Content-Type: multipart/mixed; boundary="-------------0604281504969" This is a multi-part message in MIME format. ---------------0604281504969 Content-Type: text/plain; name="06-138.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-138.keywords" Mourre, positive commutators, limit absorption principle ---------------0604281504969 Content-Type: application/postscript; name="lapsid.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lapsid.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: /home/sylvain/Jecko/chantier/1lapsid.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR10 CMCSC10 CMR12 CMSY10 CMMI12 CMTI10 CMR9 %%+ CMTI12 rsfs10 MSBM10 CMMI8 CMSY8 CMR8 MSAM10 CMSY6 CMR6 MSBM7 CMEX10 %%+ CMMI6 CMMI10 CMR7 CMBX10 CMTT10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -z 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TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: hps.pro %! /HPSdict 20 dict dup begin/braindeaddistill 50 def/rfch{dup length 1 sub 1 exch getinterval}bind def/splituri{dup(#)search{exch pop}{()exch} ifelse dup(file:)anchorsearch{pop exch pop 3 -1 roll pop false}{pop 3 -1 roll exch pop true}ifelse}bind def/lookuptarget{exch rfch dup /TargetAnchors where{pop TargetAnchors dup 3 -1 roll known{exch get true }{pop(target unknown:)print == false}ifelse}{pop pop (target dictionary unknown\012)print false}ifelse}bind def/savecount 0 def/stackstopped{count counttomark sub/savecount exch store stopped count savecount sub 1 sub dup 0 gt{{exch pop}repeat}{pop}ifelse}bind def /tempstring 256 string def/targetvalidate{1 index dup length 255 gt exch dup(/)search{pop pop pop exch pop true exch}{pop}ifelse cvn tempstring cvs token pop pop length 0 ne or not}bind def/targetdump-hook where{pop} {/targetdump-hook{dup mark exch gsave initmat setmatrix{{mark/Dest 4 2 roll targetvalidate{aload pop exch pop/Page 3 1 roll/View exch[exch /FitH exch]/DEST pdfmark}{cleartomark}ifelse}forall}stackstopped pop grestore}bind def}ifelse/baseurl{mark exch 1 dict dup 3 -1 roll/Base exch put/URI exch/DOCVIEW{pdfmark}stackstopped pop}bind def /externalhack systemdict/PDF known def/oldstyle true def/initmat matrix currentmatrix def/actiondict 2 dict dup/Subtype/URI put def /weblinkhandler{dup 3 1 roll mark 4 1 roll/Title 4 1 roll splituri 3 -1 roll dup length 0 gt{cvn/Dest exch 4 2 roll}{pop}ifelse{externalhack{ /HTTPFile exch}{actiondict dup 3 -1 roll/URI exch put/Action exch} ifelse}{externalhack{/HTTPFile exch}{/File exch/Action/GoToR}ifelse} ifelse counttomark 2 sub -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll /Color exch oldstyle{/LNK}{/Subtype/Link/ANN}ifelse gsave initmat setmatrix{pdfmark}stackstopped grestore}bind def/externalhandler where{ pop}{/externalhandler{2 copy{weblinkhandler}exec{/externalhack externalhack not store 2 copy{weblinkhandler}exec{/externalhack externalhack not store/oldstyle false store 2 copy{weblinkhandler}exec{ (WARNING: external refs disabled\012)print/externalhandler{pop pop}bind store externalhandler}{pop pop}ifelse}{pop pop/externalhack externalhack not store}ifelse}{pop pop/externalhandler{weblinkhandler pop}bind store} ifelse}bind def}ifelse/pdfmnew{dup type/stringtype eq{externalhandler}{ exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop/Rect 4 1 roll/Border 3 1 roll/Color exch/LNK gsave initmat setmatrix pdfmark grestore}{pop pop}ifelse}ifelse}bind def/pdfmold{dup type/stringtype eq{externalhandler}{exch dup rfch exch 3 -1 roll lookuptarget{mark 4 1 roll/Title 4 1 roll aload pop exch pop/Page 3 1 roll/View exch[exch/FitH exch]5 -1 roll aload pop pop 0 3 getinterval /Rect 3 1 roll/Border exch/LNK gsave initmat setmatrix pdfmark grestore} {pop pop}ifelse}ifelse}bind def/pdfm where{pop}{/pdfm /currentdistillerparams where{pop currentdistillerparams dup /CoreDistVersion known{/CoreDistVersion get}{0}ifelse dup braindeaddistill le{(WARNING: switching to old pdfm because version =) print ==/pdfmold}{pop/pdfmnew}ifelse load}{/pdfmark where{pop{dup type /stringtype eq{externalhandler}{2 copy mark 3 1 roll{pdfmnew} stackstopped{2 copy mark 3 1 roll{pdfmold}stackstopped{ (WARNING: pdfm disabled\012)print/pdfm{pop pop}store}{ (WARNING: new pdfm failed, switching to old pdfm\012)print/pdfm/pdfmold load store}ifelse}{/pdfm/pdfmnew load store}ifelse pop pop}ifelse}}{{ pop pop}}ifelse}ifelse bind def}ifelse end def %%EndProcSet %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % 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 (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8232429ED8B7CF6A3D879A2D19 38DD5C4467F9DD8C5D1A2000B3A6BF2F25629BAEC199AE8BD4BA6ED9BBF7DABF 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(C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 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 0 /.notdef 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 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 26 /braceleftbigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA17D1AFFF95F4224CF7ECEE45C BFB7C8C77C22A01C345078D28D3ECBF804CDC2FE5025FA0D05CCC5EFC0C4F87E CBED13DDDF8F34E404F471C6DD2E43331D73E89BBC71E7BF889F6293793FEF5A C9DD3792F032E37A364C70914843F7AA314413D022AE3238730B420A7E9D0CF5 D0E24F501451F9CDECE10AF7E14FF15C4F12F3FCA47DD9CD3C7AEA8D1551017D 23131C09ED104C052054520268A4FA3C6338BA6CF14C3DE3BAF2EA35296EE3D8 D6496277E11DFF6076FE64C8A8C3419FA774473D63223FFA41CBAE609C3D976B 93DFB4079ADC7C4EF07303F93808DDA9F651F61BCCF79555059A44CBAF84A711 6D98083CEF58230D54AD486C74C4A257FC703ACF918219D0A597A5F680B606E4 EF94ADF8BF91A5096A806DB64EC96636A98397D22A74932EB7346A9C4B5EE953 CB3C80AA634BFC28AA938C704BDA8DC4D13551CCFE2B2784BE8BF54502EBA9AF D49B79237B9C56310550BC30E9108BB06EAC755D6AA4E688EFE2A0AAB17F20FE 00CD0BFF1B9CB6BDA0FA3A29A3117388B6686657A150CE6421FD5D420F4F7FB5 B0DAA1BA19D638676E9CF159AC7325EF17B9F74E082BEF75E10A31C7011C0FFA 99B797CE549B5C45238DD0FADD6B99D233AC69282DF0D91EA2DBD08CE0083904 A6D968D5AE3BD159D01BDFF42D16111BC0A517C66B43972080D9DD4F3B9AE7FB 11B035CE715C1218B2D779761D8D7E9DEBE277531BD58F313EBD27E33BEF9DC5 50C7821A8BBC3B9FDF899D7EAA0B94493B97AFEAC503EB5ED7A7AB67C3039A0F BF0BA4B455D035FF3126F33A4DF59FF96752CF58DC0863924F6042B14DC2DEE4 099CAA764A8EFFEB78F7FFD276F0928BA32624C3A69ACEB8777A461298B201D3 4DA271476B8AC8966668D4E28A6FBA952F6D295AA80CC09B0CFD6BE840703B07 7ED93999EB5630C2C7CD9BA683ED457D1341C3AA702B5FA8593A51E8E9165FF6 1385693E19161FF1ADC1FE0BE16FD3AFCDC3916A309FCE6B3F2B362C9F958A16 821E1EB2DD899522DD3155C0B7A7F0876E4DA32FD30D8289266D9F05E77F1F1C D3BFD2D4D6C8B6FDE0E8957BDBBEC54458C70B96F4CD77E734EE0CF620684024 EB05DF0536FAFF10FC6F6050AB338D44EA9C38150D3779A550C505C06D417A71 62D41C2D23843E0A2A73F72D4B5286E3CB7B8695B2434479776A59CB3BC6D352 677CCABF4133E20F12D91CFD8978C09FB81DF801707194AB8FABA9A6599C20FA B36DB0A166490D20BB0BD53581A8715B099754E6FA0F7D2892F5573380DDFF1E F9C3DA9F74BDBEBBDFA5715AE23264F967BC7841CE58E5EE2764555978D904F6 E8DEF6ED66C478848FC35A9827ACEA1D7CE85C8626C162FF296869DE 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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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 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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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 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 1477F5496111E055C7491002AFF272E46ECC46422F0380D093284870022523FB DA1716CC4F2E2CCAD5F173FCBE6EDDB874AD255CD5E5C0F86214393FCB5F5C20 9C3C2BB5886E36FC3CCC21483C3AC193485A46E9D22BD7201894E4D45ADD9BF1 CC5CF6A5010B5654AC0BE0DA903DB563B13840BA3015F72E51E3BC80156388BA F83C7D393392BCBC227771CDCB976E933025324F8D38DE27D6D1FFA4E3A237AA D39326795C1F9C2107949BA74E11B8C82EE2420CBB009CC17B5B8BF19600CC2D 65603FA981FB48D7F7CB81131E0F9FB7AC1F1377E187D0EF2052EAEDA6AFB27E D7D3F43B9044178818012C5B47FD19A43D6CBAE9688CBC8BCF8970E19448AAF5 C33D86E763EDB25579832477C879AC0C2DF142CBDA71EB435EB25047B81F8873 C3D28335BEE7A3EC1CB38AB9DC3B3A56EE4ECC98F28100E4A9D8CFA801C6886B 0E9FF87145EE96FBED06141188016F5C7D03E5EAD2B7CEFFA1B5CF26220CDB6D AF66268A87EB0837B7A687FAD6956DEC0F76FA0067865CC4945C1D274496A6FB 06DF19231E4E91D08D8108FDBF449B69974F6ADE0FCE9C43E6F44488856BA474 6782B74D6DF76462C6EA0FCA8FBFC4DDDD49DC7AF71B2B2925606A5CAA2D63DB A018026FA68FF55624F3E86940DEAE968651851EB7C33AB6C7F6692DF1D77C28 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0000000000000000000000000000000000000000000000000000000000000000 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. 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All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % 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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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. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. 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%!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fs(H)8 b Fu(,)35 b(whic)m(h)i(is)e(a)g(b)s(ounded)h(op)s(erator.)50 b(This)37 b(explains)f(also)p eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 456 255 a Fq(4)660 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fu(wh)m(y)37 b(w)m(e)f(can)g(replace)h(the)f(global)f(regularit)m(y)h (assumption)h Fs(H)k Ft(2)33 b(C)3033 414 y Fk(2)3073 450 y Fu(\()p Fs(A)p Fu(\))i(b)m(y)i(a)456 566 y(lo)s(cal)d(one)h(and)f (get)h(a)f(stronger)h(result,)h(namely)g(Theorem)g(2.7.)49 b(The)35 b(later)g(is)456 683 y(co)m(v)m(ered)42 b(b)m(y)f(Sah)m (bani's)g(result)g(in)f([S)q(])g(\(cf.)66 b(Remark)41 b(2.8\).)65 b(Motiv)-5 b(ations)41 b(for)456 799 y(Theorem)36 b(2.7)f(are)g(giv)m(en)h(in)f(Remarks)h(2.8)f(and)g(2.9.)51 b(In)35 b(Subsection)i(2.5,)e(w)m(e)456 915 y(giv)m(e)25 b(a)g(sk)m(etc)m(h)i(of)d(the)i(pro)s(of)e(of)g(Theorem)i(2.7)e(and)h (presen)m(t)i(our)d(in)m(terpretation)456 1031 y(of)31 b(Mourre's)i(comm)m(utator)g(theory)-8 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b(oje)g(cte)g(d)34 b(Mourr)-5 b(e)36 b(es-)456 4983 y(timate)d Fu(\(4.28\),)27 b(that)g(is)f(w)m(eak)m(er)j(that)d (\(1.1\))o(.)42 b(In)27 b(Subsection)h(4.4,)f(w)m(e)g(illustrate)456 5099 y(this)32 b(di\013erence)h(with)f(an)f(arti\014cial)g(but)h(in)m (teresting)h(example,)g(for)e(whic)m(h)i(the)456 5216 y(reduced)h(LAP)f(holds)g(true)g(and)g(the)g(usual)g(Mourre)g(estimate) h(\(1.1\))e(is)h(false.)p eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)397 b(5)456 450 y Fy(Ac)m(kno)m(wledgmen)m(ts:)40 b Fu(W)-8 b(e)25 b(w)m(ould)h(lik)m(e)g(to)e(thank)i(Jan)f(Derezi)s (\023)-51 b(nski,)27 b(Vladimir)456 566 y(Georgescu,)41 b(and)f(Jacob)f(S.)h(M\034ller)g(for)e(helpful)i(discussions.)67 b(The)40 b(t)m(w)m(o)g(au-)456 683 y(thors)25 b(w)m(ere)i(partially)f (supp)s(orted)g(b)m(y)g(the)g(con)m(tract)g(MER)m(G-CT-2004-006375)456 799 y(funded)42 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y(In)k(the)h(text,)h(w)m(e)f(use)h(the)e(letter)h Ft(I)41 b Fu(to)33 b(denote)i(an)e(in)m(terv)-5 b(al)34 b(of)f Fn(R)p Fu(.)46 b(F)-8 b(or)33 b(suc)m(h)i(a)456 2560 y Ft(I)7 b Fu(,)34 b(w)m(e)h(denote)f(b)m(y)p 1175 2480 62 4 v 35 w Ft(I)41 b Fu(\(resp.)1553 2535 y(\027)1556 2560 y Ft(I)7 b Fu(\))34 b(its)g(closure)h(\(resp.)47 b(its)34 b(in)m(terior\).)47 b(The)35 b(scalar)456 2676 y(pro)s(duct)h Ft(h\001)p Fs(;)17 b Ft(\001i)34 b Fu(in)i Fo(H)62 b Fu(is)36 b(righ)m(t)g(linear)g(and)g Ft(k)24 b(\001)g(k)35 b Fu(denotes)i(the)f(corresp)s(onding)456 2793 y(norm)26 b(and)h(also)f(the)h(norm)f(of)g(b)s(ounded)i(op)s (erators)e(on)g Fo(H)h Fu(.)41 b(If)26 b Fs(T)41 b Fu(is)26 b(a)h(b)s(ounded)456 2909 y(op)s(erator)36 b(on)g Fo(H)63 b Fu(and)37 b Fs(k)h Ft(2)d Fn(N)p Fu(,)i(w)m(e)h(sa)m(y)g(that)e Fs(T)49 b Ft(2)35 b(C)2465 2873 y Fm(k)2507 2909 y Fu(\()p Fs(A)p Fu(\))i(if,)h(for)e(all)g Fs(f)46 b Ft(2)35 b Fo(H)26 b Fu(,)456 3025 y(the)42 b(map)g Fn(R)h Ft(3)g Fs(t)h Ft(7!)f Fs(e)1350 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b(;)17 b(A)p Fu(])42 b(if)g(and)g(only)g(if)g Fs(T)55 b Fu(preserv)m(es)45 b Ft(D)s Fu(\()p Fs(A)p Fu(\).)71 b(F)-8 b(urthermore)42 b Fs(T)57 b Ft(2)456 3616 y(C)514 3580 y Fm(k)557 3616 y Fu(\()p Fs(A)p Fu(\))34 b(if)h(and)g(only)h(if)f(the)g(iterated)g (comm)m(utator)h(ad)2524 3569 y Fm(p)2524 3643 y(A)2581 3616 y Fu(\()p Fs(T)14 b Fu(\))31 b(:=)h([ad)3024 3569 y Fm(p)p Fl(\000)p Fk(1)3024 3643 y Fm(A)3154 3616 y Fu(\()p Fs(T)14 b Fu(\))p Fs(;)j(A)p Fu(])456 3732 y(are)46 b(b)s(ounded)h(for)f Fs(p)51 b Ft(\024)h Fs(k)s Fu(.)85 b(In)47 b(particular,)j(for)c Fs(T)65 b Ft(2)51 b(C)2692 3696 y Fk(1)2732 3732 y Fu(\()p Fs(A)p Fu(\),)f Fs(T)65 b Ft(2)52 b(C)3256 3696 y Fk(2)3295 3732 y Fu(\()p Fs(A)p Fu(\))456 3848 y(if)44 b(and)h(only)g(if)g([)p Fs(T)8 b(;)17 b(A)p Fu(])48 b Ft(2)h(C)1544 3812 y Fk(1)1584 3848 y Fu(\()p Fs(A)p Fu(\).)80 b(F)-8 b(or)44 b(un)m(b)s(ounded)i (self-adjoin)m(t)f(op)s(erator,)456 3965 y(w)m(e)c(de\014ned)g(the)g Ft(C)1184 3929 y Fm(k)1227 3965 y Fu(\()p Fs(A)p Fu(\))f(regularit)m(y) h(in)f(Section)h(1.)66 b(Let)41 b Fs(H)47 b Fu(is)41 b(\(un)m(b)s(ounded\))456 4081 y(self-adjoin)m(t)46 b(op)s(erator)g (and)g Ft(I)54 b Fu(a)46 b(b)s(ounded)h(in)m(terv)-5 b(al.)85 b(Recall)47 b(that)f Fs(E)3232 4096 y Fl(I)3280 4081 y Fu(\()p Fs(H)8 b Fu(\))456 4197 y(denotes)37 b(the)f(sp)s (ectral)g(pro)5 b(jection)37 b(of)e Fs(H)44 b Fu(ab)s(o)m(v)m(e)36 b Ft(I)7 b Fu(.)54 b(If)36 b Fs(H)k Ft(2)34 b(C)2859 4161 y Fk(1)2899 4197 y Fu(\()p Fs(A)p Fu(\))h(then)i(the)456 4313 y(form)29 b Fs(E)755 4328 y Fl(I)803 4313 y Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)r(;)17 b(iA)p Fu(])p Fs(E)1327 4328 y Fl(I)1375 4313 y Fu(\()p Fs(H)8 b Fu(\))29 b(de\014ned)j(on)d (\()p Ft(D)s Fu(\()p Fs(H)8 b Fu(\))16 b Ft(\\)i(D)s Fu(\()p Fs(A)p Fu(\)\))e Ft(\002)i Fu(\()p Ft(D)s Fu(\()p Fs(H)8 b Fu(\))15 b Ft(\\)j(D)s Fu(\()p Fs(A)p Fu(\)\))456 4430 y(extend)38 b(to)e(a)h(b)s(ounded)g(self-adjoin)m(t)h(op)s (erator.)55 b(In)37 b(particular,)h(\(1.1\))f(mak)m(es)456 4546 y(sense.)43 b(A)27 b(justi\014cation)h(of)e(Remark)i(1.3)e(can)h (b)s(e)g(found)g(in)g([ABG])g(but)g(w)m(e)h(giv)m(e)456 4662 y(it)j(in)h(the)h(pro)s(of)d(of)i(Theorem)h(1.4)e(\(for)g Fs(P)42 b Fu(=)27 b(0\).)43 b(The)33 b(follo)m(wing)f(prop)s(ositions) 456 4778 y(and)g(remark)i(will)f(b)s(e)g(useful)g(later.)456 4958 y Fy(Prop)s(osition)47 b(2.1.)f Fp(F)-7 b(or)42 b Fs(f)5 b(;)17 b(g)46 b Ft(2)d(D)s Fu(\()p Fs(A)p Fu(\))p Fp(,)h(the)f(\014nite)g(r)-5 b(ank)42 b(op)-5 b(er)g(ator)43 b Ft(j)p Fs(f)11 b Ft(ih)p Fs(g)t Ft(j)40 b Fu(:)456 5074 y Fs(h)27 b Ft(!)h(h)p Fs(g)t(;)17 b(h)p Ft(i)j(\001)i Fs(f)46 b Fp(b)-5 b(elongs)34 b(to)h Ft(C)1569 5038 y Fk(1)1608 5074 y Fu(\()p Fs(A)p Fu(\))g Fp(and)g Fu([)p Ft(j)p Fs(f)11 b Ft(ih)p Fs(g)t Ft(j)p Fs(;)17 b(A)p Fu(])25 b(=)j Ft(j)p Fs(f)11 b Ft(ih)p Fs(Ag)t Ft(j)20 b(\000)i(j)p Fs(Af)11 b Ft(ih)p Fs(g)t Ft(j)p Fp(.)43 b(In)456 5191 y(p)-5 b(articular,)34 b(if)h Fs(f)5 b(;)17 b(g)31 b Ft(2)d(D)s Fu(\()p Fs(A)1486 5155 y Fk(2)1525 5191 y Fu(\))p Fp(,)35 b Ft(j)p Fs(f)11 b Ft(ih)p Fs(g)t Ft(j)25 b(2)j(C)2049 5155 y Fk(2)2089 5191 y Fu(\()p Fs(A)p Fu(\))p Fp(.)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 456 255 a Fq(6)660 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Since)34 b Fs(R)c Fu(:=)f Ft(j)p Fs(f)11 b Ft(ih)p Fs(g)t Ft(j)31 b Fu(preserv)m(es)36 b Ft(D)s Fu(\()p Fs(A)p Fu(\),)d Fs(R)d Ft(2)f(C)2485 414 y Fk(1)2525 450 y Fu(\()p Fs(A)p Fu(\).)46 b(A)33 b(direct)h(compu-)456 566 y(tation)j(giv)m(es)j(the)e(second)h(result.) 61 b(Applying)39 b(the)f(\014rst)g(result)h(to)f([)p Fs(R)q(;)17 b(A)p Fu(],)39 b(w)m(e)456 683 y(obtain)32 b(the)h(third)g(one.)2023 b Fj(\003)456 859 y Fy(Prop)s(osition)42 b(2.2.)j Fp(L)-5 b(et)39 b Fu(\()p Fs(T)1552 874 y Fm(n)1599 859 y Fu(\))1637 874 y Fm(n)1723 859 y Fp(b)-5 b(e)39 b(a)g(se)-5 b(quenc)g(e)38 b(of)h(b)-5 b(ounde)g(d)38 b(op)-5 b(er)g(ators)39 b(such)456 975 y(that,)32 b Fs(T)739 990 y Fm(n)814 975 y Ft(2)c(C)966 939 y Fk(1)1005 975 y Fu(\()p Fs(A)p Fu(\))p Fp(,)k(for)f(al)5 b(l)31 b Fs(n)p Fp(,)h(and)e(such)h(that)g(ther)-5 b(e)32 b(exist)e(b)-5 b(ounde)g(d)31 b Fs(S;)17 b(T)45 b Fp(such)456 1092 y(that)37 b Fs(T)714 1107 y Fm(n)794 1092 y Ft(!)32 b Fs(T)51 b Fp(and)37 b Fu([)p Fs(T)1310 1107 y Fm(n)1357 1092 y Fs(;)17 b(A)p Fu(])32 b Ft(!)h Fs(S)43 b Fp(in)37 b(the)g(norm)g(top)-5 b(olo)g(gy.)52 b(Then)37 b Fs(T)46 b Ft(2)33 b(C)3256 1056 y Fk(1)3295 1092 y Fu(\()p Fs(A)p Fu(\))456 1208 y Fp(and)h Fs(S)f Fu(=)28 b([)p Fs(T)8 b(;)17 b(A)p Fu(])p Fp(.)456 1437 y(Pr)-5 b(o)g(of.)41 b Fu(See)33 b(Lemma)g(2.5)g(in)f ([GG)m(\023)-46 b(eM1].)1405 b Fj(\003)456 1614 y Fp(R)-5 b(emark)55 b Fu(2.3)p Fp(.)47 b Fu(The)e(LAP)-8 b(,)45 b(resp)s(ectiv)m(ely)i(to)c(\()p Ft(I)7 b Fs(;)17 b Fu(0)p Fs(;)g(A)p Fu(\),)47 b(holds)e(true)f(for)g Fs(H)51 b Fu(if)456 1730 y(and)35 b(only)h(if)f Fs(H)43 b Fu(has)35 b(no)h(sp)s(ectrum)g(in)g Ft(I)7 b Fu(.)52 b(The)36 b(LAP)g(for)f Fs(H)8 b Fu(,)35 b(resp)s(ectiv)m(ely)k(to)456 1846 y(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\),)30 b(implies)h(the)f(LAP)g(for)g Fs(H)8 b Fu(,)30 b(resp)s(ectiv)m(ely)i(to)e(\()p Ft(I)7 b Fs(;)17 b(s)2707 1810 y Fl(0)2730 1846 y Fs(;)g(A)p Fu(\),)30 b(for)g(an)m(y)g Fs(s)3316 1810 y Fl(0)3367 1846 y Ft(\025)456 1962 y Fs(s)p Fu(.)52 b(F)-8 b(or)35 b Fs(H)41 b Fu(=)33 b Ft(\000)p Fu(\001)j(the)g(Laplace)g(op)s(erator)f (in)h Fn(R)2302 1926 y Fm(d)2378 1962 y Fu(and)g Fs(A)g Fu(the)g(m)m(ultiplication)456 2079 y(op)s(erator)c(b)m(y)h Ft(h)p Fs(x)p Ft(i)p Fu(,)g(it)f(is)h(kno)m(wn)h(that)e(LAP)h(for)f Fs(H)8 b Fu(,)32 b(resp)s(ectiv)m(ely)k(to)c(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\),)456 2195 y(holds)33 b(true)g(if)f(and)h (only)g(if)f Fs(s)c(>)g Fu(1)p Fs(=)p Fu(2)j(\(cf.)44 b([H]\).)456 2430 y(2.2.)k Fy(Lo)s(cal)43 b(regularit)m(y)f(and)h(main) g(result.)49 b Fu(In)37 b(Theorem)h(1.2,)f(the)g(LAP)456 2546 y(\(1.2\))44 b(and)i(the)g(Mourre)f(estimate)i(\(1.1\))e(are)g(lo) s(calized)h(in)g Fs(H)8 b Fu(.)81 b(It)45 b(is)h(quite)456 2662 y(natural)35 b(to)g(try)h(to)g(replace)g Fs(H)43 b Fu(and)36 b(the)g(global)f(assumption)i Fs(H)k Ft(2)33 b(C)3117 2626 y Fk(2)3157 2662 y Fu(\()p Fs(A)p Fu(\))i(b)m(y)456 2778 y(some)e(lo)s(cal)f(v)m(ersion.)45 b(By)33 b([ABG],)g(w)m(e)h(ha)m (v)m(e)456 2955 y Fy(Prop)s(osition)41 b(2.4.)j Fp(L)-5 b(et)39 b Fs(')34 b Ft(2)g(C)1711 2919 y Fl(1)1705 2980 y Fm(c)1786 2955 y Fu(\()p Fn(R)p Fu(\))p Fp(.)55 b(Supp)-5 b(ose)37 b Fs(H)42 b Ft(2)34 b(C)2674 2919 y Fm(k)2717 2955 y Fu(\()p Fs(A)p Fu(\))39 b Fp(for)f(a)g(c)-5 b(ertain)456 3071 y Fs(k)30 b Ft(2)e Fn(N)p Fp(.)45 b(Then,)34 b Fs(')p Fu(\()p Fs(H)8 b Fu(\))27 b Ft(2)h(C)1470 3035 y Fm(k)1513 3071 y Fu(\()p Fs(A)p Fu(\))p Fp(.)456 3306 y Fu(F)-8 b(or)31 b(an)m(y)j Fs(\034)39 b Ft(2)28 b(C)1048 3270 y Fl(1)1042 3331 y Fm(c)1123 3306 y Fu(\()p Fn(R)p Fu(\),)k(w)m(e)i (de\014ne)g(the)f(b)s(ounded)g(op)s(erator)456 3467 y(\(2.4\))985 b Fs(H)1723 3482 y Fm(\034)1794 3467 y Fu(:=)27 b Fs(H)8 b(\034)j Fu(\()p Fs(H)d Fu(\))p Fs(:)456 3628 y Fu(It)27 b(turns)h(out)f(that)g(w)m(e)h(can)f(deduce)i(the)f(LAP)f(for)g Fs(H)34 b Fu(resp)s(ectiv)m(ely)d(to)26 b(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))456 3744 y(from)32 b(the)i(LAP)f(for)f Fs(H)1318 3759 y Fm(\034)1394 3744 y Fu(resp)s(ectiv)m(ely)k(to)c(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\),)33 b(if)g Fs(\034)40 b Fu(=)28 b(1)k(near)h Ft(I)7 b Fu(,)34 b(as)f(seen)456 3860 y(in)44 b(Prop)s(osition)g(2.13)f(b)s(elo)m(w.)78 b(Th)m(us)46 b Fs(H)2022 3875 y Fm(\034)2108 3860 y Fu(is)f(a)e(go)s(o) s(d)g(lo)s(cal)h(\(and)g(b)s(ounded\))456 3976 y(v)m(ersion)34 b(of)e Fs(H)8 b Fu(.)43 b(F)-8 b(rom)32 b([S,)h(Prop)s(osition)g(2.1],) f(w)m(e)i(pic)m(k)g(the)f(follo)m(wing)456 4153 y Fy(Lemma)d(2.5.)36 b Fp(L)-5 b(et)29 b Ft(I)35 b Fp(b)-5 b(e)28 b(b)-5 b(ounde)g(d,)28 b(op)-5 b(en)28 b(interval.)42 b(Supp)-5 b(ose)27 b(that)h Fs(H)36 b Ft(2)28 b(C)3256 4117 y Fk(1)3295 4153 y Fu(\()p Fs(A)p Fu(\))456 4269 y Fp(and)37 b(that)i(the)f(Mourr)-5 b(e)38 b(estimate)45 b Fu(\(1.1\))38 b Fp(holds)f(true)i(on)f Ft(I)7 b Fp(.)54 b(T)-7 b(ake)37 b Fs(\022)g Ft(2)d(C)3232 4233 y Fl(1)3226 4294 y Fm(c)3307 4269 y Fu(\()p Ft(I)7 b Fu(\))456 4386 y Fp(and)34 b Fs(\034)39 b Ft(2)28 b(C)878 4349 y Fl(1)872 4410 y Fm(c)953 4386 y Fu(\()p Fn(R)p Fu(\))35 b Fp(such)g(that)g Fs(\034)11 b(\022)31 b Fu(=)d Fs(\022)s Fp(.)45 b(Then)34 b Fs(H)2248 4401 y Fm(\034)2318 4386 y Ft(2)28 b(C)2470 4349 y Fk(1)2510 4386 y Fu(\()p Fs(A)p Fu(\))35 b Fp(and)946 4547 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1267 4562 y Fm(\034)1309 4547 y Fs(;)17 b(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))28 b Ft(\025)g Fs(c\022)1922 4506 y Fk(2)1961 4547 y Fu(\()p Fs(H)8 b Fu(\))22 b(+)g Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(K)f(\022)s Fu(\()p Fs(H)h Fu(\))p Fs(:)-2333 b Fu(\(2.5\))456 4777 y Fp(Pr)-5 b(o)g(of.)41 b Fu(By)33 b(Prop)s(osition)g(2.4,)f Fs(H)1697 4792 y Fm(\034)1768 4777 y Ft(2)c(C)1920 4740 y Fk(1)1959 4777 y Fu(\()p Fs(A)p Fu(\).)44 b(F)-8 b(or)32 b Fs(f)38 b Ft(2)28 b(D)s Fu(\()p Fs(A\022)s Fu(\()p Fs(H)8 b Fu(\)\),)456 4938 y Ft(h)p Fs(H)g(\022)s Fu(\()p Fs(H)g Fu(\))p Fs(f)d(;)17 b(iA\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)j Ft(i)20 b(\000)j(h)p Fs(iA\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)d(;)17 b(H)8 b(\022)s Fu(\()p Fs(H)g Fu(\))p Fs(f)j Ft(i)26 b(\025)i Fs(c)p Ft(k)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)j Ft(k)2831 4897 y Fk(2)2891 4938 y Fu(+)22 b Ft(h)p Fs(f)5 b(;)17 b(K)7 b(f)k Ft(i)p Fs(:)456 5099 y Fu(No)m(w,)39 b(use)f(that)g Fs(H)8 b(\022)s Fu(\()p Fs(H)g Fu(\))35 b(=)h Fs(H)8 b(\034)j Fu(\()p Fs(H)d Fu(\))p Fs(\022)s Fu(\()p Fs(H)g Fu(\).)57 b(Finally)-8 b(,)39 b Ft(D)s Fu(\()p Fs(A\022)s Fu(\()p Fs(H)8 b Fu(\)\))36 b(is)i(dense)h(in)456 5216 y Fo(H)59 b Fu(since)34 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(A)32 b Fu(is)h(closed)h(with)f(a) f(dense)i(domain.)910 b Fj(\003)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)397 b(7)456 450 y Fp(R)-5 b(emark)35 b Fu(2.6)p Fp(.)f Fu(In)22 b(general,)j(one)e(should)h(not)e(exp)s(ect)i (a)e(\\real")g(Mourre)h(estimate)456 566 y(for)32 b Fs(H)686 581 y Fm(\034)761 566 y Fu(of)g(the)h(form)1084 726 y Fs(')p Fu(\()p Fs(H)1267 741 y Fm(\034)1310 726 y Fu(\)[)p Fs(H)1456 741 y Fm(\034)1499 726 y Fs(;)17 b(iA)p Fu(])p Fs(')p Fu(\()p Fs(H)1859 741 y Fm(\034)1902 726 y Fu(\))28 b Ft(\025)g Fs(c')2179 685 y Fk(2)2218 726 y Fu(\()p Fs(H)2337 741 y Fm(\034)2380 726 y Fu(\))22 b(+)g Fs(K)r(;)456 886 y Fu(for)47 b(a)g(certain)h(function)g Fs(')g Fu(whic)m(h)h (sati\014es)g(the)f(same)g(h)m(yp)s(othesis)i(as)d Fs(\022)k Fu(in)456 1002 y(Lemma)34 b(2.5.)45 b(Indeed,)35 b(since)g(0)29 b Ft(2)h Fu(supp)q Fs(\022)s Fu(,)j(there)i(is)f(no)f(suc)m(h)i (function)f Fs(')f Fu(suc)m(h)456 1118 y(that)f Fs(')p Fu(\()p Fs(t\034)11 b Fu(\()p Fs(t)p Fu(\)\))28 b(=)g Fs(\022)s Fu(\()p Fs(t)p Fu(\))33 b(for)f(all)g Fs(t)c Ft(2)g Fn(R)p Fu(.)456 1352 y(Giv)m(en)35 b(an)g(op)s(en)g(in)m(terv)-5 b(al)35 b Ft(I)43 b Fu(and)35 b Fs(k)f Ft(2)e Fn(N)p Fu(,)j(w)m(e)h(sa)m(y)g(that)f Fs(H)42 b Fu(is)35 b Fp(lo)-5 b(c)g(al)5 b(ly)37 b(of)f(class)456 1468 y Ft(C)514 1432 y Fm(k)557 1468 y Fu(\()p Fs(A)p Fu(\))d Fp(on)g Ft(I)7 b Fu(,)32 b(w)m(e)g(write)g Fs(H)j Ft(2)28 b(C)1655 1432 y Fm(k)1649 1493 y Fl(I)1698 1468 y Fu(\()p Fs(A)p Fu(\),)k(if,)f(for)g (all)g Fs(')c Ft(2)h(C)2546 1432 y Fl(1)2540 1493 y Fm(c)2621 1468 y Fu(\()p Ft(I)7 b Fu(\),)32 b Fs(')p Fu(\()p Fs(H)8 b Fu(\))27 b Ft(2)h(C)3225 1432 y Fm(k)3268 1468 y Fu(\()p Fs(A)p Fu(\).)456 1584 y(This)35 b(is)f(a)g(lo)s(cal)g(v)m(ersion)i(of) d(the)i(regularit)m(y)g Ft(C)2213 1548 y Fm(k)2256 1584 y Fu(\()p Fs(A)p Fu(\))f(whic)m(h)h(w)m(as)h(already)e(used)456 1701 y(in)e([S)q(].)456 1880 y Fy(Pro)s(of)53 b(of)i(Theorem)f(1.2:)73 b Fu(Let)47 b Ft(I)1914 1844 y Fl(00)2004 1880 y Fu(b)s(e)h(op)s(en)f (suc)m(h)h(that)p 2862 1800 62 4 v 47 w Ft(I)60 b(\032)53 b(I)3167 1844 y Fl(00)3210 1880 y Fu(.)87 b(By)456 1996 y(Lemma)33 b(2.4,)f Fs(H)k Ft(2)28 b(C)1257 1960 y Fk(2)1251 2023 y Fl(I)1295 2004 y Fi(00)1339 1996 y Fu(\()p Fs(A)p Fu(\).)44 b(Let)33 b Fs(\034)39 b Ft(2)29 b(C)1968 1960 y Fl(1)1962 2021 y Fm(c)2043 1996 y Fu(\()p Ft(I)2142 1960 y Fl(00)2184 1996 y Fu(\))k(suc)m(h)h(that)f Fs(\034)39 b Fu(=)28 b(1)k(near)h Ft(I)7 b Fu(.)44 b(Let)456 2125 y Ft(I)510 2140 y Fk(1)585 2125 y Fu(b)s(e)36 b(closed)h(suc)m(h)g (that)e Ft(I)1510 2089 y Fl(0)1567 2125 y Ft(\032)1688 2100 y Fu(\027)1678 2125 y Ft(I)1732 2140 y Fk(1)1807 2125 y Fu(and)h Ft(I)2054 2140 y Fk(1)2126 2125 y Ft(\032)e(I)7 b Fu(.)53 b(Let)36 b Fs(\022)g Ft(2)d(C)2794 2089 y Fl(1)2788 2150 y Fm(c)2869 2125 y Fu(\()p Ft(I)7 b Fu(\))36 b(suc)m(h)i(that)456 2242 y Fs(\022)30 b Fu(=)e(1)k(on)h Ft(I)906 2257 y Fk(1)945 2242 y Fu(.)44 b(By)33 b(Lemma)g(2.5)f(and)h(\(1.1\),)f(w)m(e)i(deriv)m (e)g(\(2.5\),)e(whic)m(h)i(implies)1046 2401 y Fs(E)1118 2416 y Fl(I)1157 2425 y Fh(1)1196 2401 y Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1469 2416 y Fm(\034)1511 2401 y Fs(;)17 b(iA)p Fu(])p Fs(E)1760 2416 y Fl(I)1799 2425 y Fh(1)1838 2401 y Fu(\()p Fs(H)8 b Fu(\))27 b Ft(\025)i Fs(cE)2250 2416 y Fl(I)2289 2425 y Fh(1)2327 2401 y Fu(\()p Fs(H)8 b Fu(\))22 b(+)g(0)p Fs(;)456 2561 y Fu(since)36 b Fs(\022)f Fu(=)c(1)j(on)h Ft(I)1159 2576 y Fk(1)1199 2561 y Fu(.)50 b(Th)m(us)36 b(Theorem)h(2.7)d(b)s(elo)m(w)i(applies)g(yielding)g(the)f (LAP)456 2677 y(for)d Fs(H)40 b Fu(resp)s(ectiv)m(ely)35 b(to)e(\()p Ft(I)1476 2641 y Fl(0)1499 2677 y Fs(;)17 b(s;)g(A)p Fu(\).)1596 b Fj(\003)456 2851 y Fu(So)49 b(the)h(pro)s(of)e(of)h(Theorem)i(1.2)e(reduces)j(to)d(the)h(pro)s(of)e (of)h(the)h(follo)m(wing)456 2968 y(stronger)33 b(result,)g(whic)m(h)h (is)f(our)g(main)g(result.)456 3143 y Fy(Theorem)41 b(2.7.)i Fp(L)-5 b(et)38 b Ft(I)46 b Fp(b)-5 b(e)37 b(a)g(b)-5 b(ounde)g(d,)38 b(op)-5 b(en)37 b(interval.)52 b(L)-5 b(et)38 b Ft(I)2905 3107 y Fl(00)2986 3143 y Fp(b)-5 b(e)37 b(an)g(op)-5 b(en)456 3260 y(interval)36 b(such)g(that)p 1239 3180 V 37 w Ft(I)i(\032)31 b(I)1500 3224 y Fl(00)1543 3260 y Fp(.)50 b(L)-5 b(et)37 b Fs(H)h Ft(2)31 b(C)2067 3224 y Fk(2)2061 3286 y Fl(I)2105 3267 y Fi(00)2150 3260 y Fu(\()p Fs(A)p Fu(\))36 b Fp(and)g Fs(\034)42 b Ft(2)31 b(C)2765 3224 y Fl(1)2759 3284 y Fm(c)2840 3260 y Fu(\()p Ft(I)2939 3224 y Fl(00)2982 3260 y Fu(\))37 b Fp(such)f(that)456 3376 y Fs(\034)j Fu(=)27 b(1)35 b Fp(ne)-5 b(ar)34 b Ft(I)7 b Fp(.)46 b(Supp)-5 b(ose)34 b(the)h(strict)g(Mourr)-5 b(e)35 b(estimate)920 3536 y Fs(E)992 3551 y Fl(I)1039 3536 y Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1312 3551 y Fm(\034)1355 3536 y Fs(;)17 b(iA)p Fu(])p Fs(E)1604 3551 y Fl(I)1652 3536 y Fu(\()p Fs(H)8 b Fu(\))27 b Ft(\025)h Fs(cE)2063 3551 y Fl(I)2111 3536 y Fu(\()p Fs(H)8 b Fu(\))p Fs(;)51 b Fp(with)34 b Fs(c)28 b(>)g Fu(0)p Fs(;)-2359 b Fu(\(2.6\))456 3695 y Fp(holds)40 b(true.)63 b(Then,)42 b(for)f(any)g Fs(s)e(>)f Fu(1)p Fs(=)p Fu(2)j Fp(and)f(any)h(c)-5 b(omp)g(act)40 b(interval)h Ft(I)3203 3659 y Fl(0)3268 3695 y Fp(with)456 3823 y Ft(I)517 3787 y Fl(0)568 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y(\015)1234 3481 y Fu(=)60 b(1)32 b(and)50 b Ft(kh)p Fs(A)p Ft(i)1859 3440 y Fm(s)1895 3481 y Fu(\()p Fs(H)30 b Ft(\000)22 b Fs(z)2188 3496 y Fm(n)2235 3481 y Fu(\))p Fs(f)2321 3496 y Fm(n)2368 3481 y Ft(k)60 b Fu(=)28 b(1)p Fs(=k)2731 3496 y Fm(n)2805 3481 y Ft(!)f Fu(0)p Fs(:)456 3640 y Fu(Up)36 b(to)g(a)g(subsequence,)41 b(w)m(e)c(can)f(assume)i (that)e(Re\()p Fs(z)2493 3655 y Fm(n)2540 3640 y Fu(\))e Ft(!)g Fs(\025)f Ft(2)i(I)7 b Fu(.)55 b(No)m(w,)37 b(w)m(e)456 3756 y(assume)f(the)e(LAP)h(true)g(and)g(consider)g(\()p Fs(f)2066 3771 y Fm(n)2113 3756 y Fs(;)17 b(z)2202 3771 y Fm(n)2249 3756 y Fu(\))2287 3771 y Fm(n)2334 3756 y Fu(,)35 b(a)f(sp)s(ecial)i(sequence)h(for)d Fs(H)456 3872 y Fu(asso)s(ciated)f(to)f(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\).)44 b(By)34 b(\(1.2\),)e(there)i(exists)g Fs(c)28 b(>)f Fu(0)32 b(suc)m(h)i(that)1129 3947 y Ff(\015)1129 4006 y(\015)1185 4031 y Ft(h)p Fs(A)p Ft(i)1336 3990 y Fl(\000)p Fm(s)1427 4031 y Fs(f)1475 4046 y Fm(n)1522 3947 y Ff(\015)1522 4006 y(\015)1637 4031 y Ft(\024)61 b Fs(c)p Ft(kh)p Fs(A)p Ft(i)2018 3990 y Fm(s)2054 4031 y Fu(\()p Fs(H)30 b Ft(\000)23 b Fs(z)2348 4046 y Fm(n)2395 4031 y Fu(\))p Fs(f)2481 4046 y Fm(n)2528 4031 y Ft(k)p Fs(:)456 4189 y Fu(This)33 b(implies)h Fs(\021)e Fu(=)27 b(0.)2099 b Fj(\003)456 4363 y Fu(The)33 b(previous)h(result)g(can)f(b) s(e)f(partially)h(lo)s(calized)h(in)e(energy)-8 b(.)456 4538 y Fy(Prop)s(osition)30 b(2.12.)37 b Fp(L)-5 b(et)29 b Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))29 b Fp(b)-5 b(e)28 b(a)h(triplet)g(as)g(in)36 b Fu(\(1.2\))28 b Fp(with)h Fu(0)e Ft(\024)i Fs(s)e(<)h Fu(1)p Fp(.)456 4659 y(L)-5 b(et)32 b Ft(I)682 4623 y Fl(00)758 4659 y Fp(b)-5 b(e)32 b(op)-5 b(en)31 b(such)i(that)p 1517 4579 62 4 v 32 w Ft(I)j(\032)28 b(I)1773 4623 y Fl(00)1848 4659 y Fp(and)k Fs(H)j Ft(2)28 b(C)2303 4623 y Fk(1)2297 4685 y Fl(I)2341 4666 y Fi(00)2386 4659 y Fu(\()p Fs(A)p Fu(\))p Fp(.)44 b(L)-5 b(et)32 b Fs(\022)f Ft(2)d(C)3002 4623 y Fl(1)2996 4683 y Fm(c)3077 4659 y Fu(\()p Fn(R)p Fu(\))33 b Fp(such)456 4775 y(that)39 b Fs(\022)g Fu(=)c(1)k Fp(ne)-5 b(ar)39 b Ft(I)7 b Fp(.)58 b(L)-5 b(et)39 b Fs(')c Fu(:)h Fn(R)f Ft(!)g Fn(R)k Fp(b)-5 b(or)g(elian)39 b(such)f(that,)j(for)e Fs(t)c Ft(2)h Fu(supp)q Fs(\022)s Fp(,)456 4891 y Fs(')p Fu(\()p Fs(t)p Fu(\))27 b(=)h Fs(t)p Fp(.)42 b(L)-5 b(et)26 b Fu(\()p Fs(f)1114 4906 y Fm(n)1161 4891 y Fs(;)17 b(z)1250 4906 y Fm(n)1297 4891 y Fu(\))1335 4906 y Fm(n)1409 4891 y Fp(b)-5 b(e)25 b(a)i(sp)-5 b(e)g(cial)25 b(se)-5 b(quenc)g(e)25 b(for)h Fs(H)34 b Fp(asso)-5 b(ciate)g(d)25 b(to)h Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))456 5017 y Fp(with)34 b(mass)g Fs(\021)t Fp(.)45 b(Then,)34 b(writing)1662 4991 y Fu(~)1654 5017 y Fs(\022)d Fu(=)d(1)22 b Ft(\000)g Fs(\022)s Fp(,)604 5216 y Fu(\(1\))778 5189 y(~)770 5216 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1031 5231 y Fm(n)1113 5216 y Fp(tends)34 b(to)h Fu(0)p Fp(,)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)397 b(9)604 450 y Fu(\(2\))41 b(\()p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1069 465 y Fm(n)1116 450 y Fs(;)17 b(z)1205 465 y Fm(n)1252 450 y Fu(\))1290 465 y Fm(n)1399 450 y Fp(is)63 b(a)f(sp)-5 b(e)g(cial)62 b(se)-5 b(quenc)g(e)61 b(for)i Fs(')p Fu(\()p Fs(H)8 b Fu(\))62 b Fp(asso)-5 b(ciate)g(d)61 b(to)770 566 y Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))35 b Fp(with)g(mass)f Fs(\021)t Fp(.)456 806 y(Pr)-5 b(o)g(of.)41 b Fu(Since)701 980 y Ft(k)759 953 y Fu(~)751 980 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1012 995 y Fm(n)1058 980 y Ft(k)60 b(\024)h(k)1364 953 y Fu(~)1356 980 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\)\()p Fs(H)29 b Ft(\000)23 b Fs(z)1862 995 y Fm(n)1909 980 y Fu(\))1947 938 y Fl(\000)p Fk(1)2041 980 y Ft(h)p Fs(A)p Ft(i)2192 938 y Fl(\000)p Fm(s)2284 980 y Ft(k)38 b(\001)g(kh)p Fs(A)p Ft(i)2639 938 y Fm(s)2676 980 y Fu(\()p Fs(H)29 b Ft(\000)23 b Fs(z)2969 995 y Fm(n)3016 980 y Fu(\))p Fs(f)3102 995 y Fm(n)3149 980 y Ft(k)456 1159 y Fu(and)29 b(since)j Fs(t)c Ft(7!)1077 1132 y Fu(~)1069 1159 y Fs(\022)s Fu(\()p Fs(t)p Fu(\))p Fs(=)p Fu(\()p Fs(t)16 b Ft(\000)g Fs(z)1504 1174 y Fm(n)1553 1159 y Fu(\))29 b(is)i(uniformly)f(b)s(ounded)h(in)f Fs(n)p Fu(,)g Ft(k)2837 1132 y Fu(~)2829 1159 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)3090 1174 y Fm(n)3137 1159 y Ft(k)29 b Fu(tends)456 1284 y(to)23 b(0.)41 b(Since)25 b Fs(s)i Ft(\025)h Fu(0,)e Ft(kh)p Fs(A)p Ft(i)1410 1248 y Fl(\000)p Fm(s)1509 1258 y Fu(~)1501 1284 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1762 1299 y Fm(n)1809 1284 y Ft(k)27 b(!)g Fu(0)d(and)g(therefore)h Ft(kh)p Fs(A)p Ft(i)2869 1248 y Fl(\000)p Fm(s)2960 1284 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)3221 1299 y Fm(n)3267 1284 y Ft(k)28 b(!)456 1401 y Fs(\021)t Fu(.)46 b(Since)34 b Fs(H)j Ft(2)30 b(C)1108 1364 y Fk(1)1102 1427 y Fl(I)1146 1408 y Fi(00)1190 1401 y Fu(\()p Fs(A)p Fu(\),)k Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))29 b Ft(2)h(C)1796 1364 y Fk(1)1836 1401 y Fu(\()p Fs(A)p Fu(\).)46 b(Since)34 b Fs(s)c(<)f Fu(1,)k Ft(kh)p Fs(A)p Ft(i)2804 1364 y Fm(s)2841 1401 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Ft(h)p Fs(A)p Ft(i)3205 1364 y Fl(\000)p Fm(s)3296 1401 y Ft(k)33 b Fu(is)456 1517 y(b)s(ounded,)g(b)m(y)h(Prop)s(osition)e(B.2.)44 b(No)m(w,)540 1686 y Ft(kh)p Fs(A)p Ft(i)741 1645 y Fm(s)777 1686 y Fu(\()p Fs(')p Fu(\()p Fs(H)8 b Fu(\))22 b Ft(\000)g Fs(z)1210 1701 y Fm(n)1257 1686 y Fu(\))p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1556 1701 y Fm(n)1603 1686 y Ft(k)28 b(\024)g(kh)p Fs(A)p Ft(i)1987 1645 y Fm(s)2023 1686 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Ft(h)p Fs(A)p Ft(i)2387 1645 y Fl(\000)p Fm(s)2478 1686 y Ft(k)22 b(\001)g(kh)p Fs(A)p Ft(i)2801 1645 y Fm(s)2837 1686 y Fu(\()p Fs(H)29 b Ft(\000)23 b Fs(z)3130 1701 y Fm(n)3177 1686 y Fu(\))p Fs(f)3263 1701 y Fm(n)3310 1686 y Ft(k)456 1856 y Fu(whic)m(h)34 b(tends)f(to)g(0.)2176 b Fj(\003)456 2030 y Fu(No)m(w)33 b(w)m(e)g(can)g(p)s(erform)g(the)g(reduction)g(to)g(some)g Fs(H)2415 2045 y Fm(\034)2490 2030 y Fu(\(cf.)44 b(\(2.4\))o(\).)456 2212 y Fy(Prop)s(osition)30 b(2.13.)37 b Fp(L)-5 b(et)29 b Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))29 b Fp(b)-5 b(e)28 b(a)h(triplet)g(as)g(in)36 b Fu(\(1.2\))28 b Fp(with)h Fu(0)e Ft(\024)i Fs(s)e(<)h Fu(1)p Fp(.)456 2333 y(L)-5 b(et)42 b Ft(I)692 2297 y Fl(00)776 2333 y Fp(b)-5 b(e)42 b(op)-5 b(en)41 b(such)g(that)p 1573 2253 62 4 v 42 w Ft(I)48 b(\032)41 b(I)1854 2297 y Fl(00)1939 2333 y Fp(and)g Fs(H)48 b Ft(2)41 b(C)2429 2297 y Fk(1)2423 2360 y Fl(I)2467 2341 y Fi(00)2511 2333 y Fu(\()p Fs(A)p Fu(\))p Fp(.)66 b(L)-5 b(et)42 b Fs(\034)52 b Ft(2)40 b(C)3189 2297 y Fl(1)3183 2358 y Fm(c)3264 2333 y Fu(\()p Ft(I)3363 2297 y Fl(00)3406 2333 y Fu(\))456 2450 y Fp(such)35 b(that)i Fs(\034)k Fu(=)29 b(1)36 b Fp(ne)-5 b(ar)36 b Ft(I)7 b Fp(.)48 b(If)36 b(the)g(LAP)g(r)-5 b(esp)g(e)g(ctively)36 b(to)g Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))36 b Fp(holds)f(true)456 2566 y(for)f Fs(H)692 2581 y Fm(\034)770 2566 y Fp(then)h(it)g(holds)f(true)h(for)g Fs(H)8 b Fp(.)456 2806 y(Pr)-5 b(o)g(of.)41 b Fu(By)53 b(con)m(trap)s(osition,)58 b(the)52 b(result)i(follo)m(ws)f(from)f(Prop)s(ositions)h(2.11)456 2922 y(and)32 b(2.12.)2521 b Fj(\003)456 3103 y Fp(R)-5 b(emark)57 b Fu(2.14)p Fp(.)48 b Fu(There)g(is)f(another)g(pro)s(of)e (of)h(Prop)s(osition)h(2.13.)85 b(Let)46 b Fs(\022)55 b Ft(2)456 3220 y(C)514 3183 y Fl(1)508 3244 y Fm(c)589 3220 y Fu(\()p Fn(R)p Fu(\))36 b(with)i Fs(\022)h Fu(=)c(1)i(near)g Ft(I)45 b Fu(and)37 b Fs(\034)11 b(\022)39 b Fu(=)c Fs(\022)s Fu(.)57 b(Then,)39 b(using)f(a)f(Neumann)h(serie)456 3336 y(for)g Ft(j)p Fs(z)t Ft(j)i Fu(large)f(enough)h(and)f Fs(z)44 b Ft(62)c Fn(R)p Fu(,)h(w)m(e)f(can)g(sho)m(w)g(that)f(\()p Fs(H)c Ft(\000)27 b Fs(z)t Fu(\))3022 3300 y Fl(\000)p Fk(1)3117 3336 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))39 b(=)456 3452 y(\()p Fs(H)575 3467 y Fm(\034)641 3452 y Ft(\000)24 b Fs(z)t Fu(\))829 3416 y Fl(\000)p Fk(1)924 3452 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\).)49 b(By)35 b(analyticit)m(y)-8 b(,)36 b(this)f(holds)g(true)g(for)f Fs(z)i Ft(62)31 b Fn(R)p Fu(.)49 b(Therefore,)456 3568 y(if)35 b(the)h(LAP)f(resp)s (ectiv)m(ely)k(to)c(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))35 b(is)h(true)g(for)e Fs(H)2532 3583 y Fm(\034)2611 3568 y Fu(so)h(is)h(it)f(for)g Fs(H)8 b Fu(,)36 b(since)456 3685 y Ft(h)p Fs(A)p Ft(i)607 3648 y Fm(s)643 3685 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Ft(h)p Fs(A)p Ft(i)1007 3648 y Fl(\000)p Fm(s)1130 3685 y Fu(is)33 b(b)s(ounded.)456 3924 y(2.4.)48 b Fy(A)43 b(Virial-lik)m(e)h(Theorem.)50 b Fu(In)37 b(Remark)i(1.1,)f(w)m(e)h(recalled)f(the)g(Virial)456 4041 y(Theorem.)44 b(Our)33 b(approac)m(h)g(is)g(based)h(on)e(the)h (follo)m(wing)g(Virial-lik)m(e)g(result.)456 4222 y Fy(Prop)s(osition) 41 b(2.15.)j Fp(L)-5 b(et)38 b Fu(\()p Fs(f)1596 4237 y Fm(n)1643 4222 y Fs(;)17 b(z)1732 4237 y Fm(n)1779 4222 y Fu(\))1817 4237 y Fm(n)1902 4222 y Fp(b)-5 b(e)38 b(a)g(sp)-5 b(e)g(cial)37 b(se)-5 b(quenc)g(e)37 b(for)h(a)46 b Fu(b)s(ounded)456 4338 y Fp(op)-5 b(er)g(ator)39 b Fs(H)925 4353 y Fm(b)999 4338 y Fp(r)-5 b(esp)g(e)g(ctively)40 b(to)g Fu(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))p Fp(,)41 b(as)e(in)47 b Fu(\(1.2\))40 b Fp(with)g Fs(s)c Ft(\025)i Fu(0)p Fp(.)60 b(F)-7 b(or)38 b(any)456 4455 y Fu(b)s(ounded)d Fp(b)-5 b(or)g(elian)34 b(function)h Fs(\036)p Fp(,)1330 4624 y Fu(lim)1306 4684 y Fm(n)p Fl(!1)1490 4624 y Ft(h)p Fs(f)1577 4639 y Fm(n)1624 4624 y Fs(;)17 b Fu([)p Fs(H)1776 4639 y Fm(b)1810 4624 y Fs(;)g(\036)p Fu(\()p Fs(A)p Fu(\)])p Fs(f)2136 4639 y Fm(n)2182 4624 y Ft(i)28 b Fu(=)f(0)p Fs(:)456 4901 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Since)33 b([)p Fs(H)1116 4916 y Fm(b)1151 4901 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\)])27 b(=)g([)p Fs(H)1667 4916 y Fm(b)1724 4901 y Ft(\000)c Fs(z)1869 4916 y Fm(n)1916 4901 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\)],)672 5070 y Ft(h)p Fs(f)759 5085 y Fm(n)806 5070 y Fs(;)g Fu([)p Fs(H)958 5085 y Fm(b)992 5070 y Fs(;)g(\036)p Fu(\()p Fs(A)p Fu(\)])p Fs(f)1318 5085 y Fm(n)1364 5070 y Ft(i)83 b Fu(=)g(2)p Fs(i)p Fu(Im)q(\()p Fs(z)1927 5085 y Fm(n)1974 5070 y Fu(\))p Ft(h)p Fs(f)2099 5085 y Fm(n)2145 5070 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\))p Fs(f)2444 5085 y Fm(n)2491 5070 y Ft(i)1173 5216 y Fu(+)p Ft(h)p Fu(\()p Fs(H)29 b Ft(\000)22 b Fs(z)1580 5231 y Fm(n)1628 5216 y Fu(\))p Fs(f)1714 5231 y Fm(n)1761 5216 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\))p Fs(f)2060 5231 y Fm(n)2106 5216 y Ft(i)22 b Fu(+)g Ft(h)p Fs(\036)p Fu(\()p Fs(A)p Fu(\))2511 5174 y Fl(\003)2550 5216 y Fs(f)2598 5231 y Fm(n)2645 5216 y Fs(;)17 b Fu(\()p Fs(H)29 b Ft(\000)23 b Fs(z)2982 5231 y Fm(n)3029 5216 y Fu(\))p Fs(f)3115 5231 y Fm(n)3162 5216 y Ft(i)p Fs(:)p eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 456 255 a Fq(10)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fu(By)33 b(De\014nition)g(2.10,)f(there)h(exists)h Fs(C)h(>)27 b Fu(0)33 b(suc)m(h)h(that)560 655 y Ft(jh)p Fu(\()p Fs(H)746 670 y Fm(b)802 655 y Ft(\000)22 b Fs(z)946 670 y Fm(n)994 655 y Fu(\))p Fs(f)1080 670 y Fm(n)1127 655 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\))p Fs(f)1426 670 y Fm(n)1472 655 y Ft(ij)83 b(\024)g(jhh)p Fs(A)p Ft(i)2000 614 y Fm(s)2036 655 y Fu(\()p Fs(H)2155 670 y Fm(b)2211 655 y Ft(\000)23 b Fs(z)2356 670 y Fm(n)2403 655 y Fu(\))p Fs(f)2489 670 y Fm(n)2536 655 y Fs(;)17 b Ft(h)p Fs(A)p Ft(i)2731 614 y Fl(\000)p Fm(s)2822 655 y Fs(\036)p Fu(\()p Fs(A)p Fu(\))p Fs(f)3077 670 y Fm(n)3124 655 y Ft(ij)1622 800 y(\024)83 b Fs(C)7 b Ft(k)p Fs(\036)p Fu(\()p Fs(A)p Fu(\))p Ft(k)22 b(\001)f(kh)p Fs(A)p Ft(i)2438 759 y Fm(s)2475 800 y Fu(\()p Fs(H)2594 815 y Fm(b)2650 800 y Ft(\000)i Fs(z)2795 815 y Fm(n)2842 800 y Fu(\))p Fs(f)2928 815 y Fm(n)2975 800 y Ft(k)69 b(!)3052 860 y Fm(n)p Fl(!1)3264 800 y Fu(0)p Fs(:)456 1041 y Fu(Similarly)-8 b(,)33 b(lim)q Ft(h)p Fs(\036)p Fu(\()p Fs(A)p Fu(\))1271 1005 y Fl(\003)1310 1041 y Fs(f)1358 1056 y Fm(n)1405 1041 y Fs(;)17 b Fu(\()p Fs(H)1568 1056 y Fm(b)1624 1041 y Ft(\000)22 b Fs(z)1768 1056 y Fm(n)1815 1041 y Fu(\))p Fs(f)1901 1056 y Fm(n)1948 1041 y Ft(i)28 b Fu(=)f(0.)44 b(By)33 b(De\014nition)g(2.10,)745 1246 y(Im\()p Fs(z)944 1261 y Fm(n)992 1246 y Fu(\))22 b Ft(\001)f(k)p Fs(f)1199 1261 y Fm(n)1246 1246 y Ft(k)1296 1205 y Fk(2)1419 1246 y Fu(=)82 b(Im)q Ft(h)p Fs(f)1781 1261 y Fm(n)1828 1246 y Fs(;)17 b Fu(\()p Fs(H)1991 1261 y Fm(b)2047 1246 y Ft(\000)22 b Fs(z)2191 1261 y Fm(n)2239 1246 y Fu(\))p Fs(f)2325 1261 y Fm(n)2372 1246 y Ft(i)1419 1392 y Fu(=)82 b(Im)q Ft(hh)p Fs(A)p Ft(i)1884 1351 y Fl(\000)p Fm(s)1975 1392 y Fs(f)2023 1407 y Fm(n)2070 1392 y Fs(;)17 b Ft(h)p Fs(A)p Ft(i)2265 1351 y Fm(s)2301 1392 y Fu(\()p Fs(H)2420 1407 y Fm(b)2476 1392 y Ft(\000)23 b Fs(z)2621 1407 y Fm(n)2668 1392 y Fu(\))p Fs(f)2754 1407 y Fm(n)2801 1392 y Ft(i)70 b(!)2868 1452 y Fm(n)p Fl(!1)3079 1392 y Fu(0)p Fs(:)456 1626 y Fu(Since)846 1831 y Ft(j)p Fu(Im\()p Fs(z)1073 1846 y Fm(n)1120 1831 y Fu(\))p Ft(h)p Fs(f)1245 1846 y Fm(n)1292 1831 y Fs(;)17 b(\036)p Fu(\()p Fs(A)p Fu(\))p Fs(f)1591 1846 y Fm(n)1637 1831 y Ft(ij)28 b(\024)g(j)p Fu(Im\()p Fs(z)2064 1846 y Fm(n)2111 1831 y Fu(\))p Ft(j)22 b(\001)g(k)p Fs(f)2347 1846 y Fm(n)2393 1831 y Ft(k)2443 1790 y Fk(2)2505 1831 y Ft(\001)g(k)p Fs(\036)p Fu(\()p Fs(A)p Fu(\))p Ft(k)p Fs(;)456 2035 y Fu(w)m(e)33 b(obtain)g(the)g(desired)h(result.)1700 b Fj(\003)456 2363 y Fu(2.5.)48 b Fy(Sk)m(etc)m(h)63 b(of)g(our)g(pro)s(of)g(and)h(in)m(terpretation.)49 b Fu(T)-8 b(o)55 b(pro)m(v)m(e)h(Theo-)456 2479 y(rem)48 b(2.7,)i(w)m(e)f(only)f(need)g(to)f(sho)m(w)i(the)f(LAP)f(for)g Fs(H)2564 2494 y Fm(\034)2655 2479 y Fu(on)g Ft(I)2866 2443 y Fl(0)2937 2479 y Fu(b)m(y)i(Prop)s(osi-)456 2595 y(tion)35 b(2.13.)50 b(In)36 b(view)g(of)f(Prop)s(osition)h(2.11,)f(w)m (e)h(consider)h(a)d(sp)s(ecial)j(sequence)456 2712 y(\()p Fs(f)542 2727 y Fm(n)589 2712 y Fs(;)17 b(z)678 2727 y Fm(n)725 2712 y Fu(\))763 2727 y Fm(n)863 2712 y Fu(for)54 b Fs(H)1115 2727 y Fm(\034)1212 2712 y Fu(asso)s(ciated)h(to)e(the)i (triplet)f(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\),)59 b(with)c Fs(s)64 b(>)g Fu(1)p Fs(=)p Fu(2,)456 2828 y(and)52 b(w)m(e)h(sho)m(w)g(that)f Fs(\021)65 b Fu(=)60 b(0.)102 b(By)53 b(Remark)f(2.3,)57 b(w)m(e)c(ma)m(y)g(assume)h(that)456 2944 y Fs(s)66 b Ft(2)p Fu(]1)p Fs(=)p Fu(2;)17 b(2)p Fs(=)p Fu(3[.)110 b(F)-8 b(or)54 b Fs(R)68 b(>)e Fu(1,)60 b(let)1944 2925 y Fs(\037)2005 2959 y Fm(R)2100 2944 y Fu(+)2220 2925 y(~)-56 b Fs(\037)2275 2959 y Fm(R)2398 2944 y Fu(=)67 b(1)54 b(b)s(e)i(a)f(smo)s(oth)g(par-)456 3060 y(tition)d(of)f(unit)m(y)j(on)e Fn(R)g Fu(with)1667 3041 y Fs(\037)1728 3075 y Fm(R)1838 3060 y Fu(lo)s(calized)h(in)f Ft(f)p Fs(t)61 b Ft(2)h Fn(R)p Fu(;)17 b Ft(j)p Fs(t)p Ft(j)60 b(\024)h Fu(2)p Fs(R)q Ft(g)p Fu(.)102 b(It)456 3179 y(su\016ces)53 b(to)e(sho)m(w)h(that)f(lim)1569 3194 y Fm(R)p Fl(!1)1785 3179 y Fu(lim)17 b(sup)2084 3203 y Fm(n)p Fl(!1)2288 3179 y Ft(k)2344 3160 y Fu(~)-55 b Fs(\037)2399 3194 y Fm(R)2457 3179 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2757 3143 y Fl(\000)p Fm(s)2848 3179 y Fs(f)2896 3194 y Fm(n)2943 3179 y Ft(k)59 b Fu(=)g(0)51 b(and)456 3295 y(lim)591 3310 y Fm(R)p Fl(!1)807 3295 y Fu(lim)17 b(sup)1106 3319 y Fm(n)p Fl(!1)1310 3295 y Ft(k)1360 3276 y Fs(\037)1421 3310 y Fm(R)1479 3295 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1779 3259 y Fl(\000)p Fm(s)1870 3295 y Fs(f)1918 3310 y Fm(n)1965 3295 y Ft(k)62 b Fu(=)f(0.)104 b(F)-8 b(rom)52 b(the)h(strict)g(Mourre) 456 3412 y(estimate)e(\(2.6\))o(,)k(w)m(e)c(deduce)h(\(2.5\))d(with)i Fs(K)64 b Fu(=)58 b(0.)96 b(W)-8 b(e)50 b(apply)h(the)f(later)456 3530 y(to)596 3511 y(~)-55 b Fs(\037)651 3545 y Fm(R)709 3530 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1009 3494 y Fl(\000)p Fm(s)1100 3530 y Fs(f)1148 3545 y Fm(n)1195 3530 y Fu(.)89 b(After)48 b(sev)m(eral)h(comm)m(utations,)k(the)c(use)f (of)g(Prop)s(osi-)456 3646 y(tion)35 b(2.15,)g(and)g(the)g(use)i(of)d (the)i(assumption)g Fs(s)c(>)g Fu(1)p Fs(=)p Fu(2,)j(w)m(e)h(\014nd)g (some)f Fs(\017)e(>)f Fu(0)456 3763 y(suc)m(h)i(that,)e(for)g(all)h Fs(R)28 b(>)g Fu(1,)1120 3967 y(lim)17 b(sup)1177 4046 y Fm(n)p Fl(!1)1435 3967 y Ft(k)1491 3948 y Fu(~)-55 b Fs(\037)1546 3982 y Fm(R)1604 3967 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1904 3926 y Fl(\000)p Fm(s)1995 3967 y Fs(f)2043 3982 y Fm(n)2090 3967 y Ft(k)28 b Fu(=)f Fs(O)s Fu(\()p Fs(R)2462 3926 y Fl(\000)p Fm(\017)2549 3967 y Fu(\))p Fs(:)-2158 b Fu(\(2.8\))456 4228 y(Next)42 b(w)m(e)h(apply)f(the)g(Mourre)g(estimate)h(\(2.5\))e(to)2433 4208 y Fs(\037)2494 4243 y Fm(R)2552 4228 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2749 4243 y Fm(n)2796 4228 y Fu(.)70 b(After)42 b(sev)m(eral)456 4344 y(comm)m(utations,)30 b(the)e(use)h(of)e(Prop)s (osition)h(2.15,)h(and)f(the)g(use)h(of)34 b(\(2.8\),)28 b(w)m(e)h(get)456 4460 y(lim)591 4475 y Fm(R)p Fl(!1)807 4460 y Fu(lim)17 b(sup)1106 4484 y Fm(n)p Fl(!1)1310 4460 y Ft(k)1360 4441 y Fs(\037)1421 4475 y Fm(R)1479 4460 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1676 4475 y Fm(n)1723 4460 y Ft(k)27 b Fu(=)h(0.)41 b(Since)27 b Fs(s)h Ft(\025)g Fu(0,)f(w)m(e)g(obtain)e(the)i(desired)456 4576 y(results)34 b(yielding)f Fs(\021)f Fu(=)27 b(0.)456 4751 y(This)38 b(pro)s(of)e(pro)m(vides)i(the)g(follo)m(wing)f(new)h(in)m (terpretation)g(of)e(Theorems)j(1.2)456 4867 y(and)29 b(2.7.)42 b(The)30 b(strict)g(Mourre)f(estimate)i(excludes)g(the)f (existence)i(of)c(a)h(sp)s(ecial)456 4983 y(sequence)48 b(of)d(p)s(ositiv)m(e)j(mass,)i(yielding)d(the)f(LAP)-8 b(,)46 b(in)g(a)g(similar)g(w)m(a)m(y)h(as)f(it)456 5099 y(excludes)40 b(the)f(existence)i(of)d(b)s(ound)h(state)g(in)f(Remark)h (1.1.)61 b(Our)38 b(Virial-lik)m(e)456 5216 y(Theorem)33 b(pla)m(ys)h(the)f(role)g(of)f(the)h(usual)g(Virial)g(Theorem.)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(11)1233 450 y Fu(3.)49 b Fv(A)37 b(new)h(pr)n(oof)g(of)g(the)f(LAP.)456 683 y Fu(Here)29 b(w)m(e)g(complete)h(the)e(pro)s(of)g(of)f(Theorem)j(2.7)e(sk)m(etc)m (hed)j(in)e(Subsection)h(2.5.)456 799 y(W)-8 b(e)42 b(assume)h(the)g (assumptions)g(of)f(Theorem)h(2.7)e(satis\014ed)j(and)e(tak)m(e)g(some) 456 921 y(in)m(terv)-5 b(al)43 b Ft(I)879 885 y Fl(0)947 921 y Ft(\032)1066 896 y Fu(\027)1069 921 y Ft(I)8 b Fu(.)73 b(Let)43 b Fs(\022)k Ft(2)e(C)1677 885 y Fl(1)1671 945 y Fm(c)1752 921 y Fu(\()p Ft(I)7 b Fu(\))43 b(with)g Fs(\022)48 b Fu(=)d(1)d(on)g Ft(I)2675 885 y Fl(0)2699 921 y Fu(.)73 b(Applying)44 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))456 1037 y(on)42 b(b)s(oth)g(side,)k(w)m(e)d(deduce)h(from)e (\(2.6\))g(the)h(strict)g(Mourre)f(estimate)i(\(2.5\))456 1153 y(\(i.e.)73 b(with)43 b Fs(K)52 b Fu(=)44 b(0\).)73 b(W)-8 b(e)43 b(consider)h(a)e(sp)s(ecial)h(sequence)i(\()p Fs(f)2851 1168 y Fm(n)2898 1153 y Fs(;)17 b(z)2987 1168 y Fm(n)3034 1153 y Fu(\))3072 1168 y Fm(n)3162 1153 y Fu(for)41 b Fs(H)3401 1168 y Fm(\034)456 1270 y Fu(asso)s(ciated)g(\()p Ft(I)7 b Fs(;)17 b(s;)g(A)p Fu(\))40 b(with)h Fs(s)f Ft(2)p Fu(]1)p Fs(=)p Fu(2;)17 b(2)p Fs(=)p Fu(3[.)66 b(By)40 b(Prop)s(osition)h(2.12,)g(w)m(e)h(ma)m(y)456 1386 y(assume)h(that)f Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1284 1401 y Fm(n)1375 1386 y Fu(=)44 b Fs(f)1543 1401 y Fm(n)1590 1386 y Fu(,)h(for)d(all)g Fs(n)p Fu(.)73 b(Let)42 b(us)h(\014x)g(some)g(notation.)72 b(Let)456 1483 y Fs(\037)544 1502 y Ft(2)28 b(C)696 1466 y Fl(1)690 1527 y Fm(c)771 1502 y Fu(\()p Fn(R)p Fu(\))33 b(suc)m(h)h(that)977 1640 y Fs(\037)1066 1659 y Fu(=)28 b(1)k(on)g([)p Ft(\000)p Fu(1)p Fs(;)17 b Fu(1])33 b(and)1882 1640 y Fs(\037)1970 1659 y Fu(=)28 b(0)k(on)h Fn(R)22 b Ft(n)g Fu([)p Ft(\000)p Fu(2)p Fs(;)17 b Fu(2])p Fs(:)-2301 b Fu(\(3.9\))456 1816 y(W)-8 b(e)30 b(shall)h(require)g(other)g(prop)s(erties)g (satis\014ed)h(b)m(y)2387 1797 y Fs(\037)2478 1816 y Fu(\(see)f(\(3.14\))f(b)s(elo)m(w\).)43 b(F)-8 b(or)456 1934 y Fs(R)38 b(>)f Fu(1,)i(w)m(e)h(set)1103 1915 y Fs(\037)1165 1949 y Fm(R)1222 1934 y Fu(\()p Fs(x)p Fu(\))e(=)1504 1915 y Fs(\037)1565 1934 y Fu(\()p Fs(x=R)q Fu(\))g(for)g(all)g Fs(x)g Ft(2)f Fn(R)h Fu(and)2662 1915 y(~)-55 b Fs(\037)2717 1949 y Fm(R)2812 1934 y Fu(=)37 b(1)25 b Ft(\000)3103 1915 y Fs(\037)3164 1949 y Fm(R)3222 1934 y Fu(.)60 b(W)-8 b(e)456 2051 y(denote)26 b(b)m(y)h Fs(O)967 2066 y Fm(R)1024 2051 y Fu(\()p Ft(\001)p Fu(\))e(\(resp.)42 b Fs(o)1480 2066 y Fm(R)1538 2051 y Fu(\()p Ft(\001)p Fu(\)\))25 b(the)h(Landau)g(sym)m(b)s(ol)h Fs(O)h Fu(\(resp.)42 b Fs(o)p Fu(\))26 b(where)h(the)456 2167 y(subscript)39 b Fs(R)h Fu(means)f(that)e(the)i(b)s(ound)f(\(resp.)61 b(the)38 b(limit\))h(is)f(uniform)h(w.r.t.)456 2283 y(the)33 b(other)f(v)-5 b(ariables.)456 2516 y(3.1.)48 b Fy(Pro)s(of)54 b(of)g(Theorem)h(2.7.)49 b Fu(Let)2030 2496 y Fs(\037)2143 2516 y Ft(2)k(C)2320 2479 y Fl(1)2314 2540 y Fm(c)2395 2516 y Fu(\()p Fn(R)p Fu(\))46 b(satisfying)k(\(3.9\))c(and)456 2632 y(\(3.14\))o(.)h(F)-8 b(rom)33 b(Prop)s(osition)h(3.6)f(and)h (Corollary)f(3.2)h(b)s(elo)m(w,)g(w)m(e)h(deriv)m(e)g(that,)456 2748 y(for)d(all)g Fs(")27 b(>)h Fu(0,)645 2905 y Fs(\021)87 b Fu(=)108 b(lim)940 2965 y Fm(n)p Fl(!1)1141 2905 y Ft(kh)p Fs(A)p Ft(i)1342 2864 y Fl(\000)p Fm(s)1433 2905 y Fs(f)1481 2920 y Fm(n)1528 2905 y Ft(k)780 3097 y(\024)83 b Fu(lim)17 b(sup)998 3177 y Fm(n)p Fl(!1)1256 3097 y Fu(\()p Ft(k)1350 3078 y Fu(~)-55 b Fs(\037)1404 3112 y Fm(R)1462 3097 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1762 3056 y Fl(\000)p Fm(s)1853 3097 y Fs(f)1901 3112 y Fm(n)1949 3097 y Ft(k)21 b Fu(+)h Ft(k)2168 3078 y Fs(\037)2229 3112 y Fm(R)2287 3097 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2587 3056 y Fl(\000)p Fm(s)2678 3097 y Fs(f)2726 3112 y Fm(n)2773 3097 y Ft(k)p Fu(\))780 3309 y Ft(\024)83 b Fu(lim)17 b(sup)998 3389 y Fm(n)p Fl(!1)1256 3309 y Fu(\()p Ft(k)1350 3290 y Fu(~)-55 b Fs(\037)1404 3324 y Fm(R)1462 3309 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1762 3268 y Fl(\000)p Fm(s)1853 3309 y Fs(f)1901 3324 y Fm(n)1949 3309 y Ft(k)21 b Fu(+)h Ft(k)2168 3290 y Fs(\037)2229 3324 y Fm(R)2287 3309 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2484 3324 y Fm(n)2531 3309 y Ft(k)p Fu(\))28 b(=)f Fs(O)s Fu(\()p Fs(R)2941 3268 y Fk(2)p Fm(s)p Fl(\000)p Fk(2+)p Fm(")3190 3309 y Fu(\))p Fs(:)456 3517 y Fu(Letting)38 b Fs(R)h Fu(go)f(to)g(in\014nit)m(y)-8 b(,)41 b(w)m(e)f(obtain)e(that)g Fs(\021)j Fu(=)c(0.)61 b(By)38 b(Prop)s(osition)h(2.11,)456 3633 y(the)33 b(LAP)g(holds)g (true)g(for)f Fs(H)1548 3648 y Fm(\034)1623 3633 y Fu(resp)s(ectiv)m (ely)k(to)c(\()p Ft(I)2373 3597 y Fl(0)2397 3633 y Fs(;)17 b(s;)g(A)p Fu(\).)698 b Fj(\003)456 3865 y Fu(3.2.)48 b Fy(A)40 b(\\large)h Ft(j)p Fs(A)p Ft(j)p Fy(")f(estimate.)50 b Fu(W)-8 b(e)36 b(stress)h(that,)f(in)f(this)h(subsection,)i(w)m(e)456 3982 y(supp)s(ose)33 b(that)g(1)p Fs(=)p Fu(2)27 b Fs(<)g(s)h(<)g Fu(1.)43 b(The)33 b(aim)g(of)f(this)h(part)g(is)g(to)f(sho)m(w)456 4150 y Fy(Prop)s(osition)50 b(3.1.)e Fp(L)-5 b(et)45 b Ft(I)1535 4114 y Fl(0)1604 4150 y Fp(b)-5 b(e)45 b(close)-5 b(d)45 b(with)g Ft(I)2312 4114 y Fl(0)2382 4150 y Ft(\032)2504 4125 y Fu(\027)2507 4150 y Ft(I)53 b Fp(and)44 b(let)i Fu(\()p Fs(f)3047 4165 y Fm(n)3094 4150 y Fs(;)17 b(z)3183 4165 y Fm(n)3230 4150 y Fu(\))3268 4165 y Fm(n)3360 4150 y Fp(b)-5 b(e)456 4266 y(a)41 b(sp)-5 b(e)g(cial)41 b(se)-5 b(quenc)g(e)40 b(for)i Fs(H)1509 4281 y Fm(\034)1594 4266 y Fp(r)-5 b(esp)g(e)g(ctively)41 b(to)g Fu(\()p Ft(I)2332 4230 y Fl(0)2356 4266 y Fs(;)17 b(s;)g(A)p Fu(\))41 b Fp(with)h Fu(1)e Fs(>)g(s)g(>)g Fu(1)p Fs(=)p Fu(2)p Fp(.)456 4382 y(Assume)j(that)i(the)f(Mourr)-5 b(e)44 b(estimate)51 b Fu(\(2.5\))43 b Fp(holds)g(true)i(with)f Fs(\022)j Fu(=)d(1)g Fp(on)g Ft(I)3421 4346 y Fl(0)456 4499 y Fp(and)34 b Fs(K)h Fu(=)27 b(0)p Fp(.)45 b(L)-5 b(et)1158 4479 y Fs(\037)1246 4499 y Ft(2)29 b(C)1399 4463 y Fl(1)1393 4523 y Fm(c)1474 4499 y Fu(\()p Fn(R)p Fu(\))34 b Fp(satisfying)43 b Fu(\(3.9\))34 b Fp(and)45 b Fu(\(3.14\))34 b Fp(\(b)-5 b(elow\).)43 b(Then,)456 4615 y(ther)-5 b(e)35 b(exist)h Fs(c)969 4579 y Fl(0)1022 4615 y Fs(>)29 b Fu(0)p Fp(,)35 b Fs(R)1315 4630 y Fk(1)1384 4615 y Fs(>)30 b Fu(2)p Fp(,)35 b(and)g(a)h(family)f Fu(\()p Fs(\036)2273 4630 y Fm(R)2331 4615 y Fu(\))2369 4630 y Fm(R>)p Fk(1)2552 4615 y Fp(in)h Fs(L)2739 4579 y Fl(1)2814 4615 y Fu(\()p Fn(R)p Fu(\))p Fp(,)g(such)f(that,)456 4731 y(for)f(al)5 b(l)35 b Fs(R)28 b Ft(\025)h Fs(R)1033 4746 y Fk(1)1072 4731 y Fp(,)871 4888 y Ft(h)p Fs(f)958 4903 y Fm(n)1005 4888 y Fs(;)17 b Fu([)p Fs(H)1157 4903 y Fm(\034)1200 4888 y Fs(;)g(i\036)1335 4903 y Fm(R)1392 4888 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1616 4903 y Fm(n)1663 4888 y Ft(i)83 b(\025)g Fs(c)1987 4847 y Fl(0)2011 4888 y Ft(k)2067 4869 y Fu(~)-55 b Fs(\037)2122 4903 y Fm(R)2179 4888 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2479 4847 y Fl(\000)p Fm(s)2571 4888 y Fs(f)2619 4903 y Fm(n)2666 4888 y Ft(k)2716 4847 y Fk(2)1549 5036 y Fu(+)84 b Fs(O)1784 5051 y Fm(R)1841 5036 y Fu(\()p Fs(R)1954 4995 y Fl(\000)p Fk(1)2049 5036 y Fu(\))22 b Ft(\001)g(k)2215 5016 y Fu(~)-55 b Fs(\037)2270 5051 y Fm(R)2327 5036 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2627 4995 y Fl(\000)p Fm(s)2719 5036 y Fs(f)2767 5051 y Fm(n)2814 5036 y Ft(k)-2408 b Fu(\(3.10\))1549 5183 y(+)84 b Fs(O)1784 5198 y Fm(R)1841 5183 y Fu(\()p Fs(R)1954 5142 y Fk(2)p Fm(s)p Fl(\000)p Fk(2)2117 5183 y Fu(\))22 b Ft(\001)g(k)2283 5164 y Fu(~)-55 b Fs(\037)2337 5199 y Fm(R=)p Fk(2)2466 5183 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2766 5142 y Fl(\000)p Fm(s)2857 5183 y Fs(f)2905 5198 y Fm(n)2952 5183 y Ft(k)p Fs(:)p eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 456 255 a Fq(12)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fy(Corollary)39 b(3.2.)j Fp(Under)35 b(the)g(hyp)-5 b(otheses)34 b(of)g(Pr)-5 b(op)g(osition)34 b(3.1,)h(for)f(any)h Fs(\013)29 b(>)456 566 y Fu(2)p Fs(s)21 b Ft(\000)i Fu(2)p Fp(,)1133 771 y Fu(lim)17 b(sup)1191 851 y Fm(n)p Fl(!1)1460 771 y Ft(k)1516 752 y Fu(~)-55 b Fs(\037)1571 786 y Fm(R)1629 771 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1929 730 y Fl(\000)p Fm(s)2020 771 y Fs(f)2068 786 y Fm(n)2115 771 y Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)2487 730 y Fm(\013)2536 771 y Fu(\))p Fs(:)-2145 b Fu(\(3.11\))456 1092 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Note)32 b(\014rst)i(that,)e(for)g Fs(a)c(>)f Fu(0,)33 b Fs(")27 b Ft(\025)h Fu(0,)33 b(and)f Fs(b;)17 b(c)28 b Ft(2)g Fn(R)p Fu(,)905 1296 y Fs(")f Ft(\025)i Fs(aX)1224 1255 y Fk(2)1285 1296 y Fu(+)22 b Fs(bX)31 b Fu(+)22 b Fs(c)1676 1255 y Fk(2)1770 1296 y Fu(=)-17 b Ft(\))56 b(j)p Fs(X)8 b Ft(j)26 b(\024)2262 1206 y Ff(p)p 2361 1206 146 4 v 2361 1296 a Fs("=a)c Fu(+)g Fs(O)s Fu(\()p Ft(j)p Fs(b)p Ft(j)f Fu(+)h Ft(j)p Fs(c)p Ft(j)p Fu(\))p Fs(;)-2666 b Fu(\(3.12\))456 1491 y(the)33 b(later)g(term)h(b)s(eing)f(indep)s(enden)m(t)j(of)c Fs(")p Fu(.)45 b(Since)34 b(\()p Ft(kh)p Fs(A)p Ft(i)2626 1455 y Fl(\000)p Fm(s)2717 1491 y Fs(f)2765 1506 y Fm(n)2812 1491 y Ft(k)p Fu(\))2900 1506 y Fm(n)2980 1491 y Fu(is)g(b)s(ounded)456 1607 y(b)m(y)44 b(De\014nition)h(2.10,)h(it)e(su\016ces)i(to)e(pro)m(v) m(e)h(\(3.11\))e(for)g(large)h Fs(R)q Fu(.)78 b(F)-8 b(or)43 b(\014xed)456 1723 y Fs(R)28 b Ft(\025)h Fs(R)738 1738 y Fk(1)777 1723 y Fu(,)k(w)m(e)g(com)m(bine)h(\(3.10\))e(with)i (\(3.12\))d(and)i(Prop)s(osition)g(2.15)f(to)g(get)1159 1920 y(lim)17 b(sup)1217 2000 y Fm(n)p Fl(!1)1475 1920 y Ft(k)1531 1901 y Fu(~)-55 b Fs(\037)1586 1935 y Fm(R)1643 1920 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1943 1879 y Fl(\000)p Fm(s)2035 1920 y Fs(f)2083 1935 y Fm(n)2130 1920 y Ft(k)27 b(\024)h Fs(O)s Fu(\()p Fs(R)2503 1879 y Fl(\000)p Fk(1)2597 1920 y Fu(\))1159 2138 y(+)17 b Fs(O)s Fu(\()p Fs(R)1443 2097 y Fm(s)p Fl(\000)p Fk(1)1569 2138 y Fu(\))22 b Ft(\001)g Fu(lim)17 b(sup)1736 2217 y Fm(n)p Fl(!1)1994 2138 y Ft(k)2050 2119 y Fu(~)-55 b Fs(\037)2105 2153 y Fm(R=)p Fk(2)2234 2138 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2534 2097 y Fl(\000)p Fm(s)2625 2138 y Fs(f)2673 2153 y Fm(n)2720 2138 y Ft(k)2770 2097 y Fk(1)p Fm(=)p Fk(2)2880 2138 y Fs(:)-2451 b Fu(\(3.13\))456 2390 y(W)-8 b(e)33 b(use)i(a)e(b)s(o)s(otstrapping)g(argumen)m(t.)47 b(Since)34 b(\()p Ft(kh)p Fs(A)p Ft(i)2477 2354 y Fl(\000)p Fm(s)2569 2390 y Fs(f)2617 2405 y Fm(n)2664 2390 y Ft(k)p Fu(\))2752 2405 y Fm(n)2832 2390 y Fu(is)g(b)s(ounded,)g(so)456 2509 y(is)h(\()p Ft(k)650 2489 y Fu(~)-55 b Fs(\037)705 2524 y Fm(R=)p Fk(2)833 2509 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1133 2472 y Fl(\000)p Fm(s)1225 2509 y Fs(f)1273 2524 y Fm(n)1320 2509 y Ft(k)p Fu(\))1408 2524 y Fm(n)1455 2509 y Fu(.)51 b(Then)36 b(\(3.13\))f(giv)m(es)h (\(3.11\))f(for)g Fs(\013)d Fu(=)g Fs(\013)3017 2524 y Fk(0)3089 2509 y Fu(=)g Fs(s)24 b Ft(\000)h Fu(1.)456 2625 y(No)m(w)42 b(w)m(e)i(use)f(this)g(new)g(estimate)h(in)e(\(3.13\)) g(to)f(get)i(\(3.11\))e(for)h Fs(\013)j Fu(=)f Fs(\013)3285 2640 y Fk(1)3369 2625 y Fu(=)456 2741 y(3\()p Fs(s)18 b Ft(\000)h Fu(1\))p Fs(=)p Fu(2.)42 b(By)31 b(induction,)i(w)m(e)e (get)g(\(3.11\))f(for)h(a)f(sequence)k(\()p Fs(\013)2880 2756 y Fm(n)2927 2741 y Fu(\))2965 2756 y Fm(n)3042 2741 y Fu(satisfying)456 2857 y Fs(\013)518 2872 y Fm(n)p Fk(+1)701 2857 y Fu(=)46 b Fs(\013)885 2872 y Fm(n)932 2857 y Fs(=)p Fu(2)30 b(+)f(\()p Fs(s)h Ft(\000)g Fu(1\),)46 b(for)d(all)g Fs(n)k Ft(2)f Fn(N)p Fu(.)77 b(By)44 b(a)f(\014xed)i(p)s (oin)m(t)f(argumen)m(t,)456 2974 y Fs(\013)518 2989 y Fm(n)592 2974 y Ft(!)28 b Fu(2\()p Fs(s)21 b Ft(\000)i Fu(1\).)43 b(This)34 b(yields)g(the)f(result.)1306 b Fj(\003)456 3148 y Fu(Our)37 b(strategy)g(to)g(pro)m(v)m(e)i(Prop)s (osition)e(3.1)g(is)g(the)h(follo)m(wing.)58 b(W)-8 b(e)37 b(apply)h(the)456 3264 y(strict)31 b(Mourre)g(estimate)g(\(2.5\))f (\(with)h Fs(K)k Fu(=)28 b(0\))i(to)2390 3245 y(~)-55 b Fs(\037)2445 3279 y Fm(R)2503 3264 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2803 3228 y Fl(\000)p Fm(s)2894 3264 y Fs(f)2942 3279 y Fm(n)2989 3264 y Fu(.)43 b(W)-8 b(e)31 b(mo)m(v)m(e)456 3380 y(the)k Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))35 b(to)g(the)h Fs(f)1215 3395 y Fm(n)1262 3380 y Fu(,)g(whic)m(h)h(absorb)e(them,)i(since)g Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)2707 3395 y Fm(n)2786 3380 y Fu(=)32 b Fs(f)2942 3395 y Fm(n)2989 3380 y Fu(.)52 b(W)-8 b(e)35 b(w)m(an)m(t)456 3499 y(to)e(pull)h(the)g(w)m(eigh)m(ts) 1295 3480 y(~)-55 b Fs(\037)1350 3514 y Fm(R)1408 3499 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1708 3463 y Fl(\000)p Fm(s)1833 3499 y Fu(in)m(to)34 b(the)g(comm)m(utator)g([)p Fs(H)2860 3514 y Fm(\034)2903 3499 y Fs(;)17 b(A)p Fu(],)33 b(in)h(order)456 3618 y(to)39 b(get)g(the)h(term)g(on)f(the)h(l.h.s)g (of)46 b(\(3.10\))39 b(with)h Fs(\036)2397 3633 y Fm(R)2454 3618 y Fu(\()p Fs(t)p Fu(\))g(=)f Fs(t)p Ft(h)p Fs(t)p Ft(i)2868 3581 y Fl(\000)p Fk(2)p Fm(s)3001 3598 y Fu(~)-55 b Fs(\037)3056 3633 y Fm(R)3114 3618 y Fu(\()p Fs(t)p Fu(\))3225 3581 y Fk(2)3264 3618 y Fu(.)64 b(In)456 3734 y(view)35 b(of)e(the)i(pro)s(of)e(of)g(Corollary)h(3.2,)g(w)m(e)h(need) g(2)p Fs(s)29 b Ft(\025)i Fu(1.)47 b(Our)34 b(manipulation)456 3850 y(pro)s(duces)h(of)e(course)i(error)f(terms)h(whic)m(h)g(should)g (b)s(e)f(small.)48 b(Using)34 b Fs(s)c(>)g Fu(1)p Fs(=)p Fu(2,)456 3966 y(w)m(e)46 b(actually)g(pro)m(v)m(e)h(this)f(smallness)h (if)f(w)m(e)g(only)g(mo)m(v)m(e)h(the)f Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))44 b(and)i(the)456 4085 y Ft(h)p Fs(A)p Ft(i)607 4049 y Fl(\000)p Fm(s)698 4085 y Fu(.)75 b(Cho)s(osing)43 b(appropriate)h(functions)g Fs(\036)2262 4100 y Fm(R)2319 4085 y Fu(,)i(w)m(e)e(can)f(mo)m(v)m(e)i(the)3183 4066 y(~)-55 b Fs(\037)3238 4100 y Fm(R)3295 4085 y Fu(\()p Fs(A)p Fu(\))456 4201 y(in)m(to)28 b(the)h(comm)m(utator)g(pro)s (ducing)f(an)g(error)h(term)f(whic)m(h)i(has)f(the)g(go)s(o)s(d)e (sign,)456 4317 y(up)36 b(to)g(small)i(enough)e(terms.)56 b(T)-8 b(o)37 b(this)g(end,)h(w)m(e)f(c)m(ho)s(ose)h(more)e(carefully)i (the)456 4434 y(function)838 4414 y Fs(\037)931 4434 y Fu(in)33 b(\(3.9\).)43 b(W)-8 b(e)33 b(demand)h(that)2064 4414 y Fs(\037)2158 4434 y Fu(satis\014es)g(\(3.9\))e(and)h(that)1388 4611 y(~)-55 b Fs(\037)1471 4631 y Fu(:=)27 b(1)22 b Ft(\000)1772 4611 y Fs(\037)1861 4631 y Fu(=)1971 4611 y(~)-56 b Fs(\037)2025 4646 y Fk(+)2107 4631 y Fu(+)2211 4611 y(~)h Fs(\037)2266 4646 y Fl(\000)2325 4631 y Fs(;)-1896 b Fu(\(3.14\))456 4838 y(where)752 4819 y(~)-55 b Fs(\037)807 4838 y Fu(1)856 4853 y Fg(R)904 4834 y Fe(\033)992 4838 y Fu(=)1117 4819 y(~)f Fs(\037)1172 4853 y Fm(\033)1260 4838 y Fu(for)40 b Fs(\033)47 b Ft(2)42 b(f\000)p Fs(;)17 b Fu(+)p Ft(g)p Fu(,)44 b(suc)m(h)e(that)2449 4819 y(~)-55 b Fs(\037)2504 4853 y Fm(\033)2592 4838 y Fu(and)42 b Fs(\033)2856 4819 y Fu(~)-55 b Fs(\037)2911 4779 y Fl(0)2911 4863 y Fm(\033)2999 4838 y Fu(are)41 b(square)456 4955 y(of)e(some)j(smo)s(oth)e(functions)h(\(see)g(for)f(instance)i(the)f (app)s(endix)g(in)f([DG])g(for)456 5073 y(their)k(existence\).)80 b(Let)44 b Fs(R)49 b(>)e Fu(1.)77 b(W)-8 b(e)44 b(set)2159 5054 y(~)-55 b Fs(\037)2214 5088 y Fm(\033)n(;R)2377 5073 y Fu(=)2507 5054 y(~)f Fs(\037)2561 5088 y Fm(\033)2608 5073 y Fu(\()p Fs(t=R)q Fu(\).)78 b(Notice)45 b(that)462 5192 y(~)-55 b Fs(\037)517 5226 y Fm(R)602 5211 y Fu(=)712 5192 y(~)g Fs(\037)767 5226 y Fk(+)p Fm(;R)921 5211 y Fu(+)1025 5192 y(~)g Fs(\037)1080 5226 y Fl(\000)p Fm(;R)1245 5211 y Fu(and)1441 5192 y(~)g Fs(\037)1496 5152 y Fk(2)1496 5236 y Fm(R)1581 5211 y Fu(=)1691 5192 y(~)g Fs(\037)1746 5152 y Fk(2)1746 5236 y(+)p Fm(;R)1900 5211 y Fu(+)2004 5192 y(~)g Fs(\037)2059 5152 y Fk(2)2059 5236 y Fl(\000)p Fm(;R)2191 5211 y Fu(.)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(13)456 450 y Fp(Pr)-5 b(o)g(of)34 b(of)h(Pr)-5 b(op)g(osition)34 b(3.1.)41 b Fu(Let)1541 611 y Fs(h)83 b(>)g Fu(sup)1852 692 y Fm(t)p Fl(2)p Fg(R)2002 611 y Ft(j)p Fs(t)p Ft(jh)p Fs(t)p Ft(i)2206 570 y Fl(\000)p Fk(2)p Fm(s)2332 611 y Fs(:)-1903 b Fu(\(3.15\))456 834 y(Let)32 b Fs(R)d(>)f Fu(1)k(and)h Fs(\036)1166 849 y Fm(R)1251 834 y Ft(2)28 b(C)1403 798 y Fl(1)1478 834 y Fu(\()p Fn(R)p Fu(\))k(de\014ned)i(b)m(y)1103 1016 y Fs(\036)1161 1031 y Fm(R)1219 1016 y Fu(\()p Fs(t)p Fu(\))83 b(=)1645 921 y Ff(X)1572 1137 y Fm(\033)r Fl(2f\000)p Fm(;)p Fk(+)p Fl(g)1884 997 y Fu(~)-55 b Fs(\037)1939 957 y Fk(2)1939 1041 y Fm(\033)n(;R)2055 1016 y Fu(\()p Fs(t)p Fu(\))2166 935 y Ff(\000)2212 1016 y Fs(\033)t(h)22 b Ft(\000)h Fs(t)p Ft(h)p Fs(t)p Ft(i)2597 975 y Fl(\000)p Fk(2)p Fm(s)2724 935 y Ff(\001)2770 1016 y Fs(:)-2341 b Fu(\(3.16\))456 1286 y(F)-8 b(or)31 b(all)i Fs(f)38 b Ft(2)28 b Fo(H)f Fu(,)475 1446 y Ft(h)p Fs(f)5 b(;)17 b Fu([)p Fs(H)719 1461 y Fm(\034)762 1446 y Fs(;)g(i\036)897 1461 y Fm(R)954 1446 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)11 b Ft(i)83 b Fu(=)g Ft(h)p Fs(f)5 b(;)1612 1427 y Fu(~)-55 b Fs(\037)1667 1461 y Fm(R)1724 1446 y Fu(\()p Fs(A)p Fu(\)[)p Fs(H)1981 1461 y Fm(\034)2025 1446 y Fs(;)17 b Ft(\000)p Fs(iA)p Ft(h)p Fs(A)p Ft(i)2403 1405 y Fl(\000)p Fk(2)p Fm(s)2530 1446 y Fu(])2563 1427 y(~)-55 b Fs(\037)2618 1461 y Fm(R)2676 1446 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)997 1613 y Fu(+)50 b(2)1245 1519 y Ff(X)1172 1734 y Fm(\033)r Fl(2f\000)p Fm(;)p Fk(+)p Fl(g)1478 1613 y Fu(Re)p Ft(h)p Fs(f)5 b(;)17 b Fu(\()p Fs(\033)t(h)22 b Ft(\000)h Fs(A)p Ft(h)p Fs(A)p Ft(i)2228 1572 y Fl(\000)p Fk(2)p Fm(s)2355 1613 y Fu(\))2399 1594 y(~)-55 b Fs(\037)2454 1628 y Fm(\033)n(;R)2570 1613 y Fu(\()p Fs(A)p Fu(\)[)p Fs(H)2827 1628 y Fm(\034)2870 1613 y Fs(;)17 b(i)2953 1594 y Fu(~)-55 b Fs(\037)3008 1628 y Fm(\033)n(;R)3124 1613 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)11 b Ft(i)p Fs(:)-2969 b Fu(\(3.17\))456 1888 y(W)-8 b(e)24 b(can)h(\014nd)g(some)h Fs(R)1284 1903 y Fk(1)1351 1888 y Fs(>)i Fu(2)c(\(see)h(Lemma)g(3.4)g (b)s(elo)m(w\))g(suc)m(h)h(that,)g(for)e Fs(R)k Ft(\025)h Fs(R)3378 1903 y Fk(1)3417 1888 y Fu(,)631 2051 y Ft(h)676 2032 y Fu(~)-55 b Fs(\037)731 2066 y Fm(R)788 2051 y Fu(\()p Fs(A)p Fu(\))p Fs(f)985 2066 y Fm(n)1032 2051 y Fs(;)17 b Fu([)p Fs(H)1184 2066 y Fm(\034)1227 2051 y Fs(;)g Ft(\000)p Fs(iA)p Ft(h)p Fs(A)p Ft(i)1605 2010 y Fl(\000)p Fk(2)p Fm(s)1732 2051 y Fu(])1765 2032 y(~)-55 b Fs(\037)1820 2066 y Fm(R)1878 2051 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2075 2066 y Fm(n)2122 2051 y Ft(i)28 b(\025)g Fs(c)2336 2010 y Fl(0)2359 2051 y Ft(k)2415 2032 y Fu(~)-55 b Fs(\037)2470 2066 y Fm(R)2528 2051 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2828 2010 y Fl(\000)p Fm(s)2919 2051 y Fs(f)2967 2066 y Fm(n)3014 2051 y Ft(k)3064 2010 y Fk(2)1852 2199 y Fu(+)p Fs(O)2003 2214 y Fm(R)2060 2199 y Fu(\()p Fs(R)2173 2158 y Fl(\000)p Fk(1)2267 2199 y Fu(\))23 b Ft(\001)e(k)2433 2180 y Fu(~)-55 b Fs(\037)2488 2214 y Fm(R)2546 2199 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2846 2158 y Fl(\000)p Fm(s)2937 2199 y Fs(f)2985 2214 y Fm(n)3032 2199 y Ft(k)-2626 b Fu(\(3.18\))456 2360 y(with)28 b Fs(c)715 2323 y Fl(0)766 2360 y Fu(=)f(\(2)p Fs(s)12 b Ft(\000)g Fu(1\))1190 2323 y Fl(\000)p Fk(1)1284 2360 y Fs(c=)p Fu(2)27 b Fs(>)h Fu(0.)41 b(Since)29 b(w)m(e)f(are)g (not)f(able)h(to)f(estimate)i(prop)s(erly)456 2476 y(the)e(second)h (term)g(on)e(the)i(r.h.s)g(of)33 b(\(3.17\))o(,)28 b(w)m(e)g(indend)h (to)d(use)i(some)g(p)s(ositivit)m(y)456 2592 y(argumen)m(t)40 b(to)e(get)i(rid)f(of)g(it.)64 b(In)39 b(view)i(of)k(\(B.4\),)c(w)m(e)g (c)m(ho)s(osed)f Fs(\036)2978 2607 y Fm(R)3075 2592 y Fu(in)f(\(3.16\))456 2708 y(suc)m(h)34 b(that,)e(the)h(function)g Fs( )1527 2723 y Fm(R)1613 2708 y Ft(2)28 b(C)1765 2672 y Fl(1)1759 2733 y Fm(c)1840 2708 y Fu(\()p Fn(R)p Fu(\))k(de\014ned)i (b)m(y)954 2888 y Fs( )1017 2903 y Fm(R)1075 2888 y Fu(\()p Fs(t)p Fu(\))83 b(=)g Fs(R)1503 2807 y Ff(\000)1549 2888 y Fs(\036)1607 2847 y Fl(0)1607 2912 y Fm(R)1664 2888 y Fu(\()p Fs(t)p Fu(\))22 b Ft(\000)1903 2868 y Fu(~)-55 b Fs(\037)1958 2829 y Fk(2)1958 2912 y Fm(R)2016 2888 y Fu(\()p Fs(t)p Fu(\)\()p Fs(d=dt)p Fu(\)\()p Ft(\000)p Fs(t)p Ft(h)p Fs(t)p Ft(i)2652 2847 y Fl(\000)p Fk(2)p Fm(s)2778 2888 y Fu(\))2816 2807 y Ff(\001)1269 3065 y Fu(=)1501 2970 y Ff(X)1428 3186 y Fm(\033)r Fl(2f\000)p Fm(;)p Fk(+)p Fl(g)1718 3065 y Fu(\()1762 3045 y(~)g Fs(\037)1817 3005 y Fl(0)1817 3089 y Fm(\033)1863 3065 y Fu(\))1901 3080 y Fm(R)1959 3065 y Fu(\()p Fs(t)p Fu(\)\()p Fs(\033)t(h)22 b Ft(\000)h Fs(t)p Ft(h)p Fs(t)p Ft(i)2493 3023 y Fl(\000)p Fk(2)p Fm(s)2620 3065 y Fu(\))2664 3045 y(~)-55 b Fs(\037)2719 3080 y Fm(\033)n(;R)2835 3065 y Fu(\()p Fs(t)p Fu(\))-2490 b(\(3.19\))456 3334 y(is)46 b(the)f(square)i(of)e(a)g(smo)s(oth)g(function.)83 b(W)-8 b(e)46 b(put)f(a)g(factor)g Fs(R)i Fu(in)e(fron)m(t)h(to)456 3450 y(ensure)d(that)g(the)f(family)h(\()p Fs( )1577 3465 y Fm(R)1635 3450 y Fu(\))1673 3465 y Fm(R)1773 3450 y Fu(is)g(b)s(ounded)g(in)g(some)g(sym)m(b)s(ol)h(space)f(\(see)456 3567 y(Lemma)33 b(A.4\).)43 b(Notice)33 b(that)g(supp)q Fs( )1843 3582 y Fm(R)1929 3567 y Ft(\032)28 b Fu([)p Ft(\000)p Fu(2)p Fs(R)q(;)17 b Ft(\000)p Fs(R)q Fu(])23 b Ft([)f Fu([)p Fs(R)q(;)17 b Fu(2)p Fs(R)q Fu(].)43 b(W)-8 b(e)33 b(de\014ne)845 3745 y Fs(C)915 3760 y Fm(R)1000 3745 y Fu(:=)28 b Fs( )1198 3694 y Fk(1)p Fm(=)p Fk(2)1194 3772 y Fm(R)1308 3745 y Fu(\()p Fs(A)p Fu(\))47 b(and)33 b(note)g(that)47 b Fs(C)2207 3760 y Fm(R)2271 3726 y Fu(~)-56 b Fs(\037)2326 3760 y Fm(R=)p Fk(2)2454 3745 y Fu(\()p Fs(A)p Fu(\))27 b(=)h Fs(C)2804 3760 y Fm(R)2862 3745 y Fs(:)-2433 b Fu(\(3.20\))456 3906 y(W)-8 b(e)33 b(can)f(sho)m(w)i(\(see)g(Lemma)f(3.5)f(b)s(elo)m(w\))h(that)792 4068 y Ft(h)p Fs(C)901 4083 y Fm(R)958 4068 y Fs(f)1006 4083 y Fm(n)1053 4068 y Fs(;)17 b Fu([)p Fs(H)1205 4083 y Fm(\034)1248 4068 y Fs(;)g(iA)p Fu(])p Fs(C)1495 4083 y Fm(R)1553 4068 y Fs(f)1601 4083 y Fm(n)1648 4068 y Ft(i)82 b(\025)i Fs(O)2005 4083 y Fm(R)2062 4068 y Fu(\()p Fs(R)2175 4027 y Fk(2)p Fm(s)p Fl(\000)p Fk(1)2337 4068 y Fu(\))23 b Ft(\001)e(k)2503 4049 y Fu(~)-55 b Fs(\037)2558 4084 y Fm(R=)p Fk(2)2687 4068 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2987 4027 y Fl(\000)p Fm(s)3078 4068 y Fs(f)3126 4083 y Fm(n)3173 4068 y Ft(k)p Fs(:)-2794 b Fu(\(3.21\))456 4229 y(By)33 b(Lemma)g(C.4,)785 4390 y Ft(h)p Fs(C)901 4349 y Fk(2)894 4414 y Fm(R)951 4390 y Fs(f)999 4405 y Fm(n)1046 4390 y Fs(;)17 b Fu([)p Fs(H)1198 4405 y Fm(\034)1241 4390 y Fs(;)g(iA)p Fu(])p Fs(f)1466 4405 y Fm(n)1513 4390 y Ft(i)83 b(\025)g Fs(O)1870 4405 y Fm(R)1928 4390 y Fu(\()p Fs(R)2041 4349 y Fk(2)p Fm(s)p Fl(\000)p Fk(1)2203 4390 y Fu(\))22 b Ft(\001)g(k)2369 4370 y Fu(~)-55 b Fs(\037)2424 4405 y Fm(R=)p Fk(2)2552 4390 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2852 4349 y Fl(\000)p Fm(s)2943 4390 y Fs(f)2991 4405 y Fm(n)3038 4390 y Ft(k)p Fs(;)456 4560 y Fu(since)34 b Ft(k)751 4541 y Fu(~)-55 b Fs(\037)806 4575 y Fm(R=)p Fk(2)934 4560 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1234 4524 y Fl(\000)p Fm(s)1325 4560 y Fs(f)1373 4575 y Fm(n)1420 4560 y Ft(k)28 b Fu(=)f Fs(O)1676 4575 y Fm(R)1733 4560 y Fu(\()p Fs(R)1846 4524 y Fk(0)1886 4560 y Fu(\).)43 b(No)m(w,)34 b(b)m(y)f(Lemma)g(C.2,)834 4730 y Ft(h)p Fs(f)921 4745 y Fm(n)968 4730 y Fs(;)17 b Fu(\()p Fs(\033)t(h)22 b Ft(\000)h Fs(A)p Ft(h)p Fs(A)p Ft(i)1511 4689 y Fl(\000)p Fk(2)p Fm(s)1637 4730 y Fu(\))1681 4711 y(~)-55 b Fs(\037)1736 4745 y Fm(\033)n(;R)1853 4730 y Fu(\()p Fs(A)p Fu(\)[)p Fs(H)2110 4745 y Fm(\034)2153 4730 y Fs(;)2202 4711 y Fu(~)h Fs(\037)2257 4745 y Fm(\033)n(;R)2374 4730 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)2598 4745 y Fm(n)2645 4730 y Ft(i)82 b(\025)1586 4881 y Fu(+)i Fs(O)1821 4896 y Fm(R)1878 4881 y Fu(\()p Fs(R)1991 4840 y Fk(2)p Fm(s)p Fl(\000)p Fk(2)2153 4881 y Fu(\))22 b Ft(\001)g(k)2319 4861 y Fu(~)-55 b Fs(\037)2374 4896 y Fm(R=)p Fk(2)2502 4881 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2802 4840 y Fl(\000)p Fm(s)2894 4881 y Fs(f)2942 4896 y Fm(n)2989 4881 y Ft(k)p Fs(:)456 5041 y Fu(This)33 b(yields,)i(together)d(with)i(\(3.17\))d (and)i(\(3.18\))o(,)g(the)g(result.)567 b Fj(\003)456 5216 y Fu(T)-8 b(o)32 b(pro)m(v)m(e)i(\(3.18\))e(and)h(\(3.21\))o(,)f (w)m(e)i(need)g(the)f(follo)m(wing)g(lemmata.)p eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 456 255 a Fq(14)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fy(Lemma)39 b(3.3.)j Fp(Under)34 b(the)h(assumptions)f(of)h(Pr)-5 b(op)g(osition)34 b(3.1,)662 607 y Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)996 588 y(\037)1056 622 y Fm(R)1114 607 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1338 622 y Fm(n)1385 607 y Ft(k)28 b Fu(=)f Fs(O)1641 622 y Fm(R)1699 607 y Fu(\()p Fs(R)1812 566 y Fm(s)p Fl(\000)p Fk(1)1939 607 y Fu(\))p Fs(;)33 b Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b(C)2441 622 y Fm(R)2497 607 y Fu(])p Fs(f)2572 622 y Fm(n)2620 607 y Ft(k)27 b Fu(=)h Fs(O)2876 622 y Fm(R)2933 607 y Fu(\()p Fs(R)3046 566 y Fm(s)p Fl(\000)p Fk(1)3173 607 y Fu(\))p Fs(;)1174 768 y Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b Ft(h)p Fs(A)p Ft(i)1659 726 y Fl(\000)p Fm(s)1756 748 y Fu(~)-55 b Fs(\037)1811 783 y Fm(R)1869 768 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)2093 783 y Fm(n)2140 768 y Ft(k)27 b Fu(=)h Fs(O)2396 783 y Fm(R)2453 768 y Fu(\()p Fs(R)2566 726 y Fl(\000)p Fk(1)2661 768 y Fu(\))p Fs(:)456 1010 y Fp(Pr)-5 b(o)g(of.)41 b Fu(By)63 b(Corollary)f(A.3,)70 b(the)63 b(families)h(\()2271 991 y(~)-55 b Fs(\037)2326 1025 y Fm(R)2383 1010 y Fu(\))2421 1025 y Fm(R>)p Fk(1)2632 1010 y Fu(and)62 b(\()p Fs( )2956 959 y Fk(1)p Fm(=)p Fk(2)2952 1037 y Fm(R)3066 1010 y Fu(\))3104 1025 y Fm(R>)p Fk(1)3314 1010 y Fu(are)456 1131 y(b)s(ounded)36 b(in)g Ft(S)1041 1094 y Fk(0)1081 1131 y Fu(,)h(while)g(the)f(family)h(\()p Fs(\033)1969 1146 y Fm(R)2060 1131 y Fu(:)c Fs(t)h Ft(7!)e(h)p Fs(t)p Ft(i)2434 1094 y Fl(\000)p Fm(s)2532 1111 y Fu(~)-55 b Fs(\037)2587 1146 y Fm(R)2645 1131 y Fu(\()p Fs(t)p Fu(\)\))2794 1146 y Fm(R>)p Fk(1)2977 1131 y Fu(is)37 b(b)s(ounded)456 1249 y(in)e Ft(S)639 1213 y Fl(\000)p Fm(s)732 1249 y Fu(.)52 b(F)-8 b(urthermore,)37 b(])24 b Ft(\000)g Fs(R)q(;)17 b(R)q Fu([)36 b(do)s(es)g(not)f(in)m(tersect)i (the)f(supp)s(orts)g(of)3305 1230 y(~)-56 b Fs(\037)3360 1264 y Fm(R)3417 1249 y Fu(,)456 1383 y Fs( )523 1332 y Fk(1)p Fm(=)p Fk(2)519 1410 y Fm(R)633 1383 y Fu(,)32 b(and)h Fs(\033)937 1398 y Fm(R)995 1383 y Fu(.)44 b(By)33 b(Lemma)g(B.3,)924 1545 y Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1258 1526 y(\037)1318 1560 y Fm(R)1376 1545 y Fu(\()p Fs(A)p Fu(\)])p Ft(h)p Fs(A)p Ft(i)1703 1504 y Fm(s)1739 1545 y Ft(k)28 b Fu(=)f Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)2260 1526 y Fu(~)-55 b Fs(\037)2315 1560 y Fm(R)2372 1545 y Fu(\()p Fs(A)p Fu(\)])p Ft(h)p Fs(A)p Ft(i)2699 1504 y Fm(s)2736 1545 y Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)3108 1504 y Fm(s)p Fl(\000)p Fk(1)3234 1545 y Fu(\))p Fs(;)456 1693 y Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b(C)860 1708 y Fm(R)916 1693 y Fu(])p Ft(h)p Fs(A)p Ft(i)1094 1652 y Fm(s)1131 1693 y Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)1503 1652 y Fm(s)p Fl(\000)p Fk(1)1629 1693 y Fu(\))p Fs(;)33 b Ft(k)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b Ft(h)p Fs(A)p Ft(i)2212 1652 y Fl(\000)p Fm(s)2309 1673 y Fu(~)-55 b Fs(\037)2364 1708 y Fm(R)2422 1693 y Fu(\()p Fs(A)p Fu(\)])p Ft(h)p Fs(A)p Ft(i)2749 1652 y Fm(s)2785 1693 y Ft(k)28 b Fu(=)f Fs(O)s Fu(\()p Fs(R)3157 1652 y Fl(\000)p Fk(1)3251 1693 y Fu(\))p Fs(:)456 1850 y Fu(This,)72 b(together)64 b(with)g(the)g(b)s(oundness)i(of)d(\()p Ft(kh)p Fs(A)p Ft(i)2500 1814 y Fl(\000)p Fm(s)2591 1850 y Fs(f)2639 1865 y Fm(n)2686 1850 y Ft(k)p Fu(\))2774 1865 y Fm(n)2884 1850 y Fu(\(cf.)137 b(De\014ni-)456 1966 y(tion)32 b(2.10\),)g(yields)i(the)f(results.)1695 b Fj(\003)456 2141 y Fy(Lemma)39 b(3.4.)j Fp(The)34 b(ine)-5 b(quality)43 b Fu(\(3.18\))35 b Fp(holds)f(true.)456 2368 y(Pr)-5 b(o)g(of.)41 b Fu(Applying)35 b(\(2.5\))d(\(with)h Fs(K)i Fu(=)27 b(0\))33 b(to)f(the)2305 2349 y(~)-55 b Fs(\037)2360 2383 y Fm(R)2418 2368 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2718 2332 y Fl(\000)p Fm(s)2809 2368 y Fs(f)2857 2383 y Fm(n)2904 2368 y Fu(,)784 2528 y Ft(h)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))1042 2508 y(~)-55 b Fs(\037)1096 2543 y Fm(R)1154 2528 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1454 2486 y Fl(\000)p Fm(s)1545 2528 y Fs(f)1593 2543 y Fm(n)1641 2528 y Fs(;)17 b Fu([)p Fs(H)1793 2543 y Fm(\034)1835 2528 y Fs(;)g(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))2231 2508 y(~)-55 b Fs(\037)2286 2543 y Fm(R)2344 2528 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2644 2486 y Fl(\000)p Fm(s)2735 2528 y Fs(f)2783 2543 y Fm(n)2830 2528 y Ft(i)1850 2675 y(\025)163 b Fs(c)p Ft(k)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))2401 2656 y(~)-55 b Fs(\037)2456 2690 y Fm(R)2513 2675 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2813 2634 y Fl(\000)p Fm(s)2905 2675 y Fs(f)2953 2690 y Fm(n)3000 2675 y Ft(k)3050 2634 y Fk(2)3089 2675 y Fs(:)456 2833 y Fu(Recall)33 b(that)f Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)1222 2848 y Fm(n)1296 2833 y Fu(=)28 b Fs(f)1448 2848 y Fm(n)1495 2833 y Fu(.)43 b(By)33 b(Lemma)h(3.3,)671 2908 y Ff(8)671 2998 y(>)671 3028 y(>)671 3058 y(<)671 3237 y(>)671 3267 y(>)671 3297 y(:)801 2992 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1130 2973 y Fu(~)-55 b Fs(\037)1185 3007 y Fm(R)1242 2992 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1542 2956 y Fl(\000)p Fm(s)1634 2992 y Fu(])p Fs(f)1709 3007 y Fm(n)1756 2992 y Fs(;)17 b Fu([)p Fs(H)1908 3007 y Fm(\034)1950 2992 y Fs(;)g(iA)p Fu(])2133 2973 y(~)-55 b Fs(\037)2189 3007 y Fm(R)2246 2992 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2546 2956 y Fl(\000)p Fm(s)2638 2992 y Fs(f)2686 3007 y Fm(n)2733 2992 y Ft(i)p Fs(;)801 3113 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1130 3093 y Fu(~)-55 b Fs(\037)1185 3128 y Fm(R)1242 3113 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1542 3076 y Fl(\000)p Fm(s)1634 3113 y Fu(])p Fs(f)1709 3128 y Fm(n)1756 3113 y Fs(;)17 b Fu([)p Fs(H)1908 3128 y Fm(\034)1950 3113 y Fs(;)g(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))2346 3093 y(~)-55 b Fs(\037)2401 3128 y Fm(R)2459 3113 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2759 3076 y Fl(\000)p Fm(s)2850 3113 y Fs(f)2898 3128 y Fm(n)2945 3113 y Ft(i)p Fs(;)801 3233 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1130 3214 y Fu(~)-55 b Fs(\037)1185 3248 y Fm(R)1242 3233 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1542 3197 y Fl(\000)p Fm(s)1634 3233 y Fu(])p Fs(f)1709 3248 y Fm(n)1756 3233 y Fs(;)1806 3214 y Fu(~)g Fs(\037)1861 3248 y Fm(R)1918 3233 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2218 3197 y Fl(\000)p Fm(s)2310 3233 y Fs(f)2358 3248 y Fm(n)2405 3233 y Ft(i)p Fs(;)801 3353 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1130 3334 y Fu(~)-55 b Fs(\037)1185 3368 y Fm(R)1242 3353 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1542 3317 y Fl(\000)p Fm(s)1634 3353 y Fu(])p Fs(f)1709 3368 y Fm(n)1756 3353 y Fs(;)17 b(\022)s Fu(\()p Fs(H)8 b Fu(\))2019 3334 y(~)-55 b Fs(\037)2073 3368 y Fm(R)2131 3353 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2431 3317 y Fl(\000)p Fm(s)2522 3353 y Fs(f)2570 3368 y Fm(n)2617 3353 y Ft(i)456 3523 y Fu(are)32 b(equal)i(to)e Fs(O)1070 3538 y Fm(R)1127 3523 y Fu(\()p Fs(R)1240 3487 y Fl(\000)p Fk(1)1334 3523 y Fu(\))23 b Ft(\001)e(k)1500 3504 y Fu(~)-55 b Fs(\037)1555 3538 y Fm(R)1613 3523 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1913 3487 y Fl(\000)p Fm(s)2004 3523 y Fs(f)2052 3538 y Fm(n)2099 3523 y Ft(k)p Fu(.)44 b(Therefore,)715 3683 y Ft(h)760 3663 y Fu(~)-55 b Fs(\037)815 3698 y Fm(R)873 3683 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1173 3641 y Fl(\000)p Fm(s)1264 3683 y Fs(f)1312 3698 y Fm(n)1359 3683 y Fs(;)17 b Fu([)p Fs(H)1511 3698 y Fm(\034)1554 3683 y Fs(;)g(iA)p Fu(])1737 3663 y(~)-55 b Fs(\037)1792 3698 y Fm(R)1850 3683 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2150 3641 y Fl(\000)p Fm(s)2241 3683 y Fs(f)2289 3698 y Fm(n)2336 3683 y Ft(i)1002 3830 y(\025)45 b Fs(c)p Ft(k)1222 3811 y Fu(~)-55 b Fs(\037)1277 3845 y Fm(R)1335 3830 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1635 3789 y Fl(\000)p Fm(s)1726 3830 y Fs(f)1774 3845 y Fm(n)1821 3830 y Ft(k)1871 3789 y Fk(2)1932 3830 y Fu(+)22 b Fs(O)2105 3845 y Fm(R)2163 3830 y Fu(\()p Fs(R)2276 3789 y Fl(\000)p Fk(1)2370 3830 y Fu(\))g Ft(\001)g(k)2536 3811 y Fu(~)-55 b Fs(\037)2591 3845 y Fm(R)2648 3830 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2948 3789 y Fl(\000)p Fm(s)3040 3830 y Fs(f)3088 3845 y Fm(n)3135 3830 y Ft(k)456 3988 y Fu(By)23 b(Lemma)h(C.1,)h(for)e Fs(c)1318 3951 y Fl(0)1369 3988 y Fu(=)28 b(\(2)p Fs(s)s Ft(\000)s Fu(1\))1776 3951 y Fl(\000)p Fk(1)1870 3988 y Fs(c=)p Fu(2,)c(\(3.18\))f(holds)h(true)f (for)g Fs(R)28 b Ft(\025)h Fs(R)3198 4003 y Fk(1)3265 3988 y Fs(>)f Fu(2,)456 4104 y(if)k Fs(R)619 4119 y Fk(1)691 4104 y Fu(is)h(large)g(enough.)2009 b Fj(\003)456 4278 y Fy(Lemma)39 b(3.5.)j Fp(The)34 b(ine)-5 b(quality)43 b Fu(\(3.21\))35 b Fp(holds)f(true.)456 4503 y(Pr)-5 b(o)g(of.)41 b Fu(F)-8 b(rom)45 b(\(2.5\))h(\(with)g Fs(K)58 b Fu(=)50 b(0\))c(applied)g(to)g(the)g Fs(C)2681 4518 y Fm(R)2739 4503 y Fs(f)2787 4518 y Fm(n)2834 4503 y Fu(,)j(where)e Fs(C)3275 4518 y Fm(R)3379 4503 y Fu(is)456 4619 y(de\014ned)34 b(in)e(\(3.20\),)g(w)m(e)i(deriv)m(e)g(that)1103 4777 y Ft(h)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(C)1425 4792 y Fm(R)1482 4777 y Fs(f)1530 4792 y Fm(n)1577 4777 y Fs(;)17 b Fu([)p Fs(H)1729 4792 y Fm(\034)1771 4777 y Fs(;)g(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(C)2231 4792 y Fm(R)2289 4777 y Fs(f)2337 4792 y Fm(n)2384 4777 y Ft(i)27 b(\025)h Fu(0)p Fs(:)456 4934 y Fu(Thanks)34 b(to)e(\(3.20\))g(and)g(to)h(the)g(Lemmata)g(3.3)f(and)g(C.3,)677 5013 y Ff(\032)793 5094 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b(C)1186 5109 y Fm(R)1243 5094 y Fu(])p Fs(f)1318 5109 y Fm(n)1365 5094 y Fs(;)g Fu([)p Fs(H)1517 5109 y Fm(\034)1559 5094 y Fs(;)g(iA)p Fu(])p Fs(C)1806 5109 y Fm(R)1864 5094 y Ft(h)p Fs(A)p Ft(i)2015 5057 y Fm(s)2058 5074 y Fu(~)-55 b Fs(\037)2113 5109 y Fm(R=)p Fk(2)2241 5094 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2541 5057 y Fl(\000)p Fm(s)2632 5094 y Fs(f)2680 5109 y Fm(n)2727 5094 y Ft(i)793 5214 y(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b(C)1186 5229 y Fm(R)1243 5214 y Fu(])p Fs(f)1318 5229 y Fm(n)1365 5214 y Fs(;)g Fu([)p Fs(H)1517 5229 y Fm(\034)1559 5214 y Fs(;)g(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(C)2019 5229 y Fm(R)2077 5214 y Ft(h)p Fs(A)p Ft(i)2228 5178 y Fm(s)2270 5195 y Fu(~)-55 b Fs(\037)2325 5229 y Fm(R=)p Fk(2)2454 5214 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2754 5178 y Fl(\000)p Fm(s)2845 5214 y Fs(f)2893 5229 y Fm(n)2940 5214 y Ft(i)p Fs(:)p eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(15)456 453 y Fu(are)32 b Fs(O)693 468 y Fm(R)750 453 y Fu(\()p Fs(R)863 416 y Fk(2)p Fm(s)p Fl(\000)p Fk(1)1026 453 y Fu(\))22 b Ft(\001)g(k)1192 433 y Fu(~)-55 b Fs(\037)1247 468 y Fm(R=)p Fk(2)1375 453 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1675 416 y Fl(\000)p Fm(s)1766 453 y Fs(f)1814 468 y Fm(n)1861 453 y Ft(k)p Fu(,)33 b(yielding)i(\(3.21\))o(.)753 b Fj(\003)456 693 y Fu(3.3.)48 b Fy(Absence)38 b(of)g(mass.)50 b Fu(The)33 b(aim)g(of)f(this)h(part)g(is)g(to)f(sho)m(w)456 869 y Fy(Prop)s(osition)h(3.6.)39 b Fp(Under)32 b(the)g(hyp)-5 b(otheses)31 b(of)g(Pr)-5 b(op)g(osition)31 b(3.1)g(with)h Fu(1)p Fs(=)p Fu(2)27 b Fs(<)456 986 y(s)g(<)h Fu(2)p Fs(=)p Fu(3)p Fp(,)1298 1162 y Fu(lim)1269 1224 y Fm(R)p Fl(!1)1480 1162 y Fu(lim)17 b(sup)1538 1242 y Fm(n)p Fl(!1)1796 1162 y Ft(k)1846 1143 y Fs(\037)1907 1177 y Fm(R)1964 1162 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2161 1177 y Fm(n)2208 1162 y Ft(k)28 b Fu(=)f(0)p Fs(:)-2009 b Fu(\(3.22\))456 1447 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Applying)35 b(\(2.5\))d(\(with)h Fs(K)i Fu(=)27 b(0\))33 b(to)f(the)2299 1428 y Fs(\037)2360 1462 y Fm(R)2418 1447 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2615 1462 y Fm(n)2662 1447 y Fu(,)636 1609 y Ft(h)675 1590 y Fs(\037)736 1624 y Fm(R)794 1609 y Fu(\()p Fs(A)p Fu(\))p Fs(f)991 1624 y Fm(n)1038 1609 y Fs(;)17 b(\022)s Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1403 1624 y Fm(\034)1445 1609 y Fs(;)17 b(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))1835 1590 y Fs(\037)1896 1624 y Fm(R)1954 1609 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2151 1624 y Fm(n)2198 1609 y Ft(i)83 b(\025)g Fs(c)p Ft(k)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))2785 1590 y Fs(\037)2846 1624 y Fm(R)2903 1609 y Fu(\()p Fs(A)p Fu(\))p Fs(f)3100 1624 y Fm(n)3147 1609 y Ft(k)3197 1568 y Fk(2)3237 1609 y Fs(:)456 1770 y Fu(By)33 b(Lemma)g(3.3,)914 1842 y Ff(8)914 1932 y(>)914 1961 y(>)914 1991 y(<)914 2171 y(>)914 2201 y(>)914 2230 y(:)1044 1930 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1367 1911 y(\037)1427 1945 y Fm(R)1485 1930 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1709 1945 y Fm(n)1756 1930 y Fs(;)17 b Fu([)p Fs(H)1908 1945 y Fm(\034)1950 1930 y Fs(;)g(iA)p Fu(])2127 1911 y Fs(\037)2189 1945 y Fm(R)2246 1930 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2443 1945 y Fm(n)2491 1930 y Ft(i)p Fs(;)1044 2046 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)1367 2027 y(\037)1427 2061 y Fm(R)1485 2046 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1709 2061 y Fm(n)1756 2046 y Fs(;)17 b Fu([)p Fs(H)1908 2061 y Fm(\034)1950 2046 y Fs(;)g(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))2340 2027 y Fs(\037)2401 2061 y Fm(R)2459 2046 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2656 2061 y Fm(n)2703 2046 y Ft(i)p Fs(;)1044 2162 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)g Fu(\))p Fs(;)1367 2143 y(\037)1427 2177 y Fm(R)1485 2162 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1709 2177 y Fm(n)1756 2162 y Fs(;)1800 2143 y(\037)1861 2177 y Fm(R)1918 2162 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2115 2177 y Fm(n)2162 2162 y Ft(i)p Fs(;)1044 2279 y Ft(h)p Fu([)p Fs(\022)s Fu(\()p Fs(H)g Fu(\))p Fs(;)1367 2259 y(\037)1427 2294 y Fm(R)1485 2279 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1709 2294 y Fm(n)1756 2279 y Fs(;)17 b(\022)s Fu(\()p Fs(H)8 b Fu(\))2013 2259 y Fs(\037)2073 2294 y Fm(R)2131 2279 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2328 2294 y Fm(n)2375 2279 y Ft(i)456 2452 y Fu(are)32 b(equal)i(to)e Fs(O)1070 2467 y Fm(R)1127 2452 y Fu(\()p Fs(R)1240 2416 y Fm(s)p Fl(\000)p Fk(1)1367 2452 y Fu(\))22 b Ft(\001)g(k)1527 2433 y Fs(\037)1588 2467 y Fm(R)1646 2452 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1843 2467 y Fm(n)1890 2452 y Ft(k)p Fu(.)43 b(Therefore,)456 2614 y Ft(h)495 2594 y Fs(\037)555 2629 y Fm(R)613 2614 y Fu(\()p Fs(A)p Fu(\))p Fs(f)810 2629 y Fm(n)857 2614 y Fs(;)17 b Fu([)p Fs(H)1009 2629 y Fm(\034)1052 2614 y Fs(;)g(iA)p Fu(])1229 2594 y Fs(\037)1290 2629 y Fm(R)1348 2614 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1545 2629 y Fm(n)1592 2614 y Ft(i)27 b(\025)i Fs(c)p Ft(k)1856 2594 y Fs(\037)1917 2629 y Fm(R)1974 2614 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2171 2629 y Fm(n)2218 2614 y Ft(k)2268 2573 y Fk(2)2330 2614 y Fu(+)22 b Fs(O)2503 2629 y Fm(R)2560 2614 y Fu(\()p Fs(R)2673 2573 y Fm(s)p Fl(\000)p Fk(1)2800 2614 y Fu(\))g Ft(\001)g(k)2960 2594 y Fs(\037)3021 2629 y Fm(R)3078 2614 y Fu(\()p Fs(A)p Fu(\))p Fs(f)3275 2629 y Fm(n)3323 2614 y Ft(k)p Fs(:)456 2774 y Fu(Since)33 b Fs(s)28 b(<)f Fu(2)p Fs(=)p Fu(3,)32 b(w)m(e)i(can)f(\014nd)g Fs(\014)g(>)28 b Fu(0)k(\(see)i(Lemma)f(3.7)f(b)s(elo)m(w\))h(suc)m(h)h (that)749 2939 y Ft(jh)p Fu([)p Fs(H)924 2954 y Fm(\034)966 2939 y Fs(;)1010 2920 y(\037)1071 2954 y Fm(R)1129 2939 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1353 2954 y Fm(n)1400 2939 y Fs(;)17 b(iA)1550 2920 y(\037)1611 2954 y Fm(R)1669 2939 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1866 2954 y Fm(n)1913 2939 y Ft(ij)27 b Fu(=)g Fs(O)2185 2954 y Fm(R)2243 2939 y Fu(\()p Fs(R)2356 2898 y Fl(\000)p Fm(\014)2458 2939 y Fu(\))p Ft(k)2546 2920 y Fs(\037)2607 2954 y Fm(R)2664 2939 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2861 2954 y Fm(n)2908 2939 y Ft(k)p Fs(:)-2529 b Fu(\(3.23\))456 3100 y(This)33 b(yields)619 3261 y Ft(h)p Fs(f)706 3276 y Fm(n)753 3261 y Fs(;)17 b Fu([)p Fs(H)905 3276 y Fm(\034)948 3261 y Fs(;)g(iA)1098 3242 y(\037)1159 3220 y Fk(2)1159 3286 y Fm(R)1217 3261 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1441 3276 y Fm(n)1488 3261 y Ft(i)27 b(\025)i Fs(c)p Ft(k)1752 3242 y Fs(\037)1813 3276 y Fm(R)1870 3261 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2067 3276 y Fm(n)2114 3261 y Ft(k)2164 3220 y Fk(2)2226 3261 y Fu(+)22 b Fs(o)2371 3276 y Fm(R)2428 3261 y Fu(\(1\))g Ft(\001)g(k)2675 3242 y Fs(\037)2736 3276 y Fm(R)2794 3261 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2991 3276 y Fm(n)3038 3261 y Ft(k)p Fs(:)456 3422 y Fu(No)m(w,)33 b(w)m(e)g(com)m(bine)i(\(3.12\))c(and)i(Prop)s(osition)g(2.15)f(to)g (arriv)m(e)h(at)1197 3583 y(lim)17 b(sup)1254 3662 y Fm(n)p Fl(!1)1512 3583 y Ft(k)1562 3563 y Fs(\037)1623 3598 y Fm(R)1681 3583 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1878 3598 y Fm(n)1925 3583 y Ft(k)27 b Fu(=)h Fs(o)2153 3598 y Fm(R)2211 3583 y Fu(\(1\))p Fs(:)97 b Fj(\003)456 3852 y Fu(T)-8 b(o)32 b(complete)i(the)f(pro)s(of)f(of)g(Prop)s(osition)h (3.6,)f(w)m(e)i(sho)m(w)456 4028 y Fy(Lemma)47 b(3.7.)f Fp(Under)c(the)f(assumptions)g(of)g(Pr)-5 b(op)g(osition)41 b(3.6,)i(ther)-5 b(e)42 b(exists)456 4145 y Fs(\014)33 b(>)27 b Fu(0)35 b Fp(such)g(that)44 b Fu(\(3.23\))35 b Fp(holds)f(true.)456 4374 y(Pr)-5 b(o)g(of.)41 b Fu(W)-8 b(e)33 b(decomp)s(ose)h Ft(h)p Fu([)p Fs(H)1560 4389 y Fm(\034)1603 4374 y Fs(;)1647 4354 y(\037)1707 4389 y Fm(R)1765 4374 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)1989 4389 y Fm(n)2036 4374 y Fs(;)17 b(iA)2186 4354 y(\037)2247 4389 y Fm(R)2305 4374 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2502 4389 y Fm(n)2549 4374 y Ft(i)32 b Fu(as)1148 4537 y Ft(h)p Fu([)p Fs(H)1295 4552 y Fm(\034)1338 4537 y Fs(;)1388 4517 y Fu(~)-55 b Fs(\037)1443 4552 y Fm(R)1500 4537 y Fu(\()p Fs(A)p Fu(\)])1682 4517 y(~)g Fs(\037)1738 4552 y Fm(R=)p Fk(2)1866 4537 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2063 4552 y Fm(n)2110 4537 y Fs(;)17 b(iA)2260 4517 y(\037)2321 4552 y Fm(R)2379 4537 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2576 4552 y Fm(n)2623 4537 y Ft(i)-2206 b Fu(\(3.24\))1072 4692 y(+)p Ft(h)p Fu([)p Fs(H)1295 4707 y Fm(\034)1338 4692 y Fs(;)1388 4672 y Fu(~)-55 b Fs(\037)1443 4707 y Fm(R)1500 4692 y Fu(\()p Fs(A)p Fu(\)])1676 4672 y Fs(\037)1738 4707 y Fm(R=)p Fk(2)1866 4692 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2063 4707 y Fm(n)2110 4692 y Fs(;)17 b(iA)2260 4672 y(\037)2321 4707 y Fm(R)2379 4692 y Fu(\()p Fs(A)p Fu(\))p Fs(f)2576 4707 y Fm(n)2623 4692 y Ft(i)-2206 b Fu(\(3.25\))456 4862 y(Since)28 b(\()749 4843 y(~)-55 b Fs(\037)804 4877 y Fm(R)862 4862 y Fu(\))900 4877 y Fm(R)985 4862 y Fu(is)28 b(b)s(ounded)g(in)g Ft(S)1647 4826 y Fk(0)1715 4862 y Fu(\(cf.)41 b(Corollary)28 b(A.3\))g(and)f (since)i(the)f(supp)s(ort)456 4981 y(of)576 4961 y(~)-55 b Fs(\037)631 4996 y Fm(R)724 4981 y Fu(do)s(es)36 b(not)f(in)m (tersect)j(])24 b Ft(\000)h Fs(R)q(;)17 b(R)q Fu([,)36 b(Lemma)g(B.3)f(for)g Fs(k)h Fu(=)d(1)i(ensures)j(that)456 5099 y Fs(A)p Fu([)p Fs(H)637 5114 y Fm(\034)680 5099 y Fs(;)730 5080 y Fu(~)-55 b Fs(\037)785 5114 y Fm(R)842 5099 y Fu(\()p Fs(A)p Fu(\)])p Ft(h)p Fs(A)p Ft(i)1169 5063 y Fm(s)1245 5099 y Fu(is)39 b(b)s(ounded)h(and)f(its)g(norm)g(is)g Fs(O)s Fu(\()p Fs(R)2648 5063 y Fm(s)2684 5099 y Fu(\).)62 b(Since)40 b Fs(s)f(<)f Fu(2)p Fs(=)p Fu(3,)456 5216 y(w)m(e)g(can)g(\014nd)h Fs(\013)e Ft(2)p Fu(])26 b Ft(\000)g Fs(s;)17 b Fu(2)p Fs(s)25 b Ft(\000)i Fu(2[.)59 b(This)39 b(implies,)h(using)f(Corollary)f(3.2,)h(that)p eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 456 255 a Fq(16)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fu(\(3.24\))36 b(is)i Fs(O)920 465 y Fm(R)977 450 y Fu(\()p Fs(R)1090 414 y Fm(s)p Fk(+)p Fm(\013)1227 450 y Fu(\))25 b Ft(\001)g(k)1393 431 y Fs(\037)1454 465 y Fm(R)1512 450 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1709 465 y Fm(n)1756 450 y Ft(k)p Fu(,)38 b(with)g Fs(s)25 b Fu(+)g Fs(\013)36 b(<)f Fu(0.)57 b(By)38 b(Prop)s(osition)f(B.2)456 566 y(with)c Fs(k)e Fu(=)c(2,)456 767 y([)p Fs(H)564 782 y Fm(\034)607 767 y Fs(;)657 748 y Fu(~)-55 b Fs(\037)711 782 y Fm(R)769 767 y Fu(\()p Fs(A)p Fu(\)])945 748 y Fs(\037)1006 782 y Fm(R=)p Fk(2)1135 767 y Fu(\()p Fs(A)p Fu(\))27 b(=)h([)p Fs(H)1523 782 y Fm(\034)1566 767 y Fs(;)17 b(A)p Fu(])1716 748 y(~)-55 b Fs(\037)1771 708 y Fl(0)1771 792 y Fm(R)1829 767 y Fu(\()p Fs(A)p Fu(\))1978 748 y Fs(\037)2039 782 y Fm(R=)p Fk(2)2167 767 y Fu(\()p Fs(A)p Fu(\))22 b(+)g Fs(I)2479 782 y Fk(2)2519 748 y Fs(\037)2580 782 y Fm(R=)p Fk(2)2708 767 y Fu(\()p Fs(A)p Fu(\))28 b(=)f Fs(I)3031 782 y Fk(2)3071 748 y Fs(\037)3132 782 y Fm(R=)p Fk(2)3260 767 y Fu(\()p Fs(A)p Fu(\))456 980 y(since)47 b(supp)915 961 y(~)-55 b Fs(\037)970 921 y Fl(0)970 1005 y Fm(R)1059 980 y Ft(\\)31 b Fu(supp)1364 961 y(~)-56 b Fs(\037)1418 996 y Fm(R=)p Fk(2)1597 980 y Fu(=)50 b Ft(;)p Fu(.)83 b(Lemma)46 b(B.3)f(for)h Fs(k)53 b Fu(=)d(2)45 b(implies)j(that)456 1100 y Fs(AI)572 1115 y Fk(2)611 1080 y Fs(\037)672 1115 y Fm(R=)p Fk(2)801 1100 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1101 1064 y Fm(s)1179 1100 y Fu(is)43 b(b)s(ounded)g(and)g(its)g(norm)f(is)h Fs(O)s Fu(\()p Fs(R)2604 1064 y Fm(s)p Fl(\000)p Fk(1)2730 1100 y Fu(\).)73 b(In)43 b(particular,)456 1219 y(\(3.25\))31 b(is)j Fs(O)s Fu(\()p Fs(R)1027 1183 y Fm(s)p Fl(\000)p Fk(1)1153 1219 y Fu(\))22 b Ft(\001)g(k)1313 1200 y Fs(\037)1374 1234 y Fm(R)1431 1219 y Fu(\()p Fs(A)p Fu(\))p Fs(f)1628 1234 y Fm(n)1675 1219 y Ft(k)p Fu(.)1615 b Fj(\003)934 1578 y Fu(4.)49 b Fv(The)37 b(LAP)h(f)n(or)g(the)f(reduced)h(resol)-9 b(vent.)456 1811 y Fu(4.1.)48 b Fy(Motiv)-6 b(ation.)50 b Fu(An)33 b(in)m(teresting)h(consequence)i(of)c(the)h(LAP)g(\(1.2\))g (is)g(the)456 1927 y(follo)m(wing)h(propagation)g(estimate)h(\(cf.)49 b(Kato's)34 b(lo)s(cal)g(smo)s(othness)i(in)e([ABG,)456 2043 y(JMP,)f(RS4]\):)43 b(there)34 b(exists)g Fs(C)h(>)27 b Fu(0)32 b(suc)m(h)i(that,)f(for)f(all)g Fs(f)39 b Ft(2)28 b Fo(H)f Fu(,)1063 2156 y Ff(Z)1162 2182 y Fl(1)1118 2381 y(\0001)1264 2291 y Ft(kh)p Fs(A)p Ft(i)1465 2250 y Fl(\000)p Fm(s)1556 2291 y Fs(e)1601 2250 y Fm(itH)1718 2291 y Fs(E)1790 2306 y Fl(I)1838 2291 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)j Ft(k)2112 2250 y Fk(2)2151 2291 y Fs(dt)27 b Ft(\024)h Fs(C)7 b Ft(k)p Fs(f)k Ft(k)2605 2250 y Fk(2)2644 2291 y Fs(:)-2215 b Fu(\(4.26\))456 2555 y(F)-8 b(or)35 b Fs(E)706 2570 y Fl(I)754 2555 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)44 b Ft(6)p Fu(=)33 b(0,)k(the)g(state)f Fs(e)1692 2519 y Fm(itH)1809 2555 y Fs(E)1881 2570 y Fl(I)1929 2555 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)46 b Fu(m)m(ust)37 b(mo)m(v)m(e)h(to)e(\\regions)g(where)456 2671 y Ft(j)p Fs(A)p Ft(j)45 b Fu(is)i(large")f(when)h Fs(t)k Ft(!)g(\0001)46 b Fu(and)g Fs(t)51 b Ft(!)g Fu(+)p Ft(1)p Fu(,)e(since)e(the)g(in)m (tegral)g(con-)456 2788 y(v)m(erges.)53 b(If)35 b Fs(H)8 b(f)43 b Fu(=)32 b Fs(\025f)46 b Fu(with)36 b Fs(\025)d Ft(2)f(I)43 b Fu(and)36 b Fs(f)43 b Ft(6)p Fu(=)32 b(0,)k(then)g Fs(e)2618 2751 y Fm(itH)2735 2788 y Fs(E)2807 2803 y Fl(I)2855 2788 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)42 b Fu(=)33 b Fs(e)3264 2751 y Fm(it\025)3359 2788 y Fs(f)11 b Fu(,)456 2904 y Ft(kh)p Fs(A)p Ft(i)657 2868 y Fl(\000)p Fm(s)748 2904 y Fs(e)793 2868 y Fm(itH)910 2904 y Fs(E)982 2919 y Fl(I)1029 2904 y Fu(\()p Fs(H)d Fu(\))p Fs(f)j Ft(k)40 b Fu(=)g Ft(kh)p Fs(A)p Ft(i)1660 2868 y Fl(\000)p Fm(s)1752 2904 y Fs(f)11 b Ft(k)p Fu(,)41 b(and)f(the)h(in)m(tegral)g(in)f (\(4.26\))f(div)m(erges.)456 3020 y(Therefore,)h(the)e(LAP)g(cannot)g (hold)g(true)g(near)g(an)f(eigen)m(v)-5 b(alue.)60 b(Ho)m(w)m(ev)m(er) 40 b(it)456 3136 y(is)45 b(in)m(teresting)h(to)e(\014nd)h(out)f (whether)i(the)f(estimate)g(\(4.26\))f(holds)h(true)g(for)456 3252 y(nonzero)33 b(states)h Fs(E)1169 3267 y Fl(I)1217 3252 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)43 b Fu(whic)m(h)34 b(are)f(orthogonal)g(to)f(the)i(eigen)m(v)m(ectors)h(asso-)456 3369 y(ciated)42 b(to)f(eigen)m(v)-5 b(alues)44 b(in)d Ft(I)7 b Fu(,)45 b(i.e.)71 b(nonzero)42 b(states)g Fs(P)2579 3333 y Fl(?)2638 3369 y Fs(E)2710 3384 y Fl(I)2757 3369 y Fu(\()p Fs(H)8 b Fu(\))p Fs(f)j Fu(.)70 b(No)m(w)42 b(the)456 3496 y(reduced)34 b(LAP)f(\(1.3\))f(on)g Ft(I)1481 3460 y Fl(0)1538 3496 y Fu(with)h Ft(I)i(\032)1960 3471 y Fu(\027)1954 3496 y Ft(I)2015 3468 y Fl(0)2072 3496 y Fu(implies)f(that)1025 3694 y(sup)879 3776 y Fk(Re)p Fm(z)s Fl(2I)5 b Fm(;)p Fk(Im)o Fm(z)s Fl(6)p Fk(=0)1334 3694 y Ft(kh)p Fs(A)p Ft(i)1535 3653 y Fl(\000)p Fm(s)1626 3694 y Fu(\()p Fs(H)30 b Ft(\000)22 b Fs(z)t Fu(\))1961 3653 y Fl(\000)p Fk(1)2056 3694 y Fs(P)2133 3653 y Fl(?)2192 3694 y Fs(E)2264 3709 y Fl(I)2308 3690 y Fi(0)2334 3694 y Fu(\()p Fs(H)8 b Fu(\))p Ft(h)p Fs(A)p Ft(i)2650 3653 y Fl(\000)p Fm(s)2741 3694 y Ft(k)27 b Fs(<)h Ft(1)456 3963 y Fu(since)34 b(\()p Fs(H)29 b Ft(\000)23 b Fs(z)t Fu(\))1030 3927 y Fl(\000)p Fk(1)1125 3963 y Fs(E)1197 3978 y Fg(R)p Fl(nI)1324 3960 y Fi(0)1350 3963 y Fu(\()p Fs(H)8 b Fu(\))32 b(is)h(uniformly)g(b)s(ounded,)h(yielding)h(\(4.26\)) d(with)h Fs(f)456 4086 y Fu(replaced)g(b)m(y)h Fs(P)1053 4049 y Fl(?)1111 4086 y Fs(f)44 b Fu(b)m(y)33 b(Kato's)g(lo)s(cal)f (smo)s(othness)i(\(cf.)44 b([ABG,)33 b(RS4)o(]\).)44 b(Theo-)456 4202 y(rem)33 b(1.4)f(giv)m(es)i(a)e(situation)h(where)h (the)f(later)g(estimate)g(holds)h(true.)456 4507 y(4.2.)48 b Fy(Pro)s(of)36 b(of)g(Theorem)g(1.4.)49 b Fu(As)32 b(in)f(the)h(pro)s(of)e(of)h(Theorem)h(1.2)f(in)g(Sub-)456 4623 y(section)41 b(2.2,)h Fs(H)48 b Ft(2)42 b(C)1278 4587 y Fk(2)1272 4649 y Fl(I)1316 4631 y Fi(00)1360 4623 y Fu(\()p Fs(A)p Fu(\),)h(for)c(an)m(y)i(op)s(en)g(in)m(terv)-5 b(al)41 b Ft(I)2592 4587 y Fl(00)2675 4623 y Fu(with)p 2905 4543 62 4 v 41 w Ft(I)49 b(\032)41 b(I)3187 4587 y Fl(00)3230 4623 y Fu(.)67 b(By)456 4739 y(Remark)45 b(1.1,)j(the)d(Mourre)h(estimate)g(\(1.1\))e(implies)i(that)f(Ran)p Fs(P)3025 4754 y Fl(I)3117 4739 y Fu(is)h(\014nite)456 4856 y(dimensional.)85 b(Let)46 b Fs(\022)54 b Ft(2)d(C)1533 4819 y Fl(1)1527 4880 y Fm(c)1608 4856 y Fu(\()p Ft(I)7 b Fu(\).)84 b(In)46 b(particular,)k Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))50 b Ft(2)h(C)2923 4819 y Fk(2)2962 4856 y Fu(\()p Fs(A)p Fu(\).)84 b(Since)456 4972 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)68 b Fu(=)56 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1208 4987 y Fl(I)1304 4972 y Fu(is)50 b(a)f(\014nite)h(sum)g(of)e(terms)j(lik)m(e)f Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))33 b Ft(\001)g(j)p Fs(f)11 b Ft(ih)p Fs(f)g Ft(j)p Fu(,)52 b(with)456 5088 y Fs(f)h Ft(2)43 b(D)s Fu(\()p Fs(A)857 5052 y Fk(2)896 5088 y Fu(\))e(b)m(y)i(assumption,)i Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)56 b Ft(2)43 b(C)2173 5052 y Fk(2)2212 5088 y Fu(\()p Fs(A)p Fu(\))f(b)m(y)g(Prop)s(osition)g(2.1.)69 b(Let)456 5216 y Fs(\034)40 b Ft(2)29 b(C)691 5179 y Fl(1)685 5240 y Fm(c)766 5216 y Fu(\()p Ft(I)865 5179 y Fl(00)908 5216 y Fu(\))34 b(suc)m(h)g(that)f Fs(\034)41 b Fu(=)29 b(1)k(near)g Ft(I)7 b Fu(.)46 b(Let)34 b Ft(I)2270 5179 y Fl(0)2327 5216 y Fu(b)s(e)g(closed)g(with)g Ft(I)3033 5179 y Fl(0)3086 5216 y Ft(\032)3189 5190 y Fu(\027)3192 5216 y Ft(I)41 b Fu(and)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(17)456 450 y Fs(\022)35 b Ft(2)e(C)693 414 y Fl(1)687 475 y Fm(c)768 450 y Fu(\()p Ft(I)7 b Fu(\))35 b(with)h Fs(\022)g Fu(=)c(1)j(near)g Ft(I)1718 414 y Fl(0)1742 450 y Fu(.)51 b(By)36 b(Lemma)g(2.5,)g(\(2.5\))e(holds) i(true)g(\(with)456 566 y Fs(H)537 581 y Fm(\034)612 566 y Fu(de\014ned)e(in)f(\(2.4\))o(\).)44 b(Th)m(us)871 727 y Fs(P)948 686 y Fl(?)1007 727 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1328 742 y Fm(\034)1371 727 y Fs(;)17 b(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1838 686 y Fl(?)1979 727 y Ft(\025)83 b Fs(c)p Fu(\()p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2509 686 y Fl(?)2567 727 y Fu(\))2605 686 y Fk(2)2139 877 y Fu(+)p Fs(\022)s Fu(\()p Fs(H)g Fu(\))p Fs(P)2505 836 y Fl(?)2563 877 y Fs(K)f(\022)s Fu(\()p Fs(H)h Fu(\))p Fs(P)2943 836 y Fl(?)3001 877 y Fs(:)-2572 b Fu(\(4.27\))456 1034 y(Let)24 b Fs(\022)667 1049 y Fk(1)735 1034 y Ft(2)k(C)887 998 y Fl(1)881 1059 y Fm(c)962 1034 y Fu(\()p Ft(I)1061 998 y Fl(0)1085 1034 y Fu(\).)41 b(Since)25 b Fs(P)1514 998 y Fl(?)1601 1034 y Fu(:)i(1)6 b Ft(\000)g Fs(P)39 b Fu(pro)5 b(jects)26 b(on)m(to)e(the)h(con)m(tin)m(uous)i(sp)s(ectral)456 1150 y(subspace)38 b(of)e Fs(H)8 b Fu(,)36 b Fs(\022)1179 1165 y Fk(1)1219 1150 y Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1461 1114 y Fl(?)1555 1150 y Fu(con)m(v)m(erges)39 b(strongly)e(to)f(0)g(as) g(the)h(supp)s(ort)g(of)f Fs(\022)3405 1165 y Fk(1)456 1266 y Fu(shrinks)41 b(to)e(a)g(p)s(oin)m(t.)63 b(Since)41 b Fs(K)46 b Fu(compact,)c Ft(k)p Fs(K)7 b(\022)2325 1281 y Fk(1)2364 1266 y Fu(\()p Fs(H)h Fu(\))p Fs(P)2606 1230 y Fl(?)2664 1266 y Ft(k)39 b Fu(go)s(es)h(to)e(0)h(in)h(the)456 1383 y(same)34 b(limit.)48 b(Multiplying)37 b(\(4.27\))c(b)m(y)h Fs(\022)1974 1398 y Fk(1)2014 1383 y Fu(\()p Fs(H)8 b Fu(\))33 b(on)h(b)s(oth)g(sides)h(and)f(taking)g(the)456 1499 y(supp)s(ort)f(of)f Fs(\022)973 1514 y Fk(1)1045 1499 y Fu(small)h(enough)g(inside)h Ft(I)1974 1463 y Fl(0)1998 1499 y Fu(,)f(w)m(e)g(obtain)908 1656 y Fs(P)985 1615 y Fl(?)1044 1656 y Fs(\022)1089 1671 y Fk(1)1128 1656 y Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1401 1671 y Fm(\034)1444 1656 y Fs(;)17 b(iA)p Fu(])p Fs(\022)1666 1671 y Fk(1)1706 1656 y Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1948 1615 y Fl(?)2089 1656 y Ft(\025)83 b Fu(\()p Fs(c=)p Fu(2\)\()p Fs(\022)2548 1671 y Fk(1)2588 1656 y Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2830 1615 y Fl(?)2887 1656 y Fu(\))2925 1615 y Fk(2)2965 1656 y Fs(:)456 1813 y Fu(Around)32 b(an)m(y)g(p)s(oin)m(t)g(of)g Ft(I)1419 1777 y Fl(0)1443 1813 y Fu(,)g(w)m(e)h(th)m(us)g(can)f (\014nd)g(some)h(in\014nite)g(in)m(terv)-5 b(al)32 b Ft(I)3210 1828 y Fk(1)3278 1813 y Ft(\032)c(I)456 1929 y Fu(suc)m(h)h(that)f(the)g(pro)5 b(jected)29 b(Mourre)f(estimate)h (\(4.28\))e(b)s(elo)m(w)i(holds)g(true)f(on)g Ft(I)3378 1944 y Fk(1)3417 1929 y Fu(.)456 2045 y(By)40 b(Theorem)h(4.1,)h(the)e (reduced)i(LAP)e(holds)h(true)f(on)g(an)m(y)h(closed)g Ft(I)3182 2009 y Fl(0)3175 2070 y Fk(1)3255 2045 y Fu(with)456 2172 y Ft(I)517 2136 y Fl(0)510 2197 y Fk(1)577 2172 y Ft(\032)692 2147 y Fu(\027)682 2172 y Ft(I)736 2187 y Fk(1)776 2172 y Fu(.)i(By)33 b(compacit)m(y)h(of)f Ft(I)1638 2136 y Fl(0)1661 2172 y Fu(,)g(w)m(e)h(get)e(the)h(reduced)h (LAP)f(on)g(it.)348 b Fj(\003)456 2347 y Fu(So)25 b(the)h(pro)s(of)f (of)g(Theorem)i(1.4)e(reduces)i(to)e(the)h(pro)s(of)f(of)g(a)g(lo)s (cal)g(and)h(stronger)456 2463 y(v)m(ersion)34 b(of)e(it,)g(namely)456 2633 y Fy(Theorem)41 b(4.1.)i Fp(L)-5 b(et)38 b Ft(I)46 b Fp(b)-5 b(e)37 b(a)g(b)-5 b(ounde)g(d,)38 b(op)-5 b(en)37 b(interval.)52 b(L)-5 b(et)38 b Ft(I)2905 2597 y Fl(00)2986 2633 y Fp(b)-5 b(e)37 b(an)g(op)-5 b(en)456 2749 y(interval)40 b(such)p 1047 2669 62 4 v 41 w Ft(I)47 b(\032)40 b(I)1326 2713 y Fl(00)1369 2749 y Fp(.)64 b(L)-5 b(et)42 b Fs(H)47 b Ft(2)39 b(C)1929 2713 y Fk(2)1923 2775 y Fl(I)1967 2757 y Fi(00)2012 2749 y Fu(\()p Fs(A)p Fu(\))i Fp(and)g(assume)f (that,)k(for)d(al)5 b(l)40 b Fs(\022)j Ft(2)456 2865 y(C)514 2829 y Fl(1)508 2890 y Fm(c)589 2865 y Fu(\()p Ft(I)7 b Fu(\))p Fp(,)48 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)61 b Ft(2)48 b(C)1313 2829 y Fk(2)1352 2865 y Fu(\()p Fs(A)p Fu(\))p Fp(.)77 b(L)-5 b(et)46 b Fs(\034)59 b Ft(2)48 b(C)2060 2829 y Fl(1)2054 2890 y Fm(c)2135 2865 y Fu(\()p Ft(I)2234 2829 y Fl(00)2277 2865 y Fu(\))e Fp(such)f(that)h Fs(\034)59 b Fu(=)47 b(1)f Fp(ne)-5 b(ar)45 b Ft(I)7 b Fp(.)456 2982 y(Assume)34 b(the)h Fu(pro)5 b(jected)37 b Fp(Mourr)-5 b(e)35 b(estimate)716 3138 y Fs(P)793 3097 y Fl(?)852 3138 y Fs(E)924 3153 y Fl(I)972 3138 y Fu(\()p Fs(H)8 b Fu(\)[)p Fs(H)1245 3153 y Fm(\034)1287 3138 y Fs(;)17 b(iA)p Fu(])p Fs(E)1536 3153 y Fl(I)1584 3138 y Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1826 3097 y Fl(?)1912 3138 y Ft(\025)28 b Fs(cE)2131 3153 y Fl(I)2178 3138 y Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2420 3097 y Fl(?)2478 3138 y Fs(;)52 b Fp(with)35 b Fs(c)27 b(>)h Fu(0)p Fs(;)-2562 b Fu(\(4.28\))456 3295 y Fp(holds)40 b(true.)63 b(Then,)42 b(for)f(any)g Fs(s)e(>)f Fu(1)p Fs(=)p Fu(2)j Fp(and)f(any)h(c)-5 b(omp)g(act)40 b(interval)h Ft(I)3203 3259 y Fl(0)3268 3295 y Fp(with)456 3423 y Ft(I)517 3387 y Fl(0)568 3423 y Ft(\032)670 3398 y Fu(\027)673 3423 y Ft(I)8 b Fp(,)34 b(the)h(r)-5 b(e)g(duc)g(e)g(d)34 b(LAP)45 b Fu(\(1.3\))p Fp(,)34 b(r)-5 b(esp)g(e)g(ctively)34 b(to)h Fu(\()p Ft(I)2537 3387 y Fl(0)2561 3423 y Fs(;)17 b(s;)g(A)p Fu(\))p Fp(,)34 b(holds)f(true)j(for)456 3539 y Fs(H)537 3554 y Fm(\034)614 3539 y Fp(and)f Fs(H)8 b Fp(.)456 3709 y(R)-5 b(emark)48 b Fu(4.2)p Fp(.)43 b Fu(Let)37 b Ft(I)7 b Fs(;)17 b Ft(I)1374 3673 y Fl(00)1454 3709 y Fu(and)37 b Fs(\034)48 b Fu(b)s(e)37 b(lik)m(e)h(in)f(Theorem)g (4.1.)56 b(Since)38 b Fs(\034)46 b Fu(=)34 b(1)i(on)456 3825 y Ft(I)7 b Fu(,)35 b(Ran)p Fs(P)817 3840 y Fl(I)899 3825 y Fu(is)g(exactly)g(the)g(subspace)i(of)c Fo(H)61 b Fu(generated)36 b(b)m(y)f(the)g(eigen)m(v)m(ectors)456 3942 y(of)44 b Fs(H)660 3957 y Fm(\034)748 3942 y Fu(asso)s(ciated)i (to)e(an)h(eigen)m(v)-5 b(alue)47 b(in)e Ft(I)7 b Fu(.)81 b(Assume)47 b(that)d Fs(H)57 b Ft(2)49 b(C)3186 3905 y Fk(4)3180 3968 y Fl(I)3224 3949 y Fi(00)3268 3942 y Fu(\()p Fs(A)p Fu(\).)456 4058 y(Then)c Fs(H)803 4073 y Fm(\034)894 4058 y Ft(2)k(C)1067 4022 y Fk(4)1106 4058 y Fu(\()p Fs(A)p Fu(\))c(and,)j(since)d Fs(H)1863 4073 y Fm(\034)1951 4058 y Fu(is)g(b)s(ounded,)j(w)m(e)e(can)f(use)g([C)q(,) f(CGH])456 4174 y(to)35 b(get)h(that)g(the)g(men)m(tioned)h(eigen)m(v)m (ectors)i(of)c Fs(H)2351 4189 y Fm(\034)2430 4174 y Fu(b)s(elong)h(to)f Ft(D)s Fu(\()p Fs(A)3058 4138 y Fk(2)3097 4174 y Fu(\).)54 b(As)36 b(in)456 4290 y(the)e(pro)s(of)f(of)g(Theorem)i(1.4)e(ab)s(o)m (v)m(e,)i(this)g(implies)g(that)e Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)43 b Ft(2)30 b(C)3078 4254 y Fk(2)3118 4290 y Fu(\()p Fs(A)p Fu(\),)k(for)456 4406 y Fs(\022)c Ft(2)f(C)684 4370 y Fl(1)678 4431 y Fm(c)759 4406 y Fu(\()p Ft(I)7 b Fu(\).)42 b(If)28 b(w)m(e)h(assume)g(further)g(the)f(pro)5 b(jected)29 b(Mourre)g(estimate)g(\(4.28\))o(,)456 4523 y(then)k(Theorem)h(4.1)e(applies.)456 4751 y(4.3.)48 b Fy(Pro)s(ofs)59 b(of)g(Theorem)g(4.1.)49 b Fu(W)-8 b(e)51 b(shall)h(giv)m(e)g(t)m(w)m(o)g(pro)s(ofs)f(of)f(Theo-)456 4867 y(rem)33 b(4.1.)43 b(The)33 b(\014rst)g(one)g(is)g(a)f(direct)h (generalization)g(to)f(the)h(presen)m(t)h(con)m(text)456 4983 y(of)44 b Fp(our)g Fu(pro)s(of)g(of)g(Theorem)h(2.7.)79 b(The)45 b(second)h(pro)s(of)e(is)g(close)i(to)e(the)h(cor-)456 5099 y(resp)s(onding)39 b(pro)s(ofs)f(in)h([C,)g(CGH])g(and)f(sho)m(ws) i(that)f(Theorem)h(2.7)e(actually)456 5216 y(applies)33 b(to)g Fs(H)8 b(P)1067 5179 y Fl(?)1125 5216 y Fu(.)43 b(In)33 b(Remark)g(4.4,)f(w)m(e)i(compare)f(the)g(t)m(w)m(o)g(pro)s (ofs.)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 456 255 a Fq(18)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fp(First)34 b(pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(4.1.)41 b Fu(By)66 b(Remark)g(2.3,)74 b(w)m(e)66 b(ma)m(y)g(assume)i (that)456 572 y(1)p Fs(=)p Fu(2)27 b Fs(<)g(s)h(<)f Fu(1.)41 b(Assume)27 b(the)e(reduced)i(LAP)e(for)g Fs(H)33 b Fu(false)25 b(on)g(some)h Ft(I)3016 536 y Fl(0)3067 572 y Ft(\032)3170 547 y Fu(\027)3173 572 y Ft(I)7 b Fu(.)41 b(Let)456 688 y Fs(\022)32 b Ft(2)d(C)686 652 y Fl(1)680 713 y Fm(c)761 688 y Fu(\()p Ft(I)7 b Fu(\))34 b(with)g Fs(\022)f Fu(=)c(1)k(on)g Ft(I)1617 652 y Fl(0)1641 688 y Fu(.)46 b(Notice)34 b(that,)g(since)h Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(;)17 b(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)41 b Ft(2)30 b(C)3229 652 y Fk(2)3268 688 y Fu(\()p Fs(A)p Fu(\),)456 805 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)746 768 y Fl(?)832 805 y Ft(2)30 b(C)986 768 y Fk(2)1025 805 y Fu(\()p Fs(A)p Fu(\).)46 b(Then,)35 b(using)f(the)f(pro)s(of)g(of)f(Prop)s(osition)i(2.11)f(and) g(2.12,)456 921 y(w)m(e)d(can)g(\014nd)h(a)e(sp)s(ecial)i(sequence)h (\()p Fs(f)1845 936 y Fm(n)1892 921 y Fs(;)17 b(z)1981 936 y Fm(n)2028 921 y Fu(\))2066 936 y Fm(n)2143 921 y Fu(for)29 b Fs(H)2370 936 y Fm(\034)2443 921 y Fu(with)h(p)s(ositiv)m (e)h(mass)g(suc)m(h)456 1037 y(that)h Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)928 1052 y Fm(n)1002 1037 y Fu(=)28 b Fs(f)1154 1052 y Fm(n)1229 1037 y Fu(=)f Fs(P)1409 1001 y Fl(?)1468 1037 y Fs(f)1516 1052 y Fm(n)1563 1037 y Fu(,)32 b(for)g(all)h Fs(n)p Fu(.)43 b(Since)707 1206 y Ft(h)p Fs(A)p Ft(i)858 1164 y Fl(\000)p Fm(s)949 1206 y Fs(f)997 1221 y Fm(n)1072 1206 y Fu(=)27 b Ft(h)p Fs(A)p Ft(i)1326 1164 y Fl(\000)p Fm(s)1418 1206 y Fu(\()p Fs(H)1537 1221 y Fm(\034)1602 1206 y Ft(\000)22 b Fs(z)1746 1221 y Fm(n)1794 1206 y Fu(\))1832 1164 y Fl(\000)p Fk(1)1926 1206 y Fs(P)2003 1164 y Fl(?)2061 1206 y Ft(h)p Fs(A)p Ft(i)2212 1164 y Fl(\000)p Fm(s)2304 1206 y Ft(h)p Fs(A)p Ft(i)2455 1164 y Fm(s)2491 1206 y Fu(\()p Fs(H)2610 1221 y Fm(\034)2675 1206 y Ft(\000)h Fs(z)2820 1221 y Fm(n)2867 1206 y Fu(\))p Fs(f)2953 1221 y Fm(n)3000 1206 y Fs(;)456 1370 y Fu(the)40 b(reduced)i(LAP)e(for)g Fs(H)1480 1385 y Fm(\034)1563 1370 y Fu(on)g Ft(I)1767 1334 y Fl(0)1830 1370 y Fu(m)m(ust)i(b)s(e)e(false.)66 b(So)40 b(it)g(su\016ces)i(to)e (pro)m(v)m(e)456 1486 y(the)d(reduced)h(LAP)g(for)e Fs(H)1467 1501 y Fm(\034)1547 1486 y Fu(on)h Ft(I)1748 1450 y Fl(0)1772 1486 y Fu(.)56 b(Using)38 b(Prop)s(osition)f(2.11)f(and)h(2.12)f(in)i (a)456 1602 y(similar)43 b(w)m(a)m(y)-8 b(,)46 b(w)m(e)d(can)g(sho)m(w) h(that)e(the)h(reduced)h(LAP)f(for)f Fs(H)2864 1617 y Fm(\034)2949 1602 y Fu(on)g Ft(I)3155 1566 y Fl(0)3222 1602 y Fu(holds)456 1719 y(true)c(if)g(and)h(only)g(if,)g(for)f(all)g (sp)s(ecial)i(sequence)h(\()p Fs(f)2417 1734 y Fm(n)2464 1719 y Fs(;)17 b(z)2553 1734 y Fm(n)2600 1719 y Fu(\))2638 1734 y Fm(n)2723 1719 y Fu(for)38 b Fs(H)2959 1734 y Fm(\034)3040 1719 y Fu(suc)m(h)i(that)456 1835 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(f)717 1850 y Fm(n)804 1835 y Fu(=)41 b Fs(f)969 1850 y Fm(n)1056 1835 y Fu(=)g Fs(P)1250 1799 y Fl(?)1308 1835 y Fs(f)1356 1850 y Fm(n)1403 1835 y Fu(,)i(for)c(all)h Fs(n)p Fu(,)j(its)e(mass)g(is)g(0.)66 b(No)m(w,)43 b(w)m(e)e(tak)m(e)g(suc)m(h)h(a)456 1951 y(sp)s(ecial)33 b(sequence)i(\()p Fs(f)1263 1966 y Fm(n)1310 1951 y Fs(;)17 b(z)1399 1966 y Fm(n)1446 1951 y Fu(\))1484 1966 y Fm(n)1531 1951 y Fu(.)44 b(Multiplying)35 b(\(4.28\))d(on)h(b)s (oth)f(sides)i(b)m(y)f Fs(\022)s Fu(\()p Fs(H)8 b Fu(\),)1022 2119 y Fs(P)1099 2078 y Fl(?)1158 2119 y Fs(\022)s Fu(\()p Fs(H)g Fu(\)[)p Fs(H)1479 2134 y Fm(\034)1521 2119 y Fs(;)17 b(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1988 2078 y Fl(?)2074 2119 y Ft(\025)28 b Fs(c)p Fu(\()p Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2549 2078 y Fl(?)2607 2119 y Fu(\))2645 2078 y Fk(2)2685 2119 y Fs(:)-2256 b Fu(\(4.29\))456 2284 y(Since)45 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1012 2248 y Fl(?)1119 2284 y Ft(2)48 b(C)1291 2248 y Fk(2)1331 2284 y Fu(\()p Fs(A)p Fu(\),)f(w)m(e)f(can)f(follo)m(w)g(our)f(pro)s(of)g(of)g(Theorem)i(2.7) e(in)456 2400 y(Section)33 b(3,)f(yielding)i(the)f(reduced)h(LAP)f(for) f Fs(H)2265 2415 y Fm(\034)2341 2400 y Fu(on)g Ft(I)2537 2364 y Fl(0)2561 2400 y Fu(.)779 b Fj(\003)456 2633 y Fp(Se)-5 b(c)g(ond)33 b(pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(4.1.)41 b Fu(Assume)52 b(for)d(a)h(while)h(that)f (Theorem)i(4.1)456 2749 y(holds)40 b(true)g(if)g(0)g Ft(62)p 1223 2669 62 4 v 40 w(I)8 b Fu(.)65 b(Under)41 b(the)f(assumptions)h(of)f(Theorem)h(4.1,)g(w)m(e)g(can)456 2866 y(\014nd)34 b(some)i(real)e Fs(\026)g Fu(suc)m(h)i(that)e(0)c Ft(62)h Fs(\026)23 b Fu(+)p 1976 2786 V 24 w Ft(I)7 b Fu(.)49 b(Notice)35 b(that)f Fs(H)42 b Fu(and)34 b Fs(H)d Fu(+)23 b Fs(\026)34 b Fu(ha)m(v)m(e)456 2982 y(the)28 b(same)h(eigen)m(v)-5 b(alues)30 b(and)e(eigen)m(v)m(ectors)j(and)d (that)g(the)g(eigen)m(v)-5 b(alues)30 b(of)e Fs(H)35 b Fu(in)456 3098 y Ft(I)41 b Fu(are)33 b(the)h(eigen)m(v)-5 b(alues)36 b(of)d Fs(H)d Fu(+)23 b Fs(\026)33 b Fu(in)h Fs(\026)22 b Fu(+)h Ft(I)7 b Fu(.)47 b(F)-8 b(or)32 b(an)m(y)i Fs(')c Fu(:)f Fn(R)g Ft(!)g Fn(R)34 b Fu(b)s(ounded)456 3214 y(and)f(b)s(orelian,)h Fs(')p Fu(\()p Fs(H)8 b Fu(\))29 b(=)g Fs(')p Fu(\(\()p Fs(H)h Fu(+)23 b Fs(\026)p Fu(\))f Ft(\000)i Fs(\026)p Fu(\),)33 b(a)g(function)i(of)e Fs(H)d Fu(+)23 b Fs(\026)p Fu(.)46 b(Th)m(us)35 b(the)456 3331 y(assumptions)30 b(of)e(Theorem)h(4.1)f(are)g(satis\014ed)i(if)e Fs(H)36 b Fu(is)29 b(replaced)g(b)m(y)g Fs(H)21 b Fu(+)13 b Fs(\026)28 b Fu(and)456 3447 y Ft(I)40 b Fu(b)m(y)33 b Fs(\026)22 b Fu(+)g Ft(I)7 b Fu(,)33 b(and)g(0)27 b Ft(62)h Fs(\026)22 b Fu(+)p 1524 3367 V 22 w Ft(I)8 b Fu(.)43 b(Th)m(us,)35 b(it)d(su\016ces)j(to)d(pro)m(v)m(e)i(it)e(when)i (0)28 b Ft(62)p 3265 3367 V 28 w(I)7 b Fu(.)456 3563 y(F)-8 b(or)38 b(an)m(y)i Fs(\022)j Ft(2)c(C)1079 3527 y Fl(1)1154 3563 y Fu(\()p Fn(R)27 b Ft(n)g(f)p Fu(0)p Ft(g)p Fu(\),)40 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1912 3527 y Fl(?)2010 3563 y Fu(=)39 b Fs(\022)s Fu(\()p Fs(H)8 b(P)2377 3527 y Fl(?)2435 3563 y Fu(\))39 b(b)m(y)h(Lemma)g(4.3)f(b)s(elo)m(w.)456 3679 y(Th)m(us)d Fs(H)8 b(P)871 3643 y Fl(?)960 3679 y Ft(2)31 b(C)1115 3643 y Fk(2)1109 3706 y Fl(I)1153 3687 y Fi(00)1198 3679 y Fu(\()p Fs(A)p Fu(\).)49 b(F)-8 b(urthermore,)36 b(using)f(Lemma)g (2.5,)g(w)m(e)g(deriv)m(e)i(from)456 3795 y(\(4.28\))31 b(the)i(estimate,)h(for)e Fs(\022)f Ft(2)d(C)1700 3759 y Fl(1)1694 3820 y Fm(c)1775 3795 y Fu(\()p Ft(I)7 b Fu(\))33 b(with)g Fs(\022)e Fu(=)d(1)k(near)h Ft(I)2706 3759 y Fl(0)2730 3795 y Fu(,)951 3964 y Fs(\022)s Fu(\()p Fs(H)8 b(P)1203 3923 y Fl(?)1261 3964 y Fu(\)[\()p Fs(H)g(P)1530 3923 y Fl(?)1588 3964 y Fu(\))1626 3979 y Fm(\034)1669 3964 y Fs(;)17 b(iA)p Fu(])p Fs(\022)s Fu(\()p Fs(H)8 b(P)2098 3923 y Fl(?)2156 3964 y Fu(\))28 b Ft(\025)g Fs(c\022)2417 3923 y Fk(2)2456 3964 y Fu(\()p Fs(H)8 b(P)2660 3923 y Fl(?)2718 3964 y Fu(\))p Fs(:)-2327 b Fu(\(4.30\))456 4128 y(No)m(w,)30 b(w)m(e)g(can)f(apply)h(Theorem)g (2.7)e(to)h Fs(H)8 b(P)2125 4092 y Fl(?)2212 4128 y Fu(with)29 b Ft(I)35 b Fu(=)28 b Fs(\022)2671 4092 y Fl(\000)p Fk(1)2765 4128 y Fu(\(1\),)i(yielding)g(the)456 4244 y(LAP)j(for)g Fs(H)8 b(P)1005 4208 y Fl(?)1097 4244 y Fu(on)33 b Ft(I)1294 4208 y Fl(0)1318 4244 y Fu(.)46 b(Let)33 b Fs(z)h Ft(2)c Fn(C)j Fu(with)h(Im)q(\()p Fs(z)t Fu(\))29 b Ft(6)p Fu(=)g(0.)46 b(By)34 b(F)-8 b(esh)m(bac)m(h)35 b(decom-)456 4361 y(p)s(osition)28 b(\(see)h([BFS])g(for)f(instance\),)i(\()p Fs(H)8 b(P)2074 4325 y Fl(?)2145 4361 y Ft(\000)13 b Fs(z)t Fu(\))2322 4325 y Fl(\000)p Fk(1)2418 4361 y Fs(P)2495 4325 y Fl(?)2581 4361 y Fu(=)28 b(\()p Fs(H)21 b Ft(\000)13 b Fs(z)t Fu(\))3002 4325 y Fl(\000)p Fk(1)3098 4361 y Fs(P)3175 4325 y Fl(?)3233 4361 y Fu(.)42 b(Let)456 4486 y(Re\()p Fs(z)t Fu(\))28 b Ft(2)g(I)879 4450 y Fl(0)936 4486 y Fu(and)k Fs(s)c Ft(2)g Fu([0;)17 b(1[.)43 b(Setting)1903 4460 y(~)1895 4486 y Fs(\022)31 b Fu(=)c(1)22 b Ft(\000)h Fs(\022)s Fu(,)33 b(w)m(e)g(write)645 4660 y Ft(h)p Fs(A)p Ft(i)796 4619 y Fl(\000)p Fm(s)887 4660 y Fu(\()p Fs(H)d Ft(\000)22 b Fs(z)t Fu(\))1222 4619 y Fl(\000)p Fk(1)1317 4660 y Fs(P)1394 4619 y Fl(?)1453 4660 y Ft(h)p Fs(A)p Ft(i)1604 4619 y Fl(\000)p Fm(s)1723 4660 y Fu(=)27 b Ft(h)p Fs(A)p Ft(i)1977 4619 y Fl(\000)p Fm(s)2069 4660 y Fu(\()p Fs(H)i Ft(\000)23 b Fs(z)t Fu(\))2404 4619 y Fl(\000)p Fk(1)2499 4660 y Fs(P)2576 4619 y Fl(?)2642 4634 y Fu(~)2634 4660 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Ft(h)p Fs(A)p Ft(i)2998 4619 y Fl(\000)p Fm(s)1031 4809 y Fu(+)p Ft(h)p Fs(A)p Ft(i)1258 4768 y Fl(\000)p Fm(s)1350 4809 y Fu(\()p Fs(H)g(P)1554 4768 y Fl(?)1634 4809 y Ft(\000)22 b Fs(z)t Fu(\))1820 4768 y Fl(\000)p Fk(1)1915 4809 y Ft(h)p Fs(A)p Ft(i)2066 4768 y Fl(\000)p Fm(s)2180 4809 y Ft(\001)f(h)p Fs(A)p Ft(i)2380 4768 y Fl(\000)p Fm(s)2472 4809 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2762 4768 y Fl(?)2820 4809 y Ft(h)p Fs(A)p Ft(i)2971 4768 y Fl(\000)p Fm(s)3062 4809 y Fs(:)456 4974 y Fu(Since)44 b Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1011 4938 y Fl(?)1115 4974 y Ft(2)47 b(C)1286 4938 y Fk(1)1325 4974 y Fu(\()p Fs(A)p Fu(\),)f Ft(h)p Fs(A)p Ft(i)1698 4938 y Fl(\000)p Fm(s)1790 4974 y Fs(\022)s Fu(\()p Fs(H)8 b Fu(\))p Fs(P)2080 4938 y Fl(?)2138 4974 y Ft(h)p Fs(A)p Ft(i)2289 4938 y Fl(\000)p Fm(s)2423 4974 y Fu(is)44 b(b)s(ounded)g(b)m(y)h(Prop)s(osi-)456 5099 y(tion)27 b(B.2.)42 b(This)28 b(yieds)h(the)f(reduced)h(LAP)f (\(1.3\))f(for)g Fs(H)8 b Fu(,)28 b(since)h(\()p Fs(H)19 b Ft(\000)12 b Fs(z)t Fu(\))3137 5063 y Fl(\000)p Fk(1)3240 5073 y Fu(~)3232 5099 y Fs(\022)s Fu(\()p Fs(H)c Fu(\))456 5216 y(is)33 b(uniformly)g(b)s(ounded.)1977 b Fj(\003)p eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(19)456 450 y Fu(The)29 b(second)h(pro)s(of)d(of)h (Theorem)i(4.1)e(uses)i(the)f(follo)m(wing)f(consequence)k(of)c(the)456 566 y(F)-8 b(esh)m(bac)m(h)34 b(decomp)s(osition)g(\(see)f([BFS])g(for) f(instance\).)456 748 y Fy(Lemma)j(4.3.)40 b Fp(F)-7 b(or)31 b(al)5 b(l)32 b Fs(')27 b Ft(2)h(C)1622 712 y Fl(1)1616 773 y Fm(c)1697 748 y Fu(\()p Fn(R)16 b Ft(n)g(f)p Fu(0)p Ft(g)p Fu(\))p Fp(,)32 b Fs(')p Fu(\()p Fs(H)8 b(P)2406 712 y Fl(?)2464 748 y Fu(\))p Fs(P)41 b Fu(=)28 b(0)k Fp(and)f Fs(')p Fu(\()p Fs(H)8 b(P)3245 712 y Fl(?)3303 748 y Fu(\))28 b(=)456 864 y Fs(')p Fu(\()p Fs(H)8 b Fu(\))p Fs(P)762 828 y Fl(?)819 864 y Fp(.)456 1104 y(Pr)-5 b(o)g(of.)41 b Fu(Let)23 b Fs(z)32 b Ft(2)c Fn(C)23 b Fu(with)h(Im\()p Fs(z)t Fu(\))29 b Ft(6)p Fu(=)e(0.)40 b(By)24 b(F)-8 b(esh)m(bac)m(h)24 b(decomp)s(osition,)i(\()p Fs(H)8 b(P)3306 1068 y Fl(?)3367 1104 y Ft(\000)456 1221 y Fs(z)t Fu(\))543 1184 y Fl(\000)p Fk(1)638 1221 y Fs(P)715 1184 y Fl(?)807 1221 y Fu(=)34 b(\()p Fs(H)e Ft(\000)25 b Fs(z)t Fu(\))1257 1184 y Fl(\000)p Fk(1)1352 1221 y Fs(P)1429 1184 y Fl(?)1488 1221 y Fu(.)54 b(Using)39 b(\(B.1\))o(,)f Fs(')p Fu(\()p Fs(H)8 b(P)2402 1184 y Fl(?)2460 1221 y Fu(\))p Fs(P)2575 1184 y Fl(?)2667 1221 y Fu(=)34 b Fs(')p Fu(\()p Fs(H)8 b Fu(\))p Fs(P)3083 1184 y Fl(?)3141 1221 y Fu(.)54 b(Since)456 1337 y(Ran)o Fs(P)48 b Fu(is)34 b(con)m(tained)h(in)f(the)h(k)m(ernel)g(of)f Fs(H)8 b(P)2135 1301 y Fl(?)2193 1337 y Fu(,)34 b Fs(')p Fu(\()p Fs(H)8 b(P)2522 1301 y Fl(?)2580 1337 y Fu(\))p Fs(P)43 b Fu(=)30 b Fs(')p Fu(\(0\))p Fs(P)43 b Fu(=)30 b(0,)k(b)m(y)456 1453 y(assumption)g(on)e Fs(')p Fu(.)43 b(Finally)-8 b(,)33 b Fs(')p Fu(\()p Fs(H)8 b Fu(\))p Fs(P)1898 1417 y Fl(?)1984 1453 y Fu(=)27 b Fs(')p Fu(\()p Fs(H)8 b(P)2355 1417 y Fl(?)2413 1453 y Fu(\))22 b Ft(\000)h Fu(0.)718 b Fj(\003)456 1635 y Fp(R)-5 b(emark)49 b Fu(4.4)p Fp(.)44 b Fu(In)38 b(the)h(second)g(pro)s(of,)f(the)g(idea)h(is)f(to)f (replace)i Fs(H)46 b Fu(par)37 b Fs(H)8 b(P)3359 1599 y Fl(?)3417 1635 y Fu(.)456 1751 y(Since)32 b(w)m(e)g(push)h(that)e(w)m (a)m(y)i(the)e(eigen)m(v)m(ectors)j(of)d Fs(H)39 b Fu(lea)m(ving)32 b(in)g(Ran)p Fs(E)3120 1766 y Fl(I)3167 1751 y Fu(\()p Fs(H)8 b Fu(\))31 b(in)456 1867 y(the)f(k)m(ernel)h(of)f Fs(H)8 b(P)1180 1831 y Fl(?)1238 1867 y Fu(,)30 b(they)h(are)f(no)f (longer)h(an)g(obstacle)h(to)e(the)h(strict)h(Mourre)456 1984 y(estimate)h(on)f Ft(I)7 b Fu(,)32 b(if)f(0)c Ft(62)p 1357 1903 62 4 v 28 w(I)8 b Fu(.)43 b(The)32 b(main)f(di\013erence)i(b) s(et)m(w)m(een)g(the)e(t)m(w)m(o)h(previous)456 2100 y(pro)s(ofs)40 b(is)i(probably)f(the)h(use)g(of)e(energy)i(translation) f(for)g Fs(H)48 b Fu(in)42 b(the)f(second)456 2216 y(one)32 b(to)h(a)m(v)m(oid)g(the)g(case)h(where)f(0)28 b Ft(2)p 1838 2136 V 28 w(I)8 b Fu(.)456 2456 y(4.4.)48 b Fy(An)58 b(arti\014cial)h(but)f(instructiv)m(e)g(example.)50 b Fu(In)h(this)g(section,)56 b(w)m(e)456 2572 y(construct)38 b(an)f(example)i(of)e(op)s(erators)g Fs(H)45 b Fu(and)37 b Fs(A)p Fu(,)i(for)e(whic)m(h)i(Theorems)g(1.4)456 2688 y(and)e(4.1)g(apply)h(but)g(the)g(Mourre)f(estimate)i(\(1.1\))e(cannot) g(b)s(e)h(true.)58 b(In)38 b(par-)456 2805 y(ticular,)c(Theorems)h(1.2) e(and)h(2.7)f(do)g(not)g(apply)h(to)f(this)i(example.)47 b(Our)34 b(con-)456 2921 y(truction)41 b(is)h(quite)g(arti\014cial)g (but)f(our)g(op)s(erators)h Fs(H)48 b Fu(and)42 b Fs(A)f Fu(presen)m(ts)i(some)456 3037 y(structural)33 b(similarit)m(y)h(with)f (op)s(erators)g(in)f([DJ].)456 3211 y(Let)44 b Fo(H)731 3226 y Fk(0)770 3211 y Fs(;)17 b Fo(H)903 3226 y Fk(1)986 3211 y Fu(b)s(e)45 b(in\014nite)h(dimensional)g(complex)g(Hilb)s(ert)f (spaces.)80 b(Let)45 b Fs(H)3405 3226 y Fk(0)456 3328 y Fu(and)32 b Fs(A)718 3343 y Fk(0)789 3328 y Fu(b)s(e)g(self-adjoin)m (t)g(op)s(erators)g(in)g Fo(H)2055 3343 y Fk(0)2126 3328 y Fu(suc)m(h)h(that)f Fs(H)2637 3343 y Fk(0)2708 3328 y Fu(is)g(b)s(ounded,)h Fs(H)3311 3343 y Fk(0)3378 3328 y Ft(2)456 3444 y(C)514 3408 y Fk(2)553 3444 y Fu(\()p Fs(A)664 3459 y Fk(0)704 3444 y Fu(\),)24 b(and)e(suc)m(h)h(that)e(the) i(strict)f(Mourre)g(estimate)h(\(1.1\))e(with)i Fs(K)35 b Fu(=)27 b(0)22 b(holds)456 3560 y(true)30 b(for)g Fs(H)887 3575 y Fk(0)957 3560 y Fu(and)h Fs(A)1218 3575 y Fk(0)1288 3560 y Fu(on)f(some)h(b)s(ounded,)h(in\014nite)g(in)m(terv)-5 b(al)31 b Ft(I)7 b Fu(.)43 b(F)-8 b(or)30 b(instance,)456 3676 y(w)m(e)e(can)g(tak)m(e)g(suitably)h(a)f(b)s(ounded,)h(in\014nite) g(in)m(terv)-5 b(al)28 b Ft(I)35 b Fu(included)29 b(in)f(]0;)17 b(+)p Ft(1)p Fu([,)456 3793 y Fo(H)545 3808 y Fk(0)621 3793 y Fu(=)37 b(L)795 3756 y Fk(2)835 3793 y Fu(\()p Fn(R)945 3756 y Fm(d)985 3793 y Fu(\),)j Fs(H)1171 3808 y Fk(0)1248 3793 y Fu(a)f(smo)s(oth,)h(increasing,)h(and)d(b)s(ounded)h (function)g(of)f(the)456 3909 y(Laplacian)24 b(on)h Fn(R)1093 3873 y Fm(d)1133 3909 y Fu(,)i(and)d Fs(A)1441 3924 y Fk(0)1505 3909 y Fu(the)h(generator)g(of)f(dilation)h(in)g Fn(R)2722 3873 y Fm(d)2787 3909 y Fu(\(cf.)40 b([ABG,)25 b(M]\).)456 4025 y(Let)g Fs(A)696 4040 y Fk(1)762 4025 y Fu(b)s(e)g(self-adjoin)m(t)h(op)s(erator)f(in)h Fo(H)1964 4040 y Fk(1)2003 4025 y Fu(.)41 b(Let)26 b(\()p Fs(g)2324 4040 y Fm(n)2371 4025 y Fu(\))2409 4040 y Fm(n)2481 4025 y Fu(b)s(e)g(a)f(b)s(ounded)i(sequence)456 4141 y(in)32 b Ft(D)s Fu(\()p Fs(A)760 4105 y Fk(2)760 4166 y(1)799 4141 y Fu(\))g(of)f(indep)s(enden)m(t)k(v)m(ectors)e(suc)m(h)h(that)d (it)h(is)h(b)s(ounded)g(for)e(the)i(graph)456 4257 y(norm)41 b(of)f Fs(A)911 4221 y Fk(2)911 4282 y(1)951 4257 y Fu(.)68 b(Let)41 b(\()p Fs(\013)1329 4272 y Fm(n)1376 4257 y Fu(\))1414 4272 y Fm(n)1503 4257 y Ft(2)h Fs(`)1652 4221 y Fk(1)1691 4257 y Fu(,)h(a)e(sequence)j(of)c(nonzero)h(reals.)69 b(The)42 b(serie)456 4374 y(\()494 4299 y Ff(P)599 4403 y Fm(n)p Fl(\025)p Fk(0)752 4374 y Fs(\013)814 4389 y Fm(n)861 4374 y Ft(j)p Fs(g)936 4389 y Fm(n)983 4374 y Ft(ih)p Fs(g)1108 4389 y Fm(n)1154 4374 y Ft(j)p Fu(\))1220 4389 y Fm(n)1289 4374 y Fu(con)m(v)m(erge)23 b(absolutely)h(in)e(the)g (Banac)m(h)h(space)g(of)f(b)s(ounded)456 4491 y(op)s(erators)48 b(on)g Fo(H)1143 4506 y Fk(1)1182 4491 y Fu(.)90 b(Let)49 b Fs(C)55 b Fu(b)s(e)48 b(its)h(sum.)91 b(It)49 b(is)f(a)g(self-adjoin) m(t,)53 b(compact)456 4608 y(op)s(erator)28 b(of)g(in\014nite)i(rank.) 42 b(By)29 b(Prop)s(osition)g(2.1,)h(eac)m(h)f Fs(\013)2669 4623 y Fm(n)2716 4608 y Ft(j)p Fs(g)2791 4623 y Fm(n)2838 4608 y Ft(ih)p Fs(g)2963 4623 y Fm(n)3009 4608 y Ft(j)e(2)h(C)3216 4571 y Fk(2)3256 4608 y Fu(\()p Fs(A)3367 4623 y Fk(1)3406 4608 y Fu(\))456 4724 y(and)45 b(\()696 4649 y Ff(P)801 4753 y Fm(n)p Fl(\025)p Fk(0)955 4724 y Fs(\013)1017 4739 y Fm(n)1064 4724 y Fu([)p Ft(j)p Fs(g)1166 4739 y Fm(n)1213 4724 y Ft(ih)p Fs(g)1338 4739 y Fm(n)1384 4724 y Ft(j)p Fs(;)17 b(A)1529 4739 y Fk(1)1568 4724 y Fu(]\))1633 4739 y Fm(n)1725 4724 y Fu(con)m(v)m(erges)48 b(absolutely)-8 b(,)50 b(since)d(\()p Ft(k)p Fs(g)3061 4739 y Fm(n)3107 4724 y Ft(k)p Fu(\))3195 4739 y Fm(n)3287 4724 y Fu(and)456 4851 y(\()p Ft(k)p Fs(A)617 4866 y Fk(1)656 4851 y Fs(g)703 4866 y Fm(n)750 4851 y Ft(k)p Fu(\))838 4866 y Fm(n)913 4851 y Fu(are)29 b(b)s(ounded.)43 b(By)30 b(Prop)s(osition)f(2.2,)g Fs(C)35 b Ft(2)28 b(C)2616 4815 y Fk(1)2656 4851 y Fu(\()p Fs(A)2767 4866 y Fk(1)2806 4851 y Fu(\))h(and)g([)p Fs(C)r(;)17 b(A)3275 4866 y Fk(1)3314 4851 y Fu(])28 b(=)456 4893 y Ff(P)561 4919 y Fl(1)561 4997 y Fm(n)p Fk(=0)715 4968 y Fs(\013)777 4983 y Fm(n)824 4968 y Fu([)p Ft(j)p Fs(g)926 4983 y Fm(n)972 4968 y Ft(ih)p Fs(g)1097 4983 y Fm(n)1143 4968 y Ft(j)p Fs(;)17 b(A)1288 4983 y Fk(1)1327 4968 y Fu(].)43 b(Applying)31 b(this)f(argumen)m(t)h(again,)f(this)g(implies)i(that)456 5099 y Fs(C)j Ft(2)30 b(C)715 5063 y Fk(2)754 5099 y Fu(\()p Fs(A)865 5114 y Fk(1)905 5099 y Fu(\).)45 b(Let)34 b Fs(\025)29 b Ft(2)1369 5074 y Fu(\027)1372 5099 y Ft(I)7 b Fu(.)46 b(W)-8 b(e)34 b(can)f(c)m(ho)s(ose)h(\()p Fs(\013)2264 5114 y Fm(n)2311 5099 y Fu(\))2349 5114 y Fm(n)2429 5099 y Fu(suc)m(h)h(that)e([)p Fs(\025)23 b Ft(\000)g(k)p Fs(C)7 b Ft(k)p Fu(;)17 b Fs(\025)22 b Fu(+)456 5216 y Ft(k)p Fs(C)7 b Ft(k)p Fu(])27 b Ft(\032)h(I)7 b Fu(.)42 b(Let)25 b Fs(H)1170 5231 y Fk(1)1237 5216 y Fu(=)i Fs(\025)7 b Fu(+)g Fs(C)g Fu(.)41 b(Let)25 b Fs(H)33 b Fu(b)s(e)25 b(the)h(b)s(ounded)g(self-adjoin)m(t)f(op)s(erator)p eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 456 255 a Fq(20)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fu(acting)i(in)g Fo(H)54 b Fu(:=)28 b Fo(H)1212 465 y Fk(0)1260 450 y Ft(\010)9 b Fo(H)1435 465 y Fk(1)1500 450 y Fu(b)m(y)27 b Fs(H)1710 465 y Fk(0)1758 450 y Ft(\010)9 b Fs(H)1925 465 y Fk(1)1965 450 y Fu(.)41 b(Let)27 b Fs(A)f Fu(b)s(e)g(the)h(self-adjoin)m(t)f(op)s(erator)456 566 y(acting)35 b(in)h Fo(H)63 b Fu(b)m(y)36 b Fs(A)1231 581 y Fk(0)1295 566 y Ft(\010)25 b Fs(A)1470 581 y Fk(1)1510 566 y Fu(.)53 b(Since)37 b([)p Fs(H)r(;)17 b(iA)p Fu(])33 b(=)g([)p Fs(H)2385 581 y Fk(0)2424 566 y Fs(;)17 b(iA)2574 581 y Fk(0)2614 566 y Fu(])24 b Ft(\010)h Fu([)p Fs(C)7 b Fu(;)17 b Fs(iA)3021 581 y Fk(1)3061 566 y Fu(])36 b(as)g(form)456 683 y(on)f Ft(D)s Fu(\()p Fs(A)p Fu(\))24 b Ft(\002)g(D)s Fu(\()p Fs(A)p Fu(\),)36 b(the)g(regularit)m(y)g(of)f Fs(H)2053 698 y Fk(0)2128 683 y Fu(w.r.t.)52 b Fs(A)2480 698 y Fk(0)2555 683 y Fu(and)36 b(the)g(regularit)m(y)g(of)456 799 y Fs(C)h Fu(w.r.t.)43 b Fs(A)906 814 y Fk(1)975 799 y Fu(imply)32 b(that)d Fs(H)36 b Ft(2)28 b(C)1724 763 y Fk(2)1763 799 y Fu(\()p Fs(A)p Fu(\).)43 b(Since)31 b(Ran)p Fs(C)37 b Fu(is)31 b(in\014nite)g(dimensional)456 915 y(and)41 b(the)h(sp)s(ectrum)h(of)e Fs(H)1467 930 y Fk(1)1548 915 y Fu(is)h(con)m(tained)g(in)g Ft(I)7 b Fu(,)44 b(the)e(p)s(oin)m(t)g(sp)s(ectrum)h(of)e Fs(H)456 1031 y Fu(in)g Ft(I)49 b Fu(is)42 b(in\014nite)h(therefore)f(the)f (Mourre)h(estimate)h(\(1.1\))e(cannot)g(hold)h(true)456 1147 y(on)f Ft(I)49 b Fu(b)m(y)42 b(Remark)g(1.1.)70 b(Since)43 b(the)e(strict)i(Mourre)f(estimate)g(for)f Fs(H)3141 1162 y Fk(0)3222 1147 y Fu(holds)456 1264 y(true)g(on)f Ft(I)7 b Fu(,)44 b Fs(H)1026 1279 y Fk(0)1106 1264 y Fu(has)d(no)f(eigen)m(v)-5 b(alue)42 b(in)f Ft(I)49 b Fu(b)m(y)41 b(Remark)h(1.1.)67 b(Let)41 b Fs(P)54 b Fu(b)s(e)41 b(the)456 1380 y(orthogonal)32 b(pro)5 b(jection)34 b(on)m(to)f(the)h (pure)g(p)s(oin)m(t)f(sp)s(ectral)i(subspace)g(of)e Fs(H)8 b Fu(.)45 b(By)456 1496 y(the)c(previous)h(prop)s(erties,)i Fs(P)1601 1460 y Fl(?)1660 1496 y Fu([)p Fs(H)r(;)17 b(iA)p Fu(])p Fs(P)2024 1460 y Fl(?)2124 1496 y Fu(=)42 b Fs(P)2319 1460 y Fl(?)2377 1496 y Fu(\([)p Fs(H)2523 1511 y Fk(0)2563 1496 y Fs(;)17 b(iA)2713 1511 y Fk(0)2752 1496 y Fu(])28 b Ft(\010)g Fu(0\))p Fs(P)3076 1460 y Fl(?)3134 1496 y Fu(.)69 b(Th)m(us)456 1612 y(the)43 b(strict)h(Mourre)g(estimate)g(for)f Fs(H)1897 1627 y Fk(0)1979 1612 y Fu(on)g Ft(I)51 b Fu(implies)45 b(the)e(strict,)k(pro) 5 b(jected)456 1729 y(Mourre)33 b(estimate)g(\(4.28\))f(for)g Fs(H)40 b Fu(on)33 b Ft(I)7 b Fu(.)1137 2062 y Fv(Appendix)37 b Fu(A.)49 b Fv(Symbolic)39 b(calculus.)456 2294 y Fu(In)k(this)g (section,)k(w)m(e)d(recall)g(w)m(ell)g(kno)m(wn)g(facts)f(on)g(sym)m(b) s(olic)i(calculus)f(and)456 2411 y(almost)30 b(analytic)h(extensions)h (\(see)g([DG)o(][App)s(endix)g(C]\).)e(W)-8 b(e)31 b(also)f(sho)m(w)i (that)456 2527 y(some)27 b(sequences)j(of)d(functions)h(used)g(in)f (the)g(main)g(text)h(are)f(b)s(ounded)g(in)g(some)456 2643 y(sym)m(b)s(ol)33 b(class.)456 2817 y(F)-8 b(or)31 b Fs(\032)d Ft(2)g Fn(R)p Fu(,)33 b(let)g Ft(S)1142 2781 y Fm(\032)1216 2817 y Fu(b)s(e)f(the)h(class)h(of)e(function)h Fs(')28 b Ft(2)g(C)2482 2781 y Fl(1)2557 2817 y Fu(\()p Fn(R)p Fu(;)17 b Fn(C)p Fu(\))32 b(suc)m(h)i(that)913 3006 y Ft(8)p Fs(k)d Ft(2)d Fn(N)p Fs(;)115 b(C)1428 3021 y Fm(k)1470 3006 y Fu(\()p Fs(')p Fu(\))27 b(:=)h(sup)1781 3087 y Fm(t)p Fl(2)p Fg(R)1932 3006 y Ft(h)p Fs(t)p Ft(i)2045 2964 y Fl(\000)p Fm(\032)p Fk(+)p Fm(k)2233 3006 y Ft(j)p Fs(@)2317 2964 y Fm(k)2312 3030 y(t)2360 3006 y Fs(')p Fu(\()p Fs(t)p Fu(\))p Ft(j)f Fs(<)h Ft(1)p Fs(:)-2365 b Fu(\(A.1\))456 3264 y(W)-8 b(e)38 b(also)g(write)g Fs(')1148 3227 y Fk(\()p Fm(k)r Fk(\))1283 3264 y Fu(for)g Fs(@)1494 3227 y Fm(k)1489 3288 y(t)1537 3264 y Fs(')p Fu(.)59 b(Equip)s(ed)39 b(with)g(the)f(semi-norms)h(de\014ned)g(b)m(y) 456 3380 y(\(A.1\),)f Ft(S)813 3344 y Fm(\032)891 3380 y Fu(is)g(a)f(F)-8 b(r)m(\023)-46 b(ec)m(het)38 b(space.)59 b(Leibniz')39 b(form)m(ula)e(implies)i(the)f(con)m(tin)m(uous)456 3496 y(em)m(b)s(edding:)1521 3684 y Ft(S)1588 3643 y Fm(\032)1651 3684 y Ft(\001)22 b(S)1768 3643 y Fm(\032)1804 3619 y Fi(0)1859 3684 y Ft(\032)28 b(S)2031 3643 y Fm(\032)p Fk(+)p Fm(\032)2158 3619 y Fi(0)2186 3684 y Fs(:)-1757 b Fu(\(A.2\))456 3926 y(F)-8 b(or)30 b(the)i(functional)g(calculus)h (of)e(the)h(op)s(erator)f Fs(A)h Fu(\(see)h(\(B.1)e(\)\),)h(w)m(e)g (shall)g(use)456 4042 y(the)h(follo)m(wing)f(result)i(on)e(almost)h (analytic)h(extension.)456 4232 y Fy(Lemma)j(A.1.)j Fp(L)-5 b(et)33 b Fs(')28 b Ft(2)g(S)1513 4196 y Fm(\032)1587 4232 y Fp(with)33 b Fs(\032)28 b Ft(2)g Fn(R)p Fp(.)44 b(F)-7 b(or)32 b(al)5 b(l)33 b Fs(l)d Ft(2)e Fn(N)p Fp(,)33 b(ther)-5 b(e)33 b(is)g(a)g(smo)-5 b(oth)456 4348 y(function)45 b Fs(')914 4312 y Fg(C)1013 4348 y Fu(:)i Fn(C)g Ft(!)f Fn(C)p Fp(,)i(c)-5 b(al)5 b(l)45 b(an)52 b Fu(almost-analytic)45 b(extension)i Fp(of)e Fs(')p Fp(,)j(such)456 4464 y(that:)884 4690 y Fs(')948 4649 y Fg(C)1000 4690 y Ft(j)1028 4705 y Fg(R)1108 4690 y Fu(=)27 b Fs(';)1568 4605 y Ff(\014)1568 4665 y(\014)1611 4622 y Fs(@)5 b(')1731 4586 y Fg(C)p 1611 4667 173 4 v 1644 4758 a Fs(@)p 1700 4703 50 4 v 5 w(z)1794 4690 y Fu(\()p Fs(z)t Fu(\))1919 4605 y Ff(\014)1919 4665 y(\014)1980 4690 y Ft(\024)29 b Fs(c)2128 4705 y Fk(1)2167 4690 y Ft(h)p Fu(Re\()p Fs(z)t Fu(\))p Ft(i)2485 4649 y Fm(\032)p Fl(\000)p Fk(1)p Fl(\000)p Fm(l)2692 4690 y Ft(j)p Fu(Im)q(\()p Fs(z)t Fu(\))p Ft(j)2990 4649 y Fm(l)456 4690 y Fu(\(A.3\))1568 4882 y(supp)q Fs(')1833 4841 y Fg(C)1913 4882 y Ft(\032)f(f)p Fs(x)22 b Fu(+)g Fs(iy)31 b Ft(j)d(j)p Fs(y)t Ft(j)e(\024)i Fs(c)2692 4897 y Fk(2)2731 4882 y Ft(h)p Fs(x)p Ft(ig)p Fs(;)-2485 b Fu(\(A.4\))1568 5032 y Fs(')1632 4990 y Fg(C)1684 5032 y Fu(\()p Fs(x)22 b Fu(+)g Fs(iy)t Fu(\))27 b(=)h(0)p Fs(;)51 b Fp(if)35 b Fs(')p Fu(\()p Fs(x)p Fu(\))28 b(=)f(0)p Fs(:)-2319 b Fu(\(A.5\))456 5216 y Fp(for)34 b(c)-5 b(onstants)35 b Fs(c)1082 5231 y Fk(1)1121 5216 y Fp(,)g Fs(c)1228 5231 y Fk(2)1302 5216 y Fp(dep)-5 b(ending)33 b(on)i(the)g(semi-norms) 41 b Fu(\(A.1\))35 b Fp(of)g Fs(')f Fp(in)h Ft(S)3254 5179 y Fm(\032)3295 5216 y Fp(.)p eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(21)456 450 y Fu(The)36 b(function)1045 431 y Fs(\037)1106 465 y Fm(R)1163 450 y Fu(,)h(giv)m(en)g(b)m(y)h (\(3.9\))o(,)f(b)s(elongs)f(to)g Ft(S)2432 414 y Fm(\032)2473 450 y Fu(,)g(for)g(an)m(y)g Fs(\032)g Fu(and)g(an)m(y)g Fs(R)q Fu(.)456 566 y(But)c(w)m(e)i(need)g(to)e(kno)m(w)i(that)e(the)h (family)g(\()p Fs(\037)2171 581 y Fm(R)2229 566 y Fu(\))2267 581 y Fm(R)p Fl(\025)p Fk(1)2447 566 y Fu(is)g(b)s(ounded)g(in)g(some)h Ft(S)3369 530 y Fm(\032)3410 566 y Fu(.)456 741 y Fy(Lemma)43 b(A.2.)h Fp(L)-5 b(et)38 b Fs(\034)46 b Ft(2)35 b(C)1522 705 y Fl(1)1597 741 y Fu(\()p Fn(R)p Fu(;)17 b Fn(R)p Fu(\))38 b Fp(such)g(that)h Fs(\034)2380 705 y Fl(0)2438 741 y Ft(2)34 b(C)2596 705 y Fl(1)2590 766 y Fm(c)2671 741 y Fu(\()p Fn(R)2781 705 y Fl(\003)2821 741 y Fu(;)17 b Fn(R)p Fu(\))p Fp(.)55 b(Then)37 b(the)456 857 y(family)d Fu(\()p Fs(\034)832 872 y Fm(R)890 857 y Fu(\))928 872 y Fm(R>)p Fk(1)1076 857 y Fp(,)h(with)f Fs(\034)1394 872 y Fm(R)1452 857 y Fu(\()p Fs(x)p Fu(\))28 b(:=)g Fs(\034)11 b Fu(\()p Fs(x=R)q Fu(\))p Fp(,)35 b(is)g(b)-5 b(ounde)g(d)34 b(in)h Ft(S)2778 821 y Fk(0)2818 857 y Fp(.)456 1083 y(Pr)-5 b(o)g(of.)41 b Fu(Let)34 b Fs(k)g Ft(2)d Fn(N)p Fu(.)48 b(The)36 b(semi-norm)f Fs(C)2011 1098 y Fm(k)2053 1083 y Fu(\()p Fs(\034)11 b Fu(\))35 b(\(cf.)49 b(\(A.1\))o(\))35 b(is)f(b)s(ounded)i(ab)s(o)m(v)m(e)456 1202 y(b)m(y)i(the)f(\(sup)18 b(supp)q Fs(\034)1224 1166 y Fk(\()p Fm(k)r Fk(\))1322 1202 y Fu(\))1360 1166 y Fm(k)1439 1202 y Fu(times)38 b(the)g Fs(L)1943 1166 y Fl(1)2018 1202 y Fu(-norm)f(of)f Fs(\034)2478 1166 y Fk(\()p Fm(k)r Fk(\))2576 1202 y Fu(.)57 b(F)-8 b(or)37 b(all)g Fs(R)f(>)f Fu(1)i(and)456 1318 y Fs(t)28 b Ft(2)g Fn(R)p Fu(,)855 1478 y Ft(j)p Fs(t)p Ft(j)946 1437 y Fm(k)1010 1478 y Ft(\001)22 b(j)p Fu(\()p Fs(\034)1168 1493 y Fm(R)1226 1478 y Fu(\))1264 1437 y Fk(\()p Fm(k)r Fk(\))1361 1478 y Fu(\()p Fs(t)p Fu(\))p Ft(j)28 b Fu(=)f(\()p Ft(j)p Fs(t)p Ft(j)p Fs(=R)q Fu(\))1922 1437 y Fm(k)1986 1478 y Ft(\001)22 b(j)p Fs(\034)2117 1437 y Fk(\()p Fm(k)r Fk(\))2215 1478 y Fu(\()p Fs(t=R)q Fu(\))p Ft(j)27 b(\024)h Fs(C)2680 1493 y Fm(k)2723 1478 y Fu(\()p Fs(\034)11 b Fu(\))p Fs(:)456 1636 y Fu(Th)m(us)34 b(\()p Fs(\034)783 1651 y Fm(R)841 1636 y Fu(\))879 1651 y Fm(R>)p Fk(1)1059 1636 y Fu(is)f(b)s(ounded)h(in)e Ft(S)1736 1600 y Fk(0)1777 1636 y Fu(.)1563 b Fj(\003)456 1810 y Fu(Concerning)34 b(the)f(functions)h(de\014ned)h(in)e(\(3.9\))o(,)h(\(3.14\))e(and)h (just)g(after)g(\(3.14\))o(,)456 1926 y(w)m(e)g(ha)m(v)m(e)h(the)456 2101 y Fy(Corollary)i(A.3.)k Fp(L)-5 b(emma)32 b(A.2)g(applies)g(to)h Fs(\034)39 b Fu(=)2344 2082 y Fs(\037)2405 2101 y(;)2454 2082 y Fu(~)-54 b Fs(\037)2509 2101 y(;)2559 2082 y Fu(~)f Fs(\037)2614 2116 y Fm(\033)2661 2101 y Fs(;)17 b Fu(\()2749 2082 y(~)-55 b Fs(\037)2804 2116 y Fm(\033)2851 2101 y Fu(\))2889 2065 y Fk(1)p Fm(=)p Fk(2)2999 2101 y Fp(,)33 b Fu(\()p Fs(\033)3165 2082 y Fu(~)-55 b Fs(\037)3220 2042 y Fl(0)3220 2126 y Fm(\033)3267 2101 y Fu(\))3305 2065 y Fk(1)p Fm(=)p Fk(2)3415 2101 y Fp(,)456 2217 y(for)34 b Fs(\033)e Ft(2)c(f\000)p Fu(;)17 b(+)p Ft(g)p Fp(,)35 b(and)f(also)g(to)h(their)g(derivatives.)456 2450 y Fu(W)-8 b(e)33 b(no)m(w)g(fo)s(cus)g(on)f(the)h(functions)h Fs( )1864 2465 y Fm(\033)n(;R)1980 2450 y Fu(,)f(de\014ned)h(in)f(\(3.19\))o(.) 456 2646 y Fy(Lemma)39 b(A.4.)i Fp(The)34 b(family)h Fu(\()p Fs( )1698 2595 y Fk(1)p Fm(=)p Fk(2)1694 2673 y Fm(\033)n(;R)1810 2646 y Fu(\))1848 2661 y Fm(R>)p Fk(1)2031 2646 y Fp(is)g(b)-5 b(ounde)g(d)34 b(in)g Ft(S)2693 2610 y Fk(0)2733 2646 y Fp(.)456 2878 y(Pr)-5 b(o)g(of.)41 b Fu(By)34 b(\(3.19\))o(,)746 3053 y Fs( )813 3002 y Fk(1)p Fm(=)p Fk(2)809 3080 y Fm(\033)n(;R)926 3053 y Fu(\()p Fs(x)p Fu(\))28 b(=)f(\()p Fs(\033)1285 3034 y(\037)1346 3012 y Fl(0)1346 3078 y Fm(\033)1393 3053 y Fu(\))1431 3012 y Fk(1)p Fm(=)p Fk(2)1541 3053 y Fu(\()p Fs(x=R)q Fu(\)\()p Fs(h)22 b Ft(\000)h Fs(\033)t(x)p Ft(h)p Fs(x)p Ft(i)2259 3012 y Fl(\000)p Fk(2)p Fm(s)2386 3053 y Fu(\))2424 3012 y Fk(1)p Fm(=)p Fk(2)2534 3034 y Fs(\037)2596 3012 y Fk(1)p Fm(=)p Fk(2)2596 3078 y Fm(\033)2706 3053 y Fu(\()p Fs(x=R)q Fu(\))p Fs(;)456 3218 y Fu(for)43 b(all)g Fs(x)k Ft(2)g Fn(R)d Fu(and)g(all)f Fs(R)48 b(>)e Fu(1.)77 b(By)44 b(de\014nition)g(of)g Fs(h)f Fu(\(cf.)77 b(\(3.15\))o(\),)47 b Fs(x)g Ft(7!)456 3337 y Fu(\()p Fs(h)16 b Ft(\000)g Fs(\033)t(x)p Ft(h)p Fs(x)p Ft(i)906 3301 y Fl(\000)p Fk(2)p Fm(s)1033 3337 y Fu(\))1071 3301 y Fk(1)p Fm(=)p Fk(2)1210 3337 y Fu(b)s(elongs)30 b(to)f Ft(S)1740 3301 y Fk(0)1780 3337 y Fu(.)43 b(No)m(w)30 b(the)f(result)i(follo)m(ws)f(from)f(Corollary)456 3453 y(A.3)j(and)h(\(A.2\).)2288 b Fj(\003)1006 3711 y Fv(Appendix)37 b Fu(B.)49 b Fv(Commut)-7 b(a)g(tor)40 b(exp)-7 b(ansions.)456 3944 y Fu(In)49 b(this)h(section,)55 b(w)m(e)50 b(recall)g (Hel\013er-Sj\177)-49 b(ostrand's)51 b(functional)e(calculus)i(\(cf.) 456 4060 y([HeS,)33 b(DG)o(]\))g(and)f(comm)m(utator)h(expansions)i (\(cf.)43 b([DG]\).)456 4234 y(Let)33 b Fs(\032)c(<)g Fu(0)k(and)g Fs(')28 b Ft(2)i(S)1342 4198 y Fm(\032)1383 4234 y Fu(.)45 b(The)34 b(b)s(ounded)g(op)s(erator)f Fs(')p Fu(\()p Fs(A)p Fu(\))g(can)g(b)s(e)h(reco)m(v)m(er)g(b)m(y)456 4350 y(Hel\013er-Sj\177)-49 b(ostrand's)33 b(form)m(ula:)1088 4569 y Fs(')p Fu(\()p Fs(A)p Fu(\))27 b(=)1479 4502 y Fs(i)p 1442 4546 108 4 v 1442 4637 a Fu(2)p Fs(\031)1576 4433 y Ff(Z)1631 4659 y Fg(C)1710 4502 y Fs(@)5 b(')1830 4465 y Fg(C)p 1710 4546 173 4 v 1743 4637 a Fs(@)p 1799 4582 50 4 v 5 w(z)1893 4569 y Fu(\()p Fs(z)27 b Ft(\000)22 b Fs(A)p Fu(\))2213 4528 y Fl(\000)p Fk(1)2308 4569 y Fs(dz)k Ft(^)d Fs(d)p 2570 4514 V(z)t(;)-2190 b Fu(\(B.1\))456 4783 y(where)40 b(the)h(in)m(tegral)f(exists)h(in)f(the)g(norm)g(top)s (ology)-8 b(,)40 b(b)m(y)i(\(A.3\))e(with)g Fs(l)i Fu(=)e(1.)456 4900 y(This)33 b(can)g(b)s(e)g(extended)i(as)d(sho)m(wn)i(in)456 5074 y Fy(Lemma)48 b(B.1.)e Fp(L)-5 b(et)42 b Fs(k)j Ft(2)d Fn(N)p Fp(,)i Fs(\032)e(<)f(k)k Fp(and)d Fs(')f Ft(2)h(S)2413 5038 y Fm(\032)2454 5074 y Fp(.)67 b(Str)-5 b(ongly)43 b(in)f Ft(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)3335 5038 y Fm(k)3377 5074 y Fu(\))p Fp(,)456 5191 y Fu(\(B.1\))34 b Fp(holds)g(true.)p eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 456 255 a Fq(22)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 450 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Let)33 b Fs(f)38 b Ft(2)28 b(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)1378 414 y Fm(k)1420 450 y Fu(\))f Ft(\032)i(D)s Fu(\()p Fs(')p Fu(\()p Fs(A)p Fu(\)\))j(and)2181 431 y Fs(\037)2242 465 y Fm(R)2333 450 y Fu(b)s(e)g(lik)m(e)i(in)f(\(3.9\))o(,)g(then)671 701 y Fs(')p Fu(\()p Fs(A)p Fu(\))884 682 y Fs(\037)945 716 y Fm(R)1003 701 y Fu(\()p Fs(A)p Fu(\))p Fs(f)38 b Fu(=)1389 634 y Fs(i)p 1352 678 108 4 v 1352 769 a Fu(2)p Fs(\031)1486 565 y Ff(Z)1542 791 y Fg(C)1620 634 y Fs(@)5 b Fu(\()p Fs(')1778 649 y Fm(k)1822 614 y Fs(\037)1883 649 y Fm(R)1940 634 y Fu(\))1978 597 y Fg(C)p 1620 678 411 4 v 1772 769 a Fs(@)p 1828 715 50 4 v 5 w(z)2041 701 y Fu(\()p Fs(z)26 b Ft(\000)d Fs(A)p Fu(\))2361 660 y Fl(\000)p Fk(1)2455 701 y Ft(h)p Fs(A)p Ft(i)2606 660 y Fm(k)2649 701 y Fs(f)k(dz)g Ft(^)22 b Fs(d)p 2986 646 V(z)5 b(;)-2607 b Fu(\(B.2\))456 950 y(where)37 b Fs(')805 965 y Fm(k)847 950 y Fu(\()p Fs(t)p Fu(\))d(:=)f Fs(')p Fu(\()p Fs(t)p Fu(\))p Ft(h)p Fs(t)p Ft(i)1416 914 y Fl(\000)p Fm(k)1549 950 y Fu(b)s(elongs)j(to)g Ft(S)2092 914 y Fm(\032)p Fl(\000)p Fm(k)2226 950 y Fu(.)54 b(By)36 b(Lemma)h(A.2)e(and)h(\(A.2\),)456 1066 y(\()p Fs(')558 1081 y Fm(k)600 1047 y Fs(\037)661 1081 y Fm(R)719 1066 y Fu(\))757 1081 y Fm(R)849 1066 y Fu(is)g(b)s(ounded)f(in)g Ft(S)1533 1030 y Fm(\032)p Fl(\000)p Fm(k)1668 1066 y Fu(.)50 b(Since)35 b Fs(\032)24 b Ft(\000)g Fs(k)35 b(<)c Fu(0,)k(the)g(result)h(follo)m(ws)g(from)456 1182 y(\(A.3\))c(with)h Fs(l)d Fu(=)e(1)k(and)h(the)g(dominated)g(con)m(v)m(ergence)i(theorem.) 436 b Fj(\003)456 1357 y Fu(Notice)36 b(that,)i(for)d(some)j Fs(c)c(>)f Fu(0)j(and)h Fs(s)d(<)g Fu(1,)j(there)g(exists)h(some)f Fs(C)k(>)34 b Fu(0)i(suc)m(h)456 1473 y(that,)c(for)g(all)h Fs(z)f Fu(=)c Fs(x)22 b Fu(+)g Fs(iy)31 b Ft(2)d(f)p Fs(a)22 b Fu(+)g Fs(ib)29 b Ft(j)e Fu(0)h Fs(<)f Ft(j)p Fs(b)p Ft(j)h(\024)g Fs(c)p Ft(h)p Fs(a)p Ft(ig)k Fu(\(lik)m(e)i(in)f (\(A.4\))o(\),)1171 1575 y Ff(\015)1171 1634 y(\015)1226 1659 y Ft(h)p Fs(A)p Ft(i)1377 1618 y Fm(s)1414 1659 y Fu(\()p Fs(A)22 b Ft(\000)g Fs(z)t Fu(\))1733 1618 y Fl(\000)p Fk(1)1828 1575 y Ff(\015)1828 1634 y(\015)1911 1659 y Ft(\024)29 b Fs(C)7 b Ft(h)p Fs(x)p Ft(i)2227 1618 y Fm(s)2285 1659 y Ft(\001)22 b(j)p Fs(y)t Ft(j)2443 1618 y Fl(\000)p Fk(1)2536 1659 y Fs(:)-2107 b Fu(\(B.3\))456 1902 y(Next)33 b(w)m(e)h(come)f(to)f(a)g(comm)m(utator)i(expansion.)456 2092 y Fy(Prop)s(osition)48 b(B.2.)e Fp(L)-5 b(et)44 b Fs(k)i Ft(2)e Fn(N)1772 2056 y Fl(\003)1855 2092 y Fp(and)f Fs(B)48 b Fp(b)-5 b(e)43 b(a)g(self-adjoint)f(and)h(b)-5 b(ounde)g(d)456 2208 y(op)g(er)g(ator)39 b(in)h Ft(C)1027 2172 y Fm(k)1070 2208 y Fu(\()p Fs(A)p Fu(\))p Fp(.)60 b(L)-5 b(et)41 b Fs(\032)d(<)f(k)43 b Fp(and)c Fs(')f Ft(2)f(S)2244 2172 y Fm(\032)2285 2208 y Fp(.)61 b(In)39 b(the)h(sense)g(of)f(forms)h(on)456 2325 y Ft(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)725 2289 y Fm(k)r Fl(\000)p Fk(1)857 2325 y Fu(\))22 b Ft(\002)g(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)1285 2289 y Fm(k)r Fl(\000)p Fk(1)1417 2325 y Fu(\))p Fp(:)1485 2607 y Fu([)p Fs(')p Fu(\()p Fs(A)p Fu(\))p Fs(;)17 b(B)5 b Fu(])27 b(=)2014 2482 y Fm(k)r Fl(\000)p Fk(1)2006 2512 y Ff(X)2017 2722 y Fm(j)t Fk(=1)2189 2540 y Fu(1)p 2176 2584 74 4 v 2176 2675 a Fs(j)6 b Fu(!)2259 2607 y Fs(')2323 2566 y Fk(\()p Fm(j)t Fk(\))2415 2607 y Fu(\()p Fs(A)p Fu(\)ad)2667 2560 y Fm(j)2667 2634 y(A)2724 2607 y Fu(\()p Fs(B)f Fu(\))-2423 b(\(B.4\))855 2932 y(+)978 2864 y Fs(i)p 941 2909 108 4 v 941 3000 a Fu(2)p Fs(\031)1075 2796 y Ff(Z)1131 3022 y Fg(C)1209 2864 y Fs(@)5 b(')1329 2828 y Fg(C)p 1209 2909 173 4 v 1242 3000 a Fs(@)p 1298 2945 50 4 v 5 w(z)1392 2932 y Fu(\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))1713 2891 y Fl(\000)p Fm(k)1810 2932 y Fu(ad)1913 2889 y Fm(k)1913 2956 y(A)1970 2932 y Fu(\()p Fs(B)5 b Fu(\)\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))2446 2891 y Fl(\000)p Fk(1)2540 2932 y Fs(dz)k Ft(^)22 b Fs(d)p 2802 2877 V(z)5 b(:)-2423 b Fu(\(B.5\))456 3177 y Fp(In)34 b(p)-5 b(articular,)35 b(if)f Fs(\032)28 b(<)g Fu(1)p Fp(,)34 b(then)h Fs(B)e Ft(2)28 b(C)1924 3141 y Fk(1)1964 3177 y Fu(\()p Fs(')p Fu(\()p Fs(A)p Fu(\)\))p Fp(.)456 3436 y(Pr)-5 b(o)g(of.)41 b Fu(Thanks)46 b(to)e(Lemma)i(B.1,)h(w)m(e)f(can)f(write,)k(as)c(form)f(on)h Ft(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)3257 3399 y Fm(k)3299 3436 y Fu(\))30 b Ft(\002)456 3552 y(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)725 3516 y Fm(k)766 3552 y Fu(\):)778 3803 y([)p Fs(')p Fu(\()p Fs(A)p Fu(\))p Fs(;)17 b(B)5 b Fu(])83 b(=)1457 3735 y Fs(i)p 1420 3780 108 4 v 1420 3871 a Fu(2)p Fs(\031)1554 3667 y Ff(Z)1610 3893 y Fg(C)1688 3735 y Fs(@)5 b(')1808 3699 y Fg(C)p 1688 3780 173 4 v 1722 3871 a Fs(@)p 1778 3816 50 4 v 5 w(z)1871 3803 y Fu(\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))2192 3762 y Fl(\000)p Fk(1)2286 3803 y Fu(ad)2389 3818 y Fm(A)2446 3803 y Fu(\()p Fs(B)5 b Fu(\)\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))2922 3762 y Fl(\000)p Fk(1)3016 3803 y Fs(dz)k Ft(^)22 b Fs(d)p 3278 3748 V(z)1251 4111 y Fu(=)1418 3986 y Fm(k)r Fl(\000)p Fk(1)1410 4016 y Ff(X)1421 4226 y Fm(j)t Fk(=1)1618 4043 y Fs(i)p 1581 4088 108 4 v 1581 4179 a Fu(2)p Fs(\031)1715 3975 y Ff(Z)1770 4201 y Fg(C)1849 4043 y Fs(@)5 b(')1969 4007 y Fg(C)p 1849 4088 173 4 v 1882 4179 a Fs(@)p 1938 4124 50 4 v 5 w(z)2032 4111 y Fu(\()p Fs(z)27 b Ft(\000)22 b Fs(A)p Fu(\))2352 4069 y Fl(\000)p Fm(j)t Fl(\000)p Fk(1)2534 4111 y Fu(ad)2637 4063 y Fm(j)2637 4138 y(A)2694 4111 y Fu(\()p Fs(B)5 b Fu(\))p Fs(dz)26 b Ft(^)d Fs(d)p 3111 4056 V(z)1251 4435 y Fu(+)1457 4368 y Fs(i)p 1420 4413 108 4 v 1420 4504 a Fu(2)p Fs(\031)1554 4300 y Ff(Z)1610 4525 y Fg(C)1688 4368 y Fs(@)5 b(')1808 4332 y Fg(C)p 1688 4413 173 4 v 1722 4504 a Fs(@)p 1778 4449 50 4 v 5 w(z)1871 4435 y Fu(\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))2192 4394 y Fl(\000)p Fm(k)2289 4435 y Fu(ad)2392 4393 y Fm(k)2392 4460 y(A)2449 4435 y Fu(\()p Fs(B)5 b Fu(\)\()p Fs(z)27 b Ft(\000)c Fs(A)p Fu(\))2925 4394 y Fl(\000)p Fk(1)3019 4435 y Fs(dz)k Ft(^)22 b Fs(d)p 3281 4381 V(z)5 b(:)456 4684 y Fu(This)38 b(yields)h(\(B.4\))f(on)f Ft(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)1630 4648 y Fm(k)1672 4684 y Fu(\))25 b Ft(\002)h(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)2107 4648 y Fm(k)2149 4684 y Fu(\).)58 b(Since)39 b Fs(B)i Ft(2)c(C)2808 4648 y Fm(k)2851 4684 y Fu(\()p Fs(A)p Fu(\),)i(the)e(com-)456 4809 y(m)m(utators)h(ad)979 4762 y Fm(j)979 4836 y(A)1036 4809 y Fu(\()p Fs(B)5 b Fu(\),)39 b(for)f(1)e Ft(\024)h Fs(j)43 b Ft(\024)37 b Fs(k)s Fu(,)j(are)e(b)s(ounded.)60 b(No)m(w,)40 b(as)e(in)g(the)g(pro)s(of)456 4925 y(of)i(Lemma)h(B.1,)h (w)m(e)f(see)h(that)e(this)h(form)f(extends)j(to)d(a)g(b)s(ounded)h (form)f(on)456 5041 y Ft(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)725 5005 y Fm(k)r Fl(\000)p Fk(1)857 5041 y Fu(\))22 b Ft(\002)g(D)s Fu(\()p Ft(h)p Fs(A)p Ft(i)1285 5005 y Fm(k)r Fl(\000)p Fk(1)1417 5041 y Fu(\))33 b(since)h(the)f Fs(')1959 5005 y Fk(\()p Fm(j)t Fk(\))2083 5041 y Fu(b)s(elong)f(to)g Ft(S)2580 5005 y Fm(\032)p Fl(\000)p Fk(1)2712 5041 y Fu(.)628 b Fj(\003)456 5216 y Fu(The)33 b(rest)g(of)f(the)h(previous)h (expansion)h(is)e(estimated)h(in)p eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(23)456 450 y Fy(Lemma)46 b(B.3.)f Fp(L)-5 b(et)41 b Fs(B)i Ft(2)c(C)1557 414 y Fm(k)1600 450 y Fu(\()p Fs(A)p Fu(\))i Fp(self-adjoint)e(and)i(b)-5 b(ounde)g(d.)61 b(L)-5 b(et)41 b Fs(')e Ft(2)f(S)3373 414 y Fm(\032)3415 450 y Fp(,)456 566 y(with)g Fs(\032)c(<)f(k)s Fp(.)55 b(L)-5 b(et)38 b Fs(I)1217 581 y Fm(k)1260 566 y Fu(\()p Fs(')p Fu(\))g Fp(the)g(r)-5 b(est)38 b(of)g(the)g (development)f(of)h(or)-5 b(der)38 b Fs(k)j Fu(\(B.4\))d Fp(of)456 683 y Fu([)p Fs(')p Fu(\()p Fs(A)p Fu(\))p Fs(;)17 b(B)5 b Fu(])p Fp(,)37 b(namely)45 b Fu(\(B.5\))p Fp(.)50 b(L)-5 b(et)38 b Fs(s;)17 b(s)1868 646 y Fl(0)1922 683 y Fs(<)31 b Fu(1)37 b Fp(such)g(that)g Fs(\032)24 b Fu(+)f Fs(s)h Fu(+)f Fs(s)2928 646 y Fl(0)2983 683 y Fs(<)31 b(k)s Fp(.)51 b(Then)456 803 y Ft(h)p Fs(A)p Ft(i)607 767 y Fm(s)643 803 y Fs(I)686 818 y Fm(k)729 803 y Ft(h)p Fs(A)p Ft(i)880 767 y Fm(s)913 743 y Fi(0)976 803 y Fp(is)36 b(b)-5 b(ounde)g(d)37 b(and)f(it)i(is)f(uniformly)g(b)-5 b(ounde)g(d)36 b(when)g Fs(')h Fp(stays)h(in)f(a)456 919 y(b)-5 b(ounde)g(d)37 b(subset)h(of)g Ft(S)1307 883 y Fm(\032)1348 919 y Fp(.)55 b(In)38 b(p)-5 b(articular,)39 b Fs(I)2087 934 y Fm(k)2129 919 y Fu(\()p Fs(')p Fu(\))f Fp(is)g(a)g(b)-5 b(ounde)g(d)38 b(op)-5 b(er)g(ator.)54 b(L)-5 b(et)456 1036 y Fs(R)28 b(>)g Fu(0)p Fp(.)42 b(If)29 b Fs(')g Fp(stays)g(in)g(a)f(b)-5 b(ounde)g(d)29 b(subset)g(of)f Ft(f)p Fs( )k Ft(2)c(S)2466 999 y Fm(\032)2535 1036 y Ft(j)f Fu([)p Ft(\000)p Fs(R)q Fu(;)17 b Fs(R)q Fu(])9 b Ft(\\)g Fu(supp)r(\()p Fs(')p Fu(\))28 b(=)456 1156 y Ft(;g)34 b Fp(then)h Ft(h)p Fs(R)q Ft(i)960 1120 y Fm(k)r Fl(\000)p Fm(\032)p Fl(\000)p Fm(s)p Fl(\000)p Fm(s)1266 1096 y Fi(0)1290 1156 y Ft(kh)p Fs(A)p Ft(i)1491 1120 y Fm(s)1528 1156 y Fs(I)1571 1171 y Fm(k)1613 1156 y Fu(\()p Fs(')p Fu(\))p Ft(h)p Fs(A)p Ft(i)1904 1120 y Fm(s)1937 1096 y Fi(0)1963 1156 y Ft(k)g Fp(is)f(uniformly)h(b)-5 b(ounde)g(d.)456 1404 y(Pr)g(o)g(of.)41 b Fu(W)-8 b(e)31 b(will)h(follo)m(w)f(ideas)h(from)e([DG)o(][Lemma)i(C.3.1].)43 b(In)31 b(this)h(pro)s(of,)e(all)456 1520 y(the)38 b(constan)m(ts)i (are)e(denoted)i(b)m(y)f Fs(C)7 b Fu(,)40 b(indep)s(enden)m(tly)h(of)c (their)i(v)-5 b(alue.)61 b(Giv)m(en)456 1637 y(a)34 b(complex)h(n)m(um) m(b)s(er)h Fs(z)t Fu(,)f Fs(x)g Fu(and)f Fs(y)k Fu(will)d(denote)g(its) f(real)h(and)f(imaginary)g(part,)456 1753 y(resp)s(ectiv)m(ely)-8 b(.)50 b(Since)35 b Fs(B)g Ft(2)30 b(C)1540 1717 y Fm(k)1583 1753 y Fu(\()p Fs(A)p Fu(\),)k(ad)1896 1710 y Fm(k)1896 1778 y(A)1953 1753 y Fu(\()p Fs(B)5 b Fu(\))34 b(is)g(b)s(ounded.)48 b(W)-8 b(e)35 b(start)e(with)i(the)456 1870 y(second)e(assertion.)44 b(Let)33 b Fs(')27 b Ft(2)i(S)1643 1833 y Fm(\032)1684 1870 y Fu(,)j Fs(R)d(>)e Fu(0)32 b(suc)m(h)i(that)e([)p Ft(\000)p Fs(R)q Fu(;)17 b Fs(R)q Fu(])22 b Ft(\\)f Fu(supp)q(\()p Fs(')p Fu(\))28 b(=)f Ft(;)p Fu(.)456 1986 y(Notice)41 b(that,)h(b)m(y)h(\(A.5\),)f Fs(')1521 1950 y Fg(C)1573 1986 y Fu(\()p Fs(x)28 b Fu(+)f Fs(iy)t Fu(\))41 b(=)g(0)f(for)g Ft(j)p Fs(x)p Ft(j)h(\024)g Fs(R)q Fu(.)67 b(By)41 b(de\014nition)h(of) 456 2102 y Fs(I)499 2117 y Fm(k)541 2102 y Fu(,)31 b(w)m(e)h(consider)g (\(B.5\))e(and)h(switc)m(h)h(to)e(the)h(v)-5 b(ariable)31 b(\()p Fs(x;)17 b(y)t Fu(\))29 b(b)m(y)j(noticing)f(that)456 2218 y Fs(dz)26 b Ft(^)d Fs(d)p 718 2163 50 4 v(z)32 b Fu(=)27 b Ft(\000)p Fu(2)p Fs(idx)c Ft(^)g Fs(dy)t Fu(.)42 b(By)34 b(\(B.3\))f(,)529 2466 y Ft(kh)p Fs(A)p Ft(i)730 2425 y Fm(s)766 2466 y Fs(I)809 2481 y Fm(k)851 2466 y Fu(\()p Fs(')p Fu(\))p Ft(h)p Fs(A)p Ft(i)1142 2425 y Fm(s)1175 2402 y Fi(0)1201 2466 y Ft(k)28 b(\024)1399 2399 y Fu(1)p 1394 2443 59 4 v 1394 2535 a Fs(\031)1479 2331 y Ff(Z)1595 2382 y(\014)1595 2441 y(\014)1639 2399 y Fs(@)5 b(')1759 2363 y Fg(C)p 1639 2443 173 4 v 1672 2535 a Fs(@)p 1728 2480 50 4 v 5 w(z)1821 2382 y Ff(\014)1821 2441 y(\014)1877 2466 y Ft(\001)1937 2399 y(h)p Fs(x)p Ft(i)2070 2363 y Fm(s)p 1937 2443 170 4 v 1947 2535 a Ft(j)p Fs(y)t Ft(j)2055 2506 y Fm(k)2138 2466 y Ft(\001)22 b(k)p Fu(ad)2341 2423 y Fm(k)2341 2491 y(A)2398 2466 y Fu(\()p Fs(B)5 b Fu(\))p Ft(k)22 b(\001)2685 2399 y(h)p Fs(x)p Ft(i)2818 2363 y Fm(s)2851 2339 y Fi(0)p 2685 2443 193 4 v 2727 2535 a Ft(j)p Fs(y)t Ft(j)2887 2466 y Fs(dx)g Ft(^)h Fs(dy)943 2733 y Ft(\024)28 b Fs(C)7 b Fu(\()p Fs(')p Fu(\))1282 2597 y Ff(Z)1337 2823 y Fl(j)p Fm(x)p Fl(j\025)p Fm(R)1545 2597 y Ff(Z)1600 2823 y Fl(j)p Fm(y)r Fl(j\024)p Fm(c)1763 2832 y Fh(2)1797 2823 y Fl(h)p Fm(x)p Fl(i)1895 2733 y Ft(h)p Fs(x)p Ft(i)2028 2691 y Fm(\032)p Fk(+)p Fm(s)p Fk(+)p Fm(s)2240 2668 y Fi(0)2262 2691 y Fl(\000)p Fk(1)p Fl(\000)p Fm(l)2433 2733 y Ft(j)p Fs(y)t Ft(j)2541 2691 y Fm(l)2565 2733 y Ft(j)p Fs(y)t Ft(j)2673 2691 y Fl(\000)p Fm(k)r Fl(\000)p Fk(1)2860 2733 y Fs(dx)22 b Ft(^)g Fs(dy)t(;)456 2982 y Fu(for)32 b(an)m(y)h Fs(l)r Fu(,)g(b)m(y)i(\(A.3\))o(.)44 b(W)-8 b(e)33 b(c)m(ho)s(ose)g Fs(l)d Fu(=)e Fs(k)d Fu(+)d(1.)43 b(W)-8 b(e)33 b(ha)m(v)m(e,)897 3212 y Ft(kh)p Fs(A)p Ft(i)1098 3170 y Fm(s)1134 3212 y Fs(I)1177 3227 y Fm(k)1220 3212 y Fu(\()p Fs(')p Fu(\))p Ft(h)p Fs(A)p Ft(i)1511 3147 y Fi(0)1533 3170 y Fm(s)1569 3212 y Ft(k)83 b(\024)h Fs(C)7 b Fu(\()p Fs(')p Fu(\))2097 3076 y Ff(Z)2151 3301 y Fl(j)p Fm(x)p Fl(j\025)p Fm(R)2343 3212 y Ft(h)p Fs(x)p Ft(i)2476 3170 y Fm(\032)p Fk(+)p Fm(s)p Fk(+)p Fm(s)2688 3147 y Fi(0)2709 3170 y Fl(\000)p Fm(k)r Fl(\000)p Fk(1)2897 3212 y Fs(dx)1702 3448 y Ft(\024)84 b Fs(C)7 b Fu(\()p Fs(')p Fu(\))p Ft(h)p Fs(R)q Ft(i)2233 3407 y Fm(\032)p Fk(+)p Fm(s)p Fk(+)p Fm(s)2445 3383 y Fi(0)2465 3407 y Fl(\000)p Fm(k)2563 3448 y Fs(:)456 3624 y Fu(Since)38 b Fs(C)7 b Fu(\()p Fs(')p Fu(\))37 b(is)h(b)s(ounded)g(when)h Fs(')e Fu(sta)m(ys)h(in)g(a)f(b)s(ounded)h(subset)h(of)e Ft(S)3180 3588 y Fm(\032)3221 3624 y Fu(,)i(this)456 3740 y(yields)e(the)g(second)h(assertion.)55 b(F)-8 b(or)36 b(the)g(\014rst)h(one,)h(w)m(e)f(can)f(follo)m(w)h(the)f(same)456 3857 y(lines,)d(replacing)h Fs(R)f Fu(b)m(y)h(0)e(in)h(the)g(in)m (tegrals,)g(and)g(arriv)m(e)h(at)e(the)h(result.)199 b Fj(\003)1094 4168 y Fv(Appendix)37 b Fu(C.)50 b Fv(Technical)38 b(estima)-7 b(tes.)456 4401 y Fy(Lemma)46 b(C.1.)g Fp(L)-5 b(et)41 b Fs(")e Ft(2)p Fu(]0)p Fs(;)17 b Fu(1)26 b Ft(\000)h Fs(s)p Fu([)41 b Fp(and)g(supp)-5 b(ose)40 b Fs(B)k Ft(2)c(C)2683 4365 y Fk(2)2723 4401 y Fu(\()p Fs(A)p Fu(\))h Fp(b)-5 b(ounde)g(d)40 b(and)456 4517 y(self-adjoint.)j(Then,)34 b(for)h(al)5 b(l)34 b Fs(f)39 b Ft(2)28 b Fo(H)e Fp(,)597 4694 y Ft(h)642 4674 y Fu(~)-55 b Fs(\037)697 4709 y Fm(R)754 4694 y Fu(\()p Fs(A)p Fu(\))p Fs(f)5 b(;)17 b Fu([)p Fs(B)5 b(;)17 b Ft(\000)p Fs(A)p Ft(h)p Fs(A)p Ft(i)1451 4652 y Fl(\000)p Fk(2)p Fm(s)1578 4694 y Fu(])1611 4674 y(~)-55 b Fs(\037)1667 4709 y Fm(R)1724 4694 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)83 b Fu(=)g Fs(O)2288 4709 y Fm(R)2345 4613 y Ff(\000)2391 4694 y Fs(R)2466 4652 y Fl(\000)p Fm(")2557 4694 y Fu(\))p Ft(k)2651 4674 y Fu(~)-55 b Fs(\037)2706 4709 y Fm(R)2764 4694 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)3064 4652 y Fl(\000)p Fm(s)3155 4694 y Fs(f)11 b Ft(k)3264 4652 y Fk(2)1032 4851 y Fu(+\(2)p Fs(s)21 b Ft(\000)i Fu(1\))p Ft(h)1494 4832 y Fu(~)-55 b Fs(\037)1549 4866 y Fm(R)1606 4851 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1906 4810 y Fl(\000)p Fm(s)1998 4851 y Fs(f)5 b(;)17 b Fu([)p Fs(B)5 b(;)17 b(A)p Fu(])2351 4832 y(~)-55 b Fs(\037)2406 4866 y Fm(R)2464 4851 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2764 4810 y Fl(\000)p Fm(s)2855 4851 y Fs(f)11 b Ft(i)p Fs(:)456 5099 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Let)c Fs(D)i Fu(=)d([)p Fs(B)5 b(;)17 b Ft(\000h)p Fs(A)p Ft(i)1543 5063 y Fl(\000)p Fk(2)p Fm(s)1670 5099 y Fs(A)p Fu(])26 b Ft(\000)f Fu(\(2)p Fs(sA)2104 5063 y Fk(2)2144 5099 y Ft(h)p Fs(A)p Ft(i)2295 5063 y Fl(\000)p Fk(2)p Fm(s)p Fl(\000)p Fk(2)2537 5099 y Ft(\000)h(h)p Fs(A)p Ft(i)2791 5063 y Fl(\000)p Fk(2)p Fm(s)2917 5099 y Fu(\)[)p Fs(B)5 b(;)17 b(iA)p Fu(].)59 b(By)456 5216 y(Lemma)39 b(B.3)h(for)e Fs(k)43 b Fu(=)38 b(2,)j(as)f Fs(t)f Ft(7!)g(h)p Fs(t)p Ft(i)1928 5179 y Fl(\000)p Fk(2)p Fm(s)2054 5216 y Fs(t)h Ft(2)f(S)2301 5179 y Fk(1)p Fl(\000)p Fk(2)p Fm(s)2464 5216 y Fu(,)i(one)f(has)f Ft(h)p Fs(A)p Ft(i)3049 5179 y Fm(s)p Fk(+)p Fm(")3173 5216 y Fs(D)s Ft(h)p Fs(A)p Ft(i)3408 5179 y Fm(s)p eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 456 255 a Fq(24)622 b(SYL)-9 b(V)g(AIN)24 b(GOL)1686 236 y(\023)1679 255 y(ENIA)h(AND)f(THIERR)-6 b(Y)24 b(JECK)n(O)456 453 y Fu(b)s(ounded)36 b(for)e Fs(")e(<)h Fu(1.)51 b(Then,)37 b(using)f(the)g(fact)f(that)2451 433 y(~)-55 b Fs(\037)2506 468 y Fm(R=)p Fk(2)2635 453 y Fu(\()p Fs(t)p Fu(\))32 b(=)g(1)j(for)g Fs(t)g Fu(in)h(the)456 578 y(supp)s(ort)d(of)934 559 y(~)-55 b Fs(\037)989 593 y Fm(R)1046 578 y Fu(,)456 735 y Ft(h)501 716 y Fu(~)g Fs(\037)555 750 y Fm(R)613 735 y Fu(\()p Fs(A)p Fu(\))p Fs(f)5 b(;)17 b(D)949 716 y Fu(~)-55 b Fs(\037)1004 750 y Fm(R)1062 735 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)82 b Fu(=)h Ft(hh)p Fs(A)p Ft(i)1740 694 y Fl(\000)p Fm(")1837 716 y Fu(~)-55 b Fs(\037)1892 751 y Fm(R=)p Fk(2)2020 735 y Fu(\()p Fs(A)p Fu(\))2175 716 y(~)g Fs(\037)2231 750 y Fm(R)2288 735 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2588 694 y Fl(\000)p Fm(s)2680 735 y Fs(f)5 b(;)17 b Ft(h)p Fs(A)p Ft(i)2928 694 y Fm(s)p Fk(+)p Fm(")3051 735 y Fs(D)3141 716 y Fu(~)-55 b Fs(\037)3196 750 y Fm(R)3254 735 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)1391 890 y Fu(=)83 b Fs(O)s Fu(\()p Fs(R)1741 849 y Fl(\000)p Fm(")1832 890 y Fu(\))22 b Ft(\001)g(k)1998 871 y Fu(~)-55 b Fs(\037)2053 905 y Fm(R)2111 890 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2411 849 y Fl(\000)p Fm(s)2502 890 y Fs(f)11 b Ft(k)2611 849 y Fk(2)2650 890 y Fs(:)456 1047 y Fu(Since)46 b([)p Fs(B)5 b(;)17 b(A)p Fu(])48 b Ft(2)h(C)1194 1011 y Fk(1)1234 1047 y Fu(\()p Fs(A)p Fu(\))c(and)g Fs(t)k Ft(7!)f(h)p Fs(t)p Ft(i)1975 1011 y Fl(\000)p Fm(s)2115 1047 y Ft(2)h(S)2297 1011 y Fl(\000)p Fm(s)2389 1047 y Fu(,)f(Lemma)e(B.3)e(giv)m(es)j(that) 456 1163 y Ft(h)p Fs(A)p Ft(i)607 1127 y Fm(s)p Fk(+)p Fm(")730 1163 y Fu([)p Ft(h)p Fs(A)p Ft(i)908 1127 y Fl(\000)p Fm(s)1000 1163 y Fs(;)17 b Fu([)p Fs(B)5 b(;)17 b(A)p Fu(]])p Ft(h)p Fs(A)p Ft(i)1472 1127 y Fm(s)1538 1163 y Fu(b)s(ounded)31 b(for)e Fs(")e(<)h Fu(1)16 b Ft(\000)i Fs(s)p Fu(.)42 b(Using,)31 b(lik)m(e)h(ab)s(o)m(v)m(e,)f(the) 456 1282 y(con)m(tribution)i(of)1128 1263 y(~)-55 b Fs(\037)1183 1297 y Fm(R=)p Fk(2)1312 1282 y Fu(\()p Fs(A)p Fu(\),)643 1448 y Ft(h)688 1429 y Fu(~)g Fs(\037)743 1463 y Fm(R)800 1448 y Fu(\()p Fs(A)p Fu(\))p Fs(f)5 b(;)17 b Fu(\(2)p Fs(sA)1252 1407 y Fk(2)1292 1448 y Ft(h)p Fs(A)p Ft(i)1443 1407 y Fl(\000)p Fk(2)p Fm(s)p Fl(\000)p Fk(2)1682 1448 y Ft(\000)22 b(h)p Fs(A)p Ft(i)1932 1407 y Fl(\000)p Fk(2)p Fm(s)2059 1448 y Fu(\)[)p Fs(B)5 b(;)17 b(iA)p Fu(])2386 1429 y(~)-55 b Fs(\037)2441 1463 y Fm(R)2499 1448 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)901 1596 y Fu(=)45 b Ft(h)1067 1577 y Fu(~)-55 b Fs(\037)1122 1611 y Fm(R)1179 1596 y Fu(\()p Fs(A)p Fu(\))p Fs(f)5 b(;)17 b Fu(\(2)p Fs(sA)1631 1555 y Fk(2)1670 1596 y Ft(h)p Fs(A)p Ft(i)1821 1555 y Fl(\000)p Fk(2)1938 1596 y Ft(\000)22 b Fu(1\))p Ft(h)p Fs(A)p Ft(i)2275 1555 y Fl(\000)p Fm(s)2366 1596 y Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(])p Ft(h)p Fs(A)p Ft(i)2800 1555 y Fl(\000)p Fm(s)2898 1577 y Fu(~)-55 b Fs(\037)2953 1611 y Fm(R)3011 1596 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(i)901 1742 y Fu(+)45 b Fs(O)s Fu(\()p Fs(R)1213 1701 y Fl(\000)p Fm(")1304 1742 y Fu(\))p Ft(kh)p Fs(A)p Ft(i)1543 1701 y Fl(\000)p Fm(s)1634 1723 y Fs(\037)1695 1757 y Fm(R)1752 1742 y Fu(\()p Fs(A)p Fu(\))p Fs(f)11 b Ft(k)2010 1701 y Fk(2)2049 1742 y Fs(:)456 1902 y Fu(T)-8 b(o)32 b(conclude,)i(observ)m(e)h(that)d Ft(k)1642 1882 y Fu(~)-55 b Fs(\037)1697 1917 y Fm(R=)p Fk(2)1825 1902 y Fu(\()p Fs(A)p Fu(\)\(Id)23 b Ft(\000)f Fs(A)2296 1865 y Fk(2)2336 1902 y Ft(h)p Fs(A)p Ft(i)2487 1865 y Fl(\000)p Fk(2)2581 1902 y Fu(\))p Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)2991 1865 y Fl(\000)p Fk(2)3084 1902 y Fu(\).)218 b Fj(\003)456 2078 y Fy(Lemma)31 b(C.2.)36 b Fp(L)-5 b(et)29 b Fs(B)k Ft(2)28 b(C)1501 2042 y Fk(2)1541 2078 y Fu(\()p Fs(A)p Fu(\))g Fp(b)-5 b(ounde)g(d)28 b(and)g(self-adjoint.)42 b(F)-7 b(or)27 b(al)5 b(l)28 b Fs(f)39 b Ft(2)28 b Fo(H)f Fp(,)657 2237 y Ft(h)p Fs(f)5 b(;)17 b Fu(\()p Fs(\033)t(h)22 b Ft(\000)g Fs(A)p Ft(h)p Fs(A)p Ft(i)1291 2196 y Fl(\000)p Fk(2)p Fm(s)1418 2237 y Fu(\))1462 2218 y(~)-55 b Fs(\037)1517 2252 y Fm(\033)n(;R)1633 2237 y Fu(\()p Fs(A)p Fu(\)[)p Fs(B)5 b(;)1938 2218 y Fu(~)-55 b Fs(\037)1993 2252 y Fm(\033)n(;R)2109 2237 y Fu(\()p Fs(A)p Fu(\)])p Fs(f)11 b Ft(i)83 b Fu(=)1444 2399 y Ft(h)p Fs(f)5 b(;)1586 2379 y Fu(~)-55 b Fs(\037)1641 2340 y Fl(0)1641 2423 y Fm(\033)n(;R)1757 2399 y Fu(\()p Fs(A)p Fu(\)\()p Fs(\033)t(h)22 b Ft(\000)h Fs(A)p Ft(h)p Fs(A)p Ft(i)2405 2358 y Fl(\000)p Fk(2)p Fm(s)2531 2399 y Fu(\))2575 2379 y(~)-55 b Fs(\037)2630 2414 y Fm(\033)n(;R)2747 2399 y Fu(\()p Fs(A)p Fu(\)[)p Fs(B)5 b(;)17 b(A)p Fu(])p Fs(f)11 b Ft(i)1285 2559 y Fu(+)83 b Fs(O)s Fu(\()p Fs(R)1635 2518 y Fk(2)p Fm(s)p Fl(\000)p Fk(2)1796 2559 y Fu(\))p Ft(k)1890 2540 y Fu(~)-55 b Fs(\037)1945 2574 y Fm(R)2003 2559 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2303 2518 y Fl(\000)p Fm(s)2394 2559 y Fs(f)11 b Ft(k)22 b(\001)g(kh)p Fs(A)p Ft(i)2776 2518 y Fl(\000)p Fm(s)2867 2559 y Fs(f)11 b Ft(k)p Fs(:)456 2783 y Fp(Pr)-5 b(o)g(of.)41 b Fu(By)27 b(Lemma)g(B.3,)h(w)m(e)f(dev)m(elop)i(the)e(comm)m(utator)g(and)f (denote)i(the)f(rest)456 2899 y(b)m(y)33 b Fs(I)634 2914 y Fk(2)674 2899 y Fu(.)43 b(Its)33 b(con)m(tribution)h(is)602 3056 y Ft(h)647 3037 y Fu(~)-55 b Fs(\037)702 3072 y Fm(R=)p Fk(2)830 3056 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1130 3015 y Fl(\000)p Fm(s)1221 3056 y Fs(f)5 b(;)17 b Fu(\()p Fs(\033)t(h)23 b Ft(\000)f Fs(A)p Ft(h)p Fs(A)p Ft(i)1817 3015 y Fl(\000)p Fk(2)p Fm(s)1944 3056 y Fu(\))1988 3037 y(~)-55 b Fs(\037)2043 3071 y Fm(\033)n(;R)2159 3056 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2459 3015 y Fm(s)2495 3056 y Fs(I)2538 3071 y Fk(2)2578 3056 y Ft(h)p Fs(A)p Ft(i)2729 3015 y Fm(s)2765 3056 y Ft(h)p Fs(A)p Ft(i)2916 3015 y Fl(\000)p Fm(s)3008 3056 y Fs(f)11 b Ft(i)p Fs(:)456 3223 y Fu(Note)35 b(that)914 3204 y(~)-55 b Fs(\037)969 3238 y Fm(R=)p Fk(2)1097 3223 y Fu(\()p Fs(A)p Fu(\))p Fs(f)46 b Fu(app)s(ears)35 b(freely)i(thanks)f(to)f(the) g(presence)i(of)3097 3204 y(~)-55 b Fs(\037)3152 3238 y Fm(\033)n(;R)3268 3223 y Fu(\()p Fs(A)p Fu(\).)456 3349 y(By)40 b(Corollary)g(A.3,)i(\()1316 3329 y(~)-55 b Fs(\037)1371 3364 y Fm(\033)n(;R)1487 3349 y Fu(\))1525 3364 y Fm(R)1623 3349 y Fu(is)40 b(b)s(ounded)h(in)f Ft(S)2322 3313 y Fk(0)2362 3349 y Fu(.)65 b(Note)40 b(also)g(that)g([)p Ft(\000)p Fs(R)q(;)17 b(R)q Fu(])456 3470 y(is)34 b(not)g(con)m(tained) h(in)f(the)g(supp)s(ort)g(of)1938 3451 y(~)-55 b Fs(\037)1993 3485 y Fm(\033)n(;R)2109 3470 y Fu(.)47 b(Then,)36 b(from)d(Lemma)i (B.3,)f(used)456 3586 y(with)f Fs(k)e Fu(=)c(2,)32 b(w)m(e)i(obtain)f (that)f Ft(h)p Fs(A)p Ft(i)1781 3550 y Fm(s)1817 3586 y Fs(I)1860 3601 y Fk(2)1900 3586 y Ft(h)p Fs(A)p Ft(i)2051 3550 y Fm(s)2115 3586 y Fu(=)27 b Fs(O)s Fu(\()p Fs(R)2409 3550 y Fk(2)p Fm(s)p Fl(\000)p Fk(2)2571 3586 y Fu(\).)731 b Fj(\003)456 3761 y Fu(W)-8 b(e)33 b(summarize)h(some)f(prop)s(erties) h(of)e Fs(C)1996 3776 y Fm(R)2053 3761 y Fu(.)456 3934 y Fy(Lemma)39 b(C.3.)i Fp(F)-7 b(or)34 b Fs(B)f Ft(2)28 b(C)1534 3898 y Fk(1)1574 3934 y Fu(\()p Fs(A)p Fu(\))35 b Fp(b)-5 b(ounde)g(d)34 b(and)g(self-adjoint,)604 4129 y Fu(\(1\))41 b Ft(k)p Fs(C)890 4144 y Fm(R)947 4129 y Ft(h)p Fs(A)p Ft(i)1098 4093 y Fm(\013)1148 4129 y Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)1520 4093 y Fm(\013)1568 4129 y Fu(\))p Fp(,)35 b(for)g Fs(\013)28 b Ft(2)g Fn(R)p Fp(,)604 4245 y Fu(\(2\))41 b Ft(k)p Fu([)p Fs(B)5 b(;)17 b(C)1040 4260 y Fm(R)1097 4245 y Fu(])p Ft(h)p Fs(A)p Ft(i)1275 4209 y Fm(\013)1325 4245 y Ft(k)27 b Fu(=)h Fs(O)s Fu(\()p Fs(R)1697 4209 y Fm(\013)p Fl(\000)p Fk(1)1836 4245 y Fu(\))p Fp(,)34 b(for)h Fs(\013)28 b(<)g Fu(1)p Fp(.)456 4469 y(Pr)-5 b(o)g(of.)41 b Fu(Since)f Fs( )1078 4484 y Fm(R)1136 4469 y Fu(\()p Fs(t)p Fu(\))f(=)g(0)g(for)g Ft(j)p Fs(t)p Ft(j)f(62)i Fu([)p Fs(R)q(;)17 b Fu(2)p Fs(R)q Fu(],)40 b(the)g(p)s(oin)m(t)f(\(1\))g(follo)m(ws.)64 b(Since)456 4586 y(\()p Fs( )557 4601 y Fm(R)615 4586 y Fu(\))653 4601 y Fm(R)755 4586 y Fu(is)46 b(b)s(ounded)g(in)g Ft(S)1471 4549 y Fk(0)1556 4586 y Fu(\(cf.)81 b(Lemma)46 b(A.4\))f(and)h(since)g([)p Ft(\000)p Fs(R)q(;)17 b(R)q Fu(])46 b(is)f(not)456 4702 y(con)m(tained)25 b(in)g(the)g(supp)s(ort)f (of)g Fs( )1673 4717 y Fm(R)1732 4702 y Fu(,)i(w)m(e)f(get)g(the)f(p)s (oin)m(t)h(\(2\))f(b)m(y)h(Lemma)g(B.3.)98 b Fj(\003)456 4875 y Fy(Lemma)39 b(C.4.)i Fp(L)-5 b(et)36 b Fs(B)d Ft(2)28 b(C)1521 4839 y Fk(2)1560 4875 y Fu(\()p Fs(A)p Fu(\))35 b Fp(b)-5 b(ounde)g(d,)34 b(self-adjoint.)44 b(F)-7 b(or)34 b(al)5 b(l)34 b Fs(f)k Ft(2)29 b Fo(H)d Fp(,)523 5033 y Ft(h)p Fs(C)632 5048 y Fm(R)690 5033 y Fs(f)5 b(;)17 b Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(])p Fs(C)1140 5048 y Fm(R)1198 5033 y Fs(f)11 b Ft(i)82 b Fu(=)h Ft(h)p Fs(f)5 b(;)17 b(C)1750 4992 y Fk(2)1743 5058 y Fm(R)1800 5033 y Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(])p Fs(f)11 b Ft(i)1537 5181 y Fu(+)p Fs(O)1688 5196 y Fm(R)1745 5100 y Ff(\000)1791 5181 y Fs(R)1866 5140 y Fk(2)p Fm(s)p Fl(\000)p Fk(1)2028 5181 y Fu(\))22 b Ft(\001)g(k)2194 5161 y Fu(~)-55 b Fs(\037)2249 5196 y Fm(R=)p Fk(2)2377 5181 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2677 5140 y Fl(\000)p Fm(s)2768 5181 y Fs(f)11 b Ft(k)22 b(\001)g(kh)p Fs(A)p Ft(i)3150 5140 y Fl(\000)p Fm(s)3241 5181 y Fs(f)11 b Ft(k)p Fs(:)p eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 892 251 a Fq(A)33 b(NEW)g(LOOK)g(A)-6 b(T)33 b(MOURRE'S)f(COMMUT)-6 b(A)g(TOR)33 b(THEOR)-6 b(Y)359 b(25)456 450 y Fp(Pr)-5 b(o)g(of.)41 b Fu(Giv)m(en)33 b Fs(f)39 b Ft(2)28 b Fo(H)59 b Fu(and)33 b(using)h(\(3.20\))o(,)832 621 y Ft(h)p Fs(C)941 636 y Fm(R)999 621 y Fs(f)5 b(;)17 b Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(])p Fs(C)1449 636 y Fm(R)1507 621 y Fs(f)11 b Ft(i)82 b Fu(=)h Ft(h)p Fs(f)5 b(;)17 b(C)2059 580 y Fk(2)2052 646 y Fm(R)2109 621 y Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(])p Fs(f)11 b Ft(i)741 769 y(\000)83 b(h)946 749 y Fu(~)-55 b Fs(\037)1001 784 y Fm(R=)p Fk(2)1129 769 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)1429 727 y Fl(\000)p Fm(s)1521 769 y Fs(f)5 b(;)17 b(C)1688 784 y Fm(R)1745 769 y Ft(h)p Fs(A)p Ft(i)1896 727 y Fm(s)1932 769 y Fu([)p Fs(C)2029 784 y Fm(R)2087 769 y Fs(;)g Fu([)p Fs(B)5 b(;)17 b(iA)p Fu(]])p Ft(h)p Fs(A)p Ft(i)2592 727 y Fm(s)2651 769 y Ft(\001)22 b(h)p Fs(A)p Ft(i)2852 727 y Fl(\000)p Fm(s)2943 769 y Fs(f)11 b Ft(i)p Fs(:)456 938 y Fu(The)33 b(last)g(term)g(is)g(estimated)h(ab)s(o)m(v)m(e)f(b)m (y)587 1108 y Ft(k)p Fs(C)707 1123 y Fm(R)764 1108 y Ft(h)p Fs(A)p Ft(i)915 1067 y Fm(s)952 1108 y Ft(k)22 b(\001)g(k)p Fu([)p Fs(C)1221 1123 y Fm(R)1278 1108 y Fs(;)17 b Fu([)p Fs(B)5 b(;)17 b(A)p Fu(]])p Ft(h)p Fs(A)p Ft(i)1750 1067 y Fm(s)1786 1108 y Ft(k)22 b(\001)g(k)1964 1089 y Fu(~)-55 b Fs(\037)2019 1124 y Fm(R=)p Fk(2)2147 1108 y Fu(\()p Fs(A)p Fu(\))p Ft(h)p Fs(A)p Ft(i)2447 1067 y Fl(\000)p Fm(s)2539 1108 y Fs(f)11 b Ft(k)21 b(\001)h(kh)p Fs(A)p Ft(i)2920 1067 y Fl(\000)p Fm(s)3011 1108 y Fs(f)11 b Ft(k)p Fs(:)456 1278 y Fu(No)m(w)33 b(Lemma)g(C.3)g(giv)m(es)h(the)f (result.)1490 b Fj(\003)1660 1571 y Fv(References)456 1729 y Fx([ABG])41 b(W.O.)36 b(Amrein,)h(A.)f(Boutet)f(de)h(Mon)n(v)n (el)e(and)h(V.)h(Georgescu:)51 b Fd(C)2907 1741 y Fc(0)2944 1729 y Fr(-gr)l(oups,)39 b(c)l(om-)719 1828 y(mutator)d(metho)l(ds)h (and)g(sp)l(e)l(ctr)l(al)g(the)l(ory)g(of)g Fd(N)9 b Fr(-b)l(o)l(dy)37 b(hamiltonians.)p Fx(,)i(Birkh\177)-42 b(auser)719 1928 y(1996.)456 2028 y([BFS])58 b(V.)31 b(Bac)n(h,)g(J.)g(F)-7 b(r\177)-42 b(ohlic)n(h)30 b(and)h(I.M.)g (Sigal:)43 b Fr(Quantum)31 b(ele)l(ctr)l(o)l(dynamics)j(of)g(c)l (on\014ne)l(d)719 2127 y(non-r)l(elativistic)c(p)l(articles)p Fx(.)f(Adv.)f(in)g(Math.)g Fb(137)p Fx(,)f(299{395,)d(1998.)456 2227 y([BCHM])41 b(J-F.)35 b(Bon)n(y)-7 b(,)36 b(R.)f(Carles,)h(D.)f (Hae\013ner,)i(L.)e(Mic)n(hel:)51 b Fr(Sc)l(attering)37 b(the)l(ory)g(for)g(the)719 2326 y(Schr\177)-42 b(odinger)31 b(e)l(quation)f(with)h(r)l(epulsive)g(p)l(otential.)e Fx(J.)f(Math.)g(Pures)f(Appl.)i(84,)e(no.)719 2426 y(5,)g(509-579,)d (2005.)456 2526 y([B])158 b(N.)32 b(Burq:)45 b Fr(Semiclassic)l(al)36 b(estimates)e(for)g(the)g(r)l(esolvent)g(in)g(non)f(tr)l(apping)i(ge)l (ome-)719 2625 y(tries.)28 b Fx(In)n(t.)g(Math.)f(Res.)h(Notices)f (2002,)f(no)h(5,)h(221{241.)456 2725 y([CJ])114 b(F.)40 b(Castella,)h(Th.)f(Jec)n(k)n(o:)60 b Fr(Besov)42 b(estimates)f(in)f (the)h(high-fr)l(e)l(quency)i(Helmholtz)719 2825 y(e)l(quation,)30 b(for)h(a)f(non-tr)l(apping)g(and)g Fd(C)1997 2794 y Fc(2)2064 2825 y Fr(p)l(otential.)f Fx(preprin)n(t.)456 2924 y([C])157 b(L.)35 b(Cattaneo:)52 b Fr(Mourr)l(e's)38 b(ine)l(quality)g(and)f(Emb)l(e)l(dde)l(d)i(b)l(ounde)l(d)e(states)p Fx(,)h(Bull.)d(Sci)719 3024 y(Math.)27 b Fb(129)p Fx(,)g(Issue)g(7,)h (591{614,)c(2005.)456 3123 y([CGH])42 b(L.)c(Cattaneo,)i(G.)f(M.)f (Graf)g(and)h(W.)f(Hunzik)n(er:)58 b Fr(A)40 b(gener)l(al)g(r)l (esonanc)l(e)g(the)l(ory)719 3223 y(b)l(ase)l(d)30 b(on)g(Mourr)l(e's)g (ine)l(quality)p Fx(,)f(math-ph/0507063.)456 3323 y([DG])89 b(J.)37 b(Derezi)r(\023)-44 b(nski)37 b(and)h(C.)f(G)n(\023)-39 b(erard:)55 b Fr(Sc)l(attering)39 b(the)l(ory)g(of)h(classic)l(al)h (and)f(quantum)719 3422 y(N-p)l(article)30 b(systems.)e Fx(Springer-V)-7 b(erlag)25 b(1997.)456 3522 y([DJ])111 b(J.)39 b(Derezi)r(\023)-44 b(nski)39 b(and)h(V.)g(Jak)l(\024)-37 b(si)n(\023)e(c:)58 b Fr(Sp)l(e)l(ctr)l(al)41 b(the)l(ory)h(of)g (Pauli-Fierz)g(op)l(er)l(ators.)g Fx(J.)719 3622 y(F)-7 b(unct.)28 b(Anal.)g(180,)e(no)h(2,)g(pp.)h(243-327,)d(2001.)456 3721 y([GG)n(\023)-39 b(e])49 b(V.)26 b(Georgescu)f(and)h(C.)h(G)n (\023)-39 b(erard:)34 b Fr(On)28 b(the)g(Virial)i(The)l(or)l(em)g(in)f (Quantum)e(Me)l(chan-)719 3821 y(ics)p Fx(,)h(Comm)n(un.)f(Math.)h(Ph)n (ys.)f Fb(208)p Fx(,)g(275{281,)d(\(1999\).)456 3920 y([GG)n(\023)-39 b(eM1])40 b(V.)25 b(Georgescu,)e(C.)i(G)n(\023)-39 b(erard,)23 b(and)h(J.S.)h(M\034ller:)34 b Fr(Commutators,)28 b Fd(C)2979 3932 y Fc(0)3017 3920 y Fr(-semigr)l(oups)719 4020 y(and)i(r)l(esolvent)g(estimates)p Fx(,)e(J.)f(F)-7 b(unct.)28 b(Anal.)g Fb(216)p Fx(,)f(no)g(2,)h(pp.)g(303-361,)c(2004.) 456 4120 y([GG)n(\023)-39 b(eM2])40 b(V.)c(Georgescu,)f(C.)h(G)n(\023) -39 b(erard,)34 b(and)i(J.S.)f(M\034ller:)51 b Fr(Sp)l(e)l(ctr)l(al)37 b(the)l(ory)g(of)h(massless)719 4219 y(Pauli-Fierz)31 b(mo)l(dels)p Fx(,)d(Comm.)g(Math.)g(Ph)n(ys.)e Fb(249)p Fx(,)h(no)h(1,)f(pp.)h(29-78,)d(2004.)456 4319 y([GGo])45 b(V.)32 b(Georgescu,)f(S.)g(Gol)n(\023)-39 b(enia:)44 b Fr(Isometries,)35 b(F)-6 b(o)l(ck)34 b(sp)l(ac)l(es)g(and)g(sp)l(e)l (ctr)l(al)g(analysis)h(of)719 4419 y(Schrdinger)g(op)l(er)l(ators)g(on) 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y(v)n(ol.)26 b(41,)h(no)g(2,)h(1984,)d(p.)j(207-225.)456 1247 y([M])141 b(E.)24 b(Mourre:)35 b Fr(A)n(bsenc)l(e)27 b(of)h(singular)g(c)l(ontinuous)f(sp)l(e)l(ctrum)f(for)j(c)l(ertain)e (self-adjoint)719 1347 y(op)l(er)l(ators.)i Fx(Comm)n(un.)e(in)h(Math.) g(Ph)n(ys.)e Fb(78)p Fx(,)i(391{408,)c(1981.)456 1446 y([RS4])68 b(M.)27 b(Reed,)g(B.)g(Simon)g(:)37 b Fr(Metho)l(ds)30 b(of)g(Mo)l(dern)g(Mathematic)l(al)h(Physics,)g(T)-6 b(ome)30 b(IV)f(:)719 1546 y(A)n(nalysis)h(of)g(op)l(er)l(ators.)g Fx(Academic)d(Press.)456 1646 y([S])171 b(J.)27 b(Sah)n(bani:)p Fr(The)j(c)l(onjugate)g(op)l(er)l(ator)g(metho)l(d)h(for)f(lo)l(c)l(al) t(ly)h(r)l(e)l(gular)f(Hamiltonians.)719 1745 y Fx(J.)d(Op)r(er.)g (Theory)g(38,)f(No.)i(2,)f(297{322)d(\(1997\).)456 1990 y Fw(Dep)-6 b(ar)g(tement)43 b(of)f(Ma)-6 b(thema)g(tical)43 b(Methods)f(in)g(Physics,)h(W)-10 b(arsa)j(w)42 b(Univer-)456 2090 y(sity,)31 b(Ho)806 2083 y(_)798 2090 y(za)f(74,)h(00-682)g(W)-10 b(arsza)j(w)g(a,)30 b(Poland)456 2248 y 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