%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: return.dvi %%Pages: 54 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMMI10 CMSY8 CMSY10 CMBX12 %%+ EUFM10 CMMI8 CMTI10 CMR8 CMEX10 MSBM10 CMR6 CMMI6 CMSY6 MSBM7 CMMI12 %%+ LASY10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips return.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.27:1347 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.00 %%CreationDate: 1992 Oct 23 20:19:17 % Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 8e26a39cb0ad4e25ab8f21b25ad1b106ed22af7947ea8ae4dc17b9facbc5e22e 89a91b046ba4cddde69c1997fa7f4cd949ba3ac4b6e001475daf165b9432dc70 6e30aeead8cc342d94b20a4045a331175655d6aa279627efc0fb53a4ce66b68f a45b5765f0c1368c0392eb2e0e75cae373c613a428b5e5b9f5da8553794535b5 9507cd86d2deb6fe164c4caaff59ca6352074642262e7ed0a13dafd1974444de 0f29964f712e569b04dabfa2ff96ee9df1ddaf97262d69c61dcb91ffc62a6c2e 98890fc38d0b84c6d2c9a7cc3b687062671b492ee0efcd45c26f0a03987c8de9 9c1ce65595976b24c7a23dbb3c93c2c156eec0ce9df9ff2f1ddc248597fdfa13 4a420026585c5081dc7ab11ebe0016719fc67d712f93a4e2d93e3d95acd94da8 a827993ec1e3746d5887ab1f43c65872d9b3c18904f3eca78a72b9c56cc86171 c361b1d6056ec0026057882b23f0d524d045424186d7f8b645440a35c72bccc4 c706c783462fa3d55f7bce85e9fa202b5e15b35a135446bae2d0dcaf9c18ae60 1a0f3bdb9c1ae28eb9b54ef87478b3658c27c3b8486d10119441d6baf4feb45c 1e5d48993de358458ca46f50f40fc880e7f1778b9a22181d6827e3fc6137f3b2 191dd601194119c50b7d112242e3f74aab7c3b4e8674e558507ae2fcfa876d99 69fd6c7ccee87e99e32376ef5379021fb5bdb677a098ad72f7e3c36840bec71e 88015ddfa83afd3d763343889dbe92320bfa0692cc5676a1f9df28766a0dc009 c6467e1debff45b271fc5b59b75fe25c5edd08566c5278d159a93e18713a92ed e90a5bca059a97b2851fbc9d6aec1806c2eab89b1c66b87c5b570799737b72eb efe24cf58bc6f08926f7929341c4322b13a1f647779223a94e622ffff58b1a4c 29f5dfecde4bcf55f969ec940ff5adc6af08800ff52300bb0e645934fb48aad3 7dd9e7383ca888c87b173e48ca1d44dbb48ede7556786f0c0a4e780234d6a449 09d26b814652c51dbc4fc8a3303aa7a6803243d722068cf6cf8b31a752d56efa 3be7c7f001d7f2fd1f3da0737797a16471018544b1b4e308035de8e2b1542928 1b3fc1af77ed5e177303b01f8b5a0d08ff46216795f9e2a86a136605e9db 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 26 /rho put dup 59 /comma put dup 63 /star put dup 76 /L put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 a0ff44ab6d5249e9f552a8707bb7661e242644814001c8430ebd5e5f0b944ceb 666ee64359d663e355b2f17093a964139d17287f6ca6a024767eba4fe4873855 babe2f07b91560f68300b06dfe27264c163195d446980c35bca0b48f7806626a e72636593a05ba403ce1c0f8b2cea3ecd586e90ac17d034ba4af708304f23131 3459fbbbfb97d4834d0395754ab3f22d6495d2144087d448616fa1ce27bc50d3 46543287e3860d99b433624119bb9920a2113604c0e260fd275ba55e0fd19c83 e19addc3baa1f32f6b7284038845ccee71a3311ddb17b84975f7a984bed7c6ec 2a06e5b335a763d081c6273f86a46632fd9141a27902074fc860df3a2eb59b89 774c767022dbb577e30da128bd7706a43af886d0c256b50fa968ef06776aac0b a5387e9011eb2334c1f42c090f06a1125c207ea6324e87f46414050d88070346 7b9a6e4d2045feac295d15ca55b900da594770ed9843365bbc81bff2190b731b 8790385f7c313b4f64683aededa87b8c04da713754a05b7ad08b814b94b7be0a 5f20abfa50f767a51aa084b330955da5231cf6660a76c929e56afc63797a999f 757018dd00ca061ace2eae761c8e56838535e4f0dde8afaf899c418727ddfd13 eaf22127080a73eaeb87ef529c7da9ecbeaccd055d7d8a95584db094bd443797 60d5b3c8c2c856893c71009047c00fca4e45e9fcfdef7613020da364a1d208a2 96cf44f593b8afe08c3d4c3b9e3c267aa439cdbedef4789710c8007d08c8be98 ba22f752265e1ae3a95b8a15f36276028a93e251909612bc32ba1a600631bce7 44f4d4a46670d6118480ee820405dab65be0b9ba74bc69ae615812242b171618 7be1e98a450eea2988936f304267aca0f50190f100567f61682eb29cc8288d71 686532b269ebe23d8baed4c4013d8ae35b116c1f3c65a0680e16941e4c0f6556 9530577529b8e3503bfe8f3128bd03db0975202a772a29a97fd96e387061be05 d283364eb940aaab88cf2f4f02562520ddd3c659d1ae65afd9cd115f402ef555 09cd169c368cc0197a99092418b74a59f3e7a84f34bb705d93a0eee955cd96f1 3711a7a29a0c05111709b7091ce8469044688be8ebfcee73909eb0e15e57f591 a6bb01a3cc1890fac4daf938f282a80ef0ee241f149393d860f7d613f0364a44 d4401eb4844ff90ff99e09dc5781dffc1ecbe91d9fb4e0646de36e2132c0826e 99730d734bf3c7d1544e485ea8af410d4bc1cab20b16750f12c72e313eaf0e9a 5eb7c47b4c46122eda0c8af2febfe6aec90f9b6ce7cb926b0d142ffd2e8d2e2d e232265e456aa912905ec10dbee51dfc651407773c77eee82bcc5f87cfe5a6fd 918aba2cd5e26ee5f7fe3f648c3d42c918f4938a02f9dafcd6e53f88d95a75d8 3a323c6cb129138f20f669e757b7ba82d602d40ee51f570e5cfeb5a08597bdd5 b25082bfb9d0231ee8382602d5c59cbd2c44aa10b270657ffe69c84d5b64f272 947e4abbe61f076eef06467792bced71dea9a9970e4403a83f0759de6e3fd221 d50a2fad46b6d05c1a6be29501f0bff771187d98a68f972cf33ced3378aa222c afa2cf1e28060ddbf9d213a00533160af974a114278825f9899434abaf1a624c c32d71aeb9c37a0948baf29db771069155eafdaaac79efa95c90202a1b9d4118 794c56a405ba024adcdbd7b4fea45ef769c5ca8e9d02f6e1dd01d4a9747251f7 b54e79c2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02e534d1eb98dd3f7c5d51900409354dea37de4ac74a12c3a 69320913de6a728bd887c57f4ee5dcb8772249587ce3190bb662ac853abcff77 d4bd77985a4a805f44f8cc8597416c488866ff8429f65cd0edeec1fe2f0015a3 2bb098b0b3d8c913007f5846ca593c491e1efabc8f420a5fa338c47fd2427813 3279d27f12022195e15827797fe71d898f17b8af955e5bd5ce1da5fae7b3ece8 87aa52db81daa819242c6acb7fbed098d59a27bbcb12dca67e16915a4709fa5e 16311a55ddda46359ab51be5964874f1d3612ec10d2444576c508ae6a285a57f b978458b6f5f93c00882f2227d43d42c648437715a27cefd8d4ffb490e3e9980 1d1b4f1cacbc8b9e59c04c677a04047cc4b655fd8985d8e619a99e71cad67c94 2744577e185ac8b225e5352ae93ef0d804b73b943353266c74677b899d45b286 869cbe8ce9ad231007381de74bd3c6c84958f07ffce0620e42902a01fd15c98a 421ff110e5de48cd65568060afaa1e11e9893e720f22b754c47c334a1f5d0933 9b1b1b91b036b981a6be8a3276634fd4e39af829e2f2ac2b2dde35379abd1bf5 3a1071f07537db5fff44d145fce9713d8a9e2660900c41b83f948e16731d3048 2c8fa2aef8de49e3cf649e8ff62a2748cd9e299c7b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e7a4a832f89e4f9cc 36f6027a64831992e98b1707ddc56e562de9ad0278d89f3416dfab55ea0e5b25 9cfdc4787442890a5a8986bd394cfd87942f66b3712745aa18e52c6bd1d44d48 3f75ca3a5d39391f2bb9441314fbb25ed2b6813e66bbdec80f12a88c2f94f601 fdb6b48613c26c3ac738dcd4ab05a16c34aac07eb789901b095230281fc71f04 5ac83d60c6b82255c6276398134634a8135902a31a52e84b55aece4be46e9d93 72e8c2738a3ae29ce61298d5c2e792bc098a143f3f08614489 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 20 /kappa put dup 26 /rho put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 107 /k put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 939a8995b7e0f7a6dc2a9e9ea4fd8b87fa6e5218823cee9549c1b05b26d2d3ea 739ad23f250f907abc8a7c9bc398672e2d92b77bf6d6e31dfff2723b0068293d 48a5400da0a0318bbe41aeea0baf043183a2aaf638a51e324cfa5453ed8dce97 1c45cf2518575051f4ac3cfab9ef38e6e80de1eff2328d2323eea02fedd49146 e78642af2fe8de4bfec1974af30330733e1c8783b1d03ef82361c1d037155082 7b9b9bce16d8ef3d9167f5330de9ee4a744bf0fa48f0512ce2dad48cf5bd7ec8 d488fb283819bc4a82b974bde97e353b85922553063cdaad8c4a6544272af931 9821b4fc48c60b7c06252ba2739212dc79168820e26376cd2a6e9137314bb8a5 e98334c9f342ea3dfa3f576f732f39016f2b0a660d2165a3e5df65fda8a8ce36 d4f4d07aec6f60ce30ff22ba782eb964974de8901099432f91b59bb690765676 00dfe30795cd46a7432ca76594a058daa046922d0321a24a0c59db9b5ffb7c24 47cb023f52bb50d081296ee23e9db165ea427656a1ebe52999ee8e74bdb03124 cc2e6dc566d807bed2c12176978d17ded3a0516040ff5ca259d181abf1686305 16eee5343140e44a8a4847455b416e1346a286bcfa45997b7b1ebff3a60601b7 96ca1760b4bb28e0f7d3309e4dd79d8d4f63ae1ed07e688ea4fcabdafd6c7b5c 0bafb980d90797c157d94391e7c5b2efc18fb15b93b811c3892ca899666ef39d 9192b059155fe2fb8568e2eecda1c7271aedf156b5762f3c7994f7ac1864f1e3 7f2c55220d6fdb6351f1c72142f031b17f722fd9a313239820f59f2102a93268 f2680ffc0634a81611de805fc83f5667a02eac999f0a54eac31c0b4b2611496b 67729b1e37d4b3531f9499e5590ddfad885f1586496226aa8400499a0f0d001a aedd04aa1525acfeca32cbd314016007b91d959c3f1bf7eb4b41a1dfdb0a8934 09671f82d3a0f27f920b3b8b49ed255e86cae593588d93465c46cad6e6ed6e15 6277a9207ee1d901250897895acd0692c160bdf1ef89abea2c7fdd411c23a07d 9cd2a2cb18f3d5a0fc4fefaa60dfa6608bdbaa04902285ce00fa3ed579bd3b50 ca0953b1f32449a9bd941ca83d507ec27397250ea45394bc1fbac344b8626bb3 ac96b1ad6fff47ae6d47f1a0a07865c3a37df642c7969c5026606a0fadf29a1b 54e9c5a698283fd50ad562615be7a55d116050e3402579b97ffd1f8ce03cbd12 05cae2767773115fa79203790964f4cb1ab4dc60eda1bfa4b61f0060956d29ae 7683ef9c496c68311a076089488591b8cd993d822df122dbe15dddb5b8098595 3219522325a17af662a5ec9282675504da5577bdbb865c93f99bc1707581b897 66f8035b05e3aeaa308e287e90481a0af9287b9d24df424f76b1cf02ec4a7eba 5d6bc549f0dd354bd3d70114d591237949c7b8ff3d4f1379cd45a77dfce1dbb6 3fba827673ed2bc4f184c227f7fe73225b686299c0ce387f3563de4d3b7f2a98 c625bca5e69638449e9cad5d92435870b38c768009760fd7e51626488e429ca7 95f7fb434812812e8d6a57666a06cff8afd0b06324229161f1731acc60354aea ecb7a12744acc5e5da5acb44934ec0882ace3d554effeb050d3e03f6125d602a 8058b43b5a13d09e9c3645fd9e8e9e51f6851d5f659cc68b1139375787b7f00e 3050a4de9bb61fa899662c1bf9f8cd9180d64b1f044fe855336ba57cc6023a08 dde72be61fb82a2322883f4f5daf59866b850504a38d6e4bfd415f82433bf61b 3a6daa82971ef287fe5279cf5f156c42e0df9e9d328d92b320dc1136ef22d86a 12efa0626666f13a04f3302da6106df04e15e6c3f98e838c62ceffe1c3e027f8 a7f83d38719ec24a9ac5aab779524f9a0ab92160e8e37dd59c9e49bb9bcc6bb7 586f3043944f3b55bd0566a8eb38acadb932b3561e7d68f1a2811e9c363043d3 bd2df98cbc8c794fe2a947f481cef54a5b83c0f7365c532a77a4e1458a8cbbd8 6f4b162920884bb6992df1814728411536960a931920de9ff7e6150c1b71583a d7bad13318422e9ccde698b4f69ce6b0612e7b3d76d7f024dd54742091917338 832f9733f19b748310533c75f2e5037cc7a1b48d1428796092208698d8d14886 457fccd00231e6a0db11545048a1d62ef8379c77c9a29ec4bd45b89ed8152911 242350cca0942c044a830fdc36c5f9043abafe787ebc02239438d3b5e5edf9e2 46c679557b59efa88c74331583fe364b61efb35dc619ca60b08009105ad63098 36134ebc05e4f0a3b32adf6af3bc6038e2c0bc9822ce38af61583687fd7a7386 2c6328914d7f241181cca5667c7327d015548b8189a49a3eec2ee643f11ad3c4 8ac86f1c56e402afa6b819ee2f4ec6d71efbd5c0f4f7eb0de1c6ef261f0caff9 5e56536cb5ca46a1d1816c798548cdaa5ff1dd21ef6d45f738c10b6f2f964e8c f71f88320ab4c66a2ac627226c5a39d2ed25978a366fb1ddbfddce0b6e912d83 7ccd9a02784a70b0380f0fb23331efa65144e4beac44651e59b041b897553218 429fd56ae6c8db9512e57a4f6595f95930 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 101 /e put dup 102 /f put dup 105 /i put dup 109 /m put dup 114 /r put dup 115 /s put dup 118 /v put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dc28efba4fed2f7e 6318a8ece08e180003dc883a1749f428f4e741b7465b020f836fa04b583c08d7 0970d42327e880909e0417db25a05af234b48e2bbde0fc08b66358ef587a8333 ebad8cad9d3b244d5f4ae24049c0088c6c3f38b0bed55872cf24e56c4effe497 b4412c7f1be086c934b71c55f25ae6517ddbad416ea718648193e76d7c20d319 0fc8a251cd517c21ccbab8ebe7f01e5d0cd3dc6a780cbb0e1eb5a2ae8334c610 0ab16d180b60d96354be5333e2b2cafa9664ce1af547aea406b8ebd4a0367197 d75809f4a9c807199c3e37c2c52db13d6ad1f7be47378c4a772bc28825bc9cf2 3e6fb4f61e372ba9eb760c4b216dd28febb2bf0b8d9dc1eb2820e95b0cac2a5c 736fe54b1019264e1faca7feef9ca13852d837e623420e746cf32abb031b971c 66e88128b7e0e9629845c910d19b610a17cc3631a34fbeeb3b31e98e5e5de9f0 fd341d3f16f9c12cd1f39bdb584c243c678d561bf980b33fa403afe269115fe6 928f3773245f7f306e3937618ff0e01208a91d9bb5ea745a68fb5121a9abe163 3c33c896e5b23994eac551e78a16a4cc90f34f269fdb933b58a9f31438410644 a4ea9b8b86c948aac94b4a6f10e3a9b8c977349bb0edefbb2bda72ac3f05cf76 234bb7bfa0549064b1ea6c6da82d4047ef85cfb85c1c11f345b590ab461b2263 509ca1137805e1baee91b8507f08ab41b96ccfc8b560db0a10c1f4ee767a5d04 fbf5be32f554f3b6f3ba9c34d7b87208f63d83057e48909330c80f0eca7ed9b6 39d7df9bddd26b934e74b1275704d8d3e1ca5e4dbad6fccb84d098c2094763a6 1a9c650039bd0dbe556a29bcfe3a62e58a868f9b35264829015dafdc771f953d 566ac7baf3611a74f2dce908687352323f8efd4fa8d7c06f343a61b886843c4d 93978156bb59c0d3770b2e624eb8b7a0e4b312321ef05b8eca8d0bea59c6664a 807f151628c338851cdf140741d2e83dffc3d21e11538c5a2424ce54c1212501 d1bc30de7ba3a13d9ae93f52a96fd6edbf20f3f06e8195dcfbe92ec444a38dd9 e7272d18ba39edfce249ecbde3e28428efc7143e56296bf183732f970434ef62 c4819bfc788ad06ddc67410ef1a1c756270237df6ec614d99c55de15e05c74bf b1754a187425d2e5ebb8948dbe94857265461821b30262b3ba317bd9ccd7e569 603ecc9bf0b9334f9040a8622f405171fa0bc312bbac7f545ac0c12bf703e58b 6cc2e0f1d35cc6e4a4f3575b40fdecc8daaefa079e69948fb7b24773d3f51425 99a1e106efaa9ee8046a6651c3451d3d80882e353683fca7d17f801825faf8dc ae5c252870f329f1917d9d7f15e62e2e9c23ead9062bae3de64685598b6cccd7 f6f18a55fc7fb0553016f703d108ab04fba8d0ae86562ec853a53e885fe9b3e5 e5cba94e93b07415ef80d3c45df8a591845adbe8d80648518eb9bef69912b2cb 665ea1b0cd7792ac395fc69bd15707b32d3fe93a00337f9ad9163cfb963a7c94 ec7afa6ffaaa2403e1f83aa7c7cd96eae3f5f828057d38016f8f86649995d678 4a1048aecdc9b9d80a8f4718661f5f7e5f2c1b72c37e21d07b012d251d1dcc91 2331b36cc8637b05a18ffb0a8c652223f968feb940503aef2e162047587cf1f1 7356d0ad0530264f165d314ff2426d55d54348e4396a5cd7c39d0c1dcf339078 925e58540358d069f6321117d7a1c96efcc77bf68107c8c812ca708514bd921d 2122977232a98a58db249e1d7fc2c9bfb5a8018bb30d411fe7aad5855d141728 1946e781d30baf3d3d0b36b7bf6ce8fd64da2d488d6668e2483af8442a90568d 2022c27faf0df8c65f21a3ef3c79b244fd64b64400f6487a354f616bdd6c19b1 486f1d9c43d15607e3d6e7cb483e580769e140be727c17dd277346dc536fb437 440e23edbfaa43bfeabfac6ebedcdcedeef101820c2fddf0972664c8a93932f8 b337cd76eff054f11612ba5d94b188504ebaf69e084fc655b23f31df5440b246 1967c588aab398ecb92849a3027552403b8b995692601281b4f4da557b209c7b 89b5aadc109b5393467e94d4b897fcb7416025d6ec97286220159acfb09db49d 4d20d692bc27e1c12d39adcb63505164ea80f9ed39365d9fa091d7057d3ead8b e178047cadaa2558aa7ea31185aff235f19211a84e8512b554b7f488db63dc95 d32c970f6c68a157c6bf06207471e1d1503cd401b69380bef874e74b7c44bf7c 00e59d223e9ea39113f42eaf910bb7e586a1416a31416be0a810fe6589a402e9 6d3b2b7e9f2d5e11b4ed90ff14674ecec9f74f405369305d1b73f7fe8e47b2c7 cc00fd5ed52ed2cc9c1a4bbc8b2d469d24f410dd2fdf2eabfb9a43c6d85ed6cf af0e8d6c3d811775d41b3c7554c8ec12fb0471b7f164f4ed93ab07bdd600e8c8 4807e1e0b6510c86b538f3a0ff4589d7814eb7df1246317a5cff91b232079507 fb081115904016fa6b65bb076c80e5acc6a04f7b9a4c73e8befb68d9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 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%!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 2 /Theta put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 61 /equal put dup 65 /A put dup 73 /I put dup 82 /R put dup 87 /W put dup 91 /bracketleft put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 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19d44b339fd4c726fa61419da81c51ae53ec49729c9fe556359deb19a5aaefe6 3eeb7efbe7649c0f4f2d6788b40cff6bcc30a0d7e5d4c2c1e5c7cb0560a0386d 825c9400b2c760e1b9fa34dc75f26970c6f54840de56190d47ec4b6a97e1d7a9 41bfb2e0f9e51ae4b931c96d4901f7f355a5df1b8e660cc1368e5ee7be566bbd f5b686a13261d84e4cec2bce8da988ab1b63d30b54c564182e9fa9d4263c0195 21 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 20 /caron put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204a520640fd7c4d05e35ea5486f a43d0e970787203855daccb1ed0b16325828b13b7c767a1a58c87902f0546ea5 404440fa6526975783c2d7a9572b8e5d5176bac598b8960609f49016aebf3763 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CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 15 /epsilon1 put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 69 /E put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 78 /N put dup 83 /S put dup 86 /V put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 116 /t put dup 120 /x put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def 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%%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 78 /N put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueXX 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 10 /circlemultiply put dup 20 /lessequal put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 70 /F put dup 71 /G put dup 72 /H put dup 75 /K put dup 87 /W put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 4 /Xi put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 124 /emdash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 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313daa53ccb8bbf8df46cc6f051f08f756d21f96e5ea5f6df6e4c0d6b4d38f8b 018ea485c63797c3b975bcc6975bf1bd2288c3095b3f9b682722a8a15340ca25 26cf801a874901e1aa7caf9ce0645f60a8cc2c81b6e6f829987ffc32864edfb9 4c8f5d5b9b70d9bddeca4a9173e998aa7e37cc8f91826b95707c9fca0473976c 658a2d8323a7e9ad71ec 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 74 /J put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 69 /E put dup 70 /F put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put dup 122 /z put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (return.dvi) @start /Fa 205[68 50[{}1 90.9091 /LASY10 rf /Fb 184[71 71[{}1 119.552 /CMSY10 rf /Fc 173[86 82[{}1 119.552 /MSBM10 rf /Fd 179[80 12[59 3[33 32[60 26[{}4 119.552 /CMMI12 rf /Fe 173[48 82[{}1 66.4176 /MSBM7 rf /Ff 255[48{}1 49.8132 /CMSY6 rf /Fg 148[33 25 22 3[29 32 73[32 5[36 7[35 12[{}8 49.8132 /CMMI6 rf /Fh 137[32 2[24 24 4[50 3[18 2[19 27 50[30 30 30 4[47 43[{}11 49.8132 /CMR6 rf /Fi 133[60 71 71 97 71 75 52 53 55 71 75 67 75 112 37 1[41 37 75 67 41 61 75 60 75 65 6[82 4[103 94 75 100 1[92 101 1[128 81 105 2[105 106 85 2[97 96 8[67 67 67 67 67 67 67 67 67 1[67 37 45 32[75 12[{}53 119.552 /CMBX12 rf /Fj 173[66 14[66 67[{}2 90.9091 /MSBM10 rf /Fk 142[91 91 61 61 6[131 91 51 8[101 101 51 1[131 5[43 1[96 2[137 16[81 1[81 1[81 15[73 4[53 53 7[68 6[67 67 54 54 2[51 30 10[42 42{}29 90.9091 /CMEX10 rf /Fl 137[37 1[27 28 28 1[39 1[39 59 20 2[20 2[22 31 1[31 1[35 5[20 3[72 4[52 8[25 7[53 3[55 7[35 1[35 35 35 35 4[55 1[27 27 27[39 9[55 1[44{}30 66.4176 /CMR8 rf /Fm 133[37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 9[91 68 68 65 51 66 70 62 70 68 82 57 70 1[35 68 70 59 62 69 65 64 68 5[28 28 2[46 46 46 46 46 46 46 2[28 33 28 2[37 37 25[80 1[51 56 11[{}65 90.9091 /CMTI10 rf /Fn 198[49 49 49 49 49 49 49 49 49 49 48[{}10 99.6264 /CMR12 rf /Fo 129[35 3[33 1[40 3[25 1[32 1[36 1[43 62 1[37 29 24 41 2[33 36 13[41 2[43 4[56 1[48 60 1[31 58 2[52 5[35 55 35 1[20 32[36 41 31 2[41 41 2[35 1[29 2[40 45 11[{}35 66.4176 /CMMI8 rf /Fp 177[76 95 77[{}2 90.9091 /EUFM10 rf /Fq 133[72 85 85 2[90 63 64 66 1[90 81 90 134 45 2[45 90 81 49 74 90 72 90 78 12[112 90 120 1[110 6[60 2[101 106 2[115 122 8[81 81 81 81 81 81 81 81 3[54 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b(will)f(alw)m(a)m(ys)i(assume)e(that)g Ft(h)f Fr(\025)f Fu(0.)244 5042 y(Ph)m(ysically)-8 b(,)32 b(the)f(b)s(osons)e(can)i(b)s(e)f(in)m(terpreted)g(as)h(phonons)e(or)h (photons.)380 5155 y(The)41 b(in)m(teraction)i(b)s(et)m(w)m(een)f(the)g (small)g(system)f(and)g(the)g(reserv)m(oir)h(is)f(sp)s(eci\014ed)g(b)m (y)g(a)244 5268 y(form-factor)f Ft(\025v)s Fu(,)j(where)c Ft(v)k Fr(2)e(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)28 b(\012)e(Z)7 b Fu(\))40 b(and)f Ft(\025)h Fu(is)f(a)h(real)g(coupling)g(constan)m(t) h(whic)m(h)p eop %%Page: 3 3 3 2 bop 3731 100 a Fn(3)244 407 y Fu(con)m(trols)29 b(the)g(strength)f (of)g(the)g(in)m(teraction.)42 b(Our)27 b(main)h(results)g(hold)f(for)h (su\016cien)m(tly)h(small)244 520 y(nonzero)i(v)-5 b(alues)30 b(of)h Ft(\025)p Fu(.)380 633 y(The)i(data)i(\()p Fr(K)q Ft(;)15 b(K)q(;)g Fr(Z)7 b Ft(;)15 b(h;)g(v)s Fu(\))38 b(determine)c(the)g(basic)g(P)m(auli-Fierz)i(Hamiltonian,)g(whic)m(h)e (is)244 746 y(de\014ned)29 b(as)i(the)f(self-adjoin)m(t)i(op)s(erator) 1300 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y(equilibrium)i(at)j(temp)s(erature)e(1) p Ft(=\014)5 b Fu(.)48 b(A)32 b(large)i(part)e(of)h(our)f(pap)s(er)f (is)h(not)h(restricted)g(to)g(the)244 3190 y(thermal)28 b(case)h(and)e(deals)h(with)g(an)f(arbitrary)h(radiation)g(densit)m(y) -8 b(.)41 b(These)28 b(results)f(are)h(useful)244 3303 y(in)i(the)h(study)e(of)i(non-equilibrium)e(theory)h(of)h(P)m (auli-Fierz)h(systems.)380 3416 y(F)-8 b(or)31 b(shortness,)e(in)h(the) g(remaining)f(part)h(of)g(the)g(in)m(tro)s(duction)g(w)m(e)g(restrict)g (ourselv)m(es)h(to)244 3528 y(the)d(case)h(of)f(thermal)g(densities.)40 b(The)27 b(main)h(ob)5 b(ject)28 b(of)g(our)f(study)g(is)h(a)g (1-parameter)h(family)244 3641 y(of)h Ft(W)446 3608 y Fs(\003)485 3641 y Fu(-dynamical)i(systems)1719 3754 y(\()p Fp(M)1849 3769 y Fo(\014)1897 3754 y Ft(;)15 b(\034)1977 3769 y Fo(\014)2025 3754 y Fu(\))1289 b(\(1.2\))244 3913 y(where)22 b Fp(M)594 3928 y Fo(\014)667 3913 y Fu(=)j Fr(B)s Fu(\()p Fr(K)q Fu(\))5 b Fr(\012)g Fp(M)1142 3880 y Fl(A)-8 b(W)1142 3941 y 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b(P)m(auli-Fierz)i(systems)f(\(1.2\))h(considered)e(in)g (our)g(pap)s(er)f(satisfy)i(the)g(\014rst)e(t)m(w)m(o)j(con-)244 4590 y(ditions)k(of)h(the)f(\\quasitheorem".)63 b(They)37 b(describ)s(e)g(an)g(in\014nitely)h(extended)f(system)g(\(this)244 4703 y(is)e(expressed)g(in)g(particular)h(b)m(y)g(the)f(fact)i(that)f Ft(h)f Fu(has)h(con)m(tin)m(uous)g(sp)s(ectrum\).)55 b(Since)35 b(the)244 4816 y(radiation)30 b(densit)m(y)f(of)h(the)f(b)s (osonic)h(\014eld)e(is)i(giv)m(en)g(b)m(y)f(the)h(Planc)m(k)g(la)m(w,)g (the)g(system)f(is)h(near)244 4929 y(thermal)h(equilibrium.)380 5042 y(The)36 b(information)h(on)f(the)g Ft(W)1465 5009 y Fs(\003)1504 5042 y Fu(-dynamics)h Ft(\034)1983 5057 y Fo(\014)2066 5042 y Fu(is)f(con)m(v)m(enien)m(tly)j(enco)s(ded)d(b)m (y)g(a)h(certain)244 5155 y(self-adjoin)m(t)46 b(op)s(erator)f Ft(L)1169 5170 y Fo(\014)1261 5155 y Fu(called)h(the)f(Liouvillean.)85 b(The)45 b(op)s(erator)g Ft(L)2896 5170 y Fo(\014)2988 5155 y Fu(is)g(canonically)244 5268 y(de\014ned)29 b(in)h(the)h (standard)e(represen)m(tation)j(of)e Fp(M)1992 5283 y Fo(\014)2040 5268 y Fu(.)p eop %%Page: 4 4 4 3 bop 3731 100 a Fn(4)380 407 y Fu(F)-8 b(or)31 b Ft(\014)g(<)25 b Fr(1)30 b Fu(\(p)s(ositiv)m(e)h(temp)s(eratures\),)g(under)e(quite)h (broad)g(conditions)h(one)f(can)h(sho)m(w)244 520 y(the)38 b(existence)i(of)f(a)f(\()p Ft(\034)1073 535 y Fo(\014)1121 520 y Ft(;)15 b(\014)5 b Fu(\)-KMS)39 b(v)m(ector,)j(whic)m(h)c(is)h (an)f(eigen)m(v)m(ector)j(of)e Ft(L)2990 535 y Fo(\014)3075 520 y Fu(with)f(a)h(zero)244 633 y(eigen)m(v)-5 b(alue.)65 b(This)37 b(result)h(w)m(as)g(pro)m(v)m(en)h(in)e([DJP])h(and)g(is)g (based)f(on)h(an)g(extension)g(of)g(the)244 746 y(w)m(ell-kno)m(wn)31 b(result)f(of)h(Araki)g([Ar,)f(BR2)q(].)380 858 y(F)-8 b(or)40 b Ft(\014)46 b Fu(=)40 b Fr(1)g Fu(\(the)g(zero)g(temp)s (erature\))g(in)f(man)m(y)g(cicumstances)i(one)f(can)g(sho)m(w)f(that) 244 971 y(the)31 b(P)m(auli-Fierz)i(Hamiltonian)f Ft(H)38 b Fu(has)30 b(a)i(ground)d(state)k([AH,)e(BFS1)q(,)g(Ge)q(,)g(Sp2)o(,)g (Sp3].)42 b(The)244 1084 y(ground)30 b(state)i(giv)m(es)g(rise)e(to)i (an)f(eigen)m(v)m(ector)i(of)e(the)g(corresp)s(onding)f(Liouvillean)i Ft(L)3254 1098 y Fs(1)3359 1084 y Fu(with)244 1197 y(a)f(zero)g(eigen)m (v)-5 b(alue.)380 1310 y(F)d(or)26 b Ft(\014)31 b(<)25 b Fr(1)p Fu(,)h(the)f(return)f(to)i(equilibrium)e(can)i(b)s(e)e (deduced)h(from)f(sp)s(ectral)i(prop)s(erties)e(of)244 1423 y Ft(L)306 1438 y Fo(\014)353 1423 y Fu(.)42 b(In)31 b(particular,)g(the)h(return)d(to)j(equilibrium)e(follo)m(ws)i(if)f Ft(L)2456 1438 y Fo(\014)2534 1423 y Fu(has)g(no)g(singular)g(sp)s (ectrum)244 1536 y(except)g(for)f(a)h(nondegenerate)g(eigen)m(v)-5 b(alue)33 b(at)e(zero.)380 1649 y(F)-8 b(or)40 b(a)f Ft(W)735 1616 y Fs(\003)774 1649 y Fu(-dynamical)h(system)e(\()p Fp(M)p Ft(;)15 b(\034)10 b Fu(\))41 b(with)d Fp(M)h Fu(b)s(eing)g(a)g (t)m(yp)s(e)g(I)g(factor,)j(the)d(return)244 1762 y(to)j(equilibrium)f (nev)m(er)h(holds)f(\(unless)g(the)h(algebra)g(is)g(1-dimensional\).)75 b(Therefore,)44 b(the)244 1875 y(P)m(auli-Fierz)32 b(systems)f(with)f Ft(\014)g Fu(=)25 b Fr(1)30 b Fu(do)h(not)f(ha)m(v)m(e)i(the)e(return)g (to)h(equilibrium)e(prop)s(ert)m(y)-8 b(.)380 1988 y(In)22 b(the)i(literature)g([HS1,)f(F)m(GS)q(],)i(one)e(can)g(\014nd)e(a)j (related)f(prop)s(ert)m(y)g(called)h(the)f(relaxation)244 2100 y(to)28 b(a)g(ground)e(state,)k(whic)m(h)d(in)g(some)h(cases)g (can)f(b)s(e)g(pro)m(v)m(en)h(for)f(zero)h(temp)s(erature)f(systems.) 244 2213 y(Note,)38 b(ho)m(w)m(ev)m(er,)g(that)d(to)h(pro)m(v)m(e)g (this)f(prop)s(ert)m(y)f(one)i(needs)e(to)i(consider)f(appropriate)g Ft(C)3466 2180 y Fs(\003)3505 2213 y Fu(-)244 2326 y(dynamical)c (systems,)f(whereas)h(w)m(e)f(alw)m(a)m(ys)i(consider)f Ft(W)2266 2293 y Fs(\003)2305 2326 y Fu(-dynamical)g(systems.)380 2439 y(The)d(regularit)m(y)i(assumption)e(on)g(P)m(auli-Fierz)j (systems)d(that)h(w)m(e)g(mak)m(e)g(is)g(based)f(on)g(the)244 2552 y(ideas)41 b(and)f(results)h(of)g([JP1,)g(JP2].)73 b(This)40 b(metho)s(d)g(consists)h(in)g("gluing")h(together)g(neg-)244 2665 y(ativ)m(e)g(and)f(p)s(ositiv)m(e)g(frequences)g(of)g(the)g(b)s (osons,)i(whic)m(h)d(allo)m(ws)i(to)g(de\014ne)e(a)h(\\conjugate)244 2778 y(op)s(erator"|the)c(generator)h(of)f(translations)g(in)f(the)h (sp)s(ectral)f(v)-5 b(ariables.)60 b(In)36 b(the)g(original)244 2891 y(approac)m(h)30 b(of)f([JP1)q(,)g(JP2])h(the)g(analyticit)m(y)i (of)e(the)f(form-factor)i(with)e(resp)s(ect)h(to)g(the)g(trans-)244 3004 y(lations)25 b(w)m(as)g(assumed)f(and)g(the)h(Liouvillean)g(w)m (as)g(studied)f(using)g(the)h(complex)g(deformation)244 3117 y(metho)s(d.)40 b(Here,)30 b(w)m(e)g(asssume)f(only)g(the)h (di\013eren)m(tiabilit)m(y)h(of)f(a)f(su\016cien)m(tly)h(high)f(order)g (and)244 3230 y(study)f(the)h(sp)s(ectrum)f(using)g(the)i(Mourre)e (theory)i(dev)m(elop)s(ed)f(for)g(this)f(purp)s(ose)g(in)g(our)h(pre-) 244 3342 y(vious)i(pap)s(er)e([DJ1)q(].)43 b(The)30 b(Mourre)h(theory)g (allo)m(ws)h(us)e(to)i(treat)g(P)m(auli-Fierz)h(systems)d(more)244 3455 y(e\016cien)m(tly)-8 b(,)33 b(esp)s(ecially)e(at)g(lo)m(w)g(temp)s (eratures.)380 3568 y(T)-8 b(o)27 b(express)f(our)h(e\013ectiv)m(e)i (coupling)e(assumptions)f(w)m(e)h(use)f(some)h(simple)g(algebraic)h (con-)244 3681 y(ditions)h(deriv)m(ed)g(from)f(the)h(so-called)i(F)-8 b(ermi)30 b(Golden)f(Rule,)g(whic)m(h)g(describ)s(es)f(ho)m(w)h(to)h (com-)244 3794 y(pute)i(eigen)m(v)-5 b(alues)35 b(and)d(resonances)h (to)g(the)g(second)g(order.)47 b(In)32 b(particular,)i(it)f(can)g(b)s (e)f(used)244 3907 y(to)k(predict)f(whic)m(h)g(eigen)m(v)-5 b(alues)37 b(will)f(disapp)s(ear)e(after)i(the)f(in)m(teraction)j(is)d ("switc)m(hed)h(on".)244 4020 y(The)d(information)h(obtained)g(b)m(y)f (the)h(F)-8 b(ermi)34 b(Golden)g(Rule)g(can)g(b)s(e)f(con)m(v)m(enien)m (tly)j(enco)s(ded)244 4133 y(in)30 b(the)g(so-called)j(Lev)m(el)e (Shift)f(Op)s(erator)g(\000)1783 4148 y Fo(\014)1830 4133 y Fu(|an)g(ob)5 b(ject)31 b(that)g(pla)m(ys)g(a)g(crucial)g(role)g (in)f(our)244 4246 y(pap)s(er.)380 4359 y(The)d(Liouvillean)i(of)f(a)g (P)m(auli-Fierz)i(system)d(in)h(the)g(absence)g(of)f(in)m(teraction)j (has)d(a)h(large)244 4472 y(k)m(ernel)39 b(\(of)f(dimension)g(at)g (least)i(dim)14 b Fr(K)q Fu(\).)65 b(After)38 b(the)h(in)m(teraction)h (is)e("switc)m(hed)h(on",)h(the)244 4584 y(dimension)30 b(of)h(the)g(k)m(ernel)g(of)f Ft(L)1368 4599 y Fo(\014)1446 4584 y Fu(is)h(at)g(least)h(one.)42 b(Our)29 b(aim)i(is)g(to)g(sho)m(w) g(that)g(there)g(are)g(no)244 4697 y(other)g(eigen)m(v)m(ectors)i(of)d Ft(L)1155 4712 y Fo(\014)1233 4697 y Fu(for)g(small)h(nonzero)f Ft(\025)p Fu(.)380 4810 y(F)-8 b(or)38 b(small)g(nonzero)f Ft(\025)g Fu(and)f(all)i Ft(\014)k Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([,)39 b(that)f(is)f(for)g(the)g(whole)g(range)g(of)h(p)s (ositiv)m(e)244 4923 y(temp)s(eratures,)31 b(analysis)h(of)f(the)g(Lev) m(el)i(Shift)d(Op)s(erator)h(\000)2338 4938 y Fo(\014)2416 4923 y Fu(giv)m(es)h(a)g(single)f(condition)h(that)244 5036 y(on)41 b(the)h(formal)g(lev)m(el)i(indicates)e(the)g(absence)g (of)g(the)g(singular)g(sp)s(ectrum)e(of)i Ft(L)3192 5051 y Fo(\014)3281 5036 y Fu(except)244 5149 y(for)36 b(a)g(nondegenerate)h (eigen)m(v)-5 b(alue)38 b(at)f(zero.)59 b(In)35 b(order)h(to)g(c)m(hec) m(k)i(this)e(condition)h(one)f(con-)244 5262 y(structs)g(a)h(certain)g (\(\014nite)f(dimensional\))h Fr(\003)p Fu(-algebra)h Fp(N)d Fr(\032)f(B)s Fu(\()p Fr(K)q Fu(\))j(whic)m(h)f(dep)s(ends)e (only)j(on)p eop %%Page: 5 5 5 4 bop 3731 100 a Fn(5)244 407 y Fu(\()p Fr(K)q Ft(;)15 b(K)q(;)g Fr(Z)7 b Ft(;)15 b(h;)g(v)s Fu(\))35 b(and)30 b(not)h(on)g(the)f(in)m(v)m(erse)i(temp)s(erature)f Ft(\014)5 b Fu(.)41 b(Our)30 b(p)s(ositiv)m(e)i(temp)s(erature)e(ef-)244 520 y(fectiv)m(e)f(coupling)e(assumption)f(is)h(that)g Fp(N)e Fu(=)g Fj(C)p Fu(1.)40 b(The)26 b(result)h(of)g([DJ1)q(])f(pro)m (vides)h(a)g(rigorous)244 633 y(metho)s(d)k(to)i(sho)m(w)f(that)h(the)f (ab)s(o)m(v)m(e)i(assumption)d(together)j(with)e(a)g(su\016cien)m(t)h (regularit)m(y)g(of)244 746 y(the)38 b(form-factor)h(imply)e(the)h (desired)f(sp)s(ectral)i(prop)s(erties)e(of)h Ft(L)2591 761 y Fo(\014)2638 746 y Fu(,)i(and)d(hence)h(imply)f(the)244 858 y(return)c(to)j(equilibrium.)52 b(The)34 b(result)h(describ)s(ed)e (so)i(far)g(is,)h(ho)m(w)m(ev)m(er,)h(not)e(uniform)e(in)h(the)244 971 y(temp)s(erature.)380 1084 y(Our)40 b(main)h(goal)i(is)e(to)h(sho)m (w)f(that)h(under)d(suitable)j(conditions)f(P)m(auli-Fierz)j(systems) 244 1197 y(ha)m(v)m(e)34 b(the)f(prop)s(ert)m(y)f(of)h(return)f(to)h (equilibrium)f(uniformly)g(in)h(the)g(temp)s(erature.)47 b(This)32 b(re-)244 1310 y(quires)i(a)h(detailed)g(analysis)g(of)g(the) g(zero)g(temp)s(erature)f(case,)j(whic)m(h)d(is)h(in)f(man)m(y)h(resp)s (ects)244 1423 y(di\013eren)m(t)41 b(from)f(that)h(of)g(p)s(ositiv)m(e) h(temp)s(eratures.)71 b(Analysis)41 b(of)g(the)f(Lev)m(el)i(Shift)e(Op) s(era-)244 1536 y(tor)33 b(\000)450 1550 y Fs(1)558 1536 y Fu(yields)g(natural)g(e\013ectiv)m(e)j(coupling)d(conditions)g(under) f(whic)m(h)h(one)g(should)f(exp)s(ect)244 1649 y(that)37 b(for)e(a)i(su\016cien)m(tly)g(small)f(nonzero)h Ft(\025)f Fu(the)g(P)m(auli-Fierz)i(Hamiltonian)g Ft(H)43 b Fu(has)36 b(no)g(sin-)244 1762 y(gular)c(sp)s(ectrum)f(except)i(for)f(a)h (nondegenerate)g(ground)e(state.)47 b(These)32 b(conditions)g(in)m(v)m (olv)m(e)244 1875 y(the)37 b(nondegeneracy)h(of)f(the)g(unp)s(erturb)s (ed)c(ground)j(state)j(and)d(the)h(strict)h(p)s(ositivit)m(y)g(of)f(a) 244 1988 y(certain)k(auxiliary)g(op)s(erator.)70 b(The)39 b(result)h(of)h([DJ1)q(])f(giv)m(es)h(a)g(rigorous)f(pro)s(of)f(that)i (these)244 2100 y(conditions)30 b(together)h(with)f(a)g(su\016cien)m(t) g(regularit)m(y)h(of)f(the)g(form-factor)h(imply)e(the)h(desired)244 2213 y(prop)s(erties)36 b(of)h(P)m(auli-Fierz)i(Hamiltonians.)61 b(As)36 b(an)h(immediate)h(consequence,)h(under)c(the)244 2326 y(same)40 b(conditions)g(zero)g(temp)s(erature)f(Liouvilleans)i 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b(mak)m(e)h(the)244 5033 y(pap)s(er)j(more)i(accessible,) j(in)c(this)g(section)i(w)m(e)f(restrict)g(ourselv)m(es)g(to)g(the)g (case)g(of)g(a)g(scalar)244 5146 y(massless)30 b(\014eld.)40 b(This)29 b(section)i(is)f(not)h(used)e(in)g(the)h(remaining)h(part)e (of)h(the)h(pap)s(er,)e(where)g(a)244 5259 y(more)h(general)h(class)g (of)f(mo)s(dels)f(is)h(considered)g(and)f(a)h(di\013eren)m(t)h (notation)g(is)f(used.)40 b(Section)p eop %%Page: 6 6 6 5 bop 3731 100 a Fn(6)244 407 y Fu(3)35 b(serv)m(es)h(as)g(a)f(quic)m (k)h(in)m(tro)s(duction)f(to)h(our)f(results)g(and)f(allo)m(ws)j(to)f (compare)f(them)g(easily)244 520 y(with)30 b(the)h(results)f(existing)h (in)f(the)h(literature.)380 633 y(In)37 b(Section)h(4)g(w)m(e)g(in)m (tro)s(duce)g(the)f(notation)i(and)e(review)h(some)g(basic)g(facts)g (whic)m(h)f(w)m(e)244 746 y(will)31 b(need)g(in)f(the)h(pap)s(er.)41 b(In)30 b(Section)i(5)f(w)m(e)g(in)m(tro)s(duce)g(P)m(auli-Fierz)i(op)s (erators)e(and)f(review)244 858 y(their)g(prop)s(erties)g(follo)m(wing) i([DJ1)q(].)380 971 y(In)27 b(Section)i(6)f(w)m(e)g(in)m(tro)s(duce)g (P)m(auli-Fierz)i(systems)d(at)i(densit)m(y)f Ft(\032)p Fu(,)g(compute)h(P)m(auli-Fierz)244 1084 y(Liouvilleans)23 b Ft(L)802 1098 y Fo(\032)864 1084 y Fu(and)f(study)g(their)g(prop)s (erties.)37 b(The)22 b(main)g(tec)m(hnical)j(results)d(of)g(this)g (section)244 1197 y(concern)30 b(the)h(structure)f(of)g(the)h(Lev)m(el) h(Shift)d(Op)s(erator)h(\000)2291 1211 y Fo(\032)2362 1197 y Fu(of)g(the)h(Liouvillean)g Ft(L)3152 1211 y Fo(\032)3192 1197 y Fu(.)380 1310 y(In)h(Section)h(7)f(w)m(e)h(discuss)e(thermal)i (P)m(auli-Fierz)h(systems.)46 b(In)31 b(Subsection)h(7.4)h(w)m(e)g(giv) m(e)244 1423 y(conditions)d(under)f(whic)m(h)g(thermal)h(P)m (auli-Fierz)i(systems)e(ha)m(v)m(e)i(the)e(prop)s(ert)m(y)f(of)h (return)f(to)244 1536 y(equilibrium)f(for)h(a)h(\014xed)e(in)m(v)m (erse)i(temp)s(erature.)41 b(The)28 b(result)h(uniform)f(in)h(the)h (temp)s(erature)244 1649 y(is)g(describ)s(ed)g(in)g(Subsection)g(7.7.) 380 1762 y(As)45 b(w)m(e)g(ha)m(v)m(e)h(already)g(men)m(tioned,)j(the)c (main)g(regularit)m(y)h(assumption)e(our)g(metho)s(d)244 1875 y(requires)32 b(concerns)h(the)g(gluing)g(condition)h(of)e([JP1)q (,)h(JP2,)g(DJ1].)48 b(In)32 b(Section)i(8)f(w)m(e)g(discuss)244 1988 y(the)d(gluing)h(condition)g(in)f(the)h(con)m(text)h(of)f(scalar)g (and)f(v)m(ector)i(massless)e(b)s(osons.)244 2264 y Fi(1.2)135 b(Comparison)46 b(with)f(the)h(literature)244 2436 y Fu(Hamiltonians)f(similar)g(to)g(those)g(considered)g(in)f(our)g(pap)s (er)f(app)s(ear)h(frequen)m(tly)g(in)g(the)244 2549 y(ph)m(ysics)21 b(literature,)j(see)e(e.g.[PF)r(].)38 b(In)20 b(the)i(recen)m(t)g(y)m (ears)g(there)g(has)f(b)s(een)f(a)i(reviv)-5 b(al)22 b(of)f(in)m(terest)244 2662 y(in)30 b(rigorous)h(results)g(ab)s(out)g (these)g(op)s(erators,)g(starting)h(with)e(suc)m(h)h(pap)s(ers)f(as)h ([BFS1)q(,)g(DG,)244 2775 y(HS2,)g(JP1,)f(Sk].)380 2887 y(F)-8 b(rom)23 b(the)f(tec)m(hnical)h(p)s(oin)m(t)f(of)g(view,)i(the)e (results)g(of)g(our)f(pap)s(er)g(concern)h(mainly)g(sp)s(ectral)244 3000 y(prop)s(erties)30 b(of)i(a)f(certain)h(class)g(of)g(P)m (auli-Fierz)h(op)s(erators.)43 b(A)31 b(large)i(part)e(of)g(the)g (literature)244 3113 y(on)j(sp)s(ectral)h(analysis)h(of)e(P)m (auli-Fierz)j(op)s(erators)e(can)g(b)s(e)f(divided)g(in)m(to)h(t)m(w)m (o)h(classes.)55 b(The)244 3226 y(\014rst)38 b(uses)h(the)h(generator)g (of)g(translations)g(as)f(the)h(main)f(to)s(ol)h(and)f(the)g(second)h (uses)e(the)244 3339 y(generator)31 b(of)g(dilations.)380 3452 y(In)d(the)g(con)m(text)j(of)d(P)m(auli-Fierz)j(systems,)e(the)f (generator)i(of)e(translations)h(w)m(as)g(used)f(for)244 3565 y(the)38 b(\014rst)g(time)h(in)f([JP1,)h(JP2],)h(where)e(it)h(w)m (as)g(applied)f(to)h(deform)f(analytically)j(p)s(ositiv)m(e)244 3678 y(temp)s(erature)j(P)m(auli-Fierz)j(Liouvilleans.)85 b(These)44 b(pap)s(ers)g(also)h(con)m(tain)h(the)f(\014rst)f(pro)s(of) 244 3791 y(that)34 b(thermal)h(P)m(auli-Fierz)h(systems)d(ha)m(v)m(e)j (the)e(prop)s(ert)m(y)f(of)h(return)f(to)i(equilibrium.)50 b(The)244 3904 y(in\014nitesimal)34 b(v)m(ersion)g(of)g(this)g(metho)s (d)f(based)g(on)h(the)g(Mourre)g(theory)g(w)m(as)g(dev)m(elop)s(ed)g (in)244 4017 y([DJ1)q(].)39 b(That)27 b(pap)s(er)f(w)m(as)h(a)g(tec)m (hnical)i(preparation)e(for)g(the)g(presen)m(t)g(pap)s(er.)38 b(In)26 b(fact,)j(in)d(the)244 4129 y(in)m(tro)s(duction)h(to)i([DJ1)q (])e(w)m(e)h(roughly)f(describ)s(ed)g(the)h(applications)g(con)m (tained)h(in)e(this)h(pap)s(er)244 4242 y(\(without,)j(ho)m(w)m(ev)m (er,)h(giving)f(exact)h(conditions\).)380 4355 y(The)37 b(generator)i(of)f(translations)g(is)g(also)g(the)g(main)g(to)s(ol)g (of)g(an)f(in)m(teresting)i(pap)s(er)e(b)m(y)244 4468 y(Merkli)k([M)q(],)i(whic)m(h)d(is)h(dev)m(oted)g(to)g(the)g(pro)s(of)f (of)g(return)g(to)h(equilibrium)f(in)g(the)g(mean.)244 4581 y(This)30 b(pap)s(er)f(is)i(based)f(on)g(the)h(tec)m(hnique)g(of)g 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b(sp)s(eaking,)h(ho)m(w)m(ev)m(er,) 244 1310 y(the)33 b(conditions)h(of)f([BFS2)q(])g(are)g(not)g (comparable)h(to)g(ours)e(and)h(one)g(can)g(\014nd)f(in)m(teractions) 244 1423 y(whic)m(h)i(can)h(b)s(e)f(treated)i(with)e(one)h(metho)s(d)g (and)f(not)h(b)m(y)f(the)h(other.)54 b(With)35 b(regard)g(to)g(the)244 1536 y(infrared)27 b(singularit)m(y)-8 b(,)31 b(the)d(conditions)h(of)f ([BFS2)q(])h(for)f(the)g(uniform)g(in)g(temp)s(erature)g(return)244 1649 y(to)j(equilibrium)f(are)g(somewhat)h(less)g(restrictiv)m(e)h (than)e(ours.)380 1762 y(There)35 b(is)g(a)g(v)-5 b(ast)35 b(b)s(o)s(dy)e(of)i(literature)h(dealing)g(with)f(P)m(auli-Fierz)i (systems)d(in)h(the)g(V)-8 b(an)244 1875 y(Ho)m(v)m(e)23 b(w)m(eak)g(coupling)e(limit)i Ft(s)i Fu(=)f Ft(\025)1460 1842 y Fl(2)1500 1875 y Ft(t)p Fu(,)f Ft(\025)i Fr(#)h Fu(0,)e(with)d Ft(s)g Fu(\014xed)g(\(see)i(eg.)38 b([Da)q(,)22 b(Sp2)o(]\).)39 b(In)20 b(this)i(limit)244 1988 y(one)28 b(obtains)f(an)h(irrev)m(ersible)f(Mark)m(o)m(vian)j(dynamics)d(on)g (the)h(algebra)g(of)g(the)f(small)h(system.)244 2100 y(The)d(generator)h(of)g(this)f(dynamics)h(is)f(sometimes)i(called)f (the)g(Da)m(vies)h(generator.)40 b(The)25 b(Lev)m(el)244 2213 y(Shift)g(Op)s(erator,)i(whic)m(h)f(arises)g(through)g(the)g(F)-8 b(ermi)27 b(Golden)f(Rule)g(for)g(the)g(Liouvillean)h(and)244 2326 y(is)37 b(one)g(of)f(the)h(main)g(to)s(ols)g(of)g(our)f(pap)s(er,) i(is)e(similar)i(to)f(the)g(Da)m(vies)h(generator)g(in)e(man)m(y)244 2439 y(resp)s(ects.)76 b(Let)42 b(us)g(stress,)j(ho)m(w)m(ev)m(er,)i (that)42 b(they)h(are)f(di\013eren)m(t)h(op)s(erators.)76 b(The)42 b(Da)m(vies)244 2552 y(generator)35 b(describ)s(es)e(the)i(ev) m(olution)g(of)f(observ)-5 b(ables)34 b(and)g(alw)m(a)m(ys)h(has)f(a)h (zero)f(eigen)m(v)-5 b(alue.)244 2665 y(On)31 b(the)h(other)f(hand,)h (the)g(Lev)m(el)g(Shift)f(Op)s(erator)h(describ)s(es)f(the)g(shift)h (of)g(eigen)m(v)-5 b(alues)33 b(and)244 2778 y(resonances)41 b(of)f(the)h(Liouvillean)h(and)e(often)h(do)s(es)f(not)h(ha)m(v)m(e)g (a)g(zero)g(eigen)m(v)-5 b(alue.)74 b(In)39 b(the)244 2891 y(thermal)31 b(case,)g(ho)m(w)m(ev)m(er,)h(these)f(op)s(erators)g (are)f(closely)i(related,)g(see)f([DJ2)q(].)380 3004 y(The)h(e\013ectiv)m(e)k(coupling)d(assumptions)f(for)g(return)g(to)h (equilibrium)f(used)g(in)g(our)h(pap)s(er)244 3117 y(are)d(di\013eren)m (t)h(from)e(those)i(found)e(in)g(the)i(literature.)41 b(T)-8 b(o)31 b(our)e(kno)m(wledge)i(they)g(are)f(simpler)244 3230 y(and)j(less)g(restrictiv)m(e|in)i(fact,)g(they)f(are)f(optimal)h (in)f(the)h(con)m(text)h(of)e(the)h(2nd)f(order)f(p)s(er-)244 3342 y(turbation)c(theory)-8 b(.)41 b(They)28 b(are)g(based)g(on)h(a)f (detailed)i(algebraic)g(analysis)f(of)f(the)h(Lev)m(el)g(Shift)244 3455 y(Op)s(erator)h(for)g(P)m(auli-Fierz)i(Liouvilleans)f(that)g (seems)g(to)g(app)s(ear)e(for)h(the)h(\014rst)e(time)i(in)f(the)244 3568 y(literature.)41 b(Somewhat)29 b(similar)h(e\013ectiv)m(e)h (coupling)f(conditions)f(for)g(return)f(to)h(equilibrium)244 3681 y(of)h(quan)m(tum)g(Mark)m(o)m(vian)j(semigroups)c(w)m(ere)i(giv)m (en)h(in)e([F)-8 b(r,)31 b(Sp2)o(].)380 3794 y(One)i(of)g(the)g (consequences)h(of)f(our)f(metho)s(d)h(is)g(a)g(relativ)m(ely)i(simple) e(pro)s(of)g(of)g(the)g(non-)244 3907 y(degeneracy)27 b(of)f(the)h(ground)e(state)j(of)e(P)m(auli-Fierz)i(Hamiltonians)g (under)c(certain)j(regularit)m(y)244 4020 y(and)32 b(e\013ectiv)m(e)k (coupling)d(assumptions.)48 b(The)32 b(other)i(pro)s(ofs)e(in)g(the)i (literature)g(use)e(P)m(erron-)244 4133 y(F)-8 b(rob)s(enius)38 b(t)m(yp)s(e)h(argumen)m(ts)g(and)f(are)h(restricted)h(to)f(p)s (ositivit)m(y)h(preserving)f(in)m(teractions)244 4246 y([BFS1)q(,)k(Sp1)o(,)g(Sp3].)78 b(The)42 b(only)h(exception)i(that)e (w)m(e)g(kno)m(w)g(is)g(the)g(pro)s(of)f(based)h(on)f(the)244 4359 y(\\renormalization)32 b(group")f(con)m(tained)g(in)f([BFS1)q(].) 380 4472 y(The)d(\014rst)f(result)h(ab)s(out)f(existence)i(of)f(KMS)g (states)h(for)f(P)m(auli-Fierz)h(systems)f(go)s(es)h(bac)m(k)244 4584 y(to)f([FNV)q(])f(where)g(the)g(spin-b)s(oson)f(system)h(w)m(as)g (considered.)39 b(It)27 b(w)m(as)f(also)h(pro)m(v)m(en)g(in)e([BFS2)q (])244 4697 y(under)k(a)h(more)h(restrictiv)m(e)h(infrared)e(condition) h(than)f(that)h(of)f(our)g(pap)s(er.)380 4810 y(Our)23 b(result)i(ab)s(out)f(a)h(system)f(coupled)h(to)g(sev)m(eral)h(reserv)m (oirs)f(at)g(di\013eren)m(t)g(temp)s(eratures)244 4923 y(can)k(b)s(e)f(compared)g(with)h(recen)m(t)h(w)m(orks)e(on)h (non-equilibrium)e(quan)m(tum)h(statistical)k(ph)m(ysics)244 5036 y([JP3,)45 b(Ru)o(].)83 b(Note)46 b(that)f(these)f(pap)s(ers)g (use)g Ft(C)1983 5003 y Fs(\003)2022 5036 y Fu(-dynamical)h(systems)f (rather)g(than)g Ft(W)3466 5003 y Fs(\003)3505 5036 y Fu(-)244 5149 y(dynamical)31 b(systems)f(and)g(lo)s(ok)h(for)f (stationary)i(states)f(that)g(are)g(not)g(normal.)p eop %%Page: 8 8 8 7 bop 3731 100 a Fn(8)244 407 y Fq(2)161 b(Algebraic)52 b(preliminaries)244 610 y Fu(In)27 b(this)h(section)h(w)m(e)f(review)h (some)f(elemen)m(ts)h(of)f(the)g(theory)h(of)f Ft(W)2579 577 y Fs(\003)2618 610 y Fu(-algebras)h(needed)e(in)h(our)244 723 y(pap)s(er.)40 b(F)-8 b(or)31 b(more)f(details)i(w)m(e)e(refer)h (the)f(reader)g(to)h([DJP)q(],)g(and)e(also)j([BR1)q(,)e(BR2)q(,)h(St)o (].)380 836 y(One)40 b(of)g(the)h(most)f(imp)s(ortan)m(t)h(concepts)g (of)f(the)g(mo)s(dern)f(theory)i(of)f Ft(W)3041 803 y Fs(\003)3080 836 y Fu(-algebras)i(is)244 948 y(the)36 b(so-called)h(standard)e(represen)m(tation.)57 b(W)-8 b(e)36 b(sa)m(y)h(that)f(a)g(quadruple)e(\()p Ft(\031)s(;)15 b Fr(H)q Ft(;)g(J)o(;)g Fr(H)3263 962 y Fl(+)3323 948 y Fu(\))36 b(is)f(a)244 1061 y(standard)30 b(represen)m(tation)h(of)g (a)g Ft(W)1497 1028 y Fs(\003)1535 1061 y Fu(-algebra)h Fp(M)f Fu(if)g Ft(\031)d Fu(:)e Fp(M)g Fr(!)f(B)s Fu(\()p Fr(H)q Fu(\))30 b(is)h(a)f Fr(\003)p Fu(-represen)m(tation,)244 1174 y Ft(J)41 b Fu(is)32 b(an)g(an)m(tiunitary)g(in)m(v)m(olution)h (on)f Fr(H)h Fu(and)e Fr(H)1953 1188 y Fl(+)2044 1174 y Fu(is)h(a)g(self-dual)g(cone)h(in)e Fr(H)i Fu(satisfying)f(the)244 1287 y(follo)m(wing)g(conditions:)244 1400 y(\(1\))f Ft(J)9 b(\031)s Fu(\()p Fp(M)p Fu(\))p Ft(J)36 b Fu(=)25 b Ft(\031)s Fu(\()p Fp(M)p Fu(\))1071 1367 y Fs(0)1096 1400 y Fu(;)244 1513 y(\(2\))31 b Ft(J)9 b(\031)s Fu(\()p Ft(A)p Fu(\))p Ft(J)36 b Fu(=)25 b Ft(\031)s Fu(\()p Ft(A)p Fu(\))1017 1480 y Fs(\003)1088 1513 y Fu(for)30 b Ft(A)g Fu(in)g(the)h(cen)m(ter)g(of)g Fp(M)p Fu(;)244 1626 y(\(3\))g Ft(J)9 b Fu(\011)25 b(=)g(\011)30 b(for)h(\011)25 b Fr(2)f(H)1140 1640 y Fl(+)1199 1626 y Fu(;)244 1739 y(\(4\))31 b Ft(\031)s Fu(\()p Ft(A)p Fu(\))p Ft(J)9 b(\031)s Fu(\()p Ft(A)p Fu(\))p Fr(H)912 1753 y Fl(+)999 1739 y Fr(\032)24 b(H)1171 1753 y Fl(+)1261 1739 y Fu(for)30 b Ft(A)25 b Fr(2)g Fp(M)p Fu(.)244 1852 y(Ev)m(ery)e Ft(W)597 1819 y Fs(\003)636 1852 y Fu(-algebra)i(has)d(a)i(unique)e (\(up)h(to)g(unitary)g(equiv)-5 b(alence\))25 b(standard)d(represen)m (tation.)380 1965 y(The)29 b(standard)g(represen)m(tation)h(has)f(sev)m (eral)i(imp)s(ortan)m(t)f(prop)s(erties.)39 b(First,)31 b(ev)m(ery)f(nor-)244 2078 y(mal)f(state)h Ft(!)h Fu(has)d(a)h(unique)f (v)m(ector)i(represen)m(tativ)m(e)g(in)f Fr(H)2291 2092 y Fl(+)2378 2078 y Fu(\(there)g(is)g(a)f(unique)g(normalized)244 2190 y(v)m(ector)34 b(\012)28 b Fr(2)g(H)779 2204 y Fl(+)870 2190 y Fu(suc)m(h)j(that)i Ft(!)s Fu(\()p Ft(A)p Fu(\))c(=)f(\(\012)p Fr(j)p Ft(\031)s Fu(\()p Ft(A)p Fu(\)\012\)\).)48 b(Secondly)-8 b(,)34 b(for)e(ev)m(ery)h Ft(W)3013 2157 y Fs(\003)3051 2190 y Fu(-dynamics)g Ft(\034)244 2303 y Fu(on)d Fp(M)h Fu(there)f(is)h(a)g(unique)e(self-adjoin)m(t)j(op)s(erator)e Ft(L)h Fu(on)f Fr(H)h Fu(suc)m(h)f(that)1036 2498 y Ft(\031)s Fu(\()p Ft(\034)1176 2460 y Fo(t)1206 2498 y Fu(\()p Ft(A)p Fu(\)\))d(=)e(e)1542 2460 y Fl(i)p Fo(tL)1639 2498 y Ft(\031)s Fu(\()p Ft(A)p Fu(\)e)1872 2460 y Fs(\000)p Fl(i)p Fo(tL)2026 2498 y Ft(;)137 b Fu(e)2228 2460 y Fl(i)p Fo(tL)2325 2498 y Fr(H)2402 2512 y Fl(+)2486 2498 y Fu(=)25 b Fr(H)2659 2512 y Fl(+)2718 2498 y Ft(:)3349 2496 y Fu(\(2.3\))244 2688 y(The)30 b(op)s(erator)g Ft(L)h Fu(is)f(called)i(the)e(Liouvillean)i(of)e(the)h Ft(W)2223 2655 y Fs(\003)2262 2688 y Fu(-dynamical)g(system)f(\()p Fp(M)p Ft(;)15 b(\034)10 b Fu(\).)244 2855 y Fv(Theorem)70 b(2.1)49 b Fm(L)-5 b(et)25 b Ft(!)j Fm(b)-5 b(e)25 b(a)h(normal)h (state)e(and)h Fu(\012)f Fr(2)g(H)2253 2869 y Fl(+)2337 2855 y Fm(its)g(ve)-5 b(ctor)26 b(r)-5 b(epr)g(esentative.)41 b(Then)244 2968 y Ft(!)35 b Fm(is)e Ft(\034)10 b Fm(-invariant)33 b(i\013)f Fu(\012)25 b Fr(2)g Fu(Ker)p Ft(L)p Fm(.)244 3152 y Fv(Theorem)70 b(2.2)80 b Fu(\(1\))50 b Fm(The)36 b(Liouvil)5 b(le)-5 b(an)36 b Ft(L)g Fm(has)g(no)g(eigenvalues)g(i\013) f(the)h Ft(W)3009 3119 y Fs(\003)3048 3152 y Fm(-dynamics)h Ft(\034)440 3265 y Fm(has)c(no)g(normal)i(invariant)f(states.)275 3394 y Fu(\(2\))50 b Fm(The)45 b(Liouvil)5 b(le)-5 b(an)47 b Ft(L)e Fm(has)i(exactly)f(one)g(nonde)-5 b(gener)g(ate)47 b(eigenvalue)e(at)h(zer)-5 b(o)47 b(i\013)e(the)440 3507 y Ft(W)539 3474 y Fs(\003)578 3507 y Fm(-dynamics)33 b Ft(\034)43 b Fm(has)33 b(a)g(single)g(normal)h(invariant)g(state.)275 3637 y Fu(\(3\))50 b Fm(Supp)-5 b(ose)36 b Ft(L)f Fm(has)h(no)g (singular)g(sp)-5 b(e)g(ctrum)36 b(exc)-5 b(ept)36 b(for)g(a)f(nonde)-5 b(gener)g(ate)37 b(eigenvalue)e(at)440 3750 y(zer)-5 b(o,)49 b(and)e(that)g(the)f(c)-5 b(orr)g(esp)g(onding)49 b(eigenstate)c(is)h(sep)-5 b(ar)g(ating)48 b(for)e Fp(M)p Fm(.)81 b(Then)46 b(the)440 3863 y(system)33 b Fu(\()p Fp(M)p Ft(;)15 b(\034)10 b Fu(\))34 b Fm(has)g(the)f(pr)-5 b(op)g(erty)35 b(of)e(r)-5 b(eturn)33 b(to)h(e)-5 b(quilibrium.)380 4047 y Fu(Theorem)41 b(2.1)h(follo)m(ws)g(easily)h(from)d(\(2.3\).)75 b(Theorem)41 b(2.2)h(\(3\))g(w)m(as)g(pro)m(v)m(en)f(in)g([JP1])244 4160 y(although)31 b(similar)f(results)h(can)f(b)s(e)g(traced)h(to)g(m) m(uc)m(h)g(older)f(literature)i(\(see)f([BR1)q(,)f(Ja)q(]\).)380 4273 y(W)-8 b(e)31 b(no)m(w)e(describ)s(e)f(some)i(results)f (concerning)g(p)s(erturbation)f(theory)i(of)f Ft(W)3060 4240 y Fs(\003)3099 4273 y Fu(-dynamical)244 4386 y(systems.)41 b(Our)29 b(presen)m(tation)i(follo)m(ws)h([DJP].)380 4498 y(Let)e(\()p Fp(M)p Ft(;)15 b(\034)10 b Fu(\))31 b(b)s(e)e(a)g Ft(W)1124 4465 y Fs(\003)1163 4498 y Fu(-dynamical)h (system)g(and)f(\()p Ft(\031)s(;)15 b Fr(H)q Ft(;)g(J)o(;)g Fr(H)2520 4512 y Fl(+)2580 4498 y Fu(\))30 b(a)f(standard)g(represen)m (ta-)244 4611 y(tion)g(of)g Fp(M)p Fu(.)40 b(Let)29 b Ft(L)g Fu(b)s(e)f(the)g(Liouvillean)i(of)f Ft(\034)10 b Fu(.)40 b(Let)29 b Ft(V)49 b Fu(b)s(e)28 b(a)g(self-adjoin)m(t)i(op)s (erator)f(a\016liated)244 4724 y(to)i Fp(M)p Fu(.)41 b(Let)31 b(us)f(state)h(the)g(follo)m(wing)h(assumption:)244 4908 y Fv(Assumption)k(2.A)49 b Ft(L)20 b Fu(+)g Ft(\031)s Fu(\()p Ft(V)g Fu(\))33 b Fm(is)g(essential)5 b(ly)33 b(self-adjoint)h(on)f Fr(D)s Fu(\()p Ft(L)p Fu(\))20 b Fr(\\)g(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)h Fu(\)\))33 b Fm(and)1355 5088 y Ft(L)1417 5102 y Fo(V)1503 5088 y Fu(:=)25 b Ft(L)20 b Fu(+)g Ft(\031)s Fu(\()p Ft(V)h Fu(\))f Fr(\000)g Ft(J)9 b(\031)s Fu(\()p Ft(V)21 b Fu(\))p Ft(J)244 5268 y Fm(is)32 b(essential)5 b(ly)34 b(self-adjoint)f(on)h Fr(D)s Fu(\()p Ft(L)p Fu(\))20 b Fr(\\)g(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)g Fu(\)\))h Fr(\\)f(D)s Fu(\()p Ft(J)9 b(\031)s Fu(\()p Ft(V)21 b Fu(\))p Ft(J)9 b Fu(\))p Fm(.)p eop %%Page: 9 9 9 8 bop 3731 100 a Fn(9)244 407 y Fv(Theorem)70 b(2.3)49 b([DJP)q(])32 b Fm(Assume)h(that)h(2.A)e(holds)i(and)g(set)1027 634 y Ft(\034)1077 596 y Fo(t)1067 656 y(V)1128 634 y Fu(\()p Ft(A)p Fu(\))26 b(:=)f Ft(\031)1468 596 y Fs(\000)p Fl(1)1578 533 y Fk(\020)1632 634 y Fu(e)1672 596 y Fl(i)p Fo(t)p Fl(\()p Fo(L)p Fl(+)p Fo(\031)r Fl(\()p Fo(V)17 b Fl(\)\))2034 634 y Ft(\031)s Fu(\()p Ft(A)p Fu(\)e)2267 596 y Fs(\000)p Fl(i)p Fo(t)p Fl(\()p Fo(L)p Fl(+)p Fo(\031)r Fl(\()p Fo(V)h Fl(\))2658 533 y Fk(\021)2727 634 y Ft(:)244 861 y Fm(Then)33 b Ft(\034)521 875 y Fo(V)614 861 y Fm(is)g(a)g Ft(W)890 828 y Fs(\003)929 861 y Fm(-dynamics)g(on)g Fp(M)g Fm(and)h Ft(L)1860 875 y Fo(V)1953 861 y Fm(is)e(the)h(Liouvil)5 b(le)-5 b(an)34 b(of)f Fu(\()p Fp(M)p Ft(;)15 b(\034)2987 875 y Fo(V)3048 861 y Fu(\))p Fm(.)380 1073 y Fu(Our)25 b(\014nal)g(sub)5 b(ject)26 b(is)g(the)g(p)s(erturbation)f(theory)h(of) g(KMS)f(states.)41 b(W)-8 b(e)27 b(will)f(describ)s(e)f(the)244 1186 y(results)33 b(of)g([DJP],)i(whic)m(h)e(extend)g(the)g(w)m(ell)h (kno)m(wn)f(results)g(of)g(Araki)h([Ar,)f(BR2)q(])g(v)-5 b(alid)34 b(for)244 1299 y(b)s(ounded)28 b(p)s(erturbations.)380 1412 y(Let)j Ft(!)i Fu(b)s(e)d(a)g(\()p Ft(\034)5 b(;)15 b(\014)5 b Fu(\)-KMS)32 b(state)f(and)f(\012)25 b Fr(2)g(H)1963 1426 y Fl(+)2052 1412 y Fu(its)31 b(v)m(ector)h(represen)m(tativ)m(e.) 42 b(W)-8 b(e)32 b(will)e(call)244 1525 y(\012)g(a)h(\()p Ft(\034)5 b(;)15 b(\014)5 b Fu(\)-KMS)31 b(v)m(ector)h(\(or)f(a)g Ft(\014)5 b Fu(-KMS-v)m(ector)32 b(for)e(the)h(dynamics)f Ft(\034)10 b Fu(\).)380 1638 y(W)-8 b(e)32 b(mak)m(e)f(the)g(follo)m (wing)h(additional)f(assumption)f(on)g(the)h(p)s(erturbation)e Ft(V)20 b Fu(:)244 1825 y Fv(Assumption)36 b(2.B)179 b Fr(k)p Fu(e)1243 1792 y Fs(\000)p Fo(\014)s(\031)r Fl(\()p Fo(V)17 b Fl(\))p Fo(=)p Fl(2)1571 1825 y Fu(\012)p Fr(k)26 b Ft(<)e Fr(1)p Fm(.)244 2038 y Fv(Theorem)70 b(2.4)49 b([DJP)q(])32 b Fm(Assume)h(that)h(Assumptions)f(2.A)g(and)g (2.B)g(hold.)44 b(Then)244 2151 y Fu(\(1\))c(\012)c Fr(2)g(D)s Fu(\(e)746 2118 y Fs(\000)p Fo(\014)s Fl(\()p Fo(L)p Fl(+)p Fo(\031)r Fl(\()p Fo(V)16 b Fl(\)\))p Fo(=)p Fl(2)1231 2151 y Fu(\))39 b Fm(and)h(the)f(ve)-5 b(ctor)39 b Fu(\012)1985 2165 y Fo(V)2082 2151 y Fu(:=)d(e)2254 2118 y Fs(\000)p Fo(\014)s Fl(\()p Fo(L)p Fl(+)p Fo(\031)r Fl(\()p Fo(V)17 b Fl(\)\))p Fo(=)p Fl(2)2740 2151 y Fu(\012)38 b Fm(is)h(a)g Fu(\()p Ft(\034)3108 2165 y Fo(V)3169 2151 y Ft(;)15 b(\014)5 b Fu(\))p Fm(-KMS)244 2264 y(ve)-5 b(ctor.)244 2377 y Fu(\(2\))34 b Fm(L)-5 b(et)32 b Ft(!)606 2391 y Fo(V)667 2377 y Fu(\()p Ft(A)p Fu(\))26 b(=)f(\(\012)1028 2391 y Fo(V)1089 2377 y Fr(j)p Ft(\031)s Fu(\()p Ft(A)p Fu(\)\012)1373 2391 y Fo(V)1435 2377 y Fu(\))p Ft(=)p Fr(k)p Fu(\012)1626 2391 y Fo(V)1688 2377 y Fr(k)1733 2344 y Fl(2)1773 2377 y Fm(.)41 b(Then)33 b Ft(!)2136 2391 y Fo(V)2229 2377 y Fm(is)g(a)g Fu(\()p Ft(\034)2481 2391 y Fo(V)2542 2377 y Ft(;)15 b(\014)5 b Fm(\)-KMS)32 b(state)i(on)f Fp(M)p Fm(.)380 2589 y Fu(Note)e(that)g(if)e Ft(V)46 b Fr(2)25 b Fp(M)p Fu(,)30 b(then)f(Assumptions)g(2.A)i(and)e (2.B)i(are)f(automatically)i(satis\014ed,)244 2702 y(and)e(the)g(ab)s (o)m(v)m(e)i(theorems)e(reduce)h(to)g(the)f(w)m(ell-kno)m(wn)h(results) g(of)f(Araki)h([Ar,)g(BR2].)244 3022 y Fq(3)161 b(Simpli\014ed)52 b(presen)l(tation)f(of)j(the)e(main)j(results)244 3225 y Fu(This)41 b(section)i(giv)m(es)g(a)f(self-con)m(tained)i (description)d(of)h(simpli\014ed)f(v)m(ersions)h(of)g(our)g(main)244 3338 y(results.)63 b(It)38 b(will)g(not)g(b)s(e)f(used)g(in)h(the)g (remaining)g(part)g(of)g(the)g(pap)s(er.)62 b(The)37 b(reader)h(who)244 3450 y(prefers)30 b(a)h(more)g(complete)i(exp)s (osition)e(can)h(skip)e(this)h(section)h(and)f(go)g(directly)h(to)g (Section)244 3563 y(4.)380 3676 y(In)39 b(this)g(section)i(the)e (1-particle)j(b)s(osonic)d(space)h(is)f Fr(Z)48 b Fu(=)40 b Ft(L)2555 3643 y Fl(2)2594 3676 y Fu(\()p Fj(R)2695 3643 y Fo(d)2736 3676 y Fu(\))f(and)g(the)h(1-particle)244 3789 y(energy)c Ft(h)g Fu(is)g(the)g(op)s(erator)g(of)g(m)m (ultiplication)i(b)m(y)e Fr(j)p Ft(\030)t Fr(j)p Fu(,)i(where)d Ft(\030)j Fr(2)c Fj(R)2753 3756 y Fo(d)2829 3789 y Fu(describ)s(es)h (the)h(mo-)244 3902 y(men)m(tum.)j(The)25 b(small)h(system)g(is)g (describ)s(ed)e(b)m(y)i(a)g(\014nite)f(dimensional)h(Hilb)s(ert)g (space)g Fr(K)h Fu(and)244 4015 y(a)40 b(Hamiltonian)g Ft(K)7 b Fu(.)67 b(The)39 b(in)m(teraction)i(is)f(describ)s(ed)e(b)m(y) h(a)h(measurable)f(op)s(erator-v)-5 b(alued)244 4128 y(function)30 b(\(form-factor\))i Fj(R)1212 4095 y Fo(d)1277 4128 y Fr(3)25 b Ft(\030)k Fr(7!)c Ft(v)s Fu(\()p Ft(\030)t Fu(\))i Fr(2)e(B)s Fu(\()p Fr(K)q Fu(\).)244 4405 y Fi(3.1)135 b(P)l(auli-Fierz)46 b(system)f(at)h(zero)f(temp)t(erature)244 4576 y Fu(The)27 b(Hilb)s(ert)h(space)g(of)g(the)h(P)m(auli-Fierz)g (system)f(at)h(zero)g(temp)s(erature)e(is)h Fr(K)17 b(\012)e Fu(\000)3130 4590 y Fl(s)3162 4576 y Fu(\()p Ft(L)3259 4543 y Fl(2)3298 4576 y Fu(\()p Fj(R)3399 4543 y Fo(d)3440 4576 y Fu(\)\),)244 4689 y(where)35 b(\000)569 4703 y Fl(s)601 4689 y Fu(\()p Ft(L)698 4656 y Fl(2)737 4689 y Fu(\()p Fj(R)838 4656 y Fo(d)879 4689 y Fu(\)\))h(denotes)g(the)g (symmetric)g(\(b)s(osonic\))g(F)-8 b(o)s(c)m(k)37 b(space)f(o)m(v)m(er) h(the)f(1-particle)244 4802 y(space)31 b Ft(L)549 4769 y Fl(2)588 4802 y Fu(\()p Fj(R)689 4769 y Fo(d)730 4802 y Fu(\).)41 b(The)29 b(free)i(P)m(auli-Fierz)h(Hamiltonian)g(is)1156 5048 y Ft(H)1232 5063 y Fl(fr)1310 5048 y Fu(:=)25 b Ft(K)i Fr(\012)20 b Fu(1)g(+)g(1)h Fr(\012)1939 4924 y Fk(Z)2045 5048 y Fr(j)p Ft(\030)t Fr(j)p Ft(a)2187 5011 y Fs(\003)2227 5048 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)p Fu(\()p Ft(\030)t Fu(\)d)p Ft(\030)t(;)p eop %%Page: 10 10 10 9 bop 3682 100 a Fn(10)244 407 y Fu(where)36 b Ft(a)561 374 y Fs(\003)601 407 y Fu(\()p Ft(\030)t Fu(\))p Ft(=a)p Fu(\()p Ft(\030)t Fu(\))i(are)f(the)g(creation/annihilation)j(op)s (erators)d(of)g(b)s(osons)f(of)h(momen)m(tum)244 520 y Ft(\030)29 b Fr(2)c Fj(R)465 487 y Fo(d)505 520 y Fu(.)41 b(W)-8 b(e)31 b(assume)f(that)h(the)g(form-factor)g(satis\014es)1324 637 y Fk(Z)1415 761 y Fu(\(1)21 b(+)f Fr(j)p Ft(\030)t Fr(j)1701 723 y Fs(\000)p Fl(1)1796 761 y Fu(\))p Fr(k)p Ft(v)s Fu(\()p Ft(\030)t Fu(\))p Fr(k)2082 723 y Fl(2)2123 761 y Fu(d)p Ft(\030)29 b(<)c Fr(1)p Ft(:)894 b Fu(\(3.4\))244 1002 y(The)30 b(in)m(teraction)i(is)e(giv)m(en)i(b)m(y)e(the)h(op)s (erator)1108 1243 y Ft(V)46 b Fu(:=)1328 1119 y Fk(Z)1419 1243 y Fu(\()p Ft(v)s Fu(\()p Ft(\030)t Fu(\))22 b Fr(\012)e Ft(a)1776 1205 y Fs(\003)1815 1243 y Fu(\()p Ft(\030)t Fu(\))h(+)f Ft(v)2088 1205 y Fs(\003)2128 1243 y Fu(\()p Ft(\030)t Fu(\))g Fr(\012)g Ft(a)p Fu(\()p Ft(\030)t Fu(\)\)d)p Ft(\030)t(;)244 1484 y Fu(and)30 b(the)g(full)g(P)m (auli-Fierz)j(Hamiltonian)e(equals)1587 1688 y Ft(H)h Fu(:=)25 b Ft(H)1892 1703 y Fl(fr)1965 1688 y Fu(+)20 b Ft(\025V)5 b(;)244 1892 y Fu(where)30 b Ft(\025)25 b Fr(2)g Fj(R)p Fu(.)40 b Ft(H)d Fu(is)31 b(self-adjoin)m(t)h(on)e Fr(D)s Fu(\()p Ft(H)1785 1907 y Fl(fr)1838 1892 y Fu(\))g(and)g(b)s (ounded)e(from)i(b)s(elo)m(w.)380 2005 y(W)-8 b(e)25 b(ha)m(v)m(e)h(discussed)d(in)g([DJ1)q(])h(ho)m(w)g(P)m(auli-Fierz)i (Hamiltonians)f(arise)g(as)f(an)g(appro)m(xima-)244 2118 y(tion)j(to)h(the)e(standard)g(Hamiltonian)i(of)f(the)g (non-relativistic)i(QED.)e(A)g(related)g(discussions)244 2231 y(can)k(b)s(e)e(found)g(in)h([BFS1)q(].)380 2344 y(A)23 b(simplest)h(non-trivial)g(example)f(of)h(a)f(P)m(auli-Fierz)i (Hamiltonian)g(is)e(the)g(so-called)i(spin-)244 2457 y(b)s(oson)33 b(mo)s(del)h(where)g Fr(K)f Fu(=)e Fj(C)1319 2424 y Fl(2)1358 2457 y Fu(,)k Ft(K)k Fu(=)31 b Ft(\033)1688 2471 y Fo(z)1762 2457 y Fu(and)i Ft(v)s Fu(\()p Ft(\030)t Fu(\))g(=)e Ft(\033)2290 2471 y Fo(x)2334 2457 y Ft(\013)p Fu(\()p Ft(\030)t Fu(\))k(\()p Ft(\033)2628 2471 y Fo(z)2702 2457 y Fu(and)e Ft(\033)2934 2471 y Fo(x)3012 2457 y Fu(are)i(the)f(usual)244 2570 y(P)m(auli)d(matrices)g(and)f Ft(\013)c Fr(2)f Ft(L)1253 2537 y Fl(2)1292 2570 y Fu(\()p Fj(R)1393 2537 y Fo(d)1434 2570 y Fu(\))30 b(satis\014es)h(\(3.4\)\).) 244 2846 y Fi(3.2)135 b(Bosonic)45 b(\014elds)g(at)h(non-zero)e(densit) l(y)244 3018 y Fu(Assume)38 b(that)i(the)f(radiation)h(densit)m(y)g(of) f(the)g(b)s(osonic)g(\014eld)g(is)g(describ)s(ed)f(b)m(y)h(a)h(p)s (ositiv)m(e)244 3131 y(measurable)48 b(function)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\))i(on)f Fj(R)1530 3098 y Fo(d)1570 3131 y Fu(.)93 b(The)47 b(observ)-5 b(ables)48 b(of)g(the)g(b)s(osonic) g(reserv)m(oir)g(are)244 3244 y(then)39 b(describ)s(ed)f(b)m(y)h(the)h Ft(W)1271 3211 y Fs(\003)1310 3244 y Fu(-algebra)g Fp(M)1762 3211 y Fl(A)-8 b(W)1762 3272 y Fo(\032;)p Fl(l)1885 3244 y Fu(,)42 b(the)e(\(left\))g(Araki-W)-8 b(o)s(o)s(ds)41 b(algebra)f(at)g(den-)244 3357 y(sit)m(y)k Ft(\032)p Fu(.)78 b(This)42 b(algebra)i(is)f(constructed)h(as)f(follo)m(ws.)80 b(It)43 b(is)g(represen)m(ted)g(on)f(the)i(Hilb)s(ert)244 3470 y(space)34 b(\000)547 3484 y Fl(s)579 3470 y Fu(\()p Ft(L)676 3437 y Fl(2)716 3470 y Fu(\()p Fj(R)817 3437 y Fo(d)857 3470 y Fu(\))23 b Fr(\010)f Ft(L)1070 3437 y Fl(2)1110 3470 y Fu(\()p Fj(R)1211 3437 y Fo(d)1251 3470 y Fu(\)\).)52 b(The)33 b(creation/annihilation)k(op)s(erators)d (corresp)s(onding)f(to)244 3583 y(the)e(\014rst)f Ft(L)651 3550 y Fl(2)690 3583 y Fu(\()p Fj(R)791 3550 y Fo(d)831 3583 y Fu(\))h(\(whic)m(h)g(describ)s(e)f(excitations\))k(are)d (denoted)f(b)m(y)h Ft(a)2709 3550 y Fs(\003)2709 3610 y Fl(l)2748 3583 y Fu(\()p Ft(\030)t Fu(\))p Ft(=a)2955 3598 y Fl(l)2980 3583 y Fu(\()p Ft(\030)t Fu(\),)h(and)e(those)244 3696 y(corresp)s(onding)h(to)i(the)g(second)g Ft(L)1460 3663 y Fl(2)1499 3696 y Fu(\()p Fj(R)1600 3663 y Fo(d)1640 3696 y Fu(\))g(\(describing)g(holes\))g(are)g(denoted)f(b)m(y)g Ft(a)3116 3663 y Fs(\003)3116 3718 y Fl(r)3156 3696 y Fu(\()p Ft(\030)t Fu(\))p Ft(=a)3363 3710 y Fl(r)3396 3696 y Fu(\()p Ft(\030)t Fu(\).)244 3808 y(\(l)p Ft(=)p Fu(r)f(stand)f(for)g(left/righ)m(t\).)43 b Fp(M)1363 3775 y Fl(A)-8 b(W)1363 3836 y Fo(\032;)p Fl(l)1516 3808 y Fu(is)31 b(generated)g(b)m(y)f(the)h(op)s(erators)f(of)h(the)f(form) 752 4053 y(exp)15 b(i)946 3952 y Fk(\020)1000 3980 y(R)1061 4053 y Fu(\()p Ft(f)10 b Fu(\()p Ft(\030)t Fu(\)\(1)21 b(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))1663 3990 y Fh(1)p 1664 4002 31 3 v 1664 4043 a(2)1709 4053 y Ft(a)1757 4020 y Fs(\003)1757 4081 y Fl(l)1796 4053 y Fu(\()p Ft(\030)t Fu(\))h(+)p 2022 3979 55 4 v 20 w Ft(f)9 b Fu(\()p Ft(\030)t Fu(\))p Ft(\032)p Fu(\()p Ft(\030)t Fu(\))2361 3990 y Fh(1)p 2363 4002 31 3 v 2363 4043 a(2)2407 4053 y Ft(a)2455 4067 y Fl(r)2487 4053 y Fu(\()p Ft(\030)t Fu(\))21 b(+)f(hc)o(\)d)p Ft(\030)2933 3952 y Fk(\021)3003 4053 y Ft(;)244 4291 y Fu(where)30 b Ft(f)k Fr(2)25 b Ft(L)734 4258 y Fl(2)773 4291 y Fu(\()p Fj(R)874 4258 y Fo(d)915 4291 y Fu(\))31 b(satis\014es)1315 4218 y Fk(R)1391 4291 y Fr(j)p Ft(f)10 b Fu(\()p Ft(\030)t Fu(\))p Fr(j)1610 4258 y Fl(2)1650 4291 y Ft(\032)p Fu(\()p Ft(\030)t Fu(\)d)p Ft(\030)29 b(<)c Fr(1)p Fu(.)244 4568 y Fi(3.3)135 b(P)l(auli-Fierz)46 b(systems)g(at)f(non-zero)g(densit)l(y)244 4739 y Fu(The)34 b(P)m(auli-Fierz)k(algebra)e(at)g(densit)m(y)f Ft(\032)p Fu(,)h Fp(M)1868 4753 y Fo(\032)1909 4739 y Fu(,)g(is)g(de\014ned)d(b)m (y)i Fp(M)2610 4753 y Fo(\032)2684 4739 y Fu(:=)e Fr(B)s Fu(\()p Fr(K)q Fu(\))24 b Fr(\012)f Fp(M)3229 4706 y Fl(A)-8 b(W)3229 4767 y Fo(\032;)p Fl(l)3352 4739 y Fu(.)55 b(T)-8 b(o)244 4852 y(de\014ne)30 b(the)g(dynamics,)g(w)m(e)h(need)f (the)h(follo)m(wing)h(assumption:)244 5065 y Fv(Assumption)k(3.A)1163 4992 y Fk(R)1223 5065 y Fu(\(1)21 b(+)f Fr(j)p Ft(\030)t Fr(j)1509 5032 y Fl(2)1549 5065 y Fu(\)\(1)i(+)d Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))p Fr(k)p Ft(v)s Fu(\()p Ft(\030)t Fu(\))p Fr(k)2223 5032 y Fl(2)2266 5065 y Fu(d)p Ft(\030)29 b(<)c Fr(1)p Fm(.)p eop %%Page: 11 11 11 10 bop 3682 100 a Fn(11)244 407 y Fu(Set)771 539 y Ft(L)833 501 y Fl(semi)833 562 y(fr)1000 539 y Fu(:=)25 b Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)1629 415 y Fk(Z)1720 465 y(\000)1762 539 y Fr(j)p Ft(\030)t Fr(j)p Ft(a)1904 501 y Fs(\003)1904 562 y Fl(l)1944 539 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)2106 554 y Fl(l)2130 539 y Fu(\()p Ft(\030)t Fu(\))h Fr(\000)f(j)p Ft(\030)t Fr(j)p Ft(a)2498 501 y Fs(\003)2498 561 y Fl(r)2538 539 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)2700 553 y Fl(r)2732 539 y Fu(\()p Ft(\030)t Fu(\))2846 465 y Fk(\001)2889 539 y Fu(d)p Ft(\030)t(;)794 793 y(V)847 807 y Fo(\032)913 793 y Fu(:=)1034 670 y Fk(Z)1140 793 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))h Fr(\012)1413 720 y Fk(\000)1455 793 y Fu(\(1)g(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))1853 729 y Fh(1)p 1854 741 31 3 v 1854 782 a(2)1898 793 y Ft(a)1946 756 y Fs(\003)1946 816 y Fl(l)1986 793 y Fu(\()p Ft(\030)t Fu(\))h(+)e Ft(\032)p Fu(\()p Ft(\030)t Fu(\))2382 729 y Fh(1)p 2383 741 V 2383 782 a(2)2428 793 y Ft(a)2476 807 y Fl(r)2507 793 y Fu(\()p Ft(\030)t Fu(\))2621 720 y Fk(\001)2664 793 y Fu(d)p Ft(\030)k Fu(+)d(hc)p Ft(;)1472 1006 y(L)1534 969 y Fl(semi)1534 1029 y Fo(\032)1700 1006 y Fu(:=)26 b Ft(L)1884 969 y Fl(semi)1884 1029 y(fr)2045 1006 y Fu(+)20 b Ft(\025V)2242 1020 y Fo(\032)2283 1006 y Ft(:)1041 b Fu(\(3.5\))244 1204 y Fv(Prop)s(osition)36 b(3.1)49 b Fm(Assume)35 b(that)i(Assumption)f(3.A)f(holds.)51 b(Then)36 b(the)g(op)-5 b(er)g(ator)38 b Ft(L)3294 1171 y Fl(semi)3294 1227 y Fo(\032)3471 1204 y Fm(is)244 1317 y(essential)5 b(ly)33 b(self-adjoint)h(on)f Fr(D)s Fu(\()p Ft(L)1457 1332 y Fl(fr)1510 1317 y Fu(\))21 b Fr(\\)e(D)s Fu(\()p Ft(V)1807 1331 y Fo(\032)1848 1317 y Fu(\))33 b Fm(and)1385 1518 y Ft(\034)1435 1481 y Fo(t)1425 1541 y(\032)1465 1518 y Fu(\()p Ft(A)p Fu(\))26 b(:=)f(e)1790 1481 y Fl(i)p Fo(tL)1883 1457 y Fh(semi)1883 1497 y Fg(\032)2011 1518 y Ft(A)p Fu(e)2119 1481 y Fs(\000)p Fl(i)p Fo(tL)2267 1457 y Fh(semi)2267 1497 y Fg(\032)3349 1518 y Fu(\(3.6\))244 1703 y Fm(is)32 b(a)h Ft(W)519 1670 y Fs(\003)558 1703 y Fm(-dynamics)h(on)f Fp(M)1218 1717 y Fo(\032)1259 1703 y Fm(.)380 1893 y Fu(W)-8 b(e)25 b(will)e(call)h(the)g Ft(W)1102 1860 y Fs(\003)1141 1893 y Fu(-dynamical)g(system)f(\()p Fp(M)2025 1907 y Fo(\032)2066 1893 y Ft(;)15 b(\034)2146 1907 y Fo(\032)2186 1893 y Fu(\))24 b(the)f(P)m(auli-Fierz)i(system)e (at)h(densit)m(y)244 2006 y Ft(\032)p Fu(.)41 b(In)29 b(the)i(absence)g(of)f(in)m(teraction)i(\()p Ft(\025)26 b Fu(=)f(0\))31 b(w)m(e)g(call)h(it)f(a)f(free)h(P)m(auli-Fierz)h (system.)380 2119 y(The)26 b(iden)m(tit)m(y)j(represen)m(tation)e Fp(M)1581 2133 y Fo(\032)1647 2119 y Fr(!)f(B)s Fu(\()p Fr(K)14 b(\012)f Fu(\000)2086 2133 y Fl(s)2117 2119 y Fu(\()p Ft(L)2214 2086 y Fl(2)2254 2119 y Fu(\()p Fj(R)2355 2086 y Fo(d)2395 2119 y Fu(\))g Fr(\010)g Ft(L)2589 2086 y Fl(2)2629 2119 y Fu(\()p Fj(R)2730 2086 y Fo(d)2770 2119 y Fu(\)\))27 b(will)g(b)s(e)f(called)i(the)244 2232 y(semistandard)38 b(represen)m(tation)h(of)g(the)f(P)m(auli-Fierz)j (system,)g(to)e(distinguish)e(it)i(from)f(the)244 2344 y(standard)26 b(represen)m(tation)j(describ)s(ed)d(in)h(the)h(next)f (subsection.)40 b(Similarly)-8 b(,)29 b(w)m(e)f(will)g(call)g(the)244 2457 y(op)s(erator)j Ft(L)673 2424 y Fl(semi)673 2480 y Fo(\032)844 2457 y Fu(the)g(P)m(auli-Fierz)h(semi-Liouvillean)g(at)g (densit)m(y)e Ft(\032)p Fu(.)244 2731 y Fi(3.4)135 b(P)l(auli-Fierz)46 b(systems)g(in)e(standard)h(represen)l(tation)244 2903 y Fu(Let)p 408 2830 71 4 v 32 w Fr(K)34 b Fu(b)s(e)d(a)h(Hilb)s(ert)g (space)h(complex)f(conjugate)h(to)g Fr(K)q Fu(.)46 b(The)31 b(standard)g(represen)m(tation)i(of)244 3015 y(the)d(algebra)i Fp(M)814 3029 y Fo(\032)885 3015 y Fu(is)e(realized)i(on)e(the)h(Hilb)s (ert)f(space)1170 3200 y Fr(H)c Fu(:=)f Fr(K)d(\012)p 1576 3127 V 20 w(K)g(\012)d Fu(\000)1814 3214 y Fl(s)1846 3200 y Fu(\()p Ft(L)1943 3163 y Fl(2)1983 3200 y Fu(\()p Fj(R)2084 3163 y Fo(d)2124 3200 y Fu(\))i Fr(\010)f Ft(L)2333 3163 y Fl(2)2372 3200 y Fu(\()p Fj(R)2473 3163 y Fo(d)2514 3200 y Fu(\)\))p Ft(;)244 3385 y Fu(and)30 b(for)g Ft(B)25 b Fr(\012)19 b Ft(C)32 b Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))c Fr(\012)f Fp(M)1337 3352 y Fl(A)-8 b(W)1337 3413 y Fo(\032;)p Fl(l)1460 3385 y Fu(,)1393 3570 y Ft(\031)s Fu(\()p Ft(B)25 b Fr(\012)20 b Ft(C)7 b Fu(\))25 b(=)g Ft(B)f Fr(\012)c Fu(1)p 2125 3544 59 3 v 29 x Fs(K)2204 3570 y Fr(\012)g Ft(C)q(:)244 3754 y Fu(F)-8 b(or)36 b(the)g(description)f(of)h(the)f(cone)i Fr(H)1604 3768 y Fo(\032;)p Fl(+)1753 3754 y Fu(and)e(the)h(mo)s(dular)e(conjugation)j Ft(J)44 b Fu(w)m(e)36 b(refer)f(the)244 3867 y(reader)30 b(to)h(Subsection)f(6.6.)380 3980 y(Note)i(that)f Ft(V)850 3994 y Fo(\032)921 3980 y Fu(is)f(a\016liated)i(to)f Fp(M)1592 3994 y Fo(\032)1663 3980 y Fu(and)666 4210 y Ft(\031)s Fu(\()p Ft(V)809 4224 y Fo(\032)849 4210 y Fu(\))26 b(=)1006 4087 y Fk(Z)1112 4210 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))21 b Fr(\012)f Fu(1)h Fr(\012)1542 4137 y Fk(\000)1583 4210 y Fu(\(1)g(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))1981 4146 y Fh(1)p 1982 4158 31 3 v 1982 4199 a(2)2027 4210 y Ft(a)2075 4173 y Fs(\003)2075 4234 y Fl(l)2114 4210 y Fu(\()p Ft(\030)t Fu(\))h(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\))2511 4146 y Fh(1)p 2512 4158 V 2512 4199 a(2)2556 4210 y Ft(a)2604 4224 y Fl(r)2636 4210 y Fu(\()p Ft(\030)t Fu(\))2750 4137 y Fk(\001)2792 4210 y Fu(d)p Ft(\030)k Fu(+)c(hc)p Ft(;)610 4492 y(J)9 b(\031)s Fu(\()p Ft(V)812 4506 y Fo(\032)853 4492 y Fu(\))p Ft(J)35 b Fu(=)1069 4369 y Fk(Z)1175 4492 y Fu(1)21 b Fr(\012)p 1332 4442 48 4 v 20 w Ft(v)s Fu(\()p Ft(\030)t Fu(\))g Fr(\012)1605 4419 y Fk(\000)1646 4492 y Fu(\(1)g(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))2044 4428 y Fh(1)p 2045 4440 31 3 v 2045 4481 a(2)2090 4492 y Ft(a)2138 4455 y Fs(\003)2138 4515 y Fl(r)2177 4492 y Fu(\()p Ft(\030)t Fu(\))h(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\))2574 4428 y Fh(1)p 2575 4440 V 2575 4481 a(2)2619 4492 y Ft(a)2667 4507 y Fl(l)2691 4492 y Fu(\()p Ft(\030)t Fu(\))2805 4419 y Fk(\001)2847 4492 y Fu(d)p Ft(\030)k Fu(+)c(hc)p Ft(:)380 4686 y Fu(The)30 b(Liouvillean)i(of)e(the)h(free)f (P)m(auli-Fierz)j(system)d(is)418 4908 y Ft(L)480 4923 y Fl(fr)558 4908 y Fu(=)25 b Ft(K)i Fr(\012)20 b Fu(1)g Fr(\012)g Fu(1)h Fr(\000)f Fu(1)g Fr(\012)p 1318 4835 84 4 v 20 w Ft(K)27 b Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(1)h Fr(\012)1983 4784 y Fk(Z)2074 4834 y(\000)2115 4908 y Fr(j)p Ft(\030)t Fr(j)p Ft(a)2257 4870 y Fs(\003)2257 4931 y Fl(l)2297 4908 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)2459 4923 y Fl(l)2484 4908 y Fu(\()p Ft(\030)t Fu(\))g Fr(\000)f(j)p Ft(\030)t Fr(j)p Ft(a)2852 4870 y Fs(\003)2852 4930 y Fl(r)2892 4908 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)3054 4922 y Fl(r)3086 4908 y Fu(\()p Ft(\030)t Fu(\))3200 4834 y Fk(\001)3242 4908 y Fu(d)p Ft(\030)t(:)244 5128 y Fu(Set)1258 5241 y Ft(L)1320 5255 y Fo(\032)1385 5241 y Fu(:=)26 b Ft(L)1569 5256 y Fl(fr)1642 5241 y Fu(+)20 b Ft(\025\031)s Fu(\()p Ft(V)1929 5255 y Fo(\032)1970 5241 y Fu(\))g Fr(\000)g Ft(\025J)9 b(\031)s Fu(\()p Ft(V)2371 5255 y Fo(\032)2412 5241 y Fu(\))p Ft(J)o(:)p eop %%Page: 12 12 12 11 bop 3682 100 a Fn(12)244 407 y Fv(Prop)s(osition)36 b(3.2)49 b Fm(Assume)38 b(that)h(Assumption)g(\(3.A\))e(holds.)59 b(Then)38 b(the)g(op)-5 b(er)g(ator)41 b Ft(L)3393 421 y Fo(\032)3471 407 y Fm(is)244 520 y(essential)5 b(ly)30 b(self-adjoint)g(on)g Fr(D)s Fu(\()p Ft(L)1447 535 y Fl(fr)1500 520 y Fu(\))13 b Fr(\\)g(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)1873 534 y Fo(\032)1914 520 y Fu(\)\))g Fr(\\)g(D)s Fu(\()p Ft(J)c(\031)s Fu(\()p Ft(V)2381 534 y Fo(\032)2422 520 y Fu(\))p Ft(J)g Fu(\))30 b Fm(and)g(is)g(the)f (Liouvil)5 b(le)-5 b(an)30 b(of)244 633 y(the)j(Pauli-Fierz)g(system)g Fu(\()p Fp(M)1300 647 y Fo(\032)1341 633 y Ft(;)15 b(\034)1421 647 y Fo(\032)1461 633 y Fu(\))p Fm(.)380 826 y Fu(Let)28 b(us)e(note)i(that)f(from)g(the)g(mathematical)j(p)s(oin)m(t)d(of)g (view)g(P)m(auli-Fierz)i(Hamiltonians,)244 938 y(semi-Liouvilleans)22 b(and)f(Liouvilleans)h(b)s(elong)f(to)h(the)f(class)h(of)f(op)s (erators)g(that)h(w)m(e)f(call)i(P)m(auli-)244 1051 y(Fierz)34 b(op)s(erators.)49 b(This)33 b(class)h(of)f(op)s(erators)g(has)g(b)s (een)g(studied)f(in)h(detail)i(in)d(our)h(previous)244 1164 y(pap)s(er)c([DJ1)q(].)244 1438 y Fi(3.5)135 b(Thermal)45 b(P)l(auli-Fierz)h(systems)244 1610 y Fu(Let)24 b Ft(\014)31 b(>)25 b Fu(0)f(b)s(e)f(the)h(in)m(v)m(erse)h(temp)s(erature.)38 b(A)24 b(P)m(auli-Fierz)i(system)e(whose)g(radiation)g(densit)m(y)244 1723 y(is)e(giv)m(en)i(b)m(y)e(the)h(Planc)m(k)g(la)m(w)g Ft(\032)1318 1738 y Fo(\014)1365 1723 y Fu(\()p Ft(\030)t Fu(\))j(=)f(\(e)1676 1690 y Fo(\014)s Fs(j)p Fo(\030)s Fs(j)1801 1723 y Fr(\000)t Fu(1\))1956 1690 y Fs(\000)p Fl(1)2073 1723 y Fu(is)e(called)h(a)e(thermal)h(P)m(auli-Fierz)h (system)244 1836 y(at)35 b(in)m(v)m(erse)f(temp)s(erature)g Ft(\014)5 b Fu(.)52 b(Due)34 b(to)g(the)h(sp)s(eci\014c)e(form)h(of)g (the)g(Planc)m(k)h(la)m(w)f(Assumption)244 1949 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Ft(k)g Fr(2)c Fj(R)j Fu(will)244 3650 y(b)s(e)h(denoted)g(b)m(y)h(1)898 3665 y Fo(k)941 3650 y Fu(\()p Ft(K)7 b Fu(\))35 b(and)g Ft(v)1359 3617 y Fo(k)1396 3626 y Fh(1)1431 3617 y Fo(;k)1488 3626 y Fh(2)1526 3650 y Fu(\()p Ft(\030)t Fu(\))f(=)g(1)1824 3665 y Fo(k)1861 3674 y Fh(1)1900 3650 y Fu(\()p Ft(K)7 b Fu(\))p Ft(v)s Fu(\()p Ft(\030)t Fu(\)1)2260 3665 y Fo(k)2297 3674 y Fh(2)2337 3650 y Fu(\()p Ft(K)g Fu(\).)56 b(Ob)m(viously)-8 b(,)38 b Ft(v)3071 3617 y Fo(k)3108 3626 y Fh(1)3142 3617 y Fo(;k)3199 3626 y Fh(2)3237 3650 y Fu(\()p Ft(\030)t Fu(\))d(=)e(0)244 3763 y(unless)j Ft(k)566 3777 y Fl(1)606 3763 y Ft(;)15 b(k)693 3777 y Fl(2)770 3763 y Fr(2)36 b Fu(sp)o(\()p Ft(K)7 b Fu(\).)62 b Ft(p)36 b Fr(2)h Fj(R)1440 3777 y Fl(+)1535 3763 y Fu(denotes)h(the)f(radial)h(co)s(ordinate.)62 b Ft(S)2863 3730 y Fo(d)p Fs(\000)p Fl(1)3031 3763 y Fu(is)37 b(the)g Ft(d)25 b Fr(\000)g Fu(1-)244 3876 y(dimensional)38 b(unit)g(sphere,)i Ft(!)h Fr(2)d Ft(S)1529 3843 y Fo(d)p Fs(\000)p Fl(1)1698 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Fo(k)r(;k)1599 5204 y Fu(\()p Ft(p!)s Fu(\))25 b(=)1984 5136 y Fk(P)1896 5292 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))2183 5204 y Fu(lim)2193 5266 y Fo(p)p Fs(#)p Fl(0)2324 5204 y Ft(p)2370 5171 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2535 5204 y Ft(v)2582 5171 y Fo(k)r(;k)2683 5204 y Fu(\()p Ft(p!)s Fu(\))p Ft(B)g(:)3349 5011 y Fu(\(3.7\))p eop %%Page: 13 13 13 12 bop 3682 100 a Fn(13)244 407 y Fu(Ob)m(viously)-8 b(,)31 b(1)26 b Fr(2)e Fp(N)p Fu(.)41 b(Note)31 b(also)h(that)f Fp(N)f Fu(is)g(a)h Fr(\003)p Fu(-algebra)h(in)m(v)-5 b(arian)m(t)32 b(wrt)d(e)2841 374 y Fl(i)p Fo(tK)2976 407 y Fr(\001)20 b Fu(e)3061 374 y Fs(\000)p Fl(i)p Fo(tK)3230 407 y Fu(.)380 520 y(W)-8 b(e)32 b(start)f(with)f(a)h(result)f(whic)m (h)g(do)s(es)g(not)h(hold)f(uniformly)f(in)h(the)h(temp)s(erature.)244 724 y Fv(Theorem)70 b(3.4)49 b Fm(Supp)-5 b(ose)44 b(that)g(Assumption) g(3.B)f(holds)h(and)g(the)f(fol)5 b(lowing)44 b(c)-5 b(onditions)244 837 y(ar)g(e)33 b(satis\014e)-5 b(d:)275 966 y Fu(\(1\))1225 997 y Fk(Z)1331 1121 y Fr(k)p Ft(@)1429 1083 y Fl(3)1424 1143 y Fo(p)1469 1121 y Ft(p)1515 1083 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)1771 1121 y Fr(h)p Ft(p)p Fr(i)1887 1083 y Fl(1)p Fo(=)p Fl(2)1998 1121 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Fr(k)2266 1083 y Fl(2)2306 1121 y Fu(d)p Ft(p)p Fu(d)p Ft(!)27 b(<)e Fr(1)p Ft(;)530 1362 y(@)583 1325 y Fo(j)578 1385 y(p)620 1362 y Ft(p)666 1325 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)922 1362 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))1145 1258 y Fk(\014)1145 1312 y(\014)1145 1367 y(\014)1176 1426 y Fo(p)p Fl(=0)1331 1362 y Fu(=)g(\()p Fr(\000)p Fu(1\))1613 1325 y Fo(j)1650 1362 y Ft(@)1703 1325 y Fo(j)1698 1385 y(p)1740 1362 y Ft(p)1786 1325 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)2042 1362 y Ft(v)2089 1325 y Fs(\003)2129 1362 y Fu(\()p Ft(p!)s Fu(\))2305 1258 y Fk(\014)2305 1312 y(\014)2305 1367 y(\014)2335 1426 y Fo(p)p Fl(=0)2465 1362 y Ft(;)81 b(j)31 b Fu(=)24 b(0)p Ft(;)15 b Fu(1)p Ft(;)g Fu(2)p Ft(;)83 b(!)28 b Fr(2)d Ft(S)3289 1325 y Fo(d)p Fs(\000)p Fl(1)3420 1362 y Ft(:)275 1585 y Fu(\(2\))50 b Fp(N)24 b Fu(=)h Fj(C)p Fu(1)p Fm(.)244 1714 y(Then)35 b(for)h(any)g Fu(0)30 b Ft(<)g(\014)35 b(<)29 b Fr(1)35 b Fm(ther)-5 b(e)37 b(exists)e Ft(\025)1830 1728 y Fl(0)1869 1714 y Fu(\()p Ft(\014)5 b Fu(\))31 b Ft(>)f Fu(0)35 b Fm(such)h(that)g(for)g Fu(0)30 b Ft(<)g Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)3216 1728 y Fl(0)3255 1714 y Fu(\()p Ft(\014)5 b Fu(\))36 b Fm(the)244 1827 y(Pauli-Fierz)c(Liouvil)5 b(le)-5 b(an)33 b Ft(L)1251 1842 y Fo(\014)1330 1827 y Fm(has)g(no)g(singular)g(sp)-5 b(e)g(ctrum)34 b(exc)-5 b(ept)33 b(for)f(a)h(simple)g(eigenvalue)244 1940 y(at)47 b(zer)-5 b(o.)84 b(Conse)-5 b(quently,)50 b(under)d(the)g(ab)-5 b(ove)47 b(c)-5 b(onditions)48 b(the)f(system)g Fu(\()p Fp(M)2984 1955 y Fo(\014)3032 1940 y Ft(;)15 b(\034)3112 1955 y Fo(\014)3159 1940 y Fu(\))47 b Fm(has)g(the)244 2052 y(pr)-5 b(op)g(erty)35 b(of)e(r)-5 b(eturn)34 b(to)f(e)-5 b(quilibrium.)380 2257 y Fu(Condition)25 b(\(1\))h(is)f(our)g(regularit)m(y)h (assumption.)39 b(Note)26 b(that)g(it)f(allo)m(ws)h(for)f(quite)h (singular)244 2370 y(infrared)43 b(b)s(eha)m(vior)h(of)g(the)g (form-factor.)82 b(F)-8 b(or)44 b(example,)k(assume)c(that)g Ft(v)s Fu(\()p Ft(\030)t Fu(\))h(is)f(smo)s(oth)244 2483 y(outside)31 b(of)f(zero)h(and)f(of)h(compact)g(supp)s(ort.)39 b(Then)30 b(\(1\))h(holds)f(if)g(around)g(zero)1546 2681 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))c(=)f Ft(v)1873 2695 y Fl(0)1913 2681 y Fr(j)p Ft(\030)t Fr(j)2007 2643 y Fl(1)p Fs(\000)p Fo(d=)p Fl(2)2208 2681 y Ft(;)1116 b Fu(\(3.8\))244 2878 y(where)21 b Ft(v)542 2892 y Fl(0)606 2878 y Fr(2)k(B)s Fu(\()p Fr(K)q Fu(\))d(is)g(self-adjoin)m(t.)39 b(In)20 b(particular,)k(Theorem)d(3.4)h(applies)g(to)g(mo)s(dels)f (deriv)m(ed)244 2991 y(from)30 b(QED)g(in)g(the)h(so-called)h(ohmic)f (case)g(\(see)g(Subsection)f(8.1\).)380 3104 y(Condition)36 b(\(2\))g(is)g(our)f(p)s(ositiv)m(e)h(temp)s(erature)g(e\013ectiv)m(e)i (coupling)e(assumption.)55 b(Note)244 3217 y(that)31 b(it)g(do)s(es)f(not)g(dep)s(end)f(on)h(the)h(temp)s(erature.)380 3330 y(The)f(next)h(theorem)f(holds)g(uniformly)g(in)g(the)g(temp)s (erature.)244 3534 y Fv(Theorem)70 b(3.5)49 b Fm(Supp)-5 b(ose)44 b(that)g(Assumption)g(3.B)f(holds)h(and)g(the)f(fol)5 b(lowing)44 b(c)-5 b(onditions)244 3647 y(ar)g(e)33 b(satis\014e)-5 b(d:)275 3776 y Fu(\(1\))530 3901 y Fk(R)591 3974 y Fr(h)p Ft(p)p Fr(ik)p Ft(@)805 3941 y Fl(3)800 3997 y Fo(p)846 3974 y Ft(p)892 3941 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)1147 3974 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Fr(k)1415 3941 y Fl(2)1456 3974 y Fu(d)p Ft(p)p Fu(d)p Ft(!)27 b(<)e Fr(1)p Fu(;)530 4094 y Fk(R)606 4167 y Ft(p)652 4134 y Fs(\000)p Fl(5+2)p Fo(j)869 4167 y Fr(k)p Ft(@)967 4123 y Fo(j)962 4179 y(p)1004 4167 y Ft(p)1050 4134 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)1306 4167 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Fr(k)1574 4134 y Fl(2)1614 4167 y Fu(d)p Ft(p)p Fu(d)p Ft(!)i(<)e Fr(1)p Ft(;)145 b(j)31 b Fu(=)25 b(0)p Ft(;)15 b Fu(1)p Ft(;)g Fu(2;)530 4376 y Ft(@)583 4332 y Fo(j)578 4388 y(p)620 4376 y Ft(p)666 4343 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)922 4376 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))1145 4272 y Fk(\014)1145 4326 y(\014)1145 4381 y(\014)1176 4440 y Fo(p)p Fl(=0)1331 4376 y Fu(=)25 b(\()p Fr(\000)p Fu(1\))1613 4343 y Fo(j)1650 4376 y Ft(@)1703 4332 y Fo(j)1698 4388 y(p)1740 4376 y Ft(p)1786 4343 y Fs(\000)p Fl(1+)p Fo(d=)p Fl(2)2042 4376 y Ft(v)2089 4343 y Fs(\003)2129 4376 y Fu(\()p Ft(p!)s Fu(\))2305 4272 y Fk(\014)2305 4326 y(\014)2305 4381 y(\014)2335 4440 y Fo(p)p Fl(=0)2465 4376 y Ft(;)81 b(j)31 b Fu(=)24 b(0)p Ft(;)15 b Fu(1)p Ft(;)g Fu(2)p Ft(;)83 b(!)28 b Fr(2)d Ft(S)3289 4343 y Fo(d)p Fs(\000)p Fl(1)3420 4376 y Ft(:)275 4633 y Fu(\(2\))50 b Fp(N)24 b Fu(=)h Fj(C)p Fu(1)p Fm(.)275 4766 y Fu(\(3\))50 b(dim)14 b(1)651 4781 y Fo(k)688 4790 y Fh(0)727 4766 y Fu(\()p Ft(K)7 b Fu(\))26 b(=)f(1)33 b Fm(\(the)g(op)-5 b(er)g(ator)36 b Ft(K)j Fm(has)34 b(a)f(nonde)-5 b(gener)g(ate)34 b(smal)5 b(lest)34 b(eigenvalue\).)275 4898 y Fu(\(4\))50 b Fm(Ther)-5 b(e)33 b(exists)g Ft(c)25 b(>)g Fu(0)33 b Fm(such)g(that)h(for)f(al)5 b(l)33 b Ft(k)c Fr(2)c Fu(sp)o(\()p Ft(K)7 b Fu(\))p Fm(,)33 b Ft(k)28 b Fr(6)p Fu(=)d Ft(k)2537 4912 y Fl(0)2577 4898 y Fm(,)1067 5051 y Fk(X)1070 5246 y Fo(p>)p Fl(0)1214 5014 y Fk(Z)1264 5220 y Fo(S)1311 5201 y Fg(d)p Ff(\000)p Fh(1)1430 5138 y Fu(\()p Ft(v)1512 5100 y Fs(\003)1552 5138 y Fu(\))1587 5100 y Fo(k)r(;k)r Fs(\000)p Fo(p)1778 5138 y Fu(\()p Ft(p!)s Fu(\))p Ft(v)2001 5100 y Fo(k)r Fs(\000)p Fo(p;k)2193 5138 y Fu(\()p Ft(p!)s Fu(\)d)p Ft(!)j Fr(\025)d Ft(c)p Fu(1)2685 5153 y Fo(k)2728 5138 y Fu(\()p Ft(K)7 b Fu(\))p Ft(:)p eop %%Page: 14 14 14 13 bop 3682 100 a Fn(14)244 407 y Fm(Then)42 b(for)g(any)h Fu(0)f Ft(<)g(\014)1079 421 y Fl(0)1161 407 y Ft(<)f Fr(1)h Fm(ther)-5 b(e)43 b(exists)f Ft(\025)1953 421 y Fl(0)2034 407 y Ft(>)g Fu(0)g Fm(such)g(that)h(for)f Fu(0)h Ft(<)e Fr(j)p Ft(\025)p Fr(j)i Ft(<)e(\025)3310 421 y Fl(0)3392 407 y Fm(and)244 520 y Ft(\014)k Fr(2)15 b Fu(])p Ft(\014)492 534 y Fl(0)532 520 y Ft(;)g Fr(1)p Fu([)41 b Fm(the)h(Pauli-Fierz)e(Liouvil)5 b(le)-5 b(an)42 b Ft(L)1913 535 y Fo(\014)2000 520 y Fm(has)g(no)g(singular)f(sp)-5 b(e)g(ctrum)42 b(exc)-5 b(ept)42 b(for)f(a)244 633 y(simple)k (eigenvalue)f(at)h(zer)-5 b(o.)79 b(Conse)-5 b(quently,)48 b(under)d(the)g(ab)-5 b(ove)45 b(c)-5 b(onditions)47 b(the)e(system)244 746 y Fu(\()p Fp(M)374 761 y Fo(\014)422 746 y Ft(;)15 b(\034)502 761 y Fo(\014)549 746 y Fu(\))33 b Fm(has)h(the)f(pr)-5 b(op)g(erty)35 b(of)e(r)-5 b(eturn)33 b(to)h(e)-5 b(quilibrium.)380 958 y Fu(In)21 b(comparison)g(with)h (Theorem)f(3.4,)j(in)d(Theorem)g(3.5)i(w)m(e)f(need)f(t)m(w)m(o)i (additional)f(e\013ectiv)m(e)244 1071 y(coupling)33 b(assumptions)f (\(3\))i(and)f(\(4\).)49 b(Note)34 b(also)g(that)g(the)f(regularit)m(y) h(assumption)e(\(1\))i(of)244 1184 y(Theorem)27 b(3.5)i(is)f(m)m(uc)m (h)g(stronger)g(than)g(that)g(of)g(Theorem)g(3.4.)41 b(F)-8 b(or)28 b(example,)h(assume)f(that)244 1297 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))34 b(is)g(smo)s(oth)f(a)m(w)m(a)m(y)i (from)e(zero)i(and)d(of)i(compact)h(supp)s(ort.)48 b(Then)32 b(\(1\))j(of)e(Theorem)g(3.5)244 1410 y(holds)k(if)g(around)f(zero)i(w) m(e)f(ha)m(v)m(e)i Ft(v)s Fu(\()p Ft(\030)t Fu(\))e(=)g Ft(v)1800 1424 y Fl(0)1839 1410 y Fr(j)p Ft(\030)t Fr(j)1933 1377 y Fo(\013)1983 1410 y Fu(,)i(where)e Ft(v)2361 1424 y Fl(0)2437 1410 y Fr(2)f(B)s Fu(\()p Fr(K)q Fu(\))i(and)f Ft(\013)g(>)f Fu(\(7)26 b Fr(\000)e Ft(d)p Fu(\))p Ft(=)p Fu(2)244 1523 y(\(compare)31 b(with)f(\(3.8\)\).)380 1636 y(Let)d(us)e(men)m(tion)i(that)f(our)g(formalism)g(can)g(b)s(e)g (applied)f(to)i(non-thermal)f(radiation)h(den-)244 1748 y(sities.)44 b(F)-8 b(or)32 b(instance,)h(if)e(the)h(small)f(system)h (in)m(teracts)h(with)e(sev)m(eral)h(reserv)m(oirs)g(at)g(distinct)244 1861 y(temp)s(eratures,)k(eac)m(h)h(satisfying)e(the)h(conditions)f(of) h(Theorem)f(3.4,)i(then)e(the)h(Liouvillean)244 1974 y(has)e(no)g(singular)g(sp)s(ectrum.)52 b(Consequen)m(tly)-8 b(,)36 b(under)d(these)h(assumptions)g(the)h(P)m(auli-Fierz)244 2087 y(system)30 b(has)h(no)f(normal)g(states.)244 2407 y Fq(4)161 b(Basic)53 b(notation)g(and)g(facts)244 2646 y Fi(4.1)135 b(Miscellanea)244 2818 y Fu(W)-8 b(e)32 b(set)g Fj(C)610 2832 y Fl(+)695 2818 y Fu(:=)27 b Fr(f)p Ft(z)k Fr(2)26 b Fj(C)g Fu(:)h(Im)o Ft(z)k(>)c Fu(0)p Fr(g)p Fu(.)43 b(Throughout)30 b(the)i(pap)s(er)e Ft(S)2585 2785 y Fl(cl)2671 2818 y Fu(denotes)i(the)f(closure)g(of)244 2931 y(a)g(set)g Ft(S)5 b Fu(,)30 b(so)h Fj(C)756 2898 y Fl(cl)756 2953 y(+)840 2931 y Fu(=)25 b Fr(f)p Ft(z)30 b Fr(2)24 b Fj(C)h Fu(:)h(Im)o Ft(z)k Fr(\025)25 b Fu(0)p Fr(g)p Fu(.)42 b(W)-8 b(e)31 b(will)g(use)f(the)g(shorthand)f Fr(h)p Ft(x)p Fr(i)d Fu(:=)g(\(1)21 b(+)f Ft(x)3298 2898 y Fl(2)3337 2931 y Fu(\))3372 2898 y Fl(1)p Fo(=)p Fl(2)3482 2931 y Fu(.)380 3044 y Ft(\016)s Fu(\()p Ft(p)p Fu(\))i(denotes)e(the)h (Dirac)g(delta)g(at)g(0,)i Fr(P)7 b Ft(p)1789 3011 y Fs(\000)p Fl(1)1904 3044 y Fu(the)21 b(principal)f(v)-5 b(alue)20 b(of)h Ft(p)2784 3011 y Fs(\000)p Fl(1)2878 3044 y Fu(,)h(and)e(\()p Ft(p)p Fu(+i0\))3349 3011 y Fs(\000)p Fl(1)3470 3044 y Fu(:=)244 3157 y(lim)370 3172 y Fo(\017)p Fs(#)p Fl(0)473 3157 y Fu(\()p Ft(p)h Fu(+)f(i)p Ft(\017)p Fu(\))763 3124 y Fs(\000)p Fl(1)857 3157 y Fu(.)41 b(W)-8 b(e)32 b(will)e(sometimes)i(use)e(the)g(so-called)j(So)s (c)m(ho)s(c)m(ki)e(form)m(ula:)1339 3361 y(\()p Ft(p)20 b Fu(+)g(i0\))1636 3323 y Fs(\000)p Fl(1)1757 3361 y Fu(=)25 b Fr(P)7 b Ft(p)1969 3323 y Fs(\000)p Fl(1)2084 3361 y Fr(\000)20 b Fu(i)p Ft(\031)s(\016)s Fu(\()p Ft(p)p Fu(\))p Ft(:)244 3638 y Fi(4.2)135 b(Op)t(erators)46 b(in)f(Hilb)t(ert)g(spaces)244 3809 y Fu(Let)33 b Fr(H)h Fu(b)s(e)e(a)i(Hilb)s(ert)f(space)g(with)g(the)g(scalar)h(pro)s(duct)e (\(\011)p Fr(j)p Fu(\010\),)i(\011)p Ft(;)15 b Fu(\010)29 b Fr(2)h(H)q Fu(.)48 b Fr(B)s Fu(\()p Fr(H)q Fu(\),)34 b Fr(B)3329 3823 y Fl(+)3387 3809 y Fu(\()p Fr(H)q Fu(\))244 3922 y(and)h Ft(U)10 b Fu(\()p Fr(H)q Fu(\))37 b(denote)f(the)g(set)h (of)f(b)s(ounded,)g(b)s(ounded)d(p)s(ositiv)m(e)38 b(and)d(unitary)h (op)s(erators)g(on)244 4035 y Fr(H)q Fu(.)70 b Ft(l)446 4002 y Fl(2)485 4035 y Fu(\()p Fr(H)q Fu(\))40 b(will)h(denote)g(the)f (Hilb)s(ert)g(space)h(of)f(Hilb)s(ert-Sc)m(hmidt)h(op)s(erators)f(on)g Fr(H)h Fu(with)244 4148 y(the)33 b(scalar)g(pro)s(duct)e(\()p Ft(A)p Fr(j)p Ft(B)5 b Fu(\))30 b(=)e(T)-8 b(r\()p Ft(A)1569 4115 y Fs(\003)1609 4148 y Ft(B)5 b Fu(\).)46 b Ft(l)1818 4115 y Fl(2)1816 4170 y(+)1876 4148 y Fu(\()p Fr(H)q Fu(\))32 b(is)h(the)f(set)h(of)g(p)s(ositiv)m(e)h(Hilb)s(ert-Sc)m (hmidt)244 4261 y(op)s(erators.)380 4374 y(If)c(\011)25 b Fr(2)g(H)q Fu(,)30 b(then)g Fr(j)p Fu(\011\))h(and)f(\(\011)p Fr(j)g Fu(denote)h(resp)s(ectiv)m(ely)h(the)e(op)s(erators)1052 4578 y Fj(C)25 b Fr(3)g Ft(\025)g Fr(7!)g Ft(\025)p Fu(\011)g Fr(2)g(H)q Ft(;)136 b Fr(H)26 b(3)f Fu(\010)g Fr(7!)g Fu(\(\011)p Fr(j)p Fu(\010\))h Fr(2)f Fj(C)p Ft(:)244 4782 y Fu(Ob)m(viously)-8 b(,)34 b(\(\011)p Fr(j)c Fu(:=)f Fr(j)p Fu(\011\))1109 4749 y Fs(\003)1149 4782 y Fu(.)48 b(If)33 b Fr(k)p Fu(\011)p Fr(k)d Fu(=)f(1,)34 b(then)f Fr(j)p Fu(\011\)\(\011)p Fr(j)g Fu(is)g(the)g(orthogonal)i(pro)5 b(jection)33 b(on)m(to)244 4895 y(the)d(subspace)g(spanned)f(b)m(y)i (\011.)380 5008 y(sp\()p Ft(A)p Fu(\))38 b(denotes)g(the)g(sp)s(ectrum) e(of)i(a)g(closed)g(op)s(erator)g Ft(A)g Fu(on)f Fr(H)q Fu(.)62 b(If)37 b(\002)g(is)h(an)f(isolated)244 5121 y(b)s(ounded)30 b(subset)i(of)g(sp\()p Ft(A)p Fu(\))h(\(closed)g(and)f (op)s(en)g(in)g(the)h(relativ)m(e)h(top)s(ology)g(of)e(sp\()p Ft(A)p Fu(\)\),)i(then)244 5234 y(1)289 5248 y Fl(\002)348 5234 y Fu(\()p Ft(A)p Fu(\))e(denotes)e(the)h(sp)s(ectral)g(\(Riesz\))g (pro)5 b(jection)32 b(of)e Ft(A)h Fu(on)m(to)g(\002.)p eop %%Page: 15 15 15 14 bop 3682 100 a Fn(15)380 407 y Fu(If)26 b Ft(A)h Fu(is)g(self-adjoin)m(t)g(and)f(\002)h(is)f(a)h(Borel)h(subset)e(of)g Fj(R)p Fu(,)h(then)g(1)2522 421 y Fl(\002)2581 407 y Fu(\()p Ft(A)p Fu(\))h(denotes)f(the)f(sp)s(ectral)244 520 y(pro)5 b(jection)27 b(of)f Ft(A)h Fu(on)m(to)g(\002.)39 b(1)1246 487 y Fl(p)1290 520 y Fu(\()p Ft(A)p Fu(\))27 b(denotes)g(the)g(pro)5 b(jection)27 b(on)m(to)g(the)g(subspace)f (spanned)f(b)m(y)244 633 y(the)20 b(eigen)m(v)m(ectors)k(of)c Ft(A)p Fu(.)38 b(1)1158 600 y Fl(ac)1229 633 y Fu(\()p Ft(A)p Fu(\))21 b(and)f(1)1600 600 y Fl(sc)1664 633 y Fu(\()p Ft(A)p Fu(\))26 b(:=)f(1)p Fr(\000)p Fu(1)2110 600 y Fl(ac)2182 633 y Fu(\()p Ft(A)p Fu(\))p Fr(\000)p Fu(1)2436 600 y Fl(p)2481 633 y Fu(\()p Ft(A)p Fu(\))d(denote)f(resp)s (ectiv)m(ely)g(the)244 746 y(pro)5 b(jections)32 b(on)m(to)h(the)f (absolutely)h(con)m(tin)m(uous)f(and)g(the)g(singular)f(con)m(tin)m (uous)i(part)e(of)h(the)244 858 y(sp)s(ectrum)21 b(of)i Ft(A)p Fu(.)38 b(sp)946 880 y Fl(p)990 858 y Fu(\()p Ft(A)p Fu(\),)25 b(sp)1264 880 y Fl(ac)1335 858 y Fu(\()p Ft(A)p Fu(\),)g(sp)1609 880 y Fl(sc)1673 858 y Fu(\()p Ft(A)p Fu(\))e(denote)g(resp)s(ectiv)m(ely)h(the)e(p)s(oin)m(t)h(sp)s (ectrum)e(\(the)244 971 y(set)28 b(of)g(eigen)m(v)-5 b(alues\),)30 b(the)e(absolutely)g(con)m(tin)m(uous)h(sp)s(ectrum)d (and)h(the)h(singular)f(con)m(tin)m(uous)244 1084 y(sp)s(ectrum)i(of)i Ft(A)p Fu(.)380 1197 y(If)25 b Ft(z)k Fr(2)c Fu(sp\()p Ft(A)p Fu(\))g(is)g(an)g(isolated)i(p)s(oin)m(t)e(of)g(sp)o(\()p Ft(A)p Fu(\),)j(or)d Ft(A)g Fu(is)g(self-adjoin)m(t,)i(w)m(e)f(will)f (write)h(1)3357 1211 y Fo(z)3397 1197 y Fu(\()p Ft(A)p Fu(\))244 1310 y(instead)31 b(of)f(1)706 1329 y Fs(f)p Fo(z)s Fs(g)817 1310 y Fu(\()p Ft(A)p Fu(\).)380 1423 y(W)-8 b(e)32 b(denote)f(the)f(real)h(and)f(imaginary)h(part)f(of)h Ft(A)25 b Fr(2)g(B)s Fu(\()p Fr(H)q Fu(\))30 b(b)m(y)1073 1665 y Ft(A)1141 1628 y Fl(R)1222 1665 y Fu(:=)1354 1604 y(1)p 1354 1645 46 4 v 1354 1728 a(2)1409 1665 y(\()p Ft(A)21 b Fu(+)f Ft(A)1692 1628 y Fs(\003)1731 1665 y Fu(\))p Ft(;)198 b(A)2057 1628 y Fl(I)2112 1665 y Fu(:=)2256 1604 y(1)p 2243 1645 71 4 v 2243 1728 a(2i)2324 1665 y(\()p Ft(A)20 b Fr(\000)g Ft(A)2606 1628 y Fs(\003)2646 1665 y Fu(\))p Ft(:)244 1906 y Fu(Clearly)-8 b(,)31 b Ft(A)26 b Fu(=)f Ft(A)835 1873 y Fl(R)911 1906 y Fu(+)20 b(i)p Ft(A)1095 1873 y Fl(I)1125 1906 y Fu(.)41 b Ft(A)30 b Fu(is)h(called)g(dissipativ)m(e)h(if)e Ft(A)2235 1873 y Fl(I)2290 1906 y Fr(\024)25 b Fu(0.)244 2182 y Fi(4.3)135 b(Lev)l(el)46 b(Shift)f(Op)t(erator)244 2354 y Fu(In)23 b(the)h(ph)m(ysics)f(literature,)k(the)d(form)m(ulas)f(for)h(computing) g(2nd)f(order)g(corrections)i(for)e(eigen-)244 2467 y(v)-5 b(alues)26 b(and)e(esp)s(ecially)j(resonances)f(often)f(go)i(under)c (the)j(name)f(of)h(the)f(F)-8 b(ermi)27 b(Golden)e(Rule.)244 2580 y(In)34 b(this)g(subsection)g(w)m(e)h(will)g(in)m(tro)s(duce)f(an) h(op)s(erator,)h(sometimes)f(called)h(the)f(Lev)m(el)g(Shift)244 2693 y(Op)s(erator,)30 b(that)h(can)g(b)s(e)e(used)h(to)h(formalize)h (the)e(F)-8 b(ermi)31 b(Golden)g(Rule.)380 2806 y(Supp)s(ose)h(that)i Fr(H)g Fu(is)f(a)h(Hilb)s(ert)f(space)h(with)f(a)h(distinguished)f (\014nite)g(dimensional)g(sub-)244 2919 y(space)44 b Fr(H)578 2886 y Fl(v)620 2919 y Fu(.)79 b(W)-8 b(e)45 b(set)f Fr(H)p 1127 2848 43 3 v 1127 2886 a Fl(v)1215 2919 y Fu(:=)j(\()p Fr(H)1471 2886 y Fl(v)1513 2919 y Fu(\))1548 2886 y Fs(?)1608 2919 y Fu(.)79 b(W)-8 b(e)45 b(will)e(often)h(use)f(2)29 b Fr(\002)g Fu(2)44 b(matrix)f(notation)i (for)244 3032 y(op)s(erators)30 b(on)h Fr(H)26 b Fu(=)f Fr(H)1050 2999 y Fl(v)1112 3032 y Fr(\010)20 b(H)p 1281 2961 V 1281 2999 a Fl(v)1323 3032 y Fu(.)40 b(F)-8 b(or)31 b(example,)h(an)m(y)f Ft(A)25 b Fr(2)g(B)s Fu(\()p Fr(H)q Fu(\))30 b(can)h(b)s(e)e(written)i(as)1490 3313 y Ft(A)25 b Fu(=)1679 3158 y Fk(")1774 3245 y Ft(A)1842 3212 y Fl(vv)2006 3245 y Ft(A)2074 3212 y Fl(v)p 2112 3174 39 3 v 1 w(v)1774 3381 y Ft(A)p 1842 3311 V -33 x Fl(v)q(v)2006 3381 y Ft(A)p 2074 3311 V -33 x Fl(v)p 2112 3311 V 1 w(v)2196 3158 y Fk(#)2264 3313 y Ft(:)1060 b Fu(\(4.9\))380 3589 y(Supp)s(ose)29 b(that)i Ft(L)996 3604 y Fl(fr)1079 3589 y Fu(is)g(a)f(self-adjoin)m(t)i(op)s(erator)f(that)g(lea)m(v)m(es) h Fr(H)2616 3556 y Fl(v)2689 3589 y Fu(in)m(v)-5 b(arian)m(t.)42 b(Then)1473 3866 y Ft(L)1535 3881 y Fl(fr)1614 3866 y Fu(=)1710 3710 y Fk(")1804 3797 y Ft(L)1866 3764 y Fl(vv)1866 3825 y(fr)2077 3797 y Fu(0)1852 3934 y(0)132 b Ft(L)p 2091 3863 V -33 x Fl(v)p 2129 3863 V 1 w(v)2091 3962 y(fr)2213 3710 y Fk(#)2281 3866 y Ft(:)997 b Fu(\(4.10\))244 4147 y(F)-8 b(or)31 b Ft(A)25 b Fr(2)g(B)s Fu(\()p Fr(H)762 4114 y Fl(v)804 4147 y Fu(\))31 b(and)f Ft(e)1089 4161 y Fl(1)1128 4147 y Ft(;)15 b(e)1210 4161 y Fl(2)1276 4147 y Fr(2)25 b Fj(R)30 b Fu(w)m(e)g(set)1327 4351 y Ft(A)1395 4314 y Fo(e)1428 4323 y Fh(1)1463 4314 y Fo(;e)1516 4323 y Fh(2)1579 4351 y Fu(:=)25 b(1)1745 4365 y Fo(e)1778 4374 y Fh(1)1817 4351 y Fu(\()p Ft(L)1914 4314 y Fl(vv)1914 4375 y(fr)1994 4351 y Fu(\))p Ft(A)p Fu(1)2142 4365 y Fo(e)2175 4374 y Fh(2)2215 4351 y Fu(\()p Ft(L)2312 4314 y Fl(vv)2312 4375 y(fr)2392 4351 y Fu(\))p Ft(:)380 4556 y Fu(Let)34 b Ft(Q)g Fu(b)s(e)e(a)i(self-adjoin)m(t)h(op)s(erator)f(on) f Fr(H)i Fu(suc)m(h)e(that)h Ft(Q)2419 4523 y Fl(vv)2529 4556 y Fu(=)c(0)k(and)f Ft(Q)2961 4523 y Fl(v)p 2999 4485 V 1 w(v)3075 4556 y Fu(is)g(b)s(ounded.)244 4669 y(Let)924 4781 y Ft(w)r Fu(\()p Ft(z)t Fu(\))27 b(:=)e Ft(Q)1327 4744 y Fl(v)p 1365 4706 V 1 w(v)1408 4781 y Fu(\()p Ft(z)t Fu(1)p 1534 4706 V -37 x Fl(v)p 1573 4706 V 2 w(v)1635 4781 y Fr(\000)20 b Ft(L)p 1788 4706 V -37 x Fl(v)p 1826 4706 V 1 w(v)1788 4805 y(fr)1869 4781 y Fu(\))1904 4744 y Fs(\000)p Fl(1)1999 4781 y Ft(Q)p 2071 4706 V -37 x Fl(v)q(v)2151 4781 y Ft(;)197 b(z)30 b Fr(62)25 b Fu(sp)o(\()p Ft(L)p 2714 4706 V -37 x Fl(v)p 2752 4706 V 1 w(v)2714 4805 y(fr)2795 4781 y Fu(\))p Ft(:)448 b Fu(\(4.11\))244 4948 y(Assume)30 b(that)h(for)f(all)h Ft(e)26 b Fr(2)f Fu(sp)o(\()p Ft(L)1380 4915 y Fl(vv)1380 4976 y(fr)1460 4948 y Fu(\))30 b(the)h(limit)1318 5153 y(lim)1332 5214 y Fo(\017)p Fs(#)p Fl(0)1460 5153 y Ft(w)r Fu(\()p Ft(e)21 b Fu(+)f(i)p Ft(\017)p Fu(\))1813 5115 y Fo(ee)1908 5153 y Fu(=:)25 b Ft(w)r Fu(\()p Ft(e)d Fu(+)e(i0\))2391 5115 y Fo(ee)p eop %%Page: 16 16 16 15 bop 3682 100 a Fn(16)244 407 y Fu(exists)31 b(and)f(set)1393 494 y(\000)25 b(:=)1707 426 y Fk(P)1596 582 y Fo(e)p Fs(2)p Fl(sp\()p Fo(L)1818 559 y Fh(vv)1818 606 y(fr)1887 582 y Fl(\))1930 494 y Ft(w)r Fu(\()p Ft(e)c Fu(+)f(i0\))2291 461 y Fo(ee)2362 494 y Ft(:)244 745 y Fu(W)-8 b(e)31 b(will)g(call)h(\000)e(the)h(Lev)m(el)g(Shift)f(Op)s(erator)g(asso)s (ciated)i(to)f(the)f(triple)h(\()p Fr(H)2889 712 y Fl(v)2931 745 y Ft(;)15 b(L)3033 760 y Fl(fr)3087 745 y Ft(;)g(Q)p Fu(\).)380 858 y(Note)27 b(that)g(\000)e(is)h(a)g(dissipativ)m(e)h(op)s (erator)f(\(in)g(general,)i(\000)d(is)h(not)g(self-adjoin)m(t\),)j (\000)3211 825 y Fo(e)3244 834 y Fh(1)3278 825 y Fo(;e)3331 834 y Fh(2)3394 858 y Fu(=)c(0)244 971 y(for)30 b Ft(e)425 985 y Fl(1)490 971 y Fr(6)p Fu(=)25 b Ft(e)628 985 y Fl(2)668 971 y Fu(,)30 b Ft(L)785 938 y Fl(vv)785 999 y(fr)865 971 y Fu(\000)922 938 y Fl(R)1003 971 y Fu(=)25 b(\000)1156 938 y Fl(R)1212 971 y Ft(L)1274 938 y Fl(vv)1274 999 y(fr)1384 971 y Fu(and)k Ft(L)1622 938 y Fl(vv)1622 999 y(fr)1702 971 y Fu(\000)1759 938 y Fl(I)1813 971 y Fu(=)c(\000)1966 938 y Fl(I)1996 971 y Ft(L)2058 938 y Fl(vv)2058 999 y(fr)2137 971 y Fu(.)380 1084 y(Let)31 b(us)f(no)m(w)g(describ)s(e)g(applications)i(of)e(the)h(Lev)m(el)g (Shift)f(Op)s(erator.)380 1197 y(Assume)f(that)g Ft(L)p 973 1126 39 3 v -33 x Fl(v)p 1011 1126 V 1 w(v)1071 1197 y Fu(+)17 b Ft(Q)p 1231 1126 V -33 x Fl(v)p 1269 1126 V 1 w(v)1340 1197 y Fu(is)29 b(essen)m(tially)i(self-adjoin)m(t)f(on)e Fr(D)s Fu(\()p Ft(L)p 2624 1126 V -33 x Fl(v)p 2663 1126 V 2 w(v)2705 1197 y Fu(\))17 b Fr(\\)g(D)s Fu(\()p Ft(Q)p 3015 1126 V -33 x Fl(v)p 3053 1126 V 1 w(v)3096 1197 y Fu(\).)40 b(Then)28 b(w)m(e)244 1310 y(can)j(de\014ne)e(the)i (self-adjoin)m(t)g(op)s(erator)1610 1514 y Ft(L)25 b Fu(=)g Ft(L)1855 1529 y Fl(fr)1928 1514 y Fu(+)20 b Ft(\025Q;)244 1718 y Fu(where)37 b Ft(\025)f Fr(2)h Fj(R)g Fu(is)g(a)g(coupling)h (constan)m(t.)63 b(The)36 b(Lev)m(el)j(Shift)d(Op)s(erator)h(\000)g (can)h(b)s(e)e(used)h(to)244 1831 y(describ)s(e)30 b(some)g(prop)s (erties)g(of)h(the)f(op)s(erator)h Ft(L)f Fu(for)g(a)h(small)g (coupling)g(constan)m(t.)380 1944 y(First)k(of)f(all,)i(if)e(w)m(e)g (mak)m(e)h(appropriate)f(analyticit)m(y)j(assumptions)c(similar)h(to)h (those)f(of)244 2057 y([JP1,)d(JP2],)g(then)f(the)g(op)s(erator)1689 2170 y Ft(L)1751 2132 y Fl(vv)1751 2193 y(fr)1851 2170 y Fu(+)19 b Ft(\025)1994 2132 y Fl(2)2034 2170 y Fu(\000)1212 b(\(4.12\))244 2337 y(can)24 b(b)s(e)g(used)f(to)i(predict)g(the)f (appro)m(ximate)h(lo)s(cation)h(and)e(the)g(m)m(ultiplicities)i(of)f (eigen)m(v)-5 b(alues)244 2449 y(and)30 b(resonances)g(of)h Ft(L)f Fu(for)g(small)h Ft(\025)p Fu(.)380 2562 y(If)f(w)m(e)h(do)f (not)g(mak)m(e)h(analyticit)m(y)i(assumptions,)d(then)g(w)m(e)g(cannot) h(de\014ne)e(the)i(notion)f(of)244 2675 y(resonance.)60 b(Still,)39 b(the)e(Lev)m(el)h(Shift)e(Op)s(erator)h(can)g(b)s(e)f (used)g(to)h(study)f(eigen)m(v)-5 b(alues)39 b(of)e Ft(L)p Fu(.)244 2788 y(In)29 b(particular,)h(in)g([DJ1)q(])g(w)m(e)g(pro)m(v)m (ed)g(that)g(for)g(a)g(certain)h(class)f(of)g(P)m(auli-Fierz)i(op)s (erators)e Ft(L)244 2901 y Fu(and)g(for)g(a)h(small)f(nonzero)h Ft(\025)p Fu(,)g(the)f(op)s(erator)1475 3028 y Fk(X)1356 3229 y Fo(m)p Fs(2)p Fl(sp\(\000\))p Fs(\\)p Fe(R)1741 3114 y Fu(1)1786 3128 y Fo(e)1824 3114 y Fu(\(\000\)\()p Ft(L)2048 3076 y Fl(vv)2048 3137 y(fr)2148 3114 y Fu(+)20 b Ft(\025)2292 3076 y Fl(2)2332 3114 y Fu(\000\))879 b(\(4.13\))244 3417 y(predicts)30 b(the)h(appro)m(ximate)g(lo)s(cation) h(of)f(eigen)m(v)-5 b(alues)32 b(of)e Ft(L)h Fu(and)e(that)i(the)g (estimate)1435 3622 y(dim)15 b(1)1647 3584 y Fl(p)1691 3622 y Fu(\()p Ft(L)p Fu(\))26 b Fr(\024)f Fu(dim)14 b(Ker\000)2315 3584 y Fl(I)244 3826 y Fu(giv)m(es)32 b(an)f(upp)s(er)d(b)s(ound)h(on)i(their)f(m)m(ultiplicit)m(y)-8 b(.)45 b(These)30 b(results)h(will)g(b)s(e)f(describ)s(ed)f(in)i(Sub-) 244 3939 y(section)g(5.4.)244 4215 y Fi(4.4)135 b(Space)45 b Fd(L)1011 4172 y Fw(2)1056 4215 y Fx(\()p Fc(R)p Fx(\))244 4387 y Fu(In)30 b(this)g(subsection)g(w)m(e)h(describ)s(e)f(some)h(op)s (erators)f(acting)i(on)e Ft(L)2552 4354 y Fl(2)2591 4387 y Fu(\()p Fj(R)p Fu(\).)380 4500 y(Let)h Ft(r)i Fu(denote)e(the)f (self-adjoin)m(t)i(op)s(erator)f(of)f(m)m(ultiplication)j(b)m(y)d(the)g (v)-5 b(ariable)32 b(in)e Fj(R)p Fu(,)1530 4630 y Fk(\000)1572 4704 y Ft(r)s Fu(\011)1687 4630 y Fk(\001)1728 4704 y Fu(\()p Ft(p)p Fu(\))c(:=)f Ft(p)p Fu(\011\()p Ft(p)p Fu(\))p Ft(:)244 4908 y Fu(Throughout)35 b(this)h(pap)s(er,)g(in)g(the) g(con)m(text)i(of)e(the)g(space)h Ft(L)2410 4875 y Fl(2)2449 4908 y Fu(\()p Fj(R)p Fu(\))f(the)g(generic)h(name)f(for)g(a)244 5021 y(v)-5 b(ariable)36 b(in)e Fj(R)h Fu(will)g(b)s(e)f Ft(p)p Fu(.)54 b(On)34 b(the)h(other)g(hand,)g(the)g(m)m(ultiplication) i(op)s(erator)e(on)g Ft(L)3360 4988 y Fl(2)3399 5021 y Fu(\()p Fj(R)p Fu(\))244 5134 y(b)m(y)30 b(its)h(natural)f(v)-5 b(ariable)32 b(will)e(b)s(e)g(denoted)g(b)m(y)h Ft(r)s Fu(.)p eop %%Page: 17 17 17 16 bop 3682 100 a Fn(17)380 407 y Fu(W)-8 b(e)32 b(denote)f(b)m(y)f Ft(s)g Fu(the)g(self-adjoin)m(t)i(op)s(erator)1503 596 y Ft(s)p Fu(\011\()p Ft(p)p Fu(\))26 b(:=)f Fr(\000)p Fu(i)p Ft(@)2024 610 y Fo(p)2064 596 y Fu(\011\()p Ft(p)p Fu(\))p Ft(;)1027 b Fu(\(4.14\))244 785 y(and)30 b(b)m(y)g Ft(C)7 b Fu(\()p Fj(R)p Fu(\))30 b(the)h(set)g(of)f(all)h(con)m(tin)m (uous)g(functions)f(on)h Fj(R)p Fu(.)380 898 y(F)-8 b(or)34 b Ft(p)28 b Fr(2)h Fj(R)p Fu(,)k(de\014ne)f(an)g(op)s(erator)h Ft(\031)1648 912 y Fo(p)1717 898 y Fu(:)c Ft(C)7 b Fu(\()p Fj(R)p Fu(\))29 b Fr(!)g Fj(R)j Fu(b)m(y)g Ft(\031)2406 912 y Fo(p)2446 898 y Ft(f)38 b Fu(:=)29 b Ft(f)10 b Fu(\()p Ft(p)p Fu(\).)47 b(It)33 b(is)f(w)m(ell)i(kno)m(wn)244 1011 y(that,)g(for)e Ft(\021)h(>)c Fu(1)p Ft(=)p Fu(2,)35 b Fr(D)s Fu(\()p Fr(h)p Ft(s)p Fr(i)1204 978 y Fo(\021)1246 1011 y Fu(\))e(is)g(a)g(subset)f(of)h Ft(C)7 b Fu(\()p Fj(R)p Fu(\).)47 b(Hence)34 b(the)f(op)s(erator)g Ft(\031)3006 1025 y Fo(p)3045 1011 y Fr(h)p Ft(s)p Fr(i)3158 978 y Fs(\000)p Fo(\021)3288 1011 y Fu(is)g(w)m(ell)244 1124 y(de\014ned)c(on)h Ft(L)745 1091 y Fl(2)785 1124 y Fu(\()p Fj(R)p Fu(\).)380 1237 y(The)g(follo)m(wing)i(t)m(w)m(o)g(results)e (are)h(w)m(ell)g(kno)m(wn.)244 1412 y Fv(Prop)s(osition)36 b(4.1)49 b Fm(L)-5 b(et)33 b Ft(\021)c(>)c Fu(1)p Ft(=)p Fu(2)p Fm(.)43 b(Then)244 1525 y Fu(\(1\))34 b Fm(The)e(functional)i Ft(\031)1058 1539 y Fo(p)1097 1525 y Fr(h)p Ft(s)p Fr(i)1210 1492 y Fs(\000)p Fo(\021)1340 1525 y Fm(is)f(b)-5 b(ounde)g(d)34 b(on)f Ft(L)1976 1492 y Fl(2)2015 1525 y Fu(\()p Fj(R)p Fu(\))p Fm(.)244 1638 y Fu(\(2\))h Fm(The)e(map)i Fj(R)25 b Fr(3)g Ft(p)g Fr(7!)g Ft(\031)1194 1652 y Fo(p)1234 1638 y Fr(h)p Ft(s)p Fr(i)1347 1605 y Fs(\000)p Fo(\021)1469 1638 y Fr(2)g(B)s Fu(\()p Ft(L)1715 1605 y Fl(2)1754 1638 y Fu(\()p Fj(R)p Fu(\))p Ft(;)15 b Fj(C)p Fu(\))33 b Fm(is)g(c)-5 b(ontinuous.)244 1833 y Fv(Prop)s(osition)36 b(4.2)49 b Fm(L)-5 b(et)33 b Ft(n)f Fm(b)-5 b(e)33 b(a)g(p)-5 b(ositive)34 b(inte)-5 b(ger)32 b(and)i Ft(\021)29 b(>)c(n)19 b Fr(\000)2576 1797 y Fl(1)p 2576 1812 36 4 v 2576 1864 a(2)2621 1833 y Fm(.)42 b(Then)33 b(the)g(function)1040 2022 y Fj(C)1106 2036 y Fl(+)1190 2022 y Fr(3)25 b Ft(z)k Fr(7!)d(h)p Ft(s)p Fr(i)1577 1985 y Fs(\000)p Fo(\021)1674 2022 y Fu(\()p Ft(z)f Fr(\000)20 b Ft(r)s Fu(\))1946 1985 y Fs(\000)p Fo(n)2047 2022 y Fr(h)p Ft(s)p Fr(i)2160 1985 y Fs(\000)p Fo(\021)2283 2022 y Fr(2)25 b(B)s Fu(\()p Ft(L)2529 1985 y Fl(2)2568 2022 y Fu(\()p Fj(R)p Fu(\)\))244 2212 y Fm(extends)33 b(fr)-5 b(om)34 b Fj(C)848 2226 y Fl(+)939 2212 y Fm(to)f(a)g(c)-5 b(ontinuous)34 b(function)f(on)g Fj(C)2139 2179 y Fl(cl)2139 2234 y(+)2198 2212 y Fm(.)380 2407 y Fu(Let)i Ft(\021)h(>)31 b Fu(1)p Ft(=)p Fu(2.)55 b(In)33 b(what)i(follo)m(ws,)i(the)d(functional)h Ft(\031)2266 2421 y Fo(p)2306 2407 y Fr(h)p Ft(s)p Fr(i)2419 2374 y Fs(\000)p Fo(\021)2550 2407 y Fu(will)g(b)s(e)f(often)h(used)e(in)h (the)244 2519 y(follo)m(wing)e(con)m(text.)44 b(Let)32 b Fr(G)k Fu(b)s(e)30 b(a)i(Hilb)s(ert)f(space.)43 b(Ob)m(viously)-8 b(,)31 b Fr(G)26 b(\012)20 b Fj(C)26 b Fu(=)g Fr(G)5 b Fu(.)43 b(Hence,)32 b(w)m(e)f(can)244 2632 y(in)m(tro)s(duce)f(the)h (family)g(of)f(maps)1281 2822 y(1)1326 2836 y Fs(G)1396 2822 y Fr(\012)20 b Ft(\031)1539 2836 y Fo(p)1579 2822 y Fr(h)p Ft(s)p Fr(i)1692 2784 y Fs(\000)p Fo(\021)1814 2822 y Fu(:)26 b Fr(G)f(\012)20 b Ft(L)2097 2784 y Fl(2)2136 2822 y Fu(\()p Fj(R)p Fu(\))26 b Fr(!)f(G)5 b Ft(:)805 b Fu(\(4.15\))244 3011 y(Clearly)-8 b(,)31 b(the)g(maps)f(\(4.15\))j (are)d(b)s(ounded)f(and)g(dep)s(end)g(con)m(tin)m(uously)i(on)f Ft(p)p Fu(.)244 3285 y Fi(4.5)135 b(Space)45 b Fd(L)1011 3242 y Fw(2)1056 3285 y Fx(\()p Fc(R)p Fd(;)20 b Fb(G)7 b Fx(\))244 3457 y Fu(Let)35 b Fr(G)k Fu(b)s(e)34 b(a)h(Hilb)s(ert)f (space)h(\(not)g(necessarily)g(separable\).)53 b(W)-8 b(e)35 b(sa)m(y)g(that)g(a)g(function)f Fj(R)d Fr(3)244 3570 y Ft(p)25 b Fr(7!)g Fu(\011\()p Ft(p)p Fu(\))g Fr(2)g(G)36 b Fu(b)s(elongs)30 b(to)h Fr(L)1319 3537 y Fl(2)1359 3570 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\))31 b(i\013)275 3698 y(\(1\))50 b(There)29 b(exists)i(a)g(separable)g(subspace)f Fr(G)1863 3712 y Fl(0)1933 3698 y Fu(suc)m(h)g(that)h(\011\()p Ft(p)p Fu(\))25 b Fr(2)g(G)2687 3712 y Fl(0)2757 3698 y Fu(for)30 b(all)h Ft(p)25 b Fr(2)g Fj(R)p Fu(.)275 3829 y(\(2\))50 b(F)-8 b(or)31 b(an)m(y)f(\010)25 b Fr(2)g(G)5 b Fu(,)31 b(the)g(function)f Fj(R)25 b Fr(3)g Ft(p)g Fr(7!)g Fu(\(\010)p Fr(j)p Fu(\011\()p Ft(p)p Fu(\)\))h Fr(2)f Fj(C)30 b Fu(is)g(measurable.)275 3961 y(\(3\))440 3887 y Fk(R)515 3961 y Fr(k)p Fu(\011\()p Ft(p)p Fu(\))p Fr(k)792 3928 y Fl(2)833 3961 y Fu(d)p Ft(p)24 b(<)h Fr(1)p Fu(.)380 4089 y(Let)37 b Fr(N)13 b Fu(\()p Fj(R)p Ft(;)i Fr(G)5 b Fu(\))38 b(b)s(e)e(the)h(set)g(of)f(all)i(\011)d Fr(2)g(L)1858 4056 y Fl(2)1897 4089 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\))38 b(suc)m(h)e(that)h(\011\()p Ft(p)p Fu(\))f(=)f(0)i(for)f(almost)i(all)244 4202 y Ft(p)25 b Fr(2)g Fj(R)30 b Fu(and)1281 4315 y Ft(L)1343 4277 y Fl(2)1382 4315 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\))27 b(:=)e Fr(L)1828 4277 y Fl(2)1867 4315 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\))p Ft(=)p Fr(N)13 b Fu(\()p Fj(R)p Ft(;)i Fr(G)5 b Fu(\))p Ft(:)244 4474 y Fu(There)30 b(exists)h(a)g(unique)e(unitary)h(op)s(erator)1446 4663 y Fr(G)c(\012)20 b Ft(L)1679 4626 y Fl(2)1718 4663 y Fu(\()p Fj(R)p Fu(\))26 b Fr(!)f Ft(L)2058 4626 y Fl(2)2097 4663 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\))971 b(\(4.16\))244 4853 y(suc)m(h)30 b(that)h(\011)20 b Fr(\012)g Ft(f)34 b Fr(2)25 b(G)h(\012)20 b Ft(L)1226 4820 y Fl(2)1265 4853 y Fu(\()p Fj(R)p Fu(\))31 b(is)f(mapp)s(ed)f(on)m(to)j Ft(p)25 b Fr(7!)g Ft(f)10 b Fu(\()p Ft(p)p Fu(\)\011.)380 4966 y(Let)27 b Fr(K)g Fu(b)s(e)f(another)g(Hilb)s(ert)g(space)h(and)e Ft(q)j Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(G)j(\012)12 b Ft(L)2384 4933 y Fl(2)2423 4966 y Fu(\()p Fj(R)p Fu(\)\).)39 b(Supp)s(ose)25 b(that)h(for)g(some)244 5079 y Ft(\021)j(>)24 b Fu(1)p Ft(=)p Fu(2,)32 b(1)650 5093 y Fs(G)701 5079 y Fr(\012)o(h)p Ft(s)p Fr(i)884 5046 y Fo(\021)942 5079 y Ft(q)c Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(G)26 b(\012)20 b Ft(L)1538 5046 y Fl(2)1578 5079 y Fu(\()p Fj(R)p Fu(\)\).)41 b(Then)29 b(for)h Ft(p)25 b Fr(2)g Fj(R)30 b Fu(w)m(e)h(can)g(de\014ne)858 5268 y Ft(q)s Fu(\()p Ft(p)p Fu(\))26 b(:=)f(1)1210 5282 y Fs(G)1260 5268 y Fr(\012)p Ft(\031)1383 5282 y Fo(p)1453 5268 y Ft(q)j Fu(=)d(1)1663 5282 y Fs(G)1713 5268 y Fr(\012)p Ft(\031)1836 5282 y Fo(p)1876 5268 y Fr(h)p Ft(s)p Fr(i)1989 5230 y Fs(\000)p Fo(\021)2116 5268 y Fu(1)2161 5282 y Fs(G)2211 5268 y Fr(\012h)p Ft(s)p Fr(i)2395 5230 y Fo(\021)2437 5268 y Ft(q)j Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(G)5 b Fu(\))p Ft(:)p eop %%Page: 18 18 18 17 bop 3682 100 a Fn(18)244 407 y Fu(Clearly)1420 520 y Fj(R)25 b Fr(3)g Ft(p)g Fr(7!)g Ft(q)s Fu(\()p Ft(p)p Fu(\))h Fr(2)f(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(G)5 b Fu(\))244 687 y(is)35 b(a)h(con)m(tin)m(uous)g(function.)54 b(Note)37 b(that)f(for)f(\011)e Fr(2)g(K)q Fu(,)k(the)e(v)m(ector)i Ft(q)s Fu(\011)c Fr(2)g(G)5 b(\012)p Ft(L)3060 654 y Fl(2)3099 687 y Fu(\()p Fj(R)p Fu(\))36 b(can)f(b)s(e)244 800 y(iden)m(ti\014ed)30 b(through)g(\(4.16\))j(with)d(the)g(function)g Ft(p)25 b Fr(7!)h Ft(q)s Fu(\()p Ft(p)p Fu(\)\011)k(in)g Ft(L)2583 767 y Fl(2)2622 800 y Fu(\()p Fj(R)p Ft(;)15 b Fr(G)5 b Fu(\).)380 912 y(F)-8 b(or)31 b(an)m(y)g Ft(f)k Fr(2)24 b Ft(L)942 879 y Fs(1)1017 912 y Fu(\()p Fj(R)p Fu(\))31 b(the)f(follo)m(wing)i(iden)m(tit)m(y)g(holds:)1302 1159 y Ft(q)1346 1121 y Fs(\003)1385 1159 y Ft(f)10 b Fu(\()p Ft(r)s Fu(\))p Ft(q)28 b Fu(=)1719 1035 y Fk(Z)1825 1159 y Ft(q)1869 1121 y Fs(\003)1908 1159 y Fu(\()p Ft(p)p Fu(\))p Ft(f)10 b Fu(\()p Ft(p)p Fu(\))p Ft(q)s Fu(\()p Ft(p)p Fu(\)d)p Ft(p:)826 b Fu(\(4.17\))244 1400 y(Note)32 b(the)e(estimate)1074 1623 y Fr(k)p Ft(q)1163 1585 y Fs(\003)1203 1623 y Ft(q)s Fr(k)25 b(\024)1413 1499 y Fk(Z)1519 1623 y Fr(k)p Ft(q)1608 1585 y Fs(\003)1648 1623 y Fu(\()p Ft(p)p Fu(\))p Ft(q)s Fu(\()p Ft(p)p Fu(\))p Fr(k)p Fu(d)p Ft(p)h Fu(=)2188 1499 y Fk(Z)2294 1623 y Fr(k)p Ft(q)s Fu(\()p Ft(p)p Fu(\))p Fr(k)2544 1585 y Fl(2)2584 1623 y Fu(d)p Ft(p:)597 b Fu(\(4.18\))244 1920 y Fi(4.6)135 b(Conjugate)46 b(Hilb)t(ert)g(spaces)244 2091 y Fu(Let)31 b Fr(K)i Fu(b)s(e)d(a)h(Hilb)s(ert)g(space.)43 b(The)31 b(space)p 1734 2018 71 4 v 31 w Fr(K)i Fu(conjugate)f(to)f Fr(K)i Fu(is)e(an)m(y)g(Hilb)s(ert)g(space)h(with)e(a)244 2204 y(distinguished)f(an)m(tiunitary)i(map)1554 2408 y Fr(K)c(3)e Fu(\011)g Fr(7!)p 1948 2335 V 25 w Fu(\011)g Fr(2)p 2130 2335 V 25 w(K)q Ft(:)1078 b Fu(\(4.19\))244 2613 y(The)30 b(map)g(\(4.19\))i(is)f(called)g(the)g(conjugation)g(on)g Fr(K)q Fu(.)380 2726 y(By)g(the)f(Riesz)i(lemma,)f(the)f(map)p 1417 2857 V 1417 2930 a Fr(K)d(3)p 1598 2857 V 24 w Fu(\011)e Fr(7!)h Fu(\(\011)p Fr(j)f(2)g(B)s Fu(\()p Fr(K)q Ft(;)15 b Fj(C)p Fu(\))244 3134 y(is)30 b(an)h(isomorphism)e(b)s(et)m(w)m(een)p 1344 3061 V 31 w Fr(K)j Fu(and)e Fr(B)s Fu(\()p Fr(K)q Ft(;)15 b Fj(C)p Fu(\))26 b(=)f Fr(K)2123 3101 y Fs(\003)2163 3134 y Fu(.)380 3247 y(The)31 b(in)m(v)m(erse)h(of)f(the)g(map)g (\(4.19\),)j(whic)m(h)c(is)h(a)h(conjugation)g(on)f Fr(K)q Fu(,)h(will)f(b)s(e)g(denoted)g(b)m(y)244 3377 y(the)f(same)h(sym)m(b)s (ol.)41 b(Hence)p 1245 3285 V 1245 3303 V 31 w Fr(K)27 b Fu(=)e Fr(K)32 b Fu(and)p 1714 3285 V 1714 3303 V 29 w(\011)25 b(=)g(\011.)380 3489 y(If)30 b Ft(A)c Fr(2)e(B)s Fu(\()p Fr(K)q Fu(\),)32 b(then)p 1117 3416 69 4 v 30 w Ft(A)25 b Fr(2)g(B)s Fu(\()p 1394 3416 71 4 v Fr(K)r Fu(\))30 b(is)h(de\014ned)e(b)m(y)p 2061 3416 V 30 w Fr(K)e(3)p 2243 3416 V 25 w Fu(\011)e Fr(7!)p 2455 3416 69 4 v 25 w Ft(A)p 2553 3416 71 4 v 30 w Fu(\011)g(:=)p 2770 3416 139 4 v 25 w Ft(A)p Fu(\011)h Fr(2)p 3020 3416 71 4 v 24 w(K)r Fu(.)380 3602 y(W)-8 b(e)28 b(will)e(often)g(use)g(the) h(iden)m(ti\014cation)g(of)f(the)h(set)f(of)h(Hilb)s(ert-Sc)m(hmidt)f (op)s(erators)g Ft(l)3355 3569 y Fl(2)3394 3602 y Fu(\()p Fr(K)q Fu(\))244 3715 y(with)45 b Fr(K)31 b(\012)p 667 3642 V 30 w(K)46 b Fu(so)g(that)f Fr(j)p Fu(\010)1211 3729 y Fl(1)1251 3715 y Fu(\)\(\010)1387 3729 y Fl(2)1426 3715 y Fr(j)50 b(2)g Ft(l)1641 3682 y Fl(2)1680 3715 y Fu(\()p Fr(K)q Fu(\))c(corresp)s(onds)e(to)i(\011)2577 3729 y Fl(1)2646 3715 y Fr(\012)p 2746 3642 V 29 w Fu(\011)2817 3729 y Fl(2)2906 3715 y Fr(2)j(K)32 b(\012)p 3218 3642 V 30 w(K)q Ft(:)45 b Fu(This)244 3828 y(iden)m(ti\014cation)36 b(can)e(b)s(e)f(sometimes)i(confusing.)52 b(T)-8 b(o)34 b(a)m(v)m(oid)i(misunderstanding)c(w)m(e)i(will)h(try)244 3941 y(to)h(mak)m(e)h(clear)g(whic)m(h)f(con)m(v)m(en)m(tion)i(w)m(e)e (use)f(at)i(the)f(momen)m(t.)57 b(In)35 b(particular,)j(let)f(us)e (note)244 4054 y(that)c(the)f(follo)m(wing)i(iden)m(tities)g(hold)e (for)g Ft(B)g Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))31 b(and)f Ft(C)i Fr(2)24 b(K)e(\012)p 2674 3981 V 20 w(K)27 b(')e Ft(l)2895 4021 y Fl(2)2934 4054 y Fu(\()p Fr(K)q Fu(\):)1148 4258 y Ft(B)5 b Fr(\012)o Fu(1)p 1337 4233 59 3 v 29 x Fs(K)1426 4258 y Ft(C)32 b Fu(=)25 b Ft(B)5 b(C)q(;)167 b Fu(1)1996 4272 y Fs(K)2055 4258 y Fr(\012)p 2125 4185 74 4 v -1 w Ft(B)35 b(C)d Fu(=)25 b Ft(C)7 b(B)2568 4221 y Fs(\003)2606 4258 y Ft(:)672 b Fu(\(4.20\))244 4462 y(On)31 b(the)i(left)g(hand)e(side)h Ft(C)38 b Fu(is)33 b(in)m(terpreted)f(as)h(an)f(elemen)m(t)h(of)g Fr(K)23 b(\012)p 2665 4389 71 4 v 21 w(K)34 b Fu(and)d(on)h(the)h(righ)m(t)f (as)244 4575 y(an)e(elemen)m(t)i(of)f Ft(l)839 4542 y Fl(2)878 4575 y Fu(\()p Fr(K)q Fu(\).)380 4688 y(An)j(an)m(tiunitary)h (map)f Ft(\024)h Fu(on)f Fr(K)i Fu(suc)m(h)e(that)h Ft(\024)2002 4655 y Fl(2)2073 4688 y Fu(=)d(1)j(will)f(b)s(e)g(called)h(an)g(in)m (ternal)g(conju-)244 4801 y(gation)h(on)e Fr(K)q Fu(.)54 b(Note)36 b(that)f(if)f(w)m(e)h(ha)m(v)m(e)h(a)f(\014xed)f(in)m(ternal) h(conjugation)h Ft(\024)f Fu(on)f Fr(K)q Fu(,)i(then)f Fr(K)h Fu(is)244 4914 y(naturally)31 b(iden)m(ti\014ed)f(with)p 1239 4841 V 30 w Fr(K)r Fu(.)40 b(Therefore,)31 b(in)f(this)g(case)h(w) m(e)g(do)g(not)f(need)g(to)h(in)m(tro)s(duce)p 3438 4841 V 31 w Fr(K)q Fu(.)p eop %%Page: 19 19 19 18 bop 3682 100 a Fn(19)244 407 y Fi(4.7)135 b(The)45 b Fd(?)f Fi(conjugation)244 578 y Fu(Let)27 b Fr(K)h Fu(and)e Fr(W)33 b Fu(b)s(e)26 b(Hilb)s(ert)g(spaces.)40 b(In)26 b(this)g(subsection)h(w)m(e)g(in)m(tro)s(duce)f(a)h(certain)g (an)m(tilinear)244 691 y(map)i Ft(?)g Fu(from)g(a)g(dense)g(subspace)f (of)h Fr(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)20 b(\012)d(W)7 b Fu(\))30 b(to)g(a)f(dense)g(subspace)f(of)i Fr(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)20 b(\012)p 3378 618 98 4 v 17 w(W)7 b Fu(\).)380 804 y(Let)46 b Ft(v)53 b Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)33 b(\012)c(W)7 b Fu(\).)86 b(W)-8 b(e)46 b(sa)m(y)f(that)h Ft(v)i Fu(is)e Ft(?)f Fu(conjugable)h(if)f(there)g(exists)h Ft(v)3386 771 y Fo(?)3475 804 y Fr(2)244 917 y(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)p 634 844 V 20 w(W)8 b Fu(\))30 b(suc)m(h)g(that)h (for)f(\010)p Ft(;)15 b Fu(\011)25 b Fr(2)g(K)32 b Fu(and)e Ft(w)e Fr(2)d(W)7 b Fu(,)1344 1107 y(\(\010)p Fr(\012)o Ft(w)18 b Fr(j)d Ft(v)s Fu(\011\))26 b(=)f(\()p Ft(v)1995 1070 y Fo(?)2035 1107 y Fu(\010)p Fr(j)p Fu(\011)20 b Fr(\012)p 2308 1057 68 4 v 20 w Ft(w)r Fu(\))p Ft(:)244 1297 y Fu(If)30 b Ft(v)382 1264 y Fo(?)452 1297 y Fu(exists,)h(then)f (it)h(is)g(unique.)244 1493 y Fv(Remark)k(4.3)50 b Fm(Given)25 b(an)i(orthonormal)j(b)-5 b(asis)26 b Fr(f)p Ft(w)2041 1507 y Fo(i)2121 1493 y Fu(:)51 b Ft(i)26 b Fr(2)f Ft(I)7 b Fr(g)26 b Fm(in)g Fr(W)7 b Fm(,)27 b(any)g Ft(v)i Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)7 b(\012)f(W)h Fu(\))244 1606 y Fm(c)-5 b(an)33 b(b)-5 b(e)33 b(de)-5 b(c)g(omp)g(ose)g(d)36 b(as)1539 1728 y Ft(v)28 b Fu(=)1707 1641 y Fk(X)1719 1838 y Fo(i)p Fs(2)p Fo(I)1854 1728 y Ft(B)1923 1742 y Fo(i)1971 1728 y Fr(\012)20 b(j)p Ft(w)2152 1742 y Fo(i)2180 1728 y Fu(\))p Ft(;)1063 b Fu(\(4.21\))244 1975 y Fm(wher)-5 b(e)34 b Ft(B)570 1989 y Fo(i)623 1975 y Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))33 b Fm(and)g(the)g(sum)g(should)h (b)-5 b(e)33 b(understo)-5 b(o)g(d)35 b(in)d(terms)i(of)f(the)g(str)-5 b(ong)33 b(op)-5 b(er)g(ator)244 2088 y(c)g(onver)g(genc)g(e.)42 b(Note)31 b(that)h Ft(v)1227 2055 y Fs(\003)1267 2088 y Ft(v)c Fu(=)1435 2019 y Fk(P)1531 2115 y Fo(i)p Fs(2)p Fo(I)1658 2088 y Ft(B)1732 2055 y Fs(\003)1727 2113 y Fo(i)1770 2088 y Ft(B)1839 2102 y Fo(i)1868 2088 y Fm(.)41 b(It)31 b(is)g(e)-5 b(asy)32 b(to)f(se)-5 b(e)32 b(that)g Ft(v)i Fm(is)d Ft(?)h Fm(c)-5 b(onjugable)31 b(i\013)244 2132 y Fk(P)340 2227 y Fo(i)p Fs(2)p Fo(I)466 2201 y Ft(B)535 2215 y Fo(i)563 2201 y Ft(B)637 2168 y Fs(\003)632 2226 y Fo(i)709 2201 y Fm(is)h(b)-5 b(ounde)g(d.)44 b(If)32 b(this)h(is)g(the)g(c)-5 b(ase,)1498 2413 y Ft(v)1545 2375 y Fo(?)1610 2413 y Fu(:=)1732 2326 y Fk(X)1744 2523 y Fo(i)p Fs(2)p Fo(I)1878 2413 y Ft(B)1952 2375 y Fs(\003)1947 2435 y Fo(i)2011 2413 y Fr(\012)20 b(j)p 2127 2363 94 4 v Ft(w)2192 2427 y Fo(i)2221 2413 y Fu(\))p Ft(:)244 2690 y Fm(and)33 b Ft(v)467 2657 y Fo(?)p Fs(\003)542 2690 y Ft(v)589 2657 y Fo(?)655 2690 y Fu(=)750 2622 y Fk(P)846 2717 y Fo(i)p Fs(2)p Fo(I)973 2690 y Ft(B)1042 2704 y Fo(i)1070 2690 y Ft(B)1144 2657 y Fs(\003)1139 2715 y Fo(i)1183 2690 y Fm(.)244 2886 y Fv(Prop)s(osition)j(4.4)49 b Fm(Supp)-5 b(ose)36 b(that)g(either)f Fr(K)h Fm(or)f Fr(W)42 b Fm(is)34 b(\014nite)g(dimensional.)50 b(Then)34 b(al)5 b(l)35 b Ft(v)d Fr(2)244 2999 y(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)e(W)7 b Fu(\))34 b Fm(ar)-5 b(e)33 b Ft(?)g Fm(c)-5 b(onjugable.)42 b(Mor)-5 b(e)g(over,)34 b(if)e Ft(n)25 b Fu(:=)g(min)2377 2925 y Fk(\000)2418 2999 y Fu(\(dim)15 b Fr(K)q Fu(\))2725 2966 y Fl(2)2766 2999 y Ft(;)g Fu(dim)g Fr(W)3070 2925 y Fk(\001)3112 2999 y Fm(,)32 b(then)1593 3189 y Fr(k)p Ft(v)1685 3151 y Fo(?)1726 3189 y Fr(k)25 b(\024)1892 3119 y(p)p 1968 3119 55 4 v 70 x Ft(n)p Fr(k)p Ft(v)s Fr(k)p Ft(:)244 3575 y Fv(Pro)s(of.)70 b Fu(Clearly)-8 b(,)38 b(dim)14 b Fr(B)s Fu(\()p Fr(K)q Fu(\))34 b(=)e(\(dim)15 b Fr(K)q Fu(\))1749 3542 y Fl(2)1790 3575 y Fu(.)54 b(Therefore,)36 b(w)m(e)f(can)h(c)m(ho)s(ose)f(an)g(orthonormal)244 3688 y(system)30 b Fr(f)p Ft(w)655 3702 y Fo(i)684 3688 y Fr(g)h Fu(in)f Fr(W)37 b Fu(with)31 b(at)g(most)f Ft(n)g Fu(elemen)m(ts)i(suc)m(h)e(that)h(\(4.21\))i(is)d(true.)41 b(No)m(w)444 3901 y Fr(k)p Ft(v)536 3868 y Fo(?)576 3901 y Fr(k)621 3868 y Fl(2)686 3901 y Fu(=)782 3796 y Fk(\015)782 3851 y(\015)782 3905 y(\015)848 3832 y(P)884 3982 y Fo(i)959 3901 y Ft(B)1028 3915 y Fo(i)1056 3901 y Ft(B)1130 3868 y Fs(\003)1125 3926 y Fo(i)1169 3796 y Fk(\015)1169 3851 y(\015)1169 3905 y(\015)1303 3901 y Fr(\024)1398 3832 y Fk(P)1434 3982 y Fo(i)1510 3901 y Fr(k)p Ft(B)1624 3915 y Fo(i)1652 3901 y Ft(B)1726 3868 y Fs(\003)1721 3926 y Fo(i)1765 3901 y Fr(k)26 b Fu(=)1932 3832 y Fk(P)1968 3982 y Fo(i)2043 3901 y Fr(k)p Ft(B)2162 3868 y Fs(\003)2157 3926 y Fo(i)2201 3901 y Ft(B)2270 3915 y Fo(i)2298 3901 y Fr(k)g(\024)f Ft(n)2520 3796 y Fk(\015)2520 3851 y(\015)2520 3905 y(\015)2585 3832 y(P)2621 3982 y Fo(i)2696 3901 y Ft(B)2770 3868 y Fs(\003)2765 3926 y Fo(i)2809 3901 y Ft(B)2878 3915 y Fo(i)2907 3796 y Fk(\015)2907 3851 y(\015)2907 3905 y(\015)2982 3901 y Fu(=)g Ft(n)p Fr(k)p Ft(v)s Fr(k)3270 3868 y Fl(2)3310 3901 y Ft(:)244 4117 y Fa(2)244 4358 y Fv(Remark)35 b(4.5)50 b Fm(If)23 b Fr(W)31 b Fm(and)25 b Fr(K)h Fm(ar)-5 b(e)25 b(in\014nite)e (dimensional,)28 b(then)c(it)g(is)g(e)-5 b(asy)25 b(to)f(\014nd)h(an)f (example)244 4471 y(of)33 b Ft(v)28 b Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(W)7 b Fu(\))33 b Fm(which)g(is)g(not) g Ft(?)g Fm(c)-5 b(onjugable.)244 4667 y Fv(Notation.)39 b Fu(In)28 b(what)g(follo)m(ws,)i(if)e Ft(\032)g Fu(is)h(an)f(op)s (erator)g(on)g Fr(W)36 b Fu(and)28 b Ft(v)g Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)j(\012)e(W)7 b Fu(\),)30 b(w)m(e)e(will)244 4780 y(write)i Ft(\032v)k Fu(instead)d(of)f(1)1063 4794 y Fs(K)1122 4780 y Fr(\012)p Ft(\032)20 b(v)s Fu(.)244 4976 y Fv(Prop)s(osition)36 b(4.6)80 b Fu(\(1\))50 b Fm(If)33 b Ft(v)i Fm(is)e Ft(?)g Fm(c)-5 b(onjugable,)33 b(then)g(so)g(is)g Ft(v)2468 4943 y Fo(?)2508 4976 y Fm(;)f(mor)-5 b(e)g(over,)35 b Ft(v)3036 4943 y Fo(??)3136 4976 y Fu(=)25 b Ft(v)s Fm(.)275 5107 y Fu(\(2\))50 b Fm(If)32 b Ft(\032)25 b Fr(2)g(B)s Fu(\()p Fr(W)7 b Fu(\))p Fm(,)33 b(then)g Fu(\()p Ft(\032v)s Fu(\))1350 5075 y Fo(?)1416 5107 y Fu(=)p 1512 5057 48 4 v 25 w Ft(\032v)1606 5075 y Fo(?)1646 5107 y Fm(.)275 5239 y Fu(\(3\))50 b Fm(If)32 b Ft(B)e Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))p Fm(,)33 b(then)g Fu(\()p Ft(v)s(B)5 b Fu(\))1377 5206 y Fo(?)1442 5239 y Fu(=)25 b Ft(B)1612 5206 y Fs(\003)1651 5239 y Fr(\012)p Fu(1)p 1767 5213 80 3 v 29 x Fs(W)1880 5239 y Ft(v)1927 5206 y Fo(?)1966 5239 y Fm(.)p eop %%Page: 20 20 20 19 bop 3682 100 a Fn(20)244 407 y Fi(4.8)135 b(Coupling)46 b(Hilb)t(ert-Sc)l(hmidt)f(op)t(erators)244 578 y Fu(Let)32 b Fr(K)h Fu(and)e Fr(W)38 b Fu(b)s(e)31 b(Hilb)s(ert)h(spaces.)44 b(In)31 b(this)g(subsection)h(w)m(e)g(describ)s(e)f(some)g(notation)i (and)244 691 y(iden)m(tities)f(related)f(to)g(the)g(space)g Fr(K)21 b(\012)p 1631 618 71 4 v 20 w(K)h(\012)e(W)33 b(')25 b Ft(l)2061 658 y Fl(2)2100 691 y Fu(\()p Fr(K)q Fu(\))c Fr(\012)f(W)7 b Fu(.)380 804 y(Let)37 b Fr(H)626 818 y Fl(1)665 804 y Fu(,)h Fr(H)805 818 y Fl(2)881 804 y Fu(b)s(e)d(Hilb)s(ert)i(spaces.)58 b(If)p 1758 731 74 4 v 36 w Ft(B)40 b Fr(2)34 b(B)s Fu(\()p 2060 731 71 4 v Fr(K)r Fu(\))i(and)g Ft(A)f Fr(2)g(B)s Fu(\()p Fr(K)26 b(\012)d(H)2948 818 y Fl(1)2988 804 y Ft(;)15 b Fr(K)26 b(\012)e(H)3295 818 y Fl(2)3334 804 y Fu(\),)38 b(w)m(e)244 917 y(de\014ne)p 516 1048 74 4 v 516 1121 a Ft(B)602 1107 y Fu(\024)589 1121 y Fr(\012)p Ft(A)25 b Fu(:=)h Ft(\022)921 1084 y Fs(\000)p Fl(1)1014 1121 y Fr(\012)p Fu(1)1130 1135 y Fs(H)1190 1144 y Fh(2)p 1259 1048 V 1259 1121 a Ft(B)5 b Fr(\012)p Ft(A)30 b(\022)s Fr(\012)o Fu(1)1663 1135 y Fs(H)1723 1144 y Fh(1)1788 1121 y Fr(2)24 b(B)s Fu(\()p Fr(K)e(\012)p 2153 1048 71 4 v 20 w(K)g(\012)e(H)2412 1135 y Fl(1)2451 1121 y Ft(;)15 b Fr(K)22 b(\012)p 2673 1048 V 20 w(K)g(\012)e(H)2932 1135 y Fl(2)2971 1121 y Fu(\))p Ft(;)272 b Fu(\(4.22\))244 1326 y(where)37 b Ft(\022)i Fu(:)f Fr(K)26 b(\012)p 850 1253 V 25 w(K)39 b(!)p 1086 1253 V 37 w(K)26 b(\012)f(K)39 b Fu(is)e(de\014ned)g(as)g Ft(\022)18 b Fu(\011)2054 1340 y Fl(1)2093 1326 y Fr(\012)p 2164 1253 V Fu(\011)2234 1340 y Fl(2)2311 1326 y Fu(:=)p 2444 1253 V 37 w(\011)2515 1340 y Fl(2)2554 1326 y Fr(\012)p Fu(\011)2696 1340 y Fl(1)2735 1326 y Fu(.)62 b(In)37 b(other)h(w)m(ords,)h(if)244 1439 y Ft(C)32 b Fr(2)24 b(B)s Fu(\()p Fr(K)q Fu(\),)32 b Ft(A)25 b Fr(2)g(B)s Fu(\()p Fr(H)1040 1453 y Fl(1)1079 1439 y Ft(;)15 b Fr(H)1196 1453 y Fl(2)1236 1439 y Fu(\),)31 b(w)m(e)f(set)p 1369 1570 74 4 v 1369 1643 a Ft(B)1475 1629 y Fu(\024)1462 1643 y Fr(\012)20 b Ft(C)27 b Fr(\012)20 b Ft(A)25 b Fu(:=)g Ft(C)i Fr(\012)p 2133 1570 V 20 w Ft(B)e Fr(\012)20 b Ft(A:)380 1847 y Fu(W)-8 b(e)32 b(will)f(sometimes) g(call)h(the)e(op)s(eration)1892 1833 y(\024)1879 1847 y Fr(\012)g Fu(\\tensoring)h(in)f(the)h(middle".)380 1960 y(T)-8 b(r)32 b(denotes)h(the)g(trace.)48 b(In)31 b(the)i(con)m(text)h(of)f(coupled)f(systems)h(T)-8 b(r)32 b(will)g(b)s(e)g(reserv)m(ed)h(for)244 2073 y(the)27 b(partial)i(trace)f(o)m(v)m(er)h(the)f(space)g Fr(K)q Fu(.)40 b(T)-8 b(o)28 b(denote)g(the)f(partial)i(trace)f(o)m(v)m(er)h (the)f(space)f Fr(W)35 b Fu(w)m(e)244 2186 y(will)c(use)f(tr.)40 b(Th)m(us,)30 b(if)g Ft(C)37 b Fu(is)30 b(an)h(op)s(erator)f(on)h Fr(K)21 b(\012)f(W)7 b Fu(,)31 b(then)f(tr)p Ft(C)37 b Fu(is)30 b(an)h(op)s(erator)f(on)g Fr(K)q Fu(.)380 2299 y(The)25 b(follo)m(wing)i(prop)s(ositions)d(describ)s(e)h(some)h (algebraic)h(prop)s(erties)d(of)i(tensoring)f(in)g(the)244 2412 y(middle)552 2398 y(\024)539 2412 y Fr(\012)j Fu(and)f(the)h Ft(?)g Fu(op)s(eration,)h(whic)m(h)f(w)m(e)g(will)h(use)e(in)h(our)f (computations.)41 b(They)27 b(can)h(b)s(e)244 2524 y(skipp)s(ed)h(on)h (the)h(\014rst)e(reading.)244 2737 y Fv(Prop)s(osition)36 b(4.7)49 b Fm(L)-5 b(et)45 b Ft(A)i Fr(2)f Ft(l)1411 2704 y Fl(2)1451 2737 y Fu(\()p Fr(K)q Fu(\))p Fm(,)i Ft(B)j Fr(2)46 b Ft(l)1923 2704 y Fl(2)1963 2737 y Fu(\()p Fr(K)q Ft(;)15 b Fr(K)31 b(\012)e(W)7 b Fu(\))p Fm(,)48 b Ft(v)2560 2752 y Fl(l)2630 2737 y Fr(2)e(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)31 b(\012)e(W)7 b Fu(\))p Fm(,)48 b Ft(v)3397 2751 y Fl(r)3475 2737 y Fr(2)244 2850 y(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)27 b(\012)p 643 2777 98 4 v 24 w(W)7 b Fu(\))39 b Fm(and)h(supp)-5 b(ose)40 b(that)g Ft(v)1571 2864 y Fl(r)1641 2850 y Fm(is)f Ft(?)g Fm(c)-5 b(onjugable.)60 b(Then)39 b(the)g(fol)5 b(lowing)40 b(statements)244 2963 y(hold:)1561 3050 y Fu(1)p 1606 3025 59 3 v 29 x Fs(K)1677 3036 y Fu(\024)1664 3050 y Fr(\012)p Ft(v)1779 3065 y Fl(l)1823 3050 y Ft(A)25 b Fu(=)g Ft(v)2056 3065 y Fl(l)2080 3050 y Ft(A;)1561 3234 y Fu(1)p 1606 3209 V 29 x Fs(K)1677 3220 y Fu(\024)1664 3234 y Fr(\012)p Ft(v)1782 3201 y Fs(\003)1779 3262 y Fl(l)1842 3234 y Ft(B)30 b Fu(=)25 b Ft(v)2084 3201 y Fs(\003)2081 3262 y Fl(l)2123 3234 y Ft(B)5 b(;)1561 3418 y Fu(1)1606 3432 y Fs(K)1664 3418 y Fr(\012)p 1735 3368 48 4 v Ft(v)1782 3432 y Fl(r)1834 3418 y Ft(A)26 b Fu(=)f Ft(A)p Fr(\012)o Fu(1)2207 3432 y Fs(W)2320 3418 y Ft(v)2367 3385 y Fo(?)2364 3440 y Fl(r)2407 3418 y Ft(;)1561 3601 y Fu(1)1606 3615 y Fs(K)1664 3601 y Fr(\012)p 1735 3551 V Ft(v)1782 3568 y Fs(\003)1782 3624 y Fl(r)1842 3601 y Ft(B)30 b Fu(=)25 b(tr)p Ft(B)5 b(v)2229 3568 y Fo(?)p Fs(\003)2226 3624 y Fl(r)2303 3601 y Ft(;)3303 3361 y Fu(\(4.23\))244 3849 y Fm(wher)-5 b(e)34 b(on)f(the)g(left)f(we)h(use)f(the)h Fr(K)22 b(\012)p 1565 3776 71 4 v 20 w(K)34 b Fm(notation)h(and)e(on)g (the)g(right)h(the)f Ft(l)2886 3816 y Fl(2)2925 3849 y Fu(\()p Fr(K)q Fu(\))h Fm(notation.)244 4061 y Fv(Pro)s(of.)70 b Fu(It)34 b(is)h(su\016cien)m(t)g(to)g(pro)m(v)m(e)g(the)g(statemen)m (t)h(for)f Ft(v)2319 4076 y Fl(l)2374 4061 y Fu(=)d Ft(C)e Fr(\012)22 b(j)p Ft(w)r Fu(\))36 b(and)e Ft(v)3053 4075 y Fl(r)3117 4061 y Fu(=)d Ft(C)f Fr(\012)23 b(j)p 3433 4011 68 4 v Ft(w)r Fu(\))244 4174 y(where)30 b Ft(C)h Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))31 b(and)f Ft(w)e Fr(2)d(W)7 b Fu(.)41 b(Then)1403 4376 y(1)p 1448 4350 59 3 v 29 x Fs(K)1520 4362 y Fu(\024)1507 4376 y Fr(\012)p Ft(v)1622 4391 y Fl(l)1671 4376 y Fu(=)25 b Ft(C)i Fr(\012)19 b Fu(1)p 1994 4350 V 29 x Fs(K)2073 4376 y Fr(\012)h(j)p Ft(w)r Fu(\))p Ft(;)1403 4559 y Fu(1)p 1448 4534 V 29 x Fs(K)1520 4545 y Fu(\024)1507 4559 y Fr(\012)p Ft(v)1625 4526 y Fs(\003)1622 4587 y Fl(l)1690 4559 y Fu(=)25 b Ft(C)1858 4526 y Fs(\003)1917 4559 y Fr(\012)20 b Fu(1)p 2053 4534 V 29 x Fs(K)2132 4559 y Fr(\012)g Fu(\()p Ft(w)r Fr(j)p Ft(;)1403 4744 y Fu(1)1448 4758 y Fs(K)1507 4744 y Fr(\012)p 1578 4694 48 4 v Ft(v)1625 4758 y Fl(r)1682 4744 y Fu(=)25 b(1)1823 4758 y Fs(K)1902 4744 y Fr(\012)p 1993 4671 72 4 v 20 w Ft(C)i Fr(\012)19 b(j)p Ft(w)r Fu(\))p Ft(;)1403 4934 y Fu(1)1448 4948 y Fs(K)1507 4934 y Fr(\012)p 1578 4884 48 4 v Ft(v)1625 4901 y Fs(\003)1625 4957 y Fl(r)1690 4934 y Fu(=)25 b(1)1831 4948 y Fs(K)1910 4934 y Fr(\012)p 2001 4861 72 4 v 20 w Ft(C)2072 4880 y Fs(\003)2132 4934 y Fr(\012)20 b Fu(\()p Ft(w)r Fr(j)p Ft(:)3303 4655 y Fu(\(4.24\))p eop %%Page: 21 21 21 20 bop 3682 100 a Fn(21)244 407 y Fu(W)-8 b(e)31 b(ma)m(y)g(also)h (assume)e(that)h Ft(B)f Fu(=)25 b Ft(D)e Fr(\012)c(j)p Ft(w)1764 421 y Fl(0)1804 407 y Fu(\))31 b(for)f(some)h Ft(w)2302 421 y Fl(0)2367 407 y Fr(2)24 b(W)7 b Fu(.)41 b(Using)31 b(\(4.20\))i(w)m(e)d(get)836 570 y(1)p 881 544 59 3 v 29 x Fs(K)952 556 y Fu(\024)939 570 y Fr(\012)p Ft(v)1054 585 y Fl(l)1098 570 y Ft(A)83 b Fu(=)25 b Ft(C)7 b(A)20 b Fr(\012)g(j)p Ft(w)r Fu(\))177 b(=)25 b Ft(C)7 b Fr(\012)o(j)p Ft(w)r Fu(\))22 b Ft(A;)811 753 y Fu(1)p 856 728 V 29 x Fs(K)928 739 y Fu(\024)915 753 y Fr(\012)p Ft(v)1033 720 y Fs(\003)1030 781 y Fl(l)1093 753 y Ft(B)87 b Fu(=)25 b Ft(C)1417 720 y Fs(\003)1456 753 y Ft(D)18 b Fu(\()p Ft(w)r Fr(j)p Ft(w)1741 767 y Fl(0)1782 753 y Fu(\))83 b(=)25 b Ft(C)2068 720 y Fs(\003)2107 753 y Fr(\012)p Fu(\()p Ft(w)r Fr(j)31 b Ft(D)s Fr(\012j)p Ft(w)2575 767 y Fl(0)2614 753 y Fu(\))p Ft(;)825 940 y Fu(1)870 954 y Fs(K)928 940 y Fr(\012)p 999 890 48 4 v Ft(v)1046 954 y Fl(r)1098 940 y Ft(A)83 b Fu(=)25 b Ft(AC)1485 907 y Fs(\003)1545 940 y Fr(\012)19 b(j)p Ft(w)r Fu(\))138 b(=)25 b Ft(A)p Fr(\012)p Fu(1)2180 954 y Fs(W)2290 866 y Fk(\000)2332 940 y Ft(C)7 b Fr(\012)o(j)p 2499 890 68 4 v Ft(w)s Fu(\))2602 866 y Fk(\001)2644 889 y Fo(?)2684 940 y Ft(;)811 1127 y Fu(1)856 1141 y Fs(K)915 1127 y Fr(\012)p 986 1076 48 4 v Ft(v)1033 1094 y Fs(\003)1033 1149 y Fl(r)1093 1127 y Ft(B)87 b Fu(=)25 b Ft(D)s(C)7 b Fu(\()p Ft(w)r Fr(j)p Ft(w)1687 1141 y Fl(0)1727 1127 y Fu(\))138 b(=)25 b(tr)2082 1053 y Fk(\000)2124 1127 y Ft(D)s Fr(\012)o(j)p Ft(w)2362 1141 y Fl(0)2402 1127 y Fu(\))2458 1053 y Fk(\000)2499 1127 y Ft(C)7 b Fr(\012j)p 2667 1076 68 4 v Ft(w)r Fu(\))2769 1053 y Fk(\001)2811 1076 y Fo(?)p Fs(\003)2886 1053 y Fk(\001)2943 1127 y Ft(:)3303 848 y Fu(\(4.25\))244 1262 y Fa(2)244 1466 y Fv(Prop)s(osition)36 b(4.8)49 b Fm(L)-5 b(et)34 b Ft(A)27 b Fr(2)g Ft(l)1361 1433 y Fl(2)1400 1466 y Fu(\()p Fr(K)q Fu(\))p Fm(,)35 b Ft(\032)27 b Fr(2)g(B)s Fu(\()p Fr(W)7 b Fu(\))p Fm(,)34 b Ft(v)2101 1481 y Fl(l)2151 1466 y Fr(2)27 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)e(W)7 b Fu(\))34 b Fm(and)g Ft(v)3018 1480 y Fl(r)3077 1466 y Fr(2)27 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)p 244 1506 98 4 v 244 1579 a(W)7 b Fu(\))p Fm(.)42 b(Supp)-5 b(ose)34 b(that)g Ft(v)1022 1594 y Fl(l)1046 1579 y Fm(,)e Ft(v)1150 1593 y Fl(r)1215 1579 y Fm(ar)-5 b(e)33 b Ft(?)g Fm(c)-5 b(onjugable.)42 b(Then)972 1745 y Fu(1)1017 1759 y Fs(K)1076 1745 y Fr(\012)p 1146 1695 48 4 v -1 w Ft(v)1194 1707 y Fs(\003)1194 1767 y Fl(r)1266 1745 y Ft(\032)32 b Fu(1)p 1390 1719 59 3 v 29 x Fs(K)1461 1731 y Fu(\024)1449 1745 y Fr(\012)p Ft(v)1564 1760 y Fl(l)1620 1745 y Ft(A)25 b Fu(=)g(tr)p Ft(\032v)1971 1760 y Fl(l)1995 1745 y Ft(Av)2110 1707 y Fo(?)p Fs(\003)2107 1767 y Fl(r)2211 1745 y Fu(=)g Ft(v)2354 1707 y Fo(?)p Fs(\003)2351 1768 y Fl(l)2449 1745 y Ft(A)p Fr(\012)p 2588 1695 48 4 v Ft(\032)2635 1707 y Fs(\003)2707 1745 y Ft(v)2751 1759 y Fl(r)2782 1745 y Ft(;)496 b Fu(\(4.26\))975 1910 y(1)p 1020 1885 59 3 v 29 x Fs(K)1091 1896 y Fu(\024)1079 1910 y Fr(\012)p Ft(v)1197 1873 y Fs(\003)1194 1933 y Fl(l)1257 1910 y Ft(\032)20 b Fu(1)1369 1924 y Fs(K)1427 1910 y Fr(\012)p 1498 1860 48 4 v Ft(v)1545 1924 y Fl(r)1610 1910 y Ft(A)25 b Fu(=)g(tr)p Ft(v)1917 1873 y Fo(?)1914 1933 y Fl(l)1977 1910 y Ft(A)33 b(v)2125 1873 y Fs(\003)2122 1933 y Fl(r)p 2164 1860 V 2164 1910 a Ft(\032)2211 1873 y Fs(\003)2276 1910 y Fu(=)25 b Ft(v)2419 1873 y Fs(\003)2416 1933 y Fl(l)2474 1910 y Ft(A)p Fr(\012)p Ft(\032)32 b(v)2739 1873 y Fo(?)2736 1933 y Fl(r)2779 1910 y Ft(:)499 b Fu(\(4.27\))244 2058 y Fm(If)32 b Ft(v)383 2073 y Fl(l)p Fo(;)p Fl(1)462 2058 y Ft(;)15 b(v)546 2073 y Fl(l)p Fo(;)p Fl(2)650 2058 y Fr(2)25 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(W)7 b Fu(\))33 b Fm(ar)-5 b(e)33 b Ft(?)g Fm(c)-5 b(onjugable,)33 b(then)910 2223 y Fu(1)p 955 2198 59 3 v 29 x Fs(K)1026 2209 y Fu(\024)1013 2223 y Fr(\012)p Ft(v)1131 2186 y Fs(\003)1128 2246 y Fl(l)p Fo(;)p Fl(1)1227 2223 y Ft(\032)20 b Fu(1)p 1339 2198 V 29 x Fs(K)1410 2209 y Fu(\024)1398 2223 y Fr(\012)p Ft(v)1513 2238 y Fl(l)p Fo(;)p Fl(2)1611 2223 y Ft(A)26 b Fu(=)f(tr)p Ft(v)1919 2186 y Fo(?)1916 2246 y Fl(l)p Fo(;)p Fl(1)1995 2223 y Ft(v)2042 2186 y Fo(?)p Fs(\003)2039 2246 y Fl(l)p Fo(;)p Fl(2)2137 2223 y Ft(A)p Fr(\012)p 2276 2173 48 4 v Ft(\032)2323 2186 y Fs(\003)2388 2223 y Fu(=)g Ft(v)2531 2186 y Fs(\003)2528 2246 y Fl(l)p Fo(;)p Fl(1)2607 2223 y Ft(\032v)2698 2238 y Fl(l)p Fo(;)p Fl(2)2776 2223 y Ft(A:)434 b Fu(\(4.28\))244 2401 y Fm(If)32 b Ft(v)383 2415 y Fl(r)p Fo(;)p Fl(1)470 2401 y Ft(;)15 b(v)554 2415 y Fl(r)p Fo(;)p Fl(2)666 2401 y Fr(2)25 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)p 1143 2327 98 4 v 20 w(W)7 b Fu(\))33 b Fm(ar)-5 b(e)33 b Ft(?)g Fm(c)-5 b(onjugable,)33 b(then)883 2566 y Fu(1)928 2580 y Fs(K)986 2566 y Fr(\012)p 1057 2516 48 4 v Ft(v)1104 2529 y Fs(\003)1104 2589 y Fl(r)p Fo(;)p Fl(1)1211 2566 y Ft(\032)20 b Fu(1)1323 2580 y Fs(K)1382 2566 y Fr(\012)p 1453 2516 V Ft(v)1500 2580 y Fl(r)p Fo(;)p Fl(2)1607 2566 y Ft(A)25 b Fu(=)g(tr)p Ft(A)p Fr(\012)p Ft(\032)20 b(v)2120 2529 y Fo(?)2117 2589 y Fl(r)p Fo(;)p Fl(2)2204 2566 y Ft(v)2251 2529 y Fo(?)p Fs(\003)2248 2589 y Fl(r)p Fo(;)p Fl(1)2360 2566 y Fu(=)25 b Ft(Av)2571 2529 y Fs(\003)2568 2589 y Fl(r)p Fo(;)p Fl(2)p 2655 2516 V 2655 2566 a Ft(\032)2702 2529 y Fs(\003)2741 2566 y Ft(v)2785 2580 y Fl(r)p Fo(;)p Fl(1)2872 2566 y Ft(:)406 b Fu(\(4.29\))244 2899 y Fv(Pro)s(of.)61 b Fu(W)-8 b(e)32 b(will)f(pro)m(v)m(e)g(only)f(\(4.26\).)43 b(First)31 b(note)g(that)986 3065 y(1)1031 3079 y Fs(K)1090 3065 y Fr(\012)p 1160 3015 V -1 w Ft(v)1208 3027 y Fs(\003)1208 3087 y Fl(r)1267 3065 y Ft(\032)21 b Fu(1)p 1380 3039 59 3 v 29 x Fs(K)1451 3051 y Fu(\024)1438 3065 y Fr(\012)p Ft(v)1553 3080 y Fl(l)1597 3065 y Ft(A)k Fu(=)g(1)1831 3079 y Fs(K)1890 3065 y Fr(\012)p 1961 3015 48 4 v Ft(v)2008 3027 y Fo(?)2008 3087 y Fl(r)2068 3065 y Ft(\032)20 b(v)2179 3080 y Fl(l)2203 3065 y Ft(A)25 b Fu(=)g(tr)p Ft(\032v)2554 3080 y Fl(l)2578 3065 y Ft(Av)2693 3027 y Fo(?)p Fs(\003)2690 3087 y Fl(r)2768 3065 y Ft(:)510 b Fu(\(4.30\))380 3230 y(W)-8 b(e)33 b(tak)m(e)h Ft(v)781 3245 y Fl(l)833 3230 y Fu(=)27 b Ft(C)996 3245 y Fl(l)1041 3230 y Fr(\012)21 b(j)p Ft(w)1223 3245 y Fl(l)1247 3230 y Fu(\))33 b(and)e Ft(v)1537 3244 y Fl(r)1597 3230 y Fu(=)c Ft(C)1760 3244 y Fl(r)1813 3230 y Fr(\012)21 b(j)p 1930 3180 68 4 v Ft(w)1998 3244 y Fl(r)2030 3230 y Fu(\))32 b(for)g(some)g Ft(C)2532 3245 y Fl(l)2556 3230 y Ft(;)15 b(C)2661 3244 y Fl(r)2721 3230 y Fr(2)27 b(B)s Fu(\()p Fr(K)q Fu(\),)34 b Ft(w)3136 3245 y Fl(l)3160 3230 y Ft(;)15 b(w)3265 3244 y Fl(r)3325 3230 y Fr(2)27 b(W)7 b Fu(.)244 3343 y(Then)29 b(\(4.30\))k(is)d(equal)h(to)497 3531 y(tr)568 3430 y Fk(\020)622 3531 y Ft(C)687 3546 y Fl(l)711 3531 y Fr(\012)o(j)p Ft(\032w)918 3546 y Fl(l)943 3531 y Fu(\))20 b Ft(A)1086 3458 y Fk(\000)1128 3531 y Ft(C)1193 3545 y Fl(r)1245 3531 y Fr(\012)g(j)p 1361 3481 V Ft(w)1429 3545 y Fl(r)1460 3531 y Fu(\))1495 3458 y Fk(\001)1537 3480 y Fo(?)p Fs(\003)1612 3430 y Fk(\021)1692 3531 y Fu(=)25 b(tr)1859 3430 y Fk(\020)1913 3531 y Ft(C)1978 3546 y Fl(l)2002 3531 y Fr(\012)o(j)p Ft(\032w)2209 3546 y Fl(l)2233 3531 y Fu(\))c Ft(A)f(C)2442 3545 y Fl(r)2474 3531 y Fr(\012)p Fu(\()p Ft(w)2645 3545 y Fl(r)2677 3531 y Fr(j)2702 3430 y Fk(\021)497 3715 y Fu(=)25 b Ft(C)658 3730 y Fl(l)681 3715 y Ft(AC)814 3729 y Fl(r)861 3715 y Fu(tr)932 3641 y Fk(\000)974 3715 y Fr(j)p Ft(\032w)1111 3730 y Fl(l)1135 3715 y Fu(\)\()p Ft(w)1270 3729 y Fl(r)1303 3715 y Fr(j)1328 3641 y Fk(\001)1395 3715 y Fu(=)g Ft(C)1556 3730 y Fl(l)1579 3715 y Ft(AC)1712 3729 y Fl(r)1744 3715 y Fu(\()p 1779 3665 V Ft(w)1847 3730 y Fl(l)1871 3715 y Fr(j)p 1896 3665 48 4 v Ft(\032)1943 3682 y Fs(\003)p 1983 3665 68 4 v 1983 3715 a Ft(w)2050 3729 y Fl(r)2082 3715 y Fu(\))497 3902 y(=)g Ft(C)658 3917 y Fl(l)681 3902 y Fr(\012)p Fu(\()p 787 3852 V Ft(w)855 3917 y Fl(l)879 3902 y Fr(j)30 b Ft(A)p Fr(\012)p 1073 3852 48 4 v Ft(\032)1120 3869 y Fs(\003)1190 3902 y Ft(C)1255 3916 y Fl(r)1287 3902 y Fr(\012)o(j)p 1382 3852 68 4 v Ft(w)1450 3916 y Fl(r)1482 3902 y Fu(\))25 b(=)1638 3828 y Fk(\000)1680 3902 y Ft(C)1745 3917 y Fl(l)1789 3902 y Fr(\012)20 b(j)p Ft(w)1970 3917 y Fl(l)1994 3902 y Fu(\))2029 3828 y Fk(\001)2071 3851 y Fo(?)p Fs(\003)2146 3902 y Ft(A)p Fr(\012)p 2285 3852 48 4 v Ft(\032)2332 3869 y Fs(\003)2386 3902 y Ft(C)2451 3916 y Fl(r)2483 3902 y Fr(\012)o(j)p 2578 3852 68 4 v Ft(w)2646 3916 y Fl(r)2678 3902 y Fu(\))25 b(=)g Ft(v)2881 3869 y Fo(?)p Fs(\003)2878 3929 y Fl(l)2956 3902 y Ft(A)p Fr(\012)p 3095 3852 48 4 v Ft(\032)3142 3869 y Fs(\003)3182 3902 y Ft(v)3226 3916 y Fl(r)3258 3902 y Ft(:)244 4037 y Fa(2)244 4387 y Fq(5)161 b(Abstract)53 b(P)l(auli-Fierz)f(op)t (erators)244 4590 y Fu(In)44 b(this)h(section)h(w)m(e)f(\014rst)f(in)m (tro)s(duce)h(the)g(notation)h(that)g(w)m(e)f(will)g(use)g(to)g (describ)s(e)f(the)244 4703 y(in)m(teraction)34 b(of)f(a)f(second-quan) m(tized)i(system)e(with)g(another)h(system.)47 b(Then)31 b(w)m(e)i(in)m(tro)s(duce)244 4816 y(the)27 b(class)h(of)f(P)m (auli-Fierz)j(op)s(erators.)40 b(W)-8 b(e)28 b(also)g(describ)s(e)f(a)g (n)m(um)m(b)s(er)f(of)h(results)g(ab)s(out)g(these)244 4929 y(op)s(erators)f(con)m(tained)i(in)e(the)h(literature,)i(esp)s (ecially)e(in)f([DJ1)q(],)i(whic)m(h)e(w)m(e)h(will)g(use)f(later)h (on.)380 5042 y(In)e(this)h(section)h(w)m(e)g(lo)s(ok)f(at)h(P)m (auli-Fierz)h(op)s(erators)e(just)f(as)i(a)f(certain)h(class)g(of)f (abstract)244 5155 y(self-adjoin)m(t)g(op)s(erators.)39 b(Only)24 b(in)g(the)h(next)g(t)m(w)m(o)h(sections)f(w)m(e)g(will)g (put)f(them)h(in)f(the)h(con)m(text)244 5268 y(of)30 b Ft(W)446 5235 y Fs(\003)485 5268 y Fu(-dynamical)i(systems.)p eop %%Page: 22 22 22 21 bop 3682 100 a Fn(22)244 407 y Fi(5.1)135 b (Creation/annihilation)38 b(op)t(erators)d(in)f(coupled)g(systems)244 578 y Fu(Supp)s(ose)d(that)j Fr(W)41 b Fu(is)33 b(a)g(Hilb)s(ert)h (space.)50 b(\000)1767 592 y Fl(s)1798 578 y Fu(\()p Fr(W)7 b Fu(\))31 b(=)f Fr(\010)2168 545 y Fs(1)2168 603 y Fo(n)p Fl(=0)2305 578 y Fu(\000)2362 545 y Fo(n)2362 601 y Fl(s)2409 578 y Fu(\()p Fr(W)7 b Fu(\))34 b(will)f(denote)h(the)g (b)s(osonic)244 691 y(F)-8 b(o)s(c)m(k)32 b(space)f(o)m(v)m(er)g Fr(W)38 b Fu(\(see)31 b(eg.)42 b([BR2)q(],)31 b(also)g([DJ1)q(]\).)380 804 y(Consider)i(another)h(Hilb)s(ert)g(space)h Fr(E)8 b Fu(.)51 b(In)33 b(this)h(subsection)g(w)m(e)g(discuss)f(the)h (formalism)244 917 y(of)c(the)h(coupled)f(system)h(describ)s(ed)e(b)m (y)h(the)h(Hilb)s(ert)f(space)h Fr(E)e(\012)19 b Fu(\000)2604 931 y Fl(s)2636 917 y Fu(\()p Fr(W)7 b Fu(\).)380 1030 y(Let)36 b Ft(q)f Fr(2)d(B)s Fu(\()p Fr(E)8 b Ft(;)15 b Fr(E)32 b(\012)23 b(W)7 b Fu(\).)55 b(The)34 b(creation)i(and)e (annihilation)i(op)s(erators)f(asso)s(ciated)h(to)g Ft(q)244 1143 y Fu(w)m(ere)26 b(in)m(tro)s(duced)f(in)g([DG)q(,)h(DJ1].)39 b(Since)26 b(w)m(e)g(will)g(use)f(a)h(somewhat)f(di\013eren)m(t)h (notation)h(from)244 1256 y([DJ1)q(],)k(w)m(e)f(discuss)g(these)h (notions)f(in)h(detail.)380 1369 y(W)-8 b(e)41 b(de\014ne)f(the)g (creation)h(op)s(erator)f Ft(q)s Fu(\()p Ft(a)1846 1336 y Fs(\003)1886 1369 y Fu(\))g(as)h(the)f(\(un)m(b)s(ounded\))e (quadratic)i(form)g(on)244 1482 y Fr(E)29 b(\012)20 b Fu(\000)469 1496 y Fl(s)501 1482 y Fu(\()p Fr(W)7 b Fu(\))32 b(whose)f(only)h(nonzero)f(matrix)h(elemen)m(ts)h(are)e(b)s(et)m(w)m (een)h(\011)2754 1496 y Fo(n)p Fl(+1)2918 1482 y Fr(2)26 b(E)j(\012)21 b Fu(\000)3231 1449 y Fo(n)p Fl(+1)3231 1504 y(s)3368 1482 y Fu(\()p Fr(W)7 b Fu(\))244 1595 y(and)30 b(\011)492 1609 y Fo(n)564 1595 y Fr(2)24 b(E)29 b(\012)20 b Fu(\000)874 1562 y Fo(n)874 1617 y Fl(s)920 1595 y Fu(\()p Fr(W)7 b Fu(\),)32 b(for)e Ft(n)25 b Fr(\025)g Fu(0)p Ft(;)31 b Fu(and)f(are)g(equal)992 1770 y(\(\011)1098 1784 y Fo(n)p Fl(+1)1235 1770 y Fr(j)p Ft(q)s Fu(\()p Ft(a)1387 1733 y Fs(\003)1427 1770 y Fu(\)\011)1533 1784 y Fo(n)1580 1770 y Fu(\))c(:=)1762 1694 y Fr(p)p 1838 1694 212 4 v 76 x Ft(n)19 b Fu(+)h(1)q(\(\011)2155 1784 y Fo(n)p Fl(+1)2292 1770 y Fr(j)15 b Ft(q)s Fr(\012)p Fu(1)2492 1732 y Fs(\012)p Fo(n)2492 1799 y Fs(W)2609 1770 y Fu(\011)2680 1784 y Fo(n)2727 1770 y Fu(\))p Ft(:)244 1946 y Fu(The)31 b(annihilation)h(op)s(erator)g Ft(q)1350 1913 y Fs(\003)1390 1946 y Fu(\()p Ft(a)p Fu(\))g(is)g(de\014ned)e(as)i Ft(q)s Fu(\()p Ft(a)2187 1913 y Fs(\003)2227 1946 y Fu(\))2262 1913 y Fs(\003)2329 1946 y Fu(=)27 b Ft(q)2471 1913 y Fs(\003)2510 1946 y Fu(\()p Ft(a)p Fu(\).)45 b(Note)33 b(that)g(b)s(oth)e Ft(q)s Fu(\()p Ft(a)3461 1913 y Fs(\003)3500 1946 y Fu(\))244 2059 y(and)f Ft(q)465 2026 y Fs(\003)504 2059 y Fu(\()p Ft(a)p Fu(\))h(are)g(closed.)244 2238 y Fv(Remark)k(5.1)50 b Fm(In)32 b Fu([DJ1)q(])h Ft(q)s Fu(\()p Ft(a)1314 2205 y Fs(\003)1354 2238 y Fu(\))f Fm(and)i Ft(q)1642 2205 y Fs(\003)1681 2238 y Fu(\()p Ft(a)p Fu(\))g Fm(wer)-5 b(e)33 b(denote)-5 b(d)34 b Ft(a)2423 2205 y Fs(\003)2463 2238 y Fu(\()p Ft(q)s Fu(\))f Fm(and)g Ft(a)p Fu(\()p Ft(q)s Fu(\))g Fm(r)-5 b(esp)g(e)g(ctively.)380 2417 y Fu(F)d(or)31 b(further)e(reference,)j(let)f(us)e(note)j(the)e (follo)m(wing)i(straigh)m(tforw)m(ard)f(facts:)244 2596 y Fv(Prop)s(osition)36 b(5.2)80 b Fu(\(1\))50 b Fm(L)-5 b(et)33 b Fu(\011)25 b Fr(2)g(E)j(\012)20 b Fu(\000)1750 2563 y Fo(n)1750 2619 y Fl(s)1797 2596 y Fu(\()p Fr(W)7 b Fu(\))p Fm(.)43 b(Then)33 b(the)g(fol)5 b(lowing)33 b(estimates)h(hold:)916 2772 y Fr(k)p Ft(q)s Fu(\()p Ft(a)1088 2735 y Fs(\003)1128 2772 y Fu(\)\011)p Fr(k)26 b(\024)1401 2696 y(p)p 1477 2696 V 76 x Ft(n)20 b Fu(+)f(1)q Fr(k)p Ft(q)s Fr(kk)p Fu(\011)p Fr(k)p Ft(;)114 b Fr(k)p Ft(q)2211 2735 y Fs(\003)2251 2772 y Fu(\()p Ft(a)p Fu(\)\011)p Fr(k)26 b(\024)2607 2702 y(p)p 2683 2702 55 4 v 70 x Ft(n)o Fr(k)p Ft(q)s Fr(kk)p Fu(\011)p Fr(k)p Ft(:)246 b Fu(\(5.31\))275 2964 y(\(2\))50 b Fm(If)32 b Fu(\011)606 2978 y Fl(0)670 2964 y Fr(2)25 b(E)k(\012)20 b Fu(\000)981 2931 y Fl(0)981 2986 y(s)1020 2964 y Fu(\()p Fr(W)7 b Fu(\))p Fm(,)33 b(then)g Ft(q)s Fu(\()p Ft(a)1577 2931 y Fs(\003)1617 2964 y Fu(\)\011)1723 2978 y Fl(0)1788 2964 y Fu(=)25 b Ft(q)s Fu(\011)1999 2978 y Fl(0)2063 2964 y Fr(2)g(E)j(\012)20 b Fu(\000)2373 2931 y Fl(1)2373 2986 y(s)2412 2964 y Fu(\()p Fr(W)7 b Fu(\))p Fm(.)275 3093 y Fu(\(3\))50 b Fm(If)32 b Fu(\011)606 3107 y Fl(1)670 3093 y Fr(2)25 b(E)k(\012)20 b Fu(\000)981 3060 y Fl(1)981 3115 y(s)1020 3093 y Fu(\()p Fr(W)7 b Fu(\))p Fm(,)33 b(then)g Ft(q)1494 3060 y Fs(\003)1534 3093 y Fu(\()p Ft(a)p Fu(\)\011)1723 3107 y Fl(1)1788 3093 y Fu(=)25 b Ft(q)1928 3060 y Fs(\003)1967 3093 y Fu(\011)2038 3107 y Fl(1)2102 3093 y Fr(2)g(E)k(\012)19 b Fu(\000)2412 3060 y Fl(0)2412 3115 y(s)2452 3093 y Fu(\()p Fr(W)7 b Fu(\))p Fm(.)244 3365 y Fi(5.2)135 b(P)l(auli-Fierz)46 b(op)t(erators)244 3536 y Fu(Let)40 b Fr(E)47 b Fu(and)38 b Fr(W)47 b Fu(b)s(e)38 b(as)i(ab)s(o)m(v)m(e.)68 b(F)-8 b(rom)40 b(no)m(w)f(on)g(w)m(e)h(will)g(alw)m(a)m(ys)g(assume)f(that)h Fr(E)47 b Fu(is)40 b(\014nite)244 3649 y(dimensional.)g(Let)30 b Ft(E)35 b Fu(b)s(e)29 b(a)h(self-adjoin)m(t)g(op)s(erator)g(on)f Fr(E)38 b Fu(and)29 b Ft(r)i Fu(a)f(self-adjoin)m(t)h(op)s(erator)f(on) 244 3762 y Fr(W)7 b Fu(.)41 b(A)30 b(self-adjoin)m(t)i(op)s(erator)f (on)f Fr(H)c Fu(:=)f Fr(E)j(\012)20 b Fu(\000)1914 3776 y Fl(s)1946 3762 y Fu(\()p Fr(W)7 b Fu(\))31 b(of)g(the)f(form)1400 3938 y Ft(L)1462 3953 y Fl(fr)1540 3938 y Fu(:=)25 b Ft(E)h Fr(\012)20 b Fu(1)h(+)e(1)i Fr(\012)f Fu(d\000\()p Ft(r)s Fu(\))244 4114 y(will)31 b(b)s(e)e(called)j(a)f(free)f(P)m (auli-Fierz)j(op)s(erator.)380 4226 y(F)-8 b(or)31 b(a)g(giv)m(en)h Ft(q)c Fr(2)c(B)s Fu(\()p Fr(E)8 b Ft(;)15 b Fr(E)29 b(\012)20 b(W)7 b Fu(\))31 b(the)g(P)m(auli-Fierz)h(in)m(teraction)g (is)f(de\014ned)e(b)m(y)1523 4402 y Ft(Q)c Fu(=)g Ft(q)s Fu(\()p Ft(a)1843 4365 y Fs(\003)1883 4402 y Fu(\))20 b(+)g Ft(q)2073 4365 y Fs(\003)2113 4402 y Fu(\()p Ft(a)p Fu(\))p Ft(:)244 4578 y Fu(It)k(follo)m(ws)h(from)f(Nelson's)h(theorem) f(on)g(analytic)i(v)m(ectors)f(that)g Ft(Q)f Fu(is)g(essen)m(tially)i (self-adjoin)m(t)244 4691 y(on)k Fr(E)e(\012)20 b Fu(\000)594 4658 y Fl(\014n)594 4713 y(s)677 4691 y Fu(\()p Fr(W)7 b Fu(\).)41 b(The)30 b(op)s(erator)1597 4866 y Ft(L)c Fu(:=)f Ft(L)1868 4881 y Fl(fr)1941 4866 y Fu(+)20 b Ft(\025Q;)244 5042 y Fu(where)30 b Ft(\025)25 b Fr(2)g Fj(R)p Fu(,)30 b(will)h(b)s(e)f(called)h(a)g(P)m(auli-Fierz)h(op)s (erator.)380 5155 y(W)-8 b(e)36 b(kno)m(w)g(t)m(w)m(o)g(sets)f(of)h (assumptions)e(that)i(guaran)m(tee)g(the)f(self-adjoin)m(tness)h(of)g (P)m(auli-)244 5268 y(Fierz)31 b(op)s(erators.)p eop %%Page: 23 23 23 22 bop 3682 100 a Fn(23)244 407 y Fv(Theorem)70 b(5.3)80 b Fu(\(1\))50 b Fm(If)26 b Ft(r)i Fr(\025)d Fu(0)i Fm(and)h Ft(r)1642 371 y Fs(\000)1707 344 y Fh(1)p 1706 356 31 3 v 1706 397 a(2)1751 407 y Ft(q)i Fm(is)c(b)-5 b(ounde)g(d,)30 b(then)d Ft(L)g Fm(is)f(self-adjoint)i(on)f Fr(D)s Fu(\()p Ft(L)3419 422 y Fl(fr)3472 407 y Fu(\))p Fm(.)275 540 y Fu(\(2\))50 b Fm(If)32 b Fr(j)p Ft(r)s Fr(j)p Ft(q)j Fm(is)e(b)-5 b(ounde)g(d,)34 b(then)f Ft(L)g Fm(is)f(essential)5 b(ly)34 b(self-adjoint)f(on)g Fr(D)s Fu(\()p Ft(L)2784 555 y Fl(fr)2837 540 y Fu(\))21 b Fr(\\)f(D)s Fu(\()p Ft(Q)p Fu(\))p Fm(.)380 728 y Fu(The)27 b(pro)s(of)g(of)h(\(1\))g(can)g (b)s(e)f(found)f(in)h(Prop)s(osition)h(5.2)g(of)g([DJ1)q(])f(and)g(the) h(pro)s(of)f(of)g(\(2\))i(in)244 841 y(Prop)s(osition)j(5.1)i(of)f ([DJ1)q(].)47 b(W)-8 b(e)34 b(remark)e(that)i(\(1\))f(has)g(b)s(een)e (kno)m(wn)h(for)h(a)g(long)g(time,)h(see)244 954 y(e.g.)42 b([BFS1)q(].)f(The)29 b(part)i(\(2\))g(w)m(as)g(\014rst)e(pro)m(v)m(en) i(in)f([JP1].)244 1231 y Fi(5.3)135 b(Lev)l(el)46 b(Shift)f(Op)t (erator)h(for)f(P)l(auli-Fierz)g(op)t(erators)244 1402 y Fu(W)-8 b(e)34 b(start)g(b)m(y)e(the)h(description)g(of)h(a)f (condition)g(whic)m(h)g(pla)m(ys)g(a)h(cen)m(tral)g(role)g(in)e(our)h (study)-8 b(.)244 1515 y(This)29 b(condition)i(w)m(as)g(in)m(tro)s (duced)f([JP1,)h(JP2])f(and)g(w)m(as)h(also)g(used)f(in)g([DJ1)q(,)g(M) q(].)380 1628 y(W)-8 b(e)30 b(assume)f(that)h(there)f(exists)g(a)g (Hilb)s(ert)g(space)h Fr(G)k Fu(and)28 b(a)i(unitary)e(op)s(erator)h Ft(U)36 b Fu(:)25 b Fr(W)33 b(!)244 1741 y Ft(L)306 1708 y Fl(2)345 1741 y Fu(\()p Fj(R)p Fu(\))27 b Fr(\012)e(G)45 b Fu(suc)m(h)39 b(that)g(the)h(op)s(erator)f Ft(U)10 b(r)s(U)1851 1708 y Fs(\003)1929 1741 y Fu(is)39 b(the)h(op)s(erator)f (of)g(m)m(ultiplication)i(b)m(y)e(the)244 1854 y(v)-5 b(ariable)37 b(in)f Fj(R)p Fu(.)58 b(W)-8 b(e)38 b(\014x)e(suc)m(h)g (an)g(op)s(erator)h Ft(U)46 b Fu(and)36 b(iden)m(tify)g Fr(W)43 b(\021)35 b Ft(L)2794 1821 y Fl(2)2833 1854 y Fu(\()p Fj(R)p Fu(\))25 b Fr(\012)f(G)5 b Fu(.)59 b(W)-8 b(e)37 b(will)244 1967 y(often)h(mak)m(e)h(use)f(of)g(the)g (self-adjoin)m(t)i(op)s(erator)e Ft(s)g Fu(:=)f Fr(\000)p Fu(i)p Ft(@)2374 1981 y Fo(p)2440 1967 y Fr(\012)25 b Fu(1)2581 1981 y Fs(G)2669 1967 y Fu(in)m(tro)s(duced)37 b(already)i(in)244 2080 y(Subsection)30 b(4.4.)380 2193 y(Let)j Fr(H)623 2160 y Fl(v)695 2193 y Fu(:=)c Fr(E)h(\012)21 b Fu(\000)1047 2160 y Fl(0)1047 2215 y(s)1087 2193 y Fu(\()p Fr(W)7 b Fu(\))33 b(b)s(e)f(the)h(distinguished)f(subspace)g (of)h Fr(H)d Fu(:=)f Fr(E)h(\012)22 b Fu(\000)3074 2207 y Fl(s)3105 2193 y Fu(\()p Fr(W)7 b Fu(\).)49 b(Note)244 2305 y(that)31 b(the)f(map)1448 2418 y Fr(E)k(3)24 b Fu(\011)h Fr(7!)h Fu(\011)19 b Fr(\012)h Fu(\012)25 b Fr(2)g(H)2264 2381 y Fl(v)2306 2418 y Ft(;)244 2585 y Fu(iden)m(ti\014es)k Fr(E)36 b Fu(with)28 b Fr(H)991 2552 y Fl(v)1033 2585 y Fu(.)40 b(Lik)m(ewise,)30 b(the)f(op)s(erator)g Ft(L)2069 2600 y Fl(fr)2150 2585 y Fu(preserv)m(es)g(the)f(subspace)g Fr(H)3149 2552 y Fl(v)3219 2585 y Fu(and)g Ft(L)3456 2552 y Fl(vv)3456 2613 y(fr)244 2698 y Fu(is)i(iden)m(ti\014ed)h(with)f (the)g(op)s(erator)h Ft(E)36 b Fu(on)30 b Fr(E)8 b Fu(.)41 b(Note)31 b(also)h(that)f Ft(Q)2484 2665 y Fl(vv)2588 2698 y Fu(=)25 b(0)31 b(and)f Ft(Q)p 3009 2628 39 3 v -33 x Fl(v)q(v)3115 2698 y Fu(=)25 b Ft(q)s Fu(.)380 2811 y(F)-8 b(or)31 b Ft(z)f Fr(2)25 b Fj(C)767 2825 y Fl(+)856 2811 y Fu(set)1201 2919 y Ft(w)r Fu(\()p Ft(z)t Fu(\))84 b(:=)25 b Ft(Q)1661 2886 y Fl(v)p 1699 2848 V 1 w(v)1742 2919 y Fu(\()p Ft(z)t Fu(1)p 1868 2848 V -33 x Fl(v)p 1907 2848 V 2 w(v)1970 2919 y Fr(\000)20 b Ft(L)p 2123 2848 V -33 x Fl(v)p 2161 2848 V 1 w(v)2123 2947 y(fr)2203 2919 y Fu(\))2238 2886 y Fs(\000)p Fl(1)2333 2919 y Ft(Q)p 2405 2848 V -33 x Fl(v)q(v)1468 3103 y Fu(=)25 b Ft(q)1608 3070 y Fs(\003)1647 3103 y Fu(\()p Ft(z)g Fr(\000)20 b Ft(E)26 b Fr(\012)20 b Fu(1)g Fr(\000)g Fu(1)h Fr(\012)f Ft(r)s Fu(\))2416 3070 y Fs(\000)p Fl(1)2510 3103 y Ft(q)s(:)380 3267 y Fu(The)34 b(next)g(prop)s(osition)g(follo)m (ws)h(from)f(Prop)s(osition)g(4.2.)53 b(\(It)35 b(is)f(also)h(a)g(sp)s (ecial)f(case)h(of)244 3380 y(Theorem)30 b(6.1)h(in)f([DJ1)q(]\).)244 3593 y Fv(Prop)s(osition)36 b(5.4)49 b Fm(Assume)37 b(that)h Fr(h)p Ft(s)p Fr(i)1638 3560 y Fo(\021)1680 3593 y Ft(q)e Fr(2)c(B)s Fu(\()p Fr(E)8 b Ft(;)15 b Fr(E)32 b(\012)23 b(W)7 b Fu(\))37 b Fm(for)g(some)h Ft(\021)e(>)d Fu(1)p Ft(=)p Fu(2)p Fm(.)55 b(Then)37 b(the)244 3705 y(function)c Fj(C)668 3719 y Fl(+)752 3705 y Fr(3)24 b Ft(z)30 b Fr(7!)25 b Ft(w)r Fu(\()p Ft(z)t Fu(\))34 b Fm(extends)f(by)g(c)-5 b(ontinuity)33 b(to)h(a)f(c)-5 b(ontinuous)33 b(function)g(on)g Fj(C)3311 3672 y Fl(cl)3311 3728 y(+)3370 3705 y Fm(.)380 3918 y Fu(Under)h(the)h(condition)g(of)g(this)f(prop)s(osition)g(the)h (Lev)m(el)h(Shift)e(Op)s(erator)g(for)g(the)h(triple)244 4031 y(\()p Fr(E)29 b(\012)19 b Fu(\000)503 3998 y Fl(0)503 4053 y(s)543 4031 y Fu(\()p Fr(W)7 b Fu(\))p Ft(;)15 b(L)812 4046 y Fl(fr)866 4031 y Ft(;)g(Q)p Fu(\))31 b(is)g(w)m(ell)g (de\014ned)e(and)h(is)g(equal)h(to)953 4249 y(\000)25 b(=)1255 4162 y Fk(X)1131 4364 y Fo(e)1164 4373 y Fh(1)1199 4364 y Fo(;e)1252 4373 y Fh(2)1286 4364 y Fs(2)p Fl(sp\()p Fo(E)t Fl(\))1510 4249 y Fu(\()p Ft(q)1589 4211 y Fs(\003)1629 4249 y Fu(\))1664 4211 y Fo(e)1697 4220 y Fh(1)1731 4211 y Fo(;e)1784 4220 y Fh(2)1822 4249 y Fu(\()p Ft(e)1899 4263 y Fl(1)1960 4249 y Fr(\000)20 b Ft(e)2093 4263 y Fl(2)2153 4249 y Fu(+)g(i0)h Fr(\000)e Ft(r)s Fu(\))2504 4211 y Fs(\000)p Fl(1)2599 4249 y Ft(q)2643 4211 y Fo(e)2676 4220 y Fh(2)2710 4211 y Fo(;e)2763 4220 y Fh(1)2801 4249 y Ft(;)477 b Fu(\(5.32\))244 4552 y(where)1303 4665 y Ft(q)1347 4627 y Fo(e)1380 4636 y Fh(1)1414 4627 y Fo(;e)1467 4636 y Fh(2)1530 4665 y Fu(:=)25 b(1)1696 4679 y Fo(e)1729 4688 y Fh(1)1768 4665 y Fu(\()p Ft(E)5 b Fu(\))p Fr(\012)q Fu(1)2027 4679 y Fs(W)2128 4665 y Ft(q)23 b Fu(1)2237 4679 y Fo(e)2270 4688 y Fh(2)2308 4665 y Fu(\()p Ft(E)5 b Fu(\))p Ft(:)p eop %%Page: 24 24 24 23 bop 3682 100 a Fn(24)244 407 y Fi(5.4)135 b(Sp)t(ectral)45 b(theory)h(of)f(P)l(auli-Fierz)h(op)t(erators)244 578 y Fu(The)30 b(follo)m(wing)i(theorem)e(is)h(a)g(consequence)g(of)f(the) h(main)f(results)g(of)h([DJ1)q(].)244 791 y Fv(Theorem)70 b(5.5)49 b Fm(L)-5 b(et)33 b Ft(\021)c(>)c Fu(2)p Fm(,)32 b Ft(\017)26 b(>)f Fu(0)p Fm(,)32 b Ft(C)g(>)25 b Fu(0)33 b Fm(and)h Ft(q)28 b Fr(2)d(B)s Fu(\()p Fr(E)8 b Ft(;)15 b Fr(E)28 b(\012)20 b(W)7 b Fu(\))33 b Fm(b)-5 b(e)33 b(such)g(that:)244 904 y Fu(\(a\))h Ft(L)e Fm(is)h(essential)5 b(ly)33 b(self-adjoint)h(on)f Fr(D)s Fu(\()p Ft(L)1798 919 y Fl(fr)1851 904 y Fu(\))20 b Fr(\\)g(D)s Fu(\()p Ft(Q)p Fu(\))33 b Fm(for)g(al)5 b(l)33 b Ft(\025)p Fm(.)244 1017 y Fu(\(b\))g Fr(kh)p Ft(s)p Fr(i)556 984 y Fo(\021)598 1017 y Ft(q)s Fr(k)26 b(\024)f Ft(C)7 b Fm(.)244 1130 y Fu(\(c\))34 b(\000)445 1097 y Fl(I)499 1130 y Fr(\024)25 b(\000)p Ft(\017)p Fu(\(1)c Fr(\000)f Fu(1)940 1144 y Fl(0)980 1130 y Fu(\(\000)1072 1097 y Fl(I)1101 1130 y Fu(\)\))p Fm(.)244 1243 y(Then)37 b(ther)-5 b(e)38 b(exists)g Ft(\025)1023 1257 y Fl(0)1096 1243 y Ft(>)33 b Fu(0)p Fm(,)39 b(which)f(dep)-5 b(ends)38 b(on)g Ft(q)i Fm(only)e(thr)-5 b(ough)39 b Ft(\021)s Fm(,)f Ft(\017)f Fm(and)h Ft(C)7 b Fm(,)38 b(such)f(that)244 1356 y(for)c Fu(0)26 b Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)e(\025)833 1370 y Fl(0)905 1356 y Fm(the)33 b(fol)5 b(lowing)34 b(holds:)244 1468 y Fu(\(1\))g(sp)479 1490 y Fl(sc)542 1468 y Fu(\()p Ft(L)p Fu(\))26 b(=)f Fr(;)p Fm(.)244 1581 y Fu(\(2\))34 b(dim)14 b(1)604 1548 y Fl(p)648 1581 y Fu(\()p Ft(L)p Fu(\))26 b Fr(\024)f Fu(dim)14 b(1)1113 1595 y Fl(0)1153 1581 y Fu(\(\000)1245 1548 y Fl(I)1275 1581 y Fu(\))p Fm(.)244 1794 y Fv(Remark.)59 b Fu(If)36 b(\(c\))i(is)f(replaced)g(with)f(the)h(condition)g(\000)2240 1761 y Fl(I)2305 1794 y Ft(<)e Fr(\000)p Ft(\017)p Fu(,)j(then)e(all)h (the)g(conclusions)244 1907 y(of)h(the)g(theorem)g(hold)f(under)f(the)i (w)m(eak)m(er)h(assumption)e Ft(\021)k(>)c Fu(1.)64 b(Moreo)m(v)m(er,) 42 b(in)37 b(this)h(case)244 2020 y(1)289 2034 y Fl(0)329 2020 y Fu(\(\000)421 1987 y Fl(I)451 2020 y Fu(\))25 b(=)g(0,)31 b(and)f(w)m(e)h(conclude)f(that)h Ft(L)g Fu(has)f(no)g(p)s(oin)m(t)g(sp)s(ectrum.)244 2133 y Fv(Pro)s(of)f(of)f (Theorem)g(5.5.)39 b Fu(By)24 b(Theorems)g(6.2,)j(6.3)f(and)d(6.4)j(of) e([DJ1)q(],)i(there)f(exists)g Ft(\025)3330 2147 y Fl(0)3394 2133 y Ft(>)g Fu(0)244 2246 y(suc)m(h)35 b(that)h(for)f(0)f Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)1277 2260 y Fl(0)1317 2246 y Fu(,)k(sp)1465 2267 y Fl(sc)1528 2246 y Fu(\()p Ft(L)p Fu(\))e(=)e Fr(;)i Fu(and)g(dim)15 b(1)2273 2213 y Fl(p)2317 2246 y Fu(\()p Ft(L)p Fu(\))34 b Fr(\024)f Fu(dim)15 b(1)2799 2260 y Fe(R)2851 2246 y Fu(\(\000\).)56 b(Since)36 b(\000)f(is)g(a)244 2358 y(dissipativ)m(e)e(op)s(erator,)h (dim)15 b(1)1298 2372 y Fe(R)1350 2358 y Fu(\(\000\))29 b Fr(\024)g Fu(dim)14 b(1)1817 2372 y Fl(0)1857 2358 y Fu(\(\000)1949 2325 y Fl(I)1979 2358 y Fu(\))33 b(\(see)g(Prop)s (osition)f(3.2)i(of)e([DJ1)q(]\).)48 b(Hence)244 2471 y(dim)15 b(1)456 2438 y Fl(p)499 2471 y Fu(\()p Ft(L)p Fu(\))40 b Fr(\024)f Fu(dim)15 b(1)993 2485 y Fl(0)1033 2471 y Fu(\(\000)1125 2438 y Fl(I)1155 2471 y Fu(\).)66 b(The)38 b(pro)s(ofs)g(of)h(Theorems)g(6.2,)j(6.3)e(and)e(6.4)i(yield)f (that)g(the)244 2584 y(constan)m(t)32 b Ft(\025)664 2598 y Fl(0)733 2584 y Fu(dep)s(ends)d(only)h(on)h Ft(\021)s Fu(,)g Ft(\017)f Fu(and)g Ft(C)7 b Fu(.)40 b Fa(2)380 2747 y Fu(Let)c(us)e(note)i(that)g(in)f([DJ1)q(])g(w)m(e)g(actually)i (pro)m(v)m(ed)f(m)m(uc)m(h)f(more)g(than)g(what)g(w)m(e)h(stated)244 2860 y(ab)s(o)m(v)m(e.)51 b(The)33 b(follo)m(wing)i(theorem,)f(adapted) g(from)f([DJ1)q(],)h(expresses)f(in)g(precise)h(terms)f(the)244 2973 y(in)m(tuition)e(that)g(the)g(op)s(erator)1495 3099 y Fk(X)1375 3301 y Fo(m)p Fs(2)p Fl(sp)q(\(\000\))p Fs(\\)p Fe(R)1761 3186 y Fu(1)1806 3200 y Fo(m)1873 3186 y Fu(\(\000\)\()p Ft(E)c Fu(+)19 b Ft(\025)2272 3148 y Fl(2)2312 3186 y Fu(\000\))244 3489 y(predicts)k(the)g(appro)m(ximate)i(lo)s(cation)f (of)g(eigen)m(v)-5 b(alues)25 b(of)e Ft(L)g Fu(and)g(estimates)h(from)f (ab)s(o)m(v)m(e)i(their)244 3602 y(m)m(ultiplicit)m(y)-8 b(.)380 3715 y(F)g(or)31 b Ft(x)26 b Fr(2)e Fj(R)30 b Fu(and)g Ft(\017)25 b(>)g Fu(0,)31 b(w)m(e)g(set)g Ft(I)7 b Fu(\()p Ft(x;)15 b(\017)p Fu(\))26 b(:=)g([)p Ft(x)20 b Fr(\000)g Ft(\017;)15 b(x)20 b Fu(+)g Ft(\017)p Fu(].)244 3927 y Fv(Theorem)70 b(5.6)49 b Fm(Supp)-5 b(ose)42 b(that)f Fr(kh)p Ft(s)p Fr(i)1610 3894 y Fo(\021)1653 3927 y Ft(q)s Fr(k)e Ft(<)g Fr(1)h Fm(for)g(some)h Ft(\021)h(>)d Fu(2)p Fm(,)j(that)g Ft(L)e Fm(is)g(essential)5 b(ly)244 4040 y(self-adjoint)43 b(on)h Fr(D)s Fu(\()p Ft(L)1036 4055 y Fl(fr)1089 4040 y Fu(\))27 b Fr(\\)h(D)s Fu(\()p Ft(Q)p Fu(\))42 b Fm(for)h(al)5 b(l)44 b Ft(\025)e Fm(and)i(let)e Ft(\024)i Fu(=)f(1)28 b Fr(\000)f Ft(\021)2641 4007 y Fs(\000)p Fl(1)2736 4040 y Fm(.)72 b(Then)43 b(ther)-5 b(e)43 b(exists)244 4153 y Ft(\025)297 4167 y Fl(0)362 4153 y Ft(>)25 b Fu(0)33 b Fm(and)g Ft(\013)26 b(>)f Fu(0)33 b Fm(such)g(that)g(for)h Fu(0)25 b Ft(<)g Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)1952 4167 y Fl(0)1991 4153 y Fm(,)33 b(the)g(fol)5 b(lowing)34 b(holds:)244 4266 y Fu(\(1\))g Fm(If)e Ft(e)26 b Fr(2)e Fu(sp\()p Ft(E)5 b Fu(\))33 b Fm(and)h Ft(m)25 b Fr(2)g Fu(sp)o(\(\000)1449 4233 y Fo(ee)1519 4266 y Fu(\))c Fr(\\)e Fj(R)p Fm(,)32 b(then)1109 4470 y Fu(dim)14 b(1)1320 4426 y Fl(p)1320 4505 y Fo(I)5 b Fl(\()p Fo(e)p Fl(+)p Fo(\025)1512 4486 y Fh(2)1547 4505 y Fo(m;)12 b(\013)p Fs(j)p Fo(\025)p Fs(j)1767 4486 y Fh(2+)p Fg(\024)1884 4505 y Fl(\))1916 4470 y Fu(\()p Ft(L)p Fu(\))26 b Fr(\024)f Fu(dim)14 b(1)2381 4484 y Fo(m)2448 4470 y Fu(\(\000)2540 4433 y Fo(ee)2610 4470 y Fu(\))p Ft(:)244 4689 y Fu(\(2\))918 4815 y(sp)1005 4837 y Fl(p)1048 4815 y Fu(\()p Ft(L)p Fu(\))26 b Fr(\032)1380 4729 y Fk([)1302 4930 y Fo(e)p Fs(2)p Fl(sp\()p Fo(E)t Fl(\))1773 4729 y Fk([)1607 4930 y Fo(m)p Fs(2)p Fe(R)p Fs(\\)p Fl(sp\(\000)1949 4911 y Fg(ee)2012 4930 y Fl(\))2055 4815 y Ft(I)7 b Fu(\()p Ft(e)21 b Fu(+)f Ft(\025)2344 4778 y Fl(2)2383 4815 y Ft(m;)15 b(\013)p Fr(j)p Ft(\025)p Fr(j)2664 4778 y Fl(2+)p Fo(\024)2801 4815 y Fu(\))p Ft(:)p eop %%Page: 25 25 25 24 bop 3682 100 a Fn(25)244 407 y Fq(6)161 b(P)l(auli-Fierz)53 b(systems)244 610 y Fu(In)37 b(this)i(section)g(w)m(e)g(consider)f(a)h (certain)g(class)h(of)e Ft(W)2193 577 y Fs(\003)2232 610 y Fu(-dynamical)h(systems)g(that)f(w)m(e)h(call)244 723 y(P)m(auli-Fierz)32 b(systems.)380 836 y(Subsections)f(6.1-6.4)j (are)e(dev)m(oted)g(to)g(zero)h(temp)s(erature)e(P)m(auli-Fierz)i (systems.)44 b(F)-8 b(rom)244 948 y(the)27 b(algebraic)i(p)s(oin)m(t)e (of)g(view)g(they)h(are)f(the)g(simplest)h(class)f(of)h(P)m(auli-Fierz) h(systems.)39 b(Their)244 1061 y Ft(W)343 1028 y Fs(\003)382 1061 y Fu(-algebras)25 b(are)g(t)m(yp)s(e)g(I)f(factors|they)h(are)g (just)f Fr(B)s Fu(\()p Fr(H)q Fu(\).)38 b(The)24 b(generators)h(of)g (their)f(dynam-)244 1174 y(ics)36 b(in)f(the)h(irreducible)f(represen)m (tation)i(are)e(giv)m(en)i(b)m(y)e(b)s(ounded)f(from)h(b)s(elo)m(w)h(P) m(auli-Fierz)244 1287 y(op)s(erators.)60 b(Suc)m(h)36 b(op)s(erators)g(will)h(b)s(e)f(called)i(P)m(auli-Fierz)h (Hamiltonians.)61 b(A)37 b(P)m(auli-Fierz)244 1400 y(Hamiltonian)f(is)f (completely)i(determined)d(b)m(y)h(the)g(data)h(\()p Fr(K)q Ft(;)15 b(K)q(;)g Fr(Z)7 b Ft(;)15 b(h;)g(v)s Fu(\),)40 b(where)35 b Ft(K)41 b Fu(is)35 b(the)244 1513 y(energy)26 b(op)s(erator)g(of)g(the)g(small)g(system)g(on)g(the)g 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f(on)f(a)h(Hilb)s(ert)g(space)g Fr(Z)48 b Fu(and)244 4214 y Ft(v)28 b Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(Z)7 b Fu(\).)41 b(The)30 b(self-adjoin)m(t)h(op)s (erator)1383 4384 y Ft(H)1459 4399 y Fl(fr)1537 4384 y Fu(:=)25 b Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(d\000\()p Ft(h)p Fu(\))244 4553 y(on)43 b Fr(K)31 b(\012)d Fu(\000)639 4567 y Fl(s)671 4553 y Fu(\()p Fr(Z)7 b Fu(\))44 b(will)g(b)s(e)f(called)h(a)g(free)g(P)m(auli-Fierz)h (Hamiltonian.)81 b(The)43 b(in)m(teraction)j(is)244 4666 y(describ)s(ed)29 b(b)m(y)h(the)h(self-adjoin)m(t)h(op)s(erator)1519 4836 y Ft(V)45 b Fu(=)25 b Ft(v)s Fu(\()p Ft(a)1843 4798 y Fs(\003)1883 4836 y Fu(\))c(+)f Ft(v)2077 4798 y Fs(\003)2117 4836 y Fu(\()p Ft(a)p Fu(\))p Ft(:)244 5005 y Fu(The)30 b(op)s(erator)1587 5118 y Ft(H)i Fu(:=)25 b Ft(H)1892 5133 y Fl(fr)1965 5118 y Fu(+)20 b Ft(\025V)5 b(;)244 5268 y Fu(where)30 b Ft(\025)25 b Fr(2)g Fj(R)p Fu(,)30 b(is)h(called)g(a)g(P)m(auli-Fierz)h(Hamiltonian.)42 b(W)-8 b(e)32 b(will)f(need)p eop %%Page: 26 26 26 25 bop 3682 100 a Fn(26)244 407 y Fv(Assumption)36 b(6.A)49 b Ft(h)1085 371 y Fs(\000)1150 344 y Fh(1)p 1150 356 31 3 v 1150 397 a(2)1194 407 y Ft(v)29 b Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)c(Z)7 b Fu(\))p Fm(.)380 594 y Fu(By)33 b(Theorem)g(5.3,)i(Assumption)d(6.A)i(implies)f (that)g Ft(H)40 b Fu(is)33 b(self-adjoin)m(t)h(on)f Fr(D)s Fu(\()p Ft(H)3268 609 y Fl(fr)3321 594 y Fu(\))g(and)244 707 y(b)s(ounded)28 b(from)i(b)s(elo)m(w.)244 980 y Fi(6.2)135 b(Gluing)45 b(of)g(reserv)l(oir)i(1-particle)f(spaces)244 1151 y Fu(An)30 b(imp)s(ortan)m(t)h(role)g(in)f(our)g(pap)s(er)g(will)g (b)s(e)g(pla)m(y)m(ed)i(b)m(y)e(the)h(Hilb)s(ert)f(space)i Fr(Z)27 b(\010)p 3129 1078 74 4 v 20 w(Z)37 b Fu(and)30 b(the)244 1264 y(self-adjoin)m(t)i(op)s(erator)e Ft(r)e Fu(:=)d Ft(h)c Fr(\010)f Fu(\()p Fr(\000)p 1538 1190 53 4 v Ft(h)p Fu(\).)41 b(The)30 b(follo)m(wing)i(op)s(erators:)922 1447 y Fr(Z)27 b(\010)p 1106 1374 74 4 v 20 w(Z)33 b(3)24 b Fu(\()p Ft(z)1367 1461 y Fl(1)1407 1447 y Ft(;)p 1447 1397 47 4 v 15 w(z)1494 1461 y Fl(2)1534 1447 y Fu(\))h Fr(7!)g Ft(\034)10 b Fu(\()p Ft(z)1837 1461 y Fl(1)1877 1447 y Ft(;)p 1917 1397 V 15 w(z)1964 1461 y Fl(2)2004 1447 y Fu(\))25 b(:=)g(\()p 2220 1397 V Ft(z)2267 1461 y Fl(2)2306 1447 y Ft(;)15 b(z)2388 1461 y Fl(1)2429 1447 y Fu(\))25 b Fr(2)p 2575 1374 74 4 v 25 w(Z)i(\010)20 b(Z)7 b Ft(;)446 b Fu(\(6.33\))929 1629 y Fr(Z)27 b(\010)p 1113 1556 V 20 w(Z)32 b(3)25 b Fu(\()p Ft(z)1374 1643 y Fl(1)1414 1629 y Ft(;)p 1454 1579 47 4 v 15 w(z)1501 1643 y Fl(2)1540 1629 y Fu(\))h Fr(7!)f Ft(\017)p Fu(\()p Ft(z)1831 1643 y Fl(1)1871 1629 y Ft(;)p 1911 1579 V 15 w(z)1958 1643 y Fl(2)1997 1629 y Fu(\))h(:=)f(\()p Ft(z)2256 1643 y Fl(2)2296 1629 y Ft(;)p 2336 1579 V 15 w(z)2383 1643 y Fl(1)2422 1629 y Fu(\))h Fr(2)e(Z)k(\010)p 2753 1556 74 4 v 20 w(Z)7 b Ft(;)452 b Fu(\(6.34\))244 1785 y(will)31 b(b)s(e)e(also)j(useful.)40 b(Note)31 b(that)g Ft(\034)41 b Fu(is)30 b(linear,)h Ft(\017)f 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Fu(In)j(what)g(follo)m(ws)h(w)m(e)g(assume)f (that)h(Assumption)f(6.B)h(holds)f(and)f(w)m(e)i(iden)m(tify)g Fr(Z)30 b(\010)p 3463 2425 V 24 w(Z)244 2610 y Fu(with)g Ft(L)513 2578 y Fl(2)552 2610 y Fu(\()p Fj(R)p Fu(\))21 b Fr(\012)f(G)35 b Fu(and)30 b Ft(r)j Fu(with)d(the)h(m)m (ultiplication)h(op)s(erator)f(\()p Ft(r)s Fu(\011\)\()p Ft(p)p Fu(\))25 b(:=)h Ft(p)p Fu(\011\()p Ft(p)p Fu(\).)380 2723 y(Let)g(us)f(note)g(that)h Fr(Z)33 b Fu(is)25 b(iden)m(ti\014ed)g (with)g Ft(L)1878 2690 y Fl(2)1917 2723 y Fu(\()p Fj(R)2018 2737 y Fl(+)2078 2723 y Fu(\))10 b Fr(\012)g(G)30 b Fu(and)25 b Ft(h)g Fu(with)g(1)2784 2742 y Fl([0)p Fo(;)p Fs(1)p Fl([)2954 2723 y Fu(\()p Ft(r)s Fu(\))p Ft(r)s Fu(.)38 b(Lik)m(ewise,)p 244 2774 V 244 2847 a Fr(Z)f Fu(is)31 b(iden)m(ti\014ed)f(with)g Ft(L)1104 2814 y Fl(2)1144 2847 y Fu(\()p Fj(R)1245 2861 y Fs(\000)1304 2847 y Fu(\))20 b Fr(\012)g(G)5 b Fu(.)41 b(Th)m(us)1262 3029 y(\()p Ft(\024)p Fu(\011\)\()p Ft(p)p Fu(\))27 b(:=)p 1719 2950 258 4 v 25 w(\011\()p 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Fr(Z)i(\010)p 1978 3929 V 20 w(Z)7 b Fu(\))26 b Fr(')f(K)d(\012)e Ft(L)2452 3964 y Fl(2)2491 4002 y Fu(\()p Fj(R)p Fu(\))h Fr(\012)e(G)5 b Ft(:)244 4184 y Fu(This)34 b(op)s(erator)i(can)f(b)s(e) f(also)i(written)f(as)h(a)f(function)g(de\014ned)f(for)h(almost)h(all)g Ft(p)d Fr(2)f Fj(R)j Fu(with)244 4297 y(v)-5 b(alues)31 b(in)f Fr(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)e(G)5 b Fu(\):)1331 4556 y(\()p Ft(v)s(;)15 b Fu(0\)\()p Ft(p)p Fu(\))28 b(=)1773 4401 y Fk(\()1887 4476 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)84 b(p)25 b(>)g Fu(0;)1887 4636 y(0)p Ft(;)202 b(p)25 b Fr(\024)g Fu(0)p Ft(:)3303 4556 y Fu(\(6.36\))380 4816 y(Let)31 b Ft(s)f Fu(b)s(e)g(as)g(in)g(\(4.14\).)43 b(In)30 b(the)h(follo)m(wing)g(assumption)f(w)m(e)h(ha)m(v)m(e)h Ft(\021)c Fr(\025)d Fu(0.)244 4986 y Fv(Assumption)36 b(6.C)s Fu(\()p Ft(\021)s Fu(\))1102 5000 y Fl(0)1393 4986 y Fr(h)p Ft(s)p Fr(i)1506 4953 y Fo(\021)1548 4986 y Fu(\()p Ft(v)s(;)15 b Fu(0\))28 b Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)q(\012)r Fu(\()p Fr(Z)27 b(\010)p 2433 4913 V 20 w(Z)7 b Fu(\)\))p Fm(.)244 5155 y Fu(This)32 b(assumption)g(will)h(b)s(e)g(used)f(in)g(the)h(next)g(section)h(in)f (the)g(analysis)g(of)g(the)g(Lev)m(el)h(Shift)244 5268 y(Op)s(erator)c(for)g(P)m(auli-Fierz)i(Hamiltonians.)p eop %%Page: 27 27 27 26 bop 3682 100 a Fn(27)244 407 y Fi(6.3)135 b(Lev)l(el)46 b(Shift)f(Op)t(erator)h(for)f(P)l(auli-Fierz)g(Hamiltonians)244 578 y Fu(In)19 b(this)i(subsection)f(w)m(e)h(will)f(calculate)j(the)d (Lev)m(el)i(Shift)e(Op)s(erator)f(for)h(the)h(triple)f(\()p Fr(K)q(\012)q Fu(\000)3307 545 y Fl(0)3307 601 y(s)3346 578 y Fu(\()p Fr(Z)7 b Fu(\))p Ft(;)15 b(H)3605 593 y Fl(fr)3659 578 y Ft(;)g(V)21 b Fu(\).)244 691 y(W)-8 b(e)30 b(will)g(use)f(the)g(notation)i(in)m(tro)s(duced)e(in)g (Subsection)g(4.3.)41 b(In)29 b(particular,)h(w)m(e)f(recall)i(that) 1307 868 y Ft(v)1354 830 y Fo(k)1391 839 y Fh(1)1426 830 y Fo(;k)1483 839 y Fh(2)1546 868 y Fu(=)25 b(1)1687 883 y Fo(k)1724 892 y Fh(1)1763 868 y Fu(\()p Ft(K)7 b Fu(\))p Fr(\012)p Fu(1)2033 882 y Fs(Z)2109 868 y Ft(v)19 b Fu(1)2217 883 y Fo(k)2254 892 y Fh(2)2293 868 y Fu(\()p Ft(K)7 b Fu(\))p Ft(:)380 1045 y Fu(Let)34 b Fr(F)611 1060 y Fo(k)684 1045 y Fu(:=)d Fr(f)p Ft(k)26 b Fr(\000)c Ft(k)1069 1059 y Fl(1)1139 1045 y Fu(:)30 b Ft(k)1241 1059 y Fl(1)1311 1045 y Fr(2)g Fu(sp\()p Ft(K)7 b Fu(\))p Fr(g)34 b Fu(b)s(e)f(the)g(set)h(of)g(allo)m(w)m(ed)h(transition)f (energies)g(from)244 1158 y(the)f(lev)m(el)h Ft(k)e Fr(2)d Fu(sp)o(\()p Ft(K)7 b Fu(\).)48 b(The)32 b(set)h(of)g(p)s(ositiv)m(e)g (and)f(negativ)m(e)j(transition)e(energies)h(from)e Ft(k)k Fu(is)244 1270 y(denoted)30 b(b)m(y)h Fr(F)788 1232 y Fl(+)779 1300 y Fo(k)872 1270 y Fu(:=)25 b Fr(F)1058 1285 y Fo(k)1101 1270 y Fr(\\)15 b Fu(]0)p Ft(;)g Fr(1)p Fu([)32 b(and)e Fr(F)1686 1232 y Fs(\000)1677 1300 y Fo(k)1770 1270 y Fu(:=)25 b Fr(F)1956 1285 y Fo(k)1999 1270 y Fr(\\)15 b Fu(])21 b Fr(\000)f(1)p Ft(;)15 b Fu(0[)31 b(resp)s(ectiv)m(ely)-8 b(.)42 b(W)-8 b(e)32 b(also)f(set)1108 1467 y Fr(F)k Fu(:=)1414 1381 y Fk([)1329 1582 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1616 1467 y Fr(F)1681 1482 y Fo(k)1724 1467 y Ft(;)197 b Fr(F)2020 1430 y Fl(+)2105 1467 y Fu(:=)2311 1381 y Fk([)2226 1582 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))2513 1467 y Fr(F)2587 1429 y Fl(+)2578 1497 y Fo(k)2646 1467 y Ft(:)244 1756 y Fu(Let)31 b Fr(H)485 1723 y Fl(v)552 1756 y Fu(:=)25 b Fr(K)d(\012)e Fu(\000)912 1723 y Fl(0)912 1778 y(s)951 1756 y Fu(\()p Fr(Z)7 b Fu(\))31 b(and)1187 1930 y Ft(w)r Fu(\()p Ft(z)t Fu(\))85 b(:=)25 b Ft(V)1649 1897 y Fl(v)p 1687 1859 39 3 v 1 w(v)1730 1930 y Fu(\()p Ft(z)t Fu(1)p 1856 1859 V -33 x Fl(v)p 1895 1859 V 2 w(v)1957 1930 y Fr(\000)20 b Ft(H)p 2131 1859 V 2131 1897 a Fl(v)p 2170 1859 V 2 w(v)2124 1957 y(fr)2212 1930 y Fu(\))2247 1897 y Fs(\000)p Fl(1)2342 1930 y Ft(V)p 2415 1859 V 2415 1897 a Fl(v)q(v)1455 2113 y Fu(=)k Ft(v)1597 2080 y Fs(\003)1637 2113 y Fu(\()p Ft(z)h Fr(\000)20 b Ft(K)27 b Fr(\012)20 b Fu(1)g Fr(\000)g Fu(1)h Fr(\012)f Ft(h)p Fu(\))2425 2080 y Fs(\000)p Fl(1)2520 2113 y Ft(v)s(:)244 2298 y Fv(Prop)s(osition)36 b(6.1)49 b Fm(Assume)41 b(that)h(Assumption)f(6.C)p Fu(\()p Ft(\021)s Fu(\))2297 2312 y Fl(0)2379 2298 y Fm(holds)h(with)g Ft(\021)h(>)3033 2262 y Fl(1)p 3033 2277 36 4 v 3033 2329 a(2)3078 2298 y Fm(.)66 b(Then)41 b(the)244 2411 y(function)33 b Fj(C)668 2425 y Fl(+)752 2411 y Fr(3)24 b Ft(z)30 b Fr(7!)25 b Ft(w)r Fu(\()p Ft(z)t Fu(\))34 b Fm(extends)f(to)h(a)f(c)-5 b(ontinuous)33 b(function)g(on)g Fj(C)2765 2378 y Fl(cl)2765 2433 y(+)2824 2411 y Fm(.)244 2576 y Fv(Pro)s(of.)49 b Fu(W)-8 b(e)26 b(apply)e(the)g(tric)m(k)i(of)e (\\gluing)h(non-ph)m(ysical)g(free)f(b)s(osons")g([DJ1)q(].)39 b(Consider)23 b(the)244 2688 y(extended)j(1-b)s(oson)f(space)i Fr(Z)18 b(\010)p 1362 2615 74 4 v 11 w(Z)32 b Fu(and)25 b(de\014ne)g(the)h(op)s(erators)g Ft(r)i Fu(=)d Ft(h)11 b Fr(\010)g Fu(\()p Fr(\000)p 2856 2614 53 4 v Ft(h)p Fu(\))26 b(and)f Ft(q)k Fu(=)c(\()p Ft(v)s(;)15 b Fu(0\).)244 2801 y(Then,)30 b(for)g Ft(z)f Fr(2)c Fj(C)869 2815 y Fl(+)928 2801 y Fu(,)1316 2914 y Ft(v)1363 2877 y Fs(\003)1402 2914 y Fu(\()p Ft(z)g Fr(\000)20 b Ft(h)p Fu(\))1682 2877 y Fs(\000)p Fl(1)1777 2914 y Ft(v)29 b Fu(=)c Ft(q)1990 2877 y Fs(\003)2029 2914 y Fu(\()p Ft(z)g Fr(\000)20 b Ft(r)s Fu(\))2301 2877 y Fs(\000)p Fl(1)2395 2914 y Ft(q)s(;)244 3067 y Fu(and)30 b(the)g(statemen)m(t)i(follo)m(ws)g(from) e(Prop)s(osition)g(5.4.)42 b Fa(2)380 3221 y Fu(The)30 b(Lev)m(el)i(Shift)d(Op)s(erator,)i(\000,)f(of)h(the)f(triple)h(\()p Fr(K)22 b(\012)e Fu(\000)2332 3188 y Fl(0)2332 3243 y(s)2371 3221 y Fu(\()p Fr(Z)7 b Fu(\))p Ft(;)15 b(H)2630 3236 y Fl(fr)2684 3221 y Ft(;)g(V)21 b Fu(\))30 b(is)h(equal)g(to)1576 3411 y(\000)25 b(=)1824 3325 y Fk(X)1754 3526 y Fo(k)r Fs(2)p Fl(sp)o(\()p Fo(K)5 b Fl(\))2041 3411 y Fu(\000)2098 3374 y Fo(k)r(k)2179 3411 y Ft(;)1090 3719 y Fu(\000)1147 3682 y Fo(k)r(k)1254 3719 y Fu(=)1369 3633 y Fk(X)1350 3830 y Fo(p)p Fs(2F)1483 3842 y Fg(k)1521 3719 y Fu(\()p Ft(v)1603 3682 y Fs(\003)1643 3719 y Fu(\))1678 3682 y Fo(k)r(;k)r Fs(\000)p Fo(p)1869 3719 y Fu(\()p Ft(p)21 b Fu(+)f(i0)g Fr(\000)g Ft(h)p Fu(\))2330 3682 y Fs(\000)p Fl(1)2425 3719 y Ft(v)2472 3682 y Fo(k)r Fs(\000)p Fo(p;k)2664 3719 y Ft(:)244 3979 y Fu(With)h(a)f(sligh)m(t)i(abuse)e(of)g(notation) i(w)m(e)e(set)h Ft(v)s Fu(\()p Ft(p)p Fu(\))26 b(:=)f(\()p Ft(v)s(;)15 b Fu(0\)\()p Ft(p)p Fu(\))23 b(\(recall)f(\(6.36\)\).)40 b(Let)21 b Ft(v)3164 3946 y Fo(k)3201 3955 y Fh(1)3235 3946 y Fo(;k)3292 3955 y Fh(2)3330 3979 y Fu(\()p Ft(p)p Fu(\))26 b(:=)244 4092 y(1)289 4107 y Fo(k)326 4116 y Fh(1)365 4092 y Fu(\()p Ft(K)7 b Fu(\))p Fr(\012)p Fu(1)635 4106 y Fs(G)700 4092 y Ft(v)s Fu(\()p Ft(p)p Fu(\)1)908 4107 y Fo(k)945 4116 y Fh(2)985 4092 y Fu(\()p Ft(K)g Fu(\),)45 b(\()p Ft(v)1291 4059 y Fs(\003)1331 4092 y Fu(\))1366 4059 y Fo(k)1403 4068 y Fh(1)1437 4059 y Fo(;k)1494 4068 y Fh(2)1533 4092 y Fu(\()p Ft(p)p Fu(\))e(:=)h(1)1877 4107 y Fo(k)1914 4116 y Fh(1)1952 4092 y Fu(\()p Ft(K)7 b Fu(\))p Ft(v)s Fu(\()p Ft(p)p Fu(\))2269 4059 y Fs(\003)2325 4092 y Fu(1)2370 4107 y Fo(k)2407 4116 y Fh(2)2446 4092 y Fu(\()p Ft(K)g Fu(\))p Fr(\012)p Fu(1)2716 4106 y Fs(G)2767 4092 y Fu(.)73 b(The)40 b(Assumption)244 4205 y(6.C\()p Ft(\021)s Fu(\))498 4219 y Fl(0)578 4205 y Fu(with)f Ft(\021)k(>)d Fu(1)p Ft(=)p Fu(2)h(ensures)d(that)i Ft(v)s Fu(\()p Ft(p)p Fu(\))g(is)f(a)h(con)m(tin)m(uous)g(function)f(\(see)h (Subsection)244 4318 y(4.5\).)i(Hence,)32 b(in)e(particular,)h Ft(v)s Fu(\(0\))26 b(=)f(0.)41 b(Moreo)m(v)m(er,)33 b(w)m(e)e(ha)m(v)m (e)776 4492 y(\(\000)868 4459 y Fl(R)924 4492 y Fu(\))959 4459 y Fo(k)r(k)1123 4492 y Fu(=)1257 4424 y Fk(P)1219 4575 y Fo(p)p Fs(2F)1352 4587 y Fg(k)1390 4492 y Fu(\()p Ft(v)1472 4459 y Fs(\003)1513 4492 y Fu(\))1548 4459 y Fo(k)r(;k)r Fs(\000)p Fo(p)1739 4492 y Fr(P)7 b Fu(\()p Ft(p)21 b Fr(\000)f Ft(h)p Fu(\))2089 4459 y Fs(\000)p Fl(1)2184 4492 y Ft(v)2231 4459 y Fo(k)r Fs(\000)p Fo(p;k)1123 4723 y Fu(=)1257 4655 y Fk(P)1219 4806 y Fo(p)p Fs(2F)1352 4818 y Fg(k)1406 4650 y Fk(R)1466 4723 y Fu(\()p Ft(v)1548 4690 y Fs(\003)1588 4723 y Fu(\))1623 4690 y Fo(k)r(;k)r Fs(\000)p Fo(p)1815 4723 y Fu(\()p Ft(p)1896 4737 y Fl(1)1935 4723 y Fu(\))p Fr(P)7 b Fu(\()p Ft(p)22 b Fr(\000)e Ft(p)2280 4737 y Fl(1)2319 4723 y Fu(\))2354 4690 y Fs(\000)p Fl(1)2449 4723 y Ft(v)2496 4690 y Fo(k)r Fs(\000)p Fo(p;k)2687 4723 y Fu(\()p Ft(p)2768 4737 y Fl(1)2808 4723 y Fu(\)d)p Ft(p)2940 4737 y Fl(1)2979 4723 y Ft(;)802 4954 y Fu(\(\000)894 4921 y Fl(I)924 4954 y Fu(\))959 4921 y Fo(k)r(k)1123 4954 y Fu(=)25 b Fr(\000)p Ft(\031)1408 4886 y Fk(P)1360 5054 y Fo(p)p Fs(2F)1500 5024 y Fh(+)1493 5077 y Fg(k)1551 4954 y Fu(\()p Ft(v)1633 4921 y Fs(\003)1673 4954 y Fu(\))1708 4921 y Fo(k)r(;k)r Fs(\000)p Fo(p)1899 4954 y Ft(\016)s Fu(\()p Ft(p)c Fr(\000)f Ft(h)p Fu(\))p Ft(v)2269 4921 y Fo(k)r Fs(\000)p Fo(p;k)1123 5185 y Fu(=)25 b Fr(\000)p Ft(\031)1408 5117 y Fk(P)1360 5285 y Fo(p)p Fs(2F)1500 5255 y Fh(+)1493 5309 y Fg(k)1551 5185 y Fu(\()p Ft(v)1633 5152 y Fs(\003)1673 5185 y Fu(\))1708 5152 y Fo(k)r(;k)r Fs(\000)p Fo(p)1899 5185 y Fu(\()p Ft(p)p Fu(\))p Ft(v)2062 5152 y Fo(k)r Fs(\000)p Fo(p;k)2254 5185 y Fu(\()p Ft(p)p Fu(\))p Ft(:)3303 4888 y Fu(\(6.37\))p eop %%Page: 28 28 28 27 bop 3682 100 a Fn(28)380 407 y Fu(W)-8 b(e)43 b(ha)m(v)m(e)g (describ)s(ed)e(ab)s(o)m(v)m(e)i(\000)1509 374 y Fl(I)1580 407 y Fu(and)f(\000)1826 374 y Fl(R)1923 407 y Fu(in)g(t)m(w)m(o)h (forms.)75 b(In)41 b(the)h(\014rst)f(form)h(w)m(e)g(use)244 520 y(the)37 b(self-adjoin)m(t)h(op)s(erator)f Ft(h)g Fu(on)g Fr(Z)44 b Fu(and)36 b(a)h(n)m(um)m(b)s(er)f Ft(p)f Fr(2)h Fj(R)p Fu(.)60 b(Strictly)37 b(sp)s(eaking,)i(neither)244 633 y(the)c(principal)f(v)-5 b(alue)35 b Fr(P)7 b Fu(\()p Ft(p)23 b Fr(\000)g Ft(h)p Fu(\))1382 600 y Fs(\000)p Fl(1)1511 633 y Fu(nor)34 b(the)h(delta)g(function)f Ft(\016)s Fu(\()p Ft(p)24 b Fr(\000)f Ft(h)p Fu(\))35 b(are)g(w)m(ell)g(de\014ned)e(as)244 746 y(self-adjoin)m(t)41 b(op)s(erators.)69 b(Ho)m(w)m(ev)m(er,)44 b(within)39 b(the)h(con)m(text)i(of)e(\(6.37\),)k(these)c(form)m(ulas)g(are)244 858 y(w)m(ell)35 b(de\014ned)e(b)m(y)h(the)g(in)m(tegral)i(expressions) e(using)f(the)i(represen)m(tation)g(of)f Ft(v)k Fu(in)m(to)d(a)f (direct)244 971 y(in)m(tegral)e(with)e(the)h(\014b)s(ers)d Ft(v)s Fu(\()p Ft(p)p Fu(\).)380 1084 y(Let)33 b Ft(k)592 1098 y Fl(0)664 1084 y Fu(denote)g(the)g(ground)e(state)j(energy)e(of)h Ft(K)7 b Fu(,)32 b(that)h(is,)h Ft(k)2558 1098 y Fl(0)2626 1084 y Fu(:=)28 b(inf)22 b(sp)o(\()p Ft(K)7 b Fu(\).)47 b(F)-8 b(or)34 b(later)244 1197 y(reference,)d(w)m(e)g(note)g(that)g (the)f(ground)g(states)h(of)g Ft(K)37 b Fu(b)s(elong)30 b(to)h(the)g(k)m(ernel)g(of)f(\000)3111 1164 y Fl(I)3141 1197 y Fu(:)244 1385 y Fv(Prop)s(osition)36 b(6.2)49 b Fu(Ran1)1199 1400 y Fo(k)1236 1409 y Fh(0)1275 1385 y Fu(\()p Ft(K)7 b Fu(\))26 b Fr(\032)f Fu(Ker\000)1755 1352 y Fl(I)1784 1385 y Fm(.)380 1572 y Fu(One)30 b(exp)s(ects)h(that)g (for)f(a)g("generic")j(form-factor)e Ft(v)s Fu(,)g(the)f(k)m(ernel)h (of)g(\000)2870 1539 y Fl(I)2929 1572 y Fu(should)e(coincide)244 1685 y(with)k(the)h(subspace)f(of)g(ground)g(states)i(of)e Ft(K)7 b Fu(.)50 b(This)33 b(leads)h(to)g(the)f(\014rst)g(e\013ectiv)m (e)j(coupling)244 1798 y(assumption)30 b(that)h(w)m(e)f(will)h(use)f (in)g(our)g(pap)s(er.)244 2011 y Fv(Assumption)36 b(6.D)49 b Fu(Ran1)1242 2026 y Fo(k)1279 2035 y Fh(0)1318 2011 y Fu(\()p Ft(K)7 b Fu(\))25 b(=)g(Ker\000)1797 1978 y Fl(I)1826 2011 y Fm(.)380 2223 y Fu(Our)e(second)h(e\013ectiv)m(e)i (coupling)f(assumption)e(is)h(that)g(the)h(ground)d(state)k(of)e Ft(K)30 b Fu(is)24 b(simple:)244 2436 y Fv(Assumption)36 b(6.E)48 b Fu(dim)15 b(Ran1)1397 2451 y Fo(k)1434 2460 y Fh(0)1473 2436 y Fu(\()p Ft(K)7 b Fu(\))26 b(=)e(1)p Fm(.)244 2713 y Fi(6.4)135 b(Sp)t(ectral)45 b(structure)g(of)g(P)l (auli-Fierz)h(Hamiltonians)244 2884 y Fu(In)26 b(this)g(subsection)g(w) m(e)h(form)m(ulate)g(the)g(main)f(result)h(of)f(this)g(pap)s(er)g (concerning)h(P)m(auli-Fierz)244 2997 y(Hamiltonians.)70 b(It)40 b(will)g(sa)m(y)g(that)h(if)e(the)h(in)m(teraction)i Ft(v)h Fu(is)d(su\016cien)m(tly)g(regular)g(and)f(the)244 3110 y(e\013ectiv)m(e)33 b(coupling)d(assumptions)g(6.D)h(and)f(6.E)h (hold,)f(then)g(the)h(P)m(auli-Fierz)h(Hamiltonian)244 3223 y Ft(H)37 b Fu(for)30 b(small)h(nonzero)g(coupling)f(constan)m(ts) i(has)e(purely)f(absolutely)j(con)m(tin)m(uous)f(sp)s(ectrum)244 3336 y(except)g(p)s(ossibly)f(for)g(a)h(simple)f(eigen)m(v)-5 b(alue)32 b(at)f(inf)22 b(sp\()p Ft(H)7 b Fu(\).)380 3449 y(W)-8 b(e)23 b(start)f(with)g(the)f(observ)-5 b(ation)23 b(that)f(an)f(appropriate)h(regularit)m(y)h(of)f(the)f(gluing)h (implies)244 3562 y(the)30 b(self-adjoin)m(tness)i(of)e Ft(H)7 b Fu(.)244 3774 y Fv(Theorem)70 b(6.3)49 b Fm(Assume)39 b(that)i(Assumption)g(6.C)p Fu(\()p Ft(\021)s Fu(\))2208 3788 y Fl(0)2288 3774 y Fm(holds)h(with)e Ft(\021)i(>)37 b Fu(1)p Ft(=)p Fu(2)p Fm(.)64 b(Then)40 b(As-)244 3887 y(sumption)34 b(6.A)e(holds.)244 4100 y Fv(Pro)s(of.)53 b Fu(W)-8 b(e)27 b(use)f(the)g(metho)s(d)f(describ)s(ed)g(in)h(the)g (pro)s(of)f(of)h(Prop)s(osition)g(6.1.)41 b(In)25 b(particular,)244 4212 y(w)m(e)31 b(use)f(the)g(op)s(erators)h Ft(q)i Fu(and)d Ft(r)j Fu(in)m(tro)s(duced)c(in)h(the)h(pro)s(of)f(of)g(this)g(prop)s (osition.)380 4325 y(By)42 b(Assumption)e(6.C\()p Ft(\021)s Fu(\))1302 4339 y Fl(0)1384 4325 y Fu(with)h Ft(\021)47 b(>)1818 4290 y Fl(1)p 1818 4305 36 4 v 1818 4357 a(2)1904 4325 y Fu(and)41 b(Prop)s(osition)g(4.2,)k(for)c(an)m(y)h Ft(p)h Fr(2)g Fj(R)d Fu(the)244 4438 y(op)s(erator)31 b Ft(q)655 4405 y Fs(\003)694 4438 y Fu(\()p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))1147 4405 y Fs(\000)p Fl(1)1241 4438 y Ft(q)33 b Fu(exists)e(and)f(is)g(b)s(ounded.)39 b(Setting)31 b Ft(p)25 b Fu(=)g(0)31 b(giv)m(es)1417 4643 y Ft(q)1461 4605 y Fs(\003)1501 4643 y Fu(\(i0)21 b Fr(\000)f Ft(r)s Fu(\))1797 4605 y Fs(\000)p Fl(1)1891 4643 y Ft(q)28 b Fu(=)d Ft(v)2103 4605 y Fs(\003)2143 4643 y Ft(h)2195 4605 y Fs(\000)p Fl(1)2290 4643 y Ft(v)s(:)244 4860 y Fu(Hence,)31 b Ft(h)591 4824 y Fs(\000)656 4797 y Fh(1)p 657 4809 31 3 v 657 4850 a(2)701 4860 y Ft(v)j Fu(is)c(b)s(ounded.)39 b Fa(2)380 5023 y Fu(No)m(w)31 b(w)m(e)g(deduce)f(sp)s(ectral)h(information)g(on)f Ft(H)7 b Fu(.)p eop %%Page: 29 29 29 28 bop 3682 100 a Fn(29)244 407 y Fv(Theorem)70 b(6.4)49 b Fm(Supp)-5 b(ose)41 b(that)h(Assumption)e(6.C)p Fu(\()p Ft(\021)s Fu(\))2215 421 y Fl(0)2296 407 y Fm(holds)i(with)e Ft(\021)i(>)c Fu(2)i Fm(and)h(that)g(As-)244 520 y(sumptions)c(6.D)f (and)h(6.E)f(hold.)53 b(Then)36 b(ther)-5 b(e)37 b(exists)f Ft(\025)2227 534 y Fl(0)2298 520 y Ft(>)30 b Fu(0)37 b Fm(such)f(that)h(for)f Fu(0)c Ft(<)f Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)3496 534 y Fl(0)244 633 y Fm(the)i(fol)5 b(lowing)34 b(holds:)244 746 y Fu(\(1\))g(dim)14 b(1)604 713 y Fl(p)648 746 y Fu(\()p Ft(H)7 b Fu(\))26 b Fr(\024)f Fu(1)p Ft(:)244 858 y Fu(\(2\))34 b(sp)479 880 y Fl(sc)542 858 y Fu(\()p Ft(H)7 b Fu(\))26 b(=)f Fr(;)p Ft(:)244 1176 y Fv(Pro)s(of.)81 b Fu(W)-8 b(e)41 b(again)g(use)f(the)h(metho)s (d)e(and)h(the)g(notation)h(of)g(Prop)s(osition)f(6.1.)71 b(Extend)244 1289 y(the)30 b(space)h Fr(K)21 b(\012)f Fu(\000)881 1303 y Fl(s)913 1289 y Fu(\()p Fr(Z)7 b Fu(\))30 b(to)h(the)g(space)f Fr(K)22 b(\012)d Fu(\000)1834 1303 y Fl(s)1866 1289 y Fu(\()p Fr(Z)7 b Fu(\))21 b Fr(\012)e Fu(\000)2177 1303 y Fl(s)2209 1289 y Fu(\()p 2244 1216 74 4 v Fr(Z)7 b Fu(\))26 b Fr(')f(K)c(\012)f Fu(\000)2712 1303 y Fl(s)2744 1289 y Fu(\()p Fr(Z)27 b(\010)p 2962 1216 V 19 w(Z)7 b Fu(\).)41 b(The)30 b(space)244 1402 y Fr(K)11 b(\012)f Fu(\000)462 1416 y Fl(s)493 1402 y Fu(\()p Fr(Z)d Fu(\))26 b(is)f(iden)m(ti\014ed)g(with)g(the)g(subspace) g Fr(K)11 b(\012)f Fu(\000)2085 1416 y Fl(s)2116 1402 y Fu(\()p Fr(Z)d Fu(\))j Fr(\012)g Fu(\000)2407 1369 y Fl(0)2407 1424 y(s)2446 1402 y Fu(\()p 2481 1329 V Fr(Z)e Fu(\).)39 b(Consider)24 b(the)i(extended)244 1514 y(op)s(erators)872 1619 y Ft(L)934 1634 y Fl(fr)1070 1619 y Fu(:=)f Ft(H)1267 1634 y Fl(fr)1340 1619 y Fr(\012)20 b Fu(1)g Fr(\000)g Fu(1)h Fr(\012)f Fu(d\000\()p Ft(h)p Fu(\))83 b Fr(')25 b Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(d\000\()p Ft(r)s Fu(\))p Ft(;)915 1803 y(Q)83 b Fu(:=)25 b Ft(V)40 b Fr(\012)20 b Fu(1)637 b Fr(')25 b Ft(q)s Fu(\()p Ft(a)2280 1770 y Fs(\003)2320 1803 y Fu(\))20 b(+)g Ft(q)2510 1770 y Fs(\003)2550 1803 y Fu(\()p Ft(a)p Fu(\))p Ft(;)244 1959 y Fu(and)30 b(set)1153 2072 y Ft(L)25 b Fu(:=)g Ft(L)1423 2087 y Fl(fr)1496 2072 y Fu(+)20 b Ft(\025Q)41 b Fr(')f Ft(H)27 b Fr(\012)20 b Fu(1)h Fr(\000)e Fu(1)i Fr(\012)f Fu(d\000\()p Ft(h)p Fu(\))p Ft(:)244 2235 y Fu(By)j(Theorem)g(6.3,)j Ft(H)j Fu(is)23 b(self-adjoin)m(t)i(on)e Fr(D)s Fu(\()p Ft(H)1875 2250 y Fl(fr)1927 2235 y Fu(\))h(and)e(therefore)h Ft(L)g Fu(is)g(self-adjoin)m(t)h(on)f Fr(D)s Fu(\()p Ft(L)3447 2250 y Fl(fr)3500 2235 y Fu(\))244 2348 y(\(see)31 b(Section)g(5.2)h (in)e([DJ1)q(]\).)41 b(Note)32 b(also)f(that)1125 2545 y(sp)1211 2567 y Fl(p)1254 2545 y Fu(\()p Ft(H)7 b Fu(\))26 b(=)f(sp)1616 2567 y Fl(p)1659 2545 y Fu(\()p Ft(L)p Fu(\))p Ft(;)107 b Fu(sp)2009 2567 y Fl(sc)2073 2545 y Fu(\()p Ft(H)7 b Fu(\))25 b(=)g(sp)2434 2567 y Fl(sc)2497 2545 y Fu(\()p Ft(L)p Fu(\))p Ft(:)649 b Fu(\(6.38\))380 2743 y(Clearly)-8 b(,)45 b Ft(L)d Fu(is)f(a)h(P)m(auli-Fierz)h(op)s (erator)f(suc)m(h)e(that)i Fr(kh)p Ft(s)p Fr(i)2457 2710 y Fo(\021)2500 2743 y Ft(q)s Fr(k)i Ft(<)f Fr(1)e Fu(for)g Ft(\021)47 b(>)c Fu(2.)74 b(The)244 2856 y(Lev)m(el)28 b(Shift)e(Op)s(erator)g(of)g(the)h(triple)g(\()p Fr(K)14 b(\012)f Fu(\000)1833 2823 y Fl(0)1833 2878 y(s)1872 2856 y Fu(\()p Fr(Z)20 b(\010)p 2077 2783 V 13 w(Z)6 b Fu(\))p Ft(;)15 b(L)2286 2871 y Fl(fr)2340 2856 y Ft(;)g(Q)p Fu(\))27 b(is)g(equal)g(\(after)g(the)g(ob)m(vious)244 2968 y(iden)m(ti\014cation)35 b(of)e(the)g(Hilb)s(ert)h(spaces\))g(to)g (the)f(Lev)m(el)h(Shift)f(Op)s(erator)g(of)g(the)g(triple)h(\()p Fr(K)24 b(\012)244 3081 y Fu(\000)301 3048 y Fl(0)301 3104 y(s)340 3081 y Fu(\()p Fr(Z)7 b Fu(\))p Ft(;)15 b(H)599 3096 y Fl(fr)653 3081 y Ft(;)g(V)21 b Fu(\),)34 b(whic)m(h)f(w)m(e)g(studied)g(in)f(the)i(last)g(subsection.)48 b(Assumptions)32 b(6.D)j(and)d(6.E)244 3194 y(yield)g(that)g(dim)15 b(Ker\000)1034 3161 y Fl(I)1091 3194 y Fu(=)27 b(1.)45 b(Therefore,)33 b(Theorem)e(5.5)i(implies)f(that)g(there)g(exists)h Ft(\025)3325 3208 y Fl(0)3392 3194 y Ft(>)27 b Fu(0)244 3307 y(suc)m(h)35 b(that)i(for)e(0)g Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)1282 3321 y Fl(0)1357 3307 y Fu(w)m(e)j(ha)m(v)m(e)g (dim)14 b(1)1923 3274 y Fl(p)1967 3307 y Fu(\()p Ft(L)p Fu(\))35 b Fr(\024)f Fu(1)p Ft(;)88 b Fu(sp)2483 3329 y Fl(sc)2546 3307 y Fu(\()p Ft(L)p Fu(\))35 b(=)f Fr(;)p Fu(.)58 b(By)36 b(\(6.38\),)k(this)244 3420 y(implies)30 b(dim)15 b(1)764 3387 y Fl(p)808 3420 y Fu(\()p Ft(H)7 b Fu(\))26 b Fr(\024)f Fu(1)p Ft(;)107 b Fu(sp)1346 3442 y Fl(sc)1409 3420 y Fu(\()p Ft(H)7 b Fu(\))26 b(=)f Fr(;)p Fu(.)41 b Fa(2)244 3672 y Fv(Remark)35 b(6.5)50 b Fm(F)-7 b(or)26 b(a)f(lar)-5 b(ge)26 b(class)g(of)f(inter)-5 b(actions)27 b(one)e(c)-5 b(an)26 b(show)g(that)g Fu(dim)15 b(1)2999 3690 y Fl(inf)i(sp\()p Fo(H)5 b Fl(\))3286 3672 y Fu(\()p Ft(H)i Fu(\))26 b Fr(\025)244 3785 y Fu(1)p Fm(,)51 b(namely)d(that)h(Pauli-Fierz)e(Hamiltonian)i Ft(H)54 b Fm(has)48 b(a)g(gr)-5 b(ound)49 b(state.)86 b(R)-5 b(esults)48 b(of)g(this)244 3898 y(kind)43 b(wer)-5 b(e)43 b(pr)-5 b(oven)44 b(in)f Fu([AH)q(,)f(BFS1,)g(Ge)q(,)f(Sp2,)g (Sp3])p Fm(.)73 b(If)42 b(to)i(the)f(assumptions)i(of)e(The)-5 b(o-)244 4011 y(r)g(em)40 b(6.4)h(we)f(add)h(the)f(assumptions)i(of)e (the)g(ab)-5 b(ove)41 b(r)-5 b(efer)g(enc)g(es,)42 b(then)e(we)g(c)-5 b(an)41 b(r)-5 b(eplac)g(e)41 b(\(1\))244 4124 y(with)33 b Fu(dim)15 b(1)653 4091 y Fl(p)697 4124 y Fu(\()p Ft(H)7 b Fu(\))26 b(=)f(1)33 b Fm(and)g Fu(sp)1313 4145 y Fl(p)1356 4124 y Fu(\()p Ft(H)7 b Fu(\))26 b(=)f(inf)d(sp)o(\()p Ft(H)7 b Fu(\))p Ft(:)244 4399 y Fi(6.5)135 b(P)l(auli-Fierz)46 b(systems)g(of)f(densit)l(y)g Fd(\032)244 4571 y Fu(In)36 b(this)h(subsection)f(w)m(e)i(in)m(tro)s(duce)e(P)m(auli-Fierz)j Ft(W)2118 4538 y Fs(\003)2157 4571 y Fu(-dynamical)f(systems.)60 b(They)36 b(will)h(b)s(e)244 4684 y(the)30 b(main)h(sub)5 b(ject)30 b(of)g(the)h(remaining)f(part)h(of)f(this)h(section.)380 4797 y(Let)40 b Ft(\032)h Fr(\025)f Fu(0)g(b)s(e)f(an)g(op)s(erator)h (comm)m(uting)g(with)g Ft(h)p Fu(.)68 b(It)40 b(will)g(b)s(e)e(called)j (the)f(radiation)244 4909 y(densit)m(y)-8 b(.)82 b(The)44 b(left)g(Araki-W)-8 b(o)s(o)s(ds)45 b Ft(W)1663 4876 y Fs(\003)1702 4909 y Fu(-algebra,)k(denoted)44 b(b)m(y)g Fp(M)2685 4876 y Fl(A)-8 b(W)2685 4937 y Fo(\032;)p Fl(l)2852 4909 y Fu(is)44 b(de\014ned)f(as)h(the)244 5039 y Ft(W)343 5006 y Fs(\003)382 5039 y Fu(-subalgebra)30 b(of)h Fr(B)s Fu(\(\000)1126 5053 y Fl(s)1158 5039 y Fu(\()p Fr(Z)c(\010)p 1377 4966 V 20 w(Z)7 b Fu(\)\))31 b(generated)h(b)m(y)e(the)g(op)s (erators)865 5259 y(exp)15 b(i)1044 5158 y Fk(\020)1098 5185 y(\000)1140 5259 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1414 5221 y Fl(1)p Fo(=)p Fl(2)1524 5259 y Ft(z)t(;)p 1610 5208 48 4 v 15 w(\032)1658 5221 y Fl(1)p Fo(=)p Fl(2)p 1768 5208 47 4 v 1768 5259 a Ft(z)1814 5185 y Fk(\001)1856 5259 y Fu(\()p Ft(a)1939 5221 y Fs(\003)1979 5259 y Fu(\))g(+)g(hc)2216 5158 y Fk(\021)2270 5259 y Ft(;)137 b(z)29 b Fr(2)c(D)s Fu(\()p Ft(\032)2744 5221 y Fl(1)p Fo(=)p Fl(2)2854 5259 y Fu(\))p Ft(;)p eop %%Page: 30 30 30 29 bop 3682 100 a Fn(30)244 407 y Fu(where)34 b(\()p Ft(z)588 421 y Fl(1)628 407 y Ft(;)p 668 357 47 4 v 15 w(z)715 421 y Fl(2)754 407 y Fu(\)\()p Ft(a)872 374 y Fs(\003)912 407 y Fu(\))h(denotes)g(the)g(usual)f(creation)i(op)s (erators)e(on)h(\000)2665 421 y Fl(s)2697 407 y Fu(\()p Fr(Z)30 b(\010)p 2922 334 74 4 v 23 w(Z)7 b Fu(\).)54 b(The)34 b(P)m(auli-)244 520 y(Fierz)d(algebra)h(at)f(densit)m(y)f Ft(\032)h Fu(is)f(de\014ned)f(as)1470 691 y Fp(M)1565 705 y Fo(\032)1631 691 y Fu(:=)c Fr(B)s Fu(\()p Fr(K)q Fu(\))c Fr(\012)f Fp(M)2162 653 y Fl(A)-8 b(W)2162 714 y Fo(\032;)p Fl(l)2285 691 y Ft(:)993 b Fu(\(6.39\))244 862 y(The)30 b(iden)m(tit)m(y)i(map)1476 975 y Fp(M)1571 989 y Fo(\032)1637 975 y Fr(!)25 b(B)s Fu(\(\000)1908 989 y Fl(s)1940 975 y Fu(\()p Fr(Z)j(\010)p 2160 902 V 20 w(Z)7 b Fu(\)\))1000 b(\(6.40\))244 1125 y(will)34 b(b)s(e)f(called)h(the)g(semistandard)f(represen)m(tation)i(of)e Fp(M)2325 1139 y Fo(\032)2366 1125 y Fu(.)50 b(\(The)34 b(b)s(osonic)f(part)h(of)f(\(6.40\))244 1238 y(is)d(already)h (standard,)f(the)h(part)f(in)m(v)m(olving)i Fr(K)g Fu(is)e(not|hence)h (the)f(name\).)244 1412 y Fv(Prop)s(osition)36 b(6.6)49 b Fm(Assume)33 b(that)1356 1583 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1630 1545 y Fl(1)p Fo(=)p Fl(2)1741 1583 y Ft(v)28 b Fr(2)d(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(Z)7 b Fu(\))p Ft(:)880 b Fu(\(6.41\))244 1767 y Fm(Then)p 481 1717 48 4 v 33 w Ft(\032)538 1704 y Fh(1)p 538 1716 31 3 v 538 1757 a(2)582 1767 y Ft(v)629 1734 y Fo(?)702 1767 y Fm(is)32 b(a)h(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ator)36 b(and)e(the)f(op)-5 b(er)g(ators)381 1855 y Fk(\020)435 1956 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))719 1891 y Fh(1)p 719 1903 V 719 1944 a(2)764 1956 y Ft(v)s(;)15 b Fu(0)896 1855 y Fk(\021)951 1956 y Fu(\()p Ft(a)1034 1918 y Fs(\003)1074 1956 y Fu(\))21 b(+)1221 1855 y Fk(\020)1275 1956 y Fu(0)p Ft(;)15 b(v)1407 1918 y Fo(?)p Fs(\003)p 1483 1905 48 4 v 1483 1956 a Ft(\032)1540 1891 y Fh(1)p 1540 1903 31 3 v 1540 1944 a(2)1584 1855 y Fk(\021)1639 1956 y Fu(\()p Ft(a)p Fu(\))p Ft(;)1993 1855 y Fk(\020)2047 1956 y Ft(v)2094 1918 y Fs(\003)2134 1956 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))2418 1891 y Fh(1)p 2418 1903 V 2418 1944 a(2)2463 1956 y Ft(;)15 b Fu(0)2548 1855 y Fk(\021)2603 1956 y Fu(\()p Ft(a)p Fu(\))21 b(+)2833 1855 y Fk(\020)2887 1956 y Fu(0)p Ft(;)p 2972 1905 48 4 v 15 w(\032)3030 1891 y Fh(1)p 3030 1903 31 3 v 3030 1944 a(2)3074 1956 y Ft(v)3121 1918 y Fo(?)3161 1855 y Fk(\021)3215 1956 y Fu(\()p Ft(a)3298 1918 y Fs(\003)3338 1956 y Fu(\))p Ft(;)244 2161 y Fm(which)33 b(act)g(on)h Fr(K)21 b(\012)f Fu(\000)1019 2175 y Fl(s)1051 2161 y Fu(\()p Fr(Z)28 b(\010)p 1271 2088 74 4 v 20 w(Z)7 b Fu(\))p Fm(,)32 b(ar)-5 b(e)34 b(a\016liate)-5 b(d)35 b(to)e Fp(M)2169 2175 y Fo(\032)2209 2161 y Fm(.)244 2335 y Fv(Pro)s(of.)74 b Fu(Using)37 b(Prop)s(osition)g(4.4)h(and)e (the)h(fact)h(that)f Fr(K)h Fu(has)f(a)g(\014nite)g(dimension)f(w)m(e)h (see)244 2448 y(that)h(the)g(b)s(oundedness)d(of)j(\(1)26 b(+)f Ft(\032)p Fu(\))1548 2415 y Fl(1)p Fo(=)p Fl(2)1659 2448 y Ft(v)41 b Fu(implies)d(the)g(b)s(oundedness)d(of)j(\(\(1)26 b(+)f Ft(\032)p Fu(\))3195 2415 y Fl(1)p Fo(=)p Fl(2)3305 2448 y Ft(v)s Fu(\))3387 2415 y Fo(?)3465 2448 y Fu(=)244 2561 y(\(1+)p 395 2511 48 4 v Ft(\032)q Fu(\))478 2528 y Fl(1)p Fo(=)p Fl(2)588 2561 y Ft(v)635 2528 y Fo(?)675 2561 y Fu(.)37 b(Next)21 b(note)g(that)p 1328 2511 V 21 w Ft(\032)1375 2528 y Fl(1)p Fo(=)p Fl(2)1485 2561 y Fu(\(1+)p 1636 2511 V Ft(\032)q Fu(\))1719 2528 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1905 2561 y Fu(is)f(b)s(ounded.)35 b(This)20 b(implies)h(the)f(b)s(oundedness)244 2674 y(of)p 347 2624 V 30 w Ft(\032)394 2641 y Fl(1)p Fo(=)p Fl(2)504 2674 y Ft(v)551 2641 y Fo(?)591 2674 y Fu(.)41 b Fa(2)380 2826 y Fu(In)30 b(what)g(follo)m(ws)i(w)m(e)e(assume)h(\(6.41\))h(and)e (set)991 3005 y Ft(v)1035 3019 y Fo(\032)1101 3005 y Fu(:=)25 b(\(\(1)d(+)e Ft(\032)p Fu(\))1542 2941 y Fh(1)p 1542 2953 31 3 v 1542 2995 a(2)1586 3005 y Ft(v)s(;)p 1673 2954 48 4 v 1673 3004 a(\032)1731 2941 y Fh(1)p 1731 2953 31 3 v 1731 2995 a(2)1776 3005 y Ft(v)1823 2971 y Fo(?)1862 3005 y Fu(\))26 b Fr(2)f(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)c Fu(\()p Fr(Z)28 b(\010)p 2619 2931 74 4 v 2619 3004 a(Z)2692 3005 y Fu(\)\))p Ft(:)244 3168 y Fu(Note)k(that)f(in)f(terms)g(of)g(a)h(direct)g(in)m (tegral)h(of)f(op)s(erators)f(in)g Fr(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(G)5 b Fu(\):)1194 3443 y Ft(v)1238 3457 y Fo(\032)1278 3443 y Fu(\()p Ft(p)p Fu(\))84 b(=)1574 3257 y Fk(8)1574 3339 y(<)1574 3502 y(:)1696 3363 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1980 3300 y Fh(1)p 1980 3312 31 3 v 1980 3353 a(2)2025 3363 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)84 b(p)25 b(>)g Fu(0)p Ft(;)p 1696 3489 48 4 v 1696 3539 a(\032)1753 3476 y Fh(1)p 1753 3488 31 3 v 1753 3529 a(2)1797 3539 y Ft(v)1844 3506 y Fo(?)1884 3539 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))p Ft(;)201 b(p)25 b Fr(\024)g Fu(0)p Ft(:)380 3713 y Fu(Let)796 3825 y Ft(V)849 3839 y Fo(\032)973 3825 y Fu(:=)1094 3724 y Fk(\020)1148 3825 y Fu(\(1)c(+)f Ft(\032)p Fu(\))1432 3762 y Fh(1)p 1432 3774 V 1432 3816 a(2)1477 3825 y Ft(v)s(;)p 1564 3775 48 4 v 15 w(\032)1622 3762 y Fh(1)p 1622 3774 31 3 v 1622 3816 a(2)1666 3825 y Ft(v)1713 3792 y Fo(?)1753 3724 y Fk(\021)1807 3825 y Fu(\()p Ft(a)1890 3792 y Fs(\003)1930 3825 y Fu(\))h(+)2076 3724 y Fk(\020)2131 3825 y Ft(v)2178 3792 y Fs(\003)2218 3825 y Fu(\(1)g(+)e Ft(\032)p Fu(\))2501 3762 y Fh(1)p 2502 3774 V 2502 3816 a(2)2546 3825 y Ft(;)c(v)2633 3792 y Fo(?)p Fs(\003)p 2709 3775 48 4 v 2709 3825 a Ft(\032)2766 3762 y Fh(1)p 2766 3774 31 3 v 2766 3816 a(2)2810 3724 y Fk(\021)2865 3825 y Fu(\()p Ft(a)p Fu(\))973 4009 y(=)25 b Ft(v)1113 4023 y Fo(\032)1153 4009 y Fu(\()p Ft(a)1236 3976 y Fs(\003)1276 4009 y Fu(\))20 b(+)g Ft(v)1469 3976 y Fs(\003)1466 4032 y Fo(\032)1509 4009 y Fu(\()p Ft(a)p Fu(\))p Ft(:)244 4153 y Fu(The)40 b(op)s(erator)h Ft(V)871 4167 y Fo(\032)953 4153 y Fu(is)g(essen)m(tially)h (self-adjoin)m(t)h(on)d(the)h(space)h(of)f(\014nite)g(particle)h(v)m (ectors.)244 4266 y(Moreo)m(v)m(er,)33 b(it)e(is)f(a\016liated)i(to)f Fp(M)1431 4280 y Fo(\032)1471 4266 y Fu(.)380 4379 y(The)21 b(free)g(P)m(auli-Fierz)i(semi-Liouvillean)g(is)e(the)h(self-adjoin)m (t)g(op)s(erator)g(on)f Fr(K)s(\012)r Fu(\000)3213 4393 y Fl(s)3244 4379 y Fu(\()p Fr(Z)9 b(\010)p 3427 4306 74 4 v 2 w(Z)e Fu(\))244 4492 y(de\014ned)29 b(as)1200 4582 y Ft(L)1262 4549 y Fl(semi)1262 4609 y(fr)1486 4582 y Fu(:=)c Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(d\000\()p Ft(h)h Fr(\010)f(\000)p 2493 4508 53 4 v Ft(h)p Fu(\))1486 4765 y(=)25 b Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(d\000\()p Ft(r)s Fu(\))p Ft(:)244 4908 y Fu(The)30 b(full)g(P)m(auli-Fierz)i(semi-Liouvillean)g(of)f(densit)m(y) f Ft(\032)h Fu(is)1472 5079 y Ft(L)1534 5042 y Fl(semi)1534 5102 y Fo(\032)1700 5079 y Fu(:=)26 b Ft(L)1884 5042 y Fl(semi)1884 5102 y(fr)2045 5079 y Fu(+)20 b Ft(\025V)2242 5093 y Fo(\032)2283 5079 y Ft(:)995 b Fu(\(6.42\))380 5250 y(Let)31 b(us)f(form)m(ulate)h(the)g(follo)m(wing)h(assumption:)p eop %%Page: 31 31 31 30 bop 3682 100 a Fn(31)244 407 y Fv(Assumption)36 b(6.F)974 421 y Fo(\032)1265 407 y Fu(\(1)21 b(+)f Ft(h)p Fu(\)\(1)i(+)e Ft(\032)p Fu(\))1819 374 y Fl(1)p Fo(=)p Fl(2)1929 407 y Ft(v)29 b Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)e(Z)7 b Fu(\))p Fm(.)244 619 y Fv(Theorem)70 b(6.7)80 b Fu(\(1\))50 b Ft(\034)1151 586 y Fo(t)1141 647 y Fl(fr)1194 619 y Fu(\()p Ft(A)p Fu(\))26 b(:=)f(e)1519 586 y Fl(i)p Fo(tL)1612 598 y Fh(fr)1664 619 y Ft(A)p Fu(e)1772 586 y Fs(\000)p Fl(i)p Fo(tL)1920 598 y Fh(fr)2004 619 y Fm(is)33 b(a)g Ft(W)2280 586 y Fs(\003)2319 619 y Fm(-dynamics)g(on)g Fp(M)2978 633 y Fo(\032)3019 619 y Fm(.)275 753 y Fu(\(2\))50 b Fm(Supp)-5 b(ose)37 b(that)g(Assumption) g(6.F)1619 767 y Fo(\032)1696 753 y Fm(hold.)53 b(Then)36 b Ft(L)2240 720 y Fl(semi)2240 775 y Fo(\032)2418 753 y Fm(is)g(essential)5 b(ly)36 b(self-adjoint)h(on)440 890 y Fr(D)s Fu(\()p Ft(L)610 857 y Fl(semi)610 918 y(fr)751 890 y Fu(\))21 b Fr(\\)f(D)s Fu(\()p Ft(V)1049 904 y Fo(\032)1089 890 y Fu(\))33 b Fm(and)g Ft(\034)1383 857 y Fo(t)1373 912 y(\032)1413 890 y Fu(\()p Ft(A)p Fu(\))27 b(:=)e(e)1739 854 y Fl(i)p Fo(tL)1832 830 y Fh(semi)1832 870 y Fg(\032)1959 890 y Ft(A)p Fu(e)2067 854 y Fs(\000)p Fl(i)p Fo(tL)2215 830 y Fh(semi)2215 870 y Fg(\032)2376 890 y Fm(is)32 b(a)h Ft(W)2651 857 y Fs(\003)2690 890 y Fm(-dynamics)h(on)f Fp(M)3350 904 y Fo(\032)3390 890 y Fm(.)244 1102 y Fv(Pro)s(of.)61 b Fu(P)m(art)31 b(\(1\))h(is)e(ob)m (vious.)380 1215 y(Arguing)g(as)g(in)g(the)g(pro)s(of)f(of)h(Prop)s (osition)g(6.6)h(w)m(e)f(see)g(that)h(Assumption)e(6.F)3186 1229 y Fo(\032)3257 1215 y Fu(implies)244 1345 y(that)f(\(1)15 b(+)p 619 1271 53 4 v 15 w Ft(h)q Fu(\))p 707 1295 48 4 v Ft(\032)764 1282 y Fh(1)p 764 1294 31 3 v 764 1335 a(2)809 1345 y Ft(v)856 1312 y Fo(?)923 1345 y Fu(is)28 b(b)s(ounded.)38 b(Hence)28 b(\(1)15 b(+)g Fr(j)p Ft(r)s Fr(j)p Fu(\))p Ft(v)2039 1359 y Fo(\032)2108 1345 y Fu(is)28 b(b)s(ounded.)38 b(Therefore,)28 b(Theorem)f(5.3)244 1458 y(yields)f(that)h Ft(L)750 1425 y Fl(semi)750 1481 y Fo(\032)917 1458 y Fu(is)g(essen)m(tially)h(self-adjoin)m(t)f(on)f Fr(D)s Fu(\()p Ft(L)2191 1425 y Fl(semi)2191 1486 y(fr)2333 1458 y Fu(\))12 b Fr(\\)g(D)s Fu(\()p Ft(V)2614 1472 y Fo(\032)2653 1458 y Fu(\).)40 b(Since)26 b Ft(V)3039 1472 y Fo(\032)3105 1458 y Fu(is)g(a\016liated)244 1571 y(with)k Fp(M)546 1585 y Fo(\032)587 1571 y Fu(,)g(Theorem)g(3.3)i(in)e ([DJP])h(implies)f(that)h Ft(\034)2083 1585 y Fo(\032)2154 1571 y Fu(is)f(a)h Ft(W)2420 1538 y Fs(\003)2459 1571 y Fu(-dynamics.)40 b Fa(2)380 1660 y Fk(\000)422 1734 y Fp(M)517 1748 y Fo(\032)558 1734 y Ft(;)15 b(\034)638 1748 y Fo(\032)678 1660 y Fk(\001)750 1734 y Fu(will)31 b(b)s(e)e(called)j(the)f(P)m(auli-Fierz)h Ft(W)2026 1701 y Fs(\003)2065 1734 y Fu(-dynamical)f(system)g(at)g(densit)m(y)f Ft(\032)p Fu(.)244 2010 y Fi(6.6)135 b(Standard)45 b(represen)l(tation) i(of)e(P)l(auli-Fierz)h(systems)244 2182 y Fu(Consider)29 b(the)i(represen)m(tation)g Ft(\031)i Fu(:)25 b Fp(M)1604 2196 y Fo(\032)1670 2182 y Fr(!)h(B)s Fu(\()p Fr(K)21 b(\012)p 2066 2109 71 4 v 20 w(K)h(\012)e Fu(\000)2305 2196 y Fl(s)2337 2182 y Fu(\()p Fr(Z)27 b(\010)p 2556 2109 74 4 v 20 w(Z)7 b Fu(\)\))31 b(de\014ned)e(b)m(y)1033 2386 y Ft(\031)7 b Fu(\()p Ft(A)p Fu(\))27 b(:=)e(1)p 1423 2361 59 3 v 29 x Fs(K)1494 2372 y Fu(\024)1482 2386 y Fr(\012)o Ft(A;)107 b(A)25 b Fr(2)g Fp(M)2026 2400 y Fo(\032)2092 2386 y Fu(=)g Fr(B)s Fu(\()p Fr(K)q Fu(\))c Fr(\012)f Fp(M)2598 2349 y Fl(A)-8 b(W)2598 2409 y Fo(\032;)p Fl(l)2721 2386 y Ft(;)244 2591 y Fu(where)519 2577 y(\024)507 2591 y Fr(\012)30 b Fu(w)m(as)h(in)m(tro)s(duced)e(in)h(\(4.22\).)43 b(Clearly)-8 b(,)1111 2795 y Ft(\031)7 b Fu(\()p Fr(B)s Fu(\()p Fr(K)q Fu(\))22 b Fr(\012)e Fp(M)1616 2757 y Fl(A)-8 b(W)1616 2818 y Fo(\032;)p Fl(l)1739 2795 y Fu(\))25 b(=)g Fr(B)s Fu(\()p Fr(K)q Fu(\))c Fr(\012)f Fu(1)p 2255 2769 V 29 x Fs(K)2334 2795 y Fr(\012)g Fp(M)2520 2757 y Fl(A)-8 b(W)2520 2818 y Fo(\032;)p Fl(l)2643 2795 y Ft(:)244 2999 y Fu(Set)30 b Ft(J)35 b Fu(:=)25 b Ft(J)656 3013 y Fs(K)735 2999 y Fr(\012)20 b Fu(\000\()p Ft(\017)p Fu(\),)31 b(where)1086 3203 y Ft(J)1136 3217 y Fs(K)1195 3203 y Fu(\011)1266 3217 y Fl(1)1325 3203 y Fr(\012)p 1416 3130 71 4 v 20 w Fu(\011)1486 3217 y Fl(2)1551 3203 y Fu(:=)25 b(\011)1743 3217 y Fl(2)1803 3203 y Fr(\012)p 1894 3130 V 20 w Fu(\011)1964 3217 y Fl(1)2004 3203 y Ft(;)197 b Fu(\011)2297 3217 y Fl(1)2336 3203 y Ft(;)15 b Fu(\011)2447 3217 y Fl(2)2512 3203 y Fr(2)25 b(K)q Ft(;)610 b Fu(\(6.43\))244 3407 y(and)30 b Ft(\017)g Fu(w)m(as)h(in)m(tro)s(duced)e(in)h(\(6.34\).)43 b(Note)32 b(that)f(if)f Ft(A)c Fr(2)f Fp(M)2296 3421 y Fo(\032)2336 3407 y Fu(,)31 b(then)1082 3638 y Ft(J)9 b(\031)s Fu(\()p Ft(A)p Fu(\))p Ft(J)36 b Fu(=)25 b(1)1561 3652 y Fs(K)1640 3638 y Fr(\012)1731 3537 y Fk(\020)1785 3638 y Fu(1)p 1830 3613 59 3 v 29 x Fs(K)1889 3638 y Fr(\012)o Fu(\000\()p Ft(\034)10 b Fu(\))p 2167 3565 69 4 v 31 w Ft(A)31 b Fu(1)p 2311 3613 59 3 v 29 x Fs(K)2369 3638 y Fr(\012)p Fu(\000\()p Ft(\034)10 b Fu(\))2617 3537 y Fk(\021)2672 3638 y Ft(;)244 3865 y Fu(where)30 b Ft(\034)40 b Fu(w)m(as)31 b(in)m(tro)s(duced)f(in)g(\(6.33\).)43 b(Set)912 4069 y Fr(H)989 4083 y Fo(\032;)p Fl(+)1129 4069 y Fu(:=)25 b Fr(f)p Ft(AJ)9 b(A)32 b(B)5 b Fr(\012)o Fu(\012)p Ft(;)45 b Fu(:)86 b Ft(A)26 b Fr(2)f Fp(M)2188 4083 y Fo(\032)2228 4069 y Ft(;)76 b(B)30 b Fr(2)25 b Ft(l)2543 4032 y Fl(2)2541 4092 y(+)2600 4069 y Fu(\()p Fr(K)q Fu(\))p Fr(g)2785 4032 y Fl(cl)2842 4069 y Ft(:)244 4282 y Fv(Prop)s(osition)36 b(6.8)1265 4311 y Fk(\020)1319 4412 y Ft(\031)7 b(;)36 b Fr(K)q(\012)p 1581 4339 71 4 v 1 w(K)q(\012)p Fu(\000)1779 4426 y Fl(s)1811 4412 y Fu(\()p Fr(Z)7 b(\010)p 1990 4339 74 4 v(Z)g Fu(\))p Ft(;)36 b(J)o(;)g Fr(H)2346 4426 y Fo(\032;)p Fl(+)2460 4311 y Fk(\021)244 4597 y Fm(is)c(a)h(standar)-5 b(d)36 b(r)-5 b(epr)g(esentation)35 b(of)e Fp(M)1585 4611 y Fo(\032)1626 4597 y Fm(.)380 4809 y Fu(Set)971 4922 y Ft(L)1033 4937 y Fl(fr)1111 4922 y Fu(:=)26 b Ft(K)h Fr(\012)19 b Fu(1)i Fr(\012)f Fu(1)h Fr(\000)f Fu(1)g Fr(\012)p 1897 4849 84 4 v 20 w Ft(K)27 b Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)g Fu(1)h Fr(\012)f Fu(d\000\()p Ft(r)s Fu(\))p Ft(:)244 5110 y Fv(Prop)s(osition)36 b(6.9)49 b Ft(L)1053 5125 y Fl(fr)1154 5110 y Fm(is)f(the)g(standar)-5 b(d)51 b(Liouvil)5 b(le)-5 b(an)48 b(of)g(the)g(fr)-5 b(e)g(e)48 b(Pauli-Fierz)g(system)244 5223 y Fu(\()p Fp(M)374 5237 y Fo(\032)415 5223 y Ft(;)15 b(\034)495 5238 y Fl(fr)548 5223 y Fu(\))p Fm(.)p eop %%Page: 32 32 32 31 bop 3682 100 a Fn(32)380 407 y Fu(In)30 b(what)g(follo)m(ws)i(w)m (e)e(assume)h(\(6.41\).)43 b(Note)31 b(that)600 600 y Ft(\031)s Fu(\()p Ft(V)743 614 y Fo(\032)784 600 y Fu(\))26 b(:=)f(1)p 1011 575 59 3 v 29 x Fs(K)1082 586 y Fu(\024)1070 600 y Fr(\012)o Ft(V)1193 614 y Fo(\032)845 843 y Fu(=)g(1)p 986 817 V 29 x Fs(K)1057 829 y Fu(\024)1044 843 y Fr(\012)1115 742 y Fk(\020)1169 843 y Fu(\(1)c(+)f Ft(\032)p Fu(\))1453 778 y Fh(1)p 1454 790 31 3 v 1454 831 a(2)1498 843 y Ft(v)s(;)p 1585 793 48 4 v 15 w(\032)1643 778 y Fh(1)p 1643 790 31 3 v 1643 831 a(2)1687 843 y Ft(v)1734 805 y Fo(?)1774 742 y Fk(\021)1829 843 y Fu(\()p Ft(a)1912 805 y Fs(\003)1951 843 y Fu(\))h(+)f(1)p 2143 817 59 3 v 29 x Fs(K)2214 829 y Fu(\024)2202 843 y Fr(\012)2272 742 y Fk(\020)2327 843 y Ft(v)2374 805 y Fs(\003)2413 843 y Fu(\(1)h(+)f Ft(\032)p Fu(\))2697 778 y Fh(1)p 2698 790 31 3 v 2698 831 a(2)2742 843 y Ft(;)15 b(v)2829 805 y Fo(?)p Fs(\003)p 2905 793 48 4 v 2905 843 a Ft(\032)2961 778 y Fh(1)p 2961 790 31 3 v 2961 831 a(2)3006 742 y Fk(\021)3060 843 y Fu(\()p Ft(a)p Fu(\))845 1110 y(=)941 1009 y Fk(\020)995 1110 y Fu(1)p 1040 1085 59 3 v 29 x Fs(K)1111 1096 y Fu(\024)1099 1110 y Fr(\012)o Ft(v)1213 1124 y Fo(\032)1254 1009 y Fk(\021)1308 1110 y Fu(\()p Ft(a)1391 1073 y Fs(\003)1431 1110 y Fu(\))20 b(+)1577 1009 y Fk(\020)1632 1110 y Fu(1)p 1677 1085 V 29 x Fs(K)1748 1096 y Fu(\024)1735 1110 y Fr(\012)p Ft(v)1853 1073 y Fs(\003)1850 1133 y Fo(\032)1893 1009 y Fk(\021)1947 1110 y Fu(\()p Ft(a)p Fu(\))p Ft(:)244 1341 y(\031)s Fu(\()p Ft(V)387 1355 y Fo(\032)428 1341 y Fu(\))i(is)h(essen)m(tially) h(self-adjoin)m(t)g(on)e Fr(K)6 b(\012)p 1724 1268 71 4 v 4 w(K)f(\012)t Fu(\000)1930 1308 y Fl(\014n)1930 1364 y(s)2013 1341 y Fu(\()p Fr(Z)11 b(\010)p 2200 1268 74 4 v 4 w(Z)c Fu(\),)25 b(a\016liated)f(to)f Fr(B)s Fu(\()p Fr(K)q Fu(\))t Fr(\012)t Fu(1)p 3154 1316 59 3 v 29 x Fs(K)3217 1341 y Fr(\012)t Fp(M)3387 1308 y Fl(A)-8 b(W)3387 1369 y Fo(\032;)p Fl(l)3510 1341 y Fu(,)244 1454 y(and)472 1650 y Ft(J)9 b(\031)s Fu(\()p Ft(V)674 1664 y Fo(\032)715 1650 y Fu(\))p Ft(J)92 b Fu(:=)26 b(1)1059 1664 y Fs(K)1138 1650 y Fr(\012)1228 1549 y Fk(\020)1283 1650 y Fu(1)p 1328 1625 V 29 x Fs(K)1386 1650 y Fr(\012)p Fu(\000\()p Ft(\034)10 b Fu(\))p 1665 1577 94 4 v 31 w Ft(V)1718 1664 y Fo(\032)1789 1650 y Fu(1)p 1834 1625 59 3 v 29 x Fs(K)1892 1650 y Fr(\012)p Fu(\000\()p Ft(\034)g Fu(\))2140 1549 y Fk(\021)892 1860 y Fu(=)25 b(1)1033 1874 y Fs(K)1112 1860 y Fr(\012)1203 1759 y Fk(\020)1257 1860 y Ft(\032)1314 1797 y Fh(1)p 1314 1809 31 3 v 1314 1850 a(2)p 1359 1810 48 4 v 1359 1860 a Ft(v)1406 1827 y Fo(?)1446 1860 y Ft(;)15 b Fu(\(1)21 b(+)p 1678 1810 V 20 w Ft(\032)p Fu(\))1770 1797 y Fh(1)p 1770 1809 31 3 v 1770 1850 a(2)p 1815 1810 48 4 v 1815 1860 a Ft(v)1862 1759 y Fk(\021)1917 1860 y Fu(\()p Ft(a)2000 1827 y Fs(\003)2039 1860 y Fu(\))g(+)f(1)2231 1874 y Fs(K)2310 1860 y Fr(\012)2401 1759 y Fk(\020)p 2455 1810 V 101 x Ft(v)2502 1827 y Fo(?)p Fs(\003)2577 1860 y Ft(\032)2634 1797 y Fh(1)p 2634 1809 31 3 v 2634 1850 a(2)2679 1860 y Ft(;)p 2719 1810 48 4 v 15 w(v)2766 1827 y Fs(\003)2806 1860 y Fu(\(1)h(+)p 2998 1810 V 20 w Ft(\032)p Fu(\))3090 1797 y Fh(1)p 3090 1809 31 3 v 3090 1850 a(2)3134 1759 y Fk(\021)3189 1860 y Fu(\()p Ft(a)p Fu(\))892 2069 y(=)988 1968 y Fk(\020)1043 2069 y Fu(1)1088 2083 y Fs(K)1167 2069 y Fr(\012)p 1257 2019 135 4 v 19 w Ft(\034)10 b(v)1351 2083 y Fo(\032)1392 1968 y Fk(\021)1446 2069 y Fu(\()p Ft(a)1529 2036 y Fs(\003)1569 2069 y Fu(\))21 b(+)1715 1968 y Fk(\020)1770 2069 y Fu(1)1815 2083 y Fs(K)1894 2069 y Fr(\012)p 1985 2001 137 4 v 20 w Ft(v)2032 2043 y Fs(\003)2029 2091 y Fo(\032)2071 2069 y Ft(\034)2121 1968 y Fk(\021)2176 2069 y Fu(\()p Ft(a)p Fu(\))p Ft(:)380 2282 y Fu(Set)1258 2395 y Ft(L)1320 2409 y Fo(\032)1385 2395 y Fu(:=)26 b Ft(L)1569 2410 y Fl(fr)1642 2395 y Fu(+)20 b Ft(\025\031)s Fu(\()p Ft(V)1929 2409 y Fo(\032)1970 2395 y Fu(\))g Fr(\000)g Ft(\025J)9 b(\031)s Fu(\()p Ft(V)2371 2409 y Fo(\032)2412 2395 y Fu(\))p Ft(J)o(:)782 b Fu(\(6.44\))244 2575 y Fv(Prop)s(osition)36 b(6.10)50 b Fm(Supp)-5 b(ose)43 b(that)h(Assumption)f(6.F)2249 2589 y Fo(\032)2331 2575 y Fm(holds.)72 b(Then)43 b Ft(L)2938 2589 y Fo(\032)3020 2575 y Fm(is)f(essential)5 b(ly)244 2688 y(self-adjoint)31 b(on)g Fr(D)s Fu(\()p Ft(L)1011 2703 y Fl(fr)1064 2688 y Fu(\))16 b Fr(\\)f(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)1442 2702 y Fo(\032)1483 2688 y Fu(\)\))h Fr(\\)g(D)s Fu(\()p Ft(J)9 b(\031)s Fu(\()p Ft(V)1956 2702 y Fo(\032)1996 2688 y Fu(\))p Ft(J)g Fu(\))32 b Fm(and)f(is)g(the)f(Liouvil)5 b(le)-5 b(an)32 b(of)e(the)h(Pauli-)244 2801 y(Fierz)i(system)g Fu(\()p Fp(M)908 2815 y Fo(\032)949 2801 y Ft(;)15 b(\034)1029 2815 y Fo(\032)1070 2801 y Fu(\))p Fm(.)244 2982 y Fv(Pro)s(of.)61 b Fu(The)30 b(assumptions)g (imply)g(that)728 3178 y(\(1)21 b(+)f Fr(j)p Ft(r)s Fr(j)p Fu(\)\(1)p 1129 3153 59 3 v 29 x Fs(K)1201 3164 y Fu(\024)1188 3178 y Fr(\012)p Ft(v)1303 3192 y Fo(\032)1363 3178 y Fr(\000)g Fu(1)1499 3192 y Fs(K)1558 3178 y Fr(\012)p 1629 3128 135 4 v Ft(\034)10 b(v)1723 3192 y Fo(\032)1763 3178 y Fu(\))26 b Fr(2)e(B)s Fu(\()p Fr(K)e(\012)p 2189 3105 71 4 v 20 w(K)q Ft(;)15 b Fr(K)23 b(\012)p 2482 3105 V 20 w(K)e(\012)f Fu(\()p Fr(Z)28 b(\010)p 2883 3105 74 4 v 20 w(Z)7 b Fu(\)\))p Ft(;)244 3375 y Fu(and)30 b(the)g(essen)m(tial)i(self-adjoin)m(tness)g(of)e Ft(L)1735 3389 y Fo(\032)1806 3375 y Fu(follo)m(ws)h(from)f(Theorem)g(5.3.)380 3487 y(The)g(op)s(erator)h Ft(L)996 3502 y Fl(fr)1069 3487 y Fu(+)20 b Ft(\025\031)s Fu(\()p Ft(V)1356 3501 y Fo(\032)1397 3487 y Fu(\))30 b(is)h(essen)m(tially)h(self-adjoin)m(t) g(on)e Fr(D)s Fu(\()p Ft(L)2753 3502 y Fl(fr)2806 3487 y Fu(\))21 b Fr(\\)f(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)3194 3501 y Fo(\032)3234 3487 y Fu(\)\))31 b(and)1009 3693 y Ft(\031)s Fu(\()p Ft(\034)1149 3655 y Fo(t)1139 3715 y(\032)1179 3693 y Fu(\()p Ft(A)p Fu(\)\))c(=)e(e)1515 3655 y Fl(i)p Fo(t)p Fl(\()p Fo(L)1635 3667 y Fh(fr)1683 3655 y Fl(+)p Fo(\025\031)r Fl(\()p Fo(V)1890 3663 y Fg(\032)1927 3655 y Fl(\)\))1986 3693 y Ft(\031)s Fu(\()p Ft(A)p Fu(\)e)2219 3655 y Fs(\000)p Fl(i)p Fo(t)p Fl(\()p Fo(L)2394 3667 y Fh(fr)2443 3655 y Fl(+)p Fo(\025\031)r Fl(\()p Fo(V)2650 3663 y Fg(\032)2686 3655 y Fl(\)\))2745 3693 y Ft(:)244 3889 y Fu(Hence)36 b(all)g(the)g(conditions)g(of)f (Theorem)g(2.3)i(are)e(satis\014ed)h(and)f Ft(L)2656 3903 y Fo(\032)2731 3889 y Fu(is)g(the)h(Liouvillean)g(of)244 4002 y(\()p Fp(M)374 4016 y Fo(\032)415 4002 y Ft(;)15 b(\034)495 4016 y Fo(\032)535 4002 y Fu(\).)41 b Fa(2)244 4324 y Fi(6.7)135 b(Lev)l(el)46 b(Shift)f(Op)t(erator)h(for)f(P)l (auli-Fierz)g(Liouvilleans)244 4496 y Fu(W)-8 b(e)36 b(will)e(see)i(in)e(this)g(subsection)h(that)g(the)g(Lev)m(el)h(Shift)d (Op)s(erator)h(of)h(the)g(Liouvillean)g Ft(L)3495 4510 y Fo(\032)244 4609 y Fu(has)30 b(v)m(ery)h(sp)s(ecial)g(algebraic)h (prop)s(erties.)380 4722 y(Let)g(us)e(form)m(ulate)i(the)f(follo)m (wing)i(family)e(of)g(assumptions)g(parametrized)g(b)m(y)g(the)h(radi-) 244 4835 y(ation)f(densit)m(y)g(op)s(erator)g Ft(\032)f Fu(and)g(a)g(n)m(um)m(b)s(er)f Ft(\021)g Fr(\025)c Fu(0.)244 4995 y Fv(Assumption)36 b(6.C)p Fu(\()p Ft(\021)s Fu(\))1099 5009 y Fo(\032)1382 4995 y Fr(h)p Ft(s)p Fr(i)1495 4962 y Fo(\021)1537 4995 y Ft(v)1581 5009 y Fo(\032)1647 4995 y Fr(2)25 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)c Fu(\()p Fr(Z)28 b(\010)p 2343 4922 V 20 w(Z)7 b Fu(\)\).)380 5155 y(Note)31 b(that)f(Assumption)f(6.C\()p Ft(\021)s Fu(\))1552 5169 y Fl(0)1593 5155 y Fu(,)h(in)m(tro)s(duced)f(in)g (Subsection)g(6.2,)i(is)f(the)f(sp)s(ecial)i(case)244 5268 y(of)f(Assumption)g(6.C\()p Ft(\021)s Fu(\))1105 5282 y Fo(\032)1177 5268 y Fu(for)g Ft(\032)25 b Fu(=)g(0.)p eop %%Page: 33 33 33 32 bop 3682 100 a Fn(33)380 407 y Fu(In)25 b(this)g(subsection)h(w)m (e)g(supp)s(ose)e(that)i(Assumption)f(6.C\()p Ft(\021)s Fu(\))2508 421 y Fo(\032)2575 407 y Fu(holds)g(with)g Ft(\021)k(>)3190 371 y Fl(1)p 3190 386 36 4 v 3190 438 a(2)3260 407 y Fu(and)c(w)m(e)244 520 y(study)19 b(the)i(Lev)m(el)g (Shift)f(Op)s(erator,)i(denoted)e(\000)1868 534 y Fo(\032)1908 520 y Fu(,)j(of)d(the)h(triple)2428 446 y Fk(\000)2470 520 y Fr(K)q(\012)p 2611 447 71 4 v(K)r(\012)p Fu(\000)2810 487 y Fl(0)2810 542 y(s)2849 520 y Fu(\()p Fr(Z)7 b(\010)p 3028 447 74 4 v(Z)g Fu(\))p Ft(;)15 b(L)3238 535 y Fl(fr)3292 520 y Ft(;)g(\031)s Fu(\()p Ft(V)3475 534 y Fo(\032)3516 520 y Fu(\))p Fr(\000)p Ft(J)9 b(\031)s Fu(\()p Ft(V)3824 534 y Fo(\032)3865 520 y Fu(\))p Ft(J)3959 446 y Fk(\001)4001 520 y Fu(.)380 633 y(De\014ne)31 b(the)f(follo)m(wing)i(self-adjoin)m (t)g(op)s(erators)e(on)h Fr(K)q Fu(:)802 845 y(\001)878 808 y Fl(R)878 868 y Fo(\032)960 845 y Fu(:=)1151 759 y Fk(X)1081 960 y Fo(k)r Fs(2)p Fl(sp)o(\()p Fo(K)5 b Fl(\))1387 759 y Fk(X)1368 956 y Fo(p)p Fs(2F)1501 968 y Fg(k)1539 845 y Fu(\()p Ft(v)1621 808 y Fs(\003)1618 868 y Fo(\032)1661 845 y Fu(\))1696 808 y Fo(k)r(;k)r Fs(\000)p Fo(p)1887 845 y Fr(P)i Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))2229 808 y Fs(\000)p Fl(1)2323 845 y Ft(v)2370 808 y Fo(k)r Fs(\000)p Fo(p;k)2367 868 y(\032)960 1233 y Fu(=)1126 1146 y Fk(X)1056 1348 y Fo(k)r Fs(2)p Fl(sp)o(\()p Fo(K)5 b Fl(\))1362 1146 y Fk(X)1342 1343 y Fo(p)p Fs(2F)1475 1355 y Fg(k)1513 1233 y Fu(\()p Ft(v)1595 1195 y Fs(\003)1636 1233 y Fu(\))1671 1195 y Fo(k)r(;k)r Fs(\000)p Fo(p)1862 1233 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))p Fr(P)7 b Fu(\()p Ft(p)21 b Fr(\000)f Ft(h)p Fu(\))2486 1195 y Fs(\000)p Fl(1)2581 1233 y Ft(v)2628 1195 y Fo(k)r Fs(\000)p Fo(p;k)955 1620 y Fu(+)1116 1534 y Fk(X)1045 1735 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1352 1534 y Fk(X)1332 1731 y Fo(p)p Fs(2F)1465 1743 y Fg(k)1518 1620 y Fu(tr)31 b Ft(v)1667 1583 y Fo(k)r(;k)r Fs(\000)p Fo(p)1858 1620 y Fu(\()p Ft(v)1940 1583 y Fs(\003)1980 1620 y Fu(\))2015 1583 y Fo(k)r Fs(\000)p Fo(p;k)2207 1620 y Ft(\032)p Fr(P)7 b Fu(\()p Ft(p)21 b Fu(+)f Ft(h)p Fu(\))2604 1583 y Fs(\000)p Fl(1)2699 1620 y Ft(;)768 2055 y Fu(\001)844 2017 y Fl(I)844 2077 y Fo(\032)909 2055 y Fu(:)26 b(=)f Fr(\000)p Ft(\031)1267 1969 y Fk(X)1196 2170 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1503 1969 y Fk(X)1483 2165 y Fo(p)p Fs(2F)1616 2177 y Fg(k)1654 2055 y Fu(\()p Ft(v)1736 2017 y Fs(\003)1733 2077 y Fo(\032)1776 2055 y Fu(\))1811 2017 y Fo(k)r(;k)r Fs(\000)p Fo(p)2003 2055 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(v)2365 2017 y Fo(k)r Fs(\000)p Fo(p;k)2362 2077 y(\032)960 2442 y Fu(=)25 b Fr(\000)p Ft(\031)1267 2356 y Fk(X)1196 2557 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1589 2356 y Fk(X)1483 2569 y Fo(p)p Fs(2F)1623 2539 y Fh(+)1616 2593 y Fg(k)1674 2569 y Fs([f)p Fl(0)p Fs(g)1827 2442 y Fu(\()p Ft(v)1909 2405 y Fs(\003)1949 2442 y Fu(\))1984 2405 y Fo(k)r(;k)r Fs(\000)p Fo(p)2175 2442 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))p Ft(\016)s Fu(\()p Ft(p)i Fr(\000)e Ft(h)p Fu(\))p Ft(v)2820 2405 y Fo(k)r Fs(\000)p Fo(p;k)955 2849 y Fr(\000)f Ft(\031)1186 2762 y Fk(X)1116 2964 y Fo(k)r Fs(2)p Fl(sp)o(\()p Fo(K)5 b Fl(\))1433 2762 y Fk(X)1402 2975 y Fo(p)p Fs(2F)1542 2946 y Ff(\000)1535 2999 y Fg(k)1609 2849 y Fu(tr)31 b Ft(v)1758 2811 y Fo(k)r(;k)r Fs(\000)p Fo(p)1949 2849 y Fu(\()p Ft(v)2031 2811 y Fs(\003)2071 2849 y Fu(\))2106 2811 y Fo(k)r Fs(\000)p Fo(p;k)2298 2849 y Ft(\032\016)s Fu(\()p Ft(p)21 b Fu(+)f Ft(h)p Fu(\))p Ft(:)244 3170 y Fu(Set)568 3365 y(\001)644 3379 y Fo(\032)709 3365 y Fu(:=)26 b(\001)907 3328 y Fl(R)907 3388 y Fo(\032)983 3365 y Fu(+)20 b(i\001)1175 3328 y Fl(I)1175 3388 y Fo(\032)1240 3365 y Fu(=)1406 3279 y Fk(X)1336 3480 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1643 3279 y Fk(X)1623 3476 y Fo(p)p Fs(2F)1756 3488 y Fg(k)1794 3365 y Fu(\()p Ft(v)1876 3328 y Fs(\003)1873 3388 y Fo(\032)1916 3365 y Fu(\))1951 3328 y Fo(k)r(;k)r Fs(\000)p Fo(p)2143 3365 y Fu(\()p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2596 3328 y Fs(\000)p Fl(1)2690 3365 y Ft(v)2737 3328 y Fo(k)r Fs(\000)p Fo(p;k)2734 3388 y(\032)1240 3752 y Fu(=)1406 3666 y Fk(X)1336 3867 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1643 3666 y Fk(X)1623 3863 y Fo(p)p Fs(2F)1756 3875 y Fg(k)1794 3752 y Fu(\()p Ft(v)1876 3715 y Fs(\003)1916 3752 y Fu(\))1951 3715 y Fo(k)r(;k)r Fs(\000)p Fo(p)2143 3752 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\)\()p Ft(p)g Fu(+)g(i0)h Fr(\000)f Ft(h)p Fu(\))2878 3715 y Fs(\000)p Fl(1)2973 3752 y Ft(v)3020 3715 y Fo(k)r Fs(\000)p Fo(p;k)1235 4140 y Fu(+)1396 4054 y Fk(X)1326 4255 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1633 4054 y Fk(X)1613 4250 y Fo(p)p Fs(2F)1746 4262 y Fg(k)1799 4140 y Fu(tr)30 b Ft(v)1947 4102 y Fo(k)r(;k)r Fs(\000)p Fo(p)2139 4140 y Fu(\()p Ft(v)2221 4102 y Fs(\003)2261 4140 y Fu(\))2296 4102 y Fo(k)r Fs(\000)p Fo(p;k)2487 4140 y Ft(\032)p Fu(\()p Ft(p)21 b Fu(+)f(i0)h(+)e Ft(h)p Fu(\))2995 4102 y Fs(\000)p Fl(1)3090 4140 y Ft(:)p eop %%Page: 34 34 34 33 bop 3682 100 a Fn(34)380 407 y Fu(F)-8 b(or)31 b Ft(B)f Fr(2)25 b Ft(l)757 374 y Fl(2)796 407 y Fu(\()p Fr(K)q Fu(\))32 b(set)f(also)297 626 y(\004)358 640 y Fo(\032)398 626 y Fu(\()p Ft(B)5 b Fu(\))26 b(:=)f(2)p Ft(\031)937 539 y Fk(X)805 741 y Fo(k)842 750 y Fh(1)876 741 y Fo(;k)933 750 y Fh(2)967 741 y Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1336 539 y Fk(X)1215 736 y Fo(p)p Fs(2F)1348 748 y Fg(k)1381 763 y Fh(1)1420 736 y Fs(\\F)1517 748 y Fg(k)1550 763 y Fh(2)1589 626 y Fu(\()p Ft(v)1671 588 y Fs(\003)1668 648 y Fo(\032)1711 626 y Fu(\))1746 588 y Fo(k)1783 597 y Fh(1)1818 588 y Fo(;k)1875 597 y Fh(1)1909 588 y Fs(\000)p Fo(p)2003 525 y Fk(\020)2057 626 y Ft(B)25 b Fr(\012)20 b Ft(\016)s Fu(\()p Ft(p)h Fr(\000)f Ft(r)s Fu(\))2557 525 y Fk(\021)2611 626 y Fu(\()p Ft(\034)10 b(v)2743 588 y Fo(?)2740 648 y(\032)2783 626 y Fu(\))2818 588 y Fo(k)2855 597 y Fh(2)2890 588 y Fs(\000)p Fo(p;k)3038 597 y Fh(2)568 1028 y Fu(=)25 b(2)p Ft(\031)911 941 y Fk(X)779 1143 y Fo(k)816 1152 y Fh(1)851 1143 y Fo(;k)908 1152 y Fh(2)942 1143 y Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1388 941 y Fk(X)1190 1154 y Fo(p)p Fs(2F)1330 1125 y Fh(+)1323 1178 y Fg(k)1356 1193 y Fh(1)1394 1154 y Fs(\\F)1498 1125 y Fh(+)1491 1178 y Fg(k)1524 1193 y Fh(2)1564 1154 y Fs([f)p Fl(0)p Fs(g)1716 1028 y Fu(\()p Ft(v)1798 990 y Fs(\003)1839 1028 y Fu(\))1874 990 y Fo(k)1911 999 y Fh(1)1945 990 y Fo(;k)2002 999 y Fh(1)2036 990 y Fs(\000)p Fo(p)2131 927 y Fk(\020)2185 1028 y Ft(B)25 b Fr(\012)20 b Ft(\016)s Fu(\()p Ft(p)h Fr(\000)f Ft(h)p Fu(\)\(1)h(+)f Ft(\032)p Fu(\))2977 963 y Fh(1)p 2978 975 31 3 v 2978 1016 a(2)3022 1028 y Ft(\032)3079 963 y Fh(1)p 3079 975 V 3079 1016 a(2)3124 927 y Fk(\021)3178 1028 y Ft(v)3225 990 y Fo(k)3262 999 y Fh(2)3297 990 y Fs(\000)p Fo(p;k)3445 999 y Fh(2)563 1450 y Fu(+)g(2)p Ft(\031)901 1364 y Fk(X)769 1565 y Fo(k)806 1574 y Fh(1)841 1565 y Fo(;k)898 1574 y Fh(2)931 1565 y Fs(2)p Fl(sp)q(\()p Fo(K)5 b Fl(\))1301 1364 y Fk(X)1180 1577 y Fo(p)p Fs(2F)1320 1547 y Ff(\000)1313 1601 y Fg(k)1346 1616 y Fh(1)1384 1577 y Fs(\\F)1488 1547 y Ff(\000)1481 1601 y Fg(k)1514 1616 y Fh(2)1569 1450 y Fu(tr)30 b Ft(v)1717 1413 y Fo(k)1754 1422 y Fh(1)1789 1413 y Fo(;k)1846 1422 y Fh(1)1879 1413 y Fs(\000)p Fo(p)1974 1450 y Ft(B)5 b Fu(\()p Ft(v)2130 1413 y Fs(\003)2170 1450 y Fu(\))2205 1413 y Fo(k)2242 1422 y Fh(2)2276 1413 y Fs(\000)p Fo(p;k)2424 1422 y Fh(2)2462 1450 y Ft(\016)s Fu(\()p Ft(p)21 b Fu(+)f Ft(h)p Fu(\)\(1)i(+)e Ft(\032)p Fu(\))3070 1386 y Fh(1)p 3070 1398 V 3070 1439 a(2)3114 1450 y Ft(\032)3171 1386 y Fh(1)p 3171 1398 V 3171 1439 a(2)3216 1450 y Ft(:)380 1795 y Fu(The)32 b(form)m(ulas)f(for)h(\001) 1156 1762 y Fl(R)1156 1817 y Fo(\032)1212 1795 y Fu(,)g(\001)1345 1762 y Fl(I)1345 1817 y Fo(\032)1385 1795 y Fu(,)h(\001)1519 1809 y Fo(\032)1591 1795 y Fu(and)e(\004)1830 1809 y Fo(\032)1901 1795 y Fu(are)i(written)f(in)f(t)m(w)m(o)i(equiv)-5 b(alen)m(t)34 b(forms.)44 b(The)244 1908 y(\014rst)26 b(forms)g(in)m(v)m(olv)m(e)k(the)d(op)s(erators)g Ft(v)1574 1922 y Fo(\032)1614 1908 y Fu(;)h(the)g(second)f(in)m(v)m(olv)m(e)i Ft(v)h Fu(and)c Ft(\032)p Fu(.)40 b(Altough)27 b(the)h(second)244 2020 y(forms)41 b(are)g(more)h(directly)g(related)g(to)g(the)f(basic)h (ph)m(ysical)g(quan)m(tities)h(of)e(in)m(terest,)k(they)244 2133 y(are)c(less)h(compact)g(and)f(tec)m(hnically)i(less)e(con)m(v)m (enien)m(t.)75 b(Note)43 b(in)d(particular)i(that)g(in)e(the)244 2246 y(form)m(ulas)23 b(for)g(\001)814 2213 y Fl(R)814 2269 y Fo(\032)892 2246 y Fu(and)g(\001)1138 2260 y Fo(\032)1178 2246 y Fu(,)h(the)f(terms)g(with)g Ft(p)i Fu(=)g(0)e(need)g(to)g(b)s(e) g(carefully)g(in)m(terpreted.)39 b(\(The)244 2359 y(singularit)m(y)33 b(of)f Fr(P)7 b Fu(\()p Fr(\000)p Ft(r)s Fu(\))1056 2326 y Fs(\000)p Fl(1)1183 2359 y Fu(and)32 b(\(i0)23 b Fr(\000)e Ft(r)s Fu(\))1661 2326 y Fs(\000)p Fl(1)1787 2359 y Fu(is)32 b(\\cut")i(in)m(to)f(t)m(w)m(o)g(parts)f(in)g(these)h(expressions.)244 2472 y(This)c(problem)h(is)h(absen)m(t)g(in)f(the)g(form)m(ulas)h(in)m (v)m(olving)g Ft(v)2247 2486 y Fo(\032)2288 2472 y Fu(\).)380 2585 y(In)h(the)h(expression)g(for)f(\004)1298 2599 y Fo(\032)1371 2585 y Fu(w)m(e)h(used)f(the)h(op)s(erator)g Ft(\034)10 b(v)2342 2552 y Fo(?)2339 2607 y(\032)2411 2585 y Fr(2)29 b(B)2564 2511 y Fk(\000)2605 2585 y Fr(K)q Ft(;)15 b Fr(K)24 b(\012)e Fu(\()p Fr(Z)29 b(\010)p 3123 2512 74 4 v 21 w(Z)7 b Fu(\))3231 2511 y Fk(\001)3273 2585 y Fu(.)48 b(Note)244 2698 y(that)754 2811 y Ft(\034)10 b(v)851 2773 y Fo(?)848 2833 y(\032)916 2811 y Fu(=)1012 2737 y Fk(\000)1054 2811 y Ft(\032)1111 2746 y Fh(1)p 1111 2758 31 3 v 1111 2799 a(2)1155 2811 y Ft(v)s(;)15 b Fu(\(1)22 b(+)p 1435 2761 48 4 v 20 w Ft(\032)p Fu(\))1527 2746 y Fh(1)p 1527 2758 31 3 v 1527 2799 a(2)1572 2811 y Ft(v)1619 2773 y Fo(?)1659 2737 y Fk(\001)1726 2811 y Fu(=)1822 2737 y Fk(\000)1863 2811 y Ft(\032)1920 2746 y Fh(1)p 1920 2758 V 1920 2799 a(2)1965 2811 y Fu(\(1)f(+)f Ft(\032)p Fu(\))2239 2773 y Fs(\000)2304 2746 y Fh(1)p 2304 2758 V 2304 2799 a(2)2348 2811 y Ft(;)p 2388 2761 48 4 v 15 w(\032)2436 2773 y Fs(\000)2501 2746 y Fh(1)p 2501 2758 31 3 v 2501 2799 a(2)2545 2811 y Fu(\(1)h(+)p 2737 2761 48 4 v 20 w Ft(\032)p Fu(\))2829 2746 y Fh(1)p 2829 2758 31 3 v 2829 2799 a(2)2874 2737 y Fk(\001)2916 2811 y Ft(v)2960 2825 y Fo(\032)3000 2811 y Ft(;)278 b Fu(\(6.45\))1141 3091 y Ft(\034)10 b(v)1238 3053 y Fo(?)1235 3113 y(\032)1278 3091 y Fu(\()p Ft(p)p Fu(\))26 b(=)1516 2904 y Fk(8)1516 2986 y(<)1516 3150 y(:)1638 3011 y Ft(\032)1695 2948 y Fh(1)p 1695 2960 V 1695 3001 a(2)1740 3011 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)421 b(p)25 b Fr(\025)g Fu(0)p Ft(;)1638 3187 y Fu(\(1)c(+)p 1830 3137 48 4 v 20 w Ft(\032)p Fu(\))1922 3124 y Fh(1)p 1922 3136 31 3 v 1922 3177 a(2)1967 3187 y Ft(v)2014 3154 y Fo(?)2054 3187 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))p Ft(;)83 b(p)25 b Fr(\024)g Fu(0;)244 3406 y Fv(Theorem)70 b(6.11)49 b Fm(L)-5 b(et)33 b Ft(B)d Fr(2)25 b Ft(l)1328 3373 y Fl(2)1367 3406 y Fu(\()p Fr(K)q Fu(\))p Fm(.)43 b(Then)1268 3599 y Fu(\000)1325 3613 y Fo(\032)1365 3599 y Fu(\()p Ft(B)5 b Fu(\))25 b(=)g(\001)1706 3613 y Fo(\032)1746 3599 y Ft(B)g Fr(\000)20 b Ft(B)5 b Fu(\001)2081 3562 y Fs(\003)2081 3622 y Fo(\032)2141 3599 y Fu(+)20 b(i\004)2318 3613 y Fo(\032)2358 3599 y Fu(\()p Ft(B)5 b Fu(\))p Ft(;)1252 3822 y Fu(\000)1309 3784 y Fl(R)1309 3844 y Fo(\032)1365 3822 y Fu(\()p Ft(B)g Fu(\))25 b(=)g(\001)1706 3784 y Fl(R)1706 3844 y Fo(\032)1762 3822 y Ft(B)g Fr(\000)20 b Ft(B)5 b Fu(\001)2097 3784 y Fl(R)2097 3844 y Fo(\032)2152 3822 y Ft(;)1268 4044 y Fu(\000)1325 4006 y Fl(I)1325 4066 y Fo(\032)1365 4044 y Fu(\()p Ft(B)g Fu(\))25 b(=)g(\001)1706 4006 y Fl(I)1706 4066 y Fo(\032)1746 4044 y Ft(B)g Fu(+)20 b Ft(B)5 b Fu(\001)2081 4006 y Fl(I)2081 4066 y Fo(\032)2141 4044 y Fu(+)20 b(\004)2293 4058 y Fo(\032)2333 4044 y Fu(\()p Ft(B)5 b Fu(\))p Ft(:)244 4374 y Fv(Pro)s(of.)61 b Fu(Using)31 b(\(5.32\))i(w)m(e)d(see)h(that)402 4584 y(\000)459 4598 y Fo(\032)524 4584 y Fu(=)683 4497 y Fk(X)620 4694 y Fo(e)653 4703 y Fh(1)687 4694 y Fo(;e)740 4703 y Fh(2)774 4694 y Fs(2F)499 4874 y Fk(\000)540 4948 y Fu(1)p 585 4922 59 3 v 29 x Fs(K)657 4934 y Fu(\024)644 4948 y Fr(\012)p Ft(v)762 4910 y Fs(\003)759 4970 y Fo(\032)822 4948 y Fr(\000)20 b Fu(1)958 4962 y Fs(K)1016 4948 y Fr(\012)p 1087 4880 137 4 v Ft(v)1134 4921 y Fs(\003)1131 4970 y Fo(\032)1174 4948 y Ft(\034)1224 4874 y Fk(\001)1265 4897 y Fo(e)1298 4906 y Fh(1)1333 4897 y Fo(;e)1386 4906 y Fh(2)1424 4948 y Fu(\()p Ft(e)1501 4962 y Fl(1)1561 4948 y Fr(\000)g Ft(e)1694 4962 y Fl(2)1754 4948 y Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2106 4910 y Fs(\000)p Fl(1)2200 4874 y Fk(\000)2242 4948 y Fu(1)p 2287 4922 59 3 v 29 x Fs(K)2358 4934 y Fu(\024)2345 4948 y Fr(\012)p Ft(v)2460 4962 y Fo(\032)2521 4948 y Fr(\000)g Fu(1)2657 4962 y Fs(K)2715 4948 y Fr(\012)p 2786 4898 135 4 v Ft(\034)10 b(v)2880 4962 y Fo(\032)2920 4874 y Fk(\001)2962 4897 y Fo(e)2995 4906 y Fh(2)3029 4897 y Fo(;e)3082 4906 y Fh(1)3120 4948 y Ft(;)3303 4759 y Fu(\(6.46\))244 5155 y(where)38 b(the)i(sup)s(erscripts)d Ft(e)1236 5169 y Fl(1)1276 5155 y Ft(;)15 b(e)1358 5169 y Fl(2)1437 5155 y Fu(corresp)s(ond)38 b(to)h(the)h(decomp)s(osition)f(of)h Fr(K)27 b(\012)p 3107 5082 71 4 v 26 w(K)40 b Fu(in)m(to)g(the)244 5268 y(eigenspaces)32 b(of)e Ft(K)d Fr(\012)20 b Fu(1)h Fr(\000)f Fu(1)g Fr(\012)p 1336 5195 84 4 v 20 w Ft(K)7 b Fu(.)41 b(\(Note)32 b(that)f(sp)o(\()p Ft(K)c Fr(\012)20 b Fu(1)h Fr(\000)f Fu(1)g Fr(\012)p 2567 5195 V 20 w Ft(K)7 b Fu(\))26 b(=)f Fr(F)9 b Fu(\).)p eop %%Page: 35 35 35 34 bop 3682 100 a Fn(35)380 407 y Fu(Let)21 b(us)f(switc)m(h)g(to)h (sup)s(erscripts)e(in)h(sp)o(\()p Ft(K)7 b Fu(\))21 b(and)e(to)i(the)g (decomp)s(osition)g(of)f Fr(K)i Fu(in)m(to)f(sp)s(ectral)244 520 y(subspaces)30 b(of)g Ft(K)7 b Fu(.)40 b(F)-8 b(or)32 b(a)e(\014xed)g Ft(p)25 b Fr(2)g Fj(R)30 b Fu(w)m(e)h(obtain)1048 660 y Fk(P)1027 811 y Fo(e)p Fs(2F)1164 654 y Fk(\000)1206 728 y Fu(1)p 1251 703 59 3 v 29 x Fs(K)1322 714 y Fu(\024)1309 728 y Fr(\012)p Ft(v)1427 695 y Fs(\003)1424 751 y Fo(\032)1467 654 y Fk(\001)1509 677 y Fo(e;e)p Fs(\000)p Fo(p)1821 728 y Fu(=)2005 660 y Fk(P)1917 816 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))2204 728 y Fu(1)p 2249 703 V 29 x Fs(K)2320 714 y Fu(\024)2308 728 y Fr(\012)o Fu(\()p Ft(v)2460 695 y Fs(\003)2457 751 y Fo(\032)2501 728 y Fu(\))2536 695 y Fo(k)r(;k)r Fs(\000)p Fo(p)2727 728 y Ft(;)1048 869 y Fk(P)1027 1020 y Fo(e)p Fs(2F)1164 863 y Fk(\000)1206 937 y Fu(1)1251 951 y Fs(K)1309 937 y Fr(\012)p 1380 869 137 4 v Ft(v)1427 910 y Fs(\003)1424 959 y Fo(\032)1467 937 y Ft(\034)1517 863 y Fk(\001)1559 886 y Fo(e;e)p Fs(\000)p Fo(p)1821 937 y Fu(=)2005 869 y Fk(P)1917 1024 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)g Fl(\))2204 937 y Fu(1)2249 951 y Fs(K)2308 937 y Fr(\012)p 2378 869 V -1 w Ft(v)2425 910 y Fs(\003)2422 959 y Fo(\032)2465 937 y Ft(\034)2515 888 y Fo(k)r Fs(\000)p Fo(p;k)2706 937 y Ft(;)1048 1073 y Fk(P)1027 1224 y Fo(e)p Fs(2F)1164 1068 y Fk(\000)1206 1141 y Fu(1)p 1251 1116 59 3 v 29 x Fs(K)1322 1127 y Fu(\024)1309 1141 y Fr(\012)p Ft(v)1424 1155 y Fo(\032)1464 1068 y Fk(\001)1506 1090 y Fo(e)p Fs(\000)p Fo(p;e)1821 1141 y Fu(=)2005 1073 y Fk(P)1917 1229 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)g Fl(\))2204 1141 y Fu(1)p 2249 1116 V 29 x Fs(K)2320 1127 y Fu(\024)2308 1141 y Fr(\012)o Ft(v)2425 1097 y Fo(k)r Fs(\000)p Fo(p;k)2422 1153 y(\032)2617 1141 y Ft(;)1048 1278 y Fk(P)1027 1429 y Fo(e)p Fs(2F)1164 1272 y Fk(\000)1206 1346 y Fu(1)1251 1360 y Fs(K)1309 1346 y Fr(\012)p 1380 1296 135 4 v Ft(\034)10 b(v)1474 1360 y Fo(\032)1514 1272 y Fk(\001)1556 1295 y Fo(e)p Fs(\000)p Fo(p;e)1821 1346 y Fu(=)2005 1278 y Fk(P)1917 1433 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))2204 1346 y Fu(1)2249 1360 y Fs(K)2308 1346 y Fr(\012)p 2378 1296 V -1 w Ft(\034)10 b(v)2472 1360 y Fo(\032)2513 1313 y(k)r(;k)r Fs(\000)p Fo(p)2704 1346 y Ft(:)244 1627 y Fu(The)30 b(terms)g(on)g(the)h(righ)m(t)g(can)f(b)s(e)g(nonzero)h(only)f(if)h Ft(p)25 b Fr(2)g(F)2321 1642 y Fo(k)2364 1627 y Fu(.)40 b(Therefore,)31 b(\(6.46\))h(b)s(ecomes)526 1840 y(\000)583 1854 y Fo(\032)649 1840 y Fu(=)833 1771 y Fk(P)745 1927 y Fo(k)r Fs(2)p Fl(sp)o(\()p Fo(K)5 b Fl(\))1099 1771 y Fk(P)1062 1923 y Fo(p)p Fs(2F)1195 1935 y Fg(k)1510 1766 y Fk(\000)1552 1840 y Fu(1)p 1597 1814 59 3 v 29 x Fs(K)1668 1826 y Fu(\024)1655 1840 y Fr(\012)p Fu(\()p Ft(v)1808 1807 y Fs(\003)1805 1862 y Fo(\032)1848 1840 y Fu(\))1883 1807 y Fo(k)r(;k)r Fs(\000)p Fo(p)2075 1840 y Fu(\()p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2528 1807 y Fs(\000)p Fl(1)2622 1840 y Fu(1)p 2667 1814 V 29 x Fs(K)2738 1826 y Fu(\024)2726 1840 y Fr(\012)o Ft(v)2843 1795 y Fo(k)r Fs(\000)p Fo(p;k)2840 1851 y(\032)1510 2087 y Fu(+1)1626 2101 y Fs(K)1684 2087 y Fr(\012)p 1755 2019 137 4 v Ft(v)1802 2060 y Fs(\003)1799 2109 y Fo(\032)1842 2087 y Ft(\034)1892 2038 y Fo(k)r Fs(\000)p Fo(p;k)2083 2087 y Fu(\()p Ft(p)g Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2536 2054 y Fs(\000)p Fl(1)2630 2087 y Fu(1)2675 2101 y Fs(K)2734 2087 y Fr(\012)p 2805 2037 135 4 v Ft(\034)10 b(v)2899 2101 y Fo(\032)2939 2054 y(k)r(;k)r Fs(\000)p Fo(p)3130 2013 y Fk(\001)526 2318 y Fr(\000)762 2250 y Fk(P)612 2406 y Fo(k)649 2415 y Fh(1)684 2406 y Fo(;k)741 2415 y Fh(2)775 2406 y Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1192 2250 y Fk(P)1053 2401 y Fo(p)p Fs(2F)1186 2413 y Fg(k)1219 2428 y Fh(1)1258 2401 y Fs(\\F)1355 2413 y Fg(k)1388 2428 y Fh(2)1510 2244 y Fk(\000)1552 2318 y Fu(1)p 1597 2293 59 3 v 29 x Fs(K)1668 2304 y Fu(\024)1655 2318 y Fr(\012)p Fu(\()p Ft(v)1808 2285 y Fs(\003)1805 2340 y Fo(\032)1848 2318 y Fu(\))1883 2285 y Fo(k)1920 2294 y Fh(2)1955 2285 y Fo(;k)2012 2294 y Fh(2)2046 2285 y Fs(\000)p Fo(p)2140 2318 y Fu(\()p Ft(p)21 b Fu(+)e(i0)i Fr(\000)f Ft(r)s Fu(\))2593 2285 y Fs(\000)p Fl(1)2687 2318 y Fu(1)2732 2332 y Fs(K)2791 2318 y Fr(\012)p 2862 2268 135 4 v Ft(\034)10 b(v)2956 2332 y Fo(\032)2996 2285 y(k)3033 2294 y Fh(1)3068 2285 y Fo(;k)3125 2294 y Fh(1)3159 2285 y Fs(\000)p Fo(p)1510 2565 y Fu(+1)1626 2579 y Fs(K)1684 2565 y Fr(\012)p 1755 2497 137 4 v Ft(v)1802 2539 y Fs(\003)1799 2587 y Fo(\032)1842 2565 y Ft(\034)1892 2516 y Fo(k)1929 2525 y Fh(1)1963 2516 y Fs(\000)p Fo(p;k)2111 2525 y Fh(1)2149 2565 y Fu(\()p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2602 2532 y Fs(\000)p Fl(1)2696 2565 y Fu(1)p 2741 2540 59 3 v 29 x Fs(K)2812 2551 y Fu(\024)2800 2565 y Fr(\012)o Ft(v)2917 2521 y Fo(k)2954 2530 y Fh(2)2989 2521 y Fs(\000)p Fo(p;k)3137 2530 y Fh(2)2914 2577 y Fo(\032)3174 2491 y Fk(\001)3216 2565 y Ft(:)380 2776 y Fu(No)m(w)31 b(let)h Ft(B)d Fr(2)c Ft(l)932 2743 y Fl(2)972 2776 y Fu(\()p Fr(K)q Fu(\).)42 b(W)-8 b(e)31 b(see)g(that)g(\000)1736 2790 y Fo(\032)1776 2776 y Fu(\()p Ft(B)5 b Fu(\))31 b(consists)g(of)f(four)g(t)m(yp)s(es)g (of)h(terms:)244 2889 y Fm(T)-7 b(yp)i(e)33 b(I)e Fu(Using)f(\(4.28\),) j(w)m(e)e(obtain)1028 3028 y Fk(\000)1070 3101 y Fu(1)p 1115 3076 V 29 x Fs(K)1186 3087 y Fu(\024)1173 3101 y Fr(\012)p Fu(\()p Ft(v)1326 3068 y Fs(\003)1323 3124 y Fo(\032)1366 3101 y Fu(\))1401 3068 y Fo(k)r(;k)r Fs(\000)p Fo(p)1593 3028 y Fk(\001)1634 3101 y Fu(\()p Ft(p)21 b Fu(+)f(i0)h Fr(\000)e Ft(r)s Fu(\))2087 3068 y Fs(\000)p Fl(1)2182 3028 y Fk(\000)2223 3101 y Fu(1)p 2268 3076 V 29 x Fs(K)2340 3087 y Fu(\024)2327 3101 y Fr(\012)p Ft(v)2445 3057 y Fo(k)r Fs(\000)p Fo(p;k)2442 3113 y(\032)2636 3028 y Fk(\001)2678 3101 y Ft(B)1028 3297 y Fu(=)25 b(\()p Ft(v)1206 3264 y Fs(\003)1203 3319 y Fo(\032)1246 3297 y Fu(\))1281 3264 y Fo(k)r(;k)r Fs(\000)p Fo(p)1473 3297 y Fu(\()p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))1926 3264 y Fs(\000)p Fl(1)2020 3297 y Ft(v)2067 3252 y Fo(k)r Fs(\000)p Fo(p;k)2064 3308 y(\032)2258 3297 y Ft(B)5 b(:)244 3500 y Fu(Summing)29 b(up)g(the)i(ab)s(o)m(v)m(e)g(terms)g(o)m (v)m(er)g Ft(k)e Fr(2)c Fu(sp)o(\()p Ft(K)7 b Fu(\))31 b(and)f Ft(p)24 b Fr(2)h(F)2472 3515 y Fo(k)2546 3500 y Fu(w)m(e)30 b(obtain)h(\001)3038 3514 y Fo(\032)3078 3500 y Ft(B)5 b Fu(.)244 3613 y Fm(T)-7 b(yp)i(e)32 b(II.)c Fu(W)-8 b(e)30 b(switc)m(h)f(the)f(sign)h(in)f Ft(p)g Fu(and)g(rename)h Ft(k)20 b Fr(\000)c Ft(p)28 b Fu(to)h Ft(k)s Fu(.)41 b(Using)28 b(\014rst)g(\(4.29\))j(and)d(then)244 3726 y Ft(\034)p 294 3675 44 4 v 10 w(r)r(\034)36 b Fu(=)25 b Fr(\000)p Ft(r)s Fu(,)k(w)m(e)i(get)886 3863 y Fk(\000)927 3937 y Fu(1)972 3951 y Fs(K)1031 3937 y Fr(\012)p Fu(\()p 1137 3869 137 4 v Ft(v)1184 3910 y Fs(\003)1181 3959 y Fo(\032)1224 3937 y Ft(\034)10 b Fu(\))1309 3904 y Fo(k)r(;k)r Fs(\000)p Fo(p)1500 3863 y Fk(\001)1542 3937 y Fu(\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)2031 3863 y Fk(\001)2072 3886 y Fs(\000)p Fl(1)2166 3863 y Fk(\000)2208 3937 y Fu(1)2253 3951 y Fs(K)2312 3937 y Fr(\012)o Fu(\()p 2417 3887 135 4 v Ft(\034)10 b(v)2511 3951 y Fo(\032)2552 3937 y Fu(\))2587 3904 y Fo(k)r Fs(\000)p Fo(p;k)2779 3863 y Fk(\001)2820 3937 y Ft(B)886 4132 y Fu(=)25 b Ft(B)5 b Fu(\()p Ft(v)1138 4099 y Fs(\003)1135 4154 y Fo(\032)1177 4132 y Ft(\034)10 b Fu(\))1262 4099 y Fo(k)r(;k)r Fs(\000)p Fo(p)1454 4132 y Fu(\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h Fr(\000)p 1899 4082 44 4 v 20 w Ft(r)r Fu(\))1977 4099 y Fs(\000)p Fl(1)2072 4132 y Ft(\034)10 b(v)2169 4087 y Fo(k)r Fs(\000)p Fo(p;k)2166 4144 y(\032)886 4327 y Fu(=)25 b Fr(\000)p Ft(B)5 b Fu(\()p Ft(v)1209 4294 y Fs(\003)1206 4349 y Fo(\032)1248 4327 y Fu(\))1283 4294 y Fo(k)r(;k)r Fs(\000)p Fo(p)1475 4327 y Fu(\()p Ft(p)20 b Fr(\000)g Fu(i0)h Fr(\000)f Ft(r)s Fu(\))1928 4294 y Fs(\000)p Fl(1)2022 4327 y Ft(v)2069 4283 y Fo(k)r Fs(\000)p Fo(p;k)2066 4339 y(\032)2260 4327 y Ft(:)244 4530 y Fu(Summing)29 b(up)g(the)i(ab)s(o)m(v)m(e)g(terms)g(o)m(v)m(er)g Ft(k)e Fr(2)c Fu(sp)o(\()p Ft(K)7 b Fu(\))31 b(and)f Ft(p)24 b Fr(2)h(F)2472 4545 y Fo(k)2546 4530 y Fu(w)m(e)30 b(obtain)h Fr(\000)p Ft(B)5 b Fu(\001)3183 4497 y Fs(\003)3183 4552 y Fo(\032)3223 4530 y Fu(.)244 4643 y Fm(T)-7 b(yp)i(e)33 b(III.)e Fu(W)-8 b(e)31 b(use)f(\(4.27\))j(to)e(obtain)876 4844 y(\(1)p 956 4819 59 3 v 29 x Fs(K)1028 4830 y Fu(\024)1015 4844 y Fr(\012)p Fu(\()p Ft(v)1168 4811 y Fs(\003)1165 4867 y Fo(\032)1208 4844 y Fu(\))1243 4811 y Fo(k)1280 4820 y Fh(2)1315 4811 y Fo(;k)1372 4820 y Fh(2)1405 4811 y Fs(\000)p Fo(p)1500 4844 y Fu(\)\()p Ft(p)21 b Fu(+)f(i0)g Fr(\000)g Ft(r)s Fu(\))1988 4811 y Fs(\000)p Fl(1)2083 4844 y Fu(\(1)2163 4858 y Fs(K)2242 4844 y Fr(\012)g Fu(\()p 2368 4794 135 4 v Ft(\034)10 b(v)2462 4858 y Fo(\032)2503 4844 y Fu(\))2538 4811 y Fo(k)2575 4820 y Fh(1)2609 4811 y Fo(;k)2666 4820 y Fh(1)2700 4811 y Fs(\000)p Fo(p)2795 4844 y Fu(\))p Ft(B)876 5028 y Fu(=)25 b(\()p Ft(v)1054 4995 y Fs(\003)1051 5051 y Fo(\032)1094 5028 y Fu(\))1129 4995 y Fo(k)1166 5004 y Fh(2)1201 4995 y Fo(;k)1258 5004 y Fh(2)1292 4995 y Fs(\000)p Fo(p)1386 5028 y Ft(B)5 b Fr(\012)o Fu(\()p Ft(p)21 b Fu(+)f(i0)h Fr(\000)e Ft(r)s Fu(\))1983 4995 y Fs(\000)p Fl(1)2078 5028 y Fu(\()p Ft(\034)10 b(v)2210 4995 y Fo(?)2207 5051 y(\032)2250 5028 y Fu(\))2285 4995 y Fo(k)2322 5004 y Fh(1)2357 4995 y Fs(\000)p Fo(p;k)2505 5004 y Fh(1)2542 5028 y Ft(:)p eop %%Page: 36 36 36 35 bop 3682 100 a Fn(36)244 407 y Fm(T)-7 b(yp)i(e)33 b(IV.)c Fu(W)-8 b(e)31 b(switc)m(h)f(the)g(sign)f(in)h Ft(p)f Fu(as)h(w)m(ell)g(as)g(rename)g Ft(k)2351 421 y Fl(1)2410 407 y Fr(\000)18 b Ft(p)29 b Fu(and)g Ft(k)2797 421 y Fl(2)2856 407 y Fr(\000)18 b Ft(p)30 b Fu(to)g Ft(k)3178 421 y Fl(1)3248 407 y Fu(and)f Ft(k)3471 421 y Fl(2)3510 407 y Fu(.)244 520 y(W)-8 b(e)31 b(use)g(\(4.26\))h(and)e (then)g Ft(\034)p 1254 470 44 4 v 10 w(r)s(\034)35 b Fu(=)25 b Fr(\000)p Ft(r)s Fu(:)915 732 y(\(1)995 746 y Fs(K)1054 732 y Fr(\012)o Fu(\()p 1159 664 137 4 v Ft(v)1206 706 y Fs(\003)1203 755 y Fo(\032)1247 732 y Ft(\034)10 b Fu(\))1332 699 y Fo(k)1369 708 y Fh(1)1403 699 y Fo(;k)1460 708 y Fh(1)1494 699 y Fs(\000)p Fo(p)1589 732 y Fu(\)\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h Fr(\000)f Ft(r)s Fu(\))2148 699 y Fs(\000)p Fl(1)2242 732 y Fu(\(1)p 2322 707 59 3 v 29 x Fs(K)2394 718 y Fu(\024)2381 732 y Fr(\012)p Ft(v)2499 688 y Fo(k)2536 697 y Fh(2)2570 688 y Fo(;k)2627 697 y Fh(2)2661 688 y Fs(\000)p Fo(p)2496 744 y(\032)2756 732 y Fu(\))p Ft(B)915 928 y Fu(=)25 b(\()p Ft(v)1093 895 y Fo(?)p Fs(\003)1090 950 y Fo(\032)1168 928 y Fu(\))1203 895 y Fo(k)1240 904 y Fh(2)1275 895 y Fo(;k)1332 904 y Fh(2)1366 895 y Fs(\000)p Fo(p)1460 928 y Ft(B)5 b Fr(\012)o Fu(\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h Fr(\000)p 2049 877 44 4 v 20 w Ft(r)s Fu(\))2128 895 y Fs(\000)p Fl(1)2222 928 y Ft(\034)10 b(v)2319 883 y Fo(k)2356 892 y Fh(1)2391 883 y Fs(\000)p Fo(p;k)2539 892 y Fh(1)2316 939 y Fo(\032)915 1123 y Fu(=)25 b(\()p Ft(v)1093 1090 y Fo(?)p Fs(\003)1090 1145 y Fo(\032)1168 1123 y Ft(\034)10 b Fu(\))1253 1090 y Fo(k)1290 1099 y Fh(2)1325 1090 y Fo(;k)1382 1099 y Fh(2)1416 1090 y Fs(\000)p Fo(p)1510 1123 y Ft(B)5 b Fr(\012)o Fu(\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h(+)f Ft(r)s Fu(\))2178 1090 y Fs(\000)p Fl(1)2272 1123 y Ft(v)2319 1078 y Fo(k)2356 1087 y Fh(1)2391 1078 y Fs(\000)p Fo(p;k)2539 1087 y Fh(1)2316 1134 y Fo(\032)915 1306 y Fu(=)25 b(\()p Ft(v)1093 1273 y Fs(\003)1090 1329 y Fo(\032)1133 1306 y Fu(\))1168 1273 y Fo(k)1205 1282 y Fh(2)1239 1273 y Fo(;k)1296 1282 y Fh(2)1330 1273 y Fs(\000)p Fo(p)1425 1306 y Ft(B)5 b Fr(\012)o Fu(\()p Fr(\000)p Ft(p)20 b Fu(+)g(i0)h(+)f Ft(r)s Fu(\))2093 1273 y Fs(\000)p Fl(1)2187 1306 y Fu(\()p Ft(\034)10 b(v)2319 1273 y Fo(?)2316 1329 y(\032)2359 1306 y Fu(\))2394 1273 y Fo(k)2431 1282 y Fh(1)2466 1273 y Fs(\000)p Fo(p;k)2614 1282 y Fh(1)2651 1306 y Ft(:)244 1510 y Fu(In)30 b(the)g(last)h(step)g(w)m(e)g(used)e(\(6.45\))k(and)d (the)g(fact)h(that)g Ft(\032)g Fu(comm)m(utes)g(with)f Ft(h)p Fu(.)380 1623 y(The)f(sum)g(of)h(t)m(yp)s(e)g(I)s(I)s(I)e(and)h (IV)h(terms)g(o)m(v)m(er)h Ft(k)1999 1637 y Fl(1)2039 1623 y Ft(;)15 b(k)2126 1637 y Fl(2)2191 1623 y Fr(2)25 b Fu(sp)o(\()p Ft(K)7 b Fu(\))30 b(and)g Ft(p)25 b Fr(2)f(F)2945 1638 y Fo(k)2982 1647 y Fh(1)3040 1623 y Fr(\\)19 b(F)3185 1638 y Fo(k)3222 1647 y Fh(2)3290 1623 y Fu(equals)244 1735 y(i\004)330 1749 y Fo(\032)370 1735 y Fu(\()p Ft(B)5 b Fu(\).)41 b Fa(2)380 1898 y Fu(Set)919 1997 y Fk(e)-68 b Ft(v)946 2011 y Fo(\032)986 1957 y(p)1051 1997 y Fu(:=)1260 1929 y Fk(P)1172 2085 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1459 1997 y Ft(v)1506 1953 y Fo(k)r Fs(\000)p Fo(p;k)1503 2009 y(\032)1698 1997 y Ft(;)1862 1979 y Fk(g)1859 1997 y Ft(\034)10 b(v)1956 1971 y Fo(?)1953 2019 y(\032)1996 1937 y(p)2061 1997 y Fu(:=)2270 1929 y Fk(P)2182 2085 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))2454 1997 y Fu(\()p Ft(\034)10 b(v)2586 1964 y Fo(?)2583 2019 y(\032)2626 1997 y Fu(\))2661 1964 y Fo(k)r Fs(\000)p Fo(p;k)2853 1997 y Ft(;)903 2235 y Fk(e)-52 b Ft(v)949 2202 y Fo(p)1014 2235 y Fu(:=)1223 2167 y Fk(P)1135 2323 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)5 b Fl(\))1422 2235 y Ft(v)1469 2202 y Fo(k)r Fs(\000)p Fo(p;k)1661 2235 y Ft(;)1840 2217 y Fk(e)1822 2235 y Ft(v)1869 2209 y Fo(?)1909 2177 y(p)1974 2235 y Fu(:=)2183 2167 y Fk(P)2095 2323 y Fo(k)r Fs(2)p Fl(sp\()p Fo(K)g Fl(\))2367 2235 y Fu(\()p Ft(v)2449 2202 y Fo(?)2489 2235 y Fu(\))2524 2202 y Fo(k)r Fs(\000)p Fo(p;k)2715 2235 y Ft(:)380 2489 y Fu(Here)31 b(is)g(another)f(useful)g(expression)g(for)g(\000)1915 2456 y Fl(I)1915 2511 y Fo(\032)1955 2489 y Fu(:)244 2701 y Fv(Theorem)70 b(6.12)49 b Fm(L)-5 b(et)33 b Ft(B)1183 2715 y Fl(1)1223 2701 y Ft(;)15 b(B)1332 2715 y Fl(2)1397 2701 y Fr(2)25 b Ft(l)1512 2668 y Fl(2)1551 2701 y Fu(\()p Fr(K)q Fu(\))p Fm(.)43 b(Then)244 2933 y Fr(\000)p Fu(T)-8 b(r)o Ft(B)482 2895 y Fs(\003)477 2955 y Fl(1)521 2933 y Fu(\000)578 2895 y Fl(I)578 2955 y Fo(\032)618 2933 y Fu(\()p Ft(B)722 2947 y Fl(2)762 2933 y Fu(\))26 b(=)f Ft(\031)993 2847 y Fk(X)989 3043 y Fo(p)p Fs(2F)1144 2933 y Fu(T)-8 b(r)1237 2832 y Fk(\020)1309 2933 y(e)-68 b Ft(v)1336 2947 y Fo(\032)1376 2894 y(p)1416 2933 y Ft(B)1485 2947 y Fl(1)1544 2933 y Fr(\000)20 b Ft(B)1704 2947 y Fl(1)1744 2933 y Fr(\012)o Fu(1)1878 2915 y Fk(g)1874 2933 y Ft(\034)10 b(v)1971 2907 y Fo(?)1968 2955 y(\032)2012 2873 y(p)2051 2832 y Fk(\021)2106 2855 y Fs(\003)2145 2933 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))2460 2832 y Fk(\020)2531 2933 y(e)-68 b Ft(v)2558 2947 y Fo(\032)2599 2894 y(p)2638 2933 y Ft(B)2707 2947 y Fl(2)2767 2933 y Fr(\000)20 b Ft(B)2927 2947 y Fl(2)2966 2933 y Fr(\012)p Fu(1)3100 2915 y Fk(g)3097 2933 y Ft(\034)10 b(v)3194 2907 y Fo(?)3191 2955 y(\032)3234 2873 y(p)3274 2832 y Fk(\021)340 3279 y Fu(=)25 b Ft(\031)612 3192 y Fk(X)506 3395 y Fo(p)p Fs(2F)646 3376 y Fh(+)696 3395 y Fs([f)p Fl(0)p Fs(g)864 3279 y Fu(T)-8 b(r)958 3178 y Fk(\020)1012 3279 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1296 3214 y Fh(1)p 1297 3226 31 3 v 1297 3267 a(2)1342 3279 y Fk(e)-52 b Ft(v)1388 3241 y Fo(p)1428 3279 y Ft(B)1497 3293 y Fl(1)1557 3279 y Fr(\000)20 b Ft(B)1717 3293 y Fl(1)1756 3279 y Fr(\012)p Fu(1)15 b Ft(\032)1944 3214 y Fh(1)p 1944 3226 V 1944 3267 a(2)1990 3279 y Fk(e)-52 b Ft(v)2036 3241 y Fo(p)2076 3178 y Fk(\021)2130 3200 y Fs(\003)2170 3279 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(h)p Fu(\))2493 3178 y Fk(\020)2548 3279 y Fu(\(1)h(+)f Ft(\032)p Fu(\))2832 3214 y Fh(1)p 2832 3226 V 2832 3267 a(2)2877 3279 y Fk(e)-52 b Ft(v)2923 3241 y Fo(p)2963 3279 y Ft(B)3032 3293 y Fl(2)3092 3279 y Fr(\000)20 b Ft(B)3252 3293 y Fl(2)3291 3279 y Fr(\012)p Fu(1)15 b Ft(\032)3479 3214 y Fh(1)p 3480 3226 V 3480 3267 a(2)3525 3279 y Fk(e)-52 b Ft(v)3571 3241 y Fo(p)3611 3178 y Fk(\021)335 3633 y Fu(+)20 b Ft(\031)525 3547 y Fk(X)496 3749 y Fo(p)p Fs(2F)636 3730 y Fh(+)701 3633 y Fu(T)-8 b(r)795 3533 y Fk(\020)849 3633 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1133 3569 y Fh(1)p 1134 3581 V 1134 3622 a(2)1179 3633 y Fk(e)-52 b Ft(v)1225 3596 y Fo(p)1265 3633 y Ft(B)1339 3596 y Fs(\003)1334 3656 y Fl(2)1398 3633 y Fr(\000)20 b Ft(B)1563 3596 y Fs(\003)1558 3656 y Fl(2)1602 3633 y Fr(\012)p Fu(1)15 b Ft(\032)1790 3569 y Fh(1)p 1790 3581 V 1790 3622 a(2)1836 3633 y Fk(e)-52 b Ft(v)1882 3596 y Fo(p)1922 3533 y Fk(\021)1976 3555 y Fs(\003)2016 3633 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(h)p Fu(\))2339 3533 y Fk(\020)2394 3633 y Fu(\(1)h(+)f Ft(\032)p Fu(\))2678 3569 y Fh(1)p 2678 3581 V 2678 3622 a(2)2724 3633 y Fk(e)-52 b Ft(v)2770 3596 y Fo(p)2810 3633 y Ft(B)2884 3596 y Fs(\003)2879 3656 y Fl(1)2943 3633 y Fr(\000)20 b Ft(B)3108 3596 y Fs(\003)3103 3656 y Fl(1)3147 3633 y Fr(\012)o Fu(1)15 b Ft(\032)3334 3569 y Fh(1)p 3335 3581 V 3335 3622 a(2)3380 3633 y Fk(e)-52 b Ft(v)3426 3596 y Fo(p)3466 3533 y Fk(\021)3521 3633 y Ft(:)244 4084 y Fv(Pro)s(of.)61 b Fu(Recall)32 b(that)1171 4288 y Fr(\000)p Fu(\001)1318 4250 y Fl(I)1318 4310 y Fo(\032)1383 4288 y Fu(=)25 b Ft(\031)1550 4201 y Fk(X)1568 4399 y Fo(k)r(;p)1681 4288 y Fu(\()p Ft(v)1763 4250 y Fs(\003)1760 4310 y Fo(\032)1803 4288 y Fu(\))1838 4250 y Fo(k)r(;k)r Fs(\000)p Fo(p)2029 4288 y Ft(\016)s Fu(\()p Ft(p)d Fr(\000)e Ft(r)s Fu(\))p Ft(v)2392 4250 y Fo(k)r Fs(\000)p Fo(p;k)2389 4310 y(\032)2583 4288 y Ft(:)244 4583 y Fu(Hence)701 4778 y Fr(\000)p Fu(T)-8 b(r)o Ft(B)939 4740 y Fs(\003)934 4800 y Fl(1)978 4778 y Fu(\001)1054 4740 y Fl(I)1054 4800 y Fo(\032)1094 4778 y Ft(B)1163 4792 y Fl(2)1228 4778 y Fu(=)25 b Ft(\031)1394 4692 y Fk(X)1413 4889 y Fo(k)r(;p)1540 4778 y Fu(T)-8 b(r)1634 4704 y Fk(\000)1676 4778 y Ft(v)1723 4740 y Fo(k)r Fs(\000)p Fo(p;k)1720 4800 y(\032)1914 4778 y Ft(B)1983 4792 y Fl(1)2023 4704 y Fk(\001)2064 4727 y Fs(\003)2104 4778 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(v)2466 4740 y Fo(k)r Fs(\000)p Fo(p;k)2463 4800 y(\032)2657 4778 y Ft(B)2726 4792 y Fl(2)1228 5111 y Fu(=)25 b Ft(\031)1437 5024 y Fk(X)1394 5222 y Fo(k)1431 5231 y Fh(1)1465 5222 y Fo(;k)1522 5231 y Fh(2)1556 5222 y Fo(;p)1626 5111 y Fu(T)-8 b(r)1720 5037 y Fk(\000)1762 5111 y Ft(v)1809 5073 y Fo(k)1846 5082 y Fh(1)1880 5073 y Fs(\000)p Fo(p;k)2028 5082 y Fh(1)1806 5133 y Fo(\032)2066 5111 y Ft(B)2135 5125 y Fl(1)2174 5037 y Fk(\001)2216 5060 y Fs(\003)2255 5111 y Ft(\016)s Fu(\()p Ft(p)22 b Fr(\000)d Ft(r)s Fu(\)\()p Ft(v)2652 5073 y Fo(k)2689 5082 y Fh(2)2724 5073 y Fs(\000)p Fo(p;k)2872 5082 y Fh(2)2649 5133 y Fo(\032)2910 5111 y Ft(B)2979 5125 y Fl(2)3018 5111 y Fu(\))p Ft(:)p eop %%Page: 37 37 37 36 bop 3682 100 a Fn(37)380 407 y Fu(There)30 b(is)h(an)f (alternativ)m(e)j(form)m(ula)d(for)g Fr(\000)p Fu(\001)1935 374 y Fl(I)1935 429 y Fo(\032)1975 407 y Fu(,)h(whic)m(h)f(follo)m(ws)h (from)f Ft(\034)p 2854 357 44 4 v 10 w(r)s(\034)35 b Fu(=)25 b Fr(\000)p Ft(r)s Fu(:)1006 637 y Fr(\000)p Fu(\001)1153 600 y Fl(I)1153 660 y Fo(\032)1218 637 y Fu(=)f Ft(\031)1384 551 y Fk(X)1403 749 y Fo(k)r(;p)1530 637 y Fu(tr\()p Ft(v)1683 600 y Fo(?)1680 660 y(\032)1723 637 y Fu(\))1758 600 y Fo(k)r(;k)r Fs(\000)p Fo(p)1950 637 y Fu(\()p Ft(v)2032 600 y Fo(?)p Fs(\003)2029 660 y Fo(\032)2107 637 y Fu(\))2142 600 y Fo(k)r Fs(\000)p Fo(p;k)2334 637 y Ft(\016)s Fu(\()p Ft(p)d Fr(\000)p 2570 587 V 20 w Ft(r)r Fu(\))1218 970 y(=)j Ft(\031)1384 884 y Fk(X)1403 1081 y Fo(k)r(;p)1530 970 y Fu(tr\()p Ft(\034)10 b(v)1733 933 y Fo(?)1730 993 y(\032)1773 970 y Fu(\))1808 933 y Fo(k)r(;k)r Fs(\000)p Fo(p)2000 970 y Fu(\()p Ft(v)2082 933 y Fo(?)p Fs(\003)2079 993 y Fo(\032)2157 970 y Ft(\034)g Fu(\))2242 933 y Fo(k)r Fs(\000)p Fo(p;k)2434 970 y Ft(\016)s Fu(\()p Ft(p)21 b Fu(+)f Ft(r)s Fu(\))1218 1303 y(=)k Ft(\031)1384 1217 y Fk(X)1403 1414 y Fo(k)r(;p)1530 1303 y Fu(tr\()p Ft(\034)10 b(v)1733 1265 y Fo(?)1730 1325 y(\032)1773 1303 y Fu(\))1808 1265 y Fo(k)r Fs(\000)p Fo(p;k)2000 1303 y Fu(\()p Ft(v)2082 1265 y Fo(?)p Fs(\003)2079 1325 y Fo(\032)2157 1303 y Ft(\034)g Fu(\))2242 1265 y Fo(k)r(;k)r Fs(\000)p Fo(p)2434 1303 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(:)244 1598 y Fu(Hence)576 1795 y Fr(\000)p Fu(T)-8 b(r)o Ft(B)814 1757 y Fs(\003)809 1817 y Fl(1)853 1795 y Ft(B)922 1809 y Fl(2)962 1795 y Fu(\001)1038 1757 y Fl(I)1038 1817 y Fo(\032)1103 1795 y Fu(=)25 b Ft(\031)1269 1708 y Fk(X)1288 1906 y Fo(k)r(;p)1416 1795 y Fu(T)-8 b(r)1509 1721 y Fk(\000)1551 1795 y Ft(B)1620 1809 y Fl(1)1659 1795 y Fu(\()p Ft(\034)10 b(v)1791 1757 y Fo(?)1788 1817 y(\032)1831 1721 y Fk(\001)1873 1744 y Fo(k)r Fs(\000)p Fo(p;k)2064 1795 y Fu(\))2099 1757 y Fs(\003)2139 1795 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(B)2523 1809 y Fl(2)2562 1795 y Fu(\()p Ft(\034)10 b(v)2694 1757 y Fo(?)2691 1817 y(\032)2735 1795 y Fu(\))2770 1757 y Fo(k)r Fs(\000)p Fo(p;k)1103 2127 y Fu(=)25 b Ft(\031)1312 2041 y Fk(X)1269 2239 y Fo(k)1306 2248 y Fh(1)1340 2239 y Fo(;k)1397 2248 y Fh(2)1431 2239 y Fo(;p)1502 2127 y Fu(T)-8 b(r)1595 2054 y Fk(\000)1637 2127 y Ft(B)1706 2141 y Fl(1)1745 2127 y Fu(\()p Ft(\034)10 b(v)1877 2090 y Fo(?)1874 2150 y(\032)1917 2127 y Fu(\))1952 2090 y Fo(k)1989 2099 y Fh(1)2024 2090 y Fs(\000)p Fo(p;k)2172 2099 y Fh(1)2210 2054 y Fk(\001)2251 2076 y Fs(\003)2291 2127 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(B)2675 2141 y Fl(2)2714 2127 y Fu(\()p Ft(\034)10 b(v)2846 2090 y Fo(?)2843 2150 y(\032)2886 2127 y Fu(\))2921 2090 y Fo(k)2958 2099 y Fh(2)2993 2090 y Fs(\000)p Fo(p;k)3141 2099 y Fh(2)3178 2127 y Ft(:)380 2424 y Fu(Recall)32 b(that)814 2566 y Fl(1)p 814 2581 36 4 v 814 2634 a(2)859 2602 y Fu(\004)920 2616 y Fo(\032)960 2602 y Fu(\()p Ft(B)1064 2616 y Fl(2)1103 2602 y Fu(\))26 b(=)f Ft(\031)1391 2534 y Fk(P)1330 2686 y Fo(k)1367 2695 y Fh(1)1402 2686 y Fo(;k)1459 2695 y Fh(2)1493 2686 y Fo(;p)1548 2602 y Fu(\()p Ft(v)1630 2569 y Fs(\003)1627 2625 y Fo(\032)1670 2602 y Fu(\))1705 2569 y Fo(k)1742 2578 y Fh(1)1776 2569 y Fo(;k)1833 2578 y Fh(1)1867 2569 y Fs(\000)p Fo(p)1977 2602 y Ft(B)2046 2616 y Fl(2)2086 2602 y Fr(\012)o Ft(\016)s Fu(\()p Ft(p)c Fr(\000)f Ft(r)s Fu(\))15 b(\()p Ft(\034)10 b(v)2618 2569 y Fo(?)2615 2625 y(\032)2658 2602 y Fu(\))2693 2569 y Fo(k)2730 2578 y Fh(2)2765 2569 y Fs(\000)p Fo(p;k)2913 2578 y Fh(2)2951 2602 y Ft(:)3303 2638 y Fu(\(6.47\))244 2871 y(T)-8 b(erms)30 b(coming)h(from)f(\004)1109 2885 y Fo(\032)1179 2871 y Fu(w)m(e)h(split)f(as)966 3117 y(T)-8 b(r)o Ft(B)1133 3080 y Fs(\003)1128 3140 y Fl(1)1172 3117 y Fu(\004)1233 3131 y Fo(\032)1273 3117 y Fu(\()p Ft(B)1377 3131 y Fl(2)1417 3117 y Fu(\))26 b(=)1583 3056 y(1)p 1583 3096 46 4 v 1583 3180 a(2)1639 3117 y(T)-8 b(r)p Ft(B)1807 3080 y Fs(\003)1802 3140 y Fl(1)1846 3117 y Fu(\004)1907 3131 y Fo(\032)1946 3117 y Fu(\()p Ft(B)2050 3131 y Fl(2)2090 3117 y Fu(\))21 b(+)2247 3056 y(1)p 2247 3096 V 2247 3180 a(2)2302 3117 y(T)-8 b(r)p Ft(B)2470 3080 y Fs(\003)2465 3140 y Fl(1)2509 3117 y Fu(\004)2570 3131 y Fo(\032)2610 3117 y Fu(\()p Ft(B)2714 3131 y Fl(2)2753 3117 y Fu(\))p Ft(:)490 b Fu(\(6.48\))244 3347 y(The)30 b(\014rst)f(term)i(on)f(the)h(righ)m(t)g(of)f(\(6.48\))j (w)m(e)e(treat)g(as)g(follo)m(ws:)638 3524 y Fl(1)p 638 3539 36 4 v 638 3591 a(2)684 3560 y Fu(T)-8 b(r)o Ft(B)851 3527 y Fs(\003)846 3585 y Fl(1)890 3560 y Fu(\004)951 3574 y Fo(\032)991 3560 y Fu(\()p Ft(B)1095 3574 y Fl(2)1135 3560 y Fu(\))25 b(=)g Ft(\031)1422 3492 y Fk(P)1362 3644 y Fo(k)1399 3653 y Fh(1)1433 3644 y Fo(;k)1490 3653 y Fh(2)1524 3644 y Fo(;p)1594 3560 y Fu(T)-8 b(r)1688 3487 y Fk(\000)1729 3560 y Ft(v)1776 3516 y Fo(k)1813 3525 y Fh(1)1848 3516 y Fs(\000)p Fo(p;k)1996 3525 y Fh(1)1773 3572 y Fo(\032)2034 3560 y Ft(B)2103 3574 y Fl(1)2142 3487 y Fk(\001)2184 3509 y Fs(\003)2223 3560 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))15 b Ft(B)2622 3574 y Fl(2)2662 3560 y Fu(\()p Ft(\034)10 b(v)2794 3527 y Fo(?)2791 3583 y(\032)2834 3560 y Fu(\))2869 3527 y Fo(k)2906 3536 y Fh(2)2940 3527 y Fs(\000)p Fo(p;k)3088 3536 y Fh(2)3126 3560 y Ft(:)244 3834 y Fu(Then)29 b(w)m(e)i(transform) f(the)g(form)m(ula)h(\(6.47\),)i(using)d(\(6.45\),)j(\(4.28\))f(and)e (then)g Ft(\034)p 3033 3784 44 4 v 10 w(r)s(\034)35 b Fu(=)25 b Fr(\000)p Ft(r)s Fu(:)797 4015 y(1)p 797 4055 46 4 v 797 4139 a(2)852 4076 y(\004)913 4090 y Fo(\032)953 4076 y Fu(\()p Ft(B)1057 4090 y Fl(2)1097 4076 y Fu(\))h(=)f Ft(\031)1367 3990 y Fk(X)1324 4188 y Fo(k)1361 4197 y Fh(1)1395 4188 y Fo(;k)1452 4197 y Fh(2)1486 4188 y Fo(;p)1541 4076 y Fu(\()p Ft(v)1623 4039 y Fo(?)p Fs(\003)1620 4099 y Fo(\032)1699 4076 y Ft(\034)10 b Fu(\))1784 4039 y Fo(k)1821 4048 y Fh(1)1855 4039 y Fo(;k)1912 4048 y Fh(1)1946 4039 y Fs(\000)p Fo(p)2056 4076 y Ft(B)2125 4090 y Fl(2)2164 4076 y Fr(\012)p Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))15 b Ft(v)2612 4039 y Fo(k)2649 4048 y Fh(2)2684 4039 y Fs(\000)p Fo(p;k)2832 4048 y Fh(2)2609 4099 y Fo(\032)1158 4409 y Fu(=)25 b Ft(\031)1367 4323 y Fk(X)1324 4520 y Fo(k)1361 4529 y Fh(1)1395 4520 y Fo(;k)1452 4529 y Fh(2)1486 4520 y Fo(;p)1556 4409 y Fu(tr\()p Ft(\034)10 b(v)1756 4423 y Fo(\032)1797 4409 y Fu(\))1832 4372 y Fo(k)1869 4381 y Fh(1)1904 4372 y Fo(;k)1961 4381 y Fh(1)1995 4372 y Fs(\000)p Fo(p)2104 4409 y Ft(B)2173 4423 y Fl(2)2228 4409 y Fu(\()p Ft(v)2310 4372 y Fo(?)p Fs(\003)2307 4432 y Fo(\032)2385 4409 y Fu(\))2420 4372 y Fo(k)2457 4381 y Fh(2)2492 4372 y Fs(\000)p Fo(p;k)2640 4381 y Fh(2)2678 4409 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)p 2914 4359 44 4 v 20 w Ft(r)r Fu(\))1158 4742 y(=)k Ft(\031)1367 4656 y Fk(X)1324 4853 y Fo(k)1361 4862 y Fh(1)1395 4853 y Fo(;k)1452 4862 y Fh(2)1486 4853 y Fo(;p)1556 4742 y Fu(tr)p Ft(v)1674 4704 y Fo(k)1711 4713 y Fh(1)1746 4704 y Fo(;k)1803 4713 y Fh(1)1837 4704 y Fs(\000)p Fo(p)1671 4765 y(\032)1947 4742 y Ft(B)2016 4756 y Fl(2)2070 4742 y Fu(\()p Ft(v)2152 4704 y Fo(?)p Fs(\003)2149 4765 y Fo(\032)2228 4742 y Ft(\034)10 b Fu(\))2313 4704 y Fo(k)2350 4713 y Fh(2)2384 4704 y Fs(\000)p Fo(p;k)2532 4713 y Fh(2)2570 4742 y Ft(\016)s Fu(\()p Ft(p)21 b Fu(+)f Ft(r)s Fu(\))1158 5075 y(=)25 b Ft(\031)1367 4988 y Fk(X)1324 5186 y Fo(k)1361 5195 y Fh(1)1395 5186 y Fo(;k)1452 5195 y Fh(2)1486 5186 y Fo(;p)1556 5075 y Fu(tr)p Ft(v)1674 5037 y Fo(k)1711 5046 y Fh(1)1746 5037 y Fs(\000)p Fo(p;k)1894 5046 y Fh(1)1671 5097 y Fo(\032)1947 5075 y Ft(B)2016 5089 y Fl(2)2070 5075 y Fu(\()p Ft(v)2152 5037 y Fo(?)p Fs(\003)2149 5097 y Fo(\032)2228 5075 y Ft(\034)10 b Fu(\))2313 5037 y Fo(k)2350 5046 y Fh(2)2384 5037 y Fs(\000)p Fo(p;k)2532 5046 y Fh(2)2570 5075 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))p Ft(:)p eop %%Page: 38 38 38 37 bop 3682 100 a Fn(38)380 407 y Fu(Hence)31 b(the)g(second)f(term) h(in)f(\(6.48\))j(also)e(has)f(the)h(form)638 584 y Fl(1)p 638 599 36 4 v 638 651 a(2)684 620 y Fu(T)-8 b(r)o Ft(B)851 587 y Fs(\003)846 644 y Fl(1)890 620 y Fu(\004)951 634 y Fo(\032)991 620 y Fu(\()p Ft(B)1095 634 y Fl(2)1135 620 y Fu(\))25 b(=)g Ft(\031)1422 551 y Fk(P)1362 704 y Fo(k)1399 713 y Fh(1)1433 704 y Fo(;k)1490 713 y Fh(2)1524 704 y Fo(;p)1594 620 y Fu(T)-8 b(r)1688 546 y Fk(\000)1729 620 y Ft(B)1798 634 y Fl(1)1838 620 y Fu(\()p Ft(\034)10 b(v)1970 587 y Fo(?)1967 642 y(\032)2010 620 y Fu(\))2045 587 y Fo(k)2082 596 y Fh(2)2117 587 y Fs(\000)p Fo(p;k)2265 596 y Fh(2)2302 546 y Fk(\001)2344 569 y Fs(\003)2383 620 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))15 b Ft(B)2782 634 y Fl(2)2822 620 y Ft(v)2869 575 y Fo(k)2906 584 y Fh(1)2940 575 y Fs(\000)p Fo(p;k)3088 584 y Fh(1)2866 631 y Fo(\032)3126 620 y Ft(:)244 888 y Fu(This)29 b(ends)h(the)h(pro)s (of)e(of)i(the)f(\014rst)g(iden)m(tit)m(y)i(of)e(our)g(theorem.)380 1001 y(Let)h(us)f(pro)m(v)m(e)h(the)g(second)f(iden)m(tit)m(y)-8 b(.)43 b(W)-8 b(e)31 b(ha)m(v)m(e)285 1228 y Ft(\031)377 1159 y Fk(P)356 1311 y Fo(p)p Fs(2F)510 1228 y Fu(T)-8 b(r)604 1127 y Fk(\020)675 1228 y(e)-68 b Ft(v)702 1242 y Fo(\032)743 1188 y(p)782 1228 y Ft(B)851 1242 y Fl(1)911 1228 y Fr(\000)20 b Ft(B)1071 1242 y Fl(1)1110 1228 y Fr(\012)p Fu(1)1244 1210 y Fk(g)1241 1228 y Ft(\034)10 b(v)1338 1201 y Fo(?)1335 1250 y(\032)1378 1168 y(p)1418 1127 y Fk(\021)1472 1149 y Fs(\003)1512 1228 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(r)s Fu(\))1827 1127 y Fk(\020)1898 1228 y(e)-68 b Ft(v)1925 1242 y Fo(\032)1965 1188 y(p)2005 1228 y Ft(B)2074 1242 y Fl(2)2134 1228 y Fr(\000)20 b Ft(B)2294 1242 y Fl(2)2333 1228 y Fr(\012)p Fu(1)2467 1210 y Fk(g)2464 1228 y Ft(\034)10 b(v)2561 1201 y Fo(?)2558 1250 y(\032)2601 1168 y(p)2641 1127 y Fk(\021)285 1442 y Fu(=)25 b Ft(\031)575 1374 y Fk(P)452 1531 y Fo(p)p Fs(2F)592 1512 y Fh(+)642 1531 y Fs([f)p Fl(0)p Fs(g)810 1442 y Fu(T)-8 b(r)904 1341 y Fk(\020)958 1442 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1242 1379 y Fh(1)p 1242 1391 31 3 v 1242 1432 a(2)1288 1442 y Fk(e)-52 b Ft(v)1334 1409 y Fo(p)1374 1442 y Ft(B)1443 1456 y Fl(1)1502 1442 y Fr(\000)20 b Ft(B)1662 1456 y Fl(1)1702 1442 y Fr(\012)o Fu(1)15 b Ft(\032)1889 1379 y Fh(1)p 1890 1391 V 1890 1432 a(2)1935 1442 y Fk(e)-52 b Ft(v)1981 1409 y Fo(p)2021 1341 y Fk(\021)2076 1364 y Fs(\003)2115 1442 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)f Ft(h)p Fu(\))2438 1341 y Fk(\020)2493 1442 y Fu(\(1)h(+)f Ft(\032)p Fu(\))2777 1379 y Fh(1)p 2777 1391 V 2777 1432 a(2)2823 1442 y Fk(e)-52 b Ft(v)2869 1409 y Fo(p)2909 1442 y Ft(B)2978 1456 y Fl(2)3038 1442 y Fr(\000)20 b Ft(B)3198 1456 y Fl(2)3237 1442 y Fr(\012)p Fu(1)15 b Ft(\032)3425 1379 y Fh(1)p 3425 1391 V 3425 1432 a(2)3471 1442 y Fk(e)-52 b Ft(v)3517 1409 y Fo(p)3557 1341 y Fk(\021)285 1666 y Fu(+)p Ft(\031)501 1597 y Fk(P)426 1754 y Fs(\000)p Fo(p)p Fs(2F)621 1735 y Fh(+)687 1666 y Fu(T)-8 b(r)780 1565 y Fk(\020)p 835 1615 48 4 v 835 1666 a Ft(\032)892 1603 y Fh(1)p 892 1615 31 3 v 892 1656 a(2)954 1648 y Fk(e)936 1666 y Ft(v)983 1639 y Fo(?)1023 1608 y(p)1063 1666 y Ft(B)1132 1680 y Fl(1)1191 1666 y Fr(\000)20 b Ft(B)1351 1680 y Fl(1)1391 1666 y Fr(\012)o Fu(1)15 b(\(1)22 b(+)p 1714 1615 48 4 v 20 w Ft(\032)p Fu(\))1806 1603 y Fh(1)p 1806 1615 31 3 v 1806 1656 a(2)1869 1648 y Fk(e)1851 1666 y Ft(v)1898 1639 y Fo(?)1937 1608 y(p)1977 1565 y Fk(\021)2031 1587 y Fs(\003)2071 1666 y Ft(\016)s Fu(\()p Ft(p)f Fu(+)p 2307 1592 53 4 v 20 w Ft(h)p Fu(\))2394 1565 y Fk(\020)p 2449 1615 48 4 v 2449 1666 a Ft(\032)2506 1603 y Fh(1)p 2506 1615 31 3 v 2506 1656 a(2)2569 1648 y Fk(e)2550 1666 y Ft(v)2597 1639 y Fo(?)2637 1608 y(p)2677 1666 y Ft(B)2746 1680 y Fl(2)2805 1666 y Fr(\000)f Ft(B)2965 1680 y Fl(2)3005 1666 y Fr(\012)o Fu(1)15 b(\(1)22 b(+)p 3328 1615 48 4 v 20 w Ft(\032)p Fu(\))3420 1603 y Fh(1)p 3420 1615 31 3 v 3420 1656 a(2)3483 1648 y Fk(e)3465 1666 y Ft(v)3512 1639 y Fo(?)3552 1608 y(p)3591 1565 y Fk(\021)3646 1666 y Ft(:)244 1939 y Fu(The)30 b(second)g(term)h(on)f (the)g(righ)m(t)h(side)g(can)f(b)s(e)g(transformed)g(in)m(to)285 2165 y Ft(\031)403 2097 y Fk(P)356 2254 y Fo(p)p Fs(2F)496 2235 y Fh(+)561 2165 y Fu(T)-8 b(r)655 2064 y Fk(\020)p 709 2115 48 4 v 101 x Ft(\032)766 2102 y Fh(1)p 766 2114 31 3 v 766 2156 a(2)811 2165 y Fu(\()864 2150 y Fk(e)846 2165 y Ft(v)893 2139 y Fo(p)933 2165 y Fu(\))968 2132 y Fo(?)1008 2165 y Ft(B)1077 2179 y Fl(1)1136 2165 y Fr(\000)20 b Ft(B)1296 2179 y Fl(1)1336 2165 y Fr(\012)o Fu(1)15 b(\(1)22 b(+)p 1659 2115 48 4 v 20 w Ft(\032)p Fu(\))1751 2102 y Fh(1)p 1751 2114 31 3 v 1751 2156 a(2)1796 2165 y Fu(\()1849 2150 y Fk(e)1831 2165 y Ft(v)1878 2139 y Fo(p)1918 2165 y Fu(\))1953 2132 y Fo(?)1993 2064 y Fk(\021)2047 2087 y Fs(\003)2087 2165 y Ft(\016)s Fu(\()p Ft(p)f Fr(\000)p 2323 2091 53 4 v 20 w Ft(h)p Fu(\))2410 2064 y Fk(\020)p 2465 2115 48 4 v 2465 2165 a Ft(\032)2522 2102 y Fh(1)p 2522 2114 31 3 v 2522 2156 a(2)2566 2165 y Fu(\()2620 2150 y Fk(e)2601 2165 y Ft(v)2648 2139 y Fo(p)2689 2165 y Fu(\))2724 2132 y Fo(?)2763 2165 y Ft(B)2832 2179 y Fl(2)2892 2165 y Fr(\000)f Ft(B)3052 2179 y Fl(2)3091 2165 y Fr(\012)p Fu(1)15 b(\(1)22 b(+)p 3414 2115 48 4 v 19 w Ft(\032)p Fu(\))3506 2102 y Fh(1)p 3507 2114 31 3 v 3507 2156 a(2)3551 2165 y Fu(\()3605 2150 y Fk(e)3586 2165 y Ft(v)3633 2139 y Fo(p)3674 2165 y Fu(\))3709 2132 y Fo(?)3748 2064 y Fk(\021)285 2385 y Fu(=)j Ft(\031)499 2317 y Fk(P)452 2474 y Fo(p)p Fs(2F)592 2455 y Fh(+)657 2385 y Fu(T)-8 b(r)751 2284 y Fk(\020)805 2385 y Ft(B)879 2352 y Fs(\003)874 2410 y Fl(1)918 2385 y Fr(\012)p Fu(1)15 b Ft(\032)1106 2322 y Fh(1)p 1106 2334 V 1106 2375 a(2)1152 2385 y Fk(e)-52 b Ft(v)1198 2352 y Fo(p)1258 2385 y Fr(\000)20 b Fu(\(1)h(+)f Ft(\032)p Fu(\))1633 2322 y Fh(1)p 1633 2334 V 1633 2375 a(2)1679 2385 y Fk(e)-52 b Ft(v)1725 2352 y Fo(p)1765 2385 y Ft(B)1839 2352 y Fs(\003)1834 2410 y Fl(1)1878 2284 y Fk(\021)1932 2307 y Fo(?)p Fs(\003)2007 2385 y Ft(\016)s Fu(\()p Ft(p)21 b Fr(\000)p 2243 2311 53 4 v 20 w Ft(h)p Fu(\))2330 2284 y Fk(\020)2385 2385 y Ft(B)2459 2352 y Fs(\003)2454 2410 y Fl(2)2498 2385 y Fr(\012)o Fu(1)15 b Ft(\032)2685 2322 y Fh(1)p 2686 2334 31 3 v 2686 2375 a(2)2731 2385 y Fk(e)-51 b Ft(v)2778 2352 y Fo(p)2838 2385 y Fr(\000)20 b Fu(\(1)h(+)f Ft(\032)p Fu(\))3213 2322 y Fh(1)p 3213 2334 V 3213 2375 a(2)3258 2385 y Fk(e)-52 b Ft(v)3304 2352 y Fo(p)3344 2385 y Ft(B)3418 2352 y Fs(\003)3413 2410 y Fl(2)3457 2284 y Fk(\021)3512 2307 y Fo(?)285 2605 y Fu(=)25 b Ft(\031)499 2537 y Fk(P)452 2694 y Fo(p)p Fs(2F)592 2675 y Fh(+)657 2605 y Fu(T)-8 b(r)751 2504 y Fk(\020)805 2605 y Ft(B)879 2572 y Fs(\003)874 2629 y Fl(1)918 2605 y Fr(\012)p Fu(1)15 b Ft(\032)1106 2542 y Fh(1)p 1106 2554 V 1106 2595 a(2)1152 2605 y Fk(e)-52 b Ft(v)1198 2572 y Fo(p)1258 2605 y Fr(\000)20 b Fu(\(1)h(+)f Ft(\032)p Fu(\))1633 2542 y Fh(1)p 1633 2554 V 1633 2595 a(2)1679 2605 y Fk(e)-52 b Ft(v)1725 2572 y Fo(p)1765 2605 y Ft(B)1839 2572 y Fs(\003)1834 2629 y Fl(1)1878 2504 y Fk(\021\020)1986 2605 y Ft(B)2060 2572 y Fs(\003)2055 2629 y Fl(2)2099 2605 y Fr(\012)p Fu(1)p Ft(\032)2272 2542 y Fh(1)p 2272 2554 V 2272 2595 a(2)2318 2605 y Fk(e)g Ft(v)2364 2572 y Fo(p)2424 2605 y Fr(\000)20 b Fu(\(1)h(+)f Ft(\032)p Fu(\))2799 2542 y Fh(1)p 2799 2554 V 2799 2595 a(2)2845 2605 y Fk(e)-52 b Ft(v)2891 2572 y Fo(p)2931 2605 y Ft(B)3005 2572 y Fs(\003)3000 2629 y Fl(2)3044 2504 y Fk(\021)3098 2527 y Fs(\003)3137 2605 y Ft(\016)s Fu(\()p Ft(p)22 b Fr(\000)e Ft(h)p Fu(\))285 2825 y(=)25 b Ft(\031)499 2757 y Fk(P)452 2913 y Fo(p)p Fs(2F)592 2894 y Fh(+)657 2825 y Fu(T)-8 b(r)751 2724 y Fk(\020)805 2825 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1089 2762 y Fh(1)p 1089 2774 V 1089 2815 a(2)1135 2825 y Fk(e)-52 b Ft(v)1181 2792 y Fo(p)1221 2825 y Ft(B)1295 2792 y Fs(\003)1290 2849 y Fl(2)1354 2825 y Fr(\000)20 b Ft(B)1519 2792 y Fs(\003)1514 2849 y Fl(2)1558 2825 y Fr(\012)p Fu(1)15 b Ft(\032)1746 2762 y Fh(1)p 1746 2774 V 1746 2815 a(2)1792 2825 y Fk(e)-52 b Ft(v)1838 2792 y Fo(p)1878 2724 y Fk(\021)1932 2746 y Fs(\003)1971 2825 y Ft(\016)s Fu(\()p Ft(p)22 b Fr(\000)d Ft(h)p Fu(\))2294 2724 y Fk(\020)2349 2825 y Fu(\(1)i(+)f Ft(\032)p Fu(\))2633 2762 y Fh(1)p 2634 2774 V 2634 2815 a(2)2679 2825 y Fk(e)-52 b Ft(v)2725 2792 y Fo(p)2765 2825 y Ft(B)2839 2792 y Fs(\003)2834 2849 y Fl(1)2898 2825 y Fr(\000)20 b Ft(B)3063 2792 y Fs(\003)3058 2849 y Fl(1)3102 2825 y Fr(\012)p Fu(1)15 b Ft(\032)3290 2762 y Fh(1)p 3291 2774 V 3291 2815 a(2)3336 2825 y Fk(e)-52 b Ft(v)3382 2792 y Fo(p)3422 2724 y Fk(\021)3476 2825 y Ft(:)244 3132 y Fu(In)29 b(the)g(\014rst)g(step)g(w)m(e)h(used) 1248 3114 y Fk(e)1230 3132 y Ft(v)1277 3106 y Fo(?)1317 3075 y Fs(\000)p Fo(p)1437 3132 y Fu(=)25 b(\()q Fk(e)-52 b Ft(v)1615 3099 y Fo(p)1655 3132 y Fu(\))1690 3099 y Fo(?)1730 3132 y Fu(,)29 b(then)h(Prop)s(osition)f(4.6,)i(Prop)s (osition)f(4.8)g(and)f(in)244 3245 y(the)h(last)i(step)e(w)m(e)h(used)f (the)g(cyclicit)m(y)j(of)e(trace.)42 b Fa(2)380 3408 y Fu(Since)1281 3603 y Fk(e)-68 b Ft(v)1308 3617 y Fo(\032)1349 3563 y(p)1388 3603 y Fu(\()p Ft(p)p Fu(\))26 b(=)1626 3416 y Fk(8)1626 3498 y(<)1626 3662 y(:)1748 3523 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))2032 3460 y Fh(1)p 2032 3472 V 2032 3513 a(2)2078 3523 y Fk(e)-52 b Ft(v)2124 3490 y Fo(p)2164 3523 y Fu(\()p Ft(p)p Fu(\))p Ft(;)84 b(p)25 b Fr(\025)g Fu(0)p 1748 3649 48 4 v 1748 3699 a Ft(\032)1805 3636 y Fh(1)p 1805 3648 31 3 v 1805 3689 a(2)1868 3681 y Fk(e)1850 3699 y Ft(v)1897 3673 y Fo(?)1937 3641 y(p)1976 3699 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))p Ft(;)201 b(p)25 b Fr(\024)g Fu(0;)1104 3947 y Fk(g)1102 3965 y Ft(\034)10 b(v)1199 3938 y Fo(?)1196 3987 y(\032)1238 3905 y(p)1278 3965 y Fu(\()p Ft(p)p Fu(\))26 b(=)1516 3778 y Fk(8)1516 3860 y(<)1516 4024 y(:)1638 3885 y Ft(\032)1695 3822 y Fh(1)p 1695 3834 V 1695 3875 a(2)1741 3885 y Fk(e)-52 b Ft(v)1787 3852 y Fo(p)1827 3885 y Fu(\()p Ft(p)p Fu(\))p Ft(;)421 b(p)25 b Fr(\025)g Fu(0)1638 4061 y(\(1)c(+)p 1830 4011 48 4 v 20 w Ft(\032)p Fu(\))1922 3998 y Fh(1)p 1922 4010 31 3 v 1922 4051 a(2)1985 4043 y Fk(e)1967 4061 y Ft(v)2014 4034 y Fo(?)2054 4003 y(p)2093 4061 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))p Ft(;)84 b(p)25 b Fr(\024)g Fu(0)p Ft(;)p eop %%Page: 39 39 39 38 bop 3682 100 a Fn(39)244 407 y Fu(the)30 b(iden)m(tit)m(y)i(pro)m (v)m(en)f(in)f(Theorem)g(6.12)i(can)f(b)s(e)e(also)j(written)e(as)257 633 y Fr(\000)p Fu(T)-8 b(r)o Ft(B)495 596 y Fs(\003)490 656 y Fl(1)534 633 y Fu(\000)591 596 y Fl(I)591 656 y Fo(\032)631 633 y Fu(\()p Ft(B)735 647 y Fl(2)775 633 y Fu(\))26 b(=)f Ft(\031)1006 547 y Fk(X)1002 744 y Fo(p)p Fs(2F)1157 633 y Fu(T)-8 b(r)1250 532 y Fk(\020)1322 633 y(e)-68 b Ft(v)1349 647 y Fo(\032)1389 594 y(p)1429 633 y Fu(\()p Ft(p)p Fu(\))15 b Ft(B)1629 647 y Fl(1)1689 633 y Fr(\000)20 b Ft(B)1849 647 y Fl(1)1888 633 y Fr(\012)p Fu(1)2022 615 y Fk(g)2019 633 y Ft(\034)10 b(v)2116 607 y Fo(?)2113 656 y(\032)2156 573 y(p)2196 633 y Fu(\()p Ft(p)p Fu(\))2312 532 y Fk(\021)2367 555 y Fs(\003)2406 532 y Fk(\020)2477 633 y(e)-68 b Ft(v)2504 647 y Fo(\032)2545 594 y(p)2585 633 y Fu(\()p Ft(p)p Fu(\))15 b Ft(B)2785 647 y Fl(2)2845 633 y Fr(\000)20 b Ft(B)3005 647 y Fl(2)3044 633 y Fr(\012)p Fu(1)3178 615 y Fk(g)3175 633 y Ft(\034)10 b(v)3272 607 y Fo(?)3269 656 y(\032)3312 573 y(p)3352 633 y Fu(\()p Ft(p)p Fu(\))3468 532 y Fk(\021)836 979 y Fu(=)25 b Ft(\031)1108 893 y Fk(X)1002 1095 y Fo(p)p Fs(2F)1142 1076 y Fh(+)1192 1095 y Fs([f)p Fl(0)p Fs(g)1360 979 y Fu(T)-8 b(r)1454 878 y Fk(\020)1508 979 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1792 914 y Fh(1)p 1793 926 31 3 v 1793 968 a(2)1838 979 y Fk(e)-52 b Ft(v)1884 942 y Fo(p)1924 979 y Fu(\()p Ft(p)p Fu(\))15 b Ft(B)2124 993 y Fl(1)2184 979 y Fr(\000)20 b Ft(B)2344 993 y Fl(1)2384 979 y Fr(\012)o Fu(1)15 b Ft(\032)2571 914 y Fh(1)p 2572 926 V 2572 968 a(2)2617 979 y Fk(e)-52 b Ft(v)2663 942 y Fo(p)2703 979 y Fu(\()p Ft(p)p Fu(\))2819 878 y Fk(\021)2874 901 y Fs(\003)922 1329 y Fr(\002)1013 1228 y Fk(\020)1067 1329 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1351 1265 y Fh(1)p 1351 1277 V 1351 1318 a(2)1397 1329 y Fk(e)-52 b Ft(v)1443 1292 y Fo(p)1483 1329 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1668 1343 y Fl(2)1728 1329 y Fr(\000)20 b Ft(B)1888 1343 y Fl(2)1927 1329 y Fr(\012)p Fu(1)15 b Ft(\032)2115 1265 y Fh(1)p 2115 1277 V 2115 1318 a(2)2161 1329 y Fk(e)-52 b Ft(v)2207 1292 y Fo(p)2247 1329 y Fu(\()p Ft(p)p Fu(\))2363 1228 y Fk(\021)831 1602 y Fu(+)20 b Ft(\031)1021 1515 y Fk(X)992 1717 y Fo(p)p Fs(2F)1132 1699 y Fh(+)1197 1602 y Fu(T)-8 b(r)1291 1501 y Fk(\020)1345 1602 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1629 1537 y Fh(1)p 1630 1549 V 1630 1590 a(2)1675 1602 y Fk(e)-52 b Ft(v)1721 1564 y Fo(p)1761 1602 y Fu(\()p Ft(p)p Fu(\))15 b Ft(B)1966 1564 y Fs(\003)1961 1624 y Fl(2)2026 1602 y Fr(\000)20 b Ft(B)2191 1564 y Fs(\003)2186 1624 y Fl(2)2230 1602 y Fr(\012)o Fu(1)15 b Ft(\032)2417 1537 y Fh(1)p 2418 1549 V 2418 1590 a(2)2464 1602 y Fk(e)-52 b Ft(v)2510 1564 y Fo(p)2550 1602 y Fu(\()p Ft(p)p Fu(\))2666 1501 y Fk(\021)2720 1523 y Fs(\003)922 1948 y Fr(\002)1013 1847 y Fk(\020)1067 1948 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1351 1883 y Fh(1)p 1351 1895 V 1351 1937 a(2)1397 1948 y Fk(e)-52 b Ft(v)1443 1911 y Fo(p)1483 1948 y Fu(\()p Ft(p)p Fu(\))15 b Ft(B)1688 1911 y Fs(\003)1683 1971 y Fl(1)1747 1948 y Fr(\000)20 b Ft(B)1912 1911 y Fs(\003)1907 1971 y Fl(1)1951 1948 y Fr(\012)p Fu(1)15 b Ft(\032)2139 1883 y Fh(1)p 2140 1895 V 2140 1937 a(2)2185 1948 y Fk(e)-52 b Ft(v)2231 1911 y Fo(p)2271 1948 y Fu(\()p Ft(p)p Fu(\))2387 1847 y Fk(\021)2442 1948 y Ft(:)3303 2095 y Fu(\(6.49\))380 2291 y(W)-8 b(e)32 b(are)f(no)m(w)f(ready)g(to)h(state)h(the)f(main)f (result)g(of)h(this)f(subsection.)244 2504 y Fv(Theorem)70 b(6.13)49 b Ft(B)30 b Fr(2)25 b Fu(Ker\000)1346 2471 y Fl(I)1346 2526 y Fo(\032)1418 2504 y Fm(i\013)32 b(the)h(fol)5 b(lowing)34 b(two)g(c)-5 b(ommutation)35 b(r)-5 b(elations)34 b(hold:)926 2734 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\))1210 2671 y Fh(1)p 1210 2683 V 1210 2724 a(2)1256 2734 y Fk(e)-52 b Ft(v)1302 2701 y Fo(p)1342 2734 y Fu(\()p Ft(p)p Fu(\))p Ft(B)30 b Fu(=)25 b Ft(B)5 b Fr(\012)o Fu(1)15 b Ft(\032)1914 2671 y Fh(1)p 1915 2683 V 1915 2724 a(2)1960 2734 y Fk(e)-52 b Ft(v)2006 2701 y Fo(p)2046 2734 y Fu(\()p Ft(p)p Fu(\))p Ft(;)114 b(p)25 b Fr(2)g(F)2532 2701 y Fl(+)2611 2734 y Fr([)20 b(f)p Fu(0)p Fr(g)p Ft(;)926 2934 y Fu(\(1)h(+)f Ft(\032)p Fu(\))1210 2871 y Fh(1)p 1210 2883 V 1210 2924 a(2)1256 2934 y Fk(e)-52 b Ft(v)1302 2901 y Fo(p)1342 2934 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1532 2901 y Fs(\003)1596 2934 y Fu(=)25 b Ft(B)1766 2901 y Fs(\003)1805 2934 y Fr(\012)p Fu(1)15 b Ft(\032)1993 2871 y Fh(1)p 1994 2883 V 1994 2924 a(2)2039 2934 y Fk(e)-52 b Ft(v)2085 2901 y Fo(p)2125 2934 y Fu(\()p Ft(p)p Fu(\))p Ft(;)114 b(p)25 b Fr(2)f(F)2610 2901 y Fl(+)2670 2934 y Ft(:)3303 2826 y Fu(\(6.50\))244 3280 y Fv(Pro)s(of.)51 b Fu(Note)26 b(that)g Ft(B)k Fr(2)25 b Fu(Ker)o(\000)1371 3247 y Fl(I)1371 3302 y Fo(\032)1437 3280 y Fu(i\013)g(T)-8 b(r)p Ft(B)1708 3247 y Fs(\003)1747 3280 y Fu(\000)1804 3247 y Fl(I)1804 3302 y Fo(\032)1844 3280 y Fu(\()p Ft(B)5 b Fu(\))25 b(=)g(0.)40 b(Hence)25 b Ft(B)30 b Fr(2)25 b Fu(Ker\000)2872 3247 y Fl(I)2872 3302 y Fo(\032)2937 3280 y Fu(i\013)g(all)h(the)g (terms)244 3393 y(of)k(\(6.49\))j(with)d Ft(B)886 3407 y Fl(1)951 3393 y Fu(=)25 b Ft(B)1116 3407 y Fl(2)1180 3393 y Fu(=)g Ft(B)35 b Fu(are)c(zero,)g(and)f(this)g(is)h(precisely)g (the)f(condition)h(\(6.50\).)43 b Fa(2)244 3719 y Fi(6.8)135 b(P)l(auli-Fierz)46 b(systems)g(with)f(sev)l(eral)h(reserv)l(oirs)244 3891 y Fu(Let)e(us)f(describ)s(e)h(our)f(formalism)h(in)g(the)g(case)h (of)f(a)g(small)h(system)f(coupled)g(to)g(sev)m(eral)244 4004 y(indep)s(enden)m(t)29 b(reserv)m(oirs.)380 4116 y(Let)c Fr(Z)603 4130 y Fo(i)656 4116 y Fu(b)s(e)f(Hilb)s(ert)h(spaces) g(and)f Ft(h)1578 4130 y Fo(i)1607 4116 y Fu(,)i Ft(\032)1705 4130 y Fo(i)1733 4116 y Fu(,)g(p)s(ositiv)m(e)f(comm)m(uting)h (self-adjoin)m(t)g(op)s(erators)f(on)244 4229 y Fr(Z)310 4243 y Fo(i)338 4229 y Fu(,)31 b Ft(i)c Fu(=)f(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)p Fu(.)43 b Fr(Z)985 4243 y Fo(i)1044 4229 y Fu(and)30 b Ft(h)1273 4243 y Fo(i)1333 4229 y Fu(are)h(the)h(single)f(particle)h(space)g(and)f(energy)g(op)s(erator)g (of)g(the)244 4342 y Ft(i)p Fu(-th)i(reserv)m(oir)g(and)e Ft(\032)1021 4356 y Fo(i)1082 4342 y Fu(is)h(the)h(corresp)s(onding)e (radiation)i(densit)m(y)-8 b(.)47 b(Let)33 b Ft(v)2867 4356 y Fo(i)2924 4342 y Fr(2)28 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)24 b(\012)d(Z)3472 4356 y Fo(i)3500 4342 y Fu(\))244 4455 y(b)s(e)k(the)h(form)g(factor)h(describing)f(in)m (teraction)i(of)e(the)g(small)g(system)g(with)g(the)g Ft(i)p Fu(-th)h(reserv)m(oir.)244 4568 y(Set)995 4728 y Fr(Z)33 b Fu(:=)1262 4614 y Fo(n)1215 4641 y Fk(M)1226 4837 y Fo(i)p Fl(=1)1367 4728 y Fr(Z)1433 4742 y Fo(i)1461 4728 y Ft(;)198 b(h)25 b Fu(:=)1930 4614 y Fo(n)1883 4641 y Fk(M)1894 4837 y Fo(i)p Fl(=1)2035 4728 y Ft(h)2087 4742 y Fo(i)2116 4728 y Ft(;)197 b(\032)25 b Fu(:=)2579 4614 y Fo(n)2531 4641 y Fk(M)2543 4837 y Fo(i)p Fl(=1)2684 4728 y Ft(\032)2731 4742 y Fo(i)2759 4728 y Ft(:)380 4976 y Fu(If)39 b(w)m(e)g(imp)s(ose)g(Assumption)f(6.F)1587 4990 y Fo(\032)1667 4976 y Fu(\(whic)m(h)h(is)g(equiv)-5 b(alen)m(t)40 b(to)g(imp)s(osing)f(Assumptions)244 5089 y(6.F)369 5103 y Fo(\032)405 5113 y Fg(i)462 5089 y Fu(for)26 b(all)h Ft(i)p Fu(\),)h(then)e(the)g(corresp)s(onding)f(comp)s(osite)i (P)m(auli-Fierz)h(system)e(is)g(w)m(ell-de\014ned.)p eop %%Page: 40 40 40 39 bop 3682 100 a Fn(40)380 407 y Fu(If)37 b(w)m(e)g(imp)s(ose)g (Assumptions)f(6.C\()p Ft(\021)s Fu(\))1731 421 y Fo(\032)1767 431 y Fg(i)1836 407 y Fu(with)h Ft(\021)j(>)2252 371 y Fl(1)p 2252 386 36 4 v 2252 438 a(2)2334 407 y Fu(on)d Ft(v)2511 421 y Fo(i)2576 407 y Fu(for)g(all)h Ft(i)p Fu(,)h(then)e(the)g(Lev)m(el)244 520 y(Shift)30 b(Op)s(erator)f(of)i (the)g(comp)s(osite)g(system,)g(\000)1927 534 y Fo(\032)1967 520 y Fu(,)f(is)h(w)m(ell-de\014ned)f(and)g(equal)h(to)1611 797 y(\000)1668 811 y Fo(\032)1734 797 y Fu(=)1874 684 y Fo(n)1830 711 y Fk(X)1838 906 y Fo(i)p Fl(=1)1976 797 y Fu(\000)2033 811 y Fo(i;\032)2113 821 y Fg(i)2143 797 y Ft(;)1135 b Fu(\(6.51\))244 1078 y(where)30 b(\000)564 1092 y Fo(i;\032)644 1102 y Fg(i)704 1078 y Fu(is)h(the)f(Lev)m(el)i (Shift)d(Op)s(erator)h(of)h(the)f Ft(i)p Fu(-th)h(subsystem.)244 1398 y Fq(7)161 b(Thermal)54 b(P)l(auli-Fierz)f(systems)244 1601 y Fu(Giv)m(en)46 b(a)g(P)m(auli-Fierz)i(Hamiltonian)f(w)m(e)f(can) g(de\014ne)f(a)h(family)g(of)g(P)m(auli-Fierz)i(systems)244 1714 y(parametrized)37 b(b)m(y)g Ft(\014)k Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu(])38 b(whose)f(radiation)g(densit)m(y)g(is)g(giv)m (en)h(b)m(y)f(the)g(Planc)m(k)g(la)m(w)h(at)244 1827 y(the)25 b(in)m(v)m(erse)h(temp)s(erature)f Ft(\014)5 b Fu(.)39 b(Suc)m(h)25 b(systems)g(will)g(b)s(e)g(called)h(thermal)f(P) m(auli-Fierz)i(systems.)244 1939 y(They)32 b(are)h(particularly)h(imp)s (ortan)m(t)f(from)f(the)h(ph)m(ysical)h(p)s(oin)m(t)e(of)h(view)g(and)g (enjo)m(y)g(sp)s(ecial)244 2052 y(mathematical)g(prop)s(erties.)244 2329 y Fi(7.1)135 b(Thermal)45 b(P)l(auli-Fierz)h(Liouvilleans)244 2501 y Fu(The)38 b(setup)f(of)i(this)f(section)h(is)f(v)m(ery)h (similar)f(to)h(the)g(setup)f(of)g(the)g(previous)g(section.)65 b(In)244 2614 y(particular,)30 b(the)g(op)s(erators)g Ft(K)7 b Fu(,)29 b Ft(h)p Fu(,)h Ft(v)j Fu(and)c Ft(H)7 b Fu(,)30 b(as)g(w)m(ell)g(as)g(the)g(spaces)g Fr(K)h Fu(and)e Fr(Z)36 b Fu(are)30 b(suc)m(h)f(as)244 2726 y(those)i(in)m(tro)s(duced)f(in)g(Subsection)g(6.1.)380 2839 y(Let)h(0)26 b Ft(<)f(\014)30 b Fr(\024)25 b(1)p Fu(.)41 b(In)29 b(this)i(section)g(w)m(e)g(consider)f(the)h(family)g (of)f(densities)1336 3044 y Ft(\032)1383 3059 y Fo(\014)1456 3044 y Fu(:=)25 b(\()p Ft(e)1654 3006 y Fo(\014)s(h)1763 3044 y Fr(\000)19 b Fu(1\))1933 3006 y Fs(\000)p Fl(1)2029 3044 y Ft(;)76 b(\032)2177 3058 y Fs(1)2277 3044 y Fu(=)25 b(0)p Ft(:)244 3248 y Fu(Note)32 b(that)979 3361 y(1)21 b(+)e Ft(\032)1182 3376 y Fo(\014)1255 3361 y Fu(=)25 b(\(1)c Fr(\000)f Fu(e)1583 3323 y Fs(\000)p Fo(\014)s(h)1726 3361 y Fu(\))1761 3323 y Fs(\000)p Fl(1)1881 3361 y Fu(=)25 b(e)2017 3323 y Fo(\014)s(h)2105 3361 y Ft(\032)2152 3376 y Fo(\014)2199 3361 y Ft(;)106 b Fu(1)21 b(+)f Ft(\032)2534 3375 y Fs(1)2634 3361 y Fu(=)25 b(1)p Ft(:)503 b Fu(\(7.52\))380 3528 y(W)-8 b(e)44 b(c)m(hange)g(sligh)m(tly)h(the)e(notation)h(b)m(y)f (replacing)g(the)g(subscripts)f Ft(\032)2934 3543 y Fo(\014)3024 3528 y Fu(with)g Ft(\014)5 b Fu(.)79 b(F)-8 b(or)244 3641 y(instance)35 b(w)m(e)f(will)h(write)f Ft(v)1195 3656 y Fo(\014)1242 3641 y Fu(,)i Ft(L)1365 3656 y Fo(\014)1412 3641 y Fu(,)f Ft(L)1534 3608 y Fl(semi)1534 3668 y Fo(\014)1675 3641 y Fu(,)h Fp(M)1831 3608 y Fl(A)-8 b(W)1831 3668 y Fo(\014)s(;)p Fl(l)1954 3641 y Fu(,)35 b Fp(M)2109 3656 y Fo(\014)2191 3641 y Fu(and)e Ft(\034)2421 3608 y Fo(t)2411 3668 y(\014)2492 3641 y Fu(instead)h(of)h Ft(v)2961 3655 y Fo(\032)2997 3667 y Fg(\014)3043 3641 y Fu(,)g Ft(L)3165 3655 y Fo(\032)3201 3667 y Fg(\014)3247 3641 y Fu(,)g Ft(L)3369 3608 y Fl(semi)3369 3663 y Fo(\032)3405 3675 y Fg(\014)3510 3641 y Fu(,)244 3771 y Fp(M)339 3738 y Fl(A)-8 b(W)339 3799 y Fo(\032)375 3811 y Fg(\014)417 3799 y Fo(;)p Fl(l)462 3771 y Fu(,)35 b Fp(M)617 3785 y Fo(\032)653 3797 y Fg(\014)732 3771 y Fu(and)e Ft(\034)962 3738 y Fo(t)952 3794 y(\032)988 3806 y Fg(\014)1034 3771 y Fu(.)50 b(W)-8 b(e)35 b(w)m(arn)f(the)f(reader)h(that)g(the)g(densit) m(y)g Ft(\032)d Fu(=)f(0)k(corresp)s(onds)f(no)m(w)244 3884 y(to)e(the)g(in)m(v)m(erse)g(temp)s(erature)f Ft(\014)g Fu(=)25 b Fr(1)p Fu(.)380 3997 y(Note)32 b(that)1196 4100 y Ft(v)1240 4115 y Fo(\014)1370 4100 y Fu(=)25 b Fr(j)p Fu(1)c Fr(\000)f Fu(e)1688 4067 y Fs(\000)p Fo(\014)s(r)1824 4100 y Fr(j)1849 4064 y Fs(\000)1914 4037 y Fh(1)p 1914 4049 31 3 v 1914 4090 a(2)1959 4100 y Fu(\()p Ft(v)s(;)15 b(v)2128 4067 y Fo(?)2169 4100 y Fu(\))p Ft(;)p 1146 4250 142 4 v 1146 4300 a(\034)10 b(v)1240 4315 y Fo(\014)1370 4300 y Fu(=)25 b Fr(j)p Fu(1)c Fr(\000)f Fu(e)1688 4267 y Fo(\014)s(r)1769 4300 y Fr(j)1794 4264 y Fs(\000)1859 4237 y Fh(1)p 1859 4249 31 3 v 1859 4290 a(2)1904 4300 y Fu(\()p 1939 4232 87 4 v Ft(v)1986 4274 y Fo(?)2026 4300 y Ft(;)p 2066 4250 48 4 v 15 w(v)t Fu(\))p Ft(;)1146 4500 y(\034)10 b(v)1243 4467 y Fo(?)1240 4528 y(\014)1370 4500 y Fu(=)25 b Fr(j)p Fu(1)c Fr(\000)f Fu(e)1688 4467 y Fo(\014)s(r)1769 4500 y Fr(j)1794 4464 y Fs(\000)1859 4437 y Fh(1)p 1859 4449 31 3 v 1859 4490 a(2)1904 4500 y Fu(\()p Ft(v)s(;)15 b(v)2073 4467 y Fo(?)2114 4500 y Fu(\))25 b(=)g(e)2310 4467 y Fs(\000)p Fo(\014)s(r)r(=)p Fl(2)2517 4500 y Ft(v)2561 4515 y Fo(\014)2608 4500 y Ft(:)244 4731 y Fv(Assumption)36 b(7.A)332 b Fu(\()p Ft(h)1403 4698 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1589 4731 y Fu(+)20 b Ft(h)p Fu(\))p Ft(v)29 b Fr(2)c(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d(Z)7 b Fu(\))244 4944 y Fv(Prop)s(osition)36 b(7.1)49 b Fm(Supp)-5 b(ose)40 b(Assumption)f(7.A)f(holds.)60 b(Then)38 b(for)h(any)f Fu(0)e Ft(<)f(\014)40 b Fr(\024)35 b(1)i Fm(As-)244 5057 y(sumption)d(6.F)772 5071 y Fo(\032)845 5057 y Fm(for)g Ft(\032)25 b Fu(=)g Ft(\032)1206 5072 y Fo(\014)1285 5057 y Fm(holds.)p eop %%Page: 41 41 41 40 bop 3682 100 a Fn(41)244 407 y Fv(Pro)s(of.)72 b Fu(Clearly)-8 b(,)38 b(for)e(0)e Ft(<)g(\014)39 b Fr(\024)34 b(1)p Fu(,)j(\()p Ft(h)1702 374 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1892 407 y Fu(+)23 b Ft(h)p Fu(\))2073 374 y Fs(\000)p Fl(1)2168 407 y Fu(\(1)i(+)e Ft(h)p Fu(\)\(1)i Fr(\000)f Fu(e)2694 374 y Fs(\000)p Fo(\014)s(h)2837 407 y Fu(\))2872 374 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)3073 407 y Fu(is)35 b(b)s(ounded.)244 520 y(Therefore)30 b(Assumption)g(7.A)h(implies)f (the)h(b)s(oundedness)d(of)i(\(1)21 b(+)f Ft(h)p Fu(\)\(1)i Fr(\000)e Fu(e)2943 487 y Fs(\000)p Fo(\014)s(h)3086 520 y Fu(\))3121 487 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)3286 520 y Ft(v)s Fu(.)41 b Fa(2)380 677 y Fu(The)34 b(follo)m(wing)h (theorem)g(follo)m(ws)g(immediately)g(from)f(Prop)s(osition)g(7.1)h (and)f(Theorem)244 790 y(6.10.)244 984 y Fv(Theorem)70 b(7.2)49 b Fm(Supp)-5 b(ose)30 b(that)g(Assumption)f(7.A)f(holds.)42 b(Then,)30 b(for)f(any)g Fu(0)c Ft(<)g(\014)31 b Fr(\024)25 b(1)p Fm(,)k Ft(L)3489 999 y Fo(\014)244 1097 y Fm(is)37 b(essential)5 b(ly)38 b(self-adjoint)h(on)f Fr(D)s Fu(\()p Ft(L)1574 1112 y Fl(fr)1627 1097 y Fu(\))24 b Fr(\\)f(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)2021 1112 y Fo(\014)2069 1097 y Fu(\)\))h Fr(\\)f(D)s Fu(\()p Ft(J)9 b(\031)s Fu(\()p Ft(V)2557 1112 y Fo(\014)2605 1097 y Fu(\))p Ft(J)g Fu(\))p Fm(,)39 b(the)f(thermal)h(Pauli-)244 1210 y(Fierz)33 b(system)g Fu(\()p Fp(M)908 1225 y Fo(\014)956 1210 y Ft(;)15 b(\034)1036 1225 y Fo(\014)1084 1210 y Fu(\))32 b Fm(is)h(wel)5 b(l)33 b(de\014ne)-5 b(d)34 b(and)f Ft(L)1980 1225 y Fo(\014)2060 1210 y Fm(is)f(its)h(Liouvil)5 b(le)-5 b(an.)244 1484 y Fi(7.2)135 b(KMS)45 b(states)h(for)f(thermal)h (P)l(auli-Fierz)g(systems)244 1656 y Fu(In)36 b(this)h(subsection)h(w)m (e)f(describ)s(e)g(results)g(concerning)g(the)h(existence)g(of)g(KMS)e (states)j(for)244 1769 y(P)m(auli-Fierz)j(systems)f(at)g(p)s(ositiv)m (e)h(temp)s(eratures.)70 b(Since)41 b(P)m(auli-Fierz)i Ft(W)2981 1736 y Fs(\003)3019 1769 y Fu(-algebras)f(are)244 1881 y(factors,)31 b(a)g(thermal)g(P)m(auli-Fierz)h(system)f(ma)m(y)g (ha)m(v)m(e)g(at)g(most)g(one)g(KMS-state.)380 1994 y(F)-8 b(or)31 b(0)26 b Ft(<)f(\014)31 b(<)24 b Fr(1)p Fu(,)31 b(set)1394 2107 y Ft(\015)1441 2122 y Fo(\014)1514 2107 y Fu(:=)25 b(e)1675 2070 y Fs(\000)p Fo(\014)s(K)q(=)p Fl(2)1908 2107 y Ft(=)1953 2021 y Fr(p)p 2029 2021 331 4 v 86 x Fu(T)-8 b(r)30 b(e)2193 2081 y Fs(\000)p Fo(\014)s(K)2360 2107 y Ft(:)918 b Fu(\(7.53\))244 2266 y(\()p Ft(\015)326 2281 y Fo(\014)399 2266 y Fu(is)26 b(the)g Ft(\014)5 b Fu(-KMS)25 b(v)m(ector)j(of)e(the)g(small)g(system\).)39 b(The)26 b(free)f(P)m(auli-Fierz)j(system)e(\()p Fp(M)3319 2281 y Fo(\014)3367 2266 y Ft(;)15 b(\034)3447 2281 y Fl(fr)3500 2266 y Fu(\))244 2379 y(has)21 b(a)g(unique)g Ft(\014)5 b Fu(-KMS)21 b(state)i(and)d Ft(\015)1490 2394 y Fo(\014)1539 2379 y Fr(\012)r Fu(\012)h(is)g(the)g(corresp)s(onding)g Ft(\014)5 b Fu(-KMS)21 b(v)m(ector.)39 b(Ob)m(viously)-8 b(,)244 2492 y Ft(\015)291 2507 y Fo(\014)358 2492 y Fr(\012)20 b Fu(\012)25 b Fr(2)g Fu(Ker)p Ft(L)835 2507 y Fl(fr)888 2492 y Fu(.)380 2605 y(The)g(existence)i(of)e(KMS)g(state)h (for)f(in)m(teracting)i(P)m(auli-Fierz)h(systems,)e(whic)m(h)f(is)g(a)h (some-)244 2718 y(what)45 b(delicate)i(problem)e(b)s(ecause)g(p)s (erturbation)g Ft(V)2156 2733 y Fo(\014)2248 2718 y Fu(is)g(not)h(a)f (b)s(ounded)f(op)s(erator)h(and)244 2831 y(Araki's)23 b(theory)h([Ar,)f(BR2)q(])g(cannot)h(b)s(e)e(applied)h(directly)-8 b(,)26 b(follo)m(ws)e(from)e(the)i(result)f(of)g([DJP],)244 2944 y(review)m(ed)31 b(in)f(Section)h(2.)244 3138 y Fv(Theorem)70 b(7.3)49 b Fm(Supp)-5 b(ose)41 b(that)g(Assumption)g(7.A) e(holds)i(and)g(that)g Fu(0)d Ft(<)g(\014)43 b(<)38 b Fr(1)p Fm(.)62 b(Then)244 3251 y(the)45 b(thermal)i(Pauli-Fierz)f (system)f Fu(\()p Fp(M)1683 3266 y Fo(\014)1731 3251 y Ft(;)15 b(\034)1811 3266 y Fo(\014)1859 3251 y Fu(\))45 b Fm(has)h(a)g(unique)e Ft(\014)5 b Fm(-KMS)44 b(state.)81 b(Mor)-5 b(e)g(over,)244 3363 y Ft(\015)291 3378 y Fo(\014)358 3363 y Fr(\012)20 b Fu(\012)25 b Fr(2)g(D)s Fu(\(e)774 3330 y Fs(\000)p Fo(\014)s Fl(\()p Fo(L)p Fl(+)p Fo(\025\031)r Fl(\()p Fo(V)1154 3342 y Fg(\014)1197 3330 y Fl(\)\))p Fo(=)p Fl(2)1326 3363 y Fu(\))33 b Fm(and)h(the)f(ve)-5 b(ctor)1458 3562 y Fu(e)1498 3525 y Fs(\000)p Fo(\014)s Fl(\()p Fo(L)p Fl(+)p Fo(\025\031)r Fl(\()p Fo(V)1878 3537 y Fg(\014)1921 3525 y Fl(\)\))p Fo(=)p Fl(2)2050 3562 y Ft(\015)2097 3577 y Fo(\014)2165 3562 y Fr(\012)20 b Fu(\012)981 b(\(7.54\))244 3751 y Fm(is)32 b(the)h Ft(\014)5 b Fm(-KMS)32 b(ve)-5 b(ctor)34 b(for)f Fu(\()p Fp(M)1360 3766 y Fo(\014)1408 3751 y Ft(;)15 b(\034)1488 3766 y Fo(\014)1535 3751 y Fu(\))p Fm(.)42 b(This)33 b(ve)-5 b(ctor)33 b(b)-5 b(elongs)34 b(to)f Fu(Ker)p Ft(L)2747 3766 y Fo(\014)2794 3751 y Fm(.)244 3945 y Fv(Pro)s(of.)58 b Fu(By)29 b(Theorem)f(2.4)i(\(see)f(also)h([DJP]\),)g (w)m(e)f(need)f(only)h(to)g(c)m(hec)m(k)i(that)e(for)f(all)i Ft(\025)25 b Fr(2)g Fj(R)p Fu(,)1373 4133 y Fr(k)p Fu(e)1458 4095 y Fs(\000)p Fo(\025\014)s(\031)r Fl(\()p Fo(V)1708 4107 y Fg(\014)1751 4095 y Fl(\))p Fo(=)p Fl(2)1853 4133 y Ft(\015)1900 4148 y Fo(\014)1967 4133 y Fr(\012)20 b Fu(\012)p Fr(k)26 b Ft(<)f Fr(1)p Ft(:)244 4321 y Fu(T)-8 b(o)31 b(v)m(erify)f(this,)h(it)g(su\016ces)f(to)h(sho)m(w)f(that)h (there)g(exists)g(a)g(constan)m(t)g Ft(c)g Fu(suc)m(h)f(that)h(for)f (all)h Ft(n)p Fu(,)1270 4510 y Fr(k)p Ft(\031)s Fu(\()p Ft(V)1458 4525 y Fo(\014)1506 4510 y Fu(\))1541 4472 y Fo(n)1603 4510 y Ft(\015)1650 4525 y Fo(\014)1697 4510 y Fr(\012)p Fu(\012)p Fr(k)25 b(\024)g Ft(c)2039 4472 y Fo(n)2087 4427 y Fk(p)p 2178 4427 308 4 v 83 x Fu(\()p Ft(n)20 b Fu(+)g(1\)!)q Ft(:)793 b Fu(\(7.55\))244 4698 y(Since)1257 4811 y Ft(\031)s Fu(\()p Ft(V)1400 4826 y Fo(\014)1447 4811 y Fu(\))26 b(=)f(1)p 1649 4785 59 3 v 29 x Fs(K)1720 4797 y Fu(\024)1707 4811 y Fr(\012)p Ft(v)1822 4826 y Fo(\014)1869 4811 y Fu(\()p Ft(a)1952 4773 y Fs(\003)1992 4811 y Fu(\))c(+)f(1)p 2184 4785 V 29 x Fs(K)2255 4797 y Fu(\024)2242 4811 y Fr(\012)p Ft(v)2360 4773 y Fs(\003)2357 4834 y Fo(\014)2404 4811 y Fu(\()p Ft(a)p Fu(\))244 4970 y(w)m(e)34 b(can)f(decomp)s(ose)g Ft(\031)s Fu(\()p Ft(V)1153 4985 y Fo(\014)1201 4970 y Fu(\))1236 4937 y Fo(n)1316 4970 y Fu(in)m(to)i(the)e(sum)f(of)h(2) 2009 4937 y Fo(n)2057 4970 y Fu(-terms,)h(eac)m(h)g(of)g(whic)m(h)f(is) g(a)g(pro)s(duct)f(of)244 5083 y(creation)d(and)e(annihilation)i(op)s (erators.)40 b(Applying)27 b(the)h(estimates)i(\(5.31\))g(to)e(eac)m(h) h(term)f(w)m(e)244 5196 y(deriv)m(e)j(that)g(\(7.55\))h(holds)e(with)g Ft(c)c Fu(=)f(2)p Fr(k)p Ft(v)1711 5211 y Fo(\014)1759 5196 y Fr(k)p Fu(.)41 b Fa(2)p eop %%Page: 42 42 42 41 bop 3682 100 a Fn(42)244 407 y Fi(7.3)135 b(Lev)l(el)50 b(Shift)e(Op)t(erator)h(for)f(thermal)i(P)l(auli-Fierz)f(Liou-)244 556 y(villeans)244 728 y Fu(In)35 b(this)g(subsection)h(w)m(e)g (consider)f(the)h(Lev)m(el)h(Shift)e(Op)s(erator)g(in)g(the)h(con)m (text)i(of)d(thermal)244 841 y(Liouvilleans.)380 954 y(Let)c(us)f(form)m(ulate)h(the)g(follo)m(wing)h(family)e(of)h (assumptions)f(parametrized)h(b)m(y)f Ft(\021)e Fr(\025)d Fu(0.)244 1150 y Fv(Assumption)36 b(7.B)t Fu(\()p Ft(\021)s Fu(\))252 b Fr(h)p Ft(s)p Fr(i)1466 1117 y Fo(\021)1508 1150 y Fr(j)p Ft(r)s Fr(j)1602 1117 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1767 1150 y Fr(h)p Ft(r)s Fr(i)1881 1117 y Fl(1)p Fo(=)p Fl(2)1991 1150 y Fu(\()p Ft(v)s(;)15 b(v)2160 1117 y Fo(?)2201 1150 y Fu(\))25 b Fr(2)g(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d Fu(\()p Fr(Z)28 b(\010)p 2957 1077 74 4 v 19 w(Z)8 b Fu(\)\))p Fm(.)244 1347 y Fv(Prop)s(osition)36 b(7.4)49 b Fm(Supp)-5 b(ose)33 b(that)f(Assumption)g(7.B)p Fu(\()p Ft(\021)s Fu(\))h Fm(holds.)43 b(Then)31 b Fr(h)p Ft(s)p Fr(i)2924 1314 y Fo(\021)2966 1347 y Ft(v)3010 1362 y Fo(\014)3083 1347 y Fr(2)25 b(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)k(\012)244 1460 y Fu(\()p Fr(Z)28 b(\010)p 463 1387 V 19 w(Z)8 b Fu(\)\))33 b Fm(for)g(al)5 b(l)33 b Ft(\014)e Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([)p Fm(.)244 1657 y Fv(Pro)s(of.)61 b Fu(It)31 b(is)f(easy)h(to)g(see)g(that)g(the)g(function)1128 1857 y Fj(R)25 b Fr(3)g Ft(p)g Fr(7!)g(j)p Fu(1)c Fr(\000)f Fu(e)1714 1820 y Fs(\000)p Fo(\014)s(p)1852 1857 y Fr(j)1877 1820 y Fs(\000)1942 1793 y Fh(1)p 1942 1805 31 3 v 1942 1846 a(2)1987 1857 y Fr(h)p Ft(p)p Fr(i)2103 1820 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2268 1857 y Fr(j)p Ft(p)p Fr(j)2364 1820 y Fl(1)p Fo(=)p Fl(2)2500 1857 y Fr(2)k Fj(R)652 b Fu(\(7.56\))244 2048 y(is)30 b(b)s(ounded)f(with)h(all)h(b)s(ounded)d (deriv)-5 b(ativ)m(es.)42 b(Hence)31 b(the)g(op)s(erator)1219 2249 y Fr(h)p Ft(s)p Fr(i)1332 2211 y Fo(\021)1374 2249 y Fr(j)p Fu(1)21 b Fr(\000)f Fu(e)1596 2211 y Fs(\000)p Fo(\014)s(r)1732 2249 y Fr(j)1757 2211 y Fs(\000)1822 2184 y Fh(1)p 1823 2196 V 1823 2237 a(2)1867 2249 y Fr(h)p Ft(r)s Fr(i)1981 2211 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2146 2249 y Fr(j)p Ft(r)s Fr(j)2240 2211 y Fl(1)p Fo(=)p Fl(2)2350 2249 y Fr(h)p Ft(s)p Fr(i)2463 2211 y Fs(\000)p Fo(\021)244 2439 y Fu(is)42 b(b)s(ounded)e(for)h(all)i Ft(\021)48 b Fr(\025)c Fu(0.)76 b(Therefore,)44 b(the)f(b)s(oundedness)c(of)j Fr(h)p Ft(s)p Fr(i)2765 2406 y Fo(\021)2807 2439 y Fr(j)p Ft(r)s Fr(j)2901 2406 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)3066 2439 y Fr(h)p Ft(r)s Fr(i)3180 2406 y Fl(1)p Fo(=)p Fl(2)3290 2439 y Fu(\()p Ft(v)s(;)15 b(v)3459 2406 y Fo(?)3500 2439 y Fu(\))244 2566 y(implies)30 b(the)h(b)s(oundedness)d(of)i Fr(h)p Ft(s)p Fr(i)1459 2533 y Fo(\021)1502 2566 y Fr(j)p Fu(1)21 b Fr(\000)e Fu(e)1723 2533 y Fs(\000)p Fo(\014)s(r)1860 2566 y Fr(j)1885 2530 y Fs(\000)1950 2503 y Fh(1)p 1950 2515 V 1950 2556 a(2)1994 2566 y Fu(\()p Ft(v)s(;)c(v)2163 2533 y Fo(?)2204 2566 y Fu(\).)41 b Fa(2)380 2724 y Fu(The)36 b(follo)m(wing)i(prop)s(osition)f(giv)m(es)g(a)g(condition)h(whic)m(h)e (is)h(easy)g(to)g(v)m(erify)g(in)f(practice)244 2837 y(and)30 b(whic)m(h)g(implies)g(Assumption)g(7.B\()p Ft(\021)s Fu(\):)244 3033 y Fv(Prop)s(osition)36 b(7.5)49 b Fm(L)-5 b(et)33 b Ft(n)f Fm(b)-5 b(e)33 b(a)g(non-ne)-5 b(gative)33 b(inte)-5 b(ger)33 b(and)g(assume)h(that)681 3147 y Fk(R)741 3173 y Fs(1)723 3252 y Fl(0)831 3220 y Fr(k)p Ft(@)929 3187 y Fo(n)924 3242 y(p)977 3220 y Ft(p)1023 3187 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1187 3220 y Fr(h)p Ft(p)p Fr(i)1303 3187 y Fl(1)p Fo(=)p Fl(2)1414 3220 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Fr(k)1622 3187 y Fl(2)1662 3220 y Fu(d)p Ft(p)25 b(<)g Fr(1)p Fu(;)681 3429 y Ft(@)734 3385 y Fo(j)729 3441 y(p)770 3429 y Ft(p)816 3396 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)981 3429 y Ft(v)s Fu(\()p Ft(p)p Fu(\))1144 3325 y Fk(\014)1144 3379 y(\014)1144 3434 y(\014)1175 3493 y Fo(p)p Fl(=0)1330 3429 y Fu(=)g(\()p Fr(\000)p Fu(1\))1612 3396 y Fo(j)1650 3429 y Ft(@)1703 3385 y Fo(j)1698 3441 y(p)1739 3429 y Ft(p)1785 3396 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1950 3429 y Ft(v)1997 3396 y Fo(?)2037 3429 y Fu(\()p Ft(p)p Fu(\))2153 3325 y Fk(\014)2153 3379 y(\014)2153 3434 y(\014)2183 3493 y Fo(p)p Fl(=0)2313 3429 y Ft(;)113 b(j)31 b Fu(=)25 b(0)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fr(\000)g Fu(1)p Ft(:)244 3660 y Fm(Then)33 b(Assumption)h(7.B)p Fu(\()p Ft(n)p Fu(\))f Fm(holds.)244 3856 y Fv(Pro)s(of.)61 b Fu(See)31 b(Prop)s(osition)f(7.15,)j(whic)m(h)d(has)g(a)h(similar)f (pro)s(of.)40 b Fa(2)380 4015 y Fu(Throughout)20 b(this)h(subsection)g (w)m(e)g(assume)f(that)i(Assumption)e(7.B\()p Ft(\021)s Fu(\))j(holds)d(with)h Ft(\021)28 b(>)3465 3979 y Fl(1)p 3465 3994 36 4 v 3465 4046 a(2)3510 4015 y Fu(.)244 4128 y(W)-8 b(e)37 b(will)e(describ)s(e)g(the)h(Lev)m(el)h(Shift)d(Op)s (erator)i(in)f(the)g(case)i Ft(\032)d Fu(=)f Ft(\032)2669 4143 y Fo(\014)2716 4128 y Fu(,)k(whic)m(h,)g(consisten)m(tly)244 4241 y(with)32 b(our)g(notation,)j(will)e(b)s(e)f(denoted)h(\000)1712 4256 y Fo(\014)1759 4241 y Fu(.)48 b(A)32 b(sp)s(ecial)i(atten)m(tion)g (needs)f(to)g(b)s(e)f(dev)m(oted)i(to)244 4353 y(the)c(infrared)g(term) g(in)g(\000)1122 4368 y Fo(\014)1169 4353 y Fu(.)244 4550 y Fv(Prop)s(osition)36 b(7.6)49 b Fm(Ther)-5 b(e)34 b(exists)1351 4784 y Ft(v)1395 4798 y Fl(ir)1471 4784 y Fu(:=)26 b(lim)1603 4845 y Fo(p)p Fs(#)p Fl(0)1744 4722 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p 1744 4763 164 4 v 1776 4869 a Ft(p)1832 4806 y Fh(1)p 1831 4818 31 3 v 1831 4859 a(2)1943 4784 y Fu(=)f(lim)2049 4845 y Fo(p)p Fs(#)p Fl(0)2190 4722 y Ft(v)2237 4689 y Fo(?)2277 4722 y Fu(\()p Ft(p)p Fu(\))p 2190 4763 204 4 v 2242 4869 a Ft(p)2298 4806 y Fh(1)p 2297 4818 31 3 v 2297 4859 a(2)2403 4784 y Ft(:)875 b Fu(\(7.57\))244 5057 y Fm(Set)35 b Fk(f)-93 b Ft(v)444 5071 y Fl(ir)495 5015 y(0)560 5057 y Fu(:=)681 4989 y Fk(P)777 5084 y Fo(p)832 5057 y Ft(v)879 5013 y Fo(pp)876 5085 y Fl(ir)954 5057 y Fm(.)42 b(We)32 b(have)1347 5268 y Ft(v)1391 5282 y Fl(ir)1467 5268 y Fu(=)25 b(\()p Ft(v)1642 5282 y Fl(ir)1694 5268 y Fu(\))1729 5230 y Fo(?)1769 5268 y Ft(;)113 b Fu(\()r Fk(f)-93 b Ft(v)1986 5282 y Fl(ir)2037 5226 y(0)2077 5268 y Fu(\))2112 5230 y Fo(?)2177 5268 y Fu(=)27 b Fk(f)-93 b Ft(v)2317 5282 y Fl(ir)2368 5226 y(0)2408 5268 y Ft(:)870 b Fu(\(7.58\))p eop %%Page: 43 43 43 42 bop 3682 100 a Fn(43)244 407 y Fm(Mor)-5 b(e)g(over,)34 b Fu(\000)729 374 y Fl(I)729 434 y Fo(\014)808 407 y Fm(acts)f(as)g(fol)5 b(lows:)444 651 y Fr(\000)p Fu(T)-8 b(r)o Ft(B)682 618 y Fs(\003)677 675 y Fl(1)721 651 y Fu(\000)778 618 y Fl(I)778 678 y Fo(\014)825 651 y Fu(\()p Ft(B)929 665 y Fl(2)969 651 y Fu(\))83 b(=)25 b Ft(\031)1300 582 y Fk(P)1253 739 y Fo(p)p Fs(2F)1393 720 y Fh(+)1459 651 y Fr(j)p Fu(e)1524 618 y Fo(\014)s(p)1627 651 y Fr(\000)20 b Fu(1)p Fr(j)1788 618 y Fs(\000)p Fl(1)1883 651 y Fu(T)-8 b(r)1977 550 y Fk(\020)2031 651 y Fu(e)2071 618 y Fo(\014)s(p=)p Fl(2)2226 651 y Fk(e)-52 b Ft(v)2272 618 y Fo(p)2312 651 y Fu(\()p Ft(p)p Fu(\))p Ft(B)2497 665 y Fl(1)2557 651 y Fr(\000)20 b Ft(B)2717 665 y Fl(1)2776 651 y Fr(\012)g Fu(1)34 b Fk(e)-52 b Ft(v)2992 618 y Fo(p)3032 651 y Fu(\()p Ft(p)p Fu(\))3148 550 y Fk(\021)3203 572 y Fs(\003)1282 866 y Fr(\002)1353 765 y Fk(\020)1407 866 y Fu(e)1447 833 y Fo(\014)s(p=)p Fl(2)1602 866 y Fk(e)g Ft(v)1648 833 y Fo(p)1688 866 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1873 880 y Fl(2)1933 866 y Fr(\000)20 b Ft(B)2093 880 y Fl(2)2152 866 y Fr(\012)g Fu(1)34 b Fk(e)-52 b Ft(v)2368 833 y Fo(p)2408 866 y Fu(\()p Ft(p)p Fu(\))2524 765 y Fk(\021)1087 1080 y Fu(+)p Ft(\031)1275 1012 y Fk(P)1228 1168 y Fo(p)p Fs(2F)1368 1149 y Fh(+)1433 1080 y Fr(j)p Fu(e)1498 1047 y Fs(\000)p Fo(\014)s(p)1657 1080 y Fr(\000)20 b Fu(1)p Fr(j)1818 1047 y Fs(\000)p Fl(1)1913 1080 y Fu(T)-8 b(r)2006 979 y Fk(\020)2062 1080 y(e)-52 b Ft(v)2108 1047 y Fo(p)2148 1080 y Fu(\()p Ft(p)p Fu(\))p Ft(B)2338 1047 y Fs(\003)2333 1104 y Fl(2)2397 1080 y Fr(\000)20 b Fu(e)2528 1047 y Fs(\000)p Fo(\014)s(p=)p Fl(2)2737 1080 y Ft(B)2811 1047 y Fs(\003)2806 1104 y Fl(2)2870 1080 y Fr(\012)g Fu(1)34 b Fk(e)-52 b Ft(v)3086 1047 y Fo(p)3126 1080 y Fu(\()p Ft(p)p Fu(\))3242 979 y Fk(\021)3297 1001 y Fs(\003)1282 1295 y Fr(\002)1353 1194 y Fk(\020)1408 1295 y(e)g Ft(v)1454 1262 y Fo(p)1494 1295 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1684 1262 y Fs(\003)1679 1319 y Fl(1)1744 1295 y Fr(\000)20 b Fu(e)1875 1262 y Fs(\000)p Fo(\014)s(p=)p Fl(2)2083 1295 y Ft(B)2157 1262 y Fs(\003)2152 1319 y Fl(1)2216 1295 y Fr(\012)g Fu(1)34 b Fk(e)-52 b Ft(v)2432 1262 y Fo(p)2472 1295 y Fu(\()p Ft(p)p Fu(\))2588 1194 y Fk(\021)1087 1509 y Fu(+)1168 1473 y Fo(\031)p 1168 1488 44 4 v 1168 1540 a(\014)1221 1509 y Fu(T)-8 b(r)1314 1408 y Fk(\020)1371 1509 y(f)-93 b Ft(v)1413 1523 y Fl(ir)1464 1467 y(0)1503 1509 y Ft(B)1572 1523 y Fl(1)1632 1509 y Fr(\000)20 b Ft(B)1792 1523 y Fl(1)1851 1509 y Fr(\012)g Fu(1)36 b Fk(f)-94 b Ft(v)2064 1523 y Fl(ir)2116 1467 y(0)2155 1408 y Fk(\021)2209 1431 y Fs(\003)1282 1718 y Fr(\002)1353 1617 y Fk(\020)1409 1718 y(f)h Ft(v)1451 1732 y Fl(ir)1502 1676 y(0)1542 1718 y Ft(B)1611 1732 y Fl(2)1670 1718 y Fr(\000)20 b Ft(B)1830 1732 y Fl(2)1890 1718 y Fr(\012)g Fu(1)35 b Fk(f)-93 b Ft(v)2103 1732 y Fl(ir)2154 1676 y(0)2194 1617 y Fk(\021)2248 1718 y Ft(:)3303 1854 y Fu(\(7.59\))244 2060 y Fv(Pro)s(of.)61 b Fu(F)-8 b(or)32 b Ft(p)25 b(>)g Fu(0,)1356 2181 y Ft(v)1400 2196 y Fo(\014)1447 2181 y Fu(\()p Ft(p)p Fu(\))h(=)f Fr(j)p Fu(1)c Fr(\000)f Fu(e)1907 2148 y Fs(\000)p Fo(\014)s(p)2045 2181 y Fr(j)2070 2145 y Fs(\000)2135 2118 y Fh(1)p 2135 2130 31 3 v 2135 2171 a(2)2179 2181 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)1356 2381 y(v)1400 2396 y Fo(\014)1447 2381 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))26 b(=)f Fr(j)p Fu(1)20 b Fr(\000)g Fu(e)1977 2348 y Fo(\014)s(p)2060 2381 y Fr(j)2085 2345 y Fs(\000)2150 2318 y Fh(1)p 2151 2330 V 2151 2371 a(2)2195 2381 y Ft(v)2242 2348 y Fo(?)2282 2381 y Fu(\()p Ft(p)p Fu(\))p Ft(:)244 2542 y Fu(Since)30 b(the)h(function)f Fj(R)25 b Fr(3)g Ft(p)g Fr(7!)g Ft(v)1402 2557 y Fo(\014)1449 2542 y Fu(\()p Ft(p)p Fu(\))31 b(is)f(con)m(tin)m(uous,)1102 2759 y Ft(v)1146 2774 y Fo(\014)1193 2759 y Fu(\(0\))d(=)e(lim)1441 2820 y Fo(p)p Fs(#)p Fl(0)1572 2759 y Ft(v)1616 2774 y Fo(\014)1663 2759 y Fu(\()p Ft(p)p Fu(\))h(=)f Ft(\014)1957 2723 y Fs(\000)2022 2696 y Fh(1)p 2022 2708 V 2022 2749 a(2)2082 2759 y Fu(lim)2092 2820 y Fo(p)p Fs(#)p Fl(0)2223 2759 y Ft(p)2269 2723 y Fs(\000)2334 2696 y Fh(1)p 2334 2708 V 2334 2749 a(2)2378 2759 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)1102 2959 y(v)1146 2974 y Fo(\014)1193 2959 y Fu(\(0\))i(=)e(lim)1441 3020 y Fo(p)p Fs(#)p Fl(0)1572 2959 y Ft(v)1616 2974 y Fo(\014)1663 2959 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))h(=)f Ft(\014)2028 2923 y Fs(\000)2093 2896 y Fh(1)p 2093 2908 V 2093 2949 a(2)2152 2959 y Fu(lim)2163 3020 y Fo(p)p Fs(#)p Fl(0)2294 2959 y Ft(p)2340 2923 y Fs(\000)2405 2896 y Fh(1)p 2404 2908 V 2404 2949 a(2)2449 2959 y Ft(v)2496 2926 y Fo(?)2536 2959 y Fu(\()p Ft(p)p Fu(\))p Ft(:)244 3204 y Fu(This)k(implies)i(the)g(existence)g(of)g(the) g(limits)g(in)f(\(7.57\))i(and)e(the)h(iden)m(tities)h(\(7.58\).)380 3317 y(The)42 b(iden)m(tit)m(y)i(\(7.59\))g(follo)m(ws)f(from)f (\(6.49\))i(if)e(w)m(e)h(tak)m(e)h(in)m(to)f(accoun)m(t)h(the)e(iden)m (tities)244 3430 y(\(7.52\))33 b(and)c(\(7.58\).)43 b Fa(2)380 3591 y Fu(Let)821 3704 y Fp(N)25 b Fu(:=)1043 3603 y Fk(n)1104 3704 y Ft(B)30 b Fr(2)25 b(B)s Fu(\()p Fr(K)q Fu(\))h(:)83 b Ft(B)5 b Fr(\012)o Fu(1)17 b Fk(e)-53 b Ft(v)1877 3671 y Fo(p)1918 3704 y Fu(\()p Ft(p)p Fu(\))26 b(=)f Fk(e)-52 b Ft(v)2202 3671 y Fo(p)2242 3704 y Fu(\()p Ft(p)p Fu(\))p Ft(B)5 b(;)107 b(p)25 b Fr(2)g(F)2795 3671 y Fl(+)2854 3704 y Ft(;)1626 3888 y(B)1700 3855 y Fs(\003)1739 3888 y Fr(\012)p Fu(1)16 b Fk(e)-52 b Ft(v)1917 3855 y Fo(p)1957 3888 y Fu(\()p Ft(p)p Fu(\))26 b(=)g Fk(e)-52 b Ft(v)2242 3855 y Fo(p)2282 3888 y Fu(\()p Ft(p)p Fu(\))p Ft(B)2472 3855 y Fs(\003)2511 3888 y Ft(;)107 b(p)25 b Fr(2)f(F)2873 3855 y Fl(+)2933 3888 y Ft(;)1626 4097 y(B)5 b Fr(\012)o Fu(1)18 b Fk(f)-94 b Ft(v)1874 4111 y Fl(ir)1926 4055 y(0)1991 4097 y Fu(=)27 b Fk(f)-93 b Ft(v)2131 4111 y Fl(ir)2182 4055 y(0)2222 4097 y Ft(B)2296 3996 y Fk(o)2356 4097 y Ft(:)244 4326 y Fv(Prop)s(osition)36 b(7.7)49 b Fp(N)33 b Fm(is)h(a)g Fr(\003)p Fm(-sub)-5 b(algebr)g(a)34 b(of)g Fr(B)s Fu(\()p Fr(K)q Fu(\))g Fm(c)-5 b(ontaining)35 b Fj(C)p Fu(1)2689 4340 y Fs(K)2747 4326 y Fm(.)44 b(Mor)-5 b(e)g(over,)35 b(for)f(any)244 4439 y Ft(t)25 b Fr(2)g Fj(C)32 b Fm(and)i Ft(B)29 b Fr(2)c Fp(N)32 b Fm(we)h(have)g Fu(e)1339 4406 y Fl(i)p Fo(tK)1453 4439 y Ft(B)5 b Fu(e)1567 4406 y Fs(\000)p Fl(i)p Fo(tK)1761 4439 y Fr(2)24 b Fp(N)p Fm(.)244 4645 y Fv(Pro)s(of.)59 b Fu(It)29 b(is)h(easy)f(to)h(c)m(hec)m(k)h(that)f Fp(N)f Fu(is)g(an)g(algebra)h(and)f(ob)m(viously)h(1)2780 4659 y Fs(K)2864 4645 y Fr(2)25 b Fp(N)p Fu(.)40 b(T)-8 b(o)29 b(see)h(that)244 4758 y Fp(N)c Fu(is)g(preserv)m(ed)g(b)m(y)g Fr(\003)p Fu(,)h(note)g(that)g(the)f(\014rst)f(t)m(w)m(o)j(conditions)e (are)h(manifestly)g(symmetric)f(wrt)244 4870 y Fr(\003)p Fu(.)41 b(The)30 b(relation)h(\()r Fk(f)-93 b Ft(v)954 4884 y Fl(ir)1006 4829 y(0)1046 4870 y Fu(\))1081 4837 y Fo(?)1146 4870 y Fu(=)27 b Fk(f)-93 b Ft(v)1286 4884 y Fl(ir)1337 4829 y(0)1407 4870 y Fu(and)29 b(Prop)s(osition)i(4.6)g (imply)f(that)1037 5069 y Ft(B)5 b Fr(\012)p Fu(1)17 b Fk(f)-93 b Ft(v)1286 5083 y Fl(ir)1337 5027 y(0)1402 5069 y Fu(=)27 b Fk(f)-93 b Ft(v)1542 5083 y Fl(ir)1593 5027 y(0)1633 5069 y Ft(B)90 b Fr(\))c Ft(B)2043 5032 y Fs(\003)2082 5069 y Fr(\012)p Fu(1)17 b Fk(f)-93 b Ft(v)2257 5083 y Fl(ir)2309 5027 y(0)2373 5069 y Fu(=)27 b Fk(f)-93 b Ft(v)2513 5083 y Fl(ir)2565 5027 y(0)2604 5069 y Ft(B)2678 5032 y Fs(\003)2717 5069 y Ft(;)244 5268 y Fu(and)30 b(so)g Fp(N)g Fu(is)h(a)f Fr(\003)p Fu(-algebra.)p eop %%Page: 44 44 44 43 bop 3682 100 a Fn(44)380 407 y Fu(Note)32 b(that)1065 520 y(e)1105 482 y Fl(i)p Fo(tK)1219 520 y Fr(\012)p Fu(1)16 b Fk(e)-52 b Ft(v)1397 482 y Fo(p)1437 520 y Fu(\()p Ft(p)p Fu(\)e)1593 482 y Fs(\000)p Fl(i)p Fo(tK)1788 520 y Fu(=)25 b(e)1924 482 y Fs(\000)p Fl(i)p Fo(tp)2065 520 y Fk(e)-52 b Ft(v)2111 482 y Fo(p)2151 520 y Fu(\()p Ft(p)p Fu(\))p Ft(;)107 b(p)25 b Fr(2)g(F)2630 482 y Fl(+)2689 520 y Ft(;)1435 685 y Fu(e)1475 648 y Fl(i)p Fo(tK)1589 685 y Fr(\012)o Fu(1)18 b Fk(f)-94 b Ft(v)1763 699 y Fl(ir)1815 643 y(0)1855 685 y Fu(e)1895 648 y Fs(\000)p Fl(i)p Fo(tK)2089 685 y Fu(=)27 b Fk(f)-93 b Ft(v)2229 699 y Fl(ir)2280 643 y(0)2319 685 y Ft(:)244 850 y Fu(This)29 b(implies)i(that)g Fp(N)f Fu(is)g(in)m(v)-5 b(arian)m(t)32 b(wrt)e(e)1743 817 y Fl(i)p Fo(tK)1877 850 y Fr(\001)20 b Fu(e)1962 817 y Fs(\000)p Fl(i)p Fo(tK)2131 850 y Fu(.)41 b Fa(2)244 1108 y Fv(Prop)s(osition)36 b(7.8)49 b Fu(Ker\000)1195 1075 y Fl(I)1195 1136 y Fo(\014)1275 1108 y Fm(c)-5 b(onsists)34 b(of)e(op)-5 b(er)g(ators)36 b(of)d(the)g(form)h Fu(e)2631 1075 y Fs(\000)p Fo(\014)s(K)q(=)p Fl(2)2864 1108 y Ft(C)39 b Fm(with)34 b Ft(C)d Fr(2)25 b Fp(N)p Fm(.)244 1317 y Fv(Pro)s(of.)61 b Fu(Relation)32 b(\(7.59\))h(can)e(b)s(e)e (rewritten)i(as)350 1546 y Fr(\000)p Fu(T)-8 b(r)p Ft(B)589 1513 y Fs(\003)584 1570 y Fl(1)628 1546 y Fu(\000)685 1513 y Fl(I)685 1574 y Fo(\014)732 1546 y Fu(\()p Ft(B)836 1560 y Fl(2)876 1546 y Fu(\))83 b(=)25 b Ft(\031)1207 1478 y Fk(P)1160 1635 y Fo(p)p Fs(2F)1300 1616 y Fh(+)1366 1546 y Fr(j)p Fu(e)1431 1513 y Fo(\014)s(p)1534 1546 y Fr(\000)20 b Fu(1)p Fr(j)1695 1513 y Fs(\000)p Fl(1)1790 1546 y Fu(T)-8 b(r)1884 1445 y Fk(\020)1939 1546 y(e)-52 b Ft(v)1985 1513 y Fo(p)2025 1546 y Fu(\()p Ft(p)p Fu(\)e)2181 1513 y Fo(\014)s(K)q(=)p Fl(2)2360 1546 y Ft(B)2429 1560 y Fl(1)2489 1546 y Fr(\000)19 b Fu(e)2619 1513 y Fo(\014)s(K)q(=)p Fl(2)2798 1546 y Ft(B)2867 1560 y Fl(1)2926 1546 y Fr(\012)h Fu(1)32 b Fk(e)-52 b Ft(v)3140 1513 y Fo(p)3180 1546 y Fu(\()p Ft(p)p Fu(\))3296 1445 y Fk(\021)3351 1468 y Fs(\003)1176 1761 y Fr(\002)p Fu(e)1287 1728 y Fs(\000)p Fo(\014)s(K)1473 1761 y Fr(\012)20 b Fu(1)1609 1660 y Fk(\020)1665 1761 y(e)-52 b Ft(v)1711 1728 y Fo(p)1751 1761 y Fu(\()p Ft(p)p Fu(\)e)1907 1728 y Fo(\014)s(K)q(=)p Fl(2)2086 1761 y Ft(B)2155 1775 y Fl(2)2214 1761 y Fr(\000)20 b Fu(e)2345 1728 y Fo(\014)s(K)q(=)p Fl(2)2524 1761 y Ft(B)2593 1775 y Fl(2)2652 1761 y Fr(\012)g Fu(1)32 b Fk(e)-52 b Ft(v)2866 1728 y Fo(p)2906 1761 y Fu(\()p Ft(p)p Fu(\))3022 1660 y Fk(\021)994 1975 y Fu(+)p Ft(\031)1182 1907 y Fk(P)1135 2064 y Fo(p)p Fs(2F)1275 2045 y Fh(+)1340 1975 y Fr(j)p Fu(e)1405 1942 y Fs(\000)p Fo(\014)s(p)1564 1975 y Fr(\000)20 b Fu(1)p Fr(j)1725 1942 y Fs(\000)p Fl(1)1820 1975 y Fu(T)-8 b(r)1913 1874 y Fk(\020)1969 1975 y(e)-52 b Ft(v)2015 1942 y Fo(p)2055 1975 y Fu(\()p Ft(p)p Fu(\))p Ft(B)2245 1942 y Fs(\003)2240 1999 y Fl(2)2284 1975 y Fu(e)2324 1942 y Fo(\014)s(K)q(=)p Fl(2)2523 1975 y Fr(\000)20 b Ft(B)2688 1942 y Fs(\003)2683 1999 y Fl(2)2727 1975 y Fu(e)2767 1942 y Fo(\014)s(K)q(=)p Fl(2)2965 1975 y Fr(\012)g Fu(1)32 b Fk(e)-52 b Ft(v)3179 1942 y Fo(p)3219 1975 y Fu(\()p Ft(p)p Fu(\))3335 1874 y Fk(\021)3390 1897 y Fs(\003)1176 2190 y Fr(\002)1247 2089 y Fk(\020)1302 2190 y(e)g Ft(v)1348 2157 y Fo(p)1388 2190 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1578 2157 y Fs(\003)1573 2215 y Fl(1)1617 2190 y Fu(e)1657 2157 y Fo(\014)s(K)q(=)p Fl(2)1856 2190 y Fr(\000)20 b Ft(B)2021 2157 y Fs(\003)2016 2215 y Fl(1)2060 2190 y Fu(e)2100 2157 y Fo(\014)s(K)q(=)p Fl(2)2298 2190 y Fr(\012)g Fu(1)32 b Fk(e)-52 b Ft(v)2512 2157 y Fo(p)2552 2190 y Fu(\()p Ft(p)p Fu(\))2668 2089 y Fk(\021)2723 2190 y Fu(e)2763 2157 y Fs(\000)p Fo(\014)s(K)994 2404 y Fu(+)1075 2368 y Fo(\031)p 1075 2383 44 4 v 1075 2435 a(\014)1128 2404 y Fu(T)-8 b(r)1221 2303 y Fk(\020)1278 2404 y(f)-93 b Ft(v)1320 2418 y Fl(ir)1371 2362 y(0)1410 2404 y Fu(e)1450 2371 y Fo(\014)s(K)q(=)p Fl(2)1629 2404 y Ft(B)1698 2418 y Fl(1)1757 2404 y Fr(\000)20 b Fu(e)1888 2371 y Fo(\014)s(K)q(=)p Fl(2)2067 2404 y Ft(B)2136 2418 y Fl(1)2195 2404 y Fr(\012)g Fu(1)33 b Fk(f)-93 b Ft(v)2406 2418 y Fl(ir)2457 2362 y(0)2497 2303 y Fk(\021)2551 2326 y Fs(\003)1176 2613 y Fr(\002)p Fu(e)1287 2580 y Fs(\000)p Fo(\014)s(K)1473 2613 y Fr(\012)20 b Fu(1)1609 2512 y Fk(\020)1666 2613 y(f)-93 b Ft(v)1708 2627 y Fl(ir)1759 2571 y(0)1799 2613 y Fu(e)1839 2580 y Fo(\014)s(K)q(=)p Fl(2)2017 2613 y Ft(B)2086 2627 y Fl(2)2146 2613 y Fr(\000)20 b Fu(e)2277 2580 y Fo(\014)s(K)q(=)p Fl(2)2455 2613 y Ft(B)2524 2627 y Fl(2)2584 2613 y Fr(\012)g Fu(1)33 b Fk(f)-94 b Ft(v)2794 2627 y Fl(ir)2846 2571 y(0)2885 2512 y Fk(\021)2939 2613 y Ft(:)244 2847 y Fu(Hence,)31 b Ft(B)f Fr(2)25 b Fu(Ker\000)928 2814 y Fl(I)928 2875 y Fo(\014)1005 2847 y Fu(i\013)920 3067 y Fk(e)-52 b Ft(v)966 3034 y Fo(p)1006 3067 y Fu(\()p Ft(p)p Fu(\)e)1162 3034 y Fo(\014)s(K)q(=)p Fl(2)1341 3067 y Ft(B)25 b Fr(\000)20 b Fu(e)1566 3034 y Fo(\014)s(K)q(=)p Fl(2)1744 3067 y Ft(B)5 b Fr(\012)p Fu(1)31 b Fk(e)-52 b Ft(v)2011 3034 y Fo(p)2051 3067 y Fu(\()p Ft(p)p Fu(\))26 b(=)f(0)p Ft(;)76 b(p)25 b Fr(2)g(F)2666 3034 y Fl(+)2726 3067 y Ft(;)920 3255 y Fk(e)-52 b Ft(v)966 3222 y Fo(p)1006 3255 y Fu(\()p Ft(p)p Fu(\))p Ft(B)1196 3222 y Fs(\003)1236 3255 y Fu(e)1276 3222 y Fo(\014)s(K)q(=)p Fl(2)1474 3255 y Fr(\000)20 b Ft(B)1639 3222 y Fs(\003)1678 3255 y Fu(e)1718 3222 y Fo(\014)s(K)q(=)p Fl(2)1897 3255 y Fr(\012)o Fu(1)32 b Fk(e)-52 b Ft(v)2090 3222 y Fo(p)2130 3255 y Fu(\()p Ft(p)p Fu(\))26 b(=)f(0)p Ft(;)107 b(p)25 b Fr(2)f(F)2775 3222 y Fl(+)2835 3255 y Ft(;)920 3443 y Fk(e)-52 b Ft(v)966 3410 y Fl(0)963 3469 y(ir)1015 3443 y Fu(e)1055 3410 y Fo(\014)s(K)q(=)p Fl(2)1233 3443 y Ft(B)25 b Fr(\000)20 b Fu(e)1458 3410 y Fo(\014)s(K)q(=)p Fl(2)1636 3443 y Ft(B)5 b Fr(\012)o Fu(1)32 b Fk(e)-52 b Ft(v)1903 3410 y Fl(0)1900 3469 y(ir)1977 3443 y Fu(=)25 b(0)p Ft(:)244 3656 y Fu(Therefore,)30 b Ft(B)g Fr(2)25 b Fu(Ker\000)1070 3623 y Fl(I)1070 3684 y Fo(\014)1147 3656 y Fu(i\013)30 b Ft(B)g Fu(=)25 b(e)1490 3623 y Fs(\000)p Fo(\014)s(K)q(=)p Fl(2)1723 3656 y Ft(C)37 b Fu(for)30 b(some)h Ft(C)h Fr(2)25 b Fp(N)p Fu(.)40 b Fa(2)380 3818 y Fu(Our)29 b(main)i(e\013ectiv)m(e)i(coupling)d(assumption)g(is:)244 4027 y Fv(Assumption)36 b(7.C)48 b Fp(N)25 b Fu(=)g Fj(C)p Fu(1)1337 4041 y Fs(K)1395 4027 y Fm(.)380 4236 y Fu(No)m(w)31 b(Prop)s(osition)g(7.8)g(implies)g(immediately)244 4445 y Fv(Theorem)70 b(7.9)49 b Fm(Assumption)34 b(7.C)f(is)f(satis\014e)-5 b(d)34 b(i\013)f Fu(Ker)o(\000)2340 4412 y Fl(I)2340 4472 y Fo(\014)2420 4445 y Fm(is)f(sp)-5 b(anne)g(d)35 b(by)e Ft(\015)3027 4460 y Fo(\014)3074 4445 y Fm(.)244 4721 y Fi(7.4)135 b(Return)50 b(to)g(equilibrium)g(for)f(a)h(\014xed)f (p)t(ositiv)l(e)h(temp)t(er-)244 4870 y(ature)244 5042 y Fu(In)35 b(this)g(subsection)h(w)m(e)g(describ)s(e)f(conditions)h (whic)m(h)f(ensure)g(that)h(for)g(an)m(y)g(\014xed)f(p)s(ositiv)m(e)244 5155 y(temp)s(erature)e(the)g(thermal)h(P)m(auli-Fierz)h(system)e(has)g (the)g(prop)s(ert)m(y)g(of)g(return)f(to)i(equilib-)244 5268 y(rium.)40 b(The)30 b(result)g(will)h(not)f(b)s(e)g(uniform)f(in)h (the)h(temp)s(erature.)p eop %%Page: 45 45 45 44 bop 3682 100 a Fn(45)244 407 y Fv(Theorem)70 b(7.10)49 b Fm(L)-5 b(et)44 b Ft(\021)j(>)d Fu(2)f Fm(and)h(supp)-5 b(ose)45 b(that)f(Assumptions)h(7.A,)g(7.B)p Fu(\()p Ft(\021)s Fu(\))f Fm(and)g(7.C)244 520 y(hold.)f(Then)33 b(for)g(al)5 b(l)33 b Ft(\014)e Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([)33 b Fm(ther)-5 b(e)34 b(exists)f Ft(\025)1919 534 y Fl(0)1958 520 y Fu(\()p Ft(\014)5 b Fu(\))26 b Ft(>)f Fu(0)33 b Fm(such)g(that)h(for)f Fu(0)26 b Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)3267 534 y Fl(0)3306 520 y Fu(\()p Ft(\014)5 b Fu(\))p Fm(,)811 724 y Fu(sp)897 746 y Fl(p)941 724 y Fu(\()p Ft(L)1038 739 y Fo(\014)1085 724 y Fu(\))26 b(=)f Fr(f)p Fu(0)p Fr(g)p Ft(;)179 b Fu(dim)15 b(1)1793 738 y Fl(0)1833 724 y Fu(\()p Ft(L)1930 739 y Fo(\014)1977 724 y Fu(\))26 b(=)f(1)p Ft(;)243 b Fu(sp)2533 746 y Fl(sc)2597 724 y Fu(\()p Ft(L)2694 739 y Fo(\014)2741 724 y Fu(\))26 b(=)f Fr(;)p Ft(:)244 928 y Fm(In)38 b(p)-5 b(articular,)41 b(under)d(the)g(ab)-5 b(ove)39 b(c)-5 b(onditions)39 b(the)g(Pauli-Fierz)e(system)i Fu(\()p Fp(M)3001 943 y Fo(\014)3049 928 y Ft(;)15 b(\034)3129 943 y Fo(\014)3176 928 y Fu(\))38 b Fm(has)h(the)244 1041 y(pr)-5 b(op)g(erty)35 b(of)e(r)-5 b(eturn)34 b(to)f(e)-5 b(quilibrium.)244 1254 y Fv(Pro)s(of.)64 b Fu(It)32 b(follo)m(ws)h (from)f(Theorem)f(7.9)i(that)g(dim)15 b(Ker)o(\000)2309 1221 y Fl(I)2309 1281 y Fo(\014)2384 1254 y Fu(=)27 b(1)33 b(for)e Ft(\014)i Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([.)46 b(Hence,)34 b(b)m(y)244 1367 y(Theorem)c(5.5,)i(there)e(exists)h Ft(\025)1336 1381 y Fl(0)1376 1367 y Fu(\()p Ft(\014)5 b Fu(\))31 b(suc)m(h)f(that)h(for)f(0)c Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)2519 1381 y Fl(0)2558 1367 y Fu(\()p Ft(\014)5 b Fu(\))31 b(w)m(e)g(ha)m(v)m(e)928 1571 y(dim)15 b(1)1140 1585 y Fl(p)1184 1571 y Fu(\()p Ft(L)1281 1586 y Fo(\014)1328 1571 y Fu(\))26 b Fr(\024)f Fu(dim)14 b(1)1696 1585 y Fl(0)1736 1571 y Fu(\(\000)1828 1533 y Fl(I)1828 1594 y Fo(\014)1875 1571 y Fu(\))26 b(=)f(1)p Ft(;)228 b Fu(sp)2416 1593 y Fl(sc)2480 1571 y Fu(\()p Ft(L)2577 1586 y Fo(\014)2624 1571 y Fu(\))26 b(=)f Fr(;)p Ft(:)244 1775 y Fu(By)31 b(Theorem)f(7.3,)h(dim)15 b(1)1154 1789 y Fl(0)1194 1775 y Fu(\()p Ft(L)1291 1790 y Fo(\014)1338 1775 y Fu(\))26 b Fr(\025)f Fu(1,)31 b(and)f(the)g (result)g(follo)m(ws.)42 b Fa(2)244 2101 y Fi(7.5)135 b(Zero)46 b(temp)t(erature)g(P)l(auli-Fierz)f(Liouvilleans)244 2273 y Fu(Recall)f(that)g(zero)f(temp)s(erature)g(corresp)s(onds)e(to)j Ft(\014)51 b Fu(=)46 b Fr(1)p Fu(.)78 b(All)43 b(the)g(prop)s(erties)f (of)h(the)244 2386 y(zero)32 b(temp)s(erature)f(Liouvillean,)i(denoted) e Ft(L)1857 2400 y Fs(1)1932 2386 y Fu(,)g(follo)m(w)i(easily)f(from)f (the)g(prop)s(erties)g(of)g(the)244 2499 y(P)m(auli-Fierz)h (Hamiltonian.)42 b(They)30 b(are)h(describ)s(ed)e(in)h(this)h (subsection.)380 2612 y(Note)g(that)f(after)g(the)g(iden)m (ti\014cation)h Fr(K)20 b(\012)p 1890 2539 71 4 v 18 w(K)g(\012)e Fu(\000)2125 2626 y Fl(s)2157 2612 y Fu(\()p Fr(Z)26 b(\010)p 2373 2539 74 4 v 18 w(Z)7 b Fu(\))26 b Fr(')f(K)20 b(\012)e Fu(\000)2838 2626 y Fl(s)2870 2612 y Fu(\()p Fr(Z)7 b Fu(\))19 b Fr(\012)p 3121 2533 415 4 v 18 w(K)j(\012)e Fu(\000)3360 2626 y Fl(s)3392 2612 y Fu(\()p Fr(Z)7 b Fu(\))244 2725 y(the)30 b(zero)i(temp)s (erature)e(Liouvillean)h(b)s(ecomes)1453 2929 y Ft(L)1515 2943 y Fs(1)1615 2929 y Fu(=)25 b Ft(H)i Fr(\012)20 b Fu(1)h Fr(\000)f Fu(1)g Fr(\012)p 2218 2856 83 4 v 20 w Ft(H)7 b(:)244 3133 y Fu(Hence,)31 b(the)f(Lev)m(el)i(Shift)d(Op)s (erator)h(for)g Ft(L)1746 3147 y Fs(1)1821 3133 y Fu(,)g(denoted)g (\000)2276 3147 y Fs(1)2351 3133 y Fu(,)g(can)g(b)s(e)g(expressed)g(in) f(terms)h(of)244 3246 y(the)k(Lev)m(el)h(Shift)e(Op)s(erator)h(for)f Ft(H)7 b Fu(,)35 b(denoted)f(\000,)h(as)f(follo)m(ws.)52 b(If)33 b(\000)2624 3260 y Fs(1)2733 3246 y Fu(is)h(expresed)f(in)h (terms)244 3359 y(\001)320 3373 y Fs(1)425 3359 y Fu(and)29 b(\004)662 3373 y Fs(1)767 3359 y Fu(as)i(in)f(Theorem)g(6.11,)i(then) 1466 3563 y(\001)1542 3577 y Fs(1)1642 3563 y Fu(=)25 b(\000)p Ft(;)166 b Fu(\004)2047 3577 y Fs(1)2147 3563 y Fu(=)25 b(0)p Ft(;)990 b Fu(\(7.60\))244 3767 y(and)30 b(so)1473 3880 y(\000)1530 3894 y Fs(1)1605 3880 y Fu(\()p Ft(B)5 b Fu(\))25 b(=)g(\000)p Ft(B)g Fr(\000)19 b Ft(B)5 b Fu(\000)2242 3843 y Fs(\003)2281 3880 y Ft(:)244 4047 y Fu(The)30 b(follo)m(wing)i(theorem)e(then)g(follo)m(ws)i(immediately) g(from)e(Theorem)g(6.4:)244 4260 y Fv(Theorem)70 b(7.11)49 b Fm(Under)32 b(assumptions)i(of)e(The)-5 b(or)g(em)34 b(6.4,)e(ther)-5 b(e)33 b(exists)f Ft(\025)2939 4274 y Fl(0)3003 4260 y Ft(>)25 b Fu(0)32 b Fm(such)g(that)244 4373 y(for)h Fu(0)26 b Ft(<)f Fr(j)p Ft(\025)p Fr(j)h Ft(<)e(\025)833 4387 y Fl(0)905 4373 y Fm(we)33 b(have)770 4577 y Fu(sp)856 4599 y Fl(p)899 4577 y Fu(\()p Ft(L)996 4591 y Fs(1)1071 4577 y Fu(\))26 b Fr(\032)f(f)p Fu(0)p Fr(g)p Ft(;)179 b Fu(dim)15 b(1)1779 4591 y Fl(0)1819 4577 y Fu(\()p Ft(L)1916 4591 y Fs(1)1991 4577 y Fu(\))25 b Fr(\024)g Fu(1)p Ft(;)244 b Fu(sp)2547 4599 y Fl(sc)2611 4577 y Fu(\()p Ft(L)2708 4591 y Fs(1)2783 4577 y Fu(\))25 b(=)g Fr(;)p Ft(:)p eop %%Page: 46 46 46 45 bop 3682 100 a Fn(46)244 407 y Fi(7.6)135 b(Uniform)35 b(in)g(temp)t(erature)g(estimate)h(on)f(the)f(Lev)l(el)h(Shift)244 556 y(Op)t(erator)244 728 y Fu(In)h(this)g(subsection)g(w)m(e)h (describ)s(e)f(conditions)h(under)e(whic)m(h)h(\000)2519 743 y Fo(\014)2603 728 y Fu(is)g(uniformly)g(dissipativ)m(e)244 841 y(on)30 b(the)h(orthogonal)h(complemen)m(t)g(to)g Ft(\015)1649 856 y Fo(\014)1696 841 y Fu(.)41 b(The)31 b(e\013ectiv)m(e)i(coupling)e(assumptions)f(6.D,)i(6.E,)244 954 y(and)e(7.C)g(will)h(pla)m(y)g(a)g(k)m(ey)g(role.)380 1067 y(Recall)h(that)f Ft(\015)898 1082 y Fo(\014)976 1067 y Fu(is)f(de\014ned)f(b)m(y)i(\(7.53\))h(for)e(0)c Fr(\024)f Ft(\014)30 b(<)25 b Fr(1)p Fu(.)41 b(F)-8 b(or)31 b Ft(\014)g Fu(=)25 b Fr(1)30 b Fu(w)m(e)h(set)1353 1310 y Ft(\015)1400 1324 y Fs(1)1500 1310 y Fu(:=)25 b(1)1666 1325 y Fo(k)1703 1334 y Fh(0)1742 1310 y Fu(\()p Ft(K)7 b Fu(\))p Ft(=)1941 1201 y Fk(q)p 2033 1201 370 4 v 2033 1310 a Fu(T)-8 b(r)o(1)2171 1325 y Fo(k)2208 1334 y Fh(0)2247 1310 y Fu(\()p Ft(K)7 b Fu(\))q Ft(;)244 1537 y Fu(where)29 b Ft(k)553 1551 y Fl(0)618 1537 y Fu(:=)c(inf)d(sp)o(\()p Ft(K)7 b Fu(\).)41 b(Note)30 b(that)g([0)p Ft(;)15 b Fr(1)p Fu(])27 b Fr(3)e Ft(\014)30 b Fr(7!)25 b Ft(\015)2168 1552 y Fo(\014)2241 1537 y Fr(2)g Ft(l)2356 1504 y Fl(2)2395 1537 y Fu(\()p Fr(K)q Fu(\))31 b(is)e(a)h(con)m(tin)m(uous)g(function.) 244 1749 y Fv(Theorem)70 b(7.12)49 b Fm(L)-5 b(et)33 b Ft(\021)c(>)1294 1713 y Fl(1)p 1294 1728 36 4 v 1294 1781 a(2)1372 1749 y Fm(and)k(supp)-5 b(ose)34 b(that)g(Assumption)g (7.B)p Fu(\()p Ft(\021)s Fu(\))g Fm(holds.)43 b(Then)244 1862 y Fu(\(1\))34 b Fm(The)e(function)h Fu(]0)p Ft(;)15 b Fr(1)p Fu(])27 b Fr(3)e Ft(\014)30 b Fr(7!)25 b Fu(\000)1529 1829 y Fl(I)1529 1890 y Fo(\014)1601 1862 y Fr(2)g(B)s Fu(\()p Ft(l)1814 1829 y Fl(2)1853 1862 y Fu(\()p Fr(K)q Fu(\)\))35 b Fm(is)d(c)-5 b(ontinuous.)244 1975 y Fu(\(2\))35 b Fm(Assume)f(in)g(addition)i(that)g(6.D,)e(6.E)g(and)h(7.C)g(hold.)48 b(L)-5 b(et)34 b Ft(\014)2566 1989 y Fl(0)2634 1975 y Ft(>)28 b Fu(0)p Fm(.)47 b(Then)34 b(ther)-5 b(e)35 b(exists)244 2088 y Ft(\017)25 b(>)g Fu(0)33 b Fm(such)g(that)h(for)f Ft(\014)d Fr(2)25 b Fu([)p Ft(\014)1261 2102 y Fl(0)1301 2088 y Ft(;)15 b Fr(1)p Fu(])1129 2292 y(T)-8 b(r)o Ft(B)1296 2255 y Fs(\003)1335 2292 y Fu(\000)1392 2255 y Fl(I)1392 2315 y Fo(\014)1439 2292 y Fu(\()p Ft(B)5 b Fu(\))26 b Fr(\024)f(\000)p Ft(\017)p Fu(\(T)-8 b(r)o Ft(B)2015 2255 y Fs(\003)2054 2292 y Ft(B)25 b Fr(\000)20 b(j)p Fu(T)-8 b(r)p Ft(B)5 b(\015)2479 2307 y Fo(\014)2526 2292 y Fr(j)2551 2255 y Fl(2)2590 2292 y Fu(\))p Ft(:)653 b Fu(\(7.61\))244 2709 y Fv(Pro)s(of.)61 b Fu(The)30 b(con)m(tin)m(uit)m(y)j(of)d(\000)1364 2676 y Fl(I)1364 2737 y Fo(\014)1441 2709 y Fu(in)g Ft(\014)h Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu(])31 b(follo)m(ws)h(from)e(Relation)h(\(7.59\).)380 2822 y(T)-8 b(o)34 b(pro)m(v)m(e)g(\(2\),)h(let)e(us)g(consider)g (\014rst)f(the)h(case)h Ft(\014)h Fu(=)29 b Fr(1)p Fu(.)49 b(Assumption)32 b(6.D)i(and)e(\(7.60\))244 2935 y(imply)e(that)h(there) f(exists)h Ft(\017)p Fu(\()p Fr(1)p Fu(\))26 b Ft(>)f Fu(0)31 b(suc)m(h)f(that)1355 3139 y(\001)1431 3101 y Fl(I)1431 3161 y Fs(1)1531 3139 y Fr(\024)25 b(\000)p Ft(\017)p Fu(\()p Fr(1)p Fu(\)\(1)c Fr(\000)f Fu(1)2133 3154 y Fo(k)2170 3163 y Fh(0)2209 3139 y Fu(\()p Ft(K)7 b Fu(\)\))p Ft(:)880 b Fu(\(7.62\))244 3343 y(Since)30 b(\000)538 3310 y Fl(I)538 3366 y Fs(1)613 3343 y Fu(\()p Ft(B)5 b Fu(\))25 b(=)g(\001)954 3310 y Fl(I)954 3366 y Fs(1)1029 3343 y Ft(B)f Fu(+)c Ft(B)5 b Fu(\001)1363 3310 y Fl(I)1363 3366 y Fs(1)1437 3343 y Fu(,)31 b(using)f(\(7.62\))i (and)e(Assumption)g(6.E)h(w)m(e)f(obtain)544 3545 y(T)-8 b(r)p Ft(B)712 3512 y Fs(\003)751 3545 y Fu(\000)808 3512 y Fl(I)808 3567 y Fs(1)883 3545 y Fu(\()p Ft(B)5 b Fu(\))83 b Fr(\024)25 b(\000)p Ft(\017)p Fu(\()p Fr(1)p Fu(\))1475 3471 y Fk(\000)1517 3545 y Fu(T)-8 b(r)o Ft(B)1684 3512 y Fs(\003)1723 3545 y Fu(\(1)21 b Fr(\000)f Fu(1)1960 3560 y Fo(k)1997 3569 y Fh(0)2036 3545 y Fu(\()p Ft(K)7 b Fu(\)\))p Ft(B)26 b Fu(+)20 b(T)-8 b(r)o Ft(B)2578 3512 y Fs(\003)2617 3545 y Ft(B)5 b Fu(\(1)21 b Fr(\000)f Fu(1)2928 3560 y Fo(k)2965 3569 y Fh(0)3004 3545 y Fu(\()p Ft(K)7 b Fu(\)\))3193 3471 y Fk(\001)1110 3728 y Fr(\024)25 b(\000)p Ft(\017)p Fu(\()p Fr(1)p Fu(\))1475 3655 y Fk(\000)1517 3728 y Fu(T)-8 b(r)o Ft(B)1684 3695 y Fs(\003)1723 3728 y Ft(B)25 b Fr(\000)20 b Fu(T)-8 b(r)p Ft(B)2076 3695 y Fs(\003)2115 3728 y Fu(1)2160 3743 y Fo(k)2197 3752 y Fh(0)2236 3728 y Fu(\()p Ft(K)7 b Fu(\))p Ft(B)e Fu(1)2509 3743 y Fo(k)2546 3752 y Fh(0)2584 3728 y Fu(\()p Ft(K)i Fu(\))2738 3655 y Fk(\001)1110 3912 y Fu(=)25 b Fr(\000)p Ft(\017)p Fu(\()p Fr(1)p Fu(\)\(T)-8 b(r)p Ft(B)1678 3879 y Fs(\003)1717 3912 y Ft(B)25 b Fr(\000)20 b(j)p Fu(T)-8 b(r)p Ft(B)5 b(\015)2142 3926 y Fs(1)2216 3912 y Fr(j)2241 3879 y Fl(2)2281 3912 y Fu(\))p Ft(:)380 4109 y Fu(Consider)26 b(no)m(w)h Ft(\014)j(<)25 b Fr(1)p Fu(.)39 b(It)27 b(follo)m(ws)h(from)e(Assumption)g(7.C)h(and)f(Theorem) g(7.9)i(that)f(for)244 4222 y(an)m(y)k Ft(\014)f Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([,)31 b(there)g(exists)g Ft(\017)p Fu(\()p Ft(\014)5 b Fu(\))26 b Ft(>)f Fu(0)31 b(suc)m(h)f(that)1065 4426 y(T)-8 b(r)p Ft(B)1233 4388 y Fs(\003)1272 4426 y Fu(\000)1329 4388 y Fl(I)1329 4449 y Fo(\014)1376 4426 y Fu(\()p Ft(B)5 b Fu(\))25 b Fr(\024)g(\000)p Ft(\017)p Fu(\()p Ft(\014)5 b Fu(\)\(T)-8 b(r)q Ft(B)2079 4388 y Fs(\003)2118 4426 y Ft(B)24 b Fr(\000)c(j)p Fu(T)-8 b(r)p Ft(B)5 b(\015)2542 4441 y Fo(\014)2589 4426 y Fr(j)2614 4388 y Fl(2)2654 4426 y Fu(\))p Ft(:)244 4640 y Fu(The)35 b(compactness)i(of)f([)p Ft(\014)1153 4654 y Fl(0)1193 4640 y Ft(;)15 b Fr(1)p Fu(],)38 b(the)e(con)m(tin)m(uit)m(y)i(of)e([)p Ft(\014)2194 4654 y Fl(0)2234 4640 y Ft(;)15 b Fr(1)p Fu(])35 b Fr(3)g Ft(\014)k Fr(7!)c Fu(\000)2794 4607 y Fl(I)2794 4668 y Fo(\014)2877 4640 y Fu(and)g(of)h([)p Ft(\014)3244 4654 y Fl(0)3284 4640 y Ft(;)15 b Fr(1)p Fu(])35 b Fr(3)244 4753 y Ft(\014)30 b Fr(7!)c Ft(\015)489 4768 y Fo(\014)566 4753 y Fu(imply)k(that)h(one)g(can)f(c)m(hose)i Ft(\017)25 b(>)g Fu(0)31 b(suc)m(h)f(that)h(\(7.61\))h(holds.)41 b Fa(2)p eop %%Page: 47 47 47 46 bop 3682 100 a Fn(47)244 407 y Fi(7.7)135 b(Uniform)46 b(in)f(temp)t(erature)h(return)f(to)g(equilibrium)244 578 y Fu(In)26 b(this)g(subsection)h(w)m(e)g(describ)s(e)f(the)h(main)f (result)h(of)f(this)h(pap)s(er.)38 b(W)-8 b(e)28 b(describ)s(e)e (conditions)244 691 y(under)44 b(whic)m(h)g(for)h(0)51 b Ft(<)f Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)1489 705 y Fl(0)1573 691 y Fu(and)45 b Ft(\014)55 b Fr(2)50 b Fu([)p Ft(\014)2058 705 y Fl(0)2098 691 y Ft(;)15 b Fr(1)p Fu([)46 b(the)f(Liouvillean)h Ft(L)3016 706 y Fo(\014)3108 691 y Fu(has)f(purely)244 804 y(absolutely)28 b(con)m(tin)m(uous)f(sp)s (ectrum)f(except)i(for)f(a)g(simple)g(eigen)m(v)-5 b(alue)29 b(at)f(zero.)40 b(This)26 b(implies)244 917 y(that)33 b(for)f(a)h(small)g(nonzero)g(coupling)g(constan)m(t)g(the)g(system)g (\()p Fp(M)p Ft(;)15 b(\034)2650 932 y Fo(\014)2698 917 y Fu(\))33 b(has)f(the)h(prop)s(ert)m(y)e(of)244 1030 y(return)e(to)i(equilibrium)f(uniformly)f(in)h(the)h(temp)s(erature.) 380 1143 y(One)k(of)h(the)g(ingredien)m(ts)g(of)g(our)f(pro)s(of)g(is)g (the)h(uniform)f(estimate)i(on)f(the)f(Lev)m(el)i(Shift)244 1256 y(Op)s(erator)45 b(\000)708 1271 y Fo(\014)800 1256 y Fu(obtained)h(in)f(the)h(previous)f(subsection.)86 b(The)45 b(second)h(ingredien)m(t)g(is)g(an)244 1369 y(additional)31 b(assumption)e(on)h(the)h(regularit)m(y)g(of)f(the)g (in)m(teraction,)j(whic)m(h)d(w)m(e)g(will)h(form)m(ulate)244 1482 y(b)s(elo)m(w.)380 1595 y(F)-8 b(or)31 b(an)m(y)g Ft(\021)e Fr(\025)c Fu(0)30 b(w)m(e)h(in)m(tro)s(duce)244 1772 y Fv(Assumption)36 b(7.D)s Fu(\()p Ft(\021)s Fu(\))67 b(sup)1309 1794 y Fl(1)p Fs(\024)p Fo(\014)s Fs(\0241)1587 1772 y Fr(kh)p Ft(s)p Fr(i)1745 1739 y Fo(\021)1788 1772 y Ft(v)1832 1787 y Fo(\014)1879 1772 y Fr(k)26 b Ft(<)f Fr(1)p Ft(:)244 1971 y Fv(Prop)s(osition)36 b(7.13)50 b Fm(Supp)-5 b(ose)34 b(that)g(Assumption)f(7.D)p Fu(\()p Ft(\021)s Fu(\))h Fm(holds.)44 b(Then)275 2100 y Fu(\(1\))50 b Fm(Assumption)33 b(7.B)p Fu(\()p Ft(\021)s Fu(\))h Fm(holds.)275 2231 y Fu(\(2\))50 b Fm(F)-7 b(or)33 b(any)h Ft(\014)835 2245 y Fl(0)900 2231 y Ft(>)25 b Fu(0)p Fm(,)32 b Fu(sup)1238 2253 y Fo(\014)1278 2262 y Fh(0)1313 2253 y Fs(\024)p Fo(\014)s Fs(\0241)1555 2231 y Fr(kh)p Ft(s)p Fr(i)1713 2198 y Fo(\021)1756 2231 y Ft(v)1800 2246 y Fo(\014)1847 2231 y Fr(k)26 b Ft(<)f Fr(1)p Ft(:)244 2458 y Fv(Pro)s(of.)57 b Fu(By)28 b(Assumption)f(7.D\()p Ft(\021)s Fu(\),)k Fr(kh)p Ft(s)p Fr(i)1695 2425 y Fo(\021)1737 2458 y Fr(j)p Fu(1)16 b Fr(\000)f Fu(e)1949 2425 y Fs(\000)p Fo(r)2043 2458 y Fr(j)2068 2425 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2233 2458 y Fu(\()p Ft(v)s(;)g(v)2402 2425 y Fo(?)2443 2458 y Fu(\))p Fr(k)26 b Ft(<)f Fr(1)p Fu(.)39 b(Clearly)-8 b(,)30 b Ft(\034)3182 2385 y Fk(\000)3224 2458 y Fr(h)p Ft(s)p Fr(i)3337 2425 y Fo(\021)3379 2458 y Fr(j)p Fu(1)16 b Fr(\000)244 2581 y Fu(e)284 2548 y Fs(\000)p Fo(r)377 2581 y Fr(j)402 2548 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)567 2581 y Fu(\()p Ft(v)s(;)f(v)736 2548 y Fo(?)777 2581 y Fu(\))812 2508 y Fk(\001)854 2530 y Fo(?)932 2581 y Fu(=)38 b Fr(h)p Ft(s)p Fr(i)1154 2548 y Fo(\021)1197 2581 y Fr(j)p Fu(1)26 b Fr(\000)f Fu(e)1429 2548 y Fo(r)1467 2581 y Fr(j)1492 2548 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1657 2581 y Fu(\()p Ft(v)s(;)15 b(v)1826 2548 y Fo(?)1867 2581 y Fu(\).)65 b(Th)m(us,)40 b(using)d(the)i(b)s(oundedness)d(of)i Ft(?)p Fu(,)j(w)m(e)244 2694 y(obtain)950 2807 y Fr(kh)p Ft(s)p Fr(i)1108 2770 y Fo(\021)1151 2734 y Fk(\000)1193 2807 y Fr(j)p Fu(1)21 b Fr(\000)f Fu(e)1415 2770 y Fs(\000)p Fo(r)1508 2807 y Fr(j)1533 2770 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1718 2807 y Fu(+)g Fr(j)p Fu(1)h Fr(\000)f Fu(e)2031 2770 y Fo(r)2069 2807 y Fr(j)2094 2770 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2259 2734 y Fk(\001)2301 2807 y Fu(\()p Ft(v)s(;)15 b(v)2470 2770 y Fo(?)2511 2807 y Fu(\))p Fr(k)26 b Ft(<)f Fr(1)p Ft(:)474 b Fu(\(7.63\))244 2968 y(Using)30 b(the)h(argumen)m(ts) g(of)f(the)h(pro)s(of)e(of)i(Prop)s(osition)f(7.4)i(w)m(e)f(see)g(that) 843 3160 y Fr(h)p Ft(s)p Fr(i)956 3123 y Fo(\021)999 3160 y Fr(h)p Ft(r)s Fr(i)1113 3123 y Fl(1)p Fo(=)p Fl(2)1223 3160 y Fr(j)p Ft(r)s Fr(j)1317 3123 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1482 3087 y Fk(\000)1523 3160 y Fr(j)p Fu(1)21 b Fr(\000)f Fu(e)1745 3123 y Fs(\000)p Fo(r)1838 3160 y Fr(j)1863 3123 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2049 3160 y Fu(+)g Fr(j)p Fu(1)h Fr(\000)f Fu(e)2362 3123 y Fo(r)2400 3160 y Fr(j)2425 3123 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2590 3087 y Fk(\001)2632 3110 y Fs(\000)p Fl(1)2726 3160 y Fr(h)p Ft(s)p Fr(i)2839 3123 y Fs(\000)p Fo(\021)3303 3160 y Fu(\(7.64\))244 3353 y(is)30 b(b)s(ounded.)39 b(No)m(w)31 b(\(7.63\))i(and)c(\(7.64\))k(imply)d(Assumption)g(7.B\()p Ft(\021)s Fu(\),)i(whic)m(h)e(pro)m(v)m(es)h(\(1\).)380 3466 y(F)-8 b(or)31 b(0)26 b Ft(<)f(\014)761 3480 y Fl(0)826 3466 y Fr(\024)g Fu(1,)31 b(w)m(e)g(ha)m(v)m(e)571 3658 y(sup)508 3737 y Fo(\014)548 3746 y Fh(0)582 3737 y Fs(\024)p Fo(\014)s Fs(\024)p Fl(1)785 3658 y Fr(kh)p Ft(s)p Fr(i)943 3621 y Fo(\021)986 3658 y Fr(j)p Fu(1)21 b Fr(\000)f Fu(e)1208 3621 y Fs(\000)p Fo(\014)s(r)1344 3658 y Fr(j)1369 3621 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1534 3585 y Fk(\000)1576 3658 y Fr(j)p Fu(1)h Fr(\000)f Fu(e)1798 3621 y Fs(\000)p Fo(r)1891 3658 y Fr(j)1916 3621 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2101 3658 y Fu(+)g Fr(j)p Fu(1)h Fr(\000)f Fu(e)2414 3621 y Fo(r)2452 3658 y Fr(j)2477 3621 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2643 3585 y Fk(\001)2684 3607 y Fs(\000)p Fl(1)2779 3658 y Fr(h)p Ft(s)p Fr(i)2892 3621 y Fs(\000)p Fo(\021)2989 3658 y Fr(k)25 b Ft(<)g Fr(1)p Ft(:)244 3915 y Fu(Hence)31 b(\(7.63\))i(implies)d(sup)1222 3937 y Fo(\014)1262 3946 y Fh(0)1296 3937 y Fs(\024)p Fo(\014)s Fs(\024)p Fl(1)1504 3915 y Fr(kh)p Ft(s)p Fr(i)1662 3882 y Fo(\021)1704 3915 y Ft(v)1748 3930 y Fo(\014)1796 3915 y Fr(k)25 b Ft(<)g Fr(1)p Ft(:)31 b Fu(This)e(pro)m(v)m(es)i(\(2\).)42 b Fa(2)244 4160 y Fv(Theorem)70 b(7.14)49 b Fm(Supp)-5 b(ose)39 b(Assumptions)g(7.A,)g(7.D)p Fu(\()p Ft(\021)s Fu(\))f Fm(with)h Ft(\021)f(>)c Fu(2)p Fm(,)39 b(6.D)e(6.E)h(and)g(7.C) 244 4273 y(ar)-5 b(e)34 b(satis\014e)-5 b(d.)47 b(L)-5 b(et)33 b Fu(0)28 b Ft(<)f(\014)1165 4287 y Fl(0)1231 4273 y Ft(<)g Fr(1)p Fm(.)45 b(Then)34 b(ther)-5 b(e)34 b(exists)g Ft(\025)2261 4287 y Fl(0)2327 4273 y Ft(>)27 b Fu(0)34 b Fm(such)g(that)h(for)f Fu(0)27 b Ft(<)g Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)3496 4287 y Fl(0)244 4385 y Fm(and)33 b Ft(\014)e Fr(2)25 b Fu([)p Ft(\014)664 4399 y Fl(0)704 4385 y Ft(;)15 b Fr(1)p Fu([)33 b Fm(we)g(have)811 4578 y Fu(sp)897 4600 y Fl(p)941 4578 y Fu(\()p Ft(L)1038 4593 y Fo(\014)1085 4578 y Fu(\))26 b(=)f Fr(f)p Fu(0)p Fr(g)p Ft(;)179 b Fu(dim)15 b(1)1793 4592 y Fl(0)1833 4578 y Fu(\()p Ft(L)1930 4593 y Fo(\014)1977 4578 y Fu(\))26 b(=)f(1)p Ft(;)243 b Fu(sp)2533 4600 y Fl(sc)2597 4578 y Fu(\()p Ft(L)2694 4593 y Fo(\014)2741 4578 y Fu(\))26 b(=)f Fr(;)p Ft(:)244 4770 y Fm(Henc)-5 b(e,)28 b(under)h(the)f(same)h (c)-5 b(onditions,)30 b(the)f(Pauli-Fierz)f(system)h Fu(\()p Fp(M)2670 4785 y Fo(\014)2718 4770 y Ft(;)15 b(\034)2798 4785 y Fo(\014)2845 4770 y Fu(\))28 b Fm(has)h(the)g(pr)-5 b(op)g(erty)244 4883 y(of)33 b(r)-5 b(eturn)33 b(to)g(e)-5 b(quilibrium.)42 b(Mor)-5 b(e)g(over,)34 b(for)f Fu(0)26 b Ft(<)f Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)2261 4897 y Fl(0)2301 4883 y Fm(,)770 5075 y Fu(sp)856 5097 y Fl(p)899 5075 y Fu(\()p Ft(L)996 5089 y Fs(1)1071 5075 y Fu(\))h Fr(\032)f(f)p Fu(0)p Fr(g)p Ft(;)179 b Fu(dim)15 b(1)1779 5089 y Fl(0)1819 5075 y Fu(\()p Ft(L)1916 5089 y Fs(1)1991 5075 y Fu(\))25 b Fr(\024)g Fu(1)p Ft(;)244 b Fu(sp)2547 5097 y Fl(sc)2611 5075 y Fu(\()p Ft(L)2708 5089 y Fs(1)2783 5075 y Fu(\))25 b(=)g Fr(;)p Ft(:)p eop %%Page: 48 48 48 47 bop 3682 100 a Fn(48)244 407 y Fv(Pro)s(of.)86 b Fu(By)42 b(Theorem)g(7.2)i(the)e(op)s(erator)h Ft(L)1930 422 y Fo(\014)2020 407 y Fu(is)f(essen)m(tially)i(self-adjoin)m(t)g(on) e Fr(D)s Fu(\()p Ft(L)3358 422 y Fl(fr)3411 407 y Fu(\))29 b Fr(\\)244 520 y(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)495 535 y Fo(\014)543 520 y Fu(\))15 b Fr(\000)h Ft(J)9 b(\031)s Fu(\()p Ft(V)882 535 y Fo(\014)929 520 y Fu(\))p Ft(J)g Fu(\))29 b(for)f(all)h Ft(\014)h Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([)29 b(and)f Ft(\025)d Fr(2)g Fj(R)p Fu(.)39 b(By)29 b(Theorem)e(6.3)j(the)e(op)s(erator)g Ft(L)3461 534 y Fs(1)244 633 y Fu(is)i(self-adjoin)m(t)i(on)e Fr(D)s Fu(\()p Ft(L)1099 648 y Fl(fr)1152 633 y Fu(\))c(=)f Fr(D)s Fu(\()p Ft(L)1479 648 y Fl(fr)1532 633 y Fu(\))c Fr(\\)e(D)s Fu(\()p Ft(\031)s Fu(\()p Ft(V)1919 647 y Fs(1)1995 633 y Fu(\))h Fr(\000)g Ft(J)9 b(\031)s Fu(\()p Ft(V)2343 647 y Fs(1)2418 633 y Fu(\))p Ft(J)g Fu(\))31 b(for)g(all)g Ft(\025)p Fu(.)380 746 y(By)g(Theorem)f(7.12)i(there)f (exists)g Ft(\017)25 b(>)g Fu(0)30 b(suc)m(h)g(that)h(for)g(all)g Ft(\014)f Fr(2)25 b Fu([)p Ft(\014)2727 760 y Fl(0)2767 746 y Ft(;)15 b Fr(1)p Fu(],)1467 930 y(\000)1524 892 y Fl(I)1524 953 y Fo(\014)1597 930 y Fr(\024)25 b(\000)p Ft(\017)p Fu(\(1)20 b Fr(\000)g Fu(1)2037 944 y Fl(0)2077 930 y Fu(\(\000)2169 892 y Fl(I)2169 953 y Fo(\014)2216 930 y Fu(\)\))p Ft(:)380 1115 y Fu(By)31 b(Assumption)e(7.D\()p Ft(\021)s Fu(\))k(and)d(Prop)s(osition)g(7.13)i(\(2\),)g(for)e(all)h Ft(\014)g Fr(2)25 b Fu([)p Ft(\014)2855 1129 y Fl(0)2895 1115 y Ft(;)15 b Fr(1)p Fu(])31 b(and)e Ft(\021)g(>)c Fu(2,)1243 1299 y Fr(kh)p Ft(s)p Fr(i)1401 1261 y Fo(\021)1444 1225 y Fk(\000)1485 1299 y Fu(1)p 1530 1274 59 3 v 29 x Fs(K)1602 1285 y Fu(\024)1589 1299 y Fr(\012)p Ft(v)1704 1314 y Fo(\014)1771 1299 y Fr(\000)20 b Fu(1)1907 1313 y Fs(K)1986 1299 y Fr(\012)p 2077 1249 142 4 v 20 w Ft(\034)10 b(v)2171 1314 y Fo(\014)2218 1225 y Fk(\001)2260 1299 y Fr(k)26 b(\024)e Fu(2)p Ft(c:)380 1484 y Fu(Hence,)33 b(all)f(conditions)g(of)f(Theorem)g(5.5)h(are)g(satis\014ed,)g(and)e (there)i(exists)g Ft(\025)3121 1498 y Fl(0)3187 1484 y Ft(>)26 b Fu(0)32 b(suc)m(h)244 1596 y(that)f(for)f(0)c Ft(<)f Fr(j)p Ft(\025)p Fr(j)g Ft(<)g(\025)1024 1610 y Fl(0)1094 1596 y Fu(and)30 b(all)h Ft(\014)g Fr(2)25 b Fu([)p Ft(\014)1641 1610 y Fl(0)1681 1596 y Ft(;)15 b Fr(1)p Fu(])31 b(w)m(e)f(ha)m(v)m(e)1051 1781 y(dim)15 b(1)1263 1743 y Fl(p)1307 1781 y Fu(\()p Ft(L)1404 1796 y Fo(\014)1451 1781 y Fu(\))26 b Fr(\024)f Fu(dim)14 b(\000)1831 1743 y Fl(I)1831 1804 y Fo(\014)1904 1781 y Fu(=)25 b(1)p Ft(;)228 b Fu(sp)2384 1803 y Fl(sc)2447 1781 y Fu(\()p Ft(L)2544 1796 y Fo(\014)2592 1781 y Fu(\))25 b(=)g Fr(;)p Ft(:)244 1965 y Fu(By)31 b(Theorem)f(7.3)h(dim)15 b(1)1129 1979 y Fl(0)1169 1965 y Fu(\()p Ft(L)1266 1980 y Fo(\014)1313 1965 y Fu(\))26 b Fr(\025)f Fu(1)30 b(for)h Ft(\014)f Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([,)31 b(and)f(the)h (statemen)m(t)h(follo)m(ws.)42 b Fa(2)244 2282 y Fi(7.8)135 b(Simpli\014ed)45 b(condition)h(for)f(return)g(to)g(equilibrium)244 2454 y Fu(In)35 b(this)g(subsection)h(w)m(e)g(describ)s(e)f(conditions) h(whic)m(h)f(are)h(easy)h(to)f(v)m(erify)g(in)f(practice)i(and)244 2567 y(whic)m(h)30 b(imply)g(Assumption)g(7.D\()p Ft(\021)s Fu(\).)244 2756 y Fv(Prop)s(osition)36 b(7.15)50 b Fm(Supp)-5 b(ose)34 b(that)g Ft(n)e Fm(is)g(a)h(nonne)-5 b(gative)34 b(inte)-5 b(ger)33 b(and)1317 2941 y Fr(kh)p Ft(r)s Fr(i)1476 2903 y Fl(1)p Fo(=)p Fl(2)1586 2941 y Ft(s)1629 2903 y Fo(n)1676 2941 y Fr(j)p Ft(r)s Fr(j)1770 2903 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1935 2941 y Fu(\()p Ft(v)s(;)15 b(v)2104 2903 y Fo(?)2145 2941 y Fu(\))p Fr(k)26 b Ft(<)f Fr(1)p Fu(;)840 b(\(7.65\))859 3125 y Fr(kj)p Ft(r)s Fr(j)998 3088 y Fl(1)p Fo(=)p Fl(2)p Fs(\000)p Fo(n)p Fl(+)p Fo(j)1294 3125 y Ft(s)1337 3088 y Fo(j)1373 3125 y Fr(j)p Ft(r)s Fr(j)1467 3088 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1632 3125 y Fu(\()p Ft(v)s(;)15 b(v)1801 3088 y Fo(?)1842 3125 y Fu(\))p Fr(k)26 b Ft(<)f Fr(1)p Ft(;)113 b(j)30 b Fu(=)25 b(0)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)21 b Fr(\000)f Fu(1)p Ft(:)383 b Fu(\(7.66\))244 3282 y Fm(Then)33 b(Assumption)h(7.D)p Fu(\()p Ft(n)p Fu(\))e Fm(holds.)244 3472 y Fv(Pro)s(of.)61 b Fu(Set)31 b Ft(g)s Fu(\()p Ft(p)p Fu(\))26 b(:=)g Fr(j)p Fu(1)20 b Fr(\000)g Fu(e)1275 3439 y Fs(\000)p Fo(p)1370 3472 y Fr(j)1395 3439 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1560 3472 y Fr(j)p Ft(p)p Fr(j)1656 3439 y Fl(1)p Fo(=)p Fl(2)1767 3472 y Fu(.)40 b(Note)32 b(that)f Ft(g)j Fu(is)c(smo)s(oth)h(and)1470 3656 y Fr(j)p Ft(@)1548 3619 y Fo(j)1543 3679 y(p)1585 3656 y Ft(g)s Fu(\()p Ft(p)p Fu(\))p Fr(j)c(\024)e Ft(c)1934 3670 y Fo(j)1971 3656 y Fr(h)p Ft(p)p Fr(i)2087 3619 y Fl(1)p Fo(=)p Fl(2)p Fs(\000)p Fo(j)2284 3656 y Ft(:)994 b Fu(\(7.67\))244 3841 y(No)m(w)387 4003 y Ft(s)430 3970 y Fo(n)476 4003 y Fr(j)p Fu(1)21 b Fr(\000)f Fu(e)698 3970 y Fs(\000)p Fo(\014)s(r)834 4003 y Fr(j)859 3970 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1024 4003 y Fu(\()p Ft(v)s(;)15 b(v)1193 3970 y Fo(?)1234 4003 y Fu(\))84 b(=)25 b Ft(\014)1505 3970 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1670 4003 y Ft(s)1713 3970 y Fo(n)1759 4003 y Ft(g)s Fu(\()p Ft(\014)5 b(r)s Fu(\))p Fr(j)p Ft(r)s Fr(j)2069 3970 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2235 4003 y Fu(\()p Ft(v)s(;)15 b(v)2404 3970 y Fo(?)2445 4003 y Fu(\))1353 4191 y(=)1449 4123 y Fk(P)1545 4149 y Fo(n)1545 4218 y(j)t Fl(=0)1671 4191 y Fu(\()p Fr(\000)p Fu(i\))1837 4158 y Fo(n)p Fs(\000)p Fo(j)1972 4191 y Ft(\014)2028 4158 y Fs(\000)p Fl(1)p Fo(=)p Fl(2+)p Fo(n)p Fs(\000)p Fo(j)2378 4191 y Ft(g)2424 4158 y Fl(\()p Fo(n)p Fs(\000)p Fo(j)t Fl(\))2614 4191 y Fu(\()p Ft(\014)5 b(r)s Fu(\))p Ft(s)2827 4158 y Fo(j)2864 4191 y Fr(j)p Ft(r)s Fr(j)2958 4158 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)3122 4191 y Fu(\()p Ft(v)s(;)15 b(v)3291 4158 y Fo(?)3332 4191 y Fu(\))p Ft(:)244 4374 y Fu(T)-8 b(o)31 b(estimate)h(the)e(term)h(with)f Ft(j)h Fu(=)25 b Ft(n)30 b Fu(w)m(e)g(use)h(that)g(for)f Ft(\014)g Fr(\025)25 b Fu(1)31 b(w)m(e)g(ha)m(v)m(e)1187 4558 y Ft(\014)1243 4521 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1408 4558 y Ft(g)s Fu(\()p Ft(\014)5 b(r)s Fu(\))26 b Fr(\024)f Ft(\014)1802 4521 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1967 4558 y Fr(h)p Ft(\014)5 b(r)s Fr(i)2137 4521 y Fl(1)p Fo(=)p Fl(2)2273 4558 y Fr(\024)25 b(h)p Ft(r)s Fr(i)2483 4521 y Fl(1)p Fo(=)p Fl(2)244 4743 y Fu(T)-8 b(o)31 b(estimate)h(the)e(terms)h(with)f Ft(j)h Fu(=)25 b(0)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fr(\000)g Fu(1)31 b(w)m(e)f(use)1037 4927 y Ft(\014)1093 4890 y Fs(\000)p Fl(1)p Fo(=)p Fl(2+)p Fo(n)p Fs(\000)p Fo(j)1443 4927 y Ft(g)1489 4890 y Fl(\()p Fo(n)p Fs(\000)p Fo(j)t Fl(\))1679 4927 y Fu(\()p Ft(\014)5 b(r)s Fu(\))26 b Fr(\024)f Ft(h)2023 4941 y Fo(n)p Fs(\000)p Fo(j)2157 4927 y Fu(\()p Ft(\014)5 b(r)s Fu(\))p Fr(j)p Ft(r)s Fr(j)2421 4890 y Fl(1)p Fo(=)p Fl(2)p Fs(\000)p Fo(n)p Fl(+)p Fo(j)2717 4927 y Ft(;)244 5112 y Fu(where,)30 b(b)m(y)g(\(7.67\),)j Ft(h)998 5126 y Fo(n)p Fs(\000)p Fo(j)1133 5112 y Fu(\()p Ft(p)p Fu(\))25 b(=)g Ft(g)1416 5079 y Fl(\()p Fo(n)p Fs(\000)p Fo(j)t Fl(\))1606 5112 y Fu(\()p Ft(p)p Fu(\))p Fr(j)p Ft(p)p Fr(j)1818 5079 y Fs(\000)p Fl(1)p Fo(=)p Fl(2+)p Fo(n)p Fs(\000)p Fo(j)2199 5112 y Fu(is)30 b(a)h(b)s(ounded)d(function.)41 b Fa(2)380 5268 y Fu(Here)31 b(are)g(y)m(et)g(another)g(set)g(of)f(conditions)h (implying)g(Assumption)e(7.D\()p Ft(n)p Fu(\).)p eop %%Page: 49 49 49 48 bop 3682 100 a Fn(49)244 407 y Fv(Prop)s(osition)36 b(7.16)50 b Fm(Supp)-5 b(ose)34 b(that)1279 496 y Fk(Z)1370 522 y Fs(1)1330 702 y Fl(0)1445 619 y Fr(h)p Ft(p)p Fr(ik)p Ft(@)1659 582 y Fo(n)1654 642 y(p)1707 619 y Ft(p)1753 582 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1918 619 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Fr(k)2126 582 y Fl(2)2167 619 y Fu(d)p Ft(p)24 b(<)h Fr(1)p Fu(;)803 b(\(7.68\))768 773 y Fk(Z)859 799 y Fs(1)819 979 y Fl(0)949 897 y Ft(p)995 859 y Fl(1)p Fs(\000)p Fl(2)p Fo(n)p Fl(+2)p Fo(j)1290 897 y Fr(k)p Ft(@)1388 859 y Fo(j)1383 919 y(p)1425 897 y Ft(p)1471 859 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1636 897 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Fr(k)1844 859 y Fl(2)1885 897 y Fu(d)p Ft(p)24 b(<)h Fr(1)p Ft(;)145 b(j)31 b Fu(=)25 b(0)p Ft(;)15 b(:)g(:)g(:)i(;)e(n)20 b Fr(\000)g Fu(1;)293 b(\(7.69\))513 1131 y Ft(@)566 1093 y Fo(j)561 1153 y(p)602 1131 y Ft(p)648 1093 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)813 1131 y Ft(v)s Fu(\()p Ft(p)p Fu(\))976 1026 y Fk(\014)976 1081 y(\014)976 1135 y(\014)1007 1194 y Fo(p)p Fl(=0)1162 1131 y Fu(=)25 b(\()p Fr(\000)p Fu(1\))1444 1093 y Fo(j)1482 1131 y Ft(@)1535 1093 y Fo(j)1530 1153 y(p)1571 1131 y Ft(p)1617 1093 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1782 1131 y Ft(v)1829 1093 y Fo(?)1869 1131 y Fu(\()p Ft(p)p Fu(\))1985 1026 y Fk(\014)1985 1081 y(\014)1985 1135 y(\014)2016 1194 y Fo(p)p Fl(=0)2145 1131 y Ft(;)202 b(j)31 b Fu(=)25 b(0)p Ft(;)15 b Fr(\001)g(\001)g(\001)32 b Ft(;)15 b(n)20 b Fr(\000)g Fu(1)p Ft(:)269 b Fu(\(7.70\))244 1322 y Fm(Then)33 b(\(7.65\))h(and)g(\(7.66\))g(hold,)g(and)f(henc)-5 b(e)33 b(Assumption)h(7.D)p Fu(\()p Ft(n)p Fu(\))f Fm(holds.)244 1499 y Fv(Pro)s(of.)68 b Fu(Since)34 b Fr(K)h Fu(is)f(\014nite)f (dimensional,)i(the)f Ft(?)g Fu(conjugation)h(is)f(a)g(b)s(ounded)e (linear)i(map,)244 1612 y(and)c(\(7.68\))i(and)e(\(7.69\))j(hold)d (with)g Ft(v)1579 1579 y Fo(?)1649 1612 y Fu(instead)g(of)h Ft(v)s Fu(.)380 1725 y(W)-8 b(e)32 b(will)e(pro)m(v)m(e)h(\(7.65\).)43 b(A)31 b(similar)f(argumen)m(t)h(yields)g(\(7.66\).)43 b(Using)30 b(\(7.68\))j(and)c(\(7.70\))244 1837 y(w)m(e)i(see)g(that) 532 1974 y Fk(\000)573 2047 y Fr(h)p Ft(r)s Fr(i)687 2010 y Fl(1)p Fo(=)p Fl(2)797 2047 y Ft(s)840 2010 y Fo(n)887 2047 y Fr(j)p Ft(r)s Fr(j)981 2010 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1146 2047 y Fu(\()p Ft(v)s(;)15 b(v)1315 2010 y Fo(?)1356 2047 y Fu(\))1391 1974 y Fk(\001)1433 2047 y Fu(\()p Ft(p)p Fu(\))26 b(=)f(\()p Fr(\000)p Fu(i\))1837 2010 y Fo(n)1884 2047 y Fr(h)p Ft(p)p Fr(i)2000 2010 y Fl(1)p Fo(=)p Fl(2)2126 1919 y Fk(\032)2235 1989 y Ft(@)2288 1956 y Fo(n)2283 2012 y(p)2336 1989 y Ft(p)2382 1956 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2546 1989 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Ft(;)225 b(p)25 b Fr(\025)g Fu(0)p Ft(;)2235 2107 y(@)2288 2074 y Fo(n)2283 2130 y(p)2336 2107 y Fr(j)p Ft(p)p Fr(j)2432 2074 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2597 2107 y Ft(v)2644 2074 y Fo(?)2684 2107 y Fu(\()p Fr(j)p Ft(p)p Fr(j)p Fu(\))p Ft(;)84 b(p)25 b Fr(\024)g Fu(0)p Ft(;)244 2275 y Fu(and)435 2427 y Fr(kh)p Ft(r)s Fr(i)594 2394 y Fl(1)p Fo(=)p Fl(2)704 2427 y Ft(s)747 2394 y Fo(n)794 2427 y Fr(j)p Ft(r)s Fr(j)888 2394 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1053 2427 y Fu(\()p Ft(v)s(;)15 b(v)1222 2394 y Fo(?)1263 2427 y Fu(\))p Fr(k)1343 2394 y Fl(2)1466 2427 y Fr(\024)1562 2353 y Fk(R)1623 2380 y Fs(1)1605 2458 y(\0001)1735 2427 y Fr(h)p Ft(p)p Fr(ik)p Ft(@)1949 2394 y Fo(n)1944 2449 y(p)1997 2427 y Fr(j)p Ft(p)p Fr(j)2093 2394 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2258 2427 y Fu(\()p Ft(v)s(;)g(v)2427 2394 y Fo(?)2468 2427 y Fu(\)\()p Ft(p)p Fu(\))p Fr(k)2664 2394 y Fl(2)2704 2427 y Fu(d)p Ft(p)1466 2660 y Fu(=)1571 2569 y Fs(1)1567 2587 y Fk(R)1571 2746 y Fl(0)1633 2660 y Fr(h)p Ft(p)p Fr(i)1764 2587 y Fk(\000)1806 2660 y Fr(k)p Ft(@)1904 2627 y Fo(n)1899 2683 y(p)1952 2660 y Ft(p)1998 2627 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2162 2660 y Ft(v)s Fu(\()p Ft(p)p Fu(\))p Fr(k)2370 2627 y Fl(2)2431 2660 y Fu(+)20 b Fr(k)p Ft(@)2620 2627 y Fo(n)2615 2683 y(p)2668 2660 y Ft(p)2714 2627 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)2878 2660 y Ft(v)2925 2627 y Fo(?)2965 2660 y Fu(\()p Ft(p)p Fu(\))p Fr(k)3126 2627 y Fl(2)3166 2587 y Fk(\001)3223 2660 y Fu(d)p Ft(p:)244 2865 y Fa(2)244 3176 y Fi(7.9)135 b(P)l(auli-Fierz)41 b(systems)f(with)g(sev)l(eral)h (thermal)g(reserv)l(oirs)244 3348 y Fu(In)30 b(this)g(subsection)h(w)m (e)g(pro)m(v)m(e)g(that)g(a)g(generic)h(P)m(auli-Fierz)g(system)f(with) f(a)h(small)g(nonzero)244 3460 y(coupling)21 b(constan)m(t)h(has)e(no)h (normal)g(in)m(v)-5 b(arian)m(t)22 b(states.)38 b(F)-8 b(or)22 b(shortness,)g(w)m(e)f(restrict)h(ourselv)m(es)244 3573 y(to)31 b(a)g(result)f(non-uniform)f(in)h(the)h(temp)s(erature.) 380 3686 y(W)-8 b(e)41 b(consider)e(the)g(same)g(framew)m(ork)h(as)f (in)g(Subsection)g(6.8.)68 b(Moreo)m(v)m(er,)43 b(w)m(e)d(assume)244 3799 y(that)31 b(the)f(energy)h(densit)m(y)g(of)f(the)h Ft(i)p Fu(-th)g(reserv)m(oir)g(is)f(giv)m(en)h(b)m(y)1508 3973 y Ft(\032)1555 3988 y Fo(\014)1595 3998 y Fg(i)1651 3973 y Fu(=)25 b(\(e)1822 3935 y Fo(\014)1862 3945 y Fg(i)1888 3935 y Fo(h)1929 3945 y Fg(i)1980 3973 y Fr(\000)20 b Fu(1\))2151 3935 y Fs(\000)p Fl(1)2246 3973 y Ft(;)244 4158 y Fu(where)30 b Ft(\014)558 4172 y Fo(i)613 4158 y Fr(2)p Fu(]0)p Ft(;)15 b Fr(1)p Fu([.)43 b(W)-8 b(e)32 b(set)1273 4134 y Ft(~)1267 4158 y(\014)g Fu(=)25 b(\()p Ft(\014)1532 4172 y Fl(1)1573 4158 y Ft(;)15 b(:)g(:)g(:)h(;)f(\014) 1825 4172 y Fo(N)1893 4158 y Fu(\))31 b(and,)g(after)g(replacing)h Ft(\014)k Fu(with)3068 4134 y Ft(~)3062 4158 y(\014)5 b Fu(,)31 b(w)m(e)g(adopt)244 4271 y(the)f(same)h(notational)i(con)m(v) m(en)m(tion)f(as)f(in)f(Subsection)g(7.1.)244 4448 y Fv(Theorem)70 b(7.17)49 b Fm(L)-5 b(et)42 b Ft(\021)i(>)d Fu(1)p Fm(.)68 b(Assume)41 b(that)h(Assumptions)g(7.A)f(and)i(7.B)p Fu(\()p Ft(\021)s Fu(\))f Fm(hold)h(for)244 4561 y Ft(i)j Fu(=)h(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fm(.)76 b(Supp)-5 b(ose)46 b(also)g(that)f Ft(\014)1677 4575 y Fo(j)1760 4561 y Fr(6)p Fu(=)h Ft(\014)1928 4576 y Fo(k)2016 4561 y Fm(for)f(some)f Ft(j;)15 b(k)51 b Fr(2)46 b(f)p Fu(1)p Ft(;)15 b(:)g(:)g(:)i(;)e(N)10 b Fr(g)p Fm(,)48 b(and)d(that)244 4685 y(Assumption)39 b(7.C)f(holds)h(for)f(the)g Ft(j)5 b Fm(th)39 b(and)g Ft(k)s Fm(th)f(r)-5 b(eservoir.)58 b(Then)38 b(ther)-5 b(e)38 b(exists)g Ft(\025)3184 4699 y Fl(0)3224 4685 y Fu(\()3265 4661 y Ft(~)3259 4685 y(\014)5 b Fu(\))35 b Ft(>)f Fu(0)244 4798 y Fm(such)f(that)g(for)h Fu(0)25 b Ft(<)g Fr(j)p Ft(\025)p Fr(j)h Ft(<)f(\025)1226 4812 y Fl(0)1266 4798 y Fu(\()1307 4774 y Ft(~)1301 4798 y(\014)5 b Fu(\))p Fm(,)1278 4971 y Fu(sp)1364 4993 y Fl(p)1408 4971 y Fu(\()p Ft(L)1510 4987 y Fo(~)1505 5005 y(\014)1552 4971 y Fu(\))26 b(=)f Fr(;)p Ft(;)201 b Fu(sp)2067 4993 y Fl(sc)2130 4971 y Fu(\()p Ft(L)2232 4987 y Fo(~)2227 5005 y(\014)2274 4971 y Fu(\))26 b(=)f Fr(;)p Ft(:)244 5155 y Fm(Conse)-5 b(quently,)43 b(under)d(the)h(ab)-5 b(ove)41 b(c)-5 b(onditions)41 b(the)g(system)g Fu(\()p Fp(M)2551 5171 y Fo(~)2546 5189 y(\014)2594 5155 y Ft(;)15 b(\034)2679 5171 y Fo(~)2674 5189 y(\014)2721 5155 y Fu(\))40 b Fm(has)i(no)e(normal)i(in-)244 5268 y(variant)34 b(states.)p eop %%Page: 50 50 50 49 bop 3682 100 a Fn(50)244 407 y Fv(Pro)s(of.)70 b Fu(The)34 b(pro)s(of)g(is)g(v)m(ery)h(similar)g(to)g(the)g(pro)s(of)f (of)g(Theorem)h(7.10.)54 b(By)35 b(Theorem)f(5.5)244 520 y(and)k(the)i(remark)e(after)i(it,)i(it)d(su\016ces)g(to)h(sho)m(w) f(that)g(dim)15 b(Ker\000)2612 487 y Fl(I)2617 549 y Fo(~)2612 566 y(\014)2698 520 y Fu(=)40 b Fr(f)p Fu(0)p Fr(g)p Fu(.)68 b(The)38 b(relation)244 656 y(\(6.51\))33 b(yields)1468 855 y(Ker\000)1672 817 y Fl(I)1677 880 y Fo(~)1672 897 y(\014)1744 855 y Fu(=)1865 741 y Fo(N)1846 769 y Fk(\\)1840 964 y Fo(i)p Fl(=1)1969 855 y Fu(Ker\000)2173 817 y Fl(I)2173 878 y Fo(i;\014)2257 888 y Fg(i)2286 855 y Ft(;)244 1083 y Fu(where)d(\000)564 1098 y Fo(i;\014)648 1108 y Fg(i)709 1083 y Fu(is)h(the)g(Lev)m(el)i(Shift)d(Op)s(erator)h (of)g(the)g Ft(i)p Fu(-th)h(subsystem.)41 b(By)32 b(Assumption)e(7.C,) 244 1196 y(Ker\000)448 1163 y Fl(I)448 1224 y Fo(i;\014)532 1234 y Fg(i)591 1196 y Fu(is)h(spanned)e(b)m(y)h Ft(\015)1210 1211 y Fo(\014)1250 1221 y Fg(i)1311 1196 y Fu(for)g Ft(i)25 b Fu(=)g Ft(j;)15 b(k)s Fu(.)42 b(Since)31 b Ft(\014)2085 1210 y Fo(j)2147 1196 y Fr(6)p Fu(=)25 b Ft(\014)2294 1211 y Fo(k)2337 1196 y Fu(,)30 b(Ker\000)2596 1163 y Fl(I)2596 1224 y Fo(j;\014)2685 1234 y Fg(j)2740 1196 y Fr(\\)20 b Fu(Ker\000)3025 1163 y Fl(I)3025 1224 y Fo(k)r(;\014)3124 1236 y Fg(k)3190 1196 y Fu(=)25 b Fr(f)p Fu(0)p Fr(g)p Fu(.)p Fa(2)244 1551 y Fq(8)161 b(Examples)54 b(of)f(gluing)244 1754 y Fu(As)39 b(w)m(e)h(ha)m(v)m(e)g (already)g(emphasized,)i(the)e(k)m(ey)g(ingredien)m(t)g(of)f(our)g (metho)s(d)g(is)g(the)h(Jak)-5 b(\024)-41 b(si)m(\023)e(c-)244 1867 y(Pillet)33 b(gluing)f(condition.)45 b(In)31 b(this)h(section)h(w) m(e)f(sho)m(w)g(that)g(this)g(condition)g(is)g(satis\014ed)g(in)f(a)244 1980 y(certain)23 b(class)g(of)f(ph)m(ysically)h(motiv)-5 b(ated)24 b(mo)s(dels)d(in)m(v)m(olving)j(massless)e(b)s(osons.)37 b(The)22 b(gluing)h(is)244 2093 y(accomplished)j(b)m(y)g(passing)f(to)h (the)g(radial)g(co)s(ordinates)h(in)e(the)h(momen)m(tum)f(represen)m (tation.)244 2364 y Fi(8.1)135 b(Massless)46 b(scalar)g(b)t(osons)244 2536 y Fu(In)37 b(this)h(subsection)f(w)m(e)i(consider)e(the)h(same)g (mo)s(del)g(as)g(in)g(Section)g(3.)63 b(Recall)40 b(that)e Fr(Z)45 b Fu(=)244 2649 y Ft(L)306 2616 y Fl(2)345 2649 y Fu(\()p Fj(R)446 2616 y Fo(d)487 2649 y Fu(\),)i(where)d Ft(\030)51 b Fr(2)d Fj(R)1137 2616 y Fo(d)1220 2649 y Fu(describ)s(es)43 b(the)h(momen)m(tum,)k(and)43 b(that)h Ft(h)g Fu(is)g(the)g(op)s(erator)g(of)244 2762 y(m)m(ultiplication)32 b(b)m(y)e Fr(j)p Ft(\030)t Fr(j)p Fu(.)42 b(The)29 b(gluing)i(map)f(is) h(de\014ned)e(as)814 2945 y Ft(L)876 2907 y Fl(2)916 2945 y Fu(\()p Fj(R)1017 2907 y Fo(d)1057 2945 y Fu(\))21 b Fr(\010)p 1204 2861 279 4 v 20 w Ft(L)1266 2918 y Fl(2)1305 2945 y Fu(\()p Fj(R)1406 2918 y Fo(d)1446 2945 y Fu(\))26 b Fr(3)f Fu(\()p Ft(f)1673 2959 y Fl(+)1732 2945 y Ft(;)p 1772 2871 55 4 v 15 w(f)1826 2966 y Fs(\000)1885 2945 y Fu(\))h Fr(7!)f Ft(f)35 b Fr(2)25 b Ft(L)2290 2907 y Fl(2)2329 2945 y Fu(\()p Fj(R)p Fu(\))20 b Fr(\012)g Ft(L)2638 2907 y Fl(2)2678 2945 y Fu(\()p Ft(S)2774 2907 y Fo(d)p Fs(\000)p Fl(1)2905 2945 y Fu(\))p Ft(;)338 b Fu(\(8.71\))1084 3202 y Ft(f)10 b Fu(\()p Ft(p;)15 b(!)s Fu(\))25 b(:=)1501 3046 y Fk(\()1616 3145 y Ft(p)1672 3081 y Fg(d)p Ff(\000)p Fh(1)p 1672 3094 111 3 v 1712 3135 a(2)1796 3145 y Ft(f)1841 3159 y Fl(+)1900 3145 y Fu(\()p Ft(p!)s Fu(\))p Ft(;)305 b(p)25 b(>)g Fu(0)p Ft(;)1616 3277 y Fu(\()p Fr(\000)p Ft(p)p Fu(\))1813 3214 y Fg(d)p Ff(\000)p Fh(1)p 1813 3227 V 1853 3268 a(2)p 1938 3203 55 4 v 1938 3277 a Ft(f)1992 3299 y Fs(\000)2051 3277 y Fu(\()p Fr(\000)p Ft(p!)s Fu(\))p Ft(;)83 b(p)25 b Fr(\024)g Fu(0)p Ft(:)3303 3202 y Fu(\(8.72\))244 3442 y(Here,)h(\()p Ft(p;)15 b(!)s Fu(\))26 b Fr(2)f Fj(R)8 b Fr(\002)g Ft(S)1021 3409 y Fo(d)p Fs(\000)p Fl(1)1152 3442 y Fu(.)39 b(Moreo)m(v)m(er,)28 b(the)d(conjugation)g(in)f Ft(L)2435 3409 y Fl(2)2475 3442 y Fu(\()p Fj(R)2576 3409 y Fo(d)2616 3442 y Fu(\))h(and)f Ft(L)2909 3409 y Fl(2)2948 3442 y Fu(\()p Fj(R)p Fu(\))8 b Fr(\012)g Ft(L)3233 3409 y Fl(2)3273 3442 y Fu(\()p Ft(S)3369 3409 y Fo(d)p Fs(\000)p Fl(1)3500 3442 y Fu(\))244 3555 y(is)30 b(the)h(standard)e(complex)i (conjugation.)42 b(The)30 b(map)g(\(8.71\))j(is)d(unitary)-8 b(.)380 3668 y(As)26 b(in)g(Section)h(3,)g(w)m(e)g(\014x)e(a)i (form-factor)f Ft(v)j Fu(:)d Fj(R)2009 3635 y Fo(d)2074 3668 y Fr(7!)f(B)s Fu(\()p Fr(K)q Fu(\).)40 b(Recall)28 b(that)e(the)h(corresp)s(ond-)244 3780 y(ing)j(P)m(auli-Fierz)j (Hamiltonian)f(is)1075 3947 y Ft(H)90 b Fu(:=)25 b Ft(K)i Fr(\012)20 b Fu(1)g(+)g(1)h Fr(\012)1870 3873 y Fk(R)1946 3947 y Fr(j)p Ft(\030)t Fr(j)p Ft(a)2088 3914 y Fs(\003)2128 3947 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)p Fu(\()p Ft(\030)t Fu(\))p Fr(j)p Ft(\030)t Fr(j)15 b Fu(d)p Ft(\030)1241 4130 y Fu(+)p Ft(\025)1380 4057 y Fk(R)i(\000)1482 4130 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))k Fr(\012)f Ft(a)1803 4097 y Fs(\003)1842 4130 y Fu(\()p Ft(\030)t Fu(\))h(+)f Ft(v)2115 4097 y Fs(\003)2155 4130 y Fu(\()p Ft(\030)t Fu(\))h Fr(\012)f Ft(a)p Fu(\()p Ft(\030)t Fu(\))2543 4057 y Fk(\001)2585 4130 y Fu(d)p Ft(\030)t(:)380 4313 y Fu(W)-8 b(e)29 b(\014x)f(the)g(densit)m(y)g Fj(R)1192 4280 y Fo(d)1258 4313 y Fr(3)c Ft(\030)30 b Fr(7!)25 b Ft(\032)p Fu(\()p Ft(\030)t Fu(\))h Fr(2)f Fj(R)1868 4327 y Fl(+)1954 4313 y Fu(and)j(assume)f(that)i(Assumption)e(3.A)i (holds.)244 4426 y(Recall)j(that)f(the)f(semi-Liouvillean)i(at)g (densit)m(y)e Ft(\032)g Fu(is)h(giv)m(en)g(b)m(y)716 4592 y Ft(L)778 4559 y Fl(semi)778 4614 y Fo(\032)1003 4592 y Fu(:=)25 b Ft(K)i Fr(\012)20 b Fu(1)h(+)f(1)g Fr(\012)1632 4518 y Fk(R)e(\000)1734 4592 y Fr(j)p Ft(\030)t Fr(j)p Ft(a)1876 4559 y Fs(\003)1876 4619 y Fl(l)1916 4592 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)2078 4607 y Fl(l)2103 4592 y Fu(\()p Ft(\030)t Fu(\))j Fr(\000)f(j)p Ft(\030)t Fr(j)p Ft(a)2471 4559 y Fs(\003)2471 4614 y Fl(r)2511 4592 y Fu(\()p Ft(\030)t Fu(\))p Ft(a)2673 4606 y Fl(r)2705 4592 y Fu(\()p Ft(\030)t Fu(\))2819 4518 y Fk(\001)2876 4592 y Fu(d)p Ft(\030)1003 4791 y Fu(+)p Ft(\025)1142 4718 y Fk(R)1218 4791 y Ft(v)s Fu(\()p Ft(\030)t Fu(\))h Fr(\012)1491 4718 y Fk(\000)1532 4791 y Fu(\(1)g(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\)\))1930 4728 y Fh(1)p 1931 4740 31 3 v 1931 4782 a(2)1976 4791 y Ft(a)2024 4758 y Fs(\003)2024 4819 y Fl(l)2063 4791 y Fu(\()p Ft(\030)t Fu(\))h(+)f Ft(\032)p Fu(\()p Ft(\030)t Fu(\))2460 4728 y Fh(1)p 2461 4740 V 2461 4782 a(2)2505 4791 y Ft(a)2553 4805 y Fl(r)2585 4791 y Fu(\()p Ft(\030)t Fu(\))2699 4718 y Fk(\001)2741 4791 y Fu(d)p Ft(\030)k Fu(+)c(hc)p Ft(:)244 4958 y Fu(If)30 b(w)m(e)h(use)f(the)g(glued)h(v)-5 b(ariables)31 b(and)e(in)m(tro)s(duce)908 5204 y Ft(v)952 5218 y Fo(\032)992 5204 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(:=)1355 5049 y Fk(\()1470 5147 y Ft(p)1526 5084 y Fg(d)p Ff(\000)p Fh(1)p 1526 5097 111 3 v 1566 5138 a(2)1650 5147 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\()p Ft(p!)s Fu(\)\))2110 5084 y Fh(1)p 2111 5096 31 3 v 2111 5138 a(2)2156 5147 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Ft(;)178 b(p)25 b(>)g Fu(0)1470 5280 y(\()p Fr(\000)p Ft(p)p Fu(\))1667 5216 y Fg(d)p Ff(\000)p Fh(1)p 1667 5229 111 3 v 1707 5270 a(2)1792 5280 y Ft(\032)p Fu(\()p Fr(\000)p Ft(p!)s Fu(\))2096 5217 y Fh(1)p 2096 5229 31 3 v 2096 5270 a(2)2140 5280 y Ft(v)2187 5247 y Fs(\003)2227 5280 y Fu(\()p Fr(\000)p Ft(p!)s Fu(\))p Ft(;)83 b(p)25 b Fr(\024)g Fu(0)p Ft(;)3303 5204 y Fu(\(8.73\))p eop %%Page: 51 51 51 50 bop 3682 100 a Fn(51)244 407 y Fu(then)30 b(the)h (semi-Liouvillean)h(can)e(b)s(e)g(written)g(as)728 585 y Ft(L)790 552 y Fl(semi)790 608 y Fo(\032)1015 585 y Fu(=)25 b Ft(K)i Fr(\012)20 b Fu(1)g(+)g(1)h Fr(\012)1619 512 y Fk(R)1694 585 y Ft(pa)1788 552 y Fs(\003)1828 585 y Fu(\()p Ft(p;)15 b(!)s Fu(\))p Ft(a)p Fu(\()p Ft(p;)g(!)s Fu(\))g(d)p Ft(p)p Fu(d)p Ft(!)1015 769 y Fu(+)p Ft(\025)1154 696 y Fk(R)i(\000)1256 769 y Ft(v)1300 783 y Fo(\032)1340 769 y Fu(\()p Ft(p;)e(!)s Fu(\))21 b Fr(\012)f Ft(a)1716 736 y Fs(\003)1756 769 y Fu(\()p Ft(p;)15 b(!)s Fu(\))20 b(+)g Ft(v)2130 736 y Fs(\003)2127 792 y Fo(\032)2170 769 y Fu(\()p Ft(p;)15 b(!)s Fu(\))21 b Fr(\012)f Ft(a)p Fu(\()p Ft(p;)15 b(!)s Fu(\))2762 696 y Fk(\001)2820 769 y Fu(d)p Ft(p)p Fu(d)p Ft(!)s(:)380 972 y Fu(No)m(w,)38 b(using)c(\(8.73\),)39 b(it)d(is)f(easy)h(to)g(giv)m(e)g(explicit)h (conditions)e(on)g Ft(v)s Fu(\()p Ft(\030)t Fu(\))h(needed)f(for)g(our) 244 1085 y(results.)40 b(F)-8 b(or)32 b(instance,)f(Assumption)e(6.C\() p Ft(n)p Fu(\))1875 1099 y Fo(\032)1947 1085 y Fu(is)h(satis\014ed)g (if)805 1212 y Fk(Z)911 1335 y Fr(k)p Ft(v)1000 1349 y Fo(\032)1041 1335 y Fu(\()p Ft(p;)15 b(!)s Fu(\))p Fr(k)1302 1298 y Fl(2)1343 1335 y Fu(d)p Ft(p)p Fu(d)p Ft(!)26 b(<)f Fr(1)p Ft(;)1893 1212 y Fk(Z)1999 1335 y Fr(k)p Ft(@)2097 1298 y Fo(n)2092 1358 y(p)2144 1335 y Ft(v)2188 1349 y Fo(\032)2229 1335 y Fu(\()p Ft(p;)15 b(!)s Fu(\))p Fr(k)2490 1298 y Fl(2)2530 1335 y Fu(d)p Ft(p)p Fu(d)p Ft(!)27 b(<)e Fr(1)p Ft(:)329 b Fu(\(8.74\))380 1581 y(Recall)35 b(that)g(in)e(\(6.35\))j(w)m(e)e(in)m(tro)s(duced)f (the)g(an)m(tilinear)i(map)f Ft(\024)f Fu(on)h Fr(Z)7 b Fu(.)50 b(In)33 b(the)h(con)m(text)244 1694 y(of)c(scalar)i(\014elds) e(it)g(is)h(equal)g(to)1586 1807 y Ft(\024f)10 b Fu(\()p Ft(\030)t Fu(\))26 b(=)p 1929 1733 55 4 v 25 w Ft(f)9 b Fu(\()p Fr(\000)p Ft(\030)t Fu(\))p Ft(:)244 1974 y Fu(and)30 b(satis\014es)g Ft(\024)807 1941 y Fl(2)873 1974 y Fu(=)25 b(1)30 b(\(it)i(is)e(an)g(in)m(ternal)h(conjugation\).) 42 b(Assume)30 b(that)1524 2178 y Ft(\032\024)25 b Fu(=)g Ft(\024\032;)107 b(v)2022 2141 y Fo(?)2087 2178 y Fu(=)25 b Ft(v)s(:)244 2383 y Fu(In)30 b(the)g(con)m(text)i(of)f(scalar)g (\014elds)f(this)g(means)1231 2587 y Ft(v)1278 2549 y Fs(\003)1317 2587 y Fu(\()p Ft(\030)t Fu(\))c(=)f Ft(v)s Fu(\()p Fr(\000)p Ft(\030)t Fu(\))p Ft(;)198 b(\032)p Fu(\()p Ft(\030)t Fu(\))26 b(=)f Ft(\032)p Fu(\()p Fr(\000)p Ft(\030)t Fu(\))p Ft(:)755 b Fu(\(8.75\))244 2791 y(Then)968 2958 y Ft(v)1012 2972 y Fo(\032)1052 2958 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(=)1390 2803 y Fk(\()1505 2902 y Ft(p)1561 2838 y Fg(d)p Ff(\000)p Fh(1)p 1561 2851 111 3 v 1601 2892 a(2)1685 2902 y Fu(\(1)21 b(+)f Ft(\032)p Fu(\()p Ft(p!)s Fu(\)\))2145 2839 y Fh(1)p 2146 2851 31 3 v 2146 2892 a(2)2190 2902 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Ft(;)84 b(p)25 b(>)g Fu(0)1505 3034 y(\()p Fr(\000)p Ft(p)p Fu(\))1702 2970 y Fg(d)p Ff(\000)p Fh(1)p 1702 2983 111 3 v 1742 3024 a(2)1827 3034 y Ft(\032)p Fu(\()p Ft(p!)s Fu(\))2060 2971 y Fh(1)p 2060 2983 31 3 v 2060 3024 a(2)2105 3034 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Ft(;)169 b(p)25 b Fr(\024)g Fu(0)p Ft(:)3303 2958 y Fu(\(8.76\))244 3217 y(Assume)30 b(further)f(that)i Ft(\032)1131 3232 y Fo(\014)1178 3217 y Fu(\()p Ft(\030)t Fu(\))26 b(:=)f(\(e)1514 3184 y Fo(\014)s Fs(j)p Fo(\030)s Fs(j)1655 3217 y Fr(\000)20 b Fu(1\))1826 3184 y Fs(\000)p Fl(1)1951 3217 y Fu(and)30 b(set)h Ft(v)2314 3232 y Fo(\014)2387 3217 y Fu(:=)25 b Ft(v)2552 3231 y Fo(\032)2588 3243 y Fg(\014)2634 3217 y Fu(.)40 b(Then)1151 3517 y Ft(v)1195 3532 y Fo(\014)1242 3517 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(=)1580 3388 y Fk(\022)1801 3455 y Ft(p)p 1657 3496 335 4 v 1657 3579 a Fu(1)20 b Fr(\000)g Fu(e)1853 3553 y Fs(\000)p Fo(\014)s(p)2001 3388 y Fk(\023)2078 3384 y Fh(1)p 2078 3396 31 3 v 2078 3437 a(2)2138 3517 y Fr(j)p Ft(p)p Fr(j)2244 3452 y Fg(d)p 2244 3464 33 3 v 2245 3505 a Fh(2)2286 3479 y Fs(\000)p Fl(1)2380 3517 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Ft(:)675 b Fu(\(8.77\))244 3800 y(No)m(w,)31 b(if)g Fj(R)24 b Fr(3)h Ft(p)g Fr(7!)g(j)p Ft(p)p Fr(j)1029 3737 y Fg(d)p 1030 3749 V 1031 3790 a Fh(2)1072 3764 y Fs(\000)p Fl(1)1166 3800 y Ft(v)s Fu(\()p Ft(p)15 b Fr(\001)p Fu(\))26 b Fr(2)f(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)23 b(\012)d Ft(L)1934 3767 y Fl(2)1973 3800 y Fu(\()p Ft(S)2069 3767 y Fo(d)p Fs(\000)p Fl(1)2200 3800 y Fu(\)\))31 b(is)f Ft(n)g Fu(times)h(di\013eren)m(tiable)h(and) 941 3922 y Fk(Z)1047 4046 y Fr(k)p Ft(@)1145 4009 y Fo(j)1140 4069 y(p)1182 4046 y Fr(j)p Ft(p)p Fr(j)1288 3982 y Fg(d)p 1288 3994 V 1289 4035 a Fh(2)1330 4009 y Fs(\000)p Fl(1)1424 4046 y Fr(h)p Ft(p)p Fr(i)1540 4009 y Fl(1)p Fo(=)p Fl(2)1651 4046 y Ft(v)s Fu(\()p Ft(p!)s Fu(\))p Fr(k)1919 4009 y Fl(2)1959 4046 y Fu(d)p Ft(p)p Fu(d)p Ft(!)27 b(<)e Fr(1)p Ft(;)106 b(j)31 b Fu(=)25 b(0)p Ft(;)15 b(n;)465 b Fu(\(8.78\))244 4292 y(then)26 b(\(8.74\))i(holds)d(\(the)i(\\gluing) f(condition")h(\(7.70\))i(is)d(automatically)i(satis\014ed)f(for)e Ft(\032)h Fu(=)e Ft(\032)3488 4307 y Fo(\014)244 4426 y Fu(b)m(y)34 b(the)g(di\013eren)m(tiabilit)m(y)j Fr(j)p Ft(p)p Fr(j)1281 4363 y Fg(d)p 1281 4375 V 1282 4416 a Fh(2)1323 4390 y Fs(\000)p Fl(1)1417 4426 y Ft(v)s Fu(\()p Ft(p)15 b Fr(\001)p Fu(\))35 b(at)g(zero\).)54 b(Theorem)33 b(7.10)j(\(the)f(non)m(uniform)e(result)244 4539 y(on)d(return)f(to)j(equilibrium\))d(applies)i(whenev)m(er)f (\(8.74\))j(holds)d(with)g Ft(n)25 b Fu(=)g(3.)380 4652 y(The)37 b(case)i Ft(d)e Fu(=)g(3)h(and)e Ft(v)s Fu(\()p Ft(\030)t Fu(\))j Fr(\030)d(j)p Ft(\030)t Fr(j)1633 4619 y Fs(\000)p Fl(1)p Fo(=)p Fl(2)1836 4652 y Fu(as)i Fr(j)p Ft(\030)t Fr(j)f(#)h Fu(0,)i(is)d(sometimes)i(called)f(the)g(\\ohmic) 244 4765 y(case")29 b(and)e(is)h(t)m(ypical)h(for)f(the)g(infrared)f (regime)h(of)g(QED.)g(Note)h(that)f(Theorem)g(7.10)h(co)m(v)m(ers)244 4878 y(the)h(ohmic)g(case.)41 b(Ho)m(w)m(ev)m(er,)32 b(Theorem)e(7.14)h(\(our)e(uniform)g(result\))h(do)s(es)f(not)h(apply)f (to)h(the)244 4990 y(ohmic)h(case.)p eop %%Page: 52 52 52 51 bop 3682 100 a Fn(52)244 407 y Fi(8.2)135 b(Massless)46 b(v)l(ector)g(particles)244 578 y Fu(In)e(this)h(section)h(w)m(e)g (brie\015y)e(explain)h(ho)m(w)g(Jak)-5 b(\024)-41 b(si)m(\023)e (c-Pillet)50 b(gluing)45 b(w)m(orks)g(for)g(transv)m(ersal)244 691 y(massless)30 b(v)m(ector)i(b)s(osons)e(\(e.g.)42 b(photons\).)380 804 y(Consider)31 b(\014rst)f(the)i(Hilb)s(ert)f (space)h(of)f(square)g(in)m(tegrable)i(v)m(ector)g(\014elds)e(on)g Fj(R)3180 771 y Fo(d)3220 804 y Fu(,)h(that)g(is)244 917 y Ft(L)306 884 y Fl(2)345 917 y Fu(\()p Fj(R)446 884 y Fo(d)487 917 y Fu(\))20 b Fr(\012)g Fj(C)699 884 y Fo(d)764 917 y Fr(')25 b Ft(L)922 884 y Fl(2)962 917 y Fu(\()p Fj(R)1063 884 y Fo(d)1103 917 y Ft(;)15 b Fj(C)1209 884 y Fo(d)1250 917 y Fu(\))30 b(and)g(the)h(function)1034 1107 y Fj(R)1100 1069 y Fo(d)1165 1107 y Fr(3)25 b Ft(\030)30 b Fr(7!)25 b Ft(P)1495 1121 y Fl(tr)1554 1107 y Fu(\()p Ft(\030)t Fu(\))h(:=)f(1)c Fr(\000)f(j)p Ft(\030)t Fr(j)2066 1069 y Fs(\000)p Fl(2)2161 1107 y Fr(j)p Ft(\030)t Fu(\)\()p Ft(\030)t Fr(j)26 b(2)f(B)s Fu(\()p Fj(C)2645 1069 y Fo(d)2685 1107 y Fu(\))p Ft(;)244 1296 y Fu(where)31 b Fr(j)p Ft(\030)t Fr(j)602 1263 y Fs(\000)p Fl(2)696 1296 y Fr(j)p Ft(\030)t Fu(\)\()p Ft(\030)t Fr(j)i Fu(denotes)e(the)h (orthogonal)h(pro)5 b(jection)32 b(on)m(to)g Ft(\030)t(=)p Fr(j)p Ft(\030)t Fr(j)p Fu(.)44 b(W)-8 b(e)33 b(can)e(view)h Ft(P)3364 1310 y Fl(tr)3454 1296 y Fu(as)244 1409 y(an)25 b(op)s(erator)g(in)g Ft(L)889 1376 y Fl(2)928 1409 y Fu(\()p Fj(R)1029 1376 y Fo(d)1069 1409 y Ft(;)15 b Fj(C)1175 1376 y Fo(d)1216 1409 y Fu(\).)39 b(The)25 b(Hilb)s(ert)f(space)i(of)f (square)g(in)m(tegrable)h(transv)m(ersal)g(v)m(ector)244 1522 y(\014elds)k(on)g Fj(R)669 1489 y Fo(d)739 1522 y Fu(is)h(de\014ned)e(as)i Ft(L)1318 1489 y Fl(2)1318 1544 y(tr)1377 1522 y Fu(\()p Fj(R)1478 1489 y Fo(d)1518 1522 y Ft(;)15 b Fj(C)1624 1489 y Fo(d)1665 1522 y Fu(\))25 b(:=)g(Ran)p Ft(P)2067 1536 y Fl(tr)2127 1522 y Fu(.)380 1635 y(W)-8 b(e)33 b(consider)e(a)h(P)m(auli-Fierz)h(system)f(with)f (the)h(1-particle)h(space)f Ft(L)2818 1602 y Fl(2)2818 1657 y(tr)2877 1635 y Fu(\()p Fj(R)2978 1602 y Fo(d)3018 1635 y Ft(;)15 b Fj(C)3124 1602 y Fo(d)3165 1635 y Fu(\))32 b(and)e(the)244 1748 y(1-particle)c(energy)e Fr(j)p Ft(\030)t Fr(j)p Fu(.)39 b(W)-8 b(e)25 b(assume)f(that)h(the)f(in)m(teraction)i (is)e(of)g(the)h(form)e Fj(R)2941 1715 y Fo(d)3006 1748 y Fr(3)i Ft(\030)30 b Fr(7!)25 b Ft(v)s Fu(\()p Ft(\030)t Fu(\))h(=)244 1787 y Fk(\000)286 1861 y Fu(1)20 b Fr(\000)g(j)p Ft(\030)t Fr(j)536 1828 y Fs(\000)p Fl(2)631 1861 y Fr(j)p Ft(\030)t Fu(\)\()p Ft(\030)t Fr(j)839 1787 y Fk(\001)882 1861 y Ft(v)926 1875 y Fl(0)965 1861 y Fu(\()p Ft(\030)t Fu(\),)31 b(for)g(a)f(certain)i(function)e Fj(R)2076 1828 y Fo(d)2141 1861 y Fr(3)25 b Ft(\030)k Fr(7!)c Ft(v)2456 1875 y Fl(0)2496 1861 y Fu(\()p Ft(\030)t Fu(\))h Fr(2)f(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)e Fj(C)3178 1828 y Fo(d)3218 1861 y Fu(\).)380 1973 y(W)-8 b(e)42 b(can)e(no)m(w)g(rep)s (eat)g(almost)h(v)m(erbatim)g(the)f(constructions)h(and)e(statemen)m (ts)j(of)e(the)244 2086 y(previous)g(subsection.)73 b(Note)42 b(that)f(the)h(form)m(ulas)f(\(8.73\),)46 b(\(8.76\))d(and)d(\(8.77\))j (should)d(b)s(e)244 2199 y(replaced)31 b(b)m(y)556 2461 y Ft(v)600 2475 y Fo(\032)641 2461 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(=)979 2387 y Fk(\000)1020 2461 y Fu(1)21 b Fr(\000)f(j)p Ft(!)s Fu(\)\()p Ft(!)s Fr(j)1417 2387 y Fk(\001)1475 2305 y(\()1589 2404 y Ft(p)1645 2340 y Fg(d)p Ff(\000)p Fh(1)p 1645 2353 111 3 v 1685 2394 a(2)1770 2404 y Fu(\(1)h(+)f Ft(\032)p Fu(\()p Ft(p!)s Fu(\)\))2230 2341 y Fh(1)p 2230 2353 31 3 v 2230 2394 a(2)2275 2404 y Ft(v)2319 2418 y Fl(0)2358 2404 y Fu(\()p Ft(p!)s Fu(\))p Ft(;)143 b(p)25 b(>)g Fu(0)1589 2537 y(\()p Fr(\000)p Ft(p)p Fu(\))1786 2473 y Fg(d)p Ff(\000)p Fh(1)p 1787 2486 111 3 v 1827 2527 a(2)1911 2537 y Ft(\032)p Fu(\()p Fr(\000)p Ft(p!)s Fu(\))2215 2474 y Fh(1)p 2215 2486 31 3 v 2215 2527 a(2)2260 2537 y Ft(v)2307 2504 y Fo(?)2304 2561 y Fl(0)2346 2537 y Fu(\()p Fr(\000)p Ft(p!)s Fu(\))p Ft(;)84 b(p)25 b Fr(\024)g Fu(0;)3303 2461 y(\(8.79\))586 2818 y Ft(v)630 2832 y Fo(\032)670 2818 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(=)1008 2745 y Fk(\000)1050 2818 y Fu(1)20 b Fr(\000)g(j)p Ft(!)s Fu(\)\()p Ft(!)s Fr(j)1446 2745 y Fk(\001)1504 2663 y(\()1619 2762 y Ft(p)1675 2698 y Fg(d)p Ff(\000)p Fh(1)p 1675 2711 111 3 v 1715 2752 a(2)1799 2762 y Fu(\(1)h(+)f Ft(\032)p Fu(\()p Ft(p!)s Fu(\)\))2259 2699 y Fh(1)p 2260 2711 31 3 v 2260 2752 a(2)2304 2762 y Ft(v)2348 2776 y Fl(0)2388 2762 y Fu(\()p Ft(p!)s Fu(\))p Ft(;)83 b(p)25 b(>)g Fu(0)1619 2894 y(\()p Fr(\000)p Ft(p)p Fu(\))1816 2830 y Fg(d)p Ff(\000)p Fh(1)p 1816 2843 111 3 v 1856 2884 a(2)1940 2894 y Ft(\032)p Fu(\()p Ft(p!)s Fu(\))2173 2831 y Fh(1)p 2174 2843 31 3 v 2174 2884 a(2)2218 2894 y Ft(v)2262 2908 y Fl(0)2302 2894 y Fu(\()p Ft(p!)s Fu(\))p Ft(;)169 b(p)25 b Fr(\024)g Fu(0;)3303 2818 y(\(8.80\))884 3152 y Ft(v)928 3167 y Fo(\014)976 3152 y Fu(\()p Ft(p;)15 b(!)s Fu(\))26 b(=)1314 3078 y Fk(\000)1355 3152 y Fu(1)21 b Fr(\000)f(j)p Ft(!)s Fu(\)\()p Ft(!)s Fr(j)1752 3078 y Fk(\001)1810 3023 y(\022)2031 3090 y Ft(p)p 1886 3131 335 4 v 1886 3214 a Fu(1)h Fr(\000)f Fu(e)2083 3188 y Fs(\000)p Fo(\014)s(p)2231 3023 y Fk(\023)2308 3019 y Fh(1)p 2308 3031 31 3 v 2308 3072 a(2)2368 3152 y Fr(j)p Ft(p)p Fr(j)2474 3087 y Fg(d)p 2474 3099 33 3 v 2475 3140 a Fh(2)2516 3114 y Fs(\000)p Fl(1)2610 3152 y Ft(v)2654 3166 y Fl(0)2694 3152 y Fu(\()p Ft(p!)s Fu(\))p Ft(:)408 b Fu(\(8.81\))244 3390 y(The)34 b(condition)h (\(8.78\))i(can)e(b)s(e)f(replaced)h(b)m(y)g(demanding)e(that)j Fj(R)c Fr(3)g Ft(p)g Fr(7!)g(j)p Ft(p)p Fr(j)3066 3327 y Fg(d)p 3066 3339 V 3067 3381 a Fh(2)3108 3355 y Fs(\000)p Fl(1)3202 3390 y Ft(v)3246 3404 y Fl(0)3286 3390 y Fu(\()p Ft(p)15 b Fr(\001)p Fu(\))33 b Fr(2)244 3503 y(B)s Fu(\()p Fr(K)q Ft(;)15 b Fr(K)22 b(\012)e Ft(L)696 3470 y Fl(2)736 3503 y Fu(\()p Ft(S)832 3470 y Fo(d)p Fs(\000)p Fl(1)963 3503 y Fu(\)\))31 b(is)f Ft(n)g Fu(times)h(di\013eren)m(tiable)h(and) 922 3611 y Fk(Z)1029 3735 y Fr(k)p Ft(@)1127 3697 y Fo(j)1122 3757 y(p)1164 3735 y Fr(j)p Ft(p)p Fr(j)1270 3670 y Fg(d)p 1270 3682 V 1271 3723 a Fh(2)1312 3697 y Fs(\000)p Fl(1)1406 3735 y Fr(h)p Ft(p)p Fr(i)1522 3697 y Fl(1)p Fo(=)p Fl(2)1633 3735 y Ft(v)1677 3749 y Fl(0)1716 3735 y Fu(\()p Ft(p!)s Fu(\))p Fr(k)1937 3697 y Fl(2)1977 3735 y Fu(d)p Ft(p)p Fu(d)p Ft(!)27 b(<)e Fr(1)p Ft(;)107 b(j)30 b Fu(=)25 b(0)p Ft(;)15 b(n:)447 b Fu(\(8.82\))244 4052 y Fq(References)244 4255 y Fu([AH])140 b(Arai,)36 b(A,)e(Hirok)-5 b(a)m(w)m(a,)37 b(M.:)48 b(On)33 b(the)h(existence)h(and)e(uniqueness)g(of)h(ground)e (states)570 4368 y(of)f(the)g(spin-b)s(oson)e(Hamiltonian,)j(J.)e(F)-8 b(unc.)31 b(Anal.)g Fv(151)p Fu(,)g(455)h(\(1997\).)244 4517 y([Ar])172 b(Araki,)40 b(H.:)55 b(Relativ)m(e)40 b(Hamiltonian)f(for)e(faithful)g(normal)h(states)g(of)g(a)g(v)m(on)f (Neu-)570 4630 y(mann)30 b(algebra,)i(Pub.)d(R.I.M.S.)i(Ky)m(oto)h (Univ.)e Fv(9)p Fu(,)h(165)h(\(1973\).)244 4780 y([A)-10 b(W])125 b(Araki,)50 b(H.,)g(W)-8 b(o)s(o)s(ds,)49 b(E.J.:)71 b(Represen)m(tations)47 b(of)e(the)h(canonical)h(comm)m(utation)570 4893 y(relations)34 b(describing)e(a)h(nonrelativistic)i(in\014nite)d (free)h(Bose)h(gas,)g(J.)e(Math.)i(Ph)m(ys.)570 5005 y Fv(4)p Fu(,)d(637)h(\(1963\).)244 5155 y([BFS1])57 b(Bac)m(h,)30 b(V.,)e(F)-8 b(r\177)-45 b(ohlic)m(h,)29 b(J.,)f(Sigal,)h(I.:)39 b(Quan)m(tum)27 b(electro)s(dynamics)h(of)f (con\014ned)g(non-)570 5268 y(relativistic)33 b(particles,)f(Adv.)e (Math.)h Fv(137)p Fu(,)h(299)f(\(1998\).)p eop %%Page: 53 53 53 52 bop 3682 100 a Fn(53)244 407 y Fu([BFS2])57 b(Bac)m(h,)31 b(V.,)e(F)-8 b(r\177)-45 b(ohlic)m(h,)31 b(J.,)e(Sigal,)h(I.:)40 b(Return)28 b(to)h(equilibrium,)g(Journ.)e(Math.)j(Ph)m(ys.)570 520 y Fv(41)p Fu(,)i(3985)g(\(2000\).)244 673 y([BFSS])51 b(Bac)m(h,)38 b(V.,)e(F)-8 b(r\177)-45 b(ohlic)m(h,)38 b(J.,)e(Sigal,)h(I.,)f(So\013er,)h(A.:)50 b(P)m(ositiv)m(e)37 b(comm)m(utators)f(and)e(the)570 786 y(sp)s(ectrum)22 b(of)h(P)m(auli-Fierz)i(Hamiltonian)f(of)f(atoms)g(and)f(molecules,)k (Comm.)d(Math.)570 899 y(Ph)m(ys.)31 b Fv(207)p Fu(,)g(557)h(\(1999\).) 244 1052 y([BR1])100 b(Brattelli,)34 b(O.,)d(Robinson)f(D.)i(W.:)42 b Fm(Op)-5 b(er)g(ator)35 b(A)n(lgebr)-5 b(as)33 b(and)h(Quantum)g (Statistic)-5 b(al)570 1165 y(Me)g(chanics,)33 b(V)-7 b(olume)33 b(1)p Fu(,)e(Springer-V)-8 b(erlag,)32 b(Berlin,)e(second)h (edition)g(1987.)244 1318 y([BR2])100 b(Brattelli,)34 b(O.,)d(Robinson)f(D.)i(W.:)42 b Fm(Op)-5 b(er)g(ator)35 b(A)n(lgebr)-5 b(as)33 b(and)h(Quantum)g(Statistic)-5 b(al)570 1431 y(Me)g(chanics,)33 b(V)-7 b(olume)33 b(2)p Fu(,)e(Springer-V)-8 b(erlag,)32 b(Berlin,)e(second)h(edition)g(1996.) 244 1584 y([Da])162 b(Da)m(vies,)49 b(E.)43 b(B:)h(Mark)m(o)m(vian)h (master)f(equations,)j(Comm.)c(Math.)h(Ph)m(ys.)g Fv(39)p Fu(,)j(91)570 1697 y(\(1974\).)244 1850 y([DG])136 b(Derezi)s(\023)-48 b(nski,)32 b(J.,)f(G)m(\023)-43 b(erard,)31 b(C.:)41 b(Asymptotic)31 b(completeness)g(in)f(quan)m(tum)g(\014eld)g(the-)570 1963 y(ory)-8 b(.)31 b(Massiv)m(e)h(P)m(auli-Fierz)h(Hamiltonians,)f (Rev.)e(Math.)i(Ph)m(ys.)e Fv(11)p Fu(,)h(383)h(\(1999\).)244 2116 y([DJ1])115 b(Derezi)s(\023)-48 b(nski,)34 b(J.,)f(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)35 b(V.:)44 b(Sp)s(ectral)33 b(theory)f(of)g(P)m(auli-Fierz)j(op)s(erators,)e(Journ.)570 2229 y(F)-8 b(unc.)31 b(Anal.)g Fv(180)g Fu(,)g(241)g(\(2001\).)244 2382 y([DJ2])115 b(Derezi)s(\023)-48 b(nski,)32 b(J.,)f(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)32 b(V.:)41 b(In)30 b(preparation.)244 2535 y([DJP])98 b(Derezi)s(\023)-48 b(nski,)63 b(J.,)f(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)63 b(V.,)f(Pillet,)h(C.-A.:)90 b(P)m(erturbation)56 b(theory)f(of)g Ft(W)3466 2502 y Fs(\003)3505 2535 y Fu(-)570 2648 y(dynamics,)31 b(Liouvilleans)g(and)f (KMS-states,)h(to)h(app)s(ear)d(in)h(Rev.)h(Math.)g(Ph)m(ys.)244 2801 y([FNV])81 b(F)-8 b(annes,)39 b(M.,)g(Nac)m(h)m(tergale,)i(B.,)e (V)-8 b(erb)s(eure,)38 b(A.:)53 b(The)36 b(equilibrium)f(states)j(of)e (the)570 2914 y(spin-b)s(oson)29 b(mo)s(del,)i(Comm.)f(Math.)h(Ph)m (ys.)g Fv(114)p Fu(,)g(537)h(\(1988\).)244 3067 y([F)-8 b(r])189 b(F)-8 b(rigerio,)37 b(A.:)48 b(Stationary)34 b(states)h(of)f(quan)m(tum)f(dynamical)h(semigroups,)h(Comm.)570 3180 y(Math.)d(Ph)m(ys.)e Fv(63)p Fu(,)h(269)h(\(1978\).)244 3333 y([F)m(GS])98 b(F)-8 b(r\177)-45 b(ohlic)m(h,)57 b(J.,)e(Griesemer,)h(M.,)g(Sc)m(hlein,)g(B.:)80 b(Asymptotic)52 b(completeness)f(for)570 3446 y(Ra)m(yleigh)32 b(scattering,)h(Ann.)c (Henri)i(P)m(oincar)m(\023)-43 b(e,)33 b Fv(3)p Fu(,)d(107)i(\(2002\).) 244 3599 y([Ge])165 b(Gerard,)50 b(C.:)71 b(On)45 b(the)h(existence)h (of)f(ground)e(states)j(for)f(massless)g(P)m(auli-Fierz)570 3712 y(Hamiltonians,)32 b(Ann.)e(Henri)g(P)m(oincar)m(\023)-43 b(e)63 b Fv(1)p Fu(,)31 b(443)h(\(2000\).)244 3865 y([HS1])112 b(H)s(\177)-48 b(ubner,)48 b(M.,)h(Sp)s(ohn,)e(S.:)69 b(Radiativ)m(e)47 b(deca)m(y:)71 b(non-p)s(erturbativ)m(e)45 b(approac)m(hes,)570 3978 y(Rev.)31 b(Math.)g(Ph)m(ys.)g Fv(8)p Fu(,)g(549)g(\(1996\).)244 4131 y([HS2])112 b(H)s(\177)-48 b(ubner,)28 b(M.,)i(Sp)s(ohn,)e(S.:)39 b(Sp)s(ectral)29 b(prop)s(erties)f(of)h(the)g(spin-b)s(oson)f(Hamiltonian,)570 4244 y(Ann.)i(Inst.)g(Henri)h(P)m(oincar)m(\023)-43 b(e)32 b Fv(62)p Fu(,)f(289)h(\(1995\).)244 4397 y([Ja])184 b(Jadczyk,)43 b(A.)e(Z.:)60 b(On)39 b(some)h(groups)f(of)i (automorphisms)e(of)h(v)m(on)g(Neumann)g(al-)570 4510 y(gebras)k(with)g(cyclic)h(and)e(separating)h(v)m(ectors,)49 b(Comm.)44 b(Math.)g(Ph)m(ys.)g Fv(13)p Fu(,)j(142)570 4623 y(\(1969\).)244 4776 y([JP1])122 b(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)28 b(V.,)e(Pillet,)h(C.-A.:)38 b(On)23 b(a)i(mo)s(del)e(for)h(quan)m(tum)g(friction)h(I)s(I:)e(F)-8 b(ermi's)25 b(golden)570 4889 y(rule)33 b(and)f(dynamics)h(at)h(p)s (ositiv)m(e)g(temp)s(erature,)f(Comm.)g(Math.)h(Ph)m(ys.)f Fv(176)p Fu(,)h(619)570 5002 y(\(1996\).)244 5155 y([JP2])122 b(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)49 b(V.,)e(Pillet,)i(C.-A.:)67 b(On)43 b(a)h(mo)s(del)f(for)g(quan)m(tum)g(friction)h(I)s(I)s(I:)f (Ergo)s(dic)570 5268 y(prop)s(erties)30 b(of)h(the)f(spin-b)s(oson)f (system,)i(Comm.)f(Math.)h(Ph)m(ys.)g Fv(178)p Fu(,)g(627)h(\(1996\).)p eop %%Page: 54 54 54 53 bop 3682 100 a Fn(54)244 407 y Fu([JP3])122 b(Jak)-5 b(\024)-40 b(si)m(\023)d(c,)35 b(V.,)f(Pillet,)h(C.-A.:)45 b(Non-equilibrium)33 b(steady)g(states)g(for)g(\014nite)f(quan)m(tum) 570 520 y(systems)22 b(coupled)g(to)h(thermal)f(reserv)m(oirs,)j(Comm.) c(Math.)i(Ph)m(ys.)f Fv(226)p Fu(,)j(131)e(\(2002\).)244 674 y([M])193 b(Merkli,)40 b(M.:)54 b(P)m(ositiv)m(e)39 b(comm)m(utators)f(in)e(non-equilibrium)g(quan)m(tum)g(statistical)570 787 y(mec)m(hanics,)c(Comm.)e(Math.)h(Ph)m(ys.)g Fv(223)p Fu(,)g(327)h(\(2001\).)244 941 y([PF])155 b(P)m(auli)32 b(W.,)f(Fierz,)g(M.:)42 b(Nuo)m(v)m(o)31 b(Cimen)m(to)h Fv(15)p Fu(,)f(167)g(\(1938\).)244 1096 y([O)m(Y])140 b(Ok)-5 b(amoto,)46 b(T.,)f(Y)-8 b(a)5 b(jima,)46 b(K.:)63 b(Complex)42 b(scaling)h(tec)m(hnique)f(in)f(non-relativistic)570 1209 y(qed,)31 b(Ann.)f(Inst.)g(Henri)g(P)m(oincar)m(\023)-43 b(e)33 b Fv(42)p Fu(,)e(391)g(\(1981\).)244 1363 y([Ru])158 b(Ruelle,)26 b(D.:)37 b(Natural)24 b(nonequilibrium)d(states)j(in)f (quan)m(tum)f(statistical)j(mec)m(hanics,)570 1476 y(J.)31 b(Stat.)g(Ph)m(ys.)f Fv(98)p Fu(,)h(57)g(\(2000\).)244 1631 y([Sk])177 b(Skibsted,)44 b(E.:)62 b(Sp)s(ectral)42 b(analysis)f(of)h Ft(N)10 b Fu(-b)s(o)s(dy)40 b(systems)h(coupled)g(to) h(a)g(b)s(osonic)570 1743 y(system,)31 b(Rev.)g(Math.)g(Ph)m(ys.)f Fv(10)p Fu(,)i(989)f(\(1998\).)244 1898 y([Sp1])129 b(Sp)s(ohn,)38 b(H.:)55 b(An)38 b(algebraic)h(condition)f(for)f(the)h(approac)m(h)g (to)g(equilibrium)f(of)h(an)570 2011 y(op)s(en)30 b Ft(N)10 b Fu(-lev)m(el)32 b(system,)f(Lett.)h(Math.)f(Ph)m(ys.)f Fv(2)p Fu(,)h(33)g(\(1977\).)244 2165 y([Sp2])129 b(Sp)s(ohn,)40 b(H.:)57 b(Ground)38 b(states)i(of)f(the)g(spin-b)s(oson)f (Hamiltonian,)k(Comm.)d(Math.)570 2278 y(Ph)m(ys.)31 b Fv(123)p Fu(,)g(277)h(\(1989\).)244 2433 y([Sp3])129 b(Sp)s(ohn,)37 b(H.:)54 b(Ground)36 b(state)j(of)e(a)g(quan)m(tum)g (particle)h(coupled)f(to)g(a)h(scalar)g(Bose)570 2545 y(\014eld,)31 b(Lett.)g(Math.)g(Ph)m(ys.)g Fv(44)p Fu(,)g(9)f (\(1998\).)244 2700 y([St])190 b(Stratila,)45 b(S.:)62 b Fm(Mo)-5 b(dular)44 b(The)-5 b(ory)44 b(in)e(Op)-5 b(er)g(ator)45 b(A)n(lgebr)-5 b(as)p Fu(,)44 b(Abacus)c(Press,)k(T)-8 b(urn-)570 2813 y(bridge)30 b(W)-8 b(ells)32 b(1981.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF