Content-Type: multipart/mixed; boundary="-------------0304270345299" This is a multi-part message in MIME format. ---------------0304270345299 Content-Type: text/plain; name="03-198.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-198.keywords" quantum field theory, Nelson models, spectral theory ---------------0304270345299 Content-Type: application/postscript; name="nelson-massless.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nelson-massless.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: C:\Travaux en cours\nelson massless\nelson-massless.dvi %%CreationDate: Sat Apr 26 12:19:06 2003 %%Pages: 57 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMBX10 CMR10 CMBX12 CMSY10 EUFM10 %%+ CMMI10 CMR8 CMMI8 EUFM7 CMTI10 CMSY6 CMR9 CMR6 CMEX10 CMMI6 LASY10 %%+ CMMI12 MSAM10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: c:\texmf\miktex\bin\dvips.exe %+ "C:\Travaux en cours\nelson massless\nelson-massless.dvi" %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.26:1219 %%BeginProcSet: texc.pro %! 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5280062D5FF33ECE4F5BBF3F1AABABC52E9D91F4AB2DAFA5D925DF70DA59E251 AFF0D4470FBBA03C670EC47B447F18160B23D8056A4DED6B6345254EE2A621B0 4DB1FE3DF5C56743D130503B7CBDFD2680DA7FCF6F68E02ADC4FEF3A01D30840 D0D26084000730E888DADCE068A8BB9488071A177E174A2F92F191C561383700 7FE384E141F46C52273AFA3EE0E98E1A5A9A8885008CB3E16D0C91435B1A2A7F 51A20E639B4E678EF912526BD25EB7202FB88096A51A2B7748829FF65B4474E5 F52B8DADFB490F56C2B540E4ECBA237803B9206EBD21548CB60554765105CFC4 D4D88F87CCAA2E1A3AF2B037F4B7EE12E692E50BACD8D5EF06C80CBA20D31904 8E59A181AADEFBF8AC1C0CC0D33C3834ECE6A1162B43571CEF94EC332181A845 D7AF7D418AA1521F7BADD95B2158C7D10E369818E9FC3AE7190A24A3A68AA915 21846DE7E20B4023F779030D688E9E9FC7F7B7431E2BCD9376D17FA33548EEB3 2B2C5F1799C47B252198841902304BD85340241ACCD36CBB37FABFC59D2819CF 5D0EDB35A9D5CFEAEB209E6D34E6EC4C90368B7E47FA10A7F291758BA1BDC69F CC97B1A2C73641845D47626D0015491347B13E5457EAA321E38355FE20F00FD7 4A3FA034C9BECC1C01019F19219A807F583AC4A486E30CF77913AED50735542F D0958FD64850609007C15AA0756449F4A1FF0AE6D09B48D1C20821F4BE6F736B 9AA244809E5C58C8055B53A04709528DE2EE60F0D1B145DD32E3956A8A89B618 233C7F206092D7610AD4E4D17BA552CAD6B6B19A15 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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: 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 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 33 /omega put dup 60 /less put dup 62 /greater put dup 100 /d put dup 106 /j put dup 110 /n put dup 118 /v put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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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 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put dup 113 /radicalBig put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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F72C8C3ADAAAB5D8633BAA73ADBF7836A1D8876B9FD40DBBA38FB7A59D068564 8CC0D69F48D334834567F88603BC100C22507A5685B8D7D4D09D477F2E3A1CFC 98C2959E5063F54C6903D0462E40A081567A42EFEA041D2D2B88B5FFF5706251 110E6ACDE2BD093BBF7DEEC138EAD7974CA19AD4666F3356D6C9BC99DFD45BB4 15CA1E4ABD7A5EC2F05A0848025BB394B1B6965E9D42E6ADF4FC76B97835B152 0BC27E370AA40C36B36566510C0D41229F14566542CA4F5224AD1FFDCE6B8CE6 BA49561656836206AC85AA0B099696BB54A760373CC684E381A1A3EF7C4AE37E EE9EB1DE5A55261408002B386E20E04ECFA24970D911E5CC967A744498B34F4C 427C7F8F17F94010654EFAD5370CBA0E79DADC7E4BB5F55B9FD3DB1E8F153837 3E69828D75347A05408566FF2A57B61C82E1A2C637D777F9BDE4335B4C1BDF06 7B66B75E48D77879D21AB4B2905B278381244D7C78762DCD52C3F372BF2F8280 91F0AB959AE2613B59A1091D4B2DA005C13B04E2252B12955B571E84D28BA649 030BFB03FA546F311AB131E0614E8A336F03BAADF4A2701A0222AAA97E14C7AE 8F6170D9AB1F4B3578E9B740BF58BF11044FECC081FE98D8C7234AD7C0D8AFCD F7B2C80F90DDA1203E1BEADEF0DDC13C68049E97C6A5388291E40DFF86427F80 B48B7F821E7DB7D1A9663258BA1B51F54B69F0D884FCAB6F9E615E2DC9010774 B79815FA475F38657388ABE87F163FA2F09B08F93C46682EFECB1B4FAFDBBCF1 40A7641CD3DD297C1E11483C9292AD65DD7A7FB7E7ED91885C10480F393D47E6 C80CBE43F0B2CC210D0FB53260923889835B9C47B64040007D50425F3C8ABF65 9E85197CC9D236560B7769A11C94E5C34C6CFC40CBB4FEDB58D9D8C207426D8E BB62BE2DA4F3BC2FA27DDE1C42A5C6F19572E09C525FC338F7976D5F3C82214E D0BC1249BDE8FF991E1BC9C3E66B96AFEA50635F7BCF78D5A8A54CA135A57DCE 8E3C71147F12EB6C8C21BC4C27B5524A47489BB16A8D98AC669E6E957C77ACFC 8714E01A88E33263345FD856159DFF8F58370C25F925D6556ABBF93AF386E054 3F22F08E3AAED9ACC036BB3654CB9F47C056AD398D6BCAD8BFD5D80C98AA6CFC 395CADFD735B2619E471243796FB8E2AE1E09B8F53BB2D138243B2C91EDB09EB C733DC25459B91724442844C99C46DB506EE033381E4A2F8DEDDDE404B731849 B2425E688E2F82D64A6994264A94772D52AF38A553779EEA5EC7CA832A70973A 7557B674A0028BD11E1A1EC68A075C8A090FAFC22666CCA98C5A03589783B03D 26F1F42827ECA60C7514C1115A5557977FC6D145A56DAEA3C99C23D7361C84F7 D5E62F58059157A628A28C1F185E0669FE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 115 /s put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 104 /h put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueID 5031992 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 30 /phi put dup 32 /psi put dup 34 /epsilon put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 100 /d put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 12 /fi put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 61 /equal put dup 67 /C put dup 73 /I put dup 82 /R put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 101 /e put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 112 /p put dup 115 /s put dup 116 /t put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 103 /g put dup 104 /h 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 02B150C69E56B6C84E1C8590BB610F46CFBA0A500F5819E525B9A458DFF22411 5727EE92926048828F8CFEA9365D181462844163BB8A2844D07E24DDAE03B342 E8B931B5BD41AA11ADAA4DC7CBD5800544E0821B4AE8933ABDCCD0C19C0CECDC AAF4B75FBF218201A41E0852918EE31223AEB7D62E8458EA9EC65C13E4D5319D DB2C6F732D17C1FF0ECD8861786E8A725CC52C3AD96235AF4E11D6E325B77036 C0B16EF4091E050B192D31EAFC6950EA2548019E7B9BAC490DE33C16F824FBD3 A1C6A19378E33F140A0412685C2F7ED768EFEB83E0347A9DA3DFFF729247C386 AB25D37BDF9FF029E164B0DF032E2DF6613A13307746AB0E6648C1FBAF2BADA1 76FE6E2989B4F62B209D6732B64F3855A3D3B295D8B7B7FB5E6B1DE042D95F5E 25257C14D649F0B25F9EB275534BCE2C88E0CBC31F708C1E5C57DBDDA1FC8336 BF6D0CBBE71F1D10583610987DD3D6F43C588E7DAFA77A76F1232E1010CA11D7 6950099E495D63B52453EBF6717935942141CFFADA6085B587176438D5252F48 E834F91D883180D0CE9E9D5BEF5AF2CB36CE1D9D9BDDE590DE4CB4D20CDB32DA E20819143D78CB73AB10B191AE87752268454454DECBA1ED150466FFE46EF295 6BEB8428DBB3D0910432947E826B5A71A168C61069DF9811E0AEE77EBB5FEFE5 125F4859051C4D0B84AD8B7A451EC9A5E1CDFC77652B7DA34F3C18EE89915ABD BBCF4D7A265472AD4F3ED3B857FF82D9D1DFFF7E826B5B5AF5BF19C7E43ECDCA 353D1050748CB9DFBB427F667ED3EF46155597D252411487341D030D4ED72947 9F1AC4F63AF4F92D1609355C726C083B3BB069D82C524B4BF975B612BBEF1AB0 3E4E9C00 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 5 /Pi put dup 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 22 /macron put dup 28 /oslash put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 66 /B put dup 68 /D put dup 71 /G put dup 72 /H put dup 75 /K put dup 76 /L put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put dup 121 /dagger put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict 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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. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 28 /oslash put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 74 /J put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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y Fo(3)p Fn(P)2757 1917 y Fs(\))12 b Fr(\\)g(D)s Fs(\()p Fr(j)p Fs(x)p Fr(j)3083 1884 y Fn(\013)3132 1917 y Fs(\),)28 b(whic)m(h)e(implies:)1365 2119 y Fr(j)p Fs(x)p Fr(j)1463 2082 y Fn(\013)1513 2119 y Fs(\()p Fp(K)h Fs(+)20 b Fp(b)p Fs(\))1817 2082 y Fw(\000)1882 2054 y Fh(1)p 1882 2066 V 1882 2108 a(2)1957 2119 y Fs(is)30 b(b)s(ounded)n Fp(:)-2414 b Fs(\(2.3\))0 2306 y(Note)30 b(also)f(that)g Fe(\(H0)768 2273 y Fw(0)791 2306 y Fe(\))f Fs(implies)h(that)g Fp(K)35 b Fs(has)28 b(compact)h(resolv)m(en)m(t)i(on)d Fp(L)2562 2273 y Fo(2)2601 2306 y Fs(\(I)-15 b(R)2721 2270 y Fo(3)p Fn(P)2815 2306 y Fs(\),)29 b(so)g(h)m(yp)s(othesis)f Fe(\(H0\))g Fs(in)0 2419 y(Subsection)i(2.2)i(is)e(satis\014ed.)141 2532 y(The)g(one-particle)i(space)f(for)f(b)s(osons)g(is)1564 2719 y Fq(h)25 b Fs(:=)g Fp(L)1815 2682 y Fo(2)1855 2719 y Fs(\(I)-15 b(R)1975 2682 y Fo(3)2014 2719 y Fp(;)15 b Fs(d)p Fp(k)s Fs(\))p Fp(;)0 2906 y Fs(and)30 b(the)g(b)s(osonic)h (\014eld)f(is)g(describ)s(ed)f(with)h(the)h(F)-8 b(o)s(c)m(k)32 b(space)f(\000\()p Fq(h)p Fs(\))g(and)f(the)g(Hamiltonian)i(d\000\()p Fr(j)p Fp(k)s Fr(j)p Fs(\).)141 3019 y(W)-8 b(e)32 b(assume)e(that)h (the)f(in)m(teraction)j(is)d(of)h(the)f(form)1535 3261 y Fp(V)45 b Fs(:=)1784 3155 y Fn(N)1756 3180 y Fg(X)1755 3362 y Fn(j)t Fo(=1)1892 3261 y Fp(\036)p Fs(\()8 b(\024)-53 b Fp(\032)q Fs(\(x)2112 3275 y Fn(j)2149 3261 y Fs(\)\))p Fp(;)-2244 b Fs(\(2.4\))0 3529 y(for)783 3658 y Fp(\036)p Fs(\()8 b(\024)-53 b Fp(\032)q Fs(\(x\)\))26 b(=)1243 3596 y(1)p 1205 3637 122 4 v 1205 3654 a Fr(p)p 1281 3654 46 4 v 75 x Fs(2)1352 3543 y Fg(Z)1450 3658 y Fp(\032)p Fs(\()p Fp(k)s Fs(\)e)1657 3620 y Fw(\000)p Fo(i)p Fn(k)r Fw(\001)p Fo(x)1852 3658 y Fr(\012)20 b Fp(a)1991 3620 y Fw(\003)2031 3658 y Fs(\()p Fp(k)s Fs(\))h(+)28 b(\026)-53 b Fp(\032)p Fs(\()p Fp(k)s Fs(\)e)2470 3620 y Fo(i)p Fn(k)r Fw(\001)p Fo(x)2610 3658 y Fr(\012)20 b Fp(a)p Fs(\()p Fp(k)s Fs(\)d)p Fp(k)s(;)0 3873 y Fs(where)40 b(\024)-53 b Fp(\032)32 b 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end %%Page: 8 8 TeXDict begin 8 7 bop 0 162 a Fs(for)34 b(^)-49 b Fp(r)28 b Fs(:=)d Fp(r)s Fr(h)p Fp(r)s Fr(i)487 129 y Fw(\000)p Fo(1)581 162 y Fs(.)41 b(This)29 b(yields)i(if)j(~)-49 b Fp(v)1237 176 y Fo(1)1302 162 y Fs(=)25 b(1)-20 b(l)1449 176 y Fw(K)1527 162 y Fr(\012)20 b Fp(T)13 b(v)1728 176 y Fo(1)1767 162 y Fs(:)1093 364 y Fr(j)s Fs(~)-48 b Fp(v)1162 378 y Fo(1)1202 364 y Fr(j)26 b(\024)82 b Fp(C)7 b Fr(j)h Fs(~)-53 b Fp(\032)p Fr(j)t Fs(^)k Fp(r)s Fr(h)p Fs(x)p Fr(i)p Fp(;)1007 547 y Fr(j)p Fp(@)1080 561 y Fn(r)1122 547 y Fs(~)h Fp(v)1163 561 y Fo(1)1202 547 y Fr(j)26 b(\024)82 b Fp(C)7 b Fr(j)p Fp(@)1551 561 y Fn(r)1598 547 y Fs(~)-54 b Fp(\032)p Fr(j)t Fs(^)-49 b Fp(r)s Fr(h)p Fs(x)p Fr(i)21 b Fs(+)f Fp(C)7 b Fr(h)p Fs(x)p Fr(ij)h Fs(~)-53 b Fp(\032)p Fr(j)p Fp(;)1001 731 y Fr(j)p Fp(@)1079 698 y Fo(2)1074 754 y Fn(r)1122 731 y Fs(~)-48 b Fp(v)1163 745 y Fo(1)1202 731 y Fr(j)26 b(\024)82 b Fp(C)7 b Fr(j)p Fp(@)1556 698 y Fo(2)1551 754 y Fn(r)1604 731 y Fs(~)-53 b Fp(\032)p 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y(The)f(follo)m(wing)i (notations)f(are)f(needed)g(to)h(form)m(ulate)g(the)g(limiting)g (absorption)f(principle.)40 b(Let)28 b Fr(\000)3677 2778 y Fn(@)3718 2755 y Fh(2)p 3660 2793 110 4 v 3660 2847 a Fn(@)t(r)3735 2828 y Fh(2)0 2951 y Fs(b)s(e)44 b(the)h(Laplacian)g (on)f Fp(L)940 2918 y Fo(2)980 2951 y Fs(\(I)-15 b(R)1100 2915 y Fo(+)1159 2951 y Fp(;)15 b Fs(d)p Fp(r)s Fs(\))44 b(with)g(Diric)m(hlet)i(condition)g(at)f(0,)j(and)2828 2927 y(~)2831 2951 y Fp(b)h Fs(:=)g(\()p Fr(\000)3197 2915 y Fn(@)3238 2891 y Fh(2)p 3180 2930 V 3180 2984 a Fn(@)t(r)3255 2965 y Fh(2)3299 2951 y Fs(\))3344 2888 y Fh(1)p 3345 2900 31 4 v 3345 2941 a(2)3389 2951 y Fs(.)83 b(W)-8 b(e)46 b(set)0 3077 y Fp(b)25 b Fs(:=)g(1)-20 b(l)236 3091 y Fw(K)314 3077 y Fr(\012)20 b Fp(T)471 3044 y Fw(\000)p Fo(1)562 3053 y Fs(~)565 3077 y Fp(bT)13 b Fs(,)31 b(where)f Fp(T)38 b Fs(:)25 b Fq(h)h Fr(!)1314 3059 y Fs(~)1315 3077 y Fq(h)k Fs(is)h(de\014ned)e(in)h(\(2.2\).)141 3215 y(W)-8 b(e)23 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y Fg(|)p 2476 2892 V 140 w({z)p 2690 2892 V 140 w(})2589 2975 y Fn(n)p Fw(\000)p Fn(j)2884 2850 y Fp(:)0 3164 y Fs(Since)33 b Fp(b)f Fs(is)h(closeable,) i Fp(b)851 3131 y Fw(\003)923 3164 y Fs(is)d(densely)h(de\014ned.)46 b(Moreo)m(v)m(er,)35 b(it)f(is)e(easy)h(to)h(see)f(that)g(d\000)3081 3179 y Fo(\014n)3163 3164 y Fs(\()p Fp(b)3237 3131 y Fw(\003)3277 3164 y Fs(\))c Fr(\032)f Fs(d\000)3548 3179 y Fo(\014n)3630 3164 y Fs(\()p Fp(b)p Fs(\))3739 3131 y Fw(\003)0 3277 y Fs(whic)m(h)i(implies)h(that)g(d\000)874 3292 y Fo(\014n)956 3277 y Fs(\()p Fp(b)p Fs(\))g(is)f(closeable)i(and) e(w)m(e)h(will)g(denote)g(b)m(y)f(d\000\()p Fp(b)p Fs(\))g(its)h (closure.)141 3390 y(F)-8 b(or)26 b(later)g(use)f(w)m(e)g(extend)g(the) g(meaning)g(of)h(the)f(op)s(eration)g(d\000)f(as)i(follo)m(ws.)40 b(Let)25 b Fp(S)30 b Fr(2)25 b(B)s Fs(\()p Fr(L)3297 3404 y Fo(1)3345 3390 y Fr(\012)9 b Fq(h)p Fp(;)15 b Fr(L)3571 3404 y Fo(2)3621 3390 y Fr(\012)9 b Fq(h)p Fs(\))0 3503 y(\(un)m(b)s(ounded)36 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Fr(\012)f Fs(\()p Fp(\017)h Fs(+)f Fr(j)p Fp(a)p Fr(j)p Fs(\))h(+)f(\()p Fp(\017)g Fs(+)g Fr(j)p Fp(a)p Fr(j)p Fs(\))h Fr(\012)f Fs(\()p Fp(\017)g Fs(+)g Fr(j)p Fp(a)3010 5216 y Fw(\003)3050 5249 y Fr(j)3075 5155 y Fg(\021)3125 5249 y Fp(u)p Fs(\))p Fp(:)1844 5517 y Fs(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 162 a Fs(Letting)36 b Fp(\017)e Fr(!)f Fs(0)j(w)m(e)f(obtain)h Fl(i\))p Fs(.)56 b(If)35 b Fp(a)g Fs(is)g(symmetric)h(then)f Fp(aa)2203 129 y Fw(\003)2276 162 y Fr(\024)e Fp(a)2428 129 y Fw(\003)2467 162 y Fp(a)j Fs(and)e(hence)i Fr(j)p Fp(a)3063 129 y Fw(\003)3103 162 y Fr(j)d(\024)g(j)p Fp(a)p Fr(j)p Fs(,)38 b(since)d(the)0 288 y(function)30 b Fp(\025)25 b Fr(!)h Fp(\025)614 225 y Fh(1)p 613 237 31 4 v 613 279 a(2)688 288 y Fs(is)31 b(matrix)f(monotone.)42 b(Next)291 457 y Fp(a)339 424 y Fw(\003)399 457 y Fr(\012)20 b Fp(a)g Fs(+)g Fp(a)g Fr(\012)g Fp(a)856 424 y Fw(\003)921 457 y Fr(\024)83 b(j)p Fp(a)1148 424 y Fw(\003)1187 457 y Fr(j)21 b(\012)f(j)p Fp(a)p Fr(j)h 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b(\001)g(\001)21 b(\012)f Fs(1)-20 b(l)1424 2059 y Fg(|)p 1461 2059 140 10 v 140 w({z)p 1675 2059 V 140 w(})1577 2141 y Fn(j)t Fw(\000)p Fo(1)1868 2016 y Fr(\012)p Fp(a)20 b Fr(\012)g Fs(1)-20 b(l)21 b Fr(\012)f(\001)15 b(\001)g(\001)21 b(\012)f Fs(1)-20 b(l)2098 2059 y Fg(|)p 2135 2059 V 140 w({z)p 2349 2059 V 140 w(})2248 2141 y Fn(n)p Fw(\000)p Fn(j)2528 2016 y Fp(;)0 2315 y Fs(acting)32 b(on)e Fr(\012)470 2282 y Fn(n)470 2337 y Fo(s)517 2315 y Fq(h)p Fs(.)41 b(Then)29 b(it)i(su\016ces)f(to)h(pro)m(v)m(e)887 2509 y(\()p Fp(a)970 2472 y Fw(\003)970 2532 y Fo(1)1030 2509 y Fs(+)20 b Fp(:)15 b(:)g(:)21 b Fs(+)f Fp(a)1386 2472 y Fw(\003)1386 2532 y Fn(n)1433 2509 y Fs(\)\()p Fp(a)1551 2523 y Fo(1)1611 2509 y Fs(+)g Fp(:)15 b(:)g(:)21 b Fs(+)f Fp(a)1967 2523 y Fn(n)2014 2509 y Fs(\))26 b Fr(\024)f Fs(\()p Fr(j)p Fp(a)p Fr(j)2304 2523 y Fo(1)2364 2509 y Fs(+)20 b Fp(:)15 b(:)g(:)22 b Fs(+)e Fr(j)p Fp(a)p Fr(j)2771 2523 y Fn(n)2818 2509 y Fs(\))2853 2472 y Fo(2)0 2703 y Fs(as)31 b(forms)e(on)i Fr(\012)560 2670 y Fn(n)560 2726 y(s)606 2703 y Fq(h)p Fs(.)41 b(But)312 2897 y(\()p Fp(a)395 2860 y Fw(\003)395 2920 y Fo(1)455 2897 y Fs(+)20 b Fp(:)15 b(:)g(:)21 b Fs(+)f Fp(a)811 2860 y Fw(\003)811 2920 y Fn(n)858 2897 y Fs(\)\()p Fp(a)976 2911 y Fo(1)1036 2897 y Fs(+)g Fp(:)15 b(:)g(:)21 b Fs(+)f Fp(a)1392 2911 y Fn(n)1439 2897 y Fs(\))26 b(=)1596 2817 y Fg(X)1644 2999 y Fn(i)1731 2897 y Fp(a)1779 2860 y Fw(\003)1779 2920 y Fn(i)1819 2897 y Fp(a)1867 2911 y Fn(i)1915 2897 y Fs(+)2006 2817 y Fg(X)2010 2999 y Fn(i)d Fs(0)33 b Fl(let)354 4432 y Fp(C)419 4446 y Fo(1)458 4432 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\))27 b(=)e Fr(k)p Fs(\(1)-20 b(l)908 4446 y Fw(K)986 4432 y Fr(\012)20 b Fp(!)1137 4394 y Fw(\000)1202 4367 y Fh(1)p 1202 4379 V 1202 4420 a(2)1246 4432 y Fs(\))p Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b Fp(r)s Fs(\))1638 4394 y Fw(\000)1703 4367 y Fh(1)p 1703 4379 V 1703 4420 a(2)1747 4432 y Fr(k)1792 4394 y Fo(2)1858 4432 y Fs(:=)25 b(lim)1990 4492 y Fn(")p Fw(#)p 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Fo(\000\()p Fm(h)p Fo(\))2852 4360 y Fp(f)10 b Fr(k)2952 4323 y Fo(2)2991 4360 y Fp(:)0 4632 y Fs(This)30 b(\014nishes)f(the)h (pro)s(of)g(of)g(the)h(prop)s(osition.)40 b Fd(2)0 4894 y Fe(Corollary)35 b(3.10)46 b Fl(L)-5 b(et)34 b Fp(K)39 b Fl(and)34 b Fp(!)h Fl(b)-5 b(e)33 b(p)-5 b(ositive)34 b(self-adjoint)g(op)-5 b(er)g(ators)36 b(on)d Fr(K)h Fl(and)g Fq(h)f Fl(r)-5 b(esp)g(e)g(ctively)35 b(and)e(let)0 5007 y Fp(v)j Fr(2)d(B)s Fs(\()p Fr(D)s Fs(\()p Fp(K)474 4944 y Fh(1)p 474 4956 V 474 4997 a(2)518 5007 y Fs(\))p Fp(;)15 b Fr(K)26 b(\012)d Fq(h)p Fs(\))p Fl(.)55 b(F)-7 b(or)38 b Fp(r)e(>)d Fs(0)k Fl(let)g Fp(C)1577 5021 y Fo(1)1617 5007 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\))38 b Fl(b)-5 b(e)37 b(de\014ne)-5 b(d)38 b(by)f(\(3.14\).)56 b(Then)38 b(for)f(al)5 b(l)38 b Fp(f)k Fr(2)33 b(D)s Fs(\()p Fp(a)3587 4974 y Fw(\003)3626 5007 y Fs(\()p Fp(v)s Fs(\)\))0 5120 y Fl(one)g(has)697 5233 y Fr(j)p Fs(\()p Fp(f)5 b(;)15 b(a)895 5195 y Fw(\003)935 5233 y Fs(\()p Fp(v)s Fs(\))p Fp(f)10 b Fs(\))p Fr(j)26 b(\024)f Fp(C)1354 5247 y Fo(1)1393 5233 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\))p Fr(k)p Fs(\()p Fp(K)30 b Fs(+)20 b Fp(r)s Fs(\))1955 5168 y Fh(1)p 1955 5180 V 1955 5221 a(2)2019 5233 y Fr(\012)g Fs(1)-20 b(l)2161 5251 y Fo(\000\()p Fm(h)p Fo(\))2300 5233 y Fp(f)10 b Fr(kk)p Fs(1)-20 b(l)2496 5247 y Fw(K)2574 5233 y Fr(\012)20 b Fs(d\000\()p Fp(!)s Fs(\))2913 5168 y Fh(1)p 2913 5180 V 2913 5221 a(2)2958 5233 y Fp(f)10 b Fr(k)p Fp(:)-3083 b Fs(\(3.21\))1844 5517 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 0 162 a Fi(3.5)112 b(Additional)39 b(remarks)f(on)g(the)f(spaces)h Fc(B)s Fx(\()p Fc(L)p Fb(;)17 b Fc(K)24 b(\012)e Fq(h)p Fx(\))38 b Fi(and)g Fc(B)s Fx(\()p Fc(K)q Fb(;)17 b Fc(L)2985 126 y Fw(\003)3047 162 y Fc(\012)23 b Fq(h)p Fx(\))0 334 y Fs(W)-8 b(e)33 b(\014rst)e(giv)m(e)i(an)f(alternativ)m(e)i(description)d(of)h(the)g (spaces)g Fr(B)s Fs(\()p Fr(L)2289 348 y Fo(1)2328 334 y Fp(;)15 b Fr(L)2431 348 y Fo(2)2492 334 y Fr(\012)21 b Fq(h)p Fs(\))32 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1239 y Fs(\)\))0 1352 y(is)47 b(bijectiv)m(e)h(and)e (isometric.)90 b(Note)48 b(that)f Fr(L)1663 1366 y Fo(2)1733 1352 y Fr(\012)31 b Fp(L)1897 1319 y Fo(2)1936 1352 y Fs(\(I)-15 b(R)2056 1316 y Fn(d)2097 1352 y Fs(\))52 b(=)g Fp(L)2369 1319 y Fo(2)2409 1352 y Fs(\(I)-15 b(R)2529 1316 y Fn(d)2569 1352 y Fs(;)15 b Fr(L)2672 1366 y Fo(2)2712 1352 y Fs(\).)89 b(Observ)m(e)46 b(that)i(the)e(sub-)0 1465 y(space)40 b Fp(L)314 1432 y Fo(2)354 1465 y Fs(\(I)-15 b(R)474 1429 y Fn(d)514 1465 y Fs(;)15 b Fr(B)s Fs(\()p Fr(L)715 1479 y Fo(1)754 1465 y Fp(;)g Fr(L)857 1479 y Fo(2)897 1465 y Fs(\)\))41 b(de\014ned)d(b)m(y)i(the)g(condition)2040 1396 y Fg(R)2080 1493 y Fo(I)-12 b(R)2145 1468 y Ff(d)2201 1465 y Fr(k)p Fp(v)s Fs(\()p Fp(k)s Fs(\))p Fr(k)2458 1432 y Fo(2)2499 1465 y Fs(d)p Fp(k)44 b(<)d Fr(1)e Fs(is)h(a)g(strict) h(subspace)e(of)0 1578 y Fp(L)62 1545 y Fo(2)62 1601 y Fn(w)118 1578 y Fs(\(I)-15 b(R)238 1542 y Fn(d)279 1578 y Fs(;)15 b Fr(B)s Fs(\()p Fr(L)480 1592 y Fo(1)519 1578 y Fp(;)g 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Fs(and)f(this)g(map)g(is)h(Hilb)s(ert-Sc)m(hmidt.)141 2981 y(In)f(Subsection)g(4.2,)i(w)m(e)e(will)h(need)f(to)h(consider)g (the)f(op)s(erator)h Fp(v)2435 2948 y Fw(y)2496 2981 y Fr(2)25 b(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(B)s Fs(\()p Fq(h)p Fp(;)g Fr(L)3034 2948 y Fw(\003)3074 2981 y Fs(\)\))31 b(de\014ned)e(b)m(y)1157 3173 y Fp(v)1204 3136 y Fw(y)1239 3173 y Fs(\()p Fp( )s Fs(\)\()p Fp(h)p Fs(\))f(:=)d Fp(v)1689 3136 y Fw(\003)1729 3173 y Fs(\()p Fp( )f Fr(\012)c Fp(h)p Fs(\))p Fp(;)36 b( )29 b Fr(2)c(K)q Fp(;)36 b(h)26 b Fr(2)e Fq(h)p Fp(:)0 3366 y Fs(Note)32 b(that)f(since)f Fp(v)686 3333 y Fw(\003)751 3366 y Fr(2)25 b(B)s Fs(\()p Fr(K)d(\012)e Fq(h)p Fp(;)15 b Fr(L)1263 3333 y Fw(\003)1303 3366 y Fs(\),)31 b Fp(v)1441 3333 y Fw(y)1507 3366 y Fs(b)s(elongs)f(indeed)g(to)h Fr(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(B)s Fs(\()p Fq(h)p Fp(;)g Fr(L)2684 3333 y Fw(\003)2724 3366 y Fs(\)\).)141 3479 y(Assume)44 b(no)m(w)g(additionally)h(that)g Fr(L)j(\032)g(K)d Fs(densely)f(and)g(that)h Fp(v)51 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(L)2978 3446 y Fw(\003)3047 3479 y Fr(\012)29 b Fq(h)p Fs(\).)83 b(Then)43 b Fp(v)3631 3446 y Fw(\003)3719 3479 y Fr(2)0 3592 y(B)s Fs(\()p Fr(L)20 b(\012)g Fq(h)p Fp(;)15 b Fr(K)q Fs(\))33 b(so)e(the)g(op)s(erator)g Fp(v)1176 3559 y Fw(y)1243 3592 y Fs(b)s(elongs)f(also)i(to)g Fr(B)s Fs(\()p Fr(L)p Fp(;)15 b Fr(B)s Fs(\()p Fq(h)p Fp(;)g Fr(K)q Fs(\)\).)43 b(Th)m(us)30 b Fp(v)2732 3559 y Fw(y)2799 3592 y Fs(b)s(elongs)h(to)g (the)g(last)h(space)0 3704 y(in)f(\(3.23\))j(and)d(in)g(fact)h(it)g(do) s(es)f(not,)h(in)g(general,)g(b)s(elong)g(to)g(the)g(other)f(ones,)i (as)e(the)h(follo)m(wing)h(example)0 3817 y(sho)m(ws.)39 b(Cho)s(ose)27 b Fp(')e Fr(2)g(K)j Fs(and)e Fp(J)35 b Fr(2)25 b(B)s Fs(\()p Fr(K)q Fp(;)15 b Fq(h)p Fs(\))28 b(and)e(set)h Fp(v)s Fs(\()p Fp(u)p Fs(\))f(=)f Fp(')13 b Fr(\012)g Fp(J)c Fs(\()p Fp(u)p Fs(\))28 b(for)e Fp(u)f Fr(2)g(K)q Fs(.)40 b(Then)26 b Fp(v)i Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)f(\012)f Fq(h)p Fs(\))0 3930 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Fs(d)p Fp(k)p 2673 5174 181 4 v 2673 5257 a(!)s Fs(\()p Fp(k)s Fs(\))2864 5195 y Fp(:)1844 5517 y Fs(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 0 162 a Fs(The)37 b(op)s(erator)h Fp(v)i Fs(from)d(part)g Fl(ii\))h Fs(of)f(the)h(prop)s(osition)f(is)g (iden)m(ti\014ed)h(with)f(a)h(strongly)f(measurable)h(map)0 288 y Fp(v)s Fs(\()p Fr(\001)p Fs(\))27 b(:)e(I)-15 b(R)304 253 y Fn(d)369 288 y Fr(!)26 b(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(D)s Fs(\()p Fp(K)896 225 y Fh(1)p 896 237 31 4 v 896 279 a(2)941 288 y Fs(\))976 255 y Fw(\003)1015 288 y Fs(\))31 b(and)891 517 y Fp(C)956 531 y Fo(2)995 517 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\))27 b(=)151 b(sup)1313 598 y Fn( )r Fw(2L)1457 607 y Fh(1)1492 598 y Fw(k)p Fn( )r Fw(k)p Fo(=1)1716 402 y Fg(Z)1762 590 y Fo(I)-12 b(R)1828 566 y Ff(d)1883 517 y Fr(k)p Fs(\()p Fp(K)28 b Fs(+)20 b Fp(r)s Fs(\))2238 479 y Fw(\000)2303 452 y Fh(1)p 2303 464 V 2303 505 a(2)2347 517 y Fp(v)s Fs(\()p Fp(k)s Fs(\))p Fp( )s Fr(k)2621 479 y Fo(2)2713 455 y Fs(d)p Fp(k)p 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b(w)m(e)f(\014nd)f(a)h (real)h(n)m(um)m(b)s(er)e Fp(c)p Fs(\()p Fp(")p Fs(\))i(suc)m(h)f(that) g Fp(\036)p Fs(\()p Fp(v)s Fs(\))3062 5060 y Fo(2)3133 5093 y Fr(\024)c Fp("H)3358 5060 y Fo(2)3351 5116 y(0)3420 5093 y Fs(+)22 b Fp(c)p Fs(\()p Fp(")p Fs(\))34 b(as)0 5206 y(forms)c(on)g Fr(D)s Fs(\()p Fp(H)561 5220 y Fo(0)600 5206 y Fs(\))21 b Fr(\\)e(H)813 5221 y Fo(\014n)896 5206 y Fs(.)40 b(Finally)-8 b(,)32 b(use)e(the)h(fact)g(that)g Fr(D)s Fs(\()p Fp(H)2162 5220 y Fo(0)2201 5206 y Fs(\))21 b Fr(\\)f(H)2415 5221 y Fo(\014n)2527 5206 y Fs(is)31 b(a)f(core)i(for)e Fp(H)3102 5220 y Fo(0)3141 5206 y Fs(.)40 b Fd(2)1844 5517 y Fs(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 0 162 a Fi(4.2)112 b(Essen)m(tial)38 b(sp)s(ectrum)g(of)g(abstract)f(P)m(auli-Fierz)i(Hamiltonians)0 334 y Fs(Our)24 b(next)h(purp)s(ose)e(is)i(to)h(get)g(a)f(description)g (of)h(the)f(essen)m(tial)h(sp)s(ectrum)e(of)h Fp(H)32 b Fs(under)24 b(general)i(conditions.)0 447 y(F)-8 b(or)31 b(this)g(w)m(e)g(need)f(t)m(w)m(o)i(tec)m(hnical)g(lemmas.)42 b(The)30 b(\014rst)f(one)i(con)m(tains)h(an)e(alternativ)m(e)j (description)e(of)f(the)0 560 y(condition)h Fe(\(Ia1\))e Fs(\(condition)j Fe(\(Ia2\))d Fs(can)i(b)s(e)e(expressed)h (similarly\).)0 785 y Fe(Lemma)35 b(4.5)46 b Fl(L)-5 b(et)33 b Fp(v)28 b Fr(2)d(B)s Fs(\()p Fr(D)s Fs(\()p Fp(K)1165 722 y Fh(1)p 1165 734 31 4 v 1165 776 a(2)1209 785 y Fs(\))p Fp(;)15 b Fr(K)23 b(\012)d Fq(h)p Fs(\))p Fl(.)42 b(Then)33 b Fs(lim)1978 799 y Fn(r)r Fw(!)p Fo(+)p Fw(1)2227 785 y Fp(C)2292 799 y Fo(1)2332 785 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\))27 b(=)e(0)33 b Fl(if)f(and)i(only)f(if)1030 1005 y Fs(lim)1005 1054 y Fn(r)r Fw(!1)1195 1005 y Fr(k)p Fs(\(1)-20 b(l)1327 1019 y Fw(K)1405 1005 y Fr(\012)20 b Fp(!)1556 967 y Fw(\000)1621 940 y Fh(1)p 1621 952 V 1621 993 a(2)1665 1005 y Fs(\))p Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2058 967 y Fw(\000)2123 940 y Fh(1)p 2124 952 V 2124 993 a(2)2168 1005 y Fs(1)-20 b(l)2219 1023 y Fo([)p Fn(r)n(;)p 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b(e)32 b(ha)m(v)m(e)306 1731 y Fr(k)p Fp(!)411 1694 y Fw(\000)476 1667 y Fh(1)p 476 1679 V 476 1720 a(2)521 1731 y Fp(v)s Fs(\()p Fp(K)27 b Fs(+)20 b(1\))878 1694 y Fw(\000)943 1667 y Fh(1)p 944 1679 V 944 1720 a(2)988 1731 y Fs(1)-20 b(l)1039 1745 y Fn(r)1077 1731 y Fr(k)1122 1694 y Fo(2)1187 1731 y Fs(=)25 b Fr(k)p Fp(!)1388 1694 y Fw(\000)1453 1667 y Fh(1)p 1453 1679 V 1453 1720 a(2)1498 1731 y Fp(v)s Fs(\()p Fp(K)i Fs(+)20 b(1\))1855 1694 y Fw(\000)p Fo(1)1950 1731 y Fs(1)-20 b(l)2001 1745 y Fn(r)2039 1731 y Fp(v)2086 1694 y Fw(\003)2125 1731 y Fp(!)2185 1694 y Fw(\000)2250 1667 y Fh(1)p 2250 1679 V 2250 1720 a(2)2295 1731 y Fr(k)25 b(\024)g(k)p Fp(!)2566 1694 y Fw(\000)2631 1667 y Fh(1)p 2631 1679 V 2631 1720 a(2)2676 1731 y Fp(v)s Fs(2\()p Fp(K)j Fs(+)20 b Fp(r)s Fs(\))3078 1694 y Fw(\000)p Fo(1)3172 1731 y Fp(v)3219 1694 y Fw(\003)3259 1731 y Fp(!)3319 1694 y Fw(\000)3384 1667 y Fh(1)p 3384 1679 V 3384 1720 a(2)3428 1731 y Fr(k)0 1935 y Fs(hence)31 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Fp(\033)s Fs(\()p Fp(H)7 b Fs(\))35 b(and)f(then)g(the)h(second)g (relation)g(giv)m(es)h(norm)0 5268 y(con)m(v)m(ergence)d(in)d Fr(B)s Fs(\()p Fr(D)s Fs(\()p Fp(H)908 5183 y Fh(1)p 908 5195 V 908 5236 a(2)891 5293 y Fo(0)952 5268 y Fs(\))987 5235 y Fw(\003)1027 5268 y Fp(;)15 b Fr(D)s Fs(\()p Fp(H)1268 5183 y Fh(1)p 1268 5195 V 1268 5236 a(2)1251 5293 y Fo(0)1313 5268 y Fs(\)\).)1844 5517 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 141 162 a Fs(Prop)s(osition)31 b(4.1)g(giv)m(es)832 348 y Fr(\006)p Fp(\036)p Fs(\()p Fp(v)1036 362 y Fn(\027)1079 348 y Fs(\))26 b Fr(\024)1236 269 y(p)p 1311 269 46 4 v 1311 348 a Fs(2)q Fp(C)1422 362 y Fo(1)1461 348 y Fs(\()p Fp(r)m(;)15 b(v)1618 362 y Fn(\027)1663 348 y Fs(\)\()p Fp(H)1809 362 y Fo(0)1869 348 y Fs(+)20 b Fp(r)s Fs(\))25 b Fr(\024)2160 269 y(p)p 2235 269 V 2235 348 a Fs(2)q Fp(C)2346 362 y Fo(1)2385 348 y Fs(\()p Fp(r)m(;)15 b(v)s Fs(\)\()p Fp(H)2691 362 y Fo(0)2753 348 y Fs(+)20 b Fp(r)s Fs(\))p Fp(:)0 534 y Fs(W)-8 b(e)31 b(c)m(ho)s(ose)h Fp(r)h Fs(con)m(v)m(enien)m(tly)f (and)e(\014nd)f(a)h(n)m(um)m(b)s(er)f Fp(b)h Fs(suc)m(h)g(that)h Fr(\006)p Fp(\036)p Fs(\()p Fp(v)2481 548 y Fn(\027)2525 534 y Fs(\))25 b Fr(\024)2691 498 y Fo(1)p 2691 513 36 4 v 2691 566 a(2)2736 534 y Fp(H)2812 548 y Fo(0)2871 534 y Fs(+)20 b Fp(b)30 b Fs(for)g(all)i Fp(\027)6 b Fs(.)40 b(It)30 b(follo)m(ws)0 647 y(easily)40 b(that)f(one)g(can)g(c)m (ho)s(ose)g(a)g(n)m(um)m(b)s(er)f Fp(a)g Fs(suc)m(h)g(that)i Fp(H)2122 614 y Fn(\027)2190 647 y Fs(+)26 b Fp(a)39 b Fr(\025)f Fp(H)2559 661 y Fo(0)2624 647 y Fs(+)25 b(1)39 b(for)g(all)g Fp(\027)6 b Fs(.)65 b(W)-8 b(e)40 b(shall)f(tak)m(e)0 790 y Fp(z)42 804 y Fo(0)112 790 y Fs(=)30 b Fr(\000)p Fp(a)p Fs(.)49 b(The)33 b(op)s(erator)h Fp(H)1049 757 y Fn(\027)1125 790 y Fs(has)f Fr(D)s Fs(\()p Fp(H)1491 705 y Fh(1)p 1491 717 31 4 v 1491 758 a(2)1474 814 y Fo(0)1536 790 y Fs(\))h(as)f(form)g(domain)g(so)h Fp(H)2461 757 y Fn(\027)2526 790 y Fs(+)22 b Fp(a)34 b Fs(extends)f(to)h(an)f (isomorphism)0 939 y Fr(D)s Fs(\()p Fp(H)201 855 y Fh(1)p 201 867 V 201 908 a(2)184 964 y Fo(0)245 939 y Fs(\))38 b Fr(!)e(D)s Fs(\()p Fp(H)646 855 y Fh(1)p 646 867 V 646 908 a(2)629 964 y Fo(0)691 939 y Fs(\))726 906 y Fw(\003)765 939 y Fs(.)62 b(Also)38 b Fp(H)31 b Fr(\000)25 b Fp(H)1350 906 y Fn(\027)1430 939 y Fs(=)36 b Fp(\036)p Fs(\()p Fp(v)29 b Fr(\000)24 b Fp(v)1838 953 y Fn(\027)1882 939 y Fs(\))37 b(holds)g(in)g Fr(B)s Fs(\()p Fr(D)s Fs(\()p Fp(H)2611 855 y Fh(1)p 2611 867 V 2611 908 a(2)2594 964 y Fo(0)2655 939 y Fs(\))p Fp(;)15 b Fr(D)s Fs(\()p Fp(H)2932 855 y Fh(1)p 2932 867 V 2932 908 a(2)2914 964 y Fo(0)2976 939 y Fs(\))3011 906 y Fw(\003)3051 939 y Fs(\).)62 b(Th)m(us,)38 b(if)f(w)m(e)h(set)0 1052 y Fp(R)70 1019 y Fn(\027)138 1052 y Fs(=)25 b(\()p Fp(H)352 1019 y Fn(\027)416 1052 y Fs(+)20 b Fp(a)p Fs(\))590 1019 y Fw(\000)p Fo(1)715 1052 y Fs(and)30 b Fp(R)c Fs(=)e(\()p Fp(H)k Fs(+)20 b Fp(a)p Fs(\))1395 1019 y Fw(\000)p Fo(1)1490 1052 y Fs(,)30 b(w)m(e)h(ha)m(v)m(e:)890 1273 y Fp(R)960 1235 y Fn(\027)1023 1273 y Fr(\000)20 b Fp(R)26 b Fs(=)f Fp(R)1375 1235 y Fn(\027)1418 1273 y Fp(\036)p Fs(\()p Fp(v)f Fr(\000)c Fp(v)1710 1287 y Fn(\027)1753 1273 y Fs(\))p Fp(R)79 b Fs(in)e Fr(B)s Fs(\()p Fr(D)s Fs(\()p Fp(H)2388 1188 y Fh(1)p 2388 1200 V 2388 1242 a(2)2371 1298 y Fo(0)2433 1273 y Fs(\))2468 1235 y Fw(\003)2507 1273 y Fp(;)15 b Fr(D)s Fs(\()p Fp(H)2749 1188 y Fh(1)p 2749 1200 V 2749 1242 a(2)2731 1298 y Fo(0)2793 1273 y Fs(\)\))p Fp(:)0 1474 y Fs(Let)31 b Fp(S)f Fs(=)25 b(\()p Fp(H)456 1488 y Fo(0)516 1474 y Fs(+)20 b(1\))697 1411 y Fh(1)p 697 1423 V 697 1464 a(2)742 1474 y Fs(.)41 b(W)-8 b(e)31 b(get)404 1660 y Fr(k)p Fp(S)5 b Fs(\()p Fp(R)615 1623 y Fn(\027)679 1660 y Fr(\000)19 b Fp(R)q Fs(\))p Fp(S)5 b Fr(k)26 b(\024)f(k)p Fp(S)5 b(R)1278 1623 y Fn(\027)1321 1660 y Fp(S)g Fr(kk)p Fp(S)1533 1623 y Fw(\000)p Fo(1)1629 1660 y Fp(\036)p Fs(\()p Fp(v)24 b Fr(\000)c Fp(v)1921 1674 y Fn(\027)1964 1660 y Fs(\))p Fp(S)2060 1623 y Fw(\000)p Fo(1)2155 1660 y Fr(kk)p Fp(S)5 b(R)q(S)g Fr(k)26 b(\024)f(k)p Fp(S)2710 1623 y Fw(\000)p Fo(1)2805 1660 y Fp(\036)p Fs(\()p Fp(v)f Fr(\000)c Fp(v)3097 1674 y Fn(\027)3140 1660 y Fs(\))p Fp(S)3236 1623 y Fw(\000)p Fo(1)3330 1660 y Fr(k)0 1846 y Fs(where)40 b(w)m(e)h(used)f Fp(R)704 1860 y Fn(\027)789 1846 y Fr(\024)i Fs(\()p Fp(H)1013 1860 y Fo(0)1079 1846 y Fs(+)27 b(1\))1257 1813 y Fw(\000)p Fo(1)1394 1846 y Fs(=)42 b Fp(S)1568 1813 y Fw(\000)p Fo(2)1703 1846 y Fs(hence)f(0)h Fr(\024)g Fp(S)5 b(R)2297 1813 y Fn(\027)2340 1846 y Fp(S)47 b Fr(\024)42 b Fs(1.)72 b(No)m(w)41 b(observ)m(e)g(that)g(w)m(e)g(ha)m(v)m(e)0 1959 y Fr(k)p Fp(S)106 1926 y Fw(\000)p Fo(1)201 1959 y Fp(\036)p Fs(\()p Fp(v)32 b Fr(\000)d Fp(v)510 1973 y Fn(\027)553 1959 y Fs(\))p Fp(S)649 1926 y Fw(\000)p Fo(1)744 1959 y Fr(k)47 b(\024)g Fp(\022)997 1973 y Fn(\027)1083 1959 y Fs(if)c Fr(\006)p Fp(\036)p Fs(\()p Fp(v)32 b Fr(\000)d Fp(v)1559 1973 y Fn(\027)1602 1959 y Fs(\))47 b Fr(\024)g Fp(\022)1845 1973 y Fn(\027)1888 1959 y Fs(\()p Fp(H)1999 1973 y Fo(0)2067 1959 y Fs(+)28 b(1\).)80 b(F)-8 b(rom)44 b(Prop)s(osition)g(4.1)g(w)m(e)g(get)g Fp(\022)3619 1973 y Fn(\027)3709 1959 y Fr(\024)0 1997 y(p)p 76 1997 46 4 v 75 x Fs(2)p Fp(C)186 2086 y Fo(1)226 2072 y Fs(\(1)p Fp(;)15 b(v)24 b Fr(\000)c Fp(v)549 2086 y Fn(\027)593 2072 y Fs(\))30 b(hence)h(the)f(pro)s(of)g(of)g(the)h(lemma)g(is)f (\014nished)f(if)i(w)m(e)f(sho)m(w)h(that)554 2273 y Fp(C)619 2287 y Fo(1)658 2273 y Fs(\(1)p Fp(;)15 b(v)25 b Fr(\000)20 b Fp(v)982 2287 y Fn(\027)1025 2273 y Fs(\))26 b(=)f Fr(k)p Fs(\(1)-20 b(l)1313 2287 y Fw(K)1391 2273 y Fr(\012)20 b Fp(!)1542 2235 y Fw(\000)1607 2208 y Fh(1)p 1607 2220 31 4 v 1607 2261 a(2)1651 2273 y Fs(\)\()p Fp(v)25 b Fr(\000)20 b Fp(v)1925 2287 y Fn(\027)1968 2273 y Fs(\)\()p Fp(K)27 b Fs(+)20 b(1\))2313 2235 y Fw(\000)2378 2208 y Fh(1)p 2379 2220 V 2379 2261 a(2)2424 2273 y Fr(k)2469 2235 y Fo(2)2534 2273 y Fr(!)25 b Fs(0)31 b(when)e Fp(\027)i Fr(!)25 b Fs(0)p Fp(:)-3225 b Fs(\(4.12\))0 2459 y(W)-8 b(e)32 b(shall)e(use)g(the)h(notations)g(in)m(tro)s(duced)f (in)g(the)h(pro)s(of)e(of)i(Lemma)f(4.5.)42 b(F)-8 b(or)31 b Fp(r)d(>)d Fs(0)31 b(w)m(e)g(ha)m(v)m(e)494 2653 y Fr(k)p Fp(!)599 2617 y Fw(\000)664 2590 y Fh(1)p 664 2602 V 664 2643 a(2)708 2653 y Fs(\()p Fp(v)24 b Fr(\000)c Fp(v)946 2667 y Fn(\027)990 2653 y Fs(\)\()p Fp(K)27 b Fs(+)20 b(1\))1335 2617 y Fw(\000)1400 2590 y Fh(1)p 1401 2602 V 1401 2643 a(2)1445 2653 y Fr(k)26 b Fs(=)f Fr(k)p Fp(!)1717 2617 y Fw(\000)1782 2590 y Fh(1)p 1782 2602 V 1782 2643 a(2)1827 2653 y Fp(v)s(\027)6 b(K)h Fs(\(1)20 b(+)g Fp(\027)6 b(K)h Fs(\))2370 2620 y Fw(\000)p Fo(1)2464 2653 y Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2775 2617 y Fw(\000)2840 2590 y Fh(1)p 2840 2602 V 2840 2643 a(2)2885 2653 y Fr(k)340 2782 y(\024)83 b(k)p Fp(!)599 2746 y Fw(\000)664 2719 y Fh(1)p 664 2731 V 664 2772 a(2)708 2782 y Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))1066 2746 y Fw(\000)1131 2719 y Fh(1)p 1132 2731 V 1132 2772 a(2)1176 2782 y Fs(1)-20 b(l)1227 2745 y Fw(?)1227 2804 y Fn(r)1286 2782 y Fp(\027)6 b(K)h Fs(\(1)20 b(+)g Fp(\027)6 b(K)h Fs(\))1782 2749 y Fw(\000)p Fo(1)1876 2782 y Fr(k)21 b Fs(+)f Fr(k)p Fp(!)2138 2746 y Fw(\000)2203 2719 y Fh(1)p 2203 2731 V 2203 2772 a(2)2247 2782 y Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2605 2746 y Fw(\000)2670 2719 y Fh(1)p 2671 2731 V 2671 2772 a(2)2715 2782 y Fs(1)-20 b(l)2766 2796 y Fn(r)2804 2782 y Fp(\027)6 b(K)h Fs(\(1)20 b(+)g Fp(\027)6 b(K)h Fs(\))3300 2749 y Fw(\000)p Fo(1)3394 2782 y Fr(k)340 2911 y(\024)83 b(k)p Fp(!)599 2875 y Fw(\000)664 2848 y Fh(1)p 664 2860 V 664 2901 a(2)708 2911 y Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))1066 2875 y Fw(\000)1131 2848 y Fh(1)p 1132 2860 V 1132 2901 a(2)1176 2911 y Fr(k)p Fp(\027)6 b(r)s Fs(\(1)21 b(+)f Fp(\027)6 b(r)s Fs(\))1638 2878 y Fw(\000)p Fo(1)1752 2911 y Fs(+)20 b Fr(k)p Fp(!)1948 2875 y Fw(\000)2013 2848 y Fh(1)p 2013 2860 V 2013 2901 a(2)2058 2911 y Fp(v)s Fs(\()p Fp(K)27 b Fs(+)20 b(1\))2415 2875 y Fw(\000)2480 2848 y Fh(1)p 2481 2860 V 2481 2901 a(2)2525 2911 y Fs(1)-20 b(l)2576 2925 y Fn(r)2614 2911 y Fr(k)p Fp(:)0 3089 y Fs(Th)m(us)776 3202 y(lim)15 b(sup)842 3276 y Fn(\027)t Fw(!)p Fo(0)1069 3202 y Fr(k)p Fp(!)1174 3164 y Fw(\000)1239 3137 y Fh(1)p 1239 3149 V 1239 3190 a(2)1284 3202 y Fs(\()p Fp(v)24 b Fr(\000)c Fp(v)1522 3216 y Fn(\027)1565 3202 y Fs(\)\()p Fp(K)28 b Fs(+)20 b(1\))1911 3164 y Fw(\000)1976 3137 y Fh(1)p 1976 3149 V 1976 3190 a(2)2021 3202 y Fr(k)25 b(\024)g(k)p Fp(!)2292 3164 y Fw(\000)2357 3137 y Fh(1)p 2358 3149 V 2358 3190 a(2)2402 3202 y Fp(v)s Fs(\()p Fp(K)j Fs(+)20 b(1\))2760 3164 y Fw(\000)2825 3137 y Fh(1)p 2825 3149 V 2825 3190 a(2)2870 3202 y Fs(1)-20 b(l)2920 3216 y Fn(r)2958 3202 y Fr(k)0 3406 y Fs(and)30 b(no)m(w)g(\(4.12\))j(follo)m(ws)e(from)f(Lemma)h(4.5.)41 b Fd(2)0 3613 y Fe(Remark)35 b(4.7)46 b Fs(W)-8 b(e)33 b(men)m(tion)f(the)f(follo)m(wing)i(consequence)f(of)g(\(4.7\):)44 b Fl(if)33 b Fr(f)p Fp(f)2680 3627 y Fn(n)2727 3613 y Fr(g)h Fl(is)f(a)h(se)-5 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5228 y Fw(\003)1515 5266 y Fs(\()p Fp(f)1595 5280 y Fn(n)1642 5266 y Fs(\))h Fr(\012)f Fs(1)-20 b(l)1839 5284 y Fo(\000\()p Fm(h)p Fo(\))1978 5266 y Fs(\)\()p Fp(H)28 b Fs(+)20 b Fp(c)p Fs(\))2317 5228 y Fw(\000)p Fo(1)2412 5266 y Fs(\()p Fp(H)28 b Fr(\000)20 b Fp(E)5 b Fs(\))p Fp(u)2801 5280 y Fn(")2838 5266 y Fr(k)1844 5517 y Fs(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 0 166 a Fs(whic)m(h)30 b(in)g(turn)f(is)i (smaller)g(than)f Fr(k)p Fp(v)s Fs(\()p Fp(K)e Fs(+)20 b(1\))1589 130 y Fw(\000)1654 103 y Fh(1)p 1654 115 31 4 v 1654 156 a(2)1699 166 y Fr(kk)p Fs(\()p Fp(H)28 b Fs(+)20 b Fp(c)p Fs(\))2093 133 y Fw(\000)p Fo(1)2188 166 y Fs(\()p Fp(H)28 b Fr(\000)20 b Fp(E)5 b Fs(\))p Fp(u)2577 180 y Fn(")2614 166 y Fr(k)26 b(\024)f Fp(C)2846 180 y Fo(4)2885 166 y Fp(")p Fs(.)41 b(Th)m(us)30 b(w)m(e)h(ha)m(v)m (e:)1161 367 y Fr(k)p Fs(\()p Fp(H)d Fs(+)20 b Fp(c)p Fs(\))1510 330 y Fw(\000)p Fo(1)1605 367 y Fp(a)1653 330 y Fw(\003)1692 367 y Fs(\()p Fp(f)1772 381 y Fn(n)1819 367 y Fs(\)\()p Fp(H)28 b Fr(\000)20 b Fp(E)5 b Fs(\))p Fp(u)2243 381 y Fn(")2280 367 y Fr(k)26 b(\024)f Fp(C)2512 381 y Fo(5)2551 367 y Fp(":)-2618 b Fs(\(4.16\))0 568 y(Using)31 b(\(4.7\))h(again)f(w)m(e)g(get:)532 784 y Fr(k)p Fp(a)625 747 y Fw(\003)665 784 y Fs(\(\()p Fp(!)24 b Fr(\000)c Fp(\025)p Fs(\))p Fp(f)1040 798 y Fn(n)1087 784 y Fs(\))p Fp(u)1174 798 y Fn(")1211 784 y Fr(k)26 b(\024)f Fp(C)7 b Fr(k)p Fs(\(1)21 b(+)e Fp(!)1746 747 y Fw(\000)1811 720 y Fh(1)p 1811 732 V 1811 773 a(2)1856 784 y Fs(\)\()p Fp(!)k Fr(\000)d Fp(\025)p Fs(\))p Fp(f)2230 798 y Fn(n)2277 784 y Fs(\))p Fr(kk)p Fs(\()p Fp(H)29 b Fs(+)20 b Fp(c)p Fs(\))2717 720 y Fh(1)p 2718 732 V 2718 773 a(2)2762 784 y Fp(u)2814 798 y Fn(")2851 784 y Fr(k)26 b(\024)f Fp(C)3083 798 y Fo(6)3122 784 y Fp(=n:)-3247 b Fs(\(4.17\))0 985 y(F)-8 b(rom)31 b(\(4.14\),)i(\(4.16\))f(and)e (\(4.17\))j(w)m(e)e(obtain)271 1228 y Fr(k)p Fs(\()p Fp(H)d Fs(+)20 b Fp(c)p Fs(\))620 1191 y Fw(\000)p Fo(1)715 1228 y Fs(\()p Fp(H)27 b Fr(\000)20 b Fp(E)26 b Fr(\000)20 b Fp(\025)p Fs(\))p Fp(a)1264 1191 y Fw(\003)1303 1228 y Fs(\()p Fp(f)1383 1242 y Fn(n)1430 1228 y Fs(\))p Fp(u)1517 1242 y Fn(")1554 1228 y Fr(k)26 b(\024)f Fp(C)1786 1242 y Fo(5)1825 1228 y Fp(")c Fs(+)1989 1167 y Fp(C)2054 1181 y Fo(6)p 1989 1207 105 4 v 2014 1291 a Fp(n)2123 1228 y Fs(+)2262 1167 y(1)p 2224 1207 122 4 v 2224 1224 a Fr(p)p 2300 1224 46 4 v 75 x Fs(2)2356 1228 y Fr(k)p Fs(\()p Fp(H)28 b Fs(+)19 b Fp(c)p Fs(\))2704 1191 y Fw(\000)p Fo(1)2799 1228 y Fp(v)2846 1191 y Fw(\003)2886 1228 y Fs(\()p Fp(f)2966 1242 y Fn(n)3013 1228 y Fs(\))i Fr(\012)e Fs(1)-20 b(l)3210 1247 y Fo(\000\()p Fm(h)p Fo(\))3349 1228 y Fp(u)3401 1242 y Fn(")3438 1228 y Fr(k)p Fp(:)-3508 b Fs(\(4.18\))0 1471 y(W)-8 b(e)32 b(no)m(w)f(sho)m(w)g (that)g(the)h(last)f(term)g(ab)s(o)m(v)m(e)h(con)m(v)m(erges)h(to)f (zero)g(when)e Fp(n)25 b Fr(!)h(1)p Fs(.)43 b(Since)31 b Fp(u)3173 1485 y Fn(")3236 1471 y Fr(2)25 b(E)39 b Fs(it)32 b(su\016ces)0 1584 y(to)44 b(pro)m(v)m(e)g(that)f Fr(k)p Fs(\()p Fp(H)37 b Fs(+)28 b Fp(c)p Fs(\))957 1548 y Fw(\000)1022 1521 y Fh(1)p 1022 1533 31 4 v 1022 1574 a(2)1067 1584 y Fs(\()p Fp(v)1149 1551 y Fw(\003)1189 1584 y Fs(\()p Fp(f)1269 1598 y Fn(n)1316 1584 y Fs(\))h Fr(\012)f Fs(1)-20 b(l)1530 1603 y Fo(\000\()p Fm(h)p Fo(\))1669 1584 y Fs(\)\()p Fp( )33 b Fr(\012)28 b Fp(g)s Fs(\))p Fr(k)48 b(!)e Fs(0)e(if)e Fp( )50 b Fr(2)c(D)s Fs(\()p Fp(K)2843 1521 y Fh(1)p 2843 1533 V 2843 1574 a(2)2887 1584 y Fs(\))e(and)e Fp(g)50 b Fr(2)c Fs(\000\()p Fq(h)p Fs(\).)79 b(But)0 1723 y(\()p Fp(K)28 b Fs(+)21 b(1\))h Fr(\012)f Fs(1)-20 b(l)476 1741 y Fo(\000\()p Fm(h)p Fo(\))643 1723 y Fr(\024)27 b Fp(H)817 1737 y Fo(0)877 1723 y Fs(+)21 b(1)28 b Fr(\024)f Fp(H)h Fs(+)21 b Fp(c)p Fs(,)32 b(hence)g(it)g(su\016ces)g(to)g(sho)m(w)f(that)i Fr(k)p Fs(\()p Fp(K)28 b Fs(+)21 b(1\))2983 1687 y Fw(\000)3048 1660 y Fh(1)p 3049 1672 V 3049 1713 a(2)3093 1723 y Fp(v)3140 1690 y Fw(\003)3180 1723 y Fs(\()p Fp( )k Fr(\012)c Fp(f)3436 1737 y Fn(n)3482 1723 y Fs(\))p Fr(k)28 b(!)g Fs(0.)0 1836 y(W)-8 b(e)32 b(use)e(Lemma)g(4.5)i(and)d(the)i(notations)g(from)f (its)h(pro)s(of:)216 2047 y Fr(k)p Fs(\()p Fp(K)d Fs(+)20 b(1\))572 2009 y Fw(\000)637 1982 y Fh(1)p 637 1994 V 637 2035 a(2)682 2047 y Fp(v)729 2009 y Fw(\003)769 2047 y Fs(\()p Fp( )k Fr(\012)c Fp(f)1023 2061 y Fn(n)1069 2047 y Fs(\))p Fr(k)84 b(\024)f(k)p Fs(\()p Fp(K)27 b Fs(+)20 b(1\))1742 2009 y Fw(\000)1807 1982 y Fh(1)p 1808 1994 V 1808 2035 a(2)1853 2047 y Fs(1)-20 b(l)1903 2009 y Fw(?)1903 2069 y Fn(r)1962 2047 y Fp(v)2009 2009 y Fw(\003)2049 2047 y Fs(\()p Fp( )24 b Fr(\012)c Fp(f)2303 2061 y Fn(n)2349 2047 y Fs(\))p Fr(k)h Fs(+)f Fr(k)p Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2897 2009 y Fw(\000)2962 1982 y Fh(1)p 2962 1994 V 2962 2035 a(2)3007 2047 y Fs(1)-20 b(l)3058 2061 y Fn(r)3095 2047 y Fp(v)3142 2009 y Fw(\003)3182 2047 y Fs(\()p Fp( )24 b Fr(\012)c Fp(f)3436 2061 y Fn(n)3483 2047 y Fs(\))p Fr(k)1233 2199 y(\024)83 b Fs(\()p Fp(r)23 b Fs(+)d(1\))1667 2135 y Fh(1)p 1667 2147 V 1667 2188 a(2)1712 2199 y Fr(k)p Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2068 2162 y Fw(\000)p Fo(1)2163 2199 y Fp(v)2210 2162 y Fw(\003)2250 2199 y Fs(\()p Fp( )k Fr(\012)19 b Fs(\()p Fp(!)24 b Fs(+)c(1\))2745 2162 y Fw(\000)p Fo(1)2840 2199 y Fs(\()p Fp(!)j Fs(+)d(1\))p Fp(f)3171 2213 y Fn(n)3219 2199 y Fs(\))p Fr(k)1233 2352 y Fs(+)83 b Fr(k)p Fs(\()p Fp(K)27 b Fs(+)20 b(1\))1742 2314 y Fw(\000)1807 2287 y Fh(1)p 1808 2299 V 1808 2340 a(2)1853 2352 y Fs(1)-20 b(l)1903 2366 y Fn(r)1941 2352 y Fp(v)1988 2314 y Fw(\003)2028 2352 y Fs(\(1)g(l)2114 2366 y Fw(K)2192 2352 y Fr(\012)20 b Fp(!)2343 2314 y Fw(\000)2408 2287 y Fh(1)p 2408 2299 V 2408 2340 a(2)2452 2352 y Fs(\))p Fr(kk)p Fp( )25 b Fr(\012)20 b Fs(\()p Fp(!)2857 2287 y Fh(1)p 2857 2299 V 2857 2340 a(2)2902 2352 y Fp(f)2947 2366 y Fn(n)2993 2352 y Fs(\))p Fr(k)p Fp(:)0 2566 y Fs(W)-8 b(e)28 b(ha)m(v)m(e)f Fr(k)p Fp(!)473 2503 y Fh(1)p 473 2515 V 473 2557 a(2)518 2566 y Fp(f)563 2580 y Fn(n)609 2566 y Fr(k)f(\024)f Fp(C)841 2580 y 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b(densit)m(y)f(this)h(yields:)1425 1604 y Fp(W)1524 1567 y Fw(\003)1511 1627 y Fn(t)1563 1604 y Fs(d\000\()p Fp(b)p Fs(\))p Fp(W)1866 1618 y Fn(t)1921 1604 y Fr(\024)25 b Fp(C)2082 1618 y Fn(t)2112 1604 y Fs(d\000\()p Fp(b)p Fs(\))p Fp(;)0 1761 y Fs(whic)m(h)30 b(implies)h(that)g Fp(W)852 1775 y Fn(t)906 1761 y Fs(:)26 b Fr(G)31 b(!)25 b(G)5 b Fs(.)41 b(Moreo)m(v)m(er)199 1956 y Fr(k)p Fp(W)330 1970 y Fn(t)360 1956 y Fp(u)p Fr(k)457 1919 y Fo(2)457 1979 y Fw(G)533 1956 y Fs(=)25 b(\()p Fp(W)750 1970 y Fn(t)780 1956 y Fp(u;)15 b Fs(d\000\()p Fp(b)p Fs(\))p Fp(W)1175 1970 y Fn(t)1205 1956 y Fp(u)p Fs(\))20 b(+)g(\()p Fp(u;)15 b(u)p Fs(\))27 b Fr(\024)e Fs(\()p Fp(C)1840 1970 y Fn(t)1890 1956 y Fs(+)20 b(1\))2061 1862 y Fg(\020)2111 1956 y Fs(\()p Fp(u;)15 b Fs(d\000\()p Fp(b)p Fs(\))p Fp(u)p Fs(\))21 b(+)f(\()p Fp(u;)15 b(u)p Fs(\))2868 1862 y Fg(\021)2944 1956 y Fs(=)25 b(\()p Fp(C)3140 1970 y Fn(t)3190 1956 y Fs(+)20 b(1\))p Fr(k)p Fp(u)p Fr(k)3503 1919 y Fo(2)3503 1979 y Fw(G)3555 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Fr(D)s Fs(\()p Fp(A)176 5235 y Fw(\003)176 5293 y(G)222 5275 y Fk(\003)262 5268 y Fs(\).)41 b(Then)29 b(\(4.27\))k(follo)m(ws)f(b)m(y)e(a)h (densit)m(y)f(argumen)m(t.)41 b Fd(2)1844 5517 y Fs(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 0 162 a Ft(5)135 b(The)44 b(Mourre)h(metho)t(d) 0 365 y Fs(In)31 b(this)h(section,)h(w)m(e)f(\014x)f(some)h (terminology)i(and)d(recall)i(the)e(main)h(results)g(from)f([GGM)q(].) 45 b(W)-8 b(e)33 b(refer)e(the)0 478 y(reader)f(to)h([GGM)r(])f(for)g (more)h(details)g(and)f(pro)s(ofs.)0 719 y Fi(5.1)112 b(The)38 b Fb(C)561 683 y Fo(1)600 719 y Fx(\()p Fb(A)p Fx(\))g Fi(class)0 890 y Fs(In)28 b(this)g(subsection)g(w)m(e)h(recall) h(the)e(de\014nition)h(of)f(the)h Fp(C)1983 857 y Fo(1)2022 890 y Fs(\()p Fp(A)p Fs(\))g(class)g(in)m(tro)s(duced)f(in)g([GGM)q(],) h(where)f Fp(A)h Fs(is)f(a)0 1003 y(closed)h(and)e(densely)g(de\014ned) g(op)s(erator.)40 b(This)27 b(de\014nition)g(is)h(an)f(extension)i(of)f (the)f(standard)g Fp(C)3391 970 y Fo(1)3430 1003 y Fs(\()p Fp(A)p Fs(\))i(class)0 1116 y(for)h Fp(A)h Fs(selfadjoin)m(t)g(\(see)g ([ABG)q(]\).)141 1229 y(In)f(all)h(this)f(subsection)h Fp(A)f Fs(will)h(b)s(e)f(a)g(closed)i(densely)e(de\014ned)f(op)s (erator)i(on)f(a)h(Hilb)s(ert)f(space)h Fr(H)q Fs(.)141 1342 y(W)-8 b(e)35 b(start)g(b)m(y)f(considering)g(the)h Fp(C)1365 1309 y Fo(1)1404 1342 y Fs(\()p Fp(A)p Fs(\))g(class)g(of)f (b)s(ounded)e(op)s(erators.)52 b(If)34 b Fp(S)i Fr(2)31 b(B)s Fs(\()p Fr(H)q Fs(\))j(w)m(e)h(denote)g(b)m(y)0 1455 y([)p Fp(A;)15 b(S)5 b Fs(])31 b(the)g(sesquilinear)g(form)e(on)i Fr(D)s Fs(\()p Fp(A)1413 1422 y Fw(\003)1452 1455 y Fs(\))21 b Fr(\002)f(D)s Fs(\()p Fp(A)p Fs(\))31 b(de\014ned)e(b)m(y:)701 1642 y(\()p Fp(u;)15 b Fs([)p Fp(A;)g(S)5 b Fs(])p Fp(v)s Fs(\))28 b(:=)d(\()p Fp(A)1381 1604 y Fw(\003)1421 1642 y Fp(u;)15 b(S)5 b(v)s Fs(\))22 b Fr(\000)e Fs(\()p Fp(S)1865 1604 y Fw(\003)1904 1642 y Fp(u;)15 b(Av)s Fs(\))p Fp(;)37 b(u)26 b Fr(2)e(D)s Fs(\()p Fp(A)2547 1604 y Fw(\003)2587 1642 y Fs(\))p Fp(;)36 b(v)29 b Fr(2)24 b(D)s Fs(\()p Fp(A)p Fs(\))p Fp(:)0 1829 y Fe(De\014nition)35 b(5.1)46 b Fl(A)n(n)34 b(op)-5 b(er)g(ator)37 b Fp(S)c Fr(2)28 b(B)s Fs(\()p Fr(H)q Fs(\))34 b Fl(is)g Fs(of)e(class)h Fp(C)2091 1796 y Fo(1)2130 1829 y Fs(\()p Fp(A)p Fs(\))i Fl(if)e(the)i(sesquiline)-5 b(ar)35 b(form)g Fs([)p Fp(A;)15 b(S)5 b Fs(])35 b Fl(is)f(c)-5 b(on-)0 1942 y(tinuous)35 b(for)f(the)h(top)-5 b(olo)g(gy)37 b(of)e Fr(H)22 b(\002)f(H)q Fl(.)46 b(If)35 b(this)f(is)h(the)f(c)-5 b(ase,)35 b(we)g(denote)g(by)f Fs([)p Fp(A;)15 b(S)5 b Fs(])2944 1909 y Fw(\016)3019 1942 y Fl(the)34 b(unique)g(b)-5 b(ounde)g(d)0 2055 y(op)g(er)g(ator)47 b(on)d Fr(H)g Fl(asso)-5 b(ciate)g(d)46 b(to)e(the)g(quadr)-5 b(atic)45 b(form)g Fs([)p Fp(A;)15 b(S)5 b Fs(])44 b Fl(\(note)g(that)h Fr(D)s Fs(\()p Fp(A)2873 2022 y Fw(\003)2913 2055 y Fs(\))28 b Fr(\002)g(D)s Fs(\()p Fp(A)p Fs(\))44 b Fl(is)g(dense)g(in)0 2167 y Fr(H)21 b(\002)f(H)q Fl(\).)41 b(We)32 b(denote)i(by)e Fp(C)1019 2135 y Fo(1)1058 2167 y 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v 25 w Fs(C)p Fr(n)p Fp(\033)s Fs(\()p Fp(S)5 b Fs(\))34 b Fl(such)e(that)i Fp(R)q Fs(\()p Fp(z)t Fs(\))26 b(:=)f(\()p Fp(S)h Fr(\000)20 b Fp(z)t Fs(\))2689 2907 y Fw(\000)p Fo(1)2816 2940 y Fl(is)33 b(of)g(class)g Fp(C)3311 2907 y Fo(1)3350 2940 y Fs(\()p Fp(A)p Fs(\))p Fl(.)0 3135 y Fe(De\014nition)i(5.3)46 b Fl(L)-5 b(et)29 b Fp(S)34 b Fl(b)-5 b(e)28 b(a)h(close)-5 b(d)30 b(and)g(densely)f (de\014ne)-5 b(d)30 b(op)-5 b(er)g(ator.)43 b(We)29 b(say)g(that)h Fp(S)j Fl(is)c Fs(of)d(class)k Fp(C)3602 3102 y Fo(1)3641 3135 y Fs(\()p Fp(A)p Fs(\))0 3248 y Fl(if)i(ther)-5 b(e)33 b(is)f(a)h(se)-5 b(quenc)g(e)32 b(of)h(c)-5 b(omplex)34 b(numb)-5 b(ers)33 b Fp(z)1713 3262 y Fn(\027)1781 3248 y Fr(2)25 b Fp(\032)p Fs(\()p Fp(S;)15 b(A)p Fs(\))34 b Fl(with)f Fr(j)p Fp(z)2446 3262 y Fn(\027)2490 3248 y Fr(j)25 b(!)g(1)32 b Fl(such)h(that)g Fr(k)p Fs(\()p Fp(z)3293 3262 y Fn(\027)3357 3248 y Fr(\000)19 b Fp(S)5 b Fs(\))3543 3215 y Fw(\000)p Fo(1)3638 3248 y Fr(k)26 b(\024)0 3361 y Fp(C)7 b Fr(j)p 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Fs(])1964 4851 y Fw(\016)2030 4884 y Fs(=)25 b Fp(H)2209 4851 y Fw(00)2251 4884 y Fl(.)1844 5517 y Fs(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 0 162 a Fi(5.4)112 b(Limiting)39 b(absorption)f(principle)0 334 y Fs(The)25 b(limiting)i(absorption)e (principle)g(in)g([GGM)r(])g(has)g(its)h(most)g(con)m(v)m(enien)m(t)i (form)m(ulation)e(if)g Fp(W)3325 348 y Fn(t)3380 334 y Fs(are)f(isome-)0 447 y(tries)43 b(and)f Fr(G)48 b Fs(is)43 b(b-stable)g(under)e Fr(f)p Fp(W)1380 414 y Fw(\003)1367 469 y Fn(t)1419 447 y Fr(g)p Fs(.)78 b(Then)41 b Fr(f)p Fp(W)1947 461 y Fn(t)1977 447 y Fr(g)i Fs(extends)g(to)g(a)g Fp(C)2685 461 y Fo(0)2724 447 y Fs(-semigroup)g(on)f Fr(G)3399 414 y Fw(\003)3439 447 y Fs(.)78 b(whose)0 560 y(generator)32 b(is)e(denoted)g(b)m(y)h Fp(A)1034 574 y Fw(G)1080 555 y Fk(\003)1120 560 y Fs(.)40 b(W)-8 b(e)32 b(set)f(for)f(0)c Fp(<)f(s)g(<)f Fs(1:)1256 764 y Fr(G)1315 726 y Fw(\003)1310 786 y Fn(s)1380 764 y Fs(:=)h Fr(D)s Fs(\()p Fr(j)p Fp(A)1702 778 y Fw(G)1748 759 y 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Fw(\000)p Fn(s)2669 1194 y Fs(can)i(b)s(e)e(obtained)h(b)m(y)g(complex)0 1307 y(in)m(terp)s(olation.)141 1420 y(In)h(the)g(sequel)h(w)m(e)g(set) g Fp(R)q Fs(\()p Fp(z)t Fs(\))26 b(=)f(\()p Fp(H)i Fr(\000)20 b Fp(z)t Fs(\))1576 1387 y Fw(\000)p Fo(1)1701 1420 y Fs(and)737 1624 y Fp(J)796 1586 y Fw(\006)787 1649 y Fo(0)880 1624 y Fs(=)25 b Fr(f)p Fp(\025)c Fr(\006)f Fs(i)p Fp(\026)p Fr(j)p Fp(\025)25 b Fr(2)g Fp(J)o(;)36 b(\026)25 b(>)g Fs(0)p Fr(g)p Fp(;)87 b(J)2002 1587 y Fw(\006)2086 1624 y Fs(=)25 b Fr(f)p Fp(\025)c Fr(\006)f Fs(i)p Fp(\026)p Fr(j)p Fp(\025)25 b Fr(2)g Fp(J)o(;)15 b(\026)26 b Fr(\025)f Fs(0)p Fr(g)p Fp(:)0 1837 y Fe(Theorem)70 b(5.15)46 b Fl(Assume)e(that)h(hyp)-5 b(otheses)47 b Fe(\(M1\))p Fl({)p Fe(\(M5\))d Fl(hold)h(and)g(that)g Fp(W)2949 1851 y Fn(t)3023 1837 y Fl(ar)-5 b(e)45 b(isometries)f(and)0 1950 y Fr(G)j Fl(is)42 b Fs(b-)p Fl(stable)g(under)g Fr(f)p Fp(W)963 1917 y Fw(\003)950 1972 y Fn(t)1002 1950 y Fr(g)p Fl(.)70 b(Then)42 b(if)f Fp(z)46 b Fr(2)c 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Fp(\016)948 1776 y Fw(\000)p Fo(1)1043 1809 y Fp(\037)1100 1776 y Fw(0)1123 1809 y Fs(\()1168 1773 y Fn(r)p 1168 1788 34 4 v 1168 1840 a(\016)1212 1809 y Fs(\))p Fp(d)p Fs(\()p Fp(\016)s Fs(\))e Fr(\000)d Fp(\016)1564 1776 y Fw(\000)p Fo(1)1659 1809 y Fp(\037)1716 1776 y Fw(0)1739 1809 y Fs(\()1784 1773 y Fn(r)p 1784 1788 V 1784 1840 a(\016)1828 1809 y Fs(\))p Fp(d)p Fs(\()p Fp(r)s Fs(\))h(+)f(\(1)h Fr(\000)f Fp(\037)p Fs(\)\()2465 1773 y Fn(r)p 2465 1788 V 2465 1840 a(\016)2509 1809 y Fs(\))p Fp(d)2591 1776 y Fw(0)2615 1809 y Fs(\()p Fp(r)s Fs(\))p Fp(;)458 1993 y Fs(\()p Fp(m)573 1960 y Fn(\016)573 2015 y(n)620 1993 y Fs(\()p Fp(r)s Fs(\)\))769 1960 y Fw(0)818 1993 y Fs(=)25 b Fp(d)p Fs(\()p Fp(\016)s Fs(\))p Fp(\016)1117 1960 y Fw(\000)p Fo(1)1214 1993 y Fp(\037)1271 1960 y Fw(0)1294 1993 y Fs(\()1339 1957 y Fn(r)p 1339 1972 V 1339 2024 a(\016)1383 1993 y Fs(\))1519 1957 y Fn(r)p 1428 1972 215 4 v 1428 2026 a(r)r Fo(+)p Fn(n)1560 2007 y Fk(\000)p Fh(1)1673 1993 y Fs(+)20 b(\(\(1)h Fr(\000)f Fp(\037)p Fs(\)\()2128 1957 y Fn(r)p 2128 1972 34 4 v 2128 2024 a(\016)2172 1993 y Fs(\))p Fp(d)p Fs(\()p Fp(r)s Fs(\)\))2403 1960 y Fw(0)2448 1993 y Fs(+)g Fp(\037)p Fs(\()2641 1957 y Fn(r)p 2641 1972 V 2641 2024 a(\016)2685 1993 y Fs(\))p Fp(d)p Fs(\()p Fp(\016)s Fs(\))p Fp(r)2924 1960 y Fw(\000)p Fo(1)3120 1957 y Fn(nr)p 3030 1972 257 4 v 3030 2026 a Fo(\()p Fn(nr)r Fo(+1\))3251 2007 y Fh(2)3296 1993 y Fp(:)0 1909 y Fs(\(6.7\))0 2205 y(W)-8 b(e)32 b(\014rst)d(observ)m(e)i(that)g(since)g Fp(r)d Fr(\024)d Fp(\016)34 b Fs(on)c(supp)14 b Fp(\037)p Fs(:)212 2446 y Fp(m)292 2409 y Fn(\016)330 2446 y Fs(\()p Fp(r)s Fs(\))25 b Fr(\024)g Fp(d)p Fs(\()p Fp(r)s Fs(\))p Fp(;)57 b Fr(j)p Fs(\(1)21 b Fr(\000)f Fp(\037)p Fs(\)\()1162 2385 y Fp(r)p 1162 2425 44 4 v 1162 2509 a(\016)1216 2446 y Fs(\))p Fp(d)1298 2409 y Fw(0)1322 2446 y Fs(\()p Fp(r)s Fs(\))p Fr(j)26 b(\024)1593 2385 y Fp(C)p 1593 2425 72 4 v 1607 2509 a(r)1674 2446 y(d)p Fs(\()p Fp(r)s Fs(\))p Fp(;)57 b Fr(j)p Fp(\016)1985 2409 y Fw(\000)p Fo(1)2080 2446 y Fp(\037)2137 2409 y Fw(0)2160 2446 y Fs(\()2205 2385 y Fp(r)p 2205 2425 44 4 v 2205 2509 a(\016)2259 2446 y Fs(\))p Fp(d)p Fs(\()p Fp(\016)s Fs(\))p Fr(j)28 b(\024)d Fp(\016)2646 2409 y Fw(\000)p Fo(1)2741 2446 y Fr(j)p Fp(\037)2823 2409 y Fw(0)2847 2446 y Fs(\()2892 2385 y Fp(r)p 2892 2425 V 2892 2509 a(\016)2946 2446 y Fs(\))p Fp(d)p Fs(\()p Fp(r)s Fs(\))p Fr(j)h(\024)3299 2385 y Fp(C)p 3299 2425 72 4 v 3313 2509 a(r)3381 2446 y(d)p Fs(\()p Fp(r)s Fs(\))p Fp(;)-3567 b Fs(\(6.8\))0 2687 y(uniformly)29 b(in)h(0)c Fp(<)f(\016)k Fr(\024)862 2651 y Fo(1)p 862 2666 36 4 v 862 2718 a(2)907 2687 y Fs(.)41 b(This)30 b(yields:)1018 2941 y Fp(m)1098 2904 y Fn(\016)1098 2964 y(n)1145 2941 y Fs(\()p Fp(r)s Fs(\))c Fr(\024)f Fp(m)1461 2904 y Fn(\016)1498 2941 y Fs(\()p Fp(r)s Fs(\))h Fr(\024)f Fp(d)p Fs(\()p Fp(r)s Fs(\))p Fp(;)56 b Fr(j)p Fs(\()p Fp(m)2116 2904 y Fn(\016)2116 2964 y(n)2164 2941 y Fs(\()p Fp(r)s Fs(\)\))2313 2904 y Fw(0)2336 2941 y Fr(j)26 b(\024)2493 2880 y Fp(C)p 2493 2920 72 4 v 2507 3004 a(r)2574 2941 y(d)p Fs(\()p Fp(r)s Fs(\))p Fp(:)-2760 b Fs(\(6.9\))0 3165 y(Next:)763 3278 y Fp(m)843 3241 y Fn(\016)843 3301 y(n)889 3278 y Fs(\()p Fp(r)s Fs(\))26 b Fr(!)f Fp(m)1225 3241 y Fn(\016)1263 3278 y Fs(\()p Fp(r)s Fs(\))p Fp(;)56 b Fs(\()p Fp(m)1573 3241 y Fn(\016)1573 3301 y(n)1620 3278 y Fs(\()p Fp(r)s Fs(\)\))1769 3241 y Fw(0)1818 3278 y Fr(!)25 b Fs(\()p Fp(m)2049 3241 y Fn(\016)2087 3278 y Fs(\()p Fp(r)s Fs(\)\))2236 3241 y Fw(0)2291 3278 y Fs(a.e.)41 b(when)30 b Fp(n)25 b Fr(!)g(1)p Fp(;)-3017 b Fs(\(6.10\))0 3445 y(using)30 b(\(6.2\))i(and)e(\(6.7\).) 141 3571 y(Let)h(us)f(no)m(w)h(pro)m(v)m(e)g Fl(i\))p Fs(.)42 b(W)-8 b(e)32 b(set)i(~)-48 b Fp(v)29 b Fs(=)c(\(1)-20 b(l)1542 3585 y Fw(K)1621 3571 y Fr(\012)20 b Fp(T)13 b Fs(\))p Fp(v)s Fs(,)31 b Fp(w)e Fs(=)f(~)-48 b Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2464 3536 y Fw(\000)2529 3508 y Fh(1)p 2529 3520 31 4 v 2529 3562 a(2)2574 3571 y Fs(.)42 b(It)30 b(su\016ces)h(then)f(to)h(pro)m(v)m(e)h(that)0 3697 y Fp(w)39 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)27 b(\012)d Fp(H)683 3664 y Fo(1)676 3721 y(0)723 3697 y Fs(\(I)-15 b(R)842 3662 y Fo(+)902 3697 y Fs(\))25 b Fr(\012)f Fp(L)1119 3664 y Fo(2)1158 3697 y Fs(\()p Fp(S)1254 3664 y Fn(d)p Fw(\000)p Fo(1)1385 3697 y Fs(\)\))38 b(and)e(that)j(~)-46 b Fp(a)1928 3664 y Fn(\016)1966 3697 y Fp(w)39 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)27 b(\012)d(D)s Fs(\()p Fp(r)2718 3662 y Fw(\000)2783 3634 y Fh(1)p 2782 3646 V 2782 3688 a(2)2827 3697 y Fs(\)\).)61 b(That)38 b(~)-46 b Fp(a)3265 3664 y Fn(\016)3303 3697 y Fs(\()p Fp(K)32 b Fs(+)24 b(1\))3622 3662 y Fw(\000)3687 3634 y Fh(1)p 3688 3646 V 3688 3688 a(2)3736 3697 y Fs(~)-49 b Fp(v)0 3824 y Fs(b)s(elongs)30 b(to)h Fr(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)23 b(\012)d(D)s Fs(\()p Fp(r)980 3788 y Fw(\000)1045 3761 y Fh(1)p 1044 3773 V 1044 3814 a(2)1089 3824 y Fs(\)\))31 b(can)g(b)s(e)e(pro)m(v)m(ed)i(similarly)g 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Fr(2)25 b(K)q Fs(,)j(\()p Fp( )566 4190 y Fo(2)606 4176 y Fp(;)15 b(w)r( )772 4190 y Fo(1)812 4176 y Fs(\))847 4190 y Fw(K)931 4176 y Fr(2)25 b Fp(C)1089 4143 y Fo(0)1128 4176 y Fs(\(I)-15 b(R)1248 4140 y Fo(+)1307 4176 y Fs(\))12 b Fr(\012)g Fp(L)1499 4143 y Fo(2)1538 4176 y Fs(\()p Fp(S)1634 4143 y Fn(d)p Fw(\000)p Fo(1)1765 4176 y Fs(\),)27 b(and)f(hence)g(for)g Fp(r)i Fr(\025)d Fs(0)i(the)f(expression)g(\()p Fp( )3325 4190 y Fo(2)3365 4176 y Fp(;)15 b(w)r( )3531 4190 y Fo(2)3572 4176 y Fs(\))3607 4190 y Fw(K)3665 4176 y Fs(\()p Fp(r)s Fs(\))0 4289 y(is)38 b(w)m(ell)i(de\014ned)d(as)h(an)g(elemen)m(t)i(of)f Fp(L)1384 4256 y Fo(2)1423 4289 y Fs(\()p Fp(S)1519 4256 y Fn(d)p Fw(\000)p Fo(1)1650 4289 y Fs(\).)65 b(It)38 b(su\016ces)g(to)h(sho)m (w)f(that)h(\()p Fp( )2849 4303 y Fo(2)2889 4289 y Fp(;)15 b(w)r( )3055 4303 y Fo(2)3096 4289 y Fs(\))3131 4303 y Fw(K)3189 4289 y Fs(\(0\))40 b(=)e(0)h(for)f(all)0 4402 y Fp( )59 4416 y Fo(1)99 4402 y Fp(;)15 b( )198 4416 y Fo(2)269 4402 y Fr(2)31 b(K)36 b Fs(to)f(pro)m(v)m(e)g(that)g Fp(w)f Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)25 b(\012)e Fp(H)1701 4369 y Fo(1)1694 4425 y(0)1740 4402 y Fs(\(I)-15 b(R)1860 4366 y Fo(+)1919 4402 y Fs(\))23 b Fr(\012)f Fp(L)2132 4369 y Fo(2)2172 4402 y Fs(\()p Fp(S)2268 4369 y Fn(d)p Fw(\000)p Fo(1)2399 4402 y Fs(\)\).)52 b(If)34 b(there)g(exists)h Fp( )3191 4416 y Fo(1)3231 4402 y Fp(;)15 b( )3330 4416 y Fo(2)3404 4402 y Fs(suc)m(h)34 b(that)0 4515 y(\()p Fp( )94 4529 y Fo(2)134 4515 y Fp(;)15 b(w)r( )300 4529 y Fo(2)341 4515 y Fs(\))376 4529 y Fw(K)434 4515 y Fs(\(0\))33 b Fr(6)p Fs(=)f(0,)k(then)e Fr(k)p Fs(\()p Fp( )1141 4529 y Fo(2)1181 4515 y Fp(;)15 b(w)r( )1347 4529 y Fo(1)1388 4515 y Fs(\))1423 4529 y Fw(K)1481 4515 y Fs(\()p Fp(r)s Fs(\))p Fr(k)1640 4537 y Fn(L)1688 4518 y Fh(2)1724 4537 y Fo(\()p Fn(S)1798 4518 y Ff(d)p Fk(\000)p Fh(1)1912 4537 y Fo(\))1976 4515 y Fr(\025)31 b Fp(c)i(>)e Fs(0)k(for)f(0)e Fr(\024)g Fp(r)i Fr(\034)e Fs(1.)53 b(But)35 b(this)f(con)m(tradicts)0 4657 y(h)m(yp)s(othesis)c Fe(\(I2\))g Fs(whic)m(h)g(implies)g(that)h(\(1)21 b(+)f Fp(r)1653 4621 y Fw(\000)1718 4594 y Fh(1)p 1717 4606 V 1717 4647 a(2)1762 4657 y Fs(\))p Fp(r)1841 4624 y Fw(\000)p Fo(1)1935 4657 y Fp(d)p Fs(\()p Fp(r)s Fs(\)\()p Fp( )2190 4671 y Fo(2)2231 4657 y Fp(;)15 b(w)r( )2397 4671 y Fo(1)2437 4657 y Fs(\))2472 4671 y Fw(K)2531 4657 y Fs(\()p Fp(r)s Fs(\))25 b Fr(2)g Fp(L)2818 4624 y Fo(2)2858 4657 y Fs(\(I)-15 b(R)2978 4621 y Fo(+)3037 4657 y Fs(\))20 b Fr(\012)g Fp(L)3245 4624 y Fo(2)3284 4657 y Fs(\()p Fp(S)3380 4624 y Fn(d)p Fw(\000)p Fo(1)3511 4657 y Fs(\).)141 4770 y(Since)41 b Fp(m)469 4737 y Fn(\016)506 4770 y Fs(\()p Fp(r)s Fs(\))i Fr(\024)f Fp(d)p Fs(\()p Fp(r)s Fs(\),)i Fr(j)p Fs(\()p Fp(m)1146 4737 y Fn(\016)1184 4770 y Fs(\()p Fp(r)s Fs(\)\))1333 4737 y Fw(0)1357 4770 y Fr(j)f(\024)e Fp(C)7 b Fs(\()p Fp(\016)s Fs(\),)45 b(it)c(follo)m(ws)h(then)e(from)g(h)m(yp)s(othesis)g Fe(\(I2\))g Fs(that)i(~)-46 b Fp(a)3571 4737 y Fn(\016)3609 4770 y Fp(w)45 b Fr(2)0 4896 y(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)23 b(\012)d(D)s Fs(\()p Fp(!)559 4860 y Fw(\000)624 4833 y Fh(1)p 623 4845 V 623 4886 a(2)668 4896 y Fs(\)\).)141 5009 y(Let)31 b(us)f(no)m(w)g(pro)m(v)m(e)h Fl(ii\))p Fs(.)41 b(Going)31 b(to)g(p)s(olar)f(co)s(ordinates)h(this)g(is)f (equiv)-5 b(alen)m(t)32 b(to)927 5228 y Fp(w)c Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)23 b(\012)d(D)s Fs(\()q(~)-46 b Fp(a)1653 5190 y Fn(\016)1653 5250 y(n)1700 5228 y Fs(\)\))p Fp(;)37 b Fs(~)-46 b Fp(a)1879 5190 y Fn(\016)1879 5250 y(n)1926 5228 y Fp(w)28 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)22 b(\012)e(D)s Fs(\()p Fp(r)2647 5190 y Fw(\000)2712 5163 y Fh(1)p 2712 5175 V 2712 5216 a(2)2756 5228 y Fs(\)\))p Fp(:)-2851 b Fs(\(6.11\))1844 5517 y(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 0 162 a Fs(Note)33 b(that)h(~)-46 b Fp(a)468 129 y Fn(\016)468 185 y(n)547 162 y Fs(is)32 b(the)h(closure)f(of)g(i\()p Fp(m)1349 129 y Fn(\016)1349 185 y(n)1397 162 y Fs(\()p Fp(r)s Fs(\))p Fp(@)1559 176 y Fn(r)1619 162 y Fs(+)1721 126 y Fo(1)p 1721 141 36 4 v 1721 194 a(2)1766 162 y Fs(\()p Fp(m)1881 129 y Fn(\016)1881 185 y(n)1928 162 y Fs(\()p Fp(r)s Fs(\))2042 129 y Fw(0)2066 162 y Fs(\))g(on)g Fp(C)2333 129 y Fw(1)2326 185 y Fo(0)2407 162 y Fs(\(I)-15 b(R)2527 126 y Fo(+)2586 162 y Fr(nf)p Fs(0)p Fr(g)p Fs(\))24 b Fr(\012)d Fp(C)2989 129 y Fw(1)3063 162 y Fs(\()p Fp(S)3159 129 y Fn(d)p Fw(\000)p Fo(1)3290 162 y Fs(\).)46 b(Using)32 b(the)0 275 y(fact)f(that)g Fp(m)456 242 y Fn(\016)456 298 y(n)503 275 y Fs(\()p Fp(r)s Fs(\))g(v)-5 b(anishes)30 b(at)h(0,)g(it)g(is)f(easy)h(to)g(sho) m(w)f(that)h Fr(D)s Fs(\()q(~)-46 b Fp(a)2288 242 y Fn(\016)2288 298 y(n)2335 275 y Fs(\))26 b(=)f Fr(f)p Fp(u)h Fr(2)2700 257 y Fs(~)2701 275 y Fq(h)p Fr(j)q Fs(~)-46 b Fp(a)2817 242 y Fn(\016)2817 298 y(n)2864 275 y Fp(u)25 b Fr(2)3026 257 y Fs(~)3027 275 y Fq(h)q Fr(g)p Fs(.)41 b(No)m(w)1272 520 y(~)-46 b Fp(a)1319 482 y Fn(\016)1319 542 y(n)1366 520 y Fp(w)28 b Fs(=)d(i)p Fp(m)1660 482 y Fn(\016)1660 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1348 y Fr(k)26 b(\024)f Fp(C)7 b Fr(k)p Fs(1)-20 b(l)2153 1362 y Fw(K)2231 1348 y Fr(\012)20 b Fp(!)2382 1310 y Fw(\000)2447 1283 y Fh(1)p 2447 1295 V 2447 1336 a(2)2492 1348 y Fp(h)p Fs(\()p Fp(K)27 b Fs(+)20 b(1\))2854 1310 y Fw(\000)2919 1283 y Fh(1)p 2920 1295 V 2920 1336 a(2)2964 1348 y Fr(k)-3009 b Fs(\(6.12\))0 1565 y(for)24 b Fp(h)i Fr(2)f(B)s Fs(\()p Fr(D)s Fs(\()p Fp(K)596 1502 y Fh(1)p 596 1514 V 596 1556 a(2)641 1565 y Fs(\))p Fp(;)15 b Fr(K)10 b(\012)f Fq(h)p Fs(\).)40 b(Using)24 b(\(6.9\),)k(\(6.10\))f (and)d(the)h(dominated)g(con)m(v)m(ergence)i(theorem,)f(w)m(e)f(obtain) 0 1678 y(that:)1036 1782 y(\(1)c(+)f Fp(r)1272 1746 y Fw(\000)1337 1719 y Fh(1)p 1336 1731 V 1336 1772 a(2)1381 1782 y Fs(\))p Fp(m)1496 1749 y Fn(\016)1496 1804 y(n)1543 1782 y Fs(\()p Fp(r)s Fs(\))p Fp(@)1705 1796 y Fn(r)1743 1782 y Fp(w)28 b Fr(!)d Fs(\(1)c(+)f Fp(r)2188 1746 y Fw(\000)2253 1719 y Fh(1)p 2253 1731 V 2253 1772 a(2)2297 1782 y Fs(\))p Fp(m)2412 1749 y Fn(\016)2450 1782 y Fs(\()p Fp(r)s 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y Fs(\()p Fp(r)s Fs(\))p Fr(k)43 b(\025)f Fp(c)h(>)f Fs(0)f(for)g(0)h Fr(\024)h Fp(r)h Fr(\034)f Fs(1.)72 b(But)41 b(this)f(con)m(tradicts)i(the)f (fact)h(that)0 4849 y(\(1)21 b(+)f Fp(r)236 4814 y Fw(\000)301 4786 y Fh(1)p 300 4798 31 4 v 300 4840 a(2)345 4849 y Fs(\))p Fp(d)p Fs(\()p Fp(r)s Fs(\))p Fp(@)589 4863 y Fn(r)628 4849 y Fp(w)28 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)22 b(\012)1196 4831 y Fs(~)1197 4849 y Fq(h)p Fs(\).)41 b Fd(2)1844 5517 y Fs(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 0 162 a Fi(6.4)112 b(The)38 b(semigroup)h(on)e (F)-9 b(o)s(c)m(k)38 b(space)0 334 y Fs(W)-8 b(e)28 b(no)m(w)e(extend)g (the)h Fp(C)847 348 y Fo(0)886 334 y Fs(-semigroup)g Fp(w)1415 301 y Fn(\016)1413 356 y(t)1479 334 y Fs(to)g Fr(H)g Fs(b)m(y)f(second)h(quan)m(tization,)i(as)e(in)f(Subsection)g (4.3.)40 b(W)-8 b(e)28 b(set:)1078 512 y Fp(W)1177 474 y Fn(\016)1164 534 y(t)1240 512 y Fs(:=)d(1)-20 b(l)1412 526 y Fw(K)1490 512 y Fr(\012)20 b Fs(\000\()p Fp(w)1740 474 y Fn(\016)1738 534 y(t)1779 512 y Fs(\))p Fp(;)56 b(W)1994 474 y Fn(\016)r Fw(\003)1981 534 y Fn(t)2092 512 y Fs(=)25 b(1)-20 b(l)2238 526 y Fw(K)2317 512 y Fr(\012)20 b Fs(\000\()p Fp(w)2567 474 y Fn(\016)r Fw(\003)2565 534 y Fn(t)2640 512 y Fs(\))p Fp(:)0 689 y Fs(Clearly)30 b Fp(W)414 656 y Fn(\016)401 712 y(t)482 689 y Fs(is)f(a)i Fp(C)713 703 y Fo(0)752 689 y Fs(-semigroup)f(of)g(isometries)h(on)f Fr(H)q Fs(.)40 b(W)-8 b(e)31 b(denote)f(b)m(y)g Fp(A)2656 656 y Fn(\016)2724 689 y Fs(its)g(generator.)42 b(Similarly)30 b(w)m(e)0 802 y(set)h(for)f Fp(n)25 b Fr(2)g Fs(I)-15 b(N:)1500 915 y Fp(A)1568 878 y Fn(\016)1568 938 y(n)1640 915 y Fs(:=)25 b(1)-20 b(l)1812 929 y Fw(K)1890 915 y Fr(\012)20 b Fs(d\000\()p Fp(a)2172 878 y Fn(\016)2172 938 y(n)2219 915 y Fs(\))p Fp(;)-2279 b Fs(\(6.13\))0 1078 y(whic)m(h)30 b(is)h(the)f(generator)i(of)e(the)h(unitary)f(group) g(1)-20 b(l)1802 1092 y Fw(K)1880 1078 y Fr(\012)20 b Fs(\000\(e)2103 1045 y Fo(i)p Fn(ta)2185 1022 y Ff(\016)2185 1062 y(n)2233 1078 y 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b(us)g(\014rst)f(pro)m(v)m(e)i Fl(i\))p Fs(.)40 b(T)-8 b(ak)m(e)30 b Fp(\024)1436 129 y Fw(0)1485 162 y Fr(\024)25 b Fp(\024=)p Fs(2)30 b(and)e(assume)h(the)g(con)m(trary)-8 b(.)42 b(Let)29 b Fp(u)3001 176 y Fn(n)3077 162 y Fs(b)s(e)f(eigen)m(v) m(ectors)k(of)0 275 y Fp(H)37 b Fs(with)30 b Fp(u)372 289 y Fn(n)445 275 y Fs(=)25 b(1)-20 b(l)591 294 y Fo([)p Fn(E)t Fw(\000)p Fn(\024)763 275 y Fk(0)784 294 y Fn(;E)t Fo(+)p Fn(\024)956 275 y Fk(0)977 294 y Fo(])1001 275 y Fs(\()p Fp(H)7 b Fs(\))p Fp(u)1206 289 y Fn(n)1253 275 y Fs(,)31 b Fp(u)1361 289 y Fn(n)1433 275 y Fr(!)25 b Fs(0)31 b(w)m(eakly)-8 b(.)42 b(Since)31 b Fp(f)2236 289 y Fn(E)t(;\024)2355 275 y Fs(\()p Fp(H)7 b Fs(\))p Fp(u)2560 289 y Fn(n)2633 275 y Fs(=)25 b Fp(u)2781 289 y Fn(n)2828 275 y Fs(,)31 b(w)m(e)f(obtain)h(b)m(y)g(\(7.1\):)1130 489 y(0)26 b(=)83 b(\()p Fp(u)1442 503 y Fn(n)1489 489 y Fp(;)15 b Fs(\()p Fp(M)31 b Fs(+)20 b Fp(R)q Fs(\))p Fp(u)1931 503 y Fn(n)1978 489 y Fs(\))1201 673 y(=)83 b(\()p Fp(u)1442 687 y 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y Fs(\()p Fp(H)7 b Fs(\)\))22 b(+)e(hc\))895 1551 y Fr(\025)1059 1515 y Fo(1)p 1059 1530 V 1059 1583 a(2)1104 1551 y Fs(\()p Fp(K)7 b(f)1268 1565 y Fn(E)t(;\024)1388 1551 y Fs(\()p Fp(H)g Fs(\))21 b(+)e(hc\))i Fr(\000)f Fp(\017)p Fr(k)p Fp(K)7 b Fr(k)2101 1518 y Fo(2)2161 1551 y Fr(\000)20 b Fp(\017)2289 1518 y Fw(\000)p Fo(1)2383 1551 y Fs(\(1)h Fr(\000)f Fp(f)2620 1565 y Fn(E)t(;\024)2739 1551 y Fs(\()p Fp(H)7 b Fs(\)\))2927 1518 y Fo(2)2968 1551 y Fp(;)0 1753 y Fs(whic)m(h)30 b(pro)m(v)m(es)h(the)g(statemen)m(t)h(since)f Fp(f)1388 1767 y Fn(E)t(;\024)1507 1753 y Fs(\()p Fp(H)7 b Fs(\))31 b(tends)f(strongly)h(to)g(0)g(when)e Fp(\024)d Fr(!)f Fs(0.)41 b Fd(2)141 1866 y Fs(The)30 b(follo)m(wing)i(prop)s (osition)e(is)g(an)h(abstract)g(v)m(ersion)g(of)f([MS,)h(Lemma)f(4.4].) 0 2053 y Fe(Prop)s(osition)36 b(7.6)46 b Fl(Assume)30 b(that)h(the)f(Mourr)-5 b(e)30 b(estimate)h(holds)g(at)g Fp(E)5 b Fl(.)41 b(Then)30 b Fr(8)14 b Fp(\017)2905 2067 y Fo(0)2969 2053 y Fp(>)25 b Fs(0)p Fl(,)31 b Fr(9)14 b Fp(C)q(;)h(\024)30 b Fl(such)g(that)877 2258 y Fp(M)g Fs(+)20 b Fp(f)1131 2272 y Fn(E)t(;\024)1250 2258 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1518 2272 y Fn(E)t(;\024)1638 2258 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)p Fp(\017)2021 2272 y Fo(0)2060 2258 y Fs(1)-20 b(l)21 b Fr(\000)e Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)2530 2272 y Fn(E)t(;\024)2649 2258 y Fs(\()p Fp(H)7 b Fs(\)\))2837 2220 y Fo(2)2878 2258 y Fp(:)0 2462 y Fe(Pro)s(of.)47 b Fs(If)30 b Fp(E)h Fr(62)25 b Fp(\033)656 2476 y Fo(pp)738 2462 y Fs(\()p Fp(H)7 b Fs(\))31 b(the)g(result)f(is)h(clear)g(b)m(y)f(Prop)s(osition) h(7.5)g Fl(ii\))p Fs(.)141 2592 y(Assume)g(no)m(w)h(that)h Fp(E)f Fr(2)c Fp(\033)1109 2606 y Fo(pp)1191 2592 y Fs(\()p Fp(H)7 b Fs(\))32 b(and)g(let)g Fp(P)41 b Fs(=)27 b(1)-20 b(l)1935 2610 y Fw(f)p Fn(E)t Fw(g)2065 2592 y Fs(\()p Fp(H)7 b Fs(\).)45 b(By)32 b(\(7.1\),)i Fp(P)41 b Fs(:)28 b Fr(H)g(!)f(D)s Fs(\()p Fp(M)3268 2529 y Fh(1)p 3268 2541 31 4 v 3268 2582 a(2)3312 2592 y Fs(\))32 b(and)f(hence)0 2730 y(\(1)21 b Fr(\000)f Fp(P)13 b Fs(\))26 b(:)f Fr(D)s Fs(\()p Fp(M)590 2667 y Fh(1)p 590 2679 V 590 2721 a(2)635 2730 y Fs(\))g Fr(!)g(D)s Fs(\()p Fp(M)1027 2667 y Fh(1)p 1027 2679 V 1027 2721 a(2)1072 2730 y Fs(\).)41 b(Moreo)m(v)m(er)33 b Fp(P)43 b Fs(is)30 b(compact)i(b)m(y)e(Prop)s(osition)h(7.5.)141 2843 y(Set)1192 2956 y Fp(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\))26 b(:=)g Fp(M)k Fs(+)20 b Fp(f)1902 2970 y Fn(E)t(;\024)2021 2956 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)2289 2970 y Fn(E)t(;\024)2409 2956 y Fs(\()p Fp(H)g Fs(\))p Fp(:)0 3141 y Fs(As)30 b(quadratic)h(forms)f(on)g Fr(D)s Fs(\()p Fp(M)1134 3078 y Fh(1)p 1134 3090 V 1134 3131 a(2)1179 3141 y Fs(\),)h(w)m(e)g(ha)m(v)m(e:)949 3342 y Fp(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\))26 b(=)83 b Fp(P)13 b(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\))p Fp(P)35 b Fs(+)19 b(2Re)q Fp(P)13 b(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\)\(1)23 b Fr(\000)d Fp(P)13 b Fs(\))1437 3526 y(+\(1)21 b Fr(\000)f Fp(P)13 b Fs(\))p Fp(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\)\(1)22 b Fr(\000)e Fp(P)13 b Fs(\))p Fp(:)0 3434 y Fs(\(7.2\))0 3727 y(W)-8 b(e)32 b(will)e(estimate)i(separately)g(the)f(three)f(terms)g(in)g(the) h(r.h.s)f(of)g(\(7.2\).)43 b(By)31 b(\(7.1\))h(w)m(e)f(ha)m(v)m(e)1568 3932 y Fp(P)13 b(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\))p Fp(P)40 b Fs(=)25 b(0)p Fp(:)-2211 b Fs(\(7.3\))1844 5517 y(43)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 0 162 a Fs(By)31 b(the)f(Mourre)g(estimate,)j (since)e(\(1)21 b Fr(\000)e Fp(P)13 b Fs(\)\(1)22 b Fr(\000)e Fp(f)1768 176 y Fn(E)t(;\024)1887 162 y Fs(\()p Fp(H)7 b Fs(\)\))26 b(=)f(\(1)c Fr(\000)f Fp(f)2434 176 y Fn(E)t(;\024)2554 162 y Fs(\()p Fp(H)7 b Fs(\)\):)744 351 y(\(1)21 b Fr(\000)f Fp(P)13 b Fs(\))p Fp(B)5 b Fs(\()p Fp(E)g(;)15 b(\024)p Fs(\)\(1)22 b Fr(\000)e Fp(P)13 b Fs(\))590 535 y Fr(\025)83 b(\000)p Fp(\017)852 549 y Fo(0)911 535 y Fr(\000)20 b Fs(\(1)h Fr(\000)f Fp(P)13 b Fs(\))p Fp(K)7 b Fs(\(1)21 b Fr(\000)f Fp(P)13 b Fs(\))20 b Fr(\000)g Fp(C)7 b 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Fs(\()p Fp(H)7 b Fs(\)\))p Fp(f)1770 3423 y Fn(E)t(;\024)1891 3409 y Fs(\()p Fp(H)g Fs(\)\(1)22 b Fr(\000)d Fp(P)13 b Fs(\))p Fp(;)46 b Fs(for)31 b Fp(\024)25 b Fr(\024)g Fs(2)p Fp(\024)2823 3423 y Fo(0)2863 3409 y Fp(;)0 3568 y Fs(and)36 b(use)h(that)h Fp(P)13 b(R)q(f)737 3582 y Fn(E)t(;\024)854 3591 y Fh(0)890 3568 y Fs(\()p Fp(H)7 b Fs(\))38 b(is)f(b)s(ounded,)g Fp(P)50 b Fs(is)37 b(compact)h(and)e Fp(f)2388 3582 y Fn(E)t(;\024)2508 3568 y Fs(\()p Fp(H)7 b Fs(\)\(1)26 b Fr(\000)e Fp(P)13 b Fs(\))37 b Fr(!)f Fs(0)h(strongly)g(when)0 3680 y Fp(\024)26 b Fr(!)f Fs(0)31 b(to)g(obtain:)1243 3793 y(Re)p Fp(I)1390 3807 y Fo(2)1455 3793 y Fr(\025)25 b(\000)p Fp(\017)1659 3807 y Fo(0)1698 3793 y Fp(;)46 b Fs(for)30 b Fp(\024)h Fs(small)f(enough.)-2536 b(\(7.5\))0 3952 y(Next)31 b(w)m(e)g(write)g Fp(I)629 3966 y Fo(1)698 3952 y Fs(as:)260 4127 y Fp(P)13 b(M)d Fs(\(1)21 b Fr(\000)f Fp(P)13 b Fs(\))26 b(=)82 b Fp(P)13 b Fs(1)-20 b(l)1028 4145 y Fo([)p Fn(n)1091 4154 y Fh(0)1125 4145 y Fn(;)p 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Fp(\032)2536 5201 y Fw(\000)p Fo(1)2631 5239 y Fp(M)d Fr(\000)2850 5177 y Fs(1)p 2850 5218 V 2850 5301 a(2)2905 5239 y Fp(\032)2952 5201 y Fw(\000)p Fo(1)3047 5239 y Fp(P)13 b(M)d(P)s(:)1844 5517 y Fs(44)p eop end %%Page: 45 45 TeXDict begin 45 44 bop 0 162 a Fs(W)-8 b(e)32 b(claim)f(that)1198 275 y(s-)61 b(lim)1279 325 y Fn(n)1322 334 y Fh(0)1356 325 y Fw(!1)1513 275 y Fp(P)13 b(M)1692 211 y Fh(1)p 1692 223 31 4 v 1692 264 a(2)1736 275 y Fs(1)-20 b(l)1787 294 y Fo([)p Fn(n)1850 303 y Fh(0)1884 294 y Fn(;)p Fw(1)p Fo([)1998 275 y Fs(\()p Fp(M)10 b Fs(\))p Fp(M)2275 211 y Fh(1)p 2275 223 V 2275 264 a(2)2319 275 y Fp(P)38 b Fs(=)25 b(0)p Fp(:)-2581 b Fs(\(7.6\))0 497 y(In)30 b(fact)i Fp(P)43 b Fs(sends)30 b Fr(H)i Fs(in)m(to)f Fr(D)s Fs(\()p Fp(M)1149 434 y Fh(1)p 1149 446 V 1149 487 a(2)1193 497 y Fs(\))g(hence)g Fp(M)1620 434 y Fh(1)p 1620 446 V 1620 487 a(2)1665 497 y Fp(P)43 b Fs(is)31 b(b)s(ounded)e(on)i Fr(H)g Fs(and)f(1)-20 b(l)2693 516 y Fo([)p Fn(n)2756 525 y Fh(0)2790 516 y 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Fs(is)30 b(compact,)h Fp(f)2035 1272 y Fn(E)t(;\024)2154 1258 y Fs(\()p Fp(H)7 b Fs(\)\(1)19 b Fr(\000)f Fp(P)13 b Fs(\))26 b Fr(!)f Fs(0)k(strongly)h(when)e Fp(\024)e Fr(!)f Fs(0,)30 b(so)0 1371 y(for)g(0)c Fp(<)f(\024)g(<)g(\024)531 1385 y Fo(0)571 1371 y Fs(,)31 b(w)m(e)g(ha)m(v)m(e:)1635 1484 y(Re)p Fp(R)1811 1498 y Fo(3)1876 1484 y Fr(\025)25 b(\000)p Fp(\017)2080 1498 y Fo(0)2119 1484 y Fp(:)-2144 b Fs(\(7.8\))0 1648 y(Finally)31 b(let)h(us)d(b)s(ound)g Fp(R)907 1662 y Fo(4)946 1648 y Fs(:)41 b(w)m(e)31 b(ha)m(v)m(e:)830 1880 y(ReR)1004 1894 y Fo(4)1069 1880 y Fr(\025)25 b(\000)1246 1819 y Fs(1)p 1246 1859 46 4 v 1246 1942 a(2)1301 1880 y Fp(\032P)13 b Fs(1)-20 b(l)1469 1899 y Fo([0)p Fn(;n)1587 1908 y Fh(0)1621 1899 y Fo([)1645 1880 y Fs(\()p Fp(M)10 b Fs(\))p Fp(M)1911 1843 y Fo(2)1951 1880 y Fp(P)33 b Fr(\000)2143 1819 y Fs(1)p 2143 1859 V 2143 1942 a(2)2199 1880 y Fp(\032)2246 1843 y Fw(\000)p Fo(1)2340 1880 y Fs(\(1)21 b Fr(\000)f Fp(f)2577 1894 y Fn(E)t(;\024)2696 1880 y Fs(\()p Fp(H)7 b Fs(\)\))2884 1843 y Fo(2)2925 1880 y Fp(:)-2950 b Fs(\(7.9\))0 2105 y(Collecting)36 b(\(7.3\)|\(7.9\),)k(w)m(e)35 b(obtain)g(\014nally)f(that)i(for)e(an)m (y)h Fp(\017)2221 2119 y Fo(0)2260 2105 y Fp(;)15 b(\033)36 b(>)c Fs(0)j(there)g(exist)g Fp(C)q(;)15 b(\024)3185 2119 y Fo(0)3260 2105 y Fs(suc)m(h)35 b(that)g(for)0 2217 y(0)26 b Fp(<)f(\024)g Fr(\024)g Fp(\024)392 2231 y Fo(0)432 2217 y Fs(:)747 2330 y Fp(M)30 b Fs(+)20 b Fp(f)1001 2344 y Fn(E)t(;\024)1120 2330 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1388 2344 y Fn(E)t(;\024)1508 2330 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)p Fs(8)p Fp(\017)1936 2344 y Fo(0)1996 2330 y Fr(\000)20 b Fp(\033)s(M)30 b Fr(\000)20 b Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)2660 2344 y Fn(E)t(;\024)2779 2330 y Fs(\()p Fp(H)7 b Fs(\)\))2967 2293 y Fo(2)3007 2330 y Fp(:)0 2494 y Fs(This)30 b(yields:)220 2730 y Fp(M)g Fs(+)20 b Fp(f)474 2744 y Fn(E)t(;\024)593 2730 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)861 2744 y Fn(E)t(;\024)981 2730 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)1382 2668 y Fs(8)p Fp(\017)1464 2682 y Fo(0)p 1337 2709 212 4 v 1337 2792 a Fs(1)20 b(+)g Fp(\033)1579 2730 y Fs(+)1758 2668 y Fp(\033)p 1679 2709 V 1679 2792 a Fs(1)h(+)f Fp(\033)1901 2730 y(f)1946 2744 y Fn(E)t(;\024)2066 2730 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)2334 2744 y Fn(E)t(;\024)2453 2730 y Fs(\()p Fp(H)g Fs(\))21 b Fr(\000)2798 2668 y Fp(C)p 2728 2709 V 2728 2792 a Fs(1)g(+)f Fp(\033)2950 2730 y Fs(\(1)h Fr(\000)f Fp(f)3187 2744 y Fn(E)t(;\024)3306 2730 y Fs(\()p Fp(H)7 b Fs(\)\))3494 2692 y Fo(2)3535 2730 y Fp(:)0 2962 y Fs(No)m(w)31 b(w)m(e)g(use)f(that)h Fr(k)p Fp(f)785 2976 y Fn(E)t(;\024)905 2962 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1173 2976 y Fn(E)t(;\024)1293 2962 y Fs(\()p Fp(H)g Fs(\))p Fr(k)26 b(\024)f Fp(C)1678 2976 y Fo(0)1717 2962 y Fs(,)31 b(uniformly)e(for)h(0)c Fp(<)f(\024)h Fr(\024)f Fs(1,)31 b(whic)m(h)f(yields)1357 3162 y Fp(M)g Fs(+)20 b Fp(f)1611 3176 y Fn(E)t(;\024)1730 3162 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1998 3176 y Fn(E)t(;\024)2118 3162 y Fs(\()p Fp(H)g Fs(\))1203 3349 y Fr(\025)83 b(\000)1438 3312 y Fo(8)p Fn(\017)1502 3321 y Fh(0)1536 3312 y Fo(+)p Fn(C)1641 3321 y Fh(0)1675 3312 y Fn(\033)p 1438 3328 281 4 v 1511 3380 a Fo(1+)p Fn(\033)1748 3349 y Fr(\000)1888 3313 y Fn(C)p 1849 3328 133 4 v 1849 3380 a Fo(1+)p Fn(\033)1992 3349 y Fs(\(1)21 b Fr(\000)f Fp(f)2229 3363 y Fn(E)t(;\024)2348 3349 y Fs(\()p Fp(H)7 b Fs(\)\))2536 3316 y Fo(2)1203 3533 y Fr(\025)83 b(\000)p Fs(10)p Fp(\017)1555 3547 y Fo(0)1615 3533 y Fr(\000)20 b Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2014 3547 y Fn(E)t(;\024)2133 3533 y Fs(\()p Fp(H)7 b Fs(\)\))2321 3500 y Fo(2)2362 3533 y Fp(;)0 3724 y Fs(for)30 b Fp(\033)k Fs(small)d(enough.)40 b(This)30 b(completes)h(the)g(pro)s(of)e(of)i (the)g(prop)s(osition.)40 b Fd(2)0 3930 y Fe(Lemma)35 b(7.7)46 b Fl(Assume)39 b(that)h(the)g(Mourr)-5 b(e)40 b(estimate)g(holds)h(at)f Fp(E)5 b Fl(.)62 b(Then)39 b(for)h(al)5 b(l)40 b Fp(\017)3031 3944 y Fo(0)3108 3930 y Fp(>)d Fs(0)p Fl(,)k(ther)-5 b(e)40 b(exists)0 4043 y Fp(\016)n(;)15 b(\024;)g(C)34 b(>)25 b Fs(0)p Fl(,)33 b(such)f(that)i(for)f(al)5 b(l)33 b Fp(E)1210 4010 y Fw(0)1267 4043 y Fl(with)g Fr(j)p Fp(E)26 b Fr(\000)20 b Fp(E)1745 4010 y Fw(0)1768 4043 y Fr(j)26 b(\024)f Fp(\016)s Fl(:)843 4242 y Fp(M)30 b Fs(+)20 b Fp(f)1097 4257 y Fn(E)1153 4238 y Fk(0)1175 4257 y Fn(;\024)1239 4242 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1507 4257 y Fn(E)1563 4238 y Fk(0)1585 4257 y Fn(;\024)1649 4242 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)p Fp(\017)2032 4256 y Fo(0)2071 4242 y Fs(1)-20 b(l)21 b Fr(\000)f Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2541 4257 y Fn(E)2597 4238 y Fk(0)2618 4257 y Fn(;\024)2683 4242 y Fs(\()p Fp(H)7 b Fs(\)\))2871 4204 y Fo(2)2911 4242 y Fp(:)0 4449 y Fe(Pro)s(of.)47 b Fs(this)31 b(follo)m(ws)g(immediately)h(from)e(Prop)s (osition)g(7.6)i(and)d(Lemma)i(7.3.)42 b Fd(2)141 4610 y Fs(F)-8 b(rom)45 b(Lemma)g(7.7)g(and)f(a)h(co)m(v)m(ering)i(argumen)m (t,)h(w)m(e)d(deduce)f(the)h(follo)m(wing)h(uniform)d(v)m(ersion)i(of)0 4722 y(Prop)s(osition)30 b(7.6.)0 4929 y Fe(Prop)s(osition)36 b(7.8)46 b Fl(Assume)d(that)h(the)f(Mourr)-5 b(e)43 b(estimate)h(holds) h(at)e(al)5 b(l)43 b Fp(E)49 b Fr(2)44 b Fp(I)7 b Fl(,)45 b Fp(I)50 b Fl(c)-5 b(omp)g(act)45 b(interval.)0 5042 y(Then)33 b Fr(8)20 b Fp(\017)345 5056 y Fo(0)409 5042 y Fp(>)25 b Fs(0)p Fl(,)33 b Fr(9)20 b Fp(C)q(;)15 b(\024)34 b Fl(such)e(that)i Fr(8)20 b Fp(E)30 b Fr(2)25 b Fp(I)7 b Fl(:)877 5241 y Fp(M)30 b Fs(+)20 b Fp(f)1131 5255 y Fn(E)t(;\024)1250 5241 y Fs(\()p Fp(H)7 b Fs(\))p Fp(R)q(f)1518 5255 y Fn(E)t(;\024)1638 5241 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)p Fp(\017)2021 5255 y Fo(0)2060 5241 y Fs(1)-20 b(l)21 b Fr(\000)e Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)2530 5255 y Fn(E)t(;\024)2649 5241 y Fs(\()p Fp(H)7 b Fs(\)\))2837 5203 y Fo(2)2878 5241 y Fp(:)1844 5517 y Fs(45)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 0 162 a Fi(7.2)112 b(P)m(osition)38 b(space)g(decomp)s(osition)0 334 y Fs(F)-8 b(ollo)m(wing)33 b([DG1)q(],)e(w)m(e)g(no)m(w)f(describ)s(e)g(a)h(geometric)h(decomp)s (osition)f(of)g(the)f(quadratic)h(form)1072 538 y Fp(B)1146 501 y Fn(\016)1184 538 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(=)e Fp(M)1639 501 y Fn(\016)1697 538 y Fr(\000)20 b Fp(f)1833 552 y Fn(E)t(;\024)1952 538 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2267 501 y Fn(\016)2307 538 y Fp(v)s Fs(\))p Fp(f)2434 552 y Fn(E)t(;\024)2554 538 y Fs(\()p Fp(H)g Fs(\))0 742 y(whic)m(h)23 b(will)g(b)s(e)f(useful) h(to)g(pro)m(v)m(e)h(a)g(Mourre)e(estimate.)40 b(This)23 b(decomp)s(osition)g(amoun)m(ts)g(to)h(treat)g(separately)0 855 y(the)31 b(b)s(osons)e(close)j(to)f(the)f(origin)h(and)f(those)h (close)h(to)f(in\014nit)m(y)-8 b(.)141 968 y(Let)35 b Fp(j)345 982 y Fo(0)416 968 y Fr(2)c Fp(C)580 935 y Fw(1)573 991 y Fo(0)654 968 y Fs(\(I)-15 b(R)774 932 y Fn(d)815 968 y Fs(\),)35 b Fp(j)947 982 y Fw(1)1054 968 y Fr(2)c Fp(C)1218 935 y Fw(1)1292 968 y Fs(\(I)-15 b(R)1412 932 y Fn(d)1453 968 y Fs(\),)35 b(0)d Fr(\024)g Fp(j)1765 982 y Fo(0)1804 968 y Fs(,)k(0)c Fr(\024)f Fp(j)2081 982 y Fw(1)2156 968 y Fs(,)k Fp(j)2258 935 y Fo(2)2253 991 y(0)2321 968 y Fs(+)22 b Fp(j)2456 935 y Fo(2)2451 991 y Fw(1)2558 968 y Fs(=)31 b(1,)36 b Fp(j)2803 982 y Fo(0)2874 968 y Fs(=)31 b(1)k(near)f(0)g(\(and)g(hence)0 1081 y Fp(j)37 1095 y Fw(1)137 1081 y Fs(=)25 b(0)31 b(near)f(0\).)42 b(W)-8 b(e)31 b(denote)g(again)h(b)m(y)e Fp(j)1513 1095 y Fo(0)1553 1081 y Fp(;)15 b(j)1630 1095 y Fw(1)1736 1081 y Fs(the)30 b(op)s(erators)h Fp(j)2332 1095 y Fo(0)2372 1081 y Fs(\()p Fp(x)p Fs(\))p Fp(;)15 b(j)2571 1095 y Fw(1)2647 1081 y Fs(\()p Fp(x)p Fs(\),)31 b(where)f Fp(x)25 b Fs(=)g(i)p Fr(r)3362 1096 y Fn(k)3405 1081 y Fs(.)141 1194 y(W)-8 b(e)32 b(set)f Fp(j)g Fs(:=)25 b(\()p Fp(j)701 1208 y Fo(0)741 1194 y Fp(;)15 b(j)818 1208 y Fw(1)894 1194 y Fs(\))30 b(and)g(for)g Fp(R)c Fr(\025)f Fs(1,)31 b Fp(j)1609 1161 y Fn(R)1693 1194 y Fs(=)25 b(\()p Fp(j)1866 1161 y Fn(R)1861 1217 y Fo(0)1924 1194 y Fp(;)15 b(j)2006 1161 y Fn(R)2001 1216 y Fw(1)2077 1194 y Fs(\),)31 b(where)f Fp(j)2473 1161 y Fn(R)2468 1217 y Fo(0)2531 1194 y Fs(\()p Fp(x)p Fs(\))c(=)f Fp(j)2812 1208 y Fo(0)2852 1194 y Fs(\()2904 1158 y Fn(x)p 2897 1173 54 4 v 2897 1225 a(R)2961 1194 y Fs(\),)31 b Fp(j)3094 1161 y Fn(R)3089 1216 y Fw(1)3164 1194 y Fs(\()p Fp(x)p Fs(\))26 b(=)f Fp(j)3445 1208 y Fw(1)3520 1194 y Fs(\()3572 1158 y Fn(x)p 3565 1173 V 3565 1225 a(R)3629 1194 y Fs(\).)141 1307 y(T)-8 b(o)31 b Fp(j)317 1274 y Fn(R)406 1307 y Fs(w)m(e)f(asso)s(ciate)j(the)d(isometric)i(op)s(erator)f(de\014ned)e (in)h(Subsection)g(3.2:)1292 1488 y(\024)1286 1511 y(\000\()p Fp(j)1420 1474 y Fn(R)1479 1511 y Fs(\))c(:)f Fr(H)h(!)f(H)1887 1474 y Fo(ext)2012 1511 y Fs(=)g Fr(H)c(\012)f Fs(\000\()p Fq(h)p Fs(\))p Fp(:)0 1715 y Fs(W)-8 b(e)32 b(de\014ne)d(also)j(the)e (follo)m(wing)i(Hamiltonians)g(acting)f(on)f Fr(H)2177 1682 y Fo(ext)2277 1715 y Fs(:)1217 1912 y Fp(H)1300 1879 y Fo(ext)1425 1912 y Fs(:=)c Fp(H)h Fr(\012)20 b Fs(1)-20 b(l)1791 1930 y Fo(\000\()p Fm(h)p Fo(\))1951 1912 y Fs(+)20 b(1)-20 b(l)2092 1926 y Fw(H)2177 1912 y Fr(\012)19 b Fs(d\000\()p Fp(!)s Fs(\))p Fp(;)1217 2096 y(H)1300 2063 y Fo(ext)1293 2119 y(0)1425 2096 y Fs(:=)26 b Fp(H)1623 2110 y Fo(0)1682 2096 y Fr(\012)20 b Fs(1)-20 b(l)1823 2114 y Fo(\000\()p Fm(h)p Fo(\))1983 2096 y Fs(+)20 b(1)-20 b(l)2124 2110 y Fw(H)2209 2096 y Fr(\012)19 b Fs(d\000\()p Fp(!)s Fs(\))p Fp(;)1217 2279 y(N)1300 2246 y Fo(ext)1425 2279 y Fs(:=)26 b Fp(N)k Fr(\012)20 b Fs(1)-20 b(l)1791 2298 y Fo(\000\()p Fm(h)p Fo(\))1951 2279 y Fs(+)20 b(1)-20 b(l)2092 2293 y Fw(H)2177 2279 y Fr(\012)19 b Fp(N)5 b(;)1217 2463 y(M)1315 2430 y Fn(\016)18 b Fo(ext)1490 2463 y Fs(:=)25 b Fp(M)1709 2430 y Fn(\016)1767 2463 y Fr(\012)20 b Fs(1)-20 b(l)1909 2482 y Fo(\000\()p Fm(h)p Fo(\))2068 2463 y Fs(+)20 b(1)-20 b(l)2210 2477 y Fw(H)2294 2463 y Fr(\012)20 b Fp(M)2483 2430 y Fn(\016)2521 2463 y Fp(;)1217 2647 y(M)1315 2614 y Fn(\016)1305 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Fo(ext)908 3270 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(:=)e Fp(M)1388 3233 y Fn(\016)19 b Fo(ext)1558 3270 y Fr(\000)h Fp(f)1694 3284 y Fn(E)t(;\024)1813 3270 y Fs(\()p Fp(H)1931 3233 y Fo(ext)2032 3270 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2229 3233 y Fn(\016)2268 3270 y Fp(v)s Fs(\))h Fr(\012)f Fs(1)-20 b(l)2512 3289 y Fo(\000\()p Fm(h)p Fo(\))2651 3270 y Fp(f)2696 3284 y Fn(E)t(;\024)2816 3270 y Fs(\()p Fp(H)2934 3233 y Fo(ext)3034 3270 y Fs(\))p Fp(;)0 3463 y Fs(as)31 b(a)f(quadratic)h(form)f(on)g Fr(D)s Fs(\(\()p Fp(M)1176 3430 y Fn(\016)19 b Fo(ext)1326 3463 y Fs(\))1371 3400 y Fh(1)p 1372 3412 31 4 v 1372 3453 a(2)1416 3463 y Fs(\).)0 3650 y Fe(Prop)s(osition)36 b(7.9)46 b Fl(Assume)33 b Fe(\(H0\))f Fl(and)h Fe(\(I1\))p Fl(.)41 b(L)-5 b(et)33 b Fp(w)28 b Fr(2)d(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)22 b(\012)e Fq(h)p Fs(\))33 b Fl(such)g(that)768 3870 y Fp(w)28 b Fr(2)d(B)s Fs(\()p Fr(D)s Fs(\()p Fp(K)1246 3805 y Fh(1)p 1246 3817 V 1246 3858 a(2)1291 3870 y Fs(\))p Fp(;)15 b Fr(K)22 b(\012)e(D)s Fs(\()p Fp(!)1716 3832 y Fw(\000)1781 3805 y Fh(1)p 1781 3817 V 1781 3858 a(2)1826 3870 y Fs(\)\))g Fr(\\)g(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(D)s Fs(\()p Fp(K)2408 3805 y Fh(1)p 2408 3817 V 2408 3858 a(2)2452 3870 y Fs(\))2487 3832 y Fw(\003)2547 3870 y Fr(\012)20 b(D)s Fs(\()p Fp(!)2806 3832 y Fw(\000)2871 3805 y Fh(1)p 2871 3817 V 2871 3858 a(2)2915 3870 y Fs(\)\))p Fp(;)-3010 b Fs(\(7.10\))0 4074 y Fl(and)34 b(let)e Fp(f)j Fr(2)25 b Fp(C)542 4041 y Fw(1)535 4097 y Fo(0)616 4074 y Fs(\(I)-15 b(R\))p Fl(.)42 b(Then)989 4292 y Fp(f)10 b Fs(\()p Fp(H)d Fs(\))25 b(=)1381 4269 y(\024)1376 4292 y(\000)o(\()p Fp(j)1509 4259 y Fn(R)1568 4292 y Fs(\))1603 4259 y Fw(\003)1643 4292 y Fp(f)10 b Fs(\()p Fp(H)1816 4259 y Fo(ext)1916 4292 y Fs(\))1957 4269 y(\024)1951 4292 y(\000\()p Fp(j)2085 4259 y Fn(R)2144 4292 y Fs(\))20 b(+)g Fp(N)2383 4229 y Fh(1)p 2383 4241 V 2383 4282 a(2)2427 4292 y Fp(o)2471 4307 y Fn(f)2517 4292 y Fs(\()p Fp(R)2622 4259 y Fo(0)2661 4292 y Fs(\))p Fp(N)2790 4229 y Fh(1)p 2790 4241 V 2790 4282 a(2)2834 4292 y Fp(;)341 4494 y(M)439 4461 y Fn(\016)497 4494 y Fs(+)g Fp(f)10 b Fs(\()p Fp(H)d Fs(\))p Fp(\036)p Fs(\()p Fp(w)r Fs(\))p Fp(f)j Fs(\()p Fp(H)d Fs(\))27 b(=)1381 4471 y(\024)1376 4494 y(\000)o(\()p Fp(j)1509 4461 y Fn(R)1568 4494 y Fs(\))1603 4461 y Fw(\003)1643 4400 y Fg(\020)1692 4494 y Fp(M)1790 4461 y Fn(\016)19 b Fo(ext)1960 4494 y Fs(+)h Fp(f)10 b Fs(\()p Fp(H)2224 4461 y Fo(ext)2324 4494 y Fs(\))p Fp(\036)p Fs(\()p Fp(w)r Fs(\))22 b Fr(\012)e Fs(1)-20 b(l)2713 4512 y Fo(\000\()p Fm(h)p Fo(\))2853 4494 y Fp(f)10 b Fs(\()p Fp(H)3026 4461 y Fo(ext)3126 4494 y Fs(\))3161 4400 y Fg(\021)3216 4471 y Fs(\024)3210 4494 y(\000\()p Fp(j)3344 4461 y Fn(R)3403 4494 y Fs(\))1376 4694 y(+)p Fp(N)1539 4631 y Fh(1)p 1539 4643 V 1539 4684 a(2)1584 4694 y Fp(o)1628 4709 y Fn(f)s(;\016)1723 4694 y Fs(\()p Fp(R)1828 4661 y Fo(0)1867 4694 y Fs(\))p Fp(N)1995 4631 y Fh(1)p 1995 4643 V 1995 4684 a(2)2040 4694 y Fp(:)0 4914 y Fe(Remark)35 b(7.10)47 b Fs(If)30 b(w)m(e)g(apply)g(Prop)s(osition)h(7.9)g(to)g Fp(w)d Fs(=)d Fr(\000)p Fs(i)p Fp(a)2183 4881 y Fn(\016)2221 4914 y Fp(v)s Fs(,)31 b(using)f(Lemma)g(6.2)i Fl(i\))p Fs(,)e(w)m(e)h(obtain)g(that)776 5134 y Fp(B)850 5096 y Fn(\016)888 5134 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(=)1250 5111 y(\024)1245 5134 y(\000)o(\()p Fp(j)1378 5096 y Fn(R)1437 5134 y Fs(\))1472 5096 y Fw(\003)1512 5134 y Fp(B)1586 5096 y Fn(\016)18 b Fo(ext)1735 5134 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))1975 5111 y(\024)1969 5134 y(\000)r(\()p Fp(j)2105 5096 y Fn(R)2163 5134 y Fs(\))21 b(+)f Fp(N)2403 5069 y Fh(1)p 2403 5081 V 2403 5122 a(2)2447 5134 y Fp(o)2491 5149 y Fn(\016)o(;E)t(;\024)2661 5134 y Fs(\()p Fp(R)2766 5096 y Fo(0)2805 5134 y Fs(\))p Fp(N)2933 5069 y Fh(1)p 2933 5081 V 2933 5122 a(2)2978 5134 y Fp(:)-3003 b Fs(\(7.11\))1844 5517 y(46)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 0 162 a Fe(Pro)s(of.)47 b Fs(F)-8 b(or)32 b Fp(z)d Fr(2)p 676 162 4 56 v 25 w Fs(C)p Fr(n)p Fs(I)-15 b(R)30 b(w)m(e)h(ha)m(v)m(e:)1032 356 y(\()p Fp(z)25 b Fr(\000)20 b Fp(H)1308 323 y Fo(ext)1408 356 y Fs(\))1443 323 y Fw(\000)p Fo(1)1544 333 y Fs(\024)1538 356 y(\000\()p Fp(j)1672 323 y Fn(R)1731 356 y Fs(\))g Fr(\000)1883 333 y Fs(\024)1877 356 y(\000\()p Fp(j)2011 323 y Fn(R)2070 356 y Fs(\)\()p Fp(z)25 b Fr(\000)20 b Fp(H)7 b Fs(\))2416 323 y Fw(\000)p Fo(1)879 559 y Fs(=)82 b(\()p Fp(z)25 b Fr(\000)20 b Fp(H)1308 526 y Fo(ext)1408 559 y Fs(\))1443 526 y Fw(\000)p Fo(1)1538 464 y Fg(\020)1588 559 y Fp(H)1671 526 y Fo(ext)1776 536 y Fs(\024)1771 559 y(\000\()p Fp(j)1905 526 y Fn(R)1963 559 y Fs(\))h Fr(\000)2115 536 y Fs(\024)2110 559 y(\000\()p Fp(j)2244 526 y Fn(R)2302 559 y Fs(\))p Fp(H)2420 464 y Fg(\021)2470 559 y Fs(\()p Fp(z)k Fr(\000)20 b Fp(H)7 b Fs(\))2781 526 y Fw(\000)p Fo(1)2876 559 y Fp(:)0 770 y Fs(By)31 b([DG1)q(,)g(Lemma)f(2.16])i(the)f(follo)m(wing)h(iden)m (tit)m(y)g(holds)e(on)g Fr(D)s Fs(\()p Fp(H)7 b Fs(\))20 b Fr(\\)g(D)s Fs(\()p Fp(N)10 b Fs(\):)1086 966 y Fp(H)1169 928 y Fo(ext)1162 988 y(0)1274 943 y Fs(\024)1269 966 y(\000\()p Fp(j)1403 928 y Fn(R)1461 966 y Fs(\))21 b Fr(\000)1613 943 y Fs(\024)1608 966 y(\000)o(\()p Fp(j)1741 928 y Fn(R)1800 966 y Fs(\))p Fp(H)1911 980 y Fo(0)1976 966 y Fs(=)k(d)2129 943 y(\024)2123 966 y(\000)o(\()p Fp(j)2256 928 y Fn(R)2315 966 y Fp(;)2380 942 y Fs(\024)2355 966 y(ad)2451 984 y Fw(j)p Fn(k)r Fw(j)2533 966 y Fp(j)2575 928 y Fn(R)2633 966 y Fs(\))p Fp(;)0 1173 y Fs(where)288 1149 y(\024)262 1173 y(ad)358 1192 y Fw(j)p Fn(k)r Fw(j)440 1173 y Fp(j)482 1140 y Fn(R)570 1173 y Fs(is)30 b(the)h(op)s(erator)f (\(ad)1315 1192 y Fw(j)p Fn(k)r Fw(j)1397 1173 y Fp(j)1439 1140 y Fn(R)1434 1196 y Fo(0)1497 1173 y Fp(;)15 b Fs(ad)1634 1192 y Fw(j)p Fn(k)r Fw(j)1716 1173 y Fp(j)1758 1140 y Fn(R)1753 1196 y Fw(1)1828 1173 y Fs(\).)41 b(W)-8 b(e)31 b(recall)g(also)g(the)f(follo)m(wing)h(b)s(ound)d(from)i([DG1)q (,)0 1286 y(Lemma)g(2.16]:)847 1414 y Fr(k)p Fs(\()p Fp(N)1010 1376 y Fo(ext)1131 1414 y Fs(+)20 b(1\))1302 1376 y Fw(\000)1367 1349 y Fh(1)p 1368 1361 31 4 v 1368 1403 a(2)1412 1414 y Fs(d)1469 1391 y(\024)1463 1414 y(\000\()p Fp(j)1597 1376 y Fn(R)1655 1414 y Fp(;)15 b(k)s Fs(\))p Fp(u)p Fr(k)27 b(\024)e(k)p Fs(d\000\()p Fp(k)2238 1376 y Fw(\003)2235 1436 y Fo(0)2278 1414 y Fp(k)2325 1428 y Fo(0)2385 1414 y Fs(+)20 b Fp(k)2526 1376 y Fw(\003)2523 1436 y(1)2598 1414 y Fp(k)2645 1428 y Fw(1)2720 1414 y Fs(\))2765 1349 y Fh(1)p 2765 1361 V 2765 1403 a(2)2810 1414 y Fp(u)p Fr(k)p Fp(:)-2932 b Fs(\(7.12\))0 1577 y(By)31 b(the)f(same)h(argumen)m(t)g(as)f(in)h ([G2)q(,)f(Lemma)h(5.2],)h(w)m(e)e(ha)m(v)m(e:)1274 1773 y Fr(k)p Fs([)p Fr(j)p Fp(k)s Fr(j)p Fp(;)15 b(j)1526 1735 y Fn(R)1521 1795 y(\017)1586 1773 y Fs(])p Fr(k)26 b Fs(=)f Fp(O)s Fs(\()p Fp(R)1955 1735 y Fw(\000)p Fo(1)2049 1773 y Fs(\))p Fp(;)36 b(\017)26 b Fs(=)e(0)p Fp(;)15 b Fr(1)p Fp(:)-2504 b Fs(\(7.13\))0 1968 y(Next)31 b(w)m(e)g(kno)m(w)f (b)m(y)h([G2)q(,)f(Lemma)h(3.9])h(that)e(\()p Fp(z)25 b Fr(\000)20 b Fp(H)7 b Fs(\))1921 1935 y Fw(\000)p Fo(1)2046 1968 y Fs(preserv)m(es)31 b Fr(D)s Fs(\()p Fp(N)10 b Fs(\),)30 b(and)790 2164 y Fr(k)p Fs(\()p Fp(N)h Fs(+)20 b(1\)\()p Fp(z)26 b Fr(\000)19 b Fp(H)7 b Fs(\))1456 2127 y Fw(\000)p Fo(1)1551 2164 y Fs(\()p Fp(N)31 b Fs(+)20 b(1\))1861 2127 y Fw(\000)p Fo(1)1956 2164 y Fr(k)25 b(\024)g Fp(C)7 b Fr(j)p Fs(Im)p Fp(z)t Fr(j)2399 2127 y Fw(\000)p Fo(2)2494 2164 y Fp(;)35 b(z)30 b Fr(2)24 b Fp(U)36 b Fa(b)p 2925 2164 4 56 v 25 w Fs(C)o Fp(:)-2989 b Fs(\(7.14\))0 2360 y(By)31 b(in)m(terp)s(olation)g(w)m(e)g(ha)m(v)m (e)h(also:)755 2566 y Fr(k)p Fs(\()p Fp(N)f Fs(+)20 b(1\))1120 2501 y Fh(1)p 1121 2513 31 4 v 1121 2554 a(2)1165 2566 y Fs(\()p Fp(z)25 b Fr(\000)20 b Fp(H)7 b Fs(\))1476 2528 y Fw(\000)p Fo(1)1571 2566 y Fs(\()p Fp(N)30 b Fs(+)20 b(1\))1880 2528 y Fw(\000)1945 2501 y Fh(1)p 1946 2513 V 1946 2554 a(2)1990 2566 y Fr(k)26 b(\024)f Fp(C)7 b Fr(j)p Fs(Im)o Fp(z)t Fr(j)2433 2528 y Fw(\000)p Fo(2)2528 2566 y Fp(;)36 b(z)29 b Fr(2)c Fp(U)36 b Fa(b)p 2960 2566 4 56 v 24 w Fs(C)p Fp(:)-3024 b Fs(\(7.15\))0 2762 y(Applying)30 b(\(7.12\),)j(\(7.13\))f(and)e(\(7.15\),)j(w)m(e)e (obtain:)514 2973 y Fr(k)p Fs(\()p Fp(N)677 2940 y Fo(ext)798 2973 y Fs(+)20 b(1\))969 2937 y Fw(\000)1034 2910 y Fh(1)p 1035 2922 31 4 v 1035 2963 a(2)1079 2973 y Fs(\()p Fp(z)25 b Fr(\000)20 b Fp(H)1355 2940 y Fo(ext)1455 2973 y Fs(\))1490 2940 y Fw(\000)p Fo(1)1585 2879 y Fg(\020)1634 2973 y Fp(H)1717 2940 y Fo(ext)1710 2996 y(0)1823 2950 y Fs(\024)1817 2973 y(\000\()p Fp(j)1951 2940 y Fn(R)2010 2973 y Fs(\))h Fr(\000)2162 2950 y Fs(\024)2156 2973 y(\000\()p Fp(j)2290 2940 y Fn(R)2349 2973 y Fs(\))p Fp(H)2460 2987 y Fo(0)2499 2879 y Fg(\021)2549 2973 y Fs(\()p Fp(z)k Fr(\000)20 b Fp(H)7 b Fs(\))2860 2940 y Fw(\000)p Fo(1)2954 2973 y Fs(\()p Fp(N)31 b Fs(+)20 b(1\))3264 2937 y Fw(\000)3329 2910 y Fh(1)p 3329 2922 V 3329 2963 a(2)3374 2973 y Fr(k)360 3104 y Fs(=)83 b Fp(O)s Fs(\()p Fp(R)691 3071 y Fw(\000)p Fo(1)785 3104 y Fs(\))p Fr(j)p Fs(Im)p Fp(z)t Fr(j)1025 3071 y Fw(\000)p Fo(2)1120 3104 y Fp(;)0 3029 y Fs(\(7.16\))0 3297 y(for)30 b Fp(z)g Fr(2)25 b Fp(U)35 b Fa(b)p 510 3297 4 56 v 25 w Fs(C)o(.)41 b(Next)31 b(again)h(b)m(y)e([DG1)q(,)h (Lemma)f(2.16]:)490 3491 y Fp(\036)p Fs(\()p Fp(w)r Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)844 3510 y Fo(\000\()p Fm(h)p Fo(\))989 3468 y Fs(\024)983 3491 y(\000\()p Fp(j)1117 3458 y Fn(R)1176 3491 y Fs(\))20 b Fr(\000)1328 3468 y Fs(\024)1322 3491 y(\000\()p Fp(j)1456 3458 y Fn(R)1515 3491 y Fs(\))p Fp(\036)p Fs(\()p Fp(w)r Fs(\))336 3694 y(=)529 3658 y Fo(1)p 500 3673 95 4 v 500 3682 a Fw(p)p 559 3682 36 3 v 55 x Fo(2)604 3599 y Fg(\020)653 3694 y Fs(\()p Fp(a)736 3661 y Fw(\003)776 3694 y Fs(\(\(1)i Fr(\000)e Fp(j)1046 3661 y Fn(R)1041 3717 y Fo(0)1104 3694 y Fs(\))p Fp(w)r Fs(\))h Fr(\012)f Fs(1)-20 b(l)1404 3712 y Fo(\000\()p Fm(h)p Fo(\))1563 3694 y Fr(\000)20 b Fs(1)-20 b(l)1705 3712 y Fo(\000\()p Fm(h)p Fo(\))1857 3680 y Fs(^)1844 3694 y Fr(\012)p Fp(a)1963 3661 y Fw(\003)2002 3694 y Fs(\()p Fp(j)2079 3661 y Fn(R)2074 3716 y Fw(1)2150 3694 y Fp(w)r Fs(\)\))2293 3671 y(\024)2287 3694 y(\000)q(\()p Fp(j)2422 3661 y Fn(R)2480 3694 y Fs(\))21 b Fr(\000)2632 3671 y Fs(\024)2627 3694 y(\000\()p Fp(j)2761 3661 y Fn(R)2819 3694 y Fs(\))p Fp(a)p Fs(\(\(1)h Fr(\000)e Fp(j)3172 3661 y Fn(R)3167 3717 y Fo(0)3230 3694 y Fs(\))p Fp(w)r Fs(\))3367 3599 y Fg(\021)3418 3694 y Fp(;)0 3601 y Fs(\(7.17\))0 3914 y(where)30 b(the)g(t)m(wisted)i(tensor)e(pro)s (duct)1363 3900 y(^)1350 3914 y Fr(\012)g Fs(is)h(de\014ned)e(as)h (follo)m(ws:)141 4027 y(let)h Fp(T)38 b Fs(:)26 b Fr(K)c(\012)e Fs(\000\()p Fq(h)p Fs(\))h Fr(\012)f Fs(\000\()p Fq(h)p Fs(\))26 b Fr(!)f Fs(\000\()p Fq(h)p Fs(\))c Fr(\012)f(K)h(\012)f Fs(\000\()p Fq(h)p Fs(\))31 b(b)s(e)f(the)h(unitary)f(op)s(erator)g (de\014ned)g(b)m(y)1316 4223 y Fp(T)13 b( )23 b Fr(\012)d Fp(u)1607 4237 y Fo(1)1667 4223 y Fr(\012)g Fp(u)1810 4237 y Fo(2)1875 4223 y Fs(=)25 b Fp(u)2023 4237 y Fo(1)2082 4223 y Fr(\012)20 b Fp( )k Fr(\012)c Fp(u)2399 4237 y Fo(2)2438 4223 y Fp(:)0 4418 y Fs(Then)29 b(if)i Fp(B)k Fs(is)30 b(an)g(op)s(erator)h(on)f Fr(K)22 b(\012)e Fs(\000\()p Fq(h)p Fs(\),)31 b(w)m(e)g(set)1304 4614 y(1)-20 b(l)1354 4633 y Fo(\000\()p Fm(h)p Fo(\))1506 4600 y Fs(^)1494 4614 y Fr(\012)o Fp(B)30 b Fs(:=)25 b Fp(T)1850 4577 y Fw(\000)p Fo(1)1945 4614 y Fs(\(1)-20 b(l)2030 4633 y Fo(\000\()p Fm(h)p Fo(\))2190 4614 y Fr(\012)20 b Fp(B)5 b Fs(\))p Fp(T)j(:)0 4819 y Fs(W)-8 b(e)33 b(no)m(w)g(apply)f(\(7.17\)) i(to)f Fp(w)e Fs(=)d Fp(v)s Fs(.)47 b(Note)34 b(that)e(s-)16 b(lim)1921 4833 y Fn(R)p Fw(!1)2135 4819 y Fp(j)2177 4786 y Fn(R)2172 4841 y Fw(1)2276 4819 y Fs(=)28 b(s-)15 b(lim)2582 4833 y Fn(R)p Fw(!1)2781 4819 y Fs(\(1)22 b Fr(\000)g Fp(j)3018 4786 y Fn(R)3013 4842 y Fo(0)3076 4819 y Fs(\))29 b(=)f(0)k(in)g Fr(B)s Fs(\()p Fq(h)p Fs(\))h(and)0 4945 y(hence)c(since)h Fp(v)f Fr(2)24 b(B)s Fs(\()p Fr(D)s Fs(\()p Fp(K)931 4882 y Fh(1)p 931 4894 31 4 v 931 4936 a(2)975 4945 y Fs(\))p Fp(;)15 b Fr(K)21 b(\012)c Fq(h)p Fs(\))30 b(w)m(e)g(ha)m(v)m(e)g(s-)15 b(lim)1885 4959 y Fn(R)p Fw(!1)2099 4945 y Fp(j)2141 4912 y Fn(R)2136 4968 y Fw(1)2211 4945 y Fp(v)s Fs(\()p Fp(K)25 b Fs(+)18 b(1\))2564 4910 y Fw(\000)2629 4882 y Fh(1)p 2629 4894 V 2629 4936 a(2)2699 4945 y Fs(=)25 b(s-)15 b(lim)3003 4959 y Fn(R)p Fw(!1)3202 4945 y Fs(\(1)j Fr(\000)g Fp(j)3431 4912 y Fn(R)3426 4969 y Fo(0)3489 4945 y Fs(\))p Fp(v)s Fs(\()p Fp(K)26 b Fs(+)0 5072 y(1\))80 5036 y Fw(\000)145 5009 y Fh(1)p 146 5021 V 146 5062 a(2)215 5072 y Fs(=)f(0)k(in)f Fr(B)s Fs(\()p Fr(K)q Fp(;)15 b Fr(K)j(\012)d Fq(h)p Fs(\).)41 b(Since)28 b(b)m(y)g(h)m(yp)s (othesis)g Fe(\(H0\))f Fs(\()p Fp(K)c Fs(+)15 b(1\))2363 5036 y Fw(\000)2428 5009 y Fh(1)p 2429 5021 V 2429 5062 a(2)2502 5072 y Fs(is)28 b(compact)h(on)f Fr(K)q Fs(,)i(w)m(e)e(obtain) h(that)886 5268 y Fr(k)p Fs(\(1)21 b Fr(\000)f Fp(j)1165 5230 y Fn(R)1160 5290 y Fo(0)1223 5268 y Fs(\))p Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))1616 5230 y Fw(\000)p Fo(1)1711 5268 y Fr(k)h Fs(+)f Fr(k)p Fp(j)1955 5230 y Fn(R)1950 5290 y Fw(1)2025 5268 y Fp(v)s Fs(\()p Fp(K)28 b Fs(+)20 b(1\))2383 5230 y Fw(\000)p Fo(1)2478 5268 y Fr(k)26 b Fs(=)f Fp(o)p Fs(\()p Fp(R)2794 5230 y Fo(0)2833 5268 y Fs(\))p Fp(:)-2893 b Fs(\(7.18\))1844 5517 y(47)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 0 162 a Fs(Using)31 b(also)g(\(7.15\),)i(and)d (the)g(fact)h(that)g(\()p Fp(K)d Fs(+)20 b(1\)\()p Fp(z)25 b Fr(\000)20 b Fp(H)7 b Fs(\))2057 129 y Fw(\000)p Fo(1)2182 162 y Fs(is)30 b(b)s(ounded,)f(w)m(e)i(obtain:)263 377 y Fr(k)p Fs(\()p Fp(N)426 344 y Fo(ext)548 377 y Fs(+)20 b(1\))719 342 y Fw(\000)784 314 y Fh(1)p 784 326 31 4 v 784 368 a(2)829 377 y Fs(\()p Fp(z)25 b Fr(\000)19 b Fp(H)1104 344 y Fo(ext)1205 377 y Fs(\))1240 344 y Fw(\000)p Fo(1)1334 283 y Fg(\020)1384 377 y Fp(\036)p Fs(\()p Fp(v)s Fs(\))i Fr(\012)f Fs(1)-20 b(l)1718 396 y Fo(\000\()p Fm(h)p Fo(\))1863 354 y Fs(\024)1857 377 y(\000\()p Fp(j)1991 344 y Fn(R)2050 377 y Fs(\))20 b Fr(\000)2202 354 y Fs(\024)2196 377 y(\000\()p Fp(j)2330 344 y Fn(R)2388 377 y Fs(\))p Fp(\036)p Fs(\()p Fp(v)s Fs(\))2594 283 y Fg(\021)2646 377 y Fs(\()p Fp(z)k Fr(\000)c Fp(H)7 b Fs(\))2956 344 y Fw(\000)p Fo(1)3051 377 y Fs(\()p Fp(N)31 b Fs(+)19 b(1\))3360 342 y Fw(\000)3425 314 y Fh(1)p 3426 326 V 3426 368 a(2)3471 377 y Fr(k)263 508 y Fs(=)25 b Fp(o)p Fs(\()p Fp(R)508 475 y Fo(0)548 508 y Fs(\))p Fr(j)p Fs(Im)p Fp(z)t Fr(j)788 475 y Fw(\000)p Fo(2)883 508 y Fp(;)0 434 y Fs(\(7.19\))0 705 y(for)30 b Fp(z)g Fr(2)25 b Fp(U)35 b Fa(b)p 510 705 4 56 v 25 w Fs(C)o(.)41 b(Com)m(bining)30 b(\(7.16\))j(and)c(\(7.19\))k(w)m(e)e (obtain:)586 920 y Fr(k)p Fs(\()p Fp(N)749 887 y Fo(ext)870 920 y Fs(+)20 b(1\))1041 885 y Fw(\000)1106 857 y Fh(1)p 1107 869 31 4 v 1107 911 a(2)1151 826 y Fg(\020)1201 920 y Fs(\()p Fp(z)25 b Fr(\000)20 b Fp(H)1477 887 y Fo(ext)1577 920 y Fs(\))1612 887 y Fw(\000)p Fo(1)1712 897 y Fs(\024)1707 920 y(\000)o(\()p Fp(j)1840 887 y Fn(R)1899 920 y Fs(\))h Fr(\000)2051 897 y Fs(\024)2046 920 y(\000)o(\()p Fp(j)2179 887 y Fn(R)2238 920 y Fs(\)\()p Fp(z)k Fr(\000)20 b Fp(H)7 b Fs(\))2584 887 y Fw(\000)p Fo(1)2679 826 y Fg(\021)2728 920 y Fs(\()p Fp(N)31 b Fs(+)20 b(1\))3038 885 y Fw(\000)3103 857 y Fh(1)p 3103 869 V 3103 911 a(2)3148 920 y Fr(k)586 1051 y Fs(=)25 b Fp(o)p Fs(\()p Fp(R)831 1018 y Fo(0)871 1051 y Fs(\))p Fr(j)p Fs(Im)p Fp(z)t Fr(j)1111 1018 y Fw(\000)p Fo(2)1206 1051 y Fp(;)0 977 y Fs(\(7.20\))0 1243 y(for)30 b Fp(z)g Fr(2)25 b Fp(U)35 b Fa(b)p 510 1243 4 56 v 25 w Fs(C)o(.)41 b(W)-8 b(e)31 b(recall)h(the)f(functional)f(calculus)h(form)m(ula:)1106 1464 y Fp(\037)p Fs(\()p Fp(A)p Fs(\))26 b(=)1470 1402 y(i)p 1433 1443 101 4 v 1433 1526 a(2)p Fp(\031)1558 1349 y Fg(Z)p 1625 1538 4 41 v 189 x Fo(C)1675 1464 y Fp(@)p 1735 1439 36 4 v 1735 1478 a Fn(z)1785 1464 y Fs(~)-56 b Fp(\037)p Fs(\()p Fp(z)t Fs(\)\()p Fp(z)26 b Fr(\000)20 b Fp(A)p Fs(\))2244 1426 y Fw(\000)p Fo(1)2339 1464 y Fs(d)p Fp(z)k Fr(^)c Fs(d)p 2603 1415 47 4 v 15 w Fp(z)t(;)-2674 b Fs(\(7.21\))0 1701 y(where)27 b Fp(A)h Fs(is)g(a)g(selfadjoin)m(t)h(op)s(erator)f(and)38 b(~)-55 b Fp(\037)25 b Fr(2)g Fp(C)1732 1668 y Fw(1)1725 1724 y Fo(0)1806 1701 y Fs(\()p 1868 1701 4 56 v(C\))j(is)g(an)g (almost-analytic)j(extension)d(of)g Fp(\037)g Fs(satisfying)1297 1898 y(~)-56 b Fp(\037)1343 1916 y Fw(j)p Fo(I)-12 b(R)1457 1898 y Fs(=)25 b Fp(\037;)1286 2082 y Fr(j)p Fp(@)p 1371 2057 36 4 v 1371 2095 a Fn(z)1422 2082 y Fs(~)-56 b Fp(\037)p Fs(\()p Fp(z)t Fs(\))p Fr(j)26 b(\024)f Fp(C)1796 2096 y Fn(n)1843 2082 y Fr(j)p Fs(Im)p Fp(z)t Fr(j)2048 2049 y Fn(n)2096 2082 y Fp(;)96 b(n)25 b Fr(2)f Fs(I)-15 b(N)p Fp(:)0 2278 y Fs(Using)31 b(\(7.21\))h(w)m(e)f(get:)983 2406 y Fp(f)10 b Fs(\()p Fp(H)d Fs(\))25 b(=)1318 2383 y(\024)1312 2406 y(\000\()p Fp(j)1446 2369 y Fn(R)1505 2406 y Fs(\))1540 2369 y Fw(\003)1580 2406 y Fp(f)10 b Fs(\()p Fp(H)1753 2369 y Fo(ext)1852 2406 y Fs(\))1893 2383 y(\024)1887 2406 y(\000)q(\()p Fp(j)2022 2369 y Fn(R)2080 2406 y Fs(\))21 b(+)f Fp(N)2320 2342 y Fh(1)p 2320 2354 31 4 v 2320 2395 a(2)2364 2406 y Fp(o)2408 2421 y Fn(f)2454 2406 y Fs(\()p Fp(R)2559 2369 y Fo(0)2598 2406 y Fs(\))p Fp(N)2727 2342 y Fh(1)p 2727 2354 V 2727 2395 a(2)2771 2406 y Fp(;)-2796 b Fs(\(7.22\))0 2571 y(whic)m(h)30 b(pro)m(v)m(es)h(the)g(\014rst)e(iden)m(tit)m(y)j(of)f (the)f(prop)s(osition.)141 2684 y(Let)k(us)f(no)m(w)h(pro)m(v)m(e)h (the)f(second)f(iden)m(tit)m(y)-8 b(.)53 b(W)-8 b(e)35 b(note)f(\014rst)f(that)i(applying)e(again)i(\(7.17\))h(and)d (\(7.15\),)0 2797 y(arguing)e(as)f(in)g(the)h(pro)s(of)e(of)i (\(7.19\),)i(w)m(e)e(obtain:)371 3012 y(\()p Fp(H)489 2974 y Fo(ext)610 3012 y Fs(+)20 b(i\))761 2974 y Fw(\000)p Fo(1)856 2918 y Fg(\020)905 3012 y Fp(\036)p Fs(\()p Fp(w)r Fs(\))i Fr(\012)e Fs(1)-20 b(l)1259 3030 y Fo(\000\()p Fm(h)p Fo(\))1404 2989 y Fs(\024)1399 3012 y(\000\()p Fp(j)1533 2974 y Fn(R)1591 3012 y Fs(\))21 b Fr(\000)1743 2989 y Fs(\024)1738 3012 y(\000\()p Fp(j)1872 2974 y Fn(R)1930 3012 y Fs(\))p Fp(\036)p Fs(\()p Fp(w)r Fs(\))2156 2918 y Fg(\021)2208 3012 y Fs(\()p Fp(H)27 b Fs(+)20 b(i\))2497 2974 y Fw(\000)p Fo(1)2617 3012 y Fs(=)25 b(\()p Fp(N)2831 2974 y Fo(ext)2932 3012 y Fs(\))2977 2947 y Fh(1)p 2977 2959 V 2977 3000 a(2)3022 3012 y Fp(o)p Fs(\()p Fp(R)3171 2974 y Fo(0)3210 3012 y Fs(\))p Fp(N)3338 2947 y Fh(1)p 3338 2959 V 3338 3000 a(2)3383 3012 y Fp(:)-3408 b Fs(\(7.23\))0 3238 y(Next)31 b(w)m(e)g(consider)f(the)h(term)f Fp(M)1182 3205 y Fn(\016)1220 3238 y Fs(.)41 b(By)31 b([DG1)q(,)g(Lemma)f(2.16],)j(w)m(e)d(ha)m(v)m(e:)1033 3437 y Fp(M)1131 3400 y Fn(\016)19 b Fo(ext)1287 3414 y Fs(\024)1281 3437 y(\000\()p Fp(j)1415 3400 y Fn(R)1474 3437 y Fs(\))25 b(=)1636 3414 y(\024)1630 3437 y(\000\()p Fp(j)1764 3400 y Fn(R)1823 3437 y Fs(\))p Fp(M)1956 3400 y Fn(\016)2014 3437 y Fs(+)20 b(d)2162 3414 y(\024)2156 3437 y(\000)o(\()p Fp(j)2289 3400 y Fn(R)2348 3437 y Fp(;)2414 3413 y Fs(\024)2388 3437 y(ad)2485 3459 y Fn(m)2547 3440 y Ff(\016)2585 3437 y Fp(j)2627 3400 y Fn(R)2686 3437 y Fs(\))p Fp(;)0 3637 y Fs(as)31 b(b)s(ounded)d(op)s(erators)i (from)g Fr(D)s Fs(\()p Fp(N)10 b Fs(\))31 b(in)m(to)g Fr(H)1619 3604 y Fo(ext)1719 3637 y Fs(.)141 3750 y(Applying)f (\(7.12\),)j(w)m(e)e(obtain:)301 3964 y Fr(k)p Fs(\()p Fp(N)464 3926 y Fo(ext)585 3964 y Fs(+)20 b(1\))756 3926 y Fw(\000)821 3899 y Fh(1)p 821 3911 V 821 3953 a(2)866 3964 y Fs(d)923 3941 y(\024)917 3964 y(\000)o(\()p Fp(j)1050 3926 y Fn(R)1109 3964 y Fp(;)1175 3940 y Fs(\024)1149 3964 y(ad)1245 3986 y Fn(m)1307 3967 y Ff(\016)1346 3964 y Fp(j)1388 3926 y Fn(R)1446 3964 y Fs(\)\()p Fp(N)31 b Fs(+)20 b(1\))1791 3926 y Fw(\000)1856 3899 y Fh(1)p 1857 3911 V 1857 3953 a(2)1901 3964 y Fr(k)26 b(\024)f(k)p Fs([)p Fp(m)2218 3926 y Fn(\016)2256 3964 y Fp(;)15 b(j)2338 3926 y Fn(R)2333 3986 y Fo(0)2397 3964 y Fs(])p Fr(k)21 b Fs(+)f Fr(k)p Fs([)p Fp(m)2729 3926 y Fn(\016)2767 3964 y Fp(;)15 b(j)2849 3926 y Fn(R)2844 3986 y Fw(1)2920 3964 y Fs(])p Fr(k)26 b Fs(=)f Fp(O)3181 3979 y Fn(\016)3219 3964 y Fs(\()p Fp(R)3324 3926 y Fw(\000)p Fo(1)3418 3964 y Fs(\))p Fp(:)0 4164 y Fs(This)30 b(yields:)1013 4286 y Fp(M)1111 4249 y Fn(\016)1174 4286 y Fs(=)1276 4263 y(\024)1270 4286 y(\000\()p Fp(j)1404 4249 y Fn(R)1463 4286 y Fs(\))1498 4249 y Fw(\003)1538 4286 y Fp(M)1636 4249 y Fn(\016)18 b Fo(ext)1791 4263 y Fs(\024)1785 4286 y(\000\()p Fp(j)1919 4249 y Fn(R)1978 4286 y Fs(\))i(+)g Fp(N)2217 4222 y Fh(1)p 2217 4234 V 2217 4275 a(2)2262 4286 y Fp(O)2331 4301 y Fn(\016)2369 4286 y Fs(\()p Fp(R)2474 4249 y Fw(\000)p Fo(1)2568 4286 y Fs(\))p Fp(N)2697 4222 y Fh(1)p 2697 4234 V 2697 4275 a(2)2741 4286 y Fp(:)-2766 b Fs(\(7.24\))0 4451 y(W)-8 b(e)32 b(can)e(no)m(w)h(complete)g(the)g (pro)s(of)f(of)g(the)h(prop)s(osition.)40 b(W)-8 b(e)32 b(\014rst)d(claim)j(that)1247 4660 y Fr(k)p Fp(N)1385 4595 y Fh(1)p 1385 4607 V 1385 4649 a(2)1430 4660 y Fp(\036)p Fs(\()p Fp(w)r Fs(\))p Fp(f)10 b Fs(\()p Fp(H)d Fs(\)\()p Fp(N)32 b Fs(+)20 b(1\))2140 4623 y Fw(\000)2205 4595 y Fh(1)p 2205 4607 V 2205 4649 a(2)2250 4660 y Fr(k)26 b Fp(<)e Fr(1)p Fp(:)-2532 b Fs(\(7.25\))0 4860 y(In)30 b(fact)h(this)f(follo)m(ws)i(b)m(y)e(writing)690 5067 y Fp(N)783 5004 y Fh(1)p 783 5016 V 783 5058 a(2)828 5067 y Fp(\036)p Fs(\()p Fp(w)r Fs(\))p Fp(f)10 b Fs(\()p Fp(H)d Fs(\)\()p Fp(N)32 b Fs(+)20 b(1\))1538 5031 y Fw(\000)1603 5004 y Fh(1)p 1603 5016 V 1603 5058 a(2)537 5267 y Fs(=)82 b Fp(N)783 5204 y Fh(1)p 783 5216 V 783 5257 a(2)828 5267 y Fp(\036)p Fs(\()p Fp(w)r Fs(\)\()p Fp(N)32 b Fs(+)20 b(1\))1330 5231 y Fw(\000)1395 5204 y Fh(1)p 1395 5216 V 1395 5257 a(2)1440 5267 y Fs(\()p Fp(H)27 b Fs(+)20 b(i\))1729 5234 y Fw(\000)p Fo(1)1844 5267 y Fr(\002)g Fs(\()p Fp(H)28 b Fs(+)20 b(i\)\()p Fp(N)31 b Fs(+)20 b(1\))2545 5204 y Fh(1)p 2546 5216 V 2546 5257 a(2)2590 5267 y Fp(f)10 b Fs(\()p Fp(H)d Fs(\)\()p Fp(N)31 b Fs(+)19 b(1\))3107 5231 y Fw(\000)3172 5204 y Fh(1)p 3173 5216 V 3173 5257 a(2)3218 5267 y Fp(:)1844 5517 y Fs(48)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 0 162 a Fs(The)26 b(\014rst)g(factor)i(is)f(b)s 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Fo(\000\()p Fm(h)p Fo(\))1921 1500 y Fs(\024)1915 1523 y(\000\()p Fp(j)2049 1490 y Fn(R)2108 1523 y Fs(\))p Fp(f)10 b Fs(\()p Fp(H)d Fs(\))693 1723 y(=)852 1700 y(\024)847 1723 y(\000)o(\()p Fp(j)980 1690 y Fn(R)1039 1723 y Fs(\))1074 1690 y Fw(\003)1114 1723 y Fp(f)j Fs(\()p Fp(H)1287 1690 y Fo(ext)1387 1723 y Fs(\))p Fp(\036)p Fs(\()p Fp(w)r Fs(\))22 b Fr(\012)e Fs(1)-20 b(l)1776 1741 y Fo(\000\()p Fm(h)p Fo(\))1915 1723 y Fp(f)10 b Fs(\()p Fp(H)2088 1690 y Fo(ext)2188 1723 y Fs(\))2229 1700 y(\024)2223 1723 y(\000\()p Fp(j)2357 1690 y Fn(R)2416 1723 y Fs(\))21 b(+)f Fp(N)2655 1660 y Fh(1)p 2655 1672 V 2655 1713 a(2)2700 1723 y Fp(o)p Fs(\()p Fp(R)2849 1690 y Fo(0)2889 1723 y Fs(\))p Fp(N)3017 1660 y Fh(1)p 3017 1672 V 3017 1713 a(2)3061 1723 y Fp(;)0 1920 y Fs(using)30 b(\(7.22\))j(and)c(the)i(analog)h(of)e(\(7.25\))j (for)d(the)h(Hamiltonians)g Fp(H)2425 1887 y Fo(ext)2525 1920 y Fp(;)15 b(N)2648 1887 y Fo(ext)2749 1920 y Fs(.)41 b(This)29 b(yields:)847 2111 y Fp(f)10 b Fs(\()p 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b(l)2060 3579 y Fo([)p Fn(\016)o(;)p Fw(1)p Fo([)2223 3560 y Fs(\()p Fr(j)p Fp(k)s Fr(j)p Fs(\)\))p Fp(;)928 3764 y Fq(h)971 3726 y Fn(<)971 3790 y(\016)1056 3764 y Fs(:=)25 b Fp(L)1239 3726 y Fo(2)1278 3764 y Fs(\()p Fr(fj)p Fp(k)s Fr(j)i Fp(<)e(\016)s Fr(g)p Fp(;)15 b Fs(d)p Fp(k)s Fs(\))p Fp(;)37 b Fq(h)1950 3726 y Fn(>)1950 3790 y(\016)2035 3764 y Fs(:=)25 b Fp(L)2218 3726 y Fo(2)2258 3764 y Fs(\()p Fr(fj)p Fp(k)s Fr(j)h(\025)f Fp(\016)s Fr(g)p Fp(;)15 b Fs(d)p Fp(k)s Fs(\))p Fp(:)0 3925 y Fs(The)30 b(maps)1292 4030 y Fp(F)1363 3997 y Fn(\016)1426 4030 y Fs(:)c Fq(h)f Fr(!)h Fq(h)1705 3995 y Fn(<)1705 4058 y(\016)1784 4030 y Fr(\010)20 b Fq(h)1918 3995 y Fn(>)1918 4058 y(\016)1977 4030 y Fp(;)1298 4192 y Fs(\024)1292 4215 y(\000\()p Fp(F)1455 4182 y Fn(\016)1493 4215 y Fs(\))26 b(:)f(\000\()p Fq(h)p Fs(\))h Fr(!)g Fs(\000\()p Fq(h)2052 4180 y Fn(<)2052 4243 y(\016)2111 4215 y Fs(\))21 b Fr(\012)f Fs(\000\()p Fq(h)2393 4180 y Fn(>)2393 4243 y(\016)2452 4215 y Fs(\))0 4369 y(are)31 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Fs(+)20 b(1)-20 b(l)1629 5074 y Fw(H)1714 5060 y Fr(\012)20 b Fs(d\000\()p Fr(j)p Fp(k)s Fr(j)p Fs(\))h Fr(\012)f Fs(1)-20 b(l)2245 5085 y Fo(\000\()p Fm(h)2352 5058 y Ff(>)2352 5110 y(\016)2405 5085 y Fo(\))2457 5060 y Fs(+)20 b(1)-20 b(l)2598 5074 y Fw(H)2683 5060 y Fr(\012)19 b Fs(1)-20 b(l)2824 5085 y Fo(\000\()p Fm(h)2931 5058 y Ff(<)2931 5110 y(\016)2984 5085 y Fo(\))3036 5060 y Fr(\012)19 b Fs(d\000\()p Fr(j)p Fp(k)s Fr(j)p Fs(\))p Fp(;)383 5228 y Fs(^)349 5251 y Fp(M)447 5218 y Fn(\016)g Fo(ext)622 5251 y Fs(:=)25 b Fp(M)841 5218 y Fn(\016)899 5251 y Fr(\012)20 b Fs(1)-20 b(l)1041 5276 y Fo(\000\()p Fm(h)1148 5249 y Ff(<)1148 5301 y(\016)1201 5276 y Fo(\))1252 5251 y Fr(\012)20 b Fs(1)-20 b(l)1394 5276 y Fo(\000\()p Fm(h)1501 5249 y Ff(>)1501 5301 y(\016)1554 5276 y Fo(\))1605 5251 y Fs(+)20 b(1)-20 b(l)1747 5265 y Fw(H)1831 5251 y Fr(\012)20 b Fp(M)2020 5218 y Fn(\016)2078 5251 y Fr(\012)g Fs(1)-20 b(l)2220 5276 y Fo(\000\()p Fm(h)2327 5249 y Ff(>)2327 5301 y(\016)2380 5276 y Fo(\))2431 5251 y Fs(+)20 b(1)-20 b(l)2573 5265 y Fw(H)2657 5251 y Fr(\012)20 b Fs(1)-20 b(l)2799 5276 y Fo(\000\()p Fm(h)2906 5249 y Ff(<)2906 5301 y(\016)2959 5276 y Fo(\))3010 5251 y Fr(\012)20 b Fp(M)3199 5218 y Fn(\016)3237 5251 y Fp(:)1844 5517 y Fs(49)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 0 162 a Fs(Applying)30 b(again)h(the)g(iden)m (tities)h(in)e([DG1)q(,)h(Lemma)f(2.16],)j(w)m(e)e(obtain:)840 367 y Fp(B)914 329 y Fn(\016)18 b Fo(ext)1063 367 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(=)e(\(1)-20 b(l)1506 381 y Fw(H)1591 367 y Fr(\012)1687 344 y Fs(\024)1682 367 y(\000)o(\()p Fp(F)1844 329 y Fn(\016)1883 367 y Fs(\)\))1953 329 y Fw(\003)2014 344 y Fs(^)1993 367 y Fp(B)2067 329 y Fn(\016)18 b Fo(ext)2216 367 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\)1)-20 b(l)2502 381 y Fw(H)2587 367 y Fr(\012)2683 344 y Fs(\024)2678 367 y(\000)o(\()p Fp(F)2840 329 y Fn(\016)2879 367 y Fs(\))p Fp(;)-2939 b Fs(\(7.27\))0 571 y(for)516 661 y(^)495 684 y Fp(B)569 646 y Fn(\016)18 b 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y(that)h(w)m(e)g(will)g(need)f (is:)418 1192 y Fp(M)516 1155 y Fn(\016)506 1215 y Fw(1)606 1192 y Fs(=)25 b(1)-20 b(l)752 1206 y Fw(H)837 1192 y Fr(\012)934 1169 y Fs(\024)928 1192 y(\000\()p Fp(F)1091 1155 y Fn(\016)1129 1192 y Fs(\))1164 1155 y Fw(\003)1219 1098 y Fg(\020)1268 1192 y Fs(1)g(l)1319 1206 y Fw(H)1403 1192 y Fr(\012)20 b Fp(M)1592 1155 y Fn(\016)1651 1192 y Fr(\012)f Fs(1)-20 b(l)1792 1218 y Fo(\000\()p Fm(h)1899 1190 y Ff(>)1899 1243 y(\016)1952 1218 y Fo(\))2004 1192 y Fs(+)19 b(1)-20 b(l)2145 1206 y Fw(H)2229 1192 y Fr(\012)20 b Fs(1)-20 b(l)2371 1218 y Fo(\000\()p Fm(h)2478 1190 y Ff(<)2478 1243 y(\016)2531 1218 y Fo(\))2582 1192 y Fr(\012)20 b Fp(M)2771 1155 y Fn(\016)2809 1098 y Fg(\021)2874 1192 y Fs(1)-20 b(l)2925 1206 y Fw(H)3009 1192 y Fr(\012)3106 1169 y Fs(\024)3100 1192 y(\000\()p Fp(F)3263 1155 y Fn(\016)3301 1192 y Fs(\))p Fp(:)-3361 b Fs(\(7.28\))0 1407 y(In)30 b(the)g(sequel)h(w)m(e)g(will)f(use)h(the)f(follo)m(wing)i (easy)f(observ)-5 b(ation)0 1595 y Fe(Lemma)35 b(7.11)46 b Fl(i\))33 b(We)f(have:)1220 1799 y Fs(1)-20 b(l)1270 1818 y Fo([0)p Fn(;)p Fo(1[)1404 1799 y Fs(\()p Fp(M)1537 1762 y Fn(\016)1575 1799 y Fs(\))26 b(=)f(1)-20 b(l)1782 1818 y Fw(f)p Fo(0)p Fw(g)1892 1799 y Fs(\()p Fp(M)2025 1762 y Fn(\016)2064 1799 y Fs(\))25 b(=)g(1)-20 b(l)2271 1818 y Fw(f)p Fo(0)p Fw(g)2381 1799 y Fs(\()p Fp(N)10 b Fs(\))p Fp(:)0 2003 y Fl(ii\))32 b(As)h(an)g(identity)g(on)g Fs(\000\()p Fq(h)990 1968 y Fn(<)990 2031 y(\016)1050 2003 y Fs(\))20 b Fr(\012)g Fs(\000\()p Fq(h)1331 1968 y Fn(>)1331 2031 y(\016)1391 2003 y Fs(\))p Fl(,)32 b(we)h(have:)256 2208 y Fs(1)-20 b(l)307 2226 y Fo([1)p Fn(;)p Fw(1)p Fo([)476 2208 y Fs(\()p Fp(M)609 2170 y Fn(\016)667 2208 y Fr(\012)20 b Fs(1)-20 b(l)809 2226 y Fo(\000\()p Fm(h)916 2207 y Ff(>)968 2226 y Fo(\))1020 2208 y Fs(+)20 b(1)-20 b(l)1162 2226 y Fo(\000\()p Fm(h)1269 2207 y Ff(<)1321 2226 y Fo(\))1373 2208 y Fr(\012)20 b Fp(M)1562 2170 y Fn(\016)1600 2208 y Fs(\))26 b(=)f(1)-20 b(l)1807 2226 y Fo(\000\()p Fm(h)1914 2207 y Ff(<)1967 2226 y Fo(\))2019 2208 y Fr(\012)20 b Fs(1)-20 b(l)2160 2226 y Fo([1)p Fn(;)p Fw(1)p Fo([)2329 2208 y Fs(\()p Fp(M)2462 2170 y Fn(\016)2500 2208 y Fs(\))21 b(+)f(1)-20 b(l)2697 2226 y Fo([1)p Fn(;)p Fw(1)p Fo([)2866 2208 y Fs(\()p Fp(M)2999 2170 y Fn(\016)3037 2208 y Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)3234 2226 y Fw(f)p Fo(0)p Fw(g)3344 2208 y Fs(\()p Fp(N)10 b Fs(\))p Fp(:)0 2421 y Fe(Pro)s(of.)47 b Fl(i\))31 b Fs(follo)m(ws)g(from)f(the)h(fact)g(that)g Fp(m)1551 2388 y Fn(\016)1614 2421 y Fr(\025)25 b Fs(1.)41 b(T)-8 b(o)31 b(pro)m(v)m(e)g Fl(ii\))p Fs(,)g(w)m(e)f(write:)870 2623 y(1)-20 b(l)920 2641 y Fo([1)p Fn(;)p Fw(1)p Fo([)1089 2623 y Fs(\()p Fp(M)1222 2590 y Fn(\016)1281 2623 y Fr(\012)20 b Fs(1)-20 b(l)1422 2641 y Fo(\000\()p Fm(h)1529 2622 y Ff(>)1582 2641 y Fo(\))1634 2623 y Fs(+)20 b(1)-20 b(l)1775 2641 y Fo(\000\()p Fm(h)1882 2622 y Ff(<)1935 2641 y Fo(\))1987 2623 y Fr(\012)20 b Fp(M)2176 2590 y Fn(\016)2214 2623 y Fs(\))716 2807 y(=)83 b(1)-20 b(l)920 2825 y Fo([1)p Fn(;)p Fw(1)p Fo([)1089 2807 y Fs(\()p Fp(M)1222 2774 y Fn(\016)1281 2807 y Fr(\012)20 b Fs(1)-20 b(l)1422 2825 y Fo(\000\()p Fm(h)1529 2806 y Ff(>)1582 2825 y Fo(\))1634 2807 y Fs(+)20 b(1)-20 b(l)1775 2825 y Fo(\000\()p Fm(h)1882 2806 y Ff(<)1935 2825 y Fo(\))1987 2807 y Fr(\012)20 b Fp(M)2176 2774 y Fn(\016)2214 2807 y Fs(\)1)-20 b(l)2299 2825 y Fo(\000\()p Fm(h)2406 2806 y Ff(<)2459 2825 y Fo(\))2511 2807 y Fr(\012)20 b Fs(1)-20 b(l)2652 2825 y Fo([1)p Fn(;)p Fw(1)p Fo([)2821 2807 y Fs(\()p Fp(M)2954 2774 y Fn(\016)2993 2807 y Fs(\))870 2991 y(+1)g(l)991 3009 y Fo([1)p Fn(;)p Fw(1)p Fo([)1160 2991 y Fs(\()p Fp(M)1293 2958 y Fn(\016)1351 2991 y Fr(\012)20 b Fs(1)-20 b(l)1493 3009 y Fo(\000\()p Fm(h)1600 2990 y Ff(>)1653 3009 y Fo(\))1704 2991 y Fs(+)20 b(1)-20 b(l)1846 3009 y Fo(\000\()p Fm(h)1953 2990 y Ff(<)2006 3009 y Fo(\))2057 2991 y Fr(\012)20 b Fp(M)2246 2958 y Fn(\016)2284 2991 y Fs(\)1)-20 b(l)2370 3009 y Fo(\000\()p Fm(h)2477 2990 y Ff(<)2530 3009 y Fo(\))2582 2991 y Fr(\012)20 b Fs(1)-20 b(l)2723 3009 y Fo([0)p Fn(;)p Fo(1[)2857 2991 y Fs(\()p Fp(M)2990 2958 y Fn(\016)3028 2991 y Fs(\))716 3174 y(=)83 b(1)-20 b(l)920 3193 y Fo(\000\()p Fm(h)1027 3174 y Ff(<)1080 3193 y Fo(\))1132 3174 y Fr(\012)20 b Fs(1)-20 b(l)1273 3193 y Fo([1)p Fn(;)p Fw(1)p Fo([)1442 3174 y Fs(\()p Fp(M)1575 3141 y Fn(\016)1614 3174 y Fs(\))20 b(+)g(1)-20 b(l)1810 3193 y Fo([1)p Fn(;)p Fw(1)p Fo([)1979 3174 y Fs(\()p Fp(M)2112 3141 y Fn(\016)2151 3174 y Fs(\))20 b Fr(\012)g Fs(1)-20 b(l)2348 3193 y Fw(f)p Fo(0)p Fw(g)2458 3174 y Fs(\()p Fp(N)10 b Fs(\))p Fp(;)0 3377 y Fs(using)30 b Fl(i\))p Fs(.)41 b Fd(2)0 3621 y Fi(7.4)112 b(Pro)s(of)37 b(of)h(the)f(Mourre)h(estimate)0 3792 y Fs(This)32 b(subsection)i(is)f (dev)m(oted)h(to)g(the)g(pro)s(of)e(of)i(the)f(Mourre)g(estimate)i (stated)f(in)f(Theorem)g(7.12)i(b)s(elo)m(w.)0 3905 y(Note)d(that)f (this)f(is)g(the)h(only)f(place)i(where)e(h)m(yp)s(othesis)g Fe(\(H0\))f Fs(is)i(used.)0 4093 y Fe(Theorem)70 b(7.12)46 b Fl(Assume)39 b(hyp)-5 b(otheses)41 b Fe(\(H0\))p Fl(,)e Fe(\(I1\))f Fl(and)i Fe(\(I2\))p Fl(.)59 b(F)-7 b(or)40 b(al)5 b(l)40 b Fp(E)2827 4107 y Fo(0)2902 4093 y Fp(<)d Fr(1)h Fl(ther)-5 b(e)40 b(exists)f Fs(0)e Fp(<)0 4206 y(\016)29 b Fr(\024)175 4170 y Fo(1)p 175 4185 36 4 v 175 4237 a(2)253 4206 y Fl(such)k(that:)141 4319 y(i\))g(F)-7 b(or)33 b(al)5 b(l)34 b Fp(E)c Fr(\024)25 b Fp(E)799 4333 y Fo(0)839 4319 y Fp(;)15 b(\017)916 4333 y Fo(0)981 4319 y Fp(>)25 b Fs(0)33 b Fl(ther)-5 b(e)33 b(exist)g Fp(C)q(;)15 b(\024)26 b(>)f Fs(0)33 b Fl(and)h(a)f(c)-5 b(omp)g(act)35 b(op)-5 b(er)g(ator)35 b Fp(K)2992 4333 y Fo(0)3065 4319 y Fl(such)d(that:)559 4523 y Fp(M)657 4485 y Fn(\016)715 4523 y Fr(\000)20 b Fp(f)851 4537 y Fn(E)t(;\024)970 4523 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1285 4485 y Fn(\016)1324 4523 y Fp(v)s Fs(\))p Fp(f)1451 4537 y Fn(E)t(;\024)1572 4523 y Fs(\()p Fp(H)g Fs(\))25 b Fr(\025)g Fs(\(1)c Fr(\000)f Fp(\017)2075 4537 y Fo(0)2115 4523 y Fs(\)1)-20 b(l)21 b Fr(\000)f Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2620 4537 y Fn(E)t(;\024)2739 4523 y Fs(\()p Fp(H)7 b Fs(\)\))2927 4485 y Fo(2)2988 4523 y Fr(\000)20 b Fp(K)3156 4537 y Fo(0)3196 4523 y Fp(:)-3221 b Fs(\(7.29\))0 4727 y Fl(ii\))32 b(F)-7 b(or)34 b(al)5 b(l)33 b Fp(E)e Fr(\024)25 b Fp(E)686 4741 y Fo(0)725 4727 y Fp(;)15 b(\017)802 4741 y Fo(0)867 4727 y Fp(>)25 b Fs(0)p Fl(,)33 b Fp(E)e Fr(62)25 b Fp(\033)1305 4741 y Fo(pp)1387 4727 y Fs(\()p Fp(H)7 b Fs(\))p Fl(,)33 b(ther)-5 b(e)34 b(exist)e Fp(C)q(;)15 b(\024)27 b(>)e Fs(0)33 b Fl(such)g(that:)733 4931 y Fp(M)831 4894 y Fn(\016)889 4931 y Fr(\000)20 b Fp(f)1025 4945 y Fn(E)t(;\024)1145 4931 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1460 4894 y Fn(\016)1499 4931 y Fp(v)s Fs(\))p Fp(f)1626 4945 y Fn(E)t(;\024)1746 4931 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f Fs(1)c Fr(\000)e Fp(\017)2214 4945 y Fo(0)2274 4931 y Fr(\000)h Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2673 4945 y Fn(E)t(;\024)2793 4931 y Fs(\()p Fp(H)7 b Fs(\)\))2981 4894 y Fo(2)3021 4931 y Fp(:)-3046 b Fs(\(7.30\))0 5136 y(W)-8 b(e)32 b(will)e(deduce)g(Theorem)g(7.12)i(from)e(the)h(follo)m (wing)h(prop)s(osition.)1844 5517 y(50)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 0 162 a Fe(Prop)s(osition)36 b(7.13)47 b Fl(F)-7 b(or)40 b(al)5 b(l)41 b Fp(E)1178 176 y Fo(0)1256 162 y Fp(<)d Fr(1)i Fl(ther)-5 b(e)40 b(exists)g Fs(0)f Fp(<)f(\016)43 b Fr(\024)2380 126 y Fo(1)p 2380 141 36 4 v 2380 194 a(2)2465 162 y Fl(such)d(that)h(the)f (fol)5 b(lowing)41 b(assertion)0 275 y(holds:)57 b(Assume)39 b(that)i(the)f(Mourr)-5 b(e)40 b(estimate)g(\(7.29\))h(holds)g(at)g(al) 5 b(l)40 b(ener)-5 b(gies)39 b Fp(E)k Fr(\024)37 b Fp(E)3094 289 y Fo(1)3173 275 y Fl(for)j(some)h Fp(E)3632 289 y Fo(1)3709 275 y Fr(\024)0 388 y Fp(E)67 402 y Fo(0)127 388 y Fs(+)20 b(1)p Fl(.)42 b(Then)33 b(\(7.29\))h(holds)g(at)f(al)5 b(l)34 b(ener)-5 b(gies)32 b Fp(E)f Fr(\024)25 b Fp(E)1926 402 y Fo(1)1985 388 y Fs(+)20 b Fp(\016)s Fl(.)0 576 y Fe(Pro)s(of)33 b(of)f(Theorem)g(7.12.)40 b Fs(Note)29 b(\014rst)e(that)i Fl(ii\))e Fs(follo)m(ws)i(from)e Fl(i\))h Fs(b)m(y)g(Prop)s(osition)f(7.5,)j(so)e(it)g(su\016ces)g(to)0 689 y(pro)m(v)m(e)f Fl(i\))p Fs(.)40 b(Using)27 b(Prop)s(osition)f (7.13)i(and)e(an)h(induction)f(argumen)m(t,)i(it)f(su\016ces)g(to)g (sho)m(w)g(that)g(the)g(Mourre)0 802 y(estimate)34 b(holds)d(at)i(all)g (energies)g Fp(E)g Fr(\024)28 b Fs(inf)22 b Fp(\033)s Fs(\()p Fp(H)7 b Fs(\))22 b Fr(\000)f Fs(2.)46 b(If)32 b Fp(E)h Fr(\024)28 b Fs(inf)22 b Fp(\033)s Fs(\()p Fp(H)7 b Fs(\))22 b Fr(\000)f Fs(2,)33 b Fp(\024)c Fr(\024)e Fs(1,)33 b(then)f Fp(f)3334 816 y Fn(E)t(;\024)3454 802 y Fs(\()p Fp(H)7 b Fs(\))28 b(=)g(0)0 914 y(and)870 1027 y Fp(B)944 990 y Fn(\016)982 1027 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(=)e Fp(M)1437 990 y Fn(\016)1500 1027 y Fr(\025)g Fs(0)h Fr(\025)f Fs(\(1)c Fr(\000)e Fp(\017)1991 1041 y Fo(0)2031 1027 y Fs(\)1)-20 b(l)21 b Fr(\000)f Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2536 1041 y Fn(E)t(;\024)2656 1027 y Fs(\()p Fp(H)7 b Fs(\)\))2844 990 y Fo(2)2884 1027 y Fp(;)0 1194 y Fs(for)30 b(all)h Fp(\017)302 1208 y Fo(0)367 1194 y Fp(>)25 b Fs(0)p Fp(;)15 b(C)33 b Fr(\025)24 b Fs(1.)42 b Fd(2)141 1357 y Fs(Before)31 b(starting)h(the)e(pro)s(of)g(of)g(Prop)s(osition)h(7.13,)h(w)m(e)f (state)g(an)g(auxiliary)g(lemma.)0 1545 y Fe(Lemma)k(7.14)46 b Fl(L)-5 b(et)34 b Fp(E)827 1559 y Fo(0)893 1545 y Fr(2)26 b Fs(I)-15 b(R)p Fl(.)44 b(Ther)-5 b(e)34 b(exists)g Fp(C)f(>)26 b Fs(0)34 b Fl(such)f(that)i(for)f(al)5 b(l)34 b Fp(N)2667 1559 y Fo(0)2733 1545 y Fr(2)26 b Fs(I)-15 b(N)2906 1509 y Fw(\003)2946 1545 y Fp(;)15 b(\017;)g(R)28 b(>)e Fs(0)34 b Fl(ther)-5 b(e)34 b(exists)0 1658 y Fp(K)7 b Fs(\()p Fp(N)192 1672 y Fo(0)232 1658 y Fp(;)15 b(R)q Fs(\))33 b Fl(c)-5 b(omp)g(act)34 b(such)f(that)h(for)f(al)5 b(l)33 b Fp(E)e Fr(\024)25 b Fp(E)1689 1672 y Fo(0)1728 1658 y Fp(;)15 b Fs(0)27 b Fp(<)d(\024)i Fr(\024)f Fs(1)p Fl(:)125 1861 y Fp(i)p Fs(\))41 b Fr(\000)329 1838 y Fs(\024)323 1861 y(\000\()p Fp(j)457 1828 y Fn(R)516 1861 y Fs(\))551 1828 y Fw(\003)591 1861 y Fs(1)-20 b(l)641 1879 y Fw(f)p Fo(0)p Fw(g)751 1861 y Fs(\()p Fp(M)884 1828 y Fn(\016)874 1883 y Fw(1)949 1861 y Fs(\))990 1838 y(\024)984 1861 y(\000\()p Fp(j)1118 1828 y Fn(R)1177 1861 y Fs(\))26 b Fr(\025)f(\000)p Fp(\017)p Fs(1)-20 b(l)20 b Fr(\000)g Fp(\017)1640 1828 y Fw(\000)p Fo(1)1734 1861 y Fs(\(1)h Fr(\000)f Fp(f)1971 1875 y Fn(E)t(;\024)2090 1861 y Fs(\()p Fp(H)7 b Fs(\)\))2278 1828 y Fo(2)2339 1861 y Fr(\000)20 b Fp(C)7 b(N)2585 1823 y Fw(\000)p Fo(1)2575 1886 y(0)2679 1861 y Fp(N)30 b Fr(\000)20 b Fp(K)7 b Fs(\()p Fp(N)3065 1875 y Fo(0)3104 1861 y Fp(;)15 b(R)q Fs(\))125 1977 y Fp(ii)p Fs(\))249 1954 y(\024)243 1977 y(\000\()p Fp(j)377 1944 y Fn(R)436 1977 y Fs(\))471 1944 y Fw(\003)511 1977 y Fp(B)585 1944 y Fn(\016)j Fo(ext)734 1977 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\)1)-20 b(l)1020 1996 y Fw(f)p Fo(0)p Fw(g)1130 1977 y Fs(\()p Fp(M)1263 1944 y Fn(\016)1253 2000 y Fw(1)1328 1977 y Fs(\))1369 1954 y(\024)1363 1977 y(\000)q(\()p Fp(j)1498 1944 y Fn(R)1556 1977 y Fs(\))26 b Fr(\025)f(\000)p Fp(\017)p Fs(1)-20 b(l)20 b Fr(\000)g Fp(\017)2019 1944 y Fw(\000)p Fo(1)2113 1977 y Fs(\(1)h Fr(\000)f Fp(f)2350 1991 y Fn(E)t(;\024)2470 1977 y Fs(\()p Fp(H)7 b Fs(\)\))2658 1944 y Fo(2)2718 1977 y Fr(\000)20 b Fp(C)7 b(N)2964 1939 y Fw(\000)p Fo(1)2954 2002 y(0)3058 1977 y Fp(N)30 b Fr(\000)20 b Fp(K)7 b Fs(\()p Fp(N)3444 1991 y Fo(0)3484 1977 y Fp(;)15 b(R)q Fs(\))p Fp(:)0 2194 y Fe(Pro)s(of.)53 b Fs(Let)33 b(\014rst)e Fp(j)j Fr(2)27 b Fp(C)920 2161 y Fw(1)913 2217 y Fo(0)994 2194 y Fs(\(I)-15 b(R)1114 2158 y Fn(d)1155 2194 y Fs(\),)33 b(0)28 b Fr(\024)g Fp(j)33 b Fr(\024)28 b Fs(1)33 b(and)e Fp(j)1887 2161 y Fn(R)1945 2194 y Fs(\()p Fp(x)p Fs(\))e(=)f Fp(j)5 b Fs(\()2290 2158 y Fn(x)p 2282 2173 54 4 v 2282 2225 a(R)2346 2194 y Fs(\).)46 b(W)-8 b(e)33 b(claim)g(that)g(for)f(eac)m(h) h Fp(N)3473 2208 y Fo(0)3540 2194 y Fr(2)28 b Fs(I)-15 b(N)3715 2158 y Fw(\003)3754 2194 y Fs(,)0 2307 y Fp(R)26 b Fr(\025)f Fs(1,)31 b Fp(E)359 2321 y Fo(0)424 2307 y Fr(2)25 b Fs(I)-15 b(R)30 b(there)g(exists)h(a)g(compact)g(op)s (erator)g Fp(K)7 b Fs(\()p Fp(N)2106 2321 y Fo(0)2146 2307 y Fp(;)15 b(R)q(;)g(E)2363 2321 y Fo(0)2403 2307 y Fs(\))30 b(suc)m(h)g(that)750 2511 y(\000\()p Fp(j)884 2473 y Fn(R)943 2511 y Fs(\))978 2473 y Fo(2)1043 2511 y Fr(\024)25 b Fp(K)7 b Fs(\()p Fp(N)1331 2525 y Fo(0)1370 2511 y Fp(;)15 b(R)q(;)g(E)1587 2525 y Fo(0)1627 2511 y Fs(\))21 b(+)f Fp(\017)g Fs(+)g Fp(\017)1959 2473 y Fw(\000)p Fo(1)2053 2511 y Fs(\(1)h Fr(\000)f Fp(f)2290 2525 y Fn(E)t(;\024)2409 2511 y Fs(\)\()p Fp(H)7 b Fs(\))2597 2473 y Fo(2)2658 2511 y Fs(+)20 b Fp(N)2832 2473 y Fw(\000)p Fo(1)2822 2536 y(0)2926 2511 y Fp(N)5 b(;)-3029 b Fs(\(7.31\))0 2715 y(uniformly)29 b(for)i Fp(\017)25 b(>)g Fs(0)30 b Fp(E)h Fr(\024)25 b Fp(E)1048 2729 y Fo(0)1088 2715 y Fs(,)30 b(0)c Fp(<)f(\024)g Fr(\024)g Fs(1.)41 b(In)30 b(fact)h(w)m(e)g(write:)223 2911 y(\000\()p Fp(j)357 2878 y Fn(R)416 2911 y Fs(\))451 2878 y Fo(2)516 2911 y Fs(=)83 b(\000\()p Fp(j)804 2878 y Fn(R)862 2911 y Fs(\))897 2878 y Fo(2)937 2911 y Fs(1)-20 b(l)988 2930 y Fo([0)p Fn(;N)1119 2939 y Fh(0)1153 2930 y Fo(])1176 2911 y Fs(\()p Fp(N)10 b Fs(\))21 b(+)f(\000\()p Fp(j)1575 2878 y Fn(R)1634 2911 y Fs(\))1669 2878 y Fo(2)1708 2911 y Fs(1)-20 b(l)1759 2930 y Fo(])p Fn(N)1835 2939 y Fh(0)1869 2930 y Fn(;)p Fw(1)p Fo([)1983 2911 y Fs(\()p Fp(N)10 b Fs(\))516 3097 y Fr(\024)83 b Fs(\000\()p Fp(j)804 3064 y Fn(R)862 3097 y Fs(\))897 3064 y Fo(2)937 3097 y Fs(1)-20 b(l)988 3116 y Fo([0)p Fn(;N)1119 3125 y Fh(0)1153 3116 y Fo(])1176 3097 y Fs(\()p Fp(N)10 b Fs(\))21 b(+)f Fp(N)1524 3059 y Fw(\000)p Fo(1)1514 3122 y(0)1618 3097 y Fp(N)516 3283 y Fs(=)83 b(\000\()p Fp(j)804 3250 y Fn(R)862 3283 y Fs(\))897 3250 y Fo(2)937 3283 y Fs(1)-20 b(l)988 3301 y Fo([0)p Fn(;N)1119 3310 y Fh(0)1153 3301 y Fo(])1176 3283 y Fs(\()p Fp(N)10 b Fs(\)1)-20 b(l)1381 3301 y Fo(])p Fw(\0001)p Fn(;E)1599 3310 y Fh(0)1632 3301 y Fo(+1])1745 3283 y Fs(\()p Fp(H)7 b Fs(\))21 b(+)f(\000\()p Fp(j)2144 3250 y Fn(R)2203 3283 y Fs(\))2238 3250 y Fo(2)2278 3283 y Fs(1)-20 b(l)2328 3301 y Fo([0)p Fn(;N)2459 3310 y Fh(0)2493 3301 y Fo(])2517 3283 y Fs(\()p Fp(N)10 b Fs(\)1)-20 b(l)2721 3301 y Fo(])p Fn(E)2793 3310 y Fh(0)2827 3301 y Fo(+1)p Fn(;)p Fw(1)p Fo([)3031 3283 y Fs(\()p Fp(H)7 b Fs(\))21 b(+)f Fp(N)3379 3245 y Fw(\000)p Fo(1)3369 3308 y(0)3473 3283 y Fp(N)516 3469 y Fr(\024)83 b Fp(K)7 b Fs(\()p Fp(N)862 3483 y Fo(0)901 3469 y Fp(;)15 b(R)q(;)g(E)1118 3483 y Fo(0)1158 3469 y Fs(\))21 b(+)f Fp(\017)g Fs(+)g Fp(\017)1490 3436 y Fw(\000)p Fo(1)1584 3469 y Fs(1)-20 b(l)1635 3487 y Fo(])p Fn(E)1707 3496 y Fh(0)1740 3487 y Fo(+1)p Fn(;)p Fw(1)p Fo([)1945 3469 y Fs(\()p Fp(H)7 b Fs(\))20 b(+)g Fp(N)2292 3431 y Fw(\000)p Fo(1)2282 3494 y(0)2387 3469 y Fp(N)516 3655 y Fr(\024)83 b Fp(K)7 b Fs(\()p Fp(N)862 3669 y Fo(0)901 3655 y Fp(;)15 b(R)q(;)g(E)1118 3669 y Fo(0)1158 3655 y Fs(\))21 b(+)f Fp(\017)g Fs(+)g Fp(\017)1490 3622 y Fw(\000)p Fo(1)1584 3655 y Fs(\(1)h Fr(\000)f Fp(f)1821 3669 y Fn(E)t(;\024)1940 3655 y Fs(\()p Fp(H)7 b Fs(\)\))2128 3622 y Fo(2)2189 3655 y Fs(+)20 b Fp(N)2363 3616 y Fw(\000)p Fo(1)2353 3679 y(0)2457 3655 y Fp(N)5 b(;)0 3283 y Fs(\(7.32\))0 3866 y(for)40 b Fp(K)7 b Fs(\()p Fp(N)341 3880 y Fo(0)381 3866 y Fp(;)15 b(R)q(;)g(E)598 3880 y Fo(0)638 3866 y Fs(\))43 b(=)f Fr(j)p Fs(\000\()p Fp(j)988 3833 y Fn(R)1047 3866 y Fs(\))1082 3833 y Fo(2)1122 3866 y Fs(1)-20 b(l)1172 3885 y Fo([0)p Fn(;N)1303 3894 y Fh(0)1337 3885 y Fo(])1361 3866 y Fs(\()p Fp(N)10 b Fs(\)1)-20 b(l)1565 3885 y Fo(])p Fw(\0001)p Fn(;E)1783 3894 y Fh(0)1816 3885 y Fo(+1])1930 3866 y Fs(\()p Fp(H)7 b Fs(\))p Fr(j)p Fs(.)73 b(The)40 b(op)s(erator)h Fp(K)7 b Fs(\()p Fp(N)2972 3880 y Fo(0)3011 3866 y Fp(;)15 b(R)q(;)g(E)3228 3880 y Fo(0)3268 3866 y Fs(\))41 b(is)g(compact)0 3988 y(using)30 b(the)h(fact)h(that)g(\()p Fp(K)27 b Fs(+)20 b(i\))1063 3955 y Fw(\000)p Fo(1)1189 3988 y Fs(is)31 b(compact)h(on)f Fr(K)q Fs(,)h Fp(j)1941 3955 y Fn(R)2030 3988 y Fs(has)e(compact)i(supp)s(ort)d(and)i Fp(!)s Fs(\()p Fp(k)s Fs(\))26 b(=)g Fr(j)p Fp(k)s Fr(j)h(!)f Fs(+)p Fr(1)0 4101 y Fs(when)j Fp(k)g Fr(!)c(1)p Fs(.)141 4214 y(Let)d(us)e(no)m(w)h(pro)m(v)m(e)h(the)f(lemma.)38 b(Note)22 b(\014rst)f(that)g(since)g Fp(m)2159 4181 y Fn(\016)2222 4214 y Fr(\025)k Fs(1,)f(w)m(e)d(ha)m(v)m(e)h(1)-20 b(l)2787 4233 y Fw(f)p Fo(0)p Fw(g)2897 4214 y Fs(\()p Fp(M)3030 4181 y Fn(\016)3020 4236 y Fw(1)3095 4214 y Fs(\))26 b(=)f(1)-20 b(l)3302 4228 y Fw(H)3368 4214 y Fr(\012)q Fs(1)g(l)3491 4233 y Fw(f)p Fo(0)p Fw(g)3601 4214 y Fs(\()p Fp(N)10 b Fs(\).)0 4327 y(It)30 b(follo)m(ws)i(also)f (from)f([DG1)r(,)g(Subsection)g(2.13])i(that:)417 4508 y(\024)411 4531 y(\000\()p Fp(j)545 4494 y Fn(R)604 4531 y Fs(\))639 4494 y Fw(\003)679 4531 y Fs(1)-20 b(l)729 4550 y Fw(f)p Fo(0)p Fw(g)839 4531 y Fs(\()p Fp(M)962 4545 y Fw(1)1037 4531 y Fs(\))1078 4508 y(\024)1072 4531 y(\000)q(\()p Fp(j)1207 4494 y 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Fp(j)2319 4875 y Fo(0)2359 4861 y Fs(.)40 b(T)-8 b(o)31 b(pro)m(v)m(e)g Fl(ii\))f Fs(w)m(e)h(recall)h(that:)685 5065 y Fp(B)759 5027 y Fn(\016)18 b Fo(ext)908 5065 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(:=)e Fp(M)1388 5027 y Fn(\016)19 b Fo(ext)1558 5065 y Fr(\000)h Fp(f)1694 5079 y Fn(E)t(;\024)1813 5065 y Fs(\()p Fp(H)1931 5027 y Fo(ext)2032 5065 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2229 5027 y Fn(\016)2268 5065 y Fp(v)s Fs(\))h Fr(\012)f Fs(1)-20 b(l)2512 5083 y Fo(\000\()p Fm(h)p Fo(\))2651 5065 y Fp(f)2696 5079 y Fn(E)t(;\024)2816 5065 y Fs(\()p Fp(H)2934 5027 y Fo(ext)3034 5065 y Fs(\))p Fp(:)-3094 b Fs(\(7.33\))1844 5517 y(51)p eop end %%Page: 52 52 TeXDict begin 52 51 bop 0 162 a Fs(Since)32 b Fp(M)337 129 y Fn(\016)o(;)15 b Fo(ext)530 162 y Fr(\025)28 b Fs(0)k(,)g(it)h(su\016ces)e(to)i(b)s(ound)d(from)h(b)s(elo)m(w)h(the)h (second)f(term)f(in)h(\(7.33\).)48 b(Since)32 b(b)m(y)f(Lemma)0 288 y(6.2)g Fl(iv\))g Fr(k)p Fs(\()p Fp(H)d Fs(+)20 b(i\))619 253 y Fw(\000)684 225 y Fh(1)p 684 237 31 4 v 684 279 a(2)729 288 y Fp(\036)p Fs(\(i)p Fp(a)891 255 y Fn(\016)929 288 y Fp(v)s Fs(\)\()p Fp(H)28 b Fs(+)20 b(i\))1301 253 y Fw(\000)1366 225 y Fh(1)p 1367 237 V 1367 279 a(2)1411 288 y Fr(k)26 b(\024)f Fp(C)37 b Fs(uniformly)29 b(in)h(0)c Fr(\024)f Fp(\016)k Fr(\024)2542 253 y Fo(1)p 2542 268 36 4 v 2542 320 a(2)2587 288 y Fs(,)i(w)m(e)g(ha)m(v)m(e:)1009 494 y Fr(k)p Fp(f)1099 508 y Fn(E)t(;\024)1218 494 y Fs(\()p Fp(H)1336 457 y Fo(ext)1437 494 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1634 457 y Fn(\016)1673 494 y Fp(v)s Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)1917 513 y Fo(\000\()p Fm(h)p Fo(\))2056 494 y Fp(f)2101 508 y Fn(E)t(;\024)2221 494 y Fs(\()p Fp(H)2339 457 y Fo(ext)2439 494 y Fs(\))p Fr(k)26 b(\024)f Fp(C)2706 508 y Fo(0)2746 494 y Fp(;)0 699 y Fs(uniformly)k(in)h(0)c Fp(<)f(\016)k Fr(\024)862 663 y Fo(1)p 862 678 V 862 730 a(2)907 699 y Fs(,)i Fp(E)g Fr(\024)25 b Fp(E)1224 713 y Fo(0)1263 699 y Fs(,)31 b(0)25 b Fp(<)g(\024)h Fr(\024)f Fs(1.)41 b(This)30 b(yields:)446 882 y(\024)440 905 y(\000\()p Fp(j)574 868 y Fn(R)633 905 y Fs(\))668 868 y Fw(\003)708 905 y Fs(1)-20 b(l)758 924 y Fw(f)p Fo(0)p Fw(g)868 905 y Fs(\()p Fp(M)1001 868 y Fn(\016)991 928 y Fw(1)1066 905 y Fs(\))p Fp(f)1146 919 y Fn(E)t(;\024)1266 905 y Fs(\()p Fp(H)1384 868 y Fo(ext)1485 905 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1682 868 y Fn(\016)1721 905 y Fp(v)s Fs(\))21 b Fr(\012)e Fs(1)-20 b(l)1965 924 y Fo(\000\()p Fm(h)p Fo(\))2104 905 y Fp(f)2149 919 y Fn(E)t(;\024)2269 905 y Fs(\()p Fp(H)2387 868 y Fo(ext)2487 905 y Fs(\))2528 882 y(\024)2522 905 y(\000\()p Fp(j)2656 868 y Fn(R)2715 905 y Fs(\))26 b Fr(\025)f(\000)p Fp(C)3008 919 y Fo(0)3047 905 y Fs(\000\()p Fp(j)3181 868 y Fn(R)3176 928 y Fo(1)3239 905 y Fs(\))3274 868 y Fo(2)3314 905 y Fp(:)0 1101 y Fs(Applying)30 b(then)g(\(7.31\))j (for)d Fp(j)h Fs(=)25 b Fp(j)1204 1115 y Fo(1)1244 1101 y Fs(,)30 b(w)m(e)h(obtain)g Fl(ii\))p Fs(.)40 b Fd(2)141 1261 y Fe(Pro)s(of)f(of)g(Prop)s(osition)h(7.13.)50 b Fs(Let)34 b(us)f(\014rst)g(explain)h(ho)m(w)f(to)h(\014x)f(the)h (parameter)g Fp(\016)s Fs(.)51 b(By)33 b(Lemma)0 1373 y(6.2)e Fl(iv\))p Fs(,)g(w)m(e)g(ha)m(v)m(e:)1012 1501 y(sup)966 1592 y Fo(0)p Fn(<\016)r Fw(\024)1154 1565 y Fh(1)p 1155 1577 31 4 v 1155 1618 a(2)1210 1501 y Fr(k)p Fs(\()p Fp(H)d Fs(+)20 b(i\))1545 1464 y Fw(\000)1610 1437 y Fh(1)p 1611 1449 V 1611 1490 a(2)1655 1501 y Fp(\036)p Fs(\(i)p Fp(a)1817 1464 y Fn(\016)1856 1501 y Fp(v)s Fs(\)\()p Fp(H)28 b Fs(+)20 b(i\))2228 1464 y Fw(\000)2293 1437 y Fh(1)p 2293 1449 V 2293 1490 a(2)2338 1501 y Fr(k)26 b(\024)f Fp(C)31 b(<)25 b Fr(1)p Fp(:)0 1763 y Fs(F)-8 b(or)31 b Fp(E)230 1777 y Fo(0)295 1763 y Fr(2)25 b Fs(I)-15 b(R)o(,)31 b(since)g(lim)870 1777 y Fn(t)p Fw(!)p Fo(0)1021 1763 y Fp(d)p Fs(\()p Fp(t)p Fs(\))26 b(=)f(+)p Fr(1)p Fs(,)30 b(w)m(e)h(can)f(\014x)g(0)c Fp(<)f(\016)k(<)2281 1727 y Fo(1)p 2281 1742 36 4 v 2281 1794 a(2)2356 1763 y Fs(small)i(enough)f(so)h(that:)779 1969 y Fp(d)p Fs(\()p Fp(\016)s Fs(\))c Fr(\025)71 b Fs(sup)1062 2060 y Fo(0)p Fn(<\016)r Fw(\024)1250 2033 y Fh(1)p 1251 2045 31 4 v 1251 2086 a(2)1306 1969 y Fr(k)p Fs(1)-20 b(l)1402 1987 y Fo(])p Fw(\0001)p Fn(;E)1620 1996 y Fh(0)1653 1987 y Fo(+2])1767 1969 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2082 1931 y Fn(\016)2122 1969 y Fp(v)s Fs(\)1)-20 b(l)2255 1987 y Fo(])p Fw(\0001)p Fn(;E)2473 1996 y Fh(0)2506 1987 y Fo(+2])2620 1969 y Fs(\()p Fp(H)7 b Fs(\))p Fr(k)21 b Fs(+)f(1)p Fp(:)-3000 b Fs(\(7.34\))0 2264 y(W)-8 b(e)27 b(start)g(b)m(y)e(considering)i(the)f(extended)g(ob) 5 b(jects)26 b Fp(H)1882 2231 y Fo(ext)2008 2264 y Fs(and)g Fp(B)2255 2231 y Fn(\016)18 b Fo(ext)2404 2264 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\).)41 b(By)26 b(Lemma)g(7.11)i(and)d (\(7.27\),)0 2377 y(\(7.28\))33 b(w)m(e)d(ha)m(v)m(e:)610 2573 y(1)-20 b(l)661 2591 y Fo([1)p Fn(;)p Fo(+)p Fw(1)p Fo([)884 2573 y Fs(\()p Fp(M)1017 2539 y Fn(\016)1007 2595 y Fw(1)1082 2573 y Fs(\))p Fp(B)1191 2539 y Fn(\016)19 b Fo(ext)1341 2573 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b(=)e(\(1)-20 b(l)1784 2587 y Fw(H)1868 2573 y Fr(\012)1965 2549 y Fs(\024)1959 2572 y(\000)2016 2573 y(\()p Fp(F)2122 2539 y Fn(\016)2160 2573 y Fs(\)\))2230 2539 y Fw(\003)2271 2573 y Fs(\()p Fp(I)2346 2587 y Fo(1)2406 2573 y Fs(+)19 b Fp(I)2536 2587 y Fo(2)2576 2573 y Fs(\)\(1)-20 b(l)2697 2587 y Fw(H)2782 2573 y Fr(\012)2878 2549 y Fs(\024)2872 2572 y(\000)2929 2573 y(\()p Fp(F)3035 2539 y Fn(\016)3074 2573 y Fs(\)\))p Fp(;)0 2768 y Fs(for)980 2863 y Fp(I)1020 2877 y Fo(1)1085 2863 y Fs(=)82 b(1)-20 b(l)1289 2877 y Fw(H)1373 2863 y Fr(\012)20 b Fs(1)-20 b(l)1515 2888 y Fo(\000\()p Fm(h)1622 2861 y Ff(<)1622 2913 y(\016)1674 2888 y Fo(\))1726 2863 y Fr(\012)20 b Fs(1)-20 b(l)1868 2881 y Fo([1)p Fn(;)p Fw(1)p Fo([)2036 2863 y Fs(\()p Fp(M)2169 2830 y Fn(\016)2208 2863 y Fs(\))2265 2840 y(^)2243 2863 y Fp(B)2317 2830 y Fn(\016)18 b Fo(ext)2466 2863 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))p Fp(;)980 3054 y(I)1020 3068 y Fo(2)1085 3054 y Fs(=)82 b(1)-20 b(l)1289 3068 y Fw(H)1373 3054 y Fr(\012)20 b Fs(1)-20 b(l)1515 3072 y Fo([1)p Fn(;)p Fw(1)p Fo([)1683 3054 y Fs(\()p Fp(M)1816 3021 y Fn(\016)1855 3054 y Fs(\))20 b Fr(\012)g Fs(1)-20 b(l)2052 3072 y Fw(f)p Fo(0)p Fw(g)2162 3054 y Fs(\()p Fp(N)10 b Fs(\))2337 3031 y(^)2315 3054 y Fp(B)2389 3021 y Fn(\016)18 b Fo(ext)2539 3054 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))p Fp(:)141 3210 y Fs(W)-8 b(e)32 b(start)f(b)m(y)f(b)s(ounding)e Fp(I)1083 3224 y Fo(2)1153 3210 y Fs(from)i(b)s(elo)m(w.)41 b(W)-8 b(e)32 b(can)e(write)1642 3405 y Fp(I)1682 3419 y Fo(2)1747 3405 y Fs(=)24 b Fp(I)1882 3419 y Fo(3)1942 3405 y Fs(+)c Fp(I)2073 3419 y Fo(4)2112 3405 y Fp(;)0 3601 y Fs(for)270 3778 y Fp(I)310 3792 y Fo(3)375 3778 y Fs(=)82 b(1)-20 b(l)579 3792 y Fw(H)663 3778 y Fr(\012)20 b Fs(1)-20 b(l)805 3797 y Fo([1)p Fn(;)p Fw(1)p Fo([)973 3778 y Fs(\()p Fp(M)1106 3745 y Fn(\016)1145 3778 y Fs(\))20 b Fr(\012)g Fs(1)-20 b(l)1342 3797 y Fw(f)p Fo(0)p Fw(g)1452 3778 y Fs(\()p Fp(N)10 b Fs(\))1639 3755 y(^)1605 3778 y Fp(M)1703 3745 y Fn(\016)19 b Fo(ext)375 3980 y Fr(\025)82 b Fs(\(1)21 b Fr(\000)f Fp(\017)p Fs(\)1)-20 b(l)843 3994 y Fw(H)927 3980 y Fr(\012)20 b Fs(1)-20 b(l)1069 3999 y Fo([1)p Fn(;)p Fw(1)p Fo([)1238 3980 y Fs(\()p Fp(M)1371 3947 y Fn(\016)1409 3980 y Fs(\))21 b Fr(\012)e Fs(1)-20 b(l)1606 3999 y Fw(f)p Fo(0)p Fw(g)1716 3980 y Fs(\()p Fp(N)10 b Fs(\))1869 3886 y Fg(\020)1919 3980 y Fs(1)-20 b(l)1970 3994 y Fw(H)2054 3980 y Fr(\012)20 b Fp(M)2243 3947 y Fn(\016)2301 3980 y Fr(\012)g Fs(1)-20 b(l)2443 4005 y Fo(\000\()p Fm(h)2550 3978 y Ff(>)2550 4030 y(\016)2602 4005 y Fo(\))2654 3980 y Fs(+)20 b(1)-20 b(l)2796 3994 y Fw(H)2880 3980 y Fr(\012)20 b Fs(1)-20 b(l)3022 4005 y Fo(\000\()p Fm(h)3129 3978 y Ff(<)3129 4030 y(\016)3181 4005 y Fo(\))3233 3980 y Fr(\012)20 b Fp(M)3422 3947 y Fn(\016)3460 3886 y Fg(\021)528 4164 y Fs(+)p Fp(\017d)p Fs(\()p Fp(\016)s Fs(\)1)-20 b(l)848 4178 y Fw(H)933 4164 y Fr(\012)20 b Fs(1)-20 b(l)1074 4183 y Fo([1)p Fn(;)p Fw(1)p Fo([)1243 4164 y Fs(\()p Fp(M)1376 4131 y Fn(\016)1414 4164 y Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)1611 4183 y Fw(f)p Fo(0)p Fw(g)1721 4164 y Fs(\()p Fp(N)10 b Fs(\))p Fp(:)0 3968 y Fs(\(7.35\))0 4369 y(In)30 b(fact)h Fp(m)373 4336 y Fn(\016)373 4406 y Fw(j)p Fm(h)429 4379 y Ff(<)429 4431 y(\016)510 4369 y Fr(\025)25 b Fp(d)p Fs(\()p Fp(\016)s Fs(\))32 b(whic)m(h)e(implies)h (that)535 4608 y(1)-20 b(l)585 4622 y Fw(H)670 4608 y Fr(\012)19 b Fs(1)-20 b(l)811 4627 y Fo([1)p Fn(;)p Fw(1)p Fo([)980 4608 y Fs(\()p Fp(M)1113 4571 y Fn(\016)1151 4608 y Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)1348 4627 y Fw(f)p Fo(0)p Fw(g)1458 4608 y Fs(\()p Fp(N)10 b Fs(\))1646 4585 y(^)1611 4608 y Fp(M)1709 4571 y Fn(\016)19 b Fo(ext)1885 4608 y Fr(\025)25 b Fp(d)p Fs(\()p Fp(\016)s Fs(\)1)-20 b(l)2193 4622 y Fw(H)2278 4608 y Fr(\012)19 b Fs(1)-20 b(l)2419 4627 y Fo([1)p Fn(;)p Fw(1)p Fo([)2588 4608 y Fs(\()p Fp(M)2721 4571 y Fn(\016)2759 4608 y Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)2956 4627 y Fw(f)p Fo(0)p Fw(g)3066 4608 y Fs(\()p Fp(N)10 b Fs(\))p Fp(:)0 4804 y Fs(Next:)812 4891 y Fp(I)852 4905 y Fo(4)917 4891 y Fs(=)83 b Fr(\000)p Fs(1)-20 b(l)1192 4905 y Fw(H)1276 4891 y Fr(\012)20 b Fs(1)-20 b(l)1418 4910 y Fo([1)p Fn(;)p Fw(1)p Fo([)1587 4891 y Fs(\()p Fp(M)1720 4858 y Fn(\016)1758 4891 y Fs(\))20 b Fr(\012)g Fs(1)-20 b(l)1955 4910 y Fw(f)p Fo(0)p Fw(g)2065 4891 y Fs(\()p Fp(N)10 b Fs(\))1071 5082 y Fr(\002)p Fp(f)1187 5096 y Fn(E)t(;\024)1306 5082 y Fs(\()1365 5059 y(^)1341 5082 y Fp(H)1424 5049 y Fo(ext)1524 5082 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1721 5049 y Fn(\016)1760 5082 y Fp(v)s Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)2005 5107 y Fo(\000\()p Fm(h)2112 5080 y Ff(<)2112 5132 y(\016)2164 5107 y Fo(\))2216 5082 y Fr(\012)20 b Fs(1)-20 b(l)2358 5107 y Fo(\000\()p Fm(h)2465 5080 y Ff(>)2465 5132 y(\016)2517 5107 y Fo(\))2549 5082 y Fp(f)2594 5096 y Fn(E)t(;\024)2713 5082 y Fs(\()2773 5059 y(^)2748 5082 y Fp(H)2831 5049 y Fo(ext)2932 5082 y Fs(\))917 5266 y Fr(\025)83 b(\000)p Fp(C)1207 5280 y Fo(0)1246 5266 y Fs(1)-20 b(l)1296 5280 y Fw(H)1381 5266 y Fr(\012)20 b Fs(1)-20 b(l)1522 5284 y Fo([1)p Fn(;)p Fw(1)p Fo([)1691 5266 y Fs(\()p Fp(M)1824 5233 y Fn(\016)1862 5266 y Fs(\))21 b Fr(\012)f Fs(1)-20 b(l)2059 5284 y Fw(f)p Fo(0)p Fw(g)2169 5266 y Fs(\()p Fp(N)10 b Fs(\))p Fp(;)1844 5517 y Fs(52)p eop end %%Page: 53 53 TeXDict begin 53 52 bop 0 162 a Fs(uniformly)29 b(for)i Fp(E)f Fr(\024)25 b Fp(E)814 176 y Fo(0)874 162 y Fs(+)20 b(1,)31 b(0)25 b Fp(<)g(\024)h Fr(\024)f Fs(1,)31 b(0)26 b Fp(<)f(\016)k Fr(\024)1849 126 y Fo(1)p 1849 141 36 4 v 1849 194 a(2)1894 162 y Fs(,)i(for)780 377 y Fp(C)845 391 y Fo(0)910 377 y Fs(=)71 b(sup)1006 468 y Fo(0)p Fn(<\016)r Fw(\024)1194 441 y Fh(1)p 1194 453 31 4 v 1194 494 a(2)1250 377 y Fr(k)p Fs(1)-20 b(l)1346 395 y Fo(])p Fw(\0001)p Fn(;E)1564 404 y Fh(0)1597 395 y Fo(+2])1711 377 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2026 339 y Fn(\016)2065 377 y Fp(v)s Fs(\)1)-20 b(l)2198 395 y Fo(])p Fw(\0001)p Fn(;E)2416 404 y Fh(0)2449 395 y Fo(+2])2563 377 y Fs(\()p Fp(H)7 b Fs(\))p Fr(k)26 b Fp(<)f Fr(1)p Fp(:)-2999 b Fs(\(7.36\))0 665 y(This)30 b(yields)270 880 y Fp(I)310 894 y Fo(2)375 880 y Fr(\025)82 b Fs(\(1)21 b Fr(\000)f Fp(\017)p Fs(\)1)-20 b(l)843 894 y Fw(H)927 880 y Fr(\012)20 b Fs(1)-20 b(l)1069 898 y Fo([1)p Fn(;)p Fw(1)p Fo([)1238 880 y Fs(\()p Fp(M)1371 847 y Fn(\016)1409 880 y Fs(\))21 b Fr(\012)e Fs(1)-20 b(l)1606 898 y Fw(f)p Fo(0)p Fw(g)1716 880 y Fs(\()p Fp(N)10 b Fs(\))1869 785 y Fg(\020)1919 880 y Fs(1)-20 b(l)1970 894 y Fw(H)2054 880 y Fr(\012)20 b Fp(M)2243 847 y Fn(\016)2301 880 y Fr(\012)g Fs(1)-20 b(l)2443 905 y Fo(\000\()p Fm(h)2550 878 y Ff(>)2550 930 y(\016)2602 905 y Fo(\))2654 880 y Fs(+)20 b(1)-20 b(l)2796 894 y Fw(H)2880 880 y Fr(\012)20 b Fs(1)-20 b(l)3022 905 y Fo(\000\()p Fm(h)3129 878 y Ff(<)3129 930 y(\016)3181 905 y Fo(\))3233 880 y Fr(\012)20 b Fp(M)3422 847 y Fn(\016)3460 785 y Fg(\021)528 1063 y Fs(+\()p Fp(\017d)p Fs(\()p Fp(\016)s Fs(\))i Fr(\000)e Fp(C)1009 1077 y Fo(0)1049 1063 y Fs(\)1)-20 b(l)1134 1077 y Fw(H)1219 1063 y Fr(\012)20 b Fs(1)-20 b(l)1360 1082 y Fo([1)p Fn(;)p Fw(1)p Fo([)1529 1063 y Fs(\()p Fp(M)1662 1030 y Fn(\016)1701 1063 y Fs(\))20 b Fr(\012)g Fs(1)-20 b(l)1897 1082 y Fw(f)p Fo(0)p Fw(g)2007 1063 y Fs(\()p Fp(N)10 b Fs(\)\))p Fp(;)0 963 y Fs(\(7.37\))0 1261 y(for)30 b(all)h Fp(\017)26 b(>)f Fs(0.)41 b(Let)31 b(us)e(no)m(w)i(b)s(ound)d Fp(I)1324 1275 y Fo(1)1363 1261 y Fs(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e:)571 1454 y Fp(I)611 1468 y Fo(1)675 1454 y Fs(=)83 b(1)-20 b(l)880 1468 y Fw(H)964 1454 y Fr(\012)20 b Fs(1)-20 b(l)1105 1479 y Fo(\000\()p Fm(h)1212 1452 y Ff(<)1212 1504 y(\016)1265 1479 y Fo(\))1317 1454 y Fr(\012)20 b Fs(1)-20 b(l)1458 1472 y Fo([1)p Fn(;)p Fw(1)p Fo([)1627 1454 y Fs(\()p Fp(M)1760 1421 y Fn(\016)1799 1454 y Fs(\))829 1656 y Fr(\002)900 1562 y Fg(\020)983 1633 y Fs(^)949 1656 y Fp(M)1047 1623 y Fn(\016)19 b Fo(ext)1217 1656 y Fr(\000)h Fp(f)1353 1670 y Fn(E)t(;\024)1473 1656 y Fs(\()1532 1633 y(^)1508 1656 y Fp(H)1591 1623 y Fo(ext)1691 1656 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1888 1623 y Fn(\016)1927 1656 y Fp(v)s Fs(\))h Fr(\012)f Fs(1)-20 b(l)2171 1681 y Fo(\000\()p Fm(h)2278 1654 y Ff(<)2278 1706 y(\016)2331 1681 y Fo(\))2383 1656 y Fr(\012)20 b Fs(1)-20 b(l)2524 1681 y Fo(\000\()p Fm(h)2631 1654 y Ff(>)2631 1706 y(\016)2684 1681 y Fo(\))2716 1656 y Fp(f)2761 1670 y Fn(E)t(;\024)2880 1656 y Fs(\()2939 1633 y(^)2915 1656 y Fp(H)2998 1623 y Fo(ext)3099 1656 y Fs(\))3134 1562 y Fg(\021)3184 1656 y Fp(:)0 1887 y Fs(Since)28 b(b)m(y)f(Lemma)h(7.11)i Fl(i\))d Fs(1)-20 b(l)1013 1905 y Fo([1)p Fn(;)p Fw(1)p Fo([)1182 1887 y Fs(\()p Fp(M)1315 1854 y Fn(\016)1353 1887 y Fs(\))26 b(=)f(1)-20 b(l)1560 1905 y Fo([1)p Fn(;)p Fw(1)p Fo([)1729 1887 y Fs(\()p Fp(N)10 b Fs(\),)29 b(and)e Fr(j)p Fp(k)s Fr(j)f(\025)f Fp(\016)32 b Fs(on)27 b Fq(h)2570 1851 y Fn(>)2570 1915 y(\016)2630 1887 y Fs(,)h(w)m(e)h(obtain)f(that)g (the)g(op)s(erator)0 2119 y(1)-20 b(l)51 2133 y Fw(H)126 2119 y Fr(\012)11 b Fs(1)-20 b(l)258 2144 y Fo(\000\()p Fm(h)365 2117 y Ff(<)365 2169 y(\016)418 2144 y Fo(\))460 2119 y Fr(\012)11 b Fs(1)-20 b(l)593 2137 y Fo([1)p Fn(;)p Fw(1)p Fo([)761 2119 y Fs(\()p Fp(M)894 2081 y Fn(\016)933 2119 y Fs(\))968 2025 y Fg(\020)1018 2119 y Fp(M)1116 2081 y Fn(\016)1165 2119 y Fr(\012)11 b Fs(1)-20 b(l)1297 2144 y Fo(\000\()p Fm(h)1404 2117 y Ff(<)1404 2169 y(\016)1457 2144 y Fo(\))1499 2119 y Fr(\012)11 b Fs(1)-20 b(l)1632 2144 y Fo(\000\()p Fm(h)1739 2117 y Ff(>)1739 2169 y(\016)1791 2144 y Fo(\))1834 2119 y Fr(\000)11 b Fp(f)1961 2133 y Fn(E)t(;\024)2080 2119 y Fs(\()2139 2096 y(^)2115 2119 y Fp(H)2198 2081 y Fo(ext)2299 2119 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2496 2081 y Fn(\016)2535 2119 y Fp(v)s Fs(\))g Fr(\012)g Fs(1)-20 b(l)2760 2144 y Fo(\000\()p Fm(h)2867 2117 y Ff(<)2867 2169 y(\016)2920 2144 y Fo(\))2963 2119 y Fr(\012)11 b Fs(1)-20 b(l)3095 2144 y Fo(\000\()p Fm(h)3202 2117 y Ff(>)3202 2169 y(\016)3255 2144 y Fo(\))3286 2119 y Fp(f)3331 2133 y Fn(E)t(;\024)3451 2119 y Fs(\()3510 2096 y(^)3486 2119 y Fp(H)3569 2081 y Fo(ext)3669 2119 y Fs(\))3704 2025 y Fg(\021)3754 2119 y Fp(;)0 2343 y Fs(acting)32 b(on)e Fr(H)21 b(\012)f Fs(\000\()p Fq(h)723 2308 y Fn(<)723 2371 y(\016)782 2343 y Fs(\))h Fr(\012)f Fs(\000\()p Fq(h)1064 2308 y Fn(>)1064 2371 y(\016)1123 2343 y Fs(\))31 b(is)f(unitarily)h(equiv)-5 b(alen)m(t)32 b(to)1034 2485 y Fg(Z)1117 2511 y Fw(\010)1080 2673 y Fo([)p Fn(\016)o(;)p Fw(1)p Fo([)1258 2600 y Fp(M)1356 2562 y Fn(\016)1415 2600 y Fs(+)19 b Fp(f)1550 2615 y Fn(E)t Fw(\000)p Fn(\025;\024)1766 2600 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)2081 2562 y Fn(\016)2120 2600 y Fp(v)s Fs(\))p Fp(f)2247 2615 y Fn(E)t Fw(\000)p Fn(\025;\024)2463 2600 y Fs(\()p Fp(H)g Fs(\)d)p Fp(\025:)0 2860 y Fs(By)26 b(the)g(induction)f(assumption)h(the)g(Mourre)f (estimate)j(holds)d(at)i(all)f(energies)h Fp(E)2867 2827 y Fw(0)2916 2860 y Fr(\024)e Fp(E)3079 2874 y Fo(1)3118 2860 y Fs(.)39 b(By)26 b(Prop)s(osition)0 2972 y(7.8,)32 b(for)e(an)m(y)h Fp(\017)520 2986 y Fo(0)584 2972 y Fp(>)25 b Fs(0)31 b(there)f(exists)h Fp(C)q(;)15 b(\024)27 b(>)e Fs(0)31 b(suc)m(h)f(that)h(for)f(all)h Fp(E)2335 2940 y Fw(0)2384 2972 y Fr(\024)25 b Fp(E)2547 2986 y Fo(1)672 3177 y Fp(M)770 3139 y Fn(\016)828 3177 y Fr(\000)20 b Fp(f)964 3192 y Fn(E)1020 3173 y Fk(0)1041 3192 y Fn(;\024)1106 3177 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1421 3139 y Fn(\016)1460 3177 y Fp(v)s Fs(\))p Fp(f)1587 3192 y Fn(E)1643 3173 y Fk(0)1665 3192 y Fn(;\024)1729 3177 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f(\000)p Fp(\017)2112 3191 y Fo(0)2151 3177 y Fp(=)p Fs(41)-20 b(l)22 b Fr(\000)e Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2712 3192 y Fn(E)2768 3173 y Fk(0)2790 3192 y Fn(;\024)2854 3177 y Fs(\)\()p Fp(H)7 b Fs(\))3042 3139 y Fo(2)3082 3177 y Fp(:)-3107 b Fs(\(7.38\))0 3381 y(W)-8 b(e)32 b(\014x)d Fp(\024)i Fs(suc)m(h)f(that)h(\(7.38\))i(holds)d(and)f(w)m(e)i(obtain)g(that)804 3601 y(1)-20 b(l)854 3615 y Fw(H)939 3601 y Fr(\012)20 b Fs(1)-20 b(l)1080 3626 y Fo(\000\()p Fm(h)1187 3599 y Ff(<)1187 3651 y(\016)1240 3626 y Fo(\))1292 3601 y Fr(\012)20 b Fs(1)-20 b(l)1433 3619 y Fo([1)p Fn(;)p Fw(1)p Fo([)1602 3601 y Fs(\()p Fp(M)1735 3568 y Fn(\016)1773 3601 y Fs(\))1808 3507 y Fg(\020)1858 3601 y Fp(M)1956 3568 y Fn(\016)2014 3601 y Fr(\012)20 b Fs(1)-20 b(l)2156 3626 y Fo(\000\()p Fm(h)2263 3599 y Ff(<)2263 3651 y(\016)2316 3626 y Fo(\))2367 3601 y Fr(\012)20 b Fs(1)-20 b(l)2509 3626 y Fo(\000\()p Fm(h)2616 3599 y Ff(>)2616 3651 y(\016)2669 3626 y Fo(\))650 3750 y Fr(\000)83 b Fp(f)849 3764 y Fn(E)t(;\024)968 3750 y Fs(\()1027 3727 y(^)1003 3750 y Fp(H)1086 3717 y Fo(ext)1187 3750 y Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1384 3717 y Fn(\016)1423 3750 y Fp(v)s Fs(\))21 b Fr(\012)e Fs(1)-20 b(l)1667 3775 y Fo(\000\()p Fm(h)1774 3748 y Ff(<)1774 3800 y(\016)1827 3775 y Fo(\))1879 3750 y Fr(\012)19 b Fs(1)-20 b(l)2020 3775 y Fo(\000\()p Fm(h)2127 3748 y Ff(>)2127 3800 y(\016)2180 3775 y Fo(\))2211 3750 y Fp(f)2256 3764 y Fn(E)t(;\024)2376 3750 y Fs(\()2435 3727 y(^)2411 3750 y Fp(H)2494 3717 y Fo(ext)2594 3750 y Fs(\))2629 3656 y Fg(\021)650 3952 y Fr(\025)83 b Fs(1)-20 b(l)854 3966 y Fw(H)939 3952 y Fr(\012)20 b Fs(1)-20 b(l)1080 3978 y Fo(\000\()p Fm(h)1187 3950 y Ff(<)1187 4003 y(\016)1240 3978 y Fo(\))1292 3952 y Fr(\012)20 b Fs(1)-20 b(l)1433 3971 y Fo([1)p Fn(;)p Fw(1)p Fo([)1602 3952 y Fs(\()p Fp(M)1735 3919 y Fn(\016)1773 3952 y Fs(\))1808 3858 y Fg(\020)1878 3952 y Fr(\000)20 b Fp(\017)2006 3966 y Fo(0)2046 3952 y Fp(=)p Fs(41)-20 b(l)21 b Fr(\000)f Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)2607 3966 y Fn(E)t(;\024)2726 3952 y Fs(\)\()2821 3929 y(^)2796 3952 y Fp(H)2879 3919 y Fo(ext)2980 3952 y Fs(\))3015 3919 y Fo(2)3055 3858 y Fg(\021)3104 3952 y Fp(;)0 4183 y Fs(uniformly)29 b(for)i Fp(E)f Fr(\024)25 b Fp(E)814 4197 y Fo(1)874 4183 y Fs(+)20 b Fp(\016)s Fs(.)41 b(Using)31 b(also)g(that)g(1)-20 b(l)1760 4202 y Fo([1)p Fn(;)p Fw(1)p Fo([)1928 4183 y Fs(\()p Fp(M)2061 4150 y Fn(\016)2100 4183 y Fs(\))p Fp(M)2233 4150 y Fn(\016)2296 4183 y Fr(\025)25 b Fp(d)p Fs(\()p Fp(\016)s Fs(\)1)-20 b(l)2605 4202 y Fo([1)p Fn(;)p Fw(1)p Fo([)2773 4183 y Fs(\()p Fp(M)2906 4150 y Fn(\016)2945 4183 y Fs(\),)31 b(w)m(e)g(obtain)f(\014nally:)738 4397 y Fp(I)778 4411 y Fo(1)842 4397 y Fr(\025)83 b Fs(1)-20 b(l)1047 4411 y Fw(H)1131 4397 y Fr(\012)20 b Fs(1)-20 b(l)1273 4422 y Fo(\000\()p Fm(h)1380 4395 y Ff(<)1380 4447 y(\016)1432 4422 y Fo(\))1484 4397 y Fr(\012)20 b Fs(1)-20 b(l)1626 4415 y Fo([1)p Fn(;)p Fw(1)p Fo([)1794 4397 y Fs(\()p Fp(M)1927 4364 y Fn(\016)1966 4397 y Fs(\))996 4599 y Fr(\002)1067 4505 y Fg(\020)1116 4599 y Fs(\(1)21 b Fr(\000)f Fp(\017)p Fs(\)\(1)-20 b(l)1466 4613 y Fw(H)1551 4599 y Fr(\012)20 b Fp(M)1740 4566 y Fn(\016)1798 4599 y Fr(\012)g Fs(1)-20 b(l)1939 4624 y Fo(\000\()p Fm(h)2046 4597 y Ff(>)2046 4649 y(\016)2099 4624 y Fo(\))2151 4599 y Fs(+)20 b(1)-20 b(l)2292 4613 y Fw(H)2377 4599 y Fr(\012)20 b Fs(1)-20 b(l)2518 4624 y Fo(\000\()p Fm(h)2625 4597 y Ff(<)2625 4649 y(\016)2678 4624 y Fo(\))2730 4599 y Fr(\012)20 b Fp(M)2919 4566 y Fn(\016)2957 4599 y Fs(\))2992 4505 y Fg(\021)996 4783 y Fs(+1)-20 b(l)1117 4797 y Fw(H)1202 4783 y Fr(\012)20 b Fs(1)-20 b(l)1343 4808 y Fo(\000\()p Fm(h)1450 4781 y Ff(<)1450 4833 y(\016)1503 4808 y Fo(\))1555 4783 y Fr(\012)20 b Fs(1)-20 b(l)1696 4802 y Fo([1)p Fn(;)p Fw(1)p Fo([)1865 4783 y Fs(\()p Fp(M)1998 4750 y Fn(\016)2036 4783 y Fs(\)\()p Fp(\017d)p Fs(\()p Fp(\016)s Fs(\))23 b Fr(\000)d Fp(\017)2454 4797 y Fo(0)2493 4783 y Fp(=)p Fs(4\))996 4974 y Fr(\000)p Fp(C)7 b Fs(1)-20 b(l)1189 4988 y Fw(H)1273 4974 y Fr(\012)20 b Fs(1)-20 b(l)1415 4999 y Fo(\000\()p Fm(h)1522 4972 y Ff(<)1522 5024 y(\016)1575 4999 y Fo(\))1626 4974 y Fr(\012)20 b Fs(1)-20 b(l)1768 4992 y Fo([1)p Fn(;)p Fw(1)p Fo([)1937 4974 y Fs(\()p Fp(M)2070 4941 y Fn(\016)2108 4974 y Fs(\)\(1)21 b Fr(\000)f Fp(f)2380 4988 y Fn(E)t(;\024)2500 4974 y Fs(\)\()2594 4951 y(^)2570 4974 y Fp(H)2653 4941 y Fo(ext)2753 4974 y Fs(\))2788 4941 y Fo(2)2828 4974 y Fp(;)0 4694 y Fs(\(7.39\))0 5187 y(for)30 b(all)h Fp(\017)26 b(>)f Fs(0,)31 b(uniformly)e(for)h Fp(E)h Fr(\024)25 b Fp(E)1339 5201 y Fo(1)1398 5187 y Fs(+)20 b Fp(\016)s Fs(.)1844 5517 y(53)p eop end %%Page: 54 54 TeXDict begin 54 53 bop 141 162 a Fs(Using)31 b(no)m(w)f(\(7.37\))j (and)c(the)i(functorial)g(prop)s(erties)f(of)2131 139 y(\024)2125 162 y(\000\()p Fp(F)2288 129 y Fn(\016)2326 162 y Fs(\),)h(w)m(e)g(obtain:)640 364 y(1)-20 b(l)690 382 y Fo([1)p Fn(;)p Fw(1)p Fo([)859 364 y Fs(\()p Fp(M)992 331 y Fn(\016)982 386 y Fw(1)1057 364 y Fs(\))p Fp(B)1166 331 y Fn(\016)19 b Fo(ext)1316 364 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))486 524 y Fr(\025)83 b Fs(\(1)21 b Fr(\000)f Fp(\017)p Fs(\))p Fp(M)1002 491 y Fn(\016)992 546 y Fw(1)1067 524 y Fs(1)-20 b(l)1117 542 y Fo([1)p Fn(;)p Fw(1)p Fo([)1286 524 y Fs(\()p Fp(M)1419 491 y Fn(\016)1409 546 y Fw(1)1484 524 y Fs(\))21 b(+)f(min)o Fr(f)p Fp(\017d)p Fs(\()p Fp(\016)s Fs(\))j Fr(\000)d Fp(\017)2175 538 y Fo(0)2214 524 y Fp(=)p Fs(4)p Fp(;)15 b(\017d)p Fs(\()p Fp(\016)s Fs(\))23 b Fr(\000)d Fp(C)2720 538 y Fo(0)2760 524 y Fs(\))p Fr(g)p Fs(1)-20 b(l)2891 542 y Fo([1)p Fn(;)p Fw(1)p Fo([)3060 524 y Fs(\()p Fp(M)3193 491 y Fn(\016)3183 546 y Fw(1)3258 524 y Fs(\))640 638 y Fr(\000)p Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)1019 652 y Fn(E)t(;\024)1138 638 y Fs(\)\()p Fp(H)1291 605 y Fo(ext)1392 638 y Fs(\))1427 605 y Fo(2)1467 638 y Fp(:)0 835 y Fs(Going)31 b(bac)m(k)g(to)h(the)e (original)i(Hilb)s(ert)e(space)h Fr(H)q Fs(,)f(this)g(yields:)868 1009 y(\024)863 1032 y(\000)o(\()p Fp(j)996 999 y Fn(R)1055 1032 y Fs(\))1090 999 y Fw(\003)1130 1032 y Fp(B)1204 999 y Fn(\016)18 b Fo(ext)1353 1032 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\)1)-20 b(l)1639 1051 y Fo([1)p Fn(;)p Fw(1)p Fo([)1808 1032 y Fs(\()p Fp(M)1941 999 y Fn(\016)1931 1055 y Fw(1)2006 1032 y Fs(\))2047 1009 y(\024)2041 1032 y(\000)q(\()p Fp(j)2176 999 y Fn(R)2234 1032 y Fs(\))709 1193 y Fr(\025)83 b Fs(\(1)21 b Fr(\000)e Fp(\017)p Fs(\))1132 1170 y(\024)1126 1193 y(\000)q(\()p Fp(j)1261 1160 y Fn(R)1319 1193 y Fs(\))1354 1160 y Fw(\003)1394 1193 y Fp(M)1492 1160 y Fn(\016)1482 1216 y Fw(1)1557 1193 y Fs(1)-20 b(l)1607 1212 y Fo([1)p Fn(;)p Fw(1)p Fo([)1776 1193 y Fs(\()p Fp(M)1909 1160 y Fn(\016)1899 1216 y Fw(1)1975 1193 y Fs(\))2016 1170 y(\024)2010 1193 y(\000\()p Fp(j)2144 1160 y Fn(R)2202 1193 y Fs(\))863 1355 y(+)15 b(min)o Fr(f)p Fp(\017d)p Fs(\()p Fp(\016)s Fs(\))22 b Fr(\000)e Fp(\017)1492 1369 y Fo(0)1532 1355 y Fp(=)p Fs(4)p Fp(;)15 b(\017d)p Fs(\()p Fp(\016)s Fs(\))23 b Fr(\000)d Fp(C)2038 1369 y Fo(0)2077 1355 y Fr(g)2128 1332 y Fs(\024)2122 1355 y(\000\()p Fp(j)2256 1322 y Fn(R)2315 1355 y Fs(\))2350 1322 y Fw(\003)2390 1355 y Fs(1)-20 b(l)2440 1373 y Fo([1)p Fn(;)p Fw(1)p Fo([)2609 1355 y Fs(\()p Fp(M)2742 1322 y Fn(\016)2732 1377 y Fw(1)2808 1355 y Fs(\))2849 1332 y(\024)2843 1355 y(\000\()p Fp(j)2977 1322 y Fn(R)3035 1355 y Fs(\))863 1516 y Fr(\000)p Fp(C)1011 1493 y Fs(\024)1006 1516 y(\000)o(\()p Fp(j)1139 1483 y Fn(R)1197 1516 y Fs(\))1232 1483 y Fw(\003)1272 1516 y Fs(\(1)21 b Fr(\000)f Fp(f)1509 1530 y Fn(E)t(;\024)1628 1516 y Fs(\)\()p Fp(H)1781 1483 y Fo(ext)1882 1516 y Fs(\))1917 1483 y Fo(2)1963 1493 y Fs(\024)1957 1516 y(\000\()p Fp(j)2091 1483 y Fn(R)2149 1516 y Fs(\))p Fp(;)0 1274 y Fs(\(7.40\))0 1712 y(for)29 b(all)h Fp(\017)25 b(>)g Fs(0.)41 b(W)-8 b(e)30 b(ha)m(v)m(e)h(no)m(w)e(completed)i(the)e(main)g(part)g(of)h (the)f(pro)s(of,)g(and)g(it)h(remains)f(only)g(to)h(collect)0 1825 y(the)h(error)f(terms.)141 1938 y(Let)42 b(us)e(\014x)g Fp(\017)j Fs(=)g(1)e(in)g(\(7.40\).)75 b(Using)41 b(the)g(fact)h(that)f Fp(d)p Fs(\()p Fp(\016)s Fs(\))k Fr(\025)e Fs(1,)h Fp(d)p Fs(\()p Fp(\016)s Fs(\))29 b Fr(\000)e Fp(C)2923 1952 y Fo(0)3005 1938 y Fr(\025)43 b Fs(1,)h(together)f(with)0 2051 y(Prop)s(osition)30 b(7.9)i(w)m(e)f(get:)571 2226 y(\024)566 2249 y(\000)o(\()p Fp(j)699 2216 y Fn(R)758 2249 y Fs(\))793 2216 y Fw(\003)833 2249 y Fp(B)907 2216 y Fn(\016)18 b Fo(ext)1056 2249 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\)1)-20 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Fo(0)2455 2425 y Fs(\))p Fp(N)2583 2362 y Fh(1)p 2583 2374 V 2583 2415 a(2)2628 2425 y Fp(:)0 2336 y Fs(\(7.41\))0 2627 y(By)34 b(Lemma)g(7.14)h(there) f(exists)g Fp(C)1224 2641 y Fo(1)1294 2627 y Fp(>)c Fs(0)k(suc)m(h)f (that)i(for)e(all)h Fp(N)2227 2641 y Fo(0)2298 2627 y Fr(2)c Fs(I)-15 b(N)2475 2591 y Fw(\003)2514 2627 y Fp(;)15 b(\017;)g(R)32 b(>)e Fs(0)k(there)g(exists)g Fp(K)7 b Fs(\()p Fp(N)3594 2641 y Fo(0)3634 2627 y Fp(;)15 b(R)q Fs(\))0 2739 y(compact)32 b(suc)m(h)e(that)387 2944 y Fr(\000)464 2921 y Fs(\024)458 2944 y(\000\()p Fp(j)592 2906 y Fn(R)650 2944 y Fs(\))685 2906 y Fw(\003)725 2944 y Fs(1)-20 b(l)776 2962 y Fw(f)p Fo(0)p Fw(g)886 2944 y Fs(\()p Fp(M)1019 2906 y Fn(\016)1009 2966 y Fw(1)1084 2944 y Fs(\))1125 2921 y(\024)1119 2944 y(\000\()p Fp(j)1253 2906 y Fn(R)1312 2944 y Fs(\))26 b Fr(\025)e(\000)p Fp(\017)c Fr(\000)g Fp(\017)1724 2906 y Fw(\000)p Fo(1)1818 2944 y Fs(\(1)h Fr(\000)f Fp(f)2055 2958 y Fn(E)t(;\024)2175 2944 y Fs(\()p Fp(H)7 b Fs(\)\))2363 2906 y Fo(2)2423 2944 y Fr(\000)20 b Fp(C)2579 2958 y Fo(1)2619 2944 y Fp(N)2702 2905 y Fw(\000)p Fo(1)2692 2968 y(0)2796 2944 y Fp(N)30 b Fr(\000)20 b Fp(K)7 b Fs(\()p Fp(N)3182 2958 y Fo(0)3222 2944 y Fp(;)15 b(R)q Fs(\))p Fp(;)199 3139 y Fs(\024)193 3162 y(\000\()p Fp(j)327 3125 y Fn(R)386 3162 y Fs(\))421 3125 y Fw(\003)460 3162 y Fp(B)534 3125 y Fn(\016)j Fo(ext)684 3162 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\)1)-20 b(l)970 3181 y Fw(f)p Fo(0)p Fw(g)1080 3162 y Fs(\()p Fp(M)1213 3125 y Fn(\016)1203 3185 y Fw(1)1278 3162 y Fs(\))1319 3139 y(\024)1313 3162 y(\000\()p Fp(j)1447 3125 y Fn(R)1506 3162 y Fs(\))26 b Fr(\025)f(\000)p Fp(\017)19 b Fr(\000)h Fp(\017)1918 3125 y Fw(\000)p Fo(1)2013 3162 y Fs(\(1)h Fr(\000)e Fp(f)2249 3176 y Fn(E)t(;\024)2369 3162 y Fs(\()p Fp(H)7 b Fs(\)\))2557 3125 y Fo(2)2617 3162 y Fr(\000)20 b Fp(C)2773 3176 y Fo(1)2813 3162 y Fp(N)2896 3124 y Fw(\000)p Fo(1)2886 3187 y(0)2990 3162 y Fp(N)30 b Fr(\000)20 b Fp(K)7 b Fs(\()p Fp(N)3376 3176 y Fo(0)3416 3162 y Fp(;)15 b(R)q Fs(\))p Fp(:)0 3329 y Fs(Pic)m(king)31 b Fp(\017)26 b Fs(=)f Fp(\017)520 3343 y Fo(0)559 3329 y Fp(=)p Fs(8)31 b(in)f(the)h(t)m(w)m(o)g (estimates)h(ab)s(o)m(v)m(e,)g(w)m(e)f(deduce)f(from)g(\(7.41\))j (that:)6 3525 y(\024)0 3548 y(\000\()p Fp(j)134 3511 y Fn(R)193 3548 y Fs(\))228 3511 y Fw(\003)267 3548 y Fp(B)341 3511 y Fn(\016)18 b Fo(ext)490 3548 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))730 3525 y(\024)724 3548 y(\000)r(\()p Fp(j)860 3511 y Fn(R)919 3548 y Fs(\))25 b Fr(\025)g Fs(\(1)p Fr(\000)p Fp(\017)1263 3562 y Fo(0)1303 3548 y Fp(=)p Fs(2\)1)-20 b(l)r Fr(\000)p Fp(C)7 b Fs(\(1)p Fr(\000)p Fp(f)1819 3562 y Fn(E)t(;\024)1938 3548 y Fs(\)\()p Fp(H)g Fs(\))2126 3511 y Fo(2)2167 3548 y Fr(\000)p Fp(C)g(N)2393 3510 y Fw(\000)p Fo(1)2383 3573 y(0)2486 3548 y Fp(N)j Fr(\000)p Fp(K)d Fs(\()p Fp(N)2832 3562 y Fo(0)2871 3548 y Fp(;)15 b(R)q Fs(\)+)p Fp(N)3180 3483 y Fh(1)p 3180 3495 V 3180 3537 a(2)3225 3548 y Fp(o)3269 3563 y Fn(\016)o(;E)t(;\024) 3438 3548 y Fs(\()p Fp(R)3543 3511 y Fo(0)3583 3548 y Fs(\))p Fp(N)3711 3483 y Fh(1)p 3711 3495 V 3711 3537 a(2)3756 3548 y Fp(;)0 3752 y Fs(Applying)30 b(once)h(more)g(Prop)s (osition)f(7.9,)i(w)m(e)e(obtain:)190 3966 y Fp(B)264 3929 y Fn(\016)302 3966 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b Fr(\025)e Fs(\(1)c Fr(\000)f Fp(\017)888 3980 y Fo(0)927 3966 y Fp(=)p Fs(2\))i Fr(\000)d Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)1473 3980 y Fn(E)t(;\024)1592 3966 y Fs(\()p Fp(H)7 b Fs(\)\))1780 3929 y Fo(2)1841 3966 y Fr(\000)20 b Fp(C)1997 3980 y Fo(1)2036 3966 y Fp(N)2119 3928 y Fw(\000)p Fo(1)2109 3991 y(0)2214 3966 y Fp(N)30 b Fr(\000)20 b Fp(K)7 b Fs(\()p Fp(N)2600 3980 y Fo(0)2639 3966 y Fp(;)15 b(R)q Fs(\))21 b(+)f Fp(N)2989 3902 y Fh(1)p 2989 3914 V 2989 3955 a(2)3033 3966 y Fp(o)3077 3981 y Fn(\016)o(;E)t(;\024)3247 3966 y Fs(\()p Fp(R)3352 3929 y Fo(0)3391 3966 y Fs(\))p Fp(N)3519 3902 y Fh(1)p 3519 3914 V 3519 3955 a(2)3564 3966 y Fp(:)141 4171 y Fs(Since)31 b Fp(N)k Fr(\024)25 b Fp(M)681 4138 y Fn(\016)719 4171 y Fs(,)30 b(for)g(an)m(y)h Fp(\013)26 b(>)f Fs(0)31 b(w)m(e)f(can)h(\014nally)f(\014x)g Fp(R)q(;)15 b(N)2228 4185 y Fo(0)2293 4171 y Fr(\035)25 b Fs(1)31 b(suc)m(h)f(that:)812 4375 y Fp(B)886 4337 y Fn(\016)923 4375 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b Fr(\025)e Fs(\(1)c Fr(\000)f Fp(\017)1509 4389 y Fo(0)1548 4375 y Fp(=)p Fs(2\))i Fr(\000)e Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)2094 4389 y Fn(E)t(;\024)2214 4375 y Fs(\()p Fp(H)7 b Fs(\)\))2402 4337 y Fo(2)2462 4375 y Fr(\000)20 b Fp(K)27 b Fr(\000)20 b Fp(\013M)2904 4337 y Fn(\016)2942 4375 y Fp(;)0 4579 y Fs(where)30 b Fp(K)37 b Fs(is)30 b(compact.)42 b(This)30 b(giv)m(es:)232 4783 y(\(1)21 b(+)f Fp(\013)p Fs(\))p Fp(B)591 4750 y Fn(\016)629 4783 y Fs(\()p Fp(E)5 b(;)15 b(\024)p Fs(\))27 b Fr(\025)e Fs(\(1)c Fr(\000)f Fp(\017)1215 4797 y Fo(0)1254 4783 y Fp(=)p Fs(2\))i Fr(\000)e Fp(C)7 b Fs(\(1)20 b Fr(\000)g Fp(f)1800 4797 y Fn(E)t(;\024)1919 4783 y Fs(\()p Fp(H)7 b Fs(\)\))2107 4750 y Fo(2)2168 4783 y Fr(\000)20 b Fp(K)27 b Fr(\000)20 b Fp(\013)p Fr(k)p Fp(f)2602 4797 y Fn(E)t(;\024)2722 4783 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)3037 4750 y Fn(\016)3076 4783 y Fp(v)s Fs(\))p Fp(f)3203 4797 y Fn(E)t(;\024)3323 4783 y Fs(\()p Fp(H)g Fs(\))p Fr(k)p Fp(:)0 4988 y Fs(Since)35 b(for)h Fp(E)j Fr(\024)33 b Fp(E)664 5002 y Fo(0)727 4988 y Fs(+)23 b(1)p Fp(;)15 b Fs(0)35 b Fp(<)f(\024)g Fr(\024)f Fs(1,)38 b Fr(k)p Fp(f)1479 5002 y Fn(E)t(;\024)1598 4988 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)1913 4955 y Fn(\016)1953 4988 y Fp(v)s Fs(\))p Fp(f)2080 5002 y Fn(E)t(;\024)2200 4988 y Fs(\()p Fp(H)g Fs(\))p Fr(k)35 b(\024)e Fp(C)2602 5002 y Fo(1)2641 4988 y Fs(,)k(w)m(e)f(can)g(c)m(ho) s(ose)g Fp(\013)e Fr(\034)g Fs(1)i(suc)m(h)0 5100 y(that)31 b(\(1)21 b(+)f Fp(\013)p Fs(\))482 5068 y Fw(\000)p Fo(1)577 5100 y Fs(\(1)h Fr(\000)f Fp(\017)806 5114 y Fo(0)845 5100 y Fp(=)p Fs(2)h Fr(\000)f Fp(C)1112 5114 y Fo(1)1152 5100 y Fp(\013)p Fs(\))26 b Fr(\025)f Fs(\(1)c Fr(\000)f Fp(\017)1596 5114 y Fo(0)1635 5100 y Fs(\).)41 b(This)30 b(completes)h(the)g(pro)s(of)f(of)g(the)h(prop)s(osition.)40 b Fd(2)1844 5517 y Fs(54)p eop end %%Page: 55 55 TeXDict begin 55 54 bop 0 162 a Fi(7.5)112 b(An)38 b(impro)m(v)m(ed)g (Mourre)g(estimate)0 334 y Fs(W)-8 b(e)34 b(no)m(w)f(form)m(ulate)h(an) f(impro)m(v)m(ed)h(v)m(ersion)f(of)h(the)f(Mourre)g(estimate,)j(whic)m (h)c(will)i(not)f(b)s(e)g(used)f(in)h(this)0 447 y(pap)s(er.)39 b(Nev)m(ertheless,)31 b(it)e(could)g(b)s(e)f(useful)g(to)h(pro)m(v)m(e) h(sharp)s(er)d(propagation)i(estimates)h(on)f(the)g(dynamics)0 560 y(e)40 527 y Fw(\000)p Fo(i)p Fn(tH)208 560 y Fs(.)0 743 y Fe(Theorem)70 b(7.15)46 b Fl(F)-7 b(or)28 b(al)5 b(l)28 b Fp(\017)1036 757 y Fo(1)1100 743 y Fp(>)d Fs(0)p Fl(,)k Fp(E)1365 757 y Fo(0)1429 743 y Fp(<)c Fr(1)p Fl(,)j(ther)-5 b(e)28 b(exists)g Fs(0)d Fp(<)g(\016)k Fr(\024)2478 708 y Fo(1)p 2478 723 36 4 v 2478 775 a(2)2550 743 y Fl(such)f(that)g(for)g(al)5 b(l)27 b Fp(E)k Fr(\024)25 b Fp(E)3457 757 y Fo(0)3496 743 y Fp(;)15 b(\017)3573 757 y Fo(0)3638 743 y Fp(>)25 b Fs(0)0 856 y Fl(ther)-5 b(e)34 b(exist)e Fp(C)q(;)15 b(\024;)g(R)28 b(>)d Fs(0)p Fl(,)33 b(and)g Fp(K)39 b Fl(c)-5 b(omp)g(act)35 b(such)e(that:)102 1056 y Fp(M)200 1018 y Fn(\016)258 1056 y Fr(\000)20 b Fp(f)394 1070 y Fn(E)t(;\024)513 1056 y Fs(\()p Fp(H)7 b Fs(\))p Fp(\036)p Fs(\(i)p Fp(a)828 1018 y Fn(\016)868 1056 y Fp(v)s Fs(\))p Fp(f)995 1070 y Fn(E)t(;\024)1115 1056 y Fs(\()p Fp(H)g Fs(\))26 b Fr(\025)f Fs(1)20 b Fr(\000)g Fp(\017)1583 1070 y Fo(0)1643 1056 y Fs(+)g(\(1)h Fr(\000)f Fp(\017)1963 1070 y Fo(1)2002 1056 y Fs(\))2043 1033 y(\024)2037 1056 y(\000\()p Fp(j)2171 1018 y Fn(R)2230 1056 y Fs(\))2265 1018 y Fw(\003)2305 1056 y Fp(M)2403 1018 y Fn(\016)2393 1078 y Fw(1)2473 1033 y Fs(\024)2467 1056 y(\000\()p Fp(j)2601 1018 y Fn(R)2660 1056 y Fs(\))h Fr(\000)e Fp(C)7 b Fs(\(1)21 b Fr(\000)f Fp(f)3115 1070 y Fn(E)t(;\024)3234 1056 y Fs(\()p Fp(H)7 b Fs(\)\))3422 1018 y Fo(2)3483 1056 y Fr(\000)20 b Fp(K)q(:)0 1439 y Fe(Pro)s(of.)47 b Fs(W)-8 b(e)31 b(rep)s(eat)f(the)f(pro)s(of)g(of)h (Prop)s(osition)g(7.13)h(un)m(til)f(w)m(e)g(arriv)m(e)g(at)g(\(7.40\))i (and)d(w)m(e)h(\014x)f Fp(\017)d Fs(=)f Fp(\017)3549 1453 y Fo(1)3588 1439 y Fs(.)40 b(W)-8 b(e)0 1552 y(c)m(ho)s(ose)31 b(no)m(w)g(0)25 b Fp(<)g(\016)30 b Fr(\024)24 b Fs(1)31 b(suc)m(h)f(that)1195 1752 y Fp(\017)1232 1766 y Fo(1)1271 1752 y Fp(d)p Fs(\()p Fp(\016)s Fs(\))22 b Fr(\000)e Fp(\017)1581 1766 y Fo(0)1621 1752 y Fp(=)p Fs(4)26 b Fr(\025)f Fs(1)p Fp(;)36 b(\017)1976 1766 y Fo(1)2015 1752 y Fp(d)p Fs(\()p Fp(\016)s Fs(\))22 b Fr(\000)e Fp(C)2353 1766 y Fo(0)2418 1752 y Fr(\025)25 b Fs(1)p Fp(:)0 1951 y Fs(W)-8 b(e)32 b(obtain)e(an)h(extra)g(term)1259 2126 y(\(1)21 b Fr(\000)f Fp(\017)1488 2140 y Fo(1)1528 2126 y Fs(\))1569 2104 y(\024)1563 2126 y(\000\()p Fp(j)1697 2093 y Fn(R)1755 2126 y Fs(\))1790 2093 y Fw(\003)1830 2126 y Fp(M)1928 2093 y Fn(\016)1918 2149 y Fw(1)1993 2126 y Fs(1)-20 b(l)2044 2145 y Fo([1)p Fn(;)p 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b(=)e(0,)31 b(w)m(e)h(see)f(that)0 5265 y(\()p Fp(N)g Fs(+)19 b(1\))319 5202 y Fh(1)p 320 5214 V 320 5255 a(2)365 5265 y Fs(\()p Fr(j)p Fp(A)493 5232 y Fn(\016)531 5265 y Fr(j)i Fs(+)f(1\))748 5232 y Fw(\000)p Fn(s)871 5265 y Fs(is)30 b(b)s(ounded)e(from)i Fr(H)h Fs(in)m(to)g Fr(G)1899 5232 y Fw(\003)1894 5288 y Fn(s;)p Fo(2)1986 5265 y Fs(.)1844 5517 y(55)p eop end %%Page: 56 56 TeXDict begin 56 55 bop 141 162 a Fs(Next)35 b(b)m(y)e(Prop)s(osition)h (3.4)g Fl(i\))g Fs(w)m(e)g(kno)m(w)g(that)g Fp(A)1883 129 y Fn(\016)r Fw(\003)1956 162 y Fp(A)2024 129 y Fn(\016)2093 162 y Fr(\024)c Fs(d\000\()p Fr(j)p Fp(a)2410 129 y Fn(\016)2448 162 y Fr(j)p Fs(\))2508 129 y Fo(2)2548 162 y Fs(.)51 b(Since)33 b(b)m(y)h(Lemma)f(6.1)j(~)-46 b Fp(a)3519 129 y Fn(\016)r Fw(\003)3594 162 y Fs(~)f Fp(a)3640 129 y Fn(\016)3709 162 y Fr(\024)0 275 y Fp(C)7 b Fs(\()p Fp(\016)s Fs(\))182 251 y(~)185 275 y Fp(b)224 242 y Fo(2)264 275 y Fs(,)53 b(w)m(e)c(obtain)f(using)g(Prop)s(osition)g(3.4) i Fl(ii\))e 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b(completeness)g(for)f(a)g(renormalized)g(non-relativistic)i (Hamiltonian)301 1229 y(in)24 b(quan)m(tum)h(\014eld)g(theory:)38 b(the)26 b(Nelson)f(mo)s(del.)h(Math.)g(Ph)m(ys.)f(Anal.)g(Geom.)i Fe(3)e Fs(\(2000\),)k(217{285.)0 1417 y([ABG])48 b(Amrein,)38 b(W.,)h(Boutet)f(de)f(Mon)m(v)m(el,)j(A.,)f(Georgescu,)h(V.:)54 b Fp(C)2492 1431 y Fo(0)2532 1417 y Fl(-Gr)-5 b(oups,)41 b(Commutator)g(Metho)-5 b(ds)301 1530 y(and)33 b(Sp)-5 b(e)g(ctr)g(al)35 b(The)-5 b(ory)34 b(of)f Fp(N)10 b Fl(-Bo)-5 b(dy)34 b(Hamiltonians)p Fs(,)e(Birkh\177)-45 b(auser,)30 b(Basel-Boston-Berlin,)k(1996.)0 1718 y([Ar])147 b(Arai,)45 b(A.:)65 b(Ground)41 b(State)j(of)e(the)g(Massless)i(Nelson) e(Mo)s(del)h(Without)g(Infrared)e(Cuto\013)h(in)g(a)301 1830 y(Non-F)-8 b(o)s(c)m(k)32 b(Represen)m(tation,)g(preprin)m(t)d (mp-arc)i(00-478)i(\(2000\).)0 2018 y([AH1])70 b(Arai,)23 b(A.,)h(Hirok)-5 b(a)m(w)m(a,)25 b(M.:)37 b(On)20 b(the)i(existence)g (and)f(uniqueness)f(of)h(ground)f(states)j(of)e(a)h(generalized)301 2131 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275 y(ing.)f(Ann.)g(Henri)g(P)m(oincar)m(\023)-43 b(e)33 b Fe(3)e Fs(\(2002\),)i(107{170.)0 463 y([F)m(GS3])47 b(F)-8 b(r\177)-45 b(ohlic)m(h,)29 b(J.,)f(Griesemer,)g(M.,)g(Sc)m (hlein,)g(B.:)39 b(Asymptotic)28 b(completeness)g(for)e(Compton)h (scatter-)301 576 y(ing,)j(preprin)m(t)g(mp-arc)g(01-420)j(\(2001\).)0 763 y([GGM])47 b(Georgescu,)d(V.,)g(G)m(\023)-43 b(erard,)43 b(C.,)g(M\034ller,)h(J.:)60 b(Comm)m(utators,)44 b Fp(C)2619 777 y Fo(0)2658 763 y Fr(\000)p Fs(semigroups)c(and)f(Resolv)m(en)m(t) 301 876 y(Estimates,)31 b(preprin)m(t.)0 1064 y([G1])135 b(G)m(\023)-43 b(erard,)38 b(C.)d(:)52 b(On)35 b(the)h(existence)h(of)f (ground)f(states)i(for)e(massless)h(P)m(auli-Fierz)i(Hamiltonians,)301 1177 y(Ann.)29 b(Henri)i(P)m(oincar)m(\023)-43 b(e)32 b Fe(1)f Fs(\(2000\),)i(443{459.)0 1364 y([G2])135 b(G)m(\023)-43 b(erard,)35 b(C.:)47 b(On)33 b(the)h(scattering)h(theory)f(of)g (massless)g(Nelson)g(mo)s(dels,)h(Rev.)f(Math.)g(Ph)m(ys.)g Fe(14)301 1477 y Fs(\(2002\),)f(1165-1280.)0 1665 y([HS])132 b(H)s(\177)-48 b(ubner,)34 b(M.,)h(Sp)s(ohn,)f(H.:)48 b(Sp)s(ectral)34 b(prop)s(erties)g(of)g(the)h(spin-b)s(oson)d (Hamiltonian,)37 b(Ann.)d(Inst.)301 1778 y(Henri)c(P)m(oincar)m(\023) -43 b(e)32 b Fe(62)f Fs(\(1995\),)i(289{323.)0 1966 y([Ka])135 b(Kato,)31 b(T.:)41 b Fl(Perturb)-5 b(ation)34 b(The)-5 b(ory)34 b(for)f(Line)-5 b(ar)33 b(Op)-5 b(er)g(ators)p Fs(,)34 b(Springer)29 b(V)-8 b(erlag,)32 b(1976.)0 2153 y([LMS])60 b(L\177)-45 b(orinczi,)44 b(J.,)g(Minlos,)g(R.)d(A.,)k(Sp)s (ohn,)d(H.:)62 b(The)40 b(infrared)g(b)s(eha)m(viour)h(in)g(Nelson's)g (mo)s(del)g(of)301 2266 y(a)f(quan)m(tum)f(particle)i(coupled)e(to)i(a) f(massless)g(scalar)h(\014eld.)e(Ann.)g(Henri)h(P)m(oincar)m(\023)-43 b(e)42 b Fe(3)e Fs(\(2002\),)301 2379 y(269{295.)0 2567 y([MS])117 b(M\034ller,)25 b(J.)e(S.,)i(Skibsted,)e(E.:)37 b(Sp)s(ectral)23 b(theory)g(of)h(time-p)s(erio)s(dic)f(man)m(y-b)s(o)s (dy)f(systems,)i(preprin)m(t)301 2680 y(mp)p 433 2680 28 4 v 31 w(arc)31 b(02-316)i(\(2002\).)0 2867 y([Ne])143 b(Nelson.)28 b(E:)f(In)m(teraction)i(of)e(non-relativistic)j(particles) e(with)f(a)g(quan)m(tized)h(scalar)h(\014eld,)e(J.)h(Math.)301 2980 y(Ph)m(ys.)i Fe(5)g Fs(\(1964\),)j(1190{1197.)0 3168 y([Sk])152 b(Skibsted,)25 b(E.:)38 b(Sp)s(ectral)25 b(analysis)g(of)g Fp(N)10 b Fs(-b)s(o)s(dy)24 b(systems)h(coupled)g(to) g(a)h(b)s(osonic)e(\014eld.)h(Rev.)h(Math.)301 3281 y(Ph)m(ys.)k Fe(10)h Fs(\(1998\),)i(989{1026.)0 3468 y([Sp])149 b(Sp)s(ohn,)27 b(H.:)40 b(Asymptotic)30 b(completeness)g(for)e(Ra)m(yleigh)j (scattering.)f(J.)f(Math.)h(Ph)m(ys.)e Fe(38)i Fs(\(1997\),)301 3581 y(2281{2296.)1844 5517 y(57)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0304270345299--