Content-Type: multipart/mixed; boundary="-------------0504200913200" This is a multi-part message in MIME format. ---------------0504200913200 Content-Type: text/plain; name="05-142.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-142.comments" Department of Mathematical Methods in Physics, Warsaw University, Hoza 74, 00-682, Warszawa, Poland. email: jan.derezinski@fuw.edu.pl, rafal.fruboes@fuw.edu.pl ---------------0504200913200 Content-Type: text/plain; name="05-142.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-142.keywords" strongly continuous semigroups, weakly* continuous semigroups, weak coupling limit, perturbation theory, Liouvillean, Level Shift Operator, Detailed Balance Condition, completely positive semigroups ---------------0504200913200 Content-Type: application/postscript; name="grenoble.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="grenoble.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: grenoble.dvi %%Pages: 50 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman CMMI10 CMSY7 CMR7 MSBM10 CMMI7 %%+ CMR10 CMSY10 MSBM7 CMR5 CMMI5 EUFM10 CMSY5 CMBX10 CMEX10 %%+ Times-Italic TeX-cmex7 CMMIB10 CMBSY7 CMBX7 LASY10 CMR9 CMMI9 CMSY6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips grenoble.dvi -o grenoble.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.04.14:1057 %%BeginProcSet: texc.pro %! 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TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF7499F439A6EBE9BF559D2EE0571 CE54AEC463244B0F8EAB9E96CB18BD39259CC1FEC10F47FB56A38588CE634209 8F77258607212EE1DCA4F0667B152875B2CF5AC44B930B888ACD9D4B55662542 71239286D82E14CAABE7276AB199E2429C4C3BC32713106A10F5F16C8045A580 86EE21E7783B70FAE03D8D47B5AA13A881D478232DD65DBCD1EB9811C440E362 527EF73FC86FE664ACED80DCD6806CFD932BDEE102B89C22F423992249FC2273 F39C59AEF75B2088527AA973C71A6B134D26EF1ABAB75721971A0E4E52639DA9 2E1C3B2A6FB552CA834F6443E0628DD9CE69E92DA0B9B8ACAF3641FA0A7F1126 8DF8803E683ACCCCDE88C9F6C1838BCE7E8B56A0BC8C5F0300D81479A5087FFD B8B66527B87F7977C31A54E0506C6D33EBC902841AB7B8D75BC8ADE5397905EF BCB96AE4B57D308DCF0F74A93177F2DDF3486642A43834DB5B123CFA402E4BA1 6EB4C27AF21C96932E05B79CF951354FF66668C6503CA6FD2089A91A8D 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % 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 (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 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 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBSY7 %!PS-AdobeFont-1.1: CMBSY7 001.000 %%CreationDate: 1992 Oct 22 12:18:11 % Computer Modern fonts were designed by Donald E. Knuth. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (001.000) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put readonly def /FontBBox{0 -927 1542 750}readonly def /UniqueID 5032008 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F086C44216EBE57F4BA37B479BF1E5A5139D8 91F3E6DDA157B25D359C5E7FE4CFB264DF1707BB6497D3E074CFF95D9FD269B5 0B1566D1161680C46D1548BBF384EF8653AF29FD474EA2336C876979AC00AE18 DC87DC0DDD3164B96FC6C3ED826EBAAED383BB3EC5044BA84A0426B87ED04C9F 4B3EDFC734C241D9B3D2321619F870FE68BB4BA7C060DBE8FBD12F641E7709F2 C430491944B78B1E59580798F1B40CAC4D59DE95941217EF1A861DDB0156D5C3 349AAB13FEFF6C646F6401550F5853BC09B267A6C63639228DF55BE60A99E4A5 308C616892AA0DC96ADB7CD7AB8AEFA859F69B587B61930596A46A905661E4BA DAB5E1CF15C94CF060B7FA600B17162AEC2DEE64A156B3F87248E7A7F88C9154 8C494273B33483BEDF0BEA4DF6A19941F52AA04717623ACBDE926B4851ED05A0 28698A1C5AE63A46EF473A4F3DCAF3E73C4FAF0C1077EE7A6504074C0D77947F 940B16425B3F5834763732F26D3385774A1CACA70C07F58887A0301D1BE530A7 D7AC00A0664617A0CBA9F6281FA4B9168DBA3C1EDCF915778351E6BD8A9CE7E5 3E56F2FFE0969E1CFFC83F07C01A3873EE1CE4E124565E8F493BE4FAA0A5D099 A116CEE4EC6C8CB2E93B42771FFF67680A56501A201E12AFDA8448ABE80BEB9B 80428F48753C4EFB174B693C69DB81CEF0A0B75C53A9D5B4C5F26FA58059A324 8D4E9D4E9C54AB8F9C21CD66B9B259F9C797559384A653DD43ED4B9C2110BB5B C3A6370921186AEE29EB4E62793ECD96935C3D9D89DC6AAECFB745BABDB570FD E7E6248B6ED9C09EDA896ECFCAEECE8C1E8BE20BDE6F3558EF5A32ED390ABF86 3A585DF34F2B8B9567778BF51A1BF9C1018AEEF42FF1F9AAB1F9F73627F6C7CF 2C858DF92CE5E8A8EAB1126424BA0777BB85D0498436ECB7D565A500AD4BA9FA 1BBEFB1A7B6DC9C4699E61FBFBC58C2CA12A4D9DCD8D7C6B2407CD0D2CCE800F 74C394AB27F5CDECE7237F63F0F95AFA06009AD8B44AC7BB29BB8246D684141D 7A7951CF6A1675741AF829C5139834404C5DE913CF4F572997D509030422A3AC 9F6C171106714E0DC7740989520047DA5CC6682F1FB72DFB76A070B4811A195F 10E56613B7AA14A31DB0600E4E8B48991A5B7CEB4834588811C7024C27C46E4F 375018C3689277A1F563379177AF3B2DCD5213115F40A5F023DC72D9E56D8B91 41DD44A4E9CCB7DF441B014E76D394A5251B25CB71BA29655E9C822BF66CB142 BC585892B308C800851F7802341E2957CA01F5CE297094346365B267C3548CC6 18D7F48904B32AD8394044E4216916DD8E0E03279F062F734F9572FFCF3D297D A02F2801E5A7522AE16A7F5FC52AA4D290759C0EFFD77D63E353B8307383D35F EEB5AC87972A12C4D4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % 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 (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 def currentdict end currentfile eexec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def /version (001.001) def end readonly def /FontName /TeX-cmex7 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 101 /e put readonly def /FontBBox {-14 -2954 1627 771} readonly def /UniqueID 4314415 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 101 /tildewide put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 1458C43CD471029C362871D2FB69E0E5E617BC06F3B8621E3528E4B47E731AE1 0B406A3260424830B3C86A135457A566A1A716189B86883C171DA05829CE2E37 C020AAAD5205DB1A9C1A6A56FFD9008D511B4453C4B33F6B0E977B7C760F8816 4A6F13E41191AA13358515D902E653B81AEF60FE07C4AB673435504C2C376BB2 6BAB038083E5073851178BEB6AA92FE582EFCFA3A22C6795DCAAFAFCA0F3E79D 270BB779E2071AFDF30D3F32082777AEDF90940B6DAA1F34BAC0E9AA08811281 8D4261185EC039EB930024E010615A972BBD0B8EAA09B3931D630F1215D88859 5803A82F87BF0D6B8C64E23358C903C8E022AF457F97845B948D174E063E40F3 FA1F8C2CF0C7C49EC374E22B0A865A90FDC5CE762D643812D9597AC4CA872A23 9CA95A4001607D1E336A8D3C10BBC8BA47B1DD8DAC76B3F7B4E34DA11B11F27E 5CC1B7D0249F7924540CC5E194DD545B39C060A69CFC2145AFED8242A9FEA65F ED03FC2DA7CDD3ADC394D32F22802C8EC818847BA08DE0654277ED11E5C356A7 1E590527E7A231A6AE8B779B88648B69BE95F879361588325BB6F7E01D44C6A1 8422BACAA4C2B6B4802A4452C7A95F232A5B3B9DE2E78780A95F58FF97C276B4 691C2E88AAD4FF004B752120BB3453C12CC6BEDD34165F33D1DACCE965BF81A8 8518AD8EAAD03CCF67CA8BA8A2B78749D66ED2726BACAE757CB7C3BAE0532181 AC95975182CF3EF7B32636EA7C3ABCFD2BCFD2325DE4E377278A34A790804F1B 27FBDEF77921130ED82D188D97D419AC2486AAB33B073D9EDA1ABF2B2339F56C 02B150C69E56B6C84E1C8590BB610F46CFBA0A500F5819E525B9A5670A4A0711 5C03A06ADE01EB3D493BE2C7EFD32AD2E84A3F03496290A5E90AB316719B6853 2000F2CA2EF1DFE752635BB8B157F80ABBD2A99EB5883A12F989B99A443ADB6A 6082D4E323728D937FEB276896C3EB724994AA3684D7BB762E14FA60C718C6C2 7B3D0AE951522B6AF4B37341999A484C533452976F4738ADC898679E7853EF70 C9F09173A09A269FFCC31BF21AB67770D0451A52DE52403AE8F572FB4B2E6031 5DCC551F866B43B1FB13D064D036DF9DD4E04FD15D4463606DEB05B258AE4EB0 36B37DB7A65D1AA14181C7BB752DB92BB61AC9D0F2F1D4B95EEB49B9AD361AA9 6E0B59FC58FB91EB353F690B7D16F055B1CB751F664947A467182F8E13847382 FCAE067B52B6B3231088A085B2F9EA7724E479D6AFE0F848AEBBA64FB267F1D5 1E4C334D98AAF3097E7C7C43D05BAB3403DA40FA145CF468034DC766DEEA1F23 C5EB7F9B27287BF9C18AE1C037E5BB8A46A756403F37FF2E4143B58056B73CE3 E5060D75646B93EF391F535C0A6292797E811D9401D09F3977CAFB19DBDF85B7 A55FC31B1BA0F7C93DE3652D26A41028AA495D6337AF1DDF5165953C2F05DD35 F8A8C4A048FD7C147869B6C7F4A3BD818EEFA841669423870D0D0B216A04ED2C F3697EF5C16739C1EFA8EE2FD1CB1A7F670F33CD6F7A556031F04D25AFFB91B5 BCB95E142551259660042B0B16D67F9EB5AC214B43226C440526C8A8602E7CF0 3506ED85E2763EC22F25857ACD663B294B75BB4108C989D32B9495A4CAADAE41 8839D3453D7B0E791B641F624287E799BF80EEB95BDE631C1E6F25EFA9C56690 3B278E47DC6773871AC41D89A9DFCD64019D5D33555B262B0A16E9294A00F3DD D69BEC072C7A5AA53C160309FA1F647E1416C31AF222F149DCE88A40E60CBF36 473899391068749582D2FE95B3CFBEC722B9EE11A43E6CC762B66D9E025FB485 5DBECA20CCEAB52FE07EEBA566886EE75237142EADEC7808238C75ECE203F64D 454482A882E2AA825F073612CFCB8CB1DC25BDE674DB06E82798381EA1BD3E36 5F 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % 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 (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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%%CreationDate: 1992 Feb 19 19:55:02 % 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. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 69 /E put dup 74 /J put dup 76 /L put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)130 b(8)523 2657 y(2.5)86 b(LSO)21 b(in)f(Hilbert)g(spaces)j(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)g(.)g(.)h(.)130 b(8)523 2756 y(2.6)86 b(The)20 b(choice)f(of)h(the)g(projection)f Fw(P)25 b FC(.)19 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) g(.)h(.)130 b(9)523 2856 y(2.7)86 b(Three)19 b(kinds)h(of)g(the)g (Fermi)g(Golden)f(Rule)40 b(.)18 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.) h(.)130 b(9)523 2997 y FA(3)149 b(W)-5 b(eak)20 b(coupling)g(limit)45 b FC(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g (.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.) f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(11)523 3138 y(3.1)d(Stationary)19 b(and)h(time-dependent)d(weak)j(coupling)e(limit) 35 b(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g (.)h(.)89 b(11)523 3238 y(3.2)d(Proof)19 b(of)h(the)g(stationary)g (weak)f(coupling)g(limit)35 b(.)18 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.) h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(14)523 3337 y(3.3)d(Spectral)20 b(a)n(v)o(eraging)h(.)e(.)f(.)g(.)g (.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.) f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(17)523 3437 y(3.4)d(Second)19 b(order)g(asymptotics)h(of)f(e)n(v)n(olution)g(with)h(the)h(\002rst)g (order)e(term)42 b(.)18 b(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(19)523 3537 y(3.5)d(Proof)19 b(of)h(time)g(dependent)e(weak)i (coupling)f(limit)14 b(.)k(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(21)523 3636 y(3.6)d(Proof)19 b(of)h(the)g(coincidence)f(of)h Fz(M)1720 3648 y Fx(st)1796 3636 y FC(and)g Fz(M)2018 3648 y Fx(dyn)2151 3636 y FC(with)g(the)g(LSO)12 b(.)18 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(21)523 3778 y FA(4)149 b(Completely)20 b(positi)o(v)o(e)g(semigr)o (oups)28 b FC(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.) f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g (.)h(.)89 b(22)523 3919 y(4.1)d(Completely)19 b(positi)n(v)o(e)h(maps) 13 b(.)k(.)h(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.) g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)89 b(22)523 4018 y(4.2)d(Stinespring)19 b(representation)f(of)i(a)h(completely)d(positi)n(v)o(e)i(map)44 b(.)18 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(23)523 4118 y(4.3)d(Completely)19 b(positi)n(v)o(e)h(semigroups)38 b(.)18 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(24)523 4218 y(4.4)d(Standard)19 b(Detailed)h(Balance)g(Condition)29 b(.)18 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.) g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(25)523 4317 y(4.5)d(Detailed)24 b(Balance)g(Condition)f(in)h(the)g (sense)g(of)g(Alicki-Frigerio-Gorini-)714 4417 y(K)m(ossak)o(o)n (wski-V)-9 b(erri)11 b(.)16 b(.)i(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(26)523 4558 y FA(5)149 b(Small)21 b(quantum)f(system)h(interacting)e (with)i(r)o(eser)o(v)o(oir)43 b FC(.)18 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(27)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 523 100 a FB(2)230 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FC(5.1)86 b Fz(W)804 252 y Fy(\003)842 282 y FC(-algebras)13 b(.)k(.)h(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.) g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)g(.)h(.)89 b(28)523 382 y(5.2)d(Algebraic)19 b(description)40 b(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.) g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(28)523 482 y(5.3)d (Semistandard)19 b(representation)12 b(.)j(.)j(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(29)523 581 y(5.4)d(Standard)19 b(representation)12 b(.)j(.)k(.)f(.)g(.)g(.)g (.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.) g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)89 b(29)523 722 y FA(6)149 b(T)-6 b(w)o(o)21 b(applications)e(of)h (the)h(F)n(ermi)f(Golden)g(Rule)h(to)f(open)g(quantum)h(systems)76 b FC(30)523 863 y(6.1)86 b(LSO)21 b(for)e(the)h(reduced)f(dynamics)14 b(.)j(.)h(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f (.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.) h(.)89 b(31)523 963 y(6.2)d(LSO)21 b(for)e(the)h(Liouvillean)35 b(.)19 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.) f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(33)523 1063 y(6.3)d(Relationship)26 b(between)f(the)h(Da)n(vies)h(generator)d(and)i(the)g(LSO)h(for)e(the) 714 1162 y(Liouvillean)19 b(in)h(thermal)f(case.)28 b(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) g(.)h(.)89 b(34)523 1262 y(6.4)d(Explicit)20 b(formula)e(for)i(the)g (Da)n(vies)h(generator)16 b(.)f(.)k(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g (.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(36)523 1362 y(6.5)d(Explicit)20 b(formulas)f(for)g(LSO)i(for)e(the)h (Liouvillean)k(.)18 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(38)523 1461 y(6.6)d(Identities)20 b(using)f(the)i(\002bered)e(representation)12 b(.)j(.)k(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(40)523 1602 y FA(7)149 b(F)n(ermi)20 b(Golden)g(Rule)h(f)n(or)f(a)g(composite)g(r)o (eser)o(v)o(oir)36 b FC(.)19 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(41)523 1743 y(7.1)d(LSO)21 b(for)e(a)i(sum)f(of)g(perturbations)30 b(.)18 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.) g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)89 b(41)523 1843 y(7.2)d(Multiple)20 b(reserv)n(oirs)47 b(.)18 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.) g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)89 b(43)523 1943 y(7.3)d(LSO)21 b(for)e(the)h(reduced)f(dynamics)g(in)h(the)g(case) h(of)f(a)h(composite)e(reserv)n(oir)33 b(.)18 b(.)g(.)g(.)h(.)89 b(44)523 2042 y(7.4)d(LSO)21 b(for)e(the)h(Lio)o(villean)f(in)h(the)h (case)f(of)g(a)h(composite)e(reserv)n(oir)j(.)16 b(.)j(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)89 b(44)523 2183 y FA(A)131 b(A)n(ppendix)21 b(\226)f(one-parameter)e(semigr)o(oups)47 b FC(.)19 b(.)f(.)g(.)g(.)g (.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)89 b(45)523 2325 y FA(Refer)o(ences)15 b FC(.)i(.)h(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h (.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.) g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)g(.)g(.)h(.)89 b(48)523 2640 y Fv(1)41 b(Intr)n(oduction)523 2839 y FC(These)22 b(lecture)f(notes)h(are)g(an)g(e)o(xpanded)d(v)o (ersion)i(of)g(the)h(lectures)g(gi)n(v)o(en)e(by)i(the)g(\002rst)g (author)523 2939 y(in)e(the)g(summer)f(school)g(\224Open)g(Quantum)f (Systems\224)i(held)f(in)h(Grenoble,)e(June)i(16\227July)e(4,)523 3039 y(2003.)g(W)-7 b(e)22 b(are)e(grateful)e(to)j(Stphane)e(Attal,)h (Alain)g(Jo)o(ye,)g(and)f(Claude-Alain)g(Pillet)i(for)e(their)523 3138 y(hospitality)g(and)h(in)m(vitation)f(to)h(speak.)523 3263 y FA(Ackno)o(wledgments.)41 b FC(The)g(research)f(of)i(both)e (authors)g(w)o(as)i(partly)f(supported)e(by)i(the)523 3362 y(EU)g(Postdoctoral)e(T)m(raining)g(Program)g(HPRN-CT)-8 b(-2002-0277)37 b(and)j(the)g(Polish)h(grants)523 3462 y(SPUB127)27 b(and)g(2)g(P03A)g(027)g(25.)f(A)i(part)f(of)g(this)h(w)o (ork)e(w)o(as)j(done)d(during)f(a)j(visit)g(of)f(the)523 3562 y(\002rst)21 b(author)e(to)h(Uni)n(v)o(ersity)f(of)h(Montreal)f (and)h(to)g(the)h(Schr)7 b(\250)-35 b(odinger)17 b(Institute)j(in)h(V) -5 b(ienna.)19 b(W)-7 b(e)523 3661 y(ackno)n(wledge)18 b(useful)h(con)m(v)o(ersations)e(with)k(H.)f(Spohn,)e(C.)j(A.)f (Pillet,)h(W)-8 b(.)21 b(A.)f(Maje)n(wski,)g(and)523 3761 y(especially)g(with)g(V)-11 b(.)21 b(Jak)r(\020)-30 b(si)5 b(\264)-33 b(c.)523 4002 y FA(1.1)40 b(F)n(ermi)21 b(Golden)f(Rule)h(and)f(Le)o(v)o(el)h(Shift)f(Operator)f(in)i(an)f (abstract)f(setting)523 4184 y FC(W)-7 b(e)25 b(will)g(use)f(the)g (name)f(\223the)h(Fermi)f(Golden)g(Rule\224)h(to)g(describe)f(the)h (well-kno)n(wn)e(second)523 4284 y(order)d(perturbati)n(v)o(e)e (formula)h(for)h(the)h(shift)g(of)g(eigen)m(v)n(alues)e(of)h(a)i(f)o (amily)e(of)h(operators)e Fw(L)3181 4296 y Fu(\025)3248 4284 y Ft(=)523 4384 y Fw(L)578 4396 y Fx(0)635 4384 y Ft(+)h Fz(\025)p Fw(Q)p FC(.)k(Historically)-5 b(,)20 b(the)i(Fermi)g(Golden)f(Rule)h(can)f(be)h(traced)f(back)g(to)h(the)g (early)f(years)523 4483 y(of)27 b(Quantum)f(Mechanics,)g(and)h(in)h (particular)e(to)h(the)h(f)o(amous)e(paper)h(by)g(Dirac)g([Di].)g(T)-7 b(w)o(o)523 4583 y(\223Golden)26 b(Rules\224)i(describing)d(the)i (second)f(order)g(calculations)g(for)h(scattering)f(amplitudes)523 4682 y(can)20 b(be)g(found)f(in)h(the)g(Fermi)g(lecture)g(notes)g([Fe]) g(on)g(pages)g(142)f(and)g(148.)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)230 b(3)648 282 y FC(In)17 b(its)j(traditional)d(form)g(the)h(Fermi)g(Golden)f(Rule)i(is)g (applied)e(to)h(Hamiltonians)g(of)g(quan-)523 382 y(tum)24 b(systems)h(\226)f(self-adjoint)f(operators)f(on)i(a)h(Hilbert)f (space.)g(A)g(nonzero)f(imaginary)f(shift)523 482 y(of)e(an)f(eigen)m (v)n(alue)f(of)h Fw(L)1230 494 y Fx(0)1288 482 y FC(indicates)h(that)g (the)f(eigen)m(v)n(alue)f(is)j(unstable)e(and)g(that)h(it)g(has)g (turned)523 581 y(into)g(a)h(resonance)d(under)h(the)h(in\003uence)f (of)h(the)g(perturbation)e Fz(\025)p Fw(Q)p FC(.)648 681 y(In)d(our)g(lectures)g(we)h(shall)h(use)e(the)h(term)g(Fermi)f (Golden)g(Rule)h(in)g(a)g(slightly)f(more)g(general)523 780 y(conte)o(xt,)j(not)h(restricted)f(to)i(Hilbert)f(spaces.)g(More)g (precisely)-5 b(,)18 b(we)h(shall)h(be)f(interested)g(in)g(the)523 880 y(case)f(when)g Fw(L)938 892 y Fu(\025)1000 880 y FC(is)h(a)f(generator)e(of)i(a)g(1-parameter)e(group)g(of)h(isometries) h(on)g(a)g(Banach)f(space.)523 980 y(F)o(or)31 b(e)o(xample,)f Fw(L)1057 992 y Fu(\025)1133 980 y FC(could)h(be)g(an)h (anti-self-adjoint)d(operator)h(on)h(a)h(Hilbert)f(space)h(or)f(the)523 1079 y(generator)20 b(of)i(a)g(group)f(of)h Fs(\003)p FC(-automorphisms)c(of)k(a)h Fz(W)2164 1049 y Fy(\003)2202 1079 y FC(-algebra.)d(These)i(tw)o(o)g(special)g(cases)523 1179 y(will)f(be)f(of)g(particular)f(importance)f(for)i(us.)648 1279 y(Note)g(that)h(the)f(spectrum)g(of)g(the)g(generator)f(of)h(a)h (group)e(of)h(isometries)g(is)i(purely)d(imagi-)523 1378 y(nary)-5 b(.)15 b(The)h(shift)h(computed)d(by)i(the)h(Fermi)f(Golden)f (Rule)i(may)f(ha)n(v)o(e)g(a)h(ne)o(gati)n(v)o(e)d(real)j(part)f(and) 523 1478 y(this)k(indicates)g(that)g(the)g(eigen)m(v)n(alue)d(has)j (turned)f(into)h(a)g(resonance.)e(Hence,)h(our)g(con)m(v)o(ention)523 1577 y(dif)n(fers)g(from)h(the)g(traditional)f(one)g(by)h(the)g(f)o (actor)g(of)g Ft(i)p FC(.)648 1677 y(In)g(these)h(lecture)f(notes,)h (we)g(shall)g(discuss)h(se)n(v)o(eral)e(mathematically)f(rigorous)g(v)o (ersions)523 1777 y(of)31 b(the)h(Fermi)f(Golden)f(Rule.)i(In)f(all)h (of)f(them,)g(the)g(central)g(role)g(is)i(played)d(by)h(a)h(certain)523 1876 y(operator)21 b(that)i(we)g(call)h(the)f(Le)n(v)o(el)f(Shift)h (Operator)e(\(LSO\).)i(This)g(operator)e(will)j(encode)d(the)523 1976 y(second)j(order)g(shift)i(of)f(eigen)m(v)n(alues)e(of)i Fw(L)1811 1988 y Fu(\025)1880 1976 y FC(under)f(the)h(in\003uence)g(of) f(the)i(perturbation.)c(T)-7 b(o)523 2076 y(de\002ne)21 b(the)g(LSO)h(for)f Fw(L)1224 2088 y Fu(\025)1293 2076 y Ft(=)k Fw(L)1438 2088 y Fx(0)1495 2076 y Ft(+)18 b Fz(\025)p Fw(Q)p FC(,)k(we)g(need)f(to)g(specify)g(the)g(projection)f Fw(P)h FC(commuting)523 2175 y(with)i Fw(L)749 2187 y Fx(0)810 2175 y FC(\(typically)-5 b(,)20 b(the)j(projection)e(onto)h (the)g(point)g(spectrum)g(of)g Fw(L)2615 2187 y Fx(0)2653 2175 y FC(\))h(and)f(a)h(perturbation)523 2275 y Fw(Q)p FC(.)f(F)o(or)f(the)g(most)h(part,)f(we)h(shall)g(assume)g(that)f Fw(PQP)k Ft(=)g(0)p FC(,)d(which)f(guarantees)f(the)i(absence)523 2374 y(of)c(the)f(\002rst)i(order)d(shift)i(of)g(the)g(eigen)m(v)n (alues.)d(Gi)n(v)o(en)i(the)h(datum)f Ft(\()p Fw(P)p Fz(;)d Fw(L)2609 2386 y Fx(0)2646 2374 y Fz(;)g Fw(Q)p Ft(\))p FC(,)k(we)g(shall)h(de\002ne)523 2474 y(the)h(LSO)h(as)g(a)f (certain)g(operator)e(on)i(the)g(range)g(of)f(the)i(projection)d Fw(P)p FC(.)648 2574 y(W)-7 b(e)26 b(shall)g(describe)e(se)n(v)o(eral)h (rigorous)e(applications)h(of)h(the)g(LSO)h(for)e Ft(\()p Fw(P)p Fz(;)14 b Fw(L)2956 2586 y Fx(0)2994 2574 y Fz(;)g Fw(Q)p Ft(\))p FC(.)25 b(One)523 2673 y(of)30 b(them)g(is)i(the)e (\223weak)g(coupling)f(limit\224,)i(called)f(also)h(the)f(\223v)n(an)g (Ho)o(v)o(e)f(limit\224.)i(\(W)-7 b(e)31 b(will)523 2773 y(not,)k(ho)n(we)n(v)o(er)m(,)e(use)j(the)f(latter)h(name,)f(since)g (it)i(often)d(appears)h(in)h(a)g(dif)n(ferent)e(meaning)523 2873 y(in)c(statistical)h(physics,)e(denoting)g(a)h(special)g(form)f (of)h(the)g(thermodynamical)d(limit\).)i(The)523 2972 y(time-dependent)21 b(form)i(of)h(the)g(weak)g(coupling)e(limit)i(says) h(that)f(the)g(reduced)e(and)i(rescaled)523 3080 y(dynamics)31 b Ft(e)911 3050 y Fy(\000)p Fu(t)p Fr(L)1027 3058 y Fq(0)1059 3050 y Fu(=\025)1132 3025 y Fq(2)1169 3080 y Fw(P)p Ft(e)1257 3050 y Fu(t)p Fr(L)1321 3059 y Fp(\025)1359 3050 y Fu(=\025)1432 3025 y Fq(2)1469 3080 y Fw(P)i FC(con)m(v)o(er)o(ges)c(to)j(the)g (semigroup)e(generated)g(by)i(the)g(LSO.)523 3180 y(The)24 b(time)g(dependent)d(weak)j(coupling)e(limit)i(in)g(its)h(abstract)f (form)f(w)o(as)i(pro)o(v)o(en)c(by)j(Da)n(vies)523 3280 y([Da1)o(,)d(Da2)o(,)f(Da3].)g(In)g(our)f(lectures)h(we)h(gi)n(v)o(e)e (a)i(detailed)e(e)o(xposition)g(of)h(his)g(results.)648 3379 y(W)-7 b(e)20 b(describe)e(also)h(the)g(so-called)f (\223stationary)g(weak)h(coupling)e(limit\224,)i(based)g(on)f(the)h (re-)523 3479 y(cent)e(w)o(ork)g([DF2)o(].)g(The)g(stationary)g(weak)g (coupling)e(limit)j(says)g(that)f(appropriately)e(rescaled)523 3579 y(and)20 b(reduced)e(resolv)o(ent)h(of)h Fw(L)1416 3591 y Fu(\025)1481 3579 y FC(con)m(v)o(er)o(ges)d(to)j(the)g(resolv)o (ent)f(of)h(the)g(LSO.)648 3678 y(The)29 b(LSO)h(has)g(a)h(number)d(of) h(other)h(important)e(applications.)g(It)i(can)g(be)g(used)g(to)g(de-) 523 3778 y(scribe)24 b(approximate)d(location)i(and)g(multiplicities)g (of)h(eigen)m(v)n(alues)e(and)h(resonances)f(of)i Fw(L)3269 3790 y Fu(\025)523 3877 y FC(for)g(small)i(nonzero)d Fz(\025)p FC(.)i(It)g(also)h(gi)n(v)o(es)e(an)h(upper)e(bound)g(on)i (the)g(number)e(of)h(eigen)m(v)n(alues)f(of)523 3977 y Fw(L)578 3989 y Fu(\025)643 3977 y FC(for)c(small)i(nonzero)d Fz(\025)p FC(.)523 4209 y FA(1.2)40 b(A)n(pplications)20 b(of)g(the)h(F)n(ermi)f(Golden)g(Rule)h(to)f(open)g(quantum)h(systems) 523 4384 y FC(In)e(these)h(lectures,)e(by)h(an)h(open)e(quantum)f (system)j(we)g(shall)f(mean)g(a)h(\223small\224)f(quantum)f(sys-)523 4483 y(tem)24 b Fs(S)30 b FC(interacting)22 b(with)i(a)f(lar)o(ge)g (\223en)m(vironment\224)d(or)j(\223reserv)n(oir\224)f Fs(R)p FC(.)i(The)f(small)h(quantum)523 4583 y(system)i(is)g(described) e(by)h(a)g(\002nite)h(dimensional)d(Hilbert)j(space)f Fs(K)i FC(and)e(a)g(Hamiltonian)f Fz(K)6 b FC(.)523 4682 y(The)23 b(reserv)n(oir)g(is)h(described)f(by)g(a)h Fz(W)1674 4652 y Fy(\003)1712 4682 y FC(-dynamical)d(system)j Ft(\()p Fo(M)p Fz(;)14 b(\034)9 b Ft(\))26 b FC(and)d(a)h(reference)d(state)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 523 100 a FB(4)230 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fz(!)575 294 y Fn(R)647 282 y FC(\(for)17 b(a)h(discussion)g(of)f(reference)f(states)j(see)g(the)f(lecture)f ([AJPP]\).)h(W)-7 b(e)19 b(shall)f(assume)g(that)523 382 y Fz(!)575 394 y Fn(R)649 382 y FC(is)j(normal)e(and)h Fz(\034)1158 394 y Fn(R)1212 382 y FC(-in)m(v)n(ariant.)648 482 y(If)i Fz(!)778 494 y Fn(R)855 482 y FC(is)h(a)g Ft(\()p Fz(\034)1061 494 y Fn(R)1115 482 y Fz(;)14 b(\014)t Ft(\))p FC(-KMS)24 b(state,)f(then)f(we)h(say)g(that)f(that)h(the)f (reserv)n(oir)g(at)h(in)m(v)o(erse)e(tem-)523 581 y(perature)h Fz(\014)27 b FC(and)c(that)g(the)g(open)f(quantum)f(system)i(is)h (thermal.)e(Another)g(important)f(special)523 681 y(case)33 b(is)g(when)f Fs(R)h FC(has)f(additional)f(structure,)g(namely)g (consists)i(of)f Fz(n)h FC(independent)c(parts)523 780 y Fs(R)593 792 y Fx(1)631 780 y Fz(;)14 b Fs(\001)g(\001)g(\001)27 b Fz(;)14 b Fs(R)899 792 y Fu(n)973 780 y FC(,)27 b(which)g(are)h (interpreted)d(as)k(sub-reserv)n(oirs.)c(If)i(the)h(reference)e(state)i (of)f(the)523 880 y(sub-reserv)n(oir)21 b Fs(R)1053 892 y Fu(j)1112 880 y FC(is)j Fz(\014)1238 892 y Fu(j)1273 880 y FC(-KMS)f(\(for)g Fz(j)33 b Ft(=)28 b(1)p Fz(;)14 b Fs(\001)g(\001)g(\001)27 b Fz(;)14 b(n)p Ft(\))p FC(,)24 b(then)e(we)i(shall)f(call)h(the)f(correspond-)523 980 y(ing)d(open)f(quantum)f(system)j(multi-thermal.)648 1079 y(In)26 b(the)h(literature)f(one)g(can)h(\002nd)f(at)h(least)h(tw) o(o)f(distinct)g(important)e(applications)h(of)g(the)523 1179 y(Fermi)20 b(Golden)f(Rule)i(to)f(the)g(study)g(of)g(open)f (quantum)f(systems.)648 1279 y(In)26 b(the)g(\002rst)h(application)e (one)g(considers)h(the)g(weak)g(coupling)f(limit)h(for)g(the)g (dynamics)523 1378 y(in)31 b(the)f(Heisenber)o(g)f(picture)h(reduced)f (to)i(the)f(small)h(system.)g(This)g(limit)g(turns)f(out)g(to)h(be)523 1478 y(an)e(irre)n(v)o(ersible)e(Mark)o(o)o(vian)g(dynamics\227a)h (completely)f(positi)n(v)o(e)h(semigroup)f(preserving)523 1577 y(the)e(identity)e(acting)h(on)g(the)h(observ)n(ables)e(of)h(the)g (small)h(system)g Fs(S)32 b FC(\()p Fz(n)21 b Fs(\002)g Fz(n)k FC(matrices\).)e(The)523 1677 y(generator)f(of)h(this)h (semigroup)e(is)i(gi)n(v)o(en)e(by)h(the)h(LSO)g(for)f(the)g(generator) f(of)h(the)h(dynamics.)523 1777 y(W)-7 b(e)21 b(will)g(denote)e(it)i (by)f Fz(M)9 b FC(.)648 1876 y(The)24 b(weak)g(coupling)f(limit)h(and)g (the)h(deri)n(v)n(ation)d(of)j(the)f(resulting)g(irre)n(v)o(ersible)e (Mark)o(o-)523 1976 y(vian)31 b(dynamics)g(goes)h(back)f(to)h(the)g(w)o (ork)f(of)h(P)o(auli,)g(W)m(igner)n(-W)-7 b(eissk)o(opf)31 b(and)g(v)n(an)g(Ho)o(v)o(e)523 2076 y([WW,)23 b(VH1,)f(VH2,)g(VH3])g (see)h(also)g([KTH)o(,)g(Haa].)f(In)g(the)g(mathematical)g(literature)f (it)i(w)o(as)523 2175 y(studied)h(in)g(the)g(well)h(kno)n(wn)e(papers)g (of)h(Da)n(vies)h([Da1)o(,)f(Da2,)g(Da3],)g(see)h(also)f([LeSp)o(,)h (AL)o(].)523 2275 y(Therefore,)d(the)i(operator)e Fz(M)34 b FC(is)25 b(sometimes)f(called)g(the)g(Da)n(vies)h(generator)d(in)i (the)g(Heisen-)523 2374 y(ber)o(g)19 b(picture.)648 2474 y(One)26 b(can)h(also)g(look)f(at)h(the)g(dynamics)e(in)i(the)g(Schr)7 b(\250)-35 b(odinger)24 b(picture)i(\(on)g(the)h(space)f(of)523 2574 y(density)20 b(matrices\).)h(In)f(the)h(weak)g(coupling)e(limit)j (one)e(then)h(obtains)f(a)h(completely)f(positi)n(v)o(e)523 2673 y(semigroup)e(preserving)g(the)i(trace.)g(It)g(is)h(generated)d (by)h(the)h(adjoint)g(of)f Fz(M)9 b FC(,)20 b(denoted)f(by)g Fz(M)3254 2643 y Fy(\003)3292 2673 y FC(,)523 2773 y(which)h(is)h (sometimes)f(called)g(the)g(Da)n(vies)h(generator)d(in)i(the)g(Schr)7 b(\250)-35 b(odinger)18 b(picture.)648 2873 y(The)29 b(second)h(application)f(of)h(the)g(Fermi)g(Golden)f(Rule)i(to)f(the)h (study)e(of)h(open)g(quan-)523 2972 y(tum)22 b(systems)h(is)g(relati)n (v)o(ely)e(recent.)g(It)i(has)f(appeared)f(in)h(papers)f(on)h(the)g (so-called)g(return)f(to)523 3072 y(equilibrium)26 b([JP1,)j(DJ1)o(,)g (DJ2,)g(BFS2,)f(M].)h(The)f(main)g(goal)f(of)h(these)h(papers)f(is)h (to)f(sho)n(w)523 3171 y(that)19 b(certain)g Fz(W)1003 3141 y Fy(\003)1041 3171 y FC(-dynamics)f(describing)g(open)g(quantum)g (systems)h(has)h(only)e(one)h(stationary)523 3271 y(normal)i(state)h (or)g(no)g(stationary)f(normal)f(states)j(at)g(all.)f(This)g(problem)e (can)i(be)g(reformulated)523 3371 y(into)e(a)h(question)e(about)h(the)g (point)g(spectrum)f(of)h(the)g(so-called)g(Liouvillean\227the)e (generator)523 3470 y(of)25 b(the)h(natural)e(unitary)g(implementation) g(of)h(the)g(dynamics.)f(T)-7 b(o)26 b(study)f(this)h(problem,)d(it)j (is)523 3570 y(con)m(v)o(enient)h(to)i(introduce)f(the)h(LSO)h(for)e (the)i(Liouvillean.)d(W)-7 b(e)31 b(shall)e(denote)g(it)h(by)f Ft(i)p Fz(\000)12 b FC(.)29 b(It)523 3670 y(is)e(an)e(operator)f (acting)h(on)g(Hilbert-Schmidt)f(operators)g(for)h(the)h(system)f Fs(S)6 b FC(\227again)25 b Fz(n)e Fs(\002)f Fz(n)523 3769 y FC(matrices.)648 3869 y(The)15 b(use)h(of)f Ft(i)p Fz(\000)28 b FC(in)16 b(the)g(spectral)g(theory)e(hinges)h(on)h (analytic)f(techniques)f(\(Mourre)g(theory)-5 b(,)523 3968 y(comple)o(x)25 b(deformations\),)e(which)j(we)g(shall)h(not)f (describe)f(in)i(our)e(lectures.)h(W)-7 b(e)27 b(shall)g(tak)o(e)523 4068 y(it)c(for)f(granted)f(that)i(under)e(suitable)h(technical)g (conditions)f(such)h(applications)f(are)h(possible)523 4168 y(and)31 b(we)i(will)f(focus)g(on)f(the)h(algebraic)f(properties)f (of)i Fz(M)9 b FC(,)32 b Ft(i)p Fz(\000)44 b FC(and)31 b Fz(M)2734 4138 y Fy(\003)2772 4168 y FC(.)h(T)-7 b(o)32 b(the)g(best)g(of)523 4267 y(our)24 b(kno)n(wledge,)e(some)i(of)g (these)g(properties)f(ha)n(v)o(e)h(not)g(been)g(discussed)g(pre)n (viously)f(in)h(the)523 4367 y(literature.)648 4467 y(In)e(Theorem)f (17)i(we)g(gi)n(v)o(e)f(a)h(simple)g(characterization)d(of)j(the)g(k)o (ernel)f(of)g(the)h(imaginary)523 4566 y(part)18 b(the)g(operator)e Fz(\000)c FC(.)18 b(This)h(characterization)d(implies)i(that)g Fz(\000)30 b FC(has)19 b(no)e(nontri)n(vial)g(real)h(eigen-)523 4666 y(v)n(alues)24 b(in)g(a)g(generic)f(nonthermal)f(case.)i(In)f ([DJ2],)h(this)g(result)g(w)o(as)h(pro)o(v)o(en)c(in)j(the)g(conte)o (xt)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)230 b(5)523 282 y FC(of)20 b(P)o(auli-Fierz)g(systems)g(and)g(w)o(as)h(used)f(to)h(sho)n(w)f(the)g (absence)g(of)g(normal)f(stationary)g(states)523 382 y(in)j(a)f(generic)g(multithermal)f(case.)h(In)g(our)g(lectures)g(we)h (generalize)e(the)i(result)f(of)g([DJ2])g(to)h(a)523 482 y(more)d(general)g(setting.)648 581 y(The)c(characterization)e(of)i (the)h(k)o(ernel)f(of)g(the)h(imaginary)e(part)h(of)g Fz(\000)28 b FC(in)16 b(the)f(thermal)g(case)h(is)523 681 y(gi)n(v)o(en)g(in)i(Theorem)e(18.)h(It)h(implies)g(that)f (generically)f(this)i(k)o(ernel)f(consists)h(only)f(of)g(multiples)523 780 y(of)27 b(the)g(square)f(root)h(of)f(the)h(Gibbs)g(density)g (matrix)f(for)h(the)g(small)g(system.)g(In)g([DJ2],)g(this)523 880 y(result)18 b(w)o(as)h(pro)o(v)o(en)c(in)j(the)g(more)f(restricti)n (v)o(e)g(conte)o(xt)g(of)h(P)o(auli-Fierz)f(systems)h(and)g(w)o(as)g (used)523 980 y(to)26 b(sho)n(w)g(the)f(return)g(to)h(equilibrium)e(in) i(the)g(generic)e(thermal)h(case.)h(A)g(similar)g(result)g(w)o(as)523 1079 y(obtained)19 b(earlier)h(by)f(Spohn)g([Sp].)648 1179 y(The)27 b(operators)f Fz(M)9 b FC(,)27 b Ft(i)p Fz(\000)40 b FC(and)26 b Fz(M)1631 1149 y Fy(\003)1697 1179 y FC(act)i(on)f(the)h(same)f(v)o(ector)f(space)i(\(the)f(space)g (of)g Fz(n)d Fs(\002)523 1279 y Fz(n)30 b FC(matrices\))f(and)g(ha)n(v) o(e)g(similar)h(forms.)e(Nai)n(v)o(ely)-5 b(,)28 b(one)h(may)g(e)o (xpect)g(that)g Ft(i)p Fz(\000)42 b FC(interpolates)523 1378 y(in)32 b(some)f(sense)i(between)d Fz(M)41 b FC(and)32 b Fz(M)1713 1348 y Fy(\003)1750 1378 y FC(.)g(Although)e(this)j(e)o (xpectation)c(is)k(correct,)e(its)h(full)523 1478 y(description)24 b(in)m(v)n(olv)o(es)g(some)i(adv)n(anced)e(algebraic)g(tools)i(\(the)f (so-called)g(noncommutati)n(v)o(e)523 1577 y Fz(L)580 1547 y Fu(p)618 1577 y FC(-spaces)f(associated)g(to)h(a)g(v)n(on)e (Neumann)g(algebra\),)g(and)h(for)f(reasons)h(of)g(space)h(we)f(will) 523 1677 y(not)c(discuss)g(it)h(in)g(these)f(lecture)g(notes)g(\(see)g ([DJ4,)g(JP6]\).)648 1777 y(In)g(the)g(thermal)f(case,)i(the)f (relation)f(between)h(the)g(operators)f Fz(M)9 b FC(,)20 b Ft(i)p Fz(\000)33 b FC(and)19 b Fz(M)2933 1747 y Fy(\003)2992 1777 y FC(is)i(consid-)523 1876 y(erably)e(simpler)n(\227the)o(y)g(are) h(mutually)g(similar)g(and)g(in)g(particular)f(ha)n(v)o(e)h(the)g(same) g(spectrum.)523 1976 y(This)h(result)h(has)f(been)g(recently)f(pro)o(v) o(en)e(in)k([DJ3)o(])g(and)e(we)i(will)g(describe)e(it)i(in)f(detail)h (in)f(our)523 2076 y(lectures.)648 2175 y(The)f(similarity)i(of)f Ft(i)p Fz(\000)33 b FC(and)21 b Fz(M)30 b FC(in)21 b(the)h(thermal)e (case)i(is)g(closely)f(related)g(to)g(the)g(Detailed)523 2275 y(Balance)f(Condition)f(for)g Fz(M)9 b FC(.)20 b(In)g(the)g (literature)f(one)h(can)f(\002nd)h(a)h(number)d(of)h(dif)n(ferent)g (de\002ni-)523 2374 y(tions)d(of)g(the)g(Detailed)g(Balance)g (Condition)f(applicable)g(to)h(irre)n(v)o(ersible)e(quantum)h (dynamics.)523 2474 y(In)22 b(these)g(lecture)f(notes)h(we)g(shall)g (propose)e(another)h(one)g(and)g(we)i(will)f(compare)f(it)h(with)g(the) 523 2574 y(de\002nition)d(due)h(to)g(Alicki)g([A])g(and)g (Frigerio-Gorini-K)m(ossak)o(o)n(wski-V)-9 b(erri)15 b([FGKV].)648 2673 y(F)o(or)28 b(reason)h(of)f(space)h(we)h(ha)n(v)o(e) e(omitted)h(man)o(y)e(important)h(topics)h(in)g(our)f(lectures\227)523 2773 y(the)o(y)c(are)g(treated)g(in)h(the)f(re)n(vie)n(w)g([DJ4],)g (which)g(is)i(a)e(continuation)f(of)h(these)h(lecture)f(notes.)523 2873 y(Some)18 b(additional)e(information)g(about)h(the)h(weak)g (coupling)e(limit)i(and)g(the)g(Da)n(vies)g(generator)523 2972 y(can)i(be)g(also)h(found)d(in)i(the)h(lecture)e(notes)h([AJPP)q (].)523 3288 y Fv(2)41 b(F)n(ermi)25 b(Golden)g(Rule)g(in)g(an)g (abstract)g(setting)523 3487 y FA(2.1)40 b(Notation)523 3670 y FC(Let)21 b Fz(L)g FC(be)g(an)f(operator)f(on)i(a)g(Banach)f (space)h Fs(X)12 b FC(.)22 b Ft(sp)o Fz(L)p FC(,)f Ft(sp)2239 3690 y Fx(ess)2325 3670 y Fz(L)p FC(,)g Ft(sp)2503 3690 y Fx(p)2543 3670 y Fz(L)g FC(will)h(denote)e(the)g(spec-)523 3769 y(trum,)25 b(the)i(essential)f(spectrum)g(and)f(the)i(point)e (spectrum)g(\(the)h(set)h(of)f(eigen)m(v)n(alues\))e(of)i(the)523 3869 y(operator)20 b Fz(L)p FC(.)i(If)g Fz(e)g FC(is)h(an)g(isolated)e (point)h(in)g Ft(sp)p Fz(L)p FC(,)g(then)f Fm(1)2194 3881 y Fu(e)2230 3869 y Ft(\()p Fz(L)p Ft(\))i FC(will)f(denote)f(the)h (spectral)g(pro-)523 3968 y(jection)17 b(of)g Fz(L)g FC(onto)f Fz(e)i FC(gi)n(v)o(en)e(by)g(the)i(usual)f(contour)e(inte)o (gral.)h(Sometimes)h(we)g(can)g(also)h(de\002ne)523 4068 y Fm(1)571 4080 y Fu(e)606 4068 y Ft(\()p Fz(L)p Ft(\))23 b FC(if)f Fz(e)h FC(is)g(not)e(an)h(isolated)g(point)g(in)g(the)g (spectrum.)f(This)h(is)h(well)f(kno)n(wn)f(if)h Fz(L)g FC(is)h(a)g(nor)n(-)523 4168 y(mal)j(operator)d(on)i(a)h(Hilbert)g (space.)f(The)g(de\002nition)f(of)h Fm(1)2283 4180 y Fu(e)2319 4168 y Ft(\()p Fz(L)p Ft(\))h FC(for)f(some)g(other)g (classes)h(of)523 4267 y(operators)19 b(is)i(discussed)f(in)g (Appendix,)e(see)j(\(69\),)e(\(70\).)648 4367 y(Let)i(us)h(no)n(w)f (assume)g(that)h Fz(L)f FC(is)i(a)f(self-adjoint)e(operator)f(on)i(a)h (Hilbert)f(space.)h(Let)f Fz(A;)14 b(B)523 4467 y FC(be)20 b(bounded)e(operators.)g(Suppose)h(that)i Fz(p)i Fs(2)g Fw(R)p FC(.)d(W)-7 b(e)22 b(de\002ne)1124 4649 y Fz(A)p Ft(\()p Fz(p)d Fs(\006)f Ft(i0)g Fs(\000)g Fz(L)p Ft(\))1617 4615 y Fy(\000)p Fx(1)1706 4649 y Fz(B)27 b Ft(:=)h(lim)1907 4703 y Fu(\017)p Fy(&)p Fx(0)2047 4649 y Fz(A)p Ft(\()p Fz(p)19 b Fs(\006)f Ft(i)p Fz(\017)g Fs(\000)g Fz(L)p Ft(\))2532 4615 y Fy(\000)p Fx(1)2621 4649 y Fz(B)t(;)505 b FC(\(1\))p eop end %%Page: 6 6 TeXDict begin 6 5 bop 523 100 a FB(6)230 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FC(pro)o(vided)21 b(that)i(the)h(right)f(hand)f(side) i(of)f(\(1\))g(e)o(xists.)g(W)-7 b(e)25 b(will)f(say)g(that)f Fz(A)p Ft(\()p Fz(p)e Fs(\006)g Ft(i0)f Fs(\000)h Fz(L)p Ft(\))3157 252 y Fy(\000)p Fx(1)3245 282 y Fz(B)523 382 y FC(e)o(xists)g(if)f(the)g(limit)h(in)f(\(1\))g(e)o(xists.)648 482 y(The)f(principal)g(v)n(alue)h(of)g Fz(p)e Fs(\000)g Fz(L)809 700 y(A)p Fs(P)7 b Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)p Ft(\))1200 666 y Fy(\000)p Fx(1)1289 700 y Fz(B)27 b Ft(:=)1500 644 y(1)p 1500 681 42 4 v 1500 757 a(2)1551 633 y Fl(\000)1589 700 y Fz(A)p Ft(\()p Fz(p)19 b Ft(+)f(i0)g Fs(\000)g Fz(L)p Ft(\))2082 666 y Fy(\000)p Fx(1)2171 700 y Fz(B)23 b Ft(+)18 b Fz(A)p Ft(\()p Fz(p)h Fs(\000)f Ft(i0)g Fs(\000)g Fz(L)p Ft(\))2833 666 y Fy(\000)p Fx(1)2922 700 y Fz(B)2989 633 y Fl(\001)523 905 y FC(and)i(the)g(delta)g (function)e(of)i Fz(p)f Fs(\000)f Fz(L)840 1125 y(A\016)s Ft(\()p Fz(p)h Fs(\000)f Fz(L)p Ft(\))p Fz(B)27 b Ft(:=)1452 1068 y(i)p 1418 1105 92 4 v 1418 1182 a(2)p Fz(\031)1520 1057 y Fl(\000)1558 1125 y Fz(A)p Ft(\()p Fz(p)19 b Ft(+)f(i0)g Fs(\000)g Fz(L)p Ft(\))2051 1090 y Fy(\000)p Fx(1)2140 1125 y Fz(B)k Fs(\000)c Fz(A)p Ft(\()p Fz(p)h Fs(\000)f Ft(i0)g Fs(\000)g Fz(L)p Ft(\))2801 1090 y Fy(\000)p Fx(1)2890 1125 y Fz(B)2957 1057 y Fl(\001)523 1329 y FC(are)i(then)g(well)h(de\002ned.)648 1429 y Fs(B)s Ft(\()p Fs(X)12 b Ft(\))26 b FC(denotes)g(the)g(algebra)f(of)h(bounded)d (operators)i(on)h Fs(X)12 b FC(.)27 b(If)f Fs(X)39 b FC(is)27 b(a)g(Hilbert)e(space,)523 1529 y(then)35 b Fs(B)760 1499 y Fx(1)797 1529 y Ft(\()p Fs(X)12 b Ft(\))37 b FC(denotes)e(the)g(space)h(of)f(trace)h(class)g(operators)f(and)g Fs(B)2677 1499 y Fx(2)2713 1529 y Ft(\()p Fs(X)12 b Ft(\))37 b FC(the)f(space)f(of)523 1628 y(Hilbert-Schmidt)26 b(operators)g(on)i Fs(X)12 b FC(.)29 b(By)f(a)g(density)f(matrix)h(on)f Fs(X)41 b FC(we)28 b(mean)f Fz(\032)38 b Fs(2)f(B)3140 1598 y Fx(1)3177 1628 y Ft(\()p Fs(X)12 b Ft(\))523 1728 y FC(such)20 b(that)g Fz(\032)j Fs(\025)g Ft(0)d FC(and)g Ft(T)-7 b(r)o Fz(\032)23 b Ft(=)g(1)p FC(.)d(W)-7 b(e)21 b(say)g(that)f Fz(\032)h FC(is)g(nonde)o(generate)16 b(if)21 b Ft(Ker)o Fz(\032)i Ft(=)g Fs(f)p Ft(0)p Fs(g)p FC(.)648 1828 y(F)o(or)h(more)f(background)e(material)j(useful)g(in)g (our)g(lectures)g(we)h(refer)e(the)i(reader)e(to)h(Ap-)523 1927 y(pendix.)523 2168 y FA(2.2)40 b(Le)o(v)o(el)21 b(Shift)f(Operator)523 2351 y FC(In)25 b(this)i(subsection)e(we)h (introduce)d(the)j(de\002nition)f(of)g(the)h(Le)n(v)o(el)f(Shift)h (Operator)-5 b(.)24 b(First)j(we)523 2450 y(describe)19 b(the)i(basic)f(setup)g(needed)f(to)h(mak)o(e)g(this)h(de\002nition.) 523 2616 y FA(Assumption)g(2.1)40 b Fk(W)-8 b(e)22 b(assume)e(that)f Fs(X)34 b Fk(is)21 b(a)f(Banac)o(h)e(space)o(,)h Fw(P)i Fk(is)g(pr)l(ojection)e(of)h(norm)g Ft(1)h Fk(on)523 2716 y Fs(X)33 b Fk(and)19 b Ft(e)797 2686 y Fu(t)p Fr(L)861 2694 y Fq(0)918 2716 y Fk(is)j(a)e(1-par)o(ameter)e Fz(C)1550 2728 y Fx(0)1588 2716 y Fk(-)i(gr)l(oup)g(of)g(isometries)h(commuting)e (with)i Fw(P)p Fk(.)648 2903 y FC(W)-7 b(e)17 b(set)h Fw(E)23 b Ft(:=)g Fw(L)1127 2915 y Fx(0)1164 2808 y Fl(\014)1164 2858 y(\014)1164 2907 y(\014)1192 2961 y Fx(Ran)p Fr(P)1367 2903 y FC(and)1506 2882 y Fl(e)1504 2903 y Fw(P)g Ft(:=)f Fm(1)5 b Fs(\000)g Fw(P)p FC(.)17 b(Clearly)-5 b(,)16 b Fw(E)h FC(is)h(the)e(generator)f(of)h(a)i(1-parameter)523 3065 y(group)g(of)i(isometries)f(on)h Ft(Ran)o Fw(P)p FC(.)g(and)f Fw(L)1727 3077 y Fx(0)1765 2970 y Fl(\014)1765 3020 y(\014)1765 3069 y(\014)1793 3123 y Fx(Ran)1910 3108 y Fj(e)1911 3123 y Fr(P)1971 3065 y FC(generates)f(a)j (1-parameter)c(group)h(of)i(isome-)523 3213 y(tries)h(on)f Ft(Ran)942 3192 y Fl(e)939 3213 y Fw(P)p FC(.)648 3313 y(Later)g(on,)g(we)g(will)i(often)d(write)i Fw(L)1675 3325 y Fx(0)1715 3291 y Fl(e)1713 3313 y Fw(P)f FC(instead)h(of)f Fw(L)2186 3325 y Fx(0)2223 3217 y Fl(\014)2223 3267 y(\014)2223 3317 y(\014)2251 3371 y Fx(Ran)2368 3356 y Fj(e)2369 3371 y Fr(P)2409 3313 y FC(.)h(F)o(or)f(instance,)g(in)g(\(2\))g Ft(\(\(i)p Fz(e)f Ft(+)523 3460 y Fz(\030)t Ft(\))597 3439 y Fl(e)595 3460 y Fw(P)g Fs(\000)g Fw(L)804 3472 y Fx(0)843 3439 y Fl(e)841 3460 y Fw(P)p Ft(\))924 3430 y Fy(\000)p Fx(1)1035 3460 y FC(will)i(denote)f(the)h(in)m(v)o(erse)e (of)i Ft(\(i)p Fz(e)e Ft(+)f Fz(\030)t Ft(\))p Fm(1)h Fs(\000)g Fw(L)2372 3472 y Fx(0)2430 3460 y FC(restricted)i(to)g Ft(Ran)3000 3439 y(~)2995 3460 y Fw(P)p FC(.)g(This)g(is)523 3560 y(a)g(slight)f(ab)n(use)g(of)g(notation,)f(which)g(we)i(will)g (mak)o(e)f(often)f(without)g(a)i(comment.)648 3660 y(Most)f(of)g(the)g (time)h(we)f(will)h(also)f(assume)h(that)523 3809 y FA(Assumption)g (2.2)40 b Fw(P)21 b Fk(is)g(\002nite)f(dimensional.)648 3959 y FC(Under)c(Assumption)g(2.1)h(and)g(2.2,)f(the)h(operator)f Fw(E)i FC(is)g(diagonalizable)d(and)i(we)h(can)f(write)523 4058 y(its)k(spectral)f(decomposition:)1576 4255 y Fw(E)j Ft(=)1781 4176 y Fl(X)1742 4354 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1953 4255 y Ft(i)p Fz(e)p Fm(1)2063 4267 y Fx(i)p Fu(e)2117 4255 y Ft(\()p Fw(E)p Ft(\))p Fz(:)523 4527 y FC(Note)d(that)g Fm(1)898 4539 y Fx(i)p Fu(e)953 4527 y Ft(\()p Fw(E)p Ft(\))h FC(are)f(projections)f(of)h(norm)f(one.) 648 4627 y(In)g(the)i(remaining)d(assumptions)h(we)i(impose)f(our)f (conditions)g(on)g(the)i(perturbation:)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)230 b(7)523 282 y FA(Assumption)21 b(2.3)40 b Fk(W)-8 b(e)31 b(suppose)e(that)g Fw(Q)h Fk(is)h(an)e(oper)o (ator)g(with)h Ft(Dom)p Fw(Q)40 b Fs(\033)g Ft(Dom)p Fw(L)3099 294 y Fx(0)3167 282 y Fk(and,)523 382 y(for)26 b Fs(j)p Fz(\025)p Fs(j)33 b Fz(<)f(\025)918 394 y Fx(0)956 382 y Fk(,)25 b Fw(L)1057 394 y Fu(\025)1133 382 y Ft(:=)33 b Fw(L)1309 394 y Fx(0)1368 382 y Ft(+)22 b Fz(\025)p Fw(Q)k Fk(is)g(the)g(g)o(ener)o(ator)e(of)h(a)h(1-par)o(ameter)d Fz(C)2810 394 y Fx(0)2848 382 y Fk(-semigr)l(oup)h(of)523 482 y(contr)o(actions.)648 621 y FC(Assumption)19 b(2.3)g(implies)h (that)1607 599 y Fl(e)1605 621 y Fw(P)o(QP)h FC(and)e Fw(PQ)2050 599 y Fl(e)2048 621 y Fw(P)h FC(are)g(well)h(de\002ned.)523 771 y FA(Assumption)g(2.4)40 b Fw(PQP)23 b Ft(=)f(0)p Fk(.)648 921 y FC(The)17 b(abo)o(v)o(e)g(assumption)f(is)j(needed)e(to) h(guarantee)e(that)i(the)g(\002rst)h(nontri)n(vial)d(contrib)n(ution) 523 1021 y(for)k(the)g(shift)g(of)g(eigen)m(v)n(alues)e(of)i Fw(L)1573 1033 y Fu(\025)1638 1021 y FC(is)h(2nd)e(order)g(in)i Fz(\025)p FC(.)648 1121 y(It)28 b(is)i(also)f(useful)f(to)h(note)f (that)h(if)f(Assumption)g(2.2)g(holds,)g(then)2674 1099 y Fl(e)2672 1121 y Fw(P)o(QP)h FC(and)f Fw(PQ)3134 1099 y Fl(e)3132 1121 y Fw(P)g FC(are)523 1220 y(bounded.)19 b(Note)j(also)g(that)h(in)f(the)g(de\002nition)f(of)g(LSO)i(only)e(the) h(terms)2666 1199 y Fl(e)2664 1220 y Fw(PQP)f FC(and)h Fw(PQ)3113 1199 y Fl(e)3111 1220 y Fw(P)f FC(will)523 1320 y(play)f(a)g(role)g(and)g(the)g(term)1332 1298 y Fl(e)1330 1320 y Fw(PQ)1448 1298 y Fl(e)1446 1320 y Fw(P)g FC(will)h(be)f(irrele)n(v)n(ant.)523 1470 y FA(Assumption)h(2.5)40 b Fk(W)-8 b(e)22 b(assume)e(that)g(for)g(all)h Ft(i)p Fz(e)i Fs(2)g Ft(sp)p Fw(E)e Fk(ther)m(e)f(e)n(xists)1256 1634 y Fm(1)1304 1646 y Fx(i)p Fu(e)1358 1634 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)p Ft(\(\(i)p Fz(e)g Ft(+)e(0\))1847 1613 y Fl(e)1845 1634 y Fw(P)g Fs(\000)g Fw(L)2052 1646 y Fx(0)2092 1613 y Fl(e)2089 1634 y Fw(P)p Ft(\))2172 1604 y Fy(\000)p Fx(1)2261 1634 y Fw(Q)p Fm(1)2374 1646 y Fx(i)p Fu(e)2428 1634 y Ft(\()p Fw(E)p Ft(\))1144 1816 y(:=)32 b(lim)1256 1870 y Fu(\030)r Fy(&)p Fx(0)1401 1816 y Fm(1)1449 1828 y Fx(i)p Fu(e)1504 1816 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)p Ft(\(\(i)p Fz(e)19 b Ft(+)f Fz(\030)t Ft(\))1990 1795 y Fl(e)1988 1816 y Fw(P)h Fs(\000)f Fw(L)2196 1828 y Fx(0)2235 1795 y Fl(e)2233 1816 y Fw(P)p Ft(\))2316 1786 y Fy(\000)p Fx(1)2405 1816 y Fw(Q)p Fm(1)2518 1828 y Fx(i)p Fu(e)2572 1816 y Ft(\()p Fw(E)p Ft(\))3216 1742 y FC(\(2\))648 2106 y(Under)h(Assumptions)g(2.1,)g(2.2,)h(2.3,)f(2.4)h (and)f(2.5)h(we)g(set)1055 2276 y Fz(M)32 b Ft(:=)1317 2197 y Fl(X)1278 2375 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1489 2276 y Fm(1)1537 2288 y Fx(i)p Fu(e)1591 2276 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)p Ft(\(\(i)p Fz(e)19 b Ft(+)f(0\))2079 2254 y Fl(e)2077 2276 y Fw(P)g Fs(\000)g Fw(L)2284 2288 y Fx(0)2324 2254 y Fl(e)2322 2276 y Fw(P)p Ft(\))2405 2242 y Fy(\000)p Fx(1)2494 2276 y Fw(Q)p Fm(1)2607 2288 y Fx(i)p Fu(e)2661 2276 y Ft(\()p Fw(E)p Ft(\))436 b FC(\(3\))523 2523 y(and)20 b(call)g(it)h(the)f(Le)n(v)o(el)g(Shift)g (Operator)f(\(LSO\))h(associated)g(to)g(the)g(triple)g Ft(\()p Fw(P)p Fz(;)14 b Fw(L)2871 2535 y Fx(0)2909 2523 y Fz(;)g Fw(Q)p Ft(\))p FC(.)648 2622 y(It)20 b(is)h(instructi)n(v)o(e) e(to)h(gi)n(v)o(e)g(time-dependent)d(formulas)i(for)h(the)g(LSO:)924 2777 y Fz(M)34 b Ft(=)d(lim)1126 2831 y Fu(\030)r Fy(&)p Fx(0)1326 2715 y Fl(P)1272 2851 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1482 2777 y Fm(1)1530 2789 y Fx(i)p Fu(e)1585 2777 y Ft(\()p Fw(E)p Ft(\))1718 2710 y Fl(R)1774 2730 y Fy(1)1758 2806 y Fx(0)1858 2777 y Ft(e)1895 2747 y Fy(\000)p Fu(\030)r(s)2014 2777 y Fw(QQ)p Ft(\()p Fz(s)p Ft(\))p Fm(1)2295 2789 y Fx(i)p Fu(e)2349 2777 y Ft(\()p Fw(E)p Ft(\)d)p Fz(s)1039 2996 y Ft(=)g(lim)1126 3050 y Fu(\030)r Fy(&)p Fx(0)1326 2934 y Fl(P)1272 3071 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1482 2996 y Fm(1)1530 3008 y Fx(i)p Fu(e)1585 2996 y Ft(\()p Fw(E)p Ft(\))1718 2929 y Fl(R)1774 2950 y Fy(1)1758 3026 y Fx(0)1858 2996 y Ft(e)1895 2966 y Fy(\000)p Fu(\030)r(s)2014 2996 y Fw(Q)p Ft(\()p Fs(\000)p Fz(s=)p Ft(2\))p Fw(Q)p Ft(\()p Fz(s=)p Ft(2\))p Fm(1)2631 3008 y Fx(i)p Fu(e)2683 2996 y Ft(\()p Fw(E)p Ft(\)d)p Fz(s;)523 3226 y FC(where)20 b Fw(Q)p Ft(\()p Fz(t)p Ft(\))j(:=)f(e)1076 3196 y Fu(t)p Fr(L)1140 3204 y Fq(0)1177 3226 y Fw(Q)p Ft(e)1279 3196 y Fy(\000)p Fu(t)p Fr(L)1395 3204 y Fq(0)1431 3226 y FC(.)523 3446 y FA(2.3)40 b(LSO)21 b(f)n(or)f Fi(C)1054 3416 y Fh(\003)1048 3471 y Fg(0)1096 3446 y FA(-dynamics)523 3607 y FC(In)28 b(the)g(pre)n(vious)e(subsection)h(we)i(assumed)e(that) h Fw(L)2090 3619 y Fu(\025)2163 3607 y FC(is)g(a)h(generator)d(of)i(a)g Fz(C)2876 3619 y Fx(0)2914 3607 y FC(-semigroup.)523 3707 y(In)c(one)g(of)g(our)g(applications,)f(ho)n(we)n(v)o(er)m(,)e(we) k(will)g(deal)g(with)f(another)f(type)h(of)g(semigroups,)523 3807 y(the)c(so-called)f Fz(C)1030 3777 y Fy(\003)1024 3827 y Fx(0)1069 3807 y FC(-semigroups)f(\(see)i(Appendix)e(for)h (de\002nitions)g(and)h(a)g(discussion\).)f(In)h(this)523 3906 y(case,)c(we)g(will)g(need)f(to)h(replace)f(Assumptions)g(2.1)g (and)g(2.3)g(by)g(their)g(\223dual)g(v)o(ersions\224,)g(which)523 4006 y(we)21 b(state)g(belo)n(w:)523 4145 y FA(Assumption)29 b(2.1*)f Fk(W)-8 b(e)29 b(assume)g(that)f Fs(Y)36 b Fk(is)30 b(a)e(Banac)o(h)f(space)h(and)f Fs(X)42 b Fk(is)29 b(its)h(dual,)d (that)h(is)523 4245 y Fs(X)35 b Ft(=)23 b Fs(Y)767 4214 y Fy(\003)806 4245 y Fk(,)d Fw(P)h Fk(is)g(a)f(w*)g(continuous)f(pr)l (ojection)g(of)h(norm)g Ft(1)g Fk(on)g Fs(X)33 b Fk(and)19 b Ft(e)2639 4214 y Fu(t)p Fr(L)2703 4222 y Fq(0)2760 4245 y Fk(is)j(a)e(1-par)o(ameter)523 4344 y Fz(C)588 4314 y Fy(\003)582 4365 y Fx(0)626 4344 y Fk(-)h(gr)l(oup)e(of)i (isometries)g(commuting)d(with)j Fw(P)p Fk(.)523 4483 y FA(Assumption)28 b(2.3*)e Fk(W)-8 b(e)28 b(suppose)e(that)g Fw(Q)i Fk(is)g(an)e(oper)o(ator)g(with)i Ft(Dom)p Fw(Q)35 b Fs(\033)g Ft(Dom)p Fw(L)3102 4495 y Fx(0)3167 4483 y Fk(and,)523 4583 y(for)25 b Fs(j)p Fz(\025)p Fs(j)33 b Fz(<)e(\025)916 4595 y Fx(0)954 4583 y Fk(,)25 b Fw(L)1055 4595 y Fu(\025)1131 4583 y Ft(:=)31 b Fw(L)1305 4595 y Fx(0)1365 4583 y Ft(+)21 b Fz(\025)p Fw(Q)26 b Fk(is)g(the)e(g)o (ener)o(ator)g(of)h(a)g(1-par)o(ameter)f Fz(C)2810 4553 y Fy(\003)2804 4603 y Fx(0)2848 4583 y Fk(-semigr)l(oup)g(of)523 4682 y(contr)o(actions.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 523 100 a FB(8)230 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(2.4)40 b(LSO)21 b(f)n(or)f Fi(W)1084 252 y Fh(\003)1126 282 y FA(-dynamics)523 465 y FC(The)g(formalism)g (of)g(the)h(Le)n(v)o(el)e(Shift)i(Operator)e(will)j(be)e(applied)g(to)g (open)g(quantum)f(systems)523 565 y(in)h(tw)o(o)h(distinct)f (situations.)648 664 y(In)h(the)g(\002rst)h(application,)e(the)i (Banach)f(space)g Fs(X)35 b FC(is)22 b(a)g Fz(W)2346 634 y Fy(\003)2384 664 y FC(-algebra,)e Fw(P)i FC(is)g(a)g(normal)e (con-)523 764 y(ditional)g(e)o(xpectation)e(and)h Ft(e)1377 734 y Fu(t)p Fr(L)1441 742 y Fq(0)1498 764 y FC(is)j(a)e Fz(W)1722 734 y Fy(\003)1760 764 y FC(-dynamics.)648 863 y(Note)27 b(that)g Fz(W)1079 833 y Fy(\003)1117 863 y FC(-algebras)f(are)i(usually)e(not)h(re\003e)o(xi)n(v)o(e)f(and)h Fz(W)2532 833 y Fy(\003)2570 863 y FC(-dynamics)f(are)h(usually)523 963 y(not)h Fz(C)717 975 y Fx(0)755 963 y FC(-groups.)e(Ho)n(we)n(v)o (er)m(,)g Fz(W)1493 933 y Fy(\003)1531 963 y FC(-algebras)h(are)i(dual) e(Banach)h(spaces)h(and)f Fz(W)2929 933 y Fy(\003)2967 963 y FC(-dynamics)523 1063 y(are)20 b Fz(C)710 1033 y Fy(\003)704 1083 y Fx(0)749 1063 y FC(-groups.)648 1162 y(The)k(perturbation)e(has)j(the)f(form)g Ft(i[)p Fz(V)5 b(;)14 b Fs(\001)p Ft(])25 b FC(with)g Fz(V)44 b FC(being)24 b(a)h(self-adjoint)e(element)h(of)g(the)523 1262 y Fz(W)613 1232 y Fy(\003)651 1262 y FC(-algebra.)k(Therefore,)e Ft(e)1388 1232 y Fu(t)p Fr(L)1452 1241 y Fp(\025)1525 1262 y FC(will)k(be)f(a)h Fz(W)1949 1232 y Fy(\003)1987 1262 y FC(-dynamics)e(for)g(all)i(real)f Fz(\025)h FC(\226)g(again)e(a) h Fz(C)3246 1232 y Fy(\003)3240 1283 y Fx(0)3285 1262 y FC(-)523 1362 y(group.)523 1602 y FA(2.5)40 b(LSO)21 b(in)g(Hilbert)f(spaces)523 1785 y FC(In)d(our)f(second)g(application,) f Fs(X)29 b FC(is)18 b(a)f(Hilbert)g(space.)f(Hilbert)h(spaces)g(are)g (re\003e)o(xi)n(v)o(e,)e(therefore)523 1885 y(we)21 b(do)e(not)h(need)g (to)g(distinguish)f(between)h Fz(C)1876 1897 y Fx(0)1934 1885 y FC(and)g Fz(C)2140 1854 y Fy(\003)2134 1905 y Fx(0)2178 1885 y FC(-groups.)648 1984 y(All)37 b(strongly)e(continuous) g(groups)g(of)i(isometries)f(on)h(a)g(Hilbert)f(space)h(are)f(unitary) 523 2084 y(groups.)19 b(Therefore,)f(the)j(operator)e Fw(L)1636 2096 y Fx(0)1695 2084 y FC(has)h(to)h(be)g(anti-self-adjoint) d(\(that)j(means)f Fw(L)3025 2096 y Fx(0)3087 2084 y Ft(=)j(i)p Fz(L)3255 2096 y Fx(0)3292 2084 y FC(,)523 2183 y(where)d Fz(L)804 2195 y Fx(0)861 2183 y FC(is)h(self-adjoint\).) 648 2283 y(All)29 b(projections)f(of)g(norm)g(one)h(on)f(a)i(Hilbert)f (space)g(are)f(orthogonal.)f(Therefore,)f(the)523 2383 y(distinguished)19 b(projection)f(has)i(to)h(be)f(orthogonal.)648 2482 y(In)28 b(our)f(applications)g(to)i(open)e(quantum)f(systems)j Ft(e)2247 2452 y Fu(t)p Fr(L)2311 2461 y Fp(\025)2383 2482 y FC(is)g(a)g(unitary)e(dynamics.)g(This)523 2582 y(means)20 b(in)g(particular)f(that)h Fw(Q)h FC(has)f(the)g(form)g Fw(Q)i Ft(=)h(i)p Fz(Q)p FC(,)d(where)g Fz(Q)g FC(is)h(hermitian.)648 2682 y(In)i(the)g(case)h(of)f(a)g(Hilbert)h(space)f(the)g(LSO)h(will)g (be)f(denoted)f Ft(i)p Fz(\000)12 b FC(.)23 b(Thus)g(we)h(will)g (isolate)523 2781 y(the)f(imaginary)f(unit)h(\223)p Ft(i)p FC(\224,)h(which)f(is)h(consistent)f(with)h(the)f(usual)g(con)m(v)o (entions)e(for)i(operators)523 2881 y(in)d(Hilbert)g(spaces,)h(and)e (also)i(with)f(the)g(con)m(v)o(ention)d(that)k(we)f(adopted)f(in)h ([DJ2].)523 3055 y Fk(Remark)g(1.)k FC(In)31 b([DJ2])g(we)h(used)f(a)h (formalism)e(similar)i(to)f(that)h(of)f(Subsection)f(2.2)h(in)g(the)523 3155 y(conte)o(xt)17 b(of)g(a)h(Hilbert)g(space.)f(Note,)h(ho)n(we)n(v) o(er)m(,)d(that)j(the)f(terminology)f(that)h(we)h(adopted)f(there)523 3254 y(is)j(not)e(completely)f(consistent)i(with)g(the)f(terminology)f (used)h(in)h(these)g(lectures.)f(In)g([DJ2])h(we)523 3354 y(considered)e(a)i(Hilbert)g(space)g Fs(X)12 b FC(,)20 b(an)e(orthogonal)f(projection)g Fz(P)12 b FC(,)19 b(and)f (self-adjoint)g(operators)523 3454 y Fz(L)580 3466 y Fx(0)617 3454 y Fz(;)c(Q)p FC(.)26 b(If)g Fz(\000)38 b FC(is)27 b(the)f(LSO)h(for)e(the)h(triple)g Ft(\()p Fz(P)r(;)14 b(L)1966 3466 y Fx(0)2004 3454 y Fz(;)g(Q)p Ft(\))26 b FC(according)e(to)j([DJ2)o(],)f(then)g Ft(i)p Fz(\000)38 b FC(is)27 b(the)523 3553 y(LSO)21 b(for)e Ft(\()p Fz(P)r(;)14 b Ft(i)p Fz(L)1022 3565 y Fx(0)1060 3553 y Fz(;)g Ft(i)p Fz(Q)p Ft(\))20 b FC(according)f(to)h(the)g (present)g(de\002nition.)648 3728 y(Let)g(us)h(quote)e(the)h(follo)n (wing)f(easy)h(f)o(act)g(v)n(alid)g(in)g(the)g(case)h(of)f(a)h(Hilbert) f(space.)523 3885 y FA(Theor)o(em)g(1.)k Fk(Suppose)i(that)h Fs(X)41 b Fk(is)28 b(a)g(Hilbert)f(space)o(,)g(Assumptions)g(2.1,)g (2.2,)f(2.3)h(and)g(2.5)523 3985 y(hold)19 b(and)h Fz(Q)g Fk(is)h(self-adjoint.)e(Then)h Ft(e)1643 3955 y Fx(i)p Fu(t\000)1761 3985 y Fk(is)h(contr)o(active)e(for)h Fz(t)k(>)e Ft(0)p Fk(.)523 4143 y FA(Pr)o(oof)o(.)39 b FC(W)-7 b(e)21 b(use)g(the)f(notation)f Fw(E)k Ft(=)g(i)p Fz(E)5 b FC(,)20 b Fw(L)1808 4155 y Fx(0)1869 4143 y Ft(=)j(i)p Fz(L)p FC(,)d Fw(Q)i Ft(=)h(i)p Fz(Q)p FC(.)d(W)-7 b(e)22 b(ha)n(v)o(e)1010 4304 y Ft(1)p 999 4341 65 4 v 999 4417 a(2i)1073 4360 y(\()p Fz(\000)31 b Fs(\000)18 b Fz(\000)1333 4326 y Fy(\003)1370 4360 y Ft(\))24 b(=)e Fs(\000)1627 4281 y Fl(X)1592 4459 y Fu(e)p Fy(2)p Fx(sp)p Fu(E)1797 4360 y Fm(1)1845 4372 y Fu(e)1880 4360 y Ft(\()p Fz(E)5 b Ft(\))p Fz(Q\016)s Ft(\()p Fz(e)19 b Fs(\000)f Fz(L)2346 4372 y Fx(0)2383 4360 y Ft(\))p Fz(Q)p Fm(1)2529 4372 y Fu(e)2564 4360 y Ft(\()p Fz(E)5 b Ft(\))24 b Fs(\024)e Ft(0)523 4640 y FC(Therefore,)c Ft(i)p Fz(\000)32 b FC(is)21 b(a)g(dissipati)n(v)o(e)f(operator)e(and)i Ft(e)1980 4610 y Fx(i)p Fu(t\000)2098 4640 y FC(is)h(contracti)n(v)o(e)d(for)i Fz(t)j(>)f Ft(0)p FC(.)e Ff(2)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)230 b(9)648 282 y FC(Note)28 b(that)h(in)g(Theorem)e(5)h(we)h(will)h(sho)n(w)e(that)h(the)g(LSO)g (is)g(the)g(generator)e(of)h(a)h(con-)523 382 y(tracti)n(v)o(e)21 b(semigroup)e(also)j(in)g(a)g(more)f(general)f(situation,)h(when)g Fs(X)35 b FC(is)22 b(a)g(Banach)f(space.)g(The)523 482 y(proof)16 b(of)i(this)h(f)o(act)f(will)g(be)g(ho)n(we)n(v)o(er)e(more) i(complicated)e(and)h(will)i(require)e(some)g(additional)523 581 y(technical)i(assumptions.)523 817 y FA(2.6)40 b(The)21 b(choice)g(of)e(the)i(pr)o(ojection)d Fw(P)523 996 y FC(In)23 b(typical)f(application)g(of)h(the)g(LSO,)g(the)g(operators)f Fw(L)2190 1008 y Fx(0)2251 996 y FC(and)h Fw(Q)g FC(are)g(gi)n(v)o(en)f (and)g(our)h(goal)f(is)523 1095 y(to)e(study)g(the)g(operator)1636 1195 y Fw(L)1691 1207 y Fu(\025)1758 1195 y Ft(:=)j Fw(L)1924 1207 y Fx(0)1980 1195 y Ft(+)18 b Fz(\025)p Fw(Q)p Fz(:)1017 b FC(\(4\))523 1341 y(More)18 b(precisely)-5 b(,)18 b(we)h(w)o(ant)f (to)h(kno)n(w)f(what)h(happens)e(with)i(its)h(eigen)m(v)n(alues)d(when) h(we)h(switch)523 1441 y(on)h(the)g(perturbation.)648 1540 y(Therefore,)14 b(it)j(is)g(natural)e(to)i(choose)e(the)i (projection)d Fw(P)j FC(as)g(\223the)f(projection)f(onto)h(the)g(point) 523 1640 y(spectrum)j(of)h Fw(L)993 1652 y Fx(0)1031 1640 y FC(\224,)g(that)g(is)1629 1753 y Fw(P)j Ft(=)1790 1675 y Fl(X)1791 1853 y Fu(e)p Fy(2)p Fr(R)1924 1753 y Fm(1)1972 1765 y Fx(i)p Fu(e)2026 1753 y Ft(\()p Fw(L)2113 1765 y Fx(0)2151 1753 y Ft(\))p Fz(;)1010 b FC(\(5\))523 1977 y(pro)o(vided)18 b(that)i(\(5\))g(is)h(well)f(de\002ned.)648 2077 y(More)31 b(generally)-5 b(,)30 b(if)i(we)g(were)g(interested)g (only)f(about)g(what)h(happens)e(around)g(some)523 2176 y(eigen)m(v)n(alues)25 b Fs(f)p Ft(i)p Fz(e)1041 2188 y Fx(1)1078 2176 y Fz(;)14 b(:)g(:)g(:)f(;)h Ft(i)p Fz(e)1324 2188 y Fu(n)1369 2176 y Fs(g)35 b(\032)g Ft(sp)1625 2197 y Fx(p)1666 2176 y Fw(L)1721 2188 y Fx(0)1758 2176 y FC(,)27 b(then)g(we)g(could)f(use)h(the)g(LSO)g(de\002ned)f(with)h(the) 523 2276 y(projection)1614 2439 y Fw(P)22 b Ft(=)1814 2335 y Fu(n)1775 2360 y Fl(X)1777 2537 y Fu(j)s Fx(=1)1909 2439 y Fm(1)1957 2451 y Fx(i)p Fu(e)2007 2459 y Fp(j)2042 2439 y Ft(\()p Fw(L)2129 2451 y Fx(0)2167 2439 y Ft(\))p Fz(:)994 b FC(\(6\))648 2674 y(Clearly)-5 b(,)20 b(if)i Fs(X)34 b FC(is)22 b(a)g(Hilbert)f(space)g(and)g Fw(L)1891 2686 y Fx(0)1950 2674 y FC(is)h(anti-self-adjoint,)d(then)i Fm(1)2813 2686 y Fx(i)p Fu(e)2867 2674 y Ft(\()p Fw(L)2954 2686 y Fx(0)2992 2674 y Ft(\))h FC(are)g(well)523 2774 y(de\002ned)g(for)g(all)h Fz(e)28 b Fs(2)g Fw(R)p FC(.)23 b(Moreo)o(v)o(er)m(,)c(both)j(\(5\))g(and)h(\(6\))f(are)g(projections)g (of)g(norm)g(one)g(com-)523 2873 y(muting)d(with)h Fw(L)1002 2885 y Fx(0)1040 2873 y FC(,)h(and)e(hence)h(the)o(y)f(satisfy)i (Assumption)e(2.1.)648 2973 y(There)e(is)i(no)f(guarantee)e(that)i(the) g(spectral)g(projections)f Fm(1)2353 2985 y Fx(i)p Fu(e)2407 2973 y Ft(\()p Fw(L)2494 2985 y Fx(0)2532 2973 y Ft(\))i FC(are)f(well)h(de\002ned)e(in)h(the)523 3072 y(more)26 b(general)g(case)h(when)f Fw(L)1425 3084 y Fx(0)1490 3072 y FC(is)i(the)e(generator)f(of)i(a)g(group)e(of)h(isometries)h(on) f(a)i(Banach)523 3172 y(space.)22 b(If)g(the)o(y)f(are)h(well)h (de\002ned,)e(then)g(the)o(y)h(ha)n(v)o(e)f(norm)g(one,)g(ho)n(we)n(v)o (er)m(,)f(we)i(seem)g(to)h(ha)n(v)o(e)523 3272 y(no)c(guarantee)g(that) h(their)f(sums)h(ha)n(v)o(e)g(norm)e(one.)h(In)h(Appendix)e(we)i (discuss)g(the)g(problem)e(of)523 3371 y(de\002ning)h(spectral)h (projections)f(onto)g(eigen)m(v)n(alues)f(in)j(this)f(more)g(general)f (case.)648 3471 y(Note,)i(ho)n(we)n(v)o(er)m(,)e(that)j(in)g(the)g (situation)f(considered)f(by)h(us)h(later)m(,)f(we)i(will)f(ha)n(v)o(e) f(no)g(such)523 3571 y(problems.)e(In)g(f)o(act,)i Fw(P)f FC(will)h(be)f(al)o(w)o(ays)h(gi)n(v)o(en)e(by)g(\(5\))h(and)g(will)h (al)o(w)o(ays)f(ha)n(v)o(e)g(norm)f(one.)648 3670 y(If)h Fm(1)772 3682 y Fx(i)p Fu(e)826 3670 y Ft(\()p Fw(L)913 3682 y Fx(0)951 3670 y Ft(\))h FC(is)g(well)g(de\002ned)e(for)h(all)h Fz(e)i Fs(2)g Fw(R)e FC(and)f(we)g(tak)o(e)h Fw(P)f FC(de\002ned)g(by)g (\(5\),)f(then)h Fw(P)g FC(will)523 3770 y(be)25 b(determined)e(by)i (the)h(operator)d Fw(L)1620 3782 y Fx(0)1683 3770 y FC(itself.)j(W)-7 b(e)26 b(will)g(speak)f(about)f(\223the)h(LSO)h(for)f Fw(L)3135 3782 y Fu(\025)3178 3770 y FC(\224,)h(if)523 3869 y(we)21 b(ha)n(v)o(e)e(this)i(projection)d(in)j(mind.)523 4106 y FA(2.7)40 b(Thr)o(ee)21 b(kinds)g(of)f(the)h(F)n(ermi)f(Golden)g (Rule)523 4284 y FC(Suppose)c(that)g(Assumptions)g(2.1,)f(2.2,)h(2.3,)g (2.4)g(and)g(2.5,)f(or)i(2.1*,)e(2.2,)g(2.3*,)g(2.4)h(and)g(2.5)g(are) 523 4384 y(satis\002ed.)24 b(Let)g Fw(P)g FC(be)g(gi)n(v)o(en)e(by)h (\(5\))g(and)h Fz(M)32 b FC(be)24 b(the)g(LSO)g(for)f Ft(\()p Fw(P)p Fz(;)14 b Fw(L)2550 4396 y Fx(0)2587 4384 y Fz(;)g Fw(Q)p Ft(\))p FC(.)24 b(Our)f(main)h(object)523 4483 y(of)c(interest)g(is)h(the)f(operator)f Fw(L)1429 4495 y Fu(\025)1473 4483 y FC(.)648 4583 y(The)26 b(assumption)g(2.4)g (\()p Fw(PQP)34 b Ft(=)h(0)p FC(\))27 b(guarantees)e(that)i(there)g (are)g(no)f(\002rst)i(order)d(ef)n(fects)523 4682 y(of)20 b(the)g(perturbation.)e(The)i(operator)e Fz(M)29 b FC(describes)20 b(what)g(happens)f(with)i(the)f(eigen)m(v)n(alues)e(of)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 523 100 a FB(10)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fw(L)578 294 y Fx(0)642 282 y FC(under)24 b(the)i(in\003uence)f(of)g(the)h(perturbation)d Fz(\025)p Fw(Q)k FC(at)f(the)g(second)f(order)f(of)i Fz(\025)p FC(.)h(F)o(ollo)n(wing)523 382 y(the)22 b(tradition)g(of)g(quantum)e (physics,)h(we)i(will)g(use)g(the)f(name)g(\223the)g(Fermi)g(Golden)f (Rule\224)i(to)523 482 y(describe)c(the)i(second)e(order)g(ef)n(fects)h (of)g(the)g(perturbation.)648 581 y(The)g(Fermi)g(Golden)f(Rule)i(can)f (be)g(made)g(rigorous)e(in)j(man)o(y)e(w)o(ays)i(under)d(v)n(arious)i (tech-)523 681 y(nical)25 b(assumptions.)f(W)-7 b(e)26 b(can)e(distinguish)g(at)i(least)f(three)g(v)n(arieties)g(of)f(the)h (rigorous)e(Fermi)523 780 y(Golden)c(Rule:)523 946 y Fs(\017)83 b FA(Analytic)21 b(F)n(ermi)g(Golden)h(Rule:)g Fw(E)e Ft(+)f Fz(\025)1910 916 y Fx(2)1947 946 y Fz(M)31 b Fk(pr)m(edicts)22 b(the)f(appr)l(oximate)g(location)f(\(up)648 1046 y(to)d Fz(o)p Ft(\()p Fz(\025)850 1016 y Fx(2)888 1046 y Ft(\))p Fk(\))h(and)e(the)h(multiplicity)g(of)g(the)g(r)m (esonances)f(and)h(eig)o(en)m(values)e(of)i Fw(L)2897 1058 y Fu(\025)2959 1046 y Fk(in)g(a)g(neigh-)648 1146 y(borhood)h(of)i Ft(sp)1114 1166 y Fx(p)1155 1146 y Fw(L)1210 1158 y Fx(0)1268 1146 y Fk(for)h(small)f Fz(\025)p Fk(.)648 1245 y FC(The)f(Analytic)g(Fermi)h(Golden)e(Rule)i(is)h(v)n(alid)e (under)g(some)g(analyticity)g(assumptions)g(on)648 1345 y Fw(L)703 1357 y Fu(\025)746 1345 y FC(.)k(It)f(is)h(well)g(kno)n(wn)d (and)i(follo)n(ws)f(essentially)h(by)g(the)g(standard)f(perturbation)e (theory)648 1445 y(for)e(isolated)h(eigen)m(v)n(alues)f(\([Ka)o(,)i (RS4],)f(see)h(also)f([DF1]\).)g(The)g(perturbation)d(ar)o(guments)648 1544 y(are)21 b(applied)g(not)g(to)h Fw(L)1308 1556 y Fu(\025)1374 1544 y FC(directly)-5 b(,)20 b(b)n(ut)i(to)g(the)g (analytically)e(deformed)g Fw(L)2808 1556 y Fu(\025)2852 1544 y FC(.)i(More)f(or)g(less)648 1644 y(e)o(xplicitly)-5 b(,)14 b(this)i(idea)g(w)o(as)h(applied)e(to)h(Liouvilleans)f (describing)g(open)f(quantum)h(systems)648 1743 y([JP1)o(,)32 b(JP2,)g(BFS1,)h(BFS2].)e(One)h(can)g(also)g(apply)e(it)j(to)f(the)f Fz(W)2633 1713 y Fy(\003)2672 1743 y FC(-dynamics)f(of)h(open)648 1843 y(quantum)18 b(systems)j([JP4)o(,)g(JP5].)648 1943 y(The)e FA(stationary)f(weak)i(coupling)g(\(or)f(v)o(an)h(Ho)o(v)o(e\)) f(limit)h FC(of)f([DF2],)g(described)g(in)h(The-)648 2042 y(orem)25 b(2)i(and)f(5,)g(can)g(be)h(vie)n(wed)e(as)j(an)e (in\002nitesimal)g(v)o(ersion)g(of)g(the)g(Analytic)g(Fermi)648 2142 y(Golden)19 b(Rule.)523 2242 y Fs(\017)83 b FA(Spectral)29 b(F)n(ermi)h(Golden)g(Rule:)h Fk(The)f(inter)o(section)g(of)g(the)g (spectrum)g(of)g Fw(E)c Ft(+)f Fz(\025)3185 2211 y Fx(2)3223 2242 y Fz(M)648 2341 y Fk(with)c(the)f(ima)o(ginary)g(line)h(pr)m (edicts)f(possible)h(location)e(of)i(eig)o(en)m(values)e(of)i Fw(L)2950 2353 y Fu(\025)3015 2341 y Fk(for)g(small)648 2441 y(nonzer)l(o)e Fz(\025)p Fk(.)i(It)g(also)f(gives)g(an)g(upper)f (bound)g(on)g(their)i(multiplicity)-5 b(.)648 2540 y FC(Note)18 b(that)h(if)h(the)f(Analytic)f(Fermi)h(Golden)f(Rule)h(is)h (true,)e(then)h(so)g(is)h(the)f(Spectral)f(Fermi)648 2640 y(Golden)f(Rule.)h(Ho)n(we)n(v)o(er)m(,)f(to)h(pro)o(v)o(e)f(the)h (Analytic)g(Fermi)g(Golden)g(Rule)h(we)f(need)g(strong)648 2740 y(analytic)i(assumption,)g(whereas)h(the)g(Spectral)g(Fermi)g (Golden)f(Rule)h(can)g(be)g(sho)n(wn)g(un-)648 2839 y(der)e(much)g (weak)o(er)g(conditions.)f(Roughly)g(speaking,)g(these)i(assumptions)f (should)f(allo)n(w)648 2939 y(us)i(to)g(apply)g(the)g(so-called)f (positi)n(v)o(e)h(commutator)e(method.)648 3039 y(The)30 b(Spectral)h(Fermi)g(Golden)f(Rule)h(is)h(stated)f(in)g(Theorem)e(6.7)i (of)f([DJ2],)h(which)f(is)648 3138 y(pro)o(v)o(en)22 b(in)j([DJ1)o(].)f(Strictly)h(speaking,)e(the)i(analysis)f(of)h([DJ1)o (])g(and)f([DJ2)o(])h(is)h(restricted)648 3238 y(to)20 b(P)o(auli-Fierz)f(operators,)f(b)n(ut)i(it)h(is)g(easy)f(to)h(see)f (that)g(their)g(ar)o(guments)e(e)o(xtend)h(to)h(much)648 3337 y(lar)o(ger)e(classes)k(of)e(operators.)648 3437 y(T)-7 b(o)25 b(illustrate)h(the)g(usefulness)f(of)h(the)f(Spectral)h (Fermi)f(Golden)g(Rule,)h(suppose)e(that)i Fs(X)648 3537 y FC(is)f(a)f(Hilbert)g(space,)g Fw(L)1334 3549 y Fu(\025)1409 3537 y Ft(=)30 b(i)p Fz(L)1584 3549 y Fu(\025)1652 3537 y FC(with)24 b Fz(L)1881 3549 y Fu(\025)1949 3537 y FC(self-adjoint)f (and)h Ft(i)p Fz(\000)36 b FC(is)25 b(the)f(LSO.)h(Then)e(the)648 3636 y(Spectral)c(Fermi)h(Golden)g(Rule)g(implies)g(the)h(bound)1448 3819 y Ft(dim)14 b(Ran)p Fm(1)1797 3831 y Fx(p)1837 3819 y Ft(\()p Fz(L)1926 3831 y Fu(\025)1969 3819 y Ft(\))24 b Fs(\024)e Ft(dim)15 b(Ker)o Fz(\000)2462 3785 y Fx(I)2489 3819 y Fz(;)648 4002 y FC(where)h Fz(\000)931 3972 y Fx(I)982 4002 y Ft(:=)1112 3969 y Fx(1)p 1102 3983 52 4 v 1102 4030 a(2i)1164 4002 y Ft(\()p Fz(\000)i Fs(\000)6 b Fz(\000)1399 3972 y Fy(\003)1437 4002 y Ft(\))p FC(.)17 b(Bounds)f(of)h(this)g(type)f(were)h(used)g(in)g(v)n(arious)f(papers)g (related)648 4101 y(to)k(the)g(Return)g(to)g(Equilibrium)e([JP1,)i (JP2,)h(DJ2,)f(BFS2,)h(M].)523 4209 y Fs(\017)83 b FA(Dynamical)31 b(F)n(ermi)i(Golden)f(Rule.)h Fk(The)g(oper)o(ator)e Ft(e)2339 4179 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025)2519 4154 y Fq(2)2551 4179 y Fu(M)6 b Fx(\))2684 4209 y Fk(describes)32 b(appr)l(oxi-)648 4309 y(mately)20 b(the)g(r)m(educed)f(dynamics)g Fw(P)p Ft(e)1719 4279 y Fu(t)p Fr(L)1783 4288 y Fp(\025)1826 4309 y Fw(P)h Fk(for)h(small)g Fz(\025)p Fk(.)648 4409 y FC(The)g(Dynamical)g(Fermi)h(Golden)e(Rule)j(w)o(as)f(rigorously)e(e) o(xpressed)h(in)h(the)g(form)e(of)i FA(the)648 4508 y(weak)30 b(coupling)h FC(by)f(Da)n(vies)h([Da1)o(,)f(Da2,)h(Da3)o(,)g(LeSp)o(].) f(Da)n(vies)h(sho)n(wed)f(that)h(under)648 4608 y(some)20 b(weak)g(assumptions)f(we)h(ha)n(v)o(e)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(11)1474 285 y Ft(lim)1462 340 y Fu(\025)p Fy(!)p Fx(0)1615 285 y Ft(e)1652 251 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1841 226 y Fq(2)1877 285 y Fw(P)p Ft(e)1965 251 y Fu(t)p Fr(L)2029 260 y Fp(\025)2068 251 y Fu(=\025)2141 226 y Fq(2)2178 285 y Fw(P)23 b Ft(=)f(e)2376 251 y Fu(tM)2475 285 y Fz(:)648 496 y FC(W)-7 b(e)21 b(describe)e(his)i(result)f(in)h(Theorems)d(3)j(and)e(5.)523 811 y Fv(3)41 b(W)-6 b(eak)25 b(coupling)g(limit)523 1011 y FA(3.1)40 b(Stationary)19 b(and)h(time-dependent)g(weak)h (coupling)f(limit)523 1193 y FC(In)27 b(this)g(section)g(we)h(describe) e(in)h(an)g(abstract)g(setting)g(the)g(weak)g(coupling)e(limit.)i(W)-7 b(e)28 b(will)523 1293 y(sho)n(w)j(that,)g(under)e(some)i(conditions,)e (the)i(dynamics)f(restricted)h(to)g(an)g(appropriate)e(sub-)523 1392 y(space,)c(rescaled)g(and)g(renormalized)f(by)h(the)g(free)g (dynamics,)f(con)m(v)o(er)o(ges)f(to)j(the)f(dynamics)523 1492 y(generated)18 b(by)i(the)g(LSO.)648 1592 y(W)-7 b(e)19 b(will)g(gi)n(v)o(e)e(tw)o(o)h(v)o(ersions)f(of)h(the)g(weak)g (coupling)e(limit:)j(the)f(time)g(dependent)e(and)h(the)523 1691 y(stationary)h(one.)g(The)h(time-dependent)d(v)o(ersion)i(is)h (well)h(kno)n(wn)d(and)i(in)g(its)h(rigorous)d(form)h(is)523 1791 y(due)i(to)g(Da)n(vies)h([Da1)o(,)f(Da2)o(,)h(Da3)o(].)f(Our)g(e)o (xposition)f(is)i(based)f(on)f([Da3)o(].)648 1891 y(The)h(stationary)g (weak)g(coupling)f(limit)i(describes)f(the)h(same)f(phenomenon)e(on)i (the)h(le)n(v)o(el)523 1990 y(of)26 b(the)g(resolv)o(ent.)e(Our)h(e)o (xposition)g(is)h(based)g(on)f(recent)h(w)o(ork)f([DF2)o(].)h(F)o (ormally)-5 b(,)24 b(one)h(can)523 2090 y(pass)18 b(from)e(the)i (time-dependent)c(to)k(stationary)e(weak)i(coupling)d(limit)j(by)f(the) h(Laplace)f(trans-)523 2189 y(formation.)g(Ho)n(we)n(v)o(er)m(,)f(one)j (can)g(ar)o(gue)e(that)i(the)g(assumptions)f(needed)f(to)j(pro)o(v)o(e) d(the)i(station-)523 2289 y(ary)g(weak)f(coupling)g(limit)h(are)g (sometimes)g(easier)g(to)g(v)o(erify)-5 b(.)17 b(In)i(f)o(act,)g(the)o (y)f(in)m(v)n(olv)o(e)f(the)i(e)o(xis-)523 2389 y(tence)f(of)h(certain) f(matrix)g(elements)g(of)h(the)f(resolv)o(ent)g(\(a)g(kind)g(of)h(the)f (\223Limiting)g(Absorption)523 2488 y(Principle\224\))h(only)g(at)i (the)f(spectrum)f(of)g Fw(E)p FC(,)i(a)f(discrete)g(subset)g(of)g(the)g (imaginary)e(line.)i(This)g(is)523 2588 y(often)f(possible)h(to)h(sho)n (w)f(by)f(positi)n(v)o(e)h(commutator)e(methods.)648 2688 y(Throughout)i(the)k(section)g(we)h(suppose)e(that)h(most)g(of)g (the)g(assumptions)f(of)h(Subsection)523 2787 y(2.2)e(are)g (satis\002ed.)h(W)-7 b(e)23 b(will,)g(ho)n(we)n(v)o(er)m(,)d(list)j(e)o (xplicitely)e(the)i(assumptions)e(that)h(we)h(need)f(for)523 2887 y(each)e(particular)f(result.)648 2986 y(The)g(\002rst)i(theorem)e (describes)h(the)g(stationary)f(weak)h(coupling)f(limit.)523 3161 y FA(Theor)o(em)h(2.)k Fk(Suppose)f(that)h(Assumptions)g(2.1,)f (2.2,)g(2.3)h(and)f(2.4,)h(or)h(2.1*,)d(2.2,)i(2.3*)f(and)523 3260 y(2.4)d(ar)m(e)g(true)o(.)g(W)-8 b(e)21 b(also)f(assume)g(the)h (following)e(conditions:)612 3360 y FC(1\))24 b Fk(F)-9 b(or)21 b Ft(i)p Fz(e)h Fs(2)i Ft(sp)p Fw(E)p Fk(,)d Fz(\030)27 b(>)22 b Ft(0)p Fk(,)e(we)h(have)f Ft(i)p Fz(e)e Ft(+)g Fz(\030)27 b Fs(62)d Ft(sp)2092 3339 y Fl(e)2090 3360 y Fw(P)o(L)2195 3372 y Fu(\025)2242 3339 y Fl(e)2239 3360 y Fw(P)p Fk(.)612 3460 y FC(2\))g Fk(Ther)m(e)c(e)n (xists)i(an)e(oper)o(ator)f Fz(M)1616 3472 y Fx(st)1693 3460 y Fk(on)g Ft(Ran)p Fw(P)i Fk(suc)o(h)e(that,)h(for)h(any)e Fz(\030)27 b(>)c Ft(0)p Fk(,)933 3681 y Fz(M)1014 3693 y Fx(st)1093 3681 y Ft(:=)1242 3602 y Fl(X)1204 3780 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1426 3681 y Ft(lim)1414 3735 y Fu(\025)p Fy(!)p Fx(0)1567 3681 y Fm(1)1615 3693 y Fx(i)p Fu(e)1669 3681 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1867 3588 y Fl(\020)1917 3681 y Ft(\(i)p Fz(e)c Ft(+)f Fz(\025)2161 3646 y Fx(2)2198 3681 y Fz(\030)t Ft(\))2272 3659 y Fl(e)2270 3681 y Fw(P)h Fs(\000)2425 3659 y Fl(e)2423 3681 y Fw(PL)2529 3693 y Fu(\025)2575 3659 y Fl(e)2572 3681 y Fw(P)2623 3588 y Fl(\021)2673 3606 y Fy(\000)p Fx(1)2776 3681 y Fw(Q)p Fm(1)2889 3693 y Fx(i)p Fu(e)2943 3681 y Ft(\()p Fw(E)p Ft(\))p Fz(:)131 b FC(\(7\))706 3949 y Fk(\(Note)21 b(that)f(a)h(priori)g(the)f(right)h(hand)e(side)i(of)g(\(7\))f(may)g (depend)f(on)h Fz(\030)t Fk(;)h(we)h(assume)e(that)706 4048 y(it)h(does)f(not\).)612 4148 y FC(3\))k Fk(F)-9 b(or)21 b(any)e Ft(i)p Fz(e;)14 b Ft(i)p Fz(e)1144 4118 y Fy(0)1190 4148 y Fs(2)23 b Ft(sp)p Fw(E)p Fk(,)e Fz(e)i Fs(6)p Ft(=)f Fz(e)1632 4118 y Fy(0)1676 4148 y Fk(and)d Fz(\030)28 b(>)22 b Ft(0)p Fk(,)1074 4369 y Ft(lim)1062 4423 y Fu(\025)p Fy(!)p Fx(0)1215 4369 y Fz(\025)p Fm(1)1311 4381 y Fx(i)p Fu(e)1365 4369 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1563 4277 y Fl(\020)1613 4369 y Ft(\(i)p Fz(e)d Ft(+)f Fz(\025)1857 4339 y Fx(2)1894 4369 y Fz(\030)t Ft(\))1968 4347 y Fl(e)1966 4369 y Fw(P)h Fs(\000)2121 4347 y Fl(e)2119 4369 y Fw(PL)2225 4381 y Fu(\025)2271 4347 y Fl(e)2269 4369 y Fw(P)2319 4277 y Fl(\021)2369 4294 y Fy(\000)p Fx(1)2472 4369 y Fw(Q)p Fm(1)2585 4381 y Fx(i)p Fu(e)2635 4365 y Fn(0)2661 4369 y Ft(\()p Fw(E)p Ft(\))24 b(=)f(0)p Fz(;)1074 4582 y Ft(lim)1062 4636 y Fu(\025)p Fy(!)p Fx(0)1215 4582 y Fz(\025)p Fm(1)1311 4594 y Fx(i)p Fu(e)1361 4578 y Fn(0)1388 4582 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1586 4490 y Fl(\020)1636 4582 y Ft(\(i)p Fz(e)18 b Ft(+)g Fz(\025)1879 4552 y Fx(2)1917 4582 y Fz(\030)t Ft(\))1991 4560 y Fl(e)1989 4582 y Fw(P)h Fs(\000)2144 4560 y Fl(e)2142 4582 y Fw(P)o(L)2247 4594 y Fu(\025)2293 4560 y Fl(e)2291 4582 y Fw(P)2342 4490 y Fl(\021)2391 4507 y Fy(\000)p Fx(1)2494 4582 y Fw(Q)p Fm(1)2607 4594 y Fx(i)p Fu(e)2661 4582 y Ft(\()p Fw(E)p Ft(\))24 b(=)f(0)p Fz(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 523 100 a FB(12)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fk(Then)20 b(the)g(following)g(holds:)577 432 y(1.)k Ft(e)701 402 y Fu(tM)789 410 y Fq(st)864 432 y Fk(is)d(a)f(contr)o(active)f(semigr)l(oup.)577 531 y(2.)24 b(F)-9 b(or)21 b(any)e Fz(\030)28 b(>)22 b Ft(0)1001 650 y Fl(X)963 828 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1185 728 y Ft(lim)1173 783 y Fu(\025)p Fy(!)p Fx(0)1326 728 y Fm(1)1374 740 y Fx(i)p Fu(e)1428 728 y Ft(\()p Fw(E)p Ft(\))1561 661 y Fl(\000)1600 728 y Fz(\030)g Fs(\000)c Fz(\025)1789 694 y Fy(\000)p Fx(2)1879 728 y Ft(\()p Fw(L)1966 740 y Fu(\025)2029 728 y Fs(\000)g Ft(i)p Fz(e)p Ft(\))2206 661 y Fl(\001)2244 677 y Fy(\000)p Fx(1)2347 728 y Fw(P)k Ft(=)h(\()p Fz(\030)t Fw(P)c Fs(\000)f Fz(M)2814 740 y Fx(st)2870 728 y Ft(\))2902 694 y Fy(\000)p Fx(1)2991 728 y Fz(:)577 1001 y Fk(3.)24 b(F)-9 b(or)21 b(any)e Fz(f)32 b Fs(2)23 b Fz(C)1151 1013 y Fx(0)1189 1001 y Ft(\([0)p Fz(;)14 b Fs(1)p Ft([\))p Fk(,)1073 1231 y Ft(lim)1062 1285 y Fu(\025)p Fy(!)p Fx(0)1214 1118 y Fl(Z)1297 1138 y Fy(1)1260 1306 y Fx(0)1381 1231 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\)e)1562 1196 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1751 1171 y Fq(2)1788 1231 y Fw(P)p Ft(e)1876 1196 y Fu(t)p Fr(L)1940 1205 y Fp(\025)1978 1196 y Fu(=\025)2051 1171 y Fq(2)2088 1231 y Fw(P)p Ft(d)p Fz(t)23 b Ft(=)2326 1118 y Fl(Z)2409 1138 y Fy(1)2372 1306 y Fx(0)2493 1231 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\)e)2674 1196 y Fu(tM)2762 1204 y Fq(st)2816 1231 y Ft(d)p Fz(t:)301 b FC(\(8\))648 1529 y(Ne)o(xt)20 b(we)h(describe)f(the)h(time-dependent)d(v)o(ersion)h (of)i(the)g(weak)f(coupling)f(limit)i(for)f Fz(C)3247 1541 y Fx(0)3285 1529 y FC(-)523 1629 y(groups.)523 1803 y FA(Theor)o(em)g(3.)k Fk(Suppose)f(that)i(Assumptions)f(2.1,)g(2.3)g (and)g(2.4)g(ar)m(e)h(true)o(.)g(W)-8 b(e)26 b(mak)o(e)e(also)h(the)523 1903 y(following)20 b(assumptions:)612 2002 y FC(1\))k Fw(PQ)824 1981 y Fl(e)822 2002 y Fw(P)c Fk(and)1040 1981 y Fl(e)1038 2002 y Fw(PQP)f Fk(ar)m(e)i(bounded.)c(\(Note)j(that)g (this)g(assumption)f(guar)o(antees)f(that)3115 1981 y Fl(e)3112 2002 y Fw(PL)3218 2014 y Fu(\025)3264 1981 y Fl(e)3262 2002 y Fw(P)706 2102 y Fk(is)j(the)f(g)o(ener)o(ator)g(of)g (a)g Fz(C)1457 2114 y Fx(0)1495 2102 y Fk(-semigr)l(oup)f(on)h Ft(Ran)2145 2081 y Fl(e)2142 2102 y Fw(P)p Fk(\).)612 2202 y FC(2\))k Fk(Set)1355 2370 y 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Fx(0)1888 3171 y Fz(K)1959 3183 y Fu(\025)2002 3171 y Ft(\()p Fz(t)p Ft(\))j(=)g Fz(K)706 3381 y Fk(for)e(all)f Ft(0)j Fz(<)f(t)h(<)g Fs(1)p Fk(.)612 3481 y FC(4\))h Fk(Ther)m(e)c(e)n(xists)i(an)e(oper)o(ator)f Fz(M)1616 3493 y Fx(dyn)1749 3481 y Fk(suc)o(h)g(that)1357 3724 y Ft(s)p Fs(\000)35 b Ft(lim)1469 3774 y Fu(t)p Fy(!1)1640 3724 y Fz(t)1670 3690 y Fy(\000)p Fx(1)1773 3611 y Fl(Z)1856 3632 y Fu(t)1819 3800 y Fx(0)1899 3724 y Ft(e)1936 3690 y Fu(s)p Fr(E)2010 3724 y Fz(K)6 b Ft(e)2124 3690 y Fy(\000)p Fu(s)p Fr(E)2250 3724 y Ft(d)p Fz(s)23 b Ft(=)f Fz(M)2526 3736 y Fx(dyn)2638 3724 y Fz(:)523 3948 y Fk(Then)e(the)g(following)g(holds:)577 4098 y(1.)k Ft(e)701 4067 y Fu(tM)789 4076 y Fq(dyn)910 4098 y Fk(is)e(a)e(contr)o (active)f(semigr)l(oup.)577 4197 y(2.)24 b(F)-9 b(or)21 b(any)e Fz(y)26 b Fs(2)e Ft(Ran)o Fs(Y)k Fk(and)20 b Fz(t)1494 4209 y Fx(0)1554 4197 y Fz(>)j Ft(0)p Fk(,)1164 4392 y Ft(lim)1152 4446 y Fu(\025)p Fy(!)p Fx(0)1352 4392 y Ft(sup)1305 4462 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1492 4470 y Fq(0)1538 4392 y Fs(k)p Ft(e)1617 4357 y Fy(\000)p Fr(E)p Fu(t=\025)1806 4332 y Fq(2)1842 4392 y Fw(P)p Ft(e)1930 4357 y Fu(t)p Fr(L)1994 4366 y Fp(\025)2033 4357 y Fu(=\025)2106 4332 y Fq(2)2143 4392 y Fw(P)p Fz(y)d Fs(\000)e Ft(e)2375 4357 y Fu(tM)2463 4366 y Fq(dyn)2564 4392 y Fz(y)s Fs(k)k Ft(=)h(0)p Fz(:)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(13)648 282 y FC(One)20 b(of)g(possible)f Fz(C)1254 252 y Fy(\003)1248 303 y Fx(0)1293 282 y FC(-v)o(ersions)g(of)h(the)g(abo)o(v)o(e)e(theorem)h (is)i(gi)n(v)o(en)e(belo)n(w)-5 b(.)523 459 y FA(Theor)o(em)26 b(3*)g Fk(Suppose)e(that)i(Assumptions)g(2.1*,)f(2.3*)g(and)g(2.4)h(ar) m(e)g(true)o(.)g(W)-8 b(e)27 b(mak)o(e)f(also)523 558 y(the)20 b(following)g(assumptions:)595 658 y FC(0\))41 b Ft(e)743 628 y Fu(t)p Fr(E)832 658 y Fk(is)21 b(a)f Fz(C)1029 670 y Fx(0)1067 658 y Fk(-gr)l(oup.)e(\(W)-8 b(e)21 b(alr)m(eady)f(know)g(that)g(it)g(is)i(a)e Fz(C)2365 628 y Fy(\003)2359 679 y Fx(0)2403 658 y Fk(-gr)l(oup\).)595 768 y FC(1\))41 b Fw(PQ)824 747 y Fl(e)822 768 y Fw(P)22 b Fk(and)1045 747 y Fl(e)1042 768 y Fw(PQP)g Fk(ar)m(e)h(w*)g (continuous.)d(\(Note)i(that)g(this)h(assumption)e(guar)o(antees)f (that)708 846 y Fl(e)706 868 y Fw(PL)812 880 y Fu(\025)858 846 y Fl(e)856 868 y Fw(P)g Fk(is)h(a)g(g)o(ener)o(ator)e(of)h(a)g Fz(C)1624 838 y Fy(\003)1618 889 y Fx(0)1663 868 y Fk(-semigr)l(oup)f (on)h Ft(Ran)2313 846 y Fl(e)2310 868 y Fw(P)p Fk(\).)595 968 y FC(2\))41 b Fk(In)20 b(the)g(sense)h(of)f(a)g(w*)h(inte)m(gr)o (al)e([BR1)o(])i(we)g(set)1355 1193 y Fz(K)1426 1205 y Fu(\025)1469 1193 y Ft(\()p Fz(t)p Ft(\))j(:=)1698 1080 y Fl(Z)1781 1100 y Fu(\025)1820 1075 y Fn(\000)p Fq(2)1897 1100 y Fu(t)1744 1268 y Fx(0)1940 1193 y Ft(e)1977 1158 y Fy(\000)p Fu(s)p Fr(E)2103 1193 y Fw(PQ)p Ft(e)2256 1158 y Fu(s)2286 1143 y Fj(e)2287 1158 y Fr(PL)2362 1167 y Fp(\025)2399 1143 y Fj(e)2400 1158 y Fr(P)2440 1193 y Fw(QP)p Ft(d)p Fz(s:)510 b FC(\(10\))706 1388 y Fk(W)-8 b(e)21 b(suppose)e(that)h(for)h(all)f Fz(t)1519 1400 y Fx(0)1580 1388 y Fz(>)i Ft(0)p Fk(,)e(ther)m(e)h(e)n(xists)g Fz(c)g Fk(suc)o(h)f(that)1582 1542 y Ft(sup)1544 1616 y Fy(j)p Fu(\025)p Fy(j)p Fu(<\025)1714 1624 y Fq(0)1828 1542 y Ft(sup)1781 1612 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1968 1620 y Fq(0)2014 1542 y Fs(k)p Fz(K)2127 1554 y Fu(\025)2169 1542 y Ft(\()p Fz(t)p Ft(\))p Fs(k)k(\024)e Fz(c:)595 1759 y FC(3\))41 b Fk(ther)m(e)20 b(e)n(xists)i(a)e(w*)h (continuous)d(oper)o(ator)h Fz(K)26 b Fk(on)20 b Ft(Ran)p Fw(P)g Fk(suc)o(h)g(that)1747 1913 y Ft(lim)1735 1967 y Fu(\025)p Fy(!)p Fx(0)1888 1913 y Fz(K)1959 1925 y Fu(\025)2002 1913 y Ft(\()p Fz(t)p Ft(\))j(=)g Fz(K)706 2095 y Fk(for)e(all)f Ft(0)j Fz(<)f(t)h(<)g Fs(1)p Fk(.)595 2195 y FC(4\))41 b Fk(Ther)m(e)20 b(e)n(xists)i(an)e(oper)o(ator)f Fz(M)1616 2207 y Fx(dyn)1749 2195 y Fk(suc)o(h)g(that)1400 2409 y Ft(s)p Fs(\000)34 b Ft(lim)1511 2459 y Fu(t)p Fy(!1)1682 2409 y Fz(t)1712 2375 y Fy(\000)p Fx(1)1815 2296 y Fl(Z)1898 2317 y Fu(t)1861 2485 y Fx(0)1941 2409 y Ft(e)1978 2375 y Fu(s)p Fr(E)2053 2409 y Fz(K)6 b Ft(e)2167 2375 y Fy(\000)p Fu(s)p Fr(E)2315 2409 y Ft(=)23 b Fz(M)2484 2421 y Fx(dyn)2596 2409 y Fz(:)523 2605 y Fk(Then)d(the)g(same)g (conclusions)f(as)i(in)f(Theor)m(em)g(3)g(hold.)648 2704 y FC(Theorem)14 b(3)j(is)g(due)f(to)h(Da)n(vies)g(\(we)f(put)h (together)e(Theorem)f(5.18)i(and)g(5.11)f(from)h([Da3)o(]\).)523 2804 y(Note)31 b(that,)f(follo)n(wing)f(Da)n(vies,)i(in)g(Theorems)e(3) h(and)g(3*)h(we)g(do)f(not)g(mak)o(e)g(Assumption)523 2904 y(2.2)24 b(about)f(the)i(\002nite)f(dimension)f(of)h Ft(Ran)p Fw(P)p FC(.)g(Instead,)g(we)h(mak)o(e)e(the)i(assumption)e (4\))h(about)523 3003 y(spectral)29 b(a)n(v)o(eraging.)f(If)h(we)h (impose)g(Assumption)e(2.2,)h(then)g(we)h(can)g(drop)e(4\))h(and)h(mak) o(e)523 3103 y(some)20 b(other)f(minor)h(simpli\002cations,)f(as)i(is)g (described)e(belo)n(w:)523 3249 y FA(Theor)o(em)h(4.)k Fk(Suppose)h(that)g(Assumptions)h(2.1,)f(2.2,)g(2.3)g(and)h(2.4)f(or)h (2.1*,)f(2.2,)g(2.3*)g(and)523 3348 y(2.4)20 b(ar)m(e)g(true)o(.)g(Set) 1277 3462 y Fz(K)1348 3474 y Fu(\025)1391 3462 y Ft(\()p Fz(t)p Ft(\))25 b(:=)1621 3395 y Fl(R)1676 3415 y Fu(\025)1715 3390 y Fn(\000)p Fq(2)1793 3415 y Fu(t)1660 3491 y Fx(0)1836 3462 y Ft(e)1873 3432 y Fy(\000)p Fu(s)p Fr(E)1999 3462 y Fw(PQ)p Ft(e)2152 3432 y Fu(s)2182 3417 y Fj(e)2183 3432 y Fr(P)o(L)2257 3441 y Fp(\025)2295 3417 y Fj(e)2296 3432 y Fr(P)2336 3462 y Fw(QP)p Ft(d)p Fz(s:)523 3590 y Fk(W)-8 b(e)21 b(mak)o(e)f(also)g(the)h(following)e(assumptions:)612 3689 y FC(1\))24 b Fk(W)-8 b(e)21 b(suppose)e(that)h(for)h(all)f Fz(t)1519 3701 y Fx(0)1580 3689 y Fz(>)i Ft(0)p Fk(,)e(ther)m(e)h(e)n (xists)g Fz(c)g Fk(suc)o(h)f(that)1582 3843 y Ft(sup)1544 3917 y Fy(j)p Fu(\025)p Fy(j)p Fu(<\025)1714 3925 y Fq(0)1828 3843 y Ft(sup)1781 3913 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1968 3921 y Fq(0)2014 3843 y Fs(k)p Fz(K)2127 3855 y Fu(\025)2169 3843 y Ft(\()p Fz(t)p Ft(\))p Fs(k)k(\024)e Fz(c:)612 4060 y FC(2\))i Fk(Ther)m(e)c(e)n(xists)i(an)e(oper)o(ator)f Fz(K)26 b Fk(on)20 b Ft(Ran)p Fw(P)g Fk(suc)o(h)g(that)1747 4214 y Ft(lim)1735 4268 y Fu(\025)p Fy(!)p Fx(0)1888 4214 y Fz(K)1959 4226 y Fu(\025)2002 4214 y Ft(\()p Fz(t)p Ft(\))j(=)g Fz(K)706 4396 y Fk(for)e(all)f Ft(0)j Fz(<)f(t)h(<)g Fs(1)p Fk(.)e(W)-8 b(e)21 b(set)1480 4563 y Fz(M)1561 4575 y Fx(dyn)1696 4563 y Ft(:=)1845 4484 y Fl(X)1807 4663 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)2018 4563 y Fm(1)2066 4575 y Fx(i)p Fu(e)2120 4563 y Ft(\()p Fw(E)p Ft(\))p Fz(K)6 b Fm(1)2364 4575 y Fx(i)p Fu(e)2418 4563 y Ft(\()p Fw(E)p Ft(\))p eop end %%Page: 14 14 TeXDict begin 14 13 bop 523 100 a FB(14)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fk(Then)20 b(the)g(following)g(holds:)577 429 y(1.)k Ft(e)701 399 y Fu(tM)789 408 y Fq(dyn)910 429 y Fk(is)e(a)e(contr)o(active)f(semigr)l(oup.)577 529 y(2.)24 b(F)-9 b(or)21 b(any)e Fz(t)971 541 y Fx(0)1032 529 y Fz(>)j Ft(0)p Fk(,)1208 718 y Ft(lim)1196 772 y Fu(\025)p Fy(!)p Fx(0)1396 718 y Ft(sup)1348 787 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1535 795 y Fq(0)1582 718 y Fs(k)p Ft(e)1661 683 y Fy(\000)p Fr(E)p Fu(t=\025)1850 658 y Fq(2)1886 718 y Fw(P)p Ft(e)1974 683 y Fu(t)p Fr(L)2038 692 y Fp(\025)2076 683 y Fu(=\025)2149 658 y Fq(2)2186 718 y Fw(P)c Fs(\000)g Ft(e)2375 683 y Fu(tM)2463 692 y Fq(dyn)2564 718 y Fs(k)23 b Ft(=)f(0)p Fz(:)393 b FC(\(11\))648 1017 y(Note)28 b(that)g(if)h(there)e(e)o(xists)i(an)f(operator)f Fz(M)1976 1029 y Fx(st)2060 1017 y FC(satisfying)h(\(8\),)f(and)h(an)g (operator)f Fz(M)3201 1029 y Fx(dyn)523 1117 y FC(satisfying)19 b(\(11\),)f(then)g(the)o(y)h(clearly)g(coincide.)f(In)g(our)h(last)h (theorem)e(of)h(this)h(section)f(we)g(will)523 1217 y(describe)g(a)i (connection)d(between)h Fz(M)1641 1229 y Fx(st)1697 1217 y FC(,)i Fz(M)1820 1229 y Fx(dyn)1953 1217 y FC(and)e(the)h(LSO.)523 1385 y FA(Theor)o(em)g(5.)k Fk(Suppose)f(that)h(Assumptions)g(2.1,)f (2.2,)g(2.3)h(and)f(2.4,)h(or)h(2.1*,)d(2.2,)i(2.3*)f(and)523 1485 y(2.4)d(ar)m(e)g(true)o(.)g(Suppose)e(also)j(that)e(the)i (following)e(conditions)g(hold:)612 1597 y FC(1\))706 1530 y Fl(R)761 1551 y Fy(1)745 1627 y Fx(0)884 1597 y Ft(sup)846 1671 y Fy(j)p Fu(\025)p Fy(j\024)p Fu(\025)1016 1679 y Fq(0)1062 1597 y Fs(k)p Fw(PQ)p Ft(e)1257 1567 y Fu(s)1287 1552 y Fj(e)1288 1567 y Fr(P)o(L)1362 1576 y Fp(\025)1399 1552 y Fj(e)1400 1567 y Fr(P)1440 1597 y Fw(QP)p Fs(k)p Ft(d)p Fz(s)j(<)h Fs(1)p Fz(:)612 1789 y FC(2\))h Fk(F)-9 b(or)21 b(any)e Fz(s)k(>)g Ft(0)p Fk(,)32 b Ft(lim)1216 1843 y Fu(\025)p Fy(!)p Fx(0)1368 1789 y Fw(PQ)p Ft(e)1521 1759 y Fu(s)1551 1744 y Fj(e)1552 1759 y Fr(P)o(L)1626 1768 y Fp(\025)1664 1744 y Fj(e)1665 1759 y Fr(P)1705 1789 y Fw(QP)22 b Ft(=)h Fw(PQ)p Ft(e)2084 1759 y Fu(s)2114 1744 y Fj(e)2115 1759 y Fr(P)o(L)2189 1767 y Fq(0)2225 1789 y Fw(QP)p Fz(:)523 1917 y Fk(Then)577 2064 y(1.)h(Assumption)d(2.5)h(holds,)f(and)g(hence)g(the)h(LSO)g(for)h Ft(\()p Fw(P)p Fz(;)14 b Fw(L)2376 2076 y Fx(0)2413 2064 y Fz(;)g Fw(Q)p Ft(\))p Fk(,)23 b(de\002ned)d(in)i(\(3\))f(and)h(de-) 664 2163 y(noted)d Fz(M)9 b Fk(,)21 b(e)n(xists.)577 2263 y(2.)j Ft(e)701 2233 y Fu(tM)821 2263 y Fk(is)d(a)f(contr)o (active)f(semigr)l(oup.)577 2362 y(3.)24 b(The)c(assumptions)f(of)i (Theor)m(em)e(2)h(hold)f(and)h Fz(M)31 b Ft(=)23 b Fz(M)2295 2374 y Fx(st)2351 2362 y Fk(,)d(consequently)-5 b(,)18 b(for)j(any)e Fz(\030)27 b(>)c Ft(0)998 2554 y(lim)986 2608 y Fu(\025)p Fy(!)p Fx(0)1177 2475 y Fl(X)1139 2653 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1349 2554 y Fm(1)1397 2566 y Fx(i)p Fu(e)1452 2554 y Ft(\()p Fw(E)p Ft(\))1585 2486 y Fl(\000)1623 2554 y Fz(\030)g Fs(\000)18 b Fz(\025)1813 2519 y Fy(\000)p Fx(2)1903 2554 y Ft(\()p Fw(L)1990 2566 y Fu(\025)2052 2554 y Fs(\000)g Ft(i)p Fz(e)p Ft(\))2229 2486 y Fl(\001)2267 2502 y Fy(\000)p Fx(1)2370 2554 y Fw(P)23 b Ft(=)g(\()p Fz(\030)t Fw(P)18 b Fs(\000)g Fz(M)9 b Ft(\))2878 2519 y Fy(\000)p Fx(1)2967 2554 y Fz(:)577 2815 y Fk(4.)24 b(The)c(assumptions)g(of)g(Theor)m(em)g(4)g(hold)g(and) f Fz(M)32 b Ft(=)22 b Fz(M)2296 2827 y Fx(dyn)2408 2815 y Fk(,)f(consequently)1253 3010 y Ft(lim)1241 3064 y Fu(\025)p Fy(!)p Fx(0)1441 3010 y Ft(sup)1393 3080 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1580 3088 y Fq(0)1627 3010 y Fs(k)p Ft(e)1706 2976 y Fy(\000)p Fr(E)p Fu(t=\025)1895 2951 y Fq(2)1931 3010 y Fw(P)p Ft(e)2019 2976 y Fu(t)p Fr(L)2083 2985 y Fp(\025)2121 2976 y Fu(=\025)2194 2951 y Fq(2)2231 3010 y Fw(P)d Fs(\000)g Ft(e)2420 2976 y Fu(tM)2519 3010 y Fs(k)23 b Ft(=)f(0)p Fz(:)523 3378 y FA(3.2)40 b(Pr)o(oof)19 b(of)h(the)g(stationary)f(weak)h(coupling)g (limit)523 3556 y(Pr)o(oof)f(of)g(Theor)o(em)i(2.)f FC(W)-7 b(e)21 b(follo)n(w)e([DF2].)h(Let)g Ft(i)p Fz(e)j Fs(2)g Ft(sp)p Fw(E)p FC(.)e(Set)1074 3727 y Fz(G)1139 3739 y Fu(\025)1183 3727 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))23 b(:=)h Fz(\030)t Fw(P)19 b Ft(+)f Fz(\025)1762 3697 y Fy(\000)p Fx(2)1851 3727 y Ft(\(i)p Fz(e)p Fw(P)h Fs(\000)f Fw(E)p Ft(\))1521 3940 y Fs(\000)p Fw(PQ)1716 3848 y Fl(\020)1765 3940 y Ft(\()p Fz(\025)1845 3910 y Fx(2)1883 3940 y Fz(\030)k Ft(+)c(i)p Fz(e)p Ft(\))2120 3919 y Fl(e)2118 3940 y Fw(P)h Fs(\000)2273 3919 y Fl(e)2271 3940 y Fw(P)o(L)2376 3952 y Fu(\025)2423 3919 y Fl(e)2420 3940 y Fw(P)2471 3848 y Fl(\021)2521 3865 y Fy(\000)p Fx(1)2623 3940 y Fw(QP)p Fz(:)523 4137 y FC(By)i(the)f(so-called)f (Feshbach)h(formula)e(\(see)j(e.g.)e([DJ1,)h(BFS1]\),)g(for)g Fz(\030)27 b(>)c Ft(0)d FC(we)g(ha)n(v)o(e)1212 4329 y Fz(G)1277 4341 y Fu(\025)1321 4329 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))1524 4295 y Fy(\000)p Fx(1)1636 4329 y Ft(=)23 b Fw(P)1789 4262 y Fl(\000)1826 4329 y Fz(\030)g Ft(+)18 b Fz(\025)2016 4295 y Fy(\000)p Fx(2)2105 4329 y Ft(\(i)p Fz(e)h Fs(\000)f Fw(L)2356 4341 y Fu(\025)2400 4329 y Ft(\))2432 4262 y Fl(\001)2470 4278 y Fy(\000)p Fx(1)2573 4329 y Fw(P)523 4506 y FC(This)i(and)g(the)g(dissipati)n (vity)g(of)g Fw(L)1524 4518 y Fu(\025)1588 4506 y FC(implies)h(the)f (bound)1544 4682 y Fs(k)p Fz(G)1651 4694 y Fu(\025)1694 4682 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))1897 4648 y Fy(\000)p Fx(1)1987 4682 y Fs(k)22 b(\024)h Fz(\030)2179 4648 y Fy(\000)p Fx(1)2268 4682 y Fz(:)883 b FC(\(12\))p eop end %%Page: 15 15 TeXDict begin 15 14 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(15)648 282 y FC(Write)20 b(for)g(shortness)g Fz(G)h FC(instead)f(of)g Fz(G)1802 294 y Fu(\025)1845 282 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))p FC(.)21 b(F)o(or)f Ft(i)p Fz(e)2287 252 y Fy(0)2333 282 y Fs(2)k Ft(sp)o Fw(E)p FC(,)d(set)1587 460 y Fw(P)1638 472 y Fu(e)1669 456 y Fn(0)1718 460 y Ft(:=)i Fm(1)1877 472 y Fx(i)p Fu(e)1927 456 y Fn(0)1954 460 y Ft(\()p Fw(E)p Ft(\))p Fz(;)1587 631 y Fw(P)p 1638 610 32 3 v 13 x Fu(e)1669 625 y Fn(0)1718 631 y Ft(:=)g Fw(P)18 b Fs(\000)g Fm(1)2029 643 y Fx(i)p Fu(e)2079 626 y Fn(0)2106 631 y Ft(\()p Fw(E)p Ft(\))p Fz(:)523 805 y FC(Decompose)h Fz(G)k Ft(=)g Fz(G)1177 817 y Fx(diag)1322 805 y Ft(+)18 b Fz(G)1470 817 y Fx(o\013)1572 805 y FC(into)i(its)i (diagonal)c(and)i(of)n(f-diagonal)d(part:)1171 985 y Fz(G)1236 997 y Fx(diag)1388 985 y Ft(:=)1564 923 y Fl(P)1499 1060 y Fx(i)p Fu(e)1549 1043 y Fn(0)1571 1060 y Fy(2)p Fx(sp)p Fr(E)1732 985 y Fw(P)1783 997 y Fu(e)1814 981 y Fn(0)1841 985 y Fz(G)p Fw(P)1957 997 y Fu(e)1988 981 y Fn(0)2015 985 y Fz(;)1217 1179 y(G)1282 1191 y Fx(o\013)1388 1179 y Ft(:=)1564 1117 y Fl(P)1499 1254 y Fx(i)p Fu(e)1549 1237 y Fn(0)1571 1254 y Fy(2)p Fx(sp)p Fr(E)1732 1179 y Fw(P)1783 1191 y Fu(e)1814 1175 y Fn(0)1841 1179 y Fz(G)p Fw(P)p 1957 1159 V 14 x Fu(e)1988 1173 y Fn(0)2038 1179 y Ft(=)2191 1117 y Fl(P)2125 1254 y Fx(i)p Fu(e)2175 1237 y Fn(0)2198 1254 y Fy(2)p Fx(sp)p Fr(E)2358 1179 y Fw(P)p 2409 1159 V 14 x Fu(e)2441 1173 y Fn(0)2467 1179 y Fz(G)p Fw(P)2583 1191 y Fu(e)2614 1175 y Fn(0)2641 1179 y Fz(:)648 1419 y FC(First)f(we)f(w)o(ould)g(lik)o(e)h(to)f(sho)n (w)g(that)h(for)f Fz(\030)27 b(>)22 b Ft(0)16 b FC(and)f(small)h (enough)d Fz(\025)p FC(,)j Fz(G)2764 1431 y Fx(diag)2907 1419 y FC(is)g(in)m(v)o(ertible.)523 1518 y(By)26 b(an)f(application)g (of)g(the)g(Neumann)f(series,)i Fw(P)p 2004 1497 36 3 v 13 x Fu(e)2039 1518 y Fz(G)2104 1530 y Fx(diag)2258 1518 y FC(is)g(in)m(v)o(ertible)e(on)h Ft(Ran)p Fw(P)p 2988 1497 V 13 x Fu(e)3023 1518 y FC(,)h(and)f(we)523 1618 y(ha)n(v)o(e)20 b(the)g(bound)1609 1718 y Fs(k)p Fw(P)p 1702 1697 V 12 x Fu(e)1737 1718 y Fz(G)1802 1682 y Fy(\000)p Fx(1)1802 1743 y(diag)1929 1718 y Fs(k)j(\024)g Fz(c\025)2166 1683 y Fx(2)2203 1718 y Fz(:)948 b FC(\(13\))523 1865 y(It)21 b(is)g(more)e(complicated)g(to)h(pro)o(v)o(e)e(that)j Fw(P)1778 1877 y Fu(e)1813 1865 y Fz(G)1878 1877 y Fx(diag)2026 1865 y FC(is)g(in)m(v)o(erible)d(on)i Ft(Ran)p Fw(P)2717 1877 y Fu(e)2752 1865 y FC(.)648 1964 y(W)-7 b(e)26 b(\002x)f Fz(\030)35 b(>)c Ft(0)p FC(.)25 b(W)-7 b(e)26 b(kno)n(w)e(that)h Fz(G)g FC(is)h(in)m(v)o(ertible)d(and)h Fs(k)p Fz(G)2404 1934 y Fy(\000)p Fx(1)2493 1964 y Fs(k)31 b(\024)g Fz(\030)2702 1934 y Fy(\000)p Fx(1)2792 1964 y FC(.)25 b(Hence)f(we)h(can)523 2064 y(write)1453 2163 y Fz(G)1518 2175 y Fx(diag)1645 2163 y Fz(G)1710 2129 y Fy(\000)p Fx(1)1822 2163 y Ft(=)e Fm(1)18 b Fs(\000)g Fz(G)2124 2175 y Fx(o\013)2205 2163 y Fz(G)2270 2129 y Fy(\000)p Fx(1)2360 2163 y Fz(:)523 2311 y FC(Therefore)1304 2390 y Fw(P)1355 2402 y Fu(e)1390 2390 y Fz(G)1455 2402 y Fx(diag)1582 2390 y Fz(G)1647 2359 y Fy(\000)p Fx(1)1760 2390 y Ft(=)k Fw(P)1898 2402 y Fu(e)1952 2390 y Fs(\000)c Fw(P)2086 2402 y Fu(e)2121 2390 y Fz(G)2186 2402 y Fx(o\013)2268 2390 y Fw(P)p 2319 2368 V 12 x Fu(e)2354 2390 y Fz(G)2419 2359 y Fy(\000)p Fx(1)2509 2390 y Fz(;)1304 2560 y Fw(P)p 1355 2539 V 12 x Fu(e)1390 2560 y Fz(G)1455 2572 y Fx(diag)1582 2560 y Fz(G)1647 2530 y Fy(\000)p Fx(1)1760 2560 y Ft(=)k Fw(P)p 1898 2539 V 12 x Fu(e)1952 2560 y Fs(\000)c Fw(P)p 2086 2539 V 12 x Fu(e)2121 2560 y Fz(G)2186 2572 y Fx(o\013)2268 2560 y Fz(G)2333 2530 y Fy(\000)p Fx(1)2422 2560 y Fz(:)3174 2476 y FC(\(14\))523 2703 y(The)i(latter)g(identity)g(can)g(be)g(for)f (small)i(enough)d Fz(\025)j FC(transformed)d(into)1251 2882 y Fw(P)p 1302 2861 V 12 x Fu(e)1337 2882 y Fz(G)1402 2848 y Fy(\000)p Fx(1)1515 2882 y Ft(=)k Fz(G)1667 2846 y Fy(\000)p Fx(1)1667 2907 y(diag)1794 2882 y Fw(P)p 1845 2861 V 12 x Fu(e)1899 2882 y Fs(\000)c Fz(G)2047 2846 y Fy(\000)p Fx(1)2047 2907 y(diag)2174 2882 y Fw(P)p 2225 2861 V 12 x Fu(e)2261 2882 y Fz(G)2326 2894 y Fx(o\013)2407 2882 y Fz(G)2472 2848 y Fy(\000)p Fx(1)2562 2882 y Fz(:)589 b FC(\(15\))523 3061 y(W)-7 b(e)21 b(insert)g(\(15\))e(into)h(the)g (\002rst)h(identity)e(of)h(\(14\))f(to)i(obtain)828 3239 y Fw(P)879 3251 y Fu(e)915 3239 y Fz(G)980 3251 y Fx(diag)1106 3239 y Fz(G)1171 3205 y Fy(\000)p Fx(1)1284 3239 y Ft(=)i Fw(P)1423 3251 y Fu(e)1476 3239 y Fs(\000)18 b Fw(P)1610 3251 y Fu(e)1646 3239 y Fz(G)1711 3251 y Fx(o\013)1792 3239 y Fw(P)p 1843 3218 V 13 x Fu(e)1879 3239 y Fz(G)1944 3204 y Fy(\000)p Fx(1)1944 3264 y(diag)2089 3239 y Ft(+)g Fw(P)2223 3251 y Fu(e)2259 3239 y Fz(G)2324 3251 y Fx(o\013)2405 3239 y Fw(P)p 2456 3218 V 13 x Fu(e)2492 3239 y Fz(G)2557 3204 y Fy(\000)p Fx(1)2557 3264 y(diag)2683 3239 y Fz(G)2748 3251 y Fx(o\013)2830 3239 y Fz(G)2895 3205 y Fy(\000)p Fx(1)2984 3239 y Fz(:)167 b FC(\(16\))523 3418 y(W)-7 b(e)21 b(multiply)f(\(16\))f(from)g(the)h(right)g(by)f Fw(P)1750 3430 y Fu(e)1807 3418 y FC(to)h(get)1005 3597 y Fw(P)1056 3609 y Fu(e)1091 3597 y Fz(G)1156 3609 y Fx(diag)1283 3597 y Fw(P)1334 3609 y Fu(e)1370 3597 y Fz(G)1435 3562 y Fy(\000)p Fx(1)1524 3597 y Fw(P)1575 3609 y Fu(e)1634 3597 y Ft(=)i Fw(P)1772 3609 y Fu(e)1826 3597 y Ft(+)c Fw(P)1960 3609 y Fu(e)1995 3597 y Fz(G)2060 3609 y Fx(o\013)2142 3597 y Fw(P)p 2193 3575 V 12 x Fu(e)2228 3597 y Fz(G)2293 3561 y Fy(\000)p Fx(1)2293 3622 y(diag)2420 3597 y Fz(G)2485 3609 y Fx(o\013)2567 3597 y Fz(G)2632 3562 y Fy(\000)p Fx(1)2721 3597 y Fw(P)2772 3609 y Fu(e)2808 3597 y Fz(:)343 b FC(\(17\))523 3775 y(No)n(w)-5 b(,)19 b(using)1627 3875 y Ft(lim)1615 3929 y Fu(\025)p Fy(!)p Fx(0)1767 3875 y Fz(\025)p Fs(k)p Fz(G)1922 3887 y Fx(o\013)2004 3875 y Fs(k)j Ft(=)h(0)p Fz(;)953 b FC(\(18\))523 4050 y(\(12\))19 b(and)h(\(13\))f(we)h(obtain)1316 4228 y Ft(lim)1305 4282 y Fu(\025)p Fy(!)p Fx(0)1457 4228 y Fw(P)1508 4240 y Fu(e)1543 4228 y Fz(G)1608 4240 y Fx(o\013)1690 4228 y Fw(P)p 1741 4207 V 13 x Fu(e)1776 4228 y Fz(G)1841 4193 y Fy(\000)p Fx(1)1841 4253 y(diag)1968 4228 y Fz(G)2033 4240 y Fx(o\013)2115 4228 y Fz(G)2180 4194 y Fy(\000)p Fx(1)2269 4228 y Fw(P)2320 4240 y Fu(e)2379 4228 y Ft(=)i(0)p Fz(:)523 4436 y FC(Thus,)e(for)f(small)i(enough)d Fz(\025)p FC(,)1618 4535 y Fw(P)1669 4547 y Fu(e)1705 4535 y Fz(G)1770 4547 y Fx(diag)1897 4535 y Fz(B)1960 4547 y Fx(1)2020 4535 y Ft(=)23 b Fw(P)2159 4547 y Fu(e)2194 4535 y Fz(;)523 4682 y FC(where)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 523 100 a FB(16)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)974 312 y Fz(B)1037 324 y Fx(1)1098 312 y Ft(:=)22 b Fw(P)1259 324 y Fu(e)1295 312 y Fz(G)1360 277 y Fy(\000)p Fx(1)1449 312 y Fw(P)1500 324 y Fu(e)1549 220 y Fl(\020)1599 312 y Fw(P)1650 324 y Fu(e)1704 312 y Ft(+)c Fw(P)1838 324 y Fu(e)1873 312 y Fz(G)1938 324 y Fx(o\013)2020 312 y Fw(P)p 2071 291 36 3 v 12 x Fu(e)2106 312 y Fz(G)2171 276 y Fy(\000)p Fx(1)2171 337 y(diag)2298 312 y Fz(G)2363 324 y Fx(o\013)2445 312 y Fz(G)2510 277 y Fy(\000)p Fx(1)2599 312 y Fw(P)2650 324 y Fu(e)2686 220 y Fl(\021)2735 237 y Fy(\000)p Fx(1)2838 312 y Fz(:)523 479 y FC(Similarly)-5 b(,)19 b(for)h(small)g(enough)f Fz(\025)p FC(,)i(we)f(\002nd)g Fz(B)1872 491 y Fx(2)1930 479 y FC(such)g(that)1618 659 y Fz(B)1681 671 y Fx(2)1719 659 y Fw(P)1770 671 y Fu(e)1805 659 y Fz(G)1870 671 y Fx(diag)2020 659 y Ft(=)j Fw(P)2159 671 y Fu(e)2194 659 y Fz(:)523 838 y FC(This)d(implies)h(that)f Fw(P)1153 850 y Fu(e)1188 838 y Fz(G)1253 850 y Fx(diag)1401 838 y FC(is)h(in)m(v)o(ertible)e(on)g Ft(Ran)p Fw(P)2115 850 y Fu(e)2151 838 y FC(.)648 938 y(Ne)o(xt,)g(we)i(can)f(write)895 1117 y Fz(G)960 1083 y Fy(\000)p Fx(1)1073 1117 y Ft(=)i Fz(G)1225 1082 y Fy(\000)p Fx(1)1225 1143 y(diag)1371 1117 y Fs(\000)c Fz(G)1519 1082 y Fy(\000)p Fx(1)1519 1143 y(diag)1646 1117 y Fz(G)1711 1129 y Fx(o\013)1792 1117 y Fz(G)1857 1082 y Fy(\000)p Fx(1)1857 1143 y(diag)2003 1117 y Ft(+)g Fz(G)2151 1082 y Fy(\000)p Fx(1)2151 1143 y(diag)2278 1117 y Fz(G)2343 1129 y Fx(o\013)2424 1117 y Fz(G)2489 1082 y Fy(\000)p Fx(1)2489 1143 y(diag)2616 1117 y Fz(G)2681 1129 y Fx(o\013)2763 1117 y Fz(G)2828 1083 y Fy(\000)p Fx(1)2917 1117 y Fz(:)523 1297 y FC(Hence,)827 1492 y Fw(P)878 1504 y Fu(e)914 1492 y Fz(G)979 1458 y Fy(\000)p Fx(1)1091 1492 y Ft(=)23 b Fw(P)1230 1504 y Fu(e)1265 1492 y Fz(G)1330 1457 y Fy(\000)p Fx(1)1330 1517 y(diag)1471 1400 y Fl(\020)1520 1492 y Fm(1)c Fs(\000)f Fz(G)1735 1504 y Fx(o\013)1816 1492 y Fw(P)p 1867 1471 V 13 x Fu(e)1903 1492 y Fz(G)1968 1457 y Fy(\000)p Fx(1)1968 1517 y(diag)2113 1492 y Ft(+)g Fz(G)2261 1504 y Fx(o\013)2343 1492 y Fw(P)p 2394 1471 V 13 x Fu(e)2429 1492 y Fz(G)2494 1457 y Fy(\000)p Fx(1)2494 1517 y(diag)2621 1492 y Fz(G)2686 1504 y Fx(o\013)2768 1492 y Fz(G)2833 1458 y Fy(\000)p Fx(1)2922 1400 y Fl(\021)2985 1492 y Fz(:)166 b FC(\(19\))523 1692 y(Therefore,)18 b(for)h(a)i(\002x)o(ed)f Fz(\030)t FC(,)g(by)g(\(12\),)f(\(13\))g(and)h(\(18\))f(we)h(see)h(that)f(as)h Fz(\025)j Fs(!)f Ft(0)d FC(we)h(ha)n(v)o(e)1163 1871 y Fs(\000)p Fz(G)1293 1883 y Fx(o\013)1374 1871 y Fw(P)p 1425 1850 V 13 x Fu(e)1460 1871 y Fz(G)1525 1836 y Fy(\000)p Fx(1)1525 1896 y(diag)1670 1871 y Ft(+)d Fz(G)1818 1883 y Fx(o\013)1900 1871 y Fw(P)p 1951 1850 V 13 x Fu(e)1986 1871 y Fz(G)2051 1836 y Fy(\000)p Fx(1)2051 1896 y(diag)2178 1871 y Fz(G)2243 1883 y Fx(o\013)2325 1871 y Fz(G)2390 1837 y Fy(\000)p Fx(1)2502 1871 y Fs(!)23 b Ft(0)p Fz(:)523 2051 y FC(Therefore,)g(for)h(small)h(enough)e Fz(\025)p FC(,)j(we)f(can)g(in)m(v)o(ert)f(the)g(e)o(xpression)g(in)h(the)g (brack)o(et)f(of)h(\(19\).)523 2151 y(Consequently)-5 b(,)643 2369 y Fw(P)694 2381 y Fu(e)729 2369 y Ft(\()p Fz(G)826 2334 y Fy(\000)p Fx(1)826 2394 y(diag)972 2369 y Fs(\000)18 b Fz(G)1120 2339 y Fy(\000)p Fx(1)1210 2369 y Ft(\))23 b(=)h Fw(P)1405 2381 y Fu(e)1441 2369 y Fz(G)1506 2339 y Fy(\000)p Fx(1)1609 2277 y Fl(\020)1659 2369 y Ft(1)18 b Fs(\000)g Fz(G)1867 2381 y Fx(o\013)1948 2369 y Fw(P)p 1999 2348 V 12 x Fu(e)2035 2369 y Fz(G)2100 2334 y Fy(\000)p Fx(1)2100 2394 y(diag)2245 2369 y Ft(+)g Fz(G)2393 2381 y Fx(o\013)2475 2369 y Fw(P)p 2526 2348 V 12 x Fu(e)2561 2369 y Fz(G)2626 2334 y Fy(\000)p Fx(1)2626 2394 y(diag)2753 2369 y Fz(G)2818 2381 y Fx(o\013)2899 2369 y Fz(G)2964 2339 y Fy(\000)p Fx(1)3054 2277 y Fl(\021)3103 2294 y Fy(\000)p Fx(1)1354 2565 y Fs(\002)1433 2473 y Fl(\020)1482 2565 y Fz(G)1547 2577 y Fx(o\013)1629 2565 y Fw(P)p 1680 2544 V 13 x Fu(e)1715 2565 y Fz(G)1780 2530 y Fy(\000)p Fx(1)1780 2590 y(diag)1926 2565 y Fs(\000)g Fz(G)2074 2577 y Fx(o\013)2155 2565 y Fw(P)p 2206 2544 V 13 x Fu(e)2242 2565 y Fz(G)2307 2530 y Fy(\000)p Fx(1)2307 2590 y(diag)2434 2565 y Fz(G)2499 2577 y Fx(o\013)2580 2565 y Fz(G)2645 2535 y Fy(\000)p Fx(1)2735 2473 y Fl(\021)2798 2565 y Fz(:)3174 2684 y FC(\(20\))523 2784 y(Therefore,)g(for)h(a)i (\002x)o(ed)f Fz(\030)t FC(,)g(by)g(\(12\),)f(\(13\))g(and)h(\(18\))f (we)h(see)h(that,)f(as)h Fz(\025)i Fs(!)g Ft(0)p FC(,)d(we)h(ha)n(v)o (e)1522 2963 y Fw(P)1573 2975 y Fu(e)1608 2963 y Ft(\()p Fz(G)1705 2928 y Fy(\000)p Fx(1)1705 2988 y(diag)1851 2963 y Fs(\000)d Fz(G)1999 2929 y Fy(\000)p Fx(1)2088 2963 y Ft(\))23 b Fs(!)g Ft(0)p Fz(:)860 b FC(\(21\))648 3161 y(Hence,)25 b(\(12\))h(and)g(\(21\))f(imply)h(that)h Fw(P)1809 3173 y Fu(e)1844 3161 y Fz(G)1909 3125 y Fy(\000)p Fx(1)1909 3186 y(diag)2063 3161 y FC(is)g(uniformly)e(bounded)f(as)j Fz(\025)35 b Fs(!)g Ft(0)p FC(.)26 b(W)-7 b(e)523 3260 y(kno)n(w)19 b(that)1477 3360 y Fw(P)1528 3372 y Fu(e)1563 3360 y Fz(G)1628 3372 y Fx(diag)1778 3360 y Fs(!)k Fw(P)1935 3372 y Fu(e)1971 3360 y Fz(\030)f Fs(\000)c Fw(P)2163 3372 y Fu(e)2199 3360 y Fz(M)2280 3372 y Fx(st)2336 3360 y Fz(:)815 b FC(\(22\))523 3507 y(Therefore,)18 b Fz(\030)t Fw(P)983 3519 y Fu(e)1037 3507 y Fs(\000)g Fw(P)1171 3519 y Fu(e)1206 3507 y Fz(M)1287 3519 y Fx(st)1364 3507 y FC(is)j(in)m(v)o(ertible)d(on)i Ft(Ran)p Fw(P)2078 3519 y Fu(e)2134 3507 y FC(and)1400 3687 y Fw(P)1451 3699 y Fu(e)1486 3687 y Fz(G)1551 3651 y Fy(\000)p Fx(1)1551 3712 y(diag)1701 3687 y Fs(!)k Ft(\()p Fw(P)1891 3699 y Fu(e)1926 3687 y Fz(\030)f Fs(\000)18 b Fw(P)2119 3699 y Fu(e)2154 3687 y Fz(M)2235 3699 y Fx(st)2291 3687 y Ft(\))2323 3653 y Fy(\000)p Fx(1)2413 3687 y Fz(:)523 3866 y FC(Using)i(again)f(\(21\),)g(we)i(see)f(that)1419 4046 y Fw(P)1470 4058 y Fu(e)1505 4046 y Fz(G)1570 4012 y Fy(\000)p Fx(1)1683 4046 y Fs(!)j Ft(\()p Fw(P)1872 4058 y Fu(e)1907 4046 y Fz(\030)g Fs(\000)18 b Fw(P)2100 4058 y Fu(e)2135 4046 y Fz(M)2216 4058 y Fx(st)2272 4046 y Ft(\))2304 4012 y Fy(\000)p Fx(1)2394 4046 y Fz(:)757 b FC(\(23\))523 4225 y(Summing)19 b(up)h(\(23\))f(o)o(v)o(er)f Fz(e)p FC(,)j(we)f(obtain)1290 4340 y Fl(X)1251 4518 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1462 4419 y Fw(P)1513 4431 y Fu(e)1548 4419 y Fz(G)1613 4431 y Fu(\025)1657 4419 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))1860 4384 y Fy(\000)p Fx(1)1972 4419 y Fs(!)23 b Ft(\()p Fz(\030)t Fw(P)c Fs(\000)f Fz(M)2384 4431 y Fx(st)2440 4419 y Ft(\))2472 4384 y Fy(\000)p Fx(1)2561 4419 y Fz(;)590 b FC(\(24\))523 4682 y(which)20 b(ends)g(the)g(proof)e(of)i(2.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(17)648 282 y FC(Let)20 b(us)h(no)n(w)e(pro)o(v)o(e)g(1.)h(W)-7 b(e)21 b(ha)n(v)o(e)980 423 y Fl(P)926 560 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1136 485 y Fw(P)1187 497 y Fu(e)1223 485 y Fz(G)1288 497 y Fu(\025)1331 485 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))1534 455 y Fy(\000)p Fx(1)1649 485 y Ft(=)1791 423 y Fl(P)1736 560 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1955 402 y Fy(1)1952 419 y Fl(R)1955 562 y Fx(0)2027 485 y Ft(e)2064 455 y Fy(\000)p Fu(t)p Fx(\()p Fu(\030)r Fx(+)p Fu(\025)2289 430 y Fn(\000)p Fq(2)2367 455 y Fx(i)p Fu(e)p Fx(\))2447 485 y Fw(P)2498 497 y Fu(e)2533 485 y Ft(e)2570 455 y Fu(t)p Fr(L)2634 464 y Fp(\025)2673 455 y Fu(=\025)2746 430 y Fq(2)2783 485 y Fw(P)p Ft(d)p Fz(t)1649 750 y Ft(=)1744 667 y Fy(1)1742 683 y Fl(R)1745 827 y Fx(0)1816 750 y Ft(e)1853 720 y Fy(\000)p Fu(t\030)1967 750 y Ft(e)2004 720 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)2193 695 y Fq(2)2229 750 y Fw(P)p Ft(e)2317 720 y Fu(t)p Fr(L)2381 729 y Fp(\025)2420 720 y Fu(=\025)2493 695 y Fq(2)2530 750 y Fw(P)p Ft(d)p Fz(t)3174 623 y FC(\(25\))523 986 y(Clearly)-5 b(,)20 b Fs(k)p Ft(e)883 955 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1072 930 y Fq(2)1108 986 y Fw(P)p Ft(e)1196 955 y Fu(t)p Fr(L)1260 964 y Fp(\025)1298 955 y Fu(=\025)1371 930 y Fq(2)1408 986 y Fw(P)p Fs(k)i(\024)h Ft(1)p FC(.)d(Therefore,)1391 1075 y Fl(\015)1391 1125 y(\015)1391 1174 y(\015)1391 1224 y(\015)1391 1274 y(\015)1391 1324 y(\015)1476 1166 y(X)1437 1345 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1648 1245 y Fw(P)1699 1257 y Fu(e)1734 1245 y Fz(G)1799 1257 y Fu(\025)1843 1245 y Ft(\()p Fz(\030)t(;)14 b Ft(i)p Fz(e)p Ft(\))2046 1211 y Fy(\000)p Fx(1)2135 1075 y Fl(\015)2135 1125 y(\015)2135 1174 y(\015)2135 1224 y(\015)2135 1274 y(\015)2135 1324 y(\015)2205 1245 y Fs(\024)22 b Fz(\030)2332 1211 y Fy(\000)p Fx(1)2422 1245 y Fz(:)523 1499 y FC(Hence,)e(by)f (\(24\),)1499 1528 y Fl(\015)1499 1578 y(\015)1545 1598 y Ft(\()p Fz(\030)t Fw(P)f Fs(\000)g Fz(M)1850 1610 y Fx(st)1906 1598 y Ft(\))1938 1564 y Fy(\000)p Fx(1)2028 1528 y Fl(\015)2028 1578 y(\015)2097 1598 y Fs(\024)23 b Fz(\030)2225 1564 y Fy(\000)p Fx(1)2314 1598 y Fz(;)523 1734 y FC(which)d(pro)o(v)o(es)e(1.)648 1834 y(Let)k Fz(f)36 b Fs(2)27 b Fz(C)999 1846 y Fx(0)1037 1834 y Ft(\([0)p Fz(;)14 b Fs(1)p Ft([\))23 b FC(and)e Fz(\016)30 b(>)d Ft(0)p FC(.)22 b(By)h(the)f(Stone-W)-7 b(eierstrass)22 b(Theorem,)f(we)h(can)g(\002nd)523 1933 y(a)h(\002nite)f(linear)g (combination)e(of)j(functions)e(of)h(the)g(form)f Ft(e)2288 1903 y Fy(\000)p Fu(t\030)2424 1933 y FC(for)h Fz(\030)31 b(>)c Ft(0)p FC(,)22 b(denoted)f Fz(g)s FC(,)h(such)523 2033 y(that)e Fs(k)p Ft(e)747 2003 y Fu(t\016)808 2033 y Fz(f)27 b Fs(\000)18 b Fz(g)s Fs(k)1044 2045 y Fy(1)1137 2033 y Fz(<)k(\017)p FC(.)f(Set)1096 2208 y Fz(A)1158 2220 y Fu(\025)1202 2208 y Ft(\()p Fz(t)p Ft(\))i(:=)g(e)1467 2174 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1656 2149 y Fq(2)1692 2208 y Fw(P)p Ft(e)1780 2174 y Fu(t)p Fr(L)1844 2183 y Fp(\025)1883 2174 y Fu(=\025)1956 2149 y Fq(2)1993 2208 y Fw(P)p Fz(;)96 b(A)2225 2220 y Fx(0)2263 2208 y Ft(\()p Fz(t)p Ft(\))24 b(:=)e(e)2528 2174 y Fu(tM)2616 2183 y Fq(dyn)2717 2208 y Fz(:)523 2368 y FC(Note)e(that)g Fs(k)p Fz(A)954 2380 y Fu(\025)998 2368 y Ft(\()p Fz(t)p Ft(\))p Fs(k)j(\024)g Ft(1)d FC(and)f Fs(k)p Fz(A)1551 2380 y Fx(0)1588 2368 y Ft(\()p Fz(t)p Ft(\))p Fs(k)24 b(\024)e Ft(1)p FC(.)e(No)n(w)771 2528 y Fs(k)827 2461 y Fl(R)896 2528 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\)\()p Fz(A)1134 2540 y Fu(\025)1178 2528 y Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)f Fz(A)1436 2540 y Fx(0)1473 2528 y Ft(\()p Fz(t)p Ft(\)\)d)p Fz(t)p Fs(k)170 b(\024)22 b(k)2030 2461 y Fl(R)2099 2528 y Ft(e)2136 2498 y Fy(\000)p Fu(\016)r(t)2249 2528 y Fz(g)s Ft(\()p Fz(t)p Ft(\)\()p Fz(A)2480 2540 y Fu(\025)2525 2528 y Ft(\()p Fz(t)p Ft(\))d Fs(\000)f Fz(A)2783 2540 y Fx(0)2820 2528 y Ft(\()p Fz(t)p Ft(\)\)d)p Fz(t)p Fs(k)771 2699 y Ft(+)p Fs(k)892 2632 y Fl(R)946 2699 y Ft(\()p Fz(f)9 b Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)f Ft(e)1261 2669 y Fy(\000)p Fu(\016)r(t)1375 2699 y Fz(g)s Ft(\()p Fz(t)p Ft(\)\))p Fz(A)1606 2711 y Fu(\025)1650 2699 y Ft(\()p Fz(t)p Ft(\)d)p Fz(t)p Fs(k)25 b Ft(+)p Fs(k)2008 2632 y Fl(R)2062 2699 y Ft(\()p Fz(f)9 b Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)f Ft(e)2377 2669 y Fy(\000)p Fu(\016)r(t)2490 2699 y Fz(g)s Ft(\()p Fz(t)p Ft(\)\))p Fz(A)2721 2711 y Fx(0)2759 2699 y Ft(\()p Fz(t)p Ft(\)d)p Fz(t)p Fs(k)p Fz(:)523 2854 y FC(By)25 b(2.)g(and)g(by)f(the)h(Laplace) f(transformation,)e(the)j(\002rst)h(term)f(on)f(the)h(right)f(hand)g (side)h(goes)523 2953 y(to)20 b(0)g(as)h Fz(\025)j Fs(!)f Ft(0)p FC(.)c(The)h(last)h(tw)o(o)f(terms)g(are)g(estimated)g(by)g Fz(\017)2268 2887 y Fl(R)2323 2907 y Fy(1)2306 2983 y Fx(0)2407 2953 y Ft(e)2444 2923 y Fy(\000)p Fu(\016)r(t)2557 2953 y Ft(d)p Fz(t)p FC(,)h(which)e(can)h(be)g(made)523 3053 y(arbitrarily)f(small)h(by)g(choosing)f Fz(\017)h FC(small.)h(This)f(pro)o(v)o(es)f(3.)h Ff(2)523 3316 y FA(3.3)40 b(Spectral)20 b(a)n(v)o(eraging)523 3481 y FC(Before)d(we)h(present)e(the)i(time-dependent)c(v)o(ersion)i(of)h (the)h(weak)f(coupling)e(limit,)j(we)g(discuss)523 3580 y(the)i(spectral)g(a)n(v)o(eraging)e(of)i(operators,)f(follo)n(wing)f ([Da3)o(].)648 3680 y(In)24 b(this)g(subsection,)f Fs(Y)32 b FC(is)26 b(an)e(arbitrary)e(Banach)i(space)g(and)g Ft(e)2528 3650 y Fu(t)p Fr(E)2621 3680 y FC(is)h(a)f(1-parameter)e Fz(C)3247 3692 y Fx(0)3285 3680 y FC(-)523 3779 y(group)c(of)i (isometries)g(on)g Fs(Y)7 b FC(.)21 b(F)o(or)f Fz(K)28 b Fs(2)c(B)s Ft(\()p Fs(Y)7 b Ft(\))20 b FC(we)h(de\002ne)1297 3997 y Fz(K)1374 3963 y Fu(\\)1428 3997 y Ft(:=)h(s)p Fs(\000)35 b Ft(lim)1649 4047 y Fu(t)p Fy(!1)1821 3997 y Fz(t)1851 3963 y Fy(\000)p Fx(1)1954 3884 y Fl(Z)2037 3904 y Fu(t)2000 4072 y Fx(0)2080 3997 y Ft(e)2117 3963 y Fu(s)p Fr(E)2191 3997 y Fz(K)6 b Ft(e)2305 3963 y Fy(\000)p Fu(s)p Fr(E)2430 3997 y Ft(d)p Fz(s;)636 b FC(\(26\))523 4197 y(pro)o(vided)18 b(that)i(the)g(right)g(hand)f(side)h(e)o(xists.) 523 4349 y FA(Theor)o(em)g(6.)k Fk(Suppose)19 b(that)h Fz(K)1477 4319 y Fu(\\)1528 4349 y Fk(e)n(xists.)h(Then,)f(for)g(any)g Fz(t)2248 4361 y Fx(0)2308 4349 y Fz(>)j Ft(0)p Fk(,)d Fz(y)26 b Fs(2)d(Y)7 b Fk(,)1130 4523 y Ft(lim)1119 4577 y Fu(\025)p Fy(!)p Fx(0)1318 4523 y Ft(sup)1271 4593 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1458 4601 y Fq(0)1504 4523 y Fs(k)p Ft(e)1583 4488 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1776 4523 y Ft(e)1813 4488 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025K)t Fx(\))p Fu(=\025)2156 4523 y Fz(y)22 b Fs(\000)c Ft(e)2339 4488 y Fu(tK)2424 4463 y Fp(\\)2457 4523 y Fz(y)s Fs(k)k Ft(=)g(0)p Fz(:)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 523 100 a FB(18)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(Pr)o(oof)o(.)67 b FC(Consider)33 b(the)h(space)g Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)1727 294 y Fx(0)1764 282 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))35 b FC(with)g(the)f (supremum)e(norm.)h(Set)h Fz(K)6 b Ft(\()p Fz(t)p Ft(\))49 b(=)523 382 y(e)560 352 y Fu(t)p Fr(E)p Fu(=\025)701 382 y Fz(K)6 b Ft(e)815 352 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1008 382 y FC(.)20 b(F)o(or)g Fz(f)32 b Fs(2)23 b Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)1564 394 y Fx(0)1602 382 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))p FC(,)21 b(de\002ne)1397 611 y Fz(B)1460 623 y Fu(\025)1504 611 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\))23 b(:=)1781 498 y Fl(Z)1864 519 y Fu(t)1828 687 y Fx(0)1908 611 y Fz(K)6 b Ft(\()p Fz(s=\025)p Ft(\))p Fz(f)j Ft(\()p Fz(s)p Ft(\)d)p Fz(s;)1474 895 y(B)1537 907 y Fx(0)1575 895 y Fz(f)g Ft(\()p Fz(t)p Ft(\))23 b(:=)g Fz(K)1930 861 y Fu(\\)1974 782 y Fl(Z)2057 803 y Fu(t)2020 971 y Fx(0)2100 895 y Fz(f)9 b Ft(\()p Fz(s)p Ft(\)d)p Fz(s:)648 1084 y FC(Clearly)-5 b(,)19 b Fz(B)991 1096 y Fx(0)1049 1084 y FC(and)h Fz(B)1253 1096 y Fu(\025)1317 1084 y FC(are)g(linear)g(operators)f(on)g Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)2311 1096 y Fx(0)2349 1084 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))21 b FC(satisfying)1643 1256 y Fs(k)p Fz(B)1748 1268 y Fu(\025)1791 1256 y Fs(k)h(\024)h Fz(t)1973 1268 y Fx(0)2010 1256 y Fs(k)p Fz(K)6 b Fs(k)p Fz(:)980 b FC(\(27\))523 1428 y(Moreo)o(v)o(er)1636 1527 y Ft(lim)1624 1581 y Fu(\025)p Fy(!)p Fx(0)1777 1527 y Fz(B)1840 1539 y Fu(\025)1883 1527 y Fz(f)32 b Ft(=)22 b Fz(B)2106 1539 y Fx(0)2144 1527 y Fz(f)t(:)962 b FC(\(28\))523 1709 y(T)-7 b(o)20 b(pro)o(v)o(e)f(\(28\),)g(by)g(\(27\))g(it)i(suf)n (\002ces)g(to)f(assume)g(that)g Fz(f)32 b Fs(2)23 b Fz(C)2330 1679 y Fx(1)2368 1709 y Ft(\([0)p Fz(;)14 b(t)2532 1721 y Fx(0)2569 1709 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))p FC(.)21 b(No)n(w)838 1905 y Fz(B)901 1917 y Fu(\025)945 1905 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\))25 b(=)1202 1813 y Fl(\020)1251 1838 y(R)1306 1859 y Fu(t)1290 1935 y Fx(0)1350 1905 y Fz(K)6 b Ft(\()p Fz(s=\025)p Ft(\)d)p Fz(s)1705 1813 y Fl(\021)1768 1905 y Fz(f)j Ft(\()p Fz(t)p Ft(\))18 b Fs(\000)2013 1838 y Fl(R)2069 1859 y Fu(t)2053 1935 y Fx(0)2112 1838 y Fl(\000R)2205 1859 y Fu(s)2189 1935 y Fx(0)2255 1905 y Ft(d)p Fz(s)2340 1917 y Fx(1)2377 1905 y Fz(K)6 b Ft(\()p Fz(s)2525 1917 y Fx(1)2562 1905 y Fz(=\025)p Ft(\))2684 1838 y Fl(\001)2736 1905 y Fz(f)2786 1875 y Fy(0)2809 1905 y Ft(\()p Fz(s)p Ft(\)d)p Fz(s)1114 2088 y Fs(!)23 b Fz(tK)1327 2058 y Fu(\\)1357 2088 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)1603 2021 y Fl(R)1658 2042 y Fu(t)1642 2117 y Fx(0)1701 2088 y Fz(sK)1817 2058 y Fu(\\)1848 2088 y Fz(f)1898 2058 y Fy(0)1921 2088 y Ft(\()p Fz(s)p Ft(\)d)p Fz(s)k Ft(=)g Fz(B)2283 2100 y Fx(0)2320 2088 y Fz(f)9 b Ft(\()p Fz(t)p Ft(\))p Fz(:)523 2255 y FC(W)-7 b(e)21 b(easily)g(get)1251 2392 y Fs(k)p Fz(B)1360 2358 y Fu(n)1356 2413 y(\025)1405 2392 y Fs(k)h(\024)1567 2336 y Fz(t)1597 2306 y Fu(n)1597 2357 y Fx(0)p 1567 2373 76 4 v 1568 2449 a Fz(n)p Ft(!)1652 2392 y Fs(k)p Fz(K)6 b Fs(k)1813 2358 y Fu(n)1857 2392 y Fz(;)76 b Fs(k)p Fz(B)2065 2358 y Fu(n)2061 2413 y Fx(0)2110 2392 y Fs(k)22 b(\024)2272 2336 y Fz(t)2302 2306 y Fu(n)2302 2357 y Fx(0)p 2272 2373 V 2273 2449 a Fz(n)p Ft(!)2357 2392 y Fs(k)p Fz(K)6 b Fs(k)2518 2358 y Fu(n)2562 2392 y Fz(:)589 b FC(\(29\))648 2574 y(Let)20 b Fz(y)26 b Fs(2)d(Y)7 b FC(.)21 b(Set)g Fz(y)1196 2586 y Fu(\025)1239 2574 y Ft(\()p Fz(t)p Ft(\))j(:=)e(e)1504 2544 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1698 2574 y Ft(e)1735 2544 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025K)t Fx(\))p Fu(=\025)2078 2574 y Fz(y)s FC(.)e(Note)g(that)1146 2746 y Fz(y)1187 2758 y Fu(\025)1230 2746 y Ft(\()p Fz(t)p Ft(\))k(=)f Fz(y)e Ft(+)d Fz(B)1644 2758 y Fu(\025)1687 2746 y Fz(y)1728 2758 y Fu(\025)1772 2746 y Ft(\()p Fz(t)p Ft(\))p Fz(;)76 b(y)2006 2758 y Fx(0)2043 2746 y Ft(\()p Fz(t)p Ft(\))24 b(=)e Fz(y)g Ft(+)c Fz(B)2457 2758 y Fx(0)2494 2746 y Fz(y)2535 2758 y Fx(0)2572 2746 y Ft(\()p Fz(t)p Ft(\))p Fz(:)523 2917 y FC(T)m(reating)23 b Fz(y)28 b FC(as)c(an)g(element)g(of)g Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)1702 2929 y Fx(0)1739 2917 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))25 b FC(\226)g(the)f(constant)f(function)g(equal)g(to)h Fz(y)k FC(we)c(can)523 3017 y(write)1007 3164 y Ft(\(1)19 b Fs(\000)f Fz(B)1246 3176 y Fu(\025)1289 3164 y Ft(\))1321 3129 y Fy(\000)p Fx(1)1411 3164 y Fz(y)25 b Ft(=)1594 3060 y Fy(1)1567 3085 y Fl(X)1565 3261 y Fu(n)p Fx(=0)1704 3164 y Fz(B)1771 3129 y Fu(n)1767 3184 y(\025)1816 3164 y Fz(y)s(;)76 b Ft(\(1)18 b Fs(\000)g Fz(B)2197 3176 y Fx(0)2234 3164 y Ft(\))2266 3129 y Fy(\000)p Fx(1)2356 3164 y Fz(y)26 b Ft(=)2540 3060 y Fy(1)2513 3085 y Fl(X)2510 3261 y Fu(n)p Fx(=0)2649 3164 y Fz(B)2716 3129 y Fu(n)2712 3184 y Fx(0)2762 3164 y Fz(y)s(;)523 3377 y FC(where)19 b(both)f(Neumann)f(series)j(are)f(absolutely)f(con)m(v)o(er)o(gent.)e (Therefore,)g(in)k(the)f(sense)g(of)g(the)523 3477 y(con)m(v)o(er)o (gence)d(in)21 b(in)f Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)1357 3489 y Fx(0)1394 3477 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))p FC(,)21 b(we)g(get)1355 3715 y Fz(y)1396 3727 y Fu(\025)1462 3715 y Ft(=)1579 3611 y Fy(1)1552 3636 y Fl(X)1550 3812 y Fu(n)p Fx(=0)1689 3715 y Fz(B)1756 3680 y Fu(n)1752 3735 y(\025)1801 3715 y Fz(y)26 b Fs(!)2003 3611 y Fy(1)1977 3636 y Fl(X)1974 3812 y Fu(n)p Fx(=0)2113 3715 y Fz(B)2180 3680 y Fu(n)2176 3735 y Fx(0)2225 3715 y Fz(y)g Ft(=)d Fz(y)2421 3727 y Fx(0)2458 3715 y Fz(:)523 3937 y Ff(2)523 4146 y FA(Theor)o(em)d(7.)k Fk(Let)d Fs(Y)28 b Fk(be)20 b(\002nite)g(dimesional.)f(Then)h Fz(K)2118 4115 y Fu(\\)2169 4146 y Fk(e)n(xists)i(for)e(any)g Fz(K)28 b Fs(2)c(B)s Ft(\()p Fs(Y)7 b Ft(\))20 b Fk(and)957 4329 y Fz(K)1034 4298 y Fu(\\)1088 4329 y Ft(=)1230 4266 y Fl(P)1175 4403 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1386 4329 y Fm(1)1434 4341 y Fx(i)p Fu(e)1488 4329 y Ft(\()p Fw(E)p Ft(\))p Fz(K)6 b Fm(1)1732 4341 y Fx(i)p Fu(e)1787 4329 y Ft(\()p Fw(E)p Ft(\))24 b(=)43 b(lim)2017 4378 y Fu(t)p Fy(!1)2188 4329 y Fz(t)2218 4298 y Fy(\000)p Fx(1)2321 4262 y Fl(R)2377 4282 y Fu(t)2361 4358 y Fx(0)2420 4329 y Ft(e)2457 4298 y Fu(s)p Fr(E)2531 4329 y Fz(K)6 b Ft(e)2645 4298 y Fy(\000)p Fu(s)p Fr(E)2770 4329 y Ft(d)p Fz(s;)969 4513 y Ft(lim)957 4568 y Fu(\025)p Fy(!)p Fx(0)1157 4513 y Ft(sup)1110 4583 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1297 4591 y Fq(0)1343 4513 y Fs(k)p Ft(e)1422 4483 y Fy(\000)p Fu(t)p Fr(E)p Fu(=\025)1615 4513 y Ft(e)1652 4483 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025K)t Fx(\))p Fu(=\025)2013 4513 y Fs(\000)18 b Ft(e)2133 4483 y Fu(tK)2218 4458 y Fp(\\)2251 4513 y Fs(k)23 b Ft(=)f(0)p Fz(:)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(19)523 282 y FA(Pr)o(oof)o(.)41 b FC(In)20 b(\002nite)i(dimension)d(we)j(can)f(replace)f(the)h(strong)f (limit)i(by)f(the)g(norm)f(limit.)h(More-)523 382 y(o)o(v)o(er)m(,)798 606 y Fz(t)828 572 y Fy(\000)p Fx(1)931 493 y Fl(Z)1014 513 y Fu(t)978 681 y Fx(0)1057 606 y Ft(e)1094 572 y Fu(s)p Fr(E)1169 606 y Fz(K)6 b Ft(e)1283 572 y Fy(\000)p Fu(s)p Fr(E)1408 606 y Ft(d)p Fz(s)23 b Ft(=)1710 527 y Fl(X)1604 705 y Fx(i)p Fu(e)1654 713 y Fq(1)1687 705 y Fu(;)p Fx(i)p Fu(e)1757 713 y Fq(2)1789 705 y Fy(2)p Fx(sp)p Fr(E)1949 606 y Fm(1)1997 618 y Fx(i)p Fu(e)2047 626 y Fq(1)2084 606 y Ft(\()p Fw(E)p Ft(\))p Fz(K)6 b Fm(1)2328 618 y Fx(i)p Fu(e)2378 626 y Fq(2)2415 606 y Ft(\()p Fw(E)p Ft(\))2544 550 y(e)2581 520 y Fx(i)p Fu(t)p Fx(\()p Fu(e)2682 528 y Fq(1)2715 520 y Fy(\000)p Fu(e)2798 528 y Fq(2)2831 520 y Fx(\))2880 550 y Fs(\000)18 b Ft(1)p 2544 587 460 4 v 2589 663 a(i\()p Fz(e)2683 675 y Fx(1)2739 663 y Fs(\000)g Fz(e)2861 675 y Fx(2)2898 663 y Ft(\))p Fz(t)3014 606 y(:)523 844 y Ff(2)523 1056 y Fk(Remark)i(2.)k FC(The)h(follo)n(wing)e(results)j(generalize)d(some) i(aspects)g(of)g(Theorem)e(7)j(to)f(the)g(case)523 1155 y(when)c Fw(P)h FC(is)g(not)g(necessarily)f(\002nite)g(dimensional.)f (The)o(y)h(are)g(pro)o(v)o(en)e(in)j([Da3)o(].)f(W)-7 b(e)23 b(will)f(not)523 1255 y(need)e(these)g(results.)616 1355 y(1\))k(If)c Fz(K)863 1325 y Fu(\\)914 1355 y FC(e)o(xists,)g (then)g(it)h(commutes)e(with)i Ft(e)1942 1325 y Fu(t)p Fr(E)2009 1355 y FC(.)616 1454 y(2\))j(If)19 b Fz(K)25 b FC(is)20 b(a)g(compact)e(operator)f(and)i Fs(Y)27 b FC(is)20 b(a)g(Hilbert)f(space,)g(then)f Fz(K)2683 1424 y Fu(\\)2734 1454 y FC(e)o(xists)h(and)g(we)g(can)710 1554 y(replace)g(the)h(strong)g(limit)g(in)h(\(26\))e(by)h(the)g(norm)f (limit.)616 1654 y(3\))24 b(If)c Fw(E)h FC(has)f(a)h(total)f(set)h(of)f (eigen)m(v)o(ectors,)e(then)i Fz(K)2120 1623 y Fu(\\)2171 1654 y FC(e)o(xists)g(as)h(well.)523 1887 y FA(3.4)40 b(Second)21 b(order)f(asymptotics)f(of)h(e)o(v)o(olution)f(with)i(the)f (\002rst)h(order)f(term)523 2063 y FC(In)e(this)h(subsection)f(we)g (consider)g(a)g(some)n(what)g(more)f(general)h(situation)g(than)g(in)g (Subsection)523 2162 y(3.1.)j(W)-7 b(e)22 b(mak)o(e)f(the)h (Assumptions)f(2.1,)f(2.3)h(and)g(2.4,)g(or)g(2.1*,)f(2.3*)g(and)h(2.4) g(b)n(ut)h(we)f(do)g(not)523 2262 y(assume)h(that)g Fw(P)g FC(is)h(\002nite)f(dimensional,)e(nor)h(that)h Fw(PQP)j Ft(=)h(0)p FC(.)c(Thus)f(we)h(allo)n(w)g(for)g(a)g(term)f(of)523 2361 y(\002rst)g(order)e(in)h Fz(\025)h FC(in)g(the)f(asymptotics)f(of) h(the)h(reduced)d(dynamics.)h(W)-7 b(e)21 b(again)e(follo)n(w)h([Da3)o (].)648 2461 y(W)-7 b(e)21 b(assume)g(also)g(that)g Fw(PQ)1462 2440 y Fl(e)1460 2461 y Fw(P)f FC(and)1674 2440 y Fl(e)1672 2461 y Fw(PQP)g FC(are)h(bounded)d(or)i(w*)h(continuous)d(and)i(that)h Fw(E)e Ft(+)523 2561 y Fz(\025)p Fw(PQP)h FC(generates)g(a)g Fz(C)1209 2573 y Fx(0)1247 2561 y FC(-)g(or)g Fz(C)1450 2531 y Fy(\003)1444 2581 y Fx(0)1488 2561 y FC(-group)f(of)g (isometries)i(on)e Ft(Ran)p Fw(P)p FC(.)648 2671 y(Using)32 b(the)g(boundedness)e(of)j(of)n(f-diagonal)c(elements)j Fw(PQ)2465 2650 y Fl(e)2463 2671 y Fw(P)g FC(and)2702 2650 y Fl(e)2699 2671 y Fw(PQP)p FC(,)g(we)h(see)g(that)525 2749 y Fl(e)523 2771 y Fw(PL)629 2783 y Fu(\025)675 2749 y Fl(e)673 2771 y Fw(P)20 b FC(is)h(the)f(generator)f(of)h(a)g (continuous)e(semigroup.)648 2870 y(In)g(this)g(subsection,)g(the)g (de\002nition)f(of)h Fz(K)1881 2882 y Fu(\025)1924 2870 y Ft(\()p Fz(t)p Ft(\))i FC(slightly)e(changes)f(as)i(compared)d(with)j (\(9\):)1135 3119 y Fz(K)1206 3131 y Fu(\025)1249 3119 y Ft(\()p Fz(t)p Ft(\))k(:=)1477 3006 y Fl(Z)1560 3026 y Fu(\025)1599 3001 y Fn(\000)p Fq(2)1677 3026 y Fu(t)1523 3194 y Fx(0)1720 3119 y Ft(e)1757 3084 y Fy(\000)p Fu(s)p Fx(\()p Fr(E)p Fx(+)p Fu(\025)p Fr(PQP)p Fx(\))2141 3119 y Fw(PQ)p Ft(e)2294 3084 y Fu(s)2324 3069 y Fj(e)2325 3084 y Fr(P)o(L)2399 3093 y Fp(\025)2437 3069 y Fj(e)2438 3084 y Fr(P)2478 3119 y Fw(QP)p Ft(d)p Fz(s:)523 3334 y FA(Theor)o(em)d(8.)k Fk(Suppose)19 b(that)h(the)g(following)f (assumptions)h(ar)m(e)g(true:)612 3433 y FC(1\))k Fk(F)-9 b(or)21 b(all)f Fz(t)981 3445 y Fx(0)1041 3433 y Fz(>)j Ft(0)p Fk(,)d(ther)m(e)g(e)n(xists)i Fz(c)f Fk(suc)o(h)e(that)1582 3607 y Ft(sup)1544 3681 y Fy(j)p Fu(\025)p Fy(j)p Fu(<\025)1714 3689 y Fq(0)1828 3607 y Ft(sup)1781 3677 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1968 3685 y Fq(0)2014 3607 y Fs(k)p Fz(K)2127 3619 y Fu(\025)2169 3607 y Ft(\()p Fz(t)p Ft(\))p Fs(k)24 b(\024)e Fz(c:)612 3843 y FC(2\))i Fk(Ther)m(e)h(e)n(xists)g(a) g(bounded)d(\(w*)i(continuous)e(in)j(the)f Fz(C)2326 3813 y Fy(\003)2320 3864 y Fx(0)2390 3843 y Fk(case\))g(oper)o(ator)f Fz(K)30 b Fk(on)24 b Ft(Ran)p Fw(P)706 3943 y Fk(suc)o(h)c(that)1747 4042 y Ft(lim)1735 4096 y Fu(\025)p Fy(!)p Fx(0)1888 4042 y Fz(K)1959 4054 y Fu(\025)2002 4042 y Ft(\()p Fz(t)p Ft(\))j(=)g Fz(K)706 4214 y Fk(for)e(all)f Ft(0)j Fz(<)f(t)h(<)g Fs(1)p Fk(.)523 4314 y(Then)d(for)g Fz(y)26 b Fs(2)e Ft(Ran)o Fw(P)1007 4509 y Ft(lim)995 4563 y Fu(\025)p Fy(!)p Fx(0)1195 4509 y Ft(sup)1148 4579 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1335 4587 y Fq(1)1381 4413 y Fl(\015)1381 4463 y(\015)1381 4513 y(\015)1427 4509 y Fw(P)p Ft(e)1515 4475 y Fu(t)p Fr(L)1579 4484 y Fp(\025)1618 4475 y Fu(=\025)1691 4449 y Fq(2)1728 4509 y Fw(P)p Fz(y)c Fs(\000)f Ft(e)1961 4475 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025)p Fr(PQP)p Fx(+)p Fu(\025)2348 4449 y Fq(2)2379 4475 y Fu(K)t Fx(\))p Fu(=\025)2538 4449 y Fq(2)2575 4509 y Fz(y)2619 4413 y Fl(\015)2619 4463 y(\015)2619 4513 y(\015)2688 4509 y Ft(=)k(0)p Fz(:)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 523 100 a FB(20)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(Pr)o(oof)o(.)57 b FC(Set)30 b Fs(Y)48 b Ft(:=)39 b(Ran)p Fw(P)p FC(.)30 b(Consider)e(the)i(space)f Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)2320 294 y Fx(0)2357 282 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))p FC(.)31 b(F)o(or)e Fz(f)48 b Fs(2)41 b Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)3121 294 y Fx(0)3158 282 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))523 382 y FC(de\002ne)1086 473 y Fz(H)1155 485 y Fu(\025)1198 473 y Fz(f)i Ft(\()p Fz(t)p Ft(\))24 b(:=)1476 406 y Fl(R)1532 427 y Fu(t)1516 503 y Fx(0)1575 473 y Ft(e)1612 443 y Fx(\()p Fr(E)p Fx(+)p Fr(PQP)p Fx(\)\()p Fu(t)p Fy(\000)p Fu(s)p Fx(\))p Fu(=\025)2104 418 y Fq(2)2140 473 y Fz(K)2211 485 y Fu(\025)2254 473 y Ft(\()p Fz(t)18 b Fs(\000)g Fz(s)p Ft(\))p Fz(f)9 b Ft(\()p Fz(s)p Ft(\)d)p Fz(s;)1086 656 y(G)1151 668 y Fu(\025)1195 656 y Fz(f)g Ft(\()p Fz(t)p Ft(\))23 b(:=)1473 589 y Fl(R)1528 610 y Fu(t)1512 686 y Fx(0)1571 656 y Ft(e)1608 626 y Fx(\()p Fr(E)p Fx(+)p Fr(PQP)p Fx(\)\()p Fu(t)p Fy(\000)p Fu(s)p Fx(\))p Fu(=\025)2100 601 y Fq(2)2136 656 y Fz(K)6 b(f)j Ft(\()p Fz(s)p Ft(\)d)p Fz(s:)648 805 y FC(Note)20 b(that)g Fz(H)1044 817 y Fu(\025)1108 805 y FC(and)g Fz(G)1314 817 y Fu(\025)1379 805 y FC(are)g(linear)f(operators)g(on)h Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)2373 817 y Fx(0)2410 805 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))21 b FC(satisfying)1277 985 y Fs(k)p Fz(H)1395 951 y Fu(n)1388 1005 y(\025)1439 985 y Fs(k)i(\024)f Fz(c)1627 951 y Fu(n)1672 985 y Fz(t)1702 951 y Fu(n)1702 1005 y Fx(0)1748 985 y Fz(=n)p Ft(!)p Fz(;)75 b Fs(k)p Fz(G)2068 951 y Fu(n)2068 1005 y(\025)2113 985 y Fs(k)23 b(\024)f Fz(c)2301 951 y Fu(n)2346 985 y Fz(t)2376 951 y Fu(n)2376 1005 y Fx(0)2422 985 y Fz(=n)p Ft(!)p Fz(;)523 1165 y FC(Thus)d Ft(1)14 b Fs(\000)g Fz(H)913 1177 y Fu(\025)976 1165 y FC(and)19 b Ft(1)14 b Fs(\000)g Fz(G)1316 1177 y Fu(\025)1379 1165 y FC(are)19 b(in)m(v)o(ertible.)e(In)i(f)o(act,)g(the)o(y)g(can)g(be)g(de\002ned)f (by)h(the)g(Neumann)523 1265 y(series:)1048 1364 y Ft(\(1)f Fs(\000)g Fz(H)1292 1376 y Fu(\025)1335 1364 y Ft(\))1367 1330 y Fy(\000)p Fx(1)1480 1364 y Ft(=)1568 1285 y Fl(X)1570 1462 y Fu(j)s Fx(=0)1701 1364 y Fz(H)1777 1330 y Fu(n)1770 1385 y(\025)1822 1364 y Fz(;)159 b Ft(\(1)19 b Fs(\000)f Fz(G)2245 1376 y Fu(\025)2289 1364 y Ft(\))2321 1330 y Fy(\000)p Fx(1)2433 1364 y Ft(=)2521 1285 y Fl(X)2523 1462 y Fu(j)s Fx(=0)2654 1364 y Fz(G)2719 1330 y Fu(n)2719 1385 y(\025)2765 1364 y Fz(:)523 1595 y FC(Ne)o(xt)i(we)h(note)e(that) 1116 1775 y Fs(k)p Fz(H)1234 1741 y Fu(n)1227 1796 y(\025)1297 1775 y Fs(\000)f Fz(G)1445 1741 y Fu(n)1445 1796 y(\025)1490 1775 y Fs(k)23 b(\024)g(k)p Fz(H)1754 1787 y Fu(\025)1815 1775 y Fs(\000)18 b Fz(G)1963 1787 y Fu(\025)2007 1775 y Fs(k)p Fz(c)2085 1741 y Fu(n)p Fy(\000)p Fx(1)2214 1775 y Fz(t)2244 1740 y Fu(n)p Fy(\000)p Fx(1)2244 1797 y(0)2375 1775 y Fz(=)p Ft(\()p Fz(n)g Fs(\000)g Ft(1\)!)p Fz(;)454 b FC(\(30\))523 1955 y(because)600 2170 y Fs(k)p Fz(H)718 2140 y Fu(n)711 2193 y(\025)781 2170 y Fs(\000)18 b Fz(G)929 2140 y Fu(n)929 2193 y(\025)974 2170 y Fs(k)25 b(\024)1128 2091 y Fu(n)p Fy(\000)p Fx(1)1148 2107 y Fl(P)1134 2243 y Fu(j)s Fx(=0)1268 2170 y Fs(k)p Fz(H)1386 2130 y Fu(j)1379 2195 y(\025)1422 2170 y Fs(kk)p Fz(G)1571 2130 y Fu(n)p Fy(\000)p Fu(j)s Fy(\000)p Fx(1)1571 2195 y Fu(\025)1783 2170 y Fs(kk)p Fz(H)1936 2182 y Fu(\025)1997 2170 y Fs(\000)18 b Fz(G)2145 2182 y Fu(\025)2189 2170 y Fs(k)1041 2442 y(\024)1128 2363 y Fu(n)p Fy(\000)p Fx(1)1148 2380 y Fl(P)1134 2515 y Fu(j)s Fx(=0)1307 2399 y Fu(c)1337 2373 y Fp(n)p Fn(\000)p Fq(1)1451 2399 y Fu(t)1476 2372 y Fp(n)p Fn(\000)p Fq(1)1476 2417 y(0)p 1278 2423 341 4 v 1278 2471 a Fu(k)q Fx(!\()p Fu(n)p Fy(\000)p Fu(k)q Fy(\000)p Fx(1\)!)1629 2442 y Fs(k)p Fz(H)1740 2454 y Fu(\025)1801 2442 y Fs(\000)g Fz(G)1949 2454 y Fu(\025)1993 2442 y Fs(k)23 b Ft(=)f(\(2)p Fz(ct)2285 2454 y Fx(0)2322 2442 y Ft(\))2354 2412 y Fu(n)p Fy(\000)p Fx(1)2485 2442 y Fs(k)p Fz(H)2596 2454 y Fu(\025)2657 2442 y Fs(\000)c Fz(G)2805 2454 y Fu(\025)2849 2442 y Fs(k)p Fz(=)p Ft(\()p Fz(n)g Fs(\000)g Ft(1\)!)p Fz(:)523 2680 y FC(Therefore,)1130 2780 y Fs(k)p Ft(\(1)g Fs(\000)g Fz(H)1416 2792 y Fu(\025)1460 2780 y Ft(\))1492 2746 y Fy(\000)p Fx(1)1600 2780 y Fs(\000)g Ft(\(1)g Fs(\000)g Fz(G)1923 2792 y Fu(\025)1967 2780 y Ft(\))1999 2746 y Fy(\000)p Fx(1)2088 2780 y Fs(k)23 b(\024)f Fz(c)p Fs(k)p Fz(H)2387 2792 y Fu(\025)2449 2780 y Fs(\000)c Fz(G)2597 2792 y Fu(\025)2641 2780 y Fs(k)p Fz(;)468 b FC(\(31\))523 2928 y(Ne)o(xt,)805 3157 y Ft(\()p Fz(H)906 3169 y Fu(\025)968 3157 y Fs(\000)19 b Fz(G)1117 3169 y Fu(\025)1160 3157 y Ft(\))p Fz(f)9 b Ft(\()p Fz(t)p Ft(\))24 b(=)1447 3044 y Fl(Z)1530 3064 y Fu(t)1494 3232 y Fx(0)1574 3157 y Ft(e)1611 3122 y Fx(\()p Fr(E)p Fx(+)p Fu(\025)p Fr(PQP)p Fx(\)\()p Fu(t)p Fy(\000)p Fu(s)p Fx(\))p Fu(=\025)2142 3097 y Fq(2)2178 3157 y Ft(\()p Fz(K)2281 3169 y Fu(\025)2324 3157 y Ft(\()p Fz(t)19 b Fs(\000)f Fz(s)p Ft(\))g Fs(\000)g Fz(K)6 b Ft(\))p Fz(f)j Ft(\()p Fz(s)p Ft(\)d)p Fz(s:)523 3378 y FC(and)20 b(hence)1202 3515 y Fs(k)p Fz(H)1313 3527 y Fu(\025)1374 3515 y Fs(\000)e Fz(G)1522 3527 y Fu(\025)1566 3515 y Fs(k)23 b(\024)1718 3402 y Fl(Z)1801 3423 y Fu(t)1826 3431 y Fq(0)1764 3591 y Fx(0)1877 3515 y Fs(k)p Fz(K)1990 3527 y Fu(\025)2032 3515 y Ft(\()p Fz(s)p Ft(\))c Fs(\000)f Fz(K)6 b Fs(k)p Ft(d)p Fz(s)22 b Fs(!)h Ft(0)p Fz(:)523 3704 y FC(Thus)1321 3804 y Fs(k)p Ft(\(1)18 b Fs(\000)g Fz(H)1607 3816 y Fu(\025)1651 3804 y Ft(\))1683 3769 y Fy(\000)p Fx(1)1791 3804 y Fs(\000)g Ft(\(1)g Fs(\000)g Fz(G)2114 3816 y Fu(\025)2158 3804 y Ft(\))2190 3769 y Fy(\000)p Fx(1)2279 3804 y Fs(k)23 b(!)g Ft(0)p Fz(:)659 b FC(\(32\))648 3951 y(Let)20 b Fz(y)26 b Fs(2)d(Y)7 b FC(.)21 b(De\002ne)f(the)g(follo)n(wing)f(elements)h(of)g(the)g (space)g Fz(C)6 b Ft(\([0)p Fz(;)14 b(t)2696 3963 y Fx(0)2733 3951 y Ft(])p Fz(;)g Fs(Y)7 b Ft(\))p FC(:)1403 4143 y Fz(g)1443 4155 y Fu(\025)1486 4143 y Ft(\()p Fz(t)p Ft(\))24 b(:=)e(e)1751 4113 y Fx(\()p Fr(E)p Fx(+)p Fu(\025)p Fr(PQP)p Fx(\))p Fu(t=\025)2147 4088 y Fq(2)2183 4143 y Fz(y)s(;)1403 4326 y(h)1451 4338 y Fu(\025)1494 4326 y Ft(\()p Fz(t)p Ft(\))i(:=)f Fw(P)p Ft(e)1811 4295 y Fr(L)1850 4304 y Fp(\025)1888 4295 y Fu(t=\025)1986 4270 y Fq(2)2023 4326 y Fw(P)p Fz(y)s(;)1403 4508 y(g)1443 4520 y Fu(\025)1486 4508 y Ft(\()p Fz(t)p Ft(\))h(:=)e(e)1751 4478 y Fx(\()p Fr(E)p Fx(+)p Fu(\025)p Fr(PQP)p Fx(+)p Fu(\025)2113 4453 y Fq(2)2145 4478 y Fu(K)t Fx(\))p Fu(t=\025)2329 4453 y Fq(2)2366 4508 y Fz(y)s(:)523 4682 y FC(No)n(w)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(21)1611 269 y Fz(h)1659 281 y Fu(\025)1725 269 y Ft(=)23 b Fz(g)1853 281 y Fu(\025)1915 269 y Ft(+)18 b Fz(H)2067 281 y Fu(\025)2110 269 y Fz(h)2158 281 y Fu(\025)2202 269 y Fz(;)1611 439 y(g)1651 451 y Fu(\025)1717 439 y Ft(=)23 b Fz(g)1845 451 y Fu(\025)1906 439 y Ft(+)18 b Fz(G)2054 451 y Fu(\025)2098 439 y Fz(g)2138 451 y Fu(\025)2181 439 y Fz(:)523 584 y FC(Thus)1086 684 y Fz(h)1134 696 y Fu(\025)1196 684 y Fs(\000)g Fz(g)1319 696 y Fu(\025)1385 684 y Ft(=)23 b(\(1)18 b Fs(\000)g Fz(H)1717 696 y Fu(\025)1761 684 y Ft(\))1793 649 y Fy(\000)p Fx(1)1882 684 y Fz(g)1922 696 y Fu(\025)1984 684 y Fs(\000)g Ft(\(1)g Fs(\000)g Fz(G)2307 696 y Fu(\025)2351 684 y Ft(\))2383 649 y Fy(\000)p Fx(1)2472 684 y Fz(g)2512 696 y Fu(\025)2579 684 y Fs(!)23 b Ft(0)p Fz(:)523 833 y Ff(2)523 1124 y FA(3.5)40 b(Pr)o(oof)19 b(of)h(time)g(dependent)h (weak)f(coupling)h(limit)523 1306 y(Pr)o(oof)g(of)i(Theor)o(em)g(3)g (and)h(3*.)e FC(In)h(addition)f(to)h(the)h(assumptions)e(of)h(Theorem)e (8)i(we)h(sup-)523 1406 y(pose)c(that)g Fw(PQP)i Ft(=)h(0)d FC(and)g Fz(K)1398 1376 y Fu(\\)1449 1406 y FC(e)o(xists.)648 1506 y(Theorem)e(7)i(implies)h(that)1082 1703 y Ft(lim)1070 1757 y Fu(\025)p Fy(!)p Fx(0)1270 1703 y Ft(sup)1223 1773 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1410 1781 y Fq(0)1456 1703 y Fs(k)p Ft(e)1535 1669 y Fy(\000)p Fr(E)p Fu(t=\025)1724 1644 y Fq(2)1760 1703 y Ft(e)1797 1669 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025)1977 1644 y Fq(2)2009 1669 y Fu(K)t Fx(\))p Fu(=\025)2168 1644 y Fq(2)2205 1703 y Fz(y)g Fs(\000)d Ft(e)2387 1669 y Fu(tK)2472 1644 y Fp(\\)2505 1703 y Fz(y)s Fs(k)k Ft(=)h(0)p Fz(:)523 1940 y FC(Theorem)c(8)h(yields)1115 2135 y Ft(lim)1103 2190 y Fu(\025)p Fy(!)p Fx(0)1303 2135 y Ft(sup)1256 2205 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1443 2213 y Fq(0)1489 2135 y Fs(k)p Fw(P)p Ft(e)1619 2101 y Fu(t)p Fr(L)1683 2110 y Fp(\025)1721 2101 y Fu(=\025)1794 2076 y Fq(2)1831 2135 y Fw(P)p Fz(y)h Fs(\000)d Ft(e)2064 2101 y Fu(t)p Fx(\()p Fr(E)p Fx(+)p Fu(\025)2244 2076 y Fq(2)2276 2101 y Fu(K)t Fx(\))p Fu(=\025)2435 2076 y Fq(2)2472 2135 y Fz(y)s Fs(k)k Ft(=)h(0)p Fz(:)523 2386 y FC(Using)d(that)g Ft(e)924 2356 y Fu(t)p Fr(E)1013 2386 y FC(is)h(isometric)f(we)g(obtain)1129 2583 y Ft(lim)1117 2637 y Fu(\025)p Fy(!)p Fx(0)1317 2583 y Ft(sup)1270 2653 y Fx(0)p Fy(\024)p Fu(t)p Fy(\024)p Fu(t)1457 2661 y Fq(0)1503 2583 y Fs(k)p Ft(e)1582 2549 y Fy(\000)p Fr(E)p Fu(t=\025)1771 2524 y Fq(2)1807 2583 y Fw(P)p Ft(e)1895 2549 y Fu(t)p Fr(L)1959 2558 y Fp(\025)1997 2549 y Fu(=\025)2070 2524 y Fq(2)2107 2583 y Fw(P)p Fz(y)h Fs(\000)d Ft(e)2340 2549 y Fu(tK)2425 2524 y Fp(\\)2458 2583 y Fz(y)s Fs(k)k Ft(=)h(0)p Fz(:)455 b FC(\(33\))523 2847 y(It)21 b(is)g(clear)f(from)f(\(33\))g(that)h Ft(e)1376 2817 y Fu(tK)1461 2792 y Fp(\\)1515 2847 y FC(is)h(contracti)n(v)o(e.)d Ff(2)523 2997 y FA(Pr)o(oof)j(of)h(Theor)o(em)g(4)g FC(Because)h(of)f (the)g(\002nite)g(dimension)f(all)i(operators)e(on)h Ft(Ran)o Fw(P)h FC(are)f(w*)523 3096 y(continuous)k(and)h(the)g(strong) g(and)g(norm)g(con)m(v)o(er)o(gence)c(coincide.)j(Besides,)j(we)f(can)f (apply)523 3196 y(Theorem)19 b(7)h(about)f(the)h(e)o(xistence)g(of)g Fz(K)1729 3166 y Fu(\\)1759 3196 y FC(.)h Ff(2)523 3487 y FA(3.6)40 b(Pr)o(oof)19 b(of)h(the)g(coincidence)g(of)g Fi(M)1723 3499 y Fg(st)1807 3487 y FA(and)h Fi(M)2057 3499 y Fg(dyn)2205 3487 y FA(with)f(the)h(LSO)523 3669 y(Pr)o(oof)e(of)g(Theor)o(em)i(5.)f FC(Set)1387 3852 y Fz(f)9 b Ft(\()p Fz(s)p Ft(\))23 b(:=)62 b(sup)1674 3926 y Fy(j)p Fu(\025)p Fy(j\024)p Fu(\025)1844 3934 y Fq(0)1890 3852 y Fs(k)p Fw(PQ)p Ft(e)2085 3818 y Fu(s)2115 3803 y Fj(e)2116 3818 y Fr(P)o(L)2190 3827 y Fp(\025)2228 3803 y Fj(e)2229 3818 y Fr(P)2268 3852 y Fw(QP)p Fs(k)p Fz(:)523 4102 y FC(W)-7 b(e)21 b(kno)n(w)f(that)g Fz(f)9 b Ft(\()p Fz(t)p Ft(\))21 b FC(is)g(inte)o(grable.)648 4202 y(F)o(or)e(an)o(y)h Fz(e)j Fs(2)g Fw(R)e FC(and)e Fz(\030)27 b Fs(\025)c Ft(0)d FC(we)h(can)f(dominate)f(the)h(inte)o (grand)e(in)i(the)h(inte)o(gral)1135 4398 y Fz(F)1188 4410 y Fu(\025)1232 4398 y Ft(\(i)p Fz(e;)14 b(\030)t Ft(\))25 b(:=)1571 4331 y Fl(R)1626 4352 y Fy(1)1610 4428 y Fx(0)1710 4398 y Fw(PQ)p Ft(e)1863 4368 y Fu(s)1893 4353 y Fj(e)1894 4368 y Fr(P)o(L)1968 4377 y Fp(\025)2006 4353 y Fj(e)2007 4368 y Fr(P)2047 4398 y Fw(QP)p Ft(e)2200 4368 y Fy(\000)p Fx(\(i)p Fu(e)p Fx(+)p Fu(\025)2418 4343 y Fq(2)2450 4368 y Fu(\030)r Fx(\))p Fu(s)2544 4398 y Ft(d)p Fz(s)1460 4611 y Ft(=)e Fw(PQ)1678 4519 y Fl(\020)1729 4590 y(e)1727 4611 y Fw(P)o Ft(\(i)p Fz(e)c Ft(+)f Fz(\025)2021 4581 y Fx(2)2058 4611 y Fz(\030)t Ft(\))h Fs(\000)2235 4590 y Fl(e)2232 4611 y Fw(PL)2338 4623 y Fu(\025)2384 4590 y Fl(e)2382 4611 y Fw(P)2433 4519 y Fl(\021)2482 4536 y Fy(\000)p Fx(1)2585 4611 y Fw(QP)3174 4509 y FC(\(34\))p eop end %%Page: 22 22 TeXDict begin 22 21 bop 523 100 a FB(22)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FC(by)24 b Fz(f)9 b Ft(\()p Fz(s)p Ft(\))p FC(.)25 b(Hence,)f(using)f(the)i(dominated)d(con)m(v)o (er)o(gence)f(theorem)i(we)i(see)g(that)f Fz(F)2985 294 y Fu(\025)3029 282 y Ft(\(i)p Fz(e;)14 b(\030)t Ft(\))25 b FC(is)523 382 y(continuous)18 b(at)j Fz(\025)i Ft(=)g(0)d FC(and)g Fz(\030)27 b Fs(\025)c Ft(0)p FC(.)d(But)926 497 y Fl(P)872 634 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1082 559 y Fm(1)1130 571 y Fx(i)p Fu(e)1184 559 y Ft(\()p Fw(E)p Ft(\))p Fz(F)1356 571 y Fx(0)1395 559 y Ft(\(i)p Fz(e;)14 b Ft(0\))p Fm(1)1648 571 y Fx(i)p Fu(e)1702 559 y Ft(\()p Fw(E)p Ft(\))782 820 y(=)926 757 y Fl(P)872 894 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1094 820 y Ft(lim)1082 874 y Fu(\025)p Fy(!)p Fx(0)1235 820 y Fm(1)1283 832 y Fx(i)p Fu(e)1337 820 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1535 727 y Fl(\020)1587 798 y(e)1585 820 y Fw(P)p Ft(\(i)p Fz(e)k Ft(+)g Fz(\025)1879 789 y Fx(2)1917 820 y Fz(\030)t Ft(\))h Fs(\000)2093 798 y Fl(e)2091 820 y Fw(P)o(L)2196 832 y Fu(\025)2243 798 y Fl(e)2240 820 y Fw(P)2291 727 y Fl(\021)2341 745 y Fy(\000)p Fx(1)2444 820 y Fw(QP)p Fm(1)2608 832 y Fx(i)p Fu(e)2661 820 y Ft(\()p Fw(E)p Ft(\))26 b(=)c Fz(M)2974 832 y Fx(st)3030 820 y Fz(:)648 1067 y FC(Recall)e(\(9\),)g(the)g(de\002nition)f(of)h Fz(K)1641 1079 y Fu(\025)1684 1067 y Ft(\()p Fz(t)p Ft(\))p FC(:)1264 1325 y Fz(K)1335 1337 y Fu(\025)1378 1325 y Ft(\()p Fz(t)p Ft(\))j(:=)1606 1212 y Fl(Z)1689 1232 y Fu(\025)1728 1207 y Fn(\000)p Fq(2)1806 1232 y Fu(t)1652 1400 y Fx(0)1849 1325 y Ft(e)1886 1290 y Fy(\000)p Fu(s)p Fr(E)2012 1325 y Fw(PQ)p Ft(e)2165 1290 y Fu(s)2195 1275 y Fj(e)2196 1290 y Fr(P)o(L)2270 1299 y Fp(\025)2308 1275 y Fj(e)2309 1290 y Fr(P)2348 1325 y Fw(QP)p Ft(d)p Fz(s:)523 1554 y FC(Its)e(inte)o(grand)d(can)h(also)i(be)f(dominated)e (by)i Fz(f)9 b Ft(\()p Fz(s)p Ft(\))p FC(.)20 b(Hence,)f(using)h(again) f(the)h(dominated)e(con-)523 1653 y(v)o(er)o(gence)g(theorem,)g(we)j (see)g(that,)f(for)f Fz(\025)24 b Fs(!)f Ft(0)p FC(,)d Fz(K)2014 1665 y Fu(\025)2057 1653 y Ft(\()p Fz(t)p Ft(\))h FC(is)g(con)m(v)o(er)o(gent)c(to)1400 1883 y Fz(K)29 b Ft(=)1587 1770 y Fl(Z)1670 1791 y Fy(1)1634 1959 y Fx(0)1755 1883 y Ft(e)1792 1849 y Fy(\000)p Fu(s)p Fr(E)1918 1883 y Fw(PQ)p Ft(e)2071 1849 y Fu(s)2101 1834 y Fj(e)2102 1849 y Fr(P)o(L)2176 1857 y Fq(0)2212 1883 y Fw(QP)p Ft(d)p Fz(s:)523 2107 y FC(Therefore,)1120 2214 y Fz(K)1197 2184 y Fu(\\)1252 2214 y Ft(=)1394 2151 y Fl(P)1340 2288 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1550 2214 y Fm(1)1598 2226 y Fx(i)p Fu(e)1653 2214 y Ft(\()p Fw(E)p Ft(\))1786 2147 y Fl(R)1842 2167 y Fy(1)1826 2243 y Fx(0)1926 2214 y Fw(Q)p Ft(e)2028 2184 y Fu(s)p Fr(L)2098 2192 y Fq(0)2129 2169 y Fj(e)2130 2184 y Fr(P)2170 2214 y Fw(Q)p Fm(1)2283 2226 y Fx(i)p Fu(e)2336 2214 y Ft(\()p Fw(E)p Ft(\)e)2492 2184 y Fy(\000)p Fx(i)p Fu(es)2631 2214 y Ft(d)p Fz(s)1252 2408 y Ft(=)1394 2346 y Fl(P)1340 2483 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1550 2408 y Fm(1)1598 2420 y Fx(i)p Fu(e)1653 2408 y Ft(\()p Fw(E)p Ft(\))p Fz(F)1825 2420 y Fx(0)1863 2408 y Ft(\(i)p Fz(e;)14 b Ft(0\))p Fm(1)2116 2420 y Fx(i)p Fu(e)2170 2408 y Ft(\()p Fw(E)p Ft(\))p Fz(:)523 2597 y Ff(2)523 2962 y Fv(4)41 b(Completely)25 b(positi)o(v)o(e)g(semigr)n(oups)523 3162 y FC(In)18 b(this)h(section)g(we)f(recall)h(basic)g(information)d(about)h (completely)g(positi)n(v)o(e)h(maps)g(and)g(semi-)523 3261 y(groups,)h(which)i(are)g(often)f(used)h(to)g(describe)g(irre)n(v) o(ersible)e(dynamics)h(of)h(quantum)e(systems.)523 3361 y(F)o(or)h(simplicity)-5 b(,)19 b(most)h(of)g(the)g(time)h(we)f (restrict)h(ourselv)o(es)e(to)h(the)g(\002nite)h(dimensional)d(case.) 523 3602 y FA(4.1)40 b(Completely)20 b(positi)o(v)o(e)g(maps)523 3784 y FC(The)j(follo)n(wing)f(f)o(acts)i(are)g(well)g(kno)n(wn)e(and)h (can)g(be)h(e.g.)f(found)f(in)h([BR2],)h(Notes)g(and)f(Re-)523 3884 y(marks)d(to)g(Section)g(5.3.1.)648 3984 y(Let)i Fs(K)844 3996 y Fx(1)881 3984 y Fz(;)14 b Fs(K)981 3996 y Fx(2)1042 3984 y FC(be)22 b(Hilbert)g(spaces.)g(W)-7 b(e)23 b(say)f(that)h(a)f(linear)g(map)g Fz(\004)32 b Ft(:)27 b Fs(B)s Ft(\()p Fs(K)2807 3996 y Fx(1)2844 3984 y Ft(\))g Fs(!)f(B)s Ft(\()p Fs(K)3165 3996 y Fx(2)3202 3984 y Ft(\))d FC(is)523 4083 y(positi)n(v)o(e)e(if)n(f)h Fz(A)27 b Fs(\025)f Ft(0)c FC(implies)h Fz(\004)6 b Ft(\()p Fz(A)p Ft(\))27 b Fs(\025)f Ft(0)p FC(.)c(W)-7 b(e)23 b(say)g(that)f(it)h(is)g(completely)d(positi)n(v)o(e)i(\(c.p.)f(for)523 4183 y(short\))e(if)n(f)i(for)e(an)o(y)g Fz(n)p FC(,)i Fz(\004)j Fs(\012)18 b Fm(1)1400 4198 y Fy(B)q Fx(\()p Fr(C)1513 4181 y Fp(n)1554 4198 y Fx(\))1605 4183 y FC(is)j(positi)n(v) o(e)e(as)i(a)g(map)e Fs(B)s Ft(\()p Fs(K)2425 4195 y Fx(1)2481 4183 y Fs(\012)f Fw(C)2624 4153 y Fu(n)2669 4183 y Ft(\))23 b Fs(!)g(B)s Ft(\()p Fs(K)2983 4195 y Fx(2)3039 4183 y Fs(\012)18 b Fw(C)3182 4153 y Fu(n)3227 4183 y Ft(\))p FC(.)648 4282 y(W)-7 b(e)21 b(will)g(say)f(that)h(a)f (positi)n(v)o(e)g(map)f Fz(\004)27 b FC(is)21 b(Mark)o(o)o(v)e(if)h Fz(\004)6 b Ft(\()p Fm(1)p Ft(\))23 b(=)g Fm(1)p FC(.)648 4382 y(Recall)32 b(that)g Fs(B)1107 4352 y Fx(1)1143 4382 y Ft(\()p Fs(K)1238 4394 y Fu(i)1266 4382 y Ft(\))h FC(denotes)e(the)h(space)f(of)h(trace)f(class)i(operators)d(on)i Fs(K)2976 4394 y Fu(i)3003 4382 y FC(.)g(W)-7 b(e)33 b(can)523 4482 y(de\002ne)19 b(positi)n(v)o(e)f(and)h(completely)f (positi)n(v)o(e)g(maps)i(from)e Fs(B)2268 4452 y Fx(1)2304 4482 y Ft(\()p Fs(K)2399 4494 y Fx(2)2437 4482 y Ft(\))i FC(to)g Fs(B)2632 4452 y Fx(1)2668 4482 y Ft(\()p Fs(K)2763 4494 y Fx(1)2801 4482 y Ft(\))g FC(repeating)e(v)o(er)n(-)523 4581 y(batim)23 b(the)h(de\002nition)e(for)h(the)g(algebra)f(of)i (bounded)d(operators.)g(W)-7 b(e)25 b(will)f(say)g(that)f(the)h(map)523 4681 y(is)d(Mark)o(o)o(v)e(if)h(it)h(preserv)o(es)e(the)h(trace.)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(23)648 282 y FC(W)-7 b(e)21 b(can)f(also)g(speak)g(of)g(positi)n(v)o(e)f(and)h(completely)f (positi)n(v)o(e)g(maps)h(on)g Fs(B)2816 252 y Fx(2)2852 282 y Ft(\()p Fs(K)q Ft(\))p FC(.)648 382 y(W)-7 b(e)28 b(will)h(sometimes)e(say)h(that)f(maps)h(on)f(the)h(algebra)e Fs(B)s Ft(\()p Fs(K)q Ft(\))i FC(are)g(\223in)f(the)h(Heisenber)o(g)523 482 y(picture\224,)c(maps)i(on)f Fs(B)1206 451 y Fx(1)1242 482 y Ft(\()p Fs(K)q Ft(\))i FC(are)f(\223in)f(the)h(Schr)7 b(\250)-35 b(odinger)23 b(picture\224)h(and)h(maps)h(on)f Fs(B)3020 451 y Fx(2)3056 482 y Ft(\()p Fs(K)q Ft(\))i FC(are)523 581 y(\223in)d(the)g(standard)f(picture\224)f(\(see)j(the)f (notion)e(of)i(the)g(standard)e(representation)g(later)i(on)g(and)523 681 y(in)c([DJP]\).)648 780 y(From)d(no)n(w)g(on,)f(for)h(simplicity)-5 b(,)17 b(in)g(this)i(section)e(we)h(will)g(assume)f(that)h(the)f (spaces)h Fs(K)3165 792 y Fu(i)3211 780 y FC(are)523 880 y(\002nite)25 b(dimensional.)e(Thus)h Fs(B)s Ft(\()p Fs(K)1515 892 y Fu(i)1542 880 y Ft(\))i FC(and)e Fs(B)1803 850 y Fx(2)1839 880 y Ft(\()p Fs(K)1934 892 y Fu(i)1962 880 y Ft(\))h FC(and)f Fs(B)2222 850 y Fx(1)2259 880 y Ft(\()p Fs(K)2354 892 y Fu(i)2382 880 y Ft(\))h FC(coincide)f(with)h (one)f(another)523 980 y(as)i(v)o(ector)f(spaces.)g(If)h Fz(\004)32 b FC(is)26 b(a)g(map)f(from)g(matrices)g(on)g Fs(K)2270 992 y Fx(1)2334 980 y FC(to)g(matrices)h(on)f Fs(K)2904 992 y Fx(2)2941 980 y FC(,)h(it)g(is)h(often)523 1079 y(useful)f(to)g(distinguish)g(whether)f(it)i(is)g(understood)d(as) j(a)g(map)e(from)h Fs(B)s Ft(\()p Fs(K)2752 1091 y Fx(1)2789 1079 y Ft(\))h FC(to)f Fs(B)s Ft(\()p Fs(K)3092 1091 y Fx(2)3129 1079 y Ft(\))h FC(\(we)523 1179 y(then)22 b(say)h(that)g(it)g(is)g(in)g(the)g(Heisenber)o(g)d(picture\),)h(as)j (a)f(map)f(from)f Fs(B)2593 1149 y Fx(2)2630 1179 y Ft(\()p Fs(K)2725 1191 y Fx(1)2762 1179 y Ft(\))j FC(to)f Fs(B)2964 1149 y Fx(2)3000 1179 y Ft(\()p Fs(K)3095 1191 y Fx(2)3133 1179 y Ft(\))g FC(\(we)523 1279 y(then)h(say)g(that)g(it)g(is)h(in)g (the)e(standard)g(picture\))g(or)h(as)g(a)h(map)e(from)g Fs(B)2589 1248 y Fx(1)2625 1279 y Ft(\()p Fs(K)2720 1291 y Fx(1)2758 1279 y Ft(\))i FC(to)f Fs(B)2962 1248 y Fx(1)2999 1279 y Ft(\()p Fs(K)3094 1291 y Fx(2)3131 1279 y Ft(\))h FC(\(we)523 1378 y(then)20 b(say)g(that)g(it)h(is)h(in)e(the)g(Schr)7 b(\250)-35 b(odinger)18 b(picture\).)648 1478 y(Note)24 b(that)g Fs(B)1041 1448 y Fx(1)1078 1478 y Ft(\()p Fs(K)1173 1490 y Fu(i)1201 1478 y Ft(\))h FC(and)f Fs(B)s Ft(\()p Fs(K)1556 1490 y Fu(i)1583 1478 y Ft(\))h FC(are)g(dual)f(to)g(one)g (another)-5 b(.)23 b(\(This)h(is)i(one)e(of)g(the)g(places)523 1577 y(where)17 b(we)h(use)h(one)e(of)g(propertie)f(of)i(\002nite)g (dimensional)e(spaces.)i(In)g(general,)e Fs(B)s Ft(\()p Fs(K)3013 1589 y Fu(i)3041 1577 y Ft(\))i FC(is)h(only)523 1677 y(dual)26 b(to)i Fs(B)844 1647 y Fx(1)880 1677 y Ft(\()p Fs(K)975 1689 y Fu(i)1003 1677 y Ft(\))g FC(and)e(not)h(the)g (other)f(w)o(ay)h(around.\))d(The)j(\(sesquilinear\))e(duality)h (between)523 1777 y Fs(B)581 1747 y Fx(1)617 1777 y Ft(\()p Fs(K)712 1789 y Fu(i)741 1777 y Ft(\))21 b FC(and)e Fs(B)s Ft(\()p Fs(K)1087 1789 y Fu(i)1114 1777 y Ft(\))i FC(is)h(gi)n(v)o(en)d (by)1297 1954 y Ft(T)-7 b(r)p Fz(\032)1426 1919 y Fy(\003)1464 1954 y Fz(A;)76 b(\032)23 b Fs(2)h(B)1828 1919 y Fx(1)1864 1954 y Ft(\()p Fs(K)1959 1966 y Fu(i)1987 1954 y Ft(\))p Fz(;)97 b(A)24 b Fs(2)f(B)s Ft(\()p Fs(K)2456 1966 y Fu(i)2483 1954 y Ft(\))p Fz(:)523 2131 y FC(If)31 b Fz(\004)39 b FC(is)32 b(a)g(map)f(\223in)h(the)f(Heisenber)o(g)f(picture\224,)g (then)i(its)g(adjoint)f Fz(\004)2655 2101 y Fy(\003)2693 2131 y FC(,)h(is)g(a)g(map)g(\223in)f(the)523 2230 y(Schr)7 b(\250)-35 b(odinger)20 b(picture\224)h(\(and)g(vice)h(v)o(ersa\).)f (Clearly)-5 b(,)21 b Fz(\004)29 b FC(is)23 b(a)g(Mark)o(o)o(v)d (transformation)g(in)i(the)523 2330 y(Heisenber)o(g)c(picture)i(if)n(f) g Fz(\004)1337 2300 y Fy(\003)1396 2330 y FC(is)h(Mark)o(o)o(v)d(in)j (the)f(Schr)7 b(\250)-35 b(odinger)17 b(picture.)648 2430 y(Note)h(that)g(\(in)g(a)h(\002nite)f(dimension\))e(the)i (de\002nition)f(of)h Fz(\004)2332 2399 y Fy(\003)2389 2430 y FC(does)g(not)g(depend)e(on)i(whether)523 2529 y(we)j(consider)e Fz(\004)26 b FC(in)21 b(the)f(Heisenber)o(g,)e (standard)h(or)h(Schr)7 b(\250)-35 b(odinger)17 b(picture.)523 2765 y FA(4.2)40 b(Stinespring)21 b(r)o(epr)o(esentation)d(of)i(a)g (completely)g(positi)o(v)o(e)g(map)523 2944 y FC(By)c(the)f (Stinespring)f(theorem)g([St],)h Fz(\004)29 b Ft(:)24 b Fs(B)s Ft(\()p Fs(K)1887 2956 y Fx(1)1923 2944 y Ft(\))g Fs(!)f(B)s Ft(\()p Fs(K)2238 2956 y Fx(2)2275 2944 y Ft(\))16 b FC(is)g(completely)e(positi)n(v)o(e)g(if)n(f)i(there)523 3043 y(e)o(xists)h(an)f(auxilliary)g(\002nite)g(dimensional)f(Hilbert)h (space)h Fs(H)h FC(and)e Fz(W)35 b Fs(2)23 b(B)s Ft(\()p Fs(K)2792 3055 y Fx(2)2829 3043 y Fz(;)14 b Fs(K)2929 3055 y Fx(1)2971 3043 y Fs(\012)5 b(H)q Ft(\))16 b FC(such)523 3143 y(that)1263 3242 y Fz(\004)6 b Ft(\()p Fz(B)t Ft(\))24 b(=)e Fz(W)1663 3208 y Fy(\003)1722 3242 y Fz(B)t Fs(\012)p Fm(1)1902 3254 y Fy(H)1983 3242 y Fz(W)n(;)77 b(B)27 b Fs(2)c(B)s Ft(\()p Fs(K)2480 3254 y Fx(1)2517 3242 y Ft(\))p Fz(:)602 b FC(\(35\))648 3389 y(In)25 b(practice)h(it)h(can)e (be)h(useful)g(to)g(transform)f(\(35\))g(into)g(a)i(slightly)f(dif)n (ferent)e(form.)h(Let)523 3488 y(us)h(\002x)h(an)f(orthonormal)d(basis) j Ft(\()p Fz(e)1543 3500 y Fx(1)1580 3488 y Fz(;)14 b(:)g(:)g(:)g(;)g (e)1804 3500 y Fu(n)1849 3488 y Ft(\))27 b FC(in)f Fs(H)q FC(.)g(Then)f(the)h(operator)e Fz(W)39 b FC(is)27 b(completely)523 3588 y(determined)18 b(by)i(gi)n(ving)f(a)i(f)o(amily)e(of)h(operators) f Fz(W)2044 3600 y Fx(1)2082 3588 y Fz(;)14 b(:)g(:)g(:)f(;)h(W)2344 3600 y Fu(n)2413 3588 y Fs(2)23 b(B)s Ft(\()p Fs(K)2644 3600 y Fx(2)2681 3588 y Fz(;)14 b Fs(K)2781 3600 y Fx(1)2818 3588 y Ft(\))21 b FC(such)f(that)1287 3831 y Fz(W)12 b(\011)1428 3843 y Fx(2)1488 3831 y Ft(=)1615 3728 y Fu(n)1575 3752 y Fl(X)1578 3929 y Fu(j)s Fx(=1)1695 3831 y Ft(\()p Fz(W)1805 3843 y Fu(j)1841 3831 y Fz(\011)1892 3843 y Fx(2)1929 3831 y Ft(\))19 b Fs(\012)f Fz(e)2102 3843 y Fu(j)2137 3831 y Fz(;)76 b(\011)2287 3843 y Fx(2)2347 3831 y Fs(2)23 b(K)2488 3843 y Fx(2)2526 3831 y Fz(:)523 4091 y FC(Then)1530 4238 y Fz(\004)6 b Ft(\()p Fz(B)t Ft(\))24 b(=)1880 4134 y Fu(n)1840 4159 y Fl(X)1843 4336 y Fu(j)s Fx(=1)1974 4238 y Fz(W)2064 4203 y Fy(\003)2052 4258 y Fu(j)2102 4238 y Fz(B)t(W)2247 4250 y Fu(j)2283 4238 y Fz(:)868 b FC(\(36\))648 4483 y(There)17 b(e)o(xists)i(a)g (third)f(w)o(ay)g(of)h(writing)f(\(35\),)f(which)h(is)h(sometimes)f (useful.)g(Let)p 3024 4416 71 4 v 19 w Fs(H)i FC(be)e(the)523 4583 y(space)25 b(conjugate)e(to)j Fs(H)g FC(and)f(let)h Fs(H)33 b(3)g Fz(\010)f Fs(7!)p 1917 4516 56 4 v 33 w Fz(\010)h Fs(2)p 2092 4516 71 4 v 32 w(H)27 b FC(be)e(the)g (corresponding)d(conjugation)523 4682 y(\(see)e(e.g.)g([DJ2)o(]\).)g(W) -7 b(e)22 b(de\002ne)d Fz(W)1519 4652 y Fu(?)1580 4682 y Fs(2)24 b(B)s Ft(\()p Fs(K)1812 4694 y Fx(1)1849 4682 y Fz(;)14 b Fs(K)1949 4694 y Fx(2)2005 4682 y Fs(\012)p 2088 4616 V 18 w(H)q Ft(\))21 b FC(by)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 523 100 a FB(24)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)1125 282 y Ft(\()p Fz(W)1247 248 y Fu(?)1286 282 y Fz(\011)1337 294 y Fx(1)1374 282 y Fs(j)p Fz(\011)1448 294 y Fx(2)1503 282 y Fs(\012)p 1586 216 56 4 v 18 w Fz(\010)q Ft(\))1674 308 y Fy(K)1725 316 y Fq(2)1757 308 y Fy(\012)p 1809 259 57 3 v(H)1893 282 y Ft(=)23 b(\()p Fz(\011)2064 294 y Fx(1)2120 282 y Fs(\012)18 b Fz(\010)p Fs(j)p Fz(W)12 b(\011)2422 294 y Fx(2)2459 282 y Ft(\))2491 294 y Fy(K)2542 302 y Fq(1)2574 294 y Fy(\012H)2687 282 y Fz(;)464 b FC(\(37\))523 453 y(\(see)19 b([DJ2]\).)g(\(Note)f(that)i(we)f(use)g(tw)o(o)h(dif)n(ferent)d(kinds)i (of)g(stars:)h Fs(\003)f FC(for)f(the)i(hermitian)d(conju-)523 553 y(gation)i(and)h Fz(?)g FC(for)f(\(37\)\).)g(Let)h Ft(T)-7 b(r)p 1496 530 62 3 v 25 x Fy(H)1578 553 y FC(denote)19 b(the)h(partial)g(trace)g(o)o(v)o(er)p 2523 486 71 4 v 19 w Fs(H)q FC(.)g(Then)1499 724 y Fz(\004)6 b Ft(\()p Fz(B)t Ft(\))24 b(=)e(T)-7 b(r)p 1895 701 62 3 v 25 x Fy(H)1956 724 y Fz(W)2046 689 y Fu(?)2084 724 y Fz(B)t(W)2241 689 y Fu(?)p Fy(\003)2313 724 y Fz(:)838 b FC(\(38\))648 895 y(If)20 b Fz(\004)26 b FC(is)21 b(gi)n(v)o(en)e(by)h(\(35\),)f (then)h Fz(\004)1606 864 y Fy(\003)1664 895 y FC(can)g(be)h(written)f (in)g(the)g(follo)n(wing)f(three)g(forms:)1468 1062 y Fz(\004)1536 1032 y Fy(\003)1574 1062 y Ft(\()p Fz(C)6 b Ft(\))25 b(=)e(T)-7 b(r)1902 1074 y Fy(H)1963 1062 y Fz(W)12 b(C)6 b(W)2208 1032 y Fy(\003)1728 1298 y Ft(=)1853 1219 y Fu(n)1830 1236 y Fl(P)1816 1371 y Fu(j)s Fx(=1)1945 1298 y Fz(W)2023 1310 y Fu(j)2058 1298 y Fz(C)g(W)2213 1268 y Fy(\003)2201 1320 y Fu(j)1728 1492 y Ft(=)23 b Fz(W)1906 1462 y Fu(?)p Fy(\003)1978 1492 y Fz(C)6 b Fs(\012)p Fm(1)p 2156 1470 V 26 x Fy(H)2217 1492 y Fz(W)2307 1462 y Fu(?)2345 1492 y Fz(;)523 1670 y FC(where)20 b Fz(C)29 b Fs(2)23 b(B)971 1639 y Fx(1)1008 1670 y Ft(\()p Fs(K)1103 1682 y Fx(2)1140 1670 y Ft(\))p FC(.)523 1901 y FA(4.3)40 b(Completely)20 b(positi)o(v)o(e)g(semigr)o(oups)523 2074 y FC(Let)i Fs(K)h FC(be)f(a)g(\002nite)f(dimensional)f(Hilbert)i (space)f(and)g Fz(t)k Fs(7!)h Fz(\003)p Ft(\()p Fz(t)p Ft(\))c FC(a)g(continuous)e(1-parameter)523 2174 y(semigroup)e(of)i (operators)f(on)h Fs(B)s Ft(\()p Fs(K)q Ft(\))p FC(.)g(Let)h Fz(M)29 b FC(be)20 b(its)h(generator)m(,)d(so)j(that)f Fz(\003)p Ft(\()p Fz(t)p Ft(\))j(=)g(e)2977 2144 y Fu(tM)3075 2174 y FC(.)648 2273 y(W)-7 b(e)23 b(say)g(that)g Fz(\003)p Ft(\()p Fz(t)p Ft(\))g FC(is)h(a)f(completely)e(positi)n(v)o(e)h (semigroup)f(if)n(f)h Fz(\003)p Ft(\()p Fz(t)p Ft(\))i FC(is)f(completely)f(pos-)523 2373 y(iti)n(v)o(e)k(for)e(an)o(y)h Fz(t)33 b Fs(\025)g Ft(0)p FC(.)26 b Fz(\003)p Ft(\()p Fz(t)p Ft(\))g FC(is)g(called)g(a)g(Mark)o(o)o(v)e(semigroup)g(if)n(f)h Fz(\003)p Ft(\()p Fz(t)p Ft(\))h FC(is)h(Mark)o(o)o(v)d(for)h(an)o(y) 523 2473 y Fz(t)e Fs(\025)g Ft(0)p FC(.)648 2572 y Fz(\003)p Ft(\()p Fz(t)p Ft(\))g FC(is)g(a)g(completely)d(positi)n(v)o(e)i (semigroup)e(if)n(f)i(there)g(e)o(xists)h(an)f(operator)e Fz(\001)k FC(on)d Fs(K)k FC(and)523 2672 y(a)c(completely)d(positi)n(v) o(e)i(map)f Fz(\004)27 b FC(on)20 b Fs(B)s Ft(\()p Fs(K)q Ft(\))h FC(such)f(that)1157 2843 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))23 b(=)g Fz(\001B)f Ft(+)c Fz(B)t(\001)1862 2808 y Fy(\003)1919 2843 y Ft(+)g Fz(\004)6 b Ft(\()p Fz(B)t Ft(\))p Fz(;)77 b(B)27 b Fs(2)d(B)s Ft(\()p Fs(K)q Ft(\))p Fz(:)495 b FC(\(39\))523 3014 y(Operators)19 b(of)g(the)h(form)f (\(39\))g(are)g(sometimes)h(called)g(Lindblad)e(or)h(Lindblad-K)m (ossak)o(o)n(wski)523 3113 y(generators)g([GKS)o(,)i(L].)648 3213 y(Let)f Ft([)p Fs(\001)p Fz(;)14 b Fs(\001)p Ft(])908 3225 y Fx(+)984 3213 y FC(denote)19 b(the)h(anticommutator)-5 b(.)18 b Fz(\003)p Ft(\()p Fz(t)p Ft(\))j FC(is)g(Mark)o(o)o(v)e(if)n (f)1199 3423 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))24 b(=)e(i[)p Fz(\002)r(;)14 b(B)t Ft(])20 b Fs(\000)1881 3367 y Ft(1)p 1881 3404 42 4 v 1881 3480 a(2)1933 3423 y([)p Fz(\004)6 b Ft(\(1\))p Fz(;)14 b(B)t Ft(])2257 3435 y Fx(+)2331 3423 y Ft(+)k Fz(\004)6 b Ft(\()p Fz(B)t Ft(\))p Fz(;)523 3629 y FC(where)20 b Fz(\002)25 b Ft(:=)956 3597 y Fx(1)p 956 3611 34 4 v 956 3658 a(2)999 3629 y Ft(\()p Fz(\001)19 b Ft(+)f Fz(\001)1271 3599 y Fy(\003)1309 3629 y Ft(\))p Fz(:)648 3729 y FC(If)i Fz(\004)26 b FC(is)21 b(gi)n(v)o(en)e(by)h (\(35\),)f(then)647 3897 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))25 b(=)d(i[)p Fz(\002)r(;)14 b(B)t Ft(])20 b(+)1331 3864 y Fx(1)p 1331 3878 V 1331 3926 a(2)1374 3830 y Fl(\000)1412 3897 y Fz(W)1502 3867 y Fy(\003)1540 3897 y Ft(\()p Fz(W)12 b(B)23 b Fs(\000)18 b Fz(B)t Fs(\012)p Ft(1)p Fz(W)12 b Ft(\))18 b(+)g(\()p Fz(B)t(W)2417 3867 y Fy(\003)2474 3897 y Fs(\000)g Fz(W)2647 3867 y Fy(\003)2685 3897 y Fz(B)t Fs(\012)p Ft(1\))p Fz(W)12 b Ft(\))3013 3830 y Fl(\001)893 4110 y Ft(=)22 b(i[)p Fz(\002)r(;)14 b(B)t Ft(])20 b(+)1331 4077 y Fx(1)p 1331 4091 V 1331 4139 a(2)1425 4031 y Fu(n)1401 4048 y Fl(P)1388 4183 y Fu(j)s Fx(=1)1503 4110 y Ft(\()p Fz(W)1625 4080 y Fy(\003)1613 4132 y Fu(j)1663 4110 y Ft([)p Fz(W)1764 4122 y Fu(j)1799 4110 y Fz(;)14 b(B)t Ft(])19 b(+)f([)p Fz(B)t(;)c(W)2245 4080 y Fy(\003)2233 4132 y Fu(j)2283 4110 y Ft(])p Fz(W)2384 4122 y Fu(j)2420 4110 y Ft(\))p Fz(;)3174 4035 y FC(\(40\))523 4339 y(and)876 4418 y Fz(M)966 4388 y Fy(\003)1004 4418 y Ft(\()p Fz(B)t Ft(\))26 b(=)c(i[)p Fz(\002)r(;)14 b(B)t Ft(])19 b Fs(\000)1598 4386 y Fx(1)p 1598 4400 V 1598 4447 a(2)1641 4418 y Ft([)p Fz(W)1754 4388 y Fy(\003)1793 4418 y Fz(W)n(;)14 b(B)t Ft(])1996 4430 y Fx(+)2069 4418 y Ft(+)k(T)-7 b(r)2238 4430 y Fy(H)2299 4418 y Fz(W)12 b(B)t(W)2546 4388 y Fy(\003)1161 4631 y Ft(=)22 b(i[)p Fz(\002)r(;)14 b(B)t Ft(])19 b(+)1625 4552 y Fu(n)1602 4569 y Fl(P)1588 4704 y Fu(j)s Fx(=1)1717 4564 y Fl(\000)1755 4631 y Fs(\000)1830 4599 y Fx(1)p 1830 4613 V 1830 4660 a(2)1873 4631 y Ft([)p Fz(W)1986 4601 y Fy(\003)1974 4653 y Fu(j)2024 4631 y Fz(W)2102 4643 y Fu(j)2137 4631 y Fz(;)14 b(B)t Ft(])2264 4643 y Fx(+)2338 4631 y Ft(+)k Fz(W)2511 4601 y Fu(?)p Fy(\003)2499 4653 y Fu(j)2583 4631 y Fz(B)t Fs(\012)p Ft(1)p Fz(W)2847 4601 y Fu(?)2835 4653 y(j)2885 4564 y Fl(\001)2936 4631 y Fz(:)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(25)648 282 y FC(Suppose)19 b(that)i Ft(e)1133 252 y Fu(tM)1254 282 y FC(is)g(a)h(positi)n(v)o(e)e (Mark)o(o)o(v)f(semigroup)g(in)i(the)g(Heisenber)o(g)e(picture.)h(W)-7 b(e)523 382 y(say)31 b(that)g(a)g(density)f(matrix)h Fz(\032)g FC(on)f Fs(K)j FC(is)f(stationary)e(with)h(respect)f(to)h (this)g(semiigroup)e(if)n(f)523 482 y Ft(e)560 451 y Fu(tM)654 426 y Fn(\003)693 482 y Ft(\()p Fz(\032)p Ft(\))24 b(=)f Fz(\032)p FC(.)15 b(Ev)o(ery)f(positi)n(v)o(e)h(Mark)o(o)o(v)f (semigroup)f(in)j(a)g(\002nite)g(dimension)e(has)h(a)h(stationary)523 581 y(density)k(matrix.)648 681 y(Mark)o(o)o(v)14 b(completely)g (positi)n(v)o(e)i(semigroups)e(\(both)h(in)h(the)g(Heisenber)o(g)e(and) i(Schr)7 b(\250)-35 b(odinger)523 780 y(picture\))19 b(are)g(often)g(used)h(in)f(quantum)f(physics.)h(In)g(the)h (literature,)f(the)o(y)g(are)h(called)f(by)g(man)o(y)523 880 y(names)h(such)g(as)h(quantum)d(dynamical)h(or)h(quantum)e(Mark)o (o)o(v)g(semigroups.)523 1117 y FA(4.4)40 b(Standard)20 b(Detailed)g(Balance)g(Condition)523 1297 y FC(In)k(the)g(literature)g (one)f(can)h(\002nd)g(a)h(number)d(of)i(v)n(arious)f(properties)g(that) h(are)g(called)g(the)g(De-)523 1396 y(tailed)h(Balance)g(Condition)e (\(DBC\).)i(In)g(the)g(quantum)e(conte)o(xt,)g(probably)g(the)i(best)g (kno)n(wn)523 1496 y(is)i(the)f(de\002ntion)e(due)i(to)g(Alicki)g([A])g (and)f(Frigerio-Gorini-K)m(ossak)o(o)n(wski-V)-9 b(erri)21 b([FGKV)o(],)523 1596 y(which)f(we)g(describe)g(in)g(the)g(ne)o(xt)g (subsection)f(and)h(call)g(the)g(DBC)i(in)e(the)g(sense)h(of)f(AFGKV) -11 b(.)648 1695 y(In)23 b(this)h(subsection)f(we)h(introduce)e(a)i (slightly)f(dif)n(ferent)f(property)g(that)i(we)g(think)f(is)h(the)523 1795 y(most)h(satisf)o(actory)f(generalization)e(of)j(the)f(DBC)i(from) e(the)h(clasical)g(to)g(the)f(quantum)f(case.)523 1894 y(It)e(is)h(a)g(modi\002cation)d(of)i(the)g(DBC)i(in)e(the)g(sense)g (of)g(AFGKV)-11 b(.)22 b(T)-7 b(o)21 b(distinguish)f(it)i(from)e(other) 523 1994 y(kinds)e(of)g(the)g(DBC,)h(we)f(will)h(call)g(it)g(the)f (standard)f(Detailed)h(Balance)g(Condition.)f(The)h(name)523 2094 y(is)24 b(justi\002ed)e(by)h(the)f(close)h(relationship)e(of)i (this)g(condition)e(to)h(the)h(standard)f(representation.)523 2193 y(W)-7 b(e)23 b(ha)n(v)o(e)e(not)g(seen)h(the)g(standard)e(DBC)j (in)f(the)g(literature,)f(b)n(ut)g(we)i(kno)n(w)d(that)i(it)g(belongs)f (to)523 2293 y(the)26 b(folklore)e(of)h(the)h(subject.)f(In)g (particular)m(,)f(it)j(w)o(as)f(considered)e(in)i(the)f(past)h(by)g(R.) g(Alicki)523 2393 y(and)20 b(A.)g(Maje)n(wski)g(\(pri)n(v)n(ate)f (communication\).)648 2492 y(In)28 b(the)g(literature)f(one)h(can)g (also)h(\002nd)f(other)f(properties)g(called)h(the)g(Detailed)g (Balance)523 2592 y(Condition)21 b([Ma1)o(,)h(Ma2)o(,)h(MaSt].)f(Most)g (of)g(them)g(in)m(v)n(olv)o(e)e(the)i(notion)f(of)h(the)g(time)g(re)n (v)o(ersal,)523 2692 y(which)15 b(is)h(not)g(used)f(in)h(the)f(case)h (of)f(the)h(standard)e(DBC)j(or)e(the)h(DBC)g(in)g(the)f(sense)h(of)g (AFGKV)-11 b(.)648 2791 y(Let)16 b(us)h(assume)g(that)g Fz(\032)g FC(is)g(a)g(nonde)o(generate)c(density)j(matrix)g(on)g Fs(K)q FC(.)i(\(That)e(means,)g Fz(\032)23 b(>)f Ft(0)p FC(,)523 2891 y Ft(T)-7 b(r)p Fz(\032)32 b Ft(=)g(1)p FC(,)25 b(and)g Fz(\032)1058 2861 y Fy(\000)p Fx(1)1173 2891 y FC(e)o(xists\).)g(On)h(the)f(space)g(of)g(operators)f(on)h Fs(K)i FC(we)f(introduce)e(the)h(scalar)523 2990 y(product)18 b(gi)n(v)o(en)h(by)h Fz(\032)p FC(:)1438 3090 y Ft(\()p Fz(A)p Fs(j)p Fz(B)t Ft(\))1654 3102 y Fu(\032)1716 3090 y Ft(:=)j(T)-7 b(r)p Fz(\032)1956 3056 y Fx(1)p Fu(=)p Fx(2)2060 3090 y Fz(A)2122 3056 y Fy(\003)2160 3090 y Fz(\032)2203 3056 y Fx(1)p Fu(=)p Fx(2)2308 3090 y Fz(B)t(:)776 b FC(\(41\))523 3237 y(This)23 b(space)g(equipped)d(with)j(the)g (scalar)g(product)e(\(41\))g(will)j(be)e(denoted)g(by)g Fs(B)2885 3207 y Fx(2)2882 3257 y Fu(\032)2921 3237 y Ft(\()p Fs(K)q Ft(\))p FC(.)i(Let)f Fs(\003)p Fz(\032)523 3337 y FC(denote)e(the)i(hermitian)e(conjugation)f(with)j(respect)f(to) g(this)i(scalar)e(product.)f(Thus)h(if)h Fz(M)31 b FC(is)24 b(a)523 3436 y(map)c(on)g Fs(B)s Ft(\()p Fs(K)q Ft(\))p FC(,)g(then)g Fz(M)1272 3406 y Fy(\003)p Fu(\032)1365 3436 y FC(is)h(de\002ned)e(by)1405 3615 y Ft(\()p Fz(M)1527 3580 y Fy(\003)p Fu(\032)1599 3615 y Ft(\()p Fz(A)p Ft(\))p Fs(j)p Fz(B)t Ft(\))1847 3627 y Fu(\032)1910 3615 y Ft(=)k(\()p Fz(A)p Fs(j)p Fz(M)9 b Ft(\()p Fz(B)t Ft(\)\))2368 3627 y Fu(\032)2408 3615 y Fz(:)523 3793 y FC(Explicitly)-5 b(,)1233 3893 y Fz(M)1323 3858 y Fy(\003)p Fu(\032)1395 3893 y Ft(\()p Fz(A)p Ft(\))24 b(=)e Fz(\032)1675 3858 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)1832 3893 y Fz(M)1922 3858 y Fy(\003)1959 3893 y Ft(\()p Fz(\032)2034 3858 y Fx(1)p Fu(=)p Fx(2)2139 3893 y Fz(A\032)2244 3858 y Fx(1)p Fu(=)p Fx(2)2348 3893 y Ft(\))p Fz(\032)2423 3858 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)2580 3893 y Fz(:)523 4063 y FA(De\002nition)e(1.)25 b Fk(Let)d Fz(M)31 b Fk(be)21 b(the)h(g)o(ener)o(ator)e(of)i(a)g(Mark)o(o)o(v)f(c.p.)g(semigr)l(oup)g (on)g Fs(B)s Ft(\()p Fs(K)q Ft(\))p Fk(.)h(W)-8 b(e)23 b(will)523 4163 y(say)29 b(that)f Fz(M)38 b Fk(satis\002es)29 b(the)g(standar)m(d)e(Detailed)i(Balance)e(Condition)g(with)j(r)m (espect)e(to)h Fz(\032)g Fk(if)523 4262 y(ther)m(e)20 b(e)n(xists)i(a)e(self-adjoint)f(oper)o(ator)g Fz(\002)24 b Fk(on)19 b Fs(K)k Fk(suc)o(h)c(that)1535 4420 y Ft(1)p 1524 4457 65 4 v 1524 4533 a(2i)1598 4476 y(\()p Fz(M)28 b Fs(\000)18 b Fz(M)1912 4442 y Fy(\003)p Fu(\032)1984 4476 y Ft(\))23 b(=)g([)p Fz(\002)r(;)14 b Fs(\001)p Ft(])p Fz(:)853 b FC(\(42\))523 4682 y FA(Theor)o(em)20 b(9.)k Fk(Let)d Fz(M)29 b Fk(be)21 b(the)f(g)o(ener)o(ator)f(of)h(a)h (Mark)o(o)o(v)f(c.p.)f(semigr)l(oup)h(on)g Fs(B)s Ft(\()p Fs(K)q Ft(\))p Fk(.)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 523 100 a FB(26)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)612 282 y FC(1\))24 b Fk(Let)d Fz(M)29 b Fk(satisfy)21 b(the)f(standar)m(d)f(DBC)i(with)g(r)m(espect)f(to)h Fz(\032)p Fk(.)f(Then)1145 437 y Fz(M)9 b Ft(\()p Fz(A)p Ft(\))23 b(=)g(i[)p Fz(\002)r(;)14 b(A)p Ft(])19 b(+)f Fz(M)1888 449 y Fx(d)1929 437 y Ft(\()p Fz(A)p Ft(\))p Fz(;)1145 611 y(M)1235 581 y Fy(\003)1272 611 y Ft(\()p Fz(A)p Ft(\))24 b(=)f Fs(\000)p Ft(i[)p Fz(\002)r(;)14 b(A)p Ft(])19 b(+)f Fz(\032)1953 581 y Fx(1)p Fu(=)p Fx(2)2057 611 y Fz(M)2138 623 y Fx(d)2179 611 y Ft(\()p Fz(\032)2254 581 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)2410 611 y Fz(A\032)2515 581 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)2671 611 y Ft(\))p Fz(\032)2746 581 y Fx(1)p Fu(=)p Fx(2)2851 611 y Fz(:)3174 525 y FC(\(43\))706 765 y Fk(wher)m(e)25 b Fz(M)1012 777 y Fx(d)1078 765 y Fk(is)h(a)f(g)o(ener)o(ator)f(of)h (another)f(Mark)o(o)o(v)h(c.p.)g(semigr)l(oup)f(satisfying)h Fz(M)3176 777 y Fx(d)3248 765 y Ft(=)706 865 y Fz(M)796 825 y Fy(\003)p Fu(\032)787 890 y Fx(d)893 865 y Fk(and)f Fz(\002)k Fk(is)d(a)f(self-adjoint)g(oper)o(ator)f(on)h Fs(K)q Fk(.)h(Mor)m(eo)o(ver)-9 b(,)24 b Ft([)p Fz(\002)r(;)14 b(\032)p Ft(])32 b(=)e(0)p Fk(,)25 b Fz(M)3072 835 y Fy(\003)3110 865 y Ft(\()p Fz(\032)p Ft(\))31 b(=)706 965 y Fz(M)796 934 y Fy(\003)787 988 y Fx(d)834 965 y Ft(\()p Fz(\032)p Ft(\))23 b(=)g(0)p Fk(.)612 1064 y FC(2\))h Fk(Let)19 b Fz(M)27 b Fk(be)17 b(given)h(by)f(\(40\).)g(If)h (ther)m(e)g(e)n(xists)h(a)f(unitary)f(oper)o(ator)g Fz(U)32 b Ft(:)23 b Fs(H)h(!)p 2925 998 71 4 v 23 w(H)19 b Fk(suc)o(h)e(that) 1525 1219 y Ft([)p Fz(\002)r(;)d(\032)p Ft(])23 b(=)g(0)p Fz(;)96 b Ft([)p Fz(W)2101 1188 y Fy(\003)2140 1219 y Fz(W)n(;)14 b(\032)p Ft(])23 b(=)f(0)p Fz(;)1525 1393 y(W)1615 1363 y Fu(?)1676 1393 y Ft(=)h Fz(\032)1807 1363 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)1963 1393 y Fs(\012)o Fz(U)30 b(W)12 b(\032)2247 1363 y Fx(1)p Fu(=)p Fx(2)2351 1393 y Fz(;)706 1547 y Fk(then)20 b Fz(M)29 b Fk(satis\002es)21 b(the)f(standar)m(d)f(DBC)i(wrt)g Fz(\032)p Fk(.)523 1698 y FA(Pr)o(oof)o(.)39 b FC(1\))20 b(By)g(\(42\),)1072 1857 y Ft([)p Fz(\002)r(;)14 b Fs(\001)p Ft(])24 b(=)f Fs(\000)p Ft([)p Fz(\002)r(;)14 b Fs(\001)p Ft(])1591 1822 y Fy(\003)p Fu(\032)1687 1857 y Ft(=)22 b Fs(\000)p Fz(\032)1882 1822 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)2038 1857 y Ft([)p Fz(\002)r(;)14 b(\032)2206 1822 y Fx(1)p Fu(=)p Fx(2)2329 1857 y Fs(\001)19 b Fz(\032)2414 1822 y Fx(1)p Fu(=)p Fx(2)2518 1857 y Ft(])p Fz(\032)2584 1822 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)2740 1857 y Fz(:)523 2015 y FC(Using)h Ft([)p Fz(\002)r(;)14 b Ft(1])23 b(=)g(0)p FC(,)d(we)h(obtain)e Ft([)p Fz(\002)r(;)14 b(\032)p Ft(])24 b(=)e(0)p FC(.)648 2115 y(Setting)k Fz(M)991 2127 y Fx(d)1066 2115 y Ft(:=)1198 2082 y Fx(1)p 1198 2096 34 4 v 1198 2144 a(2)1241 2115 y Ft(\()p Fz(M)32 b Ft(+)23 b Fz(M)1564 2085 y Fy(\003)p Fu(\032)1636 2115 y Ft(\))28 b FC(we)f(obtain)e(the)i (decomposition)d(\(43\).)h(Clearly)-5 b(,)26 b Ft(0)34 b(=)523 2215 y Fz(M)9 b Ft(\()p Fm(1)p Ft(\))23 b(=)g Fz(M)917 2227 y Fx(d)957 2215 y Ft(\()p Fm(1)p Ft(\))p FC(.)17 b(Hence)g Fz(M)1418 2227 y Fx(d)1475 2215 y FC(is)h(Mark)o(o)o (v)-5 b(.)14 b(Ne)o(xt)j Ft(0)22 b(=)h Fz(M)2252 2227 y Fx(d)2292 2215 y Ft(\()p Fm(1)p Ft(\))h(=)e Fz(M)2605 2175 y Fy(\003)p Fu(\032)2596 2240 y Fx(d)2677 2215 y Ft(\()p Fm(1)p Ft(\))c FC(gi)n(v)o(es)e Fz(M)3077 2227 y Fx(d)3117 2215 y Ft(\()p Fz(\032)p Ft(\))24 b(=)523 2314 y(0)p FC(.)648 2414 y(T)-7 b(o)20 b(see)h(2\))f(we)g(note)g(that)g (if)1295 2606 y Fz(M)1376 2618 y Fx(d)1439 2606 y Ft(=)1537 2550 y(1)p 1537 2587 42 4 v 1537 2663 a(2)1588 2606 y([)p Fz(W)1701 2572 y Fy(\003)1739 2606 y Fz(W)n(;)14 b(B)t Ft(])1942 2618 y Fx(+)2016 2606 y Fs(\000)k Fz(W)2189 2572 y Fy(\003)2248 2606 y Fz(B)t Fs(\012)p Ft(1)i Fz(W)n(;)523 2787 y FC(then)616 2929 y Fz(M)706 2889 y Fy(\003)p Fu(\032)697 2954 y Fx(d)778 2929 y Ft(\()p Fz(B)t Ft(\))26 b(=)d Fz(\032)1066 2899 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)1236 2862 y Fl(\000)1284 2896 y Fx(1)p 1284 2910 34 4 v 1284 2958 a(2)1327 2929 y Ft([)p Fz(W)1440 2899 y Fy(\003)1478 2929 y Fz(W)n(;)14 b(\032)1634 2899 y Fx(1)p Fu(=)p Fx(2)1738 2929 y Fz(B)t(\032)1848 2899 y Fx(1)p Fu(=)p Fx(2)1952 2929 y Ft(])1975 2941 y Fx(+)2049 2929 y Fs(\000)k Fz(W)2222 2899 y Fu(?)p Fy(\003)2315 2929 y Fz(\032)2358 2899 y Fx(1)p Fu(=)p Fx(2)2462 2929 y Fz(B)t(\032)2572 2899 y Fx(1)p Fu(=)p Fx(2)2695 2929 y Fs(\012)g Ft(1)i Fz(W)2930 2899 y Fu(?)2969 2862 y Fl(\001)3020 2929 y Fz(\032)3063 2899 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)935 3104 y Ft(=)1032 3071 y Fx(1)p 1032 3085 V 1032 3132 a(2)1076 3104 y Ft([)p Fz(W)1189 3073 y Fy(\003)1227 3104 y Fz(W)n(;)14 b(B)t Ft(])1430 3116 y Fx(+)1503 3104 y Fs(\000)k Ft(\()p Fz(\032)1661 3073 y Fx(1)p Fu(=)p Fx(2)1766 3104 y Fs(\012)p Ft(1)c Fz(W)1977 3073 y Fu(?)2014 3104 y Fz(\032)2057 3073 y Fx(1)p Fu(=)p Fx(2)2161 3104 y Ft(\))2193 3073 y Fy(\003)2253 3104 y Fz(B)t Fs(\012)o Ft(1)21 b Fz(\032)2490 3073 y Fx(1)p Fu(=)p Fx(2)2594 3104 y Fs(\012)o Ft(1)g Fz(W)2811 3073 y Fu(?)2849 3104 y Fz(\032)2892 3073 y Fy(\000)p Fx(1)p Fu(=)p Fx(2)3048 3104 y Fz(:)648 3238 y Ff(2)648 3378 y Fz(M)729 3390 y Fx(d)790 3378 y FC(is)g(called)f(the)g (dissipati)n(v)o(e)g(part)f(of)h(the)h(generator)d Fz(M)9 b FC(.)523 3599 y FA(4.5)40 b(Detailed)20 b(Balance)g(Condition)g(in)h (the)g(sense)g(of)523 3699 y(Alicki-Frigerio-Gorini-K)n(ossak)o(o)o (wski-V)-8 b(erri)523 3863 y FC(In)21 b(this)h(subsection)e(we)h (recall)g(the)g(de\002nition)f(of)h(Detailed)g(Balance)g(Condition,)f (which)g(can)523 3962 y(be)g(found)f(in)h([A,)g(FGKV].)648 4062 y(Let)g(us)h(introduce)d(the)i(scalar)g(product)1538 4221 y Ft(\()p Fz(A)p Fs(j)p Fz(B)t Ft(\))1754 4236 y Fx(\()p Fu(\032)p Fx(\))1868 4221 y Ft(:=)j(T)-7 b(r)o Fz(\032A)2169 4186 y Fy(\003)2208 4221 y Fz(B)t(:)523 4380 y FC(Let)25 b Fs(B)717 4349 y Fx(2)714 4406 y(\()p Fu(\032)p Fx(\))804 4380 y Ft(\()p Fs(K)q Ft(\))i FC(denote)d(the)h (space)g(of)f(operators)g(on)g Fs(K)j FC(equipped)c(with)j(this)f (scalar)g(product.)523 4503 y(Let)20 b Fz(M)744 4473 y Fy(\003)p Fx(\()p Fu(\032)p Fx(\))889 4503 y FC(denote)f(the)i (conjugate)d(of)i Fz(M)29 b FC(with)20 b(respect)g(to)h(this)f(scalar)h (product.)d(Explicitly:)1466 4662 y Fz(M)1556 4627 y Fy(\003)p Fx(\()p Fu(\032)p Fx(\))1680 4662 y Ft(\()p Fz(A)p Ft(\))24 b(=)e Fz(\032)1960 4627 y Fy(\000)p Fx(1)2049 4662 y Fz(M)2139 4627 y Fy(\003)2177 4662 y Ft(\()p Fz(\032A)p Ft(\))p Fz(:)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(27)523 282 y FA(De\002nition)20 b(2.)25 b Fk(W)-8 b(e)26 b(will)h(say)f(that)f Fz(M)35 b Fk(satis\002es)26 b(the)f(Detailed)g(Balance)f(Condition)h(with)h(r)m (e-)523 382 y(spect)20 b(to)h Fz(\032)f Fk(in)h(the)f(sense)h(of)f (AFGKV)g(if)h(ther)m(e)f(e)n(xists)i(a)e(self-adjoint)f(oper)o(ator)g Fz(\002)24 b Fk(suc)o(h)19 b(that)1509 508 y Ft(1)p 1498 545 65 4 v 1498 621 a(2i)1572 564 y(\()p Fz(M)28 b Fs(\000)18 b Fz(M)1886 530 y Fy(\003)p Fx(\()p Fu(\032)p Fx(\))2010 564 y Ft(\))23 b(=)g([)p Fz(\002)r(;)14 b Fs(\001)p Ft(])p Fz(:)648 734 y FC(Note)25 b(that)i(for)e(DBC)i(in)f(the)g(sense)g(of)g (AFGKV)-11 b(,)27 b(the)f(analog)e(of)i(Theorem)e(9)i(1\))g(holds,)523 833 y(where)20 b(we)g(replace)g(the)g(scalar)g(product)f Ft(\()p Fs(\001j\001)p Ft(\))1869 845 y Fu(\032)1929 833 y FC(with)h Ft(\()p Fs(\001j\001)p Ft(\))2230 848 y Fx(\()p Fu(\032)p Fx(\))2321 833 y FC(.)648 933 y(In)25 b(practical)h(applications,)e(c.p.)h(semigroups)g(usually)g(originate)g (from)f(the)i(weak)g(cou-)523 1032 y(pling)h(limit)h(of)g(reduced)e (dynamics,)g(as)i(we)h(describe)e(further)f(on)h(in)h(our)f(lectures.)g (In)h(this)523 1132 y(case)21 b(the)g(standard)f(DBC)i(is)g(equi)n(v)n (alent)d(to)i(DBC)h(in)f(the)g(sense)g(of)g(AFGKV)-11 b(,)21 b(which)g(follo)n(ws)523 1232 y(from)e(the)h(follo)n(wing)f (theorem:)523 1370 y FA(Theor)o(em)h(10.)k Fk(Suppose)18 b(that)i Fz(M)30 b Fk(satis\002es)1288 1517 y Fz(\032)1331 1483 y Fx(1)p Fu(=)p Fx(4)1435 1517 y Fz(M)9 b Ft(\()p Fz(\032)1600 1483 y Fy(\000)p Fx(1)p Fu(=)p Fx(4)1756 1517 y Fz(A\032)1861 1483 y Fx(1)p Fu(=)p Fx(4)1966 1517 y Ft(\))p Fz(\032)2041 1483 y Fy(\000)p Fx(1)p Fu(=)p Fx(4)2220 1517 y Ft(=)23 b Fz(M)9 b Ft(\()p Fz(A)p Ft(\))p Fz(:)523 1664 y Fk(Then)30 b(M)h(satis\002es)f(the)g(DBC)h(in)f(the)h (sense)f(of)g(\(42\))f(if)o(f)i(it)g(satis\002es)f(DBC)h(in)f(the)g (sense)h(of)523 1764 y(AFGKV)-11 b(.)27 b(Mor)m(eo)o(ver)-9 b(,)27 b(the)g(decompositions)f Fz(M)44 b Ft(=)36 b(i[)p Fz(\002)r(;)14 b Fs(\001)p Ft(])24 b(+)g Fz(M)2479 1776 y Fx(d)2547 1764 y Fk(obtained)h(in)i(both)g(cases)523 1864 y(concide)o(.)523 2002 y FA(Pr)o(oof)o(.)39 b FC(It)20 b(is)h(enough)e(to)h(note)g(that)g(the)g(map)1268 2149 y Fs(B)1326 2115 y Fx(2)1323 2170 y Fu(\032)1362 2149 y Ft(\()p Fs(K)q Ft(\))k Fs(3)f Fz(A)h Fs(7!)f Fz(\032)1827 2115 y Fy(\000)p Fx(1)p Fu(=)p Fx(4)1983 2149 y Fz(A\032)2088 2115 y Fx(1)p Fu(=)p Fx(4)2216 2149 y Fs(2)g(B)2352 2115 y Fx(2)2349 2172 y(\()p Fu(\032)p Fx(\))2439 2149 y Ft(\()p Fs(K)q Ft(\))523 2296 y FC(is)e(unitary)-5 b(.)18 b Ff(2)523 2619 y Fv(5)41 b(Small)25 b(quantum)h(system)f(interacting)h(with)f(r)n (eser)o(v)o(oir)523 2790 y FC(In)f(this)i(section)e(we)h(describe)f (the)h(class)g(of)g Fz(W)1945 2759 y Fy(\003)1983 2790 y FC(-dynamical)e(systems)i(that)f(we)h(consider)f(in)523 2889 y(our)f(notes.)f(The)o(y)h(are)g(meant)g(to)g(describe)g(a)g (small)h(quantum)d(system)j Fs(S)30 b FC(interacting)22 b(with)i(a)523 2989 y(lar)o(ge)d(reserv)n(oir)g Fs(R)p FC(.)h(P)o(auli-Fierz)f(systems,)h(considered)e(in)i([DJ2],)g(are)f (typical)h(e)o(xamples)e(of)523 3088 y(such)g(systems.)648 3188 y(In)36 b(Subsect.)g(5.1)h(we)g(recall)f(basic)h(elements)g(of)f (the)h(theory)f(of)g Fz(W)2799 3158 y Fy(\003)2837 3188 y FC(-algebras)g(\(see)523 3288 y([BR1,)e(BR2)q(,)g(DJP)q(])g(for)f (more)h(information\).)d(In)j(Subsect.)g(5.2)f(we)i(introduce)d(the)i (class)523 3387 y(of)d Fz(W)714 3357 y Fy(\003)753 3387 y FC(-dynamical)e(systems)j(describing)f Fs(S)i Ft(+)27 b Fs(R)32 b FC(in)g(purely)e(algebraic)h(\(representation-)523 3487 y(independent\))19 b(terms.)j(In)g(Subsect.)g(5.3)g(and)g(5.4)f (we)i(e)o(xplain)e(the)h(construction)e(of)i(tw)o(o)h(rep-)523 3587 y(resentations)k(of)g(our)g Fz(W)1274 3556 y Fy(\003)1312 3587 y FC(-dynamical)f(system:)i(the)f(semistandard)g(and)g(the)g (standard)g(rep-)523 3686 y(resentation.)21 b(Both)i(representations)e (possess)i(a)g(distinguished)e(unitary)h(implementation)e(of)523 3786 y(the)28 b(dynamics.)f(Its)h(generator)e(will)j(be)f(called)g(the) g(semi-Liouvillean)e(in)i(the)g(former)f(case)523 3885 y(and)20 b(the)g(Liouvillean)e(in)j(the)f(latter)g(case.)648 3985 y(The)k(standard)g(representation)f(and)h(the)h(Liouvillean)f(can) h(be)f(de\002ned)g(for)h(an)g(arbitrary)523 4085 y Fz(W)613 4055 y Fy(\003)651 4085 y FC(-algebra)g(\(see)i(ne)o(xt)e(subsection,)g ([DJP])i(and)f(references)f(therein\).)f(The)i(semistandard)523 4184 y(representation)17 b(and)i(the)g(semi-Liouvillean)e(are)i (concepts)g(whose)f(importance)g(is)i(limited)f(to)523 4284 y(a)k(system)g(of)g(the)g(form)f Fs(S)k Ft(+)20 b Fs(R)k FC(considered)d(in)i(these)g(notes.)g(Their)f(names)h(were)f (coined)g(in)523 4384 y([DJ2)o(].)f(The)g(adv)n(antage)d(of)j(the)g (semistandard)e(representation)g(o)o(v)o(er)g(the)i(standard)e(one)h (is)i(its)523 4483 y(simplicity)-5 b(,)23 b(and)g(this)h(is)h(the)f (reason)e(why)h(it)i(appears)e(often)f(in)i(the)g(literature)f([Da1)o (,)h(LeSp)o(].)523 4583 y(The)32 b(semistandard)f(representation)g(is)i (in)g(particular)e(well)i(adapted)e(to)i(the)f(study)g(of)g(the)523 4682 y(reduced)19 b(dynamics.)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 523 100 a FB(28)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(5.1)40 b Fi(W)772 252 y Fh(\003)815 282 y FA(-algebras)523 454 y FC(In)22 b(this)g (subsection)f(we)h(recall)g(the)g(de\002nitions)f(of)g(basic)h (concepts)f(related)g(to)h(the)g(theory)e(of)523 554 y Fz(W)613 524 y Fy(\003)651 554 y FC(-algebras)f(\(see)i([BR1)o(,)g (BR2,)g(DJP]\).)648 654 y(A)30 b Fz(W)828 623 y Fy(\003)866 654 y FC(-dynamical)e(system)j Ft(\()p Fo(M)p Fz(;)14 b(\034)9 b Ft(\))31 b FC(is)g(a)g(pair)f(consisting)f(of)h(a)g Fz(W)2734 623 y Fy(\003)2772 654 y FC(-algebra)f Fo(M)i FC(and)523 753 y(a)f(1-parameter)d(\(pointwise\))h Fz(\033)s FC(-weakly)h(continuous)e(group)h(of)h Fs(\003)p FC(-automorphisms)e (of)i Fo(M)p FC(,)523 853 y Fw(R)23 b Fs(3)g Fz(t)h Fs(7!)f Fz(\034)889 823 y Fu(t)919 853 y FC(.)648 952 y(A)d(standard)e (representation)f(of)j(a)g Fz(W)1756 922 y Fy(\003)1794 952 y FC(-algebra)e Fo(M)i FC(is)h(a)f(quadruple)d Ft(\()p Fz(\031)s(;)d Fs(H)q Fz(;)g(J)o(;)g Fs(H)3058 922 y Fx(+)3113 952 y Ft(\))20 b FC(con-)523 1052 y(sisting)28 b(of)g(a)g (representation)d Fz(\031)s FC(,)k(its)g(Hilbert)e(space)h Fs(H)q FC(,)g(an)g(antilinear)e(in)m(v)n(olution)g Fz(J)37 b FC(and)27 b(a)523 1152 y(self-dual)19 b(cone)h Fs(H)1083 1122 y Fx(+)1159 1152 y FC(satisfying)f(the)h(follo)n(wing)f (conditions:)599 1251 y(1\))41 b Fz(J)8 b(\031)s Ft(\()p Fo(M)p Ft(\))p Fz(J)32 b Ft(=)23 b Fz(\031)s Ft(\()p Fo(M)p Ft(\))1332 1221 y Fy(0)1356 1251 y FC(;)599 1351 y(2\))41 b Fz(J)8 b(\031)s Ft(\()p Fz(A)p Ft(\))p Fz(J)32 b Ft(=)23 b Fz(\031)s Ft(\()p Fz(A)p Ft(\))1282 1321 y Fy(\003)1342 1351 y FC(for)c Fz(A)i FC(in)g(the)f(center)f(of)h Fo(M)p FC(;)599 1451 y(3\))41 b Fz(J)8 b(\011)32 b Ft(=)23 b Fz(\011)29 b FC(for)20 b Fz(\011)32 b Fs(2)23 b(H)1365 1420 y Fx(+)1420 1451 y FC(;)599 1550 y(4\))41 b Fz(\031)s Ft(\()p Fz(A)p Ft(\))p Fz(J)8 b(\031)s Ft(\()p Fz(A)p Ft(\))p Fs(H)1187 1520 y Fx(+)1267 1550 y Fs(\032)23 b(H)1426 1520 y Fx(+)1502 1550 y FC(for)c Fz(A)24 b Fs(2)f Fo(M)p FC(.)523 1650 y Fz(J)34 b FC(is)27 b(called)e(the)h(modular)d (conjugation)g(and)i Fs(H)1974 1620 y Fx(+)2055 1650 y FC(the)h(modular)e(cone.)g(Ev)o(ery)g Fz(W)3002 1620 y Fy(\003)3041 1650 y FC(-algebra)523 1750 y(possesses)d(a)g(standard)e (representation,)e(unique)i(up)h(to)g(the)h(unitary)e(equi)n(v)n (alence.)648 1849 y(Suppose)26 b(that)h(we)h(are)f(gi)n(v)o(en)f(a)i(f) o(aithful)e(state)i Fz(!)j FC(on)c Fo(M)p FC(.)g(In)g(the)h (corresponding)c(GNS)523 1949 y(representation)h Fz(\031)1068 1961 y Fu(!)1152 1949 y Ft(:)36 b Fo(M)g Fs(!)g(B)s Ft(\()p Fs(H)1613 1961 y Fu(!)1660 1949 y Ft(\))p FC(,)28 b(the)f(state)h Fz(!)i FC(is)e(gi)n(v)o(en)e(by)h(a)g(c)o(yclic)g(and)f(separating)523 2048 y(v)o(ector)c Fz(\012)817 2060 y Fu(!)866 2048 y FC(.)i(The)f(T)-7 b(omita-T)g(ak)o(esaki)22 b(theory)g(yields)i(the)g (modular)d Fz(W)2616 2018 y Fy(\003)2655 2048 y FC(-dynamics)h Fz(t)29 b Fs(7!)g Fz(\033)3246 2018 y Fu(t)3243 2069 y(!)3292 2048 y FC(,)523 2148 y(the)20 b(modular)e(conjugation)f Fz(J)1400 2160 y Fu(!)1468 2148 y FC(and)j(the)g(modular)e(cone)h Fs(H)2276 2118 y Fx(+)2275 2169 y Fu(!)2354 2148 y Ft(:=)k Fs(f)p Fz(AJ)2615 2160 y Fu(!)2663 2148 y Fz(A\012)2789 2160 y Fu(!)2880 2148 y Ft(:)44 b Fz(A)23 b Fs(2)h Fo(M)p Fs(g)3240 2118 y Fx(cl)3292 2148 y FC(,)523 2248 y(where)k Ft(cl)i FC(denotes)e(the)h(closure.)g(The)f(state)i Fz(!)i FC(satis\002es)f(the)e Fs(\000)p Ft(1)p FC(-KMS)f(condition)g(for)g (the)523 2347 y(dynamics)19 b Fz(\033)909 2359 y Fu(!)958 2347 y FC(.)h(The)g(quadruple)e Ft(\()p Fz(\031)1581 2359 y Fu(!)1629 2347 y Fz(;)c Fs(H)1736 2359 y Fu(!)1784 2347 y Fz(;)g(J)1867 2359 y Fu(!)1915 2347 y Fz(;)g Fs(H)2023 2317 y Fx(+)2022 2368 y Fu(!)2078 2347 y Ft(\))21 b FC(is)g(a)g (standard)e(representation)f(of)i Fo(M)p FC(.)648 2447 y(Until)39 b(the)g(end)f(of)h(this)h(subsection,)d(we)j(suppose)e(that) h(a)g(standard)f(representation)523 2547 y Ft(\()p Fz(\031)s(;)14 b Fs(H)q Fz(;)g(J)o(;)g Fs(H)903 2516 y Fx(+)958 2547 y Ft(\))21 b FC(of)f Fo(M)h FC(is)g(gi)n(v)o(en.)648 2646 y(Let)27 b Fz(!)k FC(be)c(a)h(state)g(on)f Fo(M)p FC(.)h(Then)f(there)g(e)o(xists)h(a)g(unique)e(v)o(ector)g(in)i(the)f (modular)f(cone)523 2746 y Fz(\012)h Fs(2)d(H)764 2716 y Fx(+)840 2746 y FC(representing)18 b Fz(!)s FC(.)i Fz(\012)25 b FC(is)c(c)o(yclic)f(if)n(f)g Fz(\012)25 b FC(is)c(separating)e(if)n(f)h Fz(!)j FC(is)f(f)o(aithful.)648 2845 y(Let)j Fz(t)33 b Fs(7!)h Fz(\034)1009 2815 y Fu(t)1065 2845 y FC(be)25 b(a)h Fz(W)1322 2815 y Fy(\003)1360 2845 y FC(-dynamics)e(on)i Fo(M)p FC(.)g(The)f(Liouvillean)f Fz(L)i FC(of)f Fz(\034)36 b FC(is)26 b(a)g(self-adjoint)523 2945 y(operator)18 b(on)i Fs(H)i FC(uniquely)c(de\002ned)h(by)h (demanding)e(that)954 3114 y Fz(\031)s Ft(\()p Fz(\034)1081 3080 y Fu(t)1112 3114 y Ft(\()p Fz(A)p Ft(\)\))24 b(=)f(e)1419 3080 y Fx(i)p Fu(tL)1512 3114 y Fz(\031)s Ft(\()p Fz(A)p Ft(\)e)1725 3080 y Fy(\000)p Fx(i)p Fu(tL)1872 3114 y Fz(;)179 b Ft(e)2111 3080 y Fx(i)p Fu(tL)2205 3114 y Fs(H)2276 3080 y Fx(+)2354 3114 y Ft(=)23 b Fs(H)2513 3080 y Fx(+)2568 3114 y Fz(;)76 b(t)23 b Fs(2)g Fw(R)p Fz(:)523 3284 y FC(\()p Fz(L)16 b FC(implements)e(the)i(dynamics)f(in)h (the)f(representation)f Fz(\031)20 b FC(and)15 b(preserv)o(es)g(the)g (modular)f(cone\).)523 3384 y(It)20 b(has)f(man)o(y)f(useful)h (properties)f(that)i(mak)o(e)f(it)h(an)f(ef)n(\002cient)g(tool)g(in)g (the)h(study)f(of)g(the)g(er)o(godic)523 3483 y(properties)28 b(of)i(the)g(dynamics)f Fz(\034)9 b FC(.)31 b(In)e(particular)m(,)f Fz(L)i FC(has)g(no)g(point)f(spectrum)g(if)n(f)g Fz(\034)40 b FC(has)31 b(no)523 3583 y(normal)19 b(in)m(v)n(ariant)g(states,)j (and)e Fz(L)g FC(has)h(a)g(1-dimensional)d(k)o(ernel)i(if)n(f)h Fz(\034)31 b FC(has)20 b(a)h(single)g(in)m(v)n(ariant)523 3682 y(normal)e(state.)523 3913 y FA(5.2)40 b(Algebraic)20 b(description)523 4085 y FC(The)26 b(Hilbert)g(space)g(of)g(the)g (system)h Fs(S)33 b FC(is)27 b(denoted)e(by)h Fs(K)q FC(.)h(Throughout)22 b(the)27 b(notes)f(we)g(will)523 4184 y(assume)c(that)h Ft(dim)14 b Fs(K)28 b Fz(<)e Fs(1)p FC(.)d(Let)f(the)g(self-adjoint)f(operator)g Fz(K)28 b FC(be)22 b(the)g(Hamiltonian)f(of)h(the)523 4284 y(small)e(system.)g (The)g(free)g(dynamics)e(of)i(the)g(small)g(system)g(is)h Fz(\034)2417 4254 y Fu(t)2408 4304 y Fn(S)2452 4284 y Ft(\()p Fz(B)t Ft(\))i(:=)g(e)2754 4254 y Fx(i)p Fu(tK)2862 4284 y Fz(B)t Ft(e)2966 4254 y Fy(\000)p Fx(i)p Fu(tK)3126 4284 y FC(,)d Fz(B)27 b Fs(2)523 4384 y(B)s Ft(\()p Fs(K)q Ft(\))p FC(.)20 b(Thus)g(the)g(small)h(system)f(is)h(described)e(by)g (the)h Fz(W)2241 4353 y Fy(\003)2280 4384 y FC(-dynamical)e(system)i Ft(\()p Fs(B)s Ft(\()p Fs(K)q Ft(\))p Fz(;)14 b(\034)3216 4396 y Fn(S)3260 4384 y Ft(\))p FC(.)648 4483 y(The)21 b(reserv)n(oir)f Fs(R)i FC(is)g(described)e(by)h(a)h Fz(W)1877 4453 y Fy(\003)1915 4483 y FC(-dynamical)d(system)j Ft(\()p Fo(M)2682 4495 y Fn(R)2736 4483 y Fz(;)14 b(\034)2809 4495 y Fn(R)2863 4483 y Ft(\))p FC(.)22 b(W)-7 b(e)22 b(assume)523 4583 y(that)k(it)h(has)g(a)g(unique)e(normal)g(stationary) g(state)i Fz(!)2049 4595 y Fn(R)2129 4583 y FC(\(not)f(necessarily)g(a) g(KMS)h(state\).)f(The)523 4682 y(generator)18 b(of)i Fz(\034)983 4694 y Fn(R)1058 4682 y FC(is)h(denoted)e(by)h Fz(\016)1559 4694 y Fn(R)1633 4682 y FC(\(that)g(is)h Fz(\034)1927 4652 y Fu(t)1918 4703 y Fn(R)1995 4682 y Ft(=)i(e)2120 4652 y Fu(\016)2150 4660 y Fn(R)2204 4652 y Fu(t)2233 4682 y FC(\).)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(29)648 282 y FC(The)19 b(coupled)g(system)h Fs(S)25 b Ft(+)18 b Fs(R)j FC(is)g(described)e(by) h(the)g Fz(W)2312 252 y Fy(\003)2350 282 y FC(-algebra)f Fo(M)k Ft(:=)g Fs(B)s Ft(\()p Fs(K)q Ft(\))18 b Fs(\012)g Fo(M)3238 294 y Fn(R)3292 282 y FC(.)523 382 y(The)h(free)g(dynamics)f (is)j(gi)n(v)o(en)d(by)h(the)g(tensor)g(product)f(of)h(the)g(dynamics)f (of)h(its)i(constituents:)1320 559 y Fz(\034)1365 525 y Fu(t)1356 580 y Fx(0)1395 559 y Ft(\()p Fz(A)p Ft(\))i(:=)1655 492 y Fl(\000)1693 559 y Fz(\034)1738 525 y Fu(t)1729 580 y Fn(S)1791 559 y Fs(\012)18 b Fz(\034)1919 525 y Fu(t)1910 580 y Fn(R)1964 492 y Fl(\001)2016 559 y Ft(\()p Fz(A)p Ft(\))p Fz(;)77 b(A)23 b Fs(2)h Fo(M)p Fz(:)523 736 y FC(W)-7 b(e)21 b(will)g(denote)e(by)h Fz(\016)1185 748 y Fx(0)1243 736 y FC(the)g(generator)f(of)g Fz(\034)1825 748 y Fx(0)1863 736 y FC(.)648 836 y(Let)29 b Fz(V)49 b FC(be)29 b(a)h(self-adjoint)e(element)h(of)g Fo(M)p FC(.)h(The)f(full)h(dynamics)e Fz(t)40 b Fs(7!)g Fz(\034)2889 806 y Fu(t)2880 859 y(\025)2964 836 y Ft(:=)g(e)3129 806 y Fu(t\016)3184 815 y Fp(\025)3257 836 y FC(is)523 936 y(de\002ned)19 b(by)1596 1035 y Fz(\016)1633 1047 y Fu(\025)1700 1035 y Ft(:=)j Fz(\016)1847 1047 y Fx(0)1903 1035 y Ft(+)c(i)p Fz(\025)p Ft([)p Fz(V)5 b(;)14 b Fs(\001)p Ft(])p Fz(:)523 1181 y FC(\(One)21 b(can)h(consider)e(also)i(a)g(more)f (general)g(situation,)g(where)g Fz(V)41 b FC(is)22 b(only)f(af)n (\002lliated)h(to)f Fo(M)p FC(\227)523 1281 y(see)g([DJP])f(for)g (details\).)523 1517 y FA(5.3)40 b(Semistandard)21 b(r)o(epr)o (esentation)523 1696 y FC(Suppose)15 b(that)h Fo(M)1049 1708 y Fn(R)1119 1696 y FC(is)h(gi)n(v)o(en)d(in)i(the)g(standard)e (form)h(on)g(the)h(Hilbert)g(space)f Fs(H)2801 1708 y Fn(R)2855 1696 y FC(.)h(Let)g Fm(1)3067 1708 y Fn(R)3137 1696 y FC(stand)523 1795 y(for)24 b(the)g(identity)f(on)h Fs(H)1227 1807 y Fn(R)1281 1795 y FC(.)g(W)-7 b(e)26 b(denote)d(by)g Fs(H)1884 1765 y Fx(+)1883 1816 y Fn(R)1940 1795 y FC(,)h Fz(J)2031 1807 y Fn(R)2085 1795 y FC(,)g(and)g Fz(L)2332 1807 y Fn(R)2410 1795 y FC(the)g(corresponding)d(modular)523 1895 y(cone,)e(modular)f(conjugation,)e(and)j(standard)g(Liouvillean.)f (Let)h Fz(\012)2510 1907 y Fn(R)2585 1895 y FC(be)g(the)h(\(unique\))d (v)o(ector)523 1995 y(representati)n(v)o(e)e(in)i Fs(H)1156 1965 y Fx(+)1155 2015 y Fn(R)1229 1995 y FC(of)g(the)g(state)g Fz(!)1656 2007 y Fn(R)1710 1995 y FC(.)g(Clearly)-5 b(,)17 b Fz(\012)2090 2007 y Fn(R)2161 1995 y FC(is)h(an)f(eigen)m(v)o(ector)e (of)h Fz(L)2874 2007 y Fn(R)2928 1995 y FC(.)h Fs(j)p Fz(\012)3053 2007 y Fn(R)3107 1995 y Ft(\)\()p Fz(\012)3235 2007 y Fn(R)3290 1995 y Fs(j)523 2094 y FC(denotes)i(projection)g(on)h Fz(\012)1323 2106 y Fn(R)1377 2094 y FC(.)648 2194 y(Let)d(us)h (represent)f Fs(B)s Ft(\()p Fs(K)q Ft(\))h FC(on)f Fs(K)j FC(and)d(tak)o(e)g(the)h(representation)d(of)i Fo(M)i FC(in)e(the)h(Hilbert)f(space)523 2294 y Fs(K)c(\012)f(H)746 2306 y Fn(R)799 2294 y FC(.)19 b(W)-7 b(e)20 b(will)f(call)g(it)g(the)f (semistandard)f(representation)f(and)i(denote)g(by)g Fz(\031)2917 2263 y Fx(semi)3073 2294 y Ft(:)23 b Fo(M)h Fs(!)523 2393 y(B)s Ft(\()p Fs(K)g(\012)f(H)858 2405 y Fn(R)912 2393 y Ft(\))p FC(.)k(\(T)-7 b(o)27 b(justify)g(its)h(name,) e(note)g(that)h(it)g(is)h(standard)e(on)g(its)i(reserv)n(oir)e(part,)g (b)n(ut)523 2493 y(not)21 b(standard)g(on)g(the)h(small)g(system)g (part\).)f(W)-7 b(e)22 b(will)h(usually)e(drop)f Fz(\031)2607 2463 y Fx(semi)2763 2493 y FC(and)h(treat)h Fo(M)g FC(as)h(a)523 2592 y(subalgebra)18 b(of)i Fs(B)s Ft(\()p Fs(K)g(\012)e(H)1319 2604 y Fn(R)1373 2592 y Ft(\))p FC(.)648 2692 y(Let)i(us)h(introduce)d (the)i(so-called)g(free)f(semi-Liouvillean)1469 2869 y Fz(L)1526 2835 y Fx(semi)1526 2890 y(0)1682 2869 y Ft(=)j Fz(K)i Fs(\012)18 b Ft(1)g(+)g(1)g Fs(\012)g Fz(L)2290 2881 y Fn(R)2344 2869 y Fz(:)807 b FC(\(44\))648 3046 y(The)19 b(full)h(semi-Liouvillean)f(is)i(de\002ned)e(as)1560 3224 y Fz(L)1617 3189 y Fx(semi)1617 3244 y Fu(\025)1773 3224 y Ft(=)j Fz(L)1917 3189 y Fx(semi)1917 3244 y(0)2068 3224 y Ft(+)c Fz(\025V)5 b(:)523 3401 y FC(It)21 b(is)g(the)f (generator)e(of)i(the)g(distinguished)f(unitary)g(implementation)f(of)i (the)g(dynamics)f Fz(\034)3188 3413 y Fu(\025)3232 3401 y FC(:)1219 3592 y Fz(\034)1264 3558 y Fu(t)1255 3613 y(\025)1298 3592 y Ft(\()p Fz(A)p Ft(\))24 b(=)f(e)1573 3558 y Fx(i)p Fu(tL)1663 3533 y Fq(semi)1663 3575 y Fp(\025)1781 3592 y Fz(A)p Ft(e)1880 3558 y Fy(\000)p Fx(i)p Fu(tL)2022 3533 y Fq(semi)2022 3575 y Fp(\025)2140 3592 y Fz(;)180 b(A)23 b Fs(2)h Fo(M)p Fz(;)557 b FC(\(45\))523 3770 y(with)1651 3869 y Fz(\016)1688 3881 y Fu(\025)1755 3869 y Ft(=)22 b(i[)p Fz(L)1945 3835 y Fx(semi)1945 3890 y Fu(\025)2078 3869 y Fz(;)14 b Fs(\001)p Ft(])p Fz(:)523 4106 y FA(5.4)40 b(Standard)20 b(r)o(epr)o(esentation)523 4284 y FC(Let)26 b(us)h(recall)f(ho)n(w)g(one)f(constructs)h(the)g (standard)f(representation)f(for)h(the)i(algebra)e Fs(B)s Ft(\()p Fs(K)q Ft(\))p FC(.)523 4384 y(Recall)h(that)g Fs(B)970 4353 y Fx(2)1006 4384 y Ft(\()p Fs(K)q Ft(\))h FC(denotes)e(the)g(space)g(of)h(Hilbert-Schmidt)d(operators)h(on)h Fs(K)q FC(.)h(Equipped)523 4483 y(with)k(the)f(inner)g(product)e Ft(\()p Fz(X)7 b Fs(j)p Fz(B)t Ft(\))40 b(=)g(T)-7 b(r\()p Fz(X)1885 4453 y Fy(\003)1923 4483 y Fz(B)t Ft(\))30 b FC(it)g(is)g(a)g(Hilbert)f(space.)h(Note)f(that)g Fs(B)s Ft(\()p Fs(K)q Ft(\))523 4583 y FC(acts)21 b(naturally)e(on)h Fs(B)1151 4553 y Fx(2)1187 4583 y Ft(\()p Fs(K)q Ft(\))i FC(by)e(the)g(left)h(multiplication.)d(This)j(de\002nes)f(a)h (representation)d Fz(\031)3223 4595 y Fn(S)3290 4583 y Ft(:)523 4682 y Fs(B)s Ft(\()p Fs(K)q Ft(\))24 b Fs(!)h(B)s Ft(\()p Fs(B)989 4652 y Fx(2)1024 4682 y Ft(\()p Fs(K)q Ft(\)\))p FC(.)e(Let)e Fz(J)1406 4694 y Fn(S)1473 4682 y Ft(:)k Fs(B)1579 4652 y Fx(2)1615 4682 y Ft(\()p Fs(K)q Ft(\))g Fs(!)g(B)1934 4652 y Fx(2)1970 4682 y Ft(\()p Fs(K)q Ft(\))d FC(be)f(de\002ned)f(by)h Fz(J)2637 4694 y Fn(S)2680 4682 y Ft(\()p Fz(X)7 b Ft(\))24 b(=)g Fz(X)3009 4652 y Fy(\003)3046 4682 y FC(,)d(and)g(let)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 523 100 a FB(30)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fs(B)581 252 y Fx(2)578 303 y(+)633 282 y Ft(\()p Fs(K)q Ft(\))17 b FC(be)f(the)g(set)g(of)g (all)g(positi)n(v)o(e)f Fz(X)29 b Fs(2)24 b(B)1795 252 y Fx(2)1831 282 y Ft(\()p Fs(K)q Ft(\))p FC(.)17 b(The)f(algebra)e Fz(\031)2450 294 y Fn(S)2494 282 y Ft(\()p Fs(B)s Ft(\()p Fs(K)q Ft(\)\))j FC(is)g(in)e(the)h(standard)523 382 y(form)i(on)h(the)h(Hilbert)f(space)g Fs(B)1446 352 y Fx(2)1482 382 y Ft(\()p Fs(K)q Ft(\))p FC(,)i(and)e(its)h(modular)e (cone)g(and)h(modular)f(conjugation)e(are)523 482 y Fs(B)581 451 y Fx(2)578 502 y(+)633 482 y Ft(\()p Fs(K)q Ft(\))21 b FC(and)f Fz(J)969 494 y Fn(S)1012 482 y FC(.)648 581 y(There)f(e)o(xists)i(a)f(unique)f(representation)f Fz(\031)26 b Ft(:)d Fo(M)h Fs(!)f(B)s Ft(\()p Fs(B)2350 551 y Fx(2)2385 581 y Ft(\()p Fs(K)q Ft(\))d Fs(\012)e(H)2686 593 y Fn(R)2740 581 y Ft(\))j FC(satisfying)1484 760 y Fz(\031)s Ft(\()p Fz(B)j Fs(\012)18 b Fz(C)6 b Ft(\))23 b(=)g Fz(\031)1991 772 y Fn(S)2034 760 y Ft(\()p Fz(B)t Ft(\))c Fs(\012)f Fz(C)q(:)824 b FC(\(46\))523 939 y(The)22 b(v)n(on)f(Neumann)g(algebra) g Fz(\031)s Ft(\()p Fo(M)p Ft(\))i FC(is)g(in)f(standard)f(form)g(on)h (the)g(Hilbert)g(space)g Fs(B)3063 909 y Fx(2)3099 939 y Ft(\()p Fs(K)q Ft(\))f Fs(\012)523 1038 y(H)593 1050 y Fn(R)647 1038 y FC(.)i(The)g(modular)e(conjugation)f(is)k Fz(J)37 b Ft(=)28 b Fz(J)1860 1050 y Fn(S)1923 1038 y Fs(\012)20 b Fz(J)2054 1050 y Fn(R)2108 1038 y FC(.)k(The)e(modular)g (cone)g(can)h(be)g(obtained)523 1138 y(as)1161 1238 y Fs(H)1232 1203 y Fx(+)1310 1238 y Ft(:=)g Fs(f)p Fz(\031)s Ft(\()p Fz(A)p Ft(\))p Fz(J)8 b(\031)s Ft(\()p Fz(A)p Ft(\))23 b(\()p Fz(\032)p Fs(\012)p Fz(\012)2096 1250 y Fn(R)2149 1238 y Ft(\))45 b(:)e Fz(A)24 b Fs(2)f Fo(M)p Fs(g)2585 1196 y Fx(cl)2651 1238 y Fz(;)523 1385 y FC(where)d Fz(\032)g FC(is)h(an)g(arbitrary)d(nonde)o(generate)f(element)i(of)h Fs(B)2227 1355 y Fx(2)2224 1405 y(+)2279 1385 y Ft(\()p Fs(K)q Ft(\))p FC(.)648 1484 y(The)f(Liouvillean)g(of)h(the)g(free)g (dynamics)f(\(the)h(free)f(Liouvillean\))g(equals)1452 1663 y Fz(L)1509 1675 y Fx(0)1569 1663 y Ft(=)k([)p Fz(K)q(;)k Fs(\001)14 b Ft(])19 b Fs(\012)f Ft(1)g(+)g(1)g Fs(\012)g Fz(L)2307 1675 y Fn(R)2360 1663 y Fz(:)791 b FC(\(47\))523 1842 y(and)20 b(the)g(Liouvillean)e(of)i(the)h(full)f(dynamics)f(\(the) g(full)i(Liouvillean\))d(equals)1360 2021 y Fz(L)1417 2033 y Fu(\025)1483 2021 y Ft(=)23 b Fz(L)1628 2033 y Fx(0)1683 2021 y Ft(+)18 b Fz(\025)p Ft(\()p Fz(\031)s Ft(\()p Fz(V)i Ft(\))f Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(V)20 b Ft(\))p Fz(J)8 b Ft(\))p Fz(:)699 b FC(\(48\))648 2199 y(Sometimes)17 b(we)h(will)g(assume)f(that)h(the)f(reserv)n(oir)g (is)h(thermal.)f(By)h(this)g(we)g(mean)e(that)i Fz(!)3259 2211 y Fn(R)523 2299 y FC(is)j(a)g Fz(\014)t FC(-KMS)f(state)h(for)f (the)g(dynamics)f Fz(\034)1724 2311 y Fn(R)1778 2299 y FC(.)i(Set)1557 2478 y Fz(\011)1608 2490 y Fx(0)1668 2478 y Ft(:=)i(e)1816 2443 y Fy(\000)p Fu(\014)s(K=)p Fx(2)2054 2478 y Fs(\012)18 b Fz(\012)2201 2490 y Fn(R)2255 2478 y Fz(:)523 2656 y FC(Then)h(the)i(state)g Ft(\()p Fz(\011)1093 2668 y Fx(0)1130 2656 y Fs(j)p Fz(\031)s Ft(\()p Fs(\001)p Ft(\))p Fz(\011)1341 2668 y Fx(0)1379 2656 y Ft(\))p Fz(=)p Fs(k)p Fz(\011)1546 2668 y Fx(0)1582 2656 y Fs(k)1624 2626 y Fx(2)1682 2656 y FC(is)g(a)g Ft(\()p Fz(\034)1884 2668 y Fx(0)1921 2656 y Fz(;)14 b(\014)t Ft(\))p FC(-KMS)21 b(state.)648 2756 y(The)e(Araki)h (perturbation)e(theory)h(yields)h(that)1409 2935 y Fz(\011)1460 2947 y Fx(0)1520 2935 y Fs(2)j Ft(Dom\(e)1841 2901 y Fy(\000)p Fu(\014)s Fx(\()p Fu(L)2006 2909 y Fq(0)2038 2901 y Fx(+)p Fu(\025\031)r Fx(\()p Fu(V)15 b Fx(\)\))p Fu(=)p Fx(2)2372 2935 y Ft(\))p Fz(;)523 3114 y FC(the)20 b(v)o(ector)1476 3213 y Fz(\011)1527 3225 y Fu(\025)1594 3213 y Ft(:=)i(e)1741 3179 y Fy(\000)p Fu(\014)s Fx(\()p Fu(L)1906 3187 y Fq(0)1938 3179 y Fx(+)p Fu(\025\031)r Fx(\()p Fu(V)14 b Fx(\)\))p Fu(=)p Fx(2)2271 3213 y Fz(\011)2322 3225 y Fx(0)3174 3213 y FC(\(49\))523 3360 y(belongs)24 b(to)h Fs(H)968 3330 y Fx(+)1045 3360 y Fs(\\)d Ft(Ker)o Fz(L)1313 3372 y Fu(\025)1356 3360 y FC(,)j(and)f(that)h Ft(\()p Fz(\011)1780 3372 y Fu(\025)1824 3360 y Fs(j)p Fz(\031)s Ft(\()p Fs(\001)p Ft(\))p Fz(\011)2035 3372 y Fu(\025)2079 3360 y Ft(\))p Fz(=)p Fs(k)p Fz(\011)2246 3372 y Fu(\025)2289 3360 y Fs(k)2331 3330 y Fx(2)2393 3360 y FC(is)h(a)f Ft(\()p Fz(\034)2604 3372 y Fu(\025)2648 3360 y Fz(;)14 b(\014)t Ft(\))p FC(-KMS)25 b(state)h(\(see)523 3460 y([BR2,)c(DJP]\).)g(In)f(particular)m(,)f(zero)i(is)g(al)o(w)o (ays)h(an)f(eigen)m(v)n(alue)d(of)j Fz(L)2553 3472 y Fu(\025)2596 3460 y FC(.)g(Thus,)f(in)h(the)g(thermal)523 3560 y(case,)e Ft(\()p Fo(M)p Fz(;)14 b(\034)899 3572 y Fu(\025)944 3560 y Ft(\))21 b FC(has)f(at)h(least)g(one)e(stationary) g(state.)523 3872 y Fv(6)41 b(T)-7 b(w)o(o)25 b(applications)g(of)g (the)g(F)n(ermi)g(Golden)g(Rule)g(to)g(open)h(quantum)523 3988 y(systems)523 4184 y FC(In)31 b(this)h(section)f(we)g(k)o(eep)g (all)h(the)f(notation)f(and)h(assumtions)f(of)h(the)g(preceding)f (section.)523 4284 y(W)-7 b(e)27 b(will)f(describe)f(tw)o(o)g (applications)g(of)g(the)g(Fermi)h(Golden)e(Rule)i(to)f(the)h Fz(W)2901 4254 y Fy(\003)2939 4284 y FC(-dynamical)523 4384 y(system)20 b Ft(\()p Fo(M)p Fz(;)14 b(\034)966 4396 y Fu(\025)1011 4384 y Ft(\))21 b FC(introduced)c(in)k(the)f(pre)n (vious)f(section.)648 4483 y(In)e(the)g(\002rst)i(application)d(we)i (compute)e(the)h(LSO)h(for)f(the)h(generator)d(of)j(the)f(dynamics)f Fz(\016)3248 4495 y Fu(\025)3292 4483 y FC(.)523 4583 y(W)-7 b(e)20 b(will)f(call)h(it)f(the)g(Da)n(vies)g(generator)e(and)h (denote)f(by)i Fz(M)9 b FC(.)18 b(In)h(the)g(literature,)e Fz(M)28 b FC(appears)18 b(in)523 4682 y(the)f(conte)o(xt)f(of)h(the)g (Dynamical)f(Fermi)h(Golden)f(Rule.)h(It)h(is)g(the)f(generator)e(of)i (the)g(semigroup)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(31)523 282 y FC(obtained)16 b(by)h(the)g(weak)g(coupling)f(limit)i(to)f(the)g(reduced)f(dynamics.)g (This)i(result)f(can)g(be)g(used)523 382 y(to)24 b(partly)g(justify)g (the)g(use)h(of)f(completely)e(positi)n(v)o(e)i(semigroups)f(to)h (describe)f(dynamics)g(of)523 482 y(small)e(quantum)d(systems)j(weakly) e(interacting)g(with)h(en)m(vironment)e([Da1)o(,)i(LeSp)o(].)648 581 y(In)29 b(the)i(second)e(application)f(we)j(consider)e(the)h (standard)f(representation)f(of)i(the)g Fz(W)3247 551 y Fy(\003)3285 581 y FC(-)523 681 y(dynamical)23 b(system)h(in)h(the)f (Hilbert)g(space)h Fs(H)g FC(with)g(the)f(Liouvillean)f Fz(L)p FC(.)h(W)-7 b(e)26 b(will)f(compute)523 780 y(the)f(LSO)g(for)g Ft(i)p Fz(L)1032 792 y Fu(\025)1075 780 y FC(.)g(W)-7 b(e)25 b(denote)e(it)i(by)e Ft(i)p Fz(\000)12 b FC(.)24 b(In)g(the)g(literature,)f Ft(i)p Fz(\000)36 b FC(appears)23 b(in)h(the)g(conte)o(xt)e(of)523 880 y(the)j(Spectral)g(Golden)f(Rule.) i(It)f(is)i(used)d(to)i(study)f(the)g(point)f(spectrum)h(of)g(the)g (Liouvillean)523 980 y Fz(L)580 992 y Fu(\025)623 980 y FC(.)e(The)g(main)g(goal)f(of)h(this)g(study)g(is)g(a)h(proof)d(of)i (the)g(uniqueness)e(of)i(a)g(stationary)f(state)i(in)523 1079 y(the)18 b(thermal)f(case)i(and)e(of)h(the)g(none)o(xistence)d(of) j(a)g(stationary)f(state)i(in)f(the)g(non-thermal)d(state)523 1179 y(under)28 b(generic)g(conditions)g([DJ1)o(,)h(DJ2,)h(DJP].)f (\(See)h(also)f([JP1,)g(JP2,)h(BFS2])f(for)g(related)523 1279 y(results\).)648 1378 y(In)17 b(Subsection)f(6.3,)h(we)h(will)g (describe)f(the)h(result)f(of)h([DJ3)o(],)f(which)g(gi)n(v)o(es)h(a)g (relationship)523 1478 y(between)h(the)i(tw)o(o)f(kinds)g(of)g(LSO')-5 b(s)21 b(in)f(the)g(thermal)g(case.)648 1577 y(In)32 b(Subsections)f(6.4\2266.6)f(we)j(compute)e(both)h(LSO')-5 b(s)33 b(e)o(xplicitly)-5 b(.)31 b(In)h(the)g(case)h(of)f(the)523 1677 y(Da)n(vies)16 b(generator)m(,)d(these)j(formulas)e(are)i (essentially)f(contained)f(in)i(the)g(literature,)e(in)i(the)g(case)523 1777 y(of)22 b(the)g(LSO)g(for)g(the)g(Liouvillean,)e(the)o(y)h(are)h (generalizations)e(of)i(the)g(analoguous)d(formulas)523 1876 y(from)e([DJ2].)h(Both)h(LSO')-5 b(s)19 b(can)f(be)g(e)o(xpressed) f(in)i(a)g(number)d(of)i(distinct)h(forms,)e(each)h(ha)n(ving)523 1976 y(a)28 b(dif)n(ferent)d(adv)n(antage.)g(In)i(particular)m(,)e(as)j (a)g(result)f(of)g(our)f(computations,)f(we)i(describe)g(a)523 2076 y(simple)e(characterization)e(of)i(the)g(k)o(ernel)f(of)h (imaginary)f(part)g(of)h Fz(\000)12 b FC(,)25 b(which)g(can)g(be)g (used)g(in)523 2175 y(the)f(proof)f(of)g(the)h(return)f(to)i (equilibrium.)c(This)k(characterization)c(is)k(a)g(generalization)d(of) i(a)523 2275 y(result)c(from)f([DJ2].)523 2507 y FA(6.1)40 b(LSO)21 b(f)n(or)f(the)g(r)o(educed)h(dynamics)523 2681 y FC(It)i(is)g(easy)f(to)h(see)g(that)f(there)g(e)o(xists)h(a)f(unique) f(bounded)f(linear)i(map)g Fw(P)g FC(on)g Fo(M)h FC(such)f(that)h(for) 523 2781 y Fz(B)g Fs(\012)18 b Fz(C)29 b Fs(2)23 b Fo(M)h Fs(\032)e(B)s Ft(\()p Fs(K)d(\012)f(H)1381 2793 y Fy(R)1443 2781 y Ft(\))1424 2953 y Fw(P)p Ft(\()p Fz(B)k Fs(\012)c Fz(C)6 b Ft(\))24 b(=)e Fz(!)1935 2965 y Fn(R)1989 2953 y Ft(\()p Fz(C)6 b Ft(\))p Fz(B)23 b Fs(\012)18 b Fm(1)2335 2965 y Fn(R)2389 2953 y Fz(:)523 3125 y Fw(P)23 b Fs(2)g(B)s Ft(\()p Fo(M)p Ft(\))e FC(is)f(a)h(projection)d(of)i(norm)e(1.)i(\(It)g (is)h(an)f(e)o(xample)e(of)i(a)g Fk(conditional)e(e)n(xpectation)p FC(\).)648 3224 y(W)-7 b(e)21 b(identify)e Fs(B)s Ft(\()p Fs(K)q Ft(\))i FC(with)f Ft(Ran)p Fw(P)g FC(by)1379 3396 y Fs(B)s Ft(\()p Fs(K)q Ft(\))j Fs(3)g Fz(B)28 b Fs(7!)23 b Fz(B)f Fs(\012)c Fm(1)2079 3408 y Fn(R)2156 3396 y Fs(2)23 b Ft(Ran)p Fw(P)p Fz(:)717 b FC(\(50\))523 3593 y(Note)20 b(that)g Fz(\016)887 3605 y Fx(0)925 3498 y Fl(\014)925 3547 y(\014)925 3597 y(\014)952 3651 y Fx(Ran)p Fr(P)1131 3593 y FC(can)g(be)g(identi\002ed)f(with)i Ft(i[)p Fz(K)q(;)14 b Fs(\001)p Ft(])p FC(.)648 3722 y(W)-7 b(e)21 b(assume)f(that)g Fz(!)1239 3734 y Fn(R)1293 3722 y Ft(\()p Fz(V)f Ft(\))k(=)g(0)p FC(.)d(That)g(implies)g Fw(P)p Ft([)p Fz(V)5 b(;)14 b Fs(\001)p Ft(])p Fw(P)23 b Ft(=)f(0)p FC(.)648 3821 y(Note)f(that)g(Assumptions)f(2.1*,)g(2.2,)g (2.3*)g(and)g(2.4)g(are)h(satis\002ed)h(for)f(the)g(Banach)f(space)523 3921 y Fo(M)p FC(,)h(the)f(projection)e Fw(P)p FC(,)j(the)f Fz(C)1411 3891 y Fy(\003)1405 3942 y Fx(0)1449 3921 y FC(-group)e(of)i(isometries)g Ft(e)2180 3891 y Fu(t\016)2235 3899 y Fq(0)2272 3921 y FC(,)g(and)g(the)g(perturbation)e Ft(i[)p Fz(V)5 b(;)14 b Fs(\001)p Ft(])p FC(.)523 4085 y Fk(Remark)20 b(3.)k FC(One)g(can)g(ask)g(whether)f(the)h(abo)o(v)o(e) e(de\002ned)h(projection)f Fw(P)j FC(is)g(gi)n(v)o(en)d(by)i(the)g(for) n(-)523 4184 y(mula)f(\(5\).)f(Note)h(that)h Fo(M)f FC(is)i(not)d(a)i (re\003e)o(xi)n(v)o(e)e(Banach)h(space,)f(so)i(it)g(is)g(e)n(v)o(en)e (not)h(clear)g(if)h(this)523 4284 y(formula)19 b(mak)o(es)h(sense.)648 4384 y(Assume)h(that)h Fz(\016)1121 4396 y Fn(R)1197 4384 y FC(has)g(no)f(eigen)m(v)o(ectors)f(apart)h(from)g(scalar)g (operators.)f(Then)h(the)h(set)g(of)523 4483 y(eigen)m(v)n(alues)k(of)i Fz(\016)1073 4495 y Fx(0)1138 4483 y FC(equals)g Fs(f)p Ft(i\()p Fz(k)e Fs(\000)e Fz(k)1680 4453 y Fy(0)1703 4483 y Ft(\))66 b(:)37 b Fz(k)s(;)14 b(k)1990 4453 y Fy(0)2050 4483 y Fs(2)38 b Ft(sp)p Fz(K)6 b Fs(g)p FC(.)27 b(One)h(can)f(also)h(sho)n(w)g(that)g(for)523 4583 y(an)o(y)23 b Fz(e)28 b Fs(2)h Fw(R)p FC(,)24 b Fz(\016)959 4595 y Fx(0)1020 4583 y FC(is)g(globally)e(er)o(godic)g(at)h Ft(i)p Fz(e)29 b Fs(2)g Ft(i)p Fw(R)24 b FC(\(see)g(Appendix\))d(and)h (the)i(corresponding)523 4682 y(eigenprojection)17 b(is)k(gi)n(v)o(en)e (by)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 523 100 a FB(32)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)920 286 y Fm(1)968 298 y Fx(i)p Fu(e)1023 286 y Ft(\()p Fz(\016)1092 298 y Fx(0)1129 286 y Ft(\)\()p Fz(B)23 b Fs(\012)18 b Fz(C)6 b Ft(\))24 b(=)1613 208 y Fl(X)1571 386 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)1789 286 y Fz(!)1841 298 y Fn(R)1894 286 y Ft(\()p Fz(C)6 b Ft(\))14 b(\()q Fm(1)2118 298 y Fu(k)2159 286 y Ft(\()p Fz(K)6 b Ft(\))p Fz(B)t Fm(1)2415 298 y Fu(k)q Fy(\000)p Fu(e)2539 286 y Ft(\()p Fz(K)g Ft(\)\))14 b Fs(\012)p Fm(1)2839 298 y Fn(R)2892 286 y Fz(:)523 554 y FC(Therefore,)k(in)i(this)h(case)g (the)f(answer)g(to)g(our)f(question)g(is)j(positi)n(v)o(e)d(and)1638 750 y Fw(P)k Ft(=)1800 671 y Fl(X)1801 850 y Fu(e)p Fy(2)p Fr(R)1933 750 y Fm(1)1981 762 y Fx(i)p Fu(e)2036 750 y Ft(\()p Fz(\016)2105 762 y Fx(0)2142 750 y Ft(\))p Fz(;)523 1010 y FC(as)e(suggested)e(in)h(Subsection)g(2.6.)648 1185 y(W)-7 b(e)21 b(mak)o(e)f(the)g(follo)n(wing)e(assumption:)523 1334 y FA(Assumption)j(6.1)40 b Fk(Assumption)23 b(2.5)f(holds)h(for)g Ft(\()p Fw(P)p Fz(;)14 b(\016)2133 1346 y Fx(0)2170 1334 y Fz(;)g Ft(i[)p Fz(V)5 b(;)14 b Fs(\001)p Ft(]\))p Fk(.)24 b(This)g(means)f(that)g(ther)m(e)g(e)n(x-)523 1434 y(ists)807 1534 y Fz(M)31 b Ft(:=)23 b Fs(\000)1211 1455 y Fl(X)1109 1637 y Fu(e)p Fy(2)p Fx(sp\([)p Fu(K;)p Fy(\001)p Fx(]\))1447 1534 y Fm(1)1495 1546 y Fu(e)1530 1534 y Ft(\([)p Fz(K)q(;)14 b Fs(\001)g Ft(]\)[)p Fz(V)5 b(;)14 b Fs(\001)g Ft(]\(i)p Fz(e)19 b Ft(+)f(0)g Fs(\000)g Fz(\016)2335 1546 y Fx(0)2372 1534 y Ft(\))2404 1499 y Fy(\000)p Fx(1)2493 1534 y Ft([)p Fz(V)5 b(;)14 b Fs(\001)g Ft(])p Fm(1)2714 1546 y Fu(e)2750 1534 y Ft(\([)p Fz(K)q(;)g Fs(\001)g Ft(]\))p Fz(:)145 b FC(\(51\))648 1867 y Fz(M)37 b FC(is)29 b(the)g(LSO)f(for)g Ft(\()p Fw(P)p Fz(;)14 b(\016)1449 1879 y Fx(0)1486 1867 y Fz(;)g Ft(i[)p Fz(V)5 b(;)14 b Fs(\001)p Ft(]\))p FC(.)29 b(It)g(will)g(be)f(called)g(the)h(Da)n(vies)g(generator)d(\(in)i(the) 523 1967 y(Heisenber)o(g)18 b(picture\).)648 2066 y(T)-7 b(o)19 b(describe)f(the)i(physical)e(interpretation)f(of)i Fz(M)9 b FC(,)19 b(suppose)f(that)i(we)f(are)g(interested)g(only)523 2166 y(in)h(the)g(e)n(v)n(olution)e(of)h(the)h(observ)n(ables)e (corresponding)e(to)k(system)g Fs(S)27 b FC(\(taking,)18 b(ho)n(we)n(v)o(er)m(,)f(into)523 2265 y(account)22 b(the)h (in\003uence)f(of)h Fs(R)p FC(\).)g(W)-7 b(e)24 b(also)g(suppose)e (that)h(initially)g(the)g(reserv)n(oir)f(is)i(gi)n(v)o(en)e(by)523 2365 y(the)g(state)i Fz(!)875 2377 y Fn(R)928 2365 y FC(.)f(Let)f Fz(X)30 b FC(be)22 b(a)h(density)f(matrix)g(on)g(the)g (Hilbert)g(space)h Fs(K)q FC(,)g(such)f(that)g(the)h(initial)523 2465 y(state)c(of)f(the)g(system)h(is)g(described)e(by)h(the)g(density) g(matrix)g Fz(X)f Fs(\012)11 b(j)p Fz(\012)2529 2477 y Fn(R)2583 2465 y Ft(\)\()p Fz(\012)2711 2477 y Fn(R)2766 2465 y Fs(j)p FC(.)18 b(Let)h Fz(B)27 b Fs(2)c(B)s Ft(\()p Fs(K)q Ft(\))523 2564 y FC(be)i(an)g(observ)n(able)e(for)h(the)h (system)g Fs(S)6 b FC(,)25 b(such)g(that)g(the)g(measurement)e(at)i (the)g(\002nal)g(time)g Fz(t)g FC(is)523 2664 y(gi)n(v)o(en)i(by)h(the) g(operator)e Fz(B)j Fs(\012)24 b Fm(1)1509 2676 y Fn(R)1562 2664 y FC(.)29 b(Then)e(the)h(e)o(xpectation)e(v)n(alue)i(of)g(the)g (measurement)e(is)523 2764 y(gi)n(v)o(en)19 b(by)1387 2885 y Ft(T)-7 b(r)1472 2897 y Fy(K)1527 2793 y Fl(\020)1577 2885 y Fz(X)25 b Fs(\012)18 b(j)p Fz(\012)t Ft(\)\()p Fz(\012)t Fs(j)k Fz(\034)2067 2851 y Fu(t)2058 2906 y(\025)2101 2885 y Ft(\()p Fz(B)t Fs(\012)p Fm(1)2313 2897 y Fn(R)2367 2885 y Ft(\))2399 2793 y Fl(\021)3174 2885 y FC(\(52\))523 3054 y(Ob)o(viously)-5 b(,)18 b(\(52\))h(tensored)g(with)h Fm(1)1581 3066 y Fn(R)1656 3054 y FC(equals)1489 3237 y Ft(T)-7 b(r)1574 3249 y Fy(K)1643 3169 y Fl(\000)1681 3237 y Fz(X)7 b Fw(P)p Fz(\034)1853 3202 y Fu(t)1844 3257 y(\025)1887 3237 y Fw(P)p Ft(\()p Fz(B)22 b Fs(\012)c Fm(1)2186 3249 y Fn(R)2240 3237 y Ft(\))2272 3169 y Fl(\001)2324 3237 y Fz(:)648 3419 y FC(No)n(w)i(under)e(quite)i(general)f (conditions)g([Da1)o(,)i(Da2)o(,)f(Da3])g(we)h(ha)n(v)o(e)1389 3624 y Ft(lim)1378 3678 y Fu(\025)p Fy(!)p Fx(0)1530 3624 y Ft(e)1567 3589 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1870 3564 y Fq(2)1906 3624 y Fw(P)p Fz(\034)2002 3580 y Fu(t=\025)2100 3555 y Fq(2)1993 3649 y Fu(\025)2138 3624 y Fw(P)h Ft(=)h(e)2336 3589 y Fu(tM)2435 3624 y Fz(:)716 b FC(\(53\))548 3845 y(Thus)24 b Fz(M)34 b FC(describes)23 b(the)i(reduced)d(dynamics)i (renormalized)e(by)i Ft([)p Fz(K)q(;)14 b Fs(\001)p Ft(])p Fz(=\025)2784 3815 y Fx(2)2846 3845 y FC(in)24 b(the)h(limit)f(of)523 3945 y(the)c(weak)g(coupling,)e(where)i(we)g(rescale)h(the)f(time)g(by) g Fz(\025)2208 3915 y Fx(2)2246 3945 y FC(.)648 4044 y(Let)g(us)h(note)e(the)h(follo)n(wing)f(f)o(act:)523 4202 y FA(Theor)o(em)h(11.)k Fk(Suppose)c(Assumption)h(6.1)h(holds.)f (Then)h Fz(M)31 b Fk(is)23 b(the)f(g)o(ener)o(ator)g(of)g(a)g(Mark)o(o) o(v)523 4302 y(c.p.)e(semigr)l(oup)f(and)h(for)g(any)g Fz(z)26 b Fs(2)e Fw(C)p Fk(,)1308 4484 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))23 b(=)g(e)1677 4450 y Fu(z)r(K)1775 4484 y Fz(M)9 b Ft(\(e)1934 4450 y Fy(\000)p Fu(z)r(K)2084 4484 y Fz(B)t Ft(e)2188 4450 y Fu(z)r(K)2286 4484 y Ft(\)e)2355 4450 y Fy(\000)p Fu(z)r(K)2505 4484 y Fz(:)646 b FC(\(54\))p eop end %%Page: 33 33 TeXDict begin 33 32 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(33)523 282 y FA(Pr)o(oof)o(.)57 b FC(W)-7 b(e)31 b(kno)n(w)e(that)g(LSO)h Fz(M)39 b FC(commutes)28 b(with)i Fw(E)40 b Ft(=)g(i[)p Fz(K)q(;)14 b Fs(\001)p Ft(])p FC(.)30 b(This)g(is)g(equi)n(v)n(alent)e(to)523 382 y Ft(e)560 352 y Fu(z)r Fr(E)637 382 y Fz(M)9 b Ft(e)764 352 y Fy(\000)p Fu(z)r Fr(E)915 382 y Ft(=)23 b Fz(M)9 b FC(,)20 b(which)g(means)g(\(54\).)648 482 y(The)g(f)o(act)g(that)h Fz(M)29 b FC(is)22 b(a)e(Lindblad-K)m(ossak)o(o)n(wski)e(generator)g (and)i(annihilates)g Fm(1)h FC(will)g(fol-)523 581 y(lo)n(w)f (immediately)f(from)g(e)o(xplicit)h(formulas)f(gi)n(v)o(en)g(in)h (Subsection)f(6.4.)648 681 y(If)f(we)h(can)f(pro)o(v)o(e)e(53,)i(then)g (an)h(alternati)n(v)o(e)e(proof)g(is)i(possible:)f(we)h(immediately)e (see)i(that)523 780 y(the)h(left)h(hand)e(side)h(of)g(\(53\))f(is)i(a)g (Mark)o(o)o(v)e(c.p.)g(map)h(for)f(an)o(y)h Fz(t)h FC(and)e Fz(\025)p FC(,)i(hence)f(so)g(is)h Ft(e)3042 750 y Fu(tM)3141 780 y FC(.)f Ff(2)523 1071 y FA(6.2)40 b(LSO)21 b(f)n(or)f(the)g (Liouvillean)523 1254 y FC(Consider)g(the)g(the)g(Hilbert)g(space)g Fs(B)1607 1224 y Fx(2)1643 1254 y Ft(\()p Fs(K)q Ft(\))g Fs(\012)e(H)1944 1266 y Fn(R)2019 1254 y FC(and)h(the)h(orthogonal)e (projection)1463 1436 y Fz(P)35 b Ft(:=)23 b Fm(1)1710 1451 y Fy(B)1755 1435 y Fq(2)1787 1451 y Fx(\()p Fy(K)p Fx(\))1912 1436 y Fs(\012)18 b(j)p Fz(\012)2082 1448 y Fy(R)2144 1436 y Ft(\)\()p Fz(\012)2272 1448 y Fn(R)2326 1436 y Fs(j)p Fz(:)523 1619 y FC(W)-7 b(e)21 b(ha)n(v)o(e)f Fz(P)12 b(L)949 1631 y Fx(0)1009 1619 y Ft(=)22 b Fz(L)1153 1631 y Fx(0)1190 1619 y Fz(P)35 b Ft(=)23 b([)p Fz(K)q(;)14 b Fs(\001)p Ft(])p Fz(P)e FC(.)20 b(W)-7 b(e)21 b(identify)e Fs(B)2116 1589 y Fx(2)2152 1619 y Ft(\()p Fs(K)q Ft(\))j FC(with)f Ft(Ran)o Fz(P)33 b FC(by)1350 1802 y Fs(B)1408 1767 y Fx(2)1444 1802 y Ft(\()p Fs(K)q Ft(\))24 b Fs(3)f Fz(B)28 b Fs(7!)23 b Fz(B)f Fs(\012)c Fz(\012)2103 1814 y Fn(R)2180 1802 y Fs(2)24 b Ft(Ran)o Fz(P)r(:)689 b FC(\(55\))648 1984 y(W)-7 b(e)21 b(again)e(assume)h(that)h Fz(!)1440 1996 y Fn(R)1493 1984 y Ft(\()p Fz(V)e Ft(\))24 b(=)e(0)p FC(.)e(This)h(implies)f Fz(P)12 b(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(P)35 b Ft(=)23 b Fz(P)12 b(J)c(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(J)8 b(P)35 b Ft(=)23 b(0)p FC(.)648 2084 y(Note)28 b(that)h(Assumptions)e(2.1,)h(2.2,)f(2.3)h(and) g(2.4)g(are)g(satis\002ed)i(for)d(the)i(Hilbert)f(space)523 2183 y Fs(B)581 2153 y Fx(2)617 2183 y Ft(\()p Fs(K)q Ft(\))19 b Fs(\012)e(H)916 2195 y Fn(R)970 2183 y FC(,)k(the)f (projection)e Fz(P)12 b FC(,)20 b(the)g(strongly)f(continuous)f (unitary)g(group)h Ft(e)2904 2153 y Fx(i)p Fu(tL)2994 2161 y Fq(0)3029 2183 y FC(,)i(and)e(the)523 2283 y(perturbation)f Ft(i\()p Fz(\031)s Ft(\()p Fz(Q)p Ft(\))h Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(Q)p Ft(\))p Fz(J)g Ft(\))p FC(.)523 2457 y Fk(Remark)20 b(4.)k FC(Assume)15 b(that)h Fz(L)1370 2469 y Fn(R)1439 2457 y FC(has)f(no)g(eigen)m(v)o(ectors)e(apart)i (from)f Fz(\012)2518 2469 y Fn(R)2572 2457 y FC(.)h(Then)g(the)g(set)h (of)f(eigen-)523 2557 y(v)n(alues)20 b(of)g Fz(\016)881 2569 y Fx(0)939 2557 y FC(equals)g Fs(f)p Ft(i\()p Fz(k)h Fs(\000)d Fz(k)1462 2527 y Fy(0)1485 2557 y Ft(\))44 b(:)g Fz(k)s(;)14 b(k)1757 2527 y Fy(0)1803 2557 y Fs(2)23 b Ft(sp)p Fz(K)6 b Fs(g)20 b FC(and)932 2756 y Fm(1)980 2768 y Fu(e)1015 2756 y Ft(\()p Fz(L)1104 2768 y Fx(0)1141 2756 y Ft(\))p Fz(B)j Fs(\012)18 b Fz(\011)32 b Ft(=)23 b(\()p Fz(\012)1609 2768 y Fn(R)1663 2756 y Fs(j)p Fz(\011)9 b Ft(\))1834 2677 y Fl(X)1792 2856 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)2010 2756 y Ft(\()p Fm(1)2090 2768 y Fu(k)2131 2756 y Ft(\()p Fz(K)d Ft(\))p Fz(B)t Fm(1)2387 2768 y Fu(k)q Fy(\000)p Fu(e)2511 2756 y Ft(\()p Fz(K)g Ft(\)\))14 b Fs(\012)p Fz(\012)2827 2768 y Fn(R)2881 2756 y Fz(:)523 3024 y FC(Therefore,)1621 3123 y Fz(P)35 b Ft(=)1797 3044 y Fl(X)1798 3223 y Fu(e)p Fy(2)p Fr(R)1931 3123 y Fm(1)1979 3135 y Fx(i)p Fu(e)2033 3123 y Ft(\(i)p Fz(L)2145 3135 y Fx(0)2182 3123 y Ft(\))523 3350 y FC(is)27 b(the)f(spectral)g (projection)e(on)i(the)g(point)f(spectrum)g(of)h Ft(i)p Fz(L)2323 3362 y Fx(0)2360 3350 y FC(,)g(as)h(suggested)e(in)h (Subsection)523 3450 y(2.6.)523 3624 y FA(Assumption)21 b(6.2)40 b Fk(Assumption)23 b(2.5)g(for)g Ft(\()p Fz(P)r(;)14 b Ft(i)p Fz(L)1977 3636 y Fx(0)2015 3624 y Fz(;)g Ft(i\()p Fz(\031)s Ft(\()p Fz(V)20 b Ft(\))h Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(V)20 b Ft(\))p Fz(J)8 b Ft(\)\))24 b Fk(is)h(satis\002ed.)e(This)523 3724 y(means)d(that)g(ther)m(e)g(e)n (xists)1042 3886 y Ft(i)p Fz(\000)36 b Ft(:=)23 b Fs(\000)1460 3824 y Fl(P)1342 3964 y Fu(e)p Fy(2)p Fx(sp\([)p Fu(K;)p Fy(\001)p Fx(]\))1680 3886 y Fm(1)1728 3898 y Fu(e)1763 3886 y Ft(\([)p Fz(K)q(;)14 b Fs(\001)g Ft(]\)\()p Fz(\031)s Ft(\()p Fz(V)20 b Ft(\))f Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(J)8 b Ft(\))1152 4080 y Fs(\002)p Ft(\(i)p Fz(e)18 b Ft(+)g(0)g Fs(\000)g Ft(i)p Fz(L)1635 4092 y Fx(0)1672 4080 y Ft(\))1704 4050 y Fy(\000)p Fx(1)1794 4080 y Ft(\()p Fz(\031)s Ft(\()p Fz(V)h Ft(\))g Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(V)20 b Ft(\))p Fz(J)8 b Ft(\))p Fm(1)2479 4092 y Fu(e)2515 4080 y Ft(\([)p Fz(K)q(;)14 b Fs(\001)g Ft(]\))p Fz(:)648 4368 y Ft(i)p Fz(\000)34 b FC(is)24 b(the)f(LSO)g(for)1260 4300 y Fl(\000)1298 4368 y Fz(P)r(;)14 b Ft(i)p Fz(L)1470 4380 y Fx(0)1507 4368 y Fz(;)g Ft(i\()p Fz(\031)s Ft(\()p Fz(V)20 b Ft(\))g Fs(\000)g Fz(J)8 b(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(J)8 b Ft(\))2207 4300 y Fl(\001)2246 4368 y FC(.)23 b(W)-7 b(e)24 b(will)f(call)g(it)h(the)e(LSO)h(for)f(the)523 4467 y(Liouvillean.)g(The)i(operator)e Fz(\000)36 b FC(appeared)22 b(in)i([DJ1)o(],)g(where)g(it)g(w)o(as)h(used)f(to)g(gi)n(v)o(e)f(an)h (upper)523 4567 y(bound)18 b(on)i(the)g(point)g(spectrum)f(of)h Fz(L)1641 4579 y Fu(\025)1705 4567 y FC(for)f(small)i(nonzero)d Fz(\025)p FC(.)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 523 100 a FB(34)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(Theor)o(em)20 b(12.)k Fk(Suppose)17 b(that)i(Assumption)e(6.2)i(holds.)f(Then)g Ft(i)p Fz(\000)31 b Fk(is)20 b(the)f(g)o(ener)o(ator)f(of)h(a)g(con-) 523 382 y(tr)o(active)h(c.p.)g(semigr)l(oup)f(and)h(for)g(any)g Fz(z)26 b Fs(2)e Fw(C)p Fk(,)1335 565 y Fz(\000)12 b Ft(\()p Fz(B)t Ft(\))23 b(=)g(e)1677 530 y Fu(z)r(K)1775 565 y Fz(\000)12 b Ft(\(e)1907 530 y Fy(\000)p Fu(z)r(K)2057 565 y Fz(B)t Ft(e)2161 530 y Fu(z)r(K)2259 565 y Ft(\)e)2328 530 y Fy(\000)p Fu(z)r(K)2478 565 y Fz(:)673 b FC(\(56\))523 922 y FA(Pr)o(oof)o(.)41 b FC(The)21 b(proof)e(of)i(\(56\))f(is)i(the)g (same)f(as)h(that)f(of)g(\(54\).)f Ft(e)2325 891 y Fu(t)p Fx(i)p Fu(\000)2444 922 y FC(is)i(contracti)n(v)o(e)d(by)i(Theorem)523 1021 y(1.)f(The)g(proof)e(of)i(its)i(complete)d(positi)n(vity)g(will)i (be)f(gi)n(v)o(en)f(later)h(on)g(\(after)g(\(60\)\).)e Ff(2)523 1312 y FA(6.3)40 b(Relationship)21 b(between)f(the)g(Da)n (vies)g(generator)f(and)h(the)h(LSO)g(f)n(or)e(the)523 1411 y(Liouvillean)h(in)h(thermal)f(case.)523 1594 y FC(Ob)o(viously)-5 b(,)29 b(as)k(v)o(ector)d(spaces,)i Fs(B)s Ft(\()p Fs(K)q Ft(\))g FC(and)f Fs(B)1953 1564 y Fx(2)1989 1594 y Ft(\()p Fs(K)q Ft(\))j FC(coincide.)c(W)-7 b(e)33 b(are)e(interested)g(in)h(the)523 1694 y(relation)24 b(between)f Ft(i)p Fz(\000)37 b FC(and)24 b(generator)e Fz(M)9 b FC(.)24 b(W)-7 b(e)26 b(will)f(see)g(that)f(in)h(the)f (thermal)g(case)g(the)h(tw)o(o)523 1793 y(operators)19 b(are)h(similar)g(to)h(one)e(another)-5 b(.)648 1893 y(The)19 b(follo)n(wing)g(theorem)g(w)o(as)i(pro)o(v)o(en)d(in)i([DJ3)o (]:)523 2051 y FA(Theor)o(em)g(13.)k Fk(Suppose)d(that)h Fz(!)1498 2063 y Fn(R)1575 2051 y Fk(is)i(a)e Ft(\()p Fz(\034)1786 2063 y Fn(R)1841 2051 y Fz(;)14 b(\014)t Ft(\))p Fk(-KMS)23 b(state)o(.)g(Assumption)e(6.1)h(holds)h(if)g(and) 523 2150 y(only)18 b(if)h(Assumption)e(6.2)g(holds.)h(If)g(these)h (assumptions)e(hold,)g(then)h(for)h Fz(B)27 b Fs(2)c(B)s Ft(\()p Fs(K)q Ft(\))p Fk(,)c(we)g(have)1169 2336 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))25 b(=)d(i)p Fz(\000)12 b Ft(\()p Fz(B)t Ft(e)1724 2306 y Fy(\000)p Fu(\014)s(K=)p Fx(2)1945 2336 y Ft(\)e)2014 2306 y Fu(\014)s(K=)p Fx(2)1415 2511 y Ft(=)22 b(e)1539 2480 y Fu(\014)s(K=)p Fx(4)1708 2511 y Ft(i)p Fz(\000)12 b Ft(\(e)1863 2480 y Fy(\000)p Fu(\014)s(K=)p Fx(4)2082 2511 y Fz(B)t Ft(e)2186 2480 y Fy(\000)p Fu(\014)s(K=)p Fx(4)2407 2511 y Ft(\)e)2476 2480 y Fu(\014)s(K=)p Fx(4)2644 2511 y Fz(:)3174 2422 y FC(\(57\))523 2806 y Fk(Remark)20 b(5.)k FC(Let)k Fz(\032)37 b Ft(:=)f(e)1269 2776 y Fy(\000)p Fu(\014)s(K)1454 2806 y FC(and)27 b Fz(\015)1645 2818 y Fu(\032)1720 2806 y Ft(:)37 b Fs(B)s Ft(\()p Fs(K)q Ft(\))g Fs(!)g(B)2181 2776 y Fx(2)2217 2806 y Ft(\()p Fs(K)q Ft(\))29 b FC(be)f(the)g(linear)f(in)m(v)o(ertible)f(map)523 2906 y(de\002ned)19 b(by)1626 3005 y Fz(\015)1669 3017 y Fu(\032)1707 3005 y Ft(\()p Fz(B)t Ft(\))24 b(:=)f Fz(B)t(\032)2083 2971 y Fx(1)p Fu(=)p Fx(2)2187 3005 y Fz(:)964 b FC(\(58\))523 3155 y(Then)16 b(the)g(\002rst)i(identity)e (of)g(Theorem)f(13)h(can)g(be)h(written)f(as)i Fz(M)31 b Ft(=)23 b(i)p Fz(\015)2588 3125 y Fy(\000)p Fx(1)2583 3175 y Fu(\032)2682 3155 y Fs(\016)5 b Fz(\000)16 b Fs(\016)5 b Fz(\015)2886 3167 y Fu(\032)2924 3155 y Fz(:)18 b FC(Therefore,)523 3254 y(both)h Ft(i)p Fz(\000)33 b FC(and)19 b Fz(M)30 b FC(ha)n(v)o(e)19 b(the)i(same)f(spectrum.)648 3429 y(Theorem)j(13)h(follo)n(ws)g(from)g(the)h(e)o(xplicit)f(formulas)f (for)h Fz(M)34 b FC(and)24 b Ft(i)p Fz(\000)37 b FC(gi)n(v)o(en)24 b(in)h(Subsec-)523 3528 y(tions)d(6.4\2266.6.)d(It)j(is,)g(ho)n(we)n(v) o(er)m(,)d(instructi)n(v)o(e)i(to)h(gi)n(v)o(e)f(an)g(alternati)n(v)o (e,)f(time)i(dependent)e(proof)523 3628 y(of)g(Identity)f(\(57\),)f (which)i(a)n(v)n(oids)g(calculating)f(both)g(LSO')-5 b(s.)20 b(Strictly)g(speaking,)f(the)h(identity)523 3728 y(will)e(be)g(pro)o(v)o(en)d(for)i(the)h(\223the)g(dynamical)e(Le)n(v)o (el)h(Shift)g(Operators\224)g Fz(M)2616 3740 y Fx(dyn)2746 3728 y FC(and)g Ft(i)p Fz(\000)2958 3740 y Fx(dyn)3089 3728 y FC(which,)523 3827 y(ho)n(we)n(v)o(er)m(,)24 b(according)g(to)j (the)f(Dynamical)f(Fermi)h(Golden)g(Rule,)g(under)f(broad)g (conditions,)523 3927 y(coincide)19 b(with)h(the)h(usual)f(Le)n(v)o(el) f(Shift)h(Operators)f Fz(M)30 b FC(and)19 b Ft(i)p Fz(\000)12 b FC(.)523 4085 y FA(Theor)o(em)20 b(14.)k Fk(Suppose)14 b(that)i Fz(!)1485 4097 y Fn(R)1555 4085 y Fk(is)h(a)f Ft(\()p Fz(\034)1753 4097 y Fn(R)1807 4085 y Fz(;)e(\014)t Ft(\))p Fk(-KMS)j(state)o(.)f(Then)g(the)g(following)f(statements)523 4184 y(ar)m(e)20 b(equivalent:)612 4284 y FC(1\))k Fk(ther)m(e)c(e)n (xists)i(an)e(oper)o(ator)f Fz(M)1593 4296 y Fx(dyn)1725 4284 y Fk(satisfying)1436 4494 y Ft(lim)1424 4548 y Fu(\025)p Fy(!)p Fx(0)1577 4494 y Ft(e)1614 4460 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1917 4435 y Fq(2)1953 4494 y Fw(P)p Fz(\034)2049 4451 y Fu(t=\025)2147 4426 y Fq(2)2040 4519 y Fu(\025)2184 4494 y Fw(P)k Ft(=)g(e)2383 4460 y Fu(tM)2471 4469 y Fq(dyn)2571 4494 y Fz(:)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(35)612 282 y FC(2\))24 b Fk(ther)m(e)c(e)n(xists)i(an)e(oper)o(ator)f Fz(\000)1563 294 y Fx(dyn)1696 282 y Fk(satisfying)1351 475 y Ft(lim)1339 529 y Fu(\025)p Fy(!)p Fx(0)1492 475 y Ft(e)1529 441 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1832 416 y Fq(2)1868 475 y Fz(P)12 b Ft(e)1970 441 y Fy(\000)p Fx(i)p Fu(tL)2112 450 y Fp(\025)2150 441 y Fu(=\025)2223 416 y Fq(2)2260 475 y Fz(P)34 b Ft(=)23 b(e)2472 441 y Fx(i)p Fu(t\000)2556 450 y Fq(dyn)2656 475 y Fz(:)523 675 y Fk(Mor)m(eo)o(ver)-9 b(,)1474 775 y Fz(M)1555 787 y Fx(dyn)1690 775 y Ft(=)22 b Fz(\015)1825 741 y Fy(\000)p Fx(1)1820 796 y Fu(\032)1932 775 y Fs(\016)c Ft(i)p Fz(\000)2066 787 y Fx(dyn)2197 775 y Fs(\016)g Fz(\015)2300 787 y Fu(\032)2339 775 y Fz(:)523 1071 y FA(Pr)o(oof)o(.)42 b FC(The)21 b(Araki)h(perturbation)d(theory)h(\(see) i([DJP])g(and)g(references)e(therein\))g(yields)i(that)523 1170 y(the)33 b(v)o(ector)e Fz(\011)948 1182 y Fu(\025)991 1170 y FC(,)i(de\002ned)f(by)g(\(49\),)g(satis\002es)i Fz(\011)1979 1182 y Fu(\025)2068 1170 y Ft(=)46 b Fz(\011)2230 1182 y Fx(0)2295 1170 y Ft(+)27 b Fz(O)r Ft(\()p Fz(\025)p Ft(\))35 b FC(and)d Fz(L)2809 1182 y Fu(\025)2852 1170 y Fz(\011)2903 1182 y Fu(\025)2992 1170 y Ft(=)46 b(0)p FC(.)33 b(F)o(or)523 1270 y Fz(X)r(;)14 b(B)27 b Fs(2)c(B)s Ft(\()p Fs(K)q Ft(\))g(=)g Fs(B)1154 1240 y Fx(2)1190 1270 y Ft(\()p Fs(K)q Ft(\))p FC(,)f(using)e(the)g(identi\002cations)f (\(50\))g(and)h(\(55\),)e(we)j(ha)n(v)o(e)756 1445 y Ft(T)-7 b(r)841 1457 y Fy(K)910 1378 y Fl(\000)948 1445 y Fz(X)1024 1415 y Fy(\003)1062 1445 y Fw(P)p Fz(\034)1158 1410 y Fy(\000)p Fu(t)1149 1467 y Fx(0)1239 1445 y Fz(\034)1284 1415 y Fu(t)1275 1469 y(\025)1319 1445 y Ft(\()p Fz(B)t Fs(\012)p Fm(1)1531 1457 y Fn(R)1585 1445 y Ft(\))1617 1378 y Fl(\001)756 1642 y Ft(=)844 1549 y Fl(\020)893 1642 y Fz(X)7 b Ft(e)1006 1611 y Fu(\014)s(K=)p Fx(2)1192 1642 y Fs(\012)18 b Fz(\012)1339 1654 y Fn(R)1414 1546 y Fl(\014)1414 1596 y(\014)1414 1646 y(\014)1462 1642 y Ft(\(e)1531 1611 y Fy(\000)p Fx(i)p Fu(tL)1673 1619 y Fq(0)1709 1642 y Ft(e)1746 1611 y Fx(i)p Fu(tL)1836 1620 y Fp(\025)1899 1642 y Fz(B)t Fs(\012)p Fm(1)2079 1654 y Fn(R)2153 1642 y Ft(e)2190 1611 y Fy(\000)p Fx(i)p Fu(tL)2332 1620 y Fp(\025)2375 1642 y Ft(e)2412 1611 y Fx(i)p Fu(tL)2502 1619 y Fq(0)2538 1642 y Ft(\))j(e)2628 1611 y Fy(\000)p Fu(\014)s(K=)p Fx(2)2848 1642 y Fs(\012)o Fz(\012)2976 1654 y Fn(R)3030 1549 y Fl(\021)756 1794 y Fu(O)r Fx(\()p Fu(\025)p Fx(\))795 1848 y Ft(=)62 b(\()p Fz(X)7 b Ft(e)1067 1818 y Fu(\014)s(K=)p Fx(2)1254 1848 y Fs(\012)18 b Fz(\012)1401 1860 y Fn(R)1475 1848 y Fs(j)j Ft(e)1556 1818 y Fy(\000)p Fx(i)p Fu(tL)1698 1826 y Fq(0)1734 1848 y Ft(e)1771 1818 y Fx(i)p Fu(tL)1861 1827 y Fp(\025)1924 1848 y Fz(B)t Fs(\012)p Fm(1)2104 1860 y Fn(R)2178 1848 y Ft(e)2215 1818 y Fy(\000)p Fx(i)p Fu(tL)2357 1827 y Fp(\025)2400 1848 y Fz(\011)2451 1860 y Fu(\025)2494 1848 y Ft(\))756 2001 y Fu(O)r Fx(\()p Fu(\025)p Fx(\))795 2055 y Ft(=)62 b(\()p Fz(X)7 b Ft(e)1067 2025 y Fu(\014)s(K=)p Fx(2)1254 2055 y Fs(\012)18 b Fz(\012)1401 2067 y Fn(R)1475 2055 y Fs(j)j Ft(e)1556 2025 y Fy(\000)p Fx(i)p Fu(tL)1698 2033 y Fq(0)1734 2055 y Ft(e)1771 2025 y Fx(i)p Fu(tL)1861 2034 y Fp(\025)1924 2055 y Fz(B)t Fs(\012)p Fm(1)2104 2067 y Fn(R)2178 2055 y Ft(e)2215 2025 y Fy(\000)p Fu(\014)s(K=)p Fx(2)2435 2055 y Fs(\012)p Fz(\012)2564 2067 y Fn(R)2618 2055 y Ft(\))756 2230 y(=)844 2163 y Fl(\000)882 2230 y Fz(X)27 b Fs(j)20 b Ft(\()p Fz(P)12 b Ft(e)1155 2200 y Fy(\000)p Fx(i)p Fu(tL)1297 2208 y Fq(0)1333 2230 y Ft(e)1370 2200 y Fx(i)p Fu(tL)1460 2209 y Fp(\025)1517 2163 y Fl(\000)1555 2230 y Fz(B)t Ft(e)1659 2200 y Fy(\000)p Fu(\014)s(K=)p Fx(2)1879 2230 y Fs(\012)o Fz(\012)2007 2242 y Fn(R)2061 2230 y Ft(\))2093 2163 y Fl(\001)2145 2230 y Ft(e)2182 2200 y Fu(\014)s(K=)p Fx(2)2351 2163 y Fl(\001)523 2400 y FC(uniformly)18 b(for)h Fz(t)k Fs(\025)g Ft(0)p FC(.)d(Hence,)g(since)g Ft(dim)14 b Fs(K)25 b Fz(<)d Fs(1)p FC(,)589 2588 y Ft(e)626 2553 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)929 2528 y Fq(2)965 2588 y Fw(P)p Fz(\034)1061 2553 y Fu(t)1052 2608 y(\025)1095 2588 y Ft(\()p Fz(B)t Fs(\012)p Fm(1)1307 2600 y Fn(R)1361 2588 y Ft(\))h(=)1504 2520 y Fl(\000)1542 2588 y Ft(e)1579 2553 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1882 2528 y Fq(2)1918 2588 y Fz(P)12 b Ft(e)2020 2553 y Fx(i)p Fu(tL)2110 2562 y Fp(\025)2152 2588 y Ft(\()p Fz(B)t Ft(e)2288 2553 y Fy(\000)p Fu(\014)s(K=)p Fx(2)2509 2588 y Fs(\012)o Fz(\012)2637 2600 y Fn(R)2691 2588 y Ft(\))2723 2520 y Fl(\001)2761 2588 y Ft(e)2798 2553 y Fu(\014)s(K=)p Fx(2)2985 2588 y Ft(+)18 b Fz(O)r Ft(\()p Fz(\025)p Ft(\))523 2762 y FC(uniformly)g(for)h Fz(t)k Fs(\025)g Ft(0)p FC(.)d(W)-7 b(e)21 b(conclude)e(that)h(for)g(a)g(gi)n(v)o(en)f Fz(t)i FC(the)f(limit)1351 2949 y Ft(lim)1340 3003 y Fu(\025)p Fy(!)p Fx(0)1492 2949 y Ft(e)1529 2915 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1832 2890 y Fq(2)1868 2949 y Fz(P)12 b Ft(e)1970 2915 y Fx(i)p Fu(tL)2060 2924 y Fp(\025)2098 2915 y Fu(=\025)2171 2890 y Fq(2)2208 2949 y Fz(P)35 b Ft(=:)23 b Fz(T)2468 2915 y Fu(t)523 3152 y FC(e)o(xists)e(if)n(f)f(the)g(limit)1415 3251 y Ft(lim)1403 3305 y Fu(\025)p Fy(!)p Fx(0)1556 3251 y Ft(e)1593 3217 y Fy(\000)p Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])p Fu(=\025)1896 3192 y Fq(2)1932 3251 y Fw(P)p Fz(\034)2028 3208 y Fu(t=\025)2126 3183 y Fq(2)2019 3276 y Fu(\025)2163 3251 y Fw(P)j Ft(=:)g Fw(T)2403 3217 y Fu(t)523 3423 y FC(e)o(xists.)d(Moreo)o(v)o(er)m(,)d (if)k(the)f(limits)h(e)o(xist,)f(then)1576 3598 y Fw(T)1631 3564 y Fu(t)1684 3598 y Ft(=)j Fz(\015)1820 3564 y Fy(\000)p Fx(1)1815 3618 y Fu(\032)1927 3598 y Fs(\016)18 b Fz(T)2048 3564 y Fu(t)2095 3598 y Fs(\016)g Fz(\015)2198 3610 y Fu(\032)2236 3598 y Fz(:)523 3772 y FC(In)h(particular)m(,)e Fw(T)1022 3742 y Fu(t)1071 3772 y FC(is)i(a)h(semigroup)c(if)n(f)j Fz(T)1728 3742 y Fu(t)1776 3772 y FC(is)h(a)f(semigroup)e(and)h(their)h (generators)e(\()p Fz(M)3062 3784 y Fx(dyn)3193 3772 y FC(and)523 3872 y Ft(i)p Fz(\000)597 3884 y Fx(dyn)730 3872 y FC(respecti)n(v)o(ely\))i(satisfy)h(\(57\).)f Ff(2)648 4018 y FC(It)e(is)h(perhaps)e(interesting)g(that)h(Theorem)e (14)i(can)g(be)g(immediately)f(generalized)f(to)i(some)523 4118 y(non-thermal)h(cases.)523 4284 y FA(Theor)o(em)i(15.)k Fk(Suppose)h(that)i(instead)g(of)h(assuming)e(that)h Fz(!)2366 4296 y Fn(R)2448 4284 y Fk(is)h(KMS,)g(we)g(mak)o(e)f(the)g (fol-)523 4384 y(lowing)36 b(stability)g(assumption:)e(W)-8 b(e)37 b(suppose)f(that)f Fz(\032)i Fk(is)g(a)f(nonde)m(g)o(ener)o(ate) d(density)i(ma-)523 4483 y(trix)c(on)f Fs(K)q Fk(,)h(and)e(for)i Fs(j)p Fz(\025)p Fs(j)42 b(\024)f Fz(\025)1472 4495 y Fx(0)1540 4483 y Fk(ther)m(e)31 b(e)n(xists)g(a)g(normalized)e(vector)h Fz(\011)2718 4495 y Fu(\025)2803 4483 y Fs(2)42 b(H)32 b Fk(suc)o(h)d(that)523 4583 y Fz(\011)574 4595 y Fu(\025)650 4583 y Ft(=)k Fz(\032)791 4553 y Fx(1)p Fu(=)p Fx(2)917 4583 y Fs(\012)22 b Fz(\012)1068 4595 y Fn(R)1144 4583 y Ft(+)g Fz(o)p Ft(\()p Fz(\025)1351 4553 y Fx(0)1389 4583 y Ft(\))27 b Fk(and)d Fz(L)1655 4595 y Fu(\025)1698 4583 y Fz(\011)1749 4595 y Fu(\025)1826 4583 y Ft(=)32 b(0)p Fk(.)25 b(Then)g(all)h(the)f(statements)h(of)f(Theor)m(em)h(14) 523 4682 y(r)m(emain)20 b(true)o(,)g(with)h Fz(\032)g Fk(r)m(eplacing)e Ft(e)1551 4652 y Fy(\000)p Fu(\014)s(K)1707 4682 y Fk(.)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 523 100 a FB(36)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)648 282 y FC(Let)25 b(us)h(return)e(to)h(the)g (thermal)f(case.)i(It)f(is)h(well)g(kno)n(wn)e([A)o(,)i(FGKV])f(that)h (in)f(this)h(case)523 382 y(the)e(Da)n(vies)g(generator)e(satis\002es)j (the)f(Detailed)f(Balance)h(Condition.)e(W)-7 b(e)25 b(will)f(see)h(that)f(this)523 482 y(f)o(act)c(is)i(essentially)e(equi) n(v)n(alent)e(to)j(Relation)f(\(57\).)523 656 y FA(Theor)o(em)g(16.)k Fk(Suppose)k(that)h Fz(!)1512 668 y Fn(R)1595 656 y Fk(is)i(a)e Ft(\()p Fz(\034)1820 668 y Fn(R)1875 656 y Fz(;)14 b(\014)t Ft(\))p Fk(-KMS)30 b(state)g(and)f(Assumption)f(6.1)h(holds.)523 755 y(Then)f(the)g(Davies)g(g)o(ener)o(ator)f Fz(M)37 b Fk(satis\002es)28 b(DBC)h(for)g Ft(e)2228 725 y Fy(\000)p Fu(\014)s(K)2413 755 y Fk(both)e(in)h(the)g(standar)m(d)f(sense)523 855 y(and)19 b(in)i(the)f(sense)g(of)h(AFGKV)-11 b(.)523 1029 y FA(Pr)o(oof)o(.)37 b FC(Recall)19 b(that)h(the)f(operator)e Fz(\015)1615 1041 y Fu(\032)1673 1029 y FC(de\002ned)h(in)i(\(58\))e (is)i(unitary)d(from)i Fs(B)2749 999 y Fx(2)2746 1056 y(\()p Fu(\032)p Fx(\))2836 1029 y Ft(\()p Fs(K)q Ft(\))h FC(to)f Fs(B)3126 999 y Fx(2)3163 1029 y Ft(\()p Fs(K)q Ft(\))p FC(.)523 1129 y(Recall)i(also)f(that)h(in)f(the)g(thermal)g (case)1576 1312 y Fz(M)31 b Ft(=)23 b Fz(\015)1824 1277 y Fy(\000)p Fx(1)1819 1332 y Fu(\032)1931 1312 y Fs(\016)18 b Ft(i)p Fz(\000)30 b Fs(\016)18 b Fz(\015)2198 1324 y Fu(\032)2237 1312 y Fz(:)523 1494 y FC(Hence,)1462 1594 y Fz(M)1552 1560 y Fy(\003)p Fx(\()p Fu(\032)p Fx(\))1699 1594 y Ft(=)23 b Fs(\000)p Fz(\015)1900 1560 y Fy(\000)p Fx(1)1895 1614 y Fu(\032)2007 1594 y Fs(\016)18 b Ft(i)p Fz(\000)2153 1560 y Fy(\003)2209 1594 y Fs(\016)g Fz(\015)2312 1606 y Fu(\032)2350 1594 y Fz(:)523 1743 y FC(Thus,)1128 1804 y Fx(1)p 1119 1818 52 4 v 1119 1866 a(2i)1181 1837 y Ft(\()p Fz(M)27 b Fs(\000)18 b Fz(M)1494 1807 y Fy(\003)p Fx(\()p Fu(\032)p Fx(\))1618 1837 y Ft(\))25 b(=)e Fz(\015)1811 1807 y Fy(\000)p Fx(1)1806 1857 y Fu(\032)1918 1837 y Fs(\016)1988 1804 y Fx(1)p 1988 1818 34 4 v 1988 1866 a(2)2031 1837 y Ft(\()p Fz(\000)30 b Ft(+)18 b Fz(\000)2290 1807 y Fy(\003)2328 1837 y Ft(\))h Fs(\016)f Fz(\015)2482 1849 y Fu(\032)1675 2008 y Ft(=)23 b Fz(\015)1811 1977 y Fy(\000)p Fx(1)1806 2028 y Fu(\032)1918 2008 y Fs(\016)18 b Ft([)p Fz(\001)2070 1977 y Fx(R)2123 2008 y Fz(;)c Fs(\001)p Ft(])k Fs(\016)g Fz(\015)2327 2020 y Fu(\032)2389 2008 y Ft(=)k([)p Fz(\001)2568 1977 y Fx(R)2621 2008 y Fz(;)14 b Fs(\001)p Ft(])p Fz(;)523 2167 y FC(\(where)20 b Fz(\001)844 2137 y Fx(R)919 2167 y FC(will)i(be)f(de\002ned)f(in)h (the)h(ne)o(xt)e(subsection\).)g(This)h(pro)o(v)o(es)f(DBC)i(in)g(the)f (sense)g(of)523 2267 y(AFGKV)-11 b(.)648 2367 y(By)26 b(Theorem)d(11)j(and)f(the)g(f)o(act)h(that)g Fz(\032)g FC(is)g(proportional)d(to)j Ft(e)2495 2336 y Fy(\000)p Fu(\014)s(K)2651 2367 y FC(,)g(for)f(an)o(y)g Fz(z)36 b Fs(2)e Fw(C)26 b FC(we)523 2466 y(ha)n(v)o(e)1415 2566 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))24 b(=)e Fz(\032)1790 2532 y Fu(z)1829 2566 y Fz(M)9 b Ft(\()p Fz(\032)1994 2532 y Fy(\000)p Fu(z)2084 2566 y Fz(B)t(\032)2194 2532 y Fu(z)2232 2566 y Ft(\))p Fz(\032)2307 2532 y Fy(\000)p Fu(z)2397 2566 y Fz(:)523 2715 y FC(Therefore,)27 b(by)i(Theorem)f(10,) g(the)i(DBC)g(in)g(the)f(sense)h(of)f(AFGKV)h(is)g(equi)n(v)n(alent)e (to)h(the)523 2815 y(standard)19 b(DBC.)i Ff(2)523 3105 y FA(6.4)40 b(Explicit)21 b(f)n(ormula)e(f)n(or)h(the)h(Da)n(vies)e (generator)523 3288 y FC(In)e(this)h(subsection)f(we)g(suppose)g(that)h (Assumption)e(6.1)h(is)h(true)f(and)g(we)h(describe)e(an)i(e)o(xplicit) 523 3388 y(formula)h(for)g(the)h(Da)n(vies)h(generator)d Fz(M)9 b FC(.)648 3487 y(W)-7 b(e)21 b(introduce)e(the)h(follo)n(wing)f (notation)g(for)h(the)h(set)g(of)f(allo)n(wed)g(transition)g (frequencies)523 3587 y(and)g(the)g(set)h(of)f(allo)n(wed)f(transition) h(frequencies)e(from)h Fz(k)26 b Fs(2)e Ft(sp)o Fz(K)6 b FC(:)631 3769 y Fs(F)31 b Ft(:=)22 b Fs(f)p Fz(k)917 3781 y Fx(1)973 3769 y Fs(\000)c Fz(k)1099 3781 y Fx(2)1180 3769 y Ft(:)23 b Fz(k)1269 3781 y Fx(1)1307 3769 y Fz(;)14 b(k)1387 3781 y Fx(2)1447 3769 y Fs(2)23 b Ft(sp)p Fz(K)6 b Fs(g)22 b Ft(=)h(sp[)p Fz(K)q(;)14 b Fs(\001)p Ft(])p Fz(;)76 b Fs(F)2249 3781 y Fu(k)2313 3769 y Ft(:=)22 b Fs(f)p Fz(k)f Fs(\000)d Fz(k)2655 3781 y Fx(1)2736 3769 y Ft(:)44 b Fz(k)2846 3781 y Fx(1)2907 3769 y Fs(2)23 b Ft(sp)p Fz(K)6 b Fs(g)p Fz(:)648 3952 y FC(Let)20 b Fs(j)p Fz(\012)t Ft(\))h FC(denote)e(the)i(map)1422 4135 y Fw(C)i Fs(3)g Fz(z)k Fs(7!)c(j)p Fz(\012)t Ft(\))p Fz(z)k Ft(:=)c Fz(z)t(\012)j Fs(2)e(H)2337 4147 y Fn(R)2390 4135 y Fz(:)523 4317 y FC(Then)19 b Fm(1)762 4329 y Fy(K)817 4317 y Fs(\012j)p Fz(\012)t Ft(\))k Fs(2)h(B)s Ft(\()p Fs(K)q Fz(;)14 b Fs(K)19 b(\012)f(H)1533 4329 y Fn(R)1587 4317 y Ft(\))p FC(.)j(Set)1640 4500 y Fz(v)26 b Ft(:=)d Fz(V)40 b Fm(1)1953 4512 y Fy(K)2007 4500 y Fs(\012j)p Fz(\012)t Ft(\))523 4682 y FC(Note)20 b(that)g Fz(v)k FC(belongs)19 b(to)i Fs(B)s Ft(\()p Fs(K)q Fz(;)14 b Fs(K)19 b(\012)f(H)1705 4694 y Fn(R)1759 4682 y Ft(\))p FC(.)j(W)-7 b(e)22 b(also)e(de\002ne)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(37)1352 282 y Fz(v)1395 248 y Fu(k)1430 256 y Fq(1)1463 248 y Fu(;k)1518 256 y Fq(2)1578 282 y Ft(:=)22 b Fm(1)1736 294 y Fu(k)1771 302 y Fq(1)1808 282 y Ft(\()p Fz(K)6 b Ft(\))p Fs(\012)p Fm(1)2062 294 y Fn(R)2136 282 y Fz(v)24 b Fm(1)2248 294 y Fu(k)2283 302 y Fq(2)2319 282 y Ft(\()p Fz(K)6 b Ft(\);)1579 470 y(~)-45 b Fz(v)1619 436 y Fu(p)1681 470 y Ft(:=)1834 392 y Fl(X)1792 570 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)2010 470 y Fz(v)2053 436 y Fu(k)q(;k)q Fy(\000)p Fu(p)2236 470 y Ft(;)1016 712 y Fz(\001)25 b Ft(=)1256 649 y Fl(P)1197 787 y Fu(k)q Fy(2)p Fx(sp)q Fu(K)1453 649 y Fl(P)1416 786 y Fu(p)p Fy(2F)1543 795 y Fp(k)1578 712 y Ft(\()p Fz(v)1653 682 y Fy(\003)1692 712 y Ft(\))1724 682 y Fu(k)q(;k)q Fy(\000)p Fu(p)1908 712 y Fm(1)p Fs(\012)o Ft(\()p Fz(p)19 b Ft(+)f(i0)g Fs(\000)g Fz(L)2419 724 y Fn(R)2472 712 y Ft(\))2504 682 y Fy(\000)p Fx(1)2593 712 y Fz(v)2636 682 y Fu(k)q Fy(\000)p Fu(p;k)1110 929 y Ft(=)1220 867 y Fl(P)1197 1004 y Fu(p)p Fy(2F)1330 929 y Ft(\()s(~)-45 b Fz(v)1405 899 y Fu(p)1444 929 y Ft(\))1476 899 y Fy(\003)1514 929 y Fm(1)p Fs(\012)o Ft(\()p Fz(p)19 b Ft(+)f(i0)g Fs(\000)g Fz(L)2025 941 y Fn(R)2078 929 y Ft(\))2110 899 y Fy(\000)p Fx(1)2203 929 y Ft(~)-45 b Fz(v)2243 899 y Fu(p)2281 929 y Fz(:)523 1132 y FC(The)20 b(real)g(and)g(the)g (imaginary)e(part)i(of)g Fz(\001)h FC(are)f(gi)n(v)o(en)f(by)776 1300 y Fz(\001)845 1270 y Fx(R)920 1300 y Ft(:=)1041 1268 y Fx(1)p 1041 1282 34 4 v 1041 1329 a(2)1084 1300 y Ft(\()p Fz(\001)g Ft(+)f Fz(\001)1356 1270 y Fy(\003)1394 1300 y Ft(\))25 b(=)1597 1238 y Fl(P)1539 1375 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)1795 1238 y Fl(P)1757 1374 y Fu(p)p Fy(2F)1884 1383 y Fp(k)1920 1300 y Ft(\()p Fz(v)1995 1270 y Fy(\003)2034 1300 y Ft(\))2066 1270 y Fu(k)q(;k)q Fy(\000)p Fu(p)2249 1300 y Fm(1)p Fs(\012P)7 b Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2659 1312 y Fn(R)2712 1300 y Ft(\))2744 1270 y Fy(\000)p Fx(1)2833 1300 y Fz(v)2876 1270 y Fu(k)q Fy(\000)p Fu(p;k)1451 1518 y Ft(=)1561 1456 y Fl(P)1539 1592 y Fu(p)p Fy(2F)1671 1518 y Ft(\()s(~)-45 b Fz(v)1746 1488 y Fu(p)1785 1518 y Ft(\))1817 1488 y Fy(\003)1856 1518 y Fm(1)p Fs(\012)o(P)7 b Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2265 1530 y Fn(R)2318 1518 y Ft(\))2350 1488 y Fy(\000)p Fx(1)2443 1518 y Ft(~)-45 b Fz(v)2483 1488 y Fu(p)2521 1518 y Ft(;)803 1762 y Fz(\001)872 1732 y Fx(I)923 1762 y Ft(:=)1054 1730 y Fx(1)p 1044 1744 52 4 v 1044 1791 a(2i)1106 1762 y Ft(\()p Fz(\001)19 b Fs(\000)f Fz(\001)1378 1732 y Fy(\003)1416 1762 y Ft(\))25 b(=)e Fz(\031)1684 1700 y Fl(P)1625 1837 y Fu(k)q Fy(2)p Fx(sp)q Fu(K)1881 1700 y Fl(P)1843 1837 y Fu(p)p Fy(2F)1970 1846 y Fp(k)2006 1762 y Ft(\()p Fz(v)2081 1732 y Fy(\003)2120 1762 y Ft(\))2152 1732 y Fu(k)q(;k)q Fy(\000)p Fu(p)2335 1762 y Fm(1)p Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2720 1774 y Fn(R)2774 1762 y Ft(\))p Fz(v)2849 1732 y Fu(k)q Fy(\000)p Fu(p;k)1473 1980 y Ft(=)23 b Fz(\031)1648 1918 y Fl(P)1625 2055 y Fu(p)p Fy(2F)1758 1980 y Ft(\()s(~)-45 b Fz(v)1833 1950 y Fu(p)1871 1980 y Ft(\))1903 1950 y Fy(\003)1942 1980 y Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)2327 1992 y Fn(R)2380 1980 y Ft(\))s(~)-45 b Fz(v)2455 1950 y Fu(p)2494 1980 y Ft(;)523 2195 y FC(Note)22 b(that)g Fz(\001)923 2165 y Fx(I)977 2195 y Fs(\025)j Ft(0)p FC(.)d(Belo)n(w)g(we)g(gi)n(v)o(e)f(four)g(e)o (xplicit)g(formulas)f(for)h(the)h(Da)n(vies)g(generator)e(in)523 2295 y(the)g(Heisenber)o(g)f(picture:)535 2463 y Fz(M)9 b Ft(\()p Fz(B)t Ft(\))26 b(=)e(i\()p Fz(\001B)f Fs(\000)18 b Fz(B)t(\001)1300 2433 y Fy(\003)1339 2463 y Ft(\))871 2633 y(+2)p Fz(\031)1064 2571 y Fl(P)1041 2708 y Fu(p)p Fy(2F)1174 2633 y Ft(\()s(~)-45 b Fz(v)1249 2603 y Fu(p)1288 2633 y Ft(\))1320 2603 y Fy(\003)1379 2633 y Fz(B)t Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)1783 2645 y Fn(R)1836 2633 y Ft(\))s(~)-45 b Fz(v)1911 2603 y Fu(p)782 2875 y Ft(=)24 b(i)930 2813 y Fl(P)908 2949 y Fu(p)p Fy(2F)1040 2875 y Ft(\()s(~)-45 b Fz(v)1115 2845 y Fu(p)1154 2875 y Ft(\))1186 2845 y Fy(\003)1225 2875 y Fm(1)p Fs(\012)o Ft(\()p Fz(p)19 b Fs(\000)f Ft(i0)g Fs(\000)g Fz(L)1736 2887 y Fn(R)1789 2875 y Ft(\))1821 2845 y Fy(\000)p Fx(1)1924 2875 y Ft(\()t(~)-45 b Fz(v)2000 2845 y Fu(p)2038 2875 y Fz(B)23 b Fs(\000)18 b Fz(B)t Fs(\012)o Fm(1)2386 2887 y Fn(R)2443 2875 y Ft(~)-45 b Fz(v)2483 2845 y Fu(p)2522 2875 y Ft(\))871 3069 y Fs(\000)p Ft(i)995 3007 y Fl(P)973 3143 y Fu(p)p Fy(2F)1119 3069 y Ft(\()p Fz(B)t Ft(\()s(~)g Fz(v)1293 3039 y Fu(p)1332 3069 y Ft(\))1364 3039 y Fy(\003)1421 3069 y Fs(\000)18 b Ft(\()s(~)-45 b Fz(v)1579 3039 y Fu(p)1618 3069 y Ft(\))1650 3039 y Fy(\003)1688 3069 y Fz(B)t Fs(\012)p Fm(1)1868 3081 y Fn(R)1922 3069 y Ft(\))14 b Fm(1)p Fs(\012)o Ft(\()p Fz(p)19 b Ft(+)f(i0)g Fs(\000)g Fz(L)2479 3081 y Fn(R)2532 3069 y Ft(\))2564 3039 y Fy(\000)p Fx(1)2657 3069 y Ft(~)-46 b Fz(v)2696 3039 y Fu(p)782 3310 y Ft(=)24 b(i[)p Fz(\001)986 3280 y Fx(R)1039 3310 y Fz(;)14 b(B)t Ft(])871 3481 y(+)p Fz(\031)1022 3419 y Fl(P)1000 3555 y Fu(p)p Fy(2F)1132 3481 y Ft(\()s(~)-45 b Fz(v)1207 3451 y Fu(p)1246 3481 y Ft(\))1278 3451 y Fy(\003)1317 3481 y Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)1701 3493 y Fn(R)1755 3481 y Ft(\))c(\()p Fz(B)t Fs(\012)p Fm(1)2013 3493 y Fn(R)2070 3481 y Ft(~)-46 b Fz(v)2109 3451 y Fu(p)2167 3481 y Fs(\000)21 b Ft(~)-45 b Fz(v)2293 3451 y Fu(p)2331 3481 y Fz(B)t Ft(\))871 3651 y(+)p Fz(\031)1022 3589 y Fl(P)1000 3726 y Fu(p)p Fy(2F)1146 3651 y Ft(\(\()s(~)g Fz(v)1253 3621 y Fu(p)1292 3651 y Ft(\))1324 3621 y Fy(\003)1363 3651 y Fz(B)t Fs(\012)o Fm(1)1542 3663 y Fn(R)1614 3651 y Fs(\000)18 b Fz(B)t Ft(\()s(~)-45 b Fz(v)1839 3621 y Fu(p)1878 3651 y Ft(\))1910 3621 y Fy(\003)1949 3651 y Ft(\))14 b Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)2380 3663 y Fy(R)2441 3651 y Ft(\))s(~)-45 b Fz(v)2516 3621 y Fu(p)782 3940 y Ft(=)24 b(i)966 3877 y Fl(P)908 4015 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)1164 3877 y Fl(P)1126 4014 y Fu(p)p Fy(2F)1253 4023 y Fp(k)1311 3856 y Fy(1)1308 3873 y Fl(R)1311 4016 y Fx(0)1383 3940 y Fm(1)1431 3952 y Fu(k)1471 3940 y Ft(\()p Fz(K)6 b Ft(\)\()p Fz(\012)t Fs(j)p Fz(V)20 b Fm(1)1851 3952 y Fu(k)q Fy(\000)p Fu(p)1978 3940 y Ft(\()p Fz(K)6 b Ft(\))p Fz(\034)2164 3910 y Fu(s)2155 3960 y Fx(0)2200 3940 y Ft(\()p Fz(V)19 b Ft(\))p Fz(\012)t Ft(\))p Fm(1)2479 3952 y Fu(k)2520 3940 y Ft(\()p Fz(K)6 b Ft(\))p Fz(B)t Ft(d)p Fz(s)871 4217 y Fs(\000)p Ft(i)1031 4155 y Fl(P)973 4292 y Fu(k)q Fy(2)p Fx(sp)p Fu(K)1228 4155 y Fl(P)1191 4291 y Fu(p)p Fy(2F)1318 4300 y Fp(k)1418 4134 y Fx(0)1399 4150 y Fl(R)1359 4292 y Fy(\0001)1499 4217 y Fz(B)t Fm(1)1614 4229 y Fu(k)1655 4217 y Ft(\()p Fz(K)g Ft(\)\()p Fz(\012)t Fs(j)p Fz(V)20 b Fm(1)2035 4229 y Fu(k)q Fy(\000)p Fu(p)2161 4217 y Ft(\()p Fz(K)6 b Ft(\))p Fz(\034)2347 4187 y Fu(s)2338 4238 y Fx(0)2383 4217 y Ft(\()p Fz(V)20 b Ft(\))p Fz(\012)t Ft(\))p Fm(1)2663 4229 y Fu(k)2704 4217 y Ft(\()p Fz(K)6 b Ft(\)d)p Fz(s)871 4482 y Ft(+2)p Fz(\031)1167 4420 y Fl(P)1069 4586 y Fu(k)q(;k)1161 4561 y Fn(0)1185 4586 y Fy(2)p Fx(sp)p Fu(K)1054 4686 y(p)p Fy(2F)1181 4695 y Fp(k)1217 4686 y Fy(\\F)1310 4701 y Fp(k)1342 4689 y Fn(0)1428 4398 y Fy(1)1425 4415 y Fl(R)1386 4557 y Fy(\0001)1526 4482 y Fm(1)1574 4494 y Fu(k)1615 4482 y Ft(\()p Fz(K)g Ft(\)\()p Fz(\012)t Fs(j)p Fz(V)19 b Fm(1)1994 4494 y Fu(k)q Fy(\000)p Fu(p)2121 4482 y Ft(\()p Fz(K)6 b Ft(\))p Fz(B)t Fm(1)2377 4494 y Fu(k)2413 4478 y Fn(0)2436 4494 y Fy(\000)p Fu(p)2526 4482 y Ft(\()p Fz(K)g Ft(\))p Fz(\034)2712 4452 y Fu(s)2703 4502 y Fx(0)2748 4482 y Ft(\()p Fz(V)19 b Ft(\))p Fz(\012)t Ft(\))p Fm(1)3027 4494 y Fu(k)3063 4478 y Fn(0)3091 4482 y Ft(\()p Fz(K)6 b Ft(\)d)p Fz(s:)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 523 100 a FB(38)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FC(The)25 b(\002rst)h(e)o(xpression)e(on)h (the)g(right)g(has)h(the)f(standard)f(form)g(of)h(a)h(Lindblad-K)m (ossak)o(o)n(wski)523 382 y(generator)17 b(\(39\).)g(The)h(second)g(e)o (xpression)f(can)h(be)h(used)f(in)h(a)g(characterization)d(of)i(the)h (k)o(ernel)523 482 y(of)24 b Fz(M)9 b FC(.)24 b(In)g(particular)m(,)e (it)j(implies)f(immediately)f(that)h Fm(1)2183 494 y Fy(K)2268 482 y Fs(2)31 b Ft(Ker)o Fz(M)9 b FC(.)24 b(The)g(third)g(e)o (xpression)523 581 y(sho)n(ws)i(the)h(splitting)f(of)g Fz(M)36 b FC(into)26 b(a)h(re)n(v)o(ersible)e(part)h(and)g(an)g(irre)n (v)o(ersible)f(part.)h(The)g(fourth)523 681 y(e)o(xpression)h(uses)i (uses)g(time-dependent)d(quantities)i(and)g(is)h(analoguous)e(to)h (formulas)g(ap-)523 780 y(pearing)19 b(often)g(in)h(the)h(physics)e (literature.)523 1013 y FA(6.5)40 b(Explicit)21 b(f)n(ormulas)f(f)n(or) f(LSO)i(f)n(or)f(the)g(Liouvillean)523 1187 y FC(In)d(this)h (subsection)f(we)g(suppose)g(that)h(Assumption)e(6.2)h(is)h(true)f(and) g(we)h(describe)e(an)i(e)o(xplicit)523 1286 y(formula)h(for)g Ft(i)p Fz(\000)12 b FC(,)20 b(the)h(LSO)f(for)g(the)g(Liouvillean.)648 1386 y(Recall)e(that)g Fz(\031)j FC(denotes)c(the)h(standard)f (representation)e(of)j Fo(M)g FC(and)f Fz(L)2657 1398 y Fy(R)2736 1386 y FC(is)i(the)f(Liouvillean)523 1486 y(of)29 b(the)h(free)f(reserv)n(oir)g(dynamics)g Fz(\034)1622 1498 y Fn(R)1676 1486 y FC(.)h(Let)g Fz(L)1925 1455 y Fx(0)1925 1506 y Fn(R)2008 1486 y FC(denote)f(the)h(Liouvillean)e(of)h (the)h(modular)523 1585 y(dynamics)20 b(for)h(the)g(state)h Fz(!)1331 1597 y Fn(R)1384 1585 y FC(.)g(The)f(f)o(act)g(that)g Fz(!)1921 1597 y Fn(R)1996 1585 y FC(is)i(stationary)d(for)g Fz(\034)2586 1555 y Fu(t)2577 1606 y Fn(R)2653 1585 y FC(implies)h(that)h(the)f(tw)o(o)523 1685 y(Liouvilleans)e(commute:) 1303 1857 y Ft(e)1340 1822 y Fx(i)p Fu(tL)1430 1830 y Fn(R)1488 1857 y Ft(e)1525 1822 y Fx(i)p Fu(sL)1621 1797 y Fq(0)1621 1839 y Fn(R)1701 1857 y Ft(=)k(e)1826 1822 y Fx(i)p Fu(sL)1922 1797 y Fq(0)1922 1839 y Fn(R)1979 1857 y Ft(e)2016 1822 y Fx(i)p Fu(tL)2106 1830 y Fn(R)2164 1857 y Fz(;)55 b(t;)14 b(s)23 b Fs(2)g Fw(R)p Fz(:)648 2029 y FC(The)17 b(follo)n(wing)f(identities)i(follo)n(w)f(from)g(the)h (modular)e(theory)g(and)h(will)i(be)f(useful)f(in)h(our)523 2128 y(e)o(xplicit)i(formulas)f(for)g Fz(\000)12 b FC(:)523 2277 y FA(Pr)o(oposition)19 b(1.)24 b Fk(The)c(following)g(identities)g (ar)m(e)g(true)h(for)f Fz(B)28 b Fs(2)23 b(B)2472 2247 y Fx(2)2508 2277 y Ft(\()p Fs(K)q Ft(\))p Fk(:)1487 2449 y Fz(\031)s Ft(\()p Fz(V)c Ft(\))i Fz(B)t Fs(\012)p Fz(\012)1885 2461 y Fn(R)1964 2449 y Ft(=)h Fz(v)s(B)t(;)1379 2631 y(J)8 b(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(J)29 b(B)t Fs(\012)p Fz(\012)1885 2643 y Fn(R)1964 2631 y Ft(=)22 b Fz(B)t Fs(\012)p Ft(e)2220 2601 y Fu(L)2266 2576 y Fq(0)2266 2618 y Fn(R)2319 2601 y Fu(=)p Fx(2)2391 2631 y Fz(v)s(:)523 2809 y Fk(Mor)m(eo)o(ver)-9 b(,)20 b(if)h Fz(B)1009 2821 y Fx(1)1046 2809 y Fz(;)14 b(B)1146 2821 y Fx(2)1206 2809 y Fs(2)24 b(B)1343 2779 y Fx(2)1379 2809 y Ft(\()p Fs(K)q Ft(\))e Fk(and)d Fz(\010)k Fs(2)h(H)1901 2821 y Fn(R)1955 2809 y Fk(,)c(then)1209 2997 y Ft(\()p Fz(B)1304 3009 y Fx(1)1360 2997 y Fs(\012)e Fz(\010)p Fs(j)p Fz(v)s(B)1627 3009 y Fx(2)1665 2997 y Ft(\))23 b(=)g(\(e)1877 2962 y Fu(L)1923 2937 y Fq(0)1923 2979 y Fn(R)1976 2962 y Fu(=)p Fx(2)2048 2997 y Fz(v)s(B)2154 3009 y Fx(1)2191 2997 y Fs(j)p Fz(B)2277 3009 y Fx(2)2333 2997 y Fs(\012)18 b Fz(J)2462 3009 y Fn(R)2516 2997 y Fz(\010)p Ft(\))p Fz(:)548 b FC(\(59\))523 3169 y FA(Pr)o(oof)o(.)39 b FC(T)-7 b(o)20 b(pro)o(v)o(e)e(the)j(second)e(identity)g(we)i(note)f (that)1545 3341 y Fz(J)29 b(B)t Fs(\012)o Fz(\012)1815 3353 y Fn(R)1892 3341 y Ft(=)23 b Fz(B)2047 3306 y Fy(\003)2085 3341 y Fs(\012)p Fz(\012)2214 3353 y Fn(R)2268 3341 y Fz(;)1270 3523 y(J)8 b(\031)s Ft(\()p Fz(V)19 b Ft(\))p Fz(B)1572 3489 y Fy(\003)1611 3523 y Fs(\012)o Fz(\012)1739 3535 y Fn(R)1816 3523 y Ft(=)k(e)1941 3489 y Fu(L)1987 3464 y Fq(0)1987 3506 y Fn(R)2040 3489 y Fu(=)p Fx(2)2111 3523 y Fz(B)t Fs(\012)p Fz(\031)s Ft(\()p Fz(V)d Ft(\))p Fz(\012)2489 3535 y Fn(R)2543 3523 y Fz(:)648 3666 y FC(T)-7 b(o)26 b(see)h(\(59\),)e(we)i(note)f(that)h(it)g(is)g(enough)e (to)h(assume)h(that)g Fz(\010)35 b Ft(=)f Fz(A)2718 3636 y Fy(0)2741 3666 y Fz(\012)2805 3678 y Fn(R)2859 3666 y FC(,)27 b(where)f Fz(A)3199 3636 y Fy(0)3257 3666 y Fs(2)523 3766 y Fz(\031)s Ft(\()p Fo(M)692 3778 y Fn(R)747 3766 y Ft(\))779 3736 y Fy(0)823 3766 y FC(and)20 b Fz(\031)s Ft(\()p Fo(M)1133 3778 y Fn(R)1187 3766 y Ft(\))1219 3736 y Fy(0)1263 3766 y FC(denotes)g(the)g(commutant)e(of)i Fz(\031)s Ft(\()p Fo(M)2313 3778 y Fn(R)2368 3766 y Ft(\))p FC(.)h(Then)1103 3937 y Ft(\()p Fz(B)1198 3949 y Fx(1)1254 3937 y Fs(\012)d Fz(\010)p Fs(j)p Fz(v)s(B)1521 3949 y Fx(2)1559 3937 y Ft(\))25 b(=)e(\()p Fz(B)1799 3949 y Fx(1)1855 3937 y Fs(\012)18 b Fz(A)2000 3907 y Fy(0)2024 3937 y Fz(\012)2088 3949 y Fn(R)2141 3937 y Fs(j)p Fz(\031)s Ft(\()p Fz(V)i Ft(\))p Fz(B)2409 3949 y Fx(2)2465 3937 y Fs(\012)e Fz(\012)2612 3949 y Fn(R)2666 3937 y Ft(\))1616 4108 y(=)23 b(\()p Fz(\031)s Ft(\()p Fz(V)d Ft(\))p Fz(B)1981 4120 y Fx(1)2037 4108 y Fs(\012)e Fz(\012)2184 4120 y Fn(R)2238 4108 y Fs(j)p Fz(B)2324 4120 y Fx(2)2379 4108 y Fs(\012)g Fz(A)2524 4077 y Fy(0\003)2582 4108 y Fz(\012)2646 4120 y Fn(R)2700 4108 y Ft(\))1616 4290 y(=)23 b(\()p Fz(v)s(B)1842 4302 y Fx(1)1880 4290 y Fs(j)p Fz(B)1966 4302 y Fx(2)2021 4290 y Fs(\012)c Ft(e)2142 4260 y Fu(L)2188 4235 y Fq(0)2188 4277 y Fn(R)2241 4260 y Fu(=)p Fx(2)2312 4290 y Fz(J)2358 4302 y Fn(R)2412 4290 y Fz(A)2474 4260 y Fy(0)2497 4290 y Fz(\012)2561 4302 y Fn(R)2615 4290 y Ft(\))p Fz(:)523 4438 y Ff(2)648 4583 y FC(Note)24 b(that)h(if)h(we)f(compare)e(\(59\))h(with)h(the)g(de\002nition)f(of)h (the)g Fz(?)p FC(-operation)d(\(37\),)h(and)i(if)523 4682 y(we)c(mak)o(e)e(the)i(identi\002cation)p 1421 4616 56 4 v 18 w Fz(\010)j Ft(=)f Fz(J)1634 4694 y Fn(R)1687 4682 y Fz(\010)p FC(,)e(then)f(we)h(see)f(that)h(\(59\))e(can)h(be)g (re)n(written)f(as)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(39)1685 285 y Fz(v)1728 251 y Fu(?)1789 285 y Ft(=)23 b(e)1914 251 y Fu(L)1960 226 y Fq(0)1960 268 y Fn(R)2013 251 y Fu(=)p Fx(2)2084 285 y Fz(v)s(:)648 463 y FC(The)c(LSO)i(for)f(the)g(Liouvillean)e (equals)1142 637 y Ft(i)p Fz(\000)12 b Ft(\()p Fz(B)t Ft(\))25 b(=)e(i)p Fz(\001B)g Fs(\000)18 b Ft(i)p Fz(B)t(\001)1892 606 y Fy(\003)1384 819 y Ft(+2)p Fz(\031)1577 757 y Fl(P)1554 894 y Fu(p)p Fy(2F)1687 819 y Ft(\()s(~)-45 b Fz(v)1762 789 y Fu(p)1801 819 y Ft(\))1833 789 y Fy(\003)1892 819 y Fz(B)t Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)2296 831 y Fn(R)2349 819 y Ft(\)e)2418 789 y Fu(L)2464 764 y Fq(0)2464 806 y Fn(R)2518 789 y Fu(=)p Fx(2)2592 819 y Ft(~)-45 b Fz(v)2632 789 y Fu(p)2671 819 y Fz(:)3174 761 y FC(\(60\))523 1056 y(Note)22 b(that)g(the)f(term)h(on)f(the)h (second)f(line)g(of)h(\(60\))e(is)j(completely)d(positi)n(v)o(e.)h (Therefore,)e(\(60\))523 1156 y(is)f(in)f(the)f(Lindblad-K)m(ossak)o(o) n(wski)e(form.)i(Hence)g Ft(e)2044 1125 y Fx(i)p Fu(t\000)2158 1156 y FC(is)i(a)f(c.p.)f(semigroup.)f(This)i(completes)523 1255 y(the)j(proof)f(of)h(Theorem)e(12.)648 1355 y(Let)i(us)h(split)f Fz(\000)33 b FC(into)20 b(its)h(real)f(and)g(imaginary)e(part:)1218 1569 y Fz(\000)1281 1534 y Fx(R)1356 1569 y Ft(:=)1476 1512 y(1)p 1476 1549 42 4 v 1476 1626 a(2)1528 1569 y(\()p Fz(\000)30 b Ft(+)18 b Fz(\000)1787 1534 y Fy(\003)1825 1569 y Ft(\))p Fz(;)76 b(\000)2019 1534 y Fx(I)2070 1569 y Ft(:=)2202 1512 y(1)p 2191 1549 65 4 v 2191 1626 a(2i)2265 1569 y(\()p Fz(\000)31 b Fs(\000)18 b Fz(\000)2525 1534 y Fy(\003)2563 1569 y Ft(\))p Fz(:)523 1779 y FC(\()p Fz(\000)614 1749 y Fy(\003)672 1779 y FC(is)i(de\002ned)f(using)g(the)h (natural)f(scalar)h(product)d(in)j Fs(B)2215 1749 y Fx(2)2252 1779 y Ft(\()p Fs(K)q Ft(\))p FC(\).)g(Then)f(the)h(real)f(part)h(is)g (gi)n(v)o(en)523 1879 y(by)1592 1978 y Fz(\000)1655 1944 y Fx(R)1707 1978 y Ft(\()p Fz(B)t Ft(\))j(=)g([)p Fz(\001)2041 1944 y Fx(R)2094 1978 y Fz(;)14 b(B)t Ft(])p Fz(:)930 b FC(\(61\))523 2125 y(The)20 b(imaginary)e(part)i(equals)892 2324 y Fz(\000)955 2294 y Fx(I)1007 2324 y Ft(=)25 b Fz(\031)1183 2262 y Fl(P)1161 2399 y Fu(p)p Fy(2F)1293 2324 y Ft(\()s(~)-45 b Fz(v)1368 2294 y Fu(p)1407 2324 y Ft(\))1439 2294 y Fy(\003)1478 2324 y Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)1863 2336 y Fn(R)1916 2324 y Ft(\))1962 2232 y Fl(\020)2012 2324 y Fz(B)t Fs(\012)o Ft(e)2180 2294 y Fu(L)2226 2269 y Fq(0)2226 2311 y Fn(R)2280 2294 y Fu(=)p Fx(2)2354 2324 y Ft(~)-45 b Fz(v)2394 2294 y Fu(p)2451 2324 y Fs(\000)21 b Ft(~)-45 b Fz(v)2577 2294 y Fu(p)2616 2324 y Fz(B)2683 2232 y Fl(\021)1097 2521 y Ft(+)p Fz(\031)1248 2458 y Fl(P)1226 2595 y Fu(p)p Fy(2F)1372 2428 y Fl(\020)1421 2521 y Ft(\()s(~)g Fz(v)1496 2490 y Fu(p)1535 2521 y Ft(\))1567 2490 y Fy(\003)1606 2521 y Fz(B)t Fs(\012)o Ft(e)1774 2490 y Fu(L)1820 2465 y Fq(0)1820 2507 y Fn(R)1874 2490 y Fu(=)p Fx(2)1963 2521 y Fs(\000)18 b Fz(B)t Ft(\()s(~)-45 b Fz(v)2188 2490 y Fu(p)2227 2521 y Ft(\))2259 2490 y Fy(\003)2298 2428 y Fl(\021)2361 2521 y Fm(1)p Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2746 2533 y Fy(R)2807 2521 y Ft(\))s(~)-45 b Fz(v)2882 2490 y Fu(p)2921 2521 y Fz(:)3174 2442 y FC(\(62\))648 2770 y(Another)18 b(useful)i(formula)f(for)g Fz(\000)1628 2740 y Fx(I)1677 2770 y FC(represents)g(it)i(as)g(a)f(quadratic)f(form:)678 2944 y Ft(T)-7 b(r)o Fz(B)826 2956 y Fx(1)864 2944 y Fz(\000)927 2914 y Fx(I)954 2944 y Ft(\()p Fz(B)1049 2956 y Fx(2)1087 2944 y Ft(\))588 3127 y(=)25 b Fz(\031)764 3065 y Fl(P)742 3201 y Fu(p)p Fy(2F)888 3127 y Ft(T)-7 b(r\()s(~)-45 b Fz(v)1049 3097 y Fu(p)1088 3127 y Fz(B)1151 3139 y Fx(1)1206 3127 y Fs(\000)18 b Fz(B)1352 3139 y Fx(1)1390 3127 y Fs(\012)o Ft(e)1491 3097 y Fu(L)1537 3072 y Fq(0)1537 3114 y Fn(R)1590 3097 y Fu(=)p Fx(2)1665 3127 y Ft(~)-45 b Fz(v)1705 3097 y Fu(p)1743 3127 y Ft(\))1775 3097 y Fy(\003)1814 3127 y Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)2199 3139 y Fn(R)2252 3127 y Ft(\)\()s(~)-45 b Fz(v)2359 3097 y Fu(p)2398 3127 y Fz(B)2461 3139 y Fx(2)2517 3127 y Fs(\000)18 b Fz(B)2663 3139 y Fx(2)2700 3127 y Fs(\012)o Ft(e)2801 3097 y Fu(L)2847 3072 y Fq(0)2847 3114 y Fn(R)2901 3097 y Fu(=)p Fx(2)2975 3127 y Ft(~)-45 b Fz(v)3015 3097 y Fu(p)3054 3127 y Ft(\))p Fz(:)3174 3068 y FC(\(63\))648 3364 y(T)-7 b(o)20 b(see)h(\(63\))e(we)h (note)g(the)g(follo)n(wing)f(identities:)979 3550 y Ft(\()s(~)-45 b Fz(v)1054 3520 y Fu(p)1093 3550 y Ft(\))1125 3520 y Fy(\003)1163 3550 y Fm(1)p Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)1548 3562 y Fn(R)1602 3550 y Ft(\))s(~)-45 b Fz(v)1677 3520 y Fu(p)1740 3550 y Ft(=)23 b(T)-7 b(r)1914 3562 y Fy(H)1971 3570 y Fn(R)2028 3550 y Fm(1)p Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2413 3562 y Fn(R)2467 3550 y Ft(\)e)2536 3520 y Fu(L)2582 3495 y Fq(0)2582 3536 y Fn(R)2643 3550 y Ft(~)-46 b Fz(v)2682 3520 y Fu(p)2721 3550 y Ft(\()s(~)h Fz(v)2796 3520 y Fu(p)2835 3550 y Ft(\))2867 3520 y Fy(\003)2905 3550 y Fz(;)752 3732 y Ft(\()s(~)g Fz(v)827 3702 y Fu(p)866 3732 y Ft(\))898 3702 y Fy(\003)937 3732 y Fz(B)t Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)1341 3744 y Fn(R)1394 3732 y Ft(\)e)1463 3702 y Fu(L)1509 3677 y Fq(0)1509 3719 y Fn(R)1563 3702 y Fu(=)p Fx(2)1637 3732 y Ft(~)-45 b Fz(v)1677 3702 y Fu(p)1740 3732 y Ft(=)23 b(T)-7 b(r)1914 3744 y Fy(H)1971 3752 y Fn(R)2028 3732 y Fm(1)p Fs(\012)p Fz(\016)s Ft(\()p Fz(p)18 b Fs(\000)g Fz(L)2413 3744 y Fn(R)2467 3732 y Ft(\)e)2536 3702 y Fu(L)2582 3677 y Fq(0)2582 3719 y Fn(R)2635 3702 y Fu(=)p Fx(2)2710 3732 y Ft(~)-46 b Fz(v)2749 3702 y Fu(p)2809 3732 y Fz(B)t Ft(\()s(~)h Fz(v)2951 3702 y Fu(p)2990 3732 y Ft(\))3022 3702 y Fy(\003)3060 3732 y Fz(;)523 3905 y FC(which)20 b(follo)n(w)f(from)g(\(59\).)648 4005 y(The)j(study)h(of)f(the)h(k)o(ernel)f(of)h Fz(\000)1607 3975 y Fx(I)1658 4005 y FC(is)h(important)e(in)h(applications)e(based)i (on)g(the)g(Spectral)523 4104 y(Fermi)c(Golden)g(Rule.)g(The)g (identity)g(\(63\))f(is)i(v)o(ery)e(con)m(v)o(enient)f(for)h(this)i (purpose.)e(It)h(w)o(as)h(\002rst)523 4204 y(disco)o(v)o(ered)e(in)i (the)h(conte)o(xt)d(of)i(P)o(auli-Fierz)g(systems)g(in)h([DJ2)o(].)648 4304 y(In)e(the)i(thermal)e(case)i(\(63\))e(can)h(be)g(transformed)e (into)729 4478 y Ft(T)-7 b(r)p Fz(B)878 4490 y Fx(1)915 4478 y Fz(\000)978 4448 y Fx(I)1006 4478 y Ft(\()p Fz(B)1101 4490 y Fx(2)1138 4478 y Ft(\))24 b(=)g Fz(\031)1370 4416 y Fl(P)1347 4552 y Fu(p)p Fy(2F)1494 4478 y Ft(T)-7 b(r)o(e)1616 4448 y Fy(\000)p Fu(\014)s(K)1773 4478 y Ft(\()s(~)-45 b Fz(v)1848 4448 y Fu(p)1886 4478 y Fz(B)1949 4490 y Fx(1)1987 4478 y Ft(e)2024 4448 y Fu(\014)s(K=)p Fx(2)2210 4478 y Fs(\000)18 b Fz(B)2356 4490 y Fx(1)2393 4478 y Ft(e)2430 4448 y Fu(\014)s(K=)p Fx(2)2598 4478 y Fs(\012)p Fm(1)2711 4490 y Fn(R)2788 4478 y Ft(~)-45 b Fz(v)2828 4448 y Fu(p)2867 4478 y Ft(\))2899 4448 y Fy(\003)1283 4653 y Fs(\002)p Fm(1)p Fs(\012)o Fz(\016)s Ft(\()p Fz(p)19 b Fs(\000)f Fz(L)1733 4665 y Fn(R)1786 4653 y Ft(\)\()s(~)-45 b Fz(v)1893 4622 y Fu(p)1932 4653 y Fz(B)1995 4665 y Fx(2)2032 4653 y Ft(e)2069 4622 y Fu(\014)s(K=)p Fx(2)2256 4653 y Fs(\000)18 b Fz(B)2402 4665 y Fx(2)2439 4653 y Ft(e)2476 4622 y Fu(\014)s(K=)p Fx(2)2644 4653 y Fs(\012)p Fm(1)2757 4665 y Fn(R)2834 4653 y Ft(~)-45 b Fz(v)2874 4622 y Fu(p)2913 4653 y Ft(\))p Fz(:)3174 4564 y FC(\(64\))p eop end %%Page: 40 40 TeXDict begin 40 39 bop 523 100 a FB(40)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FA(6.6)40 b(Identities)21 b(using)g(the)g(\002ber)o(ed)f(r)o(epr)o(esentation)523 465 y FC(Using)j(the)g(decomposition)e(of)i(the)g(Hilbert)g(space)g Fs(H)2138 477 y Fn(R)2216 465 y FC(into)g(the)g(\002bered)f(inte)o (gral)h(gi)n(v)o(en)f(by)523 565 y(the)j(spectral)g(decomposition)d(of) i Fz(L)1599 577 y Fn(R)1653 565 y FC(,)h(we)g(can)g(re)n(write)f (\(63\))g(in)h(an)g(e)n(v)o(en)f(more)g(con)m(v)o(enient)523 664 y(form.)14 b(T)-7 b(o)16 b(describe)f(the)h(\002bered)f(form)f(of)i (\(63\),)e(we)i(will)g(not)g(stri)n(v)o(e)f(at)h(the)g(greatest)f (generality)-5 b(.)523 764 y(W)e(e)18 b(will)g(mak)o(e)f(the)h(follo)n (wing)d(assumptions)i(\(which)f(are)h(modelled)f(after)h(the)g(v)o (ersion)f(of)h(the)523 863 y(Jak)r(\020)-30 b(si)5 b(\264)-33 b(c-Pillet)21 b(gluing)e(condition)f(considered)g(in)j([DJ2)o(]\):)523 1013 y FA(Assumption)g(6.3)40 b Fk(Ther)m(e)33 b(e)n(xists)g(a)f (Hilbert)h(space)e Fs(G)38 b Fk(and)31 b(a)h(linear)g(isometry)h Fz(U)53 b Ft(:)45 b Fs(G)33 b(\012)523 1112 y Fz(L)580 1082 y Fx(2)617 1112 y Ft(\()p Fw(R)p Ft(\))k Fs(!)f(H)967 1124 y Fn(R)1049 1112 y Fk(suc)o(h)27 b(that)g Ft(Ran)h Fz(v)s(;)14 b Ft(Ran)28 b(e)1856 1082 y Fu(\014)s(L)1943 1057 y Fq(0)1943 1099 y Fn(R)1996 1082 y Fu(=)p Fx(2)2067 1112 y Fz(v)40 b Fs(\032)c(K)25 b(\012)f Ft(Ran)o Fz(U)37 b Fk(and)27 b Fz(U)2886 1082 y Fy(\003)2924 1112 y Fz(L)2981 1124 y Fn(R)3034 1112 y Fz(U)37 b Fk(is)28 b(the)523 1212 y(oper)o(ator)19 b(of)h(the)h(multiplication)d(by)i(the)h (variable)e(in)h Fw(R)p Fk(.)523 1362 y FC(W)-7 b(e)21 b(will)f(identify)e Ft(Ran)p Fz(U)28 b FC(with)20 b Fz(L)1538 1331 y Fx(2)1575 1362 y Ft(\()p Fw(R)p Ft(\))c Fs(\012)f(G)5 b FC(.)20 b(Note)f(that)h Fz(\011)32 b Fs(2)23 b Fz(L)2434 1331 y Fx(2)2471 1362 y Ft(\()p Fw(R)p Ft(\))16 b Fs(\012)f(G)25 b FC(can)20 b(be)f(identi\002ed)523 1461 y(with)h(an)h(almost)f(e)n(v)o (erywhere)d(de\002ned)j(function)e Fw(R)23 b Fs(3)g Fz(p)g Fs(7!)g Fz(\011)9 b Ft(\()p Fz(p)p Ft(\))24 b Fs(2)f(G)j FC(such)20 b(that)1576 1644 y Ft(\()p Fz(L)1665 1656 y Fn(R)1719 1644 y Fz(\011)9 b Ft(\)\()p Fz(p)p Ft(\))24 b(=)e Fz(p\011)9 b Ft(\()p Fz(p)p Ft(\))p Fz(;)523 1826 y FC(\(see)20 b(e.g.)g([DJ2)o(]\).)g(W)-7 b(e)22 b(can)e(\(at)g(least)h (formally\))d(write)j Fz(L)2206 1796 y Fx(0)2206 1847 y Fn(R)2280 1826 y FC(as)g(the)f(direct)g(inte)o(gral:)1489 2009 y Ft(\()p Fz(L)1578 1975 y Fx(0)1578 2030 y Fn(R)1631 2009 y Fz(\011)9 b Ft(\)\()p Fz(p)p Ft(\))24 b(=)f Fz(L)1998 1975 y Fx(0)1998 2030 y Fn(R)2051 2009 y Ft(\()p Fz(p)p Ft(\))p Fz(\011)9 b Ft(\()p Fz(p)p Ft(\))p Fz(;)523 2192 y FC(where)20 b Fz(L)804 2162 y Fx(0)804 2212 y Fn(R)857 2192 y Ft(\()p Fz(p)p Ft(\))h FC(are)f(operators)f(on)h Fs(G)5 b FC(.)648 2291 y(Lik)o(e)n(wise,)16 b Fz(v)27 b Fs(2)c(B)s Ft(\()p Fs(K)q Fz(;)14 b Fs(K)8 b(\012)f(H)1536 2303 y Fn(R)1589 2291 y Ft(\))18 b FC(can)f(be)g(interpreted)e(as)j(an) f(almost)g(e)n(v)o(erywhere)d(de\002ned)523 2391 y(function)19 b Fw(R)k Fs(3)g Fz(p)g Fs(7!)g Fz(v)s Ft(\()p Fz(p)p Ft(\))g Fs(2)h(B)s Ft(\()p Fs(K)q Fz(;)14 b Fs(K)19 b(\012)g(G)5 b Ft(\))21 b FC(such)f(that)1377 2574 y Ft(\()p Fz(L)1466 2586 y Fn(R)1520 2574 y Fz(v)s(\010)p Ft(\)\()p Fz(p)p Ft(\))k(=)f Fz(pv)s Ft(\()p Fz(p)p Ft(\))p Fz(\010;)77 b(\010)23 b Fs(2)h(K)q Fz(:)523 2756 y FA(Assumption)d(6.4)40 b Fw(R)34 b Fs(3)f Fz(p)g Fs(7!)g Fz(v)s Ft(\()p Fz(p)p Ft(\))p Fk(,)27 b Fz(L)1740 2726 y Fx(0)1740 2777 y Fn(R)1793 2756 y Ft(\()p Fz(p)p Ft(\))f Fk(ar)m(e)g(continuous)e(at)i Fz(p)33 b Fs(2)g(F)8 b Fk(,)26 b(so)g(that)f(we)i(can)523 2856 y(de\002ne)19 b(unambiguously)e Fz(v)s Ft(\()p Fz(p)p Ft(\))p Fk(,)k Fz(L)1518 2826 y Fx(0)1518 2876 y Fn(R)1572 2856 y Ft(\()p Fz(p)p Ft(\))g Fk(for)f(those)g(values)g(of)h Fz(p)p Fk(.)648 3022 y FC(Under)e(the)h(abo)o(v)o(e)f(tw)o(o)h (assumptions)f(we)i(can)f(de\002ne)1559 3205 y Fz(w)1620 3170 y Fu(p)1682 3205 y Ft(:=)26 b(~)-46 b Fz(v)1835 3170 y Fu(p)1874 3205 y Ft(\()p Fz(p)p Ft(\))63 b Fz(p)23 b Fs(2)g(F)8 b Fz(:)523 3387 y FC(Then)19 b(we)i(can)f(re)n(write)g (the)g(formula)f(\(63\))g(as)790 3563 y Ft(T)-7 b(r)p Fz(B)939 3575 y Fx(1)976 3563 y Fz(\000)1039 3533 y Fx(I)1067 3563 y Ft(\()p Fz(B)1162 3575 y Fx(2)1199 3563 y Ft(\))701 3746 y(=)24 b Fz(\031)877 3684 y Fl(P)854 3820 y Fu(p)p Fy(2F)1000 3746 y Ft(T)-7 b(r\()p Fz(w)1179 3716 y Fu(p)1218 3746 y Fz(B)1281 3758 y Fx(1)1337 3746 y Fs(\000)18 b Fz(B)1483 3758 y Fx(1)1520 3746 y Fs(\012)p Ft(e)1622 3716 y Fu(L)1668 3691 y Fq(0)1668 3732 y Fn(R)1721 3716 y Fx(\()p Fu(p)p Fx(\))p Fu(=)p Fx(2)1878 3746 y Fz(w)1939 3716 y Fu(p)1978 3746 y Ft(\))2010 3716 y Fy(\003)2049 3746 y Ft(\()p Fz(w)2142 3716 y Fu(p)2181 3746 y Fz(B)2244 3758 y Fx(2)2300 3746 y Fs(\000)g Fz(B)2446 3758 y Fx(2)2483 3746 y Fs(\012)p Ft(e)2585 3716 y Fu(L)2631 3691 y Fq(0)2631 3732 y Fn(R)2684 3716 y Fx(\()p Fu(p)p Fx(\))p Fu(=)p Fx(2)2841 3746 y Fz(w)2902 3716 y Fu(p)2941 3746 y Ft(\))p Fz(:)3174 3687 y FC(\(65\))648 3988 y(\(65\))h(implies)h(immediately) 523 4145 y FA(Theor)o(em)g(17.)k Fk(The)c(k)o(ernel)g(of)h Fz(\000)1515 4115 y Fx(I)1563 4145 y Fk(consists)g(of)f Fz(B)28 b Fs(2)23 b(B)2159 4115 y Fx(2)2195 4145 y Ft(\()p Fs(K)q Ft(\))f Fk(suc)o(h)e(that)1330 4344 y Fz(w)1391 4309 y Fu(p)1448 4344 y Fz(B)28 b Ft(=)22 b Fz(B)t Fs(\012)p Ft(e)1795 4309 y Fu(L)1841 4284 y Fq(0)1841 4326 y Fn(R)1894 4309 y Fx(\()p Fu(p)p Fx(\))p Fu(=)p Fx(2)2072 4344 y Fz(w)2133 4309 y Fu(p)2172 4344 y Fz(;)77 b(p)22 b Fs(2)i(F)8 b Fz(:)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(41)648 282 y FC(Note)26 b(that)h(Theorem)e(17)h(implies)h(that)g(generically)e Ft(Ker)o Fz(\000)2442 252 y Fx(I)2504 282 y Ft(=)35 b Fs(f)p Ft(0)p Fs(g)p FC(.)25 b(Therefore,)g(for)h(a)523 382 y(generic)i(open)h(quantum)e(system,)i(if)h(the)g(Spectral)f(Fermi) g(Golden)f(Rule)i(can)f(be)g(applied,)523 482 y(then)c(the)g (Liouvillean)f Fz(L)1285 494 y Fu(\025)1353 482 y FC(has)i(no)f(point)f (spectrum)g(for)h(small)h(nonzero)d Fz(\025)p FC(.)j(Therefore,)d(for) 523 581 y(the)d(same)h Fz(\025)p FC(,)g(the)f Fz(W)1139 551 y Fy(\003)1177 581 y FC(-dynamical)e(system)j Ft(\()p Fo(M)p Fz(;)14 b(\034)2015 593 y Fu(\025)2059 581 y Ft(\))21 b FC(has)f(no)g(in)m(v)n(ariant)e(normal)h(states.)648 681 y(Identities)29 b(\(63\),)f(\(65\))g(and)i(Theorem)d(17)j(are)f (generalizations)f(of)h(similar)h(statements)523 780 y(from)f([DJ2)o(].)h(In)f([DJ2)o(])h(the)g(reader)f(will)h(\002nd)g (their)f(rigorous)f(application)g(to)i(P)o(auli-Fierz)523 880 y(systems.)648 980 y(If)20 b Fz(!)776 992 y Fn(R)850 980 y FC(is)h(a)g Ft(\()p Fz(\034)1052 992 y Fn(R)1106 980 y Fz(;)14 b(\014)t Ft(\))p FC(-KMS)21 b(state,)g(we)f(can)g (transform)f(\(65\))g(as)i(follo)n(ws:)705 1165 y Ft(T)-7 b(r)p Fz(B)854 1177 y Fx(1)891 1165 y Fz(\000)954 1135 y Fx(I)982 1165 y Ft(\()p Fz(B)1077 1177 y Fx(2)1114 1165 y Ft(\))24 b(=)g Fz(\031)1346 1103 y Fl(P)1323 1240 y Fu(p)p Fy(2F)1470 1165 y Ft(T)-7 b(r)20 b(e)1613 1135 y Fy(\000)p Fu(\014)s(K)1769 1165 y Ft(\()p Fz(w)1862 1135 y Fu(p)1922 1165 y Fz(B)1985 1177 y Fx(1)2023 1165 y Ft(e)2060 1135 y Fu(\014)s(K=)p Fx(2)2246 1165 y Fs(\000)e Fz(B)2392 1177 y Fx(1)2429 1165 y Ft(e)2466 1135 y Fu(\014)s(K=)p Fx(2)2635 1165 y Fs(\012)o Fm(1)2747 1177 y Fn(R)2821 1165 y Fz(w)2882 1135 y Fu(p)2921 1165 y Ft(\))2953 1135 y Fy(\003)1259 1364 y Fs(\002)p Ft(\()p Fz(w)1417 1333 y Fu(p)1477 1364 y Fz(B)1540 1376 y Fx(2)1577 1364 y Ft(e)1614 1333 y Fu(\014)s(K=)p Fx(2)1801 1364 y Fs(\000)g Fz(B)1947 1376 y Fx(2)1984 1364 y Ft(e)2021 1333 y Fu(\014)s(K=)p Fx(2)2189 1364 y Fs(\012)p Fm(1)2302 1376 y Fn(R)2376 1364 y Fz(w)2437 1333 y Fu(p)2476 1364 y Ft(\))p Fz(:)3174 1263 y FC(\(66\))523 1541 y(F)o(ollo)n(wing)h([DJ2)o(],)h(de\002ne)1202 1724 y Fs(N)35 b Ft(:=)23 b Fs(f)p Fz(C)49 b Ft(:)44 b Fz(w)1694 1690 y Fu(p)1752 1724 y Fz(C)29 b Ft(=)23 b Fz(C)6 b Fs(\012)p Fm(1)2106 1736 y Fn(R)2180 1724 y Fz(w)2241 1690 y Fu(p)2280 1724 y Fz(;)55 b(p)23 b Fs(2)h(F)8 b(g)p Fz(:)539 b FC(\(67\))523 1907 y(Repeating)19 b(the)i(ar)o(guments)d(of)i([DJ2)o(])g(we)h(get)523 2064 y FA(Theor)o(em)f(18.)113 b FC(1\))24 b Fs(N)33 b Fk(is)21 b(a)f Fs(\003)p Fk(-alg)o(ebr)o(a)e(in)m(variant)h(wrt)j Ft(e)2253 2034 y Fx(i)p Fu(tK)2379 2064 y Fs(\001)d Ft(e)2458 2034 y Fy(\000)p Fx(i)p Fu(tK)2638 2064 y Fk(and)g(containing)f Fw(C)p Ft(1)p Fk(.)612 2164 y FC(2\))24 b Fk(The)c(k)o(ernel)h(of)f Fz(\000)1227 2134 y Fx(I)1276 2164 y Fk(consists)g(of)h Ft(e)1682 2134 y Fy(\000)p Fu(\014)s(K=)p Fx(2)1902 2164 y Fz(C)27 b Fk(with)20 b Fz(C)30 b Fs(2)23 b(N)12 b Fk(.)648 2322 y FC(Theorem)25 b(18)j(implies)f(that)h(in)f(a)h(thermal)f(case,)h (generically)-5 b(,)25 b Ft(Ker)o Fz(\000)2752 2292 y Fx(I)2817 2322 y Ft(=)36 b Fs(f)p Ft(0)p Fs(g)p FC(.)26 b(There-)523 2421 y(fore,)21 b(if)i(the)g(Spectral)f(Fermi)g(Golden)g (Rule)h(can)f(be)g(applied,)f(for)h(a)h(generic)f(open)f(quantum)523 2521 y(system,)27 b(for)g(small)g(nonzero)e Fz(\025)p FC(,)j(the)f(Liouvillean)f Fz(L)2123 2533 y Fu(\025)2193 2521 y FC(has)i(no)e(point)h(spectrum)f(e)o(xcept)g(for)523 2621 y(a)h(nonde)o(generate)d(eigen)m(v)n(alue)h(at)i(zero.)f (Therefore,)f(for)h(the)h(same)g Fz(\025)p FC(,)h(the)f Fz(W)2901 2591 y Fy(\003)2939 2621 y FC(-dynamical)523 2720 y(system)20 b Ft(\()p Fo(M)p Fz(;)14 b(\034)966 2732 y Fu(\025)1011 2720 y Ft(\))21 b FC(has)f(a)h(unique)d(stationary) i(normal)f(state.)648 2820 y(Again,)27 b(Identity)g(\(66\))g(and)h (Theorem)e(18)i(are)g(generalizations)f(of)h(similar)g(statements)523 2920 y(from)i([DJ2],)h(where)g(the)o(y)f(were)h(used)g(to)h(study)e (the)i(return)e(to)h(equilibrium)e(for)i(thermal)523 3019 y(P)o(auli-Fierz)19 b(systems.)523 3335 y Fv(7)41 b(F)n(ermi)25 b(Golden)g(Rule)g(f)n(or)g(a)f(composite)h(r)n(eser)o(v)o (oir)523 3534 y FC(In)i(this)g(section)g(we)h(describe)e(a)i(small)f (quantum)e(system)j(interacting)e(with)h(se)n(v)o(eral)f(reser)n(-)523 3634 y(v)n(oirs.)c(W)-7 b(e)23 b(will)f(assume)g(that)g(the)g(reserv)n (oirs)f Fs(R)1969 3646 y Fx(1)2006 3634 y Fz(;)14 b(:)g(:)g(:)g(;)g Fs(R)2261 3646 y Fu(n)2329 3634 y FC(do)21 b(not)h(interact)f (directly\227the)o(y)523 3733 y(interact)27 b(with)g(one)f(another)g (only)g(through)e(the)j(small)h(system)f Fs(S)6 b FC(.)28 b(W)-7 b(e)28 b(will)g(compute)d(both)523 3833 y(kinds)19 b(of)h(the)g(LSO)g(for)f(the)h(composite)f(system.)h(W)-7 b(e)21 b(will)g(see)f(that)g(it)h(is)f(equal)g(to)g(the)g(sum)f(of)523 3932 y(the)h(LSO')-5 b(s)21 b(corresponding)c(to)j(the)g(interaction)f (of)h Fs(S)27 b FC(with)21 b(a)f(single)h(reserv)n(oir)e Fs(R)2954 3944 y Fu(i)2982 3932 y FC(.)648 4032 y(Our)25 b(presentation)g(is)i(di)n(vided)e(into)g(3)i(subsections.)e(The)h (\002rst)h(uses)f(the)g(frame)n(w)o(ork)e(of)523 4132 y(Section)c(2,)g(the)g(second\227that)f(of)h(Section)g(5)g(and)g(the)g (third\227that)f(of)h(Section)g(6.)523 4372 y FA(7.1)40 b(LSO)21 b(f)n(or)f(a)g(sum)i(of)d(perturbations)523 4555 y FC(Let)24 b Fs(X)37 b FC(be)24 b(a)h(Banach)f(space.)f(Let)i Fw(P)1609 4525 y Fx(1)1646 4555 y Fz(;)14 b(:)g(:)g(:)f(;)h Fw(P)1881 4525 y Fu(n)1951 4555 y FC(be)24 b(projections)f(of)g(norm)g (1)h(on)g Fs(X)37 b FC(such)24 b(that)523 4655 y Fw(P)574 4625 y Fu(i)601 4655 y Fw(P)652 4625 y Fu(j)722 4655 y Ft(=)34 b Fw(P)872 4625 y Fu(i)900 4655 y Fw(P)951 4625 y Fu(j)985 4655 y FC(.)27 b(Let)g Fw(L)1226 4667 y Fx(0)1291 4655 y FC(be)f(the)h(generator)e(of)h(a)h(group)e(of)h (isometries)g(such)h(that)f Fw(L)3097 4667 y Fx(0)3135 4655 y Fw(P)3186 4625 y Fu(i)3248 4655 y Ft(=)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 523 100 a FB(42)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y Fw(P)574 252 y Fu(i)601 282 y Fw(L)656 294 y Fx(0)694 282 y FC(,)26 b Fz(i)32 b Ft(=)g(1)p Fz(;)14 b(:)g(:)g(:)f(;)h(n)p FC(.)25 b(Let)h Fw(Q)1423 252 y Fu(i)1476 282 y FC(be)f(operators)f(such)h(that)g Ft(Ran)p Fw(P)2443 252 y Fu(i)2503 282 y Fs(\032)32 b Ft(Dom)p Fw(Q)2839 252 y Fu(i)2892 282 y FC(and)25 b Fw(Q)3103 252 y Fu(i)3130 282 y Fw(P)3181 252 y Fu(j)3248 282 y Ft(=)523 382 y Fw(P)574 352 y Fu(j)609 382 y Fw(Q)674 352 y Fu(i)701 382 y FC(,)20 b Fz(i)j Fs(6)p Ft(=)g Fz(j)5 b FC(.)20 b(Set)1077 626 y Fw(Q)i Ft(:=)1314 523 y Fu(n)1275 548 y Fl(X)1277 724 y Fu(j)s Fx(=1)1409 626 y Fw(Q)1474 592 y Fu(j)1508 626 y Fz(;)97 b Fw(P)23 b Ft(:=)1850 523 y Fu(n)1817 548 y Fl(Y)1813 724 y Fu(j)s Fx(=1)1941 626 y Fw(P)1992 592 y Fu(j)2027 626 y Fz(;)97 b Fs(X)2206 638 y Fu(j)2264 626 y Ft(:=)23 b(Ran)2538 548 y Fl(Y)2538 726 y Fu(i)p Fy(6)p Fx(=)p Fu(j)2658 626 y Fw(P)2709 592 y Fu(i)2736 626 y Fz(:)523 906 y FC(Clearly)-5 b(,)15 b Fs(X)858 918 y Fu(j)909 906 y FC(is)i(left)f(in)m(v)n(ariant)e(by)h Fw(L)1568 918 y Fx(0)1606 906 y FC(,)g Fw(P)1693 876 y Fu(j)1728 906 y FC(,)h Fw(Q)1830 876 y Fu(j)1865 906 y FC(.)g(Therefore,)d(these)j(operators)e(can)h(be)h(restricted)523 1005 y(to)k Fs(X)667 1017 y Fu(j)703 1005 y FC(.)g(W)-7 b(e)21 b(set)981 1216 y Fw(L)1036 1228 y Fx(0)p Fu(;j)1147 1216 y Ft(:=)i Fw(L)1313 1228 y Fx(0)1351 1121 y Fl(\014)1351 1170 y(\014)1351 1220 y(\014)1378 1274 y Fy(X)1426 1282 y Fp(j)1461 1216 y Fz(;)97 b Fw(P)1632 1228 y Fu(j)1690 1216 y Ft(:=)22 b Fw(P)1851 1182 y Fu(j)1886 1121 y Fl(\014)1886 1170 y(\014)1886 1220 y(\014)1914 1274 y Fy(X)1962 1282 y Fp(j)2020 1216 y Ft(=)g Fw(P)2158 1121 y Fl(\014)2158 1170 y(\014)2158 1220 y(\014)2186 1274 y Fy(X)2234 1282 y Fp(j)2268 1216 y Fz(;)97 b Fw(Q)2453 1228 y Fu(j)2511 1216 y Ft(:=)23 b Fw(Q)2687 1182 y Fu(j)2721 1121 y Fl(\014)2721 1170 y(\014)2721 1220 y(\014)2749 1274 y Fy(X)2797 1282 y Fp(j)2832 1216 y Fz(:)523 1438 y FC(Clearly)-5 b(,)1218 1560 y Ft(Ran)p Fw(P)23 b Ft(=)g(Ran)o Fw(P)1728 1572 y Fu(j)1846 1560 y Fw(L)1901 1572 y Fx(0)1939 1465 y Fl(\014)1939 1515 y(\014)1939 1564 y(\014)1966 1618 y Fx(Ran)p Fr(P)2147 1560 y Ft(=)g Fw(L)2290 1572 y Fx(0)p Fu(;j)2378 1465 y Fl(\014)2378 1515 y(\014)2378 1564 y(\014)2405 1618 y Fx(Ran)p Fr(P)2559 1626 y Fp(j)2594 1560 y Fz(:)523 1788 y FC(W)-7 b(e)21 b(set)g Fw(E)j Ft(:=)e Fw(L)1009 1800 y Fx(0)1047 1693 y Fl(\014)1047 1743 y(\014)1047 1792 y(\014)1075 1846 y Fx(Ran)p Fr(P)1232 1788 y FC(.)523 1998 y FA(Theor)o(em)e(19.)k Fk(Suppose)18 b(that)i Fw(P)1492 1968 y Fu(j)1527 1998 y Fw(Q)1592 1968 y Fu(j)1626 1998 y Fw(P)1677 1968 y Fu(j)1735 1998 y Ft(=)j(0)p Fk(,)d Fz(j)28 b Ft(=)23 b(1)p Fz(;)14 b(:)g(:)g(:)f(;)h (n)p Fk(.)20 b(Then:)612 2098 y FC(1\))k Fw(PQP)e Ft(=)h(0)p Fk(,)d Fw(P)1117 2110 y Fu(j)1152 2098 y Fw(Q)1217 2110 y Fu(j)1251 2098 y Fw(P)1302 2110 y Fu(j)1360 2098 y Ft(=)j(0)p Fk(,)d Fz(j)28 b Ft(=)22 b(1)p Fz(;)14 b(:)g(:)g(:)f(;)h(n)p Fk(.)612 2197 y FC(2\))24 b Fk(Suppose)e(in)h(addition)f(that)h(the)g (LSO')m(s)g(for)h Ft(\()p Fw(P)2106 2209 y Fu(i)2133 2197 y Fz(;)14 b Fw(L)2225 2209 y Fx(0)p Fu(;i)2306 2197 y Fz(;)g Fw(Q)2408 2209 y Fu(i)2435 2197 y Ft(\))p Fk(,)24 b(denoted)e Fz(M)2880 2209 y Fu(i)2907 2197 y Fk(,)h(e)n(xist.)i(Then) 706 2297 y(the)20 b(LSO)h(for)f Ft(\()p Fw(P)p Fz(;)14 b Fw(L)1289 2309 y Fx(0)1327 2297 y Fz(;)g Fw(Q)p Ft(\))p Fk(,)20 b(denoted)f Fz(M)9 b Fk(,)20 b(e)n(xists)h(as)g(well)g(and)1777 2546 y Fz(M)31 b Ft(=)2016 2442 y Fu(n)1977 2467 y Fl(X)1983 2644 y Fu(i)p Fx(=1)2111 2546 y Fz(M)2192 2558 y Fu(i)2219 2546 y Fz(:)523 2887 y FA(Pr)o(oof)o(.)39 b FC(Set)21 b Fw(J)950 2899 y Fu(j)1007 2887 y Ft(:=)1118 2825 y Fl(Q)1196 2912 y Fu(i)p Fy(6)p Fx(=)p Fu(j)1320 2887 y Fw(P)1371 2857 y Fu(i)1398 2887 y FC(.)648 3000 y(1\))e(It)i(is)g(ob) o(vious)e(that)h Fw(P)1364 2970 y Fu(i)1391 3000 y Fw(Q)1456 2970 y Fu(i)1483 3000 y Fw(P)1534 2970 y Fu(i)1585 3000 y Ft(=)i(0)f FC(implies)f Fw(P)2051 3012 y Fu(i)2078 3000 y Fw(Q)2143 3012 y Fu(i)2171 3000 y Fw(P)2222 3012 y Fu(i)2272 3000 y Ft(=)j(0)p FC(.)648 3100 y(2\))c(W)-7 b(e)22 b(ha)n(v)o(e)1021 3318 y Fz(M)34 b Ft(=)1282 3239 y Fu(n)1259 3256 y Fl(P)1223 3391 y Fu(i;j)s Fx(=1)1450 3256 y Fl(P)1395 3393 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1606 3318 y Fm(1)1654 3330 y Fx(i)p Fu(e)1708 3318 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1892 3288 y Fu(i)1920 3318 y Ft(\(i)p Fz(e)19 b Ft(+)f(0)g Fs(\000)g Fw(L)2314 3330 y Fx(0)2351 3318 y Ft(\))2383 3288 y Fy(\000)p Fx(1)2473 3318 y Fw(Q)2538 3288 y Fu(j)2572 3318 y Fm(1)2620 3330 y Fx(i)p Fu(e)2674 3318 y Ft(\()p Fw(E)p Ft(\))p Fz(;)995 3536 y(M)1076 3548 y Fu(j)1136 3536 y Ft(=)1278 3473 y Fl(P)1223 3610 y Fx(i)p Fu(e)p Fy(2)p Fx(sp)p Fr(E)1434 3536 y Fm(1)1482 3548 y Fx(i)p Fu(e)1536 3536 y Ft(\()p Fw(E)p Ft(\))p Fw(Q)1720 3548 y Fu(j)1756 3536 y Ft(\(i)p Fz(e)g Ft(+)g(0)g Fs(\000)g Fw(L)2149 3548 y Fx(0)p Fu(;j)2237 3536 y Ft(\))2269 3506 y Fy(\000)p Fx(1)2359 3536 y Fw(Q)2424 3548 y Fu(j)2458 3536 y Fm(1)2506 3548 y Fx(i)p Fu(e)2560 3536 y Ft(\()p Fw(E)p Ft(\))p Fz(:)2830 3439 y(:)523 3779 y FC(F)o(or)i Fz(i)j Fs(6)p Ft(=)f Fz(j)5 b FC(,)891 3962 y Fw(PQ)1007 3927 y Fu(i)1034 3962 y Ft(\(i)p Fz(e)18 b Ft(+)g(0)g Fs(\000)g Fw(L)1427 3974 y Fx(0)1465 3962 y Ft(\))1497 3927 y Fy(\000)p Fx(1)1586 3962 y Fw(Q)1651 3927 y Fu(j)1686 3962 y Fw(P)23 b Ft(=)f Fw(PQ)1963 3927 y Fu(i)1990 3962 y Fw(J)2032 3974 y Fu(j)2067 3962 y Ft(\(i)p Fz(e)c Ft(+)g(0)g Fs(\000)g Fw(L)2460 3974 y Fx(0)2498 3962 y Ft(\))2530 3927 y Fy(\000)p Fx(1)2619 3962 y Fw(Q)2684 3927 y Fu(j)2719 3962 y Fw(P)23 b Ft(=)f(0)p Fz(;)523 4144 y FC(since)e Fw(PQ)830 4114 y Fu(i)857 4144 y Fw(J)899 4156 y Fu(j)957 4144 y Ft(=)j Fw(PP)1147 4114 y Fu(i)1174 4144 y Fw(Q)1239 4114 y Fu(i)1266 4144 y Fw(P)1317 4114 y Fu(i)1344 4144 y Fw(J)1386 4156 y Fu(j)1444 4144 y Ft(=)f(0)p FC(.)e(Clearly)-5 b(,)991 4327 y Fw(PQ)1107 4293 y Fu(i)1134 4327 y Ft(\(i)p Fz(e)18 b Ft(+)g(0)g Fs(\000)g Fw(L)1527 4339 y Fx(0)1565 4327 y Ft(\))1597 4293 y Fy(\000)p Fx(1)1686 4327 y Fw(Q)1751 4293 y Fu(i)1779 4327 y Fw(P)k Ft(=)h Fw(PQ)2056 4339 y Fu(i)2083 4327 y Ft(\(i)p Fz(e)18 b Ft(+)g(0)g Fs(\000)g Fw(L)2476 4339 y Fx(0)p Fu(;i)2557 4327 y Ft(\))2589 4293 y Fy(\000)p Fx(1)2679 4327 y Fw(Q)2744 4339 y Fu(i)2771 4327 y Fw(P)p Fz(:)523 4509 y Ff(2)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(43)523 282 y FA(7.2)40 b(Multiple)22 b(r)o(eser)o(v)o(oirs)523 465 y FC(Suppose)i(that)h Ft(\()p Fo(M)1099 477 y Fn(R)1149 486 y Fq(1)1186 465 y Fz(;)14 b(\034)1259 477 y Fn(R)1309 486 y Fq(1)1345 465 y Ft(\))p FC(,.)e(.)g(.)g(,)26 b Ft(\()p Fo(M)1663 477 y Fn(R)1713 486 y Fp(n)1758 465 y Fz(;)14 b(\034)1831 477 y Fn(R)1881 486 y Fp(n)1926 465 y Ft(\))26 b FC(are)f Fz(W)2201 435 y Fy(\003)2239 465 y FC(-dynamical)e(systems)i(with)h Fz(\034)3145 435 y Fu(t)3136 485 y Fn(R)3186 493 y Fp(i)3248 465 y Ft(=)523 573 y(e)560 542 y Fu(t\016)615 550 y Fn(R)665 564 y Fp(i)699 573 y FC(.)20 b(Let)f Fm(1)918 585 y Fn(R)967 594 y Fp(i)1017 573 y FC(denote)f(the)h(identity)f(on)h Fs(H)1825 585 y Fn(R)1874 594 y Fp(i)1905 573 y FC(.)g(Suppose)f(that)h Fo(M)2477 585 y Fy(R)2534 593 y Fp(i)2584 573 y FC(ha)n(v)o(e)f(a)i (standard)d(repre-)523 673 y(sentation)22 b(in)g(Hilbert)h(spaces)f Fs(H)1501 685 y Fn(R)1551 693 y Fp(i)1605 673 y FC(with)g(the)h (modular)e(conjugations)e Fz(J)2692 685 y Fn(R)2742 693 y Fp(i)2773 673 y FC(.)j(Let)h Fz(L)3007 685 y Fn(R)3056 693 y Fp(i)3110 673 y FC(be)f(the)523 772 y(Liouvillean)d(of)h(the)g (dynamics)f Fz(\034)1514 784 y Fn(R)1564 793 y Fp(i)1594 772 y FC(.)648 872 y(Let)31 b Ft(\()p Fs(B)s Ft(\()p Fs(K)q Ft(\))p Fz(;)14 b(\034)1081 884 y Fn(S)1124 872 y Ft(\))32 b FC(describe)e(the)h(small)g(quantum)e(system,)i(with)g Fz(\034)2669 842 y Fu(t)2660 892 y Fn(S)2746 872 y Ft(:=)43 b(e)2914 842 y Fx(i)p Fu(t)p Fx([)p Fu(K;)p Fy(\001)p Fx(])3095 872 y FC(,)31 b(as)h(in)523 972 y(Section)20 b(5.)g(De\002ne)g(the)g(free)g(systems)h Ft(\()p Fo(M)1793 984 y Fu(i)1821 972 y Fz(;)14 b(\034)1894 984 y Fx(0)p Fu(;i)1974 972 y Ft(\))21 b FC(where)1497 1156 y Fo(M)1584 1168 y Fu(i)1637 1156 y Ft(:=)h Fs(B)s Ft(\()p Fs(K)q Ft(\))d Fs(\012)f Fo(M)2122 1168 y Fn(R)2172 1177 y Fp(i)2202 1156 y Fz(;)1514 1327 y Fs(H)1584 1339 y Fu(i)1637 1327 y Ft(:=)k Fs(B)1805 1297 y Fx(2)1842 1327 y Ft(\()p Fs(K)q Ft(\))d Fs(\012)f(H)2142 1339 y Fn(R)2192 1348 y Fp(i)2222 1327 y Fz(;)1538 1497 y(J)1584 1509 y Fu(i)1637 1497 y Ft(:=)k Fz(J)1793 1509 y Fn(S)1855 1497 y Fs(\012)c Fz(J)1984 1509 y Fn(R)2034 1518 y Fp(i)2064 1497 y Fz(;)1495 1669 y(\034)1540 1638 y Fu(t)1531 1690 y Fx(0)p Fu(;i)1637 1669 y Ft(:=)k Fz(\034)1792 1638 y Fu(t)1783 1689 y Fn(S)1845 1669 y Fs(\012)c Fz(\034)1973 1638 y Fu(t)1964 1689 y Fn(R)2014 1697 y Fp(i)2068 1669 y Ft(=)k(e)2192 1638 y Fu(t\016)2247 1647 y Fq(0)p Fp(;\025)2338 1669 y Fz(;)1494 1839 y(\016)1531 1851 y Fx(0)p Fu(;i)1637 1839 y Ft(=)g(i[)p Fz(K)q(;)14 b Fs(\001)p Ft(])19 b(+)f Fz(\016)2064 1851 y Fn(R)2113 1860 y Fp(i)2144 1839 y Fz(;)1475 2010 y(L)1532 2022 y Fx(0)p Fu(;i)1637 2010 y Ft(=)k([)p Fz(K)q(;)14 b Fs(\001)p Ft(])19 b(+)f Fz(L)2061 2022 y Fn(R)2110 2030 y Fp(i)2140 2010 y Fz(:)523 2188 y FC(Let)24 b Fz(\031)705 2200 y Fu(i)758 2188 y FC(be)g(the)g(standard)f(representation)f(of)h Fo(M)1968 2200 y Fu(i)2021 2188 y FC(in)h Fs(H)2180 2200 y Fu(i)2232 2188 y FC(and)g Fz(J)2423 2200 y Fu(i)2475 2188 y FC(the)g(corresponding)c(conju-)523 2288 y(gations.)648 2387 y(Let)26 b Fz(V)833 2399 y Fu(i)895 2387 y Fs(2)34 b Fo(M)1071 2399 y Fu(i)1125 2387 y FC(and)25 b(de\002ne)h(the)g (perturbed)d(systems)k Ft(\()p Fo(M)2382 2399 y Fu(i)2410 2387 y Fz(;)14 b(\034)2483 2399 y Fu(\025;i)2570 2387 y Ft(\))27 b FC(where)e Fz(\034)2903 2357 y Fu(t)2894 2411 y(\025;i)3015 2387 y Ft(:=)33 b(e)3173 2357 y Fu(t\016)3228 2366 y Fp(\025;i)523 2487 y FC(and)1282 2566 y Fz(\016)1319 2578 y Fu(\025;i)1431 2566 y Ft(:=)22 b Fz(\016)1578 2578 y Fx(0)p Fu(;i)1677 2566 y Ft(+)c(i)p Fz(\025)p Ft([)p Fz(V)1902 2578 y Fu(i)1931 2566 y Fz(;)c Fs(\001)p Ft(])p Fz(;)1263 2737 y(L)1320 2749 y Fu(\025;i)1431 2737 y Ft(=)22 b Fz(L)1575 2749 y Fx(0)p Fu(;i)1674 2737 y Ft(+)c Fz(\025)p Ft(\()p Fz(\031)1884 2749 y Fu(i)1913 2737 y Ft(\()p Fz(V)1993 2749 y Fu(i)2021 2737 y Ft(\))h Fs(\000)f Fz(J)2201 2749 y Fu(i)2228 2737 y Fz(\031)2275 2749 y Fu(i)2303 2737 y Ft(\()p Fz(V)2383 2749 y Fu(i)2412 2737 y Ft(\))p Fz(J)2490 2749 y Fu(i)2518 2737 y Ft(\))p Fz(:)648 2882 y FC(Lik)o(e)n(wise,)27 b(consider)f(the)h(composite)f (reserv)n(oir)h Fs(R)h FC(described)e(by)h(the)g Fz(W)2901 2852 y Fy(\003)2939 2882 y FC(-dynamical)523 2981 y(system)20 b Ft(\()p Fo(M)893 2993 y Fn(R)948 2981 y Fz(;)14 b(\034)1021 2993 y Fn(R)1075 2981 y Ft(\))p FC(,)20 b(where)1361 3163 y Fo(M)1448 3175 y Fn(R)1527 3163 y Ft(:=)j Fo(M)1725 3175 y Fn(R)1774 3184 y Fq(1)1829 3163 y Fs(\012)18 b(\001)c(\001)g (\001)19 b(\012)f Fo(M)2198 3175 y Fn(R)2248 3184 y Fp(n)2292 3163 y Fz(;)1378 3334 y Fs(H)1448 3346 y Fn(R)1527 3334 y Ft(:=)23 b Fs(H)1708 3346 y Fn(R)1757 3354 y Fq(1)1812 3334 y Fs(\012)18 b(\001)c(\001)g(\001)19 b(\012)f(H)2164 3346 y Fn(R)2213 3354 y Fp(n)2258 3334 y Fz(;)1402 3504 y(J)1448 3516 y Fn(R)1527 3504 y Ft(:=)23 b Fz(J)1684 3516 y Fn(R)1733 3525 y Fq(1)1788 3504 y Fs(\012)18 b(\001)c(\001)g (\001)19 b(\012)f Fz(J)2116 3516 y Fn(R)2165 3525 y Fp(n)2210 3504 y Fz(;)1412 3675 y(\034)1457 3645 y Fu(t)1448 3696 y Fn(R)1527 3675 y Ft(:=)23 b Fz(\034)1683 3645 y Fu(t)1674 3696 y Fn(R)1724 3704 y Fq(1)1779 3675 y Fs(\012)18 b(\001)c(\001)g (\001)k(\012)g Fz(\034)2105 3645 y Fu(t)2096 3696 y Fn(R)2146 3704 y Fp(n)2214 3675 y Ft(=)k(e)2338 3645 y Fu(t\016)2393 3653 y Fn(R)2452 3675 y Fz(;)1411 3846 y(\016)1448 3858 y Fn(R)1527 3846 y Ft(:=)h Fz(\016)1675 3858 y Fn(R)1724 3867 y Fq(1)1779 3846 y Ft(+)18 b Fs(\001)c(\001)g(\001)k Ft(+)g Fz(\016)2097 3858 y Fn(R)2147 3867 y Fp(n)2192 3846 y Fz(;)1392 4016 y(L)1449 4028 y Fn(R)1527 4016 y Ft(=)23 b Fz(L)1672 4028 y Fn(R)1721 4037 y Fq(1)1776 4016 y Ft(+)18 b Fs(\001)c(\001)g(\001)k Ft(+)g Fz(L)2114 4028 y Fn(R)2163 4037 y Fp(n)2208 4016 y Fz(:)648 4200 y FC(De\002ne)i(the)g(free)g(composite)f(system)h Ft(\()p Fo(M)p Fz(;)14 b(\034)1967 4212 y Fx(0)2005 4200 y Ft(\))21 b FC(where)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 523 100 a FB(44)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)1543 269 y Fo(M)25 b Ft(:=)e Fs(B)s Ft(\()p Fs(K)q Ft(\))18 b Fs(\012)g Fo(M)2140 281 y Fn(R)2194 269 y Fz(;)1559 439 y Fs(H)26 b Ft(:=)d Fs(B)1824 409 y Fx(2)1860 439 y Ft(\()p Fs(K)q Ft(\))d Fs(\012)e(H)2161 451 y Fy(R)2222 439 y Fz(;)1576 610 y(J)33 b Ft(=)23 b Fz(J)1789 622 y Fn(S)1850 610 y Fs(\012)18 b Fz(J)1979 622 y Fn(R)2033 610 y Fz(;)1555 781 y(\034)1600 751 y Fu(t)1591 802 y Fx(0)1655 781 y Ft(:=)23 b Fz(\034)1811 751 y Fu(t)1802 802 y Fn(S)1864 781 y Fs(\012)18 b Fz(\034)1992 751 y Fu(t)1983 802 y Fn(R)2060 781 y Ft(=)k(e)2184 751 y Fu(t\016)2239 759 y Fq(0)2276 781 y Fz(;)1556 952 y(\016)1593 964 y Fx(0)1655 952 y Ft(=)h(i[)p Fz(K)q(;)14 b Fs(\001)p Ft(])k(+)g Fz(\016)2082 964 y Fn(R)2136 952 y Fz(;)1536 1122 y(L)1593 1134 y Fx(0)1655 1122 y Ft(=)23 b([)p Fz(K)q(;)14 b Fs(\001)p Ft(])k(+)g Fz(L)2079 1134 y Fn(R)2132 1122 y Fz(:)523 1301 y FC(Let)i Fz(\031)25 b FC(be)20 b(the)g(standard)f (representation)f(of)i Fo(M)h FC(in)f Fs(H)q FC(.)648 1400 y(Set)25 b Fz(V)51 b Ft(=)31 b Fz(V)1022 1412 y Fx(1)1082 1400 y Ft(+)21 b Fs(\001)14 b(\001)g(\001)22 b Ft(+)g Fz(V)1422 1412 y Fu(n)1467 1400 y FC(.)j(The)g(perturbed)e (composite)g(system)i(describing)f(the)h(small)523 1500 y(system)20 b Fs(S)28 b FC(interacting)19 b(with)h(the)g(composite)f (reserv)n(oir)g Fs(R)i FC(is)h Ft(\()p Fo(M)p Fz(;)14 b(\034)2554 1512 y Fu(\025)2598 1500 y Ft(\))p FC(,)20 b(where)g Fz(\034)2940 1470 y Fu(t)2931 1524 y(\025)2998 1500 y Ft(:=)j(e)3146 1470 y Fu(t\016)3201 1479 y Fp(\025)3244 1500 y FC(,)1356 1685 y Fz(\016)1393 1697 y Fu(\025)1461 1685 y Ft(:=)g Fz(\016)1609 1697 y Fx(0)1665 1685 y Ft(+)18 b(i)p Fz(\025)p Ft([)p Fz(V)5 b(;)14 b Fs(\001)p Ft(])p Fz(;)1336 1855 y(L)1393 1867 y Fu(\025)1461 1855 y Ft(:=)23 b Fz(L)1629 1867 y Fx(0)1684 1855 y Ft(+)18 b Fz(\025)p Ft(\()p Fz(\031)s Ft(\()p Fz(V)i Ft(\))f Fs(\000)f Fz(J)8 b(\031)s Ft(\()p Fz(V)20 b Ft(\))p Fz(J)8 b Ft(\))p Fz(:)2489 1771 y(:)523 2092 y FA(7.3)40 b(LSO)21 b(f)n(or)f(the)g(r)o(educed)h (dynamics)f(in)h(the)f(case)h(of)f(a)g(composite)g(r)o(eser)o(v)o(oir) 523 2274 y FC(Suppose)h(that)i(the)g(reserv)n(oir)e(dynamics)g Fz(\034)1793 2286 y Fn(R)1843 2295 y Fp(i)1897 2274 y FC(ha)n(v)o(e)h(stationary)f(states)j Fz(!)2684 2286 y Fn(R)2733 2295 y Fp(i)2764 2274 y FC(.)e(W)-7 b(e)24 b(introduce)d(a)523 2374 y(projection)d(of)i(norm)f(one)h(in)g Fo(M)p FC(,)h(denoted)e Fw(P)1856 2344 y Fu(i)1883 2374 y FC(,)h(such)g(that)552 2557 y Fw(P)603 2522 y Fu(i)631 2557 y Ft(\()p Fz(B)j Fs(\012)18 b Fz(A)894 2569 y Fx(1)931 2557 y Fs(\012)p Fz(;)c Fs(\001)g(\001)g(\001)k(\012)g Fz(A)1293 2569 y Fu(i)1339 2557 y Fs(\012)g(\001)c(\001)g(\001)19 b(\012)f Fz(A)1683 2569 y Fu(n)1728 2557 y Ft(\))23 b(=)g Fz(!)1923 2569 y Fn(R)1972 2577 y Fp(i)2003 2557 y Ft(\()p Fz(A)2097 2569 y Fu(i)2125 2557 y Ft(\))p Fz(B)g Fs(\012)18 b Fz(A)2388 2569 y Fx(1)2444 2557 y Fs(\012)g(\001)c(\001)g(\001)k (\012)g Fm(1)2773 2569 y Fn(R)2823 2577 y Fp(i)2871 2557 y Fs(\012)g(\001)c(\001)g(\001)19 b(\012)f Fz(A)3215 2569 y Fu(n)3260 2557 y Fz(:)523 2739 y FC(Set)28 b Fw(P)37 b Ft(:=)870 2677 y Fl(Q)948 2697 y Fu(n)948 2764 y(i)p Fx(=1)1074 2739 y Fw(P)1125 2709 y Fu(i)1152 2739 y FC(.)28 b(The)g(projection)d Fw(P)1774 2709 y Fu(i)1830 2739 y FC(restricted)i(to)h Fo(M)2347 2751 y Fu(i)2403 2739 y FC(\(which)f(can)h(be)f(vie)n(wed)g(as)i(a)523 2839 y(subalgebra)18 b(of)i Fo(M)p FC(\))h(is)g(denoted)e(by)g Fw(P)1643 2851 y Fu(i)1671 2839 y FC(.)h(Explicitly)-5 b(,)1359 3022 y Fw(P)1410 3034 y Fu(i)1437 3022 y Ft(\()p Fz(B)23 b Fs(\012)18 b Fz(A)1700 3034 y Fu(i)1728 3022 y Ft(\))23 b(=)g Fz(!)1923 3034 y Fn(R)1972 3042 y Fp(i)2003 3022 y Ft(\()p Fz(A)2097 3034 y Fu(i)2125 3022 y Ft(\))p Fz(B)g Fs(\012)18 b Fm(1)2374 3034 y Fn(R)2423 3042 y Fp(i)2454 3022 y Fz(:)648 3204 y FC(Assume)i(that)g Fz(!)1133 3216 y Fn(R)1182 3225 y Fp(i)1213 3204 y Ft(\()p Fz(V)1293 3216 y Fu(i)1321 3204 y Ft(\))k(=)e(0)e FC(for)g Fz(i)j Ft(=)f(1)p Fz(;)14 b(:)g(:)g(:)f(;)h(n)p FC(.)648 3304 y(Note)k(that)g(we)h(can)f(apply)f(the)h(formalism)f(of)h(Subsection)f (7.1,)h(where)f(the)h(Banach)g(space)523 3403 y(is)i Fs(X)33 b FC(is)20 b Fo(M)p FC(,)g(the)f(projections)f Fw(P)1454 3373 y Fu(i)1501 3403 y FC(are)h Fw(P)1673 3373 y Fu(i)1701 3403 y FC(,)h(the)f(generator)e(of)i(an)h(isometric)f (dynamics)f Fw(L)3106 3415 y Fx(0)3163 3403 y FC(is)i Fz(\016)3275 3415 y Fx(0)523 3503 y FC(and)f(the)g(perturbations)d Fw(Q)1306 3473 y Fu(i)1353 3503 y FC(are)j Ft(i[)p Fz(V)1568 3515 y Fu(i)1596 3503 y Fz(;)14 b Fs(\001)p Ft(])p FC(.)20 b(Clearly)-5 b(,)18 b Fs(X)2058 3515 y Fu(i)2106 3503 y FC(can)h(be)g(identi\002ed)f(with)h Fo(M)2926 3515 y Fu(i)2974 3503 y FC(and)f Ft(Ran)p Fw(P)523 3603 y FC(with)i Fs(B)s Ft(\()p Fs(K)q Ft(\))p FC(.)648 3702 y(W)-7 b(e)16 b(obtain)e(the)h(LSO)h(for)f Ft(\()p Fw(P)p Fz(;)f(\016)1555 3714 y Fx(0)1592 3702 y Fz(;)g Ft(i[)p Fz(V)5 b(;)14 b Fs(\001)p Ft(]\))p FC(,)i(denoted)d Fz(M)c FC(,)15 b(and)g(the)g(LSO')-5 b(s)16 b(for)f Ft(\()p Fw(P)2961 3714 y Fu(i)2988 3702 y Fz(;)f(\016)3062 3714 y Fx(0)p Fu(;i)3143 3702 y Fz(;)g Ft(i[)p Fz(V)3274 3714 y Fu(i)3302 3702 y Fz(;)g Fs(\001)p Ft(]\))p FC(,)523 3802 y(denoted)19 b Fz(M)888 3814 y Fu(i)915 3802 y FC(.)h(By)h (Theorem)d(19,)i(we)h(ha)n(v)o(e)1685 4048 y Fz(M)32 b Ft(=)1925 3945 y Fu(n)1885 3969 y Fl(X)1892 4146 y Fu(i)p Fx(=1)2019 4048 y Fz(M)2100 4060 y Fu(i)2127 4048 y Fz(;)523 4361 y FA(7.4)40 b(LSO)21 b(f)n(or)f(the)g(Lio)o(villean)g (in)h(the)g(case)f(of)g(a)g(composite)g(r)o(eser)o(v)o(oir)523 4544 y FC(Let)h Fz(\012)719 4556 y Fn(R)768 4565 y Fp(i)820 4544 y FC(be)f(the)h(standard)e(v)o(ector)g(representati)n(v)o(e)f(of)i Fz(!)2196 4556 y Fn(R)2246 4565 y Fp(i)2276 4544 y FC(.)h(W)-7 b(e)21 b(de\002ne)f(the)h(orthogonal)c(pro-)523 4643 y(jection)j(in)g Fs(B)s Ft(\()p Fs(H)q Ft(\))p eop end %%Page: 45 45 TeXDict begin 45 44 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(45)985 282 y Fz(P)1050 248 y Fu(i)1101 282 y Ft(:=)22 b Fm(1)1259 297 y Fy(B)1304 281 y Fq(2)1337 297 y Fx(\()p Fy(K)p Fx(\))1462 282 y Fs(\012)c Fm(1)1593 294 y Fy(R)1650 302 y Fq(1)1705 282 y Fs(\012)g(\001)c(\001)g(\001)k(\012)g(j)p Fz(\012)2073 294 y Fn(R)2123 303 y Fp(i)2153 282 y Ft(\)\()p Fz(\012)2281 294 y Fn(R)2332 303 y Fp(i)2362 282 y Fs(j)h(\012)f(\001)c(\001)g(\001) k(\012)g Fm(1)2733 294 y Fn(R)2782 303 y Fp(n)2827 282 y Fz(:)523 456 y FC(The)i(projection)e Fz(P)1095 426 y Fu(i)1144 456 y FC(restricted)h(to)i Fs(H)1629 468 y Fu(i)1677 456 y FC(is)g(denoted)e(by)h Fz(P)2194 468 y Fu(i)2242 456 y FC(and)g(equals)1447 630 y Fz(P)1500 642 y Fu(i)1551 630 y Ft(=)j(1)1681 645 y Fy(B)1726 628 y Fq(2)1758 645 y Fx(\()p Fy(K)p Fx(\))1883 630 y Fs(\012)18 b(j)p Fz(\012)2053 642 y Fn(R)2103 650 y Fp(i)2133 630 y Ft(\)\()p Fz(\012)2261 642 y Fn(R)2312 650 y Fp(i)2342 630 y Fs(j)p Fz(:)523 803 y FC(Set)j Fz(P)35 b Ft(=)825 741 y Fl(Q)904 762 y Fu(n)904 828 y(i)p Fx(=1)1029 803 y Fz(P)1094 773 y Fu(i)1122 803 y FC(.)648 903 y(W)-7 b(e)24 b(can)f(apply)g(the)g(formalism)g(of)g(Subsection)f(7.1,)h (where)g(the)g(Banach)g(space)g(is)i Fs(X)36 b FC(is)523 1003 y Fs(H)q FC(,)24 b(the)f(projections)f Fw(P)1208 972 y Fu(i)1260 1003 y FC(are)h Fz(P)1450 972 y Fu(i)1477 1003 y FC(,)h(the)g(generator)d(of)i(an)h(isometric)f(dynamics)f Fw(L)2911 1015 y Fx(0)2972 1003 y FC(is)j Ft(i)p Fz(L)3132 1015 y Fx(0)3193 1003 y FC(and)523 1102 y(the)h(perturbations)e Fw(Q)1181 1072 y Fu(i)1235 1102 y FC(are)i Ft(i\()p Fz(V)1466 1114 y Fu(i)1517 1102 y Fs(\000)c Fz(J)1650 1114 y Fu(i)1678 1102 y Fz(V)1726 1114 y Fu(i)1754 1102 y Fz(J)1800 1114 y Fu(i)1828 1102 y Ft(\))p FC(.)k(Clearly)-5 b(,)26 b Fs(X)2253 1114 y Fu(i)2307 1102 y FC(can)g(be)g(identi\002ed)f(with)h Fs(H)3138 1114 y Fu(i)3193 1102 y FC(and)523 1202 y Ft(Ran)p Fz(P)32 b FC(with)21 b Fs(B)984 1172 y Fx(2)1020 1202 y Ft(\()p Fs(K)q Ft(\))h FC(\(which)d(as)i(a)g(v)o(ector)e(space)h (coincides)f(with)h Fs(B)s Ft(\()p Fs(K)q Ft(\))p FC(\).)648 1301 y(W)-7 b(e)34 b(obtain)e(the)h(LSO)g(for)f Ft(\()p Fz(P)r(;)14 b Ft(i)p Fz(L)1690 1313 y Fx(0)1728 1301 y Fz(;)g Ft(i\()p Fz(V)47 b Fs(\000)27 b Fz(J)8 b(V)19 b(J)8 b Ft(\)\))p FC(,)34 b(denoted)d Ft(i)p Fz(\000)12 b FC(,)33 b(and)g(the)f(LSO)i(for)523 1401 y Ft(\()p Fz(P)608 1413 y Fu(i)636 1401 y Fz(;)14 b Ft(i)p Fz(L)753 1413 y Fx(0)p Fu(;i)833 1401 y Fz(;)g Ft(i\()p Fz(V)973 1413 y Fu(i)1020 1401 y Fs(\000)k Fz(J)1149 1413 y Fu(i)1177 1401 y Fz(V)1225 1413 y Fu(i)1253 1401 y Fz(J)1299 1413 y Fu(i)1326 1401 y Ft(\)\))p FC(,)j(denoted)e Ft(i)p Fz(\000)1790 1413 y Fu(i)1818 1401 y FC(.)h(By)h(Theorem)d(19,)i(we)h (ha)n(v)o(e)1690 1644 y Ft(i)p Fz(\000)35 b Ft(=)1926 1540 y Fu(n)1887 1565 y Fl(X)1893 1742 y Fu(i)p Fx(=1)2020 1644 y Ft(i)p Fz(\000)2094 1656 y Fu(i)2122 1644 y Fz(:)648 1889 y FC(The)19 b(follo)n(wing)g(theorem)g(follo)n(ws)h(from)f(ob)o (vious)f(properties)h(of)h(ne)o(gati)n(v)o(e)e(operators:)523 2040 y FA(Theor)o(em)i(20.)k Fk(Suppose)29 b(that)h(for)h(some)g Fz(i)43 b Fs(6)p Ft(=)f Fz(j)5 b Fk(,)31 b Ft(dim)14 b(Ker)o Fz(\000)2411 2010 y Fx(I)2399 2062 y Fu(i)2481 2040 y Ft(=)43 b(dim)14 b(Ker)o Fz(\000)2938 2010 y Fx(I)2926 2062 y Fu(j)3008 2040 y Ft(=)42 b(1)31 b Fk(and)523 2151 y Ft(Ker)o Fz(\000)720 2121 y Fx(I)708 2173 y Fu(i)771 2151 y Fs(6)p Ft(=)22 b(Ker)o Fz(\000)1055 2121 y Fx(I)1043 2173 y Fu(j)1083 2151 y Fk(.)f(Then)f Ft(Ker)o Fz(\000)34 b Ft(=)23 b Fs(f)p Ft(0)p Fs(g)p Fk(.)523 2316 y FA(Cor)o(ollary)18 b(1.)24 b Fk(Suppose)19 b(that)i(for)g(some)g Fz(i)i Fs(6)p Ft(=)h Fz(j)5 b Fk(,)21 b(the)g(states)g Fz(!)2341 2328 y Fn(R)2390 2337 y Fp(i)2442 2316 y Fk(and)f Fz(!)2640 2328 y Fn(R)2689 2337 y Fp(j)2745 2316 y Fk(ar)m(e)h Ft(\()p Fz(\034)2942 2328 y Fn(R)2993 2337 y Fp(i)3023 2316 y Fz(;)14 b(\014)3107 2328 y Fu(i)3135 2316 y Ft(\))21 b Fk(and)523 2416 y Ft(\()p Fz(\034)591 2428 y Fn(R)641 2437 y Fp(j)676 2416 y Fz(;)14 b(\014)760 2428 y Fu(j)795 2416 y Ft(\))p Fk(-KMS.)24 b(Let)g Fs(N)1264 2428 y Fu(i)1315 2416 y Fk(and)e Fs(N)1531 2428 y Fu(j)1590 2416 y Fk(be)h(the)h(corr)m (esponding)d Fs(\003)p Fk(-alg)o(ebr)o(as)g(de\002ned)h(as)h(in)h (\(67\).)523 2516 y(Suppose)19 b(that)g Fz(\014)1017 2528 y Fu(i)1068 2516 y Fs(6)p Ft(=)k Fz(\014)1203 2528 y Fu(j)1258 2516 y Fk(and)d Fs(N)1484 2486 y Fy(0)1472 2537 y Fu(i)1530 2516 y Ft(=)j Fs(N)1698 2486 y Fy(0)1686 2537 y Fu(j)1745 2516 y Ft(=)f Fw(C)p Ft(1)p Fk(.)e(Then)g Ft(Ker)o Fz(\000)35 b Ft(=)22 b Fs(f)p Ft(0)p Fs(g)p Fz(:)648 2681 y FC(If)g(we)i(can)f(apply)f(the)h(Spectral)f(Fermi)h (Golden)f(Rule,)h(then)g(under)e(the)i(assumptions)f(of)523 2781 y(1,)f(for)g(suf)n(\002ciently)f(small)i(nonzero)d Fz(\025)p FC(,)j Fz(L)1758 2793 y Fu(\025)1822 2781 y FC(has)g(no)f(point)f(spectrum.)g(Consequently)-5 b(,)19 b(for)h(the)523 2880 y(same)g Fz(\025)p FC(,)h(the)g(system)f Ft(\()p Fo(M)1297 2892 y Fu(\025)1341 2880 y Fz(;)14 b(\034)1414 2892 y Fu(\025)1458 2880 y Ft(\))p FC(,)20 b(has)h(no)f(in)m(v)n(ariant)e(normal)h(states.)523 3189 y Fv(A)41 b(A)n(ppendix)26 b(\226)f(one-parameter)h(semigr)n(oups)523 3381 y FC(In)32 b(this)h(section)g(we)g(w)o(ould)f(lik)o(e)h(to)f (discuss)h(some)g(concepts)e(related)h(to)h(one-parameter)523 3480 y(semigroups)e(of)h(operators)f(in)i(Banach)f(spaces,)h(which)f (are)g(used)h(in)f(our)g(lectures.)g(Ev)o(en)523 3580 y(though)22 b(the)i(material)f(that)g(we)h(present)f(is)i(quite)e (standard,)f(we)i(could)f(not)g(\002nd)h(a)g(reference)523 3679 y(that)j(presents)f(all)h(of)g(it)g(in)g(a)g(con)m(v)o(enient)d(w) o(ay)-5 b(.)26 b(Most)h(of)f(it)h(can)g(be)f(found)f(in)i([BR1].)f (Less)523 3779 y(pedantic)19 b(readers)h(may)f(skip)h(this)h(appendix)d (altogether)-5 b(.)648 3879 y(Let)31 b Fs(X)44 b FC(be)30 b(a)i(Banach)e(space.)h(Recall)g(that)g Ft([0)p Fz(;)14 b Fs(1)p Ft([)p Fs(3)43 b Fz(t)g Fs(7!)g Fz(U)9 b Ft(\()p Fz(t)p Ft(\))43 b Fs(2)g Fz(B)t Ft(\()p Fs(X)12 b Ft(\))33 b FC(is)e(called)523 3978 y(a)f(1-parameter)d(semigroup)h(if)n(f)i Fz(U)9 b Ft(\(0\))40 b(=)g(1)30 b FC(and)f Fz(U)9 b Ft(\()p Fz(t)2176 3990 y Fx(1)2213 3978 y Ft(\))p Fz(U)g Ft(\()p Fz(t)2373 3990 y Fx(2)2411 3978 y Ft(\))40 b(=)g Fz(U)9 b Ft(\()p Fz(t)2716 3990 y Fx(1)2779 3978 y Ft(+)25 b Fz(t)2899 3990 y Fx(2)2936 3978 y Ft(\))p FC(.)31 b(If)e Ft([0)p Fz(;)14 b Fs(1)p Ft([)523 4078 y FC(is)30 b(replaced)e(with)h Fw(R)p FC(,)g(then)f(we)i(speak)e(about)g(a)h(one-parameter)d(group)h (instead)i(of)g(a)g(one-)523 4178 y(parameter)19 b(semigroup.)648 4277 y(W)-7 b(e)24 b(say)g(that)g Fz(U)9 b Ft(\()p Fz(t)p Ft(\))24 b FC(is)h(a)f(strogly)f(continuous)e(semigroup)h(\(or)h(a)h Fz(C)2648 4289 y Fx(0)2685 4277 y FC(-semigroup\))d(if)n(f)j(for)523 4377 y(an)o(y)d Fz(\010)k Fs(2)h(X)12 b FC(,)22 b Fz(t)k Fs(7!)f Fz(U)9 b Ft(\()p Fz(t)p Ft(\))p Fz(\010)23 b FC(is)f(continuous.)d(Ev)o(ery)h Fz(C)2103 4389 y Fx(0)2141 4377 y FC(-semigroup)f(possesses)j(its)h(generator)m(,)523 4476 y(that)d(is)h(the)g(operator)d Fz(A)j FC(de\002ned)e(as)i(follo)n (ws:)1041 4650 y Fz(\010)i Fs(2)g Ft(Dom)q Fz(A)44 b Fs(,)k Ft(lim)1604 4704 y Fu(t)p Fy(&)p Fx(0)1743 4650 y Fz(t)1773 4616 y Fy(\000)p Fx(1)1862 4650 y Ft(\()p Fz(U)9 b Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)f Fm(1)p Ft(\))p Fz(\010)23 b Ft(=:)g Fz(A\010)42 b FC(e)o(xists)p Fz(:)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 523 100 a FB(46)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y FC(The)32 b(generator)d(is)k(al)o(w)o(ays) g(closed)e(and)g(densely)g(de\002ned)g(and)g(uniquely)f(determines)h (the)523 382 y(semigroup.)18 b(W)-7 b(e)21 b(write)g Fz(U)9 b Ft(\()p Fz(t)p Ft(\))23 b(=)g(e)1544 352 y Fu(tA)1623 382 y FC(.)648 482 y(Recall)17 b(also)h(the)f(follo)n(wing)e(well)j (kno)n(wn)e(characterization)f(of)h(contracti)n(v)o(e)g(semigroups:)523 639 y FA(Theor)o(em)k(21.)k Fk(The)c(following)g(conditions)e(ar)m(e)j (equivalent:)612 739 y FC(1\))j Ft(e)743 709 y Fu(tA)843 739 y Fk(is)d(contr)o(active)e(for)i(all)f Fz(t)j Fs(\025)g Ft(0)p Fk(.)612 839 y FC(2\))h Fz(A)17 b Fk(is)h(densely)e(de\002ned,)e Ft(sp)p Fz(A)23 b Fs(\032)g(f)p Fz(z)j Fs(2)d Fw(C)40 b Ft(:)g(Re)p Fz(z)26 b Fs(\024)d Ft(0)p Fs(g)16 b Fk(and)f Fs(k)p Ft(\()p Fz(z)8 b Fs(\000)t Fz(A)p Ft(\))2777 808 y Fy(\000)p Fx(1)2866 839 y Fs(k)23 b(\024)g Ft(\(Re)p Fz(z)t Ft(\))3224 808 y Fy(\000)p Fx(1)706 938 y Fk(for)e Ft(Re)p Fz(z)26 b(>)d Ft(0)p Fk(.)612 1038 y FC(3\))104 b(\(i\))24 b Fz(A)d Fk(is)g(densely)f(de\002ned)f(and)g(for)h(some)h Fz(z)2072 1050 y Fx(+)2147 1038 y Fk(with)g Ft(Re)p Fz(z)2448 1050 y Fx(+)2526 1038 y Fz(>)i Ft(0)p Fk(,)d Fz(z)2736 1050 y Fx(+)2813 1038 y Fs(62)k Ft(sp)p Fz(A)p Fk(,)763 1137 y FC(\(ii\))g Fz(A)30 b Fk(is)g(dissipative)o(,)f(that)g(is)h(for) f(any)g Fz(\010)39 b Fs(2)h Ft(Dom)p Fz(A)30 b Fk(ther)m(e)f(e)n(xists) i Fz(\030)44 b Fs(2)39 b(X)3101 1107 y Fy(\003)3170 1137 y Fk(with)889 1237 y Ft(\()p Fz(\030)t Fs(j)p Fz(\010)p Ft(\))24 b(=)f Fs(k)p Fz(\010)p Fs(k)d Fk(and)f Ft(\()p Fz(\030)t Fs(j)p Fz(A\010)p Ft(\))25 b Fs(\024)d Ft(0)p Fk(.)523 1337 y(Mor)m(eo)o(ver)-9 b(,)20 b(if)h Fz(A)g Fk(is)g(bounded,)c(then)j(we)h(can)f(omit)g FC(\(i\))g Fk(in)g FC(3\))p Fk(.)648 1494 y FC(There)f(e)o(xists)i(an)f(ob)o (vious)e(corollary)h(of)g(the)i(abo)o(v)o(e)d(theorem)h(for)h(groups)e (of)i(isometries:)523 1669 y FA(Theor)o(em)g(22.)k Fk(The)c(following)g (conditions)e(ar)m(e)j(equivalent:)612 1768 y FC(1\))j Ft(e)743 1738 y Fu(tA)843 1768 y Fk(is)d(isometric)g(for)f(all)h Fz(t)i Fs(2)g Fw(R)p Fk(.)612 1868 y FC(2\))h Fz(A)d Fk(is)g(densely)f(de\002ned,)e Ft(sp)p Fz(A)23 b Fs(\032)g Ft(i)p Fw(R)e Fk(and)e Fs(k)p Ft(\()p Fz(z)j Fs(\000)c Fz(A)p Ft(\))2229 1838 y Fy(\000)p Fx(1)2318 1868 y Fs(k)23 b(\024)g(j)p Ft(Re)p Fz(z)t Fs(j)2658 1838 y Fy(\000)p Fx(1)2767 1868 y Fk(for)e Ft(Re)p Fz(z)26 b Fs(6)p Ft(=)c(0)p Fk(.)612 1968 y FC(3\))104 b(\(i\))24 b Fz(A)d Fk(is)g(densely)f (de\002ned)f(and)g(for)h(some)h Fz(z)2072 1980 y Fy(\006)2148 1968 y Fk(with)g Fs(\006)p Ft(Re)p Fz(z)2514 1980 y Fy(\006)2592 1968 y Fz(>)i Ft(0)p Fk(,)d Fz(z)2802 1980 y Fy(\006)2881 1968 y Fs(62)j Ft(sp)p Fz(A)p Fk(,)763 2067 y FC(\(ii\))h Fz(A)j Fk(is)f(conservative)o(,)f(that)g(is)i(for)f(any)f Fz(\010)33 b Fs(2)g Ft(Dom)p Fz(A)27 b Fk(ther)m(e)e(e)n(xists)i Fz(\030)38 b Fs(2)33 b(X)3105 2037 y Fy(\003)3170 2067 y Fk(with)889 2167 y Ft(\()p Fz(\030)t Fs(j)p Fz(\010)p Ft(\))24 b(=)f Fs(k)p Fz(\010)p Fs(k)d Fk(and)f Ft(Re\()p Fz(\030)t Fs(j)p Fz(A\010)p Ft(\))25 b(=)d(0)p Fk(.)523 2267 y(Mor)l(o)o(ver)-9 b(,)20 b(if)h Fz(A)g Fk(is)g(bounded,)d(then)h (we)i(can)f(omit)g FC(\(i\))h Fk(in)f FC(\(3\))p Fk(.)648 2441 y FC(Not)i(all)h(semigroups)d(considered)h(in)h(our)g(lectures)g (are)g Fz(C)2374 2453 y Fx(0)2411 2441 y FC(-semigroups.)e(An)j (important)523 2540 y(role)d(in)f(our)h(lectures)f(\(and)g(in)h (applications)f(to)g(statistical)j(physics\))c(is)j(played)e(by)g(some) n(what)523 2640 y(less)e(kno)n(wn)e Fz(C)970 2610 y Fy(\003)964 2661 y Fx(0)1008 2640 y FC(-semigroups.)f(In)i(order)f(to)i(discuss)f (them,)g(\002rst)h(we)f(need)g(to)g(say)h(a)f(fe)n(w)h(w)o(ords)523 2740 y(about)i(dual)h(Banach)g(spaces.)648 2839 y(Let)h Fs(X)851 2809 y Fy(\003)910 2839 y FC(denote)f(the)h(Banach)f(space)h (dual)f(to)h Fs(X)34 b FC(\(the)20 b(space)h(of)f(continuous)f(linear)h (func-)523 2939 y(tionals)30 b(on)f Fs(X)12 b FC(\).)30 b(W)-7 b(e)31 b(will)g(use)f(the)g(sesquilinear)f(duality)f(between)h Fs(X)2675 2909 y Fy(\003)2744 2939 y FC(and)h Fs(X)12 b FC(:)30 b(the)g(form)523 3039 y Ft(\()p Fz(\030)t Fs(j)p Fz(\010)p Ft(\))22 b FC(will)f(be)f(antilinear)f(in)i Fz(\030)27 b Fs(2)c(X)1608 3008 y Fy(\003)1668 3039 y FC(and)c(linear)h(in)g Fz(\010)k Fs(2)f(X)12 b FC(.)648 3138 y(The)23 b(so-called)g(weak)p Fs(\003)g FC(\(w)p Fs(\003)p FC(\))h(topology)d(on)j Fs(X)2046 3108 y Fy(\003)2109 3138 y FC(is)h(de\002ned)d(by)i(the)g(seminorms)f Fs(j)p Ft(\()p Fs(\001j)p Fz(\010)p Ft(\))p Fs(j)p FC(,)523 3238 y(where)d Fz(\010)j Fs(2)g(X)12 b FC(.)648 3337 y(The)17 b(space)g(of)h(w)p Fs(\003)f FC(continuous)f(linear)h (operators)f(on)h Fs(X)2293 3307 y Fy(\003)2350 3337 y FC(will)h(be)f(denoted)f(by)h Fs(B)3030 3349 y Fu(w)r Fy(\003)3118 3337 y Ft(\()p Fs(X)3221 3307 y Fy(\003)3260 3337 y Ft(\))p FC(.)523 3437 y(Note)32 b(that)g Fs(B)929 3449 y Fu(w)r Fy(\003)1016 3437 y Ft(\()p Fs(X)1119 3407 y Fy(\003)1158 3437 y Ft(\))45 b Fs(\032)f(B)s Ft(\()p Fs(X)1505 3407 y Fy(\003)1543 3437 y Ft(\))p FC(.)32 b(If)g Fz(A)45 b Fs(2)g(B)s Ft(\()p Fs(X)12 b Ft(\))p FC(,)32 b(and)f Fz(A)2383 3407 y Fy(\003)2454 3437 y FC(is)i(its)g(adjoint,)e(then)g Fz(A)3174 3407 y Fy(\003)3257 3437 y Fs(2)523 3537 y Fz(B)586 3549 y Fu(w)r Fy(\003)674 3537 y Ft(\()p Fs(X)777 3507 y Fy(\003)816 3537 y Ft(\))p FC(.)g(Con)m(v)o(ersely)-5 b(,)28 b(if)k Fz(B)46 b Fs(2)e(B)1666 3549 y Fu(w)r Fy(\003)1753 3537 y Ft(\()p Fs(X)1856 3507 y Fy(\003)1895 3537 y Ft(\))p FC(,)31 b(then)f(there)h(e)o(xists)g(a)g (unique)f Fz(A)43 b Fs(2)g(B)s Ft(\()p Fs(X)12 b Ft(\))p FC(,)523 3636 y(sometimes)26 b(called)f(the)h(preadjoint)e(of)h Fz(B)t FC(,)h(such)g(that)g Fz(B)37 b Ft(=)c Fz(A)2419 3606 y Fy(\003)2458 3636 y FC(.)26 b(Lik)o(e)n(wise,)f(if)h Fz(A)h FC(is)f(closed)523 3736 y(and)20 b(densely)f(de\002ned)g(on)h Fs(X)12 b FC(,)21 b(then)f Fz(A)1646 3706 y Fy(\003)1705 3736 y FC(is)h(w)p Fs(\003)f FC(closed)g(and)g(w)p Fs(\003)g FC(densely)f(de\002ned)g(on)h Fs(X)3113 3706 y Fy(\003)3152 3736 y FC(.)648 3836 y(W)-7 b(e)25 b(say)g(that)g Ft([0)p Fz(;)14 b Fs(1)p Ft([)p Fs(3)31 b Fz(t)h Fs(7!)f Fz(W)12 b Ft(\()p Fz(t)p Ft(\))32 b Fs(2)f(B)1894 3848 y Fu(w)r Fy(\003)1982 3836 y Ft(\()p Fs(X)2085 3805 y Fy(\003)2124 3836 y Ft(\))25 b FC(is)h(a)f(w)p Fs(\003)f FC(continuous)f(semigroup)g (\(or)523 3935 y(a)32 b Fz(C)657 3905 y Fy(\003)651 3956 y Fx(0)695 3935 y FC(-semigroup\))d(if)n(f)i Fz(t)44 b Fs(3)g Fz(W)12 b Ft(\()p Fz(t)p Ft(\))p Fz(\030)37 b FC(is)32 b(w)p Fs(\003)f FC(continuous)f(for)g(an)o(y)h Fz(\030)48 b Fs(2)c(X)2821 3905 y Fy(\003)2859 3935 y FC(.)32 b(Note)f(that)h(if)523 4035 y Fz(U)9 b Ft(\()p Fz(t)p Ft(\))28 b FC(is)g(a)g Fz(C)918 4047 y Fx(0)983 4035 y FC(-semigroup,)d(then)i Fz(U)9 b Ft(\()p Fz(t)p Ft(\))1741 4005 y Fy(\003)1807 4035 y FC(is)28 b(a)g Fz(C)2020 4005 y Fy(\003)2014 4055 y Fx(0)2058 4035 y FC(-semigroup.)d(Con)m(v)o(ersely)-5 b(,)25 b(if)i Fz(W)12 b Ft(\()p Fz(t)p Ft(\))29 b FC(is)f(a)523 4134 y Fz(C)588 4104 y Fy(\003)582 4155 y Fx(0)626 4134 y FC(-semigroup)18 b(on)i Fs(X)1200 4104 y Fy(\003)1239 4134 y FC(,)g(then)g(there)f(e)o (xists)i(a)f(unique)f Fz(C)2202 4146 y Fx(0)2240 4134 y FC(-semigroup)e Fz(U)9 b Ft(\()p Fz(t)p Ft(\))21 b FC(on)f Fs(X)33 b FC(such)20 b(that)523 4234 y Fz(W)12 b Ft(\()p Fz(t)p Ft(\))24 b(=)e Fz(U)9 b Ft(\()p Fz(t)p Ft(\))978 4204 y Fy(\003)1017 4234 y FC(.)648 4334 y(Ev)o(ery)16 b Fz(C)928 4304 y Fy(\003)922 4354 y Fx(0)967 4334 y FC(-semigroup)f Fz(W)d Ft(\()p Fz(t)p Ft(\))20 b FC(possesses)e(its)h (generator)m(,)d(that)i(is)h(the)f(operator)e Fz(B)23 b FC(de\002ned)523 4433 y(as)e(follo)n(ws:)937 4616 y Fz(\030)28 b Fs(2)23 b Ft(Dom)p Fz(B)48 b Fs(,)c Ft(w)20 b Fs(\003)e(\000)f Ft(lim)1710 4670 y Fu(t)p Fy(&)p Fx(0)1847 4616 y Fz(t)1877 4582 y Fy(\000)p Fx(1)1966 4616 y Ft(\()p Fz(W)12 b Ft(\()p Fz(t)p Ft(\))19 b Fs(\000)f Fm(1)p Ft(\))p Fz(\030)28 b Ft(=:)22 b Fz(B)t(\030)46 b FC(e)o(xists)p Fz(:)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(47)523 282 y FC(The)19 b(generator)f(is)i(al)o(w)o(ays)g(w)p Fs(\003)p FC(-closed)e(and)h(w)p Fs(\003)p FC(-densely)f(de\002ned)g(and)h(uniquely)e(determines)523 382 y(the)j(semigroup.)e(W)-7 b(e)22 b(write)e Fz(W)12 b Ft(\()p Fz(t)p Ft(\))24 b(=)e(e)1690 352 y Fu(tB)1772 382 y FC(.)f(W)-7 b(e)21 b(ha)n(v)o(e)1666 565 y Ft(\(e)1735 530 y Fu(tA)1814 565 y Ft(\))1846 530 y Fy(\003)1908 565 y Ft(=)i(e)2033 530 y Fu(tA)2108 505 y Fn(\003)2146 565 y Fz(:)648 747 y FC(On)g(a)h(re\003e)o(xi)n(v)o(e)f(Banach)g (space,)g(e.g.)h(on)f(a)h(Hilbert)g(space,)f(the)h(concepts)f(of)g(a)h Fz(C)3103 759 y Fx(0)3141 747 y FC(-)g(and)523 847 y Fz(C)588 817 y Fy(\003)582 867 y Fx(0)626 847 y FC(-semigroup)19 b(coincide.)g(Unfortunately)-5 b(,)17 b Fz(W)1941 817 y Fy(\003)1979 847 y FC(-algebras)j(are)g(usually)g(not)g(re\003e)o(xi) n(v)o(e.)f(The)o(y)523 946 y(are,)29 b(ho)n(we)n(v)o(er)m(,)e(dual)j (Banach)f(spaces:)h(the)o(y)f(are)g(dual)g(to)h(the)g(space)f(of)h (normal)e(function-)523 1046 y(als.)d(In)e(the)h(conte)o(xt)f(of)g Fz(W)1331 1016 y Fy(\003)1369 1046 y FC(-algebras)g(the)h(w)p Fs(\003)p FC(-topology)d(is)k(usually)e(called)h(the)g Fz(\033)s FC(-weak)f(or)523 1146 y(ultra)o(weak)c(topology)-5 b(.)648 1245 y(Groups)18 b(of)h(automorphisms)d(of)j Fz(W)1711 1215 y Fy(\003)1749 1245 y FC(-algebras)g(are)g(rarely)f Fz(C)2467 1257 y Fx(0)2505 1245 y FC(-groups.)f(T)-7 b(o)19 b(see)h(this)g(note)523 1345 y(that)g(if)h Fz(H)27 b FC(is)21 b(a)g(self-adjoint)e(operator)f(on)i(a)h(Hilbert)f(space)g Fs(H)q FC(,)g(then)1639 1528 y Fz(t)j Fs(7!)g Ft(e)1835 1493 y Fx(i)p Fu(tH)1960 1528 y Fs(\001)18 b Ft(e)2038 1493 y Fy(\000)p Fx(i)p Fu(tH)3174 1528 y FC(\(68\))523 1710 y(is)i(al)o(w)o(ays)f(a)g Fz(C)967 1680 y Fy(\003)961 1731 y Fx(0)1005 1710 y FC(-group)e(on)h Fs(B)s Ft(\()p Fs(H)q Ft(\))p FC(.)g(It)h(is)h(a)f Fz(C)1840 1722 y Fx(0)1877 1710 y FC(-group)e(\(and)h(e)n(v)o(en)f(a)i(norm)f (continuous)e(group\))523 1810 y(if)n(f)k Fz(H)28 b FC(is)21 b(bounded,)c(which)j(is)h(usually)f(a)g(v)o(ery)f(se)n(v)o(ere)h (restriction.)648 1910 y(In)35 b(the)i(conte)o(xt)d(of)i Fz(W)1367 1879 y Fy(\003)1405 1910 y FC(-algebras,)f Fz(C)1832 1879 y Fy(\003)1826 1930 y Fx(0)1907 1910 y FC(groups)g(are)h(usually)g(called)g(\(pointwise\))f Fz(\033)s FC(-)523 2009 y(weakly)e(continuous)e(groups.)h Fz(C)1541 1979 y Fy(\003)1535 2030 y Fx(0)1579 2009 y FC(-groups)g(of)h Fs(\003)p FC(-automorphisms)d(are)j(often)g(called)g Fz(W)3247 1979 y Fy(\003)3285 2009 y FC(-)523 2109 y(dynamics.)648 2208 y(So)e(f)o(ar)m(,)f(all)h(the)g(material)g(that)g(we)g(recalled)f (can)h(be)g(found)e(e.g.)h(in)h([BR1].)g(No)n(w)g(we)523 2308 y(w)o(ould)c(lik)o(e)g(to)h(discuss)g(ho)n(w)e(to)i(de\002ne)f (the)g(spectral)g(projection)f(onto)g(a)i(\(not)f(necessarily)523 2408 y(isolated\))19 b(eigen)m(v)n(alue)f(of)h(a)h(generator)d(of)j (contracti)n(v)o(e)d(semigroup.)h(W)-7 b(e)21 b(will)f(see)g(that)g(a)g (fully)523 2507 y(satisf)o(actory)30 b(answer)g(is)i(a)n(v)n(ailable)e (for)g(purely)f(imaginary)g(eigen)m(v)n(alues)g(in)i(the)f(case)h(of)g (a)523 2607 y(re\003e)o(xi)n(v)o(e)16 b(Banach)i(spaces.)f(F)o(or)h (non-re\003e)o(xi)n(v)o(e)c(Banach)j(spaces)h(the)g(situation)f(is)i (much)e(more)523 2707 y(complicated.)h(Our)i(discussion)g(is)h(adapted) e(from)g([Zs])h(and)g(partly)f(from)g([Da3)o(].)648 2806 y(Let)d Fz(A)h FC(be)f(the)h(generator)d(of)i(a)h(contracti)n(v)o(e)d Fz(C)1979 2818 y Fx(0)2017 2806 y FC(-semigroup)g(on)i Fs(X)29 b FC(and)16 b Fz(e)23 b Fs(2)g Fw(R)p FC(.)17 b(F)o(ollo)n(wing)523 2906 y([Zs],)j(we)g(say)h(that)f Fz(A)h FC(is)g(er)o(godic)e(at)h Ft(i)p Fz(e)h FC(if)n(f)1383 3088 y Fm(1)1431 3100 y Fx(i)p Fu(e)1485 3088 y Ft(\()p Fz(A)p Ft(\))j(:=)31 b(lim)1746 3143 y Fu(\030)r Fy(&)p Fx(0)1891 3088 y Fz(\030)t Ft(\()p Fz(\030)23 b Ft(+)18 b(i)p Fz(e)h Fs(\000)f Fz(A)p Ft(\))2363 3054 y Fy(\000)p Fx(1)3174 3088 y FC(\(69\))523 3310 y(e)o(xists.)648 3410 y(Let)g Fz(B)24 b FC(be)18 b(the)g(generator)f(of)h(a)h(contracti) n(v)o(e)e Fz(C)2008 3380 y Fy(\003)2002 3430 y Fx(0)2046 3410 y FC(-semigroup)f(on)j Fs(X)2617 3380 y Fy(\003)2674 3410 y FC(and)f Fz(e)23 b Fs(2)g Fw(R)p FC(.)c(F)o(ollo)n(w-)523 3509 y(ing)h([Zs],)g(we)g(say)h(that)f Fz(B)25 b FC(is)c(globally)e(er) o(godic)f(at)j Ft(i)p Fz(e)f FC(if)n(f)1275 3692 y Fm(1)1323 3704 y Fx(i)p Fu(e)1377 3692 y Ft(\()p Fz(B)t Ft(\))j(:=)g(w)d Fs(\003)e(\000)j Ft(lim)1861 3746 y Fu(\030)r Fy(&)p Fx(0)2005 3692 y Fz(\030)t Ft(\()p Fz(\030)i Ft(+)18 b(ie)h Fs(\000)f Ft(B\))2472 3658 y Fy(\000)p Fx(1)3174 3692 y FC(\(70\))523 3914 y(e)o(xists)j(and)e(is)i(w)p Fs(\003)p FC(-continuous.)648 4013 y(As)31 b(we)h(will)g(see)f(from)f (the)i(theorem)d(belo)n(w)-5 b(,)30 b(\(69\))g(and)h(\(70\))f(can)g(be) h(called)g(spectral)523 4113 y(projections)19 b(onto)g(the)h(eigen)m(v) n(alue)e Ft(i)p Fz(e)p FC(.)523 4287 y FA(Theor)o(em)i(23.)k Fk(Let)d Fz(A)p Fk(,)f Fz(B)25 b Fk(and)20 b Fz(e)i Fs(2)i Fw(R)c Fk(be)h(as)f(abo)o(ve)o(.)612 4387 y FC(1\))k Fk(If)d Fz(A)g Fk(is)g(er)m(godic)e(at)h Ft(i)p Fz(e)p Fk(,)h(then)e Ft(1)1602 4399 y Fx(i)p Fu(e)1656 4387 y Ft(\()p Fz(A)p Ft(\))j Fk(is)f(a)f(pr)l(ojection)g(of)g(norm)g Ft(1)g Fk(suc)o(h)g(that)961 4581 y Ft(Ran)p Fm(1)1158 4593 y Fx(i)p Fu(e)1212 4581 y Ft(\()p Fz(A)p Ft(\))k(=)f(Ker)o(\()p Fz(A)c Fs(\000)f Ft(i)p Fz(e)p Ft(\))p Fz(;)76 b Ft(Ker)o Fm(1)2155 4593 y Fx(i)p Fu(e)2209 4581 y Ft(\()p Fz(A)p Ft(\))24 b(=)e(\()q(Ran)c(\()p Fz(A)h Fs(\000)f Ft(i)p Fz(e)p Ft(\)\))2968 4539 y Fx(cl)3035 4581 y Fz(:)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 523 100 a FB(48)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)612 282 y FC(2\))24 b Fk(On)d(a)g(r)m(e\003e)n(xive) g(Banac)o(h)e(space)o(,)i(we)h(have)e(always)h(the)g(er)m(godic)f(pr)l (operty)h(for)h(all)f(g)o(en-)706 382 y(er)o(ator)o(s)g(of)f(contr)o (active)f(semigr)l(oups)h(and)f(all)i Ft(i)p Fz(e)h Fs(2)i Ft(i)p Fw(R)p Fk(.)612 482 y FC(3\))g Fk(If)31 b Fz(B)k Fk(is)d(globally)d(er)m(godic)h(at)h Ft(i)p Fz(e)p Fk(,)f(then)g Ft(1)1983 494 y Fx(i)p Fu(e)2037 482 y Ft(\()p Fz(B)t Ft(\))i Fk(is)g(a)e(w)p Fs(\003)p Fk(-continuous)e(pr)l(ojection)i(of) 706 581 y(norm)20 b Ft(1)g Fk(suc)o(h)g(that)912 761 y Ft(Ran)p Fm(1)1109 773 y Fx(i)p Fu(e)1163 761 y Ft(\()p Fz(B)t Ft(\))k(=)e(Ker)o(\()p Fz(B)h Fs(\000)18 b Ft(i)p Fz(e)p Ft(\))p Fz(;)76 b Ft(Ker)o Fm(1)2115 773 y Fx(i)p Fu(e)2170 761 y Ft(\()p Fz(B)t Ft(\))23 b(=)g(\(Ran)18 b(\()p Fz(A)h Fs(\000)f Ft(i)p Fz(e)p Ft(\)\))2934 719 y Fu(w)r Fy(\003)p Fx(cl)3084 761 y Fz(:)612 940 y FC(4\))24 b Fz(A)d Fk(is)g(er)m(godic)f(at)g Ft(i)p Fz(e)g Fk(if)o(f)h Fz(A)1455 910 y Fy(\003)1514 940 y Fk(is)g(globally)e(er)m(godic)g(at)i Fs(\000)p Ft(i)p Fz(e)f Fk(and)1649 1120 y Fm(1)1697 1132 y Fx(i)p Fu(e)1752 1120 y Ft(\()p Fz(A)p Ft(\))1878 1085 y Fy(\003)1940 1120 y Ft(=)i Fm(1)2075 1132 y Fy(\000)p Fx(i)p Fu(e)2181 1120 y Ft(\()p Fz(A)2275 1085 y Fy(\003)2314 1120 y Ft(\))p Fz(:)648 1299 y FC(1\))i(and)g(2\))g(are)g(pro)o(v)o(en) f(in)h([Da3)o(])h(Theorem)e(5.1)h(and)g(Corollary)f(5.2.)g(3\))i(and)f (4\))g(can)g(be)523 1399 y(pro)o(v)o(en)18 b(by)i(adapting)e(the)j(ar)o (guments)d(of)i([Zs)o(])h(Theorem)d(3.4)i(and)f(Corollary)g(3.5.)648 1499 y(As)h(an)g(ilustration)f(of)h(the)g(abo)o(v)o(e)e(concepts)h (consider)g(the)h Fz(W)2469 1468 y Fy(\003)2507 1499 y FC(-dynamics)e(\(68\).)h(Clearly)-5 b(,)523 1598 y(it)20 b(is)g(a)f(group)e(of)i(isometries)g(and)g(the)g(spectrum)f(of)g(its)i (generator)d Ft(i[)p Fz(H)r(;)d Fs(\001)p Ft(])20 b FC(is)g(contained)e (in)h Ft(i)p Fw(R)p FC(.)523 1698 y(If)j Fz(H)29 b FC(possesses)22 b(only)f(point)g(spectrum,)g(then)g Ft(i[)p Fz(H)r(;)14 b Fs(\001)p Ft(])23 b FC(is)f(globally)f(er)o(godic)f(for)h(an)o(y)g Ft(i)p Fz(e)k Fs(2)i Ft(i)p Fw(R)p FC(.)523 1797 y(In)20 b(f)o(act,)g(we)h(ha)n(v)o(e)e(the)h(follo)n(wing)f(formula)g(for)1230 1991 y Fm(1)1278 2003 y Fx(i)p Fu(e)1332 1991 y Ft(\(i[)p Fz(H)r(;)14 b Fs(\001)p Ft(]\)\()p Fz(C)6 b Ft(\))25 b(=)1839 1912 y Fl(X)1837 2090 y Fu(x)p Fy(2)p Fr(R)1975 1991 y Fm(1)2023 2003 y Fu(x)p Fx(+)p Fu(e)2147 1991 y Ft(\()p Fz(H)7 b Ft(\))p Fz(C)f Fm(1)2400 2003 y Fu(x)2443 1991 y Ft(\()p Fz(H)h Ft(\))p Fz(:)648 2253 y FC(Note)22 b(that)h Ft(i[)p Fz(H)r(;)14 b Fs(\001)p Ft(])23 b FC(al)o(w)o(ays)g (possesses)g(an)g(eigen)m(v)n(alue)d Ft(0)j FC(and)f(the)g (corresponding)d(eigen-)523 2353 y(v)o(ectors)25 b(are)h(all)h (operators)e(commuting)f(with)i Fz(H)7 b FC(.)26 b(It)h(is)g(ne)n(v)o (er)d(globally)h(er)o(godic)g(at)h Ft(0)g FC(if)h Fz(H)523 2453 y FC(has)20 b(some)g(continuous)f(spectrum.)523 2766 y Fv(Refer)n(ences)523 2954 y FB([AJPP])40 b(Aschbacher)m(,)21 b(W)-7 b(.,)18 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)19 b(V)-10 b(.,)18 b(P)o(autrat,)h(Y)-10 b(.,)19 b(Pillet,)e(C.-A.:)h(Introduction)j(to)e(non-equilibrium)749 3045 y(quantum)h(statistical)e(mechanics.)i(Grenoble)g(lecture)f (notes.)523 3136 y([A])122 b(Alicki,)17 b(R.:)f(On)h(the)h(detailed)f (balance)h(condition)h(for)e(non-hamiltonian)i(systems,)e(Rep.)g(Math.) 749 3227 y(Phys.)h(10)i(\(1976\))g(249)523 3318 y([AL])76 b(Alicki,)15 b(R.,)f(Lendi,)h(K.:)g Fd(Quantum)h(dynamical)g(semigr)m (oups)g(and)h(applications)p FB(,)f(Lecture)f(Notes)749 3410 y(in)k(Physics)g(no)g(286,)g(Springer)g(1991)523 3500 y([BR1])41 b(Brattelli,)16 b(O.,)g(Robinson)i(D.)f(W)-7 b(.:)16 b Fd(Oper)o(ator)i(Alg)o(ebr)o(as)g(and)g(Quantum)g (Statistical)e(Mec)o(hanics,)749 3592 y(V)-8 b(olume)19 b(1)p FB(.)f(Springer)o(-V)-8 b(erlag,)19 b(Berlin,)f(second)i(edition) f(1987.)523 3682 y([BR2])41 b(Brattelli,)16 b(O.,)g(Robinson)i(D.)f(W) -7 b(.:)16 b Fd(Oper)o(ator)i(Alg)o(ebr)o(as)g(and)g(Quantum)g (Statistical)e(Mec)o(hanics,)749 3774 y(V)-8 b(olume)19 b(2)p FB(.)f(Springer)o(-V)-8 b(erlag,)19 b(Berlin,)f(second)i(edition) f(1996.)523 3864 y([BFS1])41 b(Bach,)16 b(V)-10 b(.,)16 b(Fr)6 b(\250)-31 b(ohlich,)16 b(J.,)f(Sigal,)h(I.:)f(Quantum)i (electrodynamics)h(of)f(con\002ned)g(non-relati)n(vistic)749 3956 y(particles,)i(Adv)-5 b(.)19 b(Math.)g Fc(137)p FB(,)g(299)h(\(1998\).)523 4046 y([BFS2])41 b(Bach,)16 b(V)-10 b(.,)15 b(Fr)6 b(\250)-31 b(ohlich,)16 b(J.,)f(Sigal,)g(I.:)g (Return)i(to)e(equilibrium.)i(J.)e(Math.)i(Phys.)e Fc(41)p FB(,)h(3985)i(\(2000\).)523 4137 y([Da1])52 b(Da)o(vies,)18 b(E.)g(B.:)g(Mark)o(o)o(vian)j(master)e(equations.)h(Commun.)f(Math.)g (Phys.)g Fc(39)p FB(,)g(91)g(\(1974\).)523 4228 y([Da2])52 b(Da)o(vies,)18 b(E.)g(B.:)g(Mark)o(o)o(vian)j(master)e(equations)h (II.)e(Math.)h(Ann.)g Fc(219)p FB(,)h(147)g(\(1976\).)523 4318 y([Da3])52 b(Da)o(vies,)18 b(E.)g(B.:)g(One)i(parameter)f (semigroups,)h(Academic)f(Press)g(1980)523 4409 y([DJ1])56 b(Derezi)6 b(\264)-31 b(nski,)20 b(J.,)f(Jak)l Fe(\024)-34 b(s)p FB(i)t(\264)-29 b(c,)19 b(V)-10 b(.:)20 b(Spectral)f(theory)h(of) g(P)o(auli-Fierz)e(operators.)i(J.)f(Func.)h(Anal.)f Fc(180)p FB(,)749 4500 y(243)h(\(2001\).)523 4591 y([DJ2])56 b(Derezi)6 b(\264)-31 b(nski,)25 b(J.,)e(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)24 b(V)-10 b(.:)24 b(Return)h(to)g (equilibrium)g(for)f(P)o(auli-Fierz)f(systems.)i(Ann.)f(Henri)749 4682 y(Poincar)t(\264)-29 b(e)19 b Fc(4)p FB(,)g(739)h(\(2003\).)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 1643 100 a FB(Fermi)18 b(Golden)i(Rule)f(and)g (open)h(quantum)g(systems)193 b(49)523 282 y([DJ3])56 b(Derezi)6 b(\264)-31 b(nski,)24 b(J.,)e(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)23 b(V)-10 b(.:)23 b(On)g(the)h(nature)g(of) f(Fermi)g(golden)h(rule)g(for)f(open)i(quantum)f(sys-)749 374 y(tems,)19 b(J.)f(Statist.)f(Phys.)i Fc(116)g FB(\(2004\),)h(411)g (\(2004\).)523 465 y([DJ4])56 b(Derezi)6 b(\264)-31 b(nski,)19 b(J.,)f(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)18 b(V)-10 b(.:)19 b(In)g(preparation.)523 556 y([Di])101 b(Dirac,)19 b(P)-8 b(.A.M.:)18 b(The)i(quantum)h(theory)g(of)f(the)f (emission)i(and)f(absorption)h(of)f(radiation.)g(Proc.)749 648 y(Ro)o(yal)g(Soc.)e(London,)i(Series)e(A)h Fc(114)p FB(,)g(243)h(\(1927\).)523 739 y([DJP])51 b(Derezi)6 b(\264)-31 b(nski,)22 b(J.,)f(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)21 b(V)-10 b(.,)21 b(Pillet,)f(C.)h (A.:Perturbation)h(theory)g(of)g Fb(W)2681 707 y Fa(\003)2716 739 y FB(-dynamics,)h(Liouvil-)749 830 y(leans)c(and)h(KMS-states,)e (to)h(appear)h(in)f(Re)n(v)-5 b(.)19 b(Math.)g(Phys)523 922 y([FGKV])41 b(Frigerio,)21 b(A.,)g(Gorini,)h(V)-10 b(.,)21 b(K)m(ossak)o(o)n(wski)j(A.,)d(V)-8 b(erri,)21 b(M.:)h(Quantum)g(detailed)h(balance)g(and)749 1013 y(KMS)c(condition,) g(Comm.)g(Math.)g(Phys.)g(57)g(\(1977\))h(97-110)523 1104 y([DF1])43 b(Derezi)6 b(\264)-31 b(nski,)19 b(J.,)f(Fr)6 b(\250)-31 b(uboes,)19 b(R.:)f(Le)n(v)o(el)g(Shift)g(Operator)h(and)h (2nd)f(order)g(perturbation)h(theory)-5 b(,)19 b(to)749 1196 y(appear)h(in)f(J.)f(Math.)h(Phys.)523 1287 y([DF2])43 b(Derezi)6 b(\264)-31 b(nski,)19 b(J.,)f(Fr)6 b(\250)-31 b(uboes,)19 b(R.:)f(Stationary)h(v)n(an)h(Ho)o(v)o(e)f(limit,)f(to)h (appear)g(in)g(J.)g(Math.)g(Phys.)523 1378 y([Fe])101 b(Fermi,)18 b(E.:)g(Nuclear)h(Physics,)f(Uni)n(v)o(ersity)h(of)g (Chicago)h(Press,)e(Chicago)i(1950)523 1469 y([GKS])41 b(Gorini,)18 b(V)-10 b(.,)19 b(K)m(ossak)o(o)n(wski,)h(A.,)e (Sudarshan,)i(E.C.G.)d(Journ.)i(Math.)g(Phys.)g(17)g(\(1976\))h(821)523 1561 y([Haa])56 b(Haak)o(e,)37 b(F)-6 b(.:)36 b Fd(Statistical)g(tr)m (eatment)h(of)g(open)h(systems)e(by)h(g)o(ener)o(alized)i(master)d (equation.)749 1652 y FB(Springer)19 b(T)m(racts)g(in)g(Modern)h (Physics)f Fc(66)p FB(,)g(Springer)o(-V)-8 b(erlag,)18 b(Berlin,)g(1973.)523 1743 y([JP1])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)18 b(V)-10 b(.,)18 b(Pillet,)e(C.-A.:)h(On)i (a)f(model)h(for)g(quantum)h(friction)e(III.)f(Er)o(godic)i(properties) g(of)f(the)749 1835 y(spin-boson)j(system.)e(Commun.)g(Math.)g(Phys.)g Fc(178)p FB(,)g(627)h(\(1996\).)523 1926 y([JP2])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)18 b(V)-10 b(.,)18 b(Pillet,)g(C.-A.:)f(Spectral)h(theory)i(of)f(thermal)g(relaxation.)g (J.)f(Math.)h(Phys.)f Fc(38)p FB(,)h(1757)749 2017 y(\(1997\).)523 2109 y([JP3])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)37 b(V)-10 b(.,)38 b(Pillet,)e(C.-A.:)h(From)g(resonances)j(to)e (master)g(equations.)h(Ann.)f(Inst.)f(Henri)749 2200 y(Poincar)t(\264)-29 b(e)19 b Fc(67)p FB(,)g(425)h(\(1997\).)523 2291 y([JP4])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)16 b(V)-10 b(.,)17 b(Pillet,)e(C.-A.:)h(Non-equilibrium)i(steady)g (states)f(for)g(\002nite)f(quantum)i(systems)g(cou-)749 2383 y(pled)h(to)g(thermal)g(reserv)o(oirs.)g(Commun.)g(Math.)g(Phys.)g Fc(226)p FB(,)g(131)h(\(2002\).)523 2474 y([JP5])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)24 b(V)-10 b(.,)25 b(Pillet,)e(C.-A.:)g(Mathematical)j(theory)g(of)f(non-equilibrium)h (quantum)g(statistical)749 2565 y(mechanics.)20 b(J.)e(Stat.)g(Phys.)g Fc(108)p FB(,)i(787)f(\(2002\).)523 2657 y([JP6])68 b(Jak)l Fe(\024)-34 b(s)q FB(i)t(\264)-29 b(c,)18 b(V)-10 b(.,)19 b(Pillet,)e(C.-A.:)g(In)i(preparation.)523 2748 y([Ka])89 b(Kato,)19 b(T)-6 b(.:)18 b Fd(P)-6 b(erturbation)20 b(Theory)f(for)g(Linear)g(Oper)o(ator)o(s)p FB(,)h(second)g(edition,) 523 2839 y([KTH])41 b(K)o(ubo,)18 b(R.,)e(T)-6 b(oda,)17 b(M.,)g(Hashitsume,)h(N.:)e Fd(Statistical)h(Physics)h(II.)e (Nonequilibrium)j(Statistical)749 2931 y(Mec)o(hanics.)h FB(Springer)o(-V)-8 b(erlag,)18 b(Berlin,)g(1985.)523 3022 y([L])130 b(Lindblad,)30 b(G.:)e(On)i(the)f(generators)h(of)g (quantum)g(dynamical)h(semigroups,)f(Comm.)f(Math.)749 3113 y(Phys.)18 b(48)i(\(1976\))g(119-130)523 3205 y([M])110 b(Merkli,)23 b(M.:)h(Positi)n(v)o(e)f(commutators)h(in)f (non-equilibrium)i(quantum)g(statistical)e(mechanics.)749 3296 y(Commun.)d(Math.)f(Phys.)f Fc(223)p FB(,)h(327)h(\(2001\).)523 3387 y([LeSp])41 b(Lebo)n(witz,)26 b(J.,)f(Spohn,)h(H.:)f(Irre)n(v)o (ersible)i(thermodynamics)g(for)f(quantum)h(systems)g(weakly)749 3479 y(coupled)20 b(to)f(thermal)g(reserv)o(oirs.)g(Adv)-5 b(.)19 b(Chem.)g(Phys.)f Fc(39)p FB(,)h(109)h(\(1978\).)523 3570 y([Ma1])42 b(Maje)n(wski,)29 b(W)-7 b(.)27 b(A.:)h(Dynamical)h (semigroups)g(in)g(the)f(algebraic)h(formulation)g(of)f(statistical)749 3661 y(mechanics,)20 b(F)o(ortschritte)e(der)h(Physik)g(32)g(\(1984\))h (89-133)523 3753 y([Ma2])42 b(Maje)n(wski,)22 b(W)-7 b(.)21 b(A.:)g(Journ.)h(Math.)g(Phys.)g(The)f(detailed)h(balance)h (condition)g(in)f(quantum)h(sta-)749 3844 y(tistical)18 b(mechanics)i(25)g(\(1984\))f(614)523 3935 y([MaSt])41 b(Maje)n(wski,)32 b(W)-7 b(.)30 b(A.,)g(Streater)m(,)g(R.)g(F)-6 b(.:)30 b(Detailed)h(balance)h(and)g(quantum)h(dynamical)f(maps)749 4027 y(Journ.)19 b(Phys.)g(A:)f(Math.)h(Gen.)g(31)h(\(1998\))f (7981-7995)523 4118 y([VH1])42 b(V)-8 b(an)20 b(Ho)o(v)o(e,)g(L.:)f (Quantum-mechanical)j(perturbations)g(gi)n(ving)f(rise)e(to)i(a)f (statistical)f(transport)749 4209 y(equation.)h(Physica)f Fc(21)g FB(\(1955\))h(517.)523 4301 y([VH2])42 b(V)-8 b(an)23 b(Ho)o(v)o(e,)g(L.:)f(The)h(approach)i(to)e(equilibrium)g(in)g (quantum)i(statitsics.)d(Physica)h Fc(23)g FB(\(1957\))749 4392 y(441.)523 4483 y([VH3])42 b(V)-8 b(an)21 b(Ho)o(v)o(e,)g(L.:)g (Master)g(equation)i(and)f(approach)h(to)e(equilibrium)h(for)f(quantum) i(systems.)e(In)749 4575 y Fd(Fundamental)27 b(pr)m(oblems)g(in)f (statistical)g(mec)o(hanics)p FB(,)h(compiled)g(by)f(E.G.D.)f(Cohen,)h (North-)749 4666 y(Holand,)19 b(Amsterdam)h(1962.)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 523 100 a FB(50)193 b(Jan)19 b(Derezi)6 b(\264)-31 b(nski)19 b(and)h(Raf)o(a\007)f(Fr)6 b(\250)-31 b(uboes)523 282 y([RS4])47 b(Reed,)22 b(M.,)g(Simon,)g(B.:)f Fd(Methods)i(of)f(Modern)i(Mathematical)f(Physics,)f(IV)-10 b(.)21 b(Analysis)h(of)g(Op-)749 374 y(er)o(ator)o(s)p FB(,)d(London,)h(Academic)g(Press)e(1978.)523 465 y([Sp])97 b(Spohn,)16 b(H.:)e(Entrop)o(y)i(production)g(for)f(quantum)i (dynamical)f(semigroups,)g(Journ.)g(Math.)f(Phys.)749 556 y(19)20 b(\(1978\))f(1227)523 648 y([St])113 b(Stinespring,)20 b(W)-7 b(.)19 b(F)-6 b(.:)19 b(Positi)n(v)o(e)h(functions)g(on)h Fb(C)2035 616 y Fa(\003)2071 648 y FB(-algebras,)f(Proc.)g(Am.)f(Math.) i(Soc.)e(6)h(\(1955\))749 739 y(211-216)523 830 y([WW])40 b(W)-6 b(eissk)o(opf,)29 b(V)-10 b(.,)28 b(W)m(igner)m(,)g(E.:)f (Berechnung)k(der)d(nat)6 b(\250)-31 b(urlichen)30 b(Linienbreite)e (auf)h(Grund)g(der)749 922 y(Dirakschen)20 b(Lichttheorie,)f (Zeitschrift)f(f)6 b(\250)-31 b(ur)18 b(Physik)h(63)h(\(130\))f(54)523 1013 y([Zs])101 b(Zsid)6 b(\264)-31 b(o,)15 b(L.:)e(Spectral)i(and)g (er)o(godic)g(properties)h(of)e(analytic)h(generators,)h(Journ.)f (Approximation)749 1104 y(Theory)20 b(20)f(\(1977\))h(77-138)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0504200913200--