 D<1303A51230A714C01307133F13FF1237123F481300EAFFFB13C3130312F812F01230B1 14C01307133F13FF1237123F481300EAFFFB13C3130312F812F01230A390C7FCA5123D7C AE1B>93 D97 D<13F8121FA212015BA4485A A4485AA4485AEB87E0EB9FF8EBB83C381F601E13C0EB801F497E003E1480A3141F5AA448 EB3F00A3147EA2147C5C00785B1301383803E0383C0780D81E1FC7FCEA0FFEEA03F0192E 7DAD1E>I<151FEC03FFA2EC003F153EA4157CA415F8A4EC01F0EB07E1EB1FF1EB7C1990 38F00FE03801E007EA03C01207D80F8013C0121F13005A003EEB0F80127EA348EB1F00A3 1502EC3E06A3007C137E150C383C01FE391C03BE18390E0E1E303907FC0FE03901F003C0 202E7DAD24>100 D<15FC4AB4FC9138038780EC070FEC0F1F141F021E1300EC3E0E92C7 FCA35CA690383FFFF8A2D900F8C7FCA3495AA5495AA6495AA5495AA591C8FC5BA3131EA2 133E133C1238EA7C38EAFC781370485AEA70C0EA3F806CC9FC213B7CAD22>102 D<14FC903803FE1890380F833C90381E01FCEB3C005B13F8484813F812035B12079038C0 01F0120FA3391F8003E0A4EC07C0A3000F130F15800007133F0003137F3801C1CF3900FF 9F00EB3E1F1300A2143EA400385B007C5B00FC5BEB03E038F80FC0D87FFFC7FCEA1FF81E 2A7E9C22>I<1303EB0F8014C0EB1F80130FEB060090C7FCAAEA03E0EA07F8EA0C3C1218 EA303E1260A25B12C0A2C65AA2485AA3485AA2485A144014C0EA0F80A2EB8180EA1F0138 0F030013066C5AEA03F86C5A122D7EAC18>105 D<15E0EC01F01403A2EC01E0EC00C015 00AA14FCEB03FEEB070F90380C07801318013013C01360EC0F8013C0A21300EC1F00A414 3EA45CA45CA4495AA4495AA3495A1238387C0F8000FC90C7FC133E485AEA7FF0EA3FC01C 3A81AC1D>I<133EEA07FEA2EA007E137CA45BA4485AA4485AEC01E0EC07F0EC1E183907 C0383CEC607CECC0FCEBC180390F8300F801861370018C130013B8EA1FF0A213FFEB3F80 383E07C06D7E6D7E1508007C1418A3153012F81560130015C048EB7F800060EB1F001E2E 7CAD25>I<137CEA0FFCA2120013F8A4EA01F0A4EA03E0A4EA07C0A4EA0F80A4EA1F00A4 123EA45AA31308EAF818A31330A212F0EAF860EA78C0EA3F80EA0F000E2E7DAD15>I<39 07C007E0390FE03FF83918F0783C393078C01E903879801F38607F00137E485AA25B1200 4848133EA35D485AA25D160848481418EC01F0A21630390F8003E00201136016C0913800 E18090C7EA7F000006143E251D7E9C2B>110 D<903807E03090381FF07090387C18F090 38F00DE03801E007EA03C01207D80F8013C0121F13005A003EEB0F80127EA348EB1F00A4 143EA3007C137E147C383C01FCEA1C03380E0E7C3807FCF8EA01F0C7FCA2495AA4495AA4 1307EBFFFEA21C2A7D9C20>113 D115 D<013F137C9038FFC1FF3A01C1E383803A0300F603C00006140748EBFC0F5A02F8138048 EC070092C7FCC7FC495AA4495AA35D49485A1238127C007E1406D8FC0F5BA2D878191338 397070E0F0393FE07FC0260F801FC7FC221D7E9C28>120 DI<013C136013FF48EB80C048EBC18014FF48C613000006130600045BC75A5C5C495A 495A0106C7FC5B5B5B5B4848138038038001EA060048130348EB0700381FE01E383FFFFE 38707FFC38C03FF8EB1FE0388007801B1D7C9C21>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmsy11 10.95 29 /Fn 29 115 df<007FB812FCB912FEA26C17FC3704799846>0 D<121C127FEAFF80A5EA 7F00121C0909789B19>I<006016C000F0ED01E06C15036C1507007EED0FC06CED1F806C 6CEC3F006C6C147E6C6C5C6C6C495A6C6C495A6C6C495A017E495A6D495A6D6C48C7FC90 380FC07E6D6C5A903803F1F8903801FBF06DB45A6E5A6E5AA24A7E4A7E903801FBF09038 03F1F8903807E0FC90380FC07E49487E49486C7E017E6D7E496D7E48486D7E48486D7E48 486D7E4848147E48488048C8EA1F80007EED0FC048ED07E04815034815010060ED00C02B 2C73AC46>I<150E150FB3A9007FB912C0BA12E0A26C18C0C8000FC9FCB3A6007FB912C0 BA12E0A26C18C03B3C7BBB46>6 D8 D10 D<007FB912C0BA12E0A26C18C0CDFCAE007FB912C0BA12E0A26C18C0CDFCAE007FB912C0 BA12E0A26C18C03B287BAA46>17 D<181C187EEF01FEEF07FCEF1FF0EF7FC0933801FF00 EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC049 48C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007F C0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007FC0EF1FF0EF07FCEF 01FEEF007E181C1800AE007FB812FCB912FEA26C17FC374879B846>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007F C0EC1FF0EC07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE 01FF9338007FC0EF1FF0EF07FCEF01FEA2EF07FCEF1FF0EF7FC0933801FF00EE07FCEE1F F0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07 FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB812FCB9 12FEA26C17FC374879B846>I<0203B612FC023F15FE91B7FC010316FCD90FFEC9FCEB1F E0EB7F8001FECAFCEA01F8485A485A485A5B48CBFCA2123EA25AA21278A212F8A25AA87E A21278A2127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FE0EB0FFE0103B712 FC010016FE143F020315FC373679B146>26 D<007FB67EB712F816FE6C6F7EC913E0EE0F F0EE03FCEE00FE173FEF1F80EF0FC0EF07E01703EF01F0A2EF00F8A2187CA2183CA2183E A2181EA8183EA2183CA2187CA218F8A2EF01F0A2EF03E01707EF0FC0EF1F80EF3F0017FE EE03FCEE0FF0EEFFE0007FB71280B748C7FC16F86C1580373679B146>I<0660140CDD01 F0143E050715FE4D48EB01FCDD3FC0EB07F805FFC7EA1FE0DC03FCEC7F804C48ECFF00DC 1FE0EB03FCDC7F80EB0FF0DB01FEC7EA3FC04B484A5ADB0FF0D901FEC7FCDB3FC0EB07F8 4B48495ADA01FEC7EA3FC0DA07F802FFC8FCDA0FE0EB01FCDA3FC0EB07F802FFC7EA1FE0 D903FCEC7F80D907F002FEC9FCD91FE0EB03FCD97F80EB0FF0D801FEC7EA3FC048484A5A D80FF0D901FECAFCD83FC0EB07F848C7EA0FE000FE4A5AA2007F6E7ED83FC0EB07F8D80F F0EB01FED803FC9038007F806C6C6E7E26007F80EB0FF0D91FE0EB03FCD907F0EB00FED9 03FCEC7F80D900FFEC1FE0DA3FC0EB07F86E6C6D7EDA07F8EB00FFDA01FEEC3FC0DA007F EC0FE0DB3FC0EB07F8DB0FF0EB01FEDB03FC9038007F806F6C6E7E9226007F80EB0FF0DC 1FE0EB03FCDC07F8EB00FF706CEC7F80DC00FFEC1FE0DD3FC0EB07F8716CEB03FCDD07F0 EB00FE0501153EDD0060140C4F3C7BB45A>I<197019F0851978A3197C193C193E191E19 1F8586737EA2737E737E737E1A7C1A3FF21F80F20FE0007FBB12F8BC12FEA26C1AF8CDEA 0FE0F21F80F23F001A7C624F5A4F5A4F5AA24F5A97C7FC61191E193E193C197C1978A319 F86119704F307BAE5A>33 D49 D<0203B512F0023F14F891B6FC010315F0D90FFEC8FCEB1FE0EB7F8001FEC9FCEA01F848 5A485A5B485A121F90CAFC123EA25AA21278A212F8A25AA2B812F017F8A217F000F0CAFC A27EA21278A2127CA27EA27E7F120F6C7E7F6C7E6C7EEA00FEEB7F80EB1FE0EB0FFE0103 B612F0010015F8143F020314F02D3679B13C>I<1730177817F8A2EE01F0A2EE03E0A2EE 07C0A2EE0F80A2EE1F00A2163EA25EA25EA24B5A15035E4B5AA24B5AA24BC7FCA2153EA2 5DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2 495AA249C9FCA2133E137E137C5BA2485AA2485AA2485AA2485AA248CAFCA2123EA25AA2 5AA25A12602D5474C000>54 D<92B512FC020FECFFE0027F15FC0103B8FC010F8390271F 81FC0114E0D97801D9000F7F01E003017FD803C06F6C7E260780036F7E000F170FD81F00 707E4883007E4A6E1380127C488300E019C000800107167FC7FC5D193FA3140F5DA21A80 A24A5AA2F17F00A24A5A197E19FE92C95A4A5E18016102FE4B5A1807614A4B5A01014CC7 FC181E4A5D01035E604A4A5A0107ED03804A020FC8FC171C010F15784AEB01E0011FEC07 C04A013FC9FCED07FC49B512F049148048B500FCCAFC4814E04801FCCBFC423E7EBD45> 68 D<041FB612FC4BB8FC030F1780153F5D912901F00FF000011300DA038049EB007CDA 0F001678021E1740023E94C7FC4A495A14FC495AA24948133F02C05C495A90C8FC94CAFC 5EA316FEA35E1501A25E150393B612F0614B5D614EC8FCDB0FE0CAFC5E151F5EA2153F93 CBFC5D157EA25DA24A5AA214035D14075D485C000F130FD81F805B003F131F486C48CCFC 38FFE03E387FFC7CEBFFF86C13E06C5B6C90CDFCEA01F849417FBD41>70 D72 D92 D94 D<153FEC03FFEC0FE0EC3F80EC7E00495A5C495AA2495AB3AA130F5C131F495A91C7FC13 FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE133F806D7E130F801307B3AA6D7EA26D7E 80EB007EEC3F80EC0FE0EC03FFEC003F205B7AC32D>102 D<12FCEAFFC0EA07F0EA01FC EA007E6D7E131F6D7EA26D7EB3AA801303806D7E1300147FEC1FC0EC07FEEC00FFEC07FE EC1FC0EC7F0014FC1301495A5C13075CB3AA495AA2495A133F017EC7FC485AEA07F0EAFF C000FCC8FC205B7AC32D>I<146014F01301A214E01303A214C01307A2EB0F80A214005B A2131E133EA25BA2137813F8A25B1201A25B1203A2485AA25B120FA290C7FC5AA2123EA2 123C127CA2127812F8A41278127CA2123C123EA27EA27E7FA212077FA26C7EA212017FA2 12007FA21378137CA27FA2131E131FA27F1480A2EB07C0A2130314E0A2130114F0A21300 1460145A77C323>I<126012F07EA21278127CA2123C123EA27EA27E7FA212077FA26C7E A212017FA212007FA21378137CA27FA2131E131FA27F1480A2EB07C0A2130314E0A21301 14F0A414E01303A214C01307A2EB0F80A214005BA2131E133EA25BA2137813F8A25B1201 A25B1203A2485AA25B120FA290C7FC5AA2123EA2123C127CA2127812F8A25A1260145A7B C323>I<126012F0B3B3B3B3B11260045B76C319>I<0060131800F0133CB3B3B3B3B00060 1318165A75C32D>I<1A0C1A1E1A3EA21A7CA21AF8A2F101F0A2F103E0A2F107C0A2F10F 80A21A0061A2193EA261A261A24E5AA24E5AA24E5AA24E5AA24EC7FCA2183EA260A260A2 4D5AA24D5AA201705E01F815071201D807FC4B5A120F001D4CC8FCEA79FE00E0163E487E C66C5D80013F5D80011F4A5A80010F4A5AA26E495A13076E495A13036E49C9FC13016E13 3E13005E806E5B1580023F5B15C1141FEDE3E0140FEDF7C01407EDFF80A26E90CAFCA26E 5AA26E5AA215781570475B7A834B>112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi11 10.95 61 /Fo 61 123 df11 DIII<133F14C0EB07F06D7E6D7E801300A2147FA36E7EA36E7EA36E7EA36E7EA36E7E A36E7EA26E7EA3157FA38215FF5C913803DFC0EC079FEC0F1F91381E0FE0143C147C4A6C 7EEB01F0EB03E049486C7E130FEB1F8049486C7E137E5B48486D7E1203485A4848147F48 5A4848EC3F80127F48C8FC48ED1FC05A48ED0FE0007015072B407BBE34>21 DI<011FB612FE90B7FC5A5A4816FC3B0FC03801 8000391E0030035A0038017090C7FC5A48EB600714E05AC7FC903801C006150EA2130314 80A21307151EEB0F00A349131F131E133EA2137E017C8013FCA25B000181A20003140F5B A2D801C06DC7FC2F287DA633>25 DI<021FB512FC91B6FC1303130F4915F890 3A3FC07F800090387F001F01FC6D7E485A48486D7E485A120F5B48481303A248C71207A2 5A127EA2150F00FE5D5AA24B5AA3484AC7FC153E6C147E5D007C5C4A5A6C495A4A5A6C49 5A260F803FC8FC3807C0FC3801FFF038007F802E287CA633>I<011FB612C090B7FC5A5A 481680260FC007C8FC381E000648130E12385A5A5C5AC7FC143CA35CA314F8A25CA21301 A3495AA31307A25C130FA3131FA25C6DC9FC2A287DA627>I31 D<4C7EA2160394C7FCA35E 1606A3160E160CA3161C1618A316381630A31670017C0260137048B4ED01F8260387C0EC 03FCD8070314E0000E6D13C0120C001C1601D818070101130000384B137C0030173CEA70 0F0060EBC00393C7121CEAE01F0280151800005C013F130602001538183049130E017E13 0C187001FE166049011C14E0031814C017010001EE0380490138140003305B170E00005E 6D01705B03605B017C5D017E4A5A6D9038E0038090271F80C00FC7FCD90FC0133C903903 FCC1F00100B512C0DA0FFEC8FCEC0180A2140392C9FCA35C1406A3140E140CA3141C1418 A336527EBE3B>I34 D39 D<121C127FEAFF80A5EA7F00121C 0909788819>58 D<121C127FEAFF80A213C0A3127F121C1200A512011380A2120313005A 1206120E120C121C5A5A12600A1B788819>I<181C187EEF01FEEF07FCEF1FF0EF7FC093 3801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0 EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFCA2EA7FC0 EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF 9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007FC0EF1FF0 EF07FCEF01FEEF007E181C373679B146>II<127012FCB4FCEA7FC0EA1FF0EA07FC EA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF9138007FC0ED 1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007FC0EF1FF0EF07FCEF01FE A2EF07FCEF1FF0EF7FC0933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7F C04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1F F0EA7FC048CBFC12FC1270373679B146>I<4AB4FC020F13E0023F7F91387E01FC9138F0 007ED901C07F4948EB1F8049C7EA0FC00106EC07E0EB0F8002E0EB03F0131F17F81601A2 5C0107C713FC90C8FCA7EC07F8EC3FFF9138FC0783903903F001C3903907C000E3D91F80 136349C71277017EEC37F849143F0001151F485A484815F0120F5B121F17E0485AA2EE3F C0127F5BEE7F80A200FF160090C85A5EA24B5AA2484A5A5E007E1407007F5D4B5A6C4A5A 6D49C7FC001F147E6C6C5B3907E003F83903F80FE06CB512806C6C48C8FCEB0FF02E437C C030>64 D<1706170E170F5FA25F5FA25FA25E4C7EA204067F173F160C1618A21630A216 6016E016C0DB01807FA2923803001F5D15065DA25D153815305D844B130F14015D4AC7FC A214060207B6FC5C5C0218C7EA0FF05C17075C5CA2495AA249C8FC1306A24982011C1503 131813381378D801F81507D807FC4B7EB500C00103B512F85E19F03D417DC043>I<49B7 12F818FF19E090260001FEC7EA3FF0F007F84B6E7E727E850203815D1A80A214075DA302 0F17004B5C611803021F5E4B4A5A180FF01FE0023F4B5A4B4A5ADD01FEC7FCEF07F8027F EC7FE092B6C8FC18E092C7EA07F84AEC01FE4A6E7E727E727E13014A82181FA213034A82 A301075F4A153FA261010F167F4A5E18FF4D90C7FC011F5E4A14034D5A013FED1FF04D5A 4AECFFC0017F020790C8FC007FB712FCB812F094C9FC413E7DBD44>II<49B712F818FF19C090260001FEC7EA3FF0 F00FF84BEC03FCF000FE197F020317804B153FF11FC0A20207EE0FE05D1AF0A2020F1607 5DA21AF8141F5DA2190F143F5DA21AF0147F92C9121FA34A17E04A163FA21AC00101177F 5C1A8019FF010318005C4E5A61010716034A4B5A61180F010F4C5A4A5E4E5A4EC7FC011F 16FE4AEC03FC4D5A013FED0FE0EF3FC04A49B4C8FC017FEC0FFC007FB712F0B812C004FC C9FC453E7DBD4B>I<49B91280A390260001FEC71201F0003F4B151F190F1A000203825D A314075D1906A2140F4B1306A2050E130E021F020C130C4B92C7FC171CA2023F5C5D1778 EE03F84AB55AA3ED00034A6D5A4A1300A301015D5C19180401143801034B13304A167093 C81260A2010717E04A5E180161010F16034A4BC7FCA260011F161E4A153E60013F16FC4D 5A4A140F017F15FF007FB85AB9FC60413E7DBD43>I71 D<49B6013FB512E0A3D9000190 C8383FE0004B5E4B5EA2197F14034B93C7FCA26114074B5DA21801140F4B5DA21803141F 4B5DA21807143F4B5DA2180F4AB7FC61A292C8121F5C4A5EA2183F13014A5EA2187F1303 4A93C8FCA26013074A5DA21701130F4A5DA21703131F4A5DA2013F1507A24A5D496C4A7E 007FB5D8800FB512F0B65BA24B3E7DBD4B>I<49B6FCA3D9000113005D5DA314035DA314 075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C5CA313015CA313035CA313075CA3 130F5CA3131F5CA2133FA25C497EB612C0A25D283E7DBD28>I<49B612C0A3D9000190C8 FC5D5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F92C9FCA35C5CA313015C 1808181C010316185C1838183013074A1570186018E0130F4AEC01C0A21703011FED0780 4A140F171F013FED3F0017FF4A1303017FEC1FFE007FB7FCB8FC5F363E7DBD3D>76 D<902601FFFE4AB512E081A290C76D9039000FF8004AEE03E0DBBFC05D735A039F150302 037F030F93C7FC8261EC070702066D1406150370140E140E91260C01FE140CA20300151C 021C7F02186D14188319380238133F02306E1330161F711370147002606D6C1360A20407 14E002E0804A01035C831801010114014A02FE5B1600EFFF03130391C8D87F83C8FCA2EF 3F874916C70106ED1FC618E618EE010E150F010C16FC1707A2131C01186F5AA201381501 137801FC6F5AEA03FE387FFFF0B5167018604B3E7DBD48>78 DI<49B712F018FEF0FF8090260001FEC713C0F01FF04BEC 07F81803020316FCF001FE5DA2140719FF5DA2140FF003FE5DA2021F16FC18074B15F818 0F023F16F0F01FE04BEC3FC01980027FEDFF00EF01FC92C7EA07F8EF3FE091B7128005FC C7FC4ACAFCA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C49 7E007FB57EB6FCA2403E7DBD3A>II<49B7 7E18F818FE903C0001FE0001FF809438003FC04BEC1FE0F00FF0F007F8020315034B15FC A219FE14075DA3020FED07FC5DA2F00FF8141F4B15F0F01FE0F03FC0023FED7F804BECFF 00EF01FC4D5A027FEC1FE092C7B4C7FC92B512FC17E09139FF0001F84AEB007C83830101 824A6E7EA30103825CA2171F13075CA2173F130F5CA3011F157F4A1603A2013F17071906 4A023F130E496C160C007FB56C011F131CB691380FF038050713F0CA3801FFC09438007F 0040407DBD44>II<48B912F8A33C03FE0003FC00 0701F0160148484A130001801778197048C71207000E5DA25A150F00185D003818601230 151F00705D126019E000E0023F15C0C8491400A3157F93C9FCA35D5DA314015DA314035D A314075DA3140F5DA3141F5DA3143F5DA2147FA214FF5B000FB612F0A33D3E7FBD35>I< B500FC913807FFFEA219FC000390C9EA7FC06C48EE3E0049163C18386D16301870000017 60604D5AA24DC7FC5F6D15065F7F5F5FA25F17E06E5C4C5A133F4CC8FC1606A25E161C6E 13185E131F5E5EA24B5A150302E090C9FC1506130F5D5DA25D1570ECF0605D1307ECF180 02F3CAFCA214F614FE5C5C13035C5CA25CA25C3F407BBD34>86 D<91267FFFFE0103B512 8091B55CA2020101E09039007FF0006E49EC3F806F48023EC7FC19386F6C5C1960031F5D 70495A4EC8FC6F6C13066003075C705B18706F6C5B4D5A6F6C485A4DC9FC0300130EEEFF 0C5FEE7FB017E0705AA3707EA34C7E167FEEE7F816C792380183FCED0303ED0601030C7F 15184B6C7E5D03E07F4A48804A5A4AC76C7E140E4A6E7E14184A140F4A815C49486E7E13 03010FC812034982017F15072603FF804AB4FC007F01F049B512FCB5FCA2493E7EBD4B> 88 D<143F903801FFC0903903E0E0E090390F8073F090381F003B017E131F4914E04848 130FA2485A000715C0485AA24848131F1680485AA2153F007F150090C7FCA25D48147E5A A215FEEDFC03A25A02011307EDF806127C1403007E0107130E003E010F130C021D131C6C 013813186C017013383A07C1E07C703A03FF803FE03A007E000F8028297CA72F>97 DI< EE0FE0ED07FF17C0A2ED001FA21780A2163FA21700A25EA2167EA216FEA25EA21501A25E 143F903801FFC3903803E0E390390F8073F090381F003B017E131F5B48486D5AA2485A12 0748485CA24848131FA248485CA2153F127F90C790C7FCA25D5A48147EA215FE160315FC 5A02015B1606007C14F81403007E0107130E003E010F130C021D131C6C013813186C0170 13383A07C1E07C703A03FF803FE03A007E000F802B407CBE2F>100 DI<163FEEFFC0923801E0E0923803C07092380781F0ED0F 87ED1F8F160F153FA217E092387E038093C7FCA45DA514015DA30103B512FCA390260003 F0C7FCA314075DA4140F5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413015CA35C13 03A35C121E387F07C0A212FF5C49C9FC12FEEAF81EEA603CEA7878EA1FF0EA07C02C537C BF2C>III< 143C14FEA21301A314FCEB00701400AD137E3801FF80380387C0EA0703000E13E0120C12 1CEA180712381230EA700F006013C0A2EAE01F14801200133F14005B137EA213FE5BA212 015B0003130613F0A20007130EEBE00CA2141CEBC01814381470146014E03803E3C03801 FF00EA007C173E7EBC1F>IIIIIIII<023F1318903901FFC038903903 E0E07890390F8070F090381F0039017E131B49131F4848EB0FE0A2485A1207484814C0A2 485A151F48481480A3007F143F90C71300A3485C48147EA315FE5D5AA21401007C5C1403 007E1307003E130F4A5A6C133B6C13733807C1E73903FF87E038007E071300140F5DA314 1F5DA3143F92C7FCA25C5C013F13FC5BA2253A7CA728>I115 D<14E0EB01F81303A25CA21307A25CA2130FA25CA2131FA2 5CA2007FB512E0B6FC15C0D8003FC7FC5BA2137EA213FEA25BA21201A25BA21203A25BA2 1207A25BA2120FEC018013C01403001F14005CEB8006140E140C141C5C000F5B5C3807C3 C06CB4C7FCEA00FC1B3A7FB820>I<017E143848B414FC3A0387C001FEEA0703000E13E0 120C001C1400D81807147E0038153E0030151EEA700F00605B160EEAE01F4A130C120013 3F91C7121C16185B137E163801FE14305B16701660000115E04914C01501168015031600 5D0000140E150C6D131C017C5B6D13F090381F03C0903807FF80D901FCC7FC27297EA72C >118 D<017CEE038048B4020EEB0FC0260387C0013FEB1FE0EA0703000E7F000C5D001C 037E130FD81807160700381703003003FE1301D8700F5C00605B1800D8E01F1301028049 14C01200133FDA000314014C14805B137E0307140301FE4A14005B6018060001140F494A 130E180C181C1818183818300000021F14706D5E4B6C5B017E01731301013E9039E1F007 8090281F81C0FC0EC7FC903A07FF803FFC903A00FE000FF03B297EA740>II<137E48B41407260387C0EB1F80EA0703000E7F000C153F001C1600EA180712 3800305DD8700F147E00605BA2D8E01F14FE4A5B1200133FEC00015E5B137E150313FE49 5CA2150700015D5BA2150F5EA3151F00004A5A157F6D13FFEB7E0190263F07BFC7FC9038 0FFE3FEB03F890C75A157EA215FE00075C381FC001003F5C4A5AA24848485A4A5AD83E00 5B003049C8FC0038133E6C13F8380F03F03807FFC0C648C9FC293B7EA72C>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmr9 9 29 /Fp 29 122 df45 D<123C127E12FFA4127E123C08087A8715> I48 D<13075B5B137FEA07FFEAFFBFEAF83F1200B3B3A3EBFFC0007FEBFF80A219317AB026> I<14FE903807FF8090381F81C090383C00604913709038F001F83801E003EA03C01207EA 0F80EC01F0001F90C7FC90C8FC5AA25A127EA2EB07F038FE1FFCEB381F90386007809038 C003C0D8FF8013E0EC01F0010013F8A2EC00FCA25A15FEA3127EA4123E003F14FCA27EEC 01F87E018013F00007EB03E03903C007C0D801E013803900F81F00EB3FFCEB0FF01F327D B026>54 D68 D70 D77 DI<90381FE003EBFFFC3901F01F0739078003CF48C7B4FC001E143F 121C003C804880A28112F8A281A27EA26C91C7FC127F13C0EA3FF013FF6C13F06C13FF6C 14C06C14F0C680013F7F01037FEB003FEC03FF02001380153F151F16C0150F124000C014 07A47E1680A26CEC0F00A26C141E6C141CB45CD8F3C013F039E0FC03E039C03FFF80D903 FEC7FC22357CB32B>83 D85 D87 D<3803FFC0000F13F848C67E383F803F6E7E6E7EEA1F00000E6D7EC7FCA4EB01FF13 3FEBFF873803F807EA0FE0EA1F80EA3F00127EA24815C0A4140F127C007E131B6C903833 F180391FC0E1FF2607FFC01300C6EB007C22207D9F26>97 DII<153FEC0FFFA2EC00 7F81B0EB07F0EB3FFEEBFC0F3901E001BF3907C000FF48487F90C77E5A123EA2127E127C 12FCA8127C127EA2123E7E5D6C6C5B00075B3903E003BF3A00F81E3F80D97FFC13FCEB0F E026347DB32B>II<151F90391FC0 7F8090397FF1E3C03901F07F833903C01E033A07800F018092C7FC48486C7EA24880A56C 5CA26C6C48C7FCA23803C01EEBF07C38067FF0380E1FC090C9FCA5EA0F80EBFFFE6CEBFF C06C14F06C80000780391F0001FE003EEB003F4880168048140FA400781500007C5C6C14 3E6C5C6C6C5B3903F007E0C6B51280D91FFCC7FC22317EA026>103 D105 D107 DI<3903F01FC000FFEBFFF09038F1E0F83907F300 7CD803F6137E01FC133E153F5BA25BB3A2486CEB7F80B538C7FFFCA226207E9F2B>110 DI<3903F03F8039FFF1FFF09038 F3C0F83907F6003ED803FC7F49EB0F8049EB07C0A2ED03E016F0A2150116F8A8ED03F0A2 16E0150716C0ED0F807F6DEB1F0001F6137C9038F381F89038F1FFE0D9F07FC7FC91C8FC AC487EB512C0A2252F7E9F2B>I<3803E07C00FF13FF9038E38F800007EB1FC0EA03E613 ECEC0F809038F8070091C7FCA25BB3487EB512E0A21A207F9F1E>114 DI<1330A51370A313F0A2 120112031207121FB512FEA23803F000B01403A80001130613F80000130CEB7E1CEB3FF8 EB07E0182E7FAD1E>II<3A7FFF807FF8A23A07 F8001FE00003EC0F80ED070000011406A26C6C5BA26D131C017E1318A26D5BA2EC807001 1F1360ECC0E0010F5BA2903807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A214 30A25CA214E05CA2495A1278D8FC03C8FC5B13065BEA7038EA3FF0EA0FC0252F7F9F29> 121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmsy6 6 2 /Fq 2 49 df<136013701360A20040132000E0137038F861F0387C63E0380F6F00EA03FC EA0060EA03FCEA0F6F387C63E038F861F038E060700040132000001300A2137013601415 7B9620>3 D48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr11 10.95 89 /Fr 89 128 df0 D<16C04B7EA24B7EA24B7EA24B7EA2ED 1BFEA2ED31FFA203607FA24B6C7EA24A486C7EA24A486C7EA202066D7EA24A6D7EA24A6D 7EA24A6D7EA202706D7F146002E06E7E5C01016F7E5C01036F7E91C8FC496F7E1306010E 6F7E130C011C6F7E131801386F7E133001706F7F136001E0707E5B0001717E5B0003717E 90CAFC48717E1206000E717E120C001C717E001FB9FC4884A2481980A2BB12C0A242417C C04B>I<1518153CA3157EA415FFA34A7FA34A6C7E153FA202067F151FA2020C7F150FA2 02187F1507A202307F1503A202607F1501A202C07F81A2494880167FA201038191C7123F A249811306161F010E81130C160F011C8113181607A249811603A24981160113E0830001 811880486C5CD81FFC4A13C0B56C017F13FFA338417DC03F>3 D 6 D<913803FFC0023F13FC9138FF00FFD903F8EB1FC0D90FE0EB07F0D93FC0EB03FC49C8 7E01FE157F000117804848ED3FC04848ED1FE0000F17F049150F001F17F8491507003F17 FCA34848ED03FEAA6C6CED07FCA3001F17F8A26D150F000F17F0A26C6CED1FE0A2000317 C06C6CED3F80A2000017006D5D017E157E013E157C013F15FC6D5D6D5D6E1301D8C007ED E003A2D86003EDC0066E130301011580A2010015006C170C00386D49131CD83FFFEDFFFC A36C17F8A338407CBF41>10 DII14 D<127812FCA27E7EEA7F80123FEA0FC0EA07E01203EA01F0EA00F8137C131C130E130710 1076BE2D>18 D22 D<001C130E007FEB3F8039FF807FC0A201C013 E0A3007F133F001CEB0E6000001300A5000114E04913C0A2000313010100138048130300 061400000E5B000C1306001C130E485B485B006013301B1B7DBE2D>34 D<121C127FEAFF80A213C0A3127F121C1200A512011380A2120313005A1206120E120C12 1C5A5A12600A1B78BE19>39 D<1430147014E0EB01C0EB0380EB07005B131E5BA25B5BA2 485AA2485AA212075B120FA290C7FC5AA2121E123EA3123C127CA6127812F8B21278127C A6123C123EA3121E121FA27E7FA212077F1203A26C7EA26C7EA213787FA27F7F7FEB0380 EB01C0EB00E014701430145A77C323>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7E A21378A27FA2133E131E131FA27F1480A2130714C0A3130314E0A6130114F0B214E01303 A614C01307A31480130FA214005BA2131E133E133CA25BA25BA2485A485AA2485A48C7FC 120E5A5A5A5A5A145A7BC323>I<1506150FB3A9007FB912C0BA12E0A26C18C0C8000FC9 FCB3A915063B3C7BB446>43 D<121C127FEAFF80A213C0A3127F121C1200A512011380A2 120313005A1206120E120C121C5A5A12600A1B788819>II<121C 127FEAFF80A5EA7F00121C0909788819>III< 14C013031307131F137FEA07FFB5FC139FEAF81F1200B3B3ACEB7FF0B612F8A31D3D78BC 2D>III<15 0EA2151E153EA2157E15FEA214011403A21406140E140C141814381430146014E014C0EB 0180130314001306130E130C5B133813305B13E05B485A120390C7FC1206120E120C5A12 3812305A12E0B8FCA3C8EAFE00AC4A7E49B6FCA3283E7EBD2D>I<00061403D807C0130F 01F813FE90B55AA215F05D5D92C7FC38063FF890C9FCADEB01FE90380FFF8090383E03E0 90387001F8496C7ED807C0137E497F90C713800006141FC813C0A216E0150FA316F0A412 3E127F487EA490C713E048141F12E0006015C012700030EC3F8012386CEC7F00001E14FE 6C495A3907C003F83903F00FE0C6B55A013F90C7FCEB07F8243F7CBC2D>II<12301238123E003FB612FC A316F85A16F016E00070C8FC0060EC01C0ED038016005D48140E150C151C5DC812301570 5D4A5A5D14034AC7FCA2140EA25CA2143C14381478A214F85C1301A31303A313075CA313 0FA5131FAA6D5A6D5A26407BBD2D>III<121C127FEAFF80A5EA7F00121CC7FCB3A3121C127FEAFF80A5EA7F00121C092778 A619>I<121C127FEAFF80A5EA7F00121CC7FCB3A3121C127F5A1380A4127F121D1201A5 12031300A35A1206120E120C121C121812385A1260093978A619>I<007FB912C0BA12E0 A26C18C0CDFCAE007FB912C0BA12E0A26C18C03B167BA146>61 D64 D<1507A34B7EA34B7EA24B7EA34B7E156FA2EDCFF815C7A291380187FC1583A291380301 FEA391380600FFA34A6D7EA2021C800218133FA20238800230131FA24A80160FA24A8016 07A249486D7E91B6FCA2498191C71201A201066E7EA2010E82010C157FA2011C82011815 3FA24982171FA201708201F0150F84EA03F8D80FFEED3FFEB500C00107B512F8A33D417D C044>IIIIIIIII<010FB512FCA3D900071300EC01FEB3B3A7123FEA7F80EAFFC0A44A5A1380D87F 005B006C130700705C6C495A6C495A000F495A2603C07EC7FC3800FFF8EB3FC026407DBD 2E>III< B56C93381FFFF06E5EA20001F1F80026006FE0EE6FE0A2D967F016CFA3D963F8ED018FA2 D961FCED030FA3D960FE1506A2027F150CA36E6C1418A36E6C1430A26E6C1460A36E6C14 C0A26E6CEB0180A36E6CEB0300A26E6C1306A3037F5BA26F6C5AA36F6C5AA26F6C5AA36F 6C5AA2923803F980A36FB4C7FCA26F5AA213F0486C147CD807FEEF3FF8B500F00138011F B512F0A34C3E7DBD53>IIIIIII<003FB9FC A3D9E000EBC00190C7397F80003F007EEF1F80007C170F0078170700701703A300601701 A548EF00C0A5C81600B3B24B7E4A7F0107B612F8A33A3E7DBD41>IIII<003FB712F0A349C7EA3FE013F001C0EC7FC090C8FC00 3EEDFF80003C4A1300A200384A5A00785D007014074B5AA24B5A00604A5AA24B5AA24B5A C74890C7FCA24A5A5D14074A5AA24A5A4A5AA24A5AA24A5A4990C8FCA2495A5C01071518 495AA2495A495AA2495A1738495A4890C8FCA2485A4915701207484815F0A24848140148 481403160F4848143FED01FFB8FCA32D3E7BBD37>90 DI93 D<1318133C137E13FF3801E7803803C3C0380781E0380F00F0001E137848133C48131E00 E0130700401302180D76BD2D>I97 DI<49B4FC010F13E090383F00F8017C131E4848130748 48131F0007EC3F804848137F5B121FA24848EB3F00151E007F91C7FCA290C9FC5AAB6C7E A3003F15C07F001F140116806C6C13036C6CEB0700000314066C6C131E6C6C133890383F 01F090380FFFC0D901FEC7FC222A7DA828>IIII<167E 903903F801FF903A1FFF07878090397E0FCE0F9038F803FC3903F001F8EE070048486C6C C7FC000F8049137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F803E03903FE 0FC026071FFFC8FCEB03F80006CAFC120EA3120FA26C7E7F90B512F06C14FE6C6E7E6C15 E0000315F83907C0001F001FC7EA03FC003EEC00FE007E157E007C153E00FC153F4881A5 6C5D007C153E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017FC7FC90383F FFFC010313C0293D7EA82D>III<1470EB01FCEB03FEA5EB01FCEB0070 1400AD14FE137FA313011300147EB3B3A6123C007E137CB413FCA214F8130100FE13F038 7803E0383E07C0380FFF00EA01FC175184BD1C>III<2701F803FCEB01FE00FF903B0FFF8007FFC0913B3C0F C01E07E0913BF003E07801F02607F9C0D9F0E07F3D03FB8001F9C000FC000101005C01FF D900FF147E4992C7FCA2495CA2495CB3A6486C496C14FFB528F07FFFF83F13FCA346287E A74B>I<3901F803FC00FF90380FFF8091383C0FC09138F003E02607F9C07F3A03FB8001 F80001130001FF6D7E5BA25BA25BB3A6486C497EB539F07FFFF8A32D287EA732>I<14FF 010713E090381F81F890387E007E01F8131F4848EB0F804848EB07C04848EB03E0000F15 F04848EB01F8A2003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA3001F15F86C 6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90380FFFF0 010090C7FC282A7EA82D>I<3901F807F800FFEB1FFF9138780FC09039F9E007F03A03FB 8003F86CB4486C7E496D7E49147F178049143F17C0161F17E0A3EE0FF0AAEE1FE0A4EE3F C0A217806DEC7F006D5C5E6D495A9039FB8003F89039F9C007E09039F8F81F80DA3FFEC7 FCEC07F091C9FCAD487EB512F0A32C3A7EA732>I<02FE1318903907FFC03890381F81E0 90387E00704848EB38784848131C4848EB0EF8000F140649130748481303123F1501485A A448C7FCAA6C7EA3123F6D1303121F6D1307120F6C6C130F6C6C131D6C6C13396C6C1371 90383F01E190380FFF81903801FE0190C7FCAD4B7E92B512F0A32C3A7DA730>I<3903F0 0FC000FFEB3FF0ECF0F89038F1C3FC00071383EA03F33801F70313F6EC01F89038FE0060 491300A45BB3A4487EB512FCA31E287EA723>I<90387FC0603901FFF8E03807C03D381E 000F481303481301A20070130012F01560A27EA27EB41400EA7FC013FE383FFFE06C13FC 000713FF6C1480C614C0010F13E09038007FF0140F00C01303EC01F814007E1578A27EA2 7E15F07EEC01E06C14C039F780078039F1E01F0038E0FFFC38C01FE01D2A7DA824>I<13 18A61338A41378A213F8A2120112031207001FB512C0B6FCA2D801F8C7FCB3A21560A96C 6C13E015C0A2EB7E0190383F038090381F8700EB07FEEB01F81B397EB723>IIII<3B7FFFE00FFF E0A3000390398007FE00C690380003F86D14E06D5C02805B6D6C48C7FC010F130E903807 E00C6E5A903803F83801015B6D6C5AEC7EC0EC7F80143F141F6E7E81141FEC3BF0EC71F8 EC61FCECC0FE903801807E01037FD907007F01066D7E49130F011C6D7E498001F86D7E48 6C80000F4A7EB590381FFFF8A32D277FA630>II<001FB61280A29038E0007F90C71300 001E14FE001C495A140300185C0038495A4A5A0030131F5D4A5A4AC7FCA2C712FE495A13 035C495A495A131F9138C00180EB3F80EB7F00A213FE485A000314035B48481400485A00 1F5C495B485A48C7123F4849B4FC90B6FCA221277EA628>II<01 F01308D803FC131C486C1338390FFF8070391E1FE1E0393807FFC0D870011380486C1300 0040133C1E0979BC2D>126 D<001C130E007FEB3F8039FF807FC0A5397F003F80001CEB 0E001A0977BD2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx14 14.4 37 /Fs 37 123 df11 D45 D<151E153E15FE1403140F147FEB07FF0003B5FCB6FCA3EB F87FEAFC00C7FCB3B3B3A6007FB712FCA52E4E76CD42>49 DI<913807FFC0027F13FC0103B67E010F15E090261FF80313F890267FC0007F 01FEC7EA3FFE48488148486E138013FE486C6C6D13C0804817E080A66C5B18C06C5B6C90 C75AD80038168090C8FC4C1300A24C5A5F4C5A4B5B4B13C0030F5BDB7FFEC7FC91387FFF F816C016FCEEFF80DA000313E09238007FF8EE3FFE707E70138018C07013E018F07013F8 A218FC82A218FEA3EA03C0EA0FF0EA3FFC487EA2B5FCA218FCA25E18F8A26C4816F0495C 4916E0D83FE04A13C06C485CD80FF04A1380D807FE91387FFE003B03FFE003FFFC6C90B6 5A6C6C15E0010F92C7FC010114FCD9001F1380374F7BCD42>I<17FC1601A21603160716 0FA2161F163F167FA216FF5D5DA25D5D5D167F153E157E15FC15F8EC01F01403EC07E015 C0EC0F80141FEC3F00143E5C14FC495A5C495A1307495A5C49C7FC5B137E137C5B120148 5A5B485A120F485A90C8FC123E127E5ABA1280A5C901FCC7FCAF021FB71280A5394F7CCE 42>I<171F4D7EA24D7EA34D7EA24C7FA34C7FA24C7FA34C7FA34C7F83043F80163E8304 7E80EE7C1F04FC8016F8830301814C7E03038116E0830307814C7E030F81168083031F81 93C77E4B82153E84037E82037C8003FC825D840201835D02036F7F5D8402078392B8FCA2 4A83A24A8392C9FC854A84023E82027E84147C8502FC844A820101855C850103854A8201 07855C85010F85D93FF082B600F00207B712E0A55B547CD364>65 DI<932603FFF01407047F01FF140F0307B600E0131F033F03F813 3F92B700FE137F02039126C003FF13FF020F01F8C7EA3FC1023F01C0EC0FE391B5C80003 B5FC4901FC814949814901E082011F498249498292CA7E4948834948835A4A83485B4885 A24849187FA2485B1B3FA2485B1B1FA25AA21B0091CDFCA2B5FCAE7EA280A36C1A1FA36C 7FA21B3F6C7F1B3E6C7F1B7E6C6D187C6C1AFC6E18F86C19016D6CEF03F06D7E6FEE07E0 6D6DEE0FC001076DEE1F806D01F8EE3F006D6D16FE6D01FF4B5A023F01C0EC07F8020F01 FCEC3FF00203903AFFC001FFC0020091B6C7FC033F15FC030715F0DB007F1480040301F0 C8FC505479D25F>I69 DI72 DI<93381FFF80 0303B512FC033FECFFC092B712F00207D9F80113FE021F903AC0003FFF804A48C700077F DAFFF8020113F049496E7F49496F7E49496F7E49496F7E4990C96C7F4948707F4948707F 01FF854849707F4A824886A24849717E48864A83A2481B80A248497113C0A4481BE0A291 CB7EA3B51AF0AF6C1BE0A36E5FA26C1BC0A36C1B806E5FA26C1B006E5F6C62A26C6DD903 FC4A5A6CDB0FFF5D6E49EBC0016C4B01E05C6D6C90277E07F0035B6E9039F801F807902A 3FFF01F000780F5B6D047C5C6DD981E06D4890C7FC6D01E191381F7FFE010101F1EDFFF8 6DD9F9F06D5BDA3FFF16C06E6D013F5B02079027FE01FFFEC8FC020190B612F8DA003F4B 141003071838DB001FEB83F893C7EA03FC1C7885726C14F8F2C003F2F01F97B512F084A3 1CE085A27314C01C80851C00735B735B735B735B9638003FC0556A79D263>81 DII<003FBB12FEA59126 C0007FEB000101FCC7ED001FD87FF0F007FF49844984498490C883A2007E86A3007C86A5 00FC1B80481A0FA6C994C7FCB3B3AC49B912C0A551517BD05C>I97 DI<913801FFF0021F13FF91B612C00103 15F0010F9038801FFC903A1FFC0003FED97FF8497E49485B4849491380485B485BA24890 C7FC5AA248486E1300705A705A007F92C8FC5BA312FFAD127F7FA3123F7F6CEE07C0A26C 6D140F18806C6D141F6C6D15006C6D5C6C6D147E6D6C5C6DB4EB03F8010F9038E01FF001 0390B512C0010092C7FC023F13FC020113C032387CB63B>I<943803FF80040FB5FCA5EE 003F170FB3A4913803FF80023F13F849B512FE0107ECFF8F011F9038C03FEF90273FFE00 07B5FCD97FF8130149487F484980484980484980488291C8FC5A5B123FA2127F5BA312FF AD127FA37F123FA3121F7F6C5E6C6D5C5F6C6D91B5FC6C6D5B6C6D4914E0D97FFCD90FEF EBFF80D91FFFEB7F8F010790B5120F010114FC6D6C13E00207010049C7FC41547CD249> I<913807FF80027F13F849B512FE01076E7E011F010313E0903A3FFC007FF0D97FF06D7E 49486D7E4849130F48496D7E48824890C7FC701380485AA2003F6F13C0A3485A18E082A2 12FFA290B8FCA401FCCAFCA6127FA37F123FA2EF03E06C7E17076C17C06C6D140F18806C 6D141F6C6DEC3F006C6D147ED97FFC495AD91FFFEB07F86D9038E03FF0010390B512C001 005D023F01FCC7FC020113E033387CB63C>IIII<133FEBFFC0487F487FA2487FA66C5BA26C5B6C5B013FC7FC 90C8FCAEEB1FF8B5FCA512017EB3B3A6B612F0A51C547CD324>I107 DIII<913801FFC0023F13FE91B67E010315 E0010F018013F8903A3FFC001FFED97FF0EB07FF49486D7F48496D7F48496D7F91C8127F 4883488349153F001F83A2003F8349151FA2007F83A400FF1880AC007F1800A3003F5F6D 153FA2001F5FA26C6C4B5AA26C6D4A5A6C5F6C6D495B6C6D495B6D6C4990C7FCD93FFCEB 1FFE6DB46CB45A010790B512F0010115C0D9003F49C8FC020313E039387CB642>II<90393FF0 01FCB590380FFF804B13E0037F13F09238FE1FF89138F1F83F00039138F07FFCC6EBF3E0 15C0ECF780A2ECFF00EE3FF84AEB1FF0EE0FE093C7FC5CA45CB3ABB612FEA52E367CB536 >114 D<903903FFC00E011FEBFC1E90B6127E000315FE3907FE003FD80FF0130F484813 0348481301491300127F90C8127EA248153EA27FA27F01F091C7FC13FCEBFF806C13FEEC FFF06C14FE6F7E6C15E06C816C15FC6C81C681133F010F15801301D9000F14C0EC003F03 0713E0150100F880167F6C153FA2161F7EA217C07E6D143F17807F6DEC7F0001F85C6DEB 03FE9039FF801FFC486CB512F0D8F81F14C0D8F00791C7FC39E0007FF02B387CB634>I< 147CA614FCA41301A31303A21307A2130F131F133F137F13FF1203000F90B512FEB7FCA4 26007FFCC8FCB3A9EE0F80ABEE1F006D7EA2011F143E806D6D5A6DEBC1F86DEBFFF00100 5C023F1380DA03FEC7FC294D7ECB33>II<003FB8FCA491C75B 01F8495B13E049495B49495B90C7FC4B5B007E4A5B5F5D4B90C7FC007C4A5AA24A5B4A5B 5E5CC7485B4A5BA24A5B4A90C8FC5D14FF495B495BEF0F80495B495B5D5B4949131F4990 C71300A2495A48495C5C5A48495C485B5F48495B484913074C5A4890C712FFB8FCA43135 7CB43B>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmbx10 10 7 /Ft 7 117 df65 D97 D<13FE12FFA412071203AFEC03FF021F13E0027F13F89039FFFC07FC9138E001 FFDA8000138091C7EA7FC049EC3FE0A217F0161F17F8A317FCA917F8A3EE3FF0A217E06D EC7FC017806EEBFF009039FBE001FE9039F8F80FFC9039F07FFFF0D9E01F13C09026C007 FEC7FC2E3A7DB935>I<903801FFE0010F13FC017F13FF9039FF807F803801FE00D807FC EBFFC0485AA2485A003FEC7F80ED3F004848130C92C7FCA212FFA9127FA27F123FED01E0 6C7E000FEC03C07F6C6CEB0780D801FFEB0F006CEBC07E6DB45A010F13F0010113802325 7DA42A>I<3901FC03F000FFEB0FFC4AB4FC91383C7F80147000079038E0FFC000035BEB FD80A201FFEB7F809138003F00150C92C7FC5BB3A2B512FCA422257EA427>114 D<90387FF0383903FFFEF84813FF381FC00F383F0003003E13005A157812FCA27E6C1400 13E06CB47E14F86C13FF6C14806C14E06C14F0000114F86C6C13FC13039038000FFE1403 0070130012F0157E7EA26C147C7E6C14F890388001F09038F00FE090B512C000F0140038 C01FF81F257DA426>I<131EA5133EA4137EA213FEA2120112031207001FB512E0B6FCA3 D803FEC7FCB21578A8000114F07F6CEB01E090387F83C090383FFF806D1300EB03FC1D35 7EB425>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr10 10 50 /Fu 50 123 df12 D<146014E0EB01C0EB0380EB070013 0E5B133C13385B13F05B1201485AA2485AA348C7FCA25A121EA2123EA2123CA2127CA512 7812F8B01278127CA5123CA2123EA2121EA2121F7EA26C7EA36C7EA26C7E12007F13707F 133C131C7F7FEB0380EB01C0EB00E01460135278BD20>40 D<7E7E7E12707E7E7E120F7E 6C7E7F12017F6C7EA21378A37FA2133E131EA2131FA27FA21480A5130714C0B01480130F A51400A25BA2131EA2133E133CA25BA35BA2485A5B12035B48C7FC5A120E5A5A5A5A5A5A 12527BBD20>I<121C127E127FEAFF80A213C0127FA2121C1200A4EA0180A3EA0300A312 065AA25A5A12200A19798817>44 DI<121C123E127FEAFF80A3 EA7F00123E121C0909798817>I48 D<497E1307130F133FEA01FFB5FC13DFEAFE1F1200B3B3A749 7E007FB512E0A31B3779B62A>III<1570 A215F01401A214031407A2140F141F141B1433147314E314C31301EB038314031307130E 130C131C13381330137013E013C0EA0180120313001206120E120C5A123812305A12E0B7 12F8A3C73803F000AB4A7E0103B512F0A325387EB72A>I<0006140CD80780133C9038F0 03F890B55A5D5D5D92C7FC14FC38067FE090C9FCABEB07F8EB3FFF9038F80F803907C003 C090380001F000066D7E1204C8127C157EA281A31680A3123C127EB4FCA316005A485C00 60147EA26C5C00385C00181301000E495A6C495A3903E03F806CB5C7FC38007FFCEB1FE0 21387CB62A>I<12301238123E003FB612E0A316C048158016000070C71203006014065D A25D485C5DA2C85A4A5A4AC7FCA214065CA25C143814301470A25C1301A213035C1307A3 130FA2495AA4133FA5137FA86DC8FC131E233A7BB82A>55 D57 D<1538A3157CA315FEA24A7EA34A7F153FA202077FEC061FA2020C7F150FA24A6C7EA34A 6C7EA34A6C7EA34A6C7EA34948137FA201038191C7123FA249B67EA3010EC7EA1FE0010C 140FA249811607A2496E7EA3496E7EA301E06E7E1201486C81D80FFC02031380B56C017F 13FEA3373B7DBA3E>65 DI68 D72 D<013FB512C0A39039001FF800EC07F0B3B3A3121C127FA2EAFF80A25DEB000F6C5C127C 0030495A6C49C7FC6C137E380781F86CB45A38007F80223A7CB82B>74 D76 DI79 D83 D<003FB812E0A3903AC003FE001F273E0001FC1303 48EE01F00078160000701770A300601730A400E01738481718A4C71600B3B0EC07FF011F B612C0A335397DB83C>II87 D97 DIIII<147E903803FF80903807C1C090380F 07E0011E13F0EB3E0F137C13FCEC07E09038F803C0000190C7FCADB512FCA3D801F8C7FC B3AB487E387FFFF8A31C3B7FBA19>I<90390FF003F090393FFC1FF89039F81F7C7C3901 F00FE03A03E007C0383A07C003E010160048486C7EA2001F80A6000F5CA26C6C485AA26C 6C485A6C6C485A486C48C7FC38063FFCEB0FF0000EC9FCA4120FA26C7E90B512C06C14F8 6C14FE6CECFF80000315C03A0F80007FE0001EC7120F003EEC03F048140116F8481400A5 007CEC01F0A26CEC03E06CEC07C06C6CEB0F80D807E0EB3F003901FC01FC39007FFFF001 0790C7FC26377EA42A>I II108 D<2703F00FF0EB1FE000FFD93FFCEB7FF8913AF03E01E07C903BF1C01F03803E3C0FF300 0F86001FD803F602CC14800307140F01FC02F814C0495CA3495CB3A4486C496CEB1FE0B5 00C1B50083B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03E9038F1C01F3A0FF3 000F80D803F680150701FC805BA35BB3A4486C497EB500C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9038F3801F3A07F6000F80D803FC EB07C049EB03E016F049EB01F816FC150016FEA3167E167FA8167E16FEA216FCA2ED01F8 A26DEB03F016E06DEB07C001F6EB0F8001F3EB1F009038F1E07E9038F0FFF8EC1FC091C8 FCAB487EB512C0A328357EA42E>II<3807E01F00FFEB7FC0ECE3E09038E183F0 380FE307EA03E6A29038EC03E0EC008001F81300A35BB3A3487EB512F0A31C257EA421> II< 1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A90000EB 018013FC137C90383E0300EB1F06EB0FFCEB01F81A347FB220>IIIIII<003FB512FCA290388001F8393E0003F000 3C1307003814E00030EB0FC00070131F15800060EB3F005C14FE5C495AEA00035C495A13 0F495A5C49C7FC491306137E5B1201485A49130E485A000F140C49131C485A003F143C90 C7127C007EEB03FCB6FCA21F247EA325>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmsy8 8 14 /Fv 14 108 df0 D<1238127C12FEA3127C123807077A9314>I< 130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FF E038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC03978 1E078000601301000090C7FCA5130C1A1D7C9E23>3 D<17C01603160FEE3F0016FCED03 F0ED0FC0033FC7FC15FCEC03F0EC0FC0023FC8FC14FCEB03F0EB0FC0013FC9FC13FCEA03 F0EA0FC0003FCAFC12FC12F012FC123FEA0FC0EA03F0EA00FC133FEB0FC0EB03F0EB00FC 143FEC0FC0EC03F0EC00FC153FED0FC0ED03F0ED00FC163FEE0F80EE03C0160193C7FCAD 007FB71280B812C0A22A3B7AAB37>20 D<12C012F012FC123FEA0FC0EA03F0EA00FC133F EB0FC0EB03F0EB00FC143FEC0FC0EC03F0EC00FC153FED0FC0ED03F0ED00FC163FEE0FC0 1603160FEE3F0016FCED03F0ED0FC0033FC7FC15FCEC03F0EC0FC0023FC8FC14FCEB03F0 EB0FC0013FC9FC13FCEA03F0EA0FC0003FCAFC127C12F05ACBFCAD007FB71280B812C0A2 2A3B7AAB37>I<170EA383A3717EA2717E841700187084181E84F007C0007FB912F8BA12 FC6C18F8CBEA07C0F00F00181E183860601701604D5AA24DC7FCA3170EA33E237CA147> 33 D<1338137C13FEA3EA01FCA313F81203A213F0A2EA07E0A3EA0FC0A31380121F1300 A3123EA35AA21278A212F85A12700F227EA413>48 DI<91B512C01307131F49C8FC13F848 5AEA03C0485A48C9FC120E5AA25AA25AA35AA3B712C0A300E0C9FCA31270A37EA27EA27E 120F6C7E6C7EEA01F06C7E133F6DB512C013071300222B7AA52F>I<91387FFFF00107B6 FC011F15E090B712F82701F03E017FD807809038001FFED80E00EC03FF001E017E010013 8048167F007CEE3FC00078161F4817E000C0017C140FC7FC170714FCA25CA3010116C05C A2EF0F8013034A15005F171E4948143E173C5F494814705F4C5A49C7485A040FC7FC161C 013E1478ED01E0013CEB0F80017C01FEC8FC90387FFFF848B512E04849C9FC4813E0332D 7EAC37>68 D<141F14FFEB03F0EB07C0EB0F80EB1F00131E133EB3A35BA25BEA03E0EA7F C048C7FCEA7FC0EA03E0EA00F8137CA27FB3A3131E131FEB0F80EB07C0EB03F0EB00FF14 1F18437BB123>102 D<127CB47EEA0FE0EA01F06C7E137CA27FB3A37FA2EB0F80EB07E0 EB01FEEB007FEB01FEEB07E0EB0F80EB1F00A2133EB3A35BA25B485AEA0FE0EAFF80007C C7FC18437BB123>I<12E0B3B3B3AD034378B114>106 D<0060136000E01370B3B3B3AB00 601360144379B123>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr12 12 9 /Fw 9 116 df<121E123FEA7F80EAFFC0A4EA7F80EA3F00121E0A0A78891B>46 D<121E123FEA7F80EAFFC0A4EA7F80EA3F00121EC7FCB3A5121E123FEA7F80EAFFC0A4EA 7F80EA3F00121E0A2B78AA1B>58 D71 D77 D101 D105 D<2703F801FEEC3FC000FF903B0FFFC001FFF8913B1E03E0 03C07C913B3001F006003E0007496C6C487F000349D97C186D7E2601F980D97E308001FB C75B01FEDA3EC01307043F81495DA34992C7FCB3A9486C4A6C497EB528F01FFFFE03B512 C0A34A2C7DAB51>109 D<3903F007F000FFEB1FF8EC383CEC607E0007EBC0FF3803F180 1201EBF30001F6137E153C150013FCA45BB3A7EA03FEB512FCA3202C7DAB26>114 D<90383FE0183801FFFC3907C01E38390F000378001CEB01F848130015785A153812F015 18A37E7E127ED87F801300EA3FF0EBFF806C13F8000713FE6CEBFF806C14C0D8003F13E0 010313F09038001FF814030040EB00FC00C0147C157E153E7E151EA27EA2151C7E153C6C 14386C147000FB14E039F18001C039E0F00F8039C07FFE00EB0FF01F2E7DAC26>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmbx17 17.28 27 /Fx 27 123 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 330 516 a Fx(Exp)t(onen)l(tial)49 b(Deca)l(y)j(and)f (Ionization)f(Thresholds)e(in)488 753 y(Non-Relativistic)g(Quan)l(tum)i (Electro)t(dynamics)1691 1040 y Fw(M.)33 b(Griesemer)2268 1000 y Fv(\003)730 1232 y Fu(Departmen)n(t)28 b(of)g(Mathematics,)f (Univ)n(ersit)n(y)g(of)g(Alabama)g(at)g(Birmingham,)1444 1350 y(Birmingham,)g(AL)g(35294,)f(USA)1730 1551 y(17)h(June,)h(2002) 1797 2010 y Ft(Abstract)470 2189 y Fu(Spatial)23 b(lo)r(calization)f (for)h(quan)n(tum)h(mec)n(hanical)e(particles)h(\(electrons\))g(in)n (teracting)g(with)h(quan-)345 2319 y(tized)f(radiation)f(\(photons\))h (is)g(studied)g(at)g(energies)e(b)r(elo)n(w)i(the)g(ionization)f (threshold.)35 b(W)-7 b(e)23 b(giv)n(e)f(t)n(w)n(o)345 2448 y(de\014nitions)31 b(of)g(the)h(ionization)e(threshold.)46 b(One)31 b(in)g(terms)g(of)g(minimal)g(energies)f(of)h(non-lo)r (calized)345 2578 y(states,)e(and)h(a)f(second)f(one)h(in)h(terms)f(of) g(sp)r(ectral)g(data)g(of)g(cluster)g(Hamiltonians.)42 b(W)-7 b(e)29 b(sho)n(w)g(that)345 2707 y(these)j(de\014nitions)g (agree,)g(and)g(that)h(all)e(states)h(in)g(the)h(sp)r(ectral)f (subspace)f(of)h(energies)f(b)r(elo)n(w)h(the)345 2837 y(ionization)h(threshold)h(deca)n(y)f(exp)r(onen)n(tially)g(in)h(the)g (particle)g(co)r(ordinates.)54 b(The)34 b(latter)f(result)h(is)345 2966 y(deriv)n(ed)e(from)h(a)g(new,)i(general)c(result)i(on)g(exp)r (onen)n(tial)g(deca)n(y)f(tailored)g(to)h(\014t)h(our)e(problem,)i(but) 345 3096 y(applicable)27 b(to)h(man)n(y)f(non-relativistic)f(quan)n (tum)i(systems)f(outside)g(quan)n(tum)h(electro)r(dynamics)f(as)345 3225 y(w)n(ell.)120 3556 y Fs(1)132 b(In)l(tro)t(duction)120 3793 y Fr(If)32 b(an)h(atom)g(or)g(molecule)g(is)f(in)h(a)g(state)f (with)h(total)g(energy)g(b)r(elo)n(w)g(the)g(ionization)g(threshold,)g (then)g(all)120 3940 y(electrons)k(are)f(w)n(ell)h(lo)r(calized)g(near) g(the)g(n)n(uclei.)60 b(In)37 b(non-relativistic)f(quan)n(tum)g(mec)n (hanics)g(this)g(\014nds)120 4087 y(its)31 b(mathematical)g(expression) f(in)h(the)g(fact)g(that)h(the)f(sp)r(ectrum)g(of)g(the)g(Hamilton)g (op)r(erator)h(b)r(elo)n(w)f(the)120 4233 y(ionization)f(threshold)g (is)g(discrete)g(and)g(all)h(eigenfunctions)e(deca)n(y)h(exp)r(onen)n (tially)-7 b(.)39 b(When)30 b(the)g(electrons)120 4380 y(are)35 b(coupled)h(to)f(the)g(quan)n(tized)g(radiation)g(\014eld,)i (then)e(there)g(is)g(no)g(discrete)g(sp)r(ectrum)h(an)n(ymore)e(and)120 4527 y(the)39 b(ground)g(state)g(is)g(the)g(only)g(eigenfunction.)67 b(Nev)n(ertheless,)39 b(all)g(states)f(in)h(the)h(sp)r(ectral)f (subspace)120 4674 y(of)c(energies)h(b)r(elo)n(w)f(the)h(ionization)g (threshold)f(are)h(exp)r(onen)n(tially)f(w)n(ell)g(lo)r(calized)i(as)e (functions)g(of)g(the)120 4821 y(electron)42 b(co)r(ordinates.)74 b(T)-7 b(o)41 b(pro)n(v)n(e)f(this)h(is)g(the)g(main)h(purp)r(ose)f(of) g(this)g(pap)r(er.)75 b(Lo)r(calization)42 b(of)f(the)120 4967 y(electrons)d(b)r(elo)n(w)h(the)g(ionization)f(threshold)h(is)f (necessary)f(to)i(justify)d(the)j(dip)r(ole)g(appro)n(ximation)f([2],) 120 5114 y(and)30 b(it)g(pla)n(ys)f(an)h(imp)r(ortan)n(t)h(role)f(in)g (pro)n(ving)g(existence)f(of)h(a)g(ground)g(state)g([2,)g(3,)g(7])h (and)f(for)g(Ra)n(yleigh)120 5261 y(scattering)g([5].)p 120 5322 1488 4 v 222 5375 a Fq(\003)258 5407 y Fp(W)-6 b(ork)25 b(partially)i(supp)r(orted)e(b)n(y)g(U.S.)g(National)i (Science)f(F)-6 b(oundation)25 b(gran)n(t)h(DMS)f(01-00160.)1957 5656 y Fr(1)p eop %%Page: 2 2 2 1 bop 120 -200 a Fr(2)261 99 y(The)28 b(ionization)f(threshold)h(is)f (the)g(least)g(energy)h(that)f(an)h(atom)f(or)h(molecule)g(can)f(ac)n (hiev)n(e)f(in)i(a)f(state)120 245 y(where)32 b(one)f(or)h(more)f (electrons)h(ha)n(v)n(e)e(b)r(een)i(mo)n(v)n(ed)e("in\014nitely)h(far)h (a)n(w)n(a)n(y")d(from)i(the)h(n)n(uclei.)44 b(T)-7 b(o)31 b(giv)n(e)g(a)120 392 y(more)26 b(precise)h(de\014nition)f(w)n(e)g (need)h(to)f(in)n(tro)r(duce)h(a)f(mathematical)h(mo)r(del)g(for)f (atoms)g(and)g(molecules.)39 b(A)120 539 y(\(pure\))32 b(state)f(of)g Fo(N)42 b Fr(electrons)31 b(and)h(an)f(arbitrary)h(n)n (um)n(b)r(er)f(of)g(transv)n(ersal)f(photons)h(shall)h(b)r(e)g(describ) r(ed)120 686 y(b)n(y)27 b(a)h(v)n(ector)f(in)i(the)f(Hilb)r(ert)g (space)g Fn(H)1464 701 y Fm(N)1556 686 y Fr(=)d Fn(H)1727 701 y Fl(e)p Fm(l)1800 686 y Fn(\012)16 b(F)9 b Fr(,)29 b(where)f Fn(H)2348 701 y Fl(e)p Fm(l)2433 686 y Fr(is)g(the)g(an)n (tisymmetric)f(tensor)g(pro)r(duct)120 832 y(of)d Fo(N)34 b Fr(copies)24 b(of)g Fo(L)741 801 y Fl(2)780 832 y Fr(\()p Fk(R)875 801 y Fl(3)921 832 y Fr(;)15 b Fk(C)1020 801 y Fl(2)1066 832 y Fr(\),)25 b(appropriate)g(for)f Fo(N)34 b Fr(spin-1)p Fo(=)p Fr(2)24 b(fermions,)g(and)h Fn(F)33 b Fr(is)23 b(the)i(b)r(osonic)f(F)-7 b(o)r(c)n(k)23 b(space)120 979 y(o)n(v)n(er)j Fo(L)371 948 y Fl(2)411 979 y Fr(\()p Fk(R)505 948 y Fl(3)551 979 y Fo(;)15 b Fk(C)650 948 y Fl(2)696 979 y Fr(;)g Fo(dk)s Fr(\).)39 b(The)27 b(n)n(uclei)g(are)h (static,)g(p)r(oin)n(t-lik)n(e)e(particles)i(without)f(spin.)39 b(Let)28 b Fo(H)3333 994 y Fm(N)3427 979 y Fr(denote)g(the)120 1126 y(Hamilton)h(op)r(erator)i(generating)f(the)g(time)g(ev)n(olution) f(in)h Fn(H)2251 1141 y Fm(N)2318 1126 y Fr(,)g(and)g(let)g Fo(H)2760 1095 y Fl(0)2753 1154 y Fm(N)2850 1126 y Fr(b)r(e)g(the)g (same)f(Hamiltonian)120 1273 y(without)40 b(external)f(p)r(oten)n (tials)h(\(n)n(uclei\).)68 b(W)-7 b(e)39 b(assume)g(that)h(the)f (dynamics)g(of)h(the)f(electrons)h(is)f(non-)120 1420 y(relativistic)26 b(and)g(that)g(the)h(forces)e(b)r(et)n(w)n(een)h (material)h(particles)f(\(electrons)g(and)g(n)n(uclei\))g(drop)h(o\013) f(to)g(zero)120 1566 y(with)g(increasing)f(distance.)38 b(In)25 b(view)g(of)h(the)f(latter)h(assumption)f(a)g(natural)h (de\014nition)g(for)f(the)h(ionization)120 1713 y(threshold)k Fo(\034)10 b Fr(\()p Fo(H)675 1728 y Fm(N)742 1713 y Fr(\))30 b(is)1361 1860 y Fo(\034)10 b Fr(\()p Fo(H)1520 1875 y Fm(N)1588 1860 y Fr(\))25 b(:=)38 b(min)1768 1922 y Fm(N)1831 1903 y Fq(0)1853 1922 y Fv(\025)p Fl(1)1943 1860 y Fn(f)p Fo(E)2055 1875 y Fm(N)7 b Fv(\000)p Fm(N)2236 1856 y Fq(0)2282 1860 y Fr(+)20 b Fo(E)2444 1824 y Fl(0)2439 1885 y Fm(N)2502 1866 y Fq(0)2529 1860 y Fn(g)p Fo(;)120 2074 y Fr(where)35 b Fo(E)452 2089 y Fm(N)7 b Fv(\000)p Fm(N)633 2070 y Fq(0)692 2074 y Fr(=)32 b(inf)22 b Fo(\033)s Fr(\()p Fo(H)1083 2089 y Fm(N)7 b Fv(\000)p Fm(N)1264 2070 y Fq(0)1290 2074 y Fr(\),)36 b Fo(E)1458 2042 y Fl(0)1453 2103 y Fm(N)1516 2085 y Fq(0)1575 2074 y Fr(=)d(inf)21 b Fo(\033)s Fr(\()p Fo(H)1973 2042 y Fl(0)1966 2103 y Fm(N)2029 2085 y Fq(0)2056 2074 y Fr(\),)36 b(and)f Fo(E)2399 2089 y Fm(N)7 b Fl(=0)2588 2074 y Fr(=)33 b(0.)54 b(Let)35 b Fo(m)g Fr(b)r(e)g(the)g(mass)f(of)g(the)120 2221 y(electron)h(and)f (let)h Fn(j)p Fo(x)p Fn(j)d Fr(=)h(\()1050 2152 y Fj(P)1146 2179 y Fm(n)1146 2247 y(j)t Fl(=1)1287 2221 y Fo(x)1338 2189 y Fl(2)1338 2248 y Fm(j)1378 2221 y Fr(\))1413 2189 y Fl(1)p Fm(=)p Fl(2)1558 2221 y Fr(for)h Fo(x)e Fn(2)h Fk(R)1935 2189 y Fm(n)1988 2221 y Fr(.)54 b(W)-7 b(e)33 b(pro)n(v)n(e)g(that,)j(for)e(all)h(real)f(n)n(um)n(b)r(ers)g Fo(\025)h Fr(and)f Fo(\014)120 2367 y Fr(with)c Fo(\025)21 b Fr(+)f Fo(\014)544 2336 y Fl(2)583 2367 y Fo(=)p Fr(\(2)p Fo(m)p Fr(\))25 b Fo(<)f(\034)10 b Fr(\()p Fo(H)1100 2382 y Fm(N)1168 2367 y Fr(\),)1566 2410 y Fj(\015)1566 2464 y(\015)1566 2519 y(\015)1617 2514 y Fo(e)1659 2479 y Fm(\014)s Fv(j)p Fm(x)p Fv(j)1785 2514 y Fo(E)1852 2529 y Fm(\025)1896 2514 y Fr(\()p Fo(H)2006 2529 y Fm(N)2073 2514 y Fr(\))2108 2410 y Fj(\015)2108 2464 y(\015)2108 2519 y(\015)2184 2514 y Fo(<)25 b Fn(1)p Fo(;)1331 b Fr(\(1\))120 2710 y(and)27 b(that)g(in)f(states)g(with)h(energy)f(ab)r (o)n(v)n(e)g Fo(\034)10 b Fr(\()p Fo(H)1735 2725 y Fm(N)1803 2710 y Fr(\))26 b(the)h(electrons)g(will)g(not)f(b)r(e)i(lo)r(calized)f (in)g(general.)39 b(Th)n(us)120 2857 y Fo(\034)10 b Fr(\()p Fo(H)279 2872 y Fm(N)346 2857 y Fr(\))25 b(is)f(in)h(fact)f(a)h (threshold)f(energy)g(separating)h(lo)r(calized)g(from)f(non-lo)r (calized)i(states.)37 b(The)25 b(question)120 3004 y(of)33 b(whether)h(the)g(binding)f(energy)h Fo(\034)10 b Fr(\()p Fo(H)1513 3019 y Fm(N)1580 3004 y Fr(\))23 b Fn(\000)f Fo(E)1797 3019 y Fm(N)1897 3004 y Fr(is)33 b(p)r(ositiv)n(e)g(or)h (not,)h(is)e(not)h(addressed)e(in)i(this)f(pap)r(er,)120 3151 y(see)d(ho)n(w)n(ev)n(er)e([7].)261 3297 y(The)33 b(pro)r(of)f(consists)f(of)h(t)n(w)n(o)g(indep)r(enden)n(t)g(parts.)47 b(First)32 b(w)n(e)g(giv)n(e)f(an)i(alternativ)n(e)e(de\014nition)i(of) f(the)120 3444 y(ionization)k(threshold)g(whic)n(h)g(b)r(etter)h (captures)f(the)g(idea)g(of)g(a)g(lo)r(calization)h(threshold,)g(and)g (w)n(e)e(pro)n(v)n(e)120 3591 y(exp)r(onen)n(tial)30 b(deca)n(y)f(b)r(elo)n(w)h(it.)40 b(Then)30 b(w)n(e)g(sho)n(w)f(that)h (the)g(t)n(w)n(o)f(de\014nitions)h(agree.)261 3738 y(The)j(alternativ)n (e)g(de\014nition)g(is)g(as)g(follo)n(ws.)48 b(Let)34 b Fo(D)2094 3753 y Fm(R)2181 3738 y Fr(=)d Fn(f)p Fo(')f Fn(2)g Fo(D)r Fr(\()p Fo(H)7 b Fr(\))p Fn(j)15 b Fo(')p Fr(\()p Fo(x)p Fr(\))31 b(=)f(0)p Fo(;)48 b Fr(if)33 b Fn(j)p Fo(x)p Fn(j)e Fo(<)f(R)q Fn(g)p Fr(,)k(and)120 3885 y(de\014ne)c(a)g(threshold)g(energy)f(\006\()p Fo(H)1313 3900 y Fm(N)1380 3885 y Fr(\))h(b)n(y)1245 4113 y(\006\()p Fo(H)1420 4128 y Fm(N)1487 4113 y Fr(\))25 b(=)60 b(lim)1642 4174 y Fm(R)p Fv(!1)2016 4113 y Fr(inf)1851 4178 y Fm(')p Fv(2)p Fm(D)2002 4189 y Fi(R)2053 4178 y Fm(;)11 b Fv(k)p Fm(')p Fv(k)p Fl(=1)2306 4113 y Fn(h)p Fo(';)k(H)2515 4128 y Fm(N)2581 4113 y Fo(')p Fn(i)g Fo(:)1010 b Fr(\(2\))120 4362 y(Delo)r(calization)28 b(ab)r(o)n(v)n(e)f(\006\()p Fo(H)1132 4377 y Fm(N)1199 4362 y Fr(\))h(is)f(ob)n(vious,)g(and)g(lo)r (calization)i(b)r(elo)n(w)e(\006\()p Fo(H)2774 4377 y Fm(N)2841 4362 y Fr(\))h(will)g(b)r(e)g(deriv)n(ed)f(from)g(the)120 4509 y(only)j(assumptions)e(that)i Fo(H)1099 4524 y Fm(N)1196 4509 y Fr(is)g(self-adjoin)n(t,)e(b)r(ounded)j(from)e(b)r(elo)n(w,)i (and)f(that)1538 4738 y([[)p Fo(H)1663 4753 y Fm(N)1730 4738 y Fo(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(])25 b(=)g Fn(\000)p Fr(2)p Fn(jr)p Fo(f)10 b Fn(j)2382 4702 y Fl(2)3725 4738 y Fr(\(3\))120 4967 y(for)27 b(all)h(b)r(ounded)g(smo) r(oth)g(functions)e Fo(f)10 b Fr(\()p Fo(x)p Fr(\))28 b(with)g(b)r(ounded)g(\014rst)f(deriv)-5 b(ativ)n(es.)37 b(The)28 b(latter)g(assumption)e(is)120 5114 y(satis\014ed)e(for)h(the) g(p)r(ositiv)n(e)g(Laplacian)g(\()p Fn(\000)p Fr(\001\),)i(and)e(hence) g(for)g(all)g(op)r(erators)g Fn(\000)p Fr(\001)10 b(+)g Fo(I)33 b Fr(with)25 b([[)p Fo(I)7 b(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(])25 b(=)g(0.)120 5260 y(Examples)31 b(include)h(the)f(commonly)h(traded)f(mo)r(dels)h(of)f (non-relativistic)g(atoms)g(coupled)h(to)g(quan)n(tized)120 5407 y(radiation,)e(as)g(w)n(ell)f(as)h(man)n(y)f(Sc)n(hr\177)-45 b(odinger)30 b(opartors)f(outside)h(quan)n(tum)f(electro)r(dynamics.)p eop %%Page: 3 3 3 2 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2312 b Fr(3)261 99 y(The)42 b(second)g(part)f(of)h(the)g (pro)r(of,)j(that)d Fo(\034)10 b Fr(\()p Fo(H)1890 114 y Fm(N)1957 99 y Fr(\))45 b(=)g(\006\()p Fo(H)2327 114 y Fm(N)2393 99 y Fr(\),)g(is)d(the)g(hard)g(part.)75 b(The)42 b(inequalit)n(y)120 245 y Fo(\034)10 b Fr(\()p Fo(H)279 260 y Fm(N)346 245 y Fr(\))25 b Fn(\024)g Fr(\006\()p Fo(H)676 260 y Fm(N)743 245 y Fr(\))j(requires)f(lo)r(calizing)i(b)r (oth)g(the)f(electrons)f(and)h(the)h(photons,)f(and)g(in)g(particular)g (their)120 392 y(\014eld)38 b(energy)-7 b(.)65 b(This)38 b(w)n(as)f(done)h(in)h([7,)f(6].)65 b(T)-7 b(o)38 b(sho)n(w)g(that)g Fo(\034)10 b Fr(\()p Fo(H)2424 407 y Fm(N)2491 392 y Fr(\))39 b Fn(\025)g Fr(\006\()p Fo(H)2849 407 y Fm(N)2916 392 y Fr(\))f(w)n(e)g(construct)g(suitable)120 539 y(\(compactly)43 b(supp)r(orted\))g(minimizers)f Fo(')1599 554 y Fl(0)1681 539 y Fr(and)h Fo(')1928 507 y Fm(R)1928 559 y Fv(1)2046 539 y Fr(of)f Fo(H)2236 554 y Fm(N)7 b Fv(\000)p Fm(N)2417 535 y Fq(0)2486 539 y Fr(and)43 b Fo(H)2756 507 y Fl(0)2749 569 y Fm(N)2812 550 y Fq(0)2838 539 y Fr(,)j(resp)r(ectiv)n(ely)-7 b(,)44 b(where)f Fo(')3765 507 y Fm(R)3765 559 y Fv(1)120 686 y Fr(is)38 b(lo)r(calized)g(at)h(a)f(distance)f Fo(R)j Fr(from)d(the)h(origin.)65 b(W)-7 b(e)37 b(than)h(merge)g(these)g (states)f(in)n(to)h(a)g(single)f(state)120 832 y Fo( )179 847 y Fm(R)277 832 y Fn(2)k(H)454 847 y Fm(N)521 832 y Fr(.)69 b(The)40 b(problem)g(is)f(to)h(do)f(this)g(in)h(suc)n(h)f(a)h (w)n(a)n(y)e(that)h Fn(h)p Fo( )2612 847 y Fm(R)2670 832 y Fo(;)15 b(H)2785 847 y Fm(N)2852 832 y Fo( )2911 847 y Fm(R)2968 832 y Fn(i)41 b Fr(=)g Fn(h)p Fo(')3249 847 y Fl(0)3288 832 y Fo(;)15 b(H)3403 848 y Fm(N)7 b Fv(\000)p Fm(N)3584 829 y Fq(0)3610 832 y Fo(')3669 847 y Fl(0)3709 832 y Fn(i)26 b Fr(+)120 906 y Fj(\012)163 979 y Fo(')222 948 y Fm(R)222 1000 y Fv(1)297 979 y Fo(;)15 b(H)419 948 y Fl(0)412 1009 y Fm(N)475 990 y Fq(0)501 979 y Fo(')560 948 y Fm(R)560 1000 y Fv(1)634 906 y Fj(\013)697 979 y Fr(+)20 b Fo(o)p Fr(\(1\))31 b(as)e Fo(R)d Fn(!)f(1)p Fr(.)261 1126 y(In)j(the)g(con)n(text)f(of)h(QED)g(the)h(\014rst)e (result)h(of)g(the)g(form)g(\(1\))g(is)g(due)g(to)g(Bac)n(h,)g(F)-7 b(r\177)-45 b(ohlic)n(h)27 b(and)i(Sigal)f([2],)120 1273 y(who)34 b(pro)n(v)n(ed)e(exp)r(onen)n(tial)h(binding)g(for)g(small)g (coupling)h(and)g(a)n(w)n(a)n(y)d(from)i(the)h(ionization)f(threshold)g (of)120 1420 y Fo(H)195 1435 y Fm(N)290 1420 y Fr(with)c Fg(zer)-5 b(o)34 b Fr(coupling.)39 b(The)29 b(threshold)f(energy)g Fo(\034)10 b Fr(\()p Fo(H)2102 1435 y Fm(N)2169 1420 y Fr(\))28 b(w)n(as)g(in)n(tro)r(duced)g(in)g([7)q(])g(where)g(it)h(w)n (as)e(sho)n(wn)120 1566 y(that)i Fo(E)381 1581 y Fm(N)477 1566 y Fr(is)g(an)g(eigen)n(v)-5 b(alue)28 b(of)h Fo(H)1297 1581 y Fm(N)1393 1566 y Fr(if)f Fo(\034)10 b Fr(\()p Fo(H)1633 1581 y Fm(N)1701 1566 y Fr(\))25 b Fo(>)g(E)1923 1581 y Fm(N)1989 1566 y Fr(.)40 b(The)29 b(pap)r(er)h([7])f(also)g(con) n(tains)f(an)h(easy)f(argumen)n(t)120 1713 y(sho)n(wing)j(that)h Fg(eigenve)-5 b(ctors)40 b Fr(of)31 b Fo(H)1352 1728 y Fm(N)1451 1713 y Fr(with)g(eigen)n(v)-5 b(alues)31 b(b)r(elo)n(w)g Fo(\034)10 b Fr(\()p Fo(H)2540 1728 y Fm(N)2608 1713 y Fr(\))31 b(exhibit)h(the)g(exp)r(onen)n(tial)f(deca)n (y)120 1860 y(implied)24 b(b)n(y)g(\(1\).)38 b(F)-7 b(or)23 b Fo(N)10 b Fr(-particle)24 b(Sc)n(hr\177)-45 b(odinger)24 b(op)r(erators)g(the)g(exp)r(onen)n(tial)f(deca)n(y)h(\(1\))g(with)g Fo(\034)35 b Fr(b)r(eing)24 b(the)120 2007 y(least)31 b(p)r(oin)n(t)h(of)f(the)g(essen)n(tial)g(sp)r(ectrum)g(w)n(as)g(pro)n (v)n(ed)f(b)n(y)h(O'Conner)g(in)h(1973)g([9],)g(and)f(the)h(equiv)-5 b(alence)120 2153 y(of)26 b(the)g(t)n(w)n(o)f(de\014nitions)h(for)g (the)g(ionization)h(threshold)f(is)f(sho)n(wn)h(as)f(part)i(of)e(mo)r (dern)i(pro)r(ofs)f(of)g(the)g(HVZ)120 2300 y(Theorem)35 b([8].)55 b(See)36 b(Agmon's)e(b)r(o)r(ok)h([1])g(for)g(more)g(results) f(on)i(the)f(exp)r(onen)n(tial)f(deca)n(y)h(of)f(solutions)g(of)120 2447 y(second)c(order)g(elliptic)g(equations.)261 2594 y(Section)35 b(2)g(con)n(tains)f(the)g(general)h(theorem)g(on)g(exp)r (onen)n(tial)f(deca)n(y)g(in)h(an)f(abstract)h(Hilb)r(ert)f(space)120 2741 y(setting.)40 b(In)30 b(Section)g(3)g(this)g(result)g(is)g (applied)g(to)g(quan)n(tum)g(electro)r(dynamics)f(and)i(the)f(main)g (result)g(on)120 2887 y(equalit)n(y)f(of)g(the)h(thresholds)g(is)f (form)n(ulated.)39 b(Its)29 b(pro)r(of)i(is)e(giv)n(en)g(in)h(Section)h (4.)40 b(The)30 b(App)r(endix)f(collects)120 3034 y(tec)n(hnical)h (results)f(and)h(notations)g(used)f(in)h(the)g(pro)r(ofs.)120 3362 y Fs(2)132 b(The)44 b(Abstract)g(Argumen)l(t)120 3599 y Fr(In)25 b(this)h(section)f Fo(q)k Fr(:)c Fo(D)14 b Fn(\002)e Fo(D)26 b Fn(!)f Fk(C)50 b Fr(denotes)25 b(a)h(densely)f(de\014ned,)i(closable,)g(semi-b)r(ounded)f(quadratic)f (form)120 3746 y(with)e(domain)f Fo(D)27 b Fn(\032)e(H)f Fr(in)e(a)h(Hilb)r(ert)f(space)h Fn(H)q Fr(.)37 b(W)-7 b(e)22 b(assume)f(that)i Fn(H)g Fr(is)f(a)h(closed)f(subspace)g(of)g Fo(L)3439 3714 y Fl(2)3479 3746 y Fr(\()p Fk(R)3573 3714 y Fm(n)3626 3746 y Fr(\))5 b Fn(\012)g(F)k Fr(,)120 3893 y(that)29 b(is)g(in)n(v)-5 b(arian)n(t)28 b(with)i(resp)r(ect)f(to)g(m) n(ultiplication)g(with)h(b)r(ounded)g(\(measurable\))f(functions)f (that)h(only)120 4040 y(dep)r(end)35 b(on)f Fn(j)p Fo(x)p Fn(j)p Fo(;)49 b(x)32 b Fn(2)f Fk(R)975 4008 y Fm(n)1028 4040 y Fr(.)52 b(Here)33 b Fn(F)43 b Fr(is)34 b(an)g(arbitrary)-7 b(,)34 b(additional)g(Hilb)r(ert)g(space.)52 b(In)33 b(our)h(applications)120 4186 y Fn(F)43 b Fr(will)35 b(b)r(e)f(the)h(tensor)f(pro)r(duct)h(of)f(spin)g(and)g(F)-7 b(o)r(c)n(k)33 b(space)h(and)g Fn(H)i Fr(the)e(subspace)g(with)g(the)g (symmetry)120 4333 y(required)c(b)n(y)f(the)h(nature)g(of)f(the)i (particles.)261 4480 y(On)f(the)g(quadratic)f(form)h Fo(q)j Fr(w)n(e)c(mak)n(e)g(the)h(further)g(assumption,)f(that)h(for)f (all)h Fo(f)35 b Fn(2)25 b Fo(C)3272 4448 y Fv(1)3347 4480 y Fr(\()p Fk(R)3441 4448 y Fm(n)3494 4480 y Fr(;)15 b Fk(R)s Fr(\))36 b(with)120 4627 y Fo(f)5 b(;)15 b Fn(r)p Fo(f)35 b Fn(2)25 b Fo(L)509 4595 y Fv(1)584 4627 y Fr(\()p Fk(R)678 4595 y Fm(n)731 4627 y Fr(\))30 b(and)g(with)g Fo(f)10 b Fr(\()p Fo(x)p Fr(\))26 b(=)f Fo(f)10 b Fr(\()p Fn(j)p Fo(x)p Fn(j)p Fr(\),)30 b(there)g(exist)f(constan)n(ts)g Fo(a)h Fr(and)g Fo(b)h Fr(suc)n(h)e(that)835 4845 y(\()p Fo(i)p Fr(\))166 b Fo(f)10 b(D)27 b Fn(\032)e Fo(D)804 5017 y Fr(\()p Fo(ii)p Fr(\))166 b Fn(j)p Fo(q)s Fr(\()p Fo(f)10 b(';)15 b(f)10 b(')p Fr(\))p Fn(j)25 b(\024)g Fo(aq)s Fr(\()p Fo(';)15 b(')p Fr(\))20 b(+)g Fo(b)15 b Fn(h)q Fo(';)g(')p Fn(i)773 5189 y Fr(\()p Fo(iii)p Fr(\))166 b Fo(q)s Fr(\()p Fo(f)1234 5153 y Fl(2)1274 5189 y Fo(';)15 b(')p Fr(\))k(+)h Fo(q)s Fr(\()p Fo(';)15 b(f)1807 5153 y Fl(2)1847 5189 y Fo(')p Fr(\))20 b Fn(\000)g Fr(2)p Fo(q)s Fr(\()p Fo(f)10 b(';)15 b(f)10 b(')p Fr(\))25 b(=)g Fn(\000)p Fr(2)2725 5115 y Fj(\012)2768 5189 y Fo(';)15 b Fn(jr)p Fo(f)10 b Fn(j)3046 5153 y Fl(2)3085 5189 y Fo(')3144 5115 y Fj(\013)120 5407 y Fr(for)39 b(all)h Fo(')i Fn(2)f Fo(D)r Fr(.)70 b(Requiremen)n(ts)39 b(\(i\))h(and)g(\(ii\))g(are)g(mild)g(tec)n(hnical)f(assumptions)g (whic)n(h)g(ensure)h(that)p eop %%Page: 4 4 4 3 bop 120 -200 a Fr(4)120 99 y(prop)r(ert)n(y)38 b(\(iii\))h(extends) f(to)h(all)g Fo(')f Fr(in)h(the)g(domain)g(of)f(the)h(closure)f(of)h Fo(q)s Fr(.)66 b(Equation)39 b(\(iii\))g(is)f(the)h(basis)120 245 y(of)33 b(the)h(so)g(called)g(IMS)f(\(lo)r(calization\))i(form)n (ula)e(for)g(Sc)n(hr\177)-45 b(odinger)34 b(op)r(erators.)52 b(T)-7 b(o)33 b(v)n(erify)f(it)i(in)g(the)g(case)120 392 y(where)e Fo(q)k Fr(is)c(de\014ned)g(in)h(terms)f(of)g(a)g (symmetric)f(op)r(erator)2235 369 y(~)2211 392 y Fo(H)36 b Fr(:)29 b Fo(D)i Fn(\032)e(H)g(!)g(H)k Fr(it)g(is)f(useful)f(to)i (note)f(that)120 539 y Fo(f)174 507 y Fl(2)237 516 y Fr(~)213 539 y Fo(H)27 b Fr(+)429 516 y(~)405 539 y Fo(H)7 b(f)541 507 y Fl(2)600 539 y Fn(\000)20 b Fr(2)p Fo(f)813 516 y Fr(~)789 539 y Fo(H)7 b(f)35 b Fr(=)25 b([[)1118 516 y(~)1095 539 y Fo(H)7 b(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(].)40 b(Assumption)28 b(\()p Fo(iii)p Fr(\))i(then)g(b)r (ecomes)1569 786 y([[)1642 763 y(~)1619 786 y Fo(H)6 b(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(])25 b(=)g Fn(\000)p Fr(2)p Fn(jr)p Fo(f)10 b Fn(j)2352 750 y Fl(2)120 1033 y Fr(whic)n(h)38 b(holds)g(for)g(the)h(p)r(ositiv)n(e)f(Laplacian) h(\()p Fn(\000)p Fr(\001\))f(and)h(hence)g(for)f(all)g(op)r(erators)h Fn(\000)p Fr(\001)26 b(+)g Fo(I)45 b Fr(in)38 b Fn(H)i Fr(with)120 1180 y([[)p Fo(I)7 b(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(])25 b(=)h(0.)41 b(Some)30 b(examples,)g(other)g (than)h(those)e(in)i(the)f(next)g(section,)g(are)2986 1157 y(~)2963 1180 y Fo(H)i Fr(=)25 b(\()p Fn(\000)p Fo(i)p Fn(r)20 b Fr(+)h Fo(A)p Fr(\()p Fo(x)p Fr(\)\))3710 1148 y Fl(2)3770 1180 y Fr(+)120 1326 y Fo(V)f Fr(\()p Fo(x)p Fr(\))40 b(with)f(a)g(classical)g(v)n(ector)f(p)r(oten)n(tial)h Fo(A)p Fr(\()p Fo(x)p Fr(\))h(and)f(scalar)g(p)r(oten)n(tial)g Fo(V)21 b Fr(\()p Fo(x)p Fr(\))39 b(\(c)n(ho)r(ose)g Fn(F)49 b Fr(=)40 b Fk(C)17 b Fr(\))q(,)47 b(and)120 1473 y(Sc)n(hr\177)-45 b(odinger)32 b(op)r(erators)g(with)g(restricted) g(domains)g(\012)c Fn(\032)h Fk(R)2235 1442 y Fm(n)2320 1473 y Fr(\()p Fn(H)g Fr(=)g Fo(L)2620 1442 y Fl(2)2659 1473 y Fr(\(\012\))g Fn(\032)f Fo(L)2982 1442 y Fl(2)3022 1473 y Fr(\()p Fk(R)3116 1442 y Fm(n)3169 1473 y Fr(\))22 b Fn(\012)f Fk(C)d Fr(\),)39 b(or)32 b(with)g(a)120 1620 y(translational)e(symmetry)e(\()p Fn(F)34 b Fr(=)25 b Fo(L)1356 1589 y Fl(2)1396 1620 y Fr(\()p Fk(R)r Fr(\)\))37 b(suc)n(h)29 b(as)g(in)h(w)n(a)n(v)n(e)e(guides)i(de\014ned)g(b)n(y)f (p)r(oten)n(tial)h(w)n(ells.)261 1770 y(Giv)n(en)f Fo(R)d(>)f Fr(0)30 b(let)g Fo(D)991 1785 y Fm(R)1073 1770 y Fr(=)25 b Fn(f)p Fo(')g Fn(2)g Fo(D)i Fr(:)e Fo(')p Fr(\()p Fo(x)p Fr(\))g(=)g(0)30 b(for)g Fn(j)p Fo(x)p Fn(j)25 b Fo(<)g(R)q Fn(g)30 b Fr(and)g(de\014ne)1062 2017 y(\006)1127 2032 y Fm(R)1210 2017 y Fr(=)190 b(inf)1305 2082 y Fm(')p Fv(2)p Fm(D)1456 2093 y Fi(R)1506 2082 y Fm(;)12 b Fv(k)p Fm(')p Fv(k)p Fl(=1)1759 2017 y Fo(q)s Fr(\()p Fo(';)j(')p Fr(\))90 b(and)g(\006)25 b(=)60 b(lim)2540 2077 y Fm(R)p Fv(!1)2750 2017 y Fr(\006)2815 2032 y Fm(R)2873 2017 y Fo(:)827 b Fr(\(4\))120 2284 y(The)36 b(n)n(um)n(b)r(ers)f(\006)745 2299 y Fm(R)838 2284 y Fr(are)h(\014nite,)h(b)r(ecause,)g(b)n(y)e (\(i\),)i Fo(D)1984 2299 y Fm(R)2077 2284 y Fr(is)f(not)f(empt)n(y)-7 b(.)56 b(But)36 b(\006)f(ma)n(y)g(tak)n(e)g(on)h(the)f(v)-5 b(alue)120 2431 y(+)p Fn(1)p Fr(.)120 2664 y Ff(Theorem)42 b(1.)i Fg(L)-5 b(et)39 b Fo(q)i Fg(b)-5 b(e)39 b(a)f(semi-b)-5 b(ounde)g(d,)40 b(closable,)f(quadr)-5 b(atic)39 b(form)e(on)g Fn(H)i Fg(satisfying)c(assumptions)120 2811 y(\(i\),)f(\(ii\),)g(and)g (\(iii\))f(ab)-5 b(ove,)37 b(and)d(let)h Fo(H)42 b Fg(denote)35 b(the)g(unique)g(self-adjoint)e(op)-5 b(er)g(ator)36 b(asso)-5 b(ciate)g(d)35 b(with)f(the)120 2958 y(closur)-5 b(e)33 b(of)f(the)g(form)f Fo(q)s Fg(.)41 b(If)32 b Fo(\025)g Fg(and)g Fo(\014)37 b Fg(ar)-5 b(e)32 b(r)-5 b(e)g(al)33 b(numb)-5 b(ers)33 b(with)e Fo(\025)20 b Fr(+)g Fo(\014)2542 2926 y Fl(2)2606 2958 y Fo(<)25 b Fr(\006)p Fg(,)32 b(then)1596 3100 y Fj(\015)1596 3155 y(\015)1596 3209 y(\015)1647 3205 y Fo(e)1689 3169 y Fm(\014)s Fv(j)p Fm(x)p Fv(j)1815 3205 y Fo(E)1882 3220 y Fm(\025)1926 3205 y Fr(\()p Fo(H)7 b Fr(\))2078 3100 y Fj(\015)2078 3155 y(\015)2078 3209 y(\015)2154 3205 y Fo(<)25 b Fn(1)p Fo(;)120 3452 y Fg(wher)-5 b(e)33 b Fo(E)441 3467 y Fm(\025)486 3452 y Fr(\()p Fo(H)7 b Fr(\))32 b Fg(is)f(the)h(r)-5 b(esolution)32 b(of)g(the)g(identity)e (for)i Fo(H)7 b Fg(.)261 3685 y(R)-5 b(emarks.)47 b Fr(\(1\))33 b(F)-7 b(or)32 b(Sc)n(hr\177)-45 b(odinger)32 b(op)r(erators)h Fn(\000)p Fr(\001)22 b(+)g Fo(V)53 b Fr(on)32 b(op)r(en)i(domains)e (\012)d Fn(\032)h Fk(R)3205 3654 y Fm(n)3291 3685 y Fr(with)j(Diric)n (hlet)120 3832 y(b)r(oundary)26 b(conditions)g(and)g(with)g Fo(V)1372 3847 y Fv(\000)1457 3832 y Fn(\034)e(\000)p Fr(\001)i(the)h(ab)r(o)n(v)n(e)e(theorem)h(implies)g(that)g(the)g(sp)r (ectrum)g(b)r(elo)n(w)g(\006)120 3979 y(is)i(discrete.)39 b(In)29 b(fact)f(\()p Fn(\000)p Fr(\001)17 b(+)h(1\))1227 3947 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1391 3979 y Fo(e)1433 3947 y Fv(\000)p Fm(\014)s Fv(j)p Fm(x)p Fv(j)1643 3979 y Fr(is)28 b(compact)h(and)g(hence)f(so)h(is)f Fo(E)2779 3994 y Fm(\025)2824 3979 y Fr(\()p Fo(H)7 b Fr(\))25 b(=)f([\()p Fo(H)h Fr(+)17 b Fo(i)p Fr(\))3408 3947 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)3573 3979 y Fr(\()p Fn(\000)p Fr(\001)g(+)120 4126 y(1\))200 4094 y Fl(1)p Fm(=)p Fl(2)310 4126 y Fr(][\()p Fn(\000)p Fr(\001)j(+)g(1\))730 4094 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)895 4126 y Fo(e)937 4094 y Fv(\000)p Fm(\014)s Fv(j)p Fm(x)p Fv(j)1118 4126 y Fr(][)p Fo(e)1210 4094 y Fm(\014)s Fv(j)p Fm(x)p Fv(j)1336 4126 y Fo(E)1403 4141 y Fm(\025)1447 4126 y Fr(\()p Fo(H)7 b Fr(\)][)p Fo(E)1716 4141 y Fm(\025)1761 4126 y Fr(\()p Fo(H)g Fr(\)\()p Fo(H)27 b Fr(+)20 b Fo(i)p Fr(\))2206 4094 y Fl(1)p Fm(=)p Fl(2)2316 4126 y Fr(])30 b(for)f Fo(\025)21 b Fr(+)f Fo(\014)2727 4094 y Fl(2)2791 4126 y Fo(<)25 b Fr(\006.)261 4275 y(\(2\))34 b(Ev)n(erything)e(in)i(this)f (section)g(equally)g(holds)g(for)g(an)n(y)g(norm)g Fn(j)p Fo(x)p Fn(j)h Fr(on)g Fk(R)2898 4244 y Fm(n)2985 4275 y Fr(that)g(is)f(induced)g(b)n(y)g(an)120 4422 y(inner)i(pro)r(duct)g Fo(x)24 b Fn(\001)f Fo(y)s Fr(,)36 b(if)e(\001)h(is)f(used)h(to)f (denote)h(the)g(Laplace-Beltrami)g(op)r(erator)g(with)g(resp)r(ect)g (to)g(the)120 4569 y(metric)30 b Fo(g)s Fr(\()p Fo(x;)15 b(y)s Fr(\))26 b(=)f Fo(x)20 b Fn(\001)g Fo(y)s Fr(.)261 4865 y(The)30 b(follo)n(wing)f(pro)r(of)i(is)e(inspired)h(b)n(y)f(the)h (pro)r(of)g(of)g(binding)g(in)g(Bac)n(h)f(et)h(al.[2].)120 5114 y Fg(Pr)-5 b(o)g(of.)45 b Fr(Let)33 b Fo(Q)p Fr(\()p Fo(H)7 b Fr(\))31 b Fn(\032)f(H)k Fr(denote)f(the)h(form)e(domain)i(of) f Fo(H)7 b Fr(,)33 b(i.e.,)i(the)e(domain)g(of)g(the)g(closure)g(of)g Fo(q)s Fr(.)50 b(W)-7 b(e)120 5260 y(use)26 b Fo(q)k Fr(to)c(denote)h(the)f(closure)g(of)g Fo(q)k Fr(as)c(w)n(ell.)38 b Fo(Q)p Fr(\()p Fo(H)7 b Fr(\))26 b(is)g(the)h(closure)f(of)g Fo(D)i Fr(with)f(resp)r(ect)f(to)h(the)f(form)g(norm)120 5407 y Fn(k)d(\001)g(k)281 5422 y Fm(q)354 5407 y Fr(asso)r(ciated)34 b(with)h Fo(q)s Fr(.)55 b(By)34 b(assumptions)f(\(i\))i(and)g(\(ii\),)h (m)n(ultiplication)e(with)h(a)g(b)r(ounded)g(function)p eop %%Page: 5 5 5 4 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2312 b Fr(5)120 99 y Fo(f)45 b Fn(2)35 b Fo(C)375 67 y Fv(1)450 99 y Fr(\()p Fk(R)544 67 y Fm(n)597 99 y Fr(\))h(with)g(b)r(ounded)h(deriv)-5 b(ativ)n(es)34 b(is)i(a)g(b)r(ounded)h(linear)f(op)r(erator)h(on)f(\()p Fo(D)r(;)15 b Fn(k)24 b(\001)g(k)3330 114 y Fm(q)3368 99 y Fr(\))36 b(and)g(hence)120 245 y(extends)29 b(to)h(a)g(b)r(ounded) h(linear)f(op)r(erator)h(on)f(\()p Fo(Q)p Fr(\()p Fo(H)7 b Fr(\))p Fo(;)15 b Fn(k)20 b(\001)g(k)2192 260 y Fm(q)2230 245 y Fr(\).)40 b(In)29 b(particular)1670 478 y Fo(f)10 b(Q)p Fr(\()p Fo(H)d Fr(\))25 b Fn(\032)g Fo(Q)p Fr(\()p Fo(H)7 b Fr(\))1435 b(\(5\))120 711 y(and)30 b(\(iii\))g(extends)f (from)h Fo(D)i Fr(to)e Fo(Q)p Fr(\()p Fo(H)7 b Fr(\).)261 857 y(Let)36 b Fo(E)k Fr(=)35 b(inf)21 b Fo(\033)s Fr(\()p Fo(H)7 b Fr(\))37 b(and)f(supp)r(ose)g Fo(\025)f Fn(\025)g Fo(E)5 b Fr(.)58 b(Otherwise)36 b(the)g(assertion)f(of)g(the)i(theorem) f(is)f(trivial.)120 1004 y(First)29 b(w)n(e)h(sho)n(w)f(that)1107 1151 y Fo(H)1182 1166 y Fm(R)1264 1151 y Fr(:=)c Fo(H)i Fr(+)20 b(max)o Fn(f)p Fr(\006)1853 1166 y Fm(R)1931 1151 y Fn(\000)g Fo(E)5 b(;)15 b Fr(0)p Fn(g)p Fo(\037)2279 1166 y Fl(2)p Fm(R)2397 1151 y Fn(\025)25 b Fr(\006)2557 1166 y Fm(R)2635 1151 y Fn(\000)2754 1087 y Fo(C)p 2735 1130 109 4 v 2735 1215 a(R)2804 1188 y Fl(2)3725 1151 y Fr(\(6\))120 1349 y(where)36 b Fo(\037)442 1364 y Fl(2)p Fm(R)571 1349 y Fr(denotes)f(the)g(c)n(haracteristic)g(function)g(of)g (the)h(set)f Fn(f)p Fo(x)g Fr(:)f Fn(j)p Fo(x)p Fn(j)g(\024)g Fr(2)p Fo(R)q Fn(g)i Fr(and)g Fo(C)42 b Fr(is)35 b(a)h(constan)n(t.)120 1496 y(Pic)n(k)i Fo(j)367 1511 y Fl(1)407 1496 y Fo(;)15 b(j)484 1511 y Fl(2)563 1496 y Fn(2)39 b Fo(C)733 1464 y Fv(1)808 1496 y Fr(\()p Fk(R)902 1511 y Fl(+)967 1496 y Fr(\))g(with)g Fo(j)1297 1464 y Fl(2)1292 1522 y(1)1362 1496 y Fr(+)26 b Fo(j)1500 1464 y Fl(2)1495 1522 y(2)1579 1496 y Fn(\021)40 b Fr(1,)h(supp)o(\()p Fo(j)2057 1511 y Fl(1)2097 1496 y Fr(\))f Fn(\032)f(f)p Fo(t)h Fn(\024)f Fr(2)p Fn(g)g Fr(and)g(supp\()p Fo(j)3078 1511 y Fl(2)3117 1496 y Fr(\))h Fn(\032)f(f)p Fo(t)h Fn(\025)g Fr(1)p Fn(g)p Fr(.)66 b(Let)120 1642 y Fo(j)157 1657 y Fm(i;R)258 1642 y Fr(\()p Fo(x)p Fr(\))26 b(=)f Fo(j)537 1657 y Fm(i)565 1642 y Fr(\()p Fn(j)p Fo(x)p Fn(j)p Fo(=R)q Fr(\).)40 b(Then)30 b(b)n(y)h(\(5\))f(and)g(since)g(\(iii\))g(holds)f (on)h Fo(Q)p Fr(\()p Fo(H)7 b Fr(\),)1238 1930 y Fo(H)1313 1945 y Fm(R)1454 1930 y Fr(=)1617 1866 y(1)p 1617 1909 46 4 v 1617 1994 a(2)1735 1816 y Fl(2)1687 1844 y Fj(X)1695 2039 y Fm(i)p Fl(=1)1833 1856 y Fj(\000)1875 1930 y Fo(j)1917 1894 y Fl(2)1912 1954 y Fm(i;R)2013 1930 y Fo(H)2088 1945 y Fm(R)2166 1930 y Fr(+)20 b Fo(H)2331 1945 y Fm(R)2388 1930 y Fo(j)2430 1894 y Fl(2)2425 1954 y Fm(i;R)2526 1856 y Fj(\001)1454 2247 y Fr(=)1655 2133 y Fl(2)1607 2160 y Fj(X)1615 2356 y Fm(i)p Fl(=1)1753 2247 y Fo(j)1790 2262 y Fm(i;R)1892 2247 y Fo(H)1967 2262 y Fm(R)2024 2247 y Fo(j)2061 2262 y Fm(i;R)2182 2247 y Fn(\000)2320 2133 y Fl(2)2272 2160 y Fj(X)2281 2356 y Fm(i)p Fl(=1)2419 2247 y Fn(jr)p Fo(j)2556 2262 y Fm(i;R)2657 2247 y Fn(j)2682 2211 y Fl(2)120 2527 y Fr(in)30 b(the)g(sense)f(of)h(forms)f(on)h Fo(Q)p Fr(\()p Fo(H)7 b Fr(\).)40 b(By)29 b(de\014nition)i(of)e(\006) 2084 2542 y Fm(R)2172 2527 y Fr(and)h(the)g(construction)g(of)f Fo(j)3157 2542 y Fm(i;R)3259 2527 y Fr(,)1068 2752 y Fo(j)1105 2767 y Fl(1)p Fm(;R)1232 2752 y Fo(H)1307 2767 y Fm(R)1380 2752 y Fo(j)1417 2767 y Fl(1)p Fm(;R)1554 2752 y Fn(\025)c Fo(j)1686 2767 y Fl(1)p Fm(;R)1799 2752 y Fr(\()p Fo(H)i Fr(+)20 b(\006)2091 2767 y Fm(R)2169 2752 y Fn(\000)g Fo(E)5 b Fr(\))p Fo(j)2403 2767 y Fl(1)p Fm(;R)2540 2752 y Fn(\025)25 b Fr(\006)2700 2767 y Fm(R)2758 2752 y Fo(j)2800 2716 y Fl(2)2795 2776 y(1)p Fm(;R)1068 2924 y Fo(j)1105 2939 y Fl(2)p Fm(;R)1232 2924 y Fo(H)1307 2939 y Fm(R)1380 2924 y Fo(j)1417 2939 y Fl(2)p Fm(;R)1554 2924 y Fn(\025)g Fo(j)1686 2939 y Fl(2)p Fm(;R)1799 2924 y Fo(H)7 b(j)1918 2939 y Fl(2)p Fm(;R)2056 2924 y Fn(\025)25 b Fr(\006)2216 2939 y Fm(R)2273 2924 y Fo(j)2315 2888 y Fl(2)2310 2948 y(2)p Fm(;R)2423 2924 y Fo(:)120 3130 y Fr(Hence)30 b(\(6\))g(follo)n(ws.)261 3277 y(No)n(w)23 b(let)g(\001)i(:=)g([inf)c Fo(\033)s Fr(\()p Fo(H)7 b Fr(\))p Fo(;)15 b(\025)p Fr(])25 b(with)e Fo(\025)h Fr(as)f(in)h(the)g (statemen)n(t)e(of)h(the)h(theorem)f(and)h(pic)n(k)f Fo(R)j Fn(2)f Fk(R)32 b Fr(so)23 b(large)120 3424 y(that)j Fo(\025)11 b Fr(+)g Fo(\014)511 3392 y Fl(2)575 3424 y Fo(<)25 b Fr(\006)735 3439 y Fm(R)803 3424 y Fn(\000)11 b Fo(C)t(=R)1066 3392 y Fl(2)1106 3424 y Fr(.)38 b(Keep)26 b(this)f Fo(R)h Fr(\014xed)f(in)h(the)f(follo)n(wing.)38 b(Let)26 b Fo(\016)i Fr(:=)d(\006)2948 3439 y Fm(R)3017 3424 y Fn(\000)11 b Fo(C)t(=R)3280 3392 y Fl(2)3330 3424 y Fn(\000)g Fo(\014)3467 3392 y Fl(2)3517 3424 y Fn(\000)g Fo(\025)25 b(>)g Fr(0,)120 3570 y(and)36 b(c)n(ho)r(ose)g(a)g(function) g Fo(g)1076 3585 y Fl(\001)1174 3570 y Fn(2)g Fo(C)1341 3539 y Fv(1)1334 3596 y Fl(0)1415 3570 y Fr(\()p Fk(R)s Fr(\))42 b(suc)n(h)35 b(that)i Fo(g)2041 3585 y Fl(\001)2139 3570 y Fn(\021)e Fr(1)h(on)h(\001)f(and)g(supp\()p Fo(g)3013 3585 y Fl(\001)3075 3570 y Fr(\))g Fn(\032)f Fr(\()p Fn(\0001)p Fo(;)15 b(\025)25 b Fr(+)f Fo(\016)s(=)p Fr(2].)120 3717 y(Then,)30 b(b)n(y)g(\(6\),)h Fo(g)720 3732 y Fl(\001)782 3717 y Fr(\()p Fo(H)892 3732 y Fm(R)950 3717 y Fr(\))25 b(=)g(0)30 b(and)g(therefore)1453 3950 y Fo(g)1496 3965 y Fl(\001)1559 3950 y Fr(\()p Fo(H)7 b Fr(\))25 b(=)g Fo(g)1874 3965 y Fl(\001)1937 3950 y Fr(\()p Fo(H)7 b Fr(\))20 b Fn(\000)g Fo(g)2242 3965 y Fl(\001)2305 3950 y Fr(\()p Fo(H)2415 3965 y Fm(R)2472 3950 y Fr(\))1218 b(\(7\))120 4182 y(W)-7 b(e)29 b(no)n(w)h(sho)n(w)f(that)h Fo(e)923 4151 y Fm(\014)s Fv(j)p Fm(x)p Fv(j)1049 4182 y Fr(\()p Fo(g)1127 4197 y Fl(\001)1190 4182 y Fr(\()p Fo(H)7 b Fr(\))19 b Fn(\000)h Fo(g)1494 4197 y Fl(\001)1557 4182 y Fr(\()p Fo(H)1667 4197 y Fm(R)1725 4182 y Fr(\)\))30 b(is)f(b)r(ounded.)41 b(T)-7 b(o)30 b(this)f(end,)h(w)n(e)g(de\014ne) 1182 4432 y Fo(f)10 b Fr(\()p Fo(x)p Fr(\))26 b(:=)1583 4368 y Fo(\014)5 b Fn(h)p Fo(x)p Fn(i)p 1513 4411 319 4 v 1513 4495 a Fr(1)20 b(+)g Fo(")p Fn(h)p Fo(x)p Fn(i)1841 4432 y Fo(;)195 b Fn(h)p Fo(x)p Fn(i)25 b Fr(=)g(\(1)20 b(+)g Fn(j)p Fo(x)p Fn(j)2593 4396 y Fl(2)2633 4432 y Fr(\))2668 4396 y Fl(1)p Fm(=)p Fl(2)120 4690 y Fr(and)33 b(sho)n(w)f(that,)i Fo(e)790 4658 y Fm(f)835 4690 y Fr(\()p Fo(g)913 4705 y Fl(\001)976 4690 y Fr(\()p Fo(H)7 b Fr(\))22 b Fn(\000)g Fo(g)1285 4705 y Fl(\001)1348 4690 y Fr(\()p Fo(H)1458 4705 y Fm(R)1515 4690 y Fr(\)\))33 b(is)g(b)r(ounded)g (uniformly)f(in)i Fo(")29 b(>)h Fr(0.)49 b(Note)33 b(that)g Fo(f)40 b Fn(2)30 b Fo(C)3558 4658 y Fv(1)3632 4690 y Fr(\()p Fk(R)3727 4658 y Fm(n)3780 4690 y Fr(\),)120 4837 y(is)e(b)r(ounded)i(and)e(that)h Fn(jr)p Fo(f)10 b Fn(j)25 b(\024)g Fo(\014)5 b Fr(.)39 b(Let)33 b(~)-48 b Fo(g)1566 4852 y Fl(\001)1657 4837 y Fr(b)r(e)30 b(the)e(almost)h (analytic)f(extension)j(~)-48 b Fo(g)3003 4852 y Fl(\001)3066 4837 y Fr(\()p Fo(x)18 b Fr(+)f Fo(iy)s Fr(\))25 b(=)g(\()p Fo(g)3568 4852 y Fl(\001)3631 4837 y Fr(\()p Fo(x)p Fr(\))18 b(+)120 4984 y Fo(iy)s(g)244 4952 y Fv(0)241 5012 y Fl(\001)304 4984 y Fr(\()p Fo(x)p Fr(\)\))p Fo(\015)5 b Fr(\()p Fo(y)s Fr(\))29 b(where)g Fo(\015)g Fn(2)c Fo(C)1149 4952 y Fv(1)1142 5009 y Fl(0)1224 4984 y Fr(\()p Fk(R)s Fr(\))34 b(equals)28 b(one)h(in)g(a)g(neigh)n(b)r(orho)r(o)r(d)h(of)e Fo(y)g Fr(=)d(0.)40 b(By)28 b(the)h(almost)f(analytic)120 5130 y(functional)i(calculus)f(\(see)h([4]\))1287 5375 y Fo(g)1330 5390 y Fl(\001)1393 5375 y Fr(\()p Fo(H)7 b Fr(\))25 b(=)g Fn(\000)1750 5310 y Fr(1)p 1745 5354 55 4 v 1745 5438 a Fo(\031)1824 5251 y Fj(Z)1940 5310 y Fo(@)8 b Fr(~)-48 b Fo(g)p 1940 5354 99 4 v 1940 5438 a(@)10 b Fr(\026)-50 b Fo(z)2049 5375 y Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)7 b Fr(\))2357 5339 y Fv(\000)p Fl(1)2466 5375 y Fo(dx)15 b(dy)p eop %%Page: 6 6 6 5 bop 120 -200 a Fr(6)120 99 y(and)30 b(hence,)g(using)h(\(7\))f(and) g(a)g(resolv)n(en)n(t)e(iden)n(tit)n(y)h(w)n(e)h(can)g(write)632 331 y Fo(e)674 295 y Fm(f)719 331 y Fo(g)762 346 y Fl(\001)825 331 y Fr(\()p Fo(H)7 b Fr(\))25 b(=)1112 267 y(1)p 1107 310 55 4 v 1107 395 a Fo(\031)1187 207 y Fj(Z)1302 267 y Fo(@)8 b Fr(~)-48 b Fo(g)p 1302 310 99 4 v 1302 395 a(@)10 b Fr(\026)-50 b Fo(z)1411 331 y(e)1453 295 y Fm(f)1498 331 y Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)1764 346 y Fm(R)1821 331 y Fr(\))1856 295 y Fv(\000)p Fl(1)1951 331 y Fo(e)1993 295 y Fv(\000)p Fm(f)2093 331 y Fo(e)2135 295 y Fm(f)2180 331 y Fr(\(\006)2280 346 y Fm(R)2358 331 y Fn(\000)g Fo(E)5 b Fr(\))p Fo(\037)2611 346 y Fl(2)p Fm(R)2704 331 y Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)7 b Fr(\))3012 295 y Fv(\000)p Fl(1)3121 331 y Fo(dx)15 b(dy)120 552 y Fr(whose)30 b(norm)g(w)n(e)f(estimate)h(from)f(ab)r(o)n(v)n(e)h(as) 696 773 y Fn(k)p Fo(e)783 737 y Fm(f)829 773 y Fo(g)872 788 y Fl(\001)935 773 y Fr(\()p Fo(H)7 b Fr(\))p Fn(k)82 b(\024)175 b Fr(sup)1367 855 y Fm(z)s Fv(2)p Fl(supp)q(\()r(~)-37 b Fm(g)r Fl(\))1701 773 y Fn(k)p Fo(e)1788 737 y Fm(f)1833 773 y Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)2099 788 y Fm(R)2157 773 y Fr(\))2192 737 y Fv(\000)p Fl(1)2286 773 y Fo(e)2328 737 y Fv(\000)p Fm(f)2428 773 y Fn(k)15 b(k)p Fo(e)2575 737 y Fm(f)2620 773 y Fo(\037)2676 788 y Fl(2)p Fm(R)2769 773 y Fn(k)2814 788 y Fv(1)2889 773 y Fr(\(\006)2989 788 y Fm(R)3067 773 y Fn(\000)20 b Fo(E)5 b Fr(\))1367 1046 y Fn(\002)1452 982 y Fr(1)p 1447 1025 55 4 v 1447 1110 a Fo(\031)1527 922 y Fj(Z)1633 914 y(\014)1633 969 y(\014)1633 1023 y(\014)1633 1078 y(\014)1673 982 y Fo(@)j Fr(~)-48 b Fo(g)p 1673 1025 99 4 v 1673 1110 a(@)10 b Fr(\026)-50 b Fo(z)1782 914 y Fj(\014)1782 969 y(\014)1782 1023 y(\014)1782 1078 y(\014)1827 1046 y Fn(k)p Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)7 b Fr(\))2180 1010 y Fv(\000)p Fl(1)2274 1046 y Fn(k)15 b Fo(dx)g(dy)s(:)120 1292 y Fr(The)39 b(norm)h Fn(k)p Fo(e)646 1261 y Fm(f)691 1292 y Fo(\037)747 1307 y Fl(2)p Fm(R)840 1292 y Fn(k)885 1307 y Fv(1)999 1292 y Fr(is)f(b)r(ounded)h(uniformly)f(in)g Fo(")i(>)g Fr(0)e(and)h(the)f(in)n(tegral)g(is)g(\014nite.)68 b(T)-7 b(o)39 b(estimate)120 1439 y Fn(k)p Fo(e)207 1408 y Fm(f)252 1439 y Fr(\()p Fo(z)24 b Fn(\000)c Fo(H)518 1454 y Fm(R)575 1439 y Fr(\))610 1408 y Fv(\000)p Fl(1)705 1439 y Fo(e)747 1408 y Fv(\000)p Fm(f)847 1439 y Fn(k)30 b Fr(let)g Fo(H)1127 1454 y Fm(R;f)1270 1439 y Fr(:=)25 b Fo(e)1432 1408 y Fm(f)1477 1439 y Fo(H)1552 1454 y Fm(R)1609 1439 y Fo(e)1651 1408 y Fv(\000)p Fm(f)1781 1439 y Fr(with)31 b(domain)f Fo(D)r Fr(\()p Fo(H)2494 1454 y Fm(R;f)2612 1439 y Fr(\))25 b(=)g Fo(e)2809 1408 y Fm(f)2854 1439 y Fo(D)r Fr(\()p Fo(H)7 b Fr(\))30 b(and)g(note)g (that)1323 1660 y(\()p Fo(z)23 b Fn(\000)d Fo(H)1588 1675 y Fm(R;f)1706 1660 y Fr(\))1741 1624 y Fv(\000)p Fl(1)1861 1660 y Fr(=)25 b Fo(e)1998 1624 y Fm(f)2043 1660 y Fr(\()p Fo(z)f Fn(\000)c Fo(H)2309 1675 y Fm(R)2366 1660 y Fr(\))2401 1624 y Fv(\000)p Fl(1)2495 1660 y Fo(e)2537 1624 y Fv(\000)p Fm(f)120 1881 y Fr(as)j(can)h(easily)f(b)r(e)h(seen)g (b)n(y)f(direct)g(computation.)39 b(In)23 b(particular,)i(the)f(resolv) n(en)n(t)e(sets)h Fo(\032)p Fr(\()p Fo(H)3245 1896 y Fm(R;f)3364 1881 y Fr(\))g(and)h Fo(\032)p Fr(\()p Fo(H)3747 1896 y Fm(R)3805 1881 y Fr(\))120 2028 y(coincide.)40 b(Let)31 b Fo(')24 b Fn(2)h Fo(D)r Fr(\()p Fo(H)1017 2043 y Fm(R;f)1136 2028 y Fr(\))g Fn(\032)g Fo(Q)p Fr(\()p Fo(H)7 b Fr(\))30 b(and)g Fn(k)p Fo(')p Fn(k)24 b Fr(=)h(1.)41 b(Then)937 2249 y(2)15 b(Re)1118 2175 y Fj(\012)1161 2249 y Fo(';)g(H)1335 2264 y Fm(R;f)1453 2249 y Fo(')1512 2175 y Fj(\013)1638 2249 y Fr(=)1791 2175 y Fj(\012)1834 2249 y Fo(';)g Fr(\()p Fo(e)2010 2213 y Fm(f)2055 2249 y Fo(H)2130 2264 y Fm(R)2187 2249 y Fo(e)2229 2213 y Fv(\000)p Fm(f)2349 2249 y Fr(+)20 b Fo(e)2481 2213 y Fv(\000)p Fm(f)2581 2249 y Fo(H)2656 2264 y Fm(R)2714 2249 y Fo(e)2756 2213 y Fm(f)2801 2249 y Fr(\))p Fo(')2895 2175 y Fj(\013)1638 2420 y Fr(=)1791 2347 y Fj(\012)1834 2420 y Fo(';)15 b(e)1975 2385 y Fv(\000)p Fm(f)2075 2420 y Fr(\()p Fo(e)2152 2385 y Fl(2)p Fm(f)2232 2420 y Fo(H)2307 2435 y Fm(R)2385 2420 y Fr(+)20 b Fo(H)2550 2435 y Fm(R)2607 2420 y Fo(e)2649 2385 y Fl(2)p Fm(f)2729 2420 y Fr(\))p Fo(e)2806 2385 y Fv(\000)p Fm(f)2906 2420 y Fo(')2965 2347 y Fj(\013)1638 2592 y Fr(=)83 b(2)1851 2518 y Fj(\012)1894 2592 y Fo(';)15 b Fr(\()p Fo(H)2103 2607 y Fm(R)2180 2592 y Fn(\000)20 b(jr)p Fo(f)10 b Fn(j)2449 2556 y Fl(2)2488 2592 y Fr(\))p Fo(')2582 2518 y Fj(\013)120 2813 y Fr(where)39 b(\()p Fo(iii)p Fr(\))g(w)n(as)f(used)h(in)g(the)g(last)g(equation.)67 b(In)39 b(conjunction)g(with)g(\(6\))g(this)g(sho)n(ws)f(that,)j(for)e Fo(z)44 b Fn(2)120 2960 y Fr(supp\()s(~)-48 b Fo(g)s Fr(\),)906 3106 y(Re)1027 3033 y Fj(\012)1070 3106 y Fo(';)15 b Fr(\()p Fo(H)1279 3121 y Fm(R;f)1417 3106 y Fn(\000)20 b Fo(z)t Fr(\))p Fo(')1647 3033 y Fj(\013)1714 3106 y Fn(\025)25 b Fr(\006)1874 3121 y Fm(R)1952 3106 y Fn(\000)20 b Fo(C)t(=R)2224 3071 y Fl(2)2284 3106 y Fn(\000)g Fo(\014)2430 3071 y Fl(2)2489 3106 y Fn(\000)g Fr(Re\()p Fo(z)t Fr(\))25 b Fn(\025)g Fo(\016)s(=)p Fr(2)671 b(\(8\))120 3299 y(and)30 b(hence)g(that)g Fn(k)p Fr(\()p Fo(H)895 3314 y Fm(R;f)1033 3299 y Fn(\000)20 b Fo(z)t Fr(\))p Fo(')p Fn(k)25 b(\025)g Fo(\016)s(=)p Fr(2)p Fn(k)p Fo(')p Fn(k)p Fr(.)40 b(Since)30 b Fo(\032)p Fr(\()p Fo(H)2166 3314 y Fm(R;f)2285 3299 y Fr(\))25 b(=)g Fo(\032)p Fr(\()p Fo(H)2596 3314 y Fm(R)2654 3299 y Fr(\))g Fn(\033)g Fr(supp)o(\()s(~)-48 b Fo(g)t Fr(\),)30 b(it)g(follo)n(ws)e(that)1552 3519 y Fn(k)p Fr(\()p Fo(z)c Fn(\000)c Fo(H)1863 3534 y Fm(R;f)1981 3519 y Fr(\))2016 3484 y Fv(\000)p Fl(1)2110 3519 y Fn(k)25 b(\024)g Fr(2)p Fo(=\016)120 3740 y Fr(for)30 b Fo(z)e Fn(2)d Fr(supp\()s(~)-48 b Fo(g)s Fr(\),)30 b(whic)n(h)g(completes)g(the)g(pro)r(of.)p 3775 3740 4 61 v 3779 3683 55 4 v 3779 3740 V 3832 3740 4 61 v 120 4069 a Fs(3)132 b(A)l(toms)44 b(Coupled)i(to)f(Quan)l(tized)h (Radiation)120 4306 y Fr(In)35 b(this)h(section)f(w)n(e)g(apply)h(the)g (abstract)f(result)g(of)h(the)f(previous)g(section)h(to)g(systems)d(of) j Fo(N)45 b Fr(c)n(harged,)120 4452 y(non-relativistic)30 b(quan)n(tum)g(particles,)h(in)n(teracting)f(with)h(the)g(quan)n(tized) f(radiation)g(\014eld.)43 b(Since)31 b(w)n(e)f(are)120 4599 y(mainly)e(in)n(terested)g(in)h(the)g(case)f(of)g(electrons)h(in)f (the)h(\014eld)g(of)f(static)h(n)n(uclei,)f(the)h(bulk)f(of)h(the)f (exp)r(osition)120 4746 y(deals)c(with)g(this)g(case.)38 b(A)n(t)23 b(the)h(end)h(w)n(e)e(commen)n(t)h(on)g(the)h(more)f (general)g(case)g(of)g(particles)g(from)g(di\013eren)n(t)120 4893 y(sp)r(ecies.)261 5040 y(In)e(the)g("standard)g(mo)r(del")h(of)f (non-relativistic)g(QED)g(the)h(Hilb)r(ert)f(space)g(of)g(a)g(system)f (of)h Fo(N)32 b Fr(electrons)120 5186 y(and)e(an)g(arbitrary)g(n)n(um)n (b)r(er)f(of)h(transv)n(ersal)e(photons)i(is)f(the)i(tensor)e(pro)r (duct)1439 5407 y Fn(H)1515 5422 y Fm(N)1607 5407 y Fr(=)c Fn(^)1762 5372 y Fm(N)1762 5430 y(i)p Fl(=1)1880 5407 y Fo(L)1941 5372 y Fl(2)1981 5407 y Fr(\()p Fk(R)2076 5372 y Fl(3)2121 5407 y Fr(;)15 b Fk(C)2221 5372 y Fl(2)2266 5407 y Fr(\))20 b Fn(\012)g(F)2476 5422 y Fm(f)p eop %%Page: 7 7 7 6 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2312 b Fr(7)120 99 y(of)35 b(the)h(an)n(tisymmetric)f(pro)r (duct)i(of)e Fo(N)46 b Fr(copies)36 b(of)f Fo(L)1997 67 y Fl(2)2037 99 y Fr(\()p Fk(R)2131 67 y Fl(3)2177 99 y Fr(;)15 b Fk(C)2276 67 y Fl(2)2322 99 y Fr(\))36 b(appropriate)g(for)f Fo(N)46 b Fr(spin-1/2)36 b(fermions)120 245 y(and)k Fn(F)370 260 y Fm(f)457 245 y Fr(=)i Fn(\010)639 260 y Fm(n)p Fv(\025)p Fl(0)803 245 y Fn(\012)873 214 y Fm(n)873 266 y(s)946 245 y Fo(L)1007 214 y Fl(2)1047 245 y Fr(\()p Fk(R)1141 214 y Fl(3)1187 245 y Fo(;)15 b(dk)s Fr(;)g Fk(C)1423 214 y Fl(2)1468 245 y Fr(\))41 b(is)e(the)h(b)r(osonic)h(F)-7 b(o)r(c)n(k)38 b(space)i(o)n(v)n(er)f Fo(L)2881 214 y Fl(2)2920 245 y Fr(\()p Fk(R)3015 214 y Fl(3)3061 245 y Fo(;)15 b(dk)s Fr(;)g Fk(C)3297 214 y Fl(2)3342 245 y Fr(\),)43 b(where)d(the)120 392 y(factor)28 b Fk(C)435 361 y Fl(2)509 392 y Fr(accoun)n(ts)f(for)g(the)h(t)n(w)n(o) f(p)r(ossible)h(p)r(olarizations)g(of)f(the)h(transv)n(ersal)e (photons.)39 b(Let)28 b Fn(D)3509 407 y Fm(N)3602 392 y Fn(\032)d(H)3773 407 y Fm(N)120 539 y Fr(b)r(e)31 b(the)f(subspace)f (of)g(sequences)g Fo(')c Fr(=)g(\()p Fo(')1558 554 y Fl(0)1597 539 y Fo(;)15 b(')1696 554 y Fl(1)1736 539 y Fo(;)g(:)g(:)g(:)o Fr(\))30 b(where)1094 759 y Fo(')1153 774 y Fm(n)1225 759 y Fn(2)25 b Fo(C)1381 724 y Fv(1)1374 781 y Fl(0)p Fm(;a)1470 759 y Fr(\(\()p Fk(R)1600 724 y Fl(3)1666 759 y Fn(\002)20 b(f)p Fr(1)p Fo(;)15 b Fr(2)p Fn(g)p Fr(\))2011 724 y Fm(N)2078 759 y Fr(;)g Fk(C)i Fr(\))26 b Fn(\012)20 b(\012)2398 724 y Fm(n)2398 780 y(s)2445 759 y Fo(L)2506 724 y Fl(2)2506 781 y(0)2546 759 y Fr(\()p Fk(R)2641 724 y Fl(3)2686 759 y Fo(;)15 b Fk(C)2786 724 y Fl(2)2831 759 y Fr(\))120 979 y(and)39 b Fo(')363 994 y Fm(n)451 979 y Fr(=)h(0)f(for)g(all)h(but)f (\014nitely)g(man)n(y)f Fo(n)p Fr(.)68 b(The)39 b(index)g Fo(a)h Fr(indicates)f(that)g(the)h(functions)e(are)h(an)n(ti-)120 1126 y(symmetric)d(with)g(resp)r(ect)g(to)h(p)r(erm)n(utations)f(of)g (the)g Fo(N)46 b Fr(argumen)n(ts,)38 b(and)e Fo(L)2866 1095 y Fl(2)2866 1152 y(0)2906 1126 y Fr(\()p Fk(R)3000 1095 y Fl(3)3046 1126 y Fr(;)15 b Fk(C)3145 1095 y Fl(2)3191 1126 y Fr(\))36 b(is)g(the)h(space)e(of)120 1273 y(compactly)30 b(supp)r(orted)g Fo(L)1037 1242 y Fl(2)1076 1273 y Fr(-functions.)40 b(Clearly)29 b Fn(D)1911 1288 y Fm(N)2009 1273 y Fr(is)h(a)g(dense)f (subspace)g(of)h Fn(H)2975 1288 y Fm(N)3042 1273 y Fr(.)261 1420 y(The)g(Hamilton)g(op)r(erator)1230 1397 y(~)1206 1420 y Fo(H)1281 1435 y Fm(N)1373 1420 y Fr(:)25 b Fn(D)1492 1435 y Fm(N)1585 1420 y Fn(\032)g(H)1756 1435 y Fm(N)1848 1420 y Fn(!)g(H)2039 1435 y Fm(N)2136 1420 y Fr(of)30 b(our)g(system)e(is)i(giv)n(en)f(b)n(y)912 1664 y(~)889 1687 y Fo(H)964 1702 y Fm(N)1056 1687 y Fr(=)1185 1574 y Fm(N)1151 1601 y Fj(X)1155 1796 y Fm(j)t Fl(=1)1282 1687 y Fr(\()p Fo(p)1362 1702 y Fm(j)1419 1687 y Fr(+)1509 1617 y Fn(p)p 1584 1617 58 4 v 70 x Fo(\013)q(A)p Fr(\()p Fo(x)1795 1702 y Fm(j)1832 1687 y Fr(\)\))1902 1652 y Fl(2)1961 1687 y Fr(+)2061 1623 y Fo(g)p 2061 1666 47 4 v 2062 1751 a Fr(2)2117 1617 y Fn(p)p 2192 1617 58 4 v 70 x Fo(\013)q(\033)2301 1702 y Fm(j)2358 1687 y Fn(\001)20 b Fo(B)5 b Fr(\()p Fo(x)2562 1702 y Fm(j)2599 1687 y Fr(\))20 b(+)g Fo(V)40 b Fr(+)20 b Fo(H)3001 1702 y Fm(f)3046 1687 y Fo(:)654 b Fr(\(9\))120 1968 y(where)31 b Fo(p)426 1983 y Fm(j)488 1968 y Fr(=)26 b Fn(\000)p Fo(i)p Fn(r)760 1983 y Fm(x)800 1993 y Fi(j)836 1968 y Fr(,)31 b Fo(A)p Fr(\()p Fo(x)1045 1983 y Fm(j)1082 1968 y Fr(\))g(is)f(the)h(quan)n(tized)e(v)n(ector)h(p)r(oten)n(tial)h (in)f(Coulom)n(b)g(gauge)h(ev)-5 b(aluated)30 b(at)h(the)120 2115 y(p)r(oin)n(t)24 b Fo(x)400 2130 y Fm(j)437 2115 y Fr(,)i Fo(B)5 b Fr(\()p Fo(x)647 2130 y Fm(j)683 2115 y Fr(\))25 b(=)g(curl)p Fo(A)p Fr(\()p Fo(x)1141 2130 y Fm(j)1179 2115 y Fr(\))f(is)g(the)g(magnetic)h(\014eld,)g Fo(\033)2118 2130 y Fm(j)2180 2115 y Fr(the)f(triple)g(of)g(P)n(auli)g (matrices)g(\()p Fo(\033)3332 2067 y Fl(\(1\))3329 2142 y Fm(j)3427 2115 y Fo(;)15 b(\033)3521 2067 y Fl(\(2\))3518 2142 y Fm(j)3616 2115 y Fo(;)g(\033)3710 2067 y Fl(\(3\))3707 2142 y Fm(j)3805 2115 y Fr(\))120 2262 y(acting)27 b(on)f(the)g(spin)g (degrees)g(of)g(freedom)g(of)g(the)h Fo(j)5 b Fr(th)27 b(particle,)g Fo(V)47 b Fr(is)26 b(a)g(real-v)-5 b(alued)26 b(p)r(oten)n(tial,)i(and)e Fo(H)3708 2277 y Fm(f)3779 2262 y Fr(is)120 2409 y(the)31 b(Hamilton)h(op)r(erator)g(of)f(the)g (\014eld)h(energy)-7 b(.)43 b(The)31 b(parameter)h Fo(\013)g Fr(is)f(the)h(\014ne)f(structure)g(constan)n(t)g(and)120 2556 y(the)24 b(coupling)g(constan)n(t)f Fo(g)28 b Fn(2)d Fk(R)33 b Fr(is)24 b(arbitrary)-7 b(,)24 b(to)g(allo)n(w)f(for)h(a)g (sim)n(ultaneous)f(treatmen)n(t)g(of)h(the)g(in)n(teresting)120 2702 y(cases)29 b Fo(g)f Fr(=)d(2)30 b(and)g Fo(g)f Fr(=)c(0.)261 2849 y(F)-7 b(ormally)29 b Fo(A)p Fr(\()p Fo(x)p Fr(\))i(is)e(giv)n(en) g(b)n(y)817 3069 y Fo(A)p Fr(\()p Fo(x)p Fr(\))d(=)1153 2983 y Fj(X)1126 3181 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1327 2946 y Fj(Z)1377 3152 y Fv(j)p Fm(k)r Fv(j\024)p Fl(\003)1661 3005 y Fr(1)p 1588 3048 191 4 v 1588 3067 a Fj(p)p 1679 3067 100 4 v 77 x Fn(j)p Fo(k)s Fn(j)1789 3069 y Fo(")1831 3084 y Fm(\025)1876 3069 y Fr(\()p Fo(k)s Fr(\))1996 2968 y Fj(h)2039 3069 y Fo(e)2081 3034 y Fm(ik)r Fv(\001)p Fm(x)2206 3069 y Fo(a)2253 3084 y Fm(\025)2299 3069 y Fr(\()p Fo(k)s Fr(\))20 b(+)g Fo(e)2571 3034 y Fv(\000)p Fm(ik)r Fv(\001)p Fm(x)2752 3069 y Fo(a)2799 3034 y Fv(\003)2799 3094 y Fm(\025)2844 3069 y Fr(\()p Fo(k)s Fr(\))2964 2968 y Fj(i)3007 3069 y Fo(d)3054 3034 y Fl(3)3093 3069 y Fo(k)120 3348 y Fr(where)29 b(\003)24 b Fo(<)h Fn(1)k Fr(is)f(an)h(arbitrary)f(but)h(\014xed)f(ultra)n(violett)g(cuto\013.)39 b(F)-7 b(or)28 b(ev)n(ery)f Fo(k)h Fn(6)p Fr(=)d(0)k(the)g(t)n(w)n(o)e (p)r(olarization)120 3494 y(v)n(ectors)i Fo(")468 3509 y Fm(\025)513 3494 y Fr(\()p Fo(k)s Fr(\))c Fn(2)g Fk(R)802 3463 y Fl(3)848 3494 y Fo(;)45 b(\025)25 b Fr(=)g(1)p Fo(;)15 b Fr(2)30 b(are)g(normalized,)h(orthogonal)f(to)g Fo(k)j Fr(and)d(to)g(eac)n(h)f(other.)261 3641 y(The)f(op)r(erators)h Fo(a)888 3656 y Fm(\025)934 3641 y Fr(\()p Fo(k)s Fr(\))f(and)g Fo(a)1302 3610 y Fv(\003)1302 3670 y Fm(\025)1348 3641 y Fr(\()p Fo(k)s Fr(\))g(are)g(the)h(usual)f(annihilation)g(and)g (creation)h(op)r(erators,)g(satisfying)120 3788 y(the)h(canonical)g (comm)n(utation)g(relations)939 4008 y([)p Fo(a)1011 4023 y Fm(\025)1056 4008 y Fr(\()p Fo(k)s Fr(\))p Fo(;)15 b(a)1263 3973 y Fv(\003)1263 4029 y Fm(\026)1310 4008 y Fr(\()p Fo(k)1395 3973 y Fv(0)1418 4008 y Fr(\)])25 b(=)g Fo(\016)1638 4023 y Fm(\025\026)1726 4008 y Fo(\016)s Fr(\()p Fo(k)e Fn(\000)d Fo(k)2014 3973 y Fv(0)2037 4008 y Fr(\))p Fo(;)195 b Fr([)p Fo(a)2364 3964 y Fm(])2364 4038 y(\025)2410 4008 y Fr(\()p Fo(k)s Fr(\))p Fo(;)15 b(a)2617 3973 y Fm(])2617 4029 y(\026)2663 4008 y Fr(\()p Fo(k)2748 3973 y Fv(0)2771 4008 y Fr(\)])25 b(=)g(0)p Fo(:)120 4229 y Fr(In)30 b(terms)f(of)h Fo(a)633 4244 y Fm(\025)678 4229 y Fr(\()p Fo(k)s Fr(\))g(and)g Fo(a)1050 4197 y Fv(\003)1050 4258 y Fm(\025)1096 4229 y Fr(\()p Fo(k)s Fr(\))g(the)g(\014eld)g(Hamiltonian)g(is)f(giv)n(en)g(b)n(y)1356 4463 y Fo(H)1431 4478 y Fm(f)1501 4463 y Fr(=)1624 4377 y Fj(X)1596 4575 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1797 4340 y Fj(Z)1903 4463 y Fo(d)1950 4428 y Fl(3)1989 4463 y Fo(k)18 b Fn(j)p Fo(k)s Fn(j)p Fo(a)2201 4428 y Fv(\003)2201 4488 y Fm(\025)2247 4463 y Fr(\()p Fo(k)s Fr(\))p Fo(a)2414 4478 y Fm(\025)2459 4463 y Fr(\()p Fo(k)s Fr(\))p Fo(:)120 4747 y Fr(See)30 b(App)r(endix)g(B)f(for)h(mathematically)g(more)g (prop)r(er)h(de\014nitions)e(of)h Fo(A)p Fr(\()p Fo(x)p Fr(\))h(and)f Fo(H)3082 4762 y Fm(f)3127 4747 y Fr(.)261 4893 y(The)i(p)r(oten)n(tial)h Fo(V)53 b Fr(is)31 b(the)i(sum)e(of)h (the)g(external)g(and)h(the)f(Coulom)n(b)g(t)n(w)n(o-b)r(o)r(dy)g(p)r (oten)n(tials)g(for)f(eac)n(h)120 5040 y(pair)39 b(of)g(electrons.)67 b(Ho)n(w)n(ev)n(er,)39 b(for)f(the)i(purp)r(ose)f(of)g(the)g(results)f (to)h(b)r(e)h(pro)n(v)n(ed)d(in)j(this)e(section,)j(it)e(is)120 5187 y(enough)30 b(to)g(assume)f(that)717 5407 y(\()p Fo(H)7 b Fr(1\))180 b Fo(V)45 b Fn(2)25 b Fo(L)1337 5372 y Fl(2)1337 5432 y(lo)r(c)1430 5407 y Fr(\()p Fk(R)1525 5372 y Fl(3)p Fm(N)1633 5407 y Fr(;)15 b Fk(R)s Fr(\))p Fo(;)51 b Fr(and)30 b Fo(V)2071 5422 y Fv(\000)2155 5407 y Fn(\024)25 b Fo(")p Fr(\()p Fn(\000)p Fr(\001\))20 b(+)g Fo(C)2681 5422 y Fm(")2748 5407 y Fr(for)30 b(all)g Fo(")25 b(>)g Fr(0)p Fo(;)p eop %%Page: 8 8 8 7 bop 120 -200 a Fr(8)120 99 y(and)40 b(of)g(course)h(that)f Fo(V)61 b Fr(is)40 b(symmetric)f(with)i(resp)r(ect)g(to)f(p)r(erm)n (utation)h(of)f(the)g(particle)h(co)r(ordinates.)120 245 y(The)29 b(Hamiltonian)844 222 y(~)821 245 y Fo(H)896 260 y Fm(N)992 245 y Fr(is)f(a)h(symmetric,)g(densely)f(de\014ned)h(op) r(erator)h(and)f(b)n(y)f(Lemma)h(8,)h(it)f(is)f(b)r(ounded)120 392 y(from)i(b)r(elo)n(w.)43 b(The)31 b(quadratic)g(form)f Fo(q)s Fr(\()p Fo(';)15 b( )s Fr(\))26 b(=)1826 291 y Fj(D)1881 392 y Fo(';)2004 369 y Fr(~)1980 392 y Fo(H)2055 407 y Fm(N)2122 392 y Fo( )2184 291 y Fj(E)2270 392 y Fr(with)31 b(domain)g Fo(D)e Fr(=)d Fn(D)3066 407 y Fm(N)3164 392 y Fr(is)31 b(therefore)f(semi-)120 539 y(b)r(ounded)24 b(and)g(closable)f(and)h(hence)g(the)g(theory)f(of)g(the)h(previous)f (section)g(applies,)i(once)e(w)n(e)h(ha)n(v)n(e)e(v)n(eri\014ed)120 686 y(assumptions)k(\(i\),)h(\(ii\),)h(and)f(\(iii\).)39 b(The)27 b(unique)f(self-adjoin)n(t)g(op)r(erator)h Fo(H)2696 701 y Fm(N)2790 686 y Fr(asso)r(ciated)f(with)h(the)g(closure)120 832 y(of)f(the)g(quadratic)g(form)g Fo(q)k Fr(is)c(the)g(F)-7 b(riedric)n(hs')25 b(extension)g(of)2238 809 y(~)2215 832 y Fo(H)2290 847 y Fm(N)2357 832 y Fr(.)39 b(The)26 b(thresholds)g(\006)3095 847 y Fm(R)3179 832 y Fr(and)g(\006)g(asso)r (ciated)120 979 y(with)k(the)g(form)g Fo(q)j Fr(are)d(no)n(w)g(giv)n (en)f(b)n(y)1213 1209 y(\006)1278 1224 y Fm(R)1335 1209 y Fr(\()p Fo(H)1445 1224 y Fm(N)1512 1209 y Fr(\))83 b(=)290 b(inf)1783 1274 y Fm(')p Fv(2D)1931 1285 y Fi(N)r(;R)2050 1274 y Fm(;)30 b Fv(k)p Fm(')p Fv(k)p Fl(=1)2322 1108 y Fj(D)2377 1209 y Fo(';)2500 1186 y Fr(~)2476 1209 y Fo(H)2551 1224 y Fm(N)2618 1209 y Fo(')2677 1108 y Fj(E)1270 1405 y Fr(\006\()p Fo(H)1445 1420 y Fm(N)1512 1405 y Fr(\))83 b(=)118 b(lim)1783 1466 y Fm(R)p Fv(!1)1993 1405 y Fr(\006)2058 1420 y Fm(R)2116 1405 y Fr(\()p Fo(H)2226 1420 y Fm(N)2293 1405 y Fr(\))120 1635 y(where)38 b Fn(D)457 1650 y Fm(N)s(;R)633 1635 y Fr(:=)h Fn(f)p Fo(')g Fn(2)f(D)1077 1650 y Fm(N)1184 1635 y Fr(:)h Fo(')p Fr(\()p Fo(X)7 b Fr(\))38 b(=)h(0)g(if)e Fn(j)p Fo(X)7 b Fn(j)39 b Fo(<)g(R)q Fn(g)p Fr(.)65 b(The)38 b(follo)n(wing)g(theorem)g(is)g(a)g(corollary)g (of)120 1782 y(Theorem)30 b(1.)120 1997 y Ff(Theorem)40 b(2.)k Fg(Assume)36 b(Hyp)-5 b(othesis)36 b(\(H1\))g(is)g(satis\014e)-5 b(d)36 b(and)g(let)h Fo(H)2538 2012 y Fm(N)2642 1997 y Fg(b)-5 b(e)38 b(the)f(F)-7 b(rie)i(drichs')35 b(extension)h(of)120 2144 y(the)f(symmetric)d(op)-5 b(er)g(ator)1091 2121 y Fr(~)1067 2144 y Fo(H)1142 2159 y Fm(N)1238 2144 y Fr(:)29 b Fn(D)1361 2159 y Fm(N)1458 2144 y Fn(\032)g(H)1633 2159 y Fm(N)1729 2144 y Fn(!)g(H)1924 2159 y Fm(N)2025 2144 y Fg(given)35 b(by)f(Eq.)h Fr(\(9\))p Fg(.)47 b(If)34 b Fo(\025)h Fg(and)f Fo(\014)39 b Fg(ar)-5 b(e)35 b(r)-5 b(e)g(al)35 b(numb)-5 b(ers)120 2291 y(with)31 b Fo(\025)21 b Fr(+)f Fo(\014)534 2259 y Fl(2)598 2291 y Fo(<)25 b Fr(\006\()p Fo(H)868 2306 y Fm(N)935 2291 y Fr(\))p Fg(,)31 b(then)1555 2333 y Fj(\015)1555 2387 y(\015)1555 2442 y(\015)1605 2437 y Fo(e)1647 2402 y Fm(\014)s Fv(j)p Fm(X)5 b Fv(j)1796 2437 y Fo(E)1863 2452 y Fm(\025)1908 2437 y Fr(\()p Fo(H)2018 2452 y Fm(N)2085 2437 y Fr(\))2120 2333 y Fj(\015)2120 2387 y(\015)2120 2442 y(\015)2196 2437 y Fo(<)25 b Fn(1)p Fo(:)120 2652 y Fg(Pr)-5 b(o)g(of.)45 b Fr(It)34 b(su\016ces)g(to)h(v)n(erify)e(the)i(assumptions)f(\(i\),)i (\(ii\))f(and)g(\(iii\))g(in)g(the)g(previous)f(section.)55 b(Supp)r(ose)120 2799 y Fo(f)38 b Fn(2)28 b Fo(C)361 2767 y Fv(1)436 2799 y Fr(\()p Fk(R)531 2767 y Fl(3)p Fm(N)639 2799 y Fr(\))k(with)g Fo(f)5 b(;)15 b Fn(r)p Fo(f)38 b Fn(2)28 b Fo(L)1308 2767 y Fv(1)1383 2799 y Fr(\()p Fk(R)1478 2767 y Fl(3)p Fm(N)1587 2799 y Fr(\),)k(and)g Fo(f)10 b Fr(\()p Fo(X)d Fr(\))28 b(=)g Fo(f)10 b Fr(\()p Fn(j)p Fo(X)d Fn(j)p Fr(\).)46 b(Then)32 b Fo(f)10 b Fn(D)2875 2814 y Fm(N)2971 2799 y Fn(\032)28 b(D)3138 2814 y Fm(N)3237 2799 y Fr(is)k(ob)n(vious)e(from)120 2946 y(the)g(de\014nition)g(of)g Fn(D)847 2961 y Fm(N)914 2946 y Fr(.)41 b(Prop)r(ert)n(y)29 b(\(ii\))i(follo)n(ws)d(from)774 3176 y Fo(f)10 b Fr(\()p Fo(p)908 3191 y Fm(i)957 3176 y Fr(+)1047 3106 y Fn(p)p 1122 3106 58 4 v 70 x Fo(\013A)p Fr(\()p Fo(x)1332 3191 y Fm(i)1361 3176 y Fr(\)\))1431 3140 y Fl(2)1471 3176 y Fo(f)93 b Fn(\024)83 b Fr(2)p Fn(k)p Fo(f)10 b Fn(k)1950 3140 y Fl(2)1950 3196 y Fv(1)2024 3176 y Fr(\()p Fo(p)2104 3191 y Fm(i)2153 3176 y Fr(+)2243 3106 y Fn(p)p 2318 3106 V 70 x Fo(\013)q(A)p Fr(\()p Fo(x)2529 3191 y Fm(i)2557 3176 y Fr(\)\))2627 3140 y Fl(2)2687 3176 y Fr(+)20 b(2)p Fn(kr)2942 3191 y Fm(x)2982 3201 y Fi(i)3012 3176 y Fo(f)10 b Fn(k)3111 3140 y Fl(2)3111 3196 y Fv(1)1297 3347 y Fo(f)g(H)1426 3362 y Fm(f)1471 3347 y Fo(f)93 b Fn(\024)83 b(k)p Fo(f)10 b Fn(k)1905 3312 y Fl(2)1905 3368 y Fv(1)1979 3347 y Fo(H)2054 3362 y Fm(f)1344 3519 y Fo(f)g(V)21 b(f)93 b Fn(\024)83 b(k)p Fo(f)10 b Fn(k)1905 3483 y Fl(2)1905 3539 y Fv(1)1979 3519 y Fo(V)2031 3534 y Fl(+)2091 3519 y Fo(;)120 3749 y Fr(from)34 b(Lemma)i(7)e(and)h(Lemma)g(8.)55 b(The)35 b(pro)r(of)g(of)f(\(iii\))h(is)f(a)h(straigh)n(tforw)n(ard)e (computation)i(using)g(that)120 3896 y Fo(f)174 3864 y Fl(2)237 3873 y Fr(~)213 3896 y Fo(H)288 3911 y Fm(N)375 3896 y Fr(+)489 3873 y(~)465 3896 y Fo(H)540 3911 y Fm(N)607 3896 y Fo(f)661 3864 y Fl(2)720 3896 y Fn(\000)20 b Fr(2)p Fo(f)933 3873 y Fr(~)909 3896 y Fo(H)984 3911 y Fm(N)1051 3896 y Fo(f)35 b Fr(=)25 b([[)1299 3873 y(~)1275 3896 y Fo(H)1350 3911 y Fm(N)1417 3896 y Fo(;)15 b(f)10 b Fr(])p Fo(;)15 b(f)10 b Fr(].)p 3775 3896 4 61 v 3779 3838 55 4 v 3779 3896 V 3832 3896 4 61 v 261 4116 a Fg(R)-5 b(emark.)37 b Fr(The)23 b(ab)r(o)n(v)n(e)f(theorem)g(and)h(its)f(pro)r (of)h(can)g(easily)e(b)r(e)i(generalized)g(to)g(systems)e(of)h Fo(N)32 b Fr(particles)120 4263 y(from)d Fo(n)c Fn(\024)g Fo(N)39 b Fr(sp)r(ecies,)30 b(with)f(di\013eren)n(t)g(masses)f Fo(m)1875 4278 y Fm(i)1903 4263 y Fr(,)i(c)n(harges,)e(and)i(spins.)39 b(Theorem)30 b(2)f(then)h(holds)f(again)120 4409 y(with)h(the)g(new)g (norm)g Fn(j)p Fo(X)7 b Fn(j)25 b Fr(=)1152 4336 y Fj(\000)1209 4341 y(P)1305 4368 y Fm(N)1305 4436 y(i)p Fl(=1)1438 4409 y Fr(2)p Fo(m)1562 4424 y Fm(i)1590 4409 y Fo(x)1641 4378 y Fl(2)1641 4436 y Fm(i)1681 4336 y Fj(\001)1722 4358 y Fl(1)p Fm(=)p Fl(2)1862 4409 y Fr(in)30 b(the)g(exp)r(onen)n (tial)g(factor.)261 4556 y(Our)g(next)e(goal)i(is)f(to)g(establish)g(a) g(relation)h(b)r(et)n(w)n(een)f(\006\()p Fo(H)2333 4571 y Fm(N)2400 4556 y Fr(\))g(and)g(sp)r(ectral)h(data)f(of)g(cluster)g (Hamil-)120 4703 y(tonians.)40 b(T)-7 b(o)29 b(this)h(end,)g(w)n(e)f (imp)r(ose)i(the)f(follo)n(wing)f(additional)h(assumption)f(on)h Fo(V)21 b Fr(:)560 4982 y(\()p Fo(H)7 b Fr(2\))772 4826 y Fj(\()886 4915 y Fo(V)21 b Fr(\()p Fo(X)7 b Fr(\))25 b(=)1231 4847 y Fj(P)1327 4873 y Fm(N)1327 4942 y(i)p Fl(=1)1460 4915 y Fo(v)s Fr(\()p Fo(x)1593 4930 y Fm(i)1621 4915 y Fr(\))20 b(+)1766 4847 y Fj(P)1862 4942 y Fm(i)f Fr(0)30 b(are)h(de\014ned)f(in)h(the)f(same)g(w) n(a)n(y)f(as)g Fo(H)3412 2440 y Fm(N)3510 2425 y Fr(with)h(the)120 2572 y(only)h(di\013erence)f(that)h(the)g(v)n(ector)f(p)r(oten)n(tial)h Fo(A)p Fr(\()p Fo(x)p Fr(\))h(and)f(the)g(magnetic)g(\014eld)g Fo(B)5 b Fr(\()p Fo(x)p Fr(\))31 b(in)g Fo(H)3266 2587 y Fm(N)3364 2572 y Fr(are)g(replaced)120 2719 y(b)n(y)745 2866 y Fo(A)812 2881 y Fm(\026)859 2866 y Fr(\()p Fo(x)p Fr(\))25 b(=)1128 2779 y Fj(X)1100 2977 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1301 2742 y Fj(Z)1352 2948 y Fm(\026)p Fv(\024j)p Fm(k)r Fv(j\024)p Fl(\003)1733 2801 y Fr(1)p 1660 2845 191 4 v 1660 2863 a Fj(p)p 1751 2863 100 4 v 78 x Fn(j)p Fo(k)s Fn(j)1861 2866 y Fo(")1903 2881 y Fm(\025)1948 2866 y Fr(\()p Fo(k)s Fr(\))2068 2765 y Fj(h)2111 2866 y Fo(e)2153 2830 y Fm(ik)r Fv(\001)p Fm(x)2278 2866 y Fo(a)2325 2881 y Fm(\025)2371 2866 y Fr(\()p Fo(k)s Fr(\))20 b(+)g Fo(e)2643 2830 y Fv(\000)p Fm(ik)r Fv(\001)p Fm(x)2824 2866 y Fo(a)2871 2830 y Fv(\003)2871 2891 y Fm(\025)2916 2866 y Fr(\()p Fo(k)s Fr(\))3036 2765 y Fj(i)3079 2866 y Fo(d)3126 2830 y Fl(3)3165 2866 y Fo(k)120 3111 y Fr(and)k Fo(B)357 3126 y Fm(\026)404 3111 y Fr(\()p Fo(x)p Fr(\))h(=)g(curl)p Fo(A)862 3126 y Fm(\026)909 3111 y Fr(\()p Fo(x)p Fr(\).)38 b(T)-7 b(o)24 b(separate)f(the)h(soft,)g(non-in)n(teracting)g(photons)g (from)f(the)h(in)n(teracting)f(ones)120 3257 y(w)n(e)29 b(use)h(that)g Fn(F)668 3272 y Fm(f)742 3257 y Fr(is)g(isomorphic)f(to) h Fn(F)1463 3272 y Fm(i)1511 3257 y Fn(\012)19 b(F)1665 3272 y Fm(s)1731 3257 y Fr(where)30 b Fn(F)2056 3272 y Fm(i)2114 3257 y Fr(and)g Fn(F)2354 3272 y Fm(s)2420 3257 y Fr(denote)g(the)g(b)r(osonic)g(F)-7 b(o)r(c)n(k)29 b(spaces)f(o)n(v)n(er)120 3404 y Fo(L)181 3373 y Fl(2)221 3404 y Fr(\()p Fn(j)p Fo(k)s Fn(j)33 b(\025)g Fo(\026)p Fr(\))j(and)f Fo(L)858 3373 y Fl(2)897 3404 y Fr(\()p Fn(j)p Fo(k)s Fn(j)f Fo(<)f(\026)p Fr(\))i(resp)r(ectiv)n(ely)-7 b(.)54 b(Let)35 b Fn(H)2069 3419 y Fm(i)2131 3404 y Fr(=)e Fn(^)2294 3373 y Fm(N)2361 3404 y Fo(L)2422 3373 y Fl(2)2462 3404 y Fr(\()p Fk(R)2557 3373 y Fl(3)2602 3404 y Fr(;)15 b Fk(C)2702 3373 y Fl(2)2747 3404 y Fr(\))24 b Fn(\012)f(F)2964 3419 y Fm(i)2992 3404 y Fr(.)55 b(Then)35 b(the)g(Hamilton)120 3551 y(op)r(erator)c(can)f(b)r(e)g(written)g(as)1051 3751 y Fo(H)1126 3766 y Fm(N)s(;\026)1276 3751 y Fr(=)25 b Fo(H)1453 3716 y Fm(i)1446 3772 y(\026)1512 3751 y Fn(\012)20 b Fr(1)g(+)g(1)g Fn(\012)g Fo(H)1994 3716 y Fm(s)1987 3776 y(f)2212 3751 y Fr(on)90 b Fn(H)26 b Fr(=)f Fn(H)2670 3766 y Fm(i)2718 3751 y Fn(\012)20 b(F)2873 3766 y Fm(s)3680 3751 y Fr(\(10\))120 3951 y(if)29 b(w)n(e)h(iden)n (tify)e Fn(F)38 b Fr(with)30 b Fn(F)1035 3966 y Fm(i)1083 3951 y Fn(\012)19 b(F)1237 3966 y Fm(s)1274 3951 y Fr(.)39 b(Let)31 b Fn(F)1565 3966 y Fm(s;n)1693 3951 y Fr(denote)f(the)g (subspace)f(of)g Fn(F)2681 3966 y Fm(s)2747 3951 y Fr(with)h(precisely) f Fo(n)h Fr(soft)e(b)r(osons)120 4098 y(and)i(let)g(\012)490 4113 y Fm(s)557 4098 y Fr(b)r(e)h(the)f(v)-5 b(acuum)29 b(of)h Fn(F)1335 4113 y Fm(s)1371 4098 y Fr(.)40 b(Then)30 b(\(10\))g(and)h(the)f(p)r(ositivit)n(y)e(of)i Fo(H)2777 4067 y Fm(s)2770 4127 y(f)2840 4098 y Fr(=)25 b Fo(H)3010 4113 y Fm(f)3055 4098 y Fn(j)-23 b Fr(\022)-7 b Fn(F)3160 4113 y Fm(s)3226 4098 y Fr(sho)n(w)30 b(that)1147 4305 y(inf)21 b Fo(\033)s Fr(\()p Fo(H)1435 4320 y Fm(N)s(;\026)1560 4305 y Fr(\))k(=)37 b(inf)1715 4364 y Fm(n)p Fv(\025)p Fl(0)1863 4231 y Fj(\000)1920 4305 y Fr(inf)21 b Fo(\033)s Fr(\()p Fo(H)2208 4320 y Fm(N)s(;\026)2334 4305 y Fn(j)-23 b Fr(\022)-7 b Fn(H)2450 4320 y Fm(i)2498 4305 y Fn(\012)20 b(F)2653 4320 y Fm(s;n)2752 4305 y Fr(\))2787 4231 y Fj(\001)1620 4521 y Fr(=)25 b(inf)d Fo(\033)s Fr(\()p Fo(H)2004 4536 y Fm(N)s(;\026)2129 4521 y Fn(j)-23 b Fr(\022)-7 b Fn(H)2245 4536 y Fm(i)2293 4521 y Fn(\012)20 b Fr([\012)2473 4536 y Fm(s)2510 4521 y Fr(]\))p Fo(;)3680 4406 y Fr(\(11\))120 4695 y(where)44 b([\012)484 4710 y Fm(s)521 4695 y Fr(])g(is)g(the)g(space)g(spanned)g(b)n(y)f(\012)1688 4710 y Fm(s)1725 4695 y Fr(.)82 b(This)44 b(will)g(allo)n(w)g(us)g(to)g (drop)g(the)h(soft)e(b)r(osons)h(in)g(all)120 4841 y(appro)n(ximate)29 b(energy)h(minimizers.)120 5031 y Ff(Lemma)j(4.)41 b Fg(Ther)-5 b(e)34 b(exists)e(a)g(c)-5 b(onstant)31 b Fo(C)1613 5046 y Fm(N)s(;)p Fl(\003)1745 5031 y Fg(,)h(dep)-5 b(ending)32 b(on)f Fr(\003)p Fo(;)15 b(N)5 b(;)15 b(g)35 b Fg(and)d Fo(\013)p Fg(,)g(such)h(that)650 5289 y Fn(\006)p Fr(\()p Fo(H)830 5304 y Fm(N)917 5289 y Fn(\000)20 b Fo(H)1082 5304 y Fm(N)s(;\026)1207 5289 y Fr(\))25 b Fn(\024)g Fo(\026)1416 5253 y Fl(1)p Fm(=)p Fl(2)1526 5289 y Fo(C)1590 5304 y Fm(N)s(;)p Fl(\003)1737 5134 y Fj(\()1844 5175 y Fm(N)1810 5203 y Fj(X)1819 5398 y Fm(i)p Fl(=1)1956 5289 y Fo(p)2001 5253 y Fl(2)2001 5312 y Fm(i)2061 5289 y Fr(+)20 b Fo(H)2226 5304 y Fm(f)2291 5289 y Fr(+)g(1)2426 5134 y Fj(\))2698 5289 y Fg(for)91 b Fr(0)25 b Fn(\024)g Fo(\026)g Fn(\024)g Fr(1)p Fo(:)p eop %%Page: 10 10 10 9 bop 120 -200 a Fr(10)120 99 y Fg(Pr)-5 b(o)g(of.)45 b Fr(By)29 b(de\014nition)h(of)g Fo(H)1121 114 y Fm(N)1218 99 y Fr(and)g Fo(H)1468 114 y Fm(N)s(;\026)1593 99 y Fr(,)310 354 y Fo(H)385 369 y Fm(N)472 354 y Fn(\000)20 b Fo(H)637 369 y Fm(N)s(;\026)845 354 y Fr(=)1032 240 y Fm(N)998 268 y Fj(X)1007 463 y Fm(i)p Fl(=1)1145 354 y Fr(2)1190 284 y Fn(p)p 1265 284 58 4 v 70 x Fo(\013p)1367 369 y Fm(i)1416 354 y Fn(\001)1461 280 y Fj(\000)1502 354 y Fo(A)p Fr(\()p Fo(x)1655 369 y Fm(i)1684 354 y Fr(\))g Fn(\000)g Fo(A)1896 369 y Fm(\026)1943 354 y Fr(\()p Fo(x)2029 369 y Fm(i)2058 354 y Fr(\))2093 280 y Fj(\001)2154 354 y Fr(+)g Fo(\013)2301 280 y Fj(\000)2344 354 y Fo(A)p Fr(\()p Fo(x)2497 369 y Fm(i)2526 354 y Fr(\))g Fn(\000)g Fo(A)2738 369 y Fm(\026)2785 354 y Fr(\()p Fo(x)2871 369 y Fm(i)2899 354 y Fr(\))2934 280 y Fj(\001\000)3018 354 y Fo(A)p Fr(\()p Fo(x)3171 369 y Fm(i)3199 354 y Fr(\))g(+)g Fo(A)3411 369 y Fm(\026)3458 354 y Fr(\()p Fo(x)3544 369 y Fm(i)3573 354 y Fr(\))3608 280 y Fj(\001)998 609 y Fr(+)1078 545 y Fo(g)p 1078 588 47 4 v 1079 672 a Fr(2)1135 539 y Fn(p)p 1209 539 58 4 v 1209 609 a Fo(\013)q(\033)k Fn(\001)1387 535 y Fj(\000)1428 609 y Fo(B)5 b Fr(\()p Fo(x)1587 624 y Fm(i)1616 609 y Fr(\))20 b Fn(\000)g Fo(B)1829 624 y Fm(\026)1876 609 y Fr(\()p Fo(x)1962 624 y Fm(i)1990 609 y Fr(\))2025 535 y Fj(\001)120 807 y Fr(where)27 b(w)n(e)g(used)f(that)h Fo(A)p Fr(\()p Fo(x)p Fr(\))h(and)f Fo(A)1356 822 y Fm(\026)1403 807 y Fr(\()p Fo(x)p Fr(\))h(comm)n(ute.)38 b(The)27 b(di\013erences)g Fo(A)p Fr(\()p Fo(x)p Fr(\))14 b Fn(\000)g Fo(A)2944 822 y Fm(\026)2991 807 y Fr(\()p Fo(x)p Fr(\))28 b(and)f Fo(B)5 b Fr(\()p Fo(x)p Fr(\))14 b Fn(\000)g Fo(B)3672 822 y Fm(\026)3719 807 y Fr(\()p Fo(x)p Fr(\))120 954 y(can)31 b(b)r(e)h(seen)e(as)h(a)g(v)n(ector)f(p)r(oten)n(tial)h (and)g(a)g(magnetic)g(\014eld)g(with)g(an)g Fg(ultr)-5 b(aviolett)40 b Fr(cuto\013)30 b Fo(\026)p Fr(.)43 b(Hence)31 b(the)120 1100 y(lemma)f(follo)n(ws)f(from)g(Lemma)i(7)f(with)g(\003)24 b(=)h Fo(\026)30 b Fr(and)g Fo(")25 b Fr(=)g Fo(\026)2177 1069 y Fl(1)p Fm(=)p Fl(2)2288 1100 y Fr(.)p 3775 1100 4 61 v 3779 1043 55 4 v 3779 1100 V 3832 1100 4 61 v 120 1315 a Ff(Lemma)33 b(5.)120 b Fg(\(i\))44 b Fr(\006\()p Fo(H)7 b Fr(\))25 b Fo(<)g Fn(1)31 b Fg(if)f(and)g(only)f(if)h Fr(\006\()p Fo(H)2004 1330 y Fm(\026)2051 1315 y Fr(\))25 b Fo(<)f Fn(1)31 b Fg(and)f(in)g(this)g(c)-5 b(ase)31 b(ther)-5 b(e)32 b(exists)e(a)h(c)-5 b(onstant)345 1462 y Fo(C)409 1477 y Fl(\003)495 1462 y Fg(dep)g(ending)32 b(on)f(the)i(p)-5 b(ar)g(ameters)32 b Fr(\003)p Fo(;)15 b(N)5 b(;)15 b(\013)p Fg(,)32 b(and)f Fo(g)s Fg(,)h(such)h(that)1228 1660 y Fn(j)p Fr(\006\()p Fo(H)1428 1675 y Fm(N)s(;\026)1553 1660 y Fr(\))20 b Fn(\000)g Fr(\006\()p Fo(H)1873 1675 y Fm(N)1940 1660 y Fr(\))p Fn(j)25 b(\024)g Fo(C)2184 1675 y Fl(\003)2238 1660 y Fo(\026)2292 1625 y Fl(1)p Fm(=)p Fl(2)2402 1660 y Fo(;)198 b Fg(if)32 b Fo(\026)25 b Fn(\024)g Fr(1)p Fo(:)171 1889 y Fg(\(ii\))44 b(Ther)-5 b(e)33 b(exists)f(a)g(c)-5 b(onstant)32 b Fo(C)1350 1904 y Fl(\003)1435 1889 y Fg(dep)-5 b(ending)32 b(on)g(the)g(p)-5 b(ar)g(ameters)33 b Fr(\003)p Fo(;)15 b(N)5 b(;)15 b(\013)p Fg(,)31 b(and)h Fo(g)s Fg(,)g(such)g(that)1244 2087 y Fn(j)p Fo(\034)10 b Fr(\()p Fo(H)1428 2102 y Fm(N)s(;\026)1553 2087 y Fr(\))20 b Fn(\000)g Fo(\034)10 b Fr(\()p Fo(H)1857 2102 y Fm(N)1925 2087 y Fr(\))p Fn(j)25 b(\024)g Fo(C)2169 2102 y Fl(\003)2222 2087 y Fo(\026)2276 2051 y Fl(1)p Fm(=)p Fl(2)2386 2087 y Fo(;)199 b Fg(if)31 b Fo(\026)25 b Fn(\024)g Fr(1)p Fo(:)120 2285 y Fg(Pr)-5 b(o)g(of.)45 b Fr(By)27 b(Lemma)h(4)f(and)h(Lemma)g(8,)g(there)g(exist)f(constan)n (ts)f Fo(C)35 b Fr(and)27 b Fo(D)r Fr(,)i(indep)r(enden)n(t)f(of)f Fo(\026)p Fr(,)h(suc)n(h)f(that)1066 2483 y Fo(H)1141 2498 y Fm(N)s(;\026)1291 2483 y Fn(\024)e Fo(H)1461 2498 y Fm(N)1548 2483 y Fr(+)20 b Fo(\026)1692 2448 y Fl(1)p Fm(=)p Fl(2)1802 2483 y Fr(\()p Fo(C)7 b(H)1983 2498 y Fm(N)2070 2483 y Fr(+)20 b Fo(D)r Fr(\))180 b(for)90 b Fo(\026)25 b Fn(\024)g Fr(1)p Fo(:)120 2681 y Fr(It)30 b(follo)n(ws)e(that)876 2828 y(\006\()p Fo(H)1051 2843 y Fm(N)s(;\026)1176 2828 y Fr(\))d Fn(\024)g Fr(\006\()p Fo(H)1506 2843 y Fm(N)1573 2828 y Fr(\))20 b(+)g Fo(\026)1772 2793 y Fl(1)p Fm(=)p Fl(2)1882 2828 y Fr(\()p Fo(C)7 b Fr(\006\()p Fo(H)2163 2843 y Fm(N)2230 2828 y Fr(\))20 b(+)g Fo(D)r Fr(\))180 b(for)90 b Fo(\026)25 b Fn(\024)g Fr(1)120 3009 y(and,)39 b(in)f(particular,)h(that)e(\006\()p Fo(H)1268 3024 y Fm(N)s(;\026)1393 3009 y Fr(\))h Fo(<)f Fn(1)g Fr(if)g(\006\()p Fo(H)1965 3024 y Fm(N)2032 3009 y Fr(\))g Fo(<)g Fn(1)p Fr(.)62 b(Since)38 b(the)f(roles)g(of)g Fo(H)3196 3024 y Fm(N)s(;\026)3358 3009 y Fr(and)h Fo(H)3616 3024 y Fm(N)3720 3009 y Fr(are)120 3156 y(in)n(terc)n(hangeable,)29 b(\(i\))h(follo)n(ws.)39 b(The)30 b(pro)r(of)g(of)g(\(ii\))g(is)g (similar.)p 3775 3156 V 3779 3098 55 4 v 3779 3156 V 3832 3156 4 61 v 120 3371 a Ff(Theorem)k(6.)41 b Fg(Under)32 b(the)g(assumptions)e(\(H1\),)h(\(H2\))h(and)f(\(H2\))g(on)h Fo(V)53 b Fg(one)32 b(has)1243 3569 y Fr(\006\()p Fo(H)1418 3584 y Fm(N)s(;\026)1543 3569 y Fr(\))25 b(=)g Fo(\034)10 b Fr(\()p Fo(H)1857 3584 y Fm(N)s(;\026)1982 3569 y Fr(\))184 b Fg(for)32 b(al)5 b(l)31 b Fo(\026)25 b(>)g Fr(0)p Fo(:)261 3767 y Fr(In)30 b(conjunction)f(with)h(Lemma)h(5,)f(this)f(theorem)i (pro)n(v)n(es)d(Theorem)i(3)120 3982 y Fg(Pr)-5 b(o)g(of)32 b(of)g Fr(\006\()p Fo(H)642 3997 y Fm(N)s(;\026)767 3982 y Fr(\))25 b Fn(\025)g Fo(\034)10 b Fr(\()p Fo(H)1081 3997 y Fm(N)s(;\026)1206 3982 y Fr(\))p Fg(.)45 b Fr(This)32 b(is)g(pro)n(v)n(ed)f(in)i([7,)g(6].)48 b(F)-7 b(or)31 b(the)i(sak)n(e)e(of)h(completeness)g(w)n(e)g(rep)r(eat)120 4128 y(the)26 b(argumen)n(ts)f(starting)h(from)f(Theorem)i(9.)38 b(W)-7 b(e)25 b(ma)n(y)g(assume)g(that)h(\006\()p Fo(H)2745 4143 y Fm(N)s(;\026)2870 4128 y Fr(\))f Fo(<)g Fn(1)p Fr(.)39 b(By)25 b(the)h(argumen)n(t)120 4275 y(\(11\))k(w)n(e)g(ma)n(y) f(restrict)g Fo(H)1023 4290 y Fm(N)s(;\026)1178 4275 y Fr(to)h Fn(H)1364 4290 y Fm(i)1413 4275 y Fn(\012)19 b Fr([\012)1592 4290 y Fm(s)1629 4275 y Fr(])30 b(for)g(the)g (computation)g(of)g(\006)2670 4290 y Fm(R)2727 4275 y Fr(\()p Fo(H)2837 4290 y Fm(N)s(;\026)2962 4275 y Fr(\).)41 b(By)29 b(Lemma)h(8)1050 4485 y Fo(N)1122 4500 y Fm(f)1192 4485 y Fn(\024)1302 4421 y Fr(1)p 1297 4464 55 4 v 1297 4549 a Fo(\026)1361 4485 y(H)1436 4500 y Fm(f)1506 4485 y Fn(\024)1616 4421 y Fr(1)p 1611 4464 V 1611 4549 a Fo(\026)1675 4485 y Fr(\(2)p Fo(H)1830 4500 y Fm(N)s(;\026)1975 4485 y Fr(+)20 b Fo(D)r Fr(\))180 b Fo(on)91 b Fn(H)2621 4500 y Fm(i)2669 4485 y Fn(\012)20 b Fr([\012)2849 4500 y Fm(s)2885 4485 y Fr(])120 4683 y(and)30 b(hence)g(b)n(y)f(Theorem)h (9,)702 4882 y Fo(H)777 4897 y Fm(N)s(;\026)927 4882 y Fn(\025)25 b Fo(\034)10 b Fr(\()p Fo(H)1181 4897 y Fm(N)s(;\026)1306 4882 y Fr(\))20 b Fn(\000)g Fo(o)p Fr(\()p Fo(R)1598 4846 y Fl(0)1638 4882 y Fr(\)\()p Fo(H)1783 4897 y Fm(N)s(;\026)1928 4882 y Fr(+)g Fo(C)7 b Fr(\))180 b Fo(on)90 b Fn(D)2560 4897 y Fm(N)s(;R)2717 4882 y Fn(\\)20 b Fr(\()p Fn(H)2908 4897 y Fm(i)2956 4882 y Fn(\012)g Fr([\012)3136 4897 y Fm(s)3173 4882 y Fr(]\))p Fo(:)120 5080 y Fr(It)30 b(follo)n(ws)e(that)1076 5226 y(\006)1141 5241 y Fm(R)1199 5226 y Fr(\()p Fo(H)1309 5241 y Fm(N)s(;\026)1434 5226 y Fr(\))d Fn(\025)g Fo(\034)10 b Fr(\()p Fo(H)1748 5241 y Fm(N)s(;\026)1873 5226 y Fr(\))20 b Fn(\000)g Fo(o)p Fr(\()p Fo(R)2165 5191 y Fl(0)2205 5226 y Fr(\)\(\006)2340 5241 y Fm(R)2398 5226 y Fr(\()p Fo(H)2508 5241 y Fm(N)s(;\026)2633 5226 y Fr(\))g(+)g Fo(C)7 b Fr(\))120 5407 y(and)30 b(the)g(desired)g (result)f(is)h(obtained)g(in)g(the)g(limit)g Fo(R)d Fn(!)d(1)p Fr(.)p 3775 5407 4 61 v 3779 5350 55 4 v 3779 5407 V 3832 5407 4 61 v eop %%Page: 11 11 11 10 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2267 b Fr(11)261 99 y(An)24 b(imp)r(ortan)n(t)g(role)h(in)f (the)g(follo)n(wing)g(pro)r(of)g(is)g(pla)n(y)n(ed)f(b)n(y)g(the)i (iden)n(ti\014cation)f(op)r(erator)h Fo(I)32 b Fr(:)25 b Fn(F)17 b(\012)9 b(F)33 b(!)120 245 y(F)45 b Fr(whic)n(h)37 b(collects)f(all)g(photons)h(in)f(the)h(\014rst)f(and)h(second)f (factor)g(of)g Fn(F)d(\012)25 b(F)9 b Fr(,)38 b(and)e(gathers)h(them)f (in)h(a)120 392 y(single)28 b(F)-7 b(o)r(c)n(k)27 b(space.)40 b(F)-7 b(or)27 b(the)i(precise)f(de\014nition)h(of)f Fo(I)35 b Fr(and)29 b(other)f(notations)h(in)f(the)h(follo)n(wing)e (pro)r(of)i(not)120 539 y(y)n(et)g(in)n(tro)r(duced,)h(see)g(App)r (endix)g(B.)120 758 y Fg(Pr)-5 b(o)g(of)32 b(of)g Fr(\006\()p Fo(H)642 773 y Fm(N)s(;\026)767 758 y Fr(\))25 b Fn(\024)g Fo(\034)10 b Fr(\()p Fo(H)1081 773 y Fm(N)s(;\026)1206 758 y Fr(\))p Fg(.)45 b Fr(In)38 b(the)h(follo)n(wing)f(the)h(subindex) f Fo(\026)h Fr(is)f(dropp)r(ed.)67 b(W)-7 b(e)38 b(need)h(to)g(sho)n(w) 120 904 y(that)1413 1051 y(lim)1378 1112 y Fm(R)p Fv(!1)1587 1051 y Fr(\006)1652 1066 y Fm(R)1710 1051 y Fr(\()p Fo(H)1820 1066 y Fm(N)1887 1051 y Fr(\))25 b Fn(\024)g Fo(E)2109 1066 y Fm(N)7 b Fv(\000)p Fm(N)2290 1047 y Fq(0)2336 1051 y Fr(+)20 b Fo(E)2498 1015 y Fl(0)2493 1077 y Fm(N)2556 1058 y Fq(0)120 1243 y Fr(for)41 b(all)h Fo(N)488 1212 y Fv(0)555 1243 y Fn(\025)i Fr(1.)75 b(The)42 b(strategy)f(is)g(as)g (follo)n(ws.)73 b(First)41 b(w)n(e)g(construct)h(appro)n(ximate)e (minimizers)i Fo(')3801 1258 y Fl(0)120 1390 y Fr(and)30 b Fo(')354 1405 y Fv(1)458 1390 y Fr(of)f Fo(H)635 1406 y Fm(N)7 b Fv(\000)p Fm(N)816 1387 y Fq(0)872 1390 y Fr(and)30 b Fo(H)1129 1359 y Fl(0)1122 1420 y Fm(N)1185 1401 y Fq(0)1241 1390 y Fr(resp)r(ectiv)n(ely)-7 b(,)28 b(with)h(the)h(prop)r(ert)n(y)f(that)h(the)g(electrons)f(and)h(the)f (photons)120 1537 y(describ)r(ed)d(b)n(y)e Fo(')693 1552 y Fl(0)758 1537 y Fr(and)h Fo(')987 1552 y Fv(1)1088 1537 y Fr(are)g(compactly)g(supp)r(orted.)39 b(Then,)26 b(b)n(y)f(a)g(translation)g Fo(')3075 1552 y Fv(1)3175 1537 y Fn(!)g Fo(T)3343 1552 y Fm(R)3400 1537 y Fo(')3459 1552 y Fv(1)3559 1537 y Fr(of)g(b)r(oth)120 1684 y(the)30 b(electrons)g(and)g(the)g(photons)g(in)g Fo(')1486 1699 y Fv(1)1590 1684 y Fr(w)n(e)g(ma)n(y)f(ac)n(hiev)n(e)g(\(ignoring)h (the)g(P)n(auli)g(principle\))g(that)1024 1905 y Fo( )1083 1920 y Fm(R)1165 1905 y Fr(=)25 b Fo(I)7 b Fr(\()p Fo(')1401 1920 y Fl(0)1460 1905 y Fn(\012)20 b Fo(T)1603 1920 y Fm(R)1660 1905 y Fo(')1719 1920 y Fv(1)1794 1905 y Fr(\))25 b Fn(2)g(D)2008 1920 y Fm(N)s(;R)2145 1905 y Fo(;)195 b Fr(and)30 b Fn(k)p Fo( )2644 1920 y Fm(R)2701 1905 y Fn(k)25 b Fr(=)g(1)p Fo(;)120 2127 y Fr(where)41 b Fo(T)444 2142 y Fm(R)501 2127 y Fo(')560 2142 y Fv(1)675 2127 y Fr(is)f(still)h(an)f(appro)n(ximate)g(minimizer)h(of)f Fo(H)2255 2095 y Fl(0)2248 2156 y Fm(N)2311 2137 y Fq(0)2378 2127 y Fr(b)n(y)f(the)i(translation)f(in)n(v)-5 b(ariance)40 b(of)g(this)120 2273 y(Hamiltonian.)261 2420 y(Second)30 b(w)n(e)g(sho)n(w)f(that)644 2641 y Fn(h)p Fo( )738 2656 y Fm(R)796 2641 y Fo(;)15 b(H)911 2656 y Fm(N)977 2641 y Fo( )1036 2656 y Fm(R)1094 2641 y Fn(i)25 b(\024)g(h)p Fo(')1343 2656 y Fl(0)1382 2641 y Fo(;)15 b(H)1497 2657 y Fm(N)7 b Fv(\000)p Fm(N)1678 2638 y Fq(0)1704 2641 y Fo(')1763 2656 y Fl(0)1802 2641 y Fn(i)20 b Fr(+)1947 2568 y Fj(\012)1990 2641 y Fo(')2049 2656 y Fv(1)2124 2641 y Fo(;)15 b(H)2246 2606 y Fl(0)2239 2667 y Fm(N)2302 2648 y Fq(0)2328 2641 y Fo(')2387 2656 y Fv(1)2462 2568 y Fj(\013)2525 2641 y Fr(+)20 b Fo(o)p Fr(\()p Fo(R)2762 2606 y Fl(0)2802 2641 y Fr(\))180 b Fo(R)26 b Fn(!)f(1)120 2863 y Fr(whic)n(h)37 b(concludes)h(the)g(pro)r(of.)64 b(T)-7 b(o)37 b(incorp)r(orate)i(the)f(P)n(auli)g(principle)g(one)g (needs)f(to)h(an)n(ti-symmetrize)120 3010 y Fo(I)7 b Fr(\()p Fo(')261 3025 y Fl(0)326 3010 y Fn(\012)25 b Fo(T)474 3025 y Fm(R)531 3010 y Fo(')590 3025 y Fv(1)665 3010 y Fr(\))38 b(with)h(resp)r(ect)f(to)h(the)f Fo(N)48 b Fr(electron)39 b(v)-5 b(ariables)37 b(\()p Fo(x)2486 3025 y Fm(i)2515 3010 y Fo(;)15 b(s)2597 3025 y Fm(i)2625 3010 y Fr(\))39 b Fn(2)g Fk(R)2858 2978 y Fl(3)2929 3010 y Fn(\002)25 b(f)p Fr(1)p Fo(;)15 b Fr(2)p Fn(g)p Fr(,)41 b Fo(i)d Fr(=)h(1)p Fo(;)15 b(:)g(:)g(:)g(;)g(N)10 b Fr(.)120 3156 y(After)38 b(normalization,)i(this)e(will)h(lead)f(to)h (the)f(same)g(v)-5 b(alue)39 b(for)f(the)g(energy)g Fn(h)p Fo( )2987 3171 y Fm(R)3044 3156 y Fo(;)15 b(H)3159 3171 y Fm(N)3226 3156 y Fo( )3285 3171 y Fm(R)3342 3156 y Fn(i)39 b Fr(as)f(without)120 3303 y(an)n(ti-symmetrization,)d(b)r (ecause)g(the)g(electrons)g(in)g Fo(')2020 3318 y Fl(0)2094 3303 y Fr(and)g Fo(T)2327 3318 y Fm(R)2385 3303 y Fo(')2444 3318 y Fv(1)2553 3303 y Fr(are)h(disjoin)n(tly)d(supp)r(orted)i(and)g (the)120 3450 y(Hamiltonian)30 b(is)f(lo)r(cal.)41 b(Th)n(us)29 b(one)h(do)r(es)g(not)g(need)g(to)g(an)n(ti-symmetrize.)261 3597 y(Let)39 b Fo(")g(>)h Fr(0)e(b)r(e)h(giv)n(en)f(and)h(\014xed)f (in)g(the)h(follo)n(wing)f(three)h(steps,)g(and)g(let)g Fo(y)j Fr(denote)c(the)h(p)r(osition)120 3743 y(op)r(erator)33 b Fo(y)g Fr(=)c Fo(i)p Fn(r)767 3758 y Fm(k)842 3743 y Fr(in)k(the)f(one-photon)h(Hilb)r(ert)g(space.)47 b(F)-7 b(or)32 b(simplicit)n(y)f(the)i(irrelev)-5 b(an)n(t)32 b(parameters)g Fo(\013)120 3890 y Fr(and)e Fo(g)j Fr(are)d(dropp)r(ed)h (henceforth.)261 4037 y Ff(Step)36 b(1.)j Fr(Giv)n(en)29 b Fo(")c(>)g Fr(0)30 b(there)g(are)g(normalized)g(states)g Fo(')2272 4052 y Fl(0)2336 4037 y Fn(2)25 b(D)2490 4052 y Fm(N)7 b Fv(\000)p Fm(N)2671 4033 y Fq(0)2728 4037 y Fr(and)30 b Fo(')2962 4052 y Fv(1)3061 4037 y Fn(2)25 b(D)3215 4052 y Fm(N)3278 4033 y Fq(0)3335 4037 y Fr(suc)n(h)k(that)205 4245 y(\(i\))45 b Fn(h)p Fo(')439 4260 y Fl(0)478 4245 y Fo(;)15 b(H)593 4260 y Fm(N)7 b Fv(\000)p Fm(N)774 4241 y Fq(0)801 4245 y Fo(')860 4260 y Fl(0)899 4245 y Fn(i)25 b Fo(<)g(E)1121 4260 y Fm(N)7 b Fv(\000)p Fm(N)1302 4241 y Fq(0)1348 4245 y Fr(+)20 b Fo("=)p Fr(2)90 b(and)1910 4171 y Fj(\012)1953 4245 y Fo(')2012 4260 y Fv(1)2086 4245 y Fo(;)15 b(H)2208 4213 y Fl(0)2201 4274 y Fm(N)2264 4255 y Fq(0)2291 4245 y Fo(')2350 4260 y Fv(1)2424 4171 y Fj(\013)2492 4245 y Fo(<)25 b(E)2659 4213 y Fl(0)2654 4274 y Fm(N)2717 4255 y Fq(0)2763 4245 y Fr(+)20 b Fo("=)p Fr(2.)180 4460 y(\(ii\))45 b(Both)567 4387 y Fj(\012)610 4460 y Fo(')669 4475 y Fl(0)708 4460 y Fo(;)15 b(N)820 4475 y Fm(f)866 4460 y Fo(')925 4475 y Fl(0)964 4387 y Fj(\013)1035 4460 y Fr(and)1208 4387 y Fj(\012)1251 4460 y Fo(')1310 4475 y Fv(1)1385 4460 y Fo(;)g(N)1497 4475 y Fm(f)1542 4460 y Fo(')1601 4475 y Fv(1)1676 4387 y Fj(\013)1747 4460 y Fr(are)28 b(\014nite)h(and)f(b)r(ounded)h(b)n(y)e (a)h(constan)n(t)f(that)i(is)e(indep)r(en-)345 4607 y(den)n(t)j(of)f Fo(")c(>)g Fr(0.)155 4823 y(\(iii\))45 b Fo(')404 4838 y Fl(0)475 4823 y Fr(and)31 b Fo(')710 4838 y Fv(1)816 4823 y Fr(ha)n(v)n(e)f(compact)h(supp)r(ort)h(as)e(functions)h(of)g (the)g(electronic)h(con\014gurations)f Fo(X)3546 4838 y Fm(N)7 b Fv(\000)p Fm(N)3727 4820 y Fq(0)3780 4823 y Fn(2)345 4970 y Fk(R)405 4938 y Fl(3\()p Fm(N)h Fv(\000)p Fm(N)649 4915 y Fq(0)677 4938 y Fl(\))739 4970 y Fr(and)30 b Fo(X)989 4985 y Fm(N)1052 4966 y Fq(0)1103 4970 y Fn(2)25 b Fk(R)1247 4938 y Fl(3)q Fm(N)1346 4915 y Fq(0)1408 4970 y Fr(resp)r(ectiv)n(ely)-7 b(.)157 5186 y(\(iv\))45 b(There)30 b(exists)f(an)h Fo(R)1047 5201 y Fl(0)1117 5186 y Fr(suc)n(h)f(that)1401 5407 y Fo(')1460 5422 y Fl(0)1524 5407 y Fr(=)c(\000\()p Fo(\037)1766 5422 y Fm(R)1819 5431 y Fe(0)1859 5407 y Fr(\))p Fo(')1953 5422 y Fl(0)1992 5407 y Fo(;)105 b(')2181 5422 y Fv(1)2281 5407 y Fr(=)25 b(\000\()p Fo(\037)2523 5422 y Fm(R)2576 5431 y Fe(0)2615 5407 y Fr(\))p Fo(')2709 5422 y Fv(1)p eop %%Page: 12 12 12 11 bop 120 -200 a Fr(12)345 99 y(where)30 b Fo(\037)661 114 y Fm(R)714 123 y Fe(0)783 99 y Fr(is)g(the)g(c)n(haracteristic)f (function)h(of)g(the)g(ball)g Fn(f)p Fo(y)e Fn(2)d Fk(R)2630 67 y Fl(3)2700 99 y Fr(:)40 b Fn(j)p Fo(y)s Fn(j)25 b Fo(<)g(R)3050 114 y Fl(0)3090 99 y Fn(g)p Fo(:)261 331 y Fg(Pr)-5 b(o)g(of)29 b(of)f(Step)g(1.)38 b Fr(The)27 b(prop)r(erties)f(of)g(the)g(Hamiltonians)f(that)h(are)h(relev)-5 b(an)n(t,)26 b(are)g(shared)g(b)n(y)f Fo(H)3633 347 y Fm(N)7 b Fv(\000)p Fm(N)3814 328 y Fq(0)120 478 y Fr(and)39 b Fo(H)386 447 y Fl(0)379 508 y Fm(N)442 489 y Fq(0)468 478 y Fr(.)67 b(So)39 b(it)h(su\016ces)e(to)h(pro)n(v)n(e)e(existence)i (of)f Fo(')2042 493 y Fl(0)2082 478 y Fr(.)67 b(Let)39 b Fo(H)2419 493 y Fl(0)2498 478 y Fr(:=)h Fo(H)2708 493 y Fm(N)7 b Fv(\000)p Fm(N)2889 474 y Fq(0)2954 478 y Fr(and)39 b Fo(E)3205 493 y Fl(0)3284 478 y Fr(:=)h Fo(E)3486 493 y Fm(N)7 b Fv(\000)p Fm(N)3667 474 y Fq(0)3732 478 y Fr(for)120 625 y(short.)39 b(Let)30 b Fo(\037)602 640 y Fm(P)690 625 y Fr(b)r(e)f(the)h(op)r(erator)f(of)g(m)n(ultiplication) g(with)g Fo(\037)p Fr(\()p Fn(j)p Fo(X)7 b Fn(j)p Fo(=P)12 b Fr(\))30 b(on)f Fn(H)2813 640 y Fm(N)7 b Fv(\000)p Fm(N)2994 621 y Fq(0)3049 625 y Fr(where)29 b Fo(\037)d Fn(2)f Fo(C)3546 593 y Fv(1)3620 625 y Fr(\()p Fk(R)3715 640 y Fl(+)3780 625 y Fr(\),)120 772 y Fo(\037)p Fr(\()p Fo(t)p Fr(\))j(=)g(1)j(for)g Fo(t)d Fn(\024)g Fr(1,)k Fo(\037)p Fr(\()p Fo(t)p Fr(\))c(=)f(0)32 b(for)f Fo(t)d Fn(\025)f Fr(2)32 b(and)g(0)27 b Fn(\024)g Fo(\037)h Fn(\024)g Fr(1.)44 b(Let)32 b Fo(j)2454 787 y Fm(R)2543 772 y Fr(b)r(e)g(the)g(op)r(erator)g(of)f(m)n(ultiplication)120 918 y(with)k Fo(\037)p Fr(\()p Fn(j)p Fo(y)s Fn(j)p Fo(=R)q Fr(\))g(on)f Fo(L)892 887 y Fl(2)932 918 y Fr(\()p Fk(R)1026 887 y Fl(3)1072 918 y Fo(;)15 b(dk)s Fr(\).)53 b(Existence)34 b(of)g Fo(')1897 933 y Fl(0)1970 918 y Fr(with)h(prop)r(ert)n(y)f (\(i\))h(and)f(\(ii\))h(follo)n(ws)e(from)h(the)g(fact)120 1065 y(that)c Fn(D)384 1080 y Fm(N)7 b Fv(\000)p Fm(N)565 1062 y Fq(0)622 1065 y Fr(is)30 b(a)g(form)f(core)h(of)g Fo(H)1368 1080 y Fl(0)1407 1065 y Fr(,)g(the)g(argumen)n(t)g(\(11\),)g (and)g(Lemma)g(8.)40 b(If)29 b(w)n(e)h(no)n(w)f(sho)n(w)h(that)1084 1312 y Fn(h)p Fo(\037)1175 1327 y Fm(P)1234 1312 y Fo(')1293 1327 y Fl(0)1332 1312 y Fo(;)15 b Fr(\()p Fo(H)1482 1327 y Fl(0)1541 1312 y Fn(\000)20 b Fo(E)1698 1327 y Fl(0)1737 1312 y Fr(\))p Fo(\037)1828 1327 y Fm(P)1887 1312 y Fo(')1946 1327 y Fl(0)1985 1312 y Fn(i)2103 1260 y Fm(P)10 b Fv(!1)2129 1312 y Fn(\000)-15 b(!)108 b(h)p Fo(')2476 1327 y Fl(0)2515 1312 y Fo(;)15 b Fr(\()p Fo(H)2665 1327 y Fl(0)2724 1312 y Fn(\000)20 b Fo(E)2881 1327 y Fl(0)2920 1312 y Fr(\))p Fo(')3014 1327 y Fl(0)3054 1312 y Fn(i)591 b Fr(\(12\))641 1483 y Fn(h)p Fr(\000\()p Fo(j)804 1498 y Fm(R)863 1483 y Fr(\))p Fo(\037)954 1498 y Fm(P)1013 1483 y Fo(')1072 1498 y Fl(0)1111 1483 y Fo(;)15 b Fr(\()p Fo(H)1261 1498 y Fl(0)1320 1483 y Fn(\000)20 b Fo(E)1477 1498 y Fl(0)1516 1483 y Fr(\)\000\()p Fo(j)1679 1498 y Fm(R)1737 1483 y Fr(\))p Fo(\037)1828 1498 y Fm(P)1887 1483 y Fo(')1946 1498 y Fl(0)1985 1483 y Fn(i)2104 1432 y Fm(R)p Fv(!1)2129 1483 y Fn(\000)-15 b(!)108 b(h)p Fo(\037)2473 1498 y Fm(P)2532 1483 y Fo(')2591 1498 y Fl(0)2630 1483 y Fo(;)15 b Fr(\()p Fo(H)2780 1498 y Fl(0)2839 1483 y Fn(\000)20 b Fo(E)2996 1498 y Fl(0)3035 1483 y Fr(\))p Fo(\037)3126 1498 y Fm(P)3185 1483 y Fo(')3244 1498 y Fl(0)3284 1483 y Fn(i)361 b Fr(\(13\))120 1730 y(then)32 b(\(iii\),)g(and)f(\(iv\))h (will)f(follo)n(w,)h(b)r(ecause,)g(b)n(y)e(the)i(strong)f(con)n(v)n (ergence)g Fo(\037)2815 1745 y Fm(P)2901 1730 y Fn(!)d Fr(1)j(and)h(\000\()p Fo(j)3400 1745 y Fm(R)3458 1730 y Fr(\))c Fn(!)f Fr(1)32 b(the)120 1877 y(norm)e Fn(k)p Fr(\000\()p Fo(j)528 1892 y Fm(R)586 1877 y Fr(\))p Fo(\037)677 1892 y Fm(P)736 1877 y Fo(')795 1892 y Fl(0)834 1877 y Fn(k)g Fr(is)g(close)g(to)g(1)g(for)f(large)i Fo(P)42 b Fr(and)30 b(large)g Fo(R)q Fr(.)261 2026 y(Prop)r(erties)h(\(12\))f (and)g(\(13\))g(follo)n(w)f(from)1932 2273 y(lim)1897 2333 y Fm(P)10 b Fv(!1)2092 2273 y Fr([)p Fo(H)2192 2288 y Fl(0)2231 2273 y Fo(;)15 b(\037)2327 2288 y Fm(P)2386 2273 y Fr(])p Fo(')2470 2288 y Fl(0)2593 2273 y Fr(=)83 b(0)889 b(\(14\))1204 2461 y(lim)1169 2522 y Fm(R)p Fv(!1)1364 2461 y Fr(\()p Fo(N)1471 2476 y Fm(f)1536 2461 y Fr(+)20 b(1\))1706 2425 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1871 2461 y Fr([)p Fo(H)1971 2476 y Fl(0)2010 2461 y Fo(;)15 b Fr(\000\()p Fo(j)2178 2476 y Fm(R)2236 2461 y Fr(\)])p Fo(\037)2352 2476 y Fm(P)2411 2461 y Fo(')2470 2476 y Fl(0)2593 2461 y Fr(=)83 b(0)889 b(\(15\))120 2708 y(\(to)23 b(b)r(e)h(pro)n(v)n(en)e(shortly\))h(b)n(y)f(comm)n(uting)h(the)h(op)r (erators)f Fo(\037)2167 2723 y Fm(P)2249 2708 y Fr(and)h(\000\()p Fo(j)2546 2723 y Fm(R)2604 2708 y Fr(\))f(through)h Fo(H)3071 2723 y Fl(0)3117 2708 y Fn(\000)7 b Fo(E)3261 2723 y Fl(0)3323 2708 y Fr(and)23 b(using)g(\(ii\))120 2854 y(and)j(that)g Fo(s)12 b Fn(\000)g Fr(lim)744 2869 y Fm(P)e Fv(!1)959 2854 y Fo(\037)1015 2823 y Fl(2)1015 2883 y Fm(P)1099 2854 y Fr(=)25 b(1)h(and)g Fo(s)12 b Fn(\000)g Fr(lim)1697 2869 y Fm(R)p Fv(!1)1911 2854 y Fr(\000\()p Fo(j)2039 2869 y Fm(R)2097 2854 y Fr(\))2132 2823 y Fl(2)2197 2854 y Fr(=)25 b(1.)38 b(Note)26 b(that)g(\000\()p Fo(j)2933 2869 y Fm(R)2991 2854 y Fr(\))p Fo(\037)3082 2869 y Fm(P)3141 2854 y Fn(D)3210 2870 y Fm(N)7 b Fv(\000)p Fm(N)3391 2851 y Fq(0)3443 2854 y Fn(\032)25 b(D)3607 2870 y Fm(N)7 b Fv(\000)p Fm(N)3788 2851 y Fq(0)3815 2854 y Fr(.)261 3004 y(Equation)30 b(\(14\))g(follo)n(ws)f(from)954 3309 y([)p Fo(H)1054 3324 y Fl(0)1093 3309 y Fo(;)15 b(\037)1189 3324 y Fm(P)1248 3309 y Fr(])25 b(=)1393 3195 y Fm(N)7 b Fv(\000)p Fm(N)1574 3172 y Fq(0)1429 3222 y Fj(X)1433 3418 y Fm(j)t Fl(=1)1596 3309 y Fr(\()p Fn(\000)p Fr(2)p Fo(i)p Fr(\))p Fn(r)1887 3324 y Fm(x)1927 3334 y Fi(j)1963 3309 y Fo(\037)2019 3324 y Fm(P)2098 3309 y Fn(\001)20 b Fr(\()p Fo(p)2223 3324 y Fm(j)2280 3309 y Fr(+)g Fo(A)p Fr(\()p Fo(x)2523 3324 y Fm(j)2560 3309 y Fr(\)\))g Fn(\000)g Fr(\001)2815 3324 y Fm(x)2855 3334 y Fi(j)2891 3309 y Fo(\037)2947 3324 y Fm(P)120 3622 y Fr(using)30 b Fn(r)431 3637 y Fm(x)471 3647 y Fi(j)507 3622 y Fo(\037)563 3637 y Fm(P)647 3622 y Fr(=)25 b Fo(O)r Fr(\()p Fo(P)918 3590 y Fv(\000)p Fl(1)1013 3622 y Fr(\),)30 b(\001)1178 3637 y Fm(x)1218 3647 y Fi(j)1254 3622 y Fo(\037)1310 3637 y Fm(P)1394 3622 y Fr(=)25 b Fo(O)r Fr(\()p Fo(P)1665 3590 y Fv(\000)p Fl(2)1760 3622 y Fr(\))30 b(and)g(Lemma)h(8.)261 3772 y(T)-7 b(o)30 b(pro)n(v)n(e)e(\(15\))i(w)n(e)g(write)g(the)g(comm)n(utator)g(as)520 4076 y([)p Fo(H)620 4091 y Fl(0)659 4076 y Fo(;)15 b Fr(\000\()p Fo(j)827 4091 y Fm(R)885 4076 y Fr(\)])25 b(=)1055 3963 y Fm(N)7 b Fv(\000)p Fm(N)1236 3939 y Fq(0)1091 3990 y Fj(X)1099 4185 y Fm(i)p Fl(=1)1273 3975 y Fj(n)1334 4076 y Fr(\()p Fo(p)1414 4091 y Fm(i)1462 4076 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)1705 4091 y Fm(i)1734 4076 y Fr(\)\)[)p Fo(A)p Fr(\()p Fo(x)1982 4091 y Fm(i)2011 4076 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)2214 4091 y Fm(R)2272 4076 y Fr(\)])20 b(+)g([)p Fo(A)p Fr(\()p Fo(x)2620 4091 y Fm(i)2649 4076 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)2852 4091 y Fm(R)2910 4076 y Fr(\)]\()p Fo(p)3050 4091 y Fm(i)3099 4076 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)3342 4091 y Fm(i)3370 4076 y Fr(\)\))1240 4332 y(+)g([)p Fo(\033)1406 4347 y Fm(i)1454 4332 y Fn(\001)g Fo(B)5 b Fr(\()p Fo(x)1658 4347 y Fm(i)1687 4332 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)1890 4347 y Fm(R)1948 4332 y Fr(\)])20 b(+)g([)p Fo(H)2218 4347 y Fm(f)2263 4332 y Fo(;)15 b Fr(\000\()p Fo(j)2431 4347 y Fm(R)2489 4332 y Fr(\)])2549 4231 y Fj(o)3680 4170 y Fr(\(16\))120 4574 y(Using)26 b(that)h Fo(H)637 4589 y Fm(f)707 4574 y Fr(=)e(d\000\()p Fn(j)p Fo(k)s Fn(j)p Fr(\),)i(the)g(last)f(term)h(in)g(\(16\))f(restricted)h (to)g Fn(\012)2524 4542 y Fm(n)2524 4594 y(s)2571 4574 y Fo(L)2632 4542 y Fl(2)2671 4574 y Fr(\()p Fk(R)2766 4542 y Fl(3)2811 4574 y Fr(\))g(is)f(giv)n(en)g(b)n(y)g([)p Fo(H)3414 4589 y Fm(f)3459 4574 y Fo(;)15 b Fr(\000\()p Fo(j)3627 4589 y Fm(R)3685 4574 y Fr(\)])25 b(=)120 4652 y Fj(P)216 4678 y Fm(n)216 4747 y(l)q Fl(=1)347 4720 y Fo(j)384 4735 y Fm(R)449 4720 y Fn(\012)7 b Fo(:)15 b(:)g(:)7 b Fn(\012)g Fr([)p Fn(j)p Fo(k)s Fn(j)p Fo(;)15 b(j)917 4735 y Fm(R)975 4720 y Fr(])g Fo(:)g(:)g(:)7 b Fn(\012)g Fo(j)1241 4735 y Fm(R)1299 4720 y Fr(,)25 b(the)f(comm)n(utator)f(b)r(eing)i(the)e Fo(l)r Fr(th)h(factor.)37 b(Since)24 b Fn(k)p Fr([)p Fn(j)p Fo(k)s Fn(j)p Fo(;)15 b(j)3288 4735 y Fm(R)3345 4720 y Fr(])p Fn(k)25 b Fr(=)g Fo(O)r Fr(\()p Fo(R)3710 4689 y Fv(\000)p Fl(1)3805 4720 y Fr(\))120 4867 y(it)38 b(follo)n(ws)f(that)h Fn(k)p Fr(\()p Fo(N)d Fr(+)25 b(1\))1087 4836 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1252 4867 y Fr([)p Fo(H)1352 4882 y Fm(f)1397 4867 y Fo(;)15 b Fr(\000\()p Fo(j)1565 4882 y Fm(R)1623 4867 y Fr(\)]\()p Fo(N)35 b Fr(+)25 b(1\))2000 4836 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)2165 4867 y Fn(k)38 b Fr(=)g Fo(O)r Fr(\()p Fo(R)2531 4836 y Fv(\000)p Fl(1)2626 4867 y Fr(\),)i(and)e(hence,)i(b)n(y)d(\(ii\),)j(that)e(the)120 5014 y(con)n(tribution)31 b(due)g(to)g Fo(H)991 5029 y Fm(f)1067 5014 y Fr(is)f(of)h(order)g Fo(R)1567 4982 y Fv(\000)p Fl(1)1661 5014 y Fr(.)43 b(T)-7 b(o)31 b(deal)g(with)g(the) g(\014rst)g(t)n(w)n(o)f(terms)g(in)h(\(16\))h(note)f(that,)g(b)n(y)120 5161 y(\(19\))f(and)g(\(20\),)782 5407 y([)p Fo(A)p Fr(\()p Fo(x)960 5422 y Fm(i)989 5407 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)1192 5422 y Fm(R)1250 5407 y Fr(\)])25 b(=)g Fo(a)1477 5372 y Fv(\003)1517 5407 y Fr(\(\(1)20 b Fn(\000)g Fo(j)1779 5422 y Fm(R)1837 5407 y Fr(\))p Fo(G)1943 5422 y Fm(x)1983 5432 y Fi(i)2013 5407 y Fr(\)\000\()p Fo(j)2176 5422 y Fm(R)2234 5407 y Fr(\))g Fn(\000)g Fr(\000\()p Fo(j)2507 5422 y Fm(R)2565 5407 y Fr(\))p Fo(a)p Fr(\(\(1)g Fn(\000)g Fo(j)2909 5422 y Fm(R)2967 5407 y Fr(\))p Fo(G)3073 5422 y Fm(x)3113 5432 y Fi(i)3143 5407 y Fr(\))p eop %%Page: 13 13 13 12 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2267 b Fr(13)120 99 y(where)419 336 y Fn(k)p Fo(a)511 301 y Fm(])543 336 y Fr(\(\()p Fo(j)650 351 y Fm(R)728 336 y Fn(\000)20 b Fr(1\))p Fo(G)969 351 y Fm(x)1009 361 y Fi(i)1039 336 y Fr(\))p Fo(\037)1130 351 y Fm(P)1189 336 y Fr(\()p Fo(N)30 b Fr(+)20 b(1\))1496 301 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1661 336 y Fn(k)25 b(\024)82 b Fr(sup)1826 419 y Fv(j)p Fm(x)1886 429 y Fi(i)1911 419 y Fv(j\024)p Fl(2)p Fm(P)2091 336 y Fn(k)p Fr(\()p Fo(j)2208 351 y Fm(R)2285 336 y Fn(\000)20 b Fr(1\))p Fo(G)2526 351 y Fm(x)2566 361 y Fi(i)2596 336 y Fn(k)25 b(!)g Fr(0)p Fo(;)195 b Fr(as)90 b Fo(R)26 b Fn(!)f(1)p Fo(:)120 612 y Fr(It)32 b(follo)n(ws)e(that)i(the)h(terms) e(in)h(\(16\))h(whic)n(h)e(are)h(quadratic)g(in)g Fo(A)p Fr(\()p Fo(x)2500 627 y Fm(i)2529 612 y Fr(\))g(giv)n(e)f(v)-5 b(anishing)32 b(con)n(tributions,)f(as)120 759 y(the)38 b(factors)f Fo(A)p Fr(\()p Fo(x)737 774 y Fm(i)766 759 y Fr(\))h(outside)f(the)h(comm)n(utators)f(can)h(b)r(e)g(con)n(trolled) g(b)n(y)f(\()p Fo(N)e Fr(+)25 b(1\))3050 727 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)3215 759 y Fr(.)64 b(T)-7 b(o)37 b(sho)n(w)g(that) 120 906 y(the)43 b(terms)g(with)g(an)g(op)r(erator)g Fo(p)1328 921 y Fm(i)1400 906 y Fr(v)-5 b(anish)42 b(in)h(the)g(limit)g Fo(R)48 b Fn(!)e(1)d Fr(it)h(su\016ces)e(to)h(add)g(to)g(the)g(ab)r(o)n (v)n(e)120 1053 y(argumen)n(ts)29 b(that)h Fo(p)798 1068 y Fm(i)827 1053 y Fr([)p Fo(A)p Fr(\()p Fo(x)1005 1068 y Fm(i)1034 1053 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)1237 1068 y Fm(R)1295 1053 y Fr(\)])25 b(=)g([)p Fo(A)p Fr(\()p Fo(x)1653 1068 y Fm(i)1682 1053 y Fr(\))p Fo(;)15 b Fr(\000\()p Fo(j)1885 1068 y Fm(R)1943 1053 y Fr(\)])p Fo(p)2048 1068 y Fm(i)2107 1053 y Fr(b)r(ecause)30 b Fo(p)2485 1068 y Fm(i)2543 1053 y Fr(comm)n(utes)g(with)g Fo(A)p Fr(\()p Fo(x)3325 1068 y Fm(i)3354 1053 y Fr(\))g(and)g(\000\()p Fo(j)3722 1068 y Fm(R)3780 1053 y Fr(\),)120 1199 y(that)35 b Fo(p)365 1214 y Fm(i)394 1199 y Fo(\037)450 1214 y Fm(P)542 1199 y Fr(=)f Fo(\037)702 1214 y Fm(P)761 1199 y Fo(p)806 1214 y Fm(i)857 1199 y Fn(\000)24 b Fo(i)p Fn(r)1057 1214 y Fm(i)1085 1199 y Fo(\037)1141 1214 y Fm(P)1235 1199 y Fr(and)35 b(that)g Fn(k)p Fo(p)1705 1214 y Fm(i)1734 1199 y Fo(')1793 1214 y Fl(0)1832 1199 y Fn(k)f Fo(<)f Fn(1)i Fr(b)n(y)f(Lemma)i(8.)55 b(The)35 b(term)h(in)n(v)n(olving)d Fo(B)5 b Fr(\()p Fo(x)3681 1214 y Fm(i)3709 1199 y Fr(\))35 b(is)120 1346 y(dealt)30 b(with)g(similarly)-7 b(.)261 1493 y Ff(Step)34 b(2.)39 b Fr(Let)30 b Fo(";)43 b(')946 1508 y Fl(0)986 1493 y Fr(,)29 b(and)f Fo(')1272 1508 y Fv(1)1376 1493 y Fr(b)r(e)h(as)g(in)f (Step)h(1.)40 b(Pic)n(k)28 b Fo(R)2292 1508 y Fl(0)2361 1493 y Fr(so)g(large)h(that,)g(with)g Fo(\037)3168 1508 y Fm(R)3221 1517 y Fe(0)3289 1493 y Fr(is)f(as)g(in)h(Step)g(1)120 1640 y(\(iv\),)j Fo(')379 1655 y Fl(0)447 1640 y Fr(=)c(\000\()p Fo(\037)692 1655 y Fm(R)745 1664 y Fe(0)785 1640 y Fr(\))p Fo(')879 1655 y Fl(0)918 1640 y Fr(,)k Fo(')1034 1655 y Fv(1)1137 1640 y Fr(=)d(\000\()p Fo(\037)1383 1655 y Fm(R)1436 1664 y Fe(0)1475 1640 y Fr(\))p Fo(')1569 1655 y Fv(1)1644 1640 y Fr(,)j Fo(')1760 1655 y Fl(0)1800 1640 y Fr(\()p Fo(X)1910 1655 y Fm(N)7 b Fv(\000)p Fm(N)2091 1636 y Fq(0)2117 1640 y Fr(\))28 b(=)h(0)j(if)f Fn(j)p Fo(X)2540 1655 y Fm(N)7 b Fv(\000)p Fm(N)2721 1636 y Fq(0)2747 1640 y Fn(j)29 b Fo(>)f(R)2967 1655 y Fl(0)3039 1640 y Fr(and)k Fo(')3275 1655 y Fv(1)3350 1640 y Fr(\()p Fo(X)3460 1655 y Fm(N)3523 1636 y Fq(0)3549 1640 y Fr(\))c(=)g(0)k(if) 120 1786 y Fn(j)p Fo(X)220 1802 y Fm(N)283 1783 y Fq(0)309 1786 y Fn(j)26 b Fo(>)f(R)523 1801 y Fl(0)563 1786 y Fr(.)42 b(Let)30 b Fo(R)d Fn(\025)f Fo(R)1050 1801 y Fl(0)1120 1786 y Fr(and)k(pic)n(k)g(a)g(v)n(ector)g Fo(d)25 b Fn(2)h Fk(R)2050 1755 y Fl(3)2126 1786 y Fr(with)k Fn(j)p Fo(d)p Fn(j)25 b Fr(=)h(3.)41 b(Let)31 b Fo(T)2875 1801 y Fm(R)2958 1786 y Fr(:)26 b Fn(H)3085 1802 y Fm(N)7 b Fv(\000)p Fm(N)3266 1783 y Fq(0)3318 1786 y Fn(!)26 b(H)3510 1802 y Fm(N)7 b Fv(\000)p Fm(N)3691 1783 y Fq(0)3747 1786 y Fr(b)r(e)120 1933 y(the)30 b(translation)1291 2115 y Fo(T)1344 2130 y Fm(R)1427 2115 y Fr(=)25 b(exp)1674 1960 y Fj( )1746 2115 y Fn(\000)p Fo(iR)q(d)20 b Fn(\001)2028 2014 y Fj(n)2126 2002 y Fm(N)2189 1978 y Fq(0)2103 2029 y Fj(X)2112 2224 y Fm(i)p Fl(=1)2250 2115 y Fo(p)2295 2130 y Fm(i)2343 2115 y Fr(+)g Fo(P)2491 2130 y Fm(f)2536 2014 y Fj(o)2597 1960 y(!)120 2364 y Fr(where)30 b Fo(P)438 2379 y Fm(f)508 2364 y Fr(=)25 b(d\000\()p Fo(k)s Fr(\))30 b(is)g(the)g(total)g(momen)n(tum)g(op)r(erator)g(of)g(the)g(photons.)40 b(Then)1193 2528 y Fj(\012)1236 2602 y Fo(T)1289 2617 y Fm(R)1346 2602 y Fo(')1405 2617 y Fv(1)1480 2602 y Fo(;)15 b(H)1602 2566 y Fl(0)1595 2627 y Fm(N)1658 2608 y Fq(0)1684 2602 y Fo(T)1737 2617 y Fm(R)1794 2602 y Fo(')1853 2617 y Fv(1)1928 2528 y Fj(\013)2054 2602 y Fr(=)2207 2528 y Fj(\012)2250 2602 y Fo(')2309 2617 y Fv(1)2384 2602 y Fo(;)g(H)2506 2566 y Fl(0)2499 2627 y Fm(N)2562 2608 y Fq(0)2588 2602 y Fo(')2647 2617 y Fv(1)2721 2528 y Fj(\013)2779 2602 y Fo(;)1141 2773 y( )1200 2788 y Fm(R)1282 2773 y Fr(:=)25 b Fo(I)7 b Fr(\()p Fo(')1543 2788 y Fl(0)1602 2773 y Fn(\012)20 b Fo(T)1745 2788 y Fm(R)1802 2773 y Fo(')1861 2788 y Fv(1)1936 2773 y Fr(\))88 b Fn(2)g(D)2276 2788 y Fm(N)s(;)p Fl(2)p Fm(R)2448 2773 y Fo(:)261 3011 y Ff(Step)36 b(3.)j Fr(If)29 b Fo(R)d Fn(\025)f Fo(R)965 3026 y Fl(0)1035 3011 y Fr(then)30 b Fn(k)p Fo( )1344 3026 y Fm(R)1402 3011 y Fn(k)24 b Fr(=)h(1)31 b(and)657 3249 y Fn(h)p Fo( )751 3264 y Fm(R)808 3249 y Fo(;)15 b(H)923 3264 y Fm(N)990 3249 y Fo( )1049 3264 y Fm(R)1106 3249 y Fn(i)25 b Fr(=)g Fn(h)p Fo(')1355 3264 y Fl(0)1394 3249 y Fo(;)15 b(H)1509 3264 y Fm(N)7 b Fv(\000)p Fm(N)1690 3245 y Fq(0)1717 3249 y Fo(')1776 3264 y Fl(0)1815 3249 y Fn(i)20 b Fr(+)1960 3175 y Fj(\012)2003 3249 y Fo(')2062 3264 y Fv(1)2136 3249 y Fo(;)15 b(H)2258 3213 y Fl(0)2251 3274 y Fm(N)2314 3256 y Fq(0)2341 3249 y Fo(')2400 3264 y Fv(1)2474 3175 y Fj(\013)2537 3249 y Fr(+)20 b Fo(o)p Fr(\()p Fo(R)2774 3213 y Fl(0)2814 3249 y Fr(\))p Fo(;)105 b(R)26 b Fn(!)f(1)p Fo(:)120 3487 y Fr(In)30 b(particular)g(\006)713 3502 y Fm(R)770 3487 y Fr(\()p Fo(H)880 3502 y Fm(N)947 3487 y Fr(\))25 b Fn(\024)g Fo(E)1169 3502 y Fm(N)7 b Fv(\000)p Fm(N)1350 3483 y Fq(0)1396 3487 y Fr(+)20 b Fo(E)1558 3455 y Fl(0)1553 3516 y Fm(N)1616 3498 y Fq(0)1662 3487 y Fr(+)g(2)p Fo(")30 b Fr(for)g(all)g Fo(R)q Fr(,)g(whic)n(h)g(pro)n(v)n(es)e(the)i (theorem.)261 3634 y Fg(Pr)-5 b(o)g(of)32 b(of)g(Step)f(3.)40 b Fr(By)29 b(construction)g(of)g Fo(')1738 3649 y Fl(0)1807 3634 y Fr(and)h Fo(T)2035 3649 y Fm(R)2093 3634 y Fo(')2152 3649 y Fv(1)2256 3634 y Fr(the)g(photons)f(in)h(these)f(states)g(ha)n (v)n(e)f(disjoin)n(t)120 3780 y(supp)r(ort)i(if)g Fo(R)c Fn(\025)f Fo(R)793 3795 y Fl(0)833 3780 y Fr(.)40 b(Therefore)1059 4018 y Fn(h)p Fo( )1153 4033 y Fm(R)1211 4018 y Fo(;)15 b( )1310 4033 y Fm(R)1367 4018 y Fn(i)83 b Fr(=)g Fn(h)p Fo(I)7 b Fr(\()p Fo(')1814 4033 y Fl(0)1873 4018 y Fn(\012)20 b Fo(T)2016 4033 y Fm(R)2073 4018 y Fo(')2132 4033 y Fv(1)2207 4018 y Fr(\))p Fo(;)15 b(I)7 b Fr(\()p Fo(')2423 4033 y Fl(0)2482 4018 y Fn(\012)20 b Fo(T)2625 4033 y Fm(R)2682 4018 y Fo(')2741 4033 y Fv(1)2816 4018 y Fr(\))p Fn(i)1485 4190 y Fr(=)83 b Fn(h)p Fo(')1732 4205 y Fl(0)1791 4190 y Fn(\012)20 b Fo(T)1934 4205 y Fm(R)1991 4190 y Fo(')2050 4205 y Fv(1)2125 4190 y Fo(;)15 b(')2224 4205 y Fl(0)2283 4190 y Fn(\012)20 b Fo(T)2426 4205 y Fm(R)2484 4190 y Fo(')2543 4205 y Fv(1)2617 4190 y Fn(i)1485 4362 y Fr(=)83 b Fn(h)p Fo(')1732 4377 y Fl(0)1771 4362 y Fo(;)15 b(')1870 4377 y Fl(0)1909 4362 y Fn(i)g(h)p Fo(')2053 4377 y Fv(1)2128 4362 y Fo(;)g(')2227 4377 y Fv(1)2301 4362 y Fn(i)25 b Fr(=)g(1)p Fo(:)120 4599 y Fr(In)i(the)h(follo)n(wing) f(this)g(prop)r(ert)n(y)g(of)h Fo(I)7 b Fr(,)28 b(that)f(it)h(acts)f (lik)n(e)g(an)h(isometry)f(on)g(pro)r(duct)i(states)e(with)g(photons) 120 4746 y(supp)r(orted)j(in)g Fn(fj)p Fo(y)s Fn(j)25 b(\024)g Fo(R)978 4761 y Fl(0)1018 4746 y Fn(g)30 b Fr(and)g Fn(fj)p Fo(y)23 b Fn(\000)d Fo(R)q(d)p Fn(j)25 b(\024)g Fo(R)1824 4761 y Fl(0)1864 4746 y Fn(g)30 b Fr(resp)r(ectiv)n(ely)-7 b(,)28 b(will)i(b)r(e)h(used)e(rep)r(eatedly)i(and)f(tacitly)-7 b(.)261 4893 y(W)g(riting)29 b Fo(H)666 4908 y Fm(f)736 4893 y Fr(=)831 4825 y Fj(P)927 4920 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1133 4820 y Fj(R)1208 4893 y Fn(j)p Fo(k)s Fn(j)p Fo(a)1355 4861 y Fv(\003)1355 4922 y Fm(\025)1401 4893 y Fr(\()p Fo(k)s Fr(\))p Fo(a)1568 4908 y Fm(\025)1613 4893 y Fr(\()p Fo(k)s Fr(\))p Fo(d)1780 4861 y Fl(3)1819 4893 y Fo(k)k Fr(and)d(using)h(\(21\))f(one)g(gets)540 5057 y Fj(\012)583 5131 y Fo( )642 5146 y Fm(R)699 5131 y Fo(;)15 b(H)814 5146 y Fm(f)859 5131 y Fo( )918 5146 y Fm(R)975 5057 y Fj(\013)1101 5131 y Fr(=)1254 5057 y Fj(\012)1297 5131 y Fo(')1356 5146 y Fl(0)1396 5131 y Fo(;)g(H)1511 5146 y Fm(f)1556 5131 y Fo(')1615 5146 y Fl(0)1654 5057 y Fj(\013)1717 5131 y Fr(+)1807 5057 y Fj(\012)1850 5131 y Fo(T)1903 5146 y Fm(R)1960 5131 y Fo(')2019 5146 y Fv(1)2094 5131 y Fo(;)g(H)2209 5146 y Fm(f)2254 5131 y Fo(T)2307 5146 y Fm(R)2364 5131 y Fo(')2423 5146 y Fv(1)2498 5057 y Fj(\013)1254 5320 y Fr(+2)g(Re)1533 5233 y Fj(X)1506 5431 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1707 5196 y Fj(Z)1813 5320 y Fn(j)p Fo(k)s Fn(j)g(h)o Fo(a)2009 5335 y Fm(\025)2055 5320 y Fr(\()p Fo(k)s Fr(\))p Fo(')2234 5335 y Fl(0)2273 5320 y Fo(;)g(')2372 5335 y Fl(0)2412 5320 y Fn(i)f(h)p Fo(')2555 5335 y Fv(1)2630 5320 y Fo(;)h(a)2717 5335 y Fm(\025)2763 5320 y Fr(\()p Fo(k)s Fr(\))p Fo(')2942 5335 y Fv(1)3016 5320 y Fn(i)g Fo(e)3108 5284 y Fm(iRd)p Fv(\001)p Fm(k)3284 5320 y Fo(d)3331 5284 y Fl(3)3370 5320 y Fo(k)p eop %%Page: 14 14 14 13 bop 120 -200 a Fr(14)120 99 y(where)34 b Fo(T)449 67 y Fv(\003)437 127 y Fm(R)494 99 y Fo(a)p Fr(\()p Fo(k)s Fr(\))p Fo(T)714 114 y Fm(R)803 99 y Fr(=)d Fo(e)946 67 y Fm(iRd)p Fv(\001)p Fm(k)1122 99 y Fo(a)p Fr(\()p Fo(k)s Fr(\))j(w)n(as)e(also)i(used.)50 b(The)34 b(third)g(term)g(con)n (v)n(erges)d(to)j(zero)g(as)f Fo(R)f Fn(!)f(1)j Fr(b)n(y)120 245 y(the)c(Riemann-Leb)r(esgue)h(Theorem)f(b)r(ecause)g(the)g(in)n (tegrand)g(is)f(in)h Fo(L)2573 214 y Fl(1)2613 245 y Fr(\()p Fk(R)2708 214 y Fl(3)2753 245 y Fo(;)15 b Fk(C)2853 214 y Fl(2)2898 245 y Fr(\).)261 394 y(Since)30 b(the)g(distance)f(of)h (the)f(electrons)h(describ)r(ed)g(b)n(y)f Fo(T)2210 409 y Fm(R)2267 394 y Fo(')2326 409 y Fv(1)2430 394 y Fr(to)h(the)g(origin) g(and)g(to)g(the)f(electrons)h(in)120 541 y Fo(')179 556 y Fl(0)248 541 y Fr(is)g(b)r(ounded)h(b)r(elo)n(w)f(b)n(y)f(3)p Fo(R)21 b Fn(\000)f Fr(3)p Fo(R)1424 556 y Fl(0)1464 541 y Fr(,)30 b(w)n(e)f(ha)n(v)n(e,)g(b)n(y)g(assumption)g(\(H2\),)h (that)291 786 y Fn(h)p Fo( )385 801 y Fm(R)442 786 y Fo(;)15 b(V)534 801 y Fm(N)602 786 y Fo( )661 801 y Fm(R)718 786 y Fn(i)25 b Fr(=)g Fn(h)p Fo(')967 801 y Fl(0)1006 786 y Fo(;)15 b(V)1098 801 y Fm(N)7 b Fv(\000)p Fm(N)1279 782 y Fq(0)1306 786 y Fo(')1365 801 y Fl(0)1404 786 y Fn(i)20 b Fr(+)1549 699 y Fj(X)1559 895 y Fm(iN)7 b Fv(\000)p Fm(N)2138 1797 y Fq(0)2175 1631 y Fj(\012)2218 1704 y Fo(T)2271 1719 y Fm(R)2328 1704 y Fo(')2387 1719 y Fv(1)2462 1704 y Fo(;)15 b Fr(\()p Fo(p)2582 1719 y Fm(j)2639 1704 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)2882 1719 y Fm(j)2919 1704 y Fr(\)\))2989 1669 y Fl(2)3028 1704 y Fo(T)3081 1719 y Fm(R)3139 1704 y Fo(')3198 1719 y Fv(1)3272 1631 y Fj(\013)3330 1704 y Fo(:)120 1976 y Fr(T)-7 b(o)30 b(this)f(end)h(w)n(e)g(write)g Fo(A)p Fr(\()p Fo(x)1114 1991 y Fm(j)1151 1976 y Fr(\))25 b(=)g Fo(a)p Fr(\()p Fo(G)1459 1991 y Fm(x)1499 2001 y Fi(j)1536 1976 y Fr(\))20 b(+)g Fo(a)1728 1945 y Fv(\003)1768 1976 y Fr(\()p Fo(G)1874 1991 y Fm(x)1914 2001 y Fi(j)1950 1976 y Fr(\))30 b(and)g(use)f(that)617 2221 y(\()p Fo(p)697 2236 y Fm(j)754 2221 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)997 2236 y Fm(j)1034 2221 y Fr(\)\))1104 2185 y Fl(2)1227 2221 y Fr(=)83 b Fo(p)1425 2185 y Fl(2)1425 2244 y Fm(j)1484 2221 y Fr(+)20 b(2)p Fo(p)1664 2236 y Fm(j)1721 2221 y Fn(\001)g Fo(a)p Fr(\()p Fo(G)1919 2236 y Fm(x)1959 2246 y Fi(j)1996 2221 y Fr(\))g(+)g(2)p Fo(a)2233 2185 y Fv(\003)2273 2221 y Fr(\()p Fo(G)2379 2236 y Fm(x)2419 2246 y Fi(j)2455 2221 y Fr(\))g Fn(\001)g Fo(p)2600 2236 y Fm(j)1380 2393 y Fr(+)p Fo(a)p Fr(\()p Fo(G)1603 2408 y Fm(x)1643 2418 y Fi(j)1679 2393 y Fr(\))1714 2357 y Fl(2)1774 2393 y Fr(+)g Fo(a)1911 2357 y Fv(\003)1951 2393 y Fr(\()p Fo(G)2057 2408 y Fm(x)2097 2418 y Fi(j)2133 2393 y Fr(\))2168 2357 y Fl(2)2227 2393 y Fr(+)g(2)p Fo(a)2409 2357 y Fv(\003)2449 2393 y Fr(\()p Fo(G)2555 2408 y Fm(x)2595 2418 y Fi(j)2631 2393 y Fr(\))p Fo(a)p Fr(\()p Fo(G)2819 2408 y Fm(x)2859 2418 y Fi(j)2896 2393 y Fr(\))g(+)g Fn(k)p Fo(G)3157 2408 y Fm(x)3197 2418 y Fi(j)3233 2393 y Fn(k)3278 2357 y Fl(2)3318 2393 y Fo(:)120 2637 y Fr(Let)35 b Fo(j)i Fn(\024)c Fo(N)f Fn(\000)23 b Fo(N)742 2606 y Fv(0)765 2637 y Fr(,)36 b(then)e(using)i(\(21\))e (and)h(again)f(disjoin)n(tness)e(of)i(the)h(supp)r(orts)f(of)g(the)g (photons)g(in)h Fo(')3801 2652 y Fl(0)120 2784 y Fr(and)30 b Fo(T)348 2799 y Fm(R)405 2784 y Fo(')464 2799 y Fv(1)539 2784 y Fr(,)g(one)g(\014nds)g(that)420 2955 y Fj(\012)463 3029 y Fo( )522 3044 y Fm(R)579 3029 y Fo(;)15 b Fr(\()p Fo(p)699 3044 y Fm(j)756 3029 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)999 3044 y Fm(j)1036 3029 y Fr(\)\))1106 2993 y Fl(2)1145 3029 y Fo( )1204 3044 y Fm(R)1262 2955 y Fj(\013)1388 3029 y Fr(=)1541 2955 y Fj(\012)1583 3029 y Fo(')1642 3044 y Fl(0)1682 3029 y Fo(;)15 b Fr(\()p Fo(p)1802 3044 y Fm(j)1858 3029 y Fr(+)20 b Fo(A)p Fr(\()p Fo(x)2101 3044 y Fm(j)2139 3029 y Fr(\)\))2209 2993 y Fl(2)2248 3029 y Fo(')2307 3044 y Fl(0)2346 2955 y Fj(\013)1541 3200 y Fr(+2)1671 3127 y Fj(\012)1713 3200 y Fo(')1772 3215 y Fl(0)1832 3200 y Fn(\012)g Fo(T)1975 3215 y Fm(R)2032 3200 y Fo(')2091 3215 y Fv(1)2166 3200 y Fo(;)15 b(p)2251 3215 y Fm(j)2287 3200 y Fo(')2346 3215 y Fl(0)2406 3200 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2649 3215 y Fm(x)2689 3225 y Fi(j)2725 3200 y Fr(\))p Fo(T)2813 3215 y Fm(R)2871 3200 y Fo(')2930 3215 y Fv(1)3004 3127 y Fj(\013)3097 3200 y Fr(+)50 b(h.c.)1541 3372 y(+2)1671 3298 y Fj(\012)1713 3372 y Fo(')1772 3387 y Fl(0)1832 3372 y Fn(\012)20 b Fo(T)1975 3387 y Fm(R)2032 3372 y Fo(')2091 3387 y Fv(1)2166 3372 y Fo(;)15 b(a)p Fr(\()p Fo(G)2359 3387 y Fm(x)2399 3397 y Fi(j)2435 3372 y Fr(\))p Fo(')2529 3387 y Fl(0)2589 3372 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2832 3387 y Fm(x)2872 3397 y Fi(j)2908 3372 y Fr(\))p Fo(T)2996 3387 y Fm(R)3054 3372 y Fo(')3113 3387 y Fv(1)3187 3298 y Fj(\013)3280 3372 y Fr(+)50 b(h.c.)1541 3544 y(+)1626 3470 y Fj(\012)1668 3544 y Fo(')1727 3559 y Fl(0)1787 3544 y Fn(\012)20 b Fo(T)1930 3559 y Fm(R)1987 3544 y Fo(')2046 3559 y Fv(1)2121 3544 y Fo(;)15 b(')2220 3559 y Fl(0)2279 3544 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2522 3559 y Fm(x)2562 3569 y Fi(j)2599 3544 y Fr(\))2634 3508 y Fl(2)2673 3544 y Fo(T)2726 3559 y Fm(R)2783 3544 y Fo(')2842 3559 y Fv(1)2917 3470 y Fj(\013)3010 3544 y Fr(+)50 b(h.c.)1541 3715 y(+)1626 3642 y Fj(\012)1668 3715 y Fo(a)p Fr(\()p Fo(G)1821 3730 y Fm(x)1861 3740 y Fi(j)1898 3715 y Fr(\))p Fo(')1992 3730 y Fl(0)2051 3715 y Fn(\012)20 b Fo(T)2194 3730 y Fm(R)2252 3715 y Fo(')2311 3730 y Fv(1)2385 3715 y Fo(;)15 b(')2484 3730 y Fl(0)2544 3715 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2787 3730 y Fm(x)2827 3740 y Fi(j)2863 3715 y Fr(\))p Fo(T)2951 3730 y Fm(R)3009 3715 y Fo(')3068 3730 y Fv(1)3142 3642 y Fj(\013)3235 3715 y Fr(+)50 b(h.c.)1541 3887 y(+)1626 3814 y Fj(\012)1668 3887 y Fo(')1727 3902 y Fl(0)1787 3887 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2030 3902 y Fm(x)2070 3912 y Fi(j)2106 3887 y Fr(\))p Fo(T)2194 3902 y Fm(R)2252 3887 y Fo(')2311 3902 y Fv(1)2385 3887 y Fo(;)15 b(')2484 3902 y Fl(0)2544 3887 y Fn(\012)20 b Fo(a)p Fr(\()p Fo(G)2787 3902 y Fm(x)2827 3912 y Fi(j)2863 3887 y Fr(\))p Fo(T)2951 3902 y Fm(R)3009 3887 y Fo(')3068 3902 y Fv(1)3142 3814 y Fj(\013)3200 3887 y Fo(:)120 4132 y Fr(All)30 b(terms)f(except)h(the)g(\014rst)g(one)g(v)-5 b(anish)29 b(in)h(the)g(limit)g Fo(R)c Fn(!)f(1)p Fr(.)40 b(In)30 b(fact,)1057 4377 y Fo(a)p Fr(\()p Fo(G)1210 4392 y Fm(x)1250 4402 y Fi(j)1287 4377 y Fr(\))p Fo(T)1375 4392 y Fm(R)1432 4377 y Fo(')1491 4392 y Fv(1)1649 4377 y Fr(=)83 b Fo(T)1855 4392 y Fm(R)1912 4377 y Fo(a)p Fr(\()p Fo(G)2065 4392 y Fm(x)2105 4402 y Fi(j)2138 4392 y Fv(\000)p Fm(Rd)2287 4377 y Fr(\)\000\()p Fo(\037)2469 4392 y Fm(R)2522 4401 y Fe(0)2561 4377 y Fr(\))p Fo(')2655 4392 y Fv(1)1649 4548 y Fr(=)g Fo(T)1855 4563 y Fm(R)1912 4548 y Fr(\000\()p Fo(\037)2059 4563 y Fm(R)2112 4572 y Fe(0)2152 4548 y Fr(\))p Fo(a)p Fr(\()p Fo(\037)2325 4563 y Fm(R)2378 4572 y Fe(0)2417 4548 y Fo(G)2488 4563 y Fm(x)2528 4573 y Fi(j)2560 4563 y Fv(\000)p Fm(Rd)2709 4548 y Fr(\))p Fo(')2803 4563 y Fv(1)2878 4548 y Fo(;)120 4793 y Fr(and)30 b(since)h Fn(j)p Fo(x)593 4808 y Fm(j)629 4793 y Fn(j)26 b(\024)g Fo(R)844 4808 y Fl(0)914 4793 y Fr(if)k Fo(')1056 4808 y Fl(0)1095 4793 y Fr(\()p Fo(x)1181 4808 y Fl(1)1221 4793 y Fo(;)15 b(:)g(:)g(:)g(;)g(x)1472 4808 y Fm(N)7 b Fv(\000)p Fm(N)1653 4789 y Fq(0)1680 4793 y Fr(\))25 b Fn(6)p Fr(=)h(0,)31 b(w)n(e)f(can)g(m)n(ultiply)g (this)g(in)g(all)h(the)f(ab)r(o)n(v)n(e)g(terms)g(with)120 4940 y Fo(\037)176 4955 y Fm(R)229 4964 y Fe(0)268 4940 y Fr(\()p Fo(x)354 4955 y Fm(j)391 4940 y Fr(\).)40 b(But)30 b(then,)g(b)n(y)f(Equation)h(\(18\))g(and)g(with)g Fo(G)2055 4955 y Fm(\025)2100 4940 y Fr(\()p Fo(k)s Fr(\))25 b(=)g Fn(j)p Fo(k)s Fn(j)2440 4908 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)2605 4940 y Fo(")2647 4955 y Fm(\025)2692 4940 y Fr(\()p Fo(k)s Fr(\))p Fo(\037)2868 4955 y Fl(\003)2921 4940 y Fr(\()p Fo(k)s Fr(\))120 5197 y Fn(k)p Fo(\037)221 5212 y Fm(R)274 5221 y Fe(0)313 5197 y Fr(\()p Fo(x)399 5212 y Fm(j)436 5197 y Fr(\))p Fo(a)p Fr(\()p Fo(\037)609 5212 y Fm(R)662 5221 y Fe(0)702 5197 y Fo(G)773 5212 y Fm(x)813 5222 y Fi(j)845 5212 y Fv(\000)p Fm(Rd)994 5197 y Fr(\)\()p Fo(N)1136 5212 y Fm(f)1181 5197 y Fr(+1\))1331 5162 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1496 5197 y Fn(k)1541 5162 y Fl(2)1605 5197 y Fn(\024)85 b Fr(sup)1700 5280 y Fv(j)p Fm(x)1760 5290 y Fi(j)1792 5280 y Fv(j\024)p Fm(R)1920 5289 y Fe(0)1997 5111 y Fj(X)1969 5309 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)2171 5074 y Fj(Z)2221 5280 y Fv(j)p Fm(y)r Fv(j\024)p Fm(R)2406 5289 y Fe(0)2459 5197 y Fn(j)2505 5174 y Fr(^)2484 5197 y Fo(G)2555 5212 y Fm(\025)2601 5197 y Fr(\()p Fo(x)2687 5212 y Fm(j)2723 5197 y Fn(\000)p Fo(R)q(d)p Fn(\000)p Fo(y)s Fr(\))p Fn(j)3086 5162 y Fl(2)3126 5197 y Fo(dy)28 b Fn(!)d Fr(0)180 b(\()p Fo(R)26 b Fn(!)f(1)p Fr(\))p Fo(:)3680 5407 y Fr(\(17\))p eop %%Page: 15 15 15 14 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2267 b Fr(15)261 99 y(The)45 b(case)f(where)h Fo(j)55 b(>)50 b(N)40 b Fn(\000)29 b Fo(N)1446 67 y Fv(0)1514 99 y Fr(is)45 b(dealt)g(with)g(similarly)-7 b(.)83 b(The)45 b(only)f(di\013erence)h(is)f(that)h(then)120 245 y Fn(j)p Fo(x)196 260 y Fm(j)247 245 y Fn(\000)15 b Fo(R)q(d)p Fn(j)25 b(\024)g Fo(R)661 260 y Fl(0)728 245 y Fr(in)i(the)h(supp)r (ort)f(of)g Fo(T)1466 260 y Fm(R)1524 245 y Fo(')1583 260 y Fv(1)1685 245 y Fr(and)g(the)g(photons)g(in)h Fo(')2509 260 y Fl(0)2576 245 y Fr(ha)n(v)n(e)d(supp)r(ort)j(in)f Fn(j)p Fo(y)s Fn(j)f(\024)e Fo(R)3498 260 y Fl(0)3538 245 y Fr(.)39 b(Hence)120 392 y(\(17\))30 b(will)g(b)r(e)h(replaced)f (b)n(y)120 624 y Fn(k)p Fo(\037)221 639 y Fm(R)274 648 y Fe(0)313 624 y Fr(\()p Fo(x)399 639 y Fm(j)436 624 y Fn(\000)p Fo(R)q(d)p Fr(\))p Fo(a)p Fr(\()p Fo(\037)795 639 y Fm(R)848 648 y Fe(0)887 624 y Fo(G)958 639 y Fm(x)998 649 y Fi(j)1035 624 y Fr(\)\()p Fo(N)1177 639 y Fm(f)1222 624 y Fr(+1\))1372 588 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1537 624 y Fn(k)1582 588 y Fl(2)1646 624 y Fn(\024)157 b Fr(sup)1741 707 y Fv(j)p Fm(x)1801 717 y Fi(j)1833 707 y Fv(\000)p Fm(Rd)p Fv(j\024)p Fm(R)2105 716 y Fe(0)2183 538 y Fj(X)2155 735 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)2356 500 y Fj(Z)2407 706 y Fv(j)p Fm(y)r Fv(j\024)p Fm(R)2592 715 y Fe(0)2645 624 y Fn(j)2691 601 y Fr(^)2670 624 y Fo(G)2741 639 y Fm(\025)2786 624 y Fr(\()p Fo(x)2872 639 y Fm(j)2909 624 y Fn(\000)p Fo(y)s Fr(\))p Fn(j)3086 588 y Fl(2)3126 624 y Fo(dy)28 b Fn(!)d Fr(0)180 b(\()p Fo(R)26 b Fn(!)f(1)p Fr(\))p Fo(:)120 919 y Fr(The)30 b(terms)g(in)n(v)n(olving)e Fo(B)5 b Fr(\()p Fo(x)1100 934 y Fm(i)1128 919 y Fr(\))30 b(are)g(dealt)g(with)g(similarly)-7 b(.)p 3775 919 4 61 v 3779 861 55 4 v 3779 919 V 3832 919 4 61 v 120 1249 a Fs(A)132 b(Imp)t(ortan)l(t)45 b(Estimates)120 1486 y Ff(Lemma)33 b(7.)41 b Fg(F)-7 b(or)32 b(al)5 b(l)32 b Fr(\003)24 b Fn(\025)h Fr(0)p Fg(,)32 b Fo(")25 b(>)g Fr(0)32 b Fg(and)f(al)5 b(l)32 b Fo(x)25 b Fn(2)g Fk(R)1955 1454 y Fl(3)2000 1486 y Fg(,)1476 1718 y Fo(A)p Fr(\()p Fo(x)p Fr(\))1664 1682 y Fl(2)1787 1718 y Fn(\024)83 b Fr(32)p Fo(\031)s Fr(\003\()p Fo(H)2257 1733 y Fm(f)2322 1718 y Fr(+)20 b(\003)p Fo(=)p Fr(4\))p Fo(;)1321 1901 y Fn(\006)p Fo(\033)j Fn(\001)d Fo(B)5 b Fr(\()p Fo(x)p Fr(\))83 b Fn(\024)g Fo("H)2057 1916 y Fm(f)2122 1901 y Fr(+)2222 1837 y(8)p Fo(\031)p 2222 1881 100 4 v 2251 1965 a(")2331 1901 y Fr(\003)2394 1866 y Fl(3)2433 1901 y Fo(:)261 2133 y Fr(F)-7 b(or)29 b(the)i(pro)r(of)f(see)f([7].)40 b(This)30 b(lemma)g(holds)g(equally)f(for)g Fo(A)2364 2148 y Fm(\026)2411 2133 y Fr(\()p Fo(x)p Fr(\))i(and)f Fo(B)2806 2148 y Fm(\026)2853 2133 y Fr(\()p Fo(x)p Fr(\))g(with)g Fo(\026)25 b(>)g Fr(0.)120 2350 y Ff(Lemma)33 b(8.)41 b Fg(L)-5 b(et)33 b Fo(C)f Fr(=)25 b(1)20 b(+)g(32)p Fo(\031)s(\013N)10 b Fr(\003)33 b Fg(and)e Fo(D)d Fr(=)d(8)p Fo(\031)s(\013N)10 b Fr(\003)p Fg(.)41 b(Then,)32 b(for)f(al)5 b(l)32 b Fo(\026)25 b Fn(\025)g Fr(0)p Fg(,)1091 2515 y Fm(N)1057 2542 y Fj(X)1066 2738 y Fm(i)p Fl(=1)1204 2629 y Fo(p)1249 2593 y Fl(2)1249 2651 y Fm(i)1313 2629 y Fn(\024)g Fo(C)1494 2473 y Fj(\()1601 2515 y Fm(N)1567 2542 y Fj(X)1576 2738 y Fm(i)p Fl(=1)1698 2629 y Fr(\()p Fo(p)1778 2644 y Fm(i)1827 2629 y Fr(+)1917 2559 y Fn(p)p 1992 2559 58 4 v 70 x Fo(\013)q(A)2117 2644 y Fm(\026)2164 2629 y Fr(\()p Fo(x)2250 2644 y Fm(i)2278 2629 y Fr(\)\))2348 2593 y Fl(2)2408 2629 y Fr(+)20 b Fo(H)2573 2644 y Fm(f)2618 2473 y Fj(\))2711 2629 y Fr(+)g Fo(D)r(:)120 2914 y Fg(F)-7 b(urthermor)i(e,)41 b(if)d Fo(V)822 2929 y Fv(\000)920 2914 y Fn(\024)g Fo("p)1115 2883 y Fl(2)1180 2914 y Fr(+)25 b Fo(C)1339 2929 y Fm(")1415 2914 y Fg(for)39 b(al)5 b(l)39 b Fo(")f(>)g Fr(0)p Fg(,)j(then)e(ther)-5 b(e)40 b(exist)f(c)-5 b(onstants)38 b Fo(D)r Fr(\()p Fo(")p Fr(\))p Fg(,)j(dep)-5 b(ending)40 b(on)120 3061 y Fo(\013;)15 b(g)s(;)g(N)5 b(;)15 b Fr(\003)32 b Fg(and)g Fo(")p Fg(,)g(but)g(not)g (on)f Fo(\026)p Fg(,)h(such)h(that)815 3185 y Fj(\()923 3227 y Fm(N)889 3254 y Fj(X)897 3449 y Fm(i)p Fl(=1)1020 3340 y Fr(\()p Fo(p)1100 3355 y Fm(i)1148 3340 y Fr(+)1238 3270 y Fn(p)p 1313 3270 V 70 x Fo(\013)q(A)1438 3355 y Fm(\026)1485 3340 y Fr(\()p Fo(x)1571 3355 y Fm(i)1599 3340 y Fr(\)\))1669 3305 y Fl(2)1729 3340 y Fr(+)20 b Fo(V)1871 3355 y Fl(+)1950 3340 y Fr(+)g Fo(H)2115 3355 y Fm(f)2160 3185 y Fj(\))2259 3340 y Fn(\024)25 b Fr(\(1)20 b(+)g Fo(")p Fr(\))p Fo(H)2696 3355 y Fm(N)s(;\026)2841 3340 y Fr(+)g Fo(D)r Fr(\()p Fo(")p Fr(\))p Fo(:)120 3626 y Fg(Pr)-5 b(o)g(of.)45 b Fr(The)37 b(\014rst)f(part)h(follo)n(ws) f(from)h Fo(p)1556 3595 y Fl(2)1556 3653 y Fm(i)1632 3626 y Fn(\024)f Fr(2\()p Fo(p)1863 3641 y Fm(i)1916 3626 y Fr(+)2011 3561 y Fn(p)p 2086 3561 V 65 x Fo(\013)q(A)2211 3641 y Fm(\026)2258 3626 y Fr(\()p Fo(x)2344 3641 y Fm(i)2372 3626 y Fr(\)\))2442 3595 y Fl(2)2506 3626 y Fr(+)25 b(2)p Fo(\013A)2770 3641 y Fm(\026)2818 3626 y Fr(\()p Fo(x)2904 3641 y Fm(i)2932 3626 y Fr(\))2967 3595 y Fl(2)3044 3626 y Fr(and)37 b(Lemma)g(7.)61 b(The)120 3773 y(second)30 b(b)r(ound)g(follo)n(ws)f(from)g(the)i(\014rst)e(and)h(Lemma)h(7.)p 3775 3773 4 61 v 3779 3715 55 4 v 3779 3773 V 3832 3773 4 61 v 120 3993 a Ff(Theorem)42 b(9.)j Fg(Supp)-5 b(ose)38 b(the)g(ne)-5 b(gative)39 b(p)-5 b(arts)38 b Fo(v)1822 4008 y Fv(\000)1919 3993 y Fg(and)g Fo(w)2164 4008 y Fv(\000)2261 3993 y Fg(of)g(the)g(external)h(p)-5 b(otential)37 b Fo(v)k Fg(and)d(the)g(two-)120 4140 y(p)-5 b(article)37 b(inter)-5 b(action)35 b Fo(w)k Fg(as)d(functions)e(in)i Fk(R)1696 4109 y Fl(3)1777 4140 y Fg(dr)-5 b(op)37 b(o\013)f(to)g(zer) -5 b(o)37 b(as)f Fn(j)p Fo(x)p Fn(j)d(!)f(1)p Fg(.)54 b(Then)36 b(for)g(al)5 b(l)36 b(values)g(of)120 4287 y(the)30 b(p)-5 b(ar)g(ameters)30 b Fo(N)5 b(;)15 b Fr(\003)p Fo(;)g(\013;)g(g)34 b Fg(and)29 b Fo(\026)c Fn(\025)g Fr(0)p Fg(,)30 b(ther)-5 b(e)31 b(exists)f(a)g(functions)e Fo(f)10 b Fr(\()p Fo(R)q Fr(\))31 b Fg(and)e(a)h(c)-5 b(onstant)29 b Fo(C)7 b Fg(,)30 b(dep)-5 b(ending)120 4434 y(on)32 b(these)g(p)-5 b(ar)g(ameters,)32 b(such)g(that)911 4666 y Fo(H)986 4681 y Fm(N)s(;\026)1136 4666 y Fn(\025)24 b Fo(\034)10 b Fr(\()p Fo(H)1389 4681 y Fm(N)s(;\026)1515 4666 y Fr(\))20 b Fn(\000)g Fo(f)10 b Fr(\()p Fo(R)q Fr(\)\()p Fo(H)1963 4681 y Fm(N)s(;\026)2108 4666 y Fr(+)20 b Fo(N)2270 4681 y Fm(f)2335 4666 y Fr(+)g Fo(C)7 b Fr(\))184 b Fg(on)32 b Fn(D)2913 4681 y Fm(N)s(;R)120 4897 y Fg(wher)-5 b(e)33 b Fr(lim)499 4912 y Fm(R)p Fv(!1)713 4897 y Fo(f)10 b Fr(\()p Fo(R)q Fr(\))25 b(=)g(0)p Fg(.)41 b(Her)-5 b(e)33 b Fo(\034)10 b Fr(\()p Fo(H)1514 4912 y Fm(N)s(;\026)1639 4897 y Fr(\))25 b(=)g(inf)1903 4913 y Fm(N)1966 4894 y Fq(0)1989 4913 y Fv(\025)p Fl(1)2083 4897 y Fr([inf)c Fo(\033)s Fr(\()p Fo(H)2396 4913 y Fm(N)7 b Fv(\000)p Fm(N)2577 4894 y Fq(0)2600 4913 y Fm(;\026)2666 4897 y Fr(\))20 b(+)g(inf)h Fo(\033)s Fr(\()p Fo(H)3106 4866 y Fl(0)3099 4926 y Fm(N)s(;\026)3225 4897 y Fr(\)])p Fg(.)261 5114 y Fr(This)28 b(theorem)h(is)g(a)f(v)-5 b(arian)n(t)28 b(of)h(Corollary)f(A.2)g(in)h([7])g(\(see)f(also)h ([6]\).)39 b(The)29 b(p)r(ositivit)n(y)f(of)g(the)h(photon)120 5260 y(mass)e(in)h([7])h(is)e(only)h(needed)g(to)g(estimate)g Fo(N)1687 5275 y Fm(f)1761 5260 y Fr(in)g(terms)f(of)h Fo(H)2288 5275 y Fm(N)2355 5260 y Fr(.)39 b(Th)n(us)28 b(this)f(theorem)h(holds)g(for)g(p)r(ositiv)n(e)120 5407 y(as)i(w)n(ell)f(as)h(for)f(v)-5 b(anishing)30 b(photon)g(mass.)39 b(The)30 b(pro)r(of)g(is)f(essen)n(tially)g(the)h(same)f(as)h(the)g (one)g(in)g([7].)p eop %%Page: 16 16 16 15 bop 120 -200 a Fr(16)120 99 y Fs(B)132 b(F)-11 b(o)t(c)l(k)43 b(Space)h(and)h(Second)g(Quan)l(tization)120 339 y Fr(Let)27 b Fd(h)g Fr(b)r(e)g(a)g(complex)f(Hilb)r(ert)g(space,)h (and)g(let)g Fn(\012)1830 308 y Fm(n)1830 360 y(s)1876 339 y Fd(h)g Fr(denote)g(the)g(symmetric)e(tensor)i(pro)r(duct)g(of)f Fo(n)g Fr(copies)120 486 y(of)k Fd(h)p Fr(.)40 b(Then)30 b(the)g(b)r(osonic)g(F)-7 b(o)r(c)n(k)29 b(space)h(o)n(v)n(er)e Fd(h)p Fr(:)1522 730 y Fn(F)34 b Fr(=)25 b Fn(F)9 b Fr(\()p Fd(h)p Fr(\))25 b(=)g Fn(\010)2097 745 y Fm(n)p Fv(\025)p Fl(0)2254 730 y Fn(\012)2324 695 y Fm(n)2324 751 y(s)2391 730 y Fd(h)120 975 y Fr(is)36 b(the)h(space)g(of)f(sequences)g Fo(')g Fr(=)h(\()p Fo(')1448 990 y Fm(n)1495 975 y Fr(\))1530 990 y Fm(n)p Fv(\025)p Fl(0)1667 975 y Fr(,)h(with)f Fo(')2001 990 y Fl(0)2077 975 y Fn(2)g Fk(C)17 b Fr(,)45 b Fo(')2362 990 y Fm(n)2445 975 y Fn(2)37 b(\012)2612 943 y Fm(n)2612 995 y(s)2658 975 y Fd(h)p Fr(,)j(and)c(with)h(an)g (inner)g(pro)r(duct)120 1122 y(de\014ned)30 b(b)n(y)1553 1275 y Fn(h)p Fo(';)15 b( )s Fn(i)24 b Fr(:=)1929 1188 y Fj(X)1928 1384 y Fm(n)p Fv(\025)p Fl(0)2061 1275 y Fr(\()p Fo(')2155 1290 y Fm(n)2202 1275 y Fo(;)15 b( )2301 1290 y Fm(n)2347 1275 y Fr(\))p Fo(;)120 1542 y Fr(where)33 b(\()p Fo(')477 1557 y Fm(n)524 1542 y Fo(;)15 b( )623 1557 y Fm(n)669 1542 y Fr(\))33 b(denotes)f(the)h(inner)g(pro)r(duct)g (of)f Fn(\012)1971 1511 y Fm(n)1971 1563 y(s)2018 1542 y Fd(h)p Fr(.)49 b(The)33 b(v)n(ector)e(\012)f(=)f(\(1)p Fo(;)15 b Fr(0)p Fo(;)g(:)g(:)g(:)p Fr(\))30 b Fn(2)f(F)42 b Fr(is)32 b(called)h(the)120 1689 y(v)-5 b(acuum.)43 b(By)30 b Fn(F)698 1704 y Fl(\014n)807 1689 y Fn(\032)c(F)40 b Fr(w)n(e)31 b(denote)g(the)g(dense)f(subspace)h(of)f(v)n(ectors)g Fo(')h Fr(for)f(whic)n(h)h Fo(')3169 1704 y Fm(n)3242 1689 y Fr(=)c(0,)k(for)g(all)g(but)120 1836 y(\014nitely)f(man)n(y)f Fo(n)p Fr(.)40 b(The)30 b(n)n(um)n(b)r(er)f(op)r(erator)i Fo(N)1734 1851 y Fm(f)1810 1836 y Fr(in)f Fn(F)38 b Fr(is)30 b(de\014ned)g(b)n(y)f(\()p Fo(N)2651 1851 y Fm(f)2697 1836 y Fo(')p Fr(\))2791 1851 y Fm(n)2863 1836 y Fr(=)24 b Fo(n')3070 1851 y Fm(n)3117 1836 y Fr(.)120 2126 y Fc(B.1)112 b(Creation-)37 b(and)h(Annihilation)d(Op)s(erators)120 2336 y Fr(The)30 b(creation)g(op)r(erator)h Fo(a)1060 2304 y Fv(\003)1100 2336 y Fr(\()p Fo(h)p Fr(\),)f Fo(h)24 b Fn(2)h Fd(h)p Fr(,)31 b(on)f Fn(F)1731 2351 y Fl(\014n)1838 2336 y Fn(\032)25 b(F)39 b Fr(is)29 b(de\014ned)i(b)n(y)1391 2580 y(\()p Fo(a)1473 2544 y Fv(\003)1513 2580 y Fr(\()p Fo(h)p Fr(\))p Fo(')p Fr(\))1729 2595 y Fm(n)1800 2580 y Fr(=)1895 2510 y Fn(p)p 1970 2510 54 4 v 70 x Fo(n)15 b(S)2094 2595 y Fm(n)2141 2580 y Fr(\()p Fo(h)20 b Fn(\012)g Fo(')2397 2595 y Fm(n)p Fv(\000)p Fl(1)2534 2580 y Fr(\))120 2824 y(where)38 b Fo(S)443 2839 y Fm(n)529 2824 y Fn(2)g Ff(B)p Fr(\()p Fn(\012)806 2793 y Fm(n)852 2824 y Fd(h)p Fr(\))h(denotes)e(the)i(orthogonal)f(pro)5 b(jection)37 b(on)n(to)h(the)g(symmetric)f(subspace)g Fn(\012)3637 2793 y Fm(n)3637 2845 y(s)3684 2824 y Fd(h)i Fn(\032)120 2971 y(\012)190 2940 y Fm(n)237 2971 y Fd(h)p Fr(.)54 b(The)35 b(annihilation)g(op)r(erator)g Fo(a)p Fr(\()p Fo(h)p Fr(\))g(is)f(the)h(adjoin)n(t)f(of)g Fo(a)2348 2940 y Fv(\003)2387 2971 y Fr(\()p Fo(h)p Fr(\))h(restricted)f(to)h Fn(F)3129 2986 y Fl(\014n)3211 2971 y Fr(.)54 b(Creation-)35 b(and)120 3118 y(annihilation)30 b(op)r(erators)g(satisfy)f(the)h (canonical)g(comm)n(utation)g(relations)f(\(CCR\))1092 3362 y([)p Fo(a)p Fr(\()p Fo(g)s Fr(\))p Fo(;)15 b(a)1367 3327 y Fv(\003)1407 3362 y Fr(\()p Fo(h)p Fr(\)])25 b(=)g(\()p Fo(g)s(;)15 b(h)p Fr(\))p Fo(;)285 b Fr([)p Fo(a)2264 3327 y Fm(])2296 3362 y Fr(\()p Fo(g)s Fr(\))p Fo(;)15 b(a)2499 3327 y Fm(])2531 3362 y Fr(\()p Fo(h)p Fr(\)])25 b(=)g(0)p Fo(:)120 3607 y Fr(In)30 b(particular)g([)p Fo(a)p Fr(\()p Fo(h)p Fr(\))p Fo(;)15 b(a)929 3575 y Fv(\003)969 3607 y Fr(\()p Fo(h)p Fr(\)])25 b(=)g Fn(k)p Fo(h)p Fn(k)1378 3575 y Fl(2)1417 3607 y Fr(.)40 b(F)-7 b(rom)29 b(these)h(de\014nitions)g(it)g(is)f(easy)g(to)h(see)g(that) 1455 3851 y Fn(k)p Fo(a)1547 3815 y Fm(])1579 3851 y Fr(\()p Fo(h)p Fr(\)\()p Fo(N)g Fr(+)20 b(1\))2008 3815 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)2173 3851 y Fn(k)25 b(\024)g(k)p Fo(h)p Fn(k)p Fo(:)1175 b Fr(\(18\))261 4097 y(In)44 b(the)h(case)f(where)h Fd(h)k Fr(=)h Fo(L)1315 4066 y Fl(2)1354 4097 y Fr(\()p Fk(R)1449 4066 y Fl(3)1494 4097 y Fr(;)15 b Fk(C)1594 4066 y Fl(2)1639 4097 y Fr(\),)49 b(the)44 b(one-photon)h(Hilb)r(ert)g(space,)j(the)c(annihilation)h(and) 120 4244 y(creation)39 b(op)r(erators)g(can)f(b)r(e)i(expressed)d(in)i (terms)f(of)g(the)h(op)r(erator)g(v)-5 b(alued)39 b(distributions)f Fo(a)3491 4259 y Fm(\025)3536 4244 y Fr(\()p Fo(k)s Fr(\))h(and)120 4391 y Fo(a)167 4359 y Fv(\003)167 4420 y Fm(\025)213 4391 y Fr(\()p Fo(k)s Fr(\))30 b(b)n(y)1341 4650 y Fo(a)p Fr(\()p Fo(h)p Fr(\))83 b(=)1774 4563 y Fj(X)1746 4761 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1947 4526 y Fj(Z)p 2053 4571 217 4 v 2053 4650 a Fo(h)2105 4665 y Fm(\025)2150 4650 y Fr(\()p Fo(k)s Fr(\))p Fo(a)2317 4665 y Fm(\025)2363 4650 y Fr(\()p Fo(k)s Fr(\))15 b Fo(d)2545 4614 y Fl(3)2584 4650 y Fo(k)1301 4940 y(a)1348 4904 y Fv(\003)1388 4940 y Fr(\()p Fo(h)p Fr(\))83 b(=)1774 4854 y Fj(X)1746 5051 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1947 4816 y Fj(Z)2053 4940 y Fo(h)2105 4955 y Fm(\025)2150 4940 y Fr(\()p Fo(k)s Fr(\))p Fo(a)2317 4904 y Fv(\003)2317 4965 y Fm(\025)2363 4940 y Fr(\()p Fo(k)s Fr(\))15 b Fo(d)2545 4904 y Fl(3)2584 4940 y Fo(k)s(:)120 5260 y Fr(Setting)25 b Fo(G)496 5275 y Fm(x;\025)601 5260 y Fr(\()p Fo(k)s Fr(\))f(=)h Fn(j)p Fo(k)s Fn(j)940 5229 y Fv(\000)p Fl(1)p Fm(=)p Fl(2)1105 5260 y Fo(")1147 5275 y Fm(\025)1192 5260 y Fr(\()p Fo(k)s Fr(\))p Fo(\037)1368 5279 y Fv(fj)p Fm(k)r Fv(j\024)p Fl(\003)p Fv(g)1625 5260 y Fo(e)1667 5229 y Fv(\000)p Fm(ik)r Fv(\001)p Fm(x)1873 5260 y Fr(the)g(quan)n(tized)f(v)n(ector)g (p)r(oten)n(tial)h Fo(A)p Fr(\()p Fo(x)p Fr(\))h(can)f(b)r(e)h (de\014ned)120 5407 y(as)k Fo(A)p Fr(\()p Fo(x)p Fr(\))25 b(=)g Fo(a)p Fr(\()p Fo(G)692 5422 y Fm(x)736 5407 y Fr(\))20 b(+)g Fo(a)928 5376 y Fv(\003)968 5407 y Fr(\()p Fo(G)1074 5422 y Fm(x)1118 5407 y Fr(\).)p eop %%Page: 17 17 17 16 bop 120 -200 a Fh(Griesemer,)22 b(17/June/02|Exp)r(onen)n(tial)27 b(Deca)n(y)2267 b Fr(17)120 99 y Fc(B.2)112 b(Second)38 b(Quan)m(tization)120 304 y Fr(Supp)r(ose)j Fo(b)g Fr(is)f(a)h(b)r (ounded)g(op)r(erator)g(on)g Fd(h)g Fr(and)f Fn(k)p Fo(b)p Fn(k)j(\024)g Fr(1.)72 b(One)40 b(then)h(de\014nes)f(an)h(op)r(erator)g (\000\()p Fo(b)p Fr(\))84 b(:)120 451 y Fn(F)9 b Fr(\()p Fd(h)p Fr(\))25 b Fn(!)g(F)9 b Fr(\()p Fd(h)p Fr(\))30 b(b)n(y)1623 653 y(\000\()p Fo(b)p Fr(\)\012)84 b(=)f(\012)1444 825 y(\000\()p Fo(b)p Fr(\))p Fn(j)-23 b Fr(\022)14 b Fn(\012)1739 789 y Fm(n)1739 845 y(s)1806 825 y Fd(h)83 b Fr(=)g Fo(b)21 b Fn(\012)f Fo(:)15 b(:)g(:)k Fn(\012)h Fo(b:)120 1027 y Fr(Clearly)30 b Fn(k)p Fr(\000\()p Fo(b)p Fr(\))p Fn(k)25 b(\024)g Fr(1.)40 b(F)-7 b(rom)30 b(the)g(de\014nition) g(of)f Fo(a)1858 995 y Fv(\003)1898 1027 y Fr(\()p Fo(h)p Fr(\))h(it)g(easily)f(follo)n(ws)g(that)1489 1229 y(\000\()p Fo(b)p Fr(\))p Fo(a)1700 1194 y Fv(\003)1740 1229 y Fr(\()p Fo(h)p Fr(\))83 b(=)g Fo(a)2145 1194 y Fv(\003)2185 1229 y Fr(\()p Fo(bh)p Fr(\)\000\()p Fo(b)p Fr(\))1171 b(\(19\))1450 1401 y(\000\()p Fo(b)p Fr(\))p Fo(a)p Fr(\()p Fo(b)1734 1365 y Fv(\003)1775 1401 y Fo(h)p Fr(\))83 b(=)g Fo(a)p Fr(\()p Fo(h)p Fr(\)\000\()p Fo(b)p Fr(\))p Fo(;)1224 b Fr(\(20\))120 1603 y(and)30 b(hence)g(that)g(\000\()p Fo(b)p Fr(\))p Fo(a)p Fr(\()p Fo(h)p Fr(\))c(=)f Fo(a)p Fr(\()p Fo(bh)p Fr(\)\000\()p Fo(b)p Fr(\))32 b(if)d Fo(b)1717 1572 y Fv(\003)1757 1603 y Fo(b)c Fr(=)g(1.)261 1750 y(If)k Fo(b)d Fr(:)f Fo(D)r Fr(\()p Fo(b)p Fr(\))g Fn(\032)g(H)h(!)f(H)31 b Fr(is)e(self-adjoin)n(t,)g(then)h(d\000\()p Fo(b)p Fr(\))g(in)g Fn(F)9 b Fr(\()p Fd(h)p Fr(\))31 b(is)e(de\014ned)h(b)n(y)1344 1952 y(d\000\()p Fo(b)p Fr(\)\012)83 b(=)g(0)1026 2159 y(d\000\()p Fo(b)p Fr(\))p Fn(j)-23 b Fr(\022)14 b Fn(\012)1371 2124 y Fm(n)1371 2180 y(s)1438 2159 y Fo(D)r Fr(\()p Fo(b)p Fr(\))83 b(=)1904 2046 y Fm(n)1859 2073 y Fj(X)1864 2268 y Fm(j)t Fl(=1)1991 2159 y Fr(\(1)20 b Fn(\012)g Fo(:)15 b(:)g(:)g Fr(1)2026 2202 y Fj(|)p 2067 2202 79 11 v 79 w({z)p 2228 2202 V 79 w(})2124 2286 y Fm(j)t Fv(\000)p Fl(1)2361 2159 y Fn(\012)p Fo(b)20 b Fn(\012)g Fr(1)g Fn(\012)g Fo(:)15 b(:)g(:)g Fr(1)2579 2202 y Fj(|)p 2620 2202 V 79 w({z)p 2781 2202 V 79 w(})2674 2286 y Fm(n)p Fv(\000)p Fm(j)2899 2159 y Fr(\))120 2440 y(and)39 b(b)n(y)g(linear)g(extension.)67 b(d\000\()p Fo(b)p Fr(\))40 b(is)f(essen)n(tially)e(self-adjoin)n(t)h (and,)j(denoting)f(the)f(closure)g(b)n(y)f(d\000\()p Fo(b)p Fr(\))120 2587 y(as)c(w)n(ell,)h(\000\()p Fo(e)581 2556 y Fm(ib)640 2587 y Fr(\))d(=)h Fo(e)852 2556 y Fm(i)p Fl(d\000\()p Fm(b)p Fl(\))1048 2587 y Fr(.)54 b(One)35 b(example)f(is)g(the)g(n)n(um)n(b)r(er)h(op)r(erator)g Fo(N)2705 2602 y Fm(f)2783 2587 y Fr(=)d(d\000\(1\),)k(an)f(other)g (one,)g(for)120 2734 y Fd(h)25 b Fr(=)g Fo(L)348 2702 y Fl(2)388 2734 y Fr(\()p Fk(R)483 2702 y Fl(3)528 2734 y Fr(;)15 b Fk(C)628 2702 y Fl(2)673 2734 y Fr(\))30 b(is)g(the)g(\014eld)g(energy)1158 2936 y Fo(H)1233 2951 y Fm(f)1303 2936 y Fr(=)25 b(d\000\()p Fn(j)p Fo(k)s Fn(j)p Fr(\))g(=)1822 2850 y Fj(X)1794 3047 y Fm(\025)p Fl(=1)p Fm(;)p Fl(2)1995 2812 y Fj(Z)2101 2936 y Fn(j)p Fo(k)s Fn(j)p Fo(a)2248 2900 y Fv(\003)2248 2961 y Fm(\025)2294 2936 y Fr(\()p Fo(k)s Fr(\))p Fo(a)2461 2951 y Fm(\025)2506 2936 y Fr(\()p Fo(k)s Fr(\))15 b Fo(d)2688 2900 y Fl(3)2727 2936 y Fo(k)s(:)120 3268 y Fc(B.3)112 b(The)38 b(Iden)m(ti\014cation)e (Op)s(erator)h Fb(I)e Fw(:)27 b Fa(F)32 b(\012)22 b(F)37 b(!)27 b(F)120 3474 y Fr(An)22 b(imp)r(ortan)n(t)i(role)f(in)g(the)g (pro)r(of)g(of)f(Theorem)h(6)g(is)g(pla)n(y)n(ed)f(b)n(y)g(the)h(iden)n (ti\014cation)f(op)r(erator)i Fo(I)32 b Fr(:)25 b Fn(F)15 b(\012)6 b(F)33 b(!)120 3621 y(F)39 b Fr(de\014ned)30 b(b)n(y)1355 3823 y Fo(I)7 b Fr(\()p Fo(')19 b Fn(\012)h Fr(\012\))25 b(=)g Fo(')786 3995 y(I)7 b(')20 b Fn(\012)g Fo(a)1049 3959 y Fv(\003)1088 3995 y Fr(\()p Fo(h)1175 4010 y Fl(1)1215 3995 y Fr(\))15 b Fn(\001)g(\001)g(\001)g Fo(a)1432 3959 y Fv(\003)1471 3995 y Fr(\()p Fo(h)1558 4010 y Fm(n)1605 3995 y Fr(\)\012)25 b(=)g Fo(a)1872 3959 y Fv(\003)1912 3995 y Fr(\()p Fo(h)1999 4010 y Fl(1)2038 3995 y Fr(\))15 b Fn(\001)g(\001)g(\001)g Fo(a)2255 3959 y Fv(\003)2295 3995 y Fr(\()p Fo(h)2382 4010 y Fm(n)2429 3995 y Fr(\))p Fo(';)285 b(')25 b Fn(2)g(F)3067 4010 y Fl(\014n)3149 3995 y Fo(;)120 4197 y Fr(and)j(extended)g(b)n(y)g (linearit)n(y)f(to)i Fn(F)1325 4212 y Fl(\014n)1423 4197 y Fn(\012)17 b(F)1575 4212 y Fl(\014n)1657 4197 y Fr(.)40 b(The)28 b(op)r(erator)h Fo(I)35 b Fr(is)28 b(un)n(b)r(ounded.)40 b(W)-7 b(e)27 b(need)h(the)h(imp)r(ortan)n(t)120 4344 y(comm)n(utation)h(relation)1362 4490 y Fo(a)p Fr(\()p Fo(h)p Fr(\))p Fo(I)i Fr(=)25 b Fo(I)7 b Fr(\()p Fo(a)p Fr(\()p Fo(h)p Fr(\))20 b Fn(\012)g Fr(1)g(+)g(1)g Fn(\012)g Fo(a)p Fr(\()p Fo(h)p Fr(\)\))p Fo(;)1082 b Fr(\(21\))120 4673 y(whic)n(h)30 b(is)f(in)h(con)n(trast)f(with)i Fo(a)1174 4642 y Fv(\003)1213 4673 y Fr(\()p Fo(h)p Fr(\))p Fo(I)h Fr(=)25 b Fo(I)7 b Fr(\()p Fo(a)1631 4642 y Fv(\003)1670 4673 y Fr(\()p Fo(h)p Fr(\))20 b Fn(\012)g Fr(1\))25 b(=)g Fo(I)7 b Fr(\(1)20 b Fn(\012)g Fo(a)2386 4642 y Fv(\003)2426 4673 y Fr(\()p Fo(h)p Fr(\)\).)120 4967 y Ff(Ac)m(kno)m(wledgemen)m(ts.)39 b Fr(This)29 b(w)n(ork)f(w)n(as)g (completed)h(when)h(the)f(author)g(visited)f(ETH)h(Z)r(\177)-48 b(uric)n(h)30 b(and)f(the)120 5114 y(Univ)n(ersit)n(y)c(of)i(Mainz)g (in)h(the)g(Summer)f(of)g(2002.)40 b(It)27 b(is)g(a)h(pleasure)f(to)h (thank)f(the)h(resp)r(ectiv)n(e)e(hosts,)i(J)r(\177)-48 b(urg)120 5260 y(F)-7 b(r\177)-45 b(ohlic)n(h)24 b(and)h(V)-7 b(olk)n(er)23 b(Bac)n(h,)i(for)f(their)h(hospitalit)n(y)f(and)h(useful) f(discussions.)36 b(I)25 b(also)f(thank)g(Oliv)n(er)g(Matte)120 5407 y(for)30 b(his)f(careful)h(pro)r(ofreading.)p eop %%Page: 18 18 18 17 bop 120 -200 a Fr(18)120 99 y Fs(References)120 335 y Fr([1])45 b(Shm)n(uel)23 b(Agmon.)28 b Fg(L)-5 b(e)g(ctur)g(es)29 b(on)c(exp)-5 b(onential)26 b(de)-5 b(c)g(ay)27 b(of)e(solutions)g(of)g(se)-5 b(c)g(ond-or)g(der)27 b(el)5 b(liptic)26 b(e)-5 b(quations:)260 482 y(b)g(ounds)45 b(on)f(eigenfunctions)f(of)i Fo(N)10 b Fg(-b)-5 b(o)g(dy)45 b(Schr\177)-46 b(odinger)44 b(op)-5 b(er)g(ators)p Fr(.)82 b(Princeton)44 b(Univ)n(ersit)n(y)d(Press,)260 629 y(Princeton,)31 b(NJ,)e(1982.)120 850 y([2])45 b(V.)32 b(Bac)n(h,)f(J.)i(F)-7 b(r\177)-45 b(ohlic)n(h,)31 b(and)h(I.M.)f(Sigal.)46 b(Quan)n(tum)31 b(electro)r(dynamics)h(of)g(con\014ned)g (nonrelativistic)260 997 y(particles.)40 b Fg(A)-5 b(dv.)32 b(Math.)p Fr(,)d(137\(2\):299{395,)h(1998.)120 1219 y([3])45 b(V.)31 b(Bac)n(h,)g(J.)h(F)-7 b(r\177)-45 b(ohlic)n(h,)31 b(and)g(I.M.)g(Sigal.)44 b(Sp)r(ectral)32 b(analysis)e(for)h(systems)f (of)h(atoms)g(and)g(molecules)260 1365 y(coupled)f(to)g(the)g(quan)n (tized)g(radiation)g(\014eld.)40 b Fg(Comm.)31 b(Math.)g(Phys.)p Fr(,)e(207\(2\):249{290,)h(1999.)120 1587 y([4])45 b(E.B.)30 b(Da)n(vies.)38 b(The)31 b(functional)e(calculus.)40 b Fg(J.)32 b(L)-5 b(ondon)32 b(Math.)f(So)-5 b(c.)31 b(\(2\))p Fr(,)f(52\(1\):166{176,)g(1995.)120 1808 y([5])45 b(J.)24 b(F)-7 b(r\177)-45 b(ohlic)n(h,)25 b(M.)e(Griesemer,)i(and)f (B.)g(Sc)n(hlein.)29 b(Asymptotic)23 b(completeness)g(for)h(Ra)n (yleigh)f(scattering.)260 1955 y Fg(A)n(nn.)30 b(Henri)i(Poinc)-5 b(ar)n(\023)-44 b(e)p Fr(,)30 b(3:107{170,)g(2002.)120 2177 y([6])45 b(Marcel)34 b(Griesemer,)h(Elliott)g(H.)f(Lieb,)i(and)f (Mic)n(hael)f(Loss.)52 b(Erratum)35 b(to:)49 b(Ground)35 b(states)e(in)i(non-)260 2324 y(relativistic)29 b(quan)n(tum)g(electro) r(dynamics.)40 b Fg(Invent.)31 b(Math.)p Fr(,)e(to)h(app)r(ear.)120 2545 y([7])45 b(Marcel)e(Griesemer,)i(Elliott)e(H.)f(Lieb,)47 b(and)c(Mic)n(hael)f(Loss.)77 b(Ground)43 b(states)f(in)h (non-relativistic)260 2692 y(quan)n(tum)29 b(electro)r(dynamics.)40 b Fg(Invent.)31 b(Math.)p Fr(,)e(145\(3\):557{595,)h(2001.)120 2913 y([8])45 b(W.)31 b(Hunzik)n(er)f(and)i(I.M.)f(Sigal.)45 b(The)31 b(quan)n(tum)g Fo(N)10 b Fr({b)r(o)r(dy)32 b(problem.)45 b Fg(J.)33 b(Math.)g(Phys.)p Fr(,)e(41\(6\):3448{)260 3060 y(3510,)f(2000.)120 3282 y([9])45 b(A.)35 b(J.)h(O'Connor.)57 b(Exp)r(onen)n(tial)36 b(deca)n(y)e(of)i(b)r(ound)g(state)f(w)n(a)n(v)n (e)f(functions.)56 b Fg(Comm.)36 b(Math.)g(Phys.)p Fr(,)260 3428 y(32:319{340,)30 b(1973.)120 4275 y(E-mail:)40 b (marcel@math.uab.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0206171109504--