This is a multi-part message in MIME format. ---------------0908191713173 Content-Type: text/plain; name="09-143.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-143.comments" 29 pages, 1 figure ---------------0908191713173 Content-Type: text/plain; name="09-143.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-143.keywords" time operators, canonical commutation relation, localisation operators, conjugate operator, quantum time delay ---------------0908191713173 Content-Type: application/postscript; name="integral22.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="integral22.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96dev Copyright 2007 Radical Eye Software %%Title: C:/Documents and Settings/rafael/Mes documents/science/articles/articles en cours/integral09/integral22.dvi %%CreationDate: Wed Aug 19 17:47:09 2009 %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: Times-Roman CMR8 CMSY8 CMR7 Times-Italic Times-Bold %%+ CMMI9 CMR9 CMSY9 CMR6 CMMI6 CMMI10 CMSY10 CMR10 CMSY7 CMMI7 CMSY6 %%+ CMSS10 MSBM10 CMEX10 rsfs10 MSBM7 CMMI5 CMSY5 rsfs7 CMR5 CMMIB10 %%+ CMMIB7 CMBSY7 CMBX12 CMSS8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: "C:\Program Files\MiKTeX 2.7\miktex\bin\dvips.exe" %+ "C:/Documents and Settings/rafael/Mes documents/science/articles/articles en cours/integral09/integral22.dvi" %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.08.19:1747 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. 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All Rights Reserved) readonly def /FullName (CMSS8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS8 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 76 /L put readonly def /FontBBox{-65 -250 1062 761}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 61 /equal put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBSY7 %!PS-AdobeFont-1.1: CMBSY7 001.000 %%CreationDate: 1992 Oct 22 12:18:11 % Computer Modern fonts were designed by Donald E. Knuth. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (001.000) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{0 -927 1542 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F086C44216EBE57F4BA37B479BF1E5A5139D8 91F3E6DDA157B25D359C5E7FE4CFB264DF1707BB6497D3E074CFF95D9FD269B5 0B1566D1161680C46D1548BBF384EF8653AF29FD474EA2336C876979AC00AE18 DC87DC0DDD3164B96FC6C3ED826EBAAED383BB3EC5044BA84A0426B87ED04C9F 4B3EDFC734C241D9B3D2321619F870FE68BB4BA7C060DBE8FBD12F641E7709F2 C430491944B78B1E59580798F1B40CAC4D59DE95941217EF1A861DDB0156D5C3 349AAB13FEFF6C646F6401550F5853BC09B267A6C63639228DF55BE60A99E4A5 308C616892AA0DC96ADB7CD7AB8AEFA859F69B587B61930596A46A905661E4BA DAB5E1CF15C94CF060B7FA600B17162AEC2DEE64A156B3F87248E7A7F88C9154 8C494273B33483BEDF0BEA4DF6A19941F52AA04717623ACBDE926B4851ED05A0 28698A1C5AE63A46EF473A4F3DCAF3E73C4FAF0C1077EE7A6504074C0D77947F 940B16425B3F5834763732F26D3385774A1CACA70C07F58887A0301D1BE530A7 D7AC00A0664617A0CBA9F6281FA4B9168DBA3C1EDCF915778351E6BD8A9CE7E5 3E56F2FFE0969E1CFFC83F07C01A3873EE1CE4E124565E8F493BE4FAA0A5D099 A116CEE4EC6C8CB2E93B42771FFF67680A56501A201E12AFDA8448ABE80BEB9B 80428F48753C4EFB174B693C69DB81CEF0A0B75C53A9D5B4C5F26FA58059A324 8D4E9D4E9C54AB8F9C21CD66B9B259F9C797559384A653DD43ED4B9C2110BB5B C3A6370921186AEE29EB4E62793ECD96935C3D9D89DC6AAECFB745BABDB570FD E7E6248B6ED9C09EDA896ECFCAEECE8C1E8BE20BDE6F3558EF5A32ED390ABF86 3A585DF34F2B8B9567778BF51A1BF9C1018AEEF42FF1F9AAB1F9F73627F6C7DE 05E3D013C3F77F1E74218778567C6B31B55509F0A8AC4A5E16D58D3EBF83868F 522AB82A12EAC68DFD3B94C9627D53C9486D485553956C8536C3D9DE049E7F73 47626740D55EB577B16E05D196D0998E5B67B4B9B81E1FF2B305E712D69F086B 18DC28A8CCB8960623E3F92E38BEFB2792106762D2BED9A0CE6B135F53A065D5 CB717F44CF2E53550965D1030D65AEFA3751857248C75C2FC1D5C7E56EBA689E 31140569598E70949173277F021747611768AD734B881EC3DA4102AAB60425C4 192F65F7AC85FF3F9DC2E2AD27BED08B2F99A9A18AB0FB072729783C86DA33F7 D3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put dup 80 /P put dup 84 /T put dup 104 /h put readonly def /FontBBox{-15 -250 1216 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMIB7 %!PS-AdobeFont-1.1: CMMIB7 001.100 %%CreationDate: 1996 Jul 27 07:35:50 % Computer Modern fonts were designed by Donald E. Knuth. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (001.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMMIB7 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 102 /f put readonly def /FontBBox{0 -250 1294 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 8 /Phi put dup 40 /parenleft put dup 41 /parenright put dup 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs7 %!PS-AdobeFont-1.0: rsfs7 001.000 %%CreationDate: Sat Mar 21 18:45:46 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion of metafont curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs7) readonly def /FamilyName (rsfs7) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs7 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 85 /U put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {16 -302 1349 728} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 59 /comma put dup 78 /N put dup 96 /lscript put dup 100 /d put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 120 /x put dup 121 /y put readonly def /FontBBox{37 -250 1349 750}readonly 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 82 /R put dup 83 /S put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. 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 66 /B put dup 68 /D put dup 70 /F put dup 72 /H put dup 75 /K put dup 83 /S put dup 85 /U 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 D9D66F633B846A989B9974B0179FC6CC445BCF7C3C3333173232E3FDBFF43949 1DB866C39088C203DC22FDC758584860EC7BB67FDA28CC6208249060E18FAB32 204779B5C03C0493BBBBC95CF02692CC4DEAA8D2EA90B5C2E64374E92BCB8501 429B8FAE4A76C0C6B76D6FF7CF9A7D5EDFBCA0E959541C59BD05B7DE43D25D53 FC3DDA6EF0C2743978A6D03E19CCED4A11F2EA4BCC3110BE8B8D9E2772361969 C19258EFAFDC276CB1ADE9208A941A36D18F9FB1C33DEF76AA315DD8F41F8A25 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cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % 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 (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 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 68 /D put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 82 /R put dup 84 /T put dup 101 /e put dup 109 /m put readonly def /FontBBox{-61 -250 999 759}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 74 /J put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 88 /X put dup 96 /lscript put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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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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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dup 116 /t put dup 117 /u put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 32 /arrowleft put dup 33 /arrowright put dup 38 /arrowsoutheast put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 63 /perpendicular put dup 68 /D put dup 71 /G put dup 72 /H put dup 75 /K put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P 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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 100 /d put readonly def /FontBBox{11 -250 1241 750}readonly 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 49 /one put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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 17 /equivalence put readonly def /FontBBox{-30 -958 1146 777}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /Phi put dup 40 /parenleft put dup 41 /parenright put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 58 /period put dup 59 /comma put dup 72 /H put readonly def /FontBBox{-29 -250 1075 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 6 /Sigma put dup 8 /Phi put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 59 /semicolon put dup 61 /equal put dup 68 /D put dup 70 /F put dup 78 /N put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 112 /p put readonly def /FontBBox{-27 -250 1122 750}readonly 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 /minus put dup 3 /asteriskmath put readonly def /FontBBox{-30 -955 1185 779}readonly 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. 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b(for)i(a)g(general)f(class)i(of)e (operators.)f(W)-7 b(e)24 b(refer)e(to)i(the)e(discussion)h(in)g (Section)f(6)h(for)g(more)f(information)e(on)83 5110 y(these)g(issues.)249 5209 y(Let)c(us)g(no)n(w)g(describe)f(more)g (precisely)g(the)h(content)f(of)g(this)h(paper)-5 b(.)15 b(In)h(Section)g(2)f(we)i(recall)e(the)h(necessary)f(de\002nitions)83 5309 y(from)23 b(the)g(theory)f(of)i(the)f(conjugate)f(operator)f(and)i (de\002ne)g(a)h(critical)g(set)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))24 b FI(for)f(the)h(operator)e Fy(H)7 b FI(.)23 b(In)h(the)f(more)g(usual)p Black 1905 5670 a(2)p Black eop end %%Page: 3 3 TeXDict begin 3 2 bop Black Black 83 307 a FI(setup)21 b(where)f Fy(H)31 b Fw(=)25 b Fy(h)p Fw(\()p Fy(P)12 b Fw(\))21 b FI(is)h(a)g(function)d(of)i(the)g(momentum)d(v)o(ector)i (operator)f Fy(P)34 b FI(and)20 b Fw(\010)i FI(is)g(the)f(position)f(v) o(ector)f(operator)83 407 y Fy(Q)h FI(in)h Fr(L)292 377 y FK(2)329 407 y Fw(\()p Fq(R)421 377 y Fu(d)460 407 y Fw(\))p 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b(as)i Fy(h)p Fw(\()p Fy(P)12 b Fw(\))p FI(,)23 b(the)f(deri)n(v)n(ati)n(v)o(e)f(appearing)f(in)i(the)h (de\002nition)e(of)h Fy(\024)2211 1040 y Fu(h)2277 1028 y FI(does)g(not)g(ha)n(v)o(e)g(a)h(direct)f(counterpart.)e(Ho)n(we)n(v) o(er)m(,)83 1127 y(the)25 b(identities)g Fw(\()p Fy(@)616 1139 y Fu(j)651 1127 y Fy(h)p Fw(\)\()p Fy(P)12 b Fw(\))32 b(=)g Fy(i)p Fw([)p Fy(h)p Fw(\()p Fy(P)12 b Fw(\))p Fy(;)i(Q)1321 1139 y Fu(j)1355 1127 y Fw(])26 b FI(suggest)e(to)h (de\002ne)f(the)h(set)h(of)e(critical)h(v)n(alues)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))25 b FI(in)g(terms)g(of)g(the)g(v)o(ector)83 1227 y(operator)18 b Fy(H)456 1197 y Fv(0)502 1227 y Fw(:=)613 1160 y Fp(\000)651 1227 y Fy(i)p Fw([)p Fy(H)r(;)c Fw(\010)871 1239 y FK(1)908 1227 y Fw(])p Fy(;)g(:)g(:)g(:)f(;)h(i)p Fw([)p Fy(H)r(;)g Fw(\010)1335 1239 y Fu(d)1374 1227 y Fw(])1397 1160 y Fp(\001)1435 1227 y FI(.)20 b(This)f(is)i(the)f (content)e(of)i(De\002nition)f(2.5.)f(In)i(Lemma)f(2.6)g(and)g(Theorem) f(3.6,)83 1326 y(we)i(sho)n(w)g(that)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))20 b FI(is)h(closed,)e(contains)g(the)h(set)h(of)e (eigen)m(v)n(alues)f(of)i Fy(H)7 b FI(,)20 b(and)f(that)h(the)g (spectrum)f(of)g Fy(H)27 b FI(in)20 b Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))18 b Fx(n)f Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))83 1426 y FI(is)22 b(purely)e(absolutely)g(continuous.)e(The)j (proof)e(of)i(the)g(latter)g(result)g(relies)h(on)e(the)h (construction,)e(described)h(in)h(Section)g(3,)83 1526 y(of)f(an)g(appropriate)e(conjugate)g(operator)g(for)i Fy(H)7 b FI(.)249 1625 y(In)26 b(Section)f(4,)h(we)g(recall)f(some)h (de\002nitions)f(in)h(relation)f(with)h(the)g(function)e Fy(f)35 b FI(that)26 b(appear)e(in)i(Theorem)e(1.1.)h(The)83 1725 y(function)f Fy(R)449 1737 y Fu(f)519 1725 y FI(is)j(introduced)d (and)h(some)h(of)g(its)h(properties)e(are)h(presented.)f(Section)g(5)h (is)h(the)g(core)e(of)h(the)g(paper)f(and)h(its)83 1825 y(most)18 b(technical)f(part.)g(It)i(contains)e(the)h(de\002nition)e (of)i Fy(T)1707 1837 y Fu(f)1768 1825 y FI(and)f(the)h(proof)e(of)i (the)g(precise)g(v)o(ersion)e(of)i(Theorem)e(1.1.)h(Suitable)83 1924 y(subspaces)j(of)g Fx(H)h FI(on)f(which)g(the)g(operators)f(are)h (well-de\002ned)f(and)g(on)h(which)g(the)g(equalities)g(hold)f(are)h (introduced.)249 2024 y(An)i(interpretation)f(of)h(our)f(results)i(is)g (proposed)e(in)h(Section)g(6.)h(The)f(relation)f(with)i(the)f(theory)f (of)h(time)h(operators)e(is)83 2124 y(e)o(xplained,)c(and)j(v)n(arious) e(cases)i(are)g(presented.)e(The)h(importance)f(of)h(Theorem)f(5.5)h (for)g(the)h(proof)e(of)h(the)g(e)o(xistence)g(of)h(the)83 2223 y(quantum)e(time)j(delay)e(and)h(Eisenb)n(ud-W)m(igner)d(F)o (ormula)i(is)i(also)g(sk)o(etched.)249 2323 y(In)j(Section)g(7,)g(we)g (sho)n(w)g(that)g(our)f(results)i(apply)e(to)h(man)o(y)f(operators)g Fy(H)31 b FI(appearing)22 b(in)j(physics)e(and)g(mathematics)83 2422 y(literature.)15 b(Among)g(other)h(e)o(xamples,)f(we)h(treat)h (Friedrichs)e(Hamiltonians,)g(Stark)h(Hamiltonians,)f(some)h(Jacobi)g 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(This)h(e)o(xplains)e(the)h(constant)g(use)g(of)g(abstract)g (commutators)e(methods)h(throughout)f(the)83 3718 y(paper)-5 b(.)83 3994 y Fz(2)119 b(Critical)31 b(v)o(alues)83 4179 y FI(In)e(this)g(section,)g(we)g(recall)g(some)g(standard)f(notions)g (on)h(the)g(conjugate)e(operator)g(theory)h(and)g(introduce)f(our)i (general)83 4279 y(frame)n(w)o(ork.)g(The)i(set)i(of)e(critical)g(v)n (alues)h(is)g(de\002ned)f(and)f(some)i(of)f(its)i(properties)d(are)h (outlined.)f(This)i(subset)f(of)h(the)83 4379 y(spectrum)19 b(of)h(the)g(operator)f(under)f(in)m(v)o(estigation)g(plays)i(an)g (essential)h(role)f(in)g(the)g(sequel.)249 4478 y(W)-7 b(e)26 b(\002rst)f(recall)g(some)f(f)o(acts)h(principally)e(borro)n (wed)f(from)h([1].)h(Let)h Fy(H)32 b FI(and)24 b Fy(A)h FI(be)f(tw)o(o)h(self-adjoint)e(operators)h(in)g(a)83 4578 y(Hilbert)19 b(space)g Fx(H)q FI(.)g(Their)g(respecti)n(v)o(e)f (domain)g(are)h(denoted)e(by)i Fx(D)r Fw(\()p Fy(H)7 b Fw(\))20 b FI(and)f Fx(D)r Fw(\()p Fy(A)p Fw(\))p 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FI(being)f(de\002ned)f(similarly\).)p Black 83 2586 a FA(Remark)25 b(2.1.)p Black 43 w FI(A)g(bounded)e (operator)g Fy(S)36 b Fx(2)c Fo(B)s Fw(\()p Fx(H)q Fw(\))27 b FI(belongs)d(to)h Fy(C)2120 2556 y FK(1)2157 2586 y Fw(\()p Fy(A)p Fw(\))i FI(if)e(the)g(map)f(\(2.1\))n(,)h(with)g Fy(R)3122 2598 y Fu(!)3196 2586 y FI(replaced)e(by)i Fy(S)5 b FI(,)25 b(is)83 2686 y(strongly)g(dif)n(ferentiable.)e (Similarly)-5 b(,)25 b Fy(S)39 b Fx(2)34 b Fo(B)s Fw(\()p Fx(H)q Fw(\))27 b FI(belongs)e(to)h Fy(C)2085 2656 y Fu(m)2149 2686 y Fw(\(\010\))h FI(if)f(the)g(map)f(\(2.3\))n(,)i(with)f Fy(R)3118 2698 y Fu(!)3192 2686 y FI(replaced)f(by)h Fy(S)5 b FI(,)26 b(is)83 2785 y(strongly)19 b Fy(C)441 2755 y Fu(m)504 2785 y FI(.)249 2938 y(In)29 b(the)g(sequel,)g(we)g (assume)h(that)f Fy(H)36 b FI(is)31 b(re)o(gular)c(with)j(respect)e(to) i(unitary)e(group)f Fx(f)2818 2935 y Fw(e)2855 2908 y Fu(ix)p Fv(\001)p FK(\010)2987 2938 y Fx(g)3029 2955 y Fu(x)p Fv(2)p Fn(R)3154 2939 y Fm(d)3222 2938 y FI(in)i(the)g(follo)n (wing)83 3038 y(sense.)p Black 83 3191 a FA(Assumption)22 b(2.2.)p Black 41 w FI(The)f(operator)e Fy(H)29 b FI(is)22 b(of)f(class)h Fy(C)1659 3161 y FK(3)1697 3191 y Fw(\(\010\))p FI(.)g(Furthermore,)c(for)j(each)g Fy(j)30 b Fx(2)25 b(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)p FI(,)21 b(the)g(quadratic)f(form)83 3290 y Fy(i)p Fw([)p Fy(H)r(;)14 b Fw(\010)303 3302 y Fu(j)338 3290 y Fw(])26 b FI(on)f Fx(D)r Fw(\()p Fy(H)7 b Fw(\))27 b FI(de\002nes)e(an)h(essentially)f (self-adjoint)f(operator)g(whose)h(self-adjoint)g(e)o(xtension)f(is)i (denoted)e(by)h Fy(H)3719 3260 y Fv(0)3712 3312 y Fu(j)3747 3290 y FI(.)83 3390 y(Similarly)-5 b(,)19 b(for)h(each)g Fy(k)s(;)14 b(`)23 b Fx(2)g(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)p FI(,)20 b(the)g(quadratic)f(form)g Fy(i)p Fw([)p Fy(H)2094 3360 y Fv(0)2087 3412 y Fu(j)2122 3390 y Fy(;)14 b Fw(\010)2219 3402 y Fu(k)2260 3390 y Fw(])21 b FI(on)e Fx(D)r Fw(\()p Fy(H)2581 3360 y Fv(0)2574 3412 y Fu(j)2610 3390 y Fw(\))i FI(de\002nes)f(an)h(essentially)f(self-adjoint)83 3490 y(operator)f(whose)g(self-adjoint)g(e)o(xtension)g(is)j(denoted)c (by)i Fy(H)1890 3460 y Fv(00)1883 3513 y Fu(j)s(k)1955 3490 y FI(,)g(and)g(the)g(quadratic)f(form)g Fy(i)p Fw([)p Fy(H)2899 3460 y Fv(00)2892 3513 y Fu(j)s(k)2964 3490 y Fy(;)14 b Fw(\010)3061 3502 y Fu(`)3093 3490 y Fw(])21 b FI(on)e Fx(D)r Fw(\()p Fy(H)3414 3460 y Fv(00)3407 3513 y Fu(j)s(k)3480 3490 y Fw(\))i FI(de\002nes)83 3589 y(an)f(essentially)g(self-adjoint)f(operator)g(whose)h(self-adjoint)f (e)o(xtension)f(is)j(denoted)e(by)h Fy(H)2759 3559 y Fv(000)2752 3613 y Fu(j)s(k)q(`)2851 3589 y FI(.)249 3755 y(This)d(assumption)e(implies)h(the)h(in)m(v)n(ariance)d(of)i Fx(D)r Fw(\()p Fy(H)7 b Fw(\))18 b FI(under)d(the)i(action)e(of)i(the)f (unitary)f(group)g Fx(f)3110 3752 y Fw(e)3146 3725 y Fu(ix)p Fv(\001)p FK(\010)3279 3755 y Fx(g)3321 3772 y Fu(x)p Fv(2)p Fn(R)3446 3756 y Fm(d)3484 3755 y FI(.)h(Indeed,)83 3855 y(if)k(the)g(quadratic)e(form)g Fy(i)p Fw([)p Fy(H)r(;)c Fw(\010)1006 3867 y Fu(j)1041 3855 y Fw(])20 b FI(on)g Fx(D)r Fw(\()p Fy(H)7 b Fw(\))21 b FI(de\002nes)e(an)h(essentially)f (self-adjoint)g(operator)f(in)i Fx(H)q FI(,)f(it)i(follo)n(ws)e(in)h (particular)83 3955 y(that)29 b Fx(D)r Fw(\()p Fy(H)7 b Fw(\))41 b Fx(\032)f(D)r Fw(\()p Fy(H)763 3925 y Fv(0)756 3976 y Fu(j)791 3955 y Fw(\))31 b FI(and)d(thus)i Fy(i)p Fw([)p Fy(H)r(;)14 b Fw(\010)1392 3967 y Fu(j)1426 3955 y Fw(])p Fx(D)r Fw(\()p Fy(H)7 b Fw(\))41 b Fx(\021)f Fy(H)1877 3925 y Fv(0)1870 3976 y Fu(j)1905 3955 y Fx(D)r Fw(\()p Fy(H)7 b Fw(\))41 b Fx(\032)e(H)q FI(.)30 b(It)f(follo)n(ws)g (then)g(from)g([13)n(,)h(Lemma)e(2])i(that)83 4063 y Fw(e)120 4036 y Fu(it)p FK(\010)215 4044 y Fm(j)265 4066 y Fx(D)r Fw(\()p Fy(H)7 b Fw(\))24 b Fx(\032)e(D)r Fw(\()p Fy(H)7 b Fw(\))21 b FI(for)d(all)i Fy(t)j Fx(2)h Fq(R)p FI(.)19 b(In)g(f)o(act,)g(one)g(easily)g(obtains)g(that)2270 4063 y Fw(e)2307 4036 y Fu(it)p FK(\010)2402 4044 y Fm(j)2452 4066 y Fx(D)r Fw(\()p Fy(H)7 b Fw(\))24 b(=)f Fx(D)r Fw(\()p Fy(H)7 b Fw(\))p FI(,)20 b(and)f(since)g(this)h(property)83 4165 y(holds)j(for)g(each)g Fy(j)30 b FI(one)23 b(also)h(has)1084 4162 y Fw(e)1121 4135 y Fu(ix)p Fv(\001)p FK(\010)1267 4165 y Fx(D)r Fw(\()p Fy(H)7 b Fw(\))31 b(=)e Fx(D)r Fw(\()p Fy(H)7 b Fw(\))24 b FI(for)f(all)i Fy(x)k Fx(2)h Fq(R)2278 4135 y Fu(d)2317 4165 y FI(.)24 b(As)g(a)g(consequence,)d(we) j(obtain)f(in)h(particular)83 4265 y(that)c(each)g(self-adjoint)f (operator)1518 4365 y Fy(H)7 b Fw(\()p Fy(x)p Fw(\))24 b(:=)1839 4362 y(e)1876 4330 y Fv(\000)p Fu(ix)p Fv(\001)p FK(\010)2074 4365 y Fy(H)2164 4362 y Fw(e)2201 4330 y Fu(ix)p Fv(\001)p FK(\010)3609 4365 y FI(\(2.4\))83 4504 y(\(with)c Fy(H)7 b Fw(\(0\))23 b(=)g Fy(H)7 b FI(\))20 b(has)g(domain)f Fx(D)r Fw([)p Fy(H)7 b Fw(\()p Fy(x)p Fw(\)])24 b(=)f Fx(D)r Fw(\()p Fy(H)7 b Fw(\))p FI(.)249 4604 y(Similarly)-5 b(,)15 b(the)g(domains)g Fx(D)r Fw(\()p Fy(H)1179 4574 y Fv(0)1172 4625 y Fu(j)1208 4604 y Fw(\))h FI(and)f Fx(D)r Fw(\()p Fy(H)1566 4574 y Fv(00)1559 4627 y Fu(j)s(k)1632 4604 y Fw(\))h FI(are)g(left)g(in)m(v)n(ariant)e(by)h (the)h(action)f(of)g(the)h(unitary)f(group)f Fx(f)3374 4601 y Fw(e)3410 4574 y Fu(ix)p Fv(\001)p FK(\010)3542 4604 y Fx(g)3584 4621 y Fu(x)p Fv(2)p Fn(R)3709 4604 y Fm(d)7 b FI(,)83 4717 y(and)25 b(the)h(operators)f Fy(H)769 4687 y Fv(0)762 4738 y Fu(j)797 4717 y Fw(\()p Fy(x)p Fw(\))35 b(:=)1064 4714 y(e)1101 4687 y Fv(\000)p Fu(ix)p Fv(\001)p FK(\010)1299 4717 y Fy(H)1375 4687 y Fv(0)1368 4738 y Fu(j)1417 4714 y Fw(e)1454 4687 y Fu(ix)p Fv(\001)p FK(\010)1613 4717 y FI(and)25 b Fy(H)1835 4687 y Fv(00)1828 4740 y Fu(j)s(k)1900 4717 y Fw(\()p Fy(x)p Fw(\))35 b(:=)2167 4714 y(e)2204 4687 y Fv(\000)p Fu(ix)p Fv(\001)p FK(\010)2402 4717 y Fy(H)2478 4687 y Fv(00)2471 4740 y Fu(j)s(k)2557 4714 y Fw(e)2593 4687 y Fu(ix)p Fv(\001)p FK(\010)2752 4717 y FI(are)26 b(self-adjoint)f (operators)f(with)83 4816 y(domains)19 b Fx(D)r Fw(\()p Fy(H)559 4786 y Fv(0)552 4838 y Fu(j)588 4816 y Fw(\))i FI(and)e Fx(D)r Fw(\()p Fy(H)955 4786 y Fv(00)948 4840 y Fu(j)s(k)1021 4816 y Fw(\))i FI(respecti)n(v)o(ely)-5 b(.)249 4916 y(Our)20 b(second)f(main)h(assumption)f(concerns)g(the)h (f)o(amily)g(of)g(operators)e Fy(H)7 b Fw(\()p Fy(x)p Fw(\))p FI(.)p Black 83 5069 a FA(Assumption)21 b(2.3.)p Black 40 w FI(The)f(operators)f Fx(f)p Fy(H)7 b Fw(\()p Fy(x)p Fw(\))p Fx(g)1445 5086 y Fu(x)p Fv(2)p Fn(R)1570 5069 y Fm(d)28 b FI(mutually)19 b(commute.)249 5222 y(Using)27 b(the)g(f)o(act)g(that)g(the)g(map)f Fq(R)1266 5192 y Fu(d)1340 5222 y Fx(3)36 b Fy(x)g Fx(7!)g Fy(C)1692 5234 y Fu(x)1769 5222 y Fx(2)g Fw(Aut)q([)p Fo(B)s Fw(\()p Fx(H)q Fw(\)])29 b FI(is)f(a)f(group)e(morphism,)g(one)h(easily)i(sho)n (ws)f(that)83 5321 y(Assumption)22 b(2.3)g(is)h(equi)n(v)n(alent)e(the) i(commutati)n(vity)e(of)h(each)g Fy(H)7 b Fw(\()p Fy(x)p Fw(\))24 b FI(with)f Fy(H)7 b FI(.)23 b(Furthermore,)d(Assumptions)i (2.2)g(and)g(2.3)83 5421 y(imply)e(additional)e(commutation)g (relations:)p Black 1905 5670 a(4)p Black eop end %%Page: 5 5 TeXDict begin 5 4 bop Black Black Black 83 307 a FA(Lemma)27 b(2.4.)p Black 44 w FJ(The)g(oper)o(ator)o(s)e Fy(H)7 b Fw(\()p Fy(x)p Fw(\))p FJ(,)28 b Fy(H)1363 277 y Fv(0)1356 329 y Fu(j)1391 307 y Fw(\()p Fy(y)s Fw(\))p FJ(,)f Fy(H)1623 277 y Fv(00)1616 331 y Fu(k)q(`)1685 307 y Fw(\()p Fy(z)t Fw(\))f FJ(mutually)g(commute)g(for)g(eac)o(h)g Fy(j;)14 b(k)s(;)g(`)34 b Fx(2)h(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)e(;)i(d)p Fx(g)27 b FJ(and)f(eac)o(h)83 419 y Fy(x;)14 b(y)s(;)g(z)26 b Fx(2)e Fq(R)452 389 y Fu(d)490 419 y FJ(.)p Black 83 581 a(Pr)l(oof.)p Black 41 w FI(Let)30 b Fy(!)44 b Fx(2)e Fq(C)25 b Fx(n)h Fq(R)p FI(,)k Fy(x;)14 b(y)s(;)g(z)45 b Fx(2)c Fq(R)1334 551 y Fu(d)1373 581 y FI(,)30 b Fy(j;)14 b(k)s(;)g(`;)g(m)41 b Fx(2)h(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)e(;)i(d)p Fx(g)p FI(,)30 b(and)g(set)h Fy(R)q Fw(\()p Fy(x)p Fw(\))42 b(:=)f([)p Fy(H)7 b Fw(\()p Fy(x)p Fw(\))26 b Fx(\000)g Fy(!)s Fw(])3289 551 y Fv(\000)p FK(1)3378 581 y FI(,)k Fy(R)3493 551 y Fv(0)3492 603 y Fu(j)3527 581 y Fw(\()p Fy(x)p Fw(\))42 b(:=)83 690 y([)p Fy(H)182 660 y Fv(0)175 712 y Fu(j)210 690 y Fw(\()p Fy(x)p Fw(\))20 b Fx(\000)e Fy(!)s Fw(])502 660 y Fv(\000)p FK(1)611 690 y FI(and)i Fy(R)816 660 y Fv(00)815 714 y Fu(j)s(k)887 690 y Fw(\()p Fy(x)p Fw(\))k(:=)e([)p Fy(H)1231 660 y Fv(00)1224 714 y Fu(j)s(k)1296 690 y Fw(\()p Fy(x)p Fw(\))e Fx(\000)e Fy(!)s Fw(])1588 660 y Fv(\000)p FK(1)1676 690 y FI(.)j(By)g (assumption,)d(one)i(has)g(the)h(equality)1282 898 y Fy(R)q Fw(\()p Fy(x)p Fw(\))1477 856 y Fu(R)p FK(\()p Fu("e)1615 864 y Fm(j)1647 856 y FK(\))p Fv(\000)p Fu(R)p FK(\(0\))p 1477 879 384 4 v 1653 926 a Fu(")1893 898 y Fw(=)1991 856 y Fu(R)p FK(\()p Fu("e)2129 864 y Fm(j)2161 856 y FK(\))p Fv(\000)p Fu(R)p FK(\(0\))p 1991 879 V 2167 926 a Fu(")2393 898 y Fy(R)q Fw(\()p Fy(x)p Fw(\))83 1076 y FI(for)f(each)f Fy(")k Fx(2)h Fq(R)18 b Fx(n)g(f)p Fw(0)p Fx(g)p FI(.)h(T)-7 b(aking)19 b(the)i(strong)e(limit)i(as)f Fy(")j Fx(!)g Fw(0)p FI(,)d(and)g(using)g(\(2.2\))e(and)i(Assumption)f (2.3,)g(one)h(obtains)1289 1254 y Fy(R)q Fw(\(0\))1473 1187 y Fp(\002)1507 1254 y Fy(R)q Fw(\()p Fy(x)p Fw(\))p Fy(H)1758 1220 y Fv(0)1751 1275 y Fu(j)1806 1254 y Fx(\000)e Fy(H)1965 1220 y Fv(0)1958 1275 y Fu(j)1993 1254 y Fy(R)q Fw(\()p Fy(x)p Fw(\))2168 1187 y Fp(\003)2217 1254 y Fy(R)q Fw(\(0\))23 b(=)f(0)p Fy(:)83 1433 y FI(Since)16 b(the)h(resolv)o(ent)e Fy(R)q Fw(\(0\))h FI(on)g(the)g(left)h(is)g (injecti)n(v)o(e,)e(this)i(implies)g(that)f Fy(R)q Fw(\()p Fy(x)p Fw(\))p Fy(H)2438 1403 y Fv(0)2431 1454 y Fu(j)2471 1433 y Fx(\000)t Fy(H)2616 1403 y Fv(0)2609 1454 y Fu(j)2643 1433 y Fy(R)q Fw(\()p Fy(x)p Fw(\))24 b(=)f(0)16 b FI(on)g Fx(D)r Fw(\()p Fy(H)7 b Fw(\))p FI(.)17 b(Furthermore,)83 1532 y(since)j Fx(D)r Fw(\()p Fy(H)7 b Fw(\))22 b FI(is)f(a)g(core)e (for)h Fy(H)993 1502 y Fv(0)986 1554 y Fu(j)1042 1532 y FI(the)g(last)h(equality)e(can)h(be)g(e)o(xtended)e(to)j Fx(D)r Fw(\()p Fy(H)2402 1502 y Fv(0)2395 1554 y Fu(j)2430 1532 y Fw(\))p FI(.)g(Finally)-5 b(,)20 b(by)f(multiplying)g(the)h (equation)1011 1723 y Fy(R)q Fw(\()p Fy(x)p Fw(\))k(=)e Fy(R)q Fw(\()p Fy(x)p Fw(\))1472 1656 y Fp(\000)1511 1723 y Fy(H)1587 1689 y Fv(0)1580 1743 y Fu(j)1634 1723 y Fx(\000)c Fy(!)1772 1656 y Fp(\001)1809 1723 y Fy(R)1873 1689 y Fv(0)1872 1743 y Fu(j)1907 1723 y Fw(\(0\))23 b(=)2124 1656 y Fp(\000)2162 1723 y Fy(H)2238 1689 y Fv(0)2231 1743 y Fu(j)2285 1723 y Fx(\000)18 b Fy(!)2423 1656 y Fp(\001)2460 1723 y Fy(R)q Fw(\()p Fy(x)p Fw(\))p Fy(R)2699 1689 y Fv(0)2698 1743 y Fu(j)2734 1723 y Fw(\(0\))83 1912 y FI(on)j(the)h(left)g(by)g Fy(R)615 1882 y Fv(0)614 1934 y Fu(j)649 1912 y Fw(\(0\))p FI(,)g(one)g(gets)g Fy(R)1161 1882 y Fv(0)1160 1934 y Fu(j)1195 1912 y Fw(\(0\))p Fy(R)q Fw(\()p Fy(x)p Fw(\))27 b(=)f Fy(R)q Fw(\()p Fy(x)p Fw(\))p Fy(R)1833 1882 y Fv(0)1832 1934 y Fu(j)1867 1912 y Fw(\(0\))p FI(.)d(Using)e(the)h(morphism)e(property)g(of)i(the)g(map) f Fq(R)3465 1882 y Fu(d)3530 1912 y Fx(3)26 b Fy(x)h Fx(7!)83 2012 y Fy(C)142 2024 y Fu(x)207 2012 y Fx(2)d Fw(Aut[)p Fo(B)s Fw(\()p Fx(H)q Fw(\)])p FI(,)e(one)e(infers)g(from)f (this)h(that)h Fy(H)7 b Fw(\()p Fy(x)p Fw(\))21 b FI(and)f Fy(H)1972 1982 y Fv(0)1965 2034 y Fu(j)2000 2012 y Fw(\()p Fy(y)s Fw(\))h FI(commute.)249 2112 y(A)j(similar)f(ar)o(gument)e (leads)j(to)f(the)g(commutati)n(vity)e(of)i(the)h(operators)e Fy(H)2465 2081 y Fv(0)2458 2133 y Fu(j)2493 2112 y Fw(\()p Fy(x)p Fw(\))i FI(and)f Fy(H)2848 2081 y Fv(0)2841 2135 y Fu(k)2882 2112 y Fw(\()p Fy(y)s Fw(\))h FI(by)f(considering)e(the)i (op-)83 2237 y(erators)d Fy(R)394 2206 y Fv(0)393 2258 y Fu(j)428 2237 y Fw(\()p Fy(x)p Fw(\))549 2196 y Fu(R)p FK(\()p Fu("e)687 2205 y Fm(k)725 2196 y FK(\))p Fv(\000)p Fu(R)p FK(\(0\))p 550 2218 389 4 v 729 2265 a Fu(")970 2237 y FI(and)1120 2196 y Fu(R)p FK(\()p Fu("e)1258 2205 y Fm(k)1296 2196 y FK(\))p Fv(\000)p Fu(R)p FK(\(0\))p 1120 2218 V 1299 2265 a Fu(")1519 2237 y Fy(R)1583 2206 y Fv(0)1582 2258 y Fu(j)1617 2237 y Fw(\()p Fy(x)p Fw(\))p FI(.)i(The)e(commutati)n(vity)e(of)j Fy(H)7 b Fw(\()p Fy(x)p Fw(\))21 b FI(and)f Fy(H)2940 2206 y Fv(00)2933 2260 y Fu(j)s(k)3005 2237 y Fw(\()p Fy(z)t Fw(\))h FI(is)g(obtained)e (by)i(con-)83 2378 y(sidering)c(the)i(operators)d Fy(R)q Fw(\()p Fy(x)p Fw(\))1002 2328 y Fu(R)1052 2303 y Fl(0)1052 2344 y Fm(j)1084 2328 y FK(\()p Fu("e)1172 2337 y Fm(k)1209 2328 y FK(\))p Fv(\000)p Fu(R)1337 2303 y Fl(0)1337 2344 y Fm(j)1368 2328 y FK(\(0\))p 1003 2359 450 4 v 1212 2407 a Fu(")1482 2378 y FI(and)1630 2328 y Fu(R)1680 2303 y Fl(0)1680 2344 y Fm(j)1712 2328 y FK(\()p Fu("e)1800 2337 y Fm(k)1837 2328 y FK(\))p Fv(\000)p Fu(R)1965 2303 y Fl(0)1965 2344 y Fm(j)1995 2328 y FK(\(0\))p 1630 2359 V 1840 2407 a Fu(")2090 2378 y Fy(R)q Fw(\()p Fy(x)p Fw(\))p FI(,)k(and)d(the)i(commutati)n(vity)d(of)i Fy(H)3230 2348 y Fv(0)3223 2400 y Fu(j)3258 2378 y Fw(\()p Fy(y)s Fw(\))h FI(and)f Fy(H)3600 2348 y Fv(00)3593 2402 y Fu(k)q(`)3661 2378 y Fw(\()p Fy(z)t Fw(\))83 2510 y FI(by)23 b(considering)e(the)i (operators)f Fy(R)1123 2480 y Fv(0)1122 2532 y Fu(j)1157 2510 y Fw(\()p Fy(y)s Fw(\))1275 2468 y Fu(R)1325 2443 y Fl(0)1325 2484 y Fm(k)1362 2468 y FK(\()p Fu("e)1450 2477 y Fm(`)1479 2468 y FK(\))p Fv(\000)p Fu(R)1607 2443 y Fl(0)1607 2484 y Fm(k)1643 2468 y FK(\(0\))p 1275 2491 454 4 v 1486 2539 a Fu(")1762 2510 y FI(and)1916 2468 y Fu(R)1966 2443 y Fl(0)1966 2484 y Fm(k)2002 2468 y FK(\()p Fu("e)2090 2477 y Fm(`)2120 2468 y FK(\))p Fv(\000)p Fu(R)2248 2443 y Fl(0)2248 2484 y Fm(k)2284 2468 y FK(\(0\))p 1916 2491 V 2127 2539 a Fu(")2379 2510 y Fy(R)2443 2480 y Fv(0)2442 2532 y Fu(j)2477 2510 y Fw(\()p Fy(y)s Fw(\))p FI(.)i(Finally)-5 b(,)22 b(the)h(commutation)e(between)83 2642 y Fy(H)159 2611 y Fv(00)152 2665 y Fu(j)s(k)224 2642 y Fw(\()p Fy(x)p Fw(\))g FI(and)d Fy(H)571 2611 y Fv(00)564 2665 y Fu(`m)655 2642 y Fw(\()p Fy(y)s Fw(\))i FI(is)h(obtained)d(by)g(considering)g(the)h(operators)f Fy(R)2189 2611 y Fv(00)2188 2665 y Fu(j)s(k)2260 2642 y Fw(\()p Fy(x)p Fw(\))2381 2599 y Fu(R)2431 2574 y Fl(0)2431 2616 y Fm(`)2461 2599 y FK(\()p Fu("e)2549 2607 y Fm(m)2605 2599 y FK(\))p Fv(\000)p Fu(R)2733 2574 y Fl(0)2733 2616 y Fm(`)2761 2599 y FK(\(0\))p 2382 2623 465 4 v 2598 2670 a Fu(")2876 2642 y FI(and)3026 2599 y Fu(R)3076 2574 y Fl(0)3076 2616 y Fm(`)3105 2599 y FK(\()p Fu("e)3193 2607 y Fm(m)3249 2599 y FK(\))p Fv(\000)p Fu(R)3377 2574 y Fl(0)3377 2616 y Fm(`)3406 2599 y FK(\(0\))p 3026 2623 V 3243 2670 a Fu(")3501 2642 y Fy(R)3565 2611 y Fv(00)3564 2665 y Fu(j)s(k)3635 2642 y Fw(\()p Fy(x)p Fw(\))p FI(.)83 2741 y(Details)j(are)f(left)h(to)f(the)g(reader)-5 b(.)p 3708 2741 4 57 v 3712 2689 50 4 v 3712 2741 V 3761 2741 4 57 v 249 2906 a(F)o(or)19 b(simplicity)-5 b(,)19 b(we)g(write)h Fy(H)1134 2876 y Fv(0)1177 2906 y FI(for)f(the)g(v)o(ector)f(operator)g Fw(\()p Fy(H)2046 2876 y Fv(0)2039 2927 y FK(1)2077 2906 y Fy(;)c(:)g(:)g(:)f(;)h(H)2337 2876 y Fv(0)2330 2930 y Fu(d)2369 2906 y Fw(\))p FI(,)20 b(and)f(de\002ne)f(for)h(each)g (measurable)f(function)83 3006 y Fy(f)39 b Fw(:)31 b Fq(R)277 2976 y Fu(d)347 3006 y Fx(!)f Fq(C)25 b FI(the)g(operator)d Fy(f)9 b Fw(\()p Fy(H)1131 2976 y Fv(0)1154 3006 y Fw(\))25 b FI(by)f(using)g(the)g Fy(d)p FI(-v)n(ariables)g(functional)e (calculus.)i(The)g(symbol)f Fy(E)3212 2976 y Fu(H)3276 3006 y Fw(\()p Fx(\001)p Fw(\))i FI(denotes)f(the)83 3106 y(spectral)c(measure)f(of)h Fy(H)7 b FI(.)p Black 83 3268 a FA(De\002nition)20 b(2.5.)p Black 40 w FI(A)h(number)e Fy(\025)k Fx(2)g Fq(R)e FI(is)g(called)f(a)h(re)o(gular)d(v)n(alue)i (of)g Fy(H)27 b FI(if)21 b(there)f(e)o(xists)g Fy(\016)26 b(>)d Fw(0)d FI(such)g(that)1116 3461 y Fw(lim)1108 3515 y Fu(")p Fv(&)p FK(0)1253 3390 y Fp(\015)1253 3440 y(\015)1299 3394 y(\002)1333 3461 y Fw(\()p Fy(H)1441 3426 y Fv(0)1465 3461 y Fw(\))1497 3426 y FK(2)1553 3461 y Fw(+)e Fy(")1675 3394 y Fp(\003)1709 3411 y Fv(\000)p FK(1)1798 3461 y Fy(E)1864 3426 y Fu(H)1927 3394 y Fp(\000)1965 3461 y Fw(\()p Fy(\025)h Fx(\000)f Fy(\016)o(;)c(\025)19 b Fw(+)f Fy(\016)s Fw(\))2442 3394 y Fp(\001)2480 3390 y(\015)2480 3440 y(\015)2549 3461 y Fy(<)23 b Fx(1)p Fy(:)866 b FI(\(2.5\))83 3684 y(A)23 b(number)d Fy(\025)27 b Fx(2)g Fq(R)c FI(that)f(is)i(not)d (a)i(re)o(gular)e(v)n(alue)g(of)h Fy(H)30 b FI(is)23 b(called)f(a)g(critical)h(v)n(alue)e(of)h Fy(H)7 b FI(.)23 b(W)-7 b(e)23 b(denote)e(by)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))23 b FI(the)f(set)h(of)83 3783 y(critical)d(v)n(alues)g(of)g Fy(H)7 b FI(.)249 3946 y(From)22 b(no)n(w)g(on,)f(we)i(shall)g(use)f (the)h(shorter)e(notation)g Fy(E)1914 3916 y Fu(H)1977 3946 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))24 b FI(for)e Fy(E)2376 3916 y Fu(H)2439 3879 y Fp(\000)2477 3946 y Fw(\()p Fy(\025)e Fx(\000)g Fy(\016)o(;)14 b(\025)20 b Fw(+)f Fy(\016)s Fw(\))2959 3879 y Fp(\001)2998 3946 y FI(.)j(In)g(the)h(ne)o(xt)e(lemma)h(we)83 4046 y(put)e(into)g(e)n (vidence)f(some)h(useful)f(properties)g(of)h(the)g(set)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FI(.)p Black 83 4208 a FA(Lemma)21 b(2.6.)p Black 40 w FJ(Let)g(Assumptions)e(2.2)h(and)f (2.3)h(be)g(veri\002ed.)f(Then)h(the)g(set)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))21 b FJ(possesses)g(the)f(following)g(pr)l (operties:)p Black 152 4371 a(\(a\))p Black 41 w Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))20 b FJ(is)i(closed.)p Black 152 4535 a(\(b\))p Black 41 w Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))20 b FJ(contains)g(the)g(set)h(of)f(eig)o(en)m(values)f(of)h Fy(H)7 b FJ(.)p Black 157 4716 a(\(c\))p Black 41 w(The)20 b(limit)h Fw(lim)724 4728 y Fu(")p Fv(&)p FK(0)873 4646 y Fp(\015)873 4696 y(\015)919 4649 y(\002)954 4716 y Fw(\()p Fy(H)1062 4686 y Fv(0)1085 4716 y Fw(\))1117 4686 y FK(2)1173 4716 y Fw(+)d Fy(")1295 4649 y Fp(\003)1329 4666 y Fv(\000)p FK(1)1418 4716 y Fy(E)1484 4686 y Fu(H)1547 4716 y Fw(\()p Fy(J)8 b Fw(\))1665 4646 y Fp(\015)1665 4696 y(\015)1733 4716 y FJ(is)21 b(\002nite)f(for)g(eac)o(h)g(compact)f (set)i Fy(J)31 b Fx(\032)22 b Fq(R)d Fx(n)f Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FJ(.)p Black 152 4881 a(\(d\))p Black 41 w(F)-9 b(or)31 b(eac)o(h)g(compact)f(set)j Fy(J)51 b Fx(\032)44 b Fq(R)27 b Fx(n)f Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FJ(,)32 b(ther)m(e)g(e)n(xists)g(a)g(compact)e(set)j Fy(U)52 b Fx(\032)44 b Fw(\(0)p Fy(;)14 b Fx(1)p Fw(\))32 b FJ(suc)o(h)f(that)g Fy(E)3478 4851 y Fu(H)3541 4881 y Fw(\()p Fy(J)8 b Fw(\))44 b(=)291 4980 y Fy(E)357 4950 y Fv(j)p Fu(H)435 4925 y Fl(0)458 4950 y Fv(j)482 4980 y Fw(\()p Fy(U)9 b Fw(\))p Fy(E)678 4950 y Fu(H)741 4980 y Fw(\()p Fy(J)f Fw(\))p FJ(.)p Black 83 5143 a(Pr)l(oof.)p Black 41 w FI(\(a\))22 b(Let)g Fy(\025)627 5155 y FK(0)688 5143 y FI(be)g(a)g(re)o(gular)f(v)n(alue)g(for)h Fy(H)7 b FI(,)22 b FJ(i.e)o(.)g FI(there)g(e)o(xists)h Fy(\016)2104 5155 y FK(0)2168 5143 y Fy(>)j Fw(0)c FI(such)g(that)h(\(2.5\))d(holds) i(with)h Fy(\016)i FI(replaced)c(by)h Fy(\016)3710 5155 y FK(0)3747 5143 y FI(.)83 5243 y(Let)e Fy(\025)k Fx(2)f Fw(\()p Fy(\025)444 5255 y FK(0)501 5243 y Fx(\000)18 b Fy(\016)621 5255 y FK(0)658 5243 y Fy(;)c(\025)743 5255 y FK(0)799 5243 y Fw(+)k Fy(\016)919 5255 y FK(0)956 5243 y Fw(\))j FI(and)f(let)h Fy(\016)26 b(>)c Fw(0)f FI(such)e(that)1308 5421 y Fw(\()p Fy(\025)g Fx(\000)f Fy(\016)o(;)c(\025)19 b Fw(+)f Fy(\016)s Fw(\))23 b Fx(\032)g Fw(\()p Fy(\025)1976 5433 y FK(0)2032 5421 y Fx(\000)18 b Fy(\016)2152 5433 y FK(0)2189 5421 y Fy(;)c(\025)2274 5433 y FK(0)2330 5421 y Fw(+)k Fy(\016)2450 5433 y FK(0)2487 5421 y Fw(\))p Fy(:)p Black 1905 5670 a FI(5)p Black eop end %%Page: 6 6 TeXDict begin 6 5 bop Black Black 83 307 a FI(Then,)19 b(since)i Fy(E)553 277 y Fu(H)616 307 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))24 b(=)e Fy(E)982 277 y Fu(H)1045 307 y Fw(\()p Fy(\025)1125 319 y FK(0)1163 307 y Fw(;)14 b Fy(\016)1237 319 y FK(0)1274 307 y Fw(\))p Fy(E)1372 277 y Fu(H)1436 307 y Fw(\()p Fy(\025)p Fw(;)g Fy(\016)s Fw(\))p FI(,)21 b(one)f(has)679 491 y Fw(lim)671 545 y Fu(")p Fv(&)p FK(0)816 420 y Fp(\015)816 470 y(\015)862 424 y(\002)896 491 y Fw(\()p Fy(H)1004 457 y Fv(0)1028 491 y Fw(\))1060 457 y FK(2)1116 491 y Fw(+)e Fy(")1238 424 y Fp(\003)1272 441 y Fv(\000)p FK(1)1361 491 y Fy(E)1427 457 y Fu(H)1490 491 y Fw(\()p Fy(\025)p Fw(;)c Fy(\016)s Fw(\))1679 420 y Fp(\015)1679 470 y(\015)1749 491 y Fx(\024)30 b Fw(lim)1837 545 y Fu(")p Fv(&)p FK(0)1981 420 y Fp(\015)1981 470 y(\015)2027 424 y(\002)2062 491 y Fw(\()p Fy(H)2170 457 y Fv(0)2193 491 y Fw(\))2225 457 y FK(2)2281 491 y Fw(+)18 b Fy(")2403 424 y Fp(\003)2438 441 y Fv(\000)p FK(1)2527 491 y Fy(E)2593 457 y Fu(H)2656 491 y Fw(\()p Fy(\025)2736 503 y FK(0)2774 491 y Fw(;)c Fy(\016)2848 503 y FK(0)2885 491 y Fw(\))2917 420 y Fp(\015)2917 470 y(\015)2986 491 y Fy(<)23 b Fx(1)p Fy(:)83 702 y FI(But)g(this)g(means) g(e)o(xactly)e(that)i Fy(\025)h FI(is)f(a)h(re)o(gular)d(v)n(alue)h (for)g(an)o(y)g Fy(\025)28 b Fx(2)g Fw(\()p Fy(\025)2184 714 y FK(0)2242 702 y Fx(\000)20 b Fy(\016)2364 714 y FK(0)2401 702 y Fy(;)14 b(\025)2486 714 y FK(0)2544 702 y Fw(+)20 b Fy(\016)2666 714 y FK(0)2703 702 y Fw(\))p FI(.)j(So)g(the)f(set)i(of)e(re)o(gular)f(v)n(alues)i(is)83 802 y(open,)c(and)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))20 b FI(is)i(closed.)249 902 y(\(b\))g(Let)g Fy(\025)28 b Fx(2)f Fq(R)c FI(be)f(an)g(eigen)m(v)n(alue)e(of)i Fy(H)7 b FI(,)22 b(and)g(let)h Fy(')1836 914 y Fu(\025)1903 902 y FI(be)f(an)g(associated)g(eigen)m(v)o(ector)e(with)i(norm)f(one.) h(Since)g Fy(H)30 b FI(is)83 1001 y(of)21 b(class)h Fy(C)422 971 y FK(1)459 1001 y Fw(\(\010)551 1013 y Fu(j)586 1001 y Fw(\))g FI(for)f(each)f Fy(j)5 b FI(,)21 b(we)h(kno)n(w)e(from)g(the) h(V)-5 b(irial)21 b(theorem)e([1,)i(Prop.)f(7.2.10])f(that)i Fy(E)2925 971 y Fu(H)2988 1001 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))p Fy(H)3260 971 y Fv(0)3253 1023 y Fu(j)3288 1001 y Fy(E)3354 971 y Fu(H)3417 1001 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))j(=)h(0)83 1101 y FI(for)20 b(each)f Fy(j)5 b FI(.)21 b(This,)f(together)f(with)h(Lemma)g(2.4,)f(implies)h (that)1069 1282 y Fy(E)1135 1247 y Fu(H)1198 1282 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))1394 1214 y Fp(\002)1429 1282 y Fw(\()p Fy(H)1537 1247 y Fv(0)1560 1282 y Fw(\))1592 1247 y FK(2)1648 1282 y Fw(+)e Fy(")1770 1214 y Fp(\003)1804 1232 y Fv(\000)p FK(1)1893 1282 y Fy(E)1959 1247 y Fu(H)2022 1282 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))24 b(=)e Fy(")2368 1247 y Fv(\000)p FK(1)2457 1282 y Fy(E)2523 1247 y Fu(H)2586 1282 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))83 1448 y FI(for)e(each)f Fy(")k(>)g Fw(0)p FI(.)d(In)g (particular)m(,)e(we)j(obtain)e(for)h(each)f Fy(\016)26 b(>)d Fw(0)d FI(the)h(equalities)609 1562 y Fp(\002)643 1629 y Fw(\()p Fy(H)751 1595 y Fv(0)775 1629 y Fw(\))807 1595 y FK(2)863 1629 y Fw(+)d Fy(")985 1562 y Fp(\003)1019 1579 y Fv(\000)p FK(1)1108 1629 y Fy(E)1174 1595 y Fu(H)1237 1629 y Fw(\()p Fy(\025)p Fw(;)c Fy(\016)s Fw(\))p Fy(')1480 1641 y Fu(\025)1548 1629 y Fw(=)23 b Fy(E)1702 1595 y Fu(H)1765 1629 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))1961 1562 y Fp(\002)1995 1629 y Fw(\()p Fy(H)2103 1595 y Fv(0)2127 1629 y Fw(\))2159 1595 y FK(2)2215 1629 y Fw(+)18 b Fy(")2337 1562 y Fp(\003)2371 1579 y Fv(\000)p FK(1)2460 1629 y Fy(E)2526 1595 y Fu(H)2589 1629 y Fw(\()p Fx(f)p Fy(\025)p Fx(g)p Fw(\))p Fy(')2839 1641 y Fu(\025)2906 1629 y Fw(=)23 b Fy(")3033 1595 y Fv(\000)p FK(1)3121 1629 y Fy(')3175 1641 y Fu(\025)3219 1629 y Fy(;)83 1796 y FI(and)384 1896 y Fw(lim)376 1950 y Fu(")p Fv(&)p FK(0)520 1825 y Fp(\015)520 1875 y(\015)566 1828 y(\002)601 1896 y Fw(\()p Fy(H)709 1861 y Fv(0)732 1896 y Fw(\))764 1861 y FK(2)820 1896 y Fw(+)18 b Fy(")942 1828 y Fp(\003)977 1846 y Fv(\000)p FK(1)1066 1896 y Fy(E)1132 1861 y Fu(H)1195 1896 y Fw(\()p Fy(\025)p Fw(;)c Fy(\016)s Fw(\))1384 1825 y Fp(\015)1384 1875 y(\015)1454 1896 y Fx(\025)30 b Fw(lim)1542 1950 y Fu(")p Fv(&)p FK(0)1686 1825 y Fp(\015)1686 1875 y(\015)1732 1828 y(\002)1767 1896 y Fw(\()p Fy(H)1875 1861 y Fv(0)1898 1896 y Fw(\))1930 1861 y FK(2)1986 1896 y Fw(+)18 b Fy(")2108 1828 y Fp(\003)2142 1846 y Fv(\000)p FK(1)2232 1896 y Fy(E)2298 1861 y Fu(H)2361 1896 y Fw(\()p Fy(\025)p Fw(;)c Fy(\016)s Fw(\))p Fy(')2604 1908 y Fu(\025)2648 1825 y Fp(\015)2648 1875 y(\015)2718 1896 y Fw(=)30 b(lim)2805 1950 y Fu(")p Fv(&)p FK(0)2950 1896 y Fy(")2989 1861 y Fv(\000)p FK(1)3078 1896 y Fx(k)p Fy(')3174 1908 y Fu(\025)3217 1896 y Fx(k)23 b Fw(=)f Fx(1)p Fy(:)83 2076 y FI(Since)e Fy(\016)k FI(has)c(been)g(chosen)f(arbitrarily)-5 b(,)18 b(this)j(implies)f(that)h Fy(\025)g FI(is)g(not)f(a)g(re)o (gular)f(v)n(alue)h(of)f Fy(H)7 b FI(.)249 2175 y(\(c\))20 b(This)g(follo)n(ws)g(easily)h(by)e(using)h(a)h(compacity)d(ar)o (gument.)249 2275 y(\(d\))27 b(Let)h(us)g(concentrate)e(\002rst)j(on)e (the)h(lo)n(wer)f(bound)f(of)i Fy(U)9 b FI(.)28 b(Clearly)-5 b(,)27 b(if)h Fx(j)p Fy(H)2555 2245 y Fv(0)2578 2275 y Fx(j)g FI(is)h(strictly)f(positi)n(v)o(e,)f(then)g Fy(U)37 b FI(can)28 b(be)83 2375 y(chosen)19 b(in)i Fw(\(0)p Fy(;)14 b Fx(1)p Fw(\))21 b FI(and)f(thus)g(is)h(bounded)d(from)h(belo) n(w)h(by)g(a)h(strictly)f(positi)n(v)o(e)g(number)-5 b(.)19 b(So)h(assume)g(no)n(w)g(that)h Fx(j)p Fy(H)3519 2344 y Fv(0)3542 2375 y Fx(j)g FI(is)g(not)83 2474 y(strictly)e(positi) n(v)o(e,)f(that)i(is)g Fw(0)i Fx(2)i Fy(\033)s Fw(\()p Fx(j)p Fy(H)1177 2444 y Fv(0)1200 2474 y Fx(j)p Fw(\))p FI(.)c(By)g(absurd,)d(suppose)i(that)g Fy(U)28 b FI(is)20 b(not)f(bounded)d(from)j(belo)n(w)f(by)h(a)g(strictly)g(positi)n(v)o(e) 83 2574 y(number)m(,)f FJ(i.e)o(.)i FI(there)g(does)h(not)f(e)o(xist)h Fy(a)i(>)g Fw(0)e FI(such)f(that)h Fy(U)32 b Fx(\032)23 b Fw(\()p Fy(a;)14 b Fx(1)p Fw(\))p FI(.)21 b(Then)f(for)g Fy(n)j Fw(=)h(1)p Fy(;)14 b Fw(2)p Fy(;)g(:)g(:)g(:)e FI(,)21 b(there)f(e)o(xists)h Fy( )3376 2586 y Fu(n)3445 2574 y Fx(2)j(H)e FI(such)83 2673 y(that)e Fy(E)294 2643 y Fv(j)p Fu(H)372 2618 y Fl(0)395 2643 y Fv(j)419 2606 y Fp(\000)457 2673 y Fw([0)p Fy(;)14 b Fw(1)p Fy(=n)p Fw(\))725 2606 y Fp(\001)762 2673 y Fy(E)828 2643 y Fu(H)891 2673 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )1063 2685 y Fu(n)1132 2673 y Fx(6)p Fw(=)23 b(0)p FI(,)d(and)f(the)h(v)o(ectors)1317 2918 y Fy(')1371 2930 y Fu(n)1440 2918 y Fw(:=)1602 2856 y Fy(E)1668 2826 y Fv(j)p Fu(H)1746 2801 y Fl(0)1769 2826 y Fv(j)1793 2789 y Fp(\000)1831 2856 y Fw([0)p Fy(;)14 b Fw(1)p Fy(=n)p Fw(\))2099 2789 y Fp(\001)2136 2856 y Fy(E)2202 2826 y Fu(H)2265 2856 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )2437 2868 y Fu(n)p 1560 2898 964 4 v 1560 2979 a Fx(k)p Fy(E)1668 2955 y Fv(j)p Fu(H)1746 2938 y Fl(0)1769 2955 y Fv(j)1793 2912 y Fp(\000)1831 2979 y Fw([0)p Fy(;)14 b Fw(1)p Fy(=n)p Fw(\))2099 2912 y Fp(\001)2136 2979 y Fy(E)2202 2955 y Fu(H)2265 2979 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )2437 2991 y Fu(n)2482 2979 y Fx(k)83 3162 y FI(satisfy)21 b Fx(k)p Fy(')417 3174 y Fu(n)462 3162 y Fx(k)h Fw(=)h(1)p FI(,)d(and)f Fy(E)903 3132 y Fu(H)966 3162 y Fw(\()p Fy(J)8 b Fw(\))p Fy(')1138 3174 y Fu(n)1208 3162 y Fw(=)22 b Fy(E)1361 3132 y Fv(j)p Fu(H)1439 3107 y Fl(0)1462 3132 y Fv(j)1486 3094 y Fp(\000)1524 3162 y Fw([0)p Fy(;)14 b Fw(1)p Fy(=n)p Fw(\))1792 3094 y Fp(\001)1829 3162 y Fy(')1883 3174 y Fu(n)1951 3162 y Fw(=)23 b Fy(')2093 3174 y Fu(n)2139 3162 y FI(.)d(It)h(follo)n(ws)e (by)h(point)g(\(c\))f(that)521 3354 y Fw(Const)p Fy(:)j Fx(\025)31 b Fw(lim)867 3408 y Fu(")p Fv(&)p FK(0)1011 3283 y Fp(\015)1011 3333 y(\015)1057 3286 y(\002)1092 3354 y Fw(\()p Fy(H)1200 3319 y Fv(0)1223 3354 y Fw(\))1255 3319 y FK(2)1311 3354 y Fw(+)18 b Fy(")1433 3286 y Fp(\003)1468 3304 y Fv(\000)p FK(1)1557 3354 y Fy(E)1623 3319 y Fu(H)1686 3354 y Fw(\()p Fy(J)8 b Fw(\))1804 3283 y Fp(\015)1804 3333 y(\015)1874 3354 y Fx(\025)30 b Fw(lim)1961 3408 y Fu(")p Fv(&)p FK(0)2106 3283 y Fp(\015)2106 3333 y(\015)2152 3286 y(\002)2187 3354 y Fw(\()p Fy(H)2295 3319 y Fv(0)2318 3354 y Fw(\))2350 3319 y FK(2)2406 3354 y Fw(+)18 b Fy(")2528 3286 y Fp(\003)2562 3304 y Fv(\000)p FK(1)2651 3354 y Fy(E)2717 3319 y Fu(H)2780 3354 y Fw(\()p Fy(J)8 b Fw(\))p Fy(')2952 3366 y Fu(n)2998 3283 y Fp(\015)2998 3333 y(\015)1874 3548 y Fw(=)30 b(lim)1961 3602 y Fu(")p Fv(&)p FK(0)2106 3478 y Fp(\015)2106 3527 y(\015)2152 3481 y(\002)2187 3548 y Fw(\()p Fy(H)2295 3514 y Fv(0)2318 3548 y Fw(\))2350 3514 y FK(2)2406 3548 y Fw(+)18 b Fy(")2528 3481 y Fp(\003)2562 3498 y Fv(\000)p FK(1)2651 3548 y Fy(E)2717 3514 y Fv(j)p Fu(H)2795 3489 y Fl(0)2818 3514 y Fv(j)2842 3481 y Fp(\000)2880 3548 y Fw([0)p Fy(;)c Fw(1)p Fy(=n)p Fw(\))3148 3481 y Fp(\001)3185 3548 y Fy(')3239 3560 y Fu(n)3284 3478 y Fp(\015)3284 3527 y(\015)1874 3743 y Fx(\025)30 b Fw(lim)1961 3797 y Fu(")p Fv(&)p FK(0)2106 3675 y Fp(\000)2144 3743 y Fy(n)2194 3708 y Fv(\000)p FK(2)2301 3743 y Fw(+)18 b Fy(")2423 3675 y Fp(\001)2461 3693 y Fv(\000)p FK(1)2550 3743 y Fx(k)p Fy(')2646 3755 y Fu(n)2691 3743 y Fx(k)1874 3921 y Fw(=)k Fy(n)2011 3887 y FK(2)2048 3921 y Fy(;)83 4088 y FI(which)e(leads)g(to)g(a)h(contradiction)d(when)h Fy(n)k Fx(!)g(1)p FI(.)249 4188 y(Let)f(us)g(no)n(w)f(concentrate)f(on) h(the)h(upper)e(bound)g(of)h Fy(U)9 b FI(.)22 b(Clearly)-5 b(,)21 b(if)h Fx(j)p Fy(H)2370 4158 y Fv(0)2393 4188 y Fx(j)g FI(is)h(a)f(bounded)d(operator)m(,)g(one)i(can)g(choose)g(a)83 4287 y(bounded)g(subset)j Fy(U)32 b FI(of)24 b Fq(R)g FI(and)f(thus)g Fy(U)33 b FI(is)24 b(upper)e(bounded.)f(So)j(assume)f (no)n(w)g(that)h Fx(j)p Fy(H)2704 4257 y Fv(0)2727 4287 y Fx(j)g FI(is)g(not)f(a)h(bounded)d(operator)-5 b(.)22 b(By)83 4387 y(absurd,)f(suppose)g(that)i Fy(U)31 b FI(is)24 b(not)e(bounded)d(from)j(abo)o(v)o(e,)e FJ(i.e)o(.)i FI(there)g(does)g(not)g(e)o(xist)g Fy(b)27 b(<)g Fx(1)22 b FI(such)h(that)f Fy(U)36 b Fx(\032)26 b Fw(\(0)p Fy(;)14 b(b)p Fw(\))p FI(.)22 b(Then)83 4487 y(for)e Fy(n)i Fw(=)h(1)p Fy(;)14 b Fw(2)p Fy(;)g(:)g(:)g(:)f FI(,)20 b(there)g(e)o(xists)g Fy( )1119 4499 y Fu(n)1188 4487 y Fx(2)j(H)f FI(such)e(that)g Fy(E)1742 4457 y Fv(j)p Fu(H)1820 4431 y Fl(0)1843 4457 y Fv(j)1867 4419 y Fp(\000)1905 4487 y Fw([)p Fy(n;)14 b Fx(1)p Fw(\))2130 4419 y Fp(\001)2168 4487 y Fy(E)2234 4457 y Fu(H)2297 4487 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )2469 4499 y Fu(n)2538 4487 y Fx(6)p Fw(=)23 b(0)p FI(,)d(and)g(the)g(v)o (ectors)1338 4731 y Fy(')1392 4743 y Fu(n)1460 4731 y Fw(:=)1623 4669 y Fy(E)1689 4639 y Fv(j)p Fu(H)1767 4614 y Fl(0)1790 4639 y Fv(j)1814 4602 y Fp(\000)1852 4669 y Fw([)p Fy(n;)14 b Fx(1)p Fw(\))2077 4602 y Fp(\001)2115 4669 y Fy(E)2181 4639 y Fu(H)2244 4669 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )2416 4681 y Fu(n)p 1581 4712 923 4 v 1581 4792 a Fx(k)p Fy(E)1689 4768 y Fv(j)p Fu(H)1767 4752 y Fl(0)1790 4768 y Fv(j)1814 4725 y Fp(\000)1852 4792 y Fw([)p Fy(n;)14 b Fx(1)p Fw(\))2077 4725 y Fp(\001)2115 4792 y Fy(E)2181 4768 y Fu(H)2244 4792 y Fw(\()p Fy(J)8 b Fw(\))p Fy( )2416 4804 y Fu(n)2462 4792 y Fx(k)83 4975 y FI(satisfy)19 b Fx(k)p Fy(')415 4987 y Fu(n)460 4975 y Fx(k)j Fw(=)h(1)p FI(,)18 b(and)f Fy(E)897 4945 y Fu(H)960 4975 y Fw(\()p Fy(J)8 b Fw(\))p Fy(')1132 4987 y Fu(n)1202 4975 y Fw(=)22 b Fy(E)1355 4945 y Fv(j)p Fu(H)1433 4920 y Fl(0)1456 4945 y Fv(j)1480 4908 y Fp(\000)1518 4975 y Fw([)p Fy(n;)14 b Fx(1)p Fw(\))1743 4908 y Fp(\001)1781 4975 y Fy(')1835 4987 y Fu(n)1904 4975 y Fw(=)23 b Fy(')2046 4987 y Fu(n)2091 4975 y FI(.)c(It)f(follo)n(ws)g(by)f(Assumption)h(2.2) f(and)h(Lemma)f(2.4)h(that)83 5075 y Fx(j)p Fy(H)182 5044 y Fv(0)205 5075 y Fx(j)9 b Fy(E)303 5044 y Fu(H)366 5075 y Fw(\()p Fy(J)f Fw(\))22 b FI(is)f(a)g(bounded)c(operator)m(,)h (and)495 5253 y Fw(Const)p Fy(:)23 b Fx(\025)842 5183 y Fp(\015)842 5232 y(\015)888 5253 y Fx(j)p Fy(H)987 5219 y Fv(0)1010 5253 y Fx(j)9 b Fy(E)1108 5219 y Fu(H)1171 5253 y Fw(\()p Fy(J)f Fw(\))1289 5183 y Fp(\015)1289 5232 y(\015)1359 5253 y Fx(\025)1447 5183 y Fp(\015)1447 5232 y(\015)1493 5253 y Fx(j)p Fy(H)1592 5219 y Fv(0)1615 5253 y Fx(j)h Fy(E)1713 5219 y Fu(H)1776 5253 y Fw(\()p Fy(J)f Fw(\))p Fy(')1948 5265 y Fu(n)1994 5183 y Fp(\015)1994 5232 y(\015)2064 5253 y Fw(=)2151 5183 y Fp(\015)2151 5232 y(\015)2197 5253 y Fx(j)p Fy(H)2296 5219 y Fv(0)2319 5253 y Fx(j)h Fy(E)2417 5219 y Fv(j)p Fu(H)2495 5194 y Fl(0)2519 5219 y Fv(j)2543 5186 y Fp(\000)2581 5253 y Fw([)p Fy(n;)14 b Fx(1)p Fw(\))2806 5186 y Fp(\001)2844 5253 y Fy(')2898 5265 y Fu(n)2943 5183 y Fp(\015)2943 5232 y(\015)3013 5253 y Fx(\025)22 b Fy(n)9 b Fx(k)p Fy(')3255 5265 y Fu(n)3300 5253 y Fx(k)83 5420 y FI(which)20 b(leads)g(to)g(a)h(contradiction)d(when)h Fy(n)k Fx(!)g(1)p FI(.)p 3708 5420 4 57 v 3712 5367 50 4 v 3712 5420 V 3761 5420 4 57 v Black 1905 5670 a(6)p Black eop end %%Page: 7 7 TeXDict begin 7 6 bop Black Black 83 307 a Fz(3)119 b(Locally)30 b(smooth)f(operators)g(and)i(absolute)f(continuity)83 493 y FI(In)e(this)h(section)f(we)h(e)o(xhibit)e(a)i(lar)o(ge)e(class)j (of)e(locally)g Fy(H)7 b FI(-smooth)27 b(operators.)f(W)-7 b(e)30 b(also)e(sho)n(w)h(that)f(the)g(operator)f Fy(H)36 b FI(is)83 592 y(purely)26 b(absolutely)h(continuous)e(in)j Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))24 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FI(.)27 b(These)h(results)g(are)f(obtained)f (by)h(using)g(commutators)f(methods)g(as)83 692 y(presented)19 b(in)h([1)o(].)249 792 y(In)i(order)f(to)i(moti)n(v)n(ate)e(our)h (choice)f(of)h(conjugate)f(operator)f(for)i Fy(H)7 b FI(,)22 b(we)h(present)f(\002rst)h(a)g(formal)e(calculation.)g(Let)h Fy(A)3727 804 y Fu(\021)83 891 y FI(be)e(gi)n(v)o(en)f(by)1267 991 y Fy(A)1329 1003 y Fu(\021)1392 991 y Fw(:=)1513 958 y FK(1)p 1513 972 34 4 v 1513 1019 a(2)1556 924 y Fp(\010)1605 991 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fy(H)1865 957 y Fv(0)1907 991 y Fx(\001)18 b Fw(\010)g(+)g(\010)h Fx(\001)f Fy(H)2305 957 y Fv(0)2328 991 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))2512 924 y Fp(\011)2561 991 y Fy(;)83 1131 y FI(where)24 b Fy(\021)29 b FI(is)d(some)f(real)g(function)e (with)i(a)h(suf)n(\002ciently)e(rapid)g(decrease)h(to)g Fw(0)g FI(at)g(in\002nity)-5 b(.)24 b(Then)g Fy(A)3037 1143 y Fu(\021)3104 1131 y FI(satis\002es)i(with)f Fy(H)33 b FI(the)83 1230 y(commutation)18 b(relation)678 1397 y Fy(i)p Fw([)p Fy(H)r(;)c(A)900 1409 y Fu(\021)941 1397 y Fw(])23 b(=)1089 1364 y Fu(i)p 1084 1378 V 1084 1425 a FK(2)1141 1334 y Fp(P)1229 1355 y Fu(d)1229 1421 y(j)s FK(=1)1362 1329 y Fp(\010)1410 1397 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fy(H)1670 1366 y Fv(0)1663 1418 y Fu(j)1708 1397 y Fw([)p Fy(H)r(;)14 b Fw(\010)1899 1409 y Fu(j)1934 1397 y Fw(])k(+)g([)p Fy(H)r(;)c Fw(\010)2249 1409 y Fu(j)2285 1397 y Fw(])9 b Fy(H)2393 1366 y Fv(0)2386 1418 y Fu(j)2421 1397 y Fy(\021)s Fw(\()p Fy(H)e Fw(\))2605 1329 y Fp(\011)2677 1397 y Fw(=)22 b(\()p Fy(H)2872 1366 y Fv(0)2896 1397 y Fw(\))2928 1366 y FK(2)2965 1397 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fy(;)83 1563 y FI(which)24 b(pro)o(vides)f(\(in)i(a)g(sense)g(to)g(be)g(speci\002ed\))f(a)h (Mourre)e(estimate.)i(So,)g(in)g(the)g(sequel,)f(one)g(only)g(has)h(to) g(justify)f(these)83 1663 y(formal)19 b(manipulations)f(and)i(to)g (determinate)f(an)h(appropriate)e(function)g Fy(\021)s FI(.)249 1762 y(First)29 b(of)e(all,)i(one)e(observ)o(es)g(that)h(for)f (each)g Fy(j)43 b Fx(2)37 b(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)28 b FI(and)f(each)h Fy(!)40 b Fx(2)d Fq(C)24 b Fx(n)g Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))29 b FI(the)f(operator)e Fy(H)3527 1732 y Fv(0)3520 1784 y Fu(j)3555 1762 y Fy(R)3618 1774 y Fu(!)3703 1762 y Fx(\021)83 1871 y Fy(H)159 1841 y Fv(0)152 1893 y Fu(j)187 1871 y Fw(\()p Fy(H)j Fx(\000)21 b Fy(!)s Fw(\))490 1841 y Fv(\000)p FK(1)604 1871 y FI(is)k(a)g (bounded)d(operator)-5 b(.)23 b(Indeed,)f(one)i(has)h Fw(\()p Fy(H)k Fx(\000)21 b Fy(!)s Fw(\))2230 1841 y Fv(\000)p FK(1)2319 1871 y Fx(H)32 b Fw(=)e Fx(D)r Fw(\()p Fy(H)7 b Fw(\))32 b Fx(\032)e(D)r Fw(\()p Fy(H)3023 1841 y Fv(0)3016 1893 y Fu(j)3052 1871 y Fw(\))25 b FI(by)f(Assumption)g (2.2.)83 1986 y(In)e(the)g(follo)n(wing)e(lemmas,)i(Assumptions)f(2.3)g (and)h(2.2)f(are)h(tacitly)g(assumed,)g(and)f(we)i(set)f Fx(h)p Fy(x)p Fx(i)28 b Fw(:=)e(\(1)20 b(+)f Fy(x)3333 1956 y FK(2)3371 1986 y Fw(\))3403 1956 y FK(1)p Fu(=)p FK(2)3530 1986 y FI(for)i(an)o(y)83 2086 y Fy(x)j Fx(2)f Fq(R)292 2055 y Fu(n)337 2086 y FI(.)p Black 83 2239 a FA(Lemma)e(3.1.)p Black Black 109 w FJ(\(a\))p Black 40 w(F)-9 b(or)28 b(eac)o(h)e Fy(j;)14 b(k)39 b Fx(2)d(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)e(;)i(d)p Fx(g)27 b FJ(and)g(eac)o(h)f Fy(\015)5 b(;)14 b(!)37 b Fx(2)f Fq(C)24 b Fx(n)f Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))p FJ(,)28 b(the)f(bounded)d(oper)o(ator)i Fy(R)3510 2251 y Fu(\015)3553 2239 y Fy(H)3629 2208 y Fv(0)3622 2260 y Fu(j)3657 2239 y Fy(R)3720 2251 y Fu(!)291 2347 y FJ(belongs)19 b(to)h Fy(C)720 2317 y FK(1)757 2347 y Fw(\(\010)849 2359 y Fu(k)891 2347 y Fw(\))p FJ(.)p Black 152 2507 a(\(b\))p Black 41 w(F)-9 b(or)20 b(eac)o(h)f Fy(j;)14 b(k)27 b Fx(2)c(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)20 b FJ(the)g(bounded)e(self-adjoint)h(oper)o(ator)g Fx(h)p Fy(H)7 b Fx(i)2473 2477 y Fv(\000)p FK(2)2562 2507 y Fy(H)2638 2477 y Fv(0)2631 2529 y Fu(j)2666 2507 y Fx(h)p Fy(H)g Fx(i)2806 2477 y Fv(\000)p FK(2)2917 2507 y FJ(belongs)19 b(to)h Fy(C)3346 2477 y FK(1)3383 2507 y Fw(\(\010)3475 2519 y Fu(k)3517 2507 y Fw(\))p FJ(.)p Black 157 2679 a(\(c\))p Black 41 w(F)-9 b(or)34 b(eac)o(h)g Fy(j;)14 b(k)s(;)g(`)49 b Fx(2)g(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)p FJ(,)34 b(the)h(bounded)c (self-adjoint)j(oper)o(ator)f Fy(i)2606 2611 y Fp(\002)2640 2679 y Fx(h)p Fy(H)7 b Fx(i)2780 2649 y Fv(\000)p FK(2)2870 2679 y Fy(H)2946 2649 y Fv(0)2939 2700 y Fu(j)2974 2679 y Fx(h)p Fy(H)g Fx(i)3114 2649 y Fv(\000)p FK(2)3203 2679 y Fy(;)14 b Fw(\010)3300 2691 y Fu(k)3341 2611 y Fp(\003)3410 2679 y FJ(belongs)33 b(to)291 2787 y Fy(C)356 2757 y FK(1)393 2787 y Fw(\(\010)485 2799 y Fu(`)517 2787 y Fw(\))p FJ(.)p Black 83 2940 a(Pr)l(oof.)p Black 41 w FI(Due)20 b(to)g(Assumption)f(2.2)h(one)f(has)i(for)e(each)h Fy(')k Fx(2)f(D)r Fw(\(\010)1998 2952 y Fu(k)2039 2940 y Fw(\))427 3040 y Fp(\012)466 3107 y Fw(\010)526 3119 y Fu(k)567 3107 y Fy(';)14 b(R)721 3119 y Fu(\015)764 3107 y Fy(H)840 3073 y Fv(0)833 3127 y Fu(j)868 3107 y Fy(R)931 3119 y Fu(!)979 3107 y Fy(')1033 3040 y Fp(\013)1091 3107 y Fx(\000)1174 3040 y Fp(\012)1213 3107 y Fy(R)1281 3119 y FK(\026)-38 b Fu(!)1324 3107 y Fy(H)1400 3073 y Fv(0)1393 3127 y Fu(j)1428 3107 y Fy(R)1494 3119 y FK(\026)i Fu(\015)1534 3107 y Fy(';)14 b Fw(\010)1685 3119 y Fu(k)1726 3107 y Fy(')1780 3040 y Fp(\013)450 3242 y Fw(=)538 3175 y Fp(\012)577 3242 y Fw(\010)637 3254 y Fu(k)678 3242 y Fy(';)g(R)832 3254 y Fu(\015)875 3242 y Fy(H)951 3208 y Fv(0)944 3263 y Fu(j)979 3242 y Fy(R)1042 3254 y Fu(!)1090 3242 y Fy(')1144 3175 y Fp(\013)1202 3242 y Fx(\000)1285 3175 y Fp(\012)1324 3242 y Fw(\010)1384 3254 y Fu(k)1425 3242 y Fy(R)1491 3254 y FK(\026)-36 b Fu(\015)1530 3242 y Fy(';)14 b(H)1697 3208 y Fv(0)1690 3263 y Fu(j)1726 3242 y Fy(R)1789 3254 y Fu(!)1837 3242 y Fy(')1891 3175 y Fp(\013)1949 3242 y Fw(+)2032 3175 y Fp(\012)2071 3242 y Fw(\010)2131 3254 y Fu(k)2172 3242 y Fy(R)2238 3254 y FK(\026)-36 b Fu(\015)2277 3242 y Fy(';)14 b(H)2444 3208 y Fv(0)2437 3263 y Fu(j)2472 3242 y Fy(R)2535 3254 y Fu(!)2584 3242 y Fy(')2638 3175 y Fp(\013)2696 3242 y Fx(\000)2779 3175 y Fp(\012)2818 3242 y Fy(R)2886 3254 y FK(\026)-38 b Fu(!)2929 3242 y Fy(H)3005 3208 y Fv(0)2998 3263 y Fu(j)3033 3242 y Fy(R)3099 3254 y FK(\026)i Fu(\015)3138 3242 y Fy(';)14 b Fw(\010)3289 3254 y Fu(k)3331 3242 y Fy(')3385 3175 y Fp(\013)450 3378 y Fw(=)538 3311 y Fp(\012)577 3378 y Fw([)p Fy(R)666 3390 y FK(\026)-36 b Fu(\015)706 3378 y Fy(;)14 b Fw(\010)803 3390 y Fu(k)843 3378 y Fw(])p Fy(';)g(H)1033 3344 y Fv(0)1026 3399 y Fu(j)1062 3378 y Fy(R)1125 3390 y Fu(!)1173 3378 y Fy(')1227 3311 y Fp(\013)1285 3378 y Fw(+)1368 3311 y Fp(\012)1407 3378 y Fw(\010)1467 3390 y Fu(k)1508 3378 y Fy(R)1574 3390 y FK(\026)-36 b Fu(\015)1613 3378 y Fy(';)14 b(H)1780 3344 y Fv(0)1773 3399 y Fu(j)1809 3378 y Fy(R)1872 3390 y Fu(!)1920 3378 y Fy(')1974 3311 y Fp(\013)2032 3378 y Fx(\000)2115 3311 y Fp(\012)2154 3378 y Fy(H)2230 3344 y Fv(0)2223 3399 y Fu(j)2258 3378 y Fy(R)2324 3390 y FK(\026)-36 b Fu(\015)2363 3378 y Fy(';)14 b Fw(\010)2514 3390 y Fu(k)2555 3378 y Fy(R)2618 3390 y Fu(!)2667 3378 y Fy(')2721 3311 y Fp(\013)529 3514 y Fw(+)612 3446 y Fp(\012)651 3514 y Fy(H)727 3479 y Fv(0)720 3534 y Fu(j)755 3514 y Fy(R)821 3526 y FK(\026)-36 b Fu(\015)860 3514 y Fy(';)14 b Fw(\010)1011 3526 y Fu(k)1052 3514 y Fy(R)1115 3526 y Fu(!)1163 3514 y Fy(')1217 3446 y Fp(\013)1275 3514 y Fx(\000)1358 3446 y Fp(\012)1398 3514 y Fy(R)1466 3526 y FK(\026)-38 b Fu(!)1509 3514 y Fy(H)1585 3479 y Fv(0)1578 3534 y Fu(j)1613 3514 y Fy(R)1679 3526 y FK(\026)i Fu(\015)1718 3514 y Fy(';)14 b Fw(\010)1869 3526 y Fu(k)1910 3514 y Fy(')1964 3446 y Fp(\013)450 3649 y Fw(=)538 3582 y Fp(\012)577 3649 y Fw([)p Fy(R)666 3661 y FK(\026)-36 b Fu(\015)706 3649 y Fy(;)14 b Fw(\010)803 3661 y Fu(k)843 3649 y Fw(])p Fy(';)g(H)1033 3615 y Fv(0)1026 3670 y Fu(j)1062 3649 y Fy(R)1125 3661 y Fu(!)1173 3649 y Fy(')1227 3582 y Fp(\013)1285 3649 y Fw(+)1368 3582 y Fp(\012)1407 3649 y Fw([)p Fy(H)1506 3615 y Fv(0)1499 3670 y Fu(j)1534 3649 y Fy(;)g Fw(\010)1631 3661 y Fu(k)1672 3649 y Fw(])p Fy(R)1761 3661 y FK(\026)-36 b Fu(\015)1800 3649 y Fy(';)14 b(R)1954 3661 y Fu(!)2003 3649 y Fy(')2057 3582 y Fp(\013)2115 3649 y Fw(+)2198 3582 y Fp(\012)2237 3649 y Fy(H)2313 3615 y Fv(0)2306 3670 y Fu(j)2341 3649 y Fy(R)2407 3661 y FK(\026)-36 b Fu(\015)2447 3649 y Fy(';)14 b Fw([\010)2621 3661 y Fu(k)2662 3649 y Fy(;)g(R)2762 3661 y Fu(!)2810 3649 y Fw(])p Fy(')2887 3582 y Fp(\013)2926 3649 y Fy(:)83 3816 y FI(This)20 b(implies)h(that)f(there)g(e)o(xists)i Fs(C)k Fy(<)c Fx(1)f FI(such)f(that)1016 3912 y Fp(\014)1016 3961 y(\014)1044 3915 y(\012)1083 3982 y Fw(\010)1143 3994 y Fu(k)1184 3982 y Fy(';)14 b(R)1338 3994 y Fu(\015)1381 3982 y Fy(H)1457 3948 y Fv(0)1450 4003 y Fu(j)1485 3982 y Fy(R)1548 3994 y Fu(!)1596 3982 y Fy(')1650 3915 y Fp(\013)1708 3982 y Fx(\000)1791 3915 y Fp(\012)1830 3982 y Fy(R)1898 3994 y FK(\026)-38 b Fu(!)1941 3982 y Fy(H)2017 3948 y Fv(0)2010 4003 y Fu(j)2045 3982 y Fy(R)2111 3994 y FK(\026)i Fu(\015)2151 3982 y Fy(';)14 b Fw(\010)2302 3994 y Fu(k)2343 3982 y Fy(')2397 3915 y Fp(\013)2436 3912 y(\014)2436 3961 y(\014)2487 3982 y Fx(\024)25 b Fs(C)16 b Fx(k)p Fy(')p Fx(k)2775 3948 y FK(2)2812 3982 y Fy(:)83 4148 y FI(for)k(each)f Fy(')24 b Fx(2)f(D)r Fw(\(\010)687 4160 y Fu(k)729 4148 y Fw(\))p FI(,)d(and)g(thus)g(the)g(\002rst)h(statement)f(follo)n(ws)g(from)f([1) o(,)i(Lem.)f(6.2.9].)249 4248 y(F)o(or)26 b(the)g(second)f(statement,)h (since)g Fx(h)p Fy(H)7 b Fx(i)1478 4218 y Fv(\000)p FK(2)1601 4248 y Fw(=)34 b Fy(R)1763 4260 y Fv(\000)p Fu(i)1842 4248 y Fy(R)1905 4260 y Fu(i)1933 4248 y FI(,)26 b(the)g(operator)f Fx(h)p Fy(H)7 b Fx(i)2552 4218 y Fv(\000)p FK(2)2641 4248 y Fy(H)2717 4218 y Fv(0)2710 4270 y Fu(j)2745 4248 y Fx(h)p Fy(H)g Fx(i)2885 4218 y Fv(\000)p FK(2)3001 4248 y FI(is)27 b(clearly)f(bounded)d(and)83 4348 y(self-adjoint.)c (Furthermore,)e(by)j(observing)e(that)1265 4514 y Fx(h)p Fy(H)7 b Fx(i)1405 4480 y Fv(\000)p FK(2)1495 4514 y Fy(H)1571 4480 y Fv(0)1564 4535 y Fu(j)1599 4514 y Fx(h)p Fy(H)g Fx(i)1739 4480 y Fv(\000)p FK(2)1851 4514 y Fw(=)23 b Fy(R)2002 4526 y Fu(i)2030 4447 y Fp(\000)2068 4514 y Fy(R)2131 4526 y Fv(\000)p Fu(i)2210 4514 y Fy(H)2286 4480 y Fv(0)2279 4535 y Fu(j)2314 4514 y Fy(R)2377 4526 y Fu(i)2405 4447 y Fp(\001)2443 4514 y Fy(R)2506 4526 y Fv(\000)p Fu(i)83 4681 y FI(one)h(concludes)e(from)h(\(a\))h(that)g Fx(h)p Fy(H)7 b Fx(i)1176 4650 y Fv(\000)p FK(2)1266 4681 y Fy(H)1342 4650 y Fv(0)1335 4702 y Fu(j)1370 4681 y Fx(h)p Fy(H)g Fx(i)1510 4650 y Fv(\000)p FK(2)1624 4681 y FI(is)25 b(the)f(product)e(of)i(three)f(operators)g(belonging)f (to)i Fy(C)3237 4650 y FK(1)3274 4681 y Fw(\(\010)3366 4693 y Fu(k)3407 4681 y Fw(\))p FI(,)h(and)f(thus)83 4789 y(belongs)19 b(to)h Fy(C)512 4759 y FK(1)550 4789 y Fw(\(\010)642 4801 y Fu(k)683 4789 y Fw(\))h FI(due)f(to)g([1)o(,)h (Prop.)e(5.1.5].)249 4889 y(F)o(or)h(the)g(last)h(statement,)f(one)g (gets)g(by)g(taking)f(Lemma)h(2.4)f(into)h(account)451 5055 y Fy(i)480 4988 y Fp(\002)515 5055 y Fx(h)p Fy(H)7 b Fx(i)655 5021 y Fv(\000)p FK(2)744 5055 y Fy(H)820 5021 y Fv(0)813 5076 y Fu(j)848 5055 y Fx(h)p Fy(H)g Fx(i)988 5021 y Fv(\000)p FK(2)1078 5055 y Fy(;)14 b Fw(\010)1175 5067 y Fu(k)1215 4988 y Fp(\003)1273 5055 y Fw(=)23 b Fx(\000)p Fw(2\()p Fy(R)1563 5067 y Fu(i)1590 5055 y Fy(H)1666 5021 y Fv(0)1659 5076 y Fu(k)1699 5055 y Fy(R)1762 5067 y Fu(i)1790 5055 y Fw(\)\()p Fy(R)1917 5067 y Fv(\000)p Fu(i)1997 5055 y Fy(H)2073 5021 y Fv(0)2066 5076 y Fu(j)2101 5055 y Fy(R)2164 5067 y Fv(\000)p Fu(i)2244 5055 y Fw(\)\()p Fy(R)2371 5067 y Fu(i)2418 5055 y Fw(+)18 b Fy(R)2564 5067 y Fv(\000)p Fu(i)2643 5055 y Fw(\))h(+)f Fx(h)p Fy(H)7 b Fx(i)2917 5021 y Fv(\000)p FK(2)3006 5055 y Fy(H)3082 5021 y Fv(00)3075 5076 y Fu(j)s(k)3147 5055 y Fx(h)p Fy(H)g Fx(i)3287 5021 y Fv(\000)p FK(2)3377 5055 y Fy(:)83 5222 y FI(The)20 b(\002rst)i(term)e(is)i(a)f(product)e (of)h(operators)f(which)h(belong)f(to)i Fy(C)2009 5192 y FK(1)2047 5222 y Fw(\(\010)2139 5234 y Fu(`)2171 5222 y Fw(\))p FI(,)g(and)f(thus)h(it)g(belongs)f(to)h Fy(C)3044 5192 y FK(1)3081 5222 y Fw(\(\010)3173 5234 y Fu(`)3205 5222 y Fw(\))p FI(.)h(F)o(or)e(the)h(second)83 5321 y(term,)f(a)h (calculation)f(similar)g(to)h(the)g(one)f(presented)f(for)h(the)g (statement)h(\(a\))f(using)g(Assumption)g(2.2)f(sho)n(ws)i(that)g(this) g(term)83 5421 y(also)g(belongs)e(to)h Fy(C)667 5391 y FK(1)704 5421 y Fw(\(\010)796 5433 y Fu(`)829 5421 y Fw(\))p FI(,)g(and)g(so)h(the)f(claim)g(is)h(pro)o(v)o(ed.)p 3708 5421 4 57 v 3712 5368 50 4 v 3712 5421 V 3761 5421 4 57 v Black 1905 5670 a(7)p Black eop end %%Page: 8 8 TeXDict begin 8 7 bop Black Black 249 307 a FI(W)-7 b(e)18 b(can)e(no)n(w)g(gi)n(v)o(e)g(a)h(precise)f(de\002nition)g(of)g(the)g (conjugate)f(operator)g Fy(A)i FI(we)g(will)g(use,)g(and)f(pro)o(v)o(e) f(its)i(self-adjointness.)83 407 y(F)o(or)j(that)g(purpose,)e(we)j (consider)e(the)h(f)o(amily)1185 587 y Fw(\005)1247 599 y Fu(j)1306 587 y Fw(:=)i Fx(h)q Fy(H)7 b Fx(i)1557 545 y Fv(\000)p FK(2)1660 587 y Fy(H)1736 552 y Fv(0)1729 607 y Fu(j)1778 587 y Fx(h)p Fy(H)g Fx(i)1918 545 y Fv(\000)p FK(2)2021 587 y Fy(;)180 b(j)28 b Fw(=)22 b(1)p Fy(;)14 b(:)g(:)g(:)g(;)g(d;)83 767 y FI(of)28 b(mutually)f(commuting)f (bounded)g(self-adjoint)h(operators,)g(and)g(we)i(write)f Fw(\005)39 b(:=)e(\(\005)2786 779 y FK(1)2824 767 y Fy(;)14 b(:)g(:)g(:)g(;)g Fw(\005)3071 779 y Fu(d)3110 767 y Fw(\))29 b FI(for)f(the)g(associated)83 866 y(v)o(ector)19 b(operator)-5 b(.)19 b(Due)h(to)g(Lemma)f(3.1.\(b\),)f(each)i(operator) e Fw(\005)1937 878 y Fu(j)1993 866 y FI(belongs)h(to)i Fy(C)2423 836 y FK(1)2460 866 y Fw(\(\010)2552 878 y Fu(k)2593 866 y Fw(\))p FI(.)g(Therefore)d(the)i(operator)1530 1046 y Fy(A)j Fw(:=)1736 1014 y FK(1)p 1736 1028 34 4 v 1736 1075 a(2)1779 979 y Fp(\000)1817 1046 y Fw(\005)c Fx(\001)f Fw(\010)h(+)f(\010)g Fx(\001)h Fw(\005)2283 979 y Fp(\001)83 1246 y FI(is)24 b(well-de\002ned)e(and)h(symmetric)f (on)1226 1184 y Fp(T)1295 1204 y Fu(d)1295 1271 y(j)s FK(=1)1428 1246 y Fx(D)r Fw(\(\010)1586 1258 y Fu(j)1622 1246 y Fw(\))p FI(.)h(F)o(or)g(the)h(ne)o(xt)e(lemma,)h(we)g(note)g (that)g(this)h(set)g(contains)f(the)g(domain)83 1358 y Fx(D)r Fw(\(\010)241 1328 y FK(2)279 1358 y Fw(\))e FI(of)f Fw(\010)482 1328 y FK(2)519 1358 y FI(.)p Black 83 1522 a FA(Lemma)h(3.2.)p Black 40 w FJ(The)f(oper)o(ator)f Fy(A)i FJ(is)g(essentially)g(self-adjoint)e(on)g Fx(D)r Fw(\(\010)2188 1492 y FK(2)2226 1522 y Fw(\))p FJ(.)p Black 83 1686 a(Pr)l(oof.)p Black 41 w FI(W)-7 b(e)21 b(use)f(the)h(criterion)e(of)h(essential)g(self-adjointness)f([27)o(,)h (Thm.)g(X.37].)249 1786 y(Gi)n(v)o(en)k Fy(a)30 b(>)g Fw(1)p FI(,)24 b(we)h(de\002ne)f(the)g(self-adjoint)f(operator)g Fy(N)39 b Fw(:=)31 b(\010)2196 1755 y FK(2)2254 1786 y Fw(+)21 b(\005)2402 1755 y FK(2)2461 1786 y Fw(+)g Fy(a)k FI(with)g(domain)e Fx(D)r Fw(\()p Fy(N)9 b Fw(\))31 b Fx(\021)f(D)r Fw(\(\010)3553 1755 y FK(2)3591 1786 y Fw(\))25 b FI(and)83 1885 y(observ)o(e)19 b(that)h(in)g(the)h(form)e (sense)h(on)g Fx(D)r Fw(\()p Fy(N)9 b Fw(\))22 b FI(one)d(has)691 2065 y Fy(N)767 2031 y FK(2)827 2065 y Fw(=)j(\010)974 2031 y FK(4)1030 2065 y Fw(+)c(\005)1175 2031 y FK(4)1231 2065 y Fw(+)g Fy(a)1358 2031 y FK(2)1413 2065 y Fw(+)g(2)p Fy(a)p Fw(\010)1642 2031 y FK(2)1698 2065 y Fw(+)g(2)p Fy(a)p Fw(\005)1929 2031 y FK(2)1984 2065 y Fw(+)g(\010)2127 2031 y FK(2)2164 2065 y Fw(\005)2226 2031 y FK(2)2282 2065 y Fw(+)g(\005)2427 2031 y FK(2)2465 2065 y Fw(\010)2525 2031 y FK(2)827 2215 y Fw(=)k(\010)974 2181 y FK(4)1030 2215 y Fw(+)c(\005)1175 2181 y FK(4)1231 2215 y Fw(+)g Fy(a)1358 2181 y FK(2)1413 2215 y Fw(+)g(2)p Fy(a)p Fw(\010)1642 2181 y FK(2)1698 2215 y Fw(+)g(2)p Fy(a)p Fw(\005)1929 2181 y FK(2)1984 2215 y Fw(+)2067 2137 y Fp(X)2085 2315 y Fu(j;k)2201 2148 y Fp(\010)2249 2215 y Fw(\010)2309 2227 y Fu(j)2344 2215 y Fw(\005)2406 2181 y FK(2)2406 2236 y Fu(k)2447 2215 y Fw(\010)2507 2227 y Fu(j)2561 2215 y Fw(+)g(\005)2706 2227 y Fu(k)2747 2215 y Fw(\010)2807 2181 y FK(2)2807 2236 y Fu(j)2844 2215 y Fw(\005)2906 2227 y Fu(k)2947 2148 y Fp(\011)3014 2215 y Fw(+)g Fy(R)83 2494 y FI(with)i Fy(R)k Fw(:=)449 2432 y Fp(P)536 2519 y Fu(j;k)638 2427 y Fp(\010)687 2494 y Fw(\005)749 2506 y Fu(k)790 2494 y Fw([\005)875 2506 y Fu(k)916 2494 y Fy(;)14 b Fw(\010)1013 2506 y Fu(j)1048 2494 y Fw(]\010)1131 2506 y Fu(j)1184 2494 y Fw(+)k(\010)1327 2506 y Fu(j)1362 2494 y Fw([\010)1445 2506 y Fu(j)1480 2494 y Fy(;)c Fw(\005)1579 2506 y Fu(k)1620 2494 y Fw(]\005)1705 2506 y Fu(k)1765 2494 y Fw(+)k([\005)1933 2506 y Fu(k)1974 2494 y Fy(;)c Fw(\010)2071 2506 y Fu(j)2106 2494 y Fw(])2129 2464 y FK(2)2166 2427 y Fp(\011)2215 2494 y FI(.)20 b(No)n(w)-5 b(,)19 b(the)i(follo)n(wing)d(inequality)h(holds)755 2627 y Fp(X)773 2806 y Fu(j;k)889 2639 y Fp(\010)937 2706 y Fw(\005)999 2718 y Fu(k)1040 2706 y Fw([\005)1125 2718 y Fu(k)1166 2706 y Fy(;)14 b Fw(\010)1263 2718 y Fu(j)1298 2706 y Fw(]\010)1381 2718 y Fu(j)1435 2706 y Fw(+)k(\010)1578 2718 y Fu(j)1613 2706 y Fw([\010)1696 2718 y Fu(j)1731 2706 y Fy(;)c Fw(\005)1830 2718 y Fu(k)1871 2706 y Fw(]\005)1956 2718 y Fu(k)1997 2639 y Fp(\011)2068 2706 y Fx(\025)23 b(\000)p Fy(d)p Fw(\010)2324 2672 y FK(2)2379 2706 y Fx(\000)2462 2627 y Fp(X)2481 2806 y Fu(j;k)2596 2636 y Fp(\014)2596 2686 y(\014)2624 2706 y Fw(\005)2686 2718 y Fu(k)2727 2706 y Fw([\005)2812 2718 y Fu(k)2853 2706 y Fy(;)14 b Fw(\010)2950 2718 y Fu(j)2985 2706 y Fw(])3008 2636 y Fp(\014)3008 2686 y(\014)3036 2656 y FK(2)3073 2706 y Fy(:)83 2982 y FI(Thus)20 b(there)g(e)o(xists)g Fy(c)j(>)g Fw(0)d FI(such)g(that)g Fy(R)k Fx(\025)f(\000)p Fy(d)p Fw(\010)1535 2952 y FK(2)1590 2982 y Fx(\000)18 b Fy(c)p FI(.)j(Altogether)m(,)d(we)i(ha)n(v)o(e)g(sho)n(wn)f(that)i (in)f(the)g(form)f(sense)i(on)f Fx(D)r Fw(\()p Fy(N)9 b Fw(\))556 3179 y Fy(N)632 3144 y FK(2)692 3179 y Fx(\025)23 b Fw(\010)840 3144 y FK(4)895 3179 y Fw(+)18 b(\005)1040 3144 y FK(4)1096 3179 y Fw(+)g(\()p Fy(a)1255 3144 y FK(2)1311 3179 y Fx(\000)g Fy(c)p Fw(\))h(+)f(\(2)p Fy(a)g Fx(\000)g Fy(d)p Fw(\)\010)1918 3144 y FK(2)1974 3179 y Fw(+)g(2)p Fy(a)p Fw(\005)2205 3144 y FK(2)2261 3179 y Fw(+)2344 3100 y Fp(X)2362 3278 y Fu(j;k)2477 3111 y Fp(\010)2526 3179 y Fw(\010)2586 3191 y Fu(j)2621 3179 y Fw(\005)2683 3144 y FK(2)2683 3199 y Fu(k)2724 3179 y Fw(\010)2784 3191 y Fu(j)2837 3179 y Fw(+)g(\005)2982 3191 y Fu(k)3023 3179 y Fw(\010)3083 3144 y FK(2)3083 3199 y Fu(j)3121 3179 y Fw(\005)3183 3191 y Fu(k)3224 3111 y Fp(\011)3272 3179 y Fy(;)83 3449 y FI(where)i(the)g(r)-5 b(.h.s.)20 b(is)h(a)f(sum)h(of)f(positi)n(v)o(e)f(terms)h(for)g Fy(a)g FI(lar)o(ge)g(enough.)d(In)j(particular)m(,)f(one)g(has)i(for)e Fy(')24 b Fx(2)f(D)r Fw(\()p Fy(N)9 b Fw(\))1307 3646 y Fx(k)p Fy(N)g(')p Fx(k)1521 3612 y FK(2)1580 3646 y Fx(\025)1668 3575 y Fp(\015)1668 3625 y(\015)1714 3646 y Fw(\005)1776 3658 y Fu(j)1811 3646 y Fw(\010)1871 3658 y Fu(j)1906 3646 y Fy(')1960 3575 y Fp(\015)1960 3625 y(\015)2007 3596 y FK(2)2062 3646 y Fw(+)2145 3575 y Fp(\015)2145 3625 y(\015)2191 3646 y Fw(\010)2251 3658 y Fu(j)2286 3646 y Fw(\005)2348 3658 y Fu(j)2384 3646 y Fy(')2438 3575 y Fp(\015)2438 3625 y(\015)2484 3596 y FK(2)2521 3646 y Fy(;)83 3826 y FI(which)20 b(implies)g(that)1023 3939 y Fx(k)p Fy(A')p Fx(k)j(\024)1344 3906 y FK(1)p 1344 3920 V 1344 3968 a(2)1400 3860 y Fp(X)1445 4037 y Fu(j)1534 3872 y Fp(\010)1583 3869 y(\015)1583 3918 y(\015)1629 3939 y Fw(\005)1691 3951 y Fu(j)1726 3939 y Fw(\010)1786 3951 y Fu(j)1821 3939 y Fy(')1875 3869 y Fp(\015)1875 3918 y(\015)1940 3939 y Fw(+)2023 3869 y Fp(\015)2023 3918 y(\015)2069 3939 y Fw(\010)2129 3951 y Fu(j)2164 3939 y Fw(\005)2226 3951 y Fu(j)2261 3939 y Fy(')2315 3869 y Fp(\015)2315 3918 y(\015)2362 3872 y(\011)2433 3939 y Fx(\024)g Fy(d)14 b Fx(k)p Fy(N)9 b(')p Fx(k)14 b Fy(:)249 4175 y FI(It)21 b(remains)e(to)h(estimate)h (the)f(commutator)e Fw([)p Fy(A;)c(N)9 b Fw(])p FI(.)21 b(In)f(the)g(form)f(sense)i(on)e Fx(D)r Fw(\()p Fy(N)9 b Fw(\))p FI(,)22 b(one)d(has)510 4372 y Fw(2[)p Fy(A;)14 b(N)9 b Fw(])22 b(=)883 4293 y Fp(X)901 4472 y Fu(j;k)1017 4305 y Fp(\010)1065 4372 y Fw([\005)1150 4384 y Fu(j)1185 4372 y Fy(;)14 b Fw(\010)1282 4384 y Fu(k)1323 4372 y Fw(]\010)1406 4384 y Fu(j)1441 4372 y Fw(\010)1501 4384 y Fu(k)1560 4372 y Fw(+)k(\010)1703 4384 y Fu(k)1744 4372 y Fw([\005)1829 4384 y Fu(j)1865 4372 y Fy(;)c Fw(\010)1962 4384 y Fu(k)2002 4372 y Fw(]\010)2085 4384 y Fu(j)2139 4372 y Fw(+)k(\010)2282 4384 y Fu(j)2317 4372 y Fw([\005)2402 4384 y Fu(j)2437 4372 y Fy(;)c Fw(\010)2534 4384 y Fu(k)2575 4372 y Fw(]\010)2658 4384 y Fu(k)2717 4372 y Fw(+)k(\010)2860 4384 y Fu(j)2895 4372 y Fw(\010)2955 4384 y Fu(k)2996 4372 y Fw([\005)3081 4384 y Fu(j)3116 4372 y Fy(;)c Fw(\010)3213 4384 y Fu(k)3254 4372 y Fw(])957 4595 y(+)k(\005)1102 4607 y Fu(j)1137 4595 y Fw([\010)1220 4607 y Fu(j)1255 4595 y Fy(;)c Fw(\005)1354 4607 y Fu(k)1395 4595 y Fw(]\005)1480 4607 y Fu(k)1540 4595 y Fw(+)k(\005)1685 4607 y Fu(j)1720 4595 y Fw(\005)1782 4607 y Fu(k)1823 4595 y Fw([\010)1906 4607 y Fu(j)1941 4595 y Fy(;)c Fw(\005)2040 4607 y Fu(k)2081 4595 y Fw(])19 b(+)f([\010)2289 4607 y Fu(j)2324 4595 y Fy(;)c Fw(\005)2423 4607 y Fu(k)2464 4595 y Fw(]\005)2549 4607 y Fu(j)2584 4595 y Fw(\005)2646 4607 y Fu(k)2705 4595 y Fw(+)k(\005)2850 4607 y Fu(k)2892 4595 y Fw([\010)2975 4607 y Fu(j)3010 4595 y Fy(;)c Fw(\005)3109 4607 y Fu(k)3150 4595 y Fw(]\005)3235 4607 y Fu(j)3270 4528 y Fp(\011)3318 4595 y Fy(:)83 4775 y FI(The)20 b(last)h(four)e(terms)h(are)g(bounded.) e(F)o(or)i(the)g(other)f(terms,)h(Lemma)g(3.1.\(c\),)e(together)h(with) h(the)g(bound)607 4955 y Fx(jh)p Fw(\010)722 4967 y Fu(j)758 4955 y Fy(';)14 b(B)t Fw(\010)976 4967 y Fu(k)1017 4955 y Fx(ij)23 b(\024)g(k)p Fy(B)t Fx(k)14 b(h)p Fy(';)g Fw(\010)1531 4921 y FK(2)1568 4955 y Fy(')p Fx(i)23 b(\024)g(k)p Fy(B)t Fx(k)14 b(h)p Fy(';)g(N)9 b(')p Fx(i)p Fy(;)180 b(')23 b Fx(2)g(D)r Fw(\()p Fy(N)9 b Fw(\))p Fy(;)36 b(B)27 b Fx(2)c Fo(B)s Fw(\()p Fx(H)q Fw(\))p Fy(;)83 5135 y FI(leads)d(to)h(the)f(desired)f(estimate,)i FJ(i.e)o(.)f Fx(h)p Fy(';)14 b Fw([)p Fy(A;)g(N)9 b Fw(])p Fy(')p Fx(i)24 b(\024)e Fw(Const)p Fy(:)9 b Fx(h)p Fy(';)14 b(N)9 b(')p Fx(i)p FI(.)p 3708 5135 4 57 v 3712 5082 50 4 v 3712 5135 V 3761 5135 4 57 v Black 83 5301 a FA(Lemma)29 b(3.3.)p Black 46 w FJ(The)g(oper)o(ator)f Fy(H)36 b FJ(is)30 b(of)f(class)g Fy(C)1573 5270 y FK(2)1611 5301 y Fw(\()p Fy(A)p Fw(\))h FJ(and)e(the)h(sesquilinear)f(form)h Fy(i)p Fw([)p Fy(H)r(;)14 b(A)p Fw(])29 b FJ(on)g Fx(D)r Fw(\()p Fy(H)7 b Fw(\))30 b FJ(e)n(xtends)f(to)g(the)83 5400 y(bounded)18 b(positive)i(oper)o(ator)f Fx(h)p Fy(H)7 b Fx(i)1119 5370 y Fv(\000)p FK(2)1208 5400 y Fw(\()p Fy(H)1316 5370 y Fv(0)1340 5400 y Fw(\))1372 5370 y FK(2)1409 5400 y Fx(h)p Fy(H)g Fx(i)1549 5370 y Fv(\000)p FK(2)1639 5400 y FJ(.)p Black 1905 5670 a FI(8)p Black eop end %%Page: 9 9 TeXDict begin 9 8 bop Black Black Black 83 307 a FJ(Pr)l(oof.)p Black 41 w FI(One)20 b(has)g(for)g(each)g Fy(')j Fx(2)g(D)r Fw(\(\010)1225 277 y FK(2)1263 307 y Fw(\))e FI(and)f(each)g Fy(!)25 b Fx(2)f Fq(C)18 b Fx(n)g Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))290 506 y(2)332 439 y Fp(\010\012)419 506 y Fy(R)487 518 y FK(\026)-38 b Fu(!)530 506 y Fy(';)14 b(A')737 439 y Fp(\013)796 506 y Fx(\000)879 439 y Fp(\012)918 506 y Fy(A';)g(R)1134 518 y Fu(!)1182 506 y Fy(')1236 439 y Fp(\013)q(\011)1347 506 y Fw(=)1435 428 y Fp(X)1480 604 y Fu(j)1569 439 y Fp(\010\012)1656 506 y Fy(R)1724 518 y FK(\026)-38 b Fu(!)1768 506 y Fy(';)1859 439 y Fp(\000)1897 506 y Fw(\005)1959 518 y Fu(j)1994 506 y Fw(\010)2054 518 y Fu(j)2107 506 y Fw(+)18 b(\010)2250 518 y Fu(j)2285 506 y Fw(\005)2347 518 y Fu(j)2383 439 y Fp(\001)2421 506 y Fy(')2475 439 y Fp(\013)2533 506 y Fx(\000)2616 439 y Fp(\012\000)2693 506 y Fw(\005)2755 518 y Fu(j)2790 506 y Fw(\010)2850 518 y Fu(j)2904 506 y Fw(+)g(\010)3047 518 y Fu(j)3082 506 y Fw(\005)3144 518 y Fu(j)3179 439 y Fp(\001)3217 506 y Fy(';)c(R)3371 518 y Fu(!)3419 506 y Fy(')3473 439 y Fp(\013)q(\011)1347 744 y Fw(=)1435 666 y Fp(X)1480 842 y Fu(j)1569 677 y Fp(\010\012)1656 744 y Fw(\005)1718 756 y Fu(j)1754 744 y Fy(';)g Fw([)p Fy(R)1931 756 y Fu(!)1979 744 y Fy(;)g Fw(\010)2076 756 y Fu(j)2111 744 y Fw(])g Fy(')2202 677 y Fp(\013)2260 744 y Fw(+)2343 677 y Fp(\012)2396 744 y Fw([\010)2479 756 y Fu(j)2514 744 y Fy(;)g(R)2619 756 y FK(\026)-38 b Fu(!)2662 744 y Fw(])14 b Fy(';)g Fw(\005)2852 756 y Fu(j)2887 744 y Fy(')2941 677 y Fp(\013)q(\011)3029 744 y Fy(:)557 b FI(\(3.1\))83 1021 y(Since)24 b(all)h(operators)e(in)i (the)f(last)i(equality)d(are)i(bounded)c(and)j(since)h Fx(D)r Fw(\(\010)2319 991 y FK(2)2357 1021 y Fw(\))g FI(is)g(a)g(core)f(for)g Fy(A)p FI(,)h(this)g(implies)f(that)h Fy(H)32 b FI(is)25 b(of)83 1120 y(class)c Fy(C)330 1090 y FK(1)368 1120 y Fw(\()p Fy(A)p Fw(\))g FI([1)o(,)g(Lem.)f(6.2.9].)249 1220 y(No)n(w)g(observ)o(e)f(that)h(the)g(follo)n(wing)f(equalities)h (hold)f(on)h Fx(H)573 1403 y Fy(i)p Fw([)p Fy(R)688 1415 y Fu(!)735 1403 y Fy(;)14 b(A)p Fw(])24 b(=)983 1370 y Fu(i)p 978 1384 34 4 v 978 1431 a FK(2)1035 1340 y Fp(P)1123 1428 y Fu(j)1172 1335 y Fp(\010)1220 1403 y Fw(\005)1282 1415 y Fu(j)1317 1403 y Fw([)p Fy(R)1403 1415 y Fu(!)1452 1403 y Fy(;)14 b Fw(\010)1549 1415 y Fu(j)1583 1403 y Fw(])19 b(+)f([)p Fy(R)1794 1415 y Fu(!)1842 1403 y Fy(;)c Fw(\010)1939 1415 y Fu(j)1974 1403 y Fw(]\005)2059 1415 y Fu(j)2094 1335 y Fp(\011)2166 1403 y Fw(=)22 b Fx(\000)p Fy(R)2381 1415 y Fu(!)2443 1403 y Fx(h)p Fy(H)7 b Fx(i)2583 1361 y Fv(\000)p FK(2)2686 1403 y Fw(\()p Fy(H)2794 1373 y Fv(0)2818 1403 y Fw(\))2850 1373 y FK(2)2901 1403 y Fx(h)p Fy(H)g Fx(i)3041 1361 y Fv(\000)p FK(2)3144 1403 y Fy(R)3207 1415 y Fu(!)3255 1403 y Fy(:)83 1609 y FI(Therefore)14 b(the)i(sesquilinear)f(form)g Fy(i)p Fw([)p Fy(H)r(;)f(A)p Fw(])i FI(on)g Fx(D)r Fw(\()p Fy(H)7 b Fw(\))17 b FI(e)o(xtends)e(to)h(the)g(bounded)e(positi)n(v)o(e)h (operator)f Fx(h)p Fy(H)7 b Fx(i)3200 1568 y Fv(\000)p FK(2)3303 1609 y Fw(\()p Fy(H)3411 1579 y Fv(0)3434 1609 y Fw(\))3466 1579 y FK(2)3518 1609 y Fx(h)p Fy(H)g Fx(i)3658 1568 y Fv(\000)p FK(2)3747 1609 y FI(.)83 1709 y(Finally)-5 b(,)19 b(the)i(operator)d Fy(i)p Fw([)p Fy(R)889 1721 y Fu(!)937 1709 y Fy(;)c(A)p Fw(])21 b FI(can)f(be)g(written)g(as)h(a)f (product)f(of)h(f)o(actors)f(in)i Fy(C)2479 1679 y FK(1)2516 1709 y Fw(\(\010)2608 1721 y Fu(`)2641 1709 y Fw(\))g FI(for)e(each)h Fy(`)p FI(,)g(namely)1064 1892 y Fy(i)p Fw([)p Fy(R)1179 1904 y Fu(!)1226 1892 y Fy(;)14 b(A)p Fw(])24 b(=)e Fx(\000)1538 1829 y Fp(P)1625 1917 y Fu(j)1674 1892 y Fy(R)1737 1904 y Fu(!)1799 1824 y Fp(\000)1837 1892 y Fy(R)1900 1904 y Fv(\000)p Fu(i)1980 1892 y Fy(H)2056 1862 y Fv(0)2049 1913 y Fu(j)2084 1892 y Fy(R)2147 1904 y Fu(i)2174 1824 y Fp(\001)14 b(\000)2264 1892 y Fy(R)2327 1904 y Fv(\000)p Fu(i)2407 1892 y Fy(H)2483 1862 y Fv(0)2476 1913 y Fu(j)2511 1892 y Fy(R)2574 1904 y Fu(i)2601 1824 y Fp(\001)2653 1892 y Fy(R)2716 1904 y Fu(!)2764 1892 y Fy(:)83 2087 y FI(So)28 b Fy(i)p Fw([)p Fy(R)314 2099 y Fu(!)361 2087 y Fy(;)14 b(A)p Fw(])29 b FI(also)e(belongs)f(to)i Fy(C)1117 2057 y FK(1)1154 2087 y Fw(\(\010)1246 2099 y Fu(`)1279 2087 y Fw(\))g FI(for)e(each)i Fy(`)p FI(,)f(and)g(thus)g (a)h(calculation)e(similar)h(to)h(the)f(one)g(of)g(\(3.1\))f(sho)n(ws)i (that)83 2186 y Fy(i)p Fw([)p Fy(R)198 2198 y Fu(!)246 2186 y Fy(;)14 b(A)p Fw(])21 b FI(belongs)e(to)h Fy(C)818 2156 y FK(1)856 2186 y Fw(\()p Fy(A)p Fw(\))p FI(.)h(This)f(implies)h (that)f Fy(H)27 b FI(is)22 b(of)d(class)j Fy(C)2113 2156 y FK(2)2150 2186 y Fw(\()p Fy(A)p Fw(\))p FI(.)p 3708 2186 4 57 v 3712 2134 50 4 v 3712 2186 V 3761 2186 4 57 v Black 83 2353 a FA(De\002nition)k(3.4.)p Black 44 w FI(A)h(number)d Fy(\025)34 b Fx(2)h Fq(R)26 b FI(is)h(called)f(a)h Fy(A)p FI(-re)o(gular)d(v)n(alue)h(of)h Fy(H)34 b FI(if)26 b(there)g(e)o(xist)g(numbers)e Fy(a;)14 b(\016)37 b(>)d Fw(0)26 b FI(such)g(that)83 2452 y Fw(\()p Fy(H)191 2422 y Fv(0)214 2452 y Fw(\))246 2422 y FK(2)284 2452 y Fy(E)350 2422 y Fu(H)413 2452 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))24 b Fx(\025)f Fy(a)9 b(E)833 2422 y Fu(H)896 2452 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))p FI(.)21 b(The)f(complement)e(of)i(this)h(set)g(in)f Fq(R)h FI(is)g(denoted)e (by)g Fy(\024)2729 2422 y Fu(A)2783 2452 y Fw(\()p Fy(H)7 b Fw(\))p FI(.)249 2618 y(The)25 b(set)h(of)g Fy(A)p FI(-re)o(gular)d(v)n(alues)i(corresponds)f(to)h(the)h(Mourre)e(set)i (with)f(respect)h(to)f Fy(A)p FI(.)h(Indeed,)e(if)i Fy(\025)g FI(is)g(a)g Fy(A)p FI(-re)o(gular)83 2718 y(v)n(alue,)19 b(then)h Fw(\()p Fy(H)574 2688 y Fv(0)597 2718 y Fw(\))629 2688 y FK(2)667 2718 y Fy(E)733 2688 y Fu(H)796 2718 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))24 b Fx(\025)f Fy(a)9 b(E)1216 2688 y Fu(H)1279 2718 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))21 b FI(for)f(some)g Fy(a;)14 b(\016)26 b(>)c Fw(0)f FI(and)492 2909 y Fy(E)558 2875 y Fu(H)621 2909 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))p Fy(i)p Fw([)p Fy(H)r(;)g(A)p Fw(])p Fy(E)1121 2875 y Fu(H)1185 2909 y Fw(\()p Fy(\025)p Fw(;)g Fy(\016)s Fw(\))24 b(=)f Fy(E)1552 2875 y Fu(H)1615 2909 y Fw(\()p Fy(\025)p Fw(;)14 b Fy(\016)s Fw(\))g Fx(h)q Fy(H)7 b Fx(i)1959 2867 y Fv(\000)p FK(2)2062 2909 y Fw(\()p Fy(H)2170 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Fy(g)s Fw(\)\()p Fy(x)p Fw(\))j(ad)1796 1285 y Fu(k)1831 1293 y Fj(1)1796 1346 y Fu(D)1850 1354 y Fm(y)1904 1254 y Fp(\000)1956 1319 y Fw(e)1993 1287 y Fu(ix)2054 1295 y Fj(1)2086 1287 y Fu(C)2134 1295 y Fj(1)2184 1322 y Fx(h)q Fy(C)2276 1334 y FK(1)2313 1322 y Fx(i)2345 1280 y Fv(\000)p FK(2)p Fu(k)2485 1254 y Fp(\001)2537 1322 y Fx(\001)g(\001)g(\001)g Fw(ad)2735 1285 y Fu(k)2770 1293 y Fm(n)2735 1346 y Fu(D)2789 1354 y Fm(y)2844 1254 y Fp(\000)2895 1319 y Fw(e)2932 1287 y Fu(ix)2993 1295 y Fj(2)3026 1287 y Fu(C)3074 1295 y Fm(n)3132 1322 y Fx(h)q Fy(C)3224 1334 y Fu(n)3269 1322 y Fx(i)3301 1280 y Fv(\000)p FK(2)p Fu(k)3441 1254 y Fp(\001)3479 1322 y Fy(;)83 1600 y FI(where)31 b Fs(C)364 1612 y Fu(k)399 1620 y Fj(1)432 1612 y Fv(\001\001\001)o Fu(k)526 1620 y Fm(n)612 1600 y Fy(>)40 b Fw(0)30 b FI(is)h(some)e(e)o (xplicit)g(constant.)g(Furthermore,)e(since)j Fy(C)2412 1612 y Fu(j)2477 1600 y FI(is)h(of)e(class)i Fy(C)2919 1570 y Fu(k)2960 1600 y Fw(\()p Fy(D)r Fw(\))p FI(,)g(we)f(kno)n(w)f (from)f([1)o(,)83 1700 y(Eq.)20 b(6.2.13])e(that)1193 1741 y Fp(\015)1193 1790 y(\015)1253 1811 y Fw(ad)1340 1766 y Fu(k)1375 1774 y Fm(j)1340 1836 y Fu(G)1392 1844 y Fm(y)1446 1744 y Fp(\000)1498 1808 y Fw(e)1535 1777 y Fu(ix)1596 1785 y Fm(j)1627 1777 y Fu(C)1675 1785 y Fm(j)1723 1811 y Fx(h)q Fy(C)1815 1823 y Fu(j)1850 1811 y Fx(i)1882 1769 y Fv(\000)p FK(2)p Fu(k)2022 1744 y Fp(\001)2060 1741 y(\015)2060 1790 y(\015)2129 1811 y Fx(\024)25 b Fs(C)2265 1823 y Fu(k)2300 1831 y Fm(j)2350 1811 y Fx(h)p Fy(x)2429 1823 y Fu(j)2464 1811 y Fx(i)2497 1769 y Fu(k)q FK(+1)2635 1811 y Fy(;)83 1961 y FI(where)d Fs(C)355 1973 y Fu(k)390 1981 y Fm(j)449 1961 y Fx(\025)g Fw(0)e FI(is)i(independent)17 b(of)j Fy(y)j FI(and)d Fy(x)1443 1973 y Fu(j)1479 1961 y FI(.)g(This)g(implies)h(that)625 2123 y Fp(\015)625 2173 y(\015)685 2194 y Fw(ad)773 2157 y Fu(k)773 2214 y(D)827 2222 y Fm(y)881 2127 y Fp(\000)919 2194 y Fy(f)9 b Fw(\()p Fy(C)d Fw(\))1098 2127 y Fp(\001)1136 2123 y(\015)1136 2173 y(\015)1206 2194 y Fx(\024)22 b Fw(Const)p Fy(:)1543 2081 y Fp(Z)1589 2269 y Fn(R)1631 2253 y Fm(n)1690 2194 y Fw(d)p 1690 2207 V Fy(x)14 b Fx(j)p Fw(\()p Fo(F)d Fy(g)s Fw(\)\()p Fy(x)p Fw(\))p Fx(j)j(h)r Fy(x)2241 2206 y FK(1)2279 2194 y Fx(i)2311 2152 y Fu(k)q FK(+1)2450 2194 y Fx(\001)g(\001)g(\001)g(h)p Fy(x)2640 2206 y Fu(n)2685 2194 y Fx(i)2718 2152 y Fu(k)q FK(+1)2866 2194 y Fx(\024)22 b Fw(Const)p Fy(:)14 b(;)83 2417 y FI(and)20 b(the)g(claim)g(is)h(pro)o(v)o(ed.)p 3708 2417 4 57 v 3712 2365 50 4 v 3712 2417 V 3761 2417 4 57 v 249 2583 a(In)f(Lemma)f(2.6.\(a\))g(we)h(ha)n(v)o(e)g(sho)n(wn)f (that)i(the)f(set)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))21 b FI(is)g(closed.)f(So)g(we)h(can)f(de\002ne)f(for)h(each)g Fy(t)j Fx(\025)f Fw(0)f FI(the)f(set)786 2766 y Fo(D)850 2778 y Fu(t)903 2766 y Fw(:=)1013 2699 y Fp(\010)1062 2766 y Fy(')j Fx(2)h(D)r Fw(\()p Fx(h)p Fw(\010)p Fx(i)1441 2724 y Fu(t)1470 2766 y Fw(\))f Fx(j)g Fy(')h Fw(=)e Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p 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Fw(\010)i FJ(satisfy)g(Assumptions)e(2.2)g(and)g (2.3.)g(Let)i Fy(f)32 b FJ(satisfy)24 b(Assumption)e(4.1)g(and)g (assume)h(that)83 3823 y Fy(R)146 3835 y Fu(f)210 3823 y FJ(belongs)c(to)h Fy(C)639 3793 y FK(1)677 3823 y Fw(\()p Fq(R)769 3793 y Fu(d)826 3823 y Fx(n)e(f)p Fw(0)p Fx(g)p Fw(\))p FJ(.)h(Then)h(the)g(map)413 4025 y Fy(t)443 4037 y Fu(f)509 4025 y Fw(:)k Fo(D)620 4037 y FK(1)680 4025 y Fx(!)f Fq(C)p Fy(;)97 b(')23 b Fx(7!)h Fy(t)1180 4037 y Fu(f)1223 4025 y Fw(\()p Fy(')p Fw(\))g(:=)e Fx(\000)1550 3993 y FK(1)p 1550 4007 34 4 v 1550 4054 a(2)1607 3947 y Fp(X)1651 4123 y Fu(j)1740 3958 y Fp(\010)q(\012)1828 4025 y Fw(\010)1888 4037 y Fu(j)1923 4025 y Fy(';)14 b Fw(\()p Fy(@)2090 4037 y Fu(j)2126 4025 y Fy(R)2189 4037 y Fu(f)2232 4025 y Fw(\)\()p Fy(H)2372 3991 y Fv(0)2395 4025 y Fw(\))p Fy(')2481 3958 y Fp(\013)2539 4025 y Fw(+)2623 3958 y Fp(\012\000)2700 4025 y Fy(@)2744 4037 y Fu(j)2779 4025 y Fy(R)p 2842 4003 44 3 v 26 x Fu(f)2885 3958 y Fp(\001)2923 4025 y Fw(\()p Fy(H)3031 3991 y Fv(0)3054 4025 y Fw(\))p Fy(';)g Fw(\010)3237 4037 y Fu(j)3273 4025 y Fy(')3327 3958 y Fp(\013\011)3415 4025 y Fy(;)83 4297 y FJ(is)20 b(well-de\002ned.)e(Mor)m(eo)o(ver)-9 b(,)19 b(if)h Fw(\()p Fy(@)1102 4309 y Fu(j)1137 4297 y Fy(R)1200 4309 y Fu(f)1243 4297 y Fw(\)\()p Fy(H)1383 4267 y Fv(0)1407 4297 y Fw(\))p Fy(')g FJ(belongs)f(to)g Fx(D)r Fw(\(\010)2034 4309 y Fu(j)2070 4297 y Fw(\))h FJ(for)g(eac)o(h)e Fy(j)5 b FJ(,)20 b(then)f(the)g(linear)g(oper)o (ator)g Fy(T)3355 4309 y Fu(f)3420 4297 y Fw(:)k Fo(D)3530 4309 y FK(1)3591 4297 y Fx(!)g(H)83 4396 y FJ(de\002ned)c(by)545 4600 y Fy(T)594 4612 y Fu(f)637 4600 y Fy(')k Fw(:=)g Fx(\000)900 4568 y FK(1)p 899 4582 34 4 v 899 4629 a(2)943 4508 y Fp(\020)992 4600 y Fw(\010)c Fx(\001)f Fy(R)1176 4570 y Fv(0)1175 4624 y Fu(f)1218 4600 y Fw(\()p Fy(H)1326 4570 y Fv(0)1349 4600 y Fw(\))h(+)f Fy(R)1547 4570 y Fv(0)1546 4624 y Fu(f)1589 4533 y Fp(\000)1657 4568 y Fu(H)1715 4543 y Fl(0)p 1637 4582 121 4 v 1637 4629 a Fv(j)p 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FK(1)3713 4808 y FJ(is)83 4908 y(dense)d(in)g Fx(H)q FJ(.)p Black 83 5074 a FA(Remark)g(5.3.)p Black 38 w FI(F)o(ormula)e(\(5.1\))h(is)h (a)g(priori)e(rather)h(complicated)f(and)h(one)g(could)f(be)h(tempted)g (to)h(replace)e(it)i(by)g(the)f(simpler)83 5173 y(formula)f Fx(\000)441 5141 y FK(1)p 440 5155 34 4 v 440 5202 a(2)483 5106 y Fp(\000)522 5173 y Fw(\010)d Fx(\001)h Fy(R)700 5143 y Fv(0)699 5197 y Fu(f)742 5173 y Fw(\()p Fy(H)850 5143 y Fv(0)873 5173 y Fw(\))g(+)f Fy(R)1065 5143 y Fv(0)1064 5197 y Fu(f)1107 5173 y Fw(\()p Fy(H)1215 5143 y Fv(0)1239 5173 y Fw(\))h Fx(\001)f Fw(\010)1385 5106 y Fp(\001)1423 5173 y FI(.)20 b(Unfortunately)-5 b(,)16 b(a)k(precise)f(meaning)f(of)h (this)h(e)o(xpression)e(is)i(not)g(a)n(v)n(ailable)e(in)83 5273 y(general,)h(and)h(its)h(full)f(deri)n(v)n(ation)e(can)i(only)f (be)i(justi\002ed)f(in)g(concrete)f(e)o(xamples.)p Black 1884 5670 a(12)p Black eop end %%Page: 13 13 TeXDict begin 13 12 bop Black Black Black 83 307 a FA(Remark)25 b(5.4.)p Black 43 w FI(If)f Fy(')32 b Fx(2)f Fo(D)884 319 y FK(1)947 307 y FI(and)24 b(if)h Fy(f)33 b FI(either)25 b(belongs)e(to)i Fo(S)16 b Fw(\()p Fq(R)2010 277 y Fu(d)2049 307 y Fw(\))25 b FI(or)f(is)i(radial,)e(then)g(the)h(assumption)e Fw(\()p Fy(@)3290 319 y Fu(j)3325 307 y Fy(R)3388 319 y Fu(f)3431 307 y Fw(\)\()p Fy(H)3571 277 y Fv(0)3595 307 y Fw(\))p Fy(')32 b Fx(2)83 407 y(D)r Fw(\(\010)241 419 y Fu(j)277 407 y Fw(\))26 b FI(holds)f(for)g(each)g Fy(j)5 b FI(.)26 b(Indeed,)e(by)h(Lemma)g(2.6.\(d\))f(there)h(e)o (xists)h Fy(\021)36 b Fx(2)e Fy(C)2475 377 y Fv(1)2469 427 y FK(c)2545 340 y Fp(\000)2583 407 y Fw(\(0)p Fy(;)14 b Fx(1)p Fw(\))2809 340 y Fp(\001)2874 407 y FI(such)25 b(that)h Fw(\()p Fy(@)3279 419 y Fu(j)3314 407 y Fy(R)3377 419 y Fu(f)3420 407 y Fw(\)\()p Fy(H)3560 377 y Fv(0)3584 407 y Fw(\))p Fy(')33 b Fw(=)83 515 y(\()p Fy(@)159 527 y Fu(j)194 515 y Fy(R)257 527 y Fu(f)300 515 y Fw(\)\()p Fy(H)440 485 y Fv(0)464 515 y Fw(\))p Fy(\021)540 447 y Fp(\000)579 515 y Fw(\()p Fy(H)687 485 y Fv(0)710 515 y Fw(\))742 485 y FK(2)779 447 y Fp(\001)818 515 y Fy(')p FI(.)i(By)h(Lemma)e(4.2)h(and)f(Proposition)g(5.1,)g(it)i(then)f(follo) n(ws)f(that)h Fw(\()p Fy(@)3003 527 y Fu(j)3039 515 y Fy(R)3102 527 y Fu(f)3145 515 y Fw(\)\()p Fy(H)3285 485 y Fv(0)3309 515 y Fw(\))p Fy(\021)3385 447 y Fp(\000)3423 515 y Fw(\()p Fy(H)3531 485 y Fv(0)3554 515 y Fw(\))3586 485 y FK(2)3624 447 y Fp(\001)3713 515 y Fx(2)83 614 y Fy(C)148 584 y FK(1)186 614 y Fw(\(\010)278 626 y Fu(j)313 614 y Fw(\))p FI(,)21 b(which)e(implies)h(the)h(statement.)p Black 83 780 a FJ(Pr)l(oof)f(of)h(Pr)l(oposition)e(5.2.)p Black 40 w FI(Let)c Fy(')23 b Fx(2)h Fo(D)1296 792 y FK(1)1333 780 y FI(.)16 b(Then)e(Lemma)h(2.6.\(d\))e(implies)i(that)g (there)g(e)o(xists)h(a)f(function)f Fy(\021)26 b Fx(2)e Fy(C)3405 750 y Fv(1)3399 801 y FK(c)3475 713 y Fp(\000)3513 780 y Fw(\(0)p Fy(;)14 b Fx(1)p Fw(\))3739 713 y Fp(\001)83 880 y FI(such)20 b(that)1230 980 y Fw(\()p Fy(@)1306 992 y Fu(j)1342 980 y Fy(R)1405 992 y Fu(f)1448 980 y Fw(\)\()p Fy(H)1588 945 y Fv(0)1611 980 y Fw(\))p Fy(')k Fw(=)f(\()p Fy(@)1885 992 y Fu(j)1920 980 y Fy(R)1983 992 y Fu(f)2026 980 y Fw(\)\()p Fy(H)2166 945 y Fv(0)2190 980 y Fw(\))p Fy(\021)2266 912 y Fp(\000)2304 980 y Fw(\()p Fy(H)2412 945 y Fv(0)2436 980 y Fw(\))2468 945 y FK(2)2505 912 y Fp(\001)2543 980 y Fy(':)83 1129 y FI(Thus)d Fx(k)p Fw(\()p Fy(@)388 1141 y Fu(j)423 1129 y Fy(R)486 1141 y Fu(f)529 1129 y Fw(\)\()p Fy(H)669 1099 y Fv(0)692 1129 y Fw(\))p Fy(')p Fx(k)k(\024)e Fw(Const)p Fy(:)14 b Fx(k)p Fy(')p Fx(k)p FI(,)20 b(and)f(we)i(ha)n(v)o(e)1385 1312 y Fx(j)p Fy(t)1438 1324 y Fu(f)1481 1312 y Fw(\()p Fy(')p Fw(\))p Fx(j)j(\024)f Fw(Const)p Fy(:)14 b Fx(k)p Fy(')p Fx(k)j(\001)i(kh)p Fw(\010)p Fx(i)p Fy(')p Fx(k)p Fy(;)83 1494 y FI(which)h(implies)g(the)g(\002rst)h(part)f(of)g(the)g (claim.)249 1594 y(F)o(or)g(the)g(second)f(part)h(of)g(the)g(claim,)g (it)h(is)g(suf)n(\002cient)f(to)g(sho)n(w)g(that)853 1712 y Fp(X)897 1888 y Fu(j)987 1723 y Fp(\012\000)1064 1790 y Fy(@)1108 1802 y Fu(j)1143 1790 y Fy(R)p 1206 1768 44 3 v 26 x Fu(f)1249 1723 y Fp(\001)1287 1790 y Fw(\()p Fy(H)1395 1756 y Fv(0)1418 1790 y Fw(\))p Fy(';)14 b Fw(\010)1601 1802 y Fu(j)1637 1790 y Fy(')1691 1723 y Fp(\013)862 2022 y Fw(=)950 1955 y Fp(\012)989 2022 y Fy(';)1080 1955 y Fp(\010)1128 2022 y Fy(R)1192 1992 y Fv(0)1191 2046 y Fu(f)1235 1955 y Fp(\000)1302 1989 y Fu(H)1360 1964 y Fl(0)p 1283 2003 121 4 v 1283 2051 a Fv(j)p Fu(H)1361 2034 y Fl(0)1384 2051 y Fv(j)1413 2022 y Fw(\))19 b Fx(\001)f Fw(\010)9 b Fx(j)p Fy(H)1673 1992 y Fv(0)1697 2022 y Fx(j)1720 1992 y Fv(\000)p FK(1)1827 2022 y Fw(+)18 b Fy(iR)2003 1992 y Fv(0)2002 2046 y Fu(f)2045 1955 y Fp(\000)2113 1989 y Fu(H)2171 1964 y Fl(0)p 2093 2003 V 2093 2051 a Fv(j)p Fu(H)2171 2034 y Fl(0)2194 2051 y Fv(j)2224 1955 y Fp(\001)2280 2022 y Fx(\001)h Fw(\()p Fy(H)2430 1992 y Fv(00)2472 2022 y Fy(H)2548 1992 y Fv(0)2571 2022 y Fw(\))9 b Fx(j)p Fy(H)2711 1992 y Fv(0)2735 2022 y Fx(j)2758 1992 y Fv(\000)p FK(3)2847 1955 y Fp(\011)2896 2022 y Fy(')2950 1955 y Fp(\013)2989 2022 y Fy(:)83 2213 y FI(Using)20 b(F)o(ormula)f(\(4.2\))n(,)i(Lemma)e (2.6.\(d\),)f(and)h([10)o(,)i(Eq.)e(4.3.2],)g(one)g(gets)601 2331 y Fp(X)646 2508 y Fu(j)735 2342 y Fp(\012\000)812 2410 y Fy(@)856 2422 y Fu(j)891 2410 y Fy(R)p 954 2387 44 3 v 26 x Fu(f)997 2342 y Fp(\001)1036 2410 y Fw(\()p Fy(H)1144 2375 y Fv(0)1167 2410 y Fw(\))p Fy(';)14 b Fw(\010)1350 2422 y Fu(j)1385 2410 y Fy(')1439 2342 y Fp(\013)611 2648 y Fw(=)698 2569 y Fp(X)743 2746 y Fu(j)832 2580 y Fp(\012)871 2648 y Fw(\()p Fy(@)947 2660 y Fu(j)982 2648 y Fy(R)p 1045 2625 V 25 x Fu(f)1089 2648 y Fw(\))1121 2580 y Fp(\000)1189 2615 y Fu(H)1247 2590 y Fl(0)p 1169 2629 121 4 v 1169 2676 a Fv(j)p Fu(H)1247 2660 y Fl(0)1270 2676 y Fv(j)1300 2580 y Fp(\001)1338 2648 y Fx(j)p Fy(H)1437 2617 y Fv(0)1460 2648 y Fx(j)1483 2617 y Fv(\000)p FK(1)1572 2648 y Fy(';)g Fw(\010)1723 2660 y Fu(j)1758 2648 y Fy(')1812 2580 y Fp(\013)611 2886 y Fw(=)698 2807 y Fp(X)743 2983 y Fu(j)840 2886 y Fw(lim)832 2940 y Fu(")p Fv(&)p FK(0)976 2818 y Fp(\012)q(\000)1054 2886 y Fy(@)1098 2898 y Fu(j)1133 2886 y Fy(R)p 1196 2863 44 3 v 25 x Fu(f)1239 2818 y Fp(\001\000)1345 2853 y Fu(H)1403 2828 y Fl(0)p 1325 2867 121 4 v 1325 2914 a Fv(j)p Fu(H)1403 2898 y Fl(0)1426 2914 y Fv(j)1456 2818 y Fp(\001)1494 2886 y Fy(';)g Fw([\()p Fy(H)1716 2855 y Fv(0)1739 2886 y Fw(\))1771 2855 y FK(2)1827 2886 y Fw(+)k Fy(")p Fw(])1972 2855 y Fv(\000)p FK(1)p Fu(=)p FK(2)2128 2886 y Fw(\010)2188 2898 y Fu(j)2223 2886 y Fy(')2277 2818 y Fp(\013)611 3117 y Fw(=)698 3050 y Fp(\012)737 3117 y Fy(';)c(R)892 3087 y Fv(0)891 3141 y Fu(f)935 3050 y Fp(\000)1002 3084 y Fu(H)1060 3059 y Fl(0)p 983 3098 V 983 3146 a Fv(j)p Fu(H)1061 3129 y Fl(0)1084 3146 y Fv(j)1113 3117 y Fw(\))19 b Fx(\001)g Fw(\010)9 b Fx(j)p Fy(H)1374 3087 y Fv(0)1397 3117 y Fx(j)1420 3087 y Fv(\000)p FK(1)1509 3117 y Fy(')1563 3050 y Fp(\013)772 3311 y Fw(+)18 b Fy(\031)905 3277 y Fv(\000)p FK(1)1008 3232 y Fp(X)1053 3409 y Fu(j)1150 3311 y Fw(lim)1142 3365 y Fu(")p Fv(&)p FK(0)1286 3198 y Fp(Z)1370 3218 y Fv(1)1333 3387 y FK(0)1454 3311 y Fw(d)p Fy(t)c(t)1574 3277 y Fv(\000)p FK(1)p Fu(=)p FK(2)1730 3244 y Fp(\012\000)1807 3311 y Fy(@)1851 3323 y Fu(j)1886 3311 y Fy(R)p 1949 3288 44 3 v 26 x Fu(f)1992 3244 y Fp(\001\000)2098 3278 y Fu(H)2156 3253 y Fl(0)p 2078 3292 121 4 v 2078 3340 a Fv(j)p Fu(H)2156 3323 y Fl(0)2179 3340 y Fv(j)2209 3244 y Fp(\001)2247 3311 y Fy(';)2338 3244 y Fp(\002)2373 3311 y Fw([\()p Fy(H)2504 3281 y Fv(0)2527 3311 y Fw(\))2559 3281 y FK(2)2615 3311 y Fw(+)k Fy(")g Fw(+)g Fy(t)p Fw(])2891 3281 y Fv(\000)p FK(1)2981 3311 y Fy(;)c Fw(\010)3078 3323 y Fu(j)3112 3244 y Fp(\003)3147 3311 y Fy(')3201 3244 y Fp(\013)3241 3311 y Fy(:)83 3576 y FI(No)n(w)-5 b(,)19 b(by)h(using)g(Assumption)f(2.2)h(and)f(Lemma)h (2.4)f(one)h(obtains)f(that)886 3706 y Fp(\002)921 3773 y Fw([\()p Fy(H)1052 3739 y Fv(0)1075 3773 y Fw(\))1107 3739 y FK(2)1163 3773 y Fw(+)f Fy(")h Fw(+)f Fy(t)p Fw(])1440 3739 y Fv(\000)p FK(1)1529 3773 y Fy(;)c Fw(\010)1626 3785 y Fu(j)1661 3706 y Fp(\003)1695 3773 y Fy(')24 b Fw(=)e(2)p Fy(i)1931 3706 y Fp(\002)1965 3773 y Fw(\()p Fy(H)2073 3739 y Fv(0)2096 3773 y Fw(\))2128 3739 y FK(2)2184 3773 y Fw(+)c Fy(")g Fw(+)h Fy(t)2438 3706 y Fp(\003)2472 3723 y Fv(\000)p FK(2)2561 3773 y Fw(\()p Fy(H)2669 3739 y Fv(00)2712 3773 y Fy(H)2788 3739 y Fv(0)2811 3773 y Fw(\))2843 3785 y Fu(j)2887 3773 y Fy(':)83 3956 y FI(It)i(follo)n(ws)e (that)640 4166 y Fy(\031)690 4132 y Fv(\000)p FK(1)793 4087 y Fp(X)838 4264 y Fu(j)935 4166 y Fw(lim)927 4220 y Fu(")p Fv(&)p FK(0)1071 4053 y Fp(Z)1154 4073 y Fv(1)1117 4242 y FK(0)1239 4166 y Fw(d)p Fy(t)14 b(t)1359 4132 y Fv(\000)p FK(1)p Fu(=)p FK(2)1515 4099 y Fp(\012\000)1592 4166 y Fy(@)1636 4178 y Fu(j)1671 4166 y Fy(R)p 1734 4143 44 3 v 26 x Fu(f)1777 4099 y Fp(\001\000)1883 4133 y Fu(H)1941 4108 y Fl(0)p 1863 4147 121 4 v 1863 4195 a Fv(j)p Fu(H)1941 4178 y Fl(0)1964 4195 y Fv(j)1994 4099 y Fp(\001)2032 4166 y Fy(';)g Fw(2)p Fy(i)p Fw([\()p Fy(H)2325 4136 y Fv(0)2348 4166 y Fw(\))2380 4136 y FK(2)2436 4166 y Fw(+)k Fy(")g Fw(+)g Fy(t)p Fw(])2712 4136 y Fv(\000)p FK(2)2801 4166 y Fw(\()p Fy(H)2909 4136 y Fv(00)2951 4166 y Fy(H)3027 4136 y Fv(0)3050 4166 y Fw(\))3082 4178 y Fu(j)3118 4166 y Fy(')3172 4099 y Fp(\013)663 4404 y Fw(=)751 4325 y Fp(X)795 4502 y Fu(j)892 4404 y Fw(lim)884 4458 y Fu(")p Fv(&)p FK(0)1029 4337 y Fp(\012\000)1106 4404 y Fy(@)1150 4416 y Fu(j)1185 4404 y Fy(R)p 1248 4381 44 3 v 26 x Fu(f)1291 4337 y Fp(\001\000)1397 4371 y Fu(H)1455 4346 y Fl(0)p 1377 4385 121 4 v 1377 4432 a Fv(j)p Fu(H)1455 4416 y Fl(0)1478 4432 y Fv(j)1508 4337 y Fp(\001)1546 4404 y Fy(';)c(i)p Fw([\()p Fy(H)1797 4374 y Fv(0)1820 4404 y Fw(\))1852 4374 y FK(2)1908 4404 y Fw(+)k Fy(")p Fw(])2053 4374 y Fv(\000)p FK(3)p Fu(=)p FK(2)2209 4404 y Fw(\()p Fy(H)2317 4374 y Fv(00)2360 4404 y Fy(H)2436 4374 y Fv(0)2459 4404 y Fw(\))2491 4416 y Fu(j)2526 4404 y Fy(')2580 4337 y Fp(\013)663 4635 y Fw(=)751 4568 y Fp(\012)790 4635 y Fy(';)c(iR)974 4605 y Fv(0)973 4659 y Fu(f)1016 4568 y Fp(\000)1083 4603 y Fu(H)1141 4578 y Fl(0)p 1064 4617 V 1064 4664 a Fv(j)p Fu(H)1142 4647 y Fl(0)1165 4664 y Fv(j)1194 4568 y Fp(\001)1251 4635 y Fx(\001)k Fw(\()p Fy(H)1400 4605 y Fv(00)1443 4635 y Fy(H)1519 4605 y Fv(0)1542 4635 y Fw(\))9 b Fx(j)p Fy(H)1682 4605 y Fv(0)1705 4635 y Fx(j)1728 4605 y Fv(\000)p FK(3)1818 4635 y Fy(')1872 4568 y Fp(\013)1911 4635 y Fy(;)83 4826 y FI(and)20 b(thus)400 4930 y Fp(X)445 5107 y Fu(j)534 4942 y Fp(\012\000)611 5009 y Fy(@)655 5021 y Fu(j)690 5009 y Fy(R)p 753 4986 44 3 v 26 x Fu(f)796 4942 y Fp(\001)834 5009 y Fw(\()p Fy(H)942 4975 y Fv(0)966 5009 y Fw(\))p Fy(';)14 b Fw(\010)1149 5021 y Fu(j)1184 5009 y Fy(')1238 4942 y Fp(\013)1301 5009 y Fw(=)1388 4942 y Fp(\012)1428 5009 y Fy(';)1519 4942 y Fp(\010)1567 5009 y Fy(R)1631 4979 y Fv(0)1630 5033 y Fu(f)1673 4942 y Fp(\000)1741 4976 y Fu(H)1799 4951 y Fl(0)p 1721 4990 121 4 v 1721 5038 a Fv(j)p Fu(H)1799 5021 y Fl(0)1822 5038 y Fv(j)1852 5009 y Fw(\))19 b Fx(\001)f Fw(\010)9 b Fx(j)p Fy(H)2112 4979 y Fv(0)2136 5009 y Fx(j)2159 4979 y Fv(\000)p FK(1)2266 5009 y Fw(+)18 b Fy(iR)2442 4979 y Fv(0)2441 5033 y Fu(f)2484 4942 y Fp(\000)2552 4976 y Fu(H)2610 4951 y Fl(0)p 2532 4990 V 2532 5038 a Fv(j)p Fu(H)2610 5021 y Fl(0)2633 5038 y Fv(j)2663 4942 y Fp(\001)2719 5009 y Fx(\001)h Fw(\()p Fy(H)2869 4979 y Fv(00)2911 5009 y Fy(H)2987 4979 y Fv(0)3010 5009 y Fw(\))9 b Fx(j)p Fy(H)3150 4979 y Fv(0)3174 5009 y Fx(j)3197 4979 y Fv(\000)p FK(3)3286 4942 y Fp(\011)3334 5009 y Fy(')3388 4942 y Fp(\013)3428 5009 y Fy(:)p 3708 5274 4 57 v 3712 5221 50 4 v 3712 5274 V 3761 5274 4 57 v Black 1884 5670 a FI(13)p Black eop end %%Page: 14 14 TeXDict begin 14 13 bop Black Black 249 307 a FI(Suppose)29 b(for)h(a)g(while)g(that)g Fy(f)40 b FI(is)31 b(radial.)e(Then)h(one)f (has)h Fw(\()p Fy(@)2102 319 y Fu(j)2138 307 y Fy(R)2201 319 y Fu(f)2244 307 y Fw(\)\()p Fy(x)p Fw(\))43 b(=)e Fx(\000)p Fy(x)2648 277 y Fv(\000)p FK(2)2737 307 y Fy(x)2784 319 y Fu(j)2850 307 y FI(due)29 b(to)i(Lemma)e(4.2.\(c\),)f(and)83 407 y(F)o(ormula)19 b(\(5.1\))g(holds)g(by)h(Remark)g(5.4.)f(This)h (implies)h(that)f Fy(T)1925 419 y Fu(f)1988 407 y FI(is)h(equal)f(to) 983 607 y Fy(T)34 b Fw(:=)1186 575 y FK(1)p 1186 589 34 4 v 1186 636 a(2)1230 515 y Fp(\020)1279 607 y Fw(\010)19 b Fx(\001)1451 575 y Fu(H)1509 550 y Fl(0)p 1409 589 166 4 v 1409 636 a FK(\()p Fu(H)1493 619 y Fl(0)1516 636 y FK(\))1542 619 y Fj(2)1603 607 y Fw(+)1716 575 y Fu(H)1774 550 y Fl(0)p 1696 589 121 4 v 1696 636 a Fv(j)p Fu(H)1774 619 y Fl(0)1797 636 y Fv(j)1845 607 y Fx(\001)g Fw(\010)9 b Fx(j)p Fy(H)2055 577 y Fv(0)2078 607 y Fx(j)2101 577 y Fv(\000)p FK(1)2209 607 y Fw(+)2332 575 y Fu(iH)2413 550 y Fl(0)p 2302 589 166 4 v 2302 636 a FK(\()p Fu(H)2386 619 y Fl(0)2409 636 y FK(\))2435 619 y Fj(4)2495 607 y Fx(\001)19 b Fw(\()p Fy(H)2645 577 y Fv(00)2687 607 y Fy(H)2763 577 y Fv(0)2787 607 y Fw(\))2819 515 y Fp(\021)3609 607 y FI(\(5.2\))83 801 y(on)h Fo(D)251 813 y FK(1)288 801 y FI(.)249 900 y(The)d(ne)o(xt)g (theorem)e(is)k(our)d(main)h(result;)g(it)h(relates)g(the)f(e)n(v)n (olution)f(of)h(localisation)g(operators)e Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))19 b FI(to)e(the)g(operator)83 1000 y Fy(T)132 1012 y Fu(f)175 1000 y FI(.)22 b(In)f(its)i(proof,)d(we)j(freely)e(use) h(the)g(notations)f(of)h([1)o(])g(for)f(some)h(re)o(gularity)e(classes) j(with)f(respect)g(to)g(the)g(unitary)e(group)83 1100 y(generated)e(by)i Fw(\010)p FI(.)h(F)o(or)f(us,)g(a)g(function)f Fy(f)32 b Fw(:)23 b Fq(R)1416 1070 y Fu(d)1477 1100 y Fx(!)h Fq(C)c FI(is)h(e)n(v)o(en)f(if)g Fy(f)9 b Fw(\()p Fy(x)p Fw(\))24 b(=)e Fy(f)9 b Fw(\()p Fx(\000)p Fy(x)p Fw(\))21 b FI(for)f FJ(a.e)o(.)f Fy(x)24 b Fx(2)f Fq(R)2971 1070 y Fu(d)3010 1100 y FI(.)p Black 83 1258 a FA(Theor)o(em)j(5.5.)p Black 44 w FJ(Let)g Fy(H)33 b FJ(and)25 b Fw(\010)i FJ(satisfy)f (Assumptions)f(2.2)g(and)g(2.3.)g(Let)i Fy(f)42 b Fx(2)34 b Fo(S)16 b Fw(\()p Fq(R)2674 1228 y Fu(d)2713 1258 y Fw(\))27 b FJ(be)e(an)h(e)o(ven)f(function)g(suc)o(h)g(that)83 1357 y Fy(f)32 b Fw(=)22 b(1)f FJ(on)e(a)i(neighbourhood)16 b(of)k Fw(0)p FJ(.)g(Then)g(we)h(have)e(for)i(eac)o(h)e Fy(')24 b Fx(2)f Fo(D)2159 1369 y FK(2)698 1573 y Fw(lim)673 1623 y Fu(r)r Fv(!1)862 1541 y FK(1)p 862 1555 34 4 v 862 1602 a(2)919 1460 y Fp(Z)1002 1481 y Fv(1)965 1649 y FK(0)1086 1573 y Fw(d)p Fy(t)1176 1506 y Fp(\012)1215 1573 y Fy(';)1306 1506 y Fp(\002)1355 1570 y Fw(e)1392 1539 y Fv(\000)p Fu(itH)1569 1573 y Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))1838 1570 y(e)1875 1539 y Fu(itH)2000 1573 y Fx(\000)2079 1570 y Fw(e)2116 1539 y Fu(itH)2241 1573 y Fy(f)g Fw(\(\010)p Fy(=r)r Fw(\))2510 1570 y(e)2547 1539 y Fv(\000)p Fu(itH)2724 1506 y Fp(\003)2759 1573 y Fy(')2813 1506 y Fp(\013)2876 1573 y Fw(=)22 b Fy(t)2993 1585 y Fu(f)3036 1573 y Fw(\()p Fy(')p Fw(\))p Fy(:)432 b FI(\(5.3\))249 1793 y(Note)20 b(that)h(the)f(inte)o(gral)f(on)h(the)g (l.h.s.)g(of)g(\(5.3\))e(is)k(\002nite)e(for)g(each)f Fy(r)26 b(>)d Fw(0)d FI(since)h Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))21 b FI(can)f(be)g(f)o(actorized)f(as)1054 1965 y Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))24 b Fx(\021)e(j)p Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))p Fx(j)1721 1931 y FK(1)p Fu(=)p FK(2)1845 1965 y Fx(\001)18 b Fw(sgn)o([)p Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\)])19 b Fx(\001)g(j)p Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))p Fx(j)2669 1931 y FK(1)p Fu(=)p FK(2)2774 1965 y Fy(;)83 2138 y FI(with)22 b Fx(j)p Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))p Fx(j)554 2108 y FK(1)p Fu(=)p FK(2)681 2138 y FI(locally)21 b Fy(H)7 b FI(-smooth)20 b(on)h Fq(R)e Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))22 b FI(by)f(Theorem)f(3.6.)g(Furthermore,)f (since)j(Remark)f(5.4)g(applies,)g(the)83 2237 y(r)-5 b(.h.s.)20 b(can)g(also)g(be)h(written)f(as)g(the)h(e)o(xpectation)d(v) n(alue)h Fx(h)p Fy(';)14 b(T)1893 2249 y Fu(f)1936 2237 y Fy(')p Fx(i)p FI(.)p Black 83 2401 a FJ(Pr)l(oof.)p Black 41 w FI(\(i\))24 b(Let)g Fy(')31 b Fx(2)g Fo(D)804 2413 y FK(2)841 2401 y FI(,)25 b(tak)o(e)f(a)h(real)f Fy(\021)34 b Fx(2)c Fy(C)1485 2371 y Fv(1)1479 2422 y FK(c)1556 2334 y Fp(\000)1594 2401 y Fq(R)21 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))1926 2334 y Fp(\001)1989 2401 y FI(such)24 b(that)g Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fy(')32 b Fw(=)e Fy(')p FI(,)25 b(and)e(set)i Fy(\021)3082 2413 y Fu(t)3112 2401 y Fw(\()p Fy(H)7 b Fw(\))30 b(:=)3400 2398 y(e)3437 2371 y Fu(itH)3563 2401 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p FI(.)83 2501 y(Then)19 b(we)i(ha)n(v)o(e)449 2607 y Fp(\012)488 2674 y Fy(';)579 2607 y Fp(\002)628 2671 y Fw(e)665 2639 y Fu(itH)790 2674 y Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))1059 2671 y(e)1096 2639 y Fv(\000)p Fu(itH)1274 2674 y Fx(\000)1353 2671 y Fw(e)1389 2639 y Fv(\000)p Fu(itH)1566 2674 y Fy(f)g Fw(\(\010)p Fy(=r)r Fw(\))1835 2671 y(e)1872 2639 y Fu(itH)1998 2607 y Fp(\003)2032 2674 y Fy(')2086 2607 y Fp(\013)472 2850 y Fw(=)560 2737 y Fp(Z)606 2926 y Fn(R)648 2909 y Fm(d)701 2850 y Fw(d)p 701 2863 47 4 v Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1118 2783 y Fp(\012)1159 2850 y Fy(';)1250 2783 y Fp(\002)1285 2850 y Fy(\021)1326 2862 y Fu(t)1355 2850 y Fw(\()p Fy(H)e Fw(\))1509 2847 y(e)1547 2816 y Fu(i)1580 2793 y Fm(x)p 1580 2802 33 3 v 1582 2836 a(r)1623 2816 y Fv(\001)p FK(\010)1708 2850 y Fy(\021)1749 2862 y Fv(\000)p Fu(t)1830 2850 y Fw(\()p Fy(H)g Fw(\))19 b Fx(\000)f Fy(\021)2113 2862 y Fv(\000)p Fu(t)2194 2850 y Fw(\()p Fy(H)7 b Fw(\))2348 2847 y(e)2385 2816 y Fu(i)2419 2793 y Fm(x)p 2419 2802 V 2421 2836 a(r)2462 2816 y Fv(\001)p FK(\010)2547 2850 y Fy(\021)2588 2862 y Fu(t)2617 2850 y Fw(\()p Fy(H)g Fw(\))2757 2783 y Fp(\003)2792 2850 y Fy(')2846 2783 y Fp(\013)472 3072 y Fw(=)560 2959 y Fp(Z)606 3147 y Fn(R)648 3131 y Fm(d)701 3072 y Fw(d)p 701 3085 47 4 v Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1118 3004 y Fp(\012)1159 3072 y Fy(';)1250 3004 y Fp(\002)1299 3069 y Fw(e)1336 3037 y Fu(i)1369 3015 y Fm(x)p 1369 3024 33 3 v 1371 3057 a(r)1412 3037 y Fv(\001)p FK(\010)1497 3072 y Fy(\021)1538 3084 y Fu(t)1568 3004 y Fp(\000)1606 3072 y Fy(H)e Fw(\()1724 3039 y Fu(x)p 1724 3053 38 4 v 1727 3100 a(r)1771 3072 y Fw(\))1803 3004 y Fp(\001)1842 3072 y Fy(\021)1883 3084 y Fv(\000)p Fu(t)1964 3072 y Fw(\()p Fy(H)g Fw(\))19 b Fx(\000)f Fy(\021)2247 3084 y Fv(\000)p Fu(t)2328 3072 y Fw(\()p Fy(H)7 b Fw(\))p Fy(\021)2509 3084 y Fu(t)2539 3004 y Fp(\000)2577 3072 y Fy(H)g Fw(\()p Fx(\000)2760 3039 y Fu(x)p 2759 3053 V 2762 3100 a(r)2807 3072 y Fw(\))2839 3004 y Fp(\001)2891 3069 y Fw(e)2928 3042 y Fu(i)2962 3019 y Fm(x)p 2962 3028 33 3 v 2964 3062 a(r)3004 3042 y Fv(\001)p FK(\010)3090 3004 y Fp(\003)3124 3072 y Fy(')3178 3004 y Fp(\013)472 3294 y Fw(=)560 3181 y Fp(Z)606 3369 y Fn(R)648 3353 y Fm(d)701 3294 y Fw(d)p 701 3307 47 4 v Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1118 3226 y Fp(\012)1159 3294 y Fy(';)1250 3226 y Fp(\010)q(\000)1351 3291 y Fw(e)1388 3259 y Fu(i)1421 3237 y Fm(x)p 1421 3246 33 3 v 1423 3279 a(r)1464 3259 y Fv(\001)p FK(\010)1549 3294 y Fx(\000)p Fw(1)1656 3226 y Fp(\001)1693 3294 y Fy(\021)1734 3306 y Fu(t)1764 3226 y Fp(\000)1802 3294 y Fy(H)e Fw(\()1920 3261 y Fu(x)p 1920 3275 38 4 v 1923 3322 a(r)1967 3294 y Fw(\))1999 3226 y Fp(\001)2038 3294 y Fy(\021)2079 3306 y Fv(\000)p Fu(t)2160 3294 y Fw(\()p Fy(H)g Fw(\))1309 b FI(\(5.4\))800 3473 y Fw(+)18 b Fy(\021)924 3485 y Fv(\000)p Fu(t)1005 3473 y Fw(\()p Fy(H)7 b Fw(\))1145 3406 y Fp(\002)1180 3473 y Fy(\021)1221 3485 y Fu(t)1250 3406 y Fp(\000)1288 3473 y Fy(H)g Fw(\()1406 3440 y Fu(x)p 1406 3454 V 1409 3502 a(r)1454 3473 y Fw(\))1486 3406 y Fp(\001)1543 3473 y Fx(\000)18 b Fy(\021)1667 3485 y Fu(t)1696 3406 y Fp(\000)1734 3473 y Fy(H)7 b Fw(\()p Fx(\000)1917 3440 y Fu(x)p 1917 3454 V 1920 3502 a(r)1965 3473 y Fw(\))1997 3406 y Fp(\001\003)2088 3473 y Fx(\000)18 b Fy(\021)2212 3485 y Fv(\000)p Fu(t)2293 3473 y Fw(\()p Fy(H)7 b Fw(\))p Fy(\021)2474 3485 y Fu(t)2504 3406 y Fp(\000)2542 3473 y Fy(H)g Fw(\()p Fx(\000)2725 3440 y Fu(x)p 2725 3454 V 2728 3502 a(r)2772 3473 y Fw(\))2804 3406 y Fp(\001)q(\000)2895 3470 y Fw(e)2931 3443 y Fu(i)2965 3421 y Fm(x)p 2965 3430 33 3 v 2967 3463 a(r)3008 3443 y Fv(\001)p FK(\010)3093 3473 y Fx(\000)p Fw(1)3200 3406 y Fp(\001)o(\011)3285 3473 y Fy(')3339 3406 y Fp(\013)3379 3473 y Fy(:)83 3646 y FI(Since)20 b Fy(f)30 b FI(is)21 b(e)n(v)o(en,)e Fo(F)11 b Fy(f)30 b FI(is)21 b(also)f(e)n(v)o(en,)f (and)846 3743 y Fp(Z)892 3932 y Fn(R)934 3915 y Fm(d)986 3856 y Fw(d)p 986 3869 47 4 v 1 w Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1418 3789 y Fp(\012)1459 3856 y Fy(';)14 b(\021)1591 3868 y Fv(\000)p Fu(t)1672 3856 y Fw(\()p Fy(H)7 b Fw(\))1812 3789 y Fp(\002)1847 3856 y Fy(\021)1888 3868 y Fu(t)1918 3789 y Fp(\000)1956 3856 y Fy(H)g Fw(\()2074 3823 y Fu(x)p 2074 3837 38 4 v 2077 3885 a(r)2121 3856 y Fw(\))2153 3789 y Fp(\001)2210 3856 y Fx(\000)18 b Fy(\021)2334 3868 y Fu(t)2364 3789 y Fp(\000)2402 3856 y Fy(H)7 b Fw(\()p Fx(\000)2585 3823 y Fu(x)p 2584 3837 V 2587 3885 a(r)2632 3856 y Fw(\))2664 3789 y Fp(\001\003)2737 3856 y Fy(')2791 3789 y Fp(\013)2853 3856 y Fw(=)23 b(0)p Fy(:)83 4075 y FI(Thus)d(F)o(ormula)f(\(5.4\))n(,) h(Lemma)g(2.4,)f(and)h(the)g(change)f(of)h(v)n(ariables)f Fy(\026)k Fw(:=)g Fy(t=r)r FI(,)e Fy(\027)28 b Fw(:=)23 b(1)p Fy(=r)r FI(,)d(gi)n(v)o(e)213 4295 y Fw(lim)188 4345 y Fu(r)r Fv(!1)377 4262 y FK(1)p 377 4276 34 4 v 377 4324 a(2)434 4182 y Fp(Z)517 4202 y Fv(1)480 4371 y FK(0)601 4295 y Fw(d)p Fy(t)691 4228 y Fp(\012)730 4295 y Fy(';)821 4228 y Fp(\002)870 4292 y Fw(e)907 4261 y Fv(\000)p Fu(itH)1084 4295 y Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))1353 4292 y(e)1390 4261 y Fu(itH)1516 4295 y Fx(\000)1595 4292 y Fw(e)1631 4261 y Fu(itH)1756 4295 y Fy(f)g Fw(\(\010)p Fy(=r)r Fw(\))2025 4292 y(e)2062 4261 y Fv(\000)p Fu(itH)2240 4228 y Fp(\003)2274 4295 y Fy(')2328 4228 y Fp(\013)2391 4295 y Fw(=)23 b Fx(\000)2554 4262 y FK(1)p 2553 4276 V 2553 4324 a(2)2620 4295 y Fw(lim)2610 4349 y Fu(\027)t Fv(&)p FK(0)2760 4182 y Fp(Z)2843 4202 y Fv(1)2806 4371 y FK(0)2927 4295 y Fw(d)p Fy(\026)3037 4182 y Fp(Z)3084 4371 y Fn(R)3126 4354 y Fm(d)3178 4295 y Fw(d)p 3178 4308 47 4 v Fy(x)14 b(K)6 b Fw(\()p Fy(\027)q(;)14 b(\026;)g(x)p Fw(\))p Fy(;)3609 4435 y FI(\(5.5\))83 4535 y(where)596 4708 y Fy(K)6 b Fw(\()p Fy(\027)q(;)14 b(\026;)g(x)p Fw(\))23 b(:=)g(\()p Fo(F)11 b Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1394 4641 y Fp(\012)1435 4708 y Fy(';)1526 4641 y Fp(\010)1587 4675 y FK(1)p 1585 4689 38 4 v 1585 4736 a Fu(\027)1632 4641 y Fp(\000)1684 4705 y Fw(e)1720 4678 y Fu(i\027)t(x)p Fv(\001)p FK(\010)1904 4708 y Fx(\000)p Fw(1)2011 4641 y Fp(\001)2048 4708 y Fy(\021)s Fw(\()p Fy(H)e Fw(\()p Fy(\027)e(x)p Fw(\)\))2403 4705 y(e)2442 4678 y Fu(i)2475 4651 y Fm(\026)p 2475 4665 36 3 v 2476 4698 a(\027)2520 4678 y FK([)p Fu(H)t FK(\()p Fu(\027)t(x)p FK(\))p Fv(\000)p Fu(H)t FK(])1593 4850 y Fx(\000)18 b Fy(\021)s Fw(\()p Fy(H)7 b Fw(\()p Fx(\000)p Fy(\027)e(x)p Fw(\)\))2096 4847 y(e)2134 4816 y Fu(i)2168 4789 y Fm(\026)p 2168 4803 V 2169 4836 a(\027)2213 4816 y FK([)p Fu(H)t FK(\()p Fv(\000)p Fu(\027)t(x)p FK(\))p Fv(\000)p Fu(H)t FK(])2629 4817 y(1)p 2627 4831 38 4 v 2627 4878 a Fu(\027)2674 4783 y Fp(\000)2726 4847 y Fw(e)2762 4820 y Fu(i\027)t(x)p Fv(\001)p FK(\010)2946 4850 y Fx(\000)p Fw(1)3053 4783 y Fp(\001)o(\011)3138 4850 y Fy(')3192 4783 y Fp(\013)3232 4850 y Fy(:)249 5023 y FI(\(ii\))27 b(T)-7 b(o)27 b(pro)o(v)o(e)e(the)i (statement,)g(we)g(shall)g(sho)n(w)g(that)g(one)f(may)h(interchange)d (the)j(limit)h(and)e(the)h(inte)o(grals)f(in)h(\(5.5\))n(,)83 5122 y(by)c(in)m(v)n(oking)e(Lebesgue')-5 b(s)22 b(dominated)f(con)m(v) o(er)o(gence)e(theorem.)j(This)h(will)h(be)f(done)f(in)h(\(iii\))g (belo)n(w)-5 b(.)22 b(Here)h(we)h(pursue)e(the)83 5222 y(calculations)d(assuming)h(that)g(these)g(interchanges)f(are)h (justi\002ed.)249 5321 y(W)-7 b(e)32 b(kno)n(w)e(from)g(Assumption)g (2.2)g(that)h Fy(H)39 b FI(is)32 b(of)e(class)i Fy(C)2073 5291 y FK(2)2111 5321 y Fw(\(\010)2203 5333 y Fu(j)2238 5321 y Fw(\))g FI(\(and)e(thus)h(of)g(class)h Fy(C)3010 5291 y FK(1)p Fu(;)p FK(1)3100 5321 y Fw(\(\010)3192 5333 y Fu(j)3227 5321 y Fw(\))p FI(\))g(for)e(each)h Fy(j)48 b Fx(2)83 5421 y(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(d)p Fx(g)p FI(.)32 b(Since)g(the)h(domain)e(of)h Fy(H)40 b FI(is)33 b(in)m(v)n(ariant)e(under)g(the)i(group)d(generated)h(by)h Fw(\010)2860 5433 y Fu(j)2895 5421 y FI(,)h(it)g(follo)n(ws)f(then)g (from)f([1)o(,)p Black 1884 5670 a(14)p Black eop end %%Page: 15 15 TeXDict begin 15 14 bop Black Black 83 307 a FI(Thm.)22 b(6.3.4.\(b\)])e(that)j Fy(H)31 b FI(belongs)21 b(to)j Fy(C)1302 277 y FK(1)p Fu(;)p FK(1)1392 307 y Fw(\(\010)1484 319 y Fu(j)1519 307 y Fy(;)14 b Fx(G)5 b Fy(;)14 b Fx(G)1701 277 y Fv(\003)1740 307 y Fw(\))p FI(,)24 b(where)e Fx(G)29 b FI(denotes)22 b(the)h(space)g Fx(D)r Fw(\()p Fy(H)7 b Fw(\))24 b FI(endo)n(wed)e(with)h(the)g(graph)83 407 y(topology)-5 b(.)29 b(In)i(particular)m(,)f Fy(H)39 b FI(belongs)30 b(to)i Fy(C)1455 377 y FK(1)1449 427 y Fu(u)1493 407 y Fw(\(\010)1585 419 y Fu(j)1620 407 y Fy(;)14 b Fx(G)5 b Fy(;)14 b Fx(G)1802 377 y Fv(\003)1840 407 y Fw(\))p FI(;)33 b(namely)-5 b(,)30 b(the)h(map)g Fq(R)45 b Fx(3)f Fy(\027)49 b Fx(7!)c Fy(H)7 b Fw(\()p Fy(\027)e(e)3143 419 y Fu(j)3178 407 y Fw(\))44 b Fx(2)h Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)3609 377 y Fv(\003)3648 407 y Fw(\))33 b FI(is)83 506 y(continuously)18 b(dif)n(ferentiable)g (in)i(the)g(uniform)e(topology)-5 b(.)18 b(Therefore)g(the)i(map)1171 687 y Fq(R)e Fx(n)g(f)p Fw(0)p Fx(g)k(3)h Fy(\027)29 b Fx(7!)1723 655 y FK(1)p 1721 669 38 4 v 1721 716 a Fu(\027)1768 687 y Fw([)p Fy(H)7 b Fw(\()p Fy(\027)e(e)1984 699 y Fu(j)2019 687 y Fw(\))19 b Fx(\000)f Fy(H)7 b Fw(])23 b Fx(2)g Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)2608 657 y Fv(\003)2648 687 y Fw(\))83 869 y FI(e)o(xtends)19 b(to)i(a)f(continuous)e(map)i(de\002ned)f(on)h Fq(R)h FI(and)e(taking)h(v)n(alue)f Fy(H)2141 838 y Fv(0)2134 890 y Fu(j)2190 869 y FI(at)i Fy(\027)28 b Fw(=)23 b(0)p FI(.)249 980 y(No)n(w)-5 b(,)20 b(the)g(e)o(xponential)e(map)h Fy(B)28 b Fx(7!)1334 977 y Fw(e)1370 949 y Fu(iB)1472 980 y FI(is)21 b(continuous)d(from)h Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)2370 949 y Fv(\003)2410 980 y Fw(\))21 b FI(to)f Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)2803 949 y Fv(\003)2843 980 y Fw(\))p FI(.)21 b(So,)f(the)g(composed)f(map) 1317 1174 y Fq(R)k Fx(3)g Fy(\027)29 b Fx(7!)1654 1171 y Fw(e)1706 1118 y Fm(i)p 1701 1127 33 3 v 1701 1160 a(\027)1743 1140 y FK([)p Fu(H)t FK(\()p Fu(\027)t(e)1914 1148 y Fm(j)1946 1140 y FK(\))p Fv(\000)p Fu(H)t FK(])2129 1174 y Fx(2)23 b Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)2462 1140 y Fv(\003)2502 1174 y Fw(\))83 1367 y FI(is)29 b(also)f (continuous,)e(and)i(tak)o(es)g(v)n(alue)1297 1364 y Fw(e)1333 1333 y Fu(iH)1414 1308 y Fl(0)1410 1350 y Fm(j)1475 1367 y FI(at)h Fy(\027)43 b Fw(=)37 b(0)p FI(.)28 b(By)g(linearity)g (and)f(by)h(taking)f(Lemma)g(2.4)h(into)g(account,)e(one)83 1466 y(\002nally)20 b(obtains)g(in)g Fo(B)s Fw(\()p Fx(G)5 b Fy(;)14 b Fx(G)917 1436 y Fv(\003)957 1466 y Fw(\))1439 1578 y(lim)1428 1632 y Fu(\027)t Fv(&)p FK(0)1578 1575 y Fw(e)1615 1543 y Fu(i)1649 1517 y Fm(\026)p 1649 1531 36 3 v 1650 1563 a(\027)1694 1543 y FK([)p Fu(H)t FK(\()p Fu(\027)t(x)p FK(\))p Fv(\000)p Fu(H)t FK(])2055 1578 y Fw(=)2143 1575 y(e)2179 1543 y Fu(i\026x)p Fv(\001)p Fu(H)2358 1518 y Fl(0)2400 1578 y Fy(:)83 1766 y FI(It)21 b(follo)n(ws)e(that)i(for)e(an)o(y)g Fy(';)14 b( )27 b Fx(2)c(G)5 b FI(,)21 b(one)f(has)1205 1956 y Fw(lim)1194 2011 y Fu(\027)t Fv(&)p FK(0)1345 1889 y Fp(\012)1384 1956 y Fy( )s(;)1478 1953 y Fw(e)1515 1922 y Fu(i)1548 1895 y Fm(\026)p 1548 1909 V 1549 1942 a(\027)1593 1922 y FK([)p Fu(H)t FK(\()p Fu(\027)t(x)p FK(\))p Fv(\000)p Fu(H)t FK(])1945 1956 y Fy(')1999 1889 y Fp(\013)2062 1956 y Fw(=)2149 1889 y Fp(\012)2189 1956 y Fy( )s(;)2283 1953 y Fw(e)2320 1922 y Fu(i\026x)p Fv(\001)p Fu(H)2499 1897 y Fl(0)2540 1956 y Fy(')2594 1889 y Fp(\013)2634 1956 y Fy(:)83 2182 y FI(In)h(f)o(act,)h(since)f(the)h(operators)e Fy(H)r(;)14 b(H)7 b Fw(\()p Fy(\027)e(x)p Fw(\))23 b FI(and)e Fy(H)1570 2152 y Fv(0)1563 2204 y Fu(j)1620 2182 y FI(are)g(self-adjoint)g(this)h(equality)e(e)n(v)o(en)h(holds)g (for)f Fy(';)14 b( )29 b Fx(2)d(H)q FI(,)c(b)n(ut)f(we)h(do)83 2282 y(not)c(need)g(such)h(an)f(e)o(xtension.)f(This)i(identity)-5 b(,)17 b(together)g(with)i(the)g(symmetry)e(of)h Fy(f)9 b FI(,)19 b(Lemma)f(4.2.\(a\),)e(and)i(Proposition)g(5.2,)83 2382 y(implies)i(that)h(for)e Fy(')k Fx(2)h Fo(D)831 2394 y FK(2)558 2607 y Fw(lim)534 2657 y Fu(r)r Fv(!1)722 2574 y FK(1)p 722 2588 34 4 v 722 2635 a(2)779 2494 y Fp(Z)862 2514 y Fv(1)825 2682 y FK(0)946 2607 y Fw(d)p Fy(t)1036 2540 y Fp(\012)1076 2607 y Fy(';)1167 2540 y Fp(\002)1215 2604 y Fw(e)1252 2572 y Fv(\000)p Fu(itH)1429 2607 y Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))1698 2604 y(e)1736 2572 y Fu(itH)1861 2607 y Fx(\000)1940 2604 y Fw(e)1976 2572 y Fu(itH)2102 2607 y Fy(f)g Fw(\(\010)p Fy(=r)r Fw(\))2371 2604 y(e)2408 2572 y Fv(\000)p Fu(itH)2585 2540 y Fp(\003)2620 2607 y Fy(')2674 2540 y Fp(\013)543 2833 y Fw(=)22 b Fx(\000)710 2800 y Fu(i)p 705 2814 V 705 2862 a FK(2)762 2720 y Fp(Z)845 2741 y Fv(1)808 2909 y FK(0)929 2833 y Fw(d)p Fy(\026)1053 2720 y Fp(Z)1099 2909 y Fn(R)1141 2892 y Fm(d)1194 2833 y Fw(d)p 1194 2846 47 4 v Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1611 2766 y Fp(\010)s(\012)1715 2833 y Fw(\()p Fy(x)19 b Fx(\001)f Fw(\010\))c Fy(';)2051 2830 y Fw(e)2088 2799 y Fu(i\026x)p Fv(\001)p Fu(H)2267 2774 y Fl(0)2309 2833 y Fy(')2363 2766 y Fp(\013)2421 2833 y Fx(\000)2504 2766 y Fp(\012)2543 2833 y Fy(';)2634 2830 y Fw(e)2671 2799 y Fv(\000)p Fu(i\026x)p Fv(\001)p Fu(H)2902 2774 y Fl(0)2944 2833 y Fw(\()p Fy(x)19 b Fx(\001)f Fw(\010\))c Fy(')3243 2766 y Fp(\013)q(\011)543 3059 y Fw(=)22 b Fx(\000)705 3027 y FK(1)p 705 3041 34 4 v 705 3088 a(2)762 2981 y Fp(X)806 3157 y Fu(j)896 2946 y Fp(Z)979 2967 y Fv(1)942 3135 y FK(0)1063 3059 y Fw(d)p Fy(\026)1187 2946 y Fp(Z)1233 3135 y Fn(R)1275 3119 y Fm(d)1328 3059 y Fw(d)p 1328 3072 47 4 v Fy(x)14 b Fw([)p Fo(F)d 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Fy(x)14 b(K)6 b Fw(\()p Fy(\027)q(;)14 b(\026;)g(x)p Fw(\))p FI(:)454 4070 y Fy(K)525 4082 y FK(1)562 4070 y Fw(\()p Fy(\027)q(;)g(\026)p Fw(\))23 b(:=)889 3957 y Fp(Z)935 4146 y Fn(R)977 4129 y Fm(d)1030 4070 y Fw(d)p 1030 4083 V Fy(x)14 b Fw(\()p Fo(F)d Fy(f)e Fw(\)\()p Fy(x)p Fw(\))1461 4003 y Fp(\012)1502 4070 y Fx(h)p Fw(\010)p Fx(i)1626 4036 y FK(2)1664 4070 y Fy(';)1767 4037 y FK(1)p 1765 4051 38 4 v 1765 4099 a Fu(\027)1812 4003 y Fp(\000)1864 4067 y Fw(e)1901 4040 y Fu(i\027)t(x)p Fv(\001)p FK(\010)2084 4070 y Fx(\000)p Fw(1)2191 4003 y Fp(\001)2229 4070 y Fx(h)p Fw(\010)p Fx(i)2353 4040 y Fv(\000)p FK(2)2442 4070 y Fy(\021)s Fw(\()p Fy(H)e Fw(\()p Fy(\027)e(x)p Fw(\)\))2797 4067 y(e)2836 4040 y Fu(i)2869 4013 y Fm(\026)p 2869 4027 36 3 v 2870 4060 a(\027)2915 4040 y FK([)p Fu(H)t FK(\()p Fu(\027)t(x)p FK(\))p Fv(\000)p Fu(H)t FK(])3266 4070 y Fy(')3320 4003 y Fp(\013)3374 4070 y Fy(:)83 4306 y FI(Observ)o(e)19 b(that)h(for)g(each)g(multi-inde)o(x)e Fy(\013)23 b Fx(2)h 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Fp(\012)1928 2552 y Fx(h)p Fw(\010)p Fx(i)2052 2517 y FK(2)2090 2552 y Fy(';)14 b(A)2243 2564 y Fu(j;\027)2332 2552 y Fw(\()p Fy(x)p Fw(\))2443 2484 y Fp(\000)2482 2552 y Fy(@)2526 2564 y Fu(j)2561 2552 y Fy(B)2628 2517 y Fu(J)2624 2572 y(\027;\026)2722 2484 y Fp(\001)2760 2552 y Fw(\()p Fy(x)p Fw(\))p Fy(')2925 2484 y Fp(\013)2979 2552 y Fy(:)83 2831 y FI(Moreo)o(v)o(er)m(,)i(direct)i(calculations)g(using)g (Equation)g(\(5.6\))f(and)h(Proposition)g(5.1)g(sho)n(w)g(that)h(the)g (map)f Fq(R)3110 2801 y Fu(d)3172 2831 y Fx(3)23 b Fy(x)h Fx(7!)f Fy(A)3489 2843 y Fu(j;\027)3578 2831 y Fw(\()p Fy(x)p Fw(\))h Fx(2)83 2930 y Fo(B)s Fw(\()p Fx(H)q Fw(\))e FI(is)f(twice)g(strongly)e(dif)n(ferentiable)f(and)h(satis\002es)1396 3042 y Fp(\015)1396 3092 y(\015)1442 3113 y Fw(\()p Fy(@)1518 3125 y Fu(j)1554 3113 y Fy(A)1616 3125 y Fu(j;\027)1704 3113 y Fw(\)\()p Fy(x)p Fw(\))1847 3042 y Fp(\015)1847 3092 y(\015)1918 3113 y Fx(\024)j Fw(Const)p Fy(:)9 b Fx(h)p Fy(x)p Fx(i)2361 3078 y Fv(\000)p 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b(=)g Fy(i\026)951 3828 y Fv(\000)p FK(1)1054 3783 y Fp(X)1098 3960 y Fu(j)1187 3749 y Fp(Z)1233 3938 y Fn(R)1275 3921 y Fm(d)1328 3862 y Fw(d)p 1328 3875 V Fy(x)1435 3795 y Fp(\012)1475 3862 y Fx(h)p Fw(\010)p Fx(i)1599 3828 y FK(2)1637 3862 y Fy(';)14 b Fw(\()p Fy(@)1804 3874 y Fu(j)1839 3862 y Fy(A)1901 3874 y Fu(j;\027)1990 3862 y Fw(\)\()p Fy(x)p Fw(\))p Fy(B)2200 3828 y Fu(J)2196 3883 y(\027;\026)2295 3862 y Fw(\()p Fy(x)p Fw(\))p Fy(')2460 3795 y Fp(\013)784 4126 y Fw(=)23 b Fx(\000)p Fy(\026)987 4092 y Fv(\000)p FK(2)1090 4047 y Fp(X)1112 4226 y Fu(j;`)1223 4013 y Fp(Z)1269 4202 y Fn(R)1311 4185 y Fm(d)1364 4126 y Fw(d)p 1364 4139 V Fy(x)1471 4059 y Fp(\012)1511 4126 y Fx(h)p Fw(\010)p Fx(i)1635 4092 y FK(2)1673 4126 y Fy(';)14 b(@)1808 4138 y Fu(`)1840 4059 y Fp(\010)1888 4126 y Fw(\()p Fy(@)1964 4138 y Fu(j)2000 4126 y Fy(A)2062 4138 y Fu(j;\027)2150 4126 y Fw(\))p Fy(H)2258 4092 y Fv(0)2251 4146 y Fu(`)2283 4126 y Fw(\()p Fy(\027)h Fx(\001)9 b Fw(\)\()p Fy(H)2543 4092 y 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Fp(\000)279 4857 y Fw(\(0)p Fy(;)c Fx(1)p Fw(\))p Fy(;)g Fw(d)p Fy(\026)638 4790 y Fp(\001)676 4857 y FI(.)19 b(Since)f(similar)h(ar)o(guments)d (sho)n(ws)i(that)g(the)h(same)f(holds)g(for)f(the)i(second)e(term)h(of) 3152 4790 y Fp(R)3191 4887 y Fn(R)3233 4870 y Fm(d)3286 4857 y Fw(d)p 3286 4870 V Fy(x)c(K)6 b Fw(\()p Fy(\027)q(;)14 b(\026;)g(x)p Fw(\))p FI(,)83 4957 y(one)20 b(can)g(interchange)e(the)i (limit)h Fy(\027)28 b Fx(&)23 b Fw(0)d FI(and)g(the)g(inte)o(gration)e (o)o(v)o(er)h Fy(\026)i FI(in)f(\(5.5\))n(.)249 5056 y(The)g(interchange)e(of)i(the)g(limit)h Fy(\027)28 b Fx(&)23 b Fw(0)d FI(and)g(the)g(inte)o(gration)f(o)o(v)o(er)f Fy(x)k FI(in)e(\(5.5\))f(is)i(justi\002ed)f(by)g(the)g(bound)1322 5168 y Fp(\014)1322 5218 y(\014)1350 5239 y Fy(K)6 b Fw(\()p Fy(\027)q(;)14 b(\026;)g(x)p Fw(\))1704 5168 y Fp(\014)1704 5218 y(\014)1754 5239 y Fx(\024)23 b Fw(Const)p Fy(:)2091 5168 y Fp(\014)2091 5218 y(\014)2119 5239 y Fy(x)p Fw(\()p Fo(F)11 b Fy(f)e Fw(\)\()p Fy(x)p Fw(\))2476 5168 y Fp(\014)2476 5218 y(\014)2506 5239 y Fy(;)83 5421 y FI(which)20 b(follo)n(ws)f(from)h(F)o(ormula)e(\(5.6\))o(.)p 3708 5421 4 57 v 3712 5368 50 4 v 3712 5421 V 3761 5421 4 57 v Black 1884 5670 a(16)p Black eop end %%Page: 17 17 TeXDict begin 17 16 bop Black Black 249 307 a FI(When)19 b(the)g(localisation)f(function)g Fy(f)28 b FI(is)20 b(radial,)e(the)h(operator)e Fy(T)2126 319 y Fu(f)2189 307 y FI(is)j(equal)e(to)h(the)g(operator)e Fy(T)12 b FI(,)19 b(which)f(is)i(independent)83 407 y(of)d Fy(f)9 b FI(.)18 b(The)f(ne)o(xt)g(result,)h(which)f(depicts)g(this)h (situation)g(of)f(particular)f(interest,)i(is)g(a)g(direct)g (consequence)d(of)j(Lemma)e(4.2.\(c\))83 506 y(and)k(Theorem)e(5.5)p Black 83 672 a FA(Cor)o(ollary)k(5.6.)p Black 43 w FJ(Let)j Fy(H)32 b FJ(and)23 b Fw(\010)i FJ(satisfy)g(Assumptions)f(2.2)f(and)h (2.3.)f(Let)i Fy(f)40 b Fx(2)31 b Fo(S)16 b Fw(\()p Fq(R)2674 642 y Fu(d)2713 672 y Fw(\))25 b FJ(be)f(a)g(r)o(adial)g(function)f (suc)o(h)h(that)83 772 y Fy(f)32 b Fw(=)22 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y Fu(f)3178 3613 y Fy(')p Fx(i)p FI(,)21 b(with)f Fy(T)3523 3625 y Fu(f)3587 3613 y FI(gi)n(v)o(en)83 3712 y(by)27 b(\(5.1\))n(.)g(No)n(w)-5 b(,)27 b(a)g(direct)g (calculation)f(using)h(F)o(ormulas)f(\(4.1\))n(,)h(\(4.2\))n(,)h(and)e (\(5.1\))g(sho)n(ws)h(that)h(the)f(operators)e Fy(T)3474 3724 y Fu(f)3545 3712 y FI(and)h Fy(H)83 3812 y FI(satisfy)21 b(in)f(the)g(form)f(sense)i(on)f Fo(D)1079 3824 y FK(1)1137 3812 y FI(the)g(canonical)f(commutation)f(relation)1708 3927 y Fp(\002)1742 3994 y Fy(T)1791 4006 y Fu(f)1834 3994 y Fy(;)c(H)1947 3927 y Fp(\003)2004 3994 y Fw(=)23 b Fy(i:)1465 b FI(\(6.1\))83 4177 y(Therefore,)21 b(since)i(the)g (group)e Fx(f)1033 4174 y Fw(e)1069 4147 y Fv(\000)p Fu(itH)1233 4177 y Fx(g)1275 4189 y Fu(t)p Fv(2)p Fn(R)1414 4177 y FI(lea)n(v)o(es)i Fo(D)1706 4189 y FK(1)1767 4177 y FI(in)m(v)n(ariant,)e(the)i(follo)n(wing)f(equalities)g(hold)h(in)g (the)g(form)f(sense)h(on)83 4277 y Fo(D)147 4289 y FK(1)185 4277 y FI(:)163 4509 y Fy(T)212 4521 y Fu(f)269 4506 y Fw(e)306 4474 y Fv(\000)p Fu(itH)492 4509 y Fw(=)580 4506 y(e)617 4474 y Fv(\000)p Fu(itH)794 4509 y Fy(T)843 4521 y Fu(f)904 4509 y Fw(+)987 4441 y Fp(\002)1021 4509 y Fy(T)1070 4521 y Fu(f)1113 4509 y Fy(;)1150 4506 y Fw(e)1187 4474 y Fv(\000)p Fu(itH)1364 4441 y Fp(\003)1422 4509 y Fw(=)1509 4506 y(e)1546 4474 y Fv(\000)p Fu(itH)1723 4509 y Fy(T)1772 4521 y Fu(f)1833 4509 y Fx(\000)c Fy(i)1960 4396 y Fp(Z)2042 4416 y Fu(t)2005 4584 y FK(0)2085 4509 y Fw(d)p Fy(s)2198 4506 y Fw(e)2235 4474 y Fv(\000)p Fu(i)p FK(\()p Fu(t)p Fv(\000)p Fu(s)p FK(\))p Fu(H)2547 4441 y Fp(\002)2582 4509 y Fy(T)2631 4521 y Fu(f)2673 4509 y Fy(;)14 b(H)2786 4441 y Fp(\003)2834 4506 y Fw(e)2871 4474 y Fv(\000)p Fu(isH)3064 4509 y Fw(=)3152 4506 y(e)3188 4479 y Fv(\000)p Fu(itH)3366 4441 y Fp(\000)3404 4509 y Fy(T)3453 4521 y Fu(f)3514 4509 y Fw(+)k Fy(t)3627 4441 y Fp(\001)3665 4509 y Fy(:)83 4732 y FI(In)i(other)f(terms,)h(one) g(has)1227 4765 y Fp(\012)1266 4832 y Fy( )s(;)14 b(T)1409 4844 y Fu(f)1466 4829 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Fy(T)1207 5375 y Fu(f)1263 5360 y Fw(e)1300 5333 y Fv(\000)p Fu(itH)1477 5363 y Fy(')k Fw(=)1642 5360 y(e)1679 5333 y Fv(\000)p Fu(itH)1857 5296 y Fp(\000)1895 5363 y Fy(T)1944 5375 y Fu(f)2005 5363 y Fw(+)18 b Fy(t)2118 5296 y Fp(\001)2156 5363 y Fy(';)180 b(')23 b Fx(2)h Fo(D)2633 5375 y FK(1)2670 5363 y Fy(:)916 b FI(\(6.3\))p Black 1884 5670 a(17)p Black eop end %%Page: 18 18 TeXDict begin 18 17 bop Black Black 83 307 a FI(This)20 b(relation,)f(also)h(kno)n(wn)e(as)i Fy(T)1082 319 y Fu(f)1139 307 y FI(-weak)f(W)-7 b(e)o(yl)20 b(relation)f([19)o(,)h (Def.)g(1.1],)e(has)i(deep)f(implications)g(on)g(the)h(spectral)g (nature)83 407 y(of)i Fy(H)29 b FI(and)21 b(on)h(the)g(form)f(of)h Fy(T)970 419 y Fu(f)1035 407 y FI(in)g(the)g(spectral)g(representation) d(of)j Fy(H)7 b FI(.)22 b(F)o(ormally)-5 b(,)20 b(it)j(suggests)f(that) g Fy(T)3140 419 y Fu(f)3209 407 y Fw(=)k Fy(i)3368 374 y FK(d)p 3338 388 96 4 v 3338 435 a(d)p Fu(H)3444 407 y FI(,)c(and)g(thus)83 506 y Fx(\000)p Fy(iT)226 518 y 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Fu(\025)179 2424 y FI(\).)249 2524 y FA(Case)k(2)f(\()p Fi(T)586 2536 y Fh(f)656 2524 y FA(symmetric\):)f FI(If)i(the)f(set)i Fo(D)1486 2536 y FK(1)1545 2524 y FI(is)g(dense)e(in)h Fx(H)q FI(,)f(then)h(we)f(kno)n(w)g(from)f(Proposition)g(5.2)h(and)g (Remark)g(5.4)83 2624 y(that)j Fy(T)281 2636 y Fu(f)348 2624 y FI(is)g(symmetric.)f(In)h(such)f(a)h(situation,)f(\(6.3\))g (once)g(more)g(implies)h(that)g(the)f(spectrum)g(of)h Fy(H)31 b FI(is)24 b(purely)f(absolutely)83 2723 y(continuous)18 b([19)n(,)i(Thm.)f(4.4],)g(b)n(ut)g(it)i(may)e(not)g(co)o(v)o(er)g(the) g(whole)h(real)f(line.)h(W)-7 b(e)21 b(e)o(xpect)d(that)i(the)g (operator)e Fy(T)3305 2735 y Fu(f)3368 2723 y FI(is)i(still)h(equal)83 2823 y(to)g Fy(i)227 2790 y FK(d)p 207 2804 V 207 2852 a(d)p Fu(\025)314 2823 y FI(\(on)f(a)h(suitable)g(subspace\))e(in)i (the)f(spectral)h(representation)d(of)i Fy(H)7 b FI(,)21 b(b)n(ut)f(we)h(ha)n(v)o(e)f(not)g(been)g(able)h(to)f(pro)o(v)o(e)f(it) i(in)g(this)83 2923 y(generality)-5 b(.)25 b(Ho)n(we)n(v)o(er)m(,)g (this)i(property)e(holds)h(in)h(most)g(of)g(the)g(e)o(xamples)f (presented)f(belo)n(w)-5 b(.)26 b(If)h Fy(T)3033 2935 y Fu(f)3103 2923 y FI(and)f Fy(H)34 b FI(satisfy)27 b(more)83 3022 y(assumptions,)19 b(then)h(more)f(can)h(be)g(said)h(\(see)f(for)g (instance)g([33)n(]\).)249 3122 y FA(Case)g(3)g(\()p Fi(T)583 3134 y Fh(f)650 3122 y FA(not)g(densely)g(de\002ned\):)g FI(If)g Fo(D)1543 3134 y FK(1)1601 3122 y FI(is)g(not)g(dense)f(in)h Fx(H)q FI(,)g(then)g(we)g(are)f(not)h(a)o(w)o(are)f(of)h(general)f(w)o (orks)g(using)g(a)83 3221 y(relation)d(lik)o(e)h(\(6.2\))f(to)h(deduce) e(results)j(on)e(the)h(spectral)g(nature)f(of)g Fy(H)25 b FI(or)16 b(on)h(the)g(form)e(of)i Fy(T)2761 3233 y Fu(f)2821 3221 y FI(in)g(the)g(spectral)g(representation)83 3321 y(of)22 b Fy(H)7 b FI(.)22 b(In)g(such)h(a)f(case,)h(we)f(only)g (kno)n(w)f(from)g(Theorem)g(3.6)h(that)g(the)g(spectrum)g(of)g Fy(H)29 b FI(is)24 b(purely)d(absolutely)g(continuous)83 3421 y(in)27 b Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))24 b Fx(n)f Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FI(,)27 b(b)n(ut)g(we)g(ha)n (v)o(e)f(no)g(general)g(information)e(on)j(the)f(form)g(of)g Fy(T)2501 3433 y Fu(f)2571 3421 y FI(in)h(the)g(spectral)f (representation)f(of)i Fy(H)7 b FI(.)83 3520 y(Ho)n(we)n(v)o(er)m(,)20 b(with)i(a)g(suitable)g(additional)f(assumption)g(the)h(analysis)g(can) g(be)g(continued.)d(Indeed,)h(consider)h(the)h(orthogonal)83 3620 y(decomposition)h Fx(H)34 b Fw(:=)f Fx(K)24 b(\010)e(G)5 b FI(,)26 b(with)g Fx(K)35 b(\021)d Fw(k)n(er)1592 3553 y Fp(\000)1630 3620 y Fw(\()p Fy(H)1738 3590 y Fv(0)1762 3620 y Fw(\))1794 3590 y FK(2)1831 3553 y Fp(\001)1869 3620 y FI(.)26 b(Then)f(the)h(operators)e Fy(H)7 b FI(,)25 b Fy(H)2774 3590 y Fv(0)2767 3642 y Fu(j)2802 3620 y FI(,)h(and)f Fy(H)3071 3590 y Fv(00)3064 3644 y Fu(k)q(`)3159 3620 y FI(are)h(all)g(reduced)e(by)83 3720 y(this)19 b(decomposition,)d(due)i(to)h(the)f(commutation)f(assumption)g(2.3.)h (If)g(we)h(assume)g(additionally)e(that)h Fy(T)3155 3732 y Fu(f)3198 3720 y Fo(D)3262 3732 y FK(1)3323 3720 y Fx(\032)k(G)5 b FI(,)19 b(then)g(the)83 3819 y(analysis)h(can)g(be)g (performed)e(in)i(the)g(subspace)g Fx(G)5 b FI(.)249 3919 y(Since)25 b Fo(D)523 3931 y FK(1)592 3919 y Fx(\032)31 b(G)f FI(by)24 b(Lemma)g(6.1,)g(the)h(additional)e(hypothesis)g(allo)n (ws)i(us)g(to)g(consider)e(the)i(restriction)f(of)g Fy(T)3535 3931 y Fu(f)3603 3919 y FI(to)h Fx(G)5 b FI(,)83 4019 y(which)19 b(we)g(denote)f(by)g Fr(T)811 4031 y Fu(f)854 4019 y FI(.)h(Let)g(also)h Fr(H)p FI(,)f Fr(H)1336 3988 y Fv(0)1336 4040 y Fu(j)1370 4019 y FI(,)h(and)e Fr(H)1609 3988 y Fv(00)1609 4042 y Fu(k)q(`)1697 4019 y FI(denote)g(the)h (restrictions)g(of)g(the)g(corresponding)c(operators)j(in)h Fx(G)5 b FI(.)19 b(W)-7 b(e)83 4118 y(then)20 b(set)641 4218 y Fr(D)701 4230 y Fu(t)753 4218 y Fw(:=)864 4151 y Fp(\010)912 4218 y Fy(')k Fx(2)f(D)r Fw(\()p Fx(h)p Fw(\010)p Fx(i)1290 4183 y Fu(t)1320 4218 y Fw(\))c Fx(\\)g(G)28 b(j)c Fy(')f Fw(=)g Fy(\021)s Fw(\()p Fr(H)p Fw(\))p Fy(')e FI(for)f(some)g Fy(\021)26 b Fx(2)d Fy(C)2500 4183 y Fv(1)2494 4238 y FK(c)2571 4151 y Fp(\000)2609 4218 y Fq(R)18 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))2935 4151 y Fp(\001)q(\011)3045 4218 y Fx(\032)23 b(G)5 b Fy(;)83 4363 y FI(and)19 b(observ)o(e)f(that)i(the)g(equality)f (\(6.1\))f(holds)h(in)h(the)g(form)f(sense)h(on)f Fr(D)2186 4375 y FK(1)2224 4363 y FI(.)h(In)f(other)g(w)o(ords,)g(\(6.1\))g(can)g (be)h(considered)e(in)i(the)83 4463 y(reduced)g(Hilbert)i(space)g Fx(G)27 b FI(instead)22 b(of)g Fx(H)q FI(.)g(The)f(interest)h(of)g(the) f(abo)o(v)o(e)g(decomposition)e(comes)i(from)g(the)h(follo)n(wing)e(f)o (act:)83 4563 y(If)k Fr(D)223 4575 y FK(1)284 4563 y FI(is)h(dense)e(in)h Fx(G)29 b FI(\(which)23 b(is)i(certainly)d(more)h (lik)o(ely)h(than)f(in)h Fx(H)q FI(\),)f(then)h Fr(T)2414 4575 y Fu(f)2480 4563 y FI(is)h(symmetric)d(and)i(the)f(situation)h (reduces)83 4662 y(to)f(the)f(case)h Fw(2)f FI(with)h(the)f(operators)f Fr(H)i FI(and)f Fr(T)1423 4674 y Fu(f)1465 4662 y FI(.)h(If)f(in)h (addition)e Fr(T)2016 4674 y Fu(f)2081 4662 y FI(is)j(essentially)e (self-adjoint)f(on)h Fr(D)3100 4674 y FK(1)3137 4662 y FI(,)h(the)g(situation)f(e)n(v)o(en)83 4762 y(reduces)j(to)i(the)f (case)h Fw(1)f FI(with)h(the)f(operators)f Fr(H)h FI(and)g Fr(T)1738 4774 y Fu(f)1780 4762 y FI(.)h(In)f(both)f(situations,)h(the) h(spectrum)e(of)h Fr(H)h FI(is)g(purely)e(absolutely)83 4862 y(continuous.)18 b(In)i(Section)g(7,)g(we)g(shall)h(present)e Fw(2)i FI(e)o(xamples)e(corresponding)d(to)21 b(these)f(situations.)p Black 83 5023 a FA(Remark)25 b(6.2.)p Black 43 w FI(The)g(implicit)f (condition)g Fy(T)1395 5035 y Fu(f)1437 5023 y Fo(D)1501 5035 y FK(1)1570 5023 y Fx(\032)32 b(G)f FI(can)24 b(be)h(made)f(more)g (e)o(xplicit.)g(F)o(or)h(e)o(xample,)e(if)i(the)g(collection)f Fw(\010)83 5122 y FI(is)29 b(reduced)e(by)h(the)g(decomposition)e Fx(H)39 b Fw(=)f Fx(K)26 b(\010)e(G)5 b FI(,)29 b(then)e(the)i (condition)d(holds)i(\(and)f(\(5.3\))g(also)i(holds)f(on)g Fr(D)3474 5134 y FK(2)3511 5122 y FI(\).)g(More)83 5222 y(generally)-5 b(,)23 b(if)i Fw(\010)569 5234 y Fu(j)604 5222 y Fo(D)668 5234 y FK(1)737 5222 y Fx(\032)31 b(G)f FI(for)24 b(each)h Fy(j)5 b FI(,)25 b(then)f(the)h(condition)e(holds.)h (Indeed,)f(if)i Fy(')32 b Fx(2)f Fo(D)2741 5234 y FK(1)2804 5222 y FI(one)24 b(kno)n(ws)g(from)g(Remark)g(5.4)83 5321 y(that)g Fw(\()p Fy(@)308 5333 y Fu(j)343 5321 y Fy(R)406 5333 y Fu(f)450 5321 y Fw(\)\()p Fy(H)590 5291 y Fv(0)613 5321 y Fw(\))p Fy(')31 b Fx(2)f(D)r Fw(\()p Fx(h)p Fw(\010)p Fx(i)p Fw(\))p FI(,)c(and)d(one)h(can)g(pro)o(v)o(e)e (similarly)h(that)i Fx(j)p Fy(H)2324 5291 y Fv(0)2347 5321 y Fx(j)2370 5291 y Fv(\000)p FK(1)2459 5321 y Fy(')30 b Fx(2)h(D)r Fw(\()p Fx(h)p Fw(\010)p Fx(i)p Fw(\))p FI(.)25 b(Furthermore,)d(there)h(e)o(xists)83 5421 y Fy(\021)42 b Fx(2)c Fy(C)324 5391 y Fv(1)318 5442 y FK(c)395 5354 y Fp(\000)433 5421 y Fq(R)24 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))771 5354 y Fp(\001)839 5421 y FI(such)28 b(that)g Fw(\()p Fy(@)1249 5433 y Fu(j)1285 5421 y Fy(R)1348 5433 y Fu(f)1391 5421 y Fw(\)\()p Fy(H)1531 5391 y Fv(0)1554 5421 y Fw(\))p Fy(')39 b Fw(=)f Fy(\021)s Fw(\()p Fy(H)7 b Fw(\)\()p Fy(@)2042 5433 y Fu(j)2078 5421 y Fy(R)2141 5433 y Fu(f)2184 5421 y Fw(\)\()p Fy(H)2324 5391 y Fv(0)2348 5421 y Fw(\))p Fy(')29 b FI(and)f Fx(j)p Fy(H)2711 5391 y Fv(0)2734 5421 y Fx(j)2757 5391 y Fv(\000)p FK(1)2847 5421 y Fy(')38 b Fw(=)g Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fx(j)p Fy(H)3325 5391 y Fv(0)3349 5421 y Fx(j)3372 5391 y Fv(\000)p FK(1)3461 5421 y Fy(')p FI(,)29 b(which)p Black 1884 5670 a(18)p Black eop end %%Page: 19 19 TeXDict begin 19 18 bop Black Black 83 307 a FI(means)21 b(that)g(both)g(v)o(ectors)f Fy(@)937 319 y Fu(j)972 307 y Fy(R)1035 319 y Fu(f)1078 307 y Fw(\()p Fy(H)1186 277 y Fv(0)1209 307 y Fw(\))p Fy(')j FI(and)d Fx(j)p Fy(H)1558 277 y Fv(0)1582 307 y Fx(j)1605 277 y Fv(\000)p FK(1)1694 307 y Fy(')i FI(belong)e(to)h Fo(D)2168 319 y FK(1)2206 307 y FI(.)g(It)h(follo)n(ws)e(that)i Fy(T)2785 319 y Fu(f)2827 307 y Fy(')j Fx(2)h(G)h FI(by)21 b(taking)f(the)h(e)o (xplicit)83 407 y(form)e(\(5.1\))g(of)h Fy(T)584 419 y Fu(f)647 407 y FI(into)g(account.)249 573 y(Let)i(us)f(no)n(w)g (concentrate)f(on)h(the)g(other)g(term)g(in)g(F)o(ormula)f(\(5.3\))n(.) i(If)f(we)h(consider)e(the)h(operators)f Fw(\010)3243 585 y Fu(j)3300 573 y FI(as)i(the)f(compo-)83 672 y(nents)f(of)f(an)h (abstract)f(position)g(operator)f Fw(\010)p FI(,)i(then)f(the)h(l.h.s.) g(of)f(F)o(ormula)g(\(5.3\))f(has)i(the)g(follo)n(wing)e(meaning:)g(F)o (or)h Fy(r)24 b FI(\002x)o(ed,)83 772 y(it)i(can)e(be)h(interpreted)e (as)j(the)e(dif)n(ference)f(of)i(times)g(spent)g(by)f(the)h(e)n(v)n (olving)e(state)2591 769 y Fw(e)2628 742 y Fv(\000)p Fu(itH)2805 772 y Fy(')j FI(in)f(the)g(past)g(\(\002rst)g(term\))f(and) 83 872 y(in)f(the)f(future)f(\(second)g(term\))h(within)g(the)h(re)o (gion)e(de\002ned)g(by)h(the)h(localisation)e(operator)g Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))p FI(.)23 b(Thus,)f(F)o(ormula)f (\(5.3\))83 971 y(sho)n(ws)f(that)h(this)f(dif)n(ference)e(of)i(times)h (tends)f(as)h Fy(r)26 b Fx(!)d(1)e FI(to)f(the)g(e)o(xpectation)e(v)n (alue)i(in)g Fy(')h FI(of)f(the)g(operator)f Fy(T)3327 983 y Fu(f)3369 971 y FI(.)249 1071 y(On)h(the)f(other)g(hand,)f(let)i (us)g(consider)e(a)i(quantum)d(scattering)i(pair)g Fx(f)p Fy(H)r(;)14 b(H)22 b Fw(+)16 b Fy(V)j Fx(g)p FI(,)g(with)g Fy(V)39 b FI(an)20 b(appropriate)d(perturba-)83 1171 y(tion)i(of)g Fy(H)7 b FI(.)19 b(Let)g(us)g(also)h(assume)f(that)g(the) g(corresponding)c(scattering)k(operator)e Fy(S)24 b FI(is)c(unitary)-5 b(,)17 b(and)i(recall)g(that)g Fy(S)24 b FI(commute)83 1270 y(with)h Fy(H)7 b FI(.)25 b(In)f(this)i(frame)n(w)o(ork,)c(the)j (global)f(time)h(delay)f Fy(\034)9 b Fw(\()p Fy(')p Fw(\))27 b FI(for)d(the)g(state)i Fy(')g FI(de\002ned)d(in)i(terms)g(of)f(the)h (localisation)f(op-)83 1370 y(erators)f Fy(f)9 b Fw(\(\010)p Fy(=r)r Fw(\))25 b FI(can)e(usually)g(be)g(ree)o(xpressed)f(as)i(follo) n(ws:)g(it)g(is)g(equal)f(to)h(the)g(l.h.s.)f(of)g(\(5.3\))f(minus)i (the)f(same)h(quantity)83 1469 y(with)f Fy(')h FI(replaced)e(by)g Fy(S)5 b(')p FI(.)24 b(Therefore,)c(if)k Fy(')g FI(and)e Fy(S)5 b(')24 b FI(are)f(elements)f(of)h Fo(D)2299 1481 y FK(2)2337 1469 y FI(,)g(then)g(the)g(time)g(delay)f(for)h(the)g (scattering)f(pair)83 1569 y Fx(f)p Fy(H)r(;)14 b(H)25 b Fw(+)18 b Fy(V)h Fx(g)h FI(should)f(satisfy)i(the)f(equation)1477 1752 y Fy(\034)9 b Fw(\()p Fy(')p Fw(\))25 b(=)d Fx(\000h)p Fy(';)14 b(S)1996 1717 y Fv(\003)2034 1752 y Fw([)p Fy(T)2106 1764 y Fu(f)2149 1752 y Fy(;)g(S)5 b Fw(])p Fy(')p Fx(i)p Fy(:)1235 b FI(\(6.4\))83 1934 y(In)19 b(addition,)e(if)i Fy(T)603 1946 y Fu(f)666 1934 y FI(acts)g(in)g(the)g(spectral)g (representation)e(of)i Fy(H)26 b FI(as)20 b(a)f(dif)n(ferential)e (operator)g Fy(i)2866 1902 y FK(d)p 2837 1916 96 4 v 2837 1963 a(d)p Fu(H)2943 1934 y FI(,)i(then)f Fy(\034)9 b Fw(\()p Fy(')p Fw(\))21 b FI(w)o(ould)e(v)o(erify)-5 b(,)83 2034 y(in)20 b(our)g(complete)f(abstract)h(setting,)g(the)g (Eisenb)n(ud-W)m(igner)d(formula)1509 2217 y Fy(\034)9 b Fw(\()p Fy(')p Fw(\))24 b(=)1784 2149 y Fp(\012)1823 2217 y Fy(';)14 b Fx(\000)p Fy(iS)2064 2187 y Fv(\003)2119 2184 y FK(d)p Fu(S)p 2111 2198 V 2111 2245 a FK(d)p Fu(H)2226 2217 y Fy(')2280 2149 y Fp(\013)2319 2217 y Fy(:)249 2399 y FI(Summing)29 b(up,)g(as)i(soon)e(as)i(the)f(position)f (operator)f Fw(\010)i FI(and)g(the)g(operator)e Fy(H)37 b FI(satisfy)31 b(Assumptions)e(2.2)g(and)h(2.3,)83 2499 y(then)c(our)g(study)g(establishes)h(a)g(preliminary)e(relation)h (between)g(time)h(operators)e Fy(T)2609 2511 y Fu(f)2679 2499 y FI(gi)n(v)o(en)g(by)h(\(5.1\))g(and)g(the)h(theory)e(of)83 2599 y(quantum)h(time)i(delay)-5 b(.)27 b(Man)o(y)g(concrete)f(e)o (xamples)h(discussed)h(in)g(the)g(literature)f([2)o(,)h(3,)g(4,)g(15)o (,)g(20)o(,)g(35)o(,)h(37)o(])f(turn)f(out)g(to)83 2698 y(\002t)f(in)g(the)g(present)f(frame)n(w)o(ork,)e(and)i(se)n(v)o(eral)g (old)h(or)f(ne)n(w)g(e)o(xamples)g(are)h(presented)e(in)i(the)f(follo)n (wing)f(section.)i(Further)83 2798 y(in)m(v)o(estigations)18 b(in)i(relation)g(with)g(the)g(abstract)g(F)o(ormula)f(\(6.4\))g(will)i (be)f(considered)e(else)n(where.)249 2897 y(No)n(w)-5 b(,)22 b(most)g(of)g(the)g(abo)o(v)o(e)f(discussion)h(depends)e(on)i (the)h(size)g(of)f Fo(D)2268 2909 y FK(1)2328 2897 y FI(in)g Fx(H)q FI(,)h(and)f(implicitly)f(on)h(the)h(size)f(of)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))23 b FI(in)83 2997 y Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))p FI(.)23 b(W)-7 b(e)24 b(collect)e(some)g(information)e(about)h(these)h(sets.)h(It)g(has)f (been)g(pro)o(v)o(ed)e(in)i(Lemma)f(2.6.\(d\))g(that)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))23 b FI(is)g(closed)83 3097 y(and)c(corresponds)e(to)i(the)g(complement)e(in)j Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))20 b FI(of)f(the)g(Mourre)f(set)i (\(see)f(the)h(comment)d(after)i(De\002nition)g(3.4\).)f(It)h(al)o(w)o (ays)83 3196 y(contains)f(the)g(eigen)m(v)n(alues)f(of)h Fy(H)7 b FI(.)19 b(Furthermore,)d(since)j(the)f(spectrum)g(of)g Fy(H)26 b FI(is)20 b(absolutely)d(continuous)g(on)h Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))13 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FI(,)83 3296 y(the)18 b(support)e(of)h(the)h (singularly)e(continuous)g(spectrum,)g(if)i(an)o(y)-5 b(,)16 b(is)j(contained)d(in)i Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))p FI(.)18 b(In)f(particular)m(,)f(if)i Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))19 b FI(is)f(discrete,)83 3396 y(then)f Fy(H)25 b FI(has)17 b(no)g(singularly)f(continuous)f (spectrum.)h(Thus,)h(the)g(determination)e(of)i(the)h(size)g(of)f Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))18 b FI(is)g(an)f(important)f(issue)83 3495 y(for)k(the)g(spectral)g(analysis)g(of)g Fy(H)7 b FI(.)20 b(More)g(will)h(be)f(said)g(in)h(the)f(concrete)f(e)o (xamples)g(of)h(the)g(ne)o(xt)f(section.)249 3595 y(Let)j(us)h(no)n(w)f (turn)f(to)h(the)g(density)g(properties)f(of)h(the)g(sets)h Fo(D)2055 3607 y Fu(t)2085 3595 y FI(.)f(F)o(or)g(this,)g(we)h(recall)f (that)g(a)h(subset)f 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Fy(C)1075 1223 y Fu(k)1116 1253 y Fw(\(\010\))h FI(and)h Fp(e)-49 b Fy(\021)32 b Fx(2)d Fy(C)1631 1223 y Fv(1)1625 1274 y FK(c)1702 1253 y Fw(\()p Fq(R)p Fw(\))24 b FI(\(see)g([1)o(,)g(Thm.)e (6.2.5]\).)f(So,)i(we)h(obtain)f(from)f([1)o(,)i(Prop.)e(5.3.1])83 1353 y(that)e Fx(h)q Fw(\010)p Fx(i)353 1311 y Fu(t)400 1353 y Fp(e)-50 b Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))14 b Fx(h)q Fw(\010)p Fx(i)719 1311 y Fv(\000)p Fu(t)821 1353 y FI(is)21 b(bounded)c(on)j Fx(H)q FI(,)h(which)e(implies)i(the)f (claim.)p 3708 1353 4 57 v 3712 1300 50 4 v 3712 1353 V 3761 1353 4 57 v 83 1633 a Fz(7)119 b(Examples)83 1818 y FI(In)22 b(this)h(section)g(we)g(sho)n(w)f(that)h(Assumptions)f(2.2)g (and)g(2.3)g(are)g(satis\002ed)h(in)g(v)n(arious)f(general)f (situations.)h(In)h(these)g(situa-)83 1918 y(tions)d(all)h(the)g (results)f(of)h(the)f(preceding)e(sections)j(such)f(as)h(Theorem)d(3.6) i(or)g(F)o(ormula)f(\(5.3\))g(hold.)h(Ho)n(we)n(v)o(er)m(,)e(it)j(is)g (usually)83 2017 y(impossible)c(to)h(determine)f(e)o(xplicitly)g(the)h (set)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))18 b FI(when)g(the)g(frame)n(w) o(ork)d(is)k(too)f(general.)e(Therefore,)g(we)i(also)g(illustrate)83 2117 y(our)25 b(approach)e(with)i(some)g(concrete)f(e)o(xamples)h(for)f (which)h(e)n(v)o(erything)e(can)i(be)g(computed)e(e)o(xplicitly)-5 b(.)24 b(When)h(possible,)83 2217 y(we)c(also)g(relate)f(these)h(e)o (xamples)f(with)g(the)h(dif)n(ferent)e(cases)i(presented)e(in)i (Section)f(6.)h(F)o(or)f(that)h(purpose,)d(we)j(shall)g(al)o(w)o(ays)83 2316 y(assume)f(that)g Fy(f)30 b FI(is)21 b(a)g(real)f(and)f(e)n(v)o (en)h(function)e(in)i Fo(S)c Fw(\()p Fq(R)1720 2286 y Fu(d)1759 2316 y Fw(\))21 b FI(with)g Fy(f)31 b Fw(=)23 b(1)d FI(on)g(a)h(neighbourhood)15 b(of)20 b Fw(0)p FI(.)249 2416 y(The)29 b(con\002guration)e(space)i(of)g(the)g(system)h(under)e (consideration)f(will)j(sometimes)f(be)h Fq(R)2995 2386 y Fu(n)3040 2416 y FI(,)f(and)g(the)h(correspond-)83 2516 y(ing)d(Hilbert)f(space)h Fr(L)728 2486 y FK(2)766 2516 y Fw(\()p Fq(R)858 2486 y Fu(n)903 2516 y Fw(\))p FI(.)h(In)f(that)g(case,)g(the)g(notations)f Fy(Q)35 b Fx(\021)g Fw(\()p Fy(Q)2184 2528 y FK(1)2221 2516 y Fy(;)14 b(:)g(:)g(:)g(;)g(Q)2472 2528 y Fu(n)2517 2516 y Fw(\))28 b FI(and)e Fy(P)47 b Fx(\021)36 b Fw(\()p Fy(P)3010 2528 y FK(1)3047 2516 y Fy(;)14 b(:)g(:)g(:)g(;)g(P)3285 2528 y Fu(n)3330 2516 y Fw(\))28 b FI(refer)e(to)i(the)83 2615 y(f)o(amilies)i(of)f(position)f(operators)g(and)h(momentum)e (operators.)h(More)g(precisely)-5 b(,)28 b(for)h(suitable)g Fy(')41 b Fx(2)f Fr(L)3229 2585 y FK(2)3267 2615 y Fw(\()p Fq(R)3359 2585 y Fu(n)3404 2615 y Fw(\))30 b FI(and)f(each)83 2715 y Fy(j)f Fx(2)23 b(f)p Fw(1)p Fy(;)14 b(:)g(:)g(:)f(;)h(n)p Fx(g)p FI(,)20 b(we)g(ha)n(v)o(e)g Fw(\()p Fy(Q)1014 2727 y Fu(j)1049 2715 y Fy(')p Fw(\)\()p Fy(x)p Fw(\))25 b(=)d Fy(x)1405 2727 y Fu(j)1441 2715 y Fy(')p Fw(\()p Fy(x)p Fw(\))g FI(and)d Fw(\()p Fy(P)1853 2727 y Fu(j)1889 2715 y Fy(')p Fw(\)\()p Fy(x)p Fw(\))25 b(=)d Fx(\000)p Fy(i)p Fw(\()p Fy(@)2368 2727 y Fu(j)2403 2715 y Fy(')p Fw(\)\()p Fy(x)p Fw(\))g FI(for)e(each)g Fy(x)j Fx(2)g Fq(R)3121 2685 y Fu(n)3167 2715 y FI(.)83 2952 y Fg(7.1)99 b Ff(H)413 2916 y Fe(0)466 2952 y Fg(constant)83 3108 y FI(Suppose)21 b(that)i Fy(H)30 b FI(is)23 b(of)g(class)g Fy(C)1054 3078 y FK(1)1092 3108 y Fw(\(\010\))p FI(,)g(and)f(assume)g (that)h(there)f(e)o(xists)h Fy(v)31 b Fx(2)c Fq(R)2431 3078 y Fu(d)2490 3108 y Fx(n)20 b(f)p Fw(0)p Fx(g)i FI(such)g(that)g Fy(H)3098 3078 y Fv(0)3149 3108 y Fw(=)27 b Fy(v)s FI(.)c(Then)f Fy(H)29 b FI(is)24 b(of)83 3208 y(class)d Fy(C)330 3178 y Fv(1)401 3208 y Fw(\(\010\))p FI(,)g(Assumption)e(2.2)g(is)j (directly)d(v)o(eri\002ed,)g(and)g(one)h(has)h(on)e Fx(D)r Fw(\()p Fy(H)7 b Fw(\))415 3449 y Fy(H)g Fw(\()p Fy(x)p Fw(\))24 b(=)f Fy(H)7 b Fw(\(0\))18 b(+)997 3336 y Fp(Z)1080 3356 y FK(1)1043 3525 y(0)1131 3449 y Fw(d)p Fy(t)1221 3382 y Fp(\000)1259 3449 y Fy(x)h Fx(\001)g Fy(H)1443 3415 y Fv(0)1466 3449 y Fw(\()p Fy(tx)p 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FI(reduces)e(to)h(the)g(constant)f(v)o (ector)83 3877 y Fy(R)147 3846 y Fv(0)146 3900 y Fu(f)189 3877 y Fw(\()p Fy(v)s Fw(\))p FI(.)h(Therefore,)c(we)j(ha)n(v)o(e)e (the)i(equality)e Fy(T)1478 3889 y Fu(f)1550 3877 y Fw(=)29 b Fx(\000)p Fy(R)1773 3846 y Fv(0)1772 3900 y Fu(f)1814 3877 y Fw(\()p Fy(v)s Fw(\))22 b Fx(\001)f Fw(\010)j FI(on)f Fo(D)2242 3889 y FK(1)2279 3877 y FI(,)h(and)f(it)h(is)h (easily)e(sho)n(wn)g(that)h Fy(T)3272 3889 y Fu(f)3338 3877 y FI(is)h(essentially)83 3976 y(self-adjoint)f(on)g Fo(D)659 3988 y FK(1)697 3976 y FI(.)h(It)g(follo)n(ws)f(from)g(the)h (case)g Fw(1)g FI(of)g(Section)f(6)h(that)g(the)g(spectrum)e(of)i Fy(H)32 b FI(co)o(v)o(ers)24 b(the)h(whole)f(real)h(line,)83 4076 y(and)20 b(there)f(e)o(xists)i(a)g(unitary)e(operator)f Fo(U)45 b Fw(:)23 b Fx(H)h(!)f Fr(L)1624 4046 y FK(2)1662 4076 y Fw(\()p Fq(R)p Fw(;)14 b Fq(C)1851 4046 y Fu(N)1914 4076 y Fy(;)g Fw(d)p Fy(\025)p Fw(\))21 b FI(such)f(that)1154 4300 y Fx(h)p Fy( )s(;)14 b(T)1329 4312 y Fu(f)1372 4300 y Fy(')p Fx(i)23 b Fw(=)1569 4187 y Fp(Z)1615 4376 y Fn(R)1675 4300 y Fw(d)p Fy(\025)1783 4233 y Fp(\012)1823 4300 y Fw(\()p Fo(U)f Fy( )s Fw(\)\()p Fy(\025)p Fw(\))p Fy(;)14 b(i)2234 4260 y FK(d\()p Fk(U)h Fu(')p FK(\))p 2234 4281 202 4 v 2297 4329 a(d)p Fu(\025)2445 4300 y Fw(\()p Fy(\025)p Fw(\))2557 4233 y Fp(\013)2598 4333 y Fn(C)2640 4317 y Fm(N)83 4524 y FI(for)20 b(each)f Fy( )27 b Fx(2)c(H)f FI(and)d Fy(')24 b Fx(2)f Fo(D)984 4536 y FK(1)1022 4524 y FI(.)249 4624 y(T)-7 b(ypical)20 b(e)o(xamples)f(of)h(operators)f Fy(H)28 b FI(and)19 b Fw(\010)i FI(\002tting)f(into)g(this)h(construction)e(are)h (Friedrichs-type)e(Hamiltonians)h(and)83 4724 y(position)27 b(operators.)g(F)o(or)g(illustration,)g(we)i(mention)e(the)h(case)g Fy(H)45 b Fw(:=)37 b Fy(v)27 b Fx(\001)e Fy(P)36 b Fw(+)24 b Fy(V)19 b Fw(\()p Fy(Q)p Fw(\))28 b FI(and)g Fw(\010)37 b(:=)h Fy(Q)28 b FI(in)g Fr(L)3370 4693 y FK(2)3408 4724 y Fw(\()p Fq(R)3500 4693 y Fu(d)3539 4724 y Fw(\))p FI(,)g(with)83 4823 y Fy(v)33 b Fx(2)d Fq(R)301 4793 y Fu(d)360 4823 y Fx(n)21 b(f)p Fw(0)p Fx(g)i FI(and)g Fy(V)49 b Fx(2)29 b Fr(L)934 4793 y Fv(1)1005 4823 y Fw(\()p Fq(R)1097 4793 y Fu(d)1136 4823 y Fw(;)14 b Fq(R)p Fw(\))25 b FI(\(see)f(also)g ([37)n(,)g(Sec.)g(5])g(for)f(informations)e(on)j(quantum)d(time)j (delay)f(in)h(a)g(similar)83 4923 y(case\).)249 5023 y(Stark)19 b(Hamiltonians)f(and)h(momentum)e(operators)g(also)j(\002t)g (into)e(the)i(construction,)c FJ(i.e)o(.)j Fy(H)30 b Fw(:=)23 b Fy(P)3135 4992 y FK(2)3186 5023 y Fw(+)14 b Fy(v)k Fx(\001)d Fy(Q)k FI(in)g Fr(L)3567 4992 y FK(2)3605 5023 y Fw(\()p Fq(R)3697 4992 y Fu(d)3736 5023 y Fw(\))83 5122 y FI(with)i Fy(v)29 b Fx(2)c Fq(R)461 5092 y Fu(d)519 5122 y Fx(n)19 b(f)p Fw(0)p Fx(g)p FI(,)h(and)h Fw(\010)j(:=)h Fy(P)12 b FI(.)22 b(W)-7 b(e)22 b(refer)e(to)i([25)n(,)g(29)o(,)g(30)o (])f(for)g(pre)n(vious)e(accounts)i(on)g(the)g(theory)f(of)h(time)g (operators)83 5222 y(and)f(quantum)e(time)i(delay)g(in)g(similar)h (situations.)249 5321 y(Note)f(that)g(these)h(\002rst)g(tw)o(o)f(e)o (xamples)f(are)h(interesting)g(since)g(the)g(operators)f Fy(H)27 b FI(contain)19 b(not)h(only)g(a)g(kinetic)g(part,)g(b)n(ut)83 5421 y(also)h(a)f(potential)f(perturbation.)p Black 1884 5670 a(20)p Black eop end %%Page: 21 21 TeXDict begin 21 20 bop Black Black 249 307 a FI(Another)18 b(e)o(xample)g(is)i(pro)o(vided)d(by)i(the)h(Jacobi)f(operator)e (related)i(to)h(the)f(f)o(amily)g(of)g(Hermite)g(polynomials)f(\(see)i ([32)n(,)83 407 y(Appendix)e(A])j(for)e(details\).)h(In)g(the)g (Hilbert)g(space)g Fx(H)k Fw(:=)f Fy(`)1856 377 y FK(2)1893 407 y Fw(\()p Fq(N)p Fw(\))p FI(,)e(consider)e(the)h(Jacobi)g(operator) e(gi)n(v)o(en)h(for)h Fy(')j Fx(2)h(H)d FI(by)1170 606 y Fw(\()p Fy(H)7 b(')p Fw(\)\()p Fy(n)p Fw(\))24 b(:=)1623 525 y Fv(p)p 1677 525 127 3 v 1677 569 a Fu(n)p Fv(\000)p FK(1)p 1623 586 181 4 v 1696 634 a(2)1827 606 y Fy(')p Fw(\()p Fy(n)19 b Fx(\000)f Fw(1\))g(+)2250 525 y Fv(p)p 2305 525 42 3 v 41 x Fu(n)p 2250 586 96 4 v 2281 634 a FK(2)2370 606 y Fy(')p Fw(\()p Fy(n)g Fw(+)g(1\))83 788 y FI(with)k(the)g(con)m(v)o(ention)d(that)j Fy(')p Fw(\(0\))k(=)g(0)p FI(.)c(The)g(operator)e Fy(H)29 b FI(is)23 b(essentially)f(self-adjoint)e(on)i Fy(`)2814 758 y FK(2)2814 809 y(0)2851 788 y FI(,)g(the)g(subspace)f(of)g (sequences)83 888 y(in)f Fx(H)i FI(with)e(only)g(\002nitely)g(man)o(y)f (non-zero)f(components.)f(As)k(operator)e Fw(\010)h FI(\(with)g(one)g (component\),)d(tak)o(e)1039 1070 y Fw(\(\010)p Fy(')p Fw(\)\()p Fy(n)p Fw(\))24 b(:=)f Fx(\000)p Fy(i)1560 1003 y Fp(\010)1607 1001 y Fx(p)p 1677 1001 193 4 v 1677 1070 a Fy(n)18 b Fx(\000)g Fw(1)13 b Fy(')p Fw(\()p Fy(n)19 b Fx(\000)f Fw(1\))g Fx(\000)2296 1007 y(p)p 2366 1007 50 4 v 2366 1070 a Fy(n)13 b(')p Fw(\()p Fy(n)19 b Fw(+)f(1\))2741 1003 y Fp(\011)2789 1070 y Fy(;)83 1253 y FI(which)29 b(is)i(also)f(essentially)g(self-adjoint)f(on)g Fy(`)1504 1223 y FK(2)1504 1274 y(0)1541 1253 y FI(.)h(Then)f Fy(H)38 b FI(is)30 b(of)g(class)h Fy(C)2342 1223 y FK(1)2379 1253 y Fw(\(\010\))g FI(and)f Fy(H)2761 1223 y Fv(0)2825 1253 y Fx(\021)40 b Fy(i)p Fw([)p Fy(H)r(;)14 b Fw(\010])41 b(=)f(1)p FI(,)30 b(and)f(so)i(the)83 1353 y(preceding)18 b(results)j(hold.)83 1590 y Fg(7.2)99 b Ff(H)413 1554 y Fe(0)473 1590 y Fd(=)32 b Ff(H)83 1746 y FI(Suppose)e(that)h Fw(\010)g FI(has)h(only)e(one)g(component,)e(and)j(assume)g(that)g Fy(H)38 b FI(is)32 b Fw(\010)p FI(-homogeneous)27 b(of)k(de)o(gree)e Fw(1)p FI(,)i FJ(i.e)o(.)g Fy(H)7 b Fw(\()p Fy(x)p Fw(\))43 b Fx(\021)83 1843 y Fw(e)120 1816 y Fv(\000)p Fu(ix)p FK(\010)298 1846 y Fy(H)388 1843 y Fw(e)425 1816 y Fu(ix)p FK(\010)561 1846 y Fw(=)648 1843 y(e)685 1816 y Fu(x)741 1846 y Fy(H)26 b FI(for)18 b(all)i Fy(x)j Fx(2)h Fq(R)p FI(.)19 b(This)g(implies)g(that)g Fy(H)26 b FI(is)20 b(of)e(class)i Fy(C)2383 1816 y Fv(1)2454 1846 y Fw(\(\010\))g FI(and)e(that)h Fy(H)2957 1816 y Fv(0)3003 1846 y Fw(=)k Fy(H)7 b FI(.)19 b(So,)f(Assumptions)83 1945 y(2.2)i(and)g(2.3)g(are)g (readily)g(v)o(eri\002ed.)f(Moreo)o(v)o(er)m(,)e(since)k Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))24 b(=)f Fx(f)p Fw(0)p Fx(g)p FI(,)c(Theorem)g(3.6)h(implies)h(that)f Fy(H)28 b FI(is)22 b(purely)d(absolutely)83 2045 y(continuous)f(e)o(xcept)h(at) i(the)f(origin,)f(where)h(it)h(may)e(ha)n(v)o(e)h(the)g(eigen)m(v)n (alue)e Fw(0)p FI(.)249 2145 y(No)n(w)-5 b(,)25 b(let)h(us)f(sho)n(w)g (that)h(the)f(formal)f(formula)g(of)h(Remark)g(5.3)f(holds)h(in)h(this) g(case.)f(F)o(or)g(an)o(y)g Fy(')32 b Fx(2)h Fo(D)3339 2157 y FK(1)3403 2145 y FI(one)24 b(has)i(by)83 2244 y(Remark)20 b(5.4)f(that)h Fy(R)700 2214 y Fv(0)699 2268 y Fu(f)743 2244 y Fw(\()p Fy(H)851 2214 y Fv(0)874 2244 y Fw(\))p Fy(')k Fx(\021)e Fy(R)1135 2214 y Fv(0)1134 2268 y Fu(f)1177 2244 y Fw(\()p Fy(H)7 b Fw(\))p Fy(')22 b FI(belongs)d(to)h Fx(D)r Fw(\(\010\))p FI(.)h(On)g(another)d(hand,)h (we)i(ha)n(v)o(e)1085 2442 y Fw(\010)p Fy(')j Fw(=)1310 2375 y Fp(\010)1359 2442 y Fy(H)7 b Fw(\010)18 b(+)g([\010)p Fy(;)c(H)7 b Fw(])1815 2375 y Fp(\011)1863 2442 y Fy(H)1939 2408 y Fv(\000)p FK(1)2028 2442 y Fy(')23 b Fw(=)g Fy(H)7 b Fw(\(\010)19 b(+)f Fy(i)p Fw(\))p Fy(H)2600 2408 y Fv(\000)p FK(1)2688 2442 y Fy(';)83 2635 y FI(which)i(implies)g(that)g Fy(R)781 2605 y Fv(0)780 2658 y Fu(f)823 2635 y Fw(\()p Fy(H)7 b Fw(\)\010)p Fy(')24 b Fw(=)f Fy(R)1253 2605 y Fv(0)1252 2658 y Fu(f)1295 2568 y Fp(\000)1362 2602 y Fu(H)p 1343 2616 99 4 v 1343 2663 a Fv(j)p Fu(H)t Fv(j)1451 2568 y Fp(\001)1519 2602 y Fu(H)p 1499 2616 V 1499 2663 a Fv(j)p Fu(H)t Fv(j)1607 2635 y Fw(\(\010)c(+)f Fy(i)p Fw(\))p Fy(H)1938 2605 y Fv(\000)p FK(1)2027 2635 y Fy(')23 b Fx(2)g(H)q FI(.)e(In)f(consequence,)d(the)k(operator)1370 2840 y Fy(T)1419 2852 y Fu(f)1485 2840 y Fw(=)i Fx(\000)1648 2807 y FK(1)p 1647 2821 34 4 v 1647 2869 a(2)1690 2773 y Fp(\000)1728 2840 y Fw(\010)p Fy(R)1852 2806 y Fv(0)1851 2860 y Fu(f)1894 2840 y Fw(\()p Fy(H)7 b Fw(\))19 b(+)f Fy(R)2200 2806 y Fv(0)2199 2860 y Fu(f)2242 2840 y Fw(\()p Fy(H)7 b Fw(\)\010)2442 2773 y Fp(\001)83 3023 y FI(is)28 b(well-de\002ned)d(on)i Fo(D)783 3035 y FK(1)820 3023 y FI(.)h(In)e(particular)m(,)f(if)j Fw(0)f FI(is)h(not)e(an)h(eigen)m (v)n(alue)e(of)h Fy(H)7 b FI(,)27 b(then)g Fy(T)2620 3035 y Fu(f)2690 3023 y FI(is)h(a)f(symmetric)f(operator)f(and)i(the)83 3122 y(discussion)20 b(of)g(the)g(case)g Fw(2)h FI(of)f(Section)f(6)i (is)g(rele)n(v)n(ant)e(\(if)h Fy(T)1818 3134 y Fu(f)1881 3122 y FI(is)h(essentially)g(self-adjoint,)d(the)j(case)f Fw(1)g FI(is)i(rele)n(v)n(ant\).)249 3222 y(W)-7 b(e)31 b(no)n(w)e(gi)n(v)o(e)h(tw)o(o)g(e)o(xamples)f(of)g(pairs)h Fx(f)p Fy(H)r(;)14 b Fw(\010)p Fx(g)30 b FI(satisfying)f(the)h (preceding)e(assumptions.)g(Other)i(e)o(xamples)f(are)83 3321 y(presented)24 b(in)h([8)o(,)g(Sec.)g(10].)f(Suppose)g(that)h Fy(H)38 b Fw(:=)32 b Fy(P)1704 3291 y FK(2)1766 3321 y FI(is)26 b(the)f(free)f(Schr)7 b(\250)-35 b(odinger)22 b(operator)i(in)h Fx(H)32 b Fw(:=)g Fr(L)3211 3291 y FK(2)3248 3321 y Fw(\()p Fq(R)3340 3291 y Fu(n)3386 3321 y Fw(\))25 b FI(and)g Fw(\010)31 b(:=)93 3388 y FK(1)p 93 3402 V 93 3450 a(4)136 3421 y Fw(\()p Fy(Q)26 b Fx(\001)h Fy(P)38 b Fw(+)26 b Fy(P)38 b Fx(\001)27 b Fy(Q)p Fw(\))k FI(is)h(the)f(generator)e(of)h(dilations)h(in)g Fx(H)q FI(.)g(Then)f(the)h(relation)2584 3418 y Fw(e)2621 3391 y Fv(\000)p Fu(ix)p FK(\010)2799 3421 y Fy(H)2889 3418 y Fw(e)2926 3391 y Fu(ix)p FK(\010)3081 3421 y Fw(=)3189 3418 y(e)3225 3391 y Fu(x)3281 3421 y Fy(H)38 b FI(is)32 b(satis\002ed,)83 3521 y Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))24 b(=)e Fy(\033)431 3533 y FK(ac)499 3521 y Fw(\()p Fy(H)7 b Fw(\))23 b(=)g([0)p Fy(;)14 b Fx(1)p Fw(\))p FI(.)k(Furthermore,)e (for)h Fy( )26 b Fx(2)e(H)c FI(and)d Fy(')24 b Fx(2)f Fo(F)11 b Fy(C)2269 3491 y Fv(1)2263 3541 y FK(c)2340 3453 y Fp(\000)2378 3521 y Fq(R)2438 3491 y Fu(n)2495 3521 y Fx(n)g(f)p Fw(0)p Fx(g)2674 3453 y Fp(\001)2733 3521 y Fx(\032)23 b Fo(D)2885 3533 y FK(1)2941 3521 y FI(a)c(direct)f(calculation)f(using)83 3620 y(F)o(ormula)i(\(4.1\))g (sho)n(ws)h(that)402 3844 y Fx(h)p Fy( )s(;)14 b(T)577 3856 y Fu(f)620 3844 y Fy(')p Fx(i)24 b Fw(=)818 3777 y Fp(\012)857 3844 y Fy( )s(;)961 3811 y FK(1)p 961 3825 V 961 3873 a(4)1004 3777 y Fp(\000)1042 3844 y Fy(Q)18 b Fx(\001)g Fy(P)12 b(P)1297 3810 y Fv(\000)p FK(2)1405 3844 y Fw(+)18 b Fy(P)12 b(P)1618 3810 y Fv(\000)p FK(2)1725 3844 y Fx(\001)18 b Fy(Q)1832 3777 y Fp(\001)1870 3844 y Fy(')1924 3777 y Fp(\013)1987 3844 y Fw(=)2074 3731 y Fp(Z)2157 3751 y Fv(1)2120 3920 y FK(0)2242 3844 y Fw(d)p Fy(\025)2350 3777 y Fp(\012)2389 3844 y Fw(\()p Fo(U)k Fy( )s Fw(\)\()p Fy(\025)p Fw(\))p Fy(;)14 b(i)2801 3804 y FK(d\()p Fk(U)g Fu(')p FK(\))p 2801 3825 202 4 v 2863 3873 a(d)p Fu(\025)3012 3844 y Fw(\()p Fy(\025)p Fw(\))3124 3777 y Fp(\013)3164 3877 y Fc(L)3191 3860 y Fj(2)3223 3877 y FK(\()p Fn(S)3281 3860 y Fm(n)p Fl(\000)p Fj(1)3395 3877 y FK(\))3426 3844 y Fy(;)83 4093 y FI(where)25 b Fo(U)56 b Fw(:)34 b Fx(H)h(!)713 4026 y Fp(R)768 4047 y Fv(\010)752 4123 y FK([0)p Fu(;)p Fv(1)p FK(\))934 4093 y Fw(d)p Fy(\025)14 b 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FI(with)25 b(the)g(con)m(v)o(ention)d(that)j Fy(')p Fw(\(0\))32 b(=)f(0)p FI(.)24 b(The)h(operator)e Fy(H)32 b FI(is)26 b(essentially)f(self-adjoint)e(on)i Fy(`)2860 4836 y FK(2)2860 4887 y(0)2896 4866 y FI(.)h(As)f(operator)e Fw(\010)j FI(\(with)e(one)83 4966 y(component\),)17 b(tak)o(e)1077 5066 y Fw(\(\010)p Fy(')p Fw(\)\()p Fy(n)p Fw(\))25 b(:=)d Fx(\000)1584 5033 y Fu(i)p 1579 5047 34 4 v 1579 5094 a FK(2)1622 4998 y Fp(\010)1670 5066 y Fw(\()p Fy(n)d Fx(\000)f Fw(1\))p Fy(')p Fw(\()p Fy(n)h Fx(\000)f Fw(1\))g Fx(\000)g Fy(n')p Fw(\()p Fy(n)h Fw(+)f(1\))2703 4998 y Fp(\011)2751 5066 y Fy(:)83 5215 y FI(Then)23 b(one)g(has)i Fy(H)634 5185 y Fv(0)687 5215 y Fx(\021)k Fy(i)p Fw([)p Fy(H)r(;)14 b Fw(\010])30 b(=)f Fy(H)7 b FI(,)24 b(which)g(implies)g (that)g Fy(H)31 b FI(is)25 b Fw(\010)p FI(-homogeneous)20 b(of)k(de)o(gree)e Fw(1)i FI(and)f(so)i(the)f(preceding)83 5315 y(results)d(hold.)p Black 1884 5670 a(21)p Black eop end %%Page: 22 22 TeXDict begin 22 21 bop Black Black 83 307 a Fg(7.3)99 b(Dirac)25 b(operator)83 463 y FI(In)20 b(the)g(Hilbert)g(space)g Fx(H)k Fw(:=)f Fr(L)998 433 y FK(2)1036 463 y Fw(\()p Fq(R)1128 433 y FK(3)1165 463 y Fw(;)14 b Fq(C)1262 433 y FK(4)1299 463 y Fw(\))21 b FI(we)g(consider)e(the)h(Dirac)g(operator) f(for)g(a)i(spin-)2749 430 y FK(1)p 2748 444 34 4 v 2748 491 a(2)2812 463 y FI(particle)e(of)h(mass)h Fy(m)i(>)g Fw(0)1607 626 y Fy(H)30 b Fw(:=)23 b Fy(\013)c Fx(\001)f Fy(P)30 b Fw(+)18 b Fy(\014)t(m;)83 790 y FI(where)e Fy(\013)23 b Fx(\021)g Fw(\()p Fy(\013)552 802 y FK(1)589 790 y Fy(;)14 b(\013)679 802 y FK(2)717 790 y Fy(;)g(\013)807 802 y FK(3)844 790 y Fw(\))j FI(and)e Fy(\014)21 b FI(denote)15 b(the)i(usual)f Fw(4)s Fx(\002)s Fw(4)g FI(Dirac)g(matrices.)f(It)i(is) g(kno)n(wn)d(that)j Fy(H)23 b FI(has)17 b(domain)d Fx(H)3414 760 y FK(1)3451 790 y Fw(\()p Fq(R)3543 760 y FK(3)3581 790 y Fw(;)g Fq(C)3678 760 y FK(4)3715 790 y Fw(\))p FI(,)83 890 y(that)20 b Fx(j)p Fy(H)7 b Fx(j)23 b Fw(=)g(\()p Fy(P)558 860 y FK(2)614 890 y Fw(+)18 b Fy(m)770 860 y FK(2)807 890 y Fw(\))839 860 y FK(1)p Fu(=)p FK(2)964 890 y FI(and)i(that)g Fy(\033)s Fw(\()p Fy(H)7 b Fw(\))24 b(=)f Fy(\033)1599 902 y FK(ac)1666 890 y Fw(\()p Fy(H)7 b Fw(\))23 b(=)g(\()p Fx(\0001)p Fy(;)14 b Fx(\000)p Fy(m)p Fw(])k Fx([)g Fw([)p Fy(m;)c Fx(1)p Fw(\))p FI(.)249 989 y(W)-7 b(e)29 b(also)f(let)g Fw(\010)37 b(:=)g Fo(U)969 954 y Fv(\000)p FK(1)947 1014 y(FW)1062 989 y Fy(Q)p Fo(U)1194 1001 y FK(FW)1346 989 y Fx(\021)g Fy(Q)1514 1001 y FK(NW)1662 989 y FI(be)28 b(the)g(W)m(igner)n(-Ne)n(wton)d (position)i(operator)m(,)f(with)i Fo(U)3320 1001 y FK(FW)3463 989 y FI(the)g(usual)83 1089 y(F)o(oldy-W)-7 b(outhuysen)17 b(transformation)h([34)n(,)j(Sec.)f(1.4.3].)e(Then)i(a)g(direct)g (calculation)f(sho)n(ws)h(that)1499 1285 y Fy(H)7 b Fw(\()p Fy(x)p Fw(\))24 b(=)1798 1186 y Fp(q)p 1881 1186 387 4 v 1891 1245 a FK(\()p Fu(P)9 b FK(+)p Fu(x)p FK(\))2083 1228 y Fj(2)2115 1245 y FK(+)p Fu(m)2225 1228 y Fj(2)p 1891 1266 367 4 v 1961 1314 a Fu(P)2012 1297 y Fj(2)2044 1314 y FK(+)p Fu(m)2154 1297 y Fj(2)2276 1285 y Fy(H)83 1472 y FI(for)22 b(each)g Fy(x)27 b Fx(2)g Fq(R)594 1442 y FK(3)632 1472 y FI(,)22 b(and)g(thus)g(Assumptions)g(2.2)f(and)h(2.3) g(are)g(clearly)g(satis\002ed.)h(Furthermore,)c(since)k Fy(H)3249 1442 y Fv(0)3242 1493 y Fu(j)3304 1472 y Fw(=)j Fy(P)3448 1484 y Fu(j)3483 1472 y Fy(H)3559 1442 y Fv(\000)p FK(1)3671 1472 y FI(for)83 1571 y(each)20 b Fy(j)28 b Fw(=)23 b(1)p Fy(;)14 b Fw(2)p Fy(;)g Fw(3)p FI(,)k(it)j(follo)n(ws)f (that)1292 1671 y Fw(\()p Fy(H)1400 1641 y Fv(0)1424 1671 y Fw(\))1456 1641 y FK(2)1516 1671 y Fw(=)j Fy(P)1669 1641 y FK(2)1706 1671 y Fy(H)1782 1641 y Fv(\000)p FK(2)1894 1671 y Fw(=)g(\()p Fy(H)2090 1641 y FK(2)2145 1671 y Fx(\000)18 b Fy(m)2301 1641 y FK(2)2338 1671 y Fw(\))p Fy(H)2446 1641 y Fv(\000)p FK(2)2536 1671 y Fy(:)83 1809 y FI(Clearly)-5 b(,)23 b Fw(k)n(er)492 1742 y Fp(\000)530 1809 y Fw(\()p Fy(H)638 1779 y Fv(0)661 1809 y Fw(\))693 1779 y FK(2)731 1742 y Fp(\001)798 1809 y Fw(=)28 b Fx(f)p Fw(0)p Fx(g)23 b FI(and)g(one)g(infers)g(from)f(De\002nition)h(2.5)g (that)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))30 b(=)e Fx(f\006)p Fy(m)p Fx(g)p FI(,)23 b(and)g(from)f(Lemma)h(6.3.\(b\))83 1909 y(that)d(the)h(sets)682 2008 y Fo(D)746 2020 y Fu(t)799 2008 y Fw(=)886 1941 y Fp(\010)935 2008 y Fy(')i Fx(2)h Fo(U)1179 1973 y Fv(\000)p FK(1)1157 2033 y(FW)1272 2008 y Fx(D)1338 1941 y Fp(\000)1376 2008 y Fx(h)p Fy(Q)p Fx(i)1506 1974 y Fu(t)1536 1941 y Fp(\001)1597 2008 y Fx(j)f Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fy(')24 b Fw(=)e Fy(')g FI(for)d(some)h Fy(\021)26 b Fx(2)e Fy(C)2592 1974 y Fv(1)2586 2029 y FK(c)2662 1941 y Fp(\000)2700 2008 y Fq(R)19 b Fx(n)f(f\006)p Fy(m)p Fx(g)3061 1941 y Fp(\001)n(\011)3146 2008 y Fy(;)83 2146 y FI(are)i(dense)g(in)g Fx(H)q FI(.)h(So)f(the)g(discussion)g(of)g(the)g(case)h Fw(2)f FI(of)g(Section)g(6)g(is)h(rele)n(v)n(ant.)249 2246 y(W)-7 b(e)27 b(no)n(w)d(sho)n(w)i(that)f(the)g(formal)g(formula)e (of)j(Remark)e(5.3)h(holds)g(if)h Fy(f)34 b FI(is)26 b(radial.)f(Indeed,)f(each)h Fy(')33 b Fx(2)g Fo(D)3442 2258 y FK(1)3505 2246 y FI(satis\002es)83 2346 y Fy(')23 b Fw(=)g Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fo(U)520 2310 y Fv(\000)p FK(1)498 2370 y(FW)614 2346 y Fy( )24 b FI(for)19 b(some)h Fy(\021)26 b Fx(2)e Fy(C)1216 2315 y Fv(1)1210 2366 y FK(c)1286 2278 y Fp(\000)1324 2346 y Fq(R)19 b Fx(n)f(f\006)p Fy(m)p Fx(g)1685 2278 y Fp(\001)1742 2346 y FI(and)i(some)g Fy( )26 b Fx(2)d(D)r Fw(\()p Fx(h)p Fy(Q)p Fx(i)p Fw(\))p FI(.)f(So,)e(we)h(ha)n(v)o(e)269 2520 y Fy(H)345 2486 y Fv(0)368 2520 y Fw(\()p Fy(H)476 2486 y Fv(0)500 2520 y Fw(\))532 2486 y Fv(\000)p FK(2)640 2520 y Fx(\001)d Fy(Q)747 2532 y FK(NW)868 2520 y Fy(')23 b Fw(=)g Fy(P)12 b(P)1163 2486 y Fv(\000)p FK(2)1251 2520 y Fy(H)26 b Fx(\001)18 b Fo(U)1475 2484 y Fv(\000)p FK(1)1453 2544 y(FW)1568 2520 y Fy(Q)p Fo(U)1700 2532 y FK(FW)1815 2520 y Fy(\021)s Fw(\()p Fy(H)7 b Fw(\))p Fo(U)2087 2484 y Fv(\000)p FK(1)2065 2544 y(FW)2181 2520 y Fy( )26 b Fw(=)d Fo(U)2436 2484 y Fv(\000)p FK(1)2415 2544 y(FW)2530 2520 y Fy(P)12 b(P)2660 2486 y Fv(\000)p FK(2)2749 2520 y Fy(\014)t Fx(j)p Fy(H)7 b Fx(j)18 b(\001)h Fy(Q\021)s Fw(\()p Fy(\014)t Fx(j)p Fy(H)7 b Fx(j)p Fw(\))p Fy( )26 b Fx(2)e(H)q Fy(;)83 2683 y FI(and)c(the)g(operator)e Fy(T)32 b FI(of)20 b(\(5.2\))f(is)i(symmetric)e(and)h(can)g(be)g (written)g(on)g Fo(D)2241 2695 y FK(1)2299 2683 y FI(in)g(the)h (simpler)e(form)383 2847 y Fy(T)34 b Fw(=)563 2814 y FK(1)p 563 2828 34 4 v 563 2876 a(2)606 2780 y Fp(\010)655 2847 y Fy(Q)721 2859 y FK(NW)860 2847 y Fx(\001)18 b Fy(H)977 2813 y Fv(0)1000 2847 y Fw(\()p Fy(H)1108 2813 y Fv(0)1132 2847 y Fw(\))1164 2813 y Fv(\000)p FK(2)1272 2847 y Fw(+)g Fy(H)1431 2813 y Fv(0)1454 2847 y Fw(\()p Fy(H)1562 2813 y Fv(0)1585 2847 y Fw(\))1617 2813 y Fv(\000)p FK(2)1725 2847 y Fx(\001)g Fy(Q)1832 2859 y FK(NW)1953 2780 y Fp(\011)2024 2847 y Fx(\021)2122 2814 y FK(1)p 2122 2828 V 2122 2876 a(2)2165 2780 y Fp(\010)2213 2847 y Fy(Q)2279 2859 y FK(NW)2418 2847 y Fx(\001)h 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Fv(\000)p FK(1)2396 3714 y(0)83 3855 y FI(is)h(equal)e(to)h(the) f(operator)f Fw(2)p Fy(i)952 3822 y FK(d)p 932 3836 77 4 v 932 3883 a(d)p Fu(\025)1036 3855 y FI(of)i(dif)n(ferentiation)d (with)j(respect)f(to)h(the)g(spectral)f(parameter)f Fy(\025)j FI(of)e Fy(h)p Fw(\()p Fy(P)12 b Fw(\))19 b FI(\(see)f([37)n(,)g(Lemma) 83 3954 y(3.6])i(for)h(a)h(precise)e(statement\).)h(Combining)e(the)j (preceding)d(transformations)g(we)i(obtain)g(for)f(each)h Fy( )28 b Fx(2)d(H)e FI(and)e Fy(')k Fx(2)h Fo(D)3731 3966 y FK(1)83 4054 y FI(that)1019 4176 y Fx(h)p Fy( )s(;)14 b(T)e(')p Fx(i)23 b Fw(=)1403 4063 y Fp(Z)1449 4252 y Fu(\033)r FK(\()p Fu(H)t FK(\))1618 4176 y Fw(d)p Fy(\025)1726 4109 y Fp(\012)1766 4176 y Fw(\()p Fo(U)f Fy( )s Fw(\)\()p Fy(\025)p Fw(\))p Fy(;)14 b(i)2173 4136 y FK(d\()p Fk(U)g Fu(')p FK(\))p 2173 4157 202 4 v 2235 4205 a(d)p Fu(\025)2384 4176 y Fw(\()p Fy(\025)p Fw(\))2496 4109 y Fp(\013)2536 4209 y Fc(L)2563 4193 y Fj(2)2595 4209 y FK(\()p Fn(S)2653 4193 y Fj(2)2686 4209 y 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Fw(=)f Fo(F)11 b Fw(\()p Fy(\026)p Fw(\))23 b FI(is)f(dif)n(ferentiable)d(at)j(an)o(y) e(point)g Fy(\030)26 b FI(along)20 b(the)i(one-parameter)83 506 y(subgroup)e Fw(\007)480 518 y Fu(j)514 506 y FI(,)j(and)e Fx(\000)p Fy(i)p Fo(F)11 b Fw(\(\010)971 518 y Fu(j)1006 506 y Fy(\026)p Fw(\))27 b(=)g Fy(d)1250 518 y Fu(j)1285 506 y Fy(m)22 b FI([28)o(,)g(p.)g(68].)f(This)i(implies)f(that)g(the)g (operator)f Fw(\()p Fy(H)2890 476 y Fv(0)2883 527 y Fu(\026)2927 506 y Fw(\))2959 518 y Fu(j)3017 506 y FI(is)i(mapped)e(unitarily)g(by) 83 606 y Fo(F)31 b FI(on)19 b(the)g(multiplication)e(operator)g Fy(M)1274 618 y Fu(d)1309 626 y Fm(j)1340 618 y Fu(m)1403 606 y FI(,)i(and)f(thus)h Fw(\()p Fy(H)1848 576 y Fv(0)1841 627 y Fu(\026)1886 606 y Fw(\))1918 576 y FK(2)1975 606 y FI(is)h(unitarily)e(equi)n(v)n(alent)f(to)i(the)g(operator)e(of)i (multiplication)83 706 y(by)h(the)g(function)606 643 y Fp(P)694 731 y Fu(j)729 706 y Fw(\()p Fy(d)804 718 y Fu(j)840 706 y Fy(m)p Fw(\))945 676 y FK(2)982 706 y FI(.)h(It)f(follo)n(ws)g(that)690 914 y Fy(\024)p Fw(\()p Fy(H)839 926 y Fu(\026)883 914 y Fw(\))k Fx(\033)1026 847 y Fp(\010)1075 914 y Fy(\025)f Fx(2)h Fq(R)f Fx(j)g(9)p Fy(\030)k Fx(2)1558 893 y Fp(b)1541 914 y Fy(G)21 b FI(such)f(that)h Fy(m)p Fw(\()p Fy(\030)t Fw(\))i(=)g Fy(\025)e FI(and)2458 852 y Fp(P)2545 939 y Fu(j)2580 914 y Fw(\()p Fy(d)2655 926 y Fu(j)2691 914 y Fy(m)p Fw(\)\()p Fy(\030)t Fw(\))2900 884 y FK(2)2961 914 y Fw(=)i(0)3091 847 y Fp(\011)3138 914 y Fy(:)249 1097 y FI(This)30 b(property)e(of)h Fy(\024)p Fw(\()p Fy(H)987 1109 y Fu(\026)1032 1097 y Fw(\))i FI(suggests)f(a)g (w)o(ay)g(to)g(justify)f(the)h(formal)f(formula)f(of)i(Remark)f(5.3)g (and)h(to)g(write)g(nice)83 1196 y(formulas)f(for)h(the)g(operator)e Fy(T)42 b FI(gi)n(v)o(en)29 b(by)h(\(5.2\))n(.)h(Indeed,)d(since)i Fo(F)11 b Fw(\010)2227 1208 y Fu(j)2263 1196 y Fo(F)2348 1166 y Fv(\000)p FK(1)2469 1196 y FI(acts)31 b(as)g(the)f(dif)n (ferential)e(operator)h Fy(id)3638 1208 y Fu(j)3703 1196 y FI(in)83 1309 y Fr(L)120 1279 y FK(2)158 1309 y Fw(\()206 1288 y Fp(b)190 1309 y Fy(G;)14 b Fw(d)p Fy(\032)381 1321 y Fv(^)430 1309 y Fw(\))p FI(,)22 b(it)f(follo)n(ws)f(that)h Fw(\010)1046 1321 y Fu(j)1103 1309 y FI(lea)n(v)o(es)g(in)m(v)n(ariant) e(the)i(complement)d(of)j(the)g(support)e(of)i(the)g(functions)e(on)h (which)h(it)g(acts.)83 1409 y(Therefore,)d(the)i(set)h Fw(\010)747 1421 y Fu(j)782 1409 y Fo(D)846 1421 y FK(1)907 1409 y Fx(\021)h Fo(F)1079 1379 y Fv(\000)p FK(1)1169 1409 y Fw(\()p Fy(id)1273 1421 y Fu(j)1308 1409 y Fw(\))p Fo(F)11 b(D)1489 1421 y FK(1)1548 1409 y FI(is)21 b(included)e(in)h (the)h(domain)d(of)i(the)h(operator)1449 1572 y FK(\()p Fu(H)1533 1547 y Fl(0)1529 1589 y Fm(\026)1568 1572 y FK(\))1594 1580 y Fm(j)p 1448 1604 179 4 v 1448 1651 a FK(\()p Fu(H)1532 1635 y Fl(0)1528 1668 y Fm(\026)1568 1651 y FK(\))1594 1635 y Fj(2)1659 1623 y Fx(\021)i Fo(F)1832 1593 y Fv(\000)p FK(1)2016 1567 y Fu(M)2079 1576 y Fm(d)2110 1589 y(j)2141 1576 y(m)p 1932 1604 349 4 v 1932 1651 a Fu(M)1995 1639 y Fb(P)2045 1685 y Fm(k)2081 1670 y Fj(\()p Fm(d)2134 1685 y(k)2170 1670 y(m)p Fj(\))2243 1658 y(2)2304 1623 y Fo(F)11 b Fy(:)83 1833 y FI(Thus)20 b(the)g(formula)f(\(5.2\))f(tak)o(es)j(the)f(form)1083 2034 y Fy(T)34 b Fw(=)1264 2002 y FK(1)p 1264 2016 34 4 v 1264 2063 a(2)1321 1972 y Fp(P)1409 2059 y Fu(j)1457 1942 y Fp(n)1513 2034 y Fw(\010)1573 2046 y Fu(j)1715 1979 y(H)1769 1987 y Fl(\000)p Fm(i)p Fj(\010)1876 2000 y Fm(j)1908 1987 y(\026)p 1618 2015 428 4 v 1618 2019 a Fa(P)1687 2082 y Fm(k)1724 2063 y FK(\()p Fu(H)1804 2071 y Fl(\000)p Fm(i)p Fj(\010)1911 2086 y Fm(k)1947 2071 y(\026)1987 2063 y FK(\))2013 2046 y Fj(2)2073 2034 y Fw(+)2264 1979 y Fu(H)2318 1987 y Fl(\000)p Fm(i)p Fj(\010)2425 2000 y Fm(j)2456 1987 y(\026)p 2166 2015 V 2166 2019 a Fa(P)2236 2082 y Fm(k)2272 2063 y FK(\()p Fu(H)2352 2071 y Fl(\000)p Fm(i)p Fj(\010)2459 2086 y Fm(k)2496 2071 y(\026)2535 2063 y FK(\))2561 2046 y Fj(2)2617 2034 y Fw(\010)2677 2046 y Fu(j)2712 1942 y Fp(o)83 2238 y FI(on)20 b Fo(D)251 2250 y FK(1)288 2238 y FI(,)h(or)f(alternati)n(v) o(ely)e(the)i(form)1048 2443 y Fo(F)11 b Fy(T)h Fo(F)1279 2413 y Fv(\000)p FK(1)1392 2443 y Fw(=)1494 2410 y Fu(i)p 1489 2424 34 4 v 1489 2471 a FK(2)1546 2380 y Fp(P)1634 2468 y Fu(j)1683 2351 y Fp(n)1738 2443 y Fy(d)1781 2455 y Fu(j)1911 2387 y(M)1974 2396 y Fm(d)2005 2409 y(j)2036 2396 y(m)p 1826 2424 349 4 v 1826 2471 a Fu(M)1889 2459 y Fb(P)1939 2505 y Fm(k)1976 2490 y Fj(\()p Fm(d)2029 2505 y(k)2065 2490 y(m)p Fj(\))2138 2478 y(2)2204 2443 y Fw(+)2381 2387 y Fu(M)2444 2396 y Fm(d)2475 2409 y(j)2506 2396 y(m)p 2297 2424 V 2297 2471 a Fu(M)2360 2459 y Fb(P)2410 2505 y Fm(k)2446 2490 y Fj(\()p Fm(d)2499 2505 y(k)2535 2490 y(m)p Fj(\))2608 2478 y(2)2669 2443 y Fy(d)2712 2455 y Fu(j)2748 2351 y Fp(o)3609 2443 y FI(\(7.2\))83 2664 y(on)20 b Fo(F)11 b(D)336 2676 y FK(1)396 2664 y FI(\(note)20 b(that)h(the)g(last)g(e)o(xpression)f(is)h(well-de\002ned) f(on)g Fo(F)11 b(D)2135 2676 y FK(1)2173 2664 y FI(,)21 b(since)g Fy(m)k Fw(=)e Fo(F)11 b Fw(\()p Fy(\026)p Fw(\))23 b FI(is)f(of)e(class)i Fy(C)3230 2634 y FK(2)3289 2664 y FI(in)f(the)g(sense)g(of)83 2763 y(De\002nition)f(7.1\).)249 2863 y(In)e(simple)g(situations,)g(e)n(v)o(erything)e(can)i(be)g (calculated)f(e)o(xplicitly)-5 b(.)17 b(F)o(or)g(instance,)h(when)g Fy(G)23 b Fw(=)g Fq(Z)3113 2833 y Fu(d)3152 2863 y FI(,)18 b(the)g(Haar)g(measure)83 2963 y Fy(\032)j FI(is)g(the)f(counting)e (measure,)i(and)f(the)h(most)h(natural)e(real)h(characters)f Fw(\010)2233 2975 y Fu(j)2289 2963 y FI(are)h(the)g(position)g (operators)e(gi)n(v)o(en)h(by)1273 3145 y Fw(\(\010)1365 3157 y Fu(j)1400 3145 y Fy(')p Fw(\)\()p Fy(g)s Fw(\))24 b(:=)e Fy(g)1767 3157 y Fu(j)1802 3145 y Fy(')p Fw(\()p Fy(g)s Fw(\))p Fy(;)180 b(')24 b Fx(2)f Fr(L)2359 3111 y FK(2)2397 3145 y Fw(\()p Fq(Z)2484 3111 y Fu(d)2523 3145 y Fw(\))p Fy(;)83 3328 y FI(where)15 b Fy(g)342 3340 y Fu(j)393 3328 y FI(is)i(the)f Fy(j)5 b FI(-th)16 b(component)e(of)h Fy(g)26 b Fx(2)d Fq(Z)1401 3298 y Fu(d)1440 3328 y FI(.)17 b(The)e(operators)g Fy(H)2018 3340 y Fu(\026)2079 3328 y FI(and)g Fw(\()p Fy(H)2323 3298 y Fv(0)2316 3348 y Fu(\026)2361 3328 y Fw(\))2393 3298 y FK(2)2447 3328 y FI(are)h(unitarily)f(equi)n(v)n(alent)f(to)i (multiplication)83 3449 y(operators)30 b(on)556 3428 y Fp(b)540 3449 y Fy(G)45 b Fw(=)f(\()p Fx(\000)p Fy(\031)s(;)14 b(\031)s Fw(])1016 3419 y Fu(d)1055 3449 y FI(.)33 b(Since)f(the)g (measures)f Fy(\026)i FI(and)e Fw(\010)2096 3461 y Fu(j)2131 3449 y Fy(\026)h FI(ha)n(v)o(e)g(compact)e(\(and)h(thus)h(\002nite\))g (support,)e(these)83 3549 y(operators)e(are)i(just)h(multiplication)d (operators)h(by)g(polynomials)f(of)i(\002nite)g(de)o(gree)f(in)h(the)g (v)n(ariables)3202 3546 y Fw(e)3239 3518 y Fv(\000)p Fu(i\030)3344 3526 y Fj(1)3381 3549 y Fy(;)14 b(:)g(:)g(:)g(;)3566 3546 y Fw(e)3603 3518 y Fv(\000)p Fu(i\030)3708 3527 y Fm(d)3747 3549 y FI(,)83 3648 y(with)k Fy(\030)285 3660 y Fu(j)343 3648 y Fx(2)24 b Fw(\()p Fx(\000)p Fy(\031)s(;)14 b(\031)s Fw(])p 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5240 y Fu(d)1276 5370 y Fy(H)7 b Fw(\()p Fy(x)p Fw(\))23 b(=)1574 5367 y(e)1611 5336 y Fv(\000)p Fu(ix)p Fv(\001)p Fu(Q)1814 5370 y Fy(H)1883 5382 y Fu(\026)1941 5367 y Fw(e)1978 5336 y Fu(ix)p Fv(\001)p Fu(Q)2139 5370 y Fw(=)f Fy(h)p Fw(\()p Fy(P)31 b Fw(+)18 b Fy(x)p Fw(\))p Fy(;)p Black 1884 5670 a FI(24)p Black eop end %%Page: 25 25 TeXDict begin 25 24 bop Black Black 83 307 a FI(and)22 b Fy(H)302 277 y Fv(0)353 307 y Fw(=)27 b Fy(h)493 277 y Fv(0)516 307 y Fw(\()p Fy(P)12 b Fw(\))p FI(.)23 b(So)g(Assumption)f (2.3)f(is)j(directly)e(v)o(eri\002ed)f(and)h(Assumption)g(2.2)g(follo)n (ws)g(from)g(\(7.3\))n(.)h(Therefore)d(all)83 407 y(the)j(results)h(of) f(the)g(pre)n(vious)e(sections)i(are)h(v)n(alid.)e(W)-7 b(e)24 b(do)f(not)g(gi)n(v)o(e)f(more)h(details)g(since)g(man)o(y)f (aspects)i(of)f(this)g(e)o(xample,)83 506 y(including)31 b(the)h(e)o(xistence)g(of)g(time)h(delay)-5 b(,)31 b(ha)n(v)o(e)h (already)f(been)h(e)o(xtensi)n(v)o(ely)f(discussed)h(in)h([37)n(].)g(W) -7 b(e)33 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a(\(b\))p Black 41 w(for)g(an)o(y)h Fy(g)s(;)14 b(h)22 b Fx(2)h Fy(X)7 b FI(,)20 b(one)g(has)g Fw(#)p Fx(f)p Fy(N)1352 3808 y Fv(\000)1408 3838 y Fw(\()p Fy(g)s Fw(\))e Fx(\\)h Fy(N)1683 3808 y Fv(\000)1739 3838 y Fw(\()p Fy(h)p Fw(\))p Fx(g)k Fw(=)f(#)p Fx(f)p Fy(N)2190 3808 y FK(+)2245 3838 y Fw(\()p Fy(g)s Fw(\))d Fx(\\)f Fy(N)2520 3808 y FK(+)2575 3838 y Fw(\()p Fy(h)p Fw(\))p Fx(g)p FI(.)249 4000 y(It)31 b(is)h(pro)o(v)o(ed)d(in)i([21)n(,)h(Lemma)e(5.3])g(that)h(for)f (admissible)h(graphs)e(there)i(e)o(xists)g(a)h(unique)d(\(up)h(to)h (constant\))f(map)83 4099 y Fw(\010)23 b(:)g Fy(X)30 b Fx(!)23 b Fq(Z)e FI(satisfying)e Fw(\010\()p Fy(h)p Fw(\))g(+)f(1)23 b(=)f(\010\()p Fy(g)s Fw(\))f FI(whene)n(v)o(er)d Fy(h)23 b(<)g(g)s FI(.)d(W)m(ith)g(this)h(choice)f(of)g(operator)e Fw(\010)p FI(,)i(one)g(obtains)g(that)1281 4293 y Fw([)p Fy(H)7 b Fw(\()p Fy(x)p Fw(\))p Fy(')p Fw(]\()p Fy(g)s Fw(\))24 b(=)1790 4214 y Fp(X)1787 4393 y Fu(h)p Fv(\030)p Fu(g)1926 4290 y Fw(e)1963 4259 y Fu(ix)p 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Fw(\(\010\))p FI(,)h(and)e(that)i(Assumption)e(2.2)g(holds.)g (It)i(follo)n(ws)e(that)i(the)83 5421 y(general)19 b(results)i (presented)e(before)f(apply)-5 b(.)p Black 1884 5670 a(25)p Black eop end %%Page: 26 26 TeXDict begin 26 25 bop Black Black 249 307 a FI(No)n(w)-5 b(,)22 b(the)h(operator)e Fy(H)948 277 y Fv(0)994 307 y FI(acts)j(as)f Fw(\()p Fy(H)1347 277 y Fv(0)1370 307 y Fy(')p Fw(\)\()p Fy(g)s Fw(\))29 b(=)e Fy(i)1713 240 y Fp(\000)1765 245 y(P)1852 332 y Fu(h>g)1996 307 y Fy(')p Fw(\()p Fy(h)p Fw(\))21 b Fx(\000)2268 245 y Fp(P)2355 332 y Fu(hg)1974 587 y Fy(')p Fw(\()p Fy(h)p Fw(\))f(=)g(0)g(=)2403 524 y Fp(P)2491 612 y Fu(h)c Fw(0)22 b FI(and)i Fs(C)30 b Fy(<)d Fx(1)83 829 y FI(such)20 b(that)1179 929 y Fw(lim)1171 983 y Fu(")p Fv(&)p FK(0)1316 858 y Fp(\015)1316 908 y(\015)1362 862 y(\002\000)1435 929 y Fy(H)1511 895 y Fv(0)1534 929 y Fw(\()p Fy(\030)t Fw(\))1638 862 y Fp(\001)1677 879 y FK(2)1732 929 y Fw(+)e Fy(")1854 862 y Fp(\003)1889 879 y Fv(\000)p FK(1)1978 929 y 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b(no)n(w)f(gi)n(v)o(e)g(an)g(e)o(xample)f(of)h(quantum)e(w)o(a)n (v)o(e)o(guide-type)f(\002tting)j(into)g(this)h(setting)f(\(see)h([35)n (])g(for)e(more)h(details\).)83 1439 y(Let)31 b Fw(\006)f FI(be)h(a)f(bounded)e(open)h(connected)g(set)i(in)f Fq(R)1647 1408 y Fu(m)1710 1439 y FI(,)h(and)f(consider)f(in)h(the)h(Hilbert)f (space)g Fr(L)2971 1408 y FK(2)3009 1439 y Fw(\(\006)c Fx(\002)f Fq(R)p Fw(\))32 b FI(the)e(Dirichlet)83 1538 y(Laplacian)24 b Fx(\000)p Fw(\001)570 1550 y FK(D)624 1538 y FI(.)h(The)f(partial)h(F)o(ourier)e(transformation)g(along)g (the)i(longitudinal)e(axis)i(sends)g(the)g(initial)g(Hilbert)g(space)83 1647 y(onto)31 b(the)g(direct)g(inte)o(gral)g Fx(H)45 b Fw(:=)1148 1580 y Fp(R)1204 1600 y Fv(\010)1188 1676 y Fn(R)1274 1647 y Fw(d)p Fy(\030)18 b Fx(H)1444 1659 y FK(0)1481 1647 y FI(,)32 b(with)g Fx(H)1784 1659 y FK(0)1865 1647 y Fw(:=)44 b Fr(L)2034 1617 y FK(2)2071 1647 y Fw(\(\006\))p FI(,)33 b(and)e(it)h(sends)g Fx(\000)p Fw(\001)2830 1659 y FK(D)2916 1647 y FI(onto)e(the)i(\002bered)e (operator)83 1767 y Fy(H)g Fw(:=)293 1700 y Fp(R)348 1721 y Fv(\010)332 1796 y Fn(R)418 1767 y Fw(d)p Fy(\030)18 b(H)7 b Fw(\()p Fy(\030)t Fw(\))p FI(,)20 b(with)g Fy(H)7 b Fw(\()p Fy(\030)t Fw(\))24 b(:=)e Fy(\030)1261 1737 y FK(2)1315 1767 y Fx(\000)16 b Fw(\001)1465 1737 y FK(\006)1465 1790 y(D)1520 1767 y FI(.)k(Here,)f Fx(\000)p Fw(\001)1897 1737 y FK(\006)1897 1790 y(D)1971 1767 y FI(denotes)g(the)h(Dirichlet)f (Laplacian)g(in)h Fw(\006)p FI(.)g(By)g(Choosing)e(for)83 1887 y Fw(\010)k FI(the)g(position)f(operator)f Fy(Q)i FI(along)e(the)i(longitudinal)e(axis)i(one)f(obtains)g(that)h Fy(H)7 b Fw(\()p Fy(x)p Fw(\))26 b(=)2731 1820 y Fp(R)2786 1841 y Fv(\010)2770 1917 y Fn(R)2856 1887 y Fw(d)p Fy(\030)18 b(H)7 b Fw(\()p Fy(\030)t(;)14 b(x)p Fw(\))23 b FI(with)f Fy(H)7 b Fw(\()p Fy(\030)t(;)14 b(x)p Fw(\))26 b(=)83 1987 y(\()p Fy(\030)17 b Fw(+)12 b Fy(x)p Fw(\))324 1957 y FK(2)374 1987 y Fx(\000)g Fw(\001)520 1957 y FK(\006)520 2010 y(D)574 1987 y FI(.)18 b(Clearly)-5 b(,)18 b Fy(H)7 b Fw(\()p Fy(\030)t Fw(\))20 b FI(and)e Fy(H)7 b Fw(\()p Fy(\030)t(;)14 b(x)p Fw(\))20 b FI(commute,)c(and)i(so)h(do)f Fy(H)26 b FI(and)18 b Fy(H)7 b Fw(\()p Fy(x)p Fw(\))p FI(.)20 b(Furthermore,)15 b(the)k(operator)e Fy(H)25 b FI(is)20 b(of)83 2086 y(class)g Fy(C)329 2056 y Fv(1)399 2086 y Fw(\(\010\))p FI(,)g(and)e Fy(H)779 2056 y Fv(0)822 2086 y FI(is)h(the)g(\002bered)f(operator)f(gi)n(v)o(en)g(by)i Fy(H)1942 2056 y Fv(0)1965 2086 y Fw(\()p Fy(\030)t Fw(\))24 b(=)e(2)p Fy(\030)t FI(.)d(It)g(follo)n(ws)f(that)h(both)f(Assumptions) g(2.2)g(and)g(2.3)83 2186 y(hold,)k(and)h(thus)h(the)f(general)f (theory)g(applies.)h(No)n(w)g(a)h(simple)g(calculation)e(using)h (\(7.6\))f(sho)n(ws)h(that)h Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))29 b(=)f Fy(\033)s Fw(\()p Fx(\000)p Fw(\001)3660 2156 y FK(\006)3660 2209 y(D)3715 2186 y Fw(\))p FI(.)83 2286 y(Furthermore,)d(in)i(the)g(tensorial)g(representation)e Fr(L)1617 2255 y FK(2)1655 2286 y Fw(\(\006\))f Fx(\012)f Fr(L)1928 2255 y FK(2)1966 2286 y Fw(\()p Fq(R)p Fw(\))28 b FI(of)f Fr(L)2252 2255 y FK(2)2289 2286 y Fw(\(\006)d Fx(\002)g Fq(R)p Fw(\))p FI(,)j(one)g(obtains)g(that)g Fy(T)3251 2298 y Fu(f)3330 2286 y Fw(=)35 b Fy(T)47 b Fw(=)3637 2253 y FK(1)p 3637 2267 34 4 v 3637 2314 a(4)3703 2286 y Fx(\012)83 2385 y Fw(\()p Fy(QP)246 2355 y Fv(\000)p FK(1)353 2385 y Fw(+)18 b Fy(P)501 2355 y Fv(\000)p FK(1)590 2385 y Fy(Q)p Fw(\))j FI(on)f(the)g(dense)g(set)599 2568 y Fo(D)663 2580 y FK(1)724 2568 y Fw(=)811 2501 y Fp(\010)860 2568 y Fy(')j Fx(2)g Fr(L)1052 2534 y FK(2)1090 2568 y Fw(\(\006\))c Fx(\012)f(D)r Fw(\()p Fx(h)p Fy(Q)p Fx(i)p Fw(\))24 b Fx(j)f Fy(')h Fw(=)e Fy(\021)s Fw(\()p Fx(\000)p Fw(\001)2021 2580 y FK(D)2076 2568 y Fw(\))p Fy(')f FI(for)f(some)g Fy(\021)26 b Fx(2)d Fy(C)2707 2534 y Fv(1)2701 2588 y FK(c)2778 2501 y Fp(\000)2816 2568 y Fq(R)18 b Fx(n)g Fy(\024)p Fw(\()p Fy(H)7 b Fw(\))3142 2501 y Fp(\001)q(\011)3229 2568 y Fy(;)83 2761 y FI(and)24 b Fy(T)277 2773 y Fu(f)345 2761 y FI(is)i(equal)f(to)g Fy(i)780 2729 y FK(d)p 760 2743 77 4 v 760 2790 a(d)p Fu(\025)872 2761 y FI(in)g(the)g(spectral)g (representation)d(of)j Fx(\000)p Fw(\001)2097 2773 y FK(D)2151 2761 y FI(.)g(In)g([35)o(])g(it)g(is)h(e)n(v)o(en)e(sho)n(wn) g(that)h(the)g(quantum)e(time)83 2861 y(delay)d(e)o(xists)g(and)g(is)h (gi)n(v)o(en)e(by)h(F)o(ormula)f(\(6.4\))f(for)i(appropriate)e (scattering)h(pairs)h Fx(f\000)p Fw(\001)2713 2873 y FK(D)2767 2861 y Fy(;)14 b Fx(\000)p Fw(\001)2938 2873 y FK(D)3010 2861 y Fw(+)k Fy(V)h Fx(g)p FI(.)83 3141 y Fz(Ackno)o(wledgements)83 3327 y FI(S.)j(Richard)e(is)i(supported)e (by)g(the)i(Swiss)g(National)f(Science)g(F)o(oundation.)d(R.)k(T)m (iedra)f(de)g(Aldecoa)f(is)i(partially)f(supported)83 3426 y(by)e(the)h(N)7 b(\264)-35 b(ucleo)19 b(Cient)n(\264)-26 b(\021\002co)20 b(ICM)g(P07-027-F)d(\223Mathematical)i(Theory)f(of)h (Quantum)f(and)i(Classical)h(Magnetic)d(Systems\224)83 3526 y(and)i(by)f(the)i(Chilean)f(Science)g(F)o(oundation)d(F)o(ondec)o (yt)i(under)f(the)j(Grant)e(1090008.)83 3806 y Fz(Refer)n(ences)p Black 83 3992 a FI([1])p Black 40 w(W)-8 b(.)32 b(O.)e(Amrein,)f(A.)i (Boutet)f(de)g(Mon)m(v)o(el)e(and)i(V)-11 b(.)30 b(Geor)o(gescu.)60 b Fy(C)2264 4004 y FK(0)2301 3992 y FJ(-gr)l(oups,)29 b(commutator)g(methods)h(and)f(spectr)o(al)221 4091 y(theory)20 b(of)h Fy(N)9 b FJ(-body)18 b(Hamiltonians)p FI(,)h(v)n(olume)g(135)g (of)h FJ(Pr)l(o)o(gr)m(ess)h(in)f(Math.)29 b FI(Birkh)5 b(\250)-33 b(auser)m(,)19 b(Basel,)i(1996.)p Black 83 4257 a([2])p Black 40 w(W)-8 b(.)23 b(O.)f(Amrein)g(and)f(M.)h(B.)h (Cibils.)35 b(Global)22 b(and)f(Eisenb)n(ud-Wigner)e(time)k(delay)e(in) h(scattering)f(theory)-5 b(.)33 b FJ(Helv)-6 b(.)22 b(Phys.)221 4357 y(Acta)e FI(60:)g(481\226500,)d(1987.)p Black 83 4523 a([3])p Black 40 w(W)-8 b(.)20 b(O.)e(Amrein,)g(M.)g(B.)i(Cibils)f (and)f(K.)g(B.)i(Sinha.)25 b(Con\002guration)16 b(space)i(properties)f (of)h(the)h Fy(S)5 b FI(-matrix)17 b(and)h(time)g(delay)221 4623 y(in)j(potential)e(scattering.)28 b FJ(Ann.)20 b(Inst.)g(Henri)g (P)-7 b(oincar)1813 4624 y(\264)1809 4623 y(e)20 b FI(47:)g (367\226382,)d(1987.)p Black 83 4789 a([4])p Black 40 w(W)-8 b(.)23 b(O.)e(Amrein)g(and)g(Ph.)g(Jacquet.)32 b(T)m(ime)22 b(delay)e(for)h(one-dimensional)d(quantum)i(systems)i (with)f(steplik)o(e)g(potentials.)221 4888 y FJ(Phys.)f(Re)o(v)-6 b(.)20 b(A)g FI(75\(2\):)f(022106,)f(2007.)p Black 83 5054 a([5])p Black 40 w(I.)25 b(Antoniou,)e(I.)h(Prigogine,)f(V)-11 b(.)25 b(Sado)o(vnichii)e(and)h(S.)h(A.)g(Shkarin.)42 b(T)m(ime)25 b(operator)d(for)i(dif)n(fusion.)42 b FJ(Chaos)24 b(Solitons)221 5154 y(F)-5 b(r)o(actals)21 b FI(11\(4\):)d(465\226477,) f(2000.)p Black 83 5320 a([6])p Black 40 w(A.)23 b(Arai.)35 b(Generalized)21 b(W)-7 b(e)o(yl)23 b(relation)e(and)h(decay)f(of)h (quantum)f(dynamics.)34 b FJ(Re)o(v)-6 b(.)21 b(Math.)h(Phys.)g FI(17\(9\):)f(1071\2261109,)221 5420 y(2005.)p Black 1884 5670 a(27)p Black eop end %%Page: 28 28 TeXDict begin 28 27 bop Black Black Black 83 307 a FI([7])p Black 40 w(M.)19 b(Sh.)g(Birman)f(and)g(M.)h(Z.)g(Solomjak.)24 b FJ(Spectr)o(al)18 b(theory)g(of)h(selfadjoint)f(oper)o(ator)o(s)g(in) h(Hilbert)g(space)p FI(.)25 b(Mathematics)221 407 y(and)c(its)i (Applications)d(\(So)o(viet)h(Series\).)h(D.)g(Reidel)g(Publishing)e (Co.,)h(Dordrecht,)f(1987.)32 b(T)m(ranslated)21 b(from)f(the)i(1980) 221 506 y(Russian)f(original)e(by)h(S.)g(Khrushch)5 b(\250)-33 b(ev)18 b(and)i(V)-11 b(.)21 b(Peller)-5 b(.)p Black 83 671 a([8])p Black 40 w(A.)20 b(Boutet)g(de)g(Mon)m(v)o(el)d(and)j(V) -11 b(.)20 b(Geor)o(gescu.)26 b(The)19 b(method)f(of)i(dif)n(ferential) e(inequalities.)27 b(In)19 b FJ(Recent)h(de)o(velopments)d(in)221 770 y(quantum)i(mec)o(hanics)g FI(pp.)g(279\226298.)26 b(Math.)20 b(Phys.)g(Stud.)f(12,)h(Kluwer)g(Acad.)f(Publ.,)h (Dordrecht,)e(1991.)p Black 83 935 a([9])p Black 40 w(A.)28 b(Boutet)g(de)f(Mon)m(v)o(el,)e(G.)j(Kazantse)n(v)n(a)f(and)g(M.)g(M)5 b(\013)-33 b(antoiu.)51 b(Some)28 b(anisotropic)e(Schr)7 b(\250)-35 b(odinger)24 b(operators)i(without)221 1034 y(singular)20 b(spectrum.)28 b FJ(Helv)-6 b(.)20 b(Phys.)g(Acta)g FI(69\(1\):)e(13\22625,)g(1996.)p Black 83 1199 a([10])p Black 40 w(E.)25 b(B.)i(Da)n(vies.)46 b FJ(Spectr)o(al)25 b(theory)g(and)f(dif)o(fer)m(ential)h(oper)o(ator)o(s)p FI(,)g(v)n(olume)f(42)h(of)g FJ(Cambridg)o(e)g(Studies)g(in)h(Advanced) 221 1298 y(Mathematics)j FI(Cambridge)19 b(Uni)n(v)o(ersity)g(Press,)i (Cambridge,)d(1995.)p Black 83 1462 a([11])p Black 40 w(G.)25 b(B.)h(F)o(olland.)45 b FJ(A)25 b(cour)o(se)g(in)h(abstr)o(act) f(harmonic)f(analysis)p FI(.)45 b(Studies)25 b(in)g(Adv)n(anced)f (Mathematics.)g(CRC)j(Press,)221 1562 y(Boca)21 b(Raton,)f(1995.)p Black 83 1726 a([12])p Black 40 w(E.)k(A.)g(Galapon.)41 b(P)o(auli')-5 b(s)24 b(theorem)f(and)g(quantum)g(canonical)f(pairs:)i (the)h(consistenc)o(y)d(of)i(a)h(bounded,)c(self-adjoint)221 1826 y(time)h(operator)d(canonically)h(conjugate)f(to)i(a)h (Hamiltonian)d(with)j(non-empty)c(point)j(spectrum.)31 b FJ(R.)21 b(Soc.)f(Lond.)h(Pr)l(oc.)221 1926 y(Ser)-9 b(.)20 b(A)h(Math.)f(Phys.)f(Eng)o(.)g(Sci.)h FI(458:)g(451\226472,)d (2002.)p Black 83 2090 a([13])p Black 40 w(V)-11 b(.)25 b(Geor)o(gescu)d(and)i(C.)h(G)5 b(\264)-33 b(erard.)42 b(On)24 b(the)h(virial)f(theorem)f(in)i(quantum)d(mechanics.)42 b FJ(Commun.)23 b(Math.)h(Phys.)g FI(208:)221 2189 y(275\226281,)17 b(1999.)p Black 83 2354 a([14])p Black 40 w(F)-7 b(.)21 b(G)7 b(\264)-35 b(omez.)28 b(Self-adjoint)19 b(time)h(operators)f(and) h(in)m(v)n(ariant)e(subspaces.)29 b FJ(Rep.)19 b(Math.)h(Phys.)g FI(61\(1\):)f(123\226148,)e(2008.)p Black 83 2518 a([15])p Black 40 w(C.)25 b(G)5 b(\264)-33 b(erard)23 b(and)g(R.)i(T)m(iedra)f (de)g(Aldecoa.)41 b(Generalized)22 b(de\002nition)h(of)h(time)h(delay)e (in)h(scattering)g(theory)-5 b(.)40 b FJ(J)n(.)24 b(Math.)221 2618 y(Phys.)c FI(page)g(122101,)d(2007.)p Black 83 2782 a([16])p Black 40 w(T)-6 b(.)18 b(Got)7 b(\257)-35 b(o,)17 b(K.)h(Y)-8 b(amaguchi)15 b(and)j(N.)g(Sud)7 b(\257)-35 b(o.)23 b(On)18 b(the)f(time)h(operator)e(in)i(quantum)e(mechanics.)h (Three)g(typical)g(e)o(xamples.)221 2882 y FJ(Pr)l(o)o(gr)-9 b(.)20 b(Theor)m(et.)g(Phys.)g FI(66\(5\):)f(1525\2261538,)d(1981.)p Black 83 3046 a([17])p Black 40 w(F)-7 b(.)17 b(Hiroshima,)f(S.)h(K)o (uribayashi)e(and)i(Y)-11 b(.)17 b(Matsuza)o(w)o(a.)k(Strong)16 b(time)h(operators)e(associated)i(with)g(generalized)e(Hamil-)221 3145 y(tonians.)29 b FJ(Lett.)21 b(Math.)e(Phys.)h FI(87\(1-2\):)e (115\226123,)f(2009.)p Black 83 3310 a([18])p Black 40 w(P)-9 b(.)16 b(T)-6 b(.)16 b(J\370r)o(gensen)e(and)h(P)-9 b(.)16 b(S.)g(Muhly)-5 b(.)18 b(Self)o(adjoint)d(e)o(xtensions)g (satisfying)g(the)h(We)o(yl)g(operator)e(commutation)f(relations.)221 3409 y FJ(J)n(.)21 b(Analyse)e(Math.)h FI(37:)g(46\22699,)e(1980.)p Black 83 3574 a([19])p Black 40 w(M.)f(Miyamoto.)j(A)d(generalized)e (We)o(yl)i(relation)f(approach)f(to)i(the)f(time)h(operator)e(and)i (its)g(connection)e(to)i(the)g(survi)n(v)n(al)221 3673 y(probability)-5 b(.)27 b FJ(J)n(.)20 b(Math.)g(Phys.)g FI(42\(3\):)f(1038\2261052,)d(2001.)p Black 83 3838 a([20])p Black 40 w(A.)24 b(Mohapatra,)e(K.)i(B.)h(Sinha)e(and)g(W)-8 b(.)25 b(O.)g(Amrein.)40 b(Con\002guration)21 b(space)j(properties)f (of)g(the)h(scattering)f(operator)221 3937 y(and)31 b(time)g(delay)f (for)g(potentials)h(decaying)e(lik)o(e)i Fx(j)p Fy(x)p Fx(j)1825 3907 y Fv(\000)p Fu(\013)1925 3937 y Fy(;)d(\013)43 b(>)g Fw(1)p FI(.)63 b FJ(Ann.)30 b(Inst.)h(H.)g(P)-7 b(oincar)3047 3938 y(\264)3043 3937 y(e)30 b(Phys.)h(Th)3416 3938 y(\264)3412 3937 y(eor)-9 b(.)30 b FI(57\(1\):)221 4037 y(89\226113,)18 b(1992.)p Black 83 4201 a([21])p Black 40 w(M.)24 b(M)5 b(\013)-33 b(antoiu,)23 b(S.)i(Richard)f(and)g (R.)h(T)m(iedra)f(de)g(Aldecoa.)41 b(Spectral)24 b(analysis)h(for)f (adjacenc)o(y)e(operators)h(on)h(graphs.)221 4301 y FJ(Ann.)c(Henri)g (P)-7 b(oincar)877 4302 y(\264)873 4301 y(e)20 b FI(8\(7\):)f (1401\2261423,)d(2007.)p Black 83 4465 a([22])p Black 40 w(M.)27 b(M)5 b(\013)-33 b(antoiu)27 b(and)g(R.)h(T)m(iedra)e(de)i (Aldecoa.)51 b(Spectral)27 b(analysis)h(for)e(con)m(v)n(olution)f (operators)h(on)h(locally)g(compact)221 4565 y(groups.)h FJ(J)n(.)20 b(Funct.)f(Anal.)h FI(253\(2\):)e(675\226691,)f(2007.)p Black 83 4729 a([23])p Black 40 w(J.)k(G.)f(Muga)f(and)h(C.)h(R.)g(Lea) n(v)o(ens.)28 b(Arri)n(v)n(al)19 b(time)i(in)f(quantum)e(mechanics.)28 b FJ(Phys.)20 b(Rep.)g FI(338\(4\):)e(353\226438,)f(2000.)p Black 83 4893 a([24])p Black 40 w(J.)j(G.)g(Muga,)f(R.)h(Sala)g(Mayato) f(and)1346 4875 y(\264)1346 4893 y(I.)g(L.)h(Egusquiza,)e(editors.)28 b FJ(T)-5 b(ime)21 b(in)f(quantum)d(mec)o(hanics.)i(Vol.)g(1)p FI(,)h(v)n(olume)f(734)221 4993 y(of)h FJ(Lectur)m(e)h(Notes)f(in)h (Physics)p FI(.)29 b(Springer)m(,)18 b(Berlin,)i(second)g(edition,)f (2008.)p Black 83 5157 a([25])p Black 40 w(M.)h(Raza)n(vy)-5 b(.)29 b(T)m(ime)20 b(of)g(arri)n(v)n(al)f(operator)-5 b(.)28 b FJ(Canad.)19 b(J)n(.)h(Phys.)g FI(49:)g(3075\2263081,)c(1971.) p Black 83 5321 a([26])p Black 40 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