Content-Type: multipart/mixed; boundary="-------------0308211433418" This is a multi-part message in MIME format. ---------------0308211433418 Content-Type: text/plain; name="03-377.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-377.keywords" Anderson localization, random Schr\"odinger operator, fractional moment method, ---------------0308211433418 Content-Type: application/postscript; name="localization_03.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="localization_03.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: localization_03.dvi %%CreationDate: Thu Aug 21 14:31:58 2003 %%Pages: 61 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR8 CMCSC10 CMTI8 CMR7 CMR10 CMMI10 MSBM10 %%+ CMMI7 CMSY10 CMEX10 CMSY7 CMMIB10 CMMIB7 CMTI10 CMMI5 CMR5 CMSY5 %%+ MSBM7 MSAM10 CMBSY10 CMBX7 CMMI8 CMMI6 CMSY8 CMBX8 CMTT8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf localization_03 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.08.21:1430 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 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 183 /lessequal put dup 106 /bar put dup 114 /nabla put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 112 /p put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 76 /L put dup 86 /V put dup 100 /d put dup 117 /u put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 107 /k put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBSY10 %!PS-AdobeFont-1.1: CMBSY10 1.00 %%CreationDate: 1992 Jul 23 21:21:18 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY10 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 114 /nabla put readonly def /FontBBox{-27 -940 1332 825}readonly def /UniqueID 5000762 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 164 /square put dup 46 /lessorsimilar put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: 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 161 /minus put dup 48 /prime put dup 49 /infinity put dup 73 /I put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 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 164 /Lambda put dup 173 /Omega 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 56 /eight put dup 59 /semicolon put dup 100 /d put dup 105 /i put dup 115 /s put dup 116 /t put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 179 /zeta put dup 180 /eta put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 76 /L put dup 93 /sharp put dup 96 /lscript put dup 99 /c put dup 106 /j put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 120 /x put dup 121 /y put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 175 /fi put dup 177 /ffi put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 196 /dieresis put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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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 65 /A put readonly def /FontBBox{0 -250 1294 750}readonly def /UniqueID 5087389 def currentdict end currentfile eexec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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 65 /A put dup 68 /D put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 def currentdict end currentfile eexec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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. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 167 /plusminus put dup 180 /equivalence put dup 183 /lessequal put dup 184 /greaterequal put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 67 /C put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 83 /S 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 121 /dagger put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 175 /vextendsingle put dup 176 /vextenddouble put dup 177 /slashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 189 /braceleftbigg put dup 190 /bracerightbigg put dup 195 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 101 /tildewide put dup 102 /tildewider put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 167 /plusminus put dup 180 /equivalence put dup 183 /lessequal put dup 184 /greaterequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 33 /arrowright put dup 47 /proportional put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 59 /emptyset put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 79 /O put dup 82 /R put dup 83 /S put dup 85 /U put dup 91 /union put dup 92 /intersection put dup 94 /logicaland 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 dup 114 /nabla put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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%%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 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 189 /rho put dup 191 /tau put dup 193 /phi put dup 195 /psi put dup 33 /omega put dup 34 /epsilon 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 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 90 /Z put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B B078D20B4B8A6D7F0300CF694E6956FF9C29C84FCC5C9E8890AA56B1BC60E868 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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. 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 182 /acute put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 196 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 163 /Theta put dup 164 /Lambda put dup 173 /Omega put dup 185 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 100 /d put dup 105 /i put dup 115 /s put dup 116 /t put dup 126 /tilde put dup 196 /dieresis put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 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 174 /ff put dup 182 /acute put dup 191 /oslash put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 196 /dieresis put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 162 /Delta put dup 174 /ff put dup 175 /fi put dup 177 /ffi put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 196 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 175 /fi put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put dup 123 /endash put dup 196 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (localization_03.dvi) @start /Fa 133[470 470 3[470 470 470 470 1[470 470 470 470 470 470 1[470 470 470 1[470 470 470 470 470 13[470 5[470 3[470 5[470 2[470 17[470 46[{}26 885.568 /CMTT8 rf /Fb 173[811 82[{}1 885.568 /CMBX8 rf /Fc 141[784 7[261 85[732 20[{}3 885.568 /CMSY8 rf /Fd 143[427 33[642 78[{}2 664.176 /CMMI6 rf /Fe 138[545 16[484 13[548 9[640 76[{}4 885.568 /CMMI8 rf /Ff 148[529 107[{}1 774.872 /CMBX7 rf /Fg 141[1061 114[{}1 1106.96 /CMBSY10 rf /Fh 209[861 42[861 3[{}2 1106.96 /MSAM10 rf /Fi 173[560 3[560 10[560 67[{}3 774.872 /MSBM7 rf /Fj 182[454 23[753 244 47[600{}4 553.48 /CMSY5 rf /Fk 139[300 304 9[223 4[415 40[223 2[377 5[377 377 377 4[569 1[300 300 29[530 6[504 3[{}14 553.48 /CMR5 rf /Fl 134[404 438 4[371 372 2[389 488 3[326 6[359 2[331 2[331 16[529 14[407 1[254 25[491 15[412 369 1[362 413 438 506 11[{}20 553.48 /CMMI5 rf /Fm 128[566 3[566 453 537 514 736 509 594 368 453 467 509 566 566 622 905 283 509 339 339 566 509 339 509 566 509 509 566 4[569 7[792 622 807 1[751 2[993 694 851 1[427 823 856 723 751 1[792 779 823 6[339 2[566 566 566 566 566 566 566 2[339 396 339 2[453 453 5[569 19[976 1[622 12[{}60 1106.96 /CMTI10 rf /Fn 190[756 65[{}1 774.872 /CMMIB7 rf /Fo 187[1038 2[962 65[{}2 1106.96 /CMMIB10 rf /Fp 134[406 10[454 2[454 263 358 358 454 454 9[597 597 7[544 6[613 1[606 501 758 4[481 3[692 8[0 3[597 882 255 12[454 1[882 11[692 692 2[692 10[692 2[454 692 263 692{}31 774.872 /CMSY7 rf /Fq 144[738 738 4[523 523 1[1107 615 2[615 7[615 1414 1599 5[523 1[1168 12[969 969 969 969 10[738 738 738 738 969 969 6[892 892 4[646 646 876 876 4[830 830 4[584 584 815 815 661 661 1[640 615 369 2[646 646 4[461 461 507 507{}45 1106.96 /CMEX10 rf /Fr 135[492 5[922 1[922 1[553 2[553 307 430 430 553 553 7[738 1[738 738 5[693 1[670 938 2[881 2[763 1[750 603 935 1[796 1[854 583 1[884 5[553 3[0 0 3[738 1107 1[861 13[1107 5[861 861 861 861 2[861 861 2[861 10[861 3[861 307 861{}43 1106.96 /CMSY10 rf /Fs 133[423 449 502 3[334 418 411 406 456 437 547 786 280 471 366 313 518 432 432 1[461 396 389 480 371 2[358 2[603 2[833 1[603 523 544 667 697 564 1[701 844 607 2[392 725 689 1[650 1[635 669 666 2[692 454 692 263 24[418 558 586 1[532 1[408 1[459 1[403 444 538 525 3[456 403 369 402 465 502 575 11[{}63 774.872 /CMMI7 rf /Ft 165[738 7[799 3[799 8[738 1[799 67[{}5 1106.96 /MSBM10 rf /Fu 133[515 543 633 792 537 1[400 519 499 494 557 537 664 1[330 576 456 381 638 528 542 515 576 479 475 585 461 2[430 2[756 643 917 1[646 756 647 679 841 875 711 844 889 1074 753 940 1[487 920 870 712 817 916 791 840 830 588 1[861 553 861 307 307 18[724 4[516 689 721 693 660 1[484 633 572 631 484 547 667 646 638 1[520 550 484 449 492 573 626 708 11[{}79 1106.96 /CMMI10 rf /Fv 128[553 553 2[553 492 584 584 799 584 615 430 437 434 584 615 553 615 922 307 584 338 307 615 553 338 492 615 492 615 553 307 1[553 307 553 307 676 830 1[1138 830 830 799 615 815 861 753 861 830 1015 692 861 569 400 830 869 723 753 846 799 784 830 1[523 523 861 1[307 307 553 553 553 553 553 553 553 553 553 553 553 307 369 307 861 1[430 430 307 3[922 553 307 13[553 3[922 922 615 615 646 799 861 799 4[769 861 922 692{}98 1106.96 /CMR10 rf /Fw 133[556 679 679 925 679 679 654 507 667 1[617 704 679 827 568 704 1[335 679 710 593 617 691 654 642 679 4[612 7[870 676 886 8[449 2[789 2[870 853 901 6[354 12[418 10[612 34[{}36 1106.96 /CMCSC10 rf /Fx 128[441 441 9[346 351 9[251 4[489 5[441 251 1[251 537 5[632 489 643 1[595 680 653 796 547 677 452 320 653 685 1[595 666 632 619 653 3[680 1[251 251 441 441 441 441 441 441 441 441 441 441 1[251 1[251 680 1[346 346 17[441 11[632 6[606 680 2[{}50 774.872 /CMR7 rf /Fy 128[486 4[389 462 438 632 437 510 316 389 401 437 486 486 534 777 243 437 1[292 486 437 292 437 486 437 437 486 6[583 3[705 705 680 534 693 729 644 729 705 851 596 1[498 365 705 735 620 644 717 680 668 705 6[292 486 486 4[486 486 486 2[292 340 292 2[389 389 292 10[486 8[486 7[583 11[{}63 885.568 /CMTI8 rf /Fz 128[490 4[445 543 2[543 543 523 405 533 2[563 543 661 455 563 1[268 543 568 474 494 553 523 514 543 6[593 4[721 696 541 2[657 1[721 876 2[502 1[721 754 1[657 2[683 721 18[283 1[283 44[{}37 885.568 /CMCSC10 rf /FA 128[470 3[470 418 497 497 680 497 523 366 371 367 497 523 470 523 784 261 497 288 261 523 470 288 418 523 418 523 470 3[261 1[261 575 705 705 966 705 705 680 523 692 732 640 732 705 862 588 731 483 339 705 738 614 640 718 680 666 705 6[261 470 470 470 470 470 470 470 470 470 470 1[261 314 261 2[366 366 261 18[470 470 470 3[784 1[523 549 9[784 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y(its)439 b(r)-57 b(esolven)440 b(set)f Fu(\275)p Fv(\()p Fu(A)15576 48222 y Fs(B)38 b(S)16925 48056 y Fv(\))440 b Fm(includes)g(the)f(close)-57 b(d)439 b(lower)g(half)g (plane)p 35439 47161 800 45 v 440 w Ft(C)36238 48222 y Fp(\241)36985 48056 y Fm(,)451 b(and)439 b(the)g(Birman-)6075 49385 y(Schwinger)397 b(r)-57 b(elation)16525 51251 y Fv(\()p Fu(A)17785 51417 y Fs(B)38 b(S)19379 51251 y Fr(\241)246 b Fu(\273)51 b FB(1)p Fv(\))22087 50793 y Fp(\241)p Fx(1)23584 51251 y Fv(=)308 b Fu(V)25645 50793 y Fx(1)p Fs(=)p Fx(2)27036 51251 y Fv(\()p Fu(A)28296 51417 y Fx(0)29039 51251 y Fr(\241)246 b Fu(\273)51 b(V)246 b Fv(\))32003 50793 y Fp(\241)p Fx(1)33192 51251 y Fu(V)34084 50793 y Fx(1)p Fs(=)p Fx(2)43650 51251 y Fv(\(2.4\))6075 53124 y Fm(holds)395 b(in)h Fv(\()p Fm(ker)186 b Fu(V)245 b Fv(\))13683 52722 y Fp(?)14827 53124 y Fm(for)396 b(every)h Fu(\273)359 b Fr(2)p 21411 52228 V 307 w Ft(C)22211 53290 y Fp(\241)22958 53124 y Fm(.)6075 55137 y FB(Remarks:)7957 56716 y Fv(\(1\))554 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b Fu(\273)359 b Fr(2)p 11278 14488 800 45 v 308 w Ft(C)12077 15550 y Fp(\241)13193 15384 y Fv(one)370 b(v)-31 b(eri\257es)369 b(explicitly)j(that)14633 17264 y(\()p Fu(A)15893 17430 y Fs(B)38 b(S)17487 17264 y Fr(\241)246 b Fu(\273)51 b FB(1)p Fv(\))p Fu(V)21088 16807 y Fx(1)p Fs(=)p Fx(2)22480 17264 y Fu(A)23310 16791 y Fp(\241)p Fx(1)23310 17559 y(0)24498 17264 y Fu(V)25390 16807 y Fx(1)p Fs(=)p Fx(2)27089 17264 y Fv(=)307 b FB(1)247 b Fr(\241)f Fu(\273)51 b(V)31674 16807 y Fx(1)p Fs(=)p Fx(2)33065 17264 y Fu(A)33895 16791 y Fp(\241)p Fx(1)33895 17559 y(0)35084 17264 y Fu(V)35976 16807 y Fx(1)p Fs(=)p Fx(2)6075 19126 y Fv(is)371 b(the)g(in)-31 b(v)g(erse)372 b(of)g Fu(I)333 b Fv(+)248 b Fu(\273)51 b(V)17388 18725 y Fx(1)p Fs(=)p Fx(2)18779 19126 y Fv(\()p Fu(A)20039 19292 y Fx(0)20784 19126 y Fr(\241)247 b Fu(\273)51 b(V)246 b Fv(\))23749 18725 y Fp(\241)p Fx(1)24938 19126 y Fu(V)25830 18725 y Fx(1)p Fs(=)p Fx(2)27592 19126 y Fv(in)372 b(\(k)-31 b(er)185 b Fu(V)245 b Fv(\))32301 18725 y Fp(?)33049 19126 y Fv(.)499 b(In)371 b(particular,)i(for)e Fu(\273)363 b Fv(=)310 b Fr(\241)p Fu(i)p Fv(,)6075 20455 y(this)480 b(sho)-31 b(ws)480 b(that)g(\()p Fu(A)15402 20621 y Fs(B)38 b(S)17070 20455 y Fv(+)320 b Fu(iI)87 b Fv(\))p Fr(D)31 b Fv(\()p Fu(A)21781 20621 y Fs(B)38 b(S)23130 20455 y Fv(\))492 b(=)f(\(k)-31 b(er)185 b Fu(V)246 b Fv(\))28820 20053 y Fp(?)29567 20455 y Fv(,)508 b(pro)-31 b(ving)481 b(that)f Fu(A)37831 20621 y Fs(B)38 b(S)39659 20455 y Fv(is)479 b(maximally)6075 21783 y(dissipativ)-31 b(e)371 b(\(see)e([65)q(]\).)494 b(Using)370 b(the)f(resolv)-31 b(en)g(t)370 b(iden)-31 b(tit)g(y)372 b(w)-31 b(e)370 b(also)g(get)g(\(2.4\).)6358 b Fh(\244)7403 23830 y Fv(The)378 b(argumen)-31 b(t)379 b(in)e(the)h(ab)31 b(o)-31 b(v)g(e)379 b(pro)31 b(of)377 b(whic)-31 b(h)379 b(sho)-31 b(w)g(ed)378 b(that)h Fu(V)34210 23428 y Fx(1)p Fs(=)p Fx(2)35602 23830 y Fu(A)36432 23356 y Fp(\241)p Fx(1)36432 24125 y(0)37620 23830 y Fu(V)38512 23428 y Fx(1)p Fs(=)p Fx(2)40281 23830 y Fv(is)e(in)-31 b(v)g(ertible)6075 25158 y(do)31 b(es)528 b(not)i(generally)g(carry)e(o)-31 b(v)g(er)529 b(to)h(the)f(case)f(where)h Fu(A)31553 25324 y Fx(0)32578 25158 y Fv(is)g(an)g(in)-31 b(v)g(ertible)531 b(self-adjoin)-31 b(t)6075 26486 y(op)31 b(erator.)487 b(If)353 b Fu(A)12885 26652 y Fx(0)13734 26486 y Fv(has)g(\257xed)f(sign)h(\(as)g(in)g(the)g (classical)h(BS-principle\))g(it)f(do)31 b(es,)356 b(but)d(not)h(if)f (0)6075 27815 y(is)398 b(in)g(a)h(sp)31 b(ectral)398 b(gap)h(for)f Fu(A)p Fv(.)580 b(Ho)-31 b(w)g(ev)g(er,)408 b(in)398 b(the)g(case)g(whic)-31 b(h)400 b(is)e(of)g(in)-31 b(terest)399 b(to)g(us,)406 b(namely)6075 29143 y(that)333 b Fu(A)9266 29309 y Fx(0)10071 29143 y Fv(=)307 b Fu(H)263 b Fr(\241)172 b Fu(E)396 b Fv(with)334 b Fu(H)422 b Fv(a)333 b(\257nite)g(v)-31 b(olume)334 b(Sc)-31 b(hr\304)-553 b(odinger)333 b(op)31 b(erator,)341 b(w)-31 b(e)333 b(can)f(mak)-31 b(e)334 b(use)e(of)6075 30471 y(the)311 b(fact)h(that)g Fu(H)400 b Fv(is)311 b Fm(lo)-57 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b(e)425 b(a)g(b)31 b(ounded)426 b(non-zero)f(p)31 b(oten)-31 b(tial)429 b(of)6075 37113 y(compact)371 b(supp)31 b(ort)368 b(with)j Fr(j)p Fu(@)61 b Fv(\(supp)185 b Fu(V)246 b Fv(\))p Fr(j)308 b Fv(=)g(0,)370 b(and)f(for)g Fu(\273)359 b Fr(2)307 b Ft(R)370 b Fv(let)22400 38915 y Fu(H)23320 39081 y Fs(\273)24112 38915 y Fv(=)308 b Fu(H)26201 39081 y Fx(0)26943 38915 y Fr(\241)246 b Fu(\273)51 b(V)61 b(:)14051 b Fv(\(2.5\))6075 40716 y(Then)549 b Fu(H)10065 40882 y Fs(\273)11098 40716 y Fv(is)f(self-adjoin)-31 b(t)551 b(with)f(compact)g(resolv)-31 b(en)g(t.)1033 b(Th)-31 b(us,)594 b(b)-31 b(y)548 b(Theorem)i(VI)31 b(I.3.9)550 b(of)6075 42045 y([36)q(],)558 b(its)519 b(rep)31 b(eated)519 b(eigen)-31 b(v)-61 b(alues)521 b Fu(E)21720 42211 y Fs(n)22323 42045 y Fv(\()p Fu(\273)51 b Fv(\),)558 b Fu(n)g Fr(2)f Ft(N)p Fv(,)h(and)519 b(corresp)31 b(onding)520 b(complete)g(set)g(of)6075 43373 y(orthonormal)510 b(eigenfunctions)g Fu(\303)20646 43539 y Fs(n)21248 43373 y Fv(\()p Fu(\273)51 b Fv(\))509 b(can)f(b)31 b(e)508 b(lab)31 b(eled)509 b(suc)-31 b(h)507 b(that)i Fu(E)36876 43539 y Fs(n)37479 43373 y Fv(\()p Fr(\242)p Fv(\))g(and)f Fu(\303)42167 43539 y Fs(n)42770 43373 y Fv(\()p Fr(\242)p Fv(\))h(are)6075 44701 y(holomorphic)426 b(\(note)f(that)g(crossings)e (are)h(p)31 b(ossible,)438 b(that)425 b(is,)438 b(the)425 b Fu(E)35716 44867 y Fs(n)36742 44701 y Fv(ma)-31 b(y)425 b(b)31 b(e)423 b(degenerate)6075 46030 y(and)402 b(are)g(not)g (necessarily)h(in)f(increasing)h(order\).)591 b(By)402 b(the)g(F)-92 b(eynman-Hellmann)404 b(Theorem)6075 47358 y(and)369 b(the)h(unique)f(con)-31 b(tin)g(uation)373 b(prop)31 b(ert)-31 b(y)369 b(of)h(eigenfunctions,)18413 49160 y Fu(E)19294 48703 y Fp(0)19230 49433 y Fs(n)19833 49160 y Fv(\()p Fu(\273)51 b Fv(\))308 b(=)g Fr(\241h)p Fu(V)246 b(\303)25609 49326 y Fs(n)26212 49160 y Fv(\()p Fu(\273)51 b Fv(\))p Fu(;)184 b(\303)28819 49326 y Fs(n)29423 49160 y Fv(\()p Fu(\273)51 b Fv(\))p Fr(i)310 b Fu(<)d Fv(0)p Fu(:)10064 b 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Fv(\))10608 61851 y Fp(\241)p Fx(1)11796 62253 y Fu(V)12688 61851 y Fx(1)p Fs(=)p Fx(2)14080 62253 y Fu(f)468 b Fv(=)350 b(0)395 b(on)f(supp)184 b Fu(V)640 b Fv(and,)402 b(since)394 b Fu(H)28712 62419 y Fs(\273)29591 62253 y Fv(is)h(lo)31 b(cal,)402 b Fu(V)34574 61851 y Fx(1)p Fs(=)p Fx(2)35966 62253 y Fu(f)468 b Fv(=)350 b(0)395 b(in)f(the)h(in)-31 b(terior)6075 63581 y(of)369 b(supp)184 b Fu(V)246 b Fv(.)493 b(This)369 b(implies)i(that)f Fu(f)427 b Fv(=)307 b(0)370 b(since)f Fr(j)p Fu(@)61 b Fv(\(supp)185 b Fu(V)246 b Fv(\))p Fr(j)308 b Fv(=)g(0.)7403 64942 y(W)-92 b(e)312 b(conclude)h(that)g Fu(K)17107 64468 y Fp(\241)p Fx(1)17028 65276 y Fs(\273)26 b(;E)18780 64942 y Fv(exists)313 b(in)f Fu(L)23767 64540 y Fx(2)24264 64942 y Fv(\(supp)185 b Fu(V)246 b Fv(\))312 b(as)g(an)h(un)-31 b(b)31 b(ounded)312 b(self-adjoin)-31 b(t)315 b(op)31 b(era-)6075 66306 y(tor)306 b(with)h(discrete)e(sp)31 b(ectrum.)472 b(By)305 b(the)i(same)f (argumen)-31 b(ts)306 b(used)g(in)g(the)g(pro)31 b(of)306 b(of)g(Lemma)h(2.1)6075 67634 y(it)370 b(ma)-31 b(y)370 b(b)31 b(e)369 b(sho)-31 b(wn)370 b(that)g(for)f Fu(E)j Fr(62)307 b Fu(\276)40 b Fv(\()p Fu(H)22794 67800 y Fs(\273)23279 67634 y Fv(\))246 b Fr([)g Fu(\276)40 b Fv(\()p Fu(H)26962 67800 y Fx(0)27459 67634 y Fv(\),)20435 69508 y Fu(K)21375 69674 y Fs(\273)26 b(;E)23123 69508 y Fv(=)307 b(\()p Fu(K)25740 69034 y Fp(\241)p Fx(1)25661 69833 y(0)p Fs(;E)27360 69508 y Fr(\241)246 b Fu(\273)51 b FB(1)p Fv(\))30068 69051 y Fp(\241)p Fx(1)31257 69508 y Fu(:)12086 b Fv(\(2.9\))p eop end %%Page: 12 12 TeXDict begin 12 11 bop 6075 3345 a Fx(12)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y FB(Lemma)424 b(2.2.)553 b Fm(L)-57 b(et)397 b Fu(E)371 b Fr(62)308 b Fu(\276)40 b Fv(\()p Fu(H)19290 6167 y Fx(0)19786 6001 y Fv(\))397 b Fm(and)f Fu(\273)359 b Fr(6)p Fv(=)307 b(0)p Fm(.)510 b(Then)7957 7594 y Fv(\(1\))554 b Fu(\301)493 b Fm(is)f(a)g(normalize)-57 b(d)493 b(eigenfunction)i(of) e Fu(K)28490 7121 y Fp(\241)p Fx(1)28411 7919 y(0)p Fs(;E)30356 7594 y Fm(with)f(eigenvalue)i Fu(\273)544 b Fm(if)493 b(and)f(only)h(if)9924 9164 y Fu(\303)397 b Fv(:=)357 b Fu(\273)51 b Fv(\()p Fu(H)14452 9330 y Fx(0)15215 9164 y Fr(\241)265 b Fu(E)64 b Fv(\))17652 8762 y Fp(\241)p Fx(1)18841 9164 y Fu(V)19733 8762 y Fx(1)p Fs(=)p Fx(2)21124 9164 y Fu(\301)423 b Fm(is)g(an)g(eigenfunction)j(of)d Fu(H)34154 9330 y Fs(\273)35062 9164 y Fm(with)g(eigenvalue)h Fu(E)487 b Fm(and)9924 10492 y Fr(h)p Fu(\303)40 b(;)184 b(V)247 b(\303)40 b Fr(i)308 b Fv(=)g(1)p Fm(.)7957 11852 y Fv(\(2\))554 b Fm(the)492 b(r)-57 b(ep)g(e)g(ate)g(d)493 b(eigenvalues)g(of)f Fu(K)24495 11379 y Fp(\241)p Fx(1)24416 12178 y(0)p Fs(;E)26359 11852 y Fm(ar)-57 b(e)492 b(given)h(by)g Fv(\241)33548 12018 y Fs(n)34150 11852 y Fv(\()p Fu(E)64 b Fv(\))p Fm(,)517 b Fu(n)482 b Fr(2)e Ft(N)p Fm(,)517 b(with)491 b(\(non-)9924 13422 y(normalize)-57 b(d\))396 b(c)-57 b(omplete)397 b(eigenve)-57 b(ctors)397 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b(b)31 b(oth)511 b Fu(Q)18152 34727 y Fx(0)18649 34561 y Fv(\()p Fr(J)205 b Fv(;)184 b Fu(x;)g(\256)t Fv(\))513 b(and)d Fu(Q)26472 34727 y Fx(2)26969 34561 y Fv(\()p Fr(J)205 b Fv(;)184 b Fu(x;)g(\256)t Fv(\))513 b(are)d(of)h(order)f (one,)546 b(for)511 b(a)f(\257nite)6075 35890 y(in)-31 b(terv)-61 b(al)250 b Fr(J)204 b Fv(,)274 b(in)248 b(the)h(sense)f (that)i(they)f(are)g(\257nite)g(and)g(do)g(not)h(deca)-31 b(y)249 b(for)g(increasing)g(dist)q(\()p Fu(x;)184 b(\256)t Fv(\).)6075 37218 y(In)368 b(particular,)j(if)f Fr(J)512 b(\275)308 b Fv(\()p Fr(\2411)p Fu(;)184 b(E)64 b Fv(\))371 b(and)e Fu(p)307 b(>)h(d=)p Fv(2)370 b(w)-31 b(e)370 b(ha)-31 b(v)g(e)12969 39171 y Fu(Q)13844 39337 y Fx(2)14340 39171 y Fv(\()p Fr(J)206 b Fv(;)184 b Fu(x;)g(\256)t Fv(\))309 b Fr(\267)554 b Fv(T)-92 b(r)184 b Fu(\302)22225 39337 y Fs(x)22782 39171 y Fu(P)23493 39337 y Fp(\267)p Fs(E)24932 39171 y Fv(\()p Fu(H)26372 38714 y Fx(\(\244\))27726 39171 y Fv(\))18792 41106 y Fr(\267)369 b Fv(\()p Fr(j)p 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b(eq.)h(\(2.6\))i(and)d(eq.)h(\(2.7\))h(w)-31 b(e)370 b(ha)-31 b(v)g(e)21140 68260 y Fu(d)p 20873 68755 1112 45 v 20873 69768 a(d\273)22117 69009 y(E)22934 69175 y Fs(n)23844 69009 y Fv(=)307 b Fr(\241h)p Fu(U)27059 69175 y Fs(\256)27691 69009 y Fu(\303)28412 69175 y Fs(n)29014 69009 y Fu(;)184 b(\303)30226 69175 y Fs(n)30830 69009 y Fr(i)11836 b Fv(\(3.11\))p eop end %%Page: 15 15 TeXDict begin 15 14 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(15)6075 6001 y Fv(and)369 b(for)g(\241)10613 6167 y Fs(n)11216 6001 y Fv(\()p Fu(E)64 b Fv(\),)370 b(the)g(in)-31 b(v)g(erse)369 b(function)i(of)e Fu(E)25622 6167 y Fs(n)26225 6001 y Fv(,)15994 7745 y Fu(d)p 15554 8240 1458 45 v 15554 9254 a(dE)17144 8494 y Fv(\241)17836 8660 y Fs(n)18438 8494 y Fv(\()p Fu(E)64 b Fv(\))309 b(=)e Fr(\241)28964 7745 y Fv(1)p 22650 8240 13182 45 v 22650 9254 a Fr(h)p Fu(U)23836 9420 y Fs(\256)24467 9254 y Fu(\303)25188 9420 y 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b(\242)p Fu(\264)10748 45043 y Fs(\256)11686 44877 y Fv(=)308 b Fu(\264)13405 45043 y Fs(\256)14281 44877 y Fr(\241)325 b Fv(^)-632 b Fu(\264)15938 45043 y Fs(\256)16568 44877 y Fv(.)493 b(In)369 b(particular,)i(eq.)e(\(3.12\))j(implies)f (that)14783 47015 y Fu(Q)15658 47181 y Fs(o)16150 47015 y Fv(\()p Fr(J)205 b Fv(;)184 b Fu(x;)g(\256)t Fv(\))679 b(=)23330 45963 y Fq(X)22386 48338 y Fs(n)p Fx(:)151 b Fs(E)23985 48449 y Fl(n)24527 48338 y Fp(2J)25874 47015 y Fr(h)p Fu(U)27060 47181 y Fs(\256)27691 47015 y Fu(\303)28412 47181 y Fs(n)29015 47015 y Fv(\()p Fu(E)64 b Fv(\))p Fu(;)184 b(\303)31968 47181 y Fs(n)32572 47015 y Fv(\()p Fu(E)64 b Fv(\))p Fr(i)20971 50509 y Fv(=)22386 49002 y Fq(Z)23001 51517 y Fp(J)23991 50509 y Fv(d)p Fu(E)25856 49457 y Fq(X)26382 51782 y Fs(n)27639 50509 y Fu(\261)226 b Fv(\()q(\241)29480 50675 y Fs(n)30082 50509 y Fv(\()p Fu(E)64 b Fv(\))247 b(+)f Fu(\270)184 b Fv(\242)p Fu(\264)35479 50675 y Fs(\256)36110 50509 y Fv(\))554 b Fu(;)43096 49149 y Fv(\(3.16\))6075 53360 y(and)369 b(for)g(other)h(v)-61 b(alues)369 b(of)h Fu(r)31 b Fv(:)9796 55971 y Fu(Q)10671 56137 y Fs(r)11161 55971 y Fv(\()p Fr(J)205 b Fv(;)184 b Fu(x;)g(\256)t Fv(\))679 b(=)18342 54919 y Fq(X)17397 57294 y Fs(n)p Fx(:)151 b Fs(E)18996 57405 y Fl(n)19538 57294 y Fp(2J)20885 55971 y Fr(h)p Fu(U)22071 56137 y Fs(\256)22702 55971 y Fu(\303)23423 56137 y Fs(n)24026 55971 y Fu(;)184 b(\303)25238 56137 y Fs(n)25842 55971 y Fr(i)26456 54410 y Fq(\265)27473 55222 y Fr(h)p Fu(\302)28596 55388 y Fs(x)29153 55222 y Fu(\303)29874 55388 y Fs(n)30477 55222 y Fu(;)g(\303)31689 55388 y Fs(n)32292 55222 y Fr(i)p 27404 55716 5387 45 v 27404 56730 a(h)p Fu(U)28590 56896 y Fs(\256)29221 56730 y Fu(\303)29942 56896 y Fs(n)30545 56730 y Fu(;)g(\303)31757 56896 y Fs(n)32361 56730 y Fr(i)32924 54410 y Fq(\266)33739 54639 y Fs(r)24 b(=)p Fx(2)15982 59690 y Fv(=)17397 58183 y Fq(Z)18012 60698 y Fp(J)19002 59690 y Fv(d)p Fu(E)20867 58638 y Fq(X)21393 60963 y Fs(n)22650 59690 y Fu(\261)227 b Fv(\(\241)24491 59856 y Fs(n)25093 59690 y Fv(\()p Fu(E)64 b Fv(\))247 b(+)f Fu(\270)184 b Fv(\242)p Fu(\264)30490 59856 y Fs(\256)31121 59690 y Fv(\))32238 58941 y Fr(h)p Fu(\302)33361 59107 y Fs(x)33919 58941 y Fu(\303)34640 59107 y Fs(n)35242 58941 y Fu(;)g(\303)36454 59107 y Fs(n)37058 58941 y Fr(i)37488 58539 y Fs(r)24 b(=)p Fx(2)p 32238 59435 6636 45 v 33255 60458 a Fr(k)q Fu(\301)34469 60624 y Fs(n)35071 60458 y Fv(\()p Fu(E)64 b Fv(\))p Fr(k)37366 59901 y Fs(r)39252 59690 y Fu(:)43096 57842 y Fv(\(3.17\))6075 62602 y(F)-92 b(or)368 b(\241)8753 62768 y Fs(n)9356 62602 y Fv(\()p Fu(E)64 b Fv(\))308 b(=)f Fr(\241)p Fu(\270)p Fv(\242)p Fu(\264)15552 62768 y Fs(\256)16490 62602 y Fv(=)h Fu(\273)51 b Fv(,)370 b(i.e.)g Fu(E)i Fv(=)307 b Fu(E)23828 62768 y Fs(n)24431 62602 y Fv(\()p Fu(\273)51 b Fv(\),)370 b(it)g(follo)-31 b(ws)372 b(that)17553 64616 y Fu(\303)18274 64782 y Fs(n)18877 64616 y Fv(\()p Fu(\273)51 b Fv(\))308 b(=)g Fu(\273)51 b Fv(\()p Fu(H)23703 64782 y Fx(^)-510 b Fs(\264)24090 64893 y Fl(\256)24953 64616 y Fr(\241)246 b Fu(E)64 b Fv(\))27371 64159 y Fp(\241)p Fx(1)28560 64616 y Fu(U)29437 64159 y Fx(1)p Fs(=)p Fx(2)29316 64889 y Fs(\256)30828 64616 y Fu(\301)31488 64782 y Fs(n)32090 64616 y Fv(\()p Fu(E)g Fv(\))308 b Fu(;)8650 b Fv(\(3.18\))6075 66787 y(since)369 b Fu(K)9806 66313 y Fp(\241)p Fx(1)9796 67121 y(^)-510 b Fs(\264)10183 67232 y Fl(\256)10995 66787 y Fu(\301)11655 66953 y Fs(n)12257 66787 y Fv(\()p Fu(E)64 b Fv(\))308 b(=)g(\241)16167 66953 y Fs(n)16769 66787 y Fv(\()p Fu(E)64 b Fv(\))p Fu(U)19387 66212 y Fx(1)p Fs(=)p Fx(2)19266 66916 y Fs(\256)20779 66787 y Fu(\303)21500 66953 y Fs(n)22102 66787 y Fv(\()p Fu(\273)51 b Fv(\))371 b(.)492 b(Th)-31 b(us)369 b(eq.)h(\(3.13\))i(follo)-31 b(ws)371 b(from)f(eq.)g(\(3.17\).)7403 68213 y(F)-92 b(or)267 b(\257xed)g Fu(E)64 b Fv(,)288 b(the)267 b(left-hand)i(side)e (of)h(eq.)f(\(3.14\),)291 b(up)267 b(to)h(a)g(factor)g(1)p Fu(=\270)p Fv(,)289 b(coun)-31 b(ts)268 b(the)f(n)-31 b(um)g(b)31 b(er)6075 69541 y(of)349 b Fu(n)h Fv(for)f(whic)-31 b(h)350 b(\241)13842 69707 y Fs(n)14444 69541 y Fv(\()p Fu(E)64 b Fv(\))309 b(=)e Fu(\270)p Fv(\()79 b(^)-632 b Fu(\264)19288 69707 y Fs(\256)20125 69541 y Fr(\241)206 b Fu(\264)21742 69707 y Fs(\256)22372 69541 y Fv(\))350 b(has)f(a)g(solution)i(with)f Fu(a)308 b(<)f(\264)35288 69707 y Fs(\256)36226 69541 y Fu(<)h(b)p Fv(.)486 b(This)349 b(n)-31 b(um)g(b)31 b(er)349 b(is)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 6075 3345 a Fx(16)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(exactly)398 b(the)e(decrease)g(in)g(the)g(n)-31 b(um)g(b)31 b(er)397 b(of)f(eigen)-31 b(v)-61 b(alues)398 b(of)e Fu(H)32808 6167 y Fs(\264)33264 6278 y Fl(\256)34277 6001 y Fv(b)31 b(elo)-31 b(w)397 b Fu(E)460 b Fv(as)396 b Fu(\264)40653 6167 y Fs(\256)41680 6001 y Fv(is)g(mo)-31 b(v)g(ed)6075 7330 y(from)369 b Fu(a)h Fv(to)g Fu(b)p Fv(.)492 b(That)370 b(yields)g(eq.)g(\(3.14\).)22026 b Fh(\244)6075 9805 y Fv(3.3.)555 b FB(Sp)35 b(ectral)334 b(shift)f(b)35 b(ounds.)555 b Fv(In)289 b(applying)i(the)f(in)-31 b(terp)31 b(olation)293 b(argumen)-31 b(t)291 b(seen)e(in)h(Lem-)6075 11134 y(ma)370 b(3.1,)h(w)-31 b(e)369 b(shall)h(need)f(b)31 b(ounds)369 b(on)g(the)h Fm(sp)-57 b(e)g(ctr)g(al)396 b(shift)g(function)p Fv(,)371 b(de\257ned)e(b)-31 b(y)12975 13535 y Fu(S)13654 13701 y Fs(\256;\270)15073 13535 y Fv(\()p Fu(H)16513 13078 y Fx(\(\244\))17866 13535 y Fv(;)184 b Fu(E)64 b Fv(\))678 b(:=)f(T)-92 b(r)23701 12307 y Fq(h)24224 13535 y Fu(P)24935 13701 y Fp(\267)p Fs(E)26374 13535 y Fv(\()p Fu(H)27814 12960 y Fx(\(\244\))27724 13831 y Fs(\264)28180 13942 y Fl(\256)28741 13831 y Fx(=0)29918 13535 y Fv(\))246 b Fr(\241)g Fu(P)32412 13701 y Fp(\267)p Fs(E)33851 13535 y Fv(\()p Fu(H)35291 12960 y Fx(\(\244\))35201 13831 y Fs(\264)35657 13942 y Fl(\256)36219 13831 y Fx(=1)37395 13535 y Fv(\))37825 12307 y Fq(i)38717 13535 y Fu(;)4072 b Fv(\(3.19\))6075 15937 y(and)300 b(expressing)f(ho)-31 b(w)301 b(man)-31 b(y)301 b(energy)f(lev)-31 b(els)301 b(are)e(pushed)g(o)-31 b(v)g(er)301 b Fu(E)363 b Fv(when)300 b(the)g(v)-61 b(alue)300 b(of)h(the)f(pa-)6075 17266 y(rameter)307 b Fu(\264)10689 17432 y Fs(\256)11626 17266 y Fv(is)g(increased)f(b)-31 b(y)307 b(1.)472 b(Note)308 b(that)g Fu(S)25855 17432 y Fs(\256;\270)27274 17266 y Fv(\()p Fu(H)28714 16864 y Fx(\(\244\))30067 17266 y Fv(;)184 b Fu(E)64 b Fv(\))308 b(is)f(non-negativ)-31 b(e)309 b(and)e(b)31 b(ounded)6075 18594 y(\(since)369 b Fr(j)p Fv(\244)p Fr(j)308 b Fu(<)g Fr(1)p Fv(\):)8476 20941 y(0)677 b Fr(\267)f Fu(S)11922 21107 y Fs(\256;\270)13340 20941 y Fv(\()p Fu(H)14780 20484 y Fx(\(\244\))16134 20941 y Fv(;)184 b Fu(E)64 b Fv(\))678 b Fr(\267)e Fv(T)-92 b(r)185 b Fu(P)22188 21107 y Fp(\267)p Fs(E)23627 20941 y Fv(\()p Fu(H)25067 20366 y Fx(\(\244\))24977 21236 y Fs(\264)25433 21347 y Fl(\256)25995 21236 y Fx(=0)27171 20941 y Fv(\))677 b Fr(\267)f Fu(C)30606 21107 y Fs(r)31281 20941 y Fv(\(1)247 b(+)f Fr(j)p Fu(E)310 b Fr(\241)246 b Fu(E)36976 21107 y Fs(o)37469 20941 y Fr(j)p Fv(\))38206 20484 y Fs(r)38697 20941 y Fr(j)p Fv(\244)p Fr(j)308 b Fu(;)2401 b Fv(\(3.20\))6075 23130 y(for)369 b(an)-31 b(y)370 b Fu(r)339 b(>)307 b(d=)p Fv(2.)7403 24459 y(In)378 b(the)g(discrete)g(setup,)j(where)d(the)g(role)g(of)h Fu(U)27380 24625 y Fs(\256)28389 24459 y Fv(is)f(tak)-31 b(en)379 b(b)-31 b(y)378 b(a)h(rank-one)f(op)31 b(erator,)382 b(the)6075 25787 y(shift)402 b(is)f(at)h(most)h(1.)590 b(F)-92 b(or)400 b(con)-31 b(tin)g(uum)404 b(op)31 b(erators)402 b(there)f(is)g(no)h(suc)-31 b(h)401 b(uniform)h(b)31 b(ound)402 b(inde-)6075 27116 y(p)31 b(enden)-31 b(t)412 b(of)h(the)g(v)-31 b(olume)415 b(\(see)d(ref.)622 b([37)r(]\),)424 b(ho)-31 b(w)g(ev)g(er)414 b Fu(S)29780 27282 y Fs(\256;\270)31199 27116 y Fv(\()p Fu(H)32639 26714 y Fx(\(\244\))33992 27116 y Fv(;)184 b Fu(E)64 b Fv(\))414 b(has)e(lo)31 b(cally)415 b(b)31 b(ounded)6075 28444 y Fu(L)6828 28042 y Fs(p)7709 28444 y Fv(norms)369 b(as)g(a)g(function)i(of)f(the)f (energy)g Fu(E)64 b Fv(,)370 b(as)f(w)-31 b(as)370 b(sho)-31 b(wn)369 b(in)h(ref.)f([17)r(].)6075 30630 y FB(Lemma)506 b(3.2)i Fv(\()p Fu(L)13988 30228 y Fs(p)14941 30630 y Fv(b)31 b(oundedness)441 b(of)g(the)g(sp)31 b(ectral)442 b(shift\))p FB(.)597 b Fm(L)-57 b(et)463 b Fv(1)428 b Fr(\267)f Fu(p)g(<)h Fr(1)462 b Fm(and)g Fu(n)428 b Fr(2)f Ft(N)6075 31958 y Fm(such)395 b(that)h Fu(n)307 b(>)h(dp=)p Fv(2)p Fm(.)511 b(Then)396 b(ther)-57 b(e)396 b(exists)f(a)h(c)-57 b(onstant)396 b Fu(C)30925 32124 y Fs(p;\270;n)33343 31958 y Fu(<)307 b Fr(1)396 b Fm(such)g(that)13094 33444 y Fq(Z)14201 33718 y Fs(E)14851 33829 y Fk(+)13709 35959 y Fp(\2411)15715 34951 y Fu(S)16394 35117 y Fs(\256;\270)17812 34951 y Fv(\()p Fu(H)19252 34494 y Fx(\(\244\))20606 34951 y Fv(;)184 b Fu(E)64 b Fv(\))22408 34494 y Fs(p)22921 34951 y Fv(d)p Fu(E)767 b Fr(\267)704 b Fu(C)27476 35117 y Fs(p;\270;n)29586 34951 y Fv(\(1)247 b(+)f Fr(j)p Fu(E)33047 35117 y Fx(+)34028 34951 y Fr(\241)g Fu(E)35952 35117 y Fs(o)36445 34951 y Fr(j)p Fv(\))37182 34494 y Fs(n)p Fx(+1)43096 34951 y Fv(\(3.21\))6075 37734 y Fm(uniformly)345 b(in)f(the)g(domain)g Fv(\244)f Fm(as)g(wel)57 b(l)343 b(as)g(in)h(the)g(choic)-57 b(e)344 b(of)g Fu(\256)k Fm(and)343 b(the)h(r)-57 b(andom)344 b(p)-57 b(ar)g(ameters)6075 39062 y Fu(\264)6625 39228 y Fs(\257)7618 39062 y Fm(\()p Fu(\257)365 b Fr(6)p Fv(=)308 b Fu(\256)t Fm(\).)6075 41248 y Fv(3.4.)555 b FB(F)-106 b(rom)454 b(fractional)f(momen)-35 b(t)454 b(b)35 b(ounds)455 b(to)f(lo)35 b(calization.)552 b Fv(W)-92 b(e)394 b(shall)h(no)-31 b(w)396 b(put)e(to-)6075 42576 y(gether)338 b(the)g(elemen)-31 b(ts)338 b(in)-31 b(tro)31 b(duced)339 b(in)f(the)f(previous)h(sections)g(and)g(pro)-31 b(v)g(e)339 b(that)f(rapid)g(deca)-31 b(y)6075 43905 y(of)369 b(the)h(resolv)-31 b(en)g(t)370 b(momen)-31 b(ts)371 b(implies)f(lo)31 b(calization)373 b(in)d(its)g(v)-61 b(arious)369 b(manifestations.)6075 46091 y FB(Theorem)500 b(3.2.)593 b Fm(L)-57 b(et)457 b Fu(H)17019 46257 y Fs(!)18116 46091 y Fm(b)-57 b(e)457 b(a)f(r)-57 b(andom)456 b(Schr\304)-566 b(odinger)457 b(op)-57 b(er)g(ator)455 b(which)h(satis\257es)g FB(A)p Fm(.)h(L)-57 b(et)6075 47419 y Fv(0)370 b Fu(<)g(s)g(<)f Fv(1)431 b Fm(and)g Fr(J)574 b(\275)370 b Fv(\()p Fr(\2411)p Fu(;)184 b(E)19778 47585 y Fx(+)20515 47419 y Fv(])431 b Fm(b)-57 b(e)431 b(a)f(b)-57 b(ounde)g(d)432 b(op)-57 b(en)431 b(interval.)613 b(Supp)-57 b(ose)430 b(that)g(for)g(some)6075 48747 y Fu(C)387 b(<)307 b Fr(1)p Fm(,)396 b Fu(\264)347 b(>)308 b Fv(0)396 b Fm(and)g(a)g(b)-57 b(ounde)g(d)397 b(r)-57 b(e)g(gion)398 b Fv(\244)p Fm(,)13846 51541 y Ft(E)14768 49980 y Fq(\265)15583 50035 y(Z)16198 52550 y Fp(J)17188 51541 y Fv(d)p Fu(E)18869 49936 y Fq(\260)18869 50600 y(\260)18869 51264 y(\260)18869 51929 y(\260)19484 51541 y Fu(\302)20177 51707 y Fs(x)23073 50792 y Fv(1)p 21051 51287 4597 45 v 21051 52320 a Fu(H)22061 52001 y Fx(\(\244\))23660 52320 y Fr(\241)246 b Fu(E)25965 51541 y(\302)26658 51707 y Fs(y)27190 49936 y Fq(\260)27190 50600 y(\260)27190 51264 y(\260)27190 51929 y(\260)27805 50210 y Fs(s)28278 49980 y Fq(\266)29796 51541 y Fr(\267)308 b Fu(C)79 b(e)32350 51084 y Fp(\241)p Fs(\264)180 b Fx(dist)35114 51195 y Fk(\244)35673 51084 y Fx(\()p Fs(x;y)28 b Fx(\))37847 51541 y Fu(;)4942 b Fv(\(3.22\))6075 54369 y Fm(for)458 b(al)57 b(l)457 b Fu(x;)184 b(y)460 b Fr(2)420 b Fv(\244)p Fm(.)695 b(Then,)475 b(for)458 b(any)h Fu(r)451 b(<)420 b Fv(1)p Fu(=)p Fv(\(2)293 b Fr(\241)d Fu(s)p Fv(\))p Fm(,)475 b(ther)-57 b(e)458 b(exists)f(a)h(volume)397 b(indep)-57 b(endent)p 36420 54761 9506 45 v 6075 55747 a(c)g(onstant)396 b Fu(C)11279 55913 y Fs(s;r)12450 55747 y Fv(\()p Fu(E)13697 55913 y Fx(+)14432 55747 y Fu(;)184 b(\270)p Fv(\))309 b Fu(<)e Fr(1)397 b Fm(such)e(that)8506 58914 y Ft(E)9428 57021 y Fq(\303)12565 58914 y Fv(sup)10305 59895 y Fs(f)87 b Fp(2)p Fs(C)12056 60006 y Fl(c)12470 59895 y Fx(\()p Fp(J)144 b Fx(\):)151 b 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y(\257)33369 65315 y(\257)33369 65979 y(\257)33369 66643 y(\257)34415 65923 y Fr(\267)36331 65175 y Fv(2)p 36085 65669 1046 45 v 36085 66683 a Fu(\270t)43650 65923 y Fv(\(4.4\))6075 69541 y(for)369 b(an)-31 b(y)370 b Fu(z)417 b Fv(and)13204 69261 y Fq(b)12945 69541 y Fu(H)459 b Fv(\(with)371 b Fr(j)e(\242)h(j)f Fv(denoting)i(the)e(Leb)31 b(esgue)369 b(measure\).)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 6075 3345 a Fx(22)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)7403 6001 y Fv(F)-92 b(or)307 b(general)h Fu(U)121 b Fv(,)320 b(w)-31 b(e)308 b(ma)-31 b(y)309 b(use)e(the)h(Birman-Sc)-31 b(h)g(winger)310 b(relation)g(eq.)e(\(2.9\))i(to)e(sho)-31 b(w)308 b(that,)6075 7330 y(for)369 b Fu(z)356 b Fv(=)307 b Fu(E)372 b Fr(2)307 b Ft(R)p Fv(,)9383 8271 y Fq(\257)9383 8935 y(\257)9383 9599 y(\257)9383 10264 y(\257)9752 8315 y(\275)10582 9876 y Fu(\264)347 b Fr(2)307 b Fv([0)p Fu(;)184 b Fv(1])680 b(:)16399 8271 y Fq(\260)16399 8935 y(\260)16399 9599 y(\260)16399 10264 y(\260)17014 9876 y Fu(U)17891 9419 y Fx(1)p Fs(=)p Fx(2)22492 9127 y Fv(1)p 19415 9622 6709 45 v 19674 10554 a Fq(b)19415 10834 y Fu(H)336 b Fv(+)246 b Fu(\270\264)40 b(U)365 b Fr(\241)246 b Fu(E)26256 9876 y(U)27133 9419 y Fx(1)p Fs(=)p Fx(2)28524 8271 y Fq(\260)28524 8935 y(\260)28524 9599 y(\260)28524 10264 y(\260)29446 9876 y Fu(>)308 b(t)31015 8315 y Fq(\276)31845 8271 y(\257)31845 8935 y(\257)31845 9599 y(\257)31845 10264 y(\257)32890 9876 y Fr(\267)34560 9127 y Fv(2\(1)248 b(+)e Fu(\273)37935 9293 y Fs(E)43 b(;\270)39471 9127 y Fv(\))p 34560 9622 5341 45 v 36708 10635 a Fu(\270t)43650 9876 y Fv(\(4.5\))6075 12854 y(where)400 b Fu(\273)9791 13020 y Fs(E)43 b(;\270)11727 12854 y Fv(denotes)401 b(the)f(n)-31 b(um)g(b)31 b(er)401 b(of)g(eigen)-31 b(v)-61 b(alues)401 b(of)30366 12574 y Fq(b)30102 12854 y Fu(K)31121 12381 y Fp(\241)p Fx(1)31042 13180 y Fs(E)32710 12854 y Fv(in)g(the)f(in)-31 b(terv)-61 b(al)402 b(\()p Fr(\241)p 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y Fv(t)g(yp)31 b(e)6075 69541 y(b)g(ound)496 b(for)g(the)h(b)31 b(oundary)496 b(v)-61 b(alues)496 b(of)h(resolv)-31 b(en)g(ts)497 b(of)f(maximally)k (dissipativ)-31 b(e)498 b(op)31 b(erators.)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(23)6075 6001 y Fv(W)-92 b(e)480 b(state)j(this)e(as)h(the)f(second)h(part)f(of)h (the)g(follo)-31 b(wing)485 b(Lemma,)511 b(whic)-31 b(h)482 b(is)f(essen)-31 b(tially)484 b(a)6075 7330 y(sp)31 b(ecial)418 b(case)g(of)g(results)g(pro)-31 b(v)g(en)418 b(in)g([49)q(].)639 b(F)-92 b(or)417 b(completeness,)432 b(w)-31 b(e)418 b(giv)-31 b(e)419 b(a)f(pro)31 b(of)418 b(based)g(on)6075 8658 y(prop)31 b(erties)354 b(of)g(the)h(Hilb)31 b(ert)355 b(transform)g(in)f(App)31 b(endix)355 b(B.)487 b(The)355 b(\257rst)e(part)i(of)f(the)h(Lemma)g(is)6075 9986 y(a)369 b(w)-31 b(ell)371 b(kno)-31 b(wn)370 b(result)f(in)h(scattering)g (theory)-92 b(,)370 b(e.g.)h([22)q(].)6075 12093 y FB(Lemma)544 b(4.1)i Fv(\(W)-92 b(eak)474 b Fu(L)17213 12259 y Fx(1)18184 12093 y Fv(b)31 b(ound\))p FB(.)616 b Fm(L)-57 b(et)494 b Fr(H)503 b Fm(and)492 b Fr(H)29255 12259 y Fx(1)30244 12093 y Fm(b)-57 b(e)493 b(sep)-57 b(ar)g(able)493 b(Hilb)-57 b(ert)493 b(sp)-57 b(ac)g(es,)517 b Fu(A)493 b Fm(a)6075 13422 y(maximal)57 b(ly)433 b(dissip)-57 b(ative)435 b(op)-57 b(er)g(ator)433 b(in)i Fr(H)11 b Fm(,)443 b(as)433 b(wel)57 b(l)433 b(as)h Fu(M)30646 13588 y Fx(1)31518 13422 y Fv(:)377 b Fr(H)386 b(!)376 b(H)35941 13588 y Fx(1)36872 13422 y Fm(and)433 b Fu(M)40133 13588 y Fx(2)41006 13422 y Fv(:)376 b Fr(H)42624 13588 y Fx(1)43497 13422 y Fr(!)g(H)6075 14750 y Fm(Hilb)-57 b(ert-Schmidt)397 b(op)-57 b(er)g(ators.)509 b(Then)7957 16411 y Fv(\(1\))554 b Fm(The)397 b(\\b)-57 b(oundary)397 b(values")15677 18926 y Fu(M)16751 19092 y Fx(1)19590 18177 y Fv(1)p 17380 18671 4974 45 v 17380 19685 a Fu(A)247 b Fr(\241)f Fu(v)285 b Fv(+)246 b(i0)22487 18926 y Fu(M)23561 19092 y Fx(2)24761 18926 y Fv(=)703 b(lim)26462 19648 y Fs(\262)p Fp(#)p Fx(0)28047 18926 y Fu(M)29121 19092 y Fx(1)31908 18177 y Fv(1)p 29750 18671 4870 45 v 29750 19685 a Fu(A)247 b Fr(\241)f Fu(v)285 b Fv(+)246 b(i)p Fu(\262)34752 18926 y(M)35826 19092 y Fx(2)43650 18926 y Fv(\(4.7\))9924 21454 y Fm(exist)341 b(as)g(Hilb)-57 b(ert-Schmidt)342 b(op)-57 b(er)g(ators)340 b(for)h(almost)f(every)i Fu(v)347 b Fr(2)308 b Ft(R)p Fm(,)353 b(with)340 b(c)-57 b(onver)g(genc)g(e)9924 22782 y(in)397 b(the)f(Hilb)-57 b(ert-Schmidt)397 b(norm.)7957 24111 y Fv(\(2\))554 b Fm(Ther)-57 b(e)397 b(exists)e Fu(C)16929 24277 y Fs(W)18244 24111 y Fu(<)307 b Fr(1)396 b Fm(\(indep)-57 b(endent)398 b(of)e Fu(A)p Fm(,)h Fu(M)31448 24277 y Fx(1)31944 24111 y Fm(,)g Fu(M)33754 24277 y Fx(2)34250 24111 y Fm(\))f(such)f(that)8853 25163 y Fq(\257)8853 25827 y(\257)8853 26491 y(\257)8853 27156 y(\257)9222 25207 y(\275)10053 26768 y Fu(v)742 b Fv(:)11947 25163 y Fq(\260)11947 25827 y(\260)11947 26491 y(\260)11947 27156 y(\260)12562 26768 y Fu(M)13636 26934 y Fx(1)16476 26019 y Fv(1)p 14266 26514 4974 45 v 14266 27528 a Fu(A)246 b Fr(\241)g Fu(v)285 b Fv(+)246 b(i0)19372 26768 y Fu(M)20446 26934 y Fx(2)20942 25163 y Fq(\260)20942 25827 y(\260)20942 26491 y(\260)20942 27156 y(\260)21557 27875 y Fs(H)59 b(S)23290 26768 y Fu(>)308 b(t)24859 25207 y Fq(\276)25689 25163 y(\257)25689 25827 y(\257)25689 26491 y(\257)25689 27156 y(\257)26761 26768 y Fr(\267)704 b Fu(C)29117 26934 y Fs(W)30493 26768 y Fr(k)q Fu(M)32121 26934 y Fx(1)32617 26768 y Fr(k)33171 27100 y Fs(H)59 b(S)34965 26768 y Fr(k)p Fu(M)36592 26934 y Fx(2)37089 26768 y Fr(k)37642 27100 y Fs(H)g(S)39570 26019 y Fv(1)p 39570 26514 554 45 v 39647 27528 a Fu(t)40564 26768 y(:)2779 b Fv(\(4.8\))6075 29533 y FB(Remark:)463 b Fv(If)311 b(a)h(dissipativ)-31 b(e)314 b(op)31 b(erator)312 b Fu(A)g Fv(has)g(strictly)h(p)31 b(ositiv)-31 b(e)313 b(imaginary)i(part,)324 b(suc)-31 b(h)311 b(as)h(a)6075 30861 y(Birman-Sc)-31 b(h)g(winger)345 b(op)31 b(erator)343 b Fu(A)20532 31027 y Fs(B)38 b(S)22223 30861 y Fv(of)344 b(the)f(t)-31 b(yp)31 b(e)343 b(consider)g(in)g (Lemma)h(2.1,)350 b(then)343 b(the)g Fu(i)p Fv(0)h(in)6075 32190 y(\(4.8\))378 b(is)f(not)g(needed)f(since)h(\()p Fu(\273)302 b Fr(\241)251 b Fu(A)21397 32356 y Fs(B)38 b(S)22745 32190 y Fv(\))23175 31788 y Fp(\241)p Fx(1)24740 32190 y Fv(is)377 b(norm-con)-31 b(tin)g(uous)378 b(for)f Fu(\273)427 b Fv(in)377 b(the)g(closed)g(lo)-31 b(w)g(er)6075 33518 y(half)370 b(plane.)7403 34847 y(In)440 b(our)g(applications)j (of)d(Lemma)i(4.1)f(w)-31 b(e)441 b(will)h(use)d(the)h(follo)-31 b(wing)444 b(\\o\256-diagonal")g(v)-31 b(er-)6075 36175 y(sion)327 b(whic)-31 b(h)328 b(follo)-31 b(ws)330 b(from)e(the)f (lemma)i(via)f(the)f(Birman-Sc)-31 b(h)g(winger)330 b(iden)-31 b(tit)g(y)329 b(and)f(a)f(simple)6075 37503 y(c)-31 b(hange)406 b(of)g(v)-61 b(ariables.)601 b(F)-92 b(or)405 b(ease)g(of)h(presen)-31 b(tation,)416 b(w)-31 b(e)406 b(state)g(this)g(v)-31 b(ersion)405 b(with)i(the)e(addi-)6075 38832 y(tional)465 b(assumption)f(that)g(the)g(op)31 b(erator)463 b Fu(A)g Fv(has)g(strictly)h(p)31 b(ositiv)-31 b(e)464 b(imaginary)i(part,)487 b(th)-31 b(us)6075 40160 y(a)g(v)g(oiding)372 b(the)d(issue)g(of)h (whether)f(certain)h(limits)h(exist.)6075 42267 y FB(Prop)35 b(osition)450 b(4.2.)566 b Fm(L)-57 b(et)416 b Fu(A)f Fm(b)-57 b(e)416 b(a)e(maximal)57 b(ly)415 b(dissip)-57 b(ative)416 b(op)-57 b(er)g(ator)414 b(with)h(strictly)g(p)-57 b(ositive)6075 43595 y(imaginary)379 b(p)-57 b(art)378 b(on)g(a)g(Hilb)-57 b(ert)379 b(sp)-57 b(ac)g(e)379 b Fr(H)11 b Fm(,)382 b(let)c Fu(M)26869 43761 y Fx(1)27365 43595 y Fu(;)184 b(M)28930 43761 y Fx(2)29806 43595 y Fm(b)-57 b(e)379 b(Hilb)-57 b(ert-Schmidt)379 b(op)-57 b(er)g(ators,)381 b(and)6075 44924 y(let)396 b Fu(U)8387 45090 y Fx(1)8883 44924 y Fm(,)h Fu(U)10375 45090 y Fx(2)11267 44924 y Fm(b)-57 b(e)397 b(non-ne)-57 b(gative)398 b(op)-57 b(er)g(ators.)509 b(Then)7366 46247 y Fq(\257)7366 46911 y(\257)7366 47575 y(\257)7366 48239 y(\257)7735 46291 y(\275)8565 47852 y Fr(h)p Fu(v)9532 48018 y Fx(1)10029 47852 y Fu(;)184 b(v)11057 48018 y Fx(2)11554 47852 y Fr(i)308 b(2)g Fv([0)p Fu(;)184 b Fv(1])15549 47395 y Fx(2)16752 47852 y Fv(:)17367 46247 y Fq(\260)17367 46911 y(\260)17367 47575 y(\260)17367 48239 y(\260)17982 47852 y Fu(M)19056 48018 y Fx(1)19552 47852 y Fu(U)20429 47277 y Fx(1)p Fs(=)p Fx(2)20308 48147 y(1)25730 47103 y Fv(1)p 21953 47597 8108 45 v 21953 48611 a Fu(A)246 b Fr(\241)g Fu(v)24673 48777 y Fx(1)25169 48611 y Fu(U)25925 48777 y Fx(1)26668 48611 y Fr(\241)f Fu(v)28311 48777 y Fx(2)28808 48611 y Fu(U)29564 48777 y Fx(2)30193 47852 y Fu(U)31070 47277 y Fx(1)p Fs(=)p Fx(2)30949 48147 y(2)32461 47852 y Fu(M)33535 48018 y Fx(2)34031 46247 y Fq(\260)34031 46911 y(\260)34031 47575 y(\260)34031 48239 y(\260)34954 47852 y Fu(>)307 b(t)36522 46291 y Fq(\276)37352 46247 y(\257)37352 46911 y(\257)37352 47575 y(\257)37352 48239 y(\257)28249 50808 y Fr(\267)703 b Fv(2)184 b Fu(C)31341 50974 y Fs(W)32534 50808 y Fr(k)p Fu(M)34161 50974 y Fx(1)34658 50808 y Fr(k)35211 51140 y Fs(H)59 b(S)36821 50808 y Fr(k)q Fu(M)38449 50974 y Fx(2)38945 50808 y Fr(k)39499 51140 y Fs(H)g(S)41242 50059 y Fv(1)p 41242 50554 554 45 v 41319 51568 a Fu(t)42235 50808 y(:)1108 b Fv(\(4.9\))6075 53392 y Fm(Pr)-57 b(o)g(of.)554 b Fv(The)465 b(k)-31 b(ey)465 b(to)g(the)g(pro)31 b(of)465 b(is)f(the)h(c)-31 b(hange)466 b(of)f(v)-61 b(ariables)465 b Fu(v)33991 53558 y Fp(\247)35204 53392 y Fv(=)36664 52956 y Fx(1)p 36664 53137 442 45 v 36664 53774 a(2)37238 53392 y Fv(\()p Fu(v)38205 53558 y Fx(1)39012 53392 y Fr(\247)309 b Fu(v)40719 53558 y Fx(2)41215 53392 y Fv(\),)489 b(so)465 b(that)6075 54720 y Fu(v)6612 54886 y Fx(1)7108 54720 y Fu(U)7864 54886 y Fx(1)8622 54720 y Fv(+)261 b Fu(v)10281 54886 y Fx(2)10777 54720 y Fu(U)11533 54886 y Fx(2)12376 54720 y Fv(=)346 b Fu(v)14120 54886 y Fx(+)14854 54720 y Fv(\()p Fu(U)16040 54886 y Fx(1)16798 54720 y Fv(+)262 b Fu(U)18677 54886 y Fx(2)19173 54720 y Fv(\))g(+)f Fu(v)21524 54886 y Fp(\241)22271 54720 y Fv(\()p Fu(U)23457 54886 y Fx(1)24215 54720 y Fr(\241)h Fu(U)26094 54886 y Fx(2)26590 54720 y Fv(\).)562 b(F)-92 b(rom)392 b(the)h(Birman-Sc)-31 b(h)g(winger)394 b(iden)-31 b(tit)g(y)-92 b(,)6075 56049 y(eq.)369 b(\(2.4\),)6916 57145 y Fq(\267)7500 58706 y Fv(\()p Fu(U)8686 58872 y Fx(1)9429 58706 y Fv(+)246 b Fu(U)11292 58872 y Fx(2)11789 58706 y Fv(\))12219 58249 y Fx(1)p Fs(=)p Fx(2)17520 57957 y Fv(1)p 13743 58452 8108 45 v 13743 59466 a Fu(A)h Fr(\241)f Fu(v)16464 59632 y Fx(1)16960 59466 y Fu(U)17716 59632 y Fx(1)18458 59466 y Fr(\241)g Fu(v)20102 59632 y Fx(2)20598 59466 y Fu(U)21354 59632 y Fx(2)21983 58706 y Fv(\()p Fu(U)23169 58872 y Fx(1)23912 58706 y Fv(+)g Fu(U)25775 58872 y Fx(2)26272 58706 y Fv(\))26702 58249 y Fx(1)p Fs(=)p Fx(2)28094 57145 y Fq(\270)29354 58706 y Fv(=)30892 57478 y Fq(h)31414 58706 y Fv([)q Fu(K)79 b Fv(\()p Fu(v)33708 58872 y Fp(\241)34455 58706 y Fv(\)])35193 58150 y Fp(\241)p Fx(1)36997 58706 y Fr(\241)615 b Fu(v)39010 58872 y Fx(+)39744 57478 y Fq(i)40267 57707 y Fp(\241)p Fx(1)41947 58706 y Fu(;)842 b Fv(\(4.10\))6075 61473 y(where)369 b([)p Fr(\242)p Fv(])h(denotes)g(the)f(restriction)h(of)g(an)f(op)31 b(erator)370 b(to)g(k)-31 b(er\()p Fu(U)33059 61639 y Fx(1)33802 61473 y Fv(+)246 b Fu(U)35665 61639 y Fx(2)36161 61473 y Fv(\))36591 61072 y Fp(?)37708 61473 y Fv(and)12668 63988 y Fu(K)79 b Fv(\()p Fu(v)14654 64154 y Fp(\241)15401 63988 y Fv(\))677 b(=)g(\()p Fu(U)19232 64154 y Fx(1)19975 63988 y Fv(+)246 b Fu(U)21838 64154 y Fx(2)22334 63988 y Fv(\))22764 63531 y Fx(1)p Fs(=)p Fx(2)28105 63239 y Fv(1)p 24289 63734 8186 45 v 24289 64747 a Fu(A)g Fr(\241)g Fu(v)27009 64913 y Fp(\241)27756 64747 y Fv(\()p Fu(U)28942 64913 y Fx(1)29684 64747 y Fr(\241)g Fu(U)31547 64913 y Fx(2)32044 64747 y Fv(\))32607 63988 y(\()p Fu(U)33793 64154 y Fx(1)34536 63988 y Fv(+)g Fu(U)36399 64154 y Fx(2)36895 63988 y Fv(\))37325 63531 y Fx(1)p Fs(=)p Fx(2)39025 63988 y Fu(:)3764 b Fv(\(4.11\))6075 66569 y(Th)-31 b(us,)8417 69058 y Fu(M)9491 69224 y Fx(1)9987 69058 y Fu(U)10864 68483 y Fx(1)p Fs(=)p Fx(2)10743 69353 y(1)16165 68309 y Fv(1)p 12388 68804 8108 45 v 12388 69818 a Fu(A)246 b Fr(\241)g Fu(v)15108 69984 y Fx(1)15604 69818 y Fu(U)16360 69984 y Fx(1)17103 69818 y Fr(\241)g Fu(v)18747 69984 y Fx(2)19243 69818 y Fu(U)19999 69984 y Fx(2)20628 69058 y Fu(U)21505 68483 y Fx(1)p Fs(=)p Fx(2)21384 69353 y(2)22896 69058 y Fu(M)23970 69224 y Fx(2)25143 69058 y Fv(=)26817 68778 y Fq(f)26680 69058 y Fu(M)27754 69224 y Fx(1)28435 67830 y Fq(h)28958 69058 y Fv([)q Fu(K)79 b Fv(\()p Fu(v)31252 69224 y Fp(\241)31999 69058 y Fv(\)])32737 68502 y Fp(\241)p Fx(1)34541 69058 y Fr(\241)615 b Fu(v)36554 69224 y Fx(+)37288 67830 y Fq(i)37811 68059 y Fp(\241)p Fx(1)39320 68778 y Fq(f)39184 69058 y Fu(M)40258 69224 y Fx(2)43096 69058 y Fv(\(4.12\))p eop end %%Page: 24 24 TeXDict begin 24 23 bop 6075 3345 a Fx(24)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6051 y Fv(with)8767 5771 y Fq(f)8631 6051 y Fu(M)9705 6217 y Fx(1)10566 6051 y Fv(=)365 b Fu(M)12866 6217 y Fx(1)13363 6051 y Fu(U)14240 5476 y Fx(1)p Fs(=)p Fx(2)14119 6346 y(1)15631 6051 y Fu(P)154 b Fv([)p Fu(U)17559 6217 y Fx(1)18324 6051 y Fv(+)269 b Fu(U)20210 6217 y Fx(2)20706 6051 y Fv(])21013 5649 y Fp(\241)p Fx(1)p Fs(=)p Fx(2)23097 6051 y Fu(P)557 b Fv(and)26688 5771 y Fq(f)26552 6051 y Fu(M)27626 6217 y Fx(2)28891 6051 y Fv(=)769 b Fu(P)154 b Fv([)p Fu(U)32449 6217 y Fx(1)33214 6051 y Fv(+)269 b Fu(U)35100 6217 y Fx(2)35596 6051 y Fv(])35903 5649 y Fp(\241)p Fx(1)p Fs(=)p Fx(2)37987 6051 y Fu(P)154 b(U)39729 5476 y Fx(1)p Fs(=)p Fx(2)39608 6346 y(2)41119 6051 y Fu(M)42193 6217 y Fx(2)43093 6051 y Fv(where)6075 7396 y Fu(P)522 b Fv(denotes)370 b(orthogonal)i(pro)61 b(jection)371 b(on)-31 b(to)370 b(k)-31 b(er\()p Fu(U)27266 7562 y Fx(1)28009 7396 y Fv(+)246 b Fu(U)29872 7562 y Fx(2)30369 7396 y Fv(\))30799 6995 y Fp(?)31546 7396 y Fv(.)7403 9057 y(Since)369 b Fu(U)11049 9223 y Fx(1)11915 9057 y Fv(and)g Fu(U)14823 9223 y Fx(2)15689 9057 y Fv(are)g(p)31 b(ositiv)-31 b(e,)21940 7784 y Fq(\260)21940 8448 y(\260)21940 9112 y(\260)22555 9057 y Fu(U)23432 8482 y Fx(1)p Fs(=)p Fx(2)23311 9365 y Fs(j)24823 9057 y Fu(P)154 b Fv([)p Fu(U)26751 9223 y Fx(1)27493 9057 y Fv(+)246 b Fu(U)29356 9223 y Fx(2)29852 9057 y Fv(])30159 8655 y Fp(\241)p Fx(1)p Fs(=)p Fx(2)32243 9057 y Fu(P)33108 7784 y Fq(\260)33108 8448 y(\260)33108 9112 y(\260)34030 9057 y Fr(\267)308 b Fv(1)369 b(and)g(therefore)20359 10474 y Fq(\260)20359 11138 y(\260)20359 11803 y(\260)21110 11467 y(f)20974 11747 y Fu(M)22048 11913 y Fs(j)22514 10474 y Fq(\260)22514 11138 y(\260)22514 11803 y(\260)23129 12522 y Fs(H)59 b(S)25231 11747 y Fr(\267)677 b(k)p Fu(M)28396 11913 y Fs(j)28863 11747 y Fr(k)29416 12119 y Fs(H)59 b(S)31334 11747 y Fu(:)11455 b Fv(\(4.13\))6075 14121 y(F)-92 b(urthermore)353 b([)p Fu(K)79 b Fv(\()p Fu(v)14773 14287 y Fp(\241)15521 14121 y Fv(\)])16258 13719 y Fp(\241)p Fx(1)17801 14121 y Fv(is)354 b(maximally)j(dissipativ)-31 b(e)356 b(as)e(sho)-31 b(wn)354 b(in)g(Lemma)h(2.1.)489 b(Th)-31 b(us)354 b(for)6075 15450 y(eac)-31 b(h)321 b(\257xed)f Fu(v)11680 15616 y Fp(\241)12747 15450 y Fv(w)-31 b(e)321 b(are)f(in)h(the)g(situation)i (go)-31 b(v)g(erned)321 b(b)-31 b(y)321 b(Lemma)g(4.1,)332 b(and)321 b(Prop.)g(4.2)h(follo)-31 b(ws)6075 16778 y(via)438 b(the)f(F)-92 b(ubini)437 b(Theorem.)696 b(The)437 b(factor)h(of)f(2)g (on)g(the)g(righ)-31 b(t)438 b(hand)f(side)g(results)f(from)i(the)6075 18106 y(Jacobian)370 b(of)g(the)g(transformation)h Fr(h)p Fu(v)22382 18272 y Fx(1)22879 18106 y Fu(;)184 b(v)23907 18272 y Fx(2)24404 18106 y Fr(i)308 b(7!)g(h)p Fu(v)27524 18272 y Fx(+)28259 18106 y Fu(;)184 b(v)29287 18272 y Fp(\241)30034 18106 y Fr(i)p Fv(.)14293 b Fh(\244)7403 20274 y Fv(W)-92 b(e)369 b(will)h(use)f(Prop.)h(4.2)g(to)g(conclude)g (that)h(for)e(the)g(random)h(op)31 b(erator)370 b Fu(H)90 b Fv(,)7366 21519 y Fq(\257)7366 22183 y(\257)7366 22847 y(\257)7366 23511 y(\257)7735 21563 y(\275)8565 23124 y Fr(h)p Fu(\264)9545 23290 y Fs(\256)10176 23124 y Fu(;)184 b(\264)11217 23290 y Fs(\257)11815 23124 y Fr(i)308 b(2)f Fv([0)p Fu(;)184 b Fv(1])15809 22667 y Fx(2)16985 23124 y Fv(:)17600 21519 y Fq(\260)17600 22183 y(\260)17600 22847 y(\260)17600 23511 y(\260)18215 23124 y Fu(M)19289 23290 y Fx(1)19786 23124 y Fu(U)20663 22667 y Fx(1)p Fs(=)p Fx(2)20542 23398 y Fs(\256)23373 22375 y Fv(1)p 22187 22869 2927 45 v 22187 23883 a Fu(H)336 b Fr(\241)246 b Fu(z)25246 23124 y(U)26123 22549 y Fx(1)p Fs(=)p Fx(2)26002 23458 y Fs(\257)27514 23124 y Fu(M)28588 23290 y Fx(2)29084 21519 y Fq(\260)29084 22183 y(\260)29084 22847 y(\260)29084 23511 y(\260)29699 24231 y Fs(H)59 b(S)31432 23124 y Fu(>)308 b(t)33001 21563 y Fq(\276)33831 21519 y(\257)33831 22183 y(\257)33831 22847 y(\257)33831 23511 y(\257)26492 26136 y Fr(\267)677 b Fv(2)369 b Fu(C)29743 26302 y Fs(W)31119 26136 y Fr(k)q Fu(M)32747 26302 y Fx(1)33243 26136 y Fr(k)33797 26468 y Fs(H)59 b(S)35591 26136 y Fr(k)q Fu(M)37219 26302 y Fx(2)37715 26136 y Fr(k)38269 26468 y Fs(H)g(S)40442 25387 y Fv(1)p 40196 25881 1046 45 v 40196 26895 a Fu(\270t)41682 26136 y(:)1107 b Fv(\(4.14\))6075 28362 y(Sev)-31 b(eral)370 b(commen)-31 b(ts)370 b(are)f(in)h(order:)7957 29984 y(\(1\))554 b(When)380 b(the)h(energy)f Fu(z)428 b Fv(lies)381 b(in)g(the)f(lo)-31 b(w)g(er)382 b(half)f(plane)g(\(4.14\))i(follo)-31 b(ws)382 b(directly)g(from)9924 31312 y(Prop.)370 b(4.2)h(with)f Fu(v)17798 31478 y Fx(1)18602 31312 y Fv(=)307 b Fu(\264)20320 31478 y Fs(\256)20951 31312 y Fu(;)184 b(v)21979 31478 y Fx(2)22783 31312 y Fv(=)308 b Fu(\264)24502 31478 y Fs(\257)25468 31312 y Fv(and)369 b Fu(U)28376 31478 y Fx(1)29180 31312 y Fv(=)307 b Fu(\270U)31750 31478 y Fs(\256)32381 31312 y Fu(;)184 b(U)33628 31478 y Fx(2)34433 31312 y Fv(=)307 b Fu(\270U)37003 31478 y Fs(\257)37600 31312 y Fv(.)7957 32641 y(\(2\))554 b(Eq.)330 b(\(4.14\))h(also)e (holds)f(for)h Fu(z)376 b Fv(in)329 b(the)f(upp)31 b(er)328 b(half)h(plane,)338 b(as)328 b(can)g(b)31 b(e)328 b(seen)g(b)-31 b(y)328 b(taking)9924 33969 y(conjugates.)7957 35297 y(\(3\))554 b(F)-92 b(or)303 b(real)g(energies)h(\()p Fu(z)356 b Fv(=)307 b Fu(E)372 b Fr(2)307 b Ft(R)p Fv(\),)318 b(eq.)304 b(\(4.14\))i(holds)d(also)h(in)g(the)f(limits)i Fu(z)356 b Fr(!)308 b Fu(E)178 b Fr(\247)114 b Fv(i0,)9924 36626 y(as)286 b(follo)-31 b(ws)288 b(from)f(the)f(b)31 b(ound)286 b(at)h(complex)g(energies)f(and)g(F)-92 b(atou's)286 b(lemma)i(\(compare)9924 37954 y(the)370 b(argumen)-31 b(t)371 b(for)e(pro)-31 b(ving)370 b(Lemma)h(4.1)f(in)f(App)31 b(endix)370 b(B\).)7403 39576 y(The)412 b(ab)31 b(o)-31 b(v)g(e)412 b(results)f(displa)-31 b(y)412 b(that)h(once)e(the)h (resolv)-31 b(en)g(t)412 b(of)g(the)f(con)-31 b(tin)g(uum)413 b(op)31 b(erator)412 b(is)6075 40904 y(brac)-31 b(k)g(eted)349 b(with)g(a)g(pair)g(of)f(Hilb)31 b(ert-Sc)-31 b(hmidt)351 b(op)31 b(erators)348 b Fu(M)32053 41070 y Fx(1)32550 40904 y Fv(,)353 b Fu(M)34284 41070 y Fx(2)35128 40904 y Fv(its)c(fractional)i(momen)-31 b(ts)6075 42233 y(can)361 b(b)31 b(e)361 b(handled)g(similarly)i(to)f(the)f(discrete)g(case.)490 b(In)361 b(the)g(pro)31 b(ofs)361 b(of)h(Lemmas)f(4.3)i(and)e(4.4)6075 43561 y(w)-31 b(e)404 b(use)g(an)g(argumen)-31 b(t)406 b(whic)-31 b(h)405 b(sho)-31 b(ws)404 b(that)i Fu(\302)25791 43727 y Fs(x)26348 43561 y Fv(\()p Fu(H)360 b Fr(\241)269 b Fu(z)49 b Fv(\))30182 43159 y Fp(\241)p Fx(1)31370 43561 y Fu(\302)32063 43727 y Fs(y)32999 43561 y Fv(can)404 b(b)31 b(e)404 b(presen)-31 b(ted)404 b(as)g(a)g(sum)6075 44889 y(of)369 b(a)h Fm(b)-57 b(ounde)g(d)370 b Fv(op)31 b(erator)370 b(and)f(one)g(of)h(the)g(form)19744 46805 y Fu(M)20818 46971 y Fs(x)21375 46805 y Fu(\302)22068 46971 y Fs(B)22737 47082 y Fl(x)23286 46805 y Fv(\()p Fu(H)336 b Fr(\241)246 b Fu(z)49 b Fv(\))27073 46348 y Fp(\241)p Fx(1)28262 46805 y Fu(\302)28955 46971 y Fs(B)29624 47082 y Fl(y)30158 46805 y Fu(M)31232 46971 y Fs(y)31948 46805 y Fu(;)10841 b Fv(\(4.15\))6075 48721 y(with)320 b Fu(M)9620 48887 y Fs(])10352 48721 y Fv(Hilb)31 b(ert-Sc)-31 b(hmidt)322 b(op)31 b(erators)319 b(and)h Fu(B)26248 48887 y Fs(x)26804 48721 y Fu(;)184 b(B)28135 48887 y Fs(y)28987 48721 y Fv(sets)319 b(somewhat)i(larger)e(than)h (the)f(balls)6075 50049 y(of)405 b(radius)h Fu(r)436 b Fv(around)405 b Fu(x)g Fv(and)h Fu(y)40 b Fv(.)600 b(This)405 b(will)i(allo)-31 b(w)408 b(us)d(to)h(conclude)g (\257niteness)f(of)h(fractional)6075 51377 y(momen)-31 b(ts)437 b(after)f(\\a)-31 b(v)g(eraging)438 b(o)-31 b(v)g(er)436 b(the)g(lo)31 b(cal)437 b(en)-31 b(vironmen)g(t,")455 b(i.e.,)f(a)-31 b(v)g(eraging)438 b(o)-31 b(v)g(er)436 b(all)h Fu(\264)45295 51543 y Fs(\256)6075 52706 y Fv(with)370 b Fu(U)9352 52872 y Fs(\256)10352 52706 y Fv(non-zero)f(in)h Fu(B)16976 52872 y Fs(x)17902 52706 y Fv(or)f Fu(B)20098 52872 y Fs(y)20629 52706 y Fv(.)6075 54943 y(4.3.)555 b FB(A)338 b(pair)f(of)h(fractional-momen)-35 b(t)338 b(lemmas.)553 b Fv(W)-92 b(e)293 b(no)-31 b(w)295 b(presen)-31 b(t)293 b(the)h(t)-31 b(w)g(o)296 b(basic)e(tec)-31 b(h-)6075 56271 y(nical)268 b(results)e(whic)-31 b(h)268 b(giv)-31 b(e)268 b(\257niteness)f(of)g(the)g(fractional)j(momen)-31 b(ts,)289 b(as)267 b(w)-31 b(ell)268 b(as)f(a)g(\\decoupling)6075 57600 y(argumen)-31 b(t")371 b(to)f(b)31 b(e)369 b(used)f(in)i(the)f (pro)31 b(of)369 b(of)h(Theorem)g(1.2.)6075 59662 y FB(De\257nition)425 b(4.1.)554 b Fv(F)-92 b(or)368 b Fu(\256)312 b Fr(2)307 b(I)82 b Fv(,)369 b(w)-31 b(e)370 b(denote)18316 61578 y Fr(I)18919 61744 y Fs(\256)20226 61578 y Fv(:=)677 b Fr(f)p Fu(\263)390 b Fr(2)307 b(I)389 b Fv(:)308 b(dist)q(\()p Fu(\256)t(;)184 b(\263)82 b Fv(\))309 b Fu(<)f Fv(3)p Fu(r)31 b Fr(g)9413 b Fv(\(4.16\))6075 63493 y(and)507 b(set)g Fr(I)10834 63659 y Fs(\256;\257)12806 63493 y Fv(=)537 b Fr(I)14807 63659 y Fs(\256)15775 63493 y Fr([)338 b(I)17454 63659 y Fs(\257)18051 63493 y Fv(.)906 b(Lik)-31 b(ewise,)543 b(for)507 b(an)-31 b(y)508 b(subset)e Fr(L)538 b(\275)f(I)82 b Fv(,)541 b Fr(I)36621 63659 y Fp(L)37826 63493 y Fv(:=)d Fr([)40270 63659 y Fs(\256)p Fp(2L)42111 63493 y Fr(I)42714 63659 y Fs(\256)43344 63493 y Fv(.)906 b(By)6075 64822 y Fr(F)6981 64420 y Fs(c)6871 65127 y Fp(L)7935 64822 y Fv(w)-31 b(e)396 b(denote)h(the)f Fu(\276)40 b Fv(-algebra)396 b(generated)h(b)-31 b(y)396 b(all)h Fu(\264)28814 64988 y Fs(\256)29840 64822 y Fv(with)g Fu(\256)j Fm(not)c Fv(in)g Fr(I)37369 64988 y Fp(L)38038 64822 y Fv(.)572 b(Th)-31 b(us,)403 b Ft(E)184 b Fv(\()q Fr(\242jF)44935 64420 y Fs(c)44825 65127 y Fp(L)45495 64822 y Fv(\))6075 66150 y(represen)-31 b(ts)368 b(a)-31 b(v)g(eraging)371 b(o)-31 b(v)g(er)370 b(the)g(\\lo)31 b(cal)371 b(en)-31 b(vironmen)g(t")371 b(of)f Fr(L)p Fv(.)7403 68213 y(Our)408 b(\257rst)g(b)31 b(ound)409 b(yields)g(the)g(\257niteness)g(of)g(the)g Fu(s)p Fv(-momen)-31 b(ts)410 b(after)f(a)-31 b(v)g(eraging)412 b(o)-31 b(v)g(er)409 b(the)6075 69541 y(lo)31 b(cal-en)-31 b(vironmen)g(t.)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(25)6075 6001 y FB(Lemma)576 b(4.3.)633 b Fm(L)-57 b(et)519 b Fu(H)16307 6167 y Fs(!)17466 6001 y Fm(b)-57 b(e)519 b(a)f(r)-57 b(andom)518 b(Schr\304)-566 b(odinger)519 b(op)-57 b(er)g(ator)517 b(which)h(satis\257es)f FB(A)i Fm(with)6075 7330 y(disor)-57 b(der)456 b(str)-57 b(ength)456 b Fu(\270)416 b(>)g Fv(0)p Fm(.)691 b(Then)457 b(ther)-57 b(e)456 b(exists)f Fu(C)28236 7496 y Fs(\270)29690 7330 y Fu(<)416 b Fr(1)456 b Fm(such)g(that)f(any) i(r)-57 b(estriction)457 b(of)6075 8658 y Fu(H)6995 8824 y Fs(!)8032 8658 y Fm(to)395 b(a)h(r)-57 b(e)g(gion)398 b Fv(\255)e Fm(ob)-57 b(eys)7366 11691 y Fv(Prob)9906 10130 y Fq(\265)10854 10086 y(\260)10854 10750 y(\260)10854 11414 y(\260)10854 12078 y(\260)11469 11691 y Fu(U)12225 11857 y Fs(\256)16111 10942 y Fv(1)p 12988 11436 6800 45 v 12988 12725 a Fu(H)13998 12150 y Fx(\(\255\))13908 12855 y Fs(!)15623 12725 y Fr(\241)246 b Fu(E)310 b Fr(\241)246 b Fv(i)p Fu(")19921 11691 y(U)20677 11857 y Fs(\257)21274 10086 y Fq(\260)21274 10750 y(\260)21274 11414 y(\260)21274 12078 y(\260)22196 11691 y Fu(>)308 b(t)23765 10086 y Fq(\257)23765 10750 y(\257)23765 11414 y(\257)23765 12078 y(\257)24318 11691 y Fr(F)25224 11234 y Fs(c)25114 11964 y(\256;\257)26549 10130 y Fq(\266)22854 14947 y Fr(\267)704 b Fu(C)25210 15113 y Fs(\270)26186 14947 y Fv(\(1)247 b(+)f(1)p Fu(=\270)p Fv(\)\(1)i(+)e Fr(j)p Fu(E)311 b Fr(\241)246 b Fu(E)36402 15113 y Fs(o)36894 14947 y Fr(j)p Fv(\))37631 14490 y Fs(d)p Fx(+2)39798 14198 y Fu(D)40745 13797 y Fx(2)p 39798 14693 1444 45 v 40320 15707 a Fu(t)41682 14947 y(;)1107 b Fv(\(4.17\))6075 17356 y Fm(for)396 b(any)g Fu(\256)t(;)184 b(\257)368 b Fr(2)307 b(I)82 b Fm(,)396 b(any)h Fu(E)371 b Fr(2)308 b Ft(R)396 b Fm(and)g Fu(")308 b(>)f Fv(0)p Fm(,)397 b(wher)-57 b(e)396 b(the)g(c)-57 b(o)g(e\261cient)398 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y Fs(\270)31313 27318 y Fu(C)32104 27484 y Fs(\270)32992 27318 y Fu(<)307 b Fr(1)p Fv(.)7957 28647 y(\(3\))554 b(F)-92 b(or)375 b(fractional)j(momen)-31 b(ts)377 b(w)-31 b(e)376 b(use)f(the)g(\\la)-31 b(y)g(er-cak)g(e")379 b(represen)-31 b(tation)376 b({)g Ft(E)184 b Fv(\()q Fu(X)43843 28245 y Fs(s)44316 28647 y Fv(\))318 b(=)9924 29125 y Fq(R)10662 29399 y Fp(1)10447 30411 y Fx(0)11785 30017 y Fv(Prob\()p Fu(X)395 b(>)308 b(t)17451 29615 y Fx(1)p Fs(=s)18818 30017 y Fv(\)d)p Fu(t)369 b Fv({)h(to)g(conclude)g (that)7182 33288 y Ft(E)8104 31727 y Fq(\265)9052 31683 y(\260)9052 32347 y(\260)9052 33011 y(\260)9052 33676 y(\260)9667 33288 y Fu(U)10423 33454 y Fs(\256)14309 32539 y Fv(1)p 11186 33034 6800 45 v 11186 34067 a Fu(H)12196 33748 y Fx(\(\255\))13821 34067 y Fr(\241)246 b Fu(E)310 b Fr(\241)246 b Fv(i)p Fu(")18119 33288 y(U)18875 33454 y Fs(\257)19472 31683 y Fq(\260)19472 32347 y(\260)19472 33011 y(\260)19472 33676 y(\260)20087 31957 y Fs(s)20560 31683 y Fq(\257)20560 32347 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Fs(s)23678 48866 y Fu(b)24153 48392 y Fp(\241)p Fx(2)p Fs(s)24153 49141 y Fp(\241)26128 48866 y Fv(\(1)h(+)f Fr(j)p Fu(E)311 b Fr(\241)246 b Fu(E)31824 49032 y Fx(0)32320 48866 y Fr(j)p Fv(\))33057 48409 y Fs(s)p Fx(\()p Fs(d)p Fx(+2\))36174 48866 y Fu(D)37121 48409 y Fx(2)p Fs(s)38404 48866 y Fu(N)39293 49032 y Fx(\244)40139 48866 y Fu(N)41028 49032 y Fx(\244)41634 48810 y Fj(0)43096 48866 y Fv(\(4.20\))9924 51428 y(where)372 b Fr(L)p Fv(\(\244\))313 b(:=)f Fr(f)p Fu(\260)374 b Fr(2)312 b(I)393 b Fv(:)312 b FB(1)22114 51594 y Fx(\244)22776 51428 y Fu(U)23532 51594 y Fs(\260)24412 51428 y Fr(6)p Fv(=)f(0)p Fr(g)373 b Fv(and)f Fu(N)30107 51594 y Fx(\244)31140 51428 y Fv(indicates)h(the)f(n)-31 b(um)g(b)31 b(er)372 b(of)g(p)31 b(oin)-31 b(ts)9924 52756 y(in)291 b Fr(L)p Fv(\(\244\).)467 b(In)289 b(particular,)308 b(this)290 b(pro)-31 b(v)g(es)290 b(eq.)g(\(1.7\))i(of)f(Lemma)f(1.1)i(under)d (assumption)9924 54085 y FB(A)p Fv(.)370 b(The)g(impro)-31 b(v)g(emen)g(t)372 b(under)c 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Fu(H)357 b Fr(\241)266 b Fu(z)49 b Fv(\))21322 59519 y Fp(\241)p Fx(1)22511 59921 y Fv(,)407 b(where)26716 59641 y Fq(b)26457 59921 y Fu(H)489 b Fv(is)400 b(obtained)h(from)g Fu(H)489 b Fv(b)-31 b(y)400 b(redra)-31 b(wing)401 b(the)6075 61250 y(coupling)371 b(v)-61 b(ariables)369 b Fu(\264)15643 61416 y Fs(\263)16520 61250 y Fv(for)h Fu(\263)451 b Fv(\\near")370 b Fu(\257)58 b Fv(.)493 b(The)369 b(basic)h(result)f(is)g(an)g(estimate)i(of)f(the)f(form)12551 64112 y Ft(E)13473 62551 y Fq(\265)14422 62507 y(\260)14422 63171 y(\260)14422 63835 y(\260)14422 64499 y(\260)15037 64112 y Fu(U)15793 64278 y Fs(\256)18018 63363 y Fv(1)p 16556 63857 3478 45 v 16556 64871 a Fu(H)17476 65037 y Fs(!)18363 64871 y Fr(\241)246 b Fu(z)20166 64112 y(U)20922 64278 y Fs(\257)21519 62507 y Fq(\260)21519 63171 y(\260)21519 63835 y(\260)21519 64499 y(\260)22134 62780 y Fs(s)22607 62507 y Fq(\257)22607 63171 y(\257)22607 63835 y(\257)22607 64499 y(\257)23161 64112 y Fr(F)24067 63655 y Fs(c)23957 64385 y(\257)24553 62551 y Fq(\266)26045 64112 y Fr(\267)676 b Fv(const.)30602 62507 y Fq(\260)30602 63171 y(\260)30602 63835 y(\260)30602 64499 y(\260)31217 64112 y Fu(\302)31910 64278 y Fs(\256)34135 63363 y Fv(1)p 32673 63857 V 32932 64789 a Fq(b)32673 65069 y Fu(H)33593 65235 y Fs(!)34480 65069 y Fr(\241)246 b Fu(z)36283 64112 y FB(1)36919 64278 y Fs(S)37561 62507 y Fq(\260)37561 63171 y(\260)37561 63835 y(\260)37561 64499 y(\260)38176 62780 y Fs(s)39141 64112 y Fu(;)3648 b Fv(\(4.21\))6075 66884 y(with)414 b Fu(S)477 b Fv(an)414 b(appropriate)g(neigh)-31 b(b)31 b(orho)g(o)g(d)415 b(of)f Fu(\257)58 b Fv(.)625 b(Ho)-31 b(w)g(ev)g(er,)427 b(the)414 b(result)f(p)31 b(ermits)413 b(a)h(n)-31 b(um)g(b)31 b(er)6075 68213 y(of)385 b(v)-61 b(ariations,)391 b(and)385 b(the)g(estimate)h(\(4.22\))i(stated)d(b)31 b(elo)-31 b(w)386 b(is)f(sligh)-31 b(tly)387 b(complicated)h(b)31 b(ecause)6075 69541 y(it)370 b(is)f(tailored)i(to)f(the)f(required)g (application)j(in)e(the)f(pro)31 b(of)370 b(of)f(Theorem)h(1.2.)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 6075 3345 a Fx(26)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)7403 6001 y Fv(F)-92 b(or)458 b(that)h(application)j(it)d(is)f(con)-31 b(v)g(enien)g(t)461 b(to)e(use)e(a)i(smo)31 b(oth)460 b(function)f(in)g(place)g(of)g Fu(U)45021 6167 y Fs(\257)45618 6001 y Fv(.)6075 7330 y(Th)-31 b(us)507 b(w)-31 b(e)508 b(\257x)g(a)g(c)-31 b(hoice)508 b(of)h(a)e(partition)j(of)e(unit)-31 b(y)-92 b(,)544 b Fr(f)p Fv(\243)30287 7496 y Fs(\256)31456 7330 y Fv(:)539 b Fu(\256)j Fr(2)c(I)82 b(g)p Fv(,)543 b(with)509 b(the)e(follo)-31 b(wing)6075 8658 y(prop)31 b(erties:)7957 10370 y(\(1\))554 b(Eac)-31 b(h)323 b(function)h(\243) 17779 10536 y Fs(\256)18731 10370 y Fv(is)e(non-negativ)-31 b(e)324 b(and)e(smo)31 b(oth)323 b(with)h(compact)f(supp)31 b(ort)322 b(in)g(the)9924 11698 y(ball)371 b(of)f(radius)f(4)p Fu(r)31 b(=)p Fv(3)370 b(cen)-31 b(tered)369 b(at)h Fu(\256)t Fv(.)7957 13026 y(\(2\))554 b(The)370 b(collection)i Fr(f)p Fv(\243)18534 13192 y Fs(\256)19165 13026 y Fr(g)d Fv(is)h(a)f(partition)i(of)f(unit)-31 b(y:)31320 12196 y Fq(P)32489 13358 y Fs(\256)33304 13026 y Fv(\243)34165 13192 y Fs(\256)34796 13026 y Fv(\()p Fu(q)40 b Fv(\))308 b(=)f(1.)7957 14355 y(\(3\))554 b(sup)11591 14625 y Fs(\256)12222 14355 y Fv(\()p Fr(k)q Fg(r)p Fv(\243)15128 14521 y Fs(\256)15759 14355 y Fr(k)16312 14687 y Fp(1)17434 14355 y Fu(;)184 b Fr(k)r Fv(\242\243)20263 14521 y Fs(\256)20894 14355 y Fr(k)21448 14687 y Fp(1)22385 14355 y Fv(\))677 b Fu(<)g Fr(1)p Fv(.)6075 16260 y(An)-31 b(y)490 b(c)-31 b(hoice)490 b(with)h(these)f(prop)31 b(erties)489 b(will)i(do,)521 b(although)492 b(the)d(constan)-31 b(t)38672 15980 y Fq(e)38452 16260 y Fu(C)568 b Fv(in)490 b(eq.)g(\(4.22\))6075 17588 y(b)31 b(elo)-31 b(w)370 b(will)h(dep)31 b(end)369 b(on)g(the)h(suprem)-31 b(um)369 b(in)g(\(3\).)6075 19753 y FB(Lemma)576 b(4.4.)633 b Fm(L)-57 b(et)519 b Fu(H)16307 19919 y Fs(!)17466 19753 y Fm(b)-57 b(e)519 b(a)f(r)-57 b(andom)518 b(Schr\304)-566 b(odinger)519 b(op)-57 b(er)g(ator)517 b(which)h(satis\257es)f FB(A)i Fm(with)6075 21158 y(disor)-57 b(der)408 b(str)-57 b(ength)408 b Fu(\270)331 b(>)f Fv(0)p Fm(.)548 b(Then)409 b(ther)-57 b(e)409 b(exists)27113 20879 y Fq(e)26893 21158 y Fu(C)27684 21324 y Fs(\270)28595 21158 y Fu(<)330 b Fr(1)409 b Fm(such)f(that)g(any)h(r)-57 b(estriction)409 b(of)g Fu(H)6075 22487 y Fm(to)395 b(a)h(r)-57 b(e)g(gion)398 b Fv(\255)e Fm(ob)-57 b(eys)7366 25516 y Fv(Prob)9906 23955 y Fq(\265)10854 23911 y(\260)10854 24575 y(\260)10854 25240 y(\260)10854 25904 y(\260)11469 25516 y Fu(\302)12162 25682 y Fs(x)14728 24767 y Fv(1)p 12851 25262 4306 45 v 12851 26551 a Fu(H)13861 25976 y Fx(\(\255\))13771 26680 y Fs(!)15487 26551 y Fr(\241)246 b Fu(z)17290 25516 y Fv(\243)18151 25682 y Fs(\257)18748 25516 y Fv(\(1)h(+)f Fu(H)22005 25682 y Fs(o)22743 25516 y Fr(\241)g Fu(E)24667 25682 y Fs(o)25160 25516 y Fv(\))25590 25059 y Fx(1)p Fs(=)p Fx(2)26982 23911 y Fq(\260)26982 24575 y(\260)26982 25240 y(\260)26982 25904 y(\260)27904 25516 y Fu(>)308 b(t)29473 23911 y Fq(\257)29473 24575 y(\257)29473 25240 y(\257)29473 25904 y(\257)30026 25516 y Fr(F)30932 25059 y Fs(c)30822 25790 y(\257)40 b(;\260)32194 23955 y Fq(\266)16717 28773 y Fr(\267)18501 28493 y Fq(e)18281 28773 y Fu(C)19072 28939 y Fs(\270)20049 28773 y Fv(\(1)247 b(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)25427 28939 y Fx(0)25924 28773 y Fr(j)p Fv(\))26661 28316 y Fx(\()p Fs(d)p Fx(+3\))29571 27168 y Fq(\260)29571 27832 y(\260)29571 28496 y(\260)29571 29160 y(\260)30186 28773 y Fu(\302)30879 28939 y Fs(x)33445 28024 y Fv(1)p 31569 28518 V 31828 29527 a Fq(b)31569 29807 y Fu(H)32579 29232 y Fx(\(\255\))32489 29937 y Fs(!)34204 29807 y Fr(\241)g Fu(z)36007 28773 y FB(1)36643 28939 y Fs(S)37187 29056 y Fl(\257)28 b(;\260)38469 27168 y Fq(\260)38469 27832 y(\260)38469 28496 y(\260)38469 29160 y(\260)39798 28024 y Fu(D)40745 27622 y Fx(2)p 39798 28518 1444 45 v 40320 29532 a Fu(t)41682 28773 y(;)1107 b Fv(\(4.22\))6075 31754 y Fm(for)502 b(any)h Fu(z)549 b Fr(2)501 b Ft(C)323 b Fr(n)h Ft(R)p Fm(,)530 b Fu(x)500 b Fr(2)g Ft(R)19317 31352 y Fs(d)19834 31754 y Fm(,)529 b(and)503 b Fu(\257)58 b(;)184 b(\260)563 b Fr(2)501 b(I)584 b Fm(with)502 b Fv(dist\()p Fu(x;)184 b(\257)58 b Fv(\))p Fm(,)532 b Fv(dist)q(\()p Fu(x;)184 b(\260)61 b Fv(\))502 b Fu(>)e Fv(6)p Fu(r)31 b Fm(.)830 b(The)6075 33220 y(c)-57 b(o)g(e\261cient)11486 32941 y Fq(e)11266 33220 y Fu(C)12057 33386 y Fs(\270)13033 33220 y Fm(ob)g(eys)19952 34647 y Fq(e)19732 34927 y Fu(C)20523 35093 y Fs(\270)21807 34927 y Fr(\267)704 b Fv(const)p Fu(:)397 b Fv(\(1)247 b(+)f Fu(\270)p Fv(\))30016 34470 y Fs(d)p Fx(+4)31960 34927 y Fu(:)10829 b Fv(\(4.23\))6075 36928 y Fm(Her)-57 b(e)530 b Fu(S)9535 37094 y Fs(\257)40 b(;\260)11457 36928 y Fv(=)549 b Fr(f)p Fu(q)590 b Fv(:)550 b(dist\()p Fu(q)40 b(;)184 b Fr(f)p Fu(\257)58 b(;)184 b(\260)61 b Fr(g)p Fv(\))555 b Fr(\267)549 b Fv(5)p Fu(r)31 b Fr(g)531 b Fm(and)28624 36648 y Fq(b)28365 36928 y Fu(H)620 b Fm(is)530 b(obtaine)-57 b(d)530 b(fr)-57 b(om)530 b Fu(H)620 b Fm(by)530 b(r)-57 b(eplacing)6075 38256 y Fr(f)p Fu(\264)7178 38422 y Fs(\263)7994 38256 y Fv(:)308 b Fu(\263)389 b Fr(2)308 b(I)11131 38422 y Fs(\257)40 b(;\260)12503 38256 y Fr(g)397 b Fm(with)e(arbitr)-57 b(ary)396 b(values)476 b Fv(^)-633 b Fu(\264)24342 38422 y Fs(\263)25158 38256 y Fr(2)307 b Fv([0)p Fu(;)184 b Fv(1])p Fm(:)17833 40332 y Fq(b)17574 40612 y Fu(H)793 b Fv(=)704 b Fu(H)336 b Fv(+)246 b Fu(\270)24634 39561 y Fq(X)24045 41944 y Fs(\263)50 b Fp(2I)25596 42061 y Fl(\257)28 b(;\260)26822 40612 y Fv(\()79 b(^)-632 b Fu(\264)27802 40778 y Fs(\263)28557 40612 y Fr(\241)246 b Fu(\264)30214 40778 y Fs(\263)30722 40612 y Fv(\))p Fu(U)31908 40778 y Fs(\263)32417 40612 y Fv(\()p Fu(q)40 b Fv(\))308 b Fu(:)8670 b Fv(\(4.24\))6075 43995 y FB(Remark:)504 b Fm(i.)1039 b Fv(Note)376 b(that)h(w)-31 b(e)376 b(ha)-31 b(v)g(e)377 b(resampled)f Fu(H)465 b Fv(at)376 b(a)g(second)g(site)g Fu(\260)j Fr(2)318 b(I)457 b Fv(as)376 b(w)-31 b(ell)377 b(as)e Fu(\257)58 b Fv(,)6075 45324 y(with)314 b(the)g(result)g(that)h (\243)16624 45490 y Fs(\257)17534 45324 y Fv(is)e(replaced)h(b)-31 b(y)314 b(the)g(c)-31 b(haracteristic)315 b(function)g(of)f(a)g(neigh) -31 b(b)31 b(orho)g(o)g(d)6075 46652 y(of)339 b Fu(\260)401 b Fv(and)339 b Fu(\257)58 b Fv(.)484 b(This)339 b(additional)j (resampling)f(is)e(required)g(in)g(the)h(application)i(of)e(this)f (lemma)6075 47980 y(in)435 b(the)h(pro)31 b(of)436 b(of)g(Theorem)g (1.2,)454 b(but)435 b(do)31 b(es)435 b(not)h(pla)-31 b(y)437 b(a)f(k)-31 b(ey)436 b(role)f(in)h(the)g(presen)-31 b(t)435 b(section.)6075 49309 y Fm(ii.)1220 b Fv(The)408 b(factor)g(\(1)272 b(+)f Fu(H)17141 49475 y Fs(o)17905 49309 y Fr(\241)g Fu(E)19854 49475 y Fs(o)20346 49309 y Fv(\))20776 48907 y Fx(1)p Fs(=)p Fx(2)22575 49309 y Fv(is)407 b(included)h(here)e(to)i(con)-31 b(trol)408 b(v)-61 b(arious)408 b(comm)-31 b(utators)6075 50637 y([)p Fu(H)28 b(;)184 b Fv(\243])414 b(whic)-31 b(h)413 b(app)31 b(ear)412 b(in)h(applications)i(of)d(the)h(lemma.)623 b(It)412 b(is)g(to)h(b)31 b(ound)412 b(this)h(factor)g(that)6075 51965 y(w)-31 b(e)369 b(c)-31 b(ho)31 b(ose)370 b(to)g(w)-31 b(ork)370 b(with)g(the)g(smo)31 b(oth)370 b(function)h(\243)28814 52131 y Fs(\257)29411 51965 y Fv(.)7403 53294 y(In)336 b(the)h(applications)j(of)d(Lemma)g(4.4)h(w)-31 b(e)338 b(will)g(c)-31 b(ho)31 b(ose)386 b Fq(b)-664 b Fu(\264)31334 53460 y Fs(\263)32178 53294 y Fv(to)338 b(ha)-31 b(v)g(e)337 b(the)g(same)g(probabilit)-31 b(y)6075 54622 y(distribution)395 b(as)e Fu(\264)14068 54788 y Fs(\263)14576 54622 y Fv(.)564 b(This)393 b(will)i(pro)-31 b(vide)394 b(us)f(with)h(a)f(\\decoupling)j (argumen)-31 b(t,")401 b(as)393 b(follo)-31 b(ws:)6075 56323 y(If)509 b Fu(X)596 b Fv(is)509 b(a)g(quan)-31 b(tit)g(y)512 b(for)d(whic)-31 b(h)510 b Ft(E)21781 55094 y Fq(\263)22443 56323 y Fu(X)23447 55921 y Fs(s)23920 56323 y Fr(jF)25133 55921 y Fs(c)25023 56636 y(\257)40 b(;\260)26396 55094 y Fq(\264)27598 56323 y Fr(\267)540 b Fu(A)29829 56489 y Fs(s)30843 56323 y Fu(<)h Fr(1)p Fv(,)k(then)509 b(eq.)h(\(4.22\))h(and)f(the)6075 57922 y(Cauc)-31 b(h)g(y-Sc)g(h)g(w)g(arz)372 b(inequalit)-31 b(y)372 b(imply)e(that)7182 61283 y Ft(E)8104 59390 y Fq(\303)9113 61283 y Fu(X)10117 60826 y Fs(s=)p Fx(2)11854 59678 y Fq(\260)11854 60342 y(\260)11854 61006 y(\260)11854 61671 y(\260)12469 61283 y Fu(\302)13162 61449 y Fs(x)15728 60534 y Fv(1)p 13852 61029 4306 45 v 13852 62062 a Fu(H)14862 61743 y Fx(\(\255\))16487 62062 y Fr(\241)246 b Fu(z)18290 61283 y Fv(\243)19151 61449 y Fs(\257)19748 61283 y Fv(\(1)h(+)f Fu(H)23005 61449 y Fs(o)23744 61283 y Fr(\241)g Fu(E)25668 61449 y Fs(o)26160 61283 y Fv(\))26590 60826 y Fx(1)p Fs(=)p Fx(2)27982 59678 y Fq(\260)27982 60342 y(\260)27982 61006 y(\260)27982 61671 y(\260)28597 59952 y Fs(s=)p Fx(2)29965 59346 y Fq(\257)29965 60010 y(\257)29965 60674 y(\257)29965 61339 y(\257)29965 62003 y(\257)30518 61283 y 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Fx(2)19419 68763 y Fs(s)21364 66884 y Fq(\260)21364 67548 y(\260)21364 68213 y(\260)21364 68877 y(\260)21979 68489 y Fu(\302)22672 68655 y Fs(x)25238 67740 y Fv(1)p 23362 68235 V 23621 69167 a Fq(b)23362 69447 y Fu(H)24372 69127 y Fx(\(\255\))25997 69447 y Fr(\241)246 b Fu(z)27800 68489 y FB(1)28436 68655 y Fs(S)29078 66884 y Fq(\260)29078 67548 y(\260)29078 68213 y(\260)29078 68877 y(\260)29693 67158 y Fs(s=)p Fx(2)31553 68489 y Fu(:)43096 66602 y Fv(\(4.25\))p eop end %%Page: 27 27 TeXDict begin 27 26 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(27)6075 6001 y Fv(If)417 b Fq(b)-663 b Fu(\264)7732 6167 y Fs(\263)8609 6001 y Fv(is)369 b(distributed)h(iden)-31 b(tically)372 b(to)e Fu(\264)22642 6167 y Fs(\263)23519 6001 y Fv(then,)g(up)31 b(on)369 b(taking)i(exp)31 b(ectations,)372 b(w)-31 b(e)369 b(obtain)7182 9231 y Ft(E)8104 7338 y Fq(\303)8980 9231 y Fu(X)9984 8774 y Fs(s=)p Fx(2)11721 7626 y Fq(\260)11721 8290 y(\260)11721 8954 y(\260)11721 9618 y(\260)12336 9231 y Fu(\302)13029 9397 y Fs(x)15595 8482 y Fv(1)p 13719 8976 4306 45 v 13719 10010 a Fu(H)14729 9690 y Fx(\(\255\))16354 10010 y Fr(\241)246 b Fu(z)18158 9231 y Fv(\243)19019 9397 y Fs(\257)19616 9231 y Fv(\(1)h(+)f Fu(H)22873 9397 y Fs(o)23611 9231 y Fr(\241)g Fu(E)25535 9397 y Fs(o)26027 9231 y Fv(\))26457 8774 y Fx(1)p Fs(=)p Fx(2)27849 7626 y Fq(\260)27849 8290 y(\260)27849 8954 y(\260)27849 9618 y(\260)28464 7899 y Fs(s=)p Fx(2)29832 7338 y Fq(!)21486 12995 y Fr(\267)676 b Fv(const.)370 b Fu(A)27057 12538 y Fx(1)p Fs(=)p Fx(2)27057 13268 y Fs(s)28818 12995 y Ft(E)29740 11102 y Fq(\303)30617 11389 y(\260)30617 12054 y(\260)30617 12718 y(\260)30617 13382 y(\260)31232 12995 y Fu(\302)31925 13161 y Fs(x)34491 12246 y Fv(1)p 32614 12740 V 32614 13774 a Fu(H)33624 13454 y Fx(\(\255\))35250 13774 y Fr(\241)246 b Fu(z)37053 12995 y FB(1)37689 13161 y Fs(S)38331 11389 y Fq(\260)38331 12054 y(\260)38331 12718 y(\260)38331 13382 y(\260)38946 11663 y Fs(s=)p Fx(2)40314 11102 y Fq(!)41682 12995 y Fu(:)1107 b Fv(\(4.26\))6075 16101 y(In)514 b(the)h(\257nal)g (expression,)551 b(w)-31 b(e)515 b(ha)-31 b(v)g(e)516 b(replaced)27332 15821 y Fq(b)27073 16101 y Fu(H)604 b Fv(b)-31 b(y)515 b Fu(H)605 b Fv(since)514 b(the)h(t)-31 b(w)g(o)517 b(are)d(iden)-31 b(tically)6075 17429 y(distributed.)6075 19558 y(4.4.)555 b FB(Av)-35 b(eraging)323 b(o)-35 b(v)g(er)323 b(lo)35 b(cal)321 b(en)-35 b(vironmen)g(ts.)554 b Fv(W)-92 b(e)280 b(no)-31 b(w)281 b(deriv)-31 b(e)280 b(the)h(t)-31 b(w)g(o)281 b(tec)-31 b(hnical)282 b(lem-)6075 20886 y(mas)351 b(stated)g(ab)31 b(o)-31 b(v)g(e.)487 b(This)351 b(subsection)g(is)f(essen)-31 b(tial)352 b(for)f(our)f(analysis,)356 b(but)350 b(the)h(reader)f(ma)-31 b(y)6075 22215 y(wish)342 b(to)h(skip)g(it)f(at)h(\257rst)f(reading,)349 b(as)342 b(the)g(argumen)-31 b(ts)344 b(here)d(are)h(not)h(required)f(elsewhere) g(in)6075 23543 y(the)369 b(article.)6075 25632 y Fm(Pr)-57 b(o)g(of)396 b(of)h(L)-57 b(emma)397 b(4.3.)554 b Fv(W)-92 b(e)328 b(will)i(write)f Fu(z)356 b Fv(=)308 b Fu(E)229 b Fv(+)165 b Fu(i")328 b Fv(and)h(assume)g(without)i(loss)d(that)i Fu(")308 b(<)f Fv(1.)6075 26961 y(Th)-31 b(us)433 b(1)289 b(+)f Fr(j)p Fu(E)353 b Fr(\241)289 b Fu(E)14378 27127 y Fx(0)14875 26961 y Fr(j)414 b(\273)g Fv(1)289 b(+)g Fr(j)p Fu(z)337 b Fr(\241)289 b Fu(E)21989 27127 y Fx(0)22486 26961 y Fr(j)p Fv(.)684 b(Before)433 b(applying)i(Prop.)f(4.2)g(and)g (the)f(asso)31 b(ciated)6075 28289 y(eq.)425 b(\(4.14\))s(,)439 b(w)-31 b(e)425 b(m)-31 b(ust)425 b(in)-31 b(tro)31 b(duce)425 b(Hilb)31 b(ert-Sc)-31 b(hmidt)427 b(op)31 b(erators)425 b(to)h(the)f(left)g(and)g(the)g(righ)-31 b(t)6075 29617 y(of)441 b Fu(U)8163 29783 y Fs(\256)8794 29617 y Fv(\()p Fu(H)384 b Fr(\241)294 b Fu(z)49 b Fv(\))12677 29216 y Fp(\241)p Fx(1)13865 29617 y Fu(U)14621 29783 y Fs(\257)15218 29617 y Fv(.)707 b(The)441 b(pro)31 b(cedure)440 b(w)-31 b(e)441 b(use)g(to)g(insert)g(these)g(op)31 b(erators)441 b(is)f(somewhat)6075 30946 y(in)-31 b(v)g(olv)g(ed,)372 b(so)d(let)h(us)e(\257rst)h(outline)i(the)e(argumen)-31 b(t:)7957 32541 y(\(1\))554 b(W)-92 b(e)369 b(replace)g Fu(U)16341 32707 y Fs(\256)16972 32541 y Fu(;)g(U)18404 32707 y Fs(\257)19370 32541 y Fv(b)-31 b(y)370 b(the)f(upp)31 b(er)368 b(b)31 b(ounds)369 b Fu(b)30309 32707 y Fx(+)31044 32541 y Fv(\243)31905 32139 y Fx(2)32402 32541 y Fu(;)g(b)33553 32707 y Fx(+)34288 32541 y Fv(\252)35149 32139 y Fx(2)35646 32541 y Fv(,)16800 33501 y Fq(\260)16800 34165 y(\260)16800 34829 y(\260)16800 35493 y(\260)17415 35106 y Fu(U)18171 35272 y Fs(\256)20121 34357 y Fv(1)p 18935 34851 2927 45 v 18935 35865 a Fu(H)336 b Fr(\241)246 b Fu(z)21994 35106 y(U)22750 35272 y Fs(\257)23347 33501 y Fq(\260)23347 34165 y(\260)23347 34829 y(\260)23347 35493 y(\260)24639 35106 y Fr(\267)676 b Fu(b)26651 34649 y Fx(2)26651 35380 y(+)27570 33501 y Fq(\260)27570 34165 y(\260)27570 34829 y(\260)27570 35493 y(\260)28185 35106 y Fv(\243)29046 34649 y Fx(2)30862 34357 y Fv(1)p 29676 34851 V 29676 35865 a Fu(H)336 b Fr(\241)246 b Fu(z)32735 35106 y Fv(\252)33596 34649 y Fx(2)34093 33501 y Fq(\260)34093 34165 y(\260)34093 34829 y(\260)34093 35493 y(\260)34892 35106 y Fu(;)7897 b Fv(\(4.27\))9924 37570 y(where)397 b(\243)p Fu(;)184 b Fv(\252)397 b(denote)g(smo)31 b(othed)397 b(c)-31 b(haracteristic)398 b(functions)f(for)g(the)f(balls)i Fu(B)43204 37168 y Fs(r)43148 37843 y(\256)43778 37570 y Fv(,)404 b Fu(B)45385 37168 y Fs(r)45329 37883 y(\257)9924 38913 y Fv(resp)31 b(ectiv)-31 b(ely)-92 b(.)7957 40242 y(\(2\))554 b(The)370 b(smo)31 b(othness)369 b(of)h(\243)f(allo)-31 b(ws)371 b(us)e(to)h(pro)-31 b(v)g(e)370 b(an)f Fm(identity)18142 42624 y Fv(\243)19003 42167 y Fx(2)20819 41875 y Fv(1)p 19633 42370 V 19633 43384 a Fu(H)336 b Fr(\241)246 b Fu(z)23369 42624 y Fv(=)676 b Fu(B)670 b Fv(+)615 b Fu(T)28816 42344 y Fq(e)28693 42624 y Fv(\243)29554 42167 y Fx(2)31370 41875 y Fv(1)p 30183 42370 V 30183 43384 a Fu(H)336 b Fr(\241)246 b Fu(z)33550 42624 y(;)9239 b Fv(\(4.28\))9924 45199 y(where)13319 44919 y Fq(e)13196 45199 y Fv(\243)440 b(is)f(a)i(\\fattened")h(v)-31 b(ersion)440 b(of)g(\243)g(and)g Fu(T)593 b Fv(is)440 b(Hilb)31 b(ert-Sc)-31 b(hmidt)442 b(with)f(HS-)9924 46527 y(norm)388 b(whic)-31 b(h)388 b(is)f(uniformly)i(b)31 b(ounded)387 b(with)h(resp)31 b(ect)387 b(to)h(the)f(disorder.)546 b(The)388 b(op)31 b(er-)9924 47855 y(ator)370 b Fu(B)425 b Fv(is)369 b(norm)g(b)31 b(ounded)369 b(\(uniformly)j(with)e(resp)31 b(ect)368 b(to)i(the)g(disorder\).)7957 49184 y(\(3\))554 b(W)-92 b(e)369 b(rep)31 b(eat)369 b(this)h(pro)31 b(cedure)368 b(to)i(the)f(righ)-31 b(t)371 b(of)e(the)h(resolv)-31 b(en)g(t)370 b(to)g(obtain)15769 51566 y(\243)16630 51109 y Fx(2)18446 50817 y Fv(1)p 17259 51312 V 17259 52325 a Fu(H)337 b Fr(\241)246 b Fu(z)20319 51566 y Fv(\252)21180 51109 y Fx(2)22353 51566 y Fv(=)676 b Fu(T)24537 51732 y Fs(\256)25291 51286 y Fq(e)25168 51566 y Fv(\243)26029 51109 y Fx(2)27845 50817 y Fv(1)p 26658 51312 V 26658 52325 a Fu(H)336 b Fr(\241)246 b Fu(z)29841 51286 y Fq(e)29718 51566 y Fv(\252)30579 51109 y Fx(2)31075 51566 y Fu(T)31722 51732 y Fs(\257)32934 51566 y Fv(+)615 b Fu(B)35306 51109 y Fp(0)35923 51566 y Fu(;)6866 b Fv(\(4.29\))9924 53893 y(where)369 b Fu(B)14021 53492 y Fp(0)14700 53893 y Fv(is)g(norm)h(b)31 b(ounded.)7957 55222 y(\(4\))554 b(W)-92 b(e)439 b(in)-31 b(tro)31 b(duce)439 b(the)h(partition)h(of)e(unit)-31 b(y)27845 54392 y Fq(P)29198 55222 y Fu(U)29954 55388 y Fs(\256)30585 55222 y Fu(=F)593 b Fv(b)31 b(et)-31 b(w)g(een)36910 54942 y Fq(e)36787 55222 y Fv(\243)37648 54820 y Fx(2)38144 55222 y Fu(;)38759 54942 y Fq(e)38635 55222 y Fv(\252)39496 54820 y Fx(2)40432 55222 y Fv(and)440 b(the)f(re-)9924 56550 y(solv)-31 b(en)g(t,)454 b(and)435 b(then)h(apply)g(Prop.)g(4.2)g(to)g(eac)-31 b(h)435 b(term)h(in)f(the)g (resulting)h(sum)f(\(the)9924 57879 y(actual)510 b(argumen)-31 b(t)509 b(is)f(complicated)i(somewhat)g(b)-31 b(y)508 b(the)g(fact)h(that)g Fu(T)40554 58045 y Fs(\256)41185 57879 y Fu(;)184 b(T)42323 58045 y Fs(\257)43428 57879 y Fv(carry)9924 59207 y(some)370 b(dep)31 b(endence)369 b(on)g(the)g(randomness\).)7403 60802 y(W)-92 b(e)463 b(no)-31 b(w)465 b(turn)f(to)g(sp)31 b(eci\257cs.)776 b(A)464 b(go)31 b(o)g(d)465 b(deal)f(rests)f(on)h(the)g(pro)31 b(of)465 b(of)f(eq.)g(\(4.28\).)780 b(This)6075 62130 y(iden)-31 b(tit)g(y)265 b(is)d(a)g(sp)31 b(ecial)263 b(case)g(of)g(the)f(follo)-31 b(wing:)443 b(let)263 b(\243)27970 62296 y Fs(j)28699 62130 y Fv(b)31 b(e)262 b(an)-31 b(y)263 b(sequence)f(of)h(smo)31 b(oth)263 b(functions)6075 63458 y(suc)-31 b(h)369 b(that)h(\243)11831 63624 y Fs(j)12666 63458 y Fv(tak)-31 b(es)370 b(v)-61 b(alue)370 b(1)g(on)f(the)g(supp)31 b(ort)369 b(of)h(\243)28948 63624 y Fs(j)45 b Fp(\241)p Fx(1)30547 63458 y Fv(,)370 b(then)f(for)h(eac)-31 b(h)369 b Fu(n)308 b Fr(\270)g Fv(1)17246 65881 y(\243)18107 65424 y Fx(2)18107 66154 y(1)19923 65132 y Fv(1)p 18737 65626 V 18737 66640 a Fu(H)336 b Fr(\241)246 b Fu(z)22473 65881 y Fv(=)676 b Fu(T)24657 66047 y Fs(n)25444 65881 y Fv(\243)26305 65424 y Fx(2)26305 66154 y Fs(n)p Fx(+1)29348 65132 y Fv(1)p 28161 65626 V 28161 66640 a Fu(H)336 b Fr(\241)246 b Fu(z)31835 65881 y Fv(+)615 b Fu(B)34151 66047 y Fs(n)43096 65881 y Fv(\(4.30\))6075 68145 y(where)21874 69541 y Fu(T)22521 69707 y Fs(n)23800 69541 y Fv(=)676 b Fu(A)26167 69707 y Fx(1)27033 69541 y Fr(\242)184 b(\242)g(\242)371 b Fu(A)29523 69707 y Fs(n)43096 69541 y Fv(\(4.31\))p eop end %%Page: 28 28 TeXDict begin 28 27 bop 6075 3345 a Fx(28)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(with)13894 7797 y Fu(A)14724 7963 y Fs(j)15867 7797 y Fv(=)677 b(\243)18266 7963 y Fs(j)20263 7048 y Fv(1)p 18865 7543 3351 45 v 18865 8557 a Fu(H)19785 8723 y Fs(o)20523 8557 y Fv(+)246 b Fu(a)22533 7797 y Fv(\(\243)23824 7963 y Fs(j)24291 7797 y Fv(\()p Fu(a)g Fv(+)g Fu(z)295 b Fr(\241)246 b Fu(\270V)29868 7963 y Fs(!)30508 7797 y Fv(\))615 b Fr(\241)g Fv([)q(\243)34198 7963 y Fs(j)34664 7797 y Fu(;)184 b(H)36075 7963 y Fs(o)36569 7797 y Fv(]\))492 b Fu(;)4991 b Fv(\(4.32\))6075 10057 y(and)7302 12318 y Fu(B)8142 12484 y Fs(n)9421 12318 y Fv(=)676 b(\243)11819 12484 y Fx(1)13847 11569 y Fv(1)p 12448 12063 V 12448 13077 a Fu(H)13368 13243 y Fs(o)14107 13077 y Fv(+)246 b Fu(a)15932 12318 y Fv(\243)16793 12484 y Fx(1)17904 12318 y Fv(+)615 b Fu(T)20027 12484 y Fx(1)20524 12318 y Fv(\243)21385 12484 y Fx(2)23413 11569 y Fv(1)p 22014 12063 V 22014 13077 a Fu(H)22934 13243 y Fs(o)23673 13077 y Fv(+)246 b Fu(a)25498 12318 y Fv(\243)26359 12484 y Fx(2)27470 12318 y Fv(+)615 b Fr(\242)184 b(\242)g(\242)617 b Fv(+)e Fu(T)32975 12484 y Fs(n)p Fp(\241)p Fx(1)34711 12318 y Fv(\243)35572 12484 y Fs(n)37706 11569 y Fv(1)p 36307 12063 V 36307 13077 a Fu(H)37227 13243 y Fs(o)37966 13077 y Fv(+)246 b Fu(a)39791 12318 y Fv(\243)40652 12484 y Fs(n)41562 12318 y Fu(:)1227 b Fv(\(4.33\))6075 14798 y(Here)368 b Fu(a)308 b Fv(:=)g(1)246 b Fr(\241)g Fu(E)13783 14964 y Fx(0)14280 14798 y Fv(,)370 b(so)f Fu(H)17236 14964 y Fs(o)17974 14798 y Fv(+)246 b Fu(a)308 b Fr(\270)f Fv(1.)7403 16127 y(T)-92 b(o)488 b(v)-31 b(erify)504 b(\(4.30\))s(,)518 b(use)487 b(induction)i(on)f Fu(n)g Fv(with)h(the)f(induction)i(step)d(pro)-31 b(vided)489 b(b)-31 b(y)488 b(the)6075 17455 y(follo)-31 b(wing)373 b(\\comm)-31 b(utator)372 b(argumen)-31 b(t":)7391 19931 y(\243)8252 19474 y Fx(2)8252 20205 y Fs(j)10068 19182 y Fv(1)p 8882 19676 2927 45 v 8882 20690 a Fu(H)336 b Fr(\241)246 b Fu(z)12617 19931 y Fv(=)369 b(\243)14708 20097 y Fs(j)16706 19182 y Fv(1)p 15308 19676 3351 45 v 15308 20690 a Fu(H)16228 20856 y Fs(o)16966 20690 y Fv(+)246 b Fu(a)18975 19931 y Fv(\()p Fu(H)20325 20097 y Fs(o)21064 19931 y Fv(+)g Fu(a)p Fv(\)\243)24047 20097 y Fs(j)25834 19182 y Fv(1)p 24647 19676 2927 45 v 24647 20690 a Fu(H)336 b Fr(\241)246 b Fu(z)12617 22761 y Fv(=)369 b(\243)14708 22927 y Fs(j)16706 22013 y Fv(1)p 15308 22507 3351 45 v 15308 23521 a Fu(H)16228 23687 y Fs(o)16966 23521 y Fv(+)246 b Fu(a)18791 22761 y Fv(\243)19652 22927 y Fs(j)20118 22761 y Fv(\()p Fu(H)21468 22927 y Fs(o)22207 22761 y Fv(+)g Fu(a)p Fv(\))25649 22013 y(1)p 24462 22507 2927 45 v 24462 23521 a Fu(H)337 b Fr(\241)246 b Fu(z)28137 22761 y Fr(\241)615 b Fv(\243)30474 22927 y Fs(j)32472 22013 y Fv(1)p 31073 22507 3351 45 v 31073 23521 a Fu(H)31993 23687 y Fs(o)32732 23521 y Fv(+)246 b Fu(a)34741 22761 y Fv([\243)35909 22927 y Fs(j)36376 22761 y Fu(;)184 b(H)37787 22927 y Fs(o)38280 22761 y Fv(])40092 22013 y(1)p 38905 22507 2927 45 v 38905 23521 a Fu(H)336 b Fr(\241)246 b Fu(z)12617 25592 y Fv(=\243)14339 25758 y Fs(j)16337 24843 y Fv(1)p 14939 25337 3351 45 v 14939 26351 a Fu(H)15859 26517 y Fs(o)16597 26351 y Fv(+)g Fu(a)18422 25592 y Fv(\243)19283 25758 y Fs(j)15938 28422 y Fv(+)615 b(\243)18275 28588 y Fs(j)20273 27673 y Fv(1)p 18874 28168 V 18874 29182 a Fu(H)19794 29348 y Fs(o)20533 29182 y Fv(+)246 b Fu(a)22542 28422 y 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Fx(2)39642 33205 y Fs(j)45 b Fx(+1)41537 32931 y Fu(;)1252 b Fv(\(4.35\))6075 34984 y(since)345 b(\243)9624 35150 y Fs(j)45 b Fx(+1)11557 34984 y Fv(is)345 b(iden)-31 b(tically)349 b(one)c(throughout)j(the)d(supp)31 b(ort)345 b(of)h(\243)33794 35150 y Fs(j)34606 34984 y Fv(and)g(\243)37596 35150 y Fs(j)38062 34984 y Fv(\()p Fu(a)199 b Fv(+)g Fu(z)247 b Fr(\241)199 b Fu(\270V)43450 35150 y Fs(!)44090 34984 y Fv(\))544 b Fr(\241)6075 36312 y Fv([\243)7243 36478 y Fs(j)7710 36312 y Fu(;)184 b(H)9121 36478 y Fs(o)9614 36312 y Fv(])370 b(is)f(a)g(di\256eren)-31 b(tial)371 b({)e(hence)g(lo)31 b(cal)371 b({)e(op)31 b(erator.)7403 37641 y(The)369 b(represen)-31 b(tation)371 b(pro)-31 b(vided)370 b(b)-31 b(y)369 b(eq.)h(\(4.30\))i(has)d(t)-31 b(w)g(o)371 b(k)-31 b(ey)369 b(features:)7957 39313 y(\(1\))554 b(The)317 b(norm)f(of)h Fu(B)17035 39479 y Fs(n)17953 39313 y Fv(is)f(b)31 b(ounded)317 b(uniformly)g(in)g Fu(!)356 b Fv(and)316 b(lo)31 b(cally)318 b(uniformly)g(in)f(energy)-92 b(.)7957 40641 y(\(2\))554 b Fu(T)10571 40807 y Fs(n)11549 40641 y Fv(is)375 b(in)h Fr(I)14569 40807 y Fs(p)15455 40641 y Fv(for)g(arbitrary)g Fu(p)317 b(>)h(d=n)376 b Fv(with)h Fr(k)250 b(\242)h(k)30540 40807 y Fs(p)31052 40641 y Fv(-norm)376 b(b)31 b(ounded)375 b(uniformly)i(in)f Fu(!)9924 41970 y Fv(and)370 b(lo)31 b(cally)371 b(uniformly)g(in)f (energy)-92 b(.)7403 43642 y(T)g(o)401 b(see)f(this)h(w)-31 b(e)402 b(\257rst)e(observ)-31 b(e)400 b(that)i Fu(A)24289 43808 y Fs(j)25116 43642 y Fr(2)360 b(I)26817 43808 y Fs(p)27729 43642 y Fv(for)401 b Fu(p)360 b(>)g(d)400 b Fv(with)i(uniform)g Fr(k)267 b(\242)h(k)41256 43808 y Fs(p)41768 43642 y Fv(-b)31 b(ounds.)6075 45052 y(This)420 b(follo)-31 b(ws)423 b(since)d Fu(\265)15622 45218 y Fs(j)16088 45052 y Fv(\()p Fu(H)17438 45218 y Fx(0)18215 45052 y Fv(+)280 b Fu(a)p Fv(\))20371 44651 y Fp(\241)p Fx(1)21953 45052 y Fr(2)393 b(I)23687 45218 y Fs(p)24618 45052 y Fv(for)420 b Fu(p)393 b(>)f(d=)p Fv(2,)435 b Fu(\265)31510 45218 y Fs(j)31976 45052 y 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b(\257xed)d(the)h(sequence)f(\243)21437 68379 y Fs(j)22328 68213 y Fv(w)-31 b(e)425 b(obtain)h(a)f(Hilb)31 b(ert-Sc)-31 b(hmidt)427 b(op)31 b(erator)425 b Fu(T)41864 68379 y Fs(n)42466 68213 y Fv(.)659 b(More)6075 69541 y(precisely)-92 b(,)334 b Fu(T)11431 69707 y Fs(n)12357 69541 y Fv(is)325 b(a)g Fm(r)-57 b(andom)324 b Fv(Hilb)31 b(ert-Sc)-31 b(hmidt)327 b(op)31 b(erator)325 b(since)g Fu(A)34288 69707 y Fs(j)35079 69541 y Fv(dep)31 b(end)324 b(on)h(the)f(random)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(29)6075 6001 y Fv(p)31 b(oten)-31 b(tial)448 b(through)f(the)g(terms)f(\243)21093 6167 y Fs(j)21559 6001 y Fu(V)22205 6167 y Fs(!)22846 6001 y Fv(.)723 b(Ho)-31 b(w)g(ev)g(er,)467 b Fu(T)29388 6167 y Fs(n)30436 6001 y Fv(is)447 b(a)f(pro)31 b(duct)446 b(of)h Fu(n)f Fv(terms)g(eac)-31 b(h)447 b(of)6075 7330 y(whic)-31 b(h)370 b(is)f(linear)h(in)f(\243)15585 7496 y Fs(j)16052 7330 y Fu(V)16698 7496 y Fs(!)17338 7330 y Fv(.)493 b(Since)20675 9855 y(\243)21536 10021 y Fs(j)22002 9855 y Fu(V)22648 10021 y Fs(!)23596 9855 y Fv(=)25021 8803 y Fq(X)24765 11186 y Fs(\263)50 b Fp(2I)26316 11297 y Fl(\256)27061 9855 y Fu(\264)27611 10021 y Fs(\263)28119 9855 y Fv(\243)28980 10021 y Fs(j)29446 9855 y Fu(U)30202 10021 y Fs(\263)31018 9855 y Fu(;)11771 b Fv(\(4.38\))6075 13324 y(w)-31 b(e)341 b(conclude)h(that)g Fu(T)15216 13490 y Fs(n)16160 13324 y Fv(is)f(a)g(p)31 b(olynomial)345 b(of)c(degree)g Fu(n)g Fv(in)h(the)f(v)-61 b(ariables)342 b Fr(f)p Fu(\264)38295 13490 y Fs(\263)39111 13324 y Fv(:)308 b Fu(\263)389 b Fr(2)308 b(I)42248 13490 y Fs(\256)42878 13324 y Fr(g)342 b Fv(with)6075 14652 y(Hilb)31 b(ert-Sc)-31 b(hmidt)364 b(v)-61 b(alued)362 b(co)31 b(e\261cien)-31 b(ts.)492 b(Recall)362 b(that)i Fr(I)30040 14818 y Fs(\256)31032 14652 y Fv(is)d(the)i(set)e(of)i(lattice)h(sites)d(within)6075 15980 y(distance)370 b(3)p Fu(r)400 b Fv(of)370 b 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b(There)392 b(is)f Fu(C)15074 27317 y Fs(d)15936 27151 y Fu(<)345 b Fr(1)p Fv(,)397 b(whic)-31 b(h)393 b(dep)31 b(ends)390 b(on)i(the)g(parameters)g(of)g(the)g(mo)31 b(del)393 b(and)f(the)9924 28480 y(c)-31 b(hoice)371 b(of)e(\243)15336 28646 y Fs(j)16172 28480 y Fv(\(but)h(not)g(on)f(the) g(coupling)i(v)-61 b(ariables)370 b Fu(\264)33612 28646 y Fs(\263)34120 28480 y Fv(\),)g(suc)-31 b(h)369 b(that)16936 30799 y Fr(k)p Fu(B)56 b Fr(k)676 b(\267)369 b Fu(C)21635 30965 y Fs(d)22521 30799 y Fv(\(1)247 b(+)e Fu(\270)h Fv(+)g Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)29897 30965 y Fx(0)30394 30799 y Fr(j)p Fv(\))31131 30342 y Fx(2)p Fs(n)19614 32652 y Fr(\267)369 b Fu(C)21635 32818 y Fs(d)22521 32652 y Fv(\(1)247 b(+)e Fu(\270)p Fv(\))25933 32195 y Fx(2)p Fs(n)26977 32652 y Fv(\(1)i(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)32355 32818 y Fx(0)32852 32652 y Fr(j)p Fv(\))33589 32195 y Fx(2)p Fs(n)34941 32652 y Fu(:)43096 31751 y Fv(\(4.40\))7957 34982 y(\(2\))554 b(Eac)-31 b(h)285 b Fu(T)13238 35148 y Fs(])13936 34982 y Fv(is)e(a)i(p)31 b(olynomial)287 b(of)d(degree)g Fu(n)308 b Fr(\267)f Fv(\()p Fu(d)76 b Fv(+)g(2\))p Fu(=)p Fv(2)286 b(in)e(the)h(v)-61 b(ariables)284 b Fu(\264)40589 35148 y Fs(\263)41381 34982 y Fv(for)g Fu(\263)389 b Fr(2)308 b(I)45512 35148 y Fs(])9924 36310 y Fv(with)371 b(co)31 b(e\261cien)-31 b(ts)370 b(whic)-31 b(h)370 b(are)g(\(non-random\))h (Hilb)31 b(ert-Sc)-31 b(hmidt)371 b(op)31 b(erators:)19350 38795 y Fu(T)19997 38961 y Fs(])21087 38795 y Fv(=)22624 37744 y Fq(X)22833 40170 y Fx(\()p Fs(k)19 b Fx(\))24408 38795 y Fu(T)25209 38220 y Fx(\()p Fs(k)g Fx(\))25055 39129 y Fs(])26890 37744 y Fq(Y)26629 40127 y Fs(\263)50 b Fp(2I)28180 40244 y Fl(])28750 38795 y Fu(\270)29396 38338 y Fp(j)p Fs(k)19 b Fp(j)30466 38795 y Fu(\264)31056 38189 y Fs(k)31527 38306 y Fl(\263)31016 39129 y Fs(\263)32343 38795 y Fu(;)10446 b Fv(\(4.41\))9924 42451 y(where)278 b(the)g(summation)i(is)d(o)-31 b(v)g(er)279 b(m)-31 b(ulti-indices)280 b(\()p Fu(k)35 b Fv(\))308 b 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b(b)31 b(ound)369 b(for)g(the)h(join)-31 b(t)371 b(densit)-31 b(y)370 b(of)f Fu(\264)40622 34409 y Fs(\263)41499 34243 y Fv(and)h Fu(\264)44202 34409 y Fs(\263)44655 34187 y Fj(0)45009 34243 y Fv(.)7403 35571 y(Putting)h(this)f(all)g(together)h(w)-31 b(e)369 b(\257nd)g(that)7366 38613 y(Prob)9906 37052 y Fq(\265)10854 37008 y(\260)10854 37672 y(\260)10854 38336 y(\260)10854 39000 y(\260)11469 38613 y Fu(U)12225 38779 y Fs(\256)14175 37864 y Fv(1)p 12988 38358 2927 45 v 12988 39372 a Fu(H)336 b Fr(\241)246 b Fu(z)16048 38613 y(U)16804 38779 y Fs(\257)17400 37008 y Fq(\260)17400 37672 y(\260)17400 38336 y(\260)17400 39000 y(\260)18323 38613 y Fu(>)307 b(t)19891 37008 y Fq(\257)19891 37672 y(\257)19891 38336 y(\257)19891 39000 y(\257)20445 38613 y Fr(F)21351 38156 y Fs(c)21241 38887 y(\256;\257)22676 37052 y Fq(\266)15283 41757 y Fr(\267)369 b Fu(M)18261 40196 y Fq(\265)19076 41757 y Fu(C)19867 41923 y Fs(d)20998 41757 y Fv(+)615 b(\()p Fu(M)367 b Fr(\241)246 b Fv(1\))h Fr(\243)27789 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15215 69244 a Fu(H)336 b Fr(\241)246 b Fu(z)18767 68205 y Fq(e)18644 68485 y Fv(\243)19505 68028 y Fx(2)19505 68759 y Fs(\257)20655 68485 y Fv(\()q(\()p Fu(z)295 b Fv(+)246 b Fu(a)g Fr(\241)g Fu(\270V)26663 68651 y Fs(!)27303 68485 y Fv(\)\243)28594 68651 y Fs(\257)29437 68485 y Fv(+)g([)q(\243)31713 68651 y Fs(\257)32310 68485 y Fu(;)184 b(H)33721 68651 y Fs(o)34214 68485 y Fv(])q(\))38163 67736 y(1)p 35639 68230 5603 45 v 35639 69264 a(\()p Fu(a)246 b Fv(+)g Fu(H)38927 69430 y Fs(o)39420 69264 y Fv(\))39850 68944 y Fx(1)p Fs(=)p Fx(2)41682 68485 y Fu(;)1107 b Fv(\(4.48\))p eop end %%Page: 31 31 TeXDict begin 31 30 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(31)6075 6001 y Fv(where)435 b(w)-31 b(e)437 b(ha)-31 b(v)g(e)437 b(used)e(that)i Fu(\302)19410 6167 y Fs(x)19967 6001 y Fv(\243)20828 6167 y Fs(\257)21843 6001 y Fv(=)419 b(0.)693 b(Here)27482 5722 y Fq(e)27359 6001 y Fv(\243)28220 6167 y Fs(\257)29253 6001 y Fv(ma)-31 b(y)437 b(b)31 b(e)435 b(an)-31 b(y)437 b(function)g(whic)-31 b(h)437 b(is)f(one)6075 7330 y(throughout)371 b(the)e(supp)31 b(ort)369 b(of)h(\243)19986 7496 y Fs(\257)20583 7330 y Fv(.)492 b(As)369 b(a)g(result,)7249 8328 y Fq(\260)7249 8992 y(\260)7249 9656 y(\260)7249 10320 y(\260)7864 9933 y Fu(\302)8557 10099 y Fs(x)10433 9184 y Fv(1)p 9247 9678 2927 45 v 9247 10692 a Fu(H)336 b Fr(\241)246 b Fu(z)12306 9933 y Fv(\243)13167 10099 y Fs(\257)13764 9933 y Fv(\()p Fu(a)h Fv(+)f Fu(H)17053 10099 y Fs(o)17545 9933 y Fv(\))17975 9476 y Fx(1)p Fs(=)p Fx(2)19367 8328 y Fq(\260)19367 8992 y(\260)19367 9656 y(\260)19367 10320 y(\260)20659 9933 y Fr(\267)676 b Fu(c)369 b Fv(\(1)247 b(+)f Fu(\270)p Fv(\)\(1)h(+)f Fr(j)q Fu(z)294 b Fr(\241)246 b Fu(E)31835 10099 y Fx(0)32332 9933 y Fr(j)p Fv(\))33623 8328 y Fq(\260)33623 8992 y(\260)33623 9656 y(\260)33623 10320 y(\260)34238 9933 y Fu(\302)34931 10099 y Fs(x)36808 9184 y Fv(1)p 35621 9678 V 35621 10692 a Fu(H)336 b Fr(\241)246 b Fu(z)39172 9653 y Fq(e)39049 9933 y Fv(\243)39910 9476 y Fx(2)39910 10207 y Fs(\257)40507 8328 y Fq(\260)40507 8992 y(\260)40507 9656 y(\260)40507 10320 y(\260)41614 9933 y Fu(;)1175 b Fv(\(4.49\))6075 12745 y(with)337 b Fu(c)307 b(<)h Fr(1)335 b Fv(\(dep)31 b(ending)337 b(on)f(the)f(size)h(of)g Fg(r)p Fv(\243)26266 12911 y Fs(\257)26863 12745 y Fv(\).)482 b(W)-92 b(e)335 b(c)-31 b(ho)31 b(ose)33556 12465 y Fq(e)33433 12745 y Fv(\243)34294 12911 y Fs(\257)35226 12745 y Fv(to)337 b(ha)-31 b(v)g(e)337 b(supp)31 b(ort)335 b(in)h(the)6075 14074 y(ball)370 b(of)g(radius)f(5)p Fu(r)31 b(=)p Fv(3)371 b(cen)-31 b(tered)369 b(at)h Fu(\257)58 b Fv(.)7403 15540 y(Th)-31 b(us)350 b(it)g(su\261ces)f(to)i(pro)-31 b(v)g(e)350 b(eq.)g(\(4.22\))j(with)e(\243)27594 15706 y Fs(\257)28191 15540 y Fv(\()p Fu(a)207 b Fv(+)g Fu(H)31401 15706 y Fs(o)31894 15540 y Fv(\))32324 15139 y Fx(1)p Fs(=)p Fx(2)34065 15540 y Fv(replaced)350 b(b)-31 b(y)40056 15260 y Fq(e)39933 15540 y Fv(\243)40794 15139 y Fx(2)40794 15854 y Fs(\257)41740 15540 y Fv(pro)g(vided)6075 16945 y(the)344 b(p)31 b(o)-31 b(w)g(er)345 b(of)g(\(1)197 b(+)f Fr(j)q Fu(z)294 b Fr(\241)246 b Fu(E)17677 17111 y Fx(0)18174 16945 y Fr(j)q Fv(\))344 b(on)h(the)f(righ)-31 b(t)346 b(hand)e(side)g(and)h(the)f(p)31 b(ermitted)346 b(p)31 b(o)-31 b(w)g(er)345 b(gro)-31 b(wth)6075 18412 y(of)7555 18132 y Fq(e)7335 18412 y Fu(C)8126 18578 y Fs(\270)9076 18412 y Fv(are)369 b(eac)-31 b(h)370 b(reduced)e(b)-31 b(y)370 b(one.)492 b(I.e.,)371 b(w)-31 b(e)369 b(m)-31 b(ust)370 b(sho)-31 b(w)7366 21246 y(Prob)9906 19685 y Fq(\265)10854 19641 y(\260)10854 20305 y(\260)10854 20969 y(\260)10854 21633 y(\260)11469 21246 y Fu(\302)12162 21412 y Fs(x)14728 20497 y Fv(1)p 12851 20991 4306 45 v 12851 22280 a Fu(H)13861 21705 y Fx(\(\255\))13771 22410 y Fs(!)15487 22280 y Fr(\241)246 b Fu(z)17413 20966 y Fq(e)17290 21246 y Fv(\243)18151 20789 y Fx(2)18151 21520 y Fs(\257)18748 19641 y Fq(\260)18748 20305 y(\260)18748 20969 y(\260)18748 21633 y(\260)19670 21246 y Fu(>)308 b(t)21239 19641 y Fq(\257)21239 20305 y(\257)21239 20969 y(\257)21239 21633 y(\257)21792 21246 y Fr(F)22698 20789 y Fs(c)22588 21520 y(\257)40 b(;\260)23960 19685 y Fq(\266)12276 24502 y Fr(\267)677 b Fu(C)79 b Fv(\(1)247 b(+)f Fu(\270)p Fv(\))18097 24045 y Fs(d)p Fx(+3)20103 24502 y Fv(\(1)h(+)f Fr(j)q Fu(z)294 b Fr(\241)246 b Fu(E)25481 24668 y Fx(0)25978 24502 y Fr(j)p Fv(\))26715 24045 y Fx(\()p Fs(d)p Fx(+2\))29598 22897 y Fq(\260)29598 23561 y(\260)29598 24226 y(\260)29598 24890 y(\260)30213 24502 y Fu(\302)30906 24668 y Fs(x)33472 23754 y Fv(1)p 31596 24248 V 31855 25257 a Fq(b)31596 25537 y Fu(H)32606 24962 y Fx(\(\255\))32516 25666 y Fs(!)34231 25537 y Fr(\241)g Fu(z)36034 24502 y FB(1)36670 24668 y Fs(S)37214 24785 y Fl(\257)28 b(;\260)38497 22897 y Fq(\260)38497 23561 y(\260)38497 24226 y(\260)38497 24890 y(\260)39798 23754 y Fu(D)40745 23352 y Fx(2)p 39798 24248 1444 45 v 40320 25262 a Fu(t)41682 24502 y(:)1107 b Fv(\(4.50\))7403 27116 y(W)-92 b(e)326 b(w)-31 b(an)g(t)328 b(to)f(a)-31 b(v)g(erage)328 b(o)-31 b(v)g(er)327 b Fu(\263)389 b Fr(2)307 b(I)22016 27282 y Fs(\257)40 b(;\260)23389 27116 y Fv(,)335 b(using)327 b(the)f(Birman)i(Sc)-31 b(h)g(winger)327 b(iden)-31 b(tit)g(y)329 b(together)6075 28445 y(with)g(the)f(w)-31 b(eak)329 b(1-1)f(inequalit)-31 b(y)-92 b(,)339 b(and)328 b(compare)h(the)f(result)f(to)i(the)f (\\re-sampled")h(op)31 b(erator:)9193 30804 y Fu(\302)9886 30970 y Fs(x)11763 30055 y Fv(1)p 10576 30549 2927 45 v 10576 31563 a Fu(H)336 b Fr(\241)246 b Fu(z)13758 30524 y Fq(e)13635 30804 y Fv(\243)14496 30346 y Fx(2)14496 31077 y Fs(\257)15770 30804 y Fv(=)676 b Fu(\302)18000 30970 y Fs(x)19877 30055 y Fv(1)p 18690 30549 V 18949 31481 a Fq(b)18690 31761 y Fu(H)336 b Fr(\241)246 b Fu(z)21872 30524 y Fq(e)21749 30804 y Fv(\243)22610 30346 y Fx(2)22610 31077 y Fs(\257)23822 30804 y Fv(+)615 b Fu(\302)25991 30970 y Fs(x)27868 30055 y Fv(1)p 26681 30549 V 26940 31481 a Fq(b)26681 31761 y Fu(H)336 b Fr(\241)246 b 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Fu(T)562 b Fv(is)408 b(Hilb)31 b(ert-Sc)-31 b(hmidt)410 b(with)g(uniformly)f(b)31 b(ounded)409 b(norm:)570 b Fr(k)35162 53007 y Fq(b)34976 53286 y Fu(T)155 b Fr(k)36331 53452 y Fx(2)37200 53286 y Fr(\267)38653 53007 y Fq(e)38433 53286 y Fu(C)39224 53452 y Fs(d)39741 53286 y Fv(\(1)273 b(+)f Fu(\270)p Fv(\))43206 52885 y Fs(n)43808 53286 y Fv(\(1)h(+)6075 54615 y Fr(j)p Fu(z)158 b Fr(\241)109 b Fu(E)8842 54781 y Fx(0)9339 54615 y Fr(j)p Fv(\))10076 54213 y Fs(n)10980 54615 y Fv(\()p Fu(n)302 b Fv(the)f(smallest)h(in)-31 b(teger)302 b(greater)f(than)g Fu(d=)p Fv(2\).)472 b(The)301 b(b)31 b(ounded)300 b(term,)316 b(\\)p Fu(B)56 b Fv(",)315 b(drops)6075 56082 y(out)425 b(of)f(this)h(represen)-31 b(tation)425 b(b)31 b(ecause)424 b Fu(\302)23764 56248 y Fs(x)24444 55802 y Fq(e)24321 56082 y Fv(\252)25182 55680 y Fx(2)26077 56082 y Fv(=)399 b(0)424 b(and)h(m)-31 b(ultipliers)426 b(of)f(supp)31 b(ort)423 b(no)i(larger)6075 57487 y(than)347 b(that)h(of)12372 57207 y Fq(e)12249 57487 y 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Fu(H)336 b Fr(\241)246 b Fu(z)32244 62361 y Fq(b)32121 62641 y Fv(\243)32982 62184 y Fx(2)32982 62914 y Fs(\257)33579 62641 y Fu(T)8870 b Fv(\(4.55\))6075 65075 y(with)271 b Fu(B)325 b Fv(b)31 b(ounded,)290 b Fr(k)p Fu(B)56 b Fr(k)307 b(\267)h Fu(C)18679 65241 y Fs(d)19195 65075 y Fv(\(1)47 b(+)g Fu(\270)g Fv(+)g Fr(j)s Fu(z)294 b Fr(\241)246 b Fu(E)25777 65241 y Fx(0)26274 65075 y Fr(j)q Fv(\))27012 64673 y Fx(2)p Fs(n)28056 65075 y Fv(,)290 b(and)270 b Fu(T)423 b Fv(a)270 b(p)31 b(olynomial)273 b(in)e Fr(f)p Fu(\264)40515 65241 y Fs(\263)41331 65075 y Fv(:)308 b Fu(\263)389 b Fr(2)307 b(I)44467 65241 y Fs(\257)45064 65075 y Fr(g)p Fv(,)18528 67235 y Fu(T)830 b Fv(=)22426 66184 y Fq(X)21542 68791 y Fx(\()p Fs(k)19 b Fx(\))p Fp(2)p Fi(N)23881 68407 y Fj(I)24335 68601 y Fl(\257)22099 69679 y Fp(j)p Fs(k)g Fp(j)o(\267)p Fs(n)25094 67235 y Fu(\270)25740 66778 y Fp(j)p Fs(k)g Fp(j)26810 67235 y Fu(T)27611 66778 y Fx(\()p Fs(k)g Fx(\))29361 66184 y Fq(Y)29032 68567 y Fs(\263)50 b Fp(2I)30583 68684 y Fl(\257)31288 67235 y Fu(\264)31878 66629 y Fs(k)32349 66746 y Fl(\263)31838 67569 y Fs(\263)33165 67235 y Fu(;)9624 b Fv(\(4.56\))p eop end %%Page: 32 32 TeXDict begin 32 31 bop 6075 3345 a Fx(32)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(with)370 b(Hilb)31 b(ert-Sc)-31 b(hmidt)372 b(co)31 b(e\261cien)-31 b(ts,)17978 6792 y Fq(\260)17978 7456 y(\260)17978 8121 y(\260)18593 8065 y Fu(T)19394 7608 y Fx(\()p Fs(k)19 b Fx(\))20630 6792 y Fq(\260)20630 7456 y(\260)20630 8121 y(\260)21245 8840 y Fs(H)59 b(S)23347 8065 y Fr(\267)25104 7785 y Fq(e)24885 8065 y Fu(C)25676 8231 y Fs(d)26192 8065 y Fv(\(1)247 b(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)31570 8231 y Fx(0)32066 8065 y Fr(j)q Fv(\))32804 7608 y Fs(n)33714 8065 y Fu(:)9075 b Fv(\(4.57\))6075 10495 y(Here)507 b(the)h(function)15472 10215 y Fq(b)15349 10495 y Fv(\243)16210 10661 y Fs(\257)17315 10495 y Fv(is)f(b)31 b(ounded)508 b(b)-31 b(y)508 b(one)g(and)g(supp)31 b(orted)507 b(in)h Fr(f)p Fu(q)348 b Fv(:)308 b(dist\()p Fu(q)40 b(;)184 b(\257)58 b Fv(\))310 b Fu(<)e Fv(2)p Fu(r)31 b Fr(g)q Fv(.)6075 12002 y(Note)370 b(that)g Fu(U)11904 12168 y Fs(\263)12536 11722 y Fq(b)12413 12002 y Fv(\243)13274 12168 y Fs(\257)14178 12002 y Fv(=)308 b(0)369 b(for)g Fu(\263)390 b Fr(62)307 b(I)20485 12168 y Fs(\257)21082 12002 y Fv(.)7403 13330 y(Putting)371 b(this)f(all)g(together,)h(w)-31 b(e)370 b(obtain)7366 15117 y Fq(\260)7366 15781 y(\260)7366 16446 y(\260)7366 17110 y(\260)7981 16722 y Fu(\302)8674 16888 y Fs(x)10550 15973 y Fv(1)p 9364 16468 2927 45 v 9364 17482 a Fu(H)336 b Fr(\241)246 b Fu(z)12546 16443 y Fq(e)12423 16722 y Fv(\243)13284 16265 y Fx(2)13284 16996 y Fs(\257)13881 15117 y Fq(\260)13881 15781 y(\260)13881 16446 y(\260)13881 17110 y(\260)14804 16722 y Fr(\267)16341 15117 y Fq(\260)16341 15781 y(\260)16341 16446 y(\260)16341 17110 y(\260)16956 16722 y Fu(\302)17649 16888 y Fs(x)19526 15973 y Fv(1)p 18339 16468 V 18598 17400 a Fq(b)18339 17680 y Fu(H)336 b Fr(\241)246 b Fu(z)21521 16443 y Fq(e)21398 16722 y Fv(\252)22259 15117 y Fq(\260)22259 15781 y(\260)22259 16446 y(\260)22259 17110 y(\260)23120 16722 y Fr(\243)24227 14497 y Fq(0)24227 16490 y(@)25196 16722 y Fv(1)615 b(+)27840 15449 y Fq(\260)27840 16113 y(\260)27840 16778 y(\260)28455 16722 y Fv(\()29145 16443 y Fq(b)28885 16722 y Fu(H)337 b Fr(\241)246 b Fu(H)90 b Fv(\))32812 16443 y Fq(b)32689 16722 y Fv(\243)33550 16888 y Fs(\257)34147 16722 y Fu(B)35043 15449 y Fq(\260)35043 16113 y(\260)35043 16778 y(\260)18240 21150 y Fv(+)20733 20099 y Fq(X)19849 22706 y Fx(\()p Fs(k)19 b Fx(\))p Fp(2)p Fi(N)22188 22322 y Fj(I)22642 22516 y Fl(\257)23401 21150 y Fu(\270)24047 20693 y Fp(j)p Fs(k)g Fp(j)25301 19545 y Fq(\260)25301 20209 y(\260)25301 20873 y(\260)25301 21538 y(\260)26039 20870 y(e)25916 21150 y Fv(\252)27147 20870 y Fq(b)26962 21150 y Fu(T)338 b Fv(\()28636 20870 y Fq(b)28377 21150 y Fu(H)705 b Fr(\241)615 b Fu(H)90 b Fv(\))34238 20401 y(1)p 33051 20896 V 33051 21910 a Fu(H)337 b Fr(\241)246 b Fu(z)36234 20870 y Fq(b)36111 21150 y Fv(\243)36972 20693 y Fx(2)36972 21424 y Fs(\257)37569 21150 y Fu(T)38370 20693 y Fx(\()p Fs(k)19 b Fx(\))39606 19545 y Fq(\260)39606 20209 y(\260)39606 20873 y(\260)39606 21538 y(\260)40221 18925 y(1)40221 20918 y(A)41682 21150 y Fu(:)1107 b Fv(\(4.58\))6075 24355 y(Th)-31 b(us,)7366 26807 y Fq(\260)7366 27471 y(\260)7366 28135 y(\260)7366 28799 y(\260)7981 28412 y Fu(\302)8674 28578 y Fs(x)10550 27663 y Fv(1)p 9364 28157 V 9364 29171 a Fu(H)336 b Fr(\241)246 b Fu(z)12546 28132 y Fq(e)12423 28412 y Fv(\243)13284 27955 y Fx(2)13284 28685 y Fs(\257)13881 26807 y Fq(\260)13881 27471 y(\260)13881 28135 y(\260)13881 28799 y(\260)14804 28412 y Fr(\267)16710 26807 y Fq(\260)16710 27471 y(\260)16710 28135 y(\260)16710 28799 y(\260)17325 28412 y Fu(\302)18018 28578 y Fs(x)19895 27663 y Fv(1)p 18708 28157 V 18967 29089 a Fq(b)18708 29369 y Fu(H)336 b Fr(\241)246 b Fu(z)21890 28132 y Fq(e)21767 28412 y Fv(\252)22628 26807 y Fq(\260)22628 27471 y(\260)22628 28135 y(\260)22628 28799 y(\260)23489 28412 y Fr(\243)24596 25522 y Fq(0)24596 27471 y(B)24596 28135 y(B)24596 28843 y(@)25565 28412 y Fv(1)615 b(+)g(2)184 b Fu(b)29421 28578 y Fx(+)30157 28412 y Fu(\270)g Fr(k)q Fu(B)56 b Fr(k)614 b Fv(+)17148 34654 y(+)20100 33603 y Fq(X)18193 36089 y Fs(\263)50 b Fp(2I)19744 36206 y Fl(\257)28 b(;\260)20971 36089 y Fs(;\263)21687 35755 y Fj(0)21986 36089 y Fp(2I)23084 36206 y Fl(\257)19215 37190 y Fx(\()p Fs(k)19 b Fx(\))p Fp(2)p Fi(N)21554 36806 y Fj(I)22008 37000 y Fl(\257)23789 34654 y Fv(2)p Fu(\270)24988 34197 y Fx(1+)p Fp(j)p Fs(k)g Fp(j)27364 33049 y Fq(\260)27364 33714 y(\260)27364 34378 y(\260)27364 35042 y(\260)28102 34375 y(e)27979 34654 y Fv(\252)29210 34375 y Fq(b)29025 34654 y Fu(T)153 b(U)30581 34820 y Fs(\263)32409 33906 y Fv(1)p 31223 34400 V 31223 35414 a Fu(H)336 b Fr(\241)246 b Fu(z)34415 33906 y(U)35171 34072 y Fs(\263)35624 33850 y Fj(0)p 34415 34400 1564 45 v 34764 35414 a Fu(F)36234 34375 y Fq(b)36111 34654 y Fv(\243)36972 34197 y Fx(2)36972 34928 y Fs(\257)37569 34654 y Fu(T)38370 34197 y Fx(\()p Fs(k)19 b Fx(\))39606 33049 y Fq(\260)39606 33714 y(\260)39606 34378 y(\260)39606 35042 y(\260)40221 31433 y(1)40221 33381 y(C)40221 34046 y(C)40221 34710 y(C)40221 35418 y(A)41682 34654 y Fu(:)1107 b Fv(\(4.59\))6075 38901 y(>F)-92 b(rom)389 b(\(4.59\))s(,)394 b(using)c(Prop.)h(4.2)f(and)g (the)g(argumen)-31 b(ts)391 b(from)f(the)f(pro)31 b(of)390 b(of)g(Lemma)h(4.3,)396 b(it)6075 40229 y(follo)-31 b(ws)371 b(that)7366 42742 y(Prob)9906 41181 y Fq(\265)10854 41137 y(\260)10854 41801 y(\260)10854 42465 y(\260)10854 43129 y(\260)11469 42742 y Fu(\302)12162 42908 y Fs(x)14038 41993 y Fv(1)p 12851 42487 2927 45 v 12851 43501 a Fu(H)337 b Fr(\241)246 b Fu(z)16034 42462 y Fq(e)15911 42742 y Fv(\243)16772 42285 y Fx(2)16772 43015 y Fs(\257)17369 41137 y Fq(\260)17369 41801 y(\260)17369 42465 y(\260)17369 43129 y(\260)18291 42742 y Fu(>)308 b(t)19860 41137 y Fq(\257)19860 41801 y(\257)19860 42465 y(\257)19860 43129 y(\257)20413 42742 y Fr(F)21319 42285 y Fs(c)21209 43015 y(\257)40 b(;\260)22581 41181 y Fq(\266)10390 45841 y Fr(\267)11804 44236 y Fq(\260)11804 44900 y(\260)11804 45564 y(\260)11804 46229 y(\260)12419 45841 y Fu(\302)13112 46007 y Fs(x)14988 45092 y Fv(1)p 13802 45587 V 14061 46519 a Fq(b)13802 46799 y Fu(H)336 b Fr(\241)246 b Fu(z)16984 45561 y Fq(e)16861 45841 y Fv(\252)17722 44236 y Fq(\260)17722 44900 y(\260)17722 45564 y(\260)17722 46229 y(\260)18583 45841 y Fr(\243)19690 44280 y Fq(\265)20505 45841 y Fv(1)615 b(+)g(2)p Fu(b)24177 46007 y Fx(+)24913 45841 y Fu(C)25704 46007 y Fs(d)26220 45841 y Fu(\270)p Fv(\(1)247 b(+)f Fu(\270)p Fv(\))30279 45384 y Fx(2)p Fs(n)31323 45841 y Fv(\(1)h(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)36701 46007 y Fx(0)37197 45841 y Fr(j)q Fv(\))37935 45384 y Fx(2)p Fs(n)14889 48841 y Fv(+)369 b(\()p Fu(M)e Fr(\241)246 b Fv(2\))184 b(4)p Fu(C)21608 49007 y Fs(W)22837 48561 y Fq(e)22617 48841 y Fu(C)23487 48384 y Fx(2)23408 49115 y Fs(d)23984 48841 y Fv(\(1)247 b(+)f Fu(\270)p Fv(\))27397 48384 y Fx(2)p Fs(n)28441 48841 y Fv(\(1)h(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)33819 49007 y Fx(0)34315 48841 y Fr(j)q Fv(\))35053 48384 y Fx(2)p Fs(n)36282 48841 y Fu(b)36757 49007 y Fx(+)37676 48841 y Fu(D)38623 48384 y Fx(2)39120 47612 y Fq(\264)40467 48092 y Fu(M)p 40467 48586 1195 45 v 40865 49600 a(t)10390 51648 y Fr(\267)11804 50043 y Fq(\260)11804 50707 y(\260)11804 51371 y(\260)11804 52036 y(\260)12419 51648 y Fu(\302)13112 51814 y Fs(x)14988 50899 y Fv(1)p 13802 51394 2927 45 v 14061 52326 a Fq(b)13802 52606 y Fu(H)336 b Fr(\241)246 b Fu(z)16984 51368 y Fq(e)16861 51648 y Fv(\252)17722 50043 y Fq(\260)17722 50707 y(\260)17722 51371 y(\260)17722 52036 y(\260)18583 51648 y Fr(\243)g Fu(M)21069 50419 y Fq(\263)21730 51648 y Fv(1)616 b(+)f(2)p Fu(b)25403 51814 y Fx(+)26138 51648 y Fu(C)26929 51814 y Fs(d)27445 51648 y Fv(\()p Fu(M)368 b Fr(\241)246 b Fv(2\))184 b(4)p Fu(C)32935 51814 y Fs(W)34164 51368 y Fq(e)33944 51648 y Fu(C)34814 51191 y Fx(2)34735 51922 y Fs(d)35311 50419 y Fq(\264)25324 54792 y Fr(\243)246 b Fv(\(1)h(+)f Fu(\270)p Fv(\))29844 54335 y Fx(2)p Fs(n)p Fx(+1)32009 54792 y Fv(\(1)h(+)f Fr(j)p Fu(z)294 b Fr(\241)246 b Fu(E)37386 54958 y Fx(0)37883 54792 y Fr(j)q Fv(\))38621 54335 y Fx(2)p Fs(n)39798 54044 y Fu(D)40745 53642 y Fx(2)p 39798 54538 1444 45 v 40320 55552 a Fu(t)41682 54792 y(;)1107 b Fv(\(4.60\))6075 56958 y(where)480 b Fu(M)601 b Fv(is)481 b(the)g(n)-31 b(um)g(b)31 b(er)481 b(of)g(terms)g(b)31 b(et)-31 b(w)g(een)482 b(the)f(brac)-31 b(k)g(ets)481 b(in)g(\(4.59\))s(.)827 b(By)481 b FB(A2)p Fv(,)510 b Fu(M)601 b Fv(is)6075 58286 y(b)31 b(ounded)369 b(uniformly)i(in)e Fu(x;)184 b(\257)58 b(;)184 b(\260)61 b Fv(.)495 b(This)370 b(completes)g(the)g(pro)31 b(of)369 b(of)h(Lemma)g(4.4.)5027 b Fh(\244)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(33)18842 6001 y Fv(5.)555 b Fw(Finite-v)-25 b(olume)417 b(criteria)6075 7994 y Fv(5.1.)555 b FB(The)481 b(\257nite-v)-35 b(olume)481 b(inequalit)-35 b(y.)552 b Fv(With)418 b(the)g(b)31 b(ounds)418 b(pro)-31 b(vided)418 b(b)-31 b(y)418 b(Lemmas)h(4.3)6075 9322 y(and)311 b(4.4,)324 b(the)311 b(pro)31 b(of)311 b(of)g(Theorem)h(1.2)g(pro)31 b(ceeds)310 b(according)i(to)g(a)f(set)f (of)i(argumen)-31 b(ts)312 b(familiar)6075 10651 y(from)437 b(the)g(fractional)i(momen)-31 b(t)439 b(metho)31 b(d)437 b(for)g(discrete)g(op)31 b(erators)437 b([6)q(],)454 b(and)437 b(related)g(to)h(the)6075 11979 y(m)-31 b(ulti-scale)419 b(analysis)e(of)g(random)h(Sc)-31 b(hr\304)-553 b(odinger)417 b(op)31 b(erators,)429 b(e.g.)635 b([67)q(,)417 b(16)q(],)429 b(as)416 b(w)-31 b(ell)418 b(as)f(the)6075 13307 y(analysis)402 b(of)f(a)g(n)-31 b(um)g(b)31 b(er)401 b(of)g(lattice)i(mo)31 b(dels)402 b(in)f(statistical)i(mec)-31 b(hanics,)411 b(e.g.)588 b([31)q(,)401 b(59)q(,)g(45)q(,)g(7)q(,)6075 14636 y(5,)341 b(23)q(].)484 b(W)-92 b(e)339 b(b)31 b(egin)342 b(b)-31 b(y)340 b(pro)-31 b(ving)342 b(a)f(\\correlation)i(inequalit) -31 b(y")344 b({)d(eq.)g(\(5.2\))i(b)31 b(elo)-31 b(w)342 b({)f(and)g(then)6075 15964 y(iterate)370 b(this)g(inequalit)-31 b(y)372 b(to)e(obtain)h(exp)31 b(onen)-31 b(tial)371 b(deca)-31 b(y)370 b(of)g(the)f(bulk)h(Green-function.)6075 17994 y FB(Lemma)493 b(5.1.)590 b Fm(L)-57 b(et)452 b Fu(H)16114 18160 y Fs(!)17206 17994 y Fm(b)-57 b(e)453 b(a)e(r)-57 b(andom)451 b(Schr\304)-566 b(odinger)452 b(op)-57 b(er)g(ator)451 b(with)g(disor)-57 b(der)451 b(p)-57 b(ar)g(ameter)6075 19399 y Fu(\270)307 b(>)g Fv(0)367 b Fm(which)e(ob)-57 b(eys)367 b FB(A)f Fm(and)f FB(IAD)p Fm(.)i(Then,)373 b(for)366 b(e)-57 b(ach)366 b Fu(s)307 b(<)h Fv(1)p Fu(=)p Fv(3)367 b Fm(ther)-57 b(e)366 b(exists)39023 19119 y Fq(e)38803 19399 y Fu(C)39594 19565 y Fs(\270;s)41162 19399 y Fu(<)308 b Fr(1)366 b Fm(such)6075 20728 y(that)395 b(if)i(we)f(de\257ne,)h(for)f Fu(L)308 b(>)f Fv(23)p Fu(r)31 b Fm(,)14169 23249 y Fu(a)p Fv(\()p Fu(x)p Fv(;)184 b Fu(L)p Fv(\))706 b(:=)e Fu(L)20822 22792 y Fs(d)p Fp(\241)p Fx(1)23476 22197 y Fq(X)22840 24580 y Fs(\263)50 b Fp(2S)24434 24691 y Fl(x;L)25895 23249 y Ft(E)26817 21688 y Fq(\265)27632 21644 y(\260)27632 22308 y(\260)27632 22972 y(\260)27632 23636 y(\260)28247 23249 y Fu(\302)28940 23415 y Fs(x)31835 22500 y Fv(1)p 29630 22994 4965 45 v 29630 24067 a Fu(H)30640 23727 y Fx(\()p Fs(B)31693 23505 y Fl(L)31655 23948 y(x)32276 23727 y Fx(\))32924 24067 y Fr(\241)246 b Fu(z)34727 23249 y(\302)35420 23415 y Fs(\263)35928 21644 y Fq(\260)35928 22308 y(\260)35928 22972 y(\260)35928 23636 y(\260)36543 21917 y Fs(s)37016 21688 y Fq(\266)43650 23249 y Fv(\(5.1\))6075 26482 y Fm(wher)-57 b(e)396 b Fr(S)9871 26648 y Fs(x;L)11606 26482 y Fv(=)307 b Fr(f)q Fu(\263)389 b Fr(2)308 b(I)389 b Fv(:)308 b Fu(L)246 b Fr(\241)g Fv(23)p Fu(r)340 b(<)307 b Fr(j)q Fu(\263)328 b Fr(\241)245 b Fu(x)p Fr(j)308 b Fu(<)g(L)246 b Fr(\241)g Fv(3)p Fu(r)31 b Fr(g)q Fm(,)396 b(then)h(for)f(any)g(r)-57 b(e)g(gion)398 b Fv(\255)308 b Fr(\276)f Fu(B)44012 26081 y Fs(L)43956 26756 y(x)44674 26482 y Fm(,)7182 29437 y Ft(E)8104 27876 y Fq(\265)8919 27832 y(\260)8919 28496 y(\260)8919 29161 y(\260)8919 29825 y(\260)9534 29437 y Fu(\302)10227 29603 y Fs(x)12793 28688 y Fv(1)p 10917 29183 4306 45 v 10917 30216 a Fu(H)11927 29897 y Fx(\(\255\))13552 30216 y Fr(\241)246 b Fu(z)15355 29437 y(\302)16048 29603 y Fs(y)16580 27832 y Fq(\260)16580 28496 y(\260)16580 29161 y(\260)16580 29825 y(\260)17195 28106 y Fs(s)17668 27876 y Fq(\266)18790 29437 y Fr(\267)20575 29157 y Fq(e)20355 29437 y Fu(C)21146 29603 y Fs(\270;s)22407 29437 y Fv(\(1)h(+)f(1)p Fu(=\270)p Fv(\))26926 28980 y Fx(2)p Fs(s)27842 29437 y Fu(D)28789 28980 y Fx(10)p Fs(s)30329 29437 y Fv(\(1)h(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)35707 29603 y Fx(0)36204 29437 y Fr(j)p Fv(\))36941 28980 y Fx(5)p Fs(s)p Fx(\()p Fs(d)p Fx(+2\))23040 32597 y Fr(\243)g Fu(a)p Fv(\()p Fu(x)p Fv(;)184 b Fu(L)p Fv(\))28687 31545 y Fq(X)28052 33964 y Fs(\263)50 b Fp(2S)29702 33698 y Fj(0)29646 34226 y Fl(x;L)31291 32597 y Ft(E)32213 31036 y Fq(\265)33028 30992 y(\260)33028 31656 y(\260)33028 32320 y(\260)33028 32984 y(\260)33643 32597 y Fu(\302)34336 32763 y Fs(\263)36853 31848 y Fv(1)p 34977 32342 V 34977 33376 a Fu(H)35987 33056 y Fx(\(\255\))37612 33376 y Fr(\241)246 b Fu(z)39415 32597 y(\302)40108 32763 y Fs(y)40640 30992 y Fq(\260)40640 31656 y(\260)40640 32320 y(\260)40640 32984 y(\260)41255 31265 y Fs(s)41728 31036 y Fq(\266)43650 32597 y Fv(\(5.2\))6075 35918 y Fm(for)462 b(any)h Fu(y)502 b Fm(with)462 b Fv(dist)q(\()p Fu(x;)184 b(y)40 b Fv(\))429 b Fu(>)f(L)294 b Fv(+)h Fu(r)22390 36084 y Fs(o)23176 35918 y Fv(+)g(23)p Fu(r)31 b Fm(,)480 b(wher)-57 b(e)463 b Fr(S)30733 35516 y Fp(0)30650 36223 y Fs(x;L)32506 35918 y Fv(=)427 b Fr(f)p Fu(\263)511 b Fr(2)428 b(I)509 b Fv(:)429 b Fu(L)294 b Fv(+)g Fu(r)41057 36084 y Fs(o)41844 35918 y Fr(\241)g Fv(13)p Fu(r)460 b(<)6075 37314 y Fr(j)p Fu(x)246 b Fr(\241)g Fu(\263)82 b Fr(j)308 b Fu(<)f(L)247 b Fv(+)e Fu(r)13322 37480 y Fs(o)14061 37314 y Fv(+)h(23)p Fu(r)31 b Fr(g)398 b Fm(\(r)-57 b(e)g(c)g(al)57 b(l)396 b(that)f Fu(r)23869 37480 y Fs(o)24758 37314 y Fm(is)h(the)g(length)g (sc)-57 b(ale)396 b(app)-57 b(e)g(aring)397 b(in)g FB(IAD)p Fm(\),)g(with)18550 39839 y Fq(e)18330 40119 y Fu(C)19121 40285 y Fs(\270;s)21086 40119 y Fr(\267)703 b Fv(Const)p Fu(:)26321 39370 y Fv(\(1)247 b(+)f Fu(\270)p Fv(\))p 26321 39864 3414 45 v 26539 40878 a(1)g Fr(\241)g Fv(3)p Fu(s)29867 38813 y Fx(5)p Fs(s)p Fx(\()p Fs(d)p Fx(+4\))33363 40119 y Fu(:)9980 b Fv(\(5.3\))6075 42444 y Fm(If)396 b Fu(H)486 b Fm(satis\257es)396 b(the)f(str)-57 b(onger)396 b(c)-57 b(ondition)398 b FB(A3)25376 42042 y Fp(0)26083 42444 y Fm(then)20361 44482 y Fq(e)20141 44762 y Fu(C)20932 44928 y Fs(\270;s)22897 44762 y Fr(\267)703 b Fv(Const)p Fu(:)29345 44013 y Fv(1)p 28133 44508 2979 45 v 28133 45522 a(1)246 b Fr(\241)g Fv(3)p Fu(s)31552 44762 y(:)11791 b Fv(\(5.4\))7403 47195 y(The)354 b(main)h(to)31 b(ols)354 b(in)g(the)g(pro)31 b(of)354 b(are)f(the)h(momen)-31 b(t)356 b(b)31 b(ounds)353 b(presen)-31 b(ted)353 b(ab)31 b(o)-31 b(v)g(e)355 b(and)f(a)g(w)-31 b(ell)6075 48523 y(kno)g(wn)428 b(analog)g(of)g(the)f(resolv)-31 b(en)g(t)427 b(expansion)h(commonly)h(used)d(for)h(discrete)g(Sc)-31 b(hr\304)-553 b(odinger)6075 49851 y(op)31 b(erators,)370 b(sometimes)g(called)h(the)e Fm(ge)-57 b(ometric)397 b(r)-57 b(esolvent)396 b(identity)p Fv(.)6075 51881 y FB(Lemma)408 b(5.2.)541 b Fm(L)-57 b(et)384 b Fu(H)473 b Fm(b)-57 b(e)384 b(a)f(Schr\304)-566 b(odinger)384 b(op)-57 b(er)g(ator.)505 b(Consider)383 b(a)g(se)-57 b(quenc)g(e)385 b(of)f(thr)-57 b(e)g(e)384 b(op)-57 b(en)6075 53210 y(sets)471 b Fv(\244)9098 53376 y Fs(o)10035 53210 y Fr(\275)445 b Fv(\244)g Fr(\275)g Fv(\255)472 b Fm(with)g Fv(dist\(\244)20601 53376 y Fs(o)21094 53210 y Fu(;)184 b Fv(\244)22354 52808 y Fs(c)22805 53210 y Fv(\))446 b Fu(>)f Fv(0)472 b Fm(and)g(let)g Fv(\243)g Fm(b)-57 b(e)473 b(a)f(smo)-57 b(oth)471 b(function)i(which)f(is)6075 54538 y(identic)-57 b(al)57 b(ly)409 b Fv(1)g Fm(in)f(a)g(neighb)-57 b(orho)g(o)g(d)409 b(of)g Fv(\244)23382 54704 y Fs(o)24282 54538 y Fm(and)f(identic)-57 b(al)57 b(ly)409 b(zer)-57 b(o)408 b(in)h(a)f(neighb)-57 b(orho)g(o)g(d)409 b(of)f Fv(\244)45135 54136 y Fs(c)45586 54538 y Fm(.)6075 55866 y(Given)397 b(any)g(r)-57 b(estrictions)396 b Fu(H)18238 55465 y Fx(\(\255\))20013 55866 y Fm(and)g Fu(H)23173 55465 y Fx(\(\244\))24922 55866 y Fm(of)h Fu(H)486 b Fm(to)395 b Fv(\255)i Fm(and)f Fv(\244)g Fm(r)-57 b(esp)g(e)g(ctively,) 8761 58185 y FB(1)9397 58351 y Fx(\244)10003 58462 y Fl(o)12512 57436 y Fv(1)p 10636 57930 4306 45 v 10636 58964 a Fu(H)11646 58644 y Fx(\(\255\))13271 58964 y Fr(\241)246 b Fu(z)15778 58185 y Fv(=)703 b FB(1)17978 58351 y Fx(\244)18584 58462 y Fl(o)21264 57436 y Fv(1)p 19401 57930 4280 45 v 19401 58964 a Fu(H)20411 58644 y Fx(\(\244\))22010 58964 y Fr(\241)246 b Fu(z)23814 58185 y Fv(\243)642 b(+)g FB(1)27456 58351 y Fx(\244)28062 58462 y Fl(o)30558 57436 y Fv(1)p 28695 57930 V 28695 58964 a Fu(H)29705 58644 y Fx(\(\244\))31304 58964 y Fr(\241)246 b Fu(z)33291 58185 y Fv([)q Fu(H)28 b(;)184 b Fv(\243])38401 57436 y(1)p 36525 57930 4306 45 v 36525 58964 a Fu(H)37535 58644 y Fx(\(\255\))39160 58964 y Fr(\241)246 b Fu(z)43650 58185 y Fv(\(5.5\))6075 60466 y Fm(for)396 b(any)g Fu(z)445 b Fm(at)396 b(which)f(b)-57 b(oth)396 b(r)-57 b(esolvents)396 b(exist.)7403 61795 y(If)416 b(furthermor)-57 b(e,)421 b Fv(\244)15814 61393 y Fp(0)15814 62068 y Fs(o)16648 61795 y Fr(\275)343 b Fv(\244)18621 61393 y Fp(0)19274 61795 y Fr(\275)f Fv(\255)416 b Fm(with)f Fv(dist)q(\(\244)27105 61393 y Fp(0)27105 62068 y Fs(o)27597 61795 y Fu(;)184 b Fv(\244)28857 61393 y Fp(0)29169 61236 y Fs(c)29620 61795 y Fv(\))343 b Fu(>)f Fv(0)416 b Fm(and)f Fv(dist)q(\(\244)37723 61393 y Fp(0)38034 61795 y Fu(;)184 b Fv(\244\))344 b Fu(>)e Fv(0)p Fm(,)421 b(and)416 b Fv(\243)45615 61393 y Fp(0)6075 63123 y Fm(is)432 b(a)g(function)i(which)e(is)g(identic)-57 b(al)57 b(ly)434 b Fv(1)f Fm(in)g(a)f(neighb)-57 b(orho)g(o)g(d)433 b(of)g Fv(\244)34515 62721 y Fp(0)34515 63397 y Fs(o)35439 63123 y Fm(and)f(identic)-57 b(al)57 b(ly)434 b Fv(0)e Fm(in)i(a)6075 64451 y(neighb)-57 b(orho)g(o)g(d)397 b(of)f Fv(\244)14722 64050 y Fp(0)15032 63893 y Fs(c)15879 64451 y Fm(then)6800 66770 y FB(1)7436 66936 y Fx(\244)8042 67047 y Fl(o)10550 66021 y Fv(1)p 8674 66515 V 8674 67549 a Fu(H)9684 67229 y Fx(\(\255\))11309 67549 y Fr(\241)246 b Fu(z)13113 66770 y FB(1)13749 66936 y Fx(\244)14355 66715 y Fj(0)14355 67157 y Fl(o)15558 66770 y Fv(=)704 b Fr(\241)p FB(1)18620 66936 y Fx(\244)19226 67047 y Fl(o)21721 66021 y Fv(1)p 19858 66515 4280 45 v 19858 67549 a Fu(H)20868 67229 y Fx(\(\244\))22467 67549 y Fr(\241)246 b Fu(z)24455 66770 y Fv([)q Fu(H)28 b(;)184 b Fv(\243])29565 66021 y(1)p 27689 66515 4306 45 v 27689 67549 a Fu(H)28699 67229 y Fx(\(\255\))30324 67549 y Fr(\241)246 b Fu(z)32312 66770 y Fv([)p Fu(H)28 b(;)184 b Fv(\243)34919 66313 y Fp(0)35232 66770 y Fv(])37869 66021 y(1)p 35856 66515 4579 45 v 35856 67549 a Fu(H)36866 67229 y Fx(\(\244)37818 67008 y Fj(0)38118 67229 y Fx(\))38765 67549 y Fr(\241)246 b Fu(z)40568 66770 y FB(1)41204 66936 y Fx(\244)41810 66715 y Fj(0)41810 67157 y Fl(o)42618 66770 y Fu(;)725 b Fv(\(5.6\))6075 69326 y Fm(with)395 b Fu(H)9489 68924 y Fx(\(\244)10441 68590 y Fj(0)10740 68924 y Fx(\))11538 69326 y Fm(any)h(r)-57 b(estriction)397 b(of)f Fu(H)486 b Fm(to)396 b Fv(\244)23812 68924 y Fp(0)24518 69326 y Fm(and)g Fu(z)445 b Fm(such)396 b(that)f(al)57 b(l)395 b(thr)-57 b(e)g(e)396 b(inverses)h(exist.)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 6075 3345 a Fx(34)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fm(Pr)-57 b(o)g(of.)554 b Fv(The)328 b(second)f(iden)-31 b(tit)g(y)-92 b(,)339 b(eq.)328 b(\(5.6\))r(,)336 b(is)328 b(a)f(consequence)h(of)g(the)g(\257rst,)335 b(eq.)328 b(\(5.5\))s(,)336 b(and)328 b(its)6075 7330 y(transp)31 b(ose)401 b(\(eq.)h(\(5.8\))i(b)31 b(elo)-31 b(w\).)591 b(A)402 b(n)-31 b(um)g(b)31 b(er)401 b(of)h(terms)g(drop)f(out)h(b)31 b(ecause)401 b(\244)267 b Fr(\\)h Fv(\244)41321 6928 y Fp(0)41993 7330 y Fv(=)361 b Fr(;)p Fv(.)590 b(T)-92 b(o)6075 8658 y(v)-31 b(erify)370 b(eq.)f(\(5.5\))j(use)c(the)i(iden) -31 b(tit)g(y)16744 11192 y([)p Fu(H)28 b(;)184 b Fv(\243])679 b(=)d(\()p Fu(H)23314 10735 y Fx(\(\244\))24914 11192 y Fr(\241)246 b Fu(z)49 b Fv(\)\243)246 b Fr(\241)g Fv(\243\()p Fu(H)31530 10735 y Fx(\(\255\))33155 11192 y Fr(\241)g Fu(z)49 b Fv(\))8394 b(\(5.7\))6075 13671 y(on)413 b Fr(D)31 b Fv(\()p Fu(H)9981 13269 y Fx(\(\255\))11361 13671 y Fv(\),)425 b(whic)-31 b(h)414 b(follo)-31 b(ws)416 b(from)e(the)g(fact)g(that)h Fu(H)29705 13269 y Fx(\(\244\))31058 13671 y Fu(f)501 b Fv(=)381 b Fu(H)34353 13269 y Fx(\(\255\))35732 13671 y Fu(f)532 b Fv(if)414 b(the)g(supp)31 b(ort)413 b(of)h Fu(f)6075 15041 y Fv(is)397 b(strictly)i(con)-31 b(tained)399 b(in)f(\244.)577 b(Multiplying)400 b(on)e(the)f(left)i(b) -31 b(y)398 b FB(1)32772 15207 y Fx(\244)33378 15318 y Fl(o)33878 15041 y Fv(\()p Fu(H)35318 14639 y Fx(\(\244\))36936 15041 y Fr(\241)265 b Fu(z)49 b Fv(\))39056 14639 y Fp(\241)p Fx(1)40641 15041 y Fv(and)398 b(on)g(the)6075 16411 y(righ)-31 b(t)370 b(b)-31 b(y)369 b(\()p Fu(H)11730 16010 y Fx(\(\255\))13356 16411 y Fr(\241)246 b Fu(z)49 b Fv(\))15457 16010 y Fp(\241)p Fx(1)17014 16411 y Fv(yields)370 b(eq.)g(\(5.5\))r(.)20603 b Fh(\244)6075 19323 y FB(Remarks:)7957 21205 y Fv(\(1\))554 b(The)370 b(iden)-31 b(tit)g(y)-92 b(,)10516 23414 y(1)p 8640 23909 4306 45 v 8640 24942 a Fu(H)9650 24623 y Fx(\(\255\))11276 24942 y Fr(\241)246 b Fu(z)13079 24163 y FB(1)13715 24329 y Fx(\244)14321 24440 y Fl(o)15497 24163 y Fv(=)861 b(\243)20076 23414 y(1)p 18213 23909 4280 45 v 18213 24942 a Fu(H)19223 24623 y Fx(\(\244\))20822 24942 y Fr(\241)246 b Fu(z)22625 24163 y FB(1)23261 24329 y Fx(\244)23867 24440 y Fl(o)24982 24163 y Fr(\241)28467 23414 y Fv(1)p 26591 23909 4306 45 v 26591 24942 a Fu(H)27601 24623 y Fx(\(\255\))29226 24942 y Fr(\241)g Fu(z)31214 24163 y Fv([)p Fu(H)28 b(;)184 b Fv(\243])36311 23414 y(1)p 34448 23909 4280 45 v 34448 24942 a Fu(H)35458 24623 y Fx(\(\244\))37057 24942 y Fr(\241)246 b Fu(z)38860 24163 y FB(1)39496 24329 y Fx(\244)40102 24440 y Fl(o)40909 24163 y Fu(;)2434 b Fv(\(5.8\))9924 27084 y(follo)-31 b(ws)372 b(from)e(the)f(transp)31 b(ose)369 b(of)h(eq.)g(\(5.5\))h(\(at)f(conjugate)h Fu(z)49 b Fv(\).)7957 28412 y(\(2\))554 b(Eq.)405 b(\(5.6\))g(holds)f(in)g(a)g(n)-31 b(um)g(b)31 b(er)403 b(of)h(other)g(con)-31 b(texts.)596 b(In)403 b(particular,)414 b(it)404 b(is)g(true)f(for)9924 29741 y(discrete)429 b(Sc)-31 b(hr\304)-553 b(odinger)428 b(op)31 b(erators)429 b(and)f(in)h(that)g(case)f(giv)-31 b(es)429 b(the)g(usual)f(geometric)9924 31069 y(resolv)-31 b(en)g(t)371 b(expansion)f(\(see)f([6)q(,)h(eq.)f(\(2.16\)]\).)6075 33981 y Fm(Pr)-57 b(o)g(of)396 b(of)h(L)-57 b(emma)397 b(5.1.)554 b Fv(W)-92 b(e)425 b(shall)i(assume)f FB(A3)p Fv(.)664 b(F)-92 b(or)425 b(the)i(extension)g(of)f(the)h(argumen)-31 b(t)427 b(to)6075 35309 y(couplings)370 b(whic)-31 b(h)370 b(satisfy)h FB(A3)19161 34907 y Fp(0)19841 35309 y Fv(see)e(Prop.)g (5.4)i(b)31 b(elo)-31 b(w.)7403 36637 y(W)-92 b(e)391 b(start)h(b)-31 b(y)391 b(using)h(the)g(geometric)h(resolv)-31 b(en)g(t)392 b(iden)-31 b(tit)g(y)394 b(eq.)e(\(5.6\))i(of)e(Lemma)g (5.2)h(with)6075 37966 y(the)369 b(sets)15511 39651 y(\244)16280 39817 y Fs(o)17449 39651 y Fv(=)676 b Fu(B)19882 39249 y Fs(L)p Fp(\241)p Fx(11)p Fs(r)19826 39925 y(x)22861 39651 y Fu(;)4475 b Fv(\244)676 b(=)g Fu(B)31521 39249 y Fs(L)31465 39925 y(x)32491 39651 y Fu(;)15511 40981 y Fv(\244)16280 40579 y Fp(0)16280 41254 y Fs(o)17449 40981 y Fv(=)g(\255)247 b Fr(n)f Fu(B)21727 40579 y Fs(L)p Fx(+)p Fs(r)23425 40690 y Fl(o)23869 40579 y Fx(+12)p Fs(r)21671 41254 y(x)26228 40981 y Fu(;)1108 b Fv(\244)28412 40579 y Fp(0)29399 40981 y Fv(=)676 b(\255)246 b Fr(n)h Fu(B)33677 40579 y Fs(L)p Fx(+)p Fs(r)35375 40690 y Fl(o)33621 41254 y Fs(x)36181 40981 y Fu(;)43650 40326 y Fv(\(5.9\))6075 43198 y(and)359 b Fg(r)p Fv(\243)p Fu(;)544 b Fg(r)o Fv(\243)12911 42796 y Fp(0)13581 43198 y Fv(supp)31 b(orted)358 b(in)i Fr(f)p Fu(q)347 b Fv(:)308 b Fu(L)226 b Fr(\241)g Fv(11)p Fu(r)340 b(<)307 b Fr(j)p Fu(q)266 b Fr(\241)226 b Fu(x)p Fr(j)307 b Fu(<)h(L)226 b Fr(\241)f Fv(10)p Fu(r)31 b Fr(g)p Fv(,)363 b Fr(f)p Fu(q)348 b Fv(:)308 b Fu(L)226 b Fv(+)g Fu(r)41315 43364 y Fs(o)42033 43198 y Fv(+)g(11)p Fu(r)339 b(<)6075 44526 y Fr(j)p Fu(q)286 b Fr(\241)246 b Fu(x)p Fr(j)308 b Fu(<)f(L)246 b Fv(+)g Fu(r)13290 44692 y Fs(o)14029 44526 y Fv(+)g(12)p Fu(r)31 b Fr(g)371 b Fv(resp)31 b(ectiv)-31 b(ely)-92 b(.)7403 45855 y(The)347 b(op)31 b(erators)14535 44958 y Fq(\243)14996 45855 y Fu(H)d(;)184 b Fv(\243)17296 45453 y Fs(])17711 44958 y Fq(\244)18519 45855 y Fv(are)347 b Fm(lo)-57 b(c)g(al)347 b Fv(and)g(supp)31 b(orted)346 b(in)h(the)g(set)g(where)f Fg(r)p Fv(\243)h(is)f(non-zero.)6075 47183 y(A)369 b(particular)h (consequence)g(of)f(this)h(observ)-61 b(ation)370 b(is)f(that)16348 50123 y([)q Fu(H)28 b(;)184 b Fv(\243])678 b(=)21766 49071 y Fq(X)21356 51454 y Fs(\263)21759 51565 y Fk(1)22191 51454 y Fp(2S)23332 51565 y Fl(o)25225 49071 y Fq(X)25607 51454 y Fs(\263)26010 51565 y Fk(2)23960 52256 y Fs(U)24563 52373 y Fl(\263)24932 52539 y Fk(2)25419 52256 y Fx(\243)26099 52373 y Fl(\263)26468 52539 y Fk(1)26955 52256 y Fp(6\264)p Fx(0)28273 50123 y Fv(\243)29134 50289 y Fs(\263)29537 50400 y Fk(1)30209 50123 y Fv([)p Fu(H)28 b(;)184 b Fv(\243])33442 49374 y Fu(U)34198 49540 y Fs(\263)34601 49651 y Fk(2)p 33442 49868 1646 45 v 33833 50882 a Fu(F)16038 54714 y Fv([)p Fu(H)28 b(;)184 b Fv(\243)18645 54257 y Fp(0)18957 54714 y Fv(])677 b(=)21766 53663 y Fq(X)21356 56082 y Fs(\263)21809 55815 y Fj(0)21759 56322 y Fk(1)22191 56082 y Fp(2S)23388 55860 y Fj(0)23332 56303 y Fl(o)25225 53663 y Fq(X)25607 56149 y Fs(\263)26060 55815 y Fj(0)26010 56370 y Fk(2)23960 57021 y Fs(U)24563 57253 y Fl(\263)24961 57062 y Fj(0)24932 57526 y Fk(2)25419 57021 y Fx(\243)26099 57253 y Fl(\263)26497 57062 y Fj(0)26468 57526 y Fk(1)26955 57021 y Fp(6\264)p Fx(0)28406 53849 y Fu(U)29162 54042 y Fs(\263)29615 53775 y Fj(0)29565 54282 y Fk(2)p 28406 54460 V 28796 55473 a Fu(F)30369 54714 y Fv([)q Fu(H)28 b(;)184 b Fv(\243)32977 54257 y Fp(0)33289 54714 y Fv(])h(\243)34642 54907 y Fs(\263)35095 54640 y Fj(0)35045 55148 y Fk(1)35839 54714 y Fu(;)43096 53404 y Fv(\(5.10\))6075 59688 y(with)346 b Fr(S)9242 59854 y Fs(o)10042 59688 y Fv(=)307 b Fr(f)q Fu(q)347 b Fr(2)308 b(I)758 b Fv(:)677 b Fu(L)246 b Fr(\241)g Fv(13)p Fu(r)339 b(<)308 b Fr(j)p Fu(q)286 b Fr(\241)246 b Fu(x)p Fr(j)308 b Fu(<)f(L)247 b Fr(\241)f Fv(8)p Fu(r)31 b Fr(g)345 b Fv(and)h Fr(S)32795 59286 y Fp(0)32712 59962 y Fs(o)33512 59688 y Fv(=)307 b Fr(f)p Fu(q)348 b Fr(2)307 b(I)734 b Fv(:)653 b Fu(L)197 b Fv(+)h Fu(r)41925 59854 y Fs(o)42615 59688 y Fv(+)f(9)p Fu(r)339 b(<)6075 61016 y Fr(j)p Fu(q)136 b Fr(\241)96 b Fu(x)p Fr(j)308 b Fu(<)g(L)96 b Fv(+)g Fu(r)12691 61182 y Fs(o)13281 61016 y Fv(+)g(14)p Fu(r)31 b Fr(g)p Fv(.)470 b(Here)293 b(\243)20606 61182 y Fs(\263)21409 61016 y Fv(is)h(the)h(smo)31 b(oth)295 b(partition)i(of)e(unit)-31 b(y)296 b(used)d(in)i(Lemma)g(4.4.)7403 62345 y(Recall)370 b(that)12385 63564 y Fq(\260)12385 64228 y(\260)12385 64892 y(\260)12385 65556 y(\260)15658 64420 y Fv(1)p 13133 64914 5603 45 v 13133 65948 a(\()p Fu(a)247 b Fv(+)e Fu(H)16421 66114 y Fs(o)16914 65948 y Fv(\))17344 65628 y Fx(1)p Fs(=)p Fx(2)19053 64272 y Fq(\243)19514 65169 y Fu(H)28 b(;)184 b Fv(\243)21814 64712 y Fs(])22229 64272 y Fq(\244)22691 63564 y(\260)22691 64228 y(\260)22691 64892 y(\260)22691 65556 y(\260)23982 65169 y Fr(\267)677 b Fv(\()25950 64228 y Fq(\260)25950 64892 y(\260)26565 65169 y Fv(\242\243)28348 64712 y Fs(])28762 64228 y Fq(\260)28762 64892 y(\260)29377 65612 y Fp(1)30930 65169 y Fv(+)32406 64199 y Fr(p)p 33328 64199 554 45 v 970 x Fv(2)34066 64228 y Fq(\260)34066 64892 y(\260)34681 65169 y Fg(r)p Fv(\243)36603 64712 y Fs(])37017 64228 y Fq(\260)37017 64892 y(\260)37632 65612 y Fp(1)38569 65169 y Fv(\))308 b Fu(;)3482 b Fv(\(5.11\))6075 68213 y(whic)-31 b(h)407 b(is)f(b)31 b(ounded)406 b(for)g(smo)31 b(oth)407 b(\243.)603 b(W)-92 b(e)406 b(ma)-31 b(y)407 b(c)-31 b(ho)31 b(ose)406 b(\243)g(so)g(that)h(the)g(righ)-31 b(t)407 b(hand)f(side)g(is)6075 69541 y(b)31 b(ounded)369 b(uniformly)i(in)e Fu(L)p Fv(,)h(b)31 b(eing)370 b(no)f(larger)h(than)g (\(const.)q(\))184 b Fu(d)33063 69139 y Fx(2)33561 69541 y Fu(=r)34644 69139 y Fx(2)35141 69541 y Fv(.)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(35)7403 6001 y Fv(With)370 b(these)f(observ)-61 b(ations,)370 b(w)-31 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68392 y Fk(1)29043 68119 y Fl(;\263)29695 67928 y Fj(0)29666 68392 y Fk(2)33782 66972 y Fv(1)p 30341 67467 7435 45 v 30600 68651 a Fq(b)30341 68930 y Fu(H)31351 68355 y Fx(\(\255)p Fp(n)p Fs(B)33490 67978 y Fl(L)p Fk(+)p Fl(r)34960 68089 y(o)33452 68477 y(x)35458 68355 y Fx(\))31261 69301 y Fs(\263)31714 69034 y Fj(0)31664 69541 y Fk(1)32096 69301 y Fs(;\263)32812 69034 y Fj(0)32762 69541 y Fk(2)36105 68930 y Fr(\241)g Fu(z)37909 67721 y(\302)38602 67887 y Fs(y)39133 65452 y Fq(\260)39133 66116 y(\260)39133 66780 y(\260)39133 67445 y(\260)39133 68109 y(\260)39133 68773 y(\260)39748 65726 y Fs(s)40221 65496 y Fq(1)40221 67489 y(A)41682 67721 y Fu(:)1107 b Fv(\(5.14\))p eop end %%Page: 36 36 TeXDict begin 36 35 bop 6075 3345 a Fx(36)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(Here)9063 5722 y Fq(b)8804 6001 y Fu(H)9724 6167 y Fs(\263)10127 6278 y Fk(1)10559 6167 y Fs(;\263)11225 6278 y Fk(2)12194 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b(pic)-31 b(k)444 b(the)g(re-sampled)g (v)-61 b(ariables)445 b(with)g(the)f(same)g(distribution)h(as)f(the)g Fr(f)p Fu(\264)44556 10198 y Fs(\263)45064 10032 y Fr(g)p Fv(,)6075 11361 y(and)362 b(include)i(a)-31 b(v)g(eraging)365 b(o)-31 b(v)g(er)363 b(these)f(v)-61 b(ariables)363 b(in)g(the)g(exp)31 b(ectations.)492 b(Since)363 b Fu(S)40328 11527 y Fs(\263)40731 11638 y Fk(1)41162 11527 y Fs(;\263)41828 11638 y Fk(2)42623 11361 y Fr(\275)307 b Fu(B)44687 10959 y Fx(8)p Fs(r)44631 11674 y(\263)45034 11785 y Fk(1)45618 11361 y Fv(,)6075 12837 y Fu(S)6754 13030 y Fs(\263)7207 12763 y Fj(0)7157 13270 y Fk(1)7588 13030 y Fs(;\263)8304 12763 y Fj(0)8254 13270 y Fk(2)9049 12837 y Fr(\275)g Fu(B)11113 12435 y Fx(8)p Fs(r)11057 13187 y(\263)11510 12920 y Fj(0)11460 13427 y Fk(1)12400 12837 y Fv(and)357 b(for)f(eac)-31 b(h)357 b Fu(\263)19183 13003 y Fx(1)19680 12837 y Fv(,)i Fu(\263)20912 12435 y Fp(0)20830 13112 y Fx(1)21683 12837 y Fv(there)d(is)g(only)h(a)g(\257xed)f(\257nite)h(n)-31 b(um)g(b)31 b(er)356 b(of)h(v)-61 b(alues)357 b(for)f Fu(\263)45121 13003 y Fx(2)45618 12837 y Fv(,)6075 14370 y Fu(\263)6641 13968 y Fp(0)6559 14645 y Fx(2)7425 14370 y Fv(\(b)-31 b(y)370 b FB(A2)p Fv(\),)g(w)-31 b(e)370 b(infer)f(from)h(eq.)g(\(5.14\),)i(after)e(adjusting)h(the)e(constan) -31 b(t,)7182 17835 y Ft(E)8104 16274 y Fq(\265)8919 16230 y(\260)8919 16894 y(\260)8919 17558 y(\260)8919 18223 y(\260)9534 17835 y Fu(\302)10227 18001 y Fs(x)12793 17086 y Fv(1)p 10917 17581 4306 45 v 10917 18614 a Fu(H)11927 18294 y Fx(\(\255\))13552 18614 y Fr(\241)246 b Fu(z)15355 17835 y(\302)16048 18001 y Fs(y)16580 16230 y Fq(\260)16580 16894 y(\260)16580 17558 y(\260)16580 18223 y(\260)17195 16504 y Fs(s)17668 16274 y Fq(\266)18790 17835 y Fr(\267)677 b Fv(\(const.)q(\))24526 17086 y Fu(C)25396 16685 y Fs(s)25317 17400 y(\270)26117 16806 y Fq(e)25897 17086 y Fu(C)26767 16685 y Fx(2)p Fs(s)26688 17400 y(\270)p 24526 17581 3157 45 v 24615 18594 a Fv(1)246 b Fr(\241)g Fv(3)p Fu(s)27815 17835 y Fv(\(1)h(+)f(1)p Fu(=\270)p Fv(\))32334 17378 y Fs(s)32808 17835 y Fv(\(1)h(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)38186 18001 y Fs(o)38678 17835 y Fr(j)p Fv(\))39415 17378 y Fx(3)p Fs(s)p Fx(\()p Fs(d)p Fx(+3\))42973 17835 y Fu(D)43920 17378 y Fx(6)p Fs(s)17789 20995 y Fr(\243)19306 19943 y Fq(X)18896 22326 y Fs(\263)19299 22437 y Fk(1)19731 22326 y Fp(2)p Fs(S)20872 22437 y Fl(o)18896 23231 y Fs(\263)19349 22897 y Fj(0)19299 23453 y Fk(1)19731 23231 y Fp(2)p Fs(S)20914 22897 y Fj(0)20872 23453 y Fl(o)21500 20995 y Ft(E)22422 19434 y Fq(\265)23237 19389 y(\260)23237 20054 y(\260)23237 20718 y(\260)23237 21382 y(\260)23852 20995 y Fu(\302)24545 21161 y Fs(x)27441 20246 y Fv(1)p 25235 20740 4965 45 v 25235 21813 a Fu(H)26245 21472 y Fx(\()p Fs(B)27298 21251 y Fl(L)27260 21694 y(x)27882 21472 y Fx(\))28529 21813 y Fr(\241)g Fu(z)30332 20995 y FB(1)30968 21237 y Fs(B)31675 20970 y Fk(8)p Fl(r)31637 21505 y(\263)32006 21671 y Fk(1)32554 19389 y Fq(\260)32554 20054 y(\260)32554 20718 y(\260)32554 21382 y(\260)33169 19663 y Fs(s)33642 19434 y Fq(\266)24681 25671 y Fr(\243)g Ft(E)26710 24110 y Fq(\265)27526 24066 y(\260)27526 24730 y(\260)27526 25394 y(\260)27526 26059 y(\260)28141 25671 y FB(1)28777 25913 y Fs(B)29484 25646 y Fk(8)p Fl(r)29446 26295 y(\263)29844 26104 y Fj(0)29815 26569 y Fk(1)33935 24922 y Fv(1)p 30495 25416 7435 45 v 30495 26625 a Fu(H)31505 26305 y Fx(\(\255)p Fp(n)p Fs(B)33644 25928 y Fl(L)p Fk(+)p Fl(r)35114 26039 y(o)33606 26427 y(x)35612 26305 y Fx(\))36259 26625 y Fr(\241)g Fu(z)38062 25671 y(\302)38755 25837 y Fs(y)39287 24066 y Fq(\260)39287 24730 y(\260)39287 25394 y(\260)39287 26059 y(\260)39902 24340 y Fs(s)40375 24110 y Fq(\266)41682 25671 y Fu(:)1107 b Fv(\(5.15\))7403 28865 y(T)-92 b(o)248 b(complete)i(the)e(pro)31 b(of)248 b(of)h(eq.)f(\(5.2\))i(w)-31 b(e)249 b(need)e(a)i(b)31 b(ound)247 b(for)h(the)h(resolv)-31 b(en)g(t)249 b(of)f Fu(H)41417 28463 y Fx(\(\255)p Fp(n)p Fs(B)43556 28129 y Fl(L)p Fk(+)p Fl(r)45026 28240 y(o)43518 28685 y(x)45524 28463 y Fx(\))6075 30235 y Fv(in)440 b(terms)g(of)g(the)g(resolv)-31 b(en)g(t)441 b(of)g Fu(H)20965 29834 y Fx(\(\255\))22344 30235 y Fv(.)705 b(F)-92 b(or)439 b(this)h(w)-31 b(e)440 b(will)i(apply)f(eq.)f(\(5.5\))i(of)f(Lemma)g(5.2)6075 31564 y(with)16661 33345 y(\244)308 b(=)f(\255)247 b Fr(n)f Fu(B)21647 32888 y Fs(L)p Fx(+)p Fs(r)23345 32999 y Fl(o)21591 33618 y Fs(x)23844 33345 y Fu(;)2399 b Fv(\244)27319 33511 y Fs(o)28118 33345 y Fv(=)308 b(\255)246 b Fr(n)g Fu(B)32027 32888 y Fs(L)p Fx(+)p Fs(r)33725 32999 y Fl(o)34169 32888 y Fx(+)p Fs(r)31971 33618 y(x)43096 33345 y Fv(\(5.16\))6075 35358 y(and)397 b(a)h(smo)31 b(oth)399 b(function)g(\243)18371 34956 y Fp(00)19334 35358 y Fv(suc)-31 b(h)397 b(that)i Fr(r)p Fv(\243)26069 34956 y Fp(00)27032 35358 y Fv(is)e(supp)31 b(orted)398 b(in)f Fr(f)p Fu(q)e Fv(:)355 b Fu(L)266 b Fv(+)f Fu(r)39475 35524 y Fs(o)40322 35358 y Fu(<)354 b Fr(j)p Fu(q)306 b Fr(\241)265 b Fu(x)p Fr(j)354 b Fu(<)6075 36686 y(L)285 b Fv(+)g Fu(r)8758 36852 y Fs(o)9536 36686 y Fv(+)g Fu(r)31 b Fr(g)p Fv(.)668 b(Note)429 b(that)g(\243)18792 36285 y Fp(00)19358 36686 y Fu(\302)20051 36852 y Fs(y)20987 36686 y Fv(=)405 b Fu(\302)22946 36852 y Fs(y)23905 36686 y Fv(and)428 b Fu(B)27012 36285 y Fx(8)p Fs(r)26956 37036 y(\263)27409 36769 y Fj(0)27359 37277 y Fk(1)28348 36686 y Fr(\275)405 b Fv(\244)30383 36852 y Fs(o)31302 36686 y Fv(for)428 b(ev)-31 b(ery)428 b Fu(\263)36604 36285 y Fp(0)36522 36961 y Fx(1)37423 36686 y Fr(2)405 b(S)39319 36285 y Fp(0)39236 36960 y Fs(o)39729 36686 y Fv(.)668 b(In)427 b(place)i(of)6075 38218 y(\(5.12\))372 b(w)-31 b(e)369 b(obtain)7366 39855 y Fq(\260)7366 40519 y(\260)7366 41183 y(\260)7366 41847 y(\260)7981 41460 y FB(1)8617 41702 y Fs(B)9324 41435 y Fk(8)p Fl(r)9286 42084 y(\263)9684 41893 y Fj(0)9655 42357 y Fk(1)13776 40711 y Fv(1)p 10335 41205 V 10335 42414 a Fu(H)11345 42094 y Fx(\(\255)p Fp(n)p Fs(B)13484 41716 y Fl(L)p Fk(+)p Fl(r)14954 41827 y(o)13446 42216 y(x)15452 42094 y Fx(\))16099 42414 y Fr(\241)246 b Fu(z)17903 41460 y(\302)18596 41626 y Fs(y)19127 39855 y Fq(\260)19127 40519 y(\260)19127 41183 y(\260)19127 41847 y(\260)20050 41460 y Fr(\267)21218 39855 y Fq(\260)21218 40519 y(\260)21218 41183 y(\260)21218 41847 y(\260)21833 41460 y FB(1)22469 41702 y Fs(B)23176 41435 y Fk(8)p Fl(r)23138 42084 y(\263)23536 41893 y Fj(0)23507 42357 y Fk(1)26063 40711 y Fv(1)p 24187 41205 4306 45 v 24187 42239 a Fu(H)25197 41919 y Fx(\(\255\))26823 42239 y Fr(\241)g Fu(z)28626 41460 y(\302)29319 41626 y Fs(y)29851 39855 y Fq(\260)29851 40519 y(\260)29851 41183 y(\260)29851 41847 y(\260)13815 44559 y Fv(+)615 b(\(const.)r(\))21799 43508 y Fq(X)21230 45994 y Fs(\263)21683 45660 y Fj(00)21633 46216 y Fk(1)22227 45994 y Fp(2S)23424 45660 y Fj(00)23368 46216 y Fl(o)19541 47076 y Fs(\263)19994 46742 y Fj(00)19944 47297 y Fk(2)20537 47076 y Fx(:)20788 47159 y Fr(j)21095 47076 y Fs(\263)21548 46742 y Fj(00)21498 47297 y Fk(2)22092 47076 y Fp(\241)p Fs(\263)23237 46742 y Fj(00)23187 47297 y Fk(1)23780 47159 y Fr(j)24088 47076 y Fs(<)p Fx(3)p Fs(r)25840 42954 y Fq(\260)25840 43618 y(\260)25840 44282 y(\260)25840 44947 y(\260)26455 44559 y FB(1)27091 44802 y Fs(B)27798 44535 y Fk(8)p Fl(r)27760 45183 y(\263)28158 44992 y Fj(0)28129 45457 y Fk(1)32250 43810 y Fv(1)p 28809 44305 7435 45 v 28809 45513 a Fu(H)29819 45193 y Fx(\(\255)p Fp(n)p Fs(B)31958 44816 y Fl(L)p Fk(+)p Fl(r)33428 44927 y(o)31920 45315 y(x)33926 45193 y Fx(\))34574 45513 y Fr(\241)246 b Fu(z)36377 44559 y(U)37133 44752 y Fs(\263)37586 44485 y Fj(00)37536 44993 y Fk(2)38185 42954 y Fq(\260)38185 43618 y(\260)38185 44282 y(\260)38185 44947 y(\260)25541 49594 y Fr(\243)26648 47989 y Fq(\260)26648 48653 y(\260)26648 49317 y(\260)26648 49981 y(\260)27263 49594 y Fv(\()p Fu(a)h Fv(+)f Fu(H)30552 49760 y Fs(o)31044 49594 y Fv(\))31474 49137 y Fx(1)p Fs(=)p Fx(2)32866 49594 y Fv(\243)33727 49787 y Fs(\263)34180 49520 y Fj(00)34130 50027 y Fk(1)36788 48845 y Fv(1)p 34912 49339 4306 45 v 34912 50373 a Fu(H)35922 50053 y Fx(\(\255\))37547 50373 y Fr(\241)g Fu(z)39350 49594 y(\302)40043 49760 y Fs(y)40575 47989 y Fq(\260)40575 48653 y(\260)40575 49317 y(\260)40575 49981 y(\260)41682 49594 y Fu(;)1107 b Fv(\(5.17\))6075 52506 y(where)369 b Fr(S)10029 52104 y Fp(00)9946 52780 y Fs(o)10902 52506 y Fv(:=)308 b Fr(f)p Fu(q)348 b Fr(2)308 b(I)388 b Fv(:)616 b Fu(L)246 b Fv(+)g Fu(r)19338 52672 y Fs(o)20077 52506 y Fr(\241)g Fv(2)p Fu(r)339 b(<)307 b Fr(j)p Fu(q)287 b Fr(\241)246 b Fu(x)p Fr(j)307 b Fu(<)h(L)246 b Fv(+)g Fu(r)30959 52672 y Fs(o)31698 52506 y Fv(+)g(3)p Fu(r)31 b Fr(g)p Fv(.)7403 53834 y(This)370 b(yields)f(the)h(exp)31 b(ectation)371 b(b)31 b(ound)7182 57075 y Ft(E)8104 55514 y Fq(\265)8919 55470 y(\260)8919 56134 y(\260)8919 56798 y(\260)8919 57463 y(\260)9534 57075 y FB(1)10170 57317 y Fs(B)10877 57050 y Fk(8)p Fl(r)10839 57699 y(\263)11237 57508 y Fj(0)11208 57973 y Fk(1)15329 56326 y Fv(1)p 11888 56820 7435 45 v 11888 58029 a Fu(H)12898 57709 y Fx(\(\255)p Fp(n)p Fs(B)15037 57332 y Fl(L)p Fk(+)p Fl(r)16507 57443 y(o)14999 57831 y(x)17005 57709 y Fx(\))17652 58029 y Fr(\241)246 b Fu(z)19456 57075 y(\302)20149 57241 y Fs(y)20680 55470 y Fq(\260)20680 56134 y(\260)20680 56798 y(\260)20680 57463 y(\260)21295 55744 y Fs(s)21768 55514 y Fq(\266)22891 57075 y Fr(\267)307 b Ft(E)24981 55514 y Fq(\265)25796 55470 y(\260)25796 56134 y(\260)25796 56798 y(\260)25796 57463 y(\260)26411 57075 y FB(1)27047 57317 y Fs(B)27754 57050 y Fk(8)p Fl(r)27716 57699 y(\263)28114 57508 y Fj(0)28085 57973 y Fk(1)30642 56326 y Fv(1)p 28766 56820 4306 45 v 28766 57854 a Fu(H)29776 57534 y Fx(\(\255\))31401 57854 y Fr(\241)246 b Fu(z)33204 57075 y(\302)33897 57241 y Fs(y)34429 55470 y Fq(\260)34429 56134 y(\260)34429 56798 y(\260)34429 57463 y(\260)35044 55744 y Fs(s)35517 55514 y Fq(\266)12448 60235 y Fv(+)615 b(\(const.)q(\))18932 59183 y Fq(X)18173 61858 y Fr(h)18604 61775 y Fs(\263)19057 61508 y Fj(00)19007 62015 y Fk(1)19600 61775 y Fs(;\263)20316 61508 y Fj(00)20266 62015 y Fk(2)20860 61858 y Fr(i)22756 59183 y Fq(X)22755 61566 y Fs(\263)50 b Fp(2I)21475 62660 y Fr(j)21782 62577 y Fs(\263)g Fp(\241)p Fs(\263)23380 62243 y Fj(0)23330 62798 y Fk(1)23762 62660 y Fr(j)24069 62577 y Fs(<)p Fx(9)p Fs(r)25822 60235 y Ft(E)26744 58674 y Fq(\265)27559 58629 y(\260)27559 59294 y(\260)27559 59958 y(\260)27559 60622 y(\260)28174 60235 y Fu(U)28930 60401 y Fs(\263)33012 59486 y Fv(1)p 29571 59980 7435 45 v 29571 61188 a Fu(H)30581 60869 y Fx(\(\255)p Fp(n)p Fs(B)32720 60491 y Fl(L)p Fk(+)p Fl(r)34190 60602 y(o)32682 60991 y(x)34688 60869 y Fx(\))35335 61188 y Fr(\241)246 b Fu(z)37139 60235 y(U)37895 60428 y Fs(\263)38348 60161 y Fj(00)38298 60668 y Fk(2)38946 58629 y Fq(\260)38946 59294 y(\260)38946 59958 y(\260)38946 60622 y(\260)39561 58903 y Fs(s)24315 65155 y Fr(\243)25360 63550 y Fq(\260)25360 64214 y(\260)25360 64878 y(\260)25360 65543 y(\260)25975 65155 y Fv(\()p Fu(a)h Fv(+)f Fu(H)29264 65321 y Fs(o)29756 65155 y Fv(\))30186 64698 y Fx(1)p Fs(=)p Fx(2)31578 65155 y Fv(\243)32439 65348 y Fs(\263)32892 65081 y Fj(00)32842 65589 y Fk(1)35500 64406 y Fv(1)p 33624 64900 4306 45 v 33624 65934 a Fu(H)34634 65614 y Fx(\(\255\))36259 65934 y Fr(\241)g Fu(z)38062 65155 y(\302)38755 65321 y Fs(y)39287 63550 y Fq(\260)39287 64214 y(\260)39287 64878 y(\260)39287 65543 y(\260)39902 63824 y Fs(s)40375 63594 y Fq(\266)41682 65155 y Fu(:)1107 b Fv(\(5.18\))6075 68065 y(F)-92 b(or)464 b(the)h(terms)g(in)g(the)g (sum)g(of)h(eq.)f(\(5.18\))j(w)-31 b(e)465 b(\257rst)g(tak)-31 b(e)466 b(the)f(exp)31 b(ectation)467 b(conditioned)6075 69394 y(on)390 b Fr(F)8539 68992 y Fs(c)8429 69743 y(\263)8882 69476 y Fj(00)8832 69984 y Fk(1)9426 69743 y Fs(;\263)10142 69476 y Fj(00)10092 69984 y Fk(2)10685 69743 y Fs(;\263)11847 69394 y Fv(and)g(use)g(the)h(Cauc)-31 b(h)g(y-Sc)g(h)g(w)g(arz)393 b(inequalit)-31 b(y)393 b(on)e(the)f(pro)31 b(duct.)557 b(The)390 b(resulting)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(37)6075 6001 y Fv(term)15550 8142 y Ft(E)16472 6249 y Fq(\303)17481 6537 y(\260)17481 7201 y(\260)17481 7865 y(\260)17481 8529 y(\260)18096 8142 y Fu(U)18852 8308 y Fs(\263)22934 7393 y Fv(1)p 19493 7887 7435 45 v 19493 9096 a Fu(H)20503 8776 y Fx(\(\255)p Fp(n)p Fs(B)22642 8398 y Fl(L)p Fk(+)p Fl(r)24112 8509 y(o)22604 8898 y(x)24610 8776 y Fx(\))25258 9096 y Fr(\241)246 b Fu(z)27061 8142 y(U)27817 8335 y Fs(\263)28270 8068 y Fj(00)28220 8575 y Fk(2)28869 6537 y Fq(\260)28869 7201 y(\260)28869 7865 y(\260)28869 8529 y(\260)29484 6810 y Fx(2)p Fs(s)30398 6205 y Fq(\257)30398 6869 y(\257)30398 7533 y(\257)30398 8197 y(\257)30398 8861 y(\257)30952 8142 y Fr(F)31858 7685 y Fs(c)31748 8436 y(\263)32201 8169 y Fj(00)32151 8676 y Fk(1)32744 8436 y Fs(;\263)33460 8169 y Fj(00)33410 8676 y Fk(2)34003 8436 y Fs(;\263)34774 6249 y Fq(!)36143 8142 y Fu(;)6646 b Fv(\(5.19\))6075 10791 y(is)369 b(treated)h(b)-31 b(y)369 b(eq.)h(\(4.19\),)i(while)14680 13673 y Ft(E)15602 11780 y Fq(\303)16611 12068 y(\260)16611 12732 y(\260)16611 13397 y(\260)16611 14061 y(\260)17226 13673 y Fv(\()p Fu(a)247 b Fv(+)f Fu(H)20515 13839 y Fs(o)21007 13673 y Fv(\))21437 13216 y Fx(1)p Fs(=)p Fx(2)22829 13673 y Fv(\243)23690 13866 y Fs(\263)24143 13599 y Fj(00)24093 14107 y Fk(1)26751 12924 y Fv(1)p 24875 13419 4306 45 v 24875 14452 a Fu(H)25885 14133 y Fx(\(\255\))27510 14452 y Fr(\241)g Fu(z)29313 13673 y(\302)30006 13839 y Fs(y)30538 12068 y Fq(\260)30538 12732 y(\260)30538 13397 y(\260)30538 14061 y(\260)31153 12342 y Fx(2)p Fs(s)32067 11736 y Fq(\257)32067 12400 y(\257)32067 13064 y(\257)32067 13729 y(\257)32067 14393 y(\257)32621 13673 y Fr(F)33527 13216 y Fs(c)33417 13968 y(\263)33870 13701 y Fj(00)33820 14208 y Fk(1)34413 13968 y Fs(;\263)35129 13701 y Fj(00)35079 14208 y Fk(2)35673 13968 y Fs(;\263)36444 11780 y Fq(!)43096 13673 y Fv(\(5.20\))6075 16494 y(can)293 b(b)31 b(e)293 b(estimated)i(through)f(a)f(straigh)-31 b(tforw)g(ard)296 b(v)-61 b(arian)-31 b(t)294 b(of)g(Lemma)g(4.4,)310 b(with)295 b(re-sampling)6075 17822 y(done)369 b(o)-31 b(v)g(er)370 b(all)g(v)-61 b(ariables)370 b(in)g Fr(I)19113 18015 y Fs(\263)19566 17748 y Fj(00)19516 18256 y Fk(1)20109 18015 y Fs(;\263)20825 17748 y Fj(00)20775 18256 y Fk(2)21368 18015 y Fs(;\263)22140 17822 y Fv(.)7403 19150 y(Av)-31 b(eraging)371 b(o)-31 b(v)g(er)370 b(the)f(re-sampled)h(v)-61 b(ariables)370 b(yields)7182 22031 y Ft(E)8104 20470 y Fq(\265)8919 20426 y(\260)8919 21090 y(\260)8919 21754 y(\260)8919 22418 y(\260)9534 22031 y FB(1)10170 22273 y Fs(B)10877 22006 y Fk(8)p Fl(r)10839 22655 y(\263)11237 22464 y Fj(0)11208 22929 y Fk(1)15329 21282 y Fv(1)p 11888 21776 7435 45 v 11888 22985 a Fu(H)12898 22665 y Fx(\(\255)p Fp(n)p Fs(B)15037 22287 y Fl(L)p Fk(+)p Fl(r)16507 22398 y(o)14999 22787 y(x)17005 22665 y Fx(\))17652 22985 y Fr(\241)246 b Fu(z)19456 22031 y(\302)20149 22197 y Fs(y)20680 20426 y 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(of)f Fu(f)496 b Fv(=)377 b Fu(f)24350 65442 y Fs(\256)24981 65276 y Fv(,)421 b(at)412 b(a)f(sp)31 b(eci\257ed)410 b(v)-61 b(alue)411 b(of)h Fu(a)37363 65442 y Fs(\256)37994 65276 y Fv(,)421 b(is)411 b(giv)-31 b(en)412 b(b)-31 b(y)411 b(the)6075 66604 y(follo)-31 b(wing)373 b(expression:)21939 67853 y Fu(\275)p Fv(\()p Fu(f)119 b(=\270)247 b Fv(+)f Fu(a)26740 68019 y Fs(\256)27371 67853 y Fv(\))p 21939 68347 5863 45 v 22057 69372 a Fu(\270)22887 68480 y Fq(R)23410 69765 y Fs(I)23802 69876 y Fl(j)24456 69372 y Fu(\275)p Fv(\()p Fu(\264)40 b Fv(\)d)p Fu(\264)28303 68602 y Fv(d)p Fu(f)427 b(;)12902 b Fv(\(5.24\))p eop end %%Page: 38 38 TeXDict begin 38 37 bop 6075 3345 a Fx(38)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(where)313 b Fu(\275)g Fv(is)g(the)h(common)h(densit)-31 b(y)314 b(for)g Fu(\264)23337 6167 y Fs(\263)23845 6001 y Fv(.)474 b(The)313 b(denominator)j(is)d(just)g(the)h(probabilit)-31 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Fs(\256)34198 35423 y(j)34916 35134 y Fv(\()p Fu(!)40 b Fv(\))248 b(and)f Fu(f)39325 35300 y Fs(\263)40141 35134 y Fv(=)307 b Fu(\270\264)42505 35300 y Fs(\263)43015 35134 y Fr(\241)r Fu(\270a)45109 35300 y Fs(\263)45618 35134 y Fv(.)6075 36463 y(The)369 b(densit)-31 b(y)370 b(of)g(the)f(distribution)i(of)f Fu(f)427 b Fv(=)307 b Fu(f)25312 36629 y Fs(\256)26312 36463 y Fv(conditioned)371 b(on)f Fu(a)34400 36629 y Fs(\256)35400 36463 y Fv(and)f Fr(F)38458 36061 y Fs(c)38348 36736 y(\256)39347 36463 y Fv(is)21928 38393 y Fu(\275)22500 38559 y Fs(\256)23131 38393 y Fv(\()p Fu(f)119 b(=\270)247 b Fv(+)f Fu(a)27360 38559 y Fs(\256)27991 38393 y Fr(j)p Fu(!)40 b Fv(\))p 21304 38888 8778 45 v 21304 39912 a Fu(\270)22134 39020 y Fq(R)22657 40306 y Fs(I)23109 40039 y Fl(\256)23049 40558 y(j)23670 40306 y Fx(\()p Fs(!)28 b Fx(\))25187 39912 y Fu(\275)25759 40078 y Fs(\256)26390 39912 y Fv(\()p Fu(\264)40 b Fr(j)p Fu(!)g Fv(\)d)p Fu(\264)30522 39142 y(;)12267 b Fv(\(5.27\))6075 42323 y(whic)-31 b(h)370 b(is)f(a)g(densit)-31 b(y)370 b(supp)31 b(orted)369 b(in)h([0)p Fu(;)184 b Fv(1])372 b(and)d(b)31 b(ounded)369 b(b)-31 b(y)370 b Fu(D)31 b Fv(.)7403 43651 y(With)449 b(these)e(notions,)470 b(it)449 b(is)f(an)g(easy)g(exercise)g(in)g(conditional)j(exp)31 b(ectations)450 b(to)f(pro)-31 b(v)g(e)6075 44980 y(the)369 b(follo)-31 b(wing)373 b(extension)d(of)g(Lemma)g(4.3:)6075 47116 y FB(Prop)35 b(osition)485 b(5.3.)584 b Fm(L)-57 b(et)444 b Fu(H)18458 47282 y Fs(!)19542 47116 y Fm(b)-57 b(e)444 b(a)f(r)-57 b(andom)443 b(Schr\304)-566 b(odinger)443 b(op)-57 b(er)g(ator)443 b(which)f(satis\257es)h FB(A1)p Fm(,)6075 48444 y FB(A2)p Fm(,)530 b(and)503 b FB(A3)12397 48042 y Fp(0)12708 48444 y Fm(.)829 b(Then)503 b(e)-57 b(q.)504 b(\(4.17\))d(of)i(L)-57 b(emma)504 b(4.3)e(holds)g(with)g(a)g (c)-57 b(o)g(e\261cient)504 b Fu(C)42126 48610 y Fs(\270)43210 48444 y Fm(which)6075 49772 y(satis\257es)401 b Fv(sup)11906 50043 y Fs(\270)12670 49772 y Fu(C)13461 49938 y Fs(\270)14360 49772 y Fu(<)318 b Fr(1)p Fm(,)404 b(pr)-57 b(ovide)g(d)402 b(we)g(use)g(for)g Fu(D)433 b Fm(the)401 b(c)-57 b(onstant)402 b(app)-57 b(e)g(aring)403 b(in)g FB(A3)42717 49371 y Fp(0)43430 49772 y Fm({)f(i.e,)6075 51101 y(the)395 b(blow-up)h(norm)g (of)g(the)g(distribution.)7403 53237 y Fv(T)-92 b(urning)422 b(no)-31 b(w)422 b(to)h(Lemma)f(4.4,)436 b(w)-31 b(e)422 b(note)g(that)h(the)e(disorder)g(strength)h(pla)-31 b(ys)422 b(a)g(role)g(in)6075 54565 y(the)403 b(re-sampling)i(pro)31 b(cedure)403 b(in)h(addition)h(to)f(the)g(a)-31 b(v)g(eraging.)598 b(Hence,)413 b(it)404 b(is)f(most)i(natural)6075 55893 y(to)399 b(use)e(the)h(lemma)i(as)e(stated)h(and)f(re-sample)h(only)g (the)f(v)-61 b(ariables)398 b Fu(f)36428 56059 y Fs(\263)37335 55893 y Fv(with)h(the)f Fu(a)42405 56059 y Fs(\263)43312 55893 y Fv(\257xed.)6075 57222 y(An)363 b(argumen)-31 b(t)365 b(along)g(these)e(lines)g(will)i(b)31 b(e)363 b(used)g(to)h(complete)h(the)e(pro)31 b(of)364 b(of)f(eq.)h(\(5.4\))i (from)6075 58550 y(Lemma)k(5.1:)6075 60686 y FB(Prop)35 b(osition)392 b(5.4.)526 b Fm(L)-57 b(et)369 b Fu(H)18232 60852 y Fs(!)19242 60686 y Fm(b)-57 b(e)369 b(a)f(r)-57 b(andom)369 b(Schr\304)-566 b(odinger)369 b(op)-57 b(er)g(ator)367 b(which)i(satis\257es)e FB(IAD)p Fm(,)6075 62091 y FB(A1)p Fm(,)356 b FB(A2)p Fm(,)g(and)344 b FB(A3)14357 61690 y Fp(0)14669 62091 y Fm(.)492 b(Then)345 b(e)-57 b(q.)346 b(\(5.2\))d(of)i(L)-57 b(emma)346 b(5.1)e(holds)f(with)h(a)h(c)-57 b(o)g(e\261cient)41033 61812 y Fq(e)40813 62091 y Fu(C)41604 62257 y Fs(s;\270)43210 62091 y Fm(which)6075 63420 y(ob)g(eys)396 b(e)-57 b(q.)398 b(\(5.4\).)6075 65556 y(Pr)-57 b(o)g(of.)554 b Fv(The)383 b(pro)31 b(of)383 b(follo)-31 b(ws)384 b(closely)g(the)f (deriv)-61 b(ation)383 b(of)g(eq.)g(\(5.2\))i(giv)-31 b(en)384 b(ab)31 b(o)-31 b(v)g(e)383 b(with)h(a)f(few)6075 66884 y(mo)31 b(di\257cations)371 b(whic)-31 b(h)371 b(w)-31 b(e)369 b(no)-31 b(w)370 b(indicate.)7403 68213 y(Let)h Fr(A)g Fv(denote)h(the)g(sigma-algebra)i(generated)e(b)-31 b(y)371 b Fu(a)30315 68379 y Fs(\263)30824 68213 y Fv(,)h Fu(\263)393 b Fr(2)310 b(I)82 b Fv(.)498 b(Conditioning)375 b(on)d Fr(A)p Fv(,)g(w)-31 b(e)6075 69541 y(obtain)416 b(a)f(random)g(Sc)-31 b(hr\304)-553 b(odinger)415 b(op)31 b(erator)416 b(with)f Fu(\270)383 b Fv(=)h(1)414 b(and)h(couplings)h Fu(f)38716 69707 y Fs(\263)39225 69541 y Fv(,)426 b Fu(\263)465 b Fr(2)383 b(I)496 b Fv(whic)-31 b(h)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(39)6075 6001 y Fv(ob)31 b(ey)293 b FB(IAD)g Fv(and)g FB(A3)p Fv(.)468 b(Th)-31 b(us,)309 b(follo)-31 b(wing)297 b(the)c(pro)31 b(of)293 b(of)h(Lemma)g(5.1)g(w)-31 b(e)294 b(obtain)g(the)f(follo)-31 b(wing)6075 7330 y(analog)371 b(of)f(eq.)f(\(5.14\):)7182 10210 y Ft(E)8104 8649 y Fq(\265)9052 8605 y(\260)9052 9269 y(\260)9052 9933 y(\260)9052 10597 y(\260)9667 10210 y Fu(\302)10360 10376 y Fs(x)12926 9461 y Fv(1)p 11050 9955 4306 45 v 11050 10989 a Fu(H)12060 10669 y Fx(\(\255\))13685 10989 y Fr(\241)246 b Fu(z)15488 10210 y(\302)16181 10376 y Fs(y)16713 8605 y Fq(\260)16713 9269 y(\260)16713 9933 y(\260)16713 10597 y(\260)17328 8879 y Fs(s)17801 8605 y Fq(\257)17801 9269 y(\257)17801 9933 y(\257)17801 10597 y(\257)18354 10210 y Fr(A)19238 8649 y Fq(\266)20361 10210 y Fr(\267)676 b Fv(\(const.)r(\))27308 9461 y(1)p 26096 9955 2979 45 v 26096 10969 a(1)246 b Fr(\241)g Fv(3)p Fu(s)29207 10210 y Fv(\(1)h(+)f Fr(j)p Fu(z)295 b Fr(\241)246 b Fu(E)34585 10376 y Fs(o)35078 10210 y Fr(j)p Fv(\))35815 9753 y Fx(3)p Fs(s)p Fx(\()p Fs(d)p Fx(+3\))39373 10210 y Fu(D)40320 9753 y Fx(6)p Fs(s)17004 13370 y Fr(\243)18521 12318 y Fq(X)18111 14701 y Fs(\263)18514 14812 y Fk(1)18946 14701 y Fp(2)p Fs(S)20087 14812 y Fl(o)18111 15606 y Fs(\263)18564 15272 y Fj(0)18514 15827 y Fk(1)18946 15606 y Fp(2)p Fs(S)20129 15272 y Fj(0)20087 15827 y Fl(o)20715 13370 y Ft(E)21637 11809 y Fq(\265)22585 11764 y(\260)22585 12429 y(\260)22585 13093 y(\260)22585 13757 y(\260)23200 13370 y Fu(\302)23893 13536 y Fs(x)26788 12621 y Fv(1)p 24583 13115 4965 45 v 24583 14188 a Fu(H)25593 13847 y Fx(\()p Fs(B)26646 13626 y Fl(L)26608 14069 y(x)27230 13847 y Fx(\))27877 14188 y Fr(\241)g Fu(z)29680 13370 y FB(1)30316 13612 y Fs(B)31023 13345 y Fk(8)p Fl(r)30985 13880 y(\263)31354 14046 y Fk(1)31901 11764 y Fq(\260)31901 12429 y(\260)31901 13093 y(\260)31901 13757 y(\260)32516 12038 y Fs(s)32990 11764 y Fq(\257)32990 12429 y(\257)32990 13093 y(\257)32990 13757 y(\257)33543 13370 y Fr(A)34427 11809 y Fq(\266)23111 18046 y Fr(\243)g Ft(E)25140 16485 y Fq(\265)26088 16441 y(\260)26088 17105 y(\260)26088 17769 y(\260)26088 18433 y(\260)26703 18046 y FB(1)27339 18288 y Fs(B)28046 18021 y Fk(8)p Fl(r)28008 18670 y(\263)28406 18479 y Fj(0)28377 18944 y Fk(1)32498 17297 y Fv(1)p 29057 17791 7435 45 v 29057 19000 a Fu(H)30067 18680 y Fx(\(\255)p Fp(n)p Fs(B)32206 18302 y Fl(L)p Fk(+)p Fl(r)33676 18413 y(o)32168 18802 y(x)34174 18680 y Fx(\))34822 19000 y Fr(\241)g Fu(z)36625 18046 y(\302)37318 18212 y Fs(y)37850 16441 y Fq(\260)37850 17105 y(\260)37850 17769 y(\260)37850 18433 y(\260)38465 16715 y Fs(s)38938 16441 y Fq(\257)38938 17105 y(\257)38938 17769 y(\257)38938 18433 y(\257)39491 18046 y Fr(A)40375 16485 y Fq(\266)41682 18046 y Fu(:)1107 b Fv(\(5.28\))6075 20818 y(W)-92 b(e)301 b(no)-31 b(w)304 b(a)-31 b(v)g(erage)303 b(with)h(resp)31 b(ect)301 b(to)i Fr(A)p Fv(,)316 b(noting)304 b(that)g Fu(H)29747 20416 y Fx(\()p Fs(B)30800 20082 y Fl(L)30762 20638 y(x)31384 20416 y Fx(\))32087 20818 y Fv(and)f Fu(H)35183 20416 y Fx(\(\255)p Fp(n)p Fs(B)37322 20082 y Fl(L)p Fk(+)p Fl(r)38792 20193 y(o)37284 20638 y(x)39290 20416 y Fx(\))39993 20818 y Fv(are)f(indep)31 b(en-)6075 22146 y(den)-31 b(t)414 b(b)-31 b(y)415 b FB(IAD)p Fv(,)g(and)g(this)f (indep)31 b(endence)414 b(is)h(inherited)g(b)-31 b(y)414 b(conditional)k(a)-31 b(v)g(erages)415 b(o)-31 b(v)g(er)415 b Fr(A)p Fv(.)6075 23475 y(The)407 b(resulting)g(b)31 b(ound)407 b(is)g(iden)-31 b(tical)409 b(to)e(eq.)h(\(5.14\))h(with)f Fu(\270)370 b Fv(=)g(1.)606 b(T)-92 b(o)407 b(complete)h(the)f(pro)31 b(of,)6075 24803 y(use)352 b(a)i(similar)g(argumen)-31 b(t)355 b(to)e(pro)-31 b(v)g(e)354 b(the)f(analog)j(of)d(eq.)h (\(5.21\),)359 b(\257rst)353 b(conditioning)j(on)e Fr(A)e Fv(to)6075 26131 y(obtain)371 b(co)31 b(e\261cien)-31 b(ts)370 b(with)g Fu(\270)308 b Fv(=)f(1.)24440 b Fh(\244)6075 28168 y Fv(5.3.)555 b FB(The)425 b(criteria)f({)h(pro)35 b(of)426 b(of)f(Theorem)g(1.2.)554 b Fv(Let)17326 30709 y Fu(M)18400 30875 y Fs(s;\270)20337 30709 y Fv(:=)22402 30429 y Fq(e)22182 30709 y Fu(C)22973 30875 y Fs(\270;s)24234 30709 y Fu(D)25181 30252 y Fx(10)p Fs(s)26783 30709 y Fr(\243)721 b Fv(sup)28337 31638 y Fs(\256)p Fp(2I)27890 32339 y Fs(L>)p Fx(23)p Fs(r)30824 29822 y Fv(#)p Fu(S)32489 29420 y Fp(0)32425 30127 y Fs(\256;L)p 30824 30455 3103 45 v 31174 31468 a Fu(L)31927 31149 y Fs(d)p Fp(\241)p Fx(1)34367 30709 y Fu(;)8422 b Fv(\(5.29\))6075 33947 y(with)478 b(the)g(sets)f Fr(S)13745 33546 y Fp(0)13662 34252 y Fs(\256;L)15640 33947 y Fv(as)h(in)f(Lemma)i(5.1,)506 b(and)477 b(supp)31 b(ose)477 b(eq.)h(\(1.14\))i(holds)e(for)f(some)h Fu(s)487 b Fr(2)6075 35344 y Fv(\(0)p Fu(;)184 b Fv(1)p Fu(=)p Fv(3\))373 b(and)c Fu(E)i Fr(2)308 b Ft(R)p Fv(.)7403 36672 y(F)-92 b(or)368 b(eac)-31 b(h)370 b(op)31 b(en)369 b(set)g(\255)308 b Fr(\275)g Ft(R)19358 36270 y Fs(d)20243 36672 y Fv(and)370 b Fu(\261)349 b(>)308 b Fv(0,)370 b(w)-31 b(e)369 b(de\257ne)14709 39159 y Fu(G)15579 38702 y Fs(\261)15579 39433 y Fx(\255)16266 39159 y Fv(\()p Fu(\256)t(;)184 b(\257)58 b Fv(\))310 b(:=)508 b(sup)20799 40140 y Fp(j)p Fs(")p Fp(j)p Fs(<\261)23050 39159 y Ft(E)23972 37598 y Fq(\265)24787 37554 y(\260)24787 38218 y(\260)24787 38882 y(\260)24787 39546 y(\260)25402 39159 y Fu(\302)26095 39325 y Fs(\256)29982 38410 y Fv(1)p 26859 38904 6800 45 v 26859 39938 a Fu(H)27869 39618 y Fx(\(\255\))29494 39938 y Fr(\241)246 b Fu(E)310 b Fr(\241)246 b Fv(i)p Fu(")33791 39159 y(\302)34484 39325 y Fs(\257)35081 37554 y Fq(\260)35081 38218 y(\260)35081 38882 y(\260)35081 39546 y(\260)35696 37827 y Fs(s)36169 37598 y Fq(\266)43096 39159 y Fv(\(5.30\))6075 41982 y(with)330 b Fu(\256)t(;)184 b(\257)367 b Fr(2)307 b(I)247 b(\\)165 b Fv(\255.)479 b(Whenev)-31 b(er)328 b Fu(\256)t(;)184 b(\257)368 b Fr(2)307 b(I)246 b(\\)165 b Fv(\255)329 b(with)h Fu(B)29793 41580 y Fs(L)29737 42255 y(\256)30763 41982 y Fr(\275)307 b Fv(\255)329 b(and)g Fr(j)p Fu(\256)169 b Fr(\241)c Fu(\257)58 b Fr(j)309 b Fu(>)e(L)165 b Fv(+)g Fu(r)42292 42148 y Fx(0)42955 41982 y Fv(+)g(23)p Fu(r)31 b Fv(,)6075 43310 y(Lemma)370 b(5.1)g(implies)h(that)15578 45594 y Fu(G)16448 45137 y Fs(\261)16448 45868 y Fx(\255)17135 45594 y Fv(\()p Fu(\256)t(;)184 b(\257)58 b Fv(\))679 b Fr(\267)e Fv(e)22591 45137 y Fp(\241)p Fs(\260)48 b Fx(\()p Fs(\261)28 b Fx(\))26380 44845 y Fv(1)p 25106 45339 3103 45 v 25106 46353 a(#)p Fu(S)26771 45972 y Fp(0)26707 46679 y Fs(\256;L)29194 44542 y Fq(X)28525 46962 y Fs(\263)50 b Fp(2)p Fs(S)30161 46695 y Fj(0)30119 47224 y Fl(\256;L)31646 45594 y Fu(G)32516 45137 y Fs(\261)32516 45868 y Fx(\255)33204 45594 y Fv(\()p Fu(\263)82 b(;)184 b(\257)58 b Fv(\))310 b Fu(;)6674 b Fv(\(5.31\))6075 48817 y(with)7182 50973 y(e)7674 50516 y Fp(\241)p Fs(\260)48 b Fx(\()p Fs(\261)28 b Fx(\))10732 50973 y Fv(:=)677 b Fu(M)13651 51139 y Fs(s;\270)14912 50973 y Fv(\(1)247 b(+)f(1)p Fu(=\270)p Fv(\))19431 50516 y Fx(2)p Fs(s)20347 50973 y Fv(\(1)h(+)f Fr(j)p Fu(E)310 b Fr(\241)246 b Fu(E)26042 51139 y Fx(0)26539 50973 y Fr(j)p Fv(\))27276 50516 y Fx(5)p Fs(s)p Fx(\()p Fs(d)p Fx(+2\))30465 50973 y Fv(\(1)h(+)e Fu(L)p Fv(\))33984 50516 y Fx(2\()p Fs(d)p Fp(\241)p Fx(1\))18689 53203 y Fr(\243)h Fv(sup)19859 54141 y Fs("<\261)21675 53203 y Fv(sup)21647 54132 y Fs(\256)p Fp(2I)23555 53203 y Ft(E)24477 51974 y Fq(\263)25138 51930 y(\260)25138 52594 y(\260)25138 53258 y(\260)25753 53203 y Fu(\302)26446 53369 y Fs(\256)27261 53203 y Fv(\()p Fu(H)28701 52746 y Fx(\()p Fs(B)29754 52412 y Fl(L)29716 52967 y(\256)30339 52746 y Fx(\))30986 53203 y Fr(\241)g Fu(E)310 b Fr(\241)246 b Fu(i")p Fv(\))35654 52746 y Fp(\241)p Fx(1)37028 53203 y FB(1)37664 53421 y Fs(\261)28 b(B)38801 53200 y Fl(L)38763 53643 y(\256)39441 51930 y Fq(\260)39441 52594 y(\260)39441 53258 y(\260)40056 52204 y Fs(s)40529 51974 y Fq(\264)41682 53203 y Fu(:)1107 b Fv(\(5.32\))6075 55747 y(Note)391 b(that)h Fu(\260)61 b Fv(\()p Fu(\261)42 b Fv(\))392 b(is)e(a)h(decreasing)g(function)h(of) f Fu(\261)432 b Fv(and)391 b(the)g(assumed)g(\257nite)g(v)-31 b(olume)392 b(b)31 b(ound)6075 57075 y(eq.)484 b(\(1.14\))h(sho)-31 b(ws)484 b(that)h(lim)18597 57241 y Fs(\261)28 b Fp(!)p Fx(0)20591 57075 y Fu(\260)61 b Fv(\()p Fu(\261)42 b Fv(\))499 b Fu(>)f Fv(0.)835 b(In)483 b(particular,)513 b(there)483 b(is)h Fu(\261)37896 57241 y Fx(0)38890 57075 y Fu(>)497 b Fv(0)484 b(suc)-31 b(h)483 b(that)6075 58403 y Fu(\260)61 b Fv(\()p Fu(\261)42 b Fv(\))309 b Fu(>)e Fv(0)370 b(for)f Fu(\261)349 b(<)308 b(\261)14699 58569 y Fs(o)15191 58403 y Fv(.)7403 59732 y(Let)369 b(us)g(de\257ne,)15474 61592 y Fu(F)16340 61135 y Fs(\261)16186 61865 y Fx(\255)16873 61592 y Fv(\()p Fu(\272)70 b Fv(\))678 b(:=)1722 b(sup)20873 62529 y Fs(\256;\257)40 b Fp(2I)50 b(\\)p Fx(\255)24814 61592 y Fv(e)25306 61135 y Fs(\272)202 b Fx(dist)27388 61246 y Fk(\255)27974 61135 y Fx(\()p Fs(\256;\257)40 b Fx(\))p Fs(=)p Fx(2)p Fs(L)31604 61592 y Fu(G)32474 61135 y Fs(\261)32474 61865 y Fx(\255)33161 61592 y Fv(\()p Fu(\256)t(;)184 b(\257)58 b Fv(\))310 b Fu(:)6571 b Fv(\(5.33\))6075 64227 y(The)433 b(pro)31 b(of)434 b(will)h(pro)31 b(ceed)433 b(as)g(follo)-31 b(ws.)688 b(First,)449 b(for)434 b Fu(\272)485 b(<)414 b(\260)61 b Fv(\()p Fu(\261)42 b Fv(\))435 b(and)e(b)31 b(ounded)434 b(\255,)450 b(w)-31 b(e)434 b(deriv)-31 b(e)6075 65556 y(an)439 b(\255)g(indep)31 b(enden)-31 b(t)440 b(b)31 b(ound)439 b(on)g Fu(F)21219 65154 y Fs(\261)21065 65861 y Fx(\255)21752 65556 y Fv(\()p Fu(\272)70 b Fv(\).)703 b(Second,)457 b(un)-31 b(b)31 b(ounded)439 b(\255)g(are)g(handled)g (via)h(\257nite)6075 66884 y(v)-31 b(olume)371 b(appro)-31 b(ximations.)496 b(Finally)-92 b(,)371 b(w)-31 b(e)369 b(pass)g(to)h(the)f(limits)i Fu(\261)350 b Fr(!)307 b Fv(0)370 b(and)f Fu(\272)378 b Fr(!)307 b Fu(\260)61 b Fv(.)7403 68213 y(No)-31 b(w,)525 b(consider)493 b(a)g(b)31 b(ounded)492 b(region)i(\255)e(and)h Fu(\256)t(;)184 b(\257)573 b Fr(2)513 b(I)410 b(\\)328 b Fv(\255)493 b(with)h(dist)38987 68379 y Fx(\255)39674 68213 y Fv(\()p Fu(\256)t(;)184 b(\257)58 b Fv(\))516 b Fu(>)d Fv(2)p Fu(L)p Fv(.)6075 69541 y(Observ)-31 b(e)406 b(that)j(either)e(eq.)h (\(5.31\))i(or)d(its)g(conjugate)i(holds)f(for)f(this)h(pair,)417 b(since)407 b Fr(j)p Fu(\256)276 b Fr(\241)271 b Fu(\257)58 b Fr(j)372 b Fu(>)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 6075 3345 a Fx(40)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 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Fx(dist)36853 52922 y Fk(\255)37439 52811 y Fx(\()p Fs(\256;\257)40 b Fx(\))p Fs(=)p Fx(2)p Fs(L)43096 53268 y Fv(\(5.40\))6075 55726 y(for)369 b(an)-31 b(y)370 b Fu(\272)378 b(<)307 b(\260)61 b Fv(.)493 b(Finally)-92 b(,)371 b(taking)g Fu(\272)378 b Fr(!)308 b Fu(\260)61 b Fv(,)370 b(w)-31 b(e)369 b(obtain)i(eq.)f(\(1.15\).)10672 b Fh(\244)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(41)11230 6001 y Fv(6.)555 b Fw(Applica)-74 b(tions:)547 b(resul)-74 b(ts)418 b(f)-25 b(or)418 b(distinct)g(ener)-25 b(gy)418 b(ranges)7403 7994 y Fv(T)-92 b(o)258 b(apply)h(the)e(ab)31 b(o)-31 b(v)g(e)259 b(results)e(on)h(lo)31 b(calization)261 b(w)-31 b(e)259 b(need)e(to)h(v)-31 b(erify)258 b(the)g(su\261ciency)g (criteria)6075 9322 y(of)413 b(Theorem)g(1.2)h(in)f(sp)31 b(eci\257c)412 b(disorder)g(regimes)h(and)g(energy)g(ranges.)623 b(In)412 b(this)h(section)h(w)-31 b(e)6075 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(,)332 b(Theorem)323 b(4.4]\).)480 b(In)322 b(the)h(w)-31 b(ords)323 b(of)g(the)g(authors)6075 65556 y(of)448 b([29)r(],)468 b(it)449 b(is)f(a)h(region)f(\\p)31 b(ossessing)449 b(ev)-31 b(ery)448 b(p)31 b(ossible)448 b(virtue)h(w)-31 b(e)448 b(can)h(imagine!")732 b(F)-92 b(or)447 b(the)6075 66884 y(random)376 b(op)31 b(erators)376 b(considered)g(here,)h(one)f(ma)-31 b(y)377 b(add)f(to)g(that)h(list)g(of)f(virtues)g(exp)31 b(onen)-31 b(tial)6075 68213 y(lo)31 b(calization)278 b(of)c(Green)f(function)j(momen)-31 b(ts)275 b(and)f(dynamical)j(lo)31 b(calization)278 b(in)c(the)g(\(stronger\))6075 69541 y(sense)368 b(of)i(Corollary)h(1.1.)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(43)6075 6001 y Fm(Pr)-57 b(o)g(of.)554 b Fv(F)-92 b(ollo)-31 b(wing)421 b(the)d(argumen)-31 b(t)420 b(for)e(the)h(lattice)h(case)e(from)h (Section)g(4.4)h(of)e(ref.)g([6)q(],)432 b(w)-31 b(e)6075 7330 y(de\257ne)369 b(complemen)-31 b(tary)371 b(\\go)31 b(o)g(d")371 b(and)f(\\bad")g(subsets)e(of)i(the)g(probabilit)-31 b(y)371 b(space)e(\255)h(b)-31 b(y)14021 9783 y(\255)14820 9949 y Fs(G)16241 9783 y Fv(:=)18086 8554 y Fq(n)18824 9783 y Fu(!)224 b Fr(j)308 b(k)p Fu(\302)21598 9949 y Fs(\256)22229 9783 y Fv(\()p Fu(H)23669 9326 y Fx(\()p Fs(B)24722 8992 y Fl(L)24684 9547 y(\256)25306 9326 y Fx(\))25954 9783 y Fr(\241)246 b Fu(E)64 b Fv(\))28372 9326 y Fp(\241)p Fx(1)29561 9783 y FB(1)30197 10001 y Fs(\261)28 b(B)31334 9780 y Fl(L)31296 10222 y(\256)31974 9783 y Fr(k)308 b(\267)f Fu(Ae)35348 9326 y Fp(\241)p Fs(\271L)37241 8554 y Fq(o)43650 9783 y Fv(\(6.9\))6075 12236 y(and)369 b(\255)9026 12402 y Fs(B)10096 12236 y Fv(:=)308 b(\255)12371 11834 y Fs(c)12371 12541 y(G)13116 12236 y Fv(.)492 b(Then)7182 15110 y Ft(E)8104 13882 y Fq(\263)8765 15110 y Fr(k)p Fu(\302)10011 15276 y Fs(\256)10642 15110 y Fv(\()p Fu(H)12082 14653 y Fx(\()p Fs(B)13135 14319 y Fl(L)13097 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b(v)g(e)371 b(used)d(that)j Fu(\261)42 b(B)16613 31608 y Fs(L)16557 32284 y Fx(0)17643 32009 y Fv(con)-31 b(tains)371 b Fr(O)31 b Fv(\()p Fu(L)24080 31608 y Fs(d)p Fp(\241)p Fx(1)25730 32009 y Fv(\))370 b(p)31 b(oin)-31 b(ts)370 b(in)f Fr(I)82 b Fv(.)7403 33338 y(In)369 b(summary)-92 b(,)370 b(w)-31 b(e)370 b(get)f(the)h(b)31 b(ound)7599 35791 y Ft(E)8521 34562 y Fq(\263)9182 35791 y Fr(k)p Fu(\302)10428 35957 y Fs(\256)11059 35791 y Fv(\()p Fu(H)12499 35334 y Fx(\()p Fs(B)13552 34999 y Fl(L)13514 35555 y(\256)14137 35334 y Fx(\))14784 35791 y Fr(\241)246 b Fu(E)64 b Fv(\))17202 35334 y Fp(\241)p Fx(1)18391 35791 y FB(1)19027 36009 y Fs(\261)28 b(B)20164 35787 y Fl(L)20126 36230 y(\256)20804 35791 y Fr(k)21357 35334 y Fs(s)21831 34562 y Fq(\264)22799 35791 y Fr(\267)308 b Fu(A)24798 35334 y Fs(s)25271 35791 y Fu(e)25786 35334 y Fp(\241)p Fs(s\271L)28343 35791 y Fv(+)246 b Fu(C)79 b Fv(\()p Fu(t;)184 b(E)64 b(;)184 b(\270)p Fv(\))34089 35334 y Fs(s=t)35537 35791 y Fu(L)36290 35334 y Fx(\()p Fs(s)p Fx(\()p Fs(d)p Fp(\241)p Fx(1\))p Fp(\241)p Fs(\273)26 b Fx(\()p Fs(t)p Fp(\241)p Fs(s)p Fx(\)\))p Fs(=t)44094 35791 y Fu(;)43096 37451 y Fv(\(6.12\))6075 38779 y(uniformly)322 b(in)f Fu(\256)t Fv(.)476 b(Since)321 b Fu(\273)358 b(>)308 b Fv(2\()p Fu(d)148 b Fr(\241)g Fv(1\))323 b(w)-31 b(e)321 b(can)f(c)-31 b(ho)31 b(ose)321 b Fu(s)f Fv(su\261cien)-31 b(tly)322 b(close)f(to)g(0)g(and)g Fu(t)f Fv(close)6075 40108 y(to)370 b(1)f(to)h(guaran)-31 b(tee)371 b(\()p Fu(\273)51 b Fv(\()p Fu(t)247 b Fr(\241)f Fu(s)p Fv(\))g Fr(\241)g Fu(s)p Fv(\()p Fu(d)g Fr(\241)g Fv(1\)\))p Fu(=t)310 b(>)d Fv(2\()p Fu(d)247 b Fr(\241)f Fv(1\).)494 b(Th)-31 b(us)12077 42622 y(lim)184 b(sup)12586 43551 y Fs(L)p Fp(!1)15835 42622 y Fu(L)16588 42165 y Fx(2\()p Fs(d)p Fp(\241)p Fx(1\))19768 42622 y Fv(sup)19740 43551 y Fs(\256)p Fp(2I)21647 42622 y Ft(E)22569 41393 y Fq(\263)23231 42622 y Fr(k)p Fu(\302)24477 42788 y Fs(\256)25108 42622 y Fv(\()p Fu(H)26548 42165 y Fx(\()p Fs(B)27601 41831 y Fl(L)27563 42387 y(\256)28185 42165 y Fx(\))28833 42622 y Fr(\241)245 b Fu(E)64 b Fv(\))31250 42165 y Fp(\241)p Fx(1)32439 42622 y FB(1)33075 42841 y Fs(\261)28 b(B)34212 42619 y Fl(L)34174 43062 y(\256)34852 42622 y Fr(k)35405 42165 y Fs(s)35879 41393 y Fq(\264)37217 42622 y Fv(=)676 b(0)308 b Fu(;)3174 b Fv(\(6.13\))6075 45516 y(and)369 b(w)-31 b(e)370 b(ma)-31 b(y)370 b(c)-31 b(ho)31 b(ose)370 b Fu(L)f Fv(large)h(enough)g(that)g(eq.)g(\(1.14\))i(holds)e(at)f Fu(E)64 b Fv(.)7403 46844 y(W)-92 b(orking)301 b(in)g(the)f(\257nite)h (v)-31 b(olumes)302 b Fu(B)22693 46442 y Fs(L)22637 47118 y(\256)23355 46844 y Fv(,)314 b(w)-31 b(e)301 b(can)g(use)f(the)h(con) -31 b(tin)g(uit)g(y)303 b(giv)-31 b(en)302 b(b)-31 b(y)300 b(Lemma)i(6.1)6075 48172 y(and)410 b(the)h(con)-31 b(tin)g(uit)g(y)413 b(of)e Fu(b)17261 48338 y Fs(s)17734 48172 y Fv(\()p Fu(\270;)184 b(E)64 b Fv(\))411 b(in)g Fu(E)474 b Fv(to)411 b(conclude)g(the)f(existence)h(of)f(a)h(complex)g(neigh-)6075 49501 y(b)31 b(orho)g(o)g(d)313 b Fr(U)422 b Fv(of)314 b Fu(E)377 b Fv(suc)-31 b(h)313 b(that)h(eq.)g(\(1.14\))i(holds)d(for)g (ev)-31 b(ery)313 b Fu(E)31782 49099 y Fp(0)32227 49501 y Fv(+)134 b Fu(i")308 b Fr(2)f(U)110 b Fv(.)474 b(Then)313 b(Theorem)h(1.2)6075 50829 y(applies)370 b(at)g(all)g(real)g Fu(E)15698 50427 y Fp(0)16316 50829 y Fr(2)307 b(U)110 b Fv(,)369 b(whic)-31 b(h)370 b(concludes)g(the)f(pro)31 b(of)370 b(b)-31 b(y)369 b(Theorem)h(6.1.)5335 b Fh(\244)6075 53378 y Fv(6.2.)555 b FB(Large)323 b(disorder.)555 b Fv(P)-31 b(erhaps)280 b(the)h(easiest)f(lo)31 b(calization)284 b(regime)d(to)h(understand)e(is)g(that)6075 54706 y(induced)270 b(at)g(the)h(b)31 b(ottom)271 b(of)g(the)f(sp)31 b(ectrum)270 b(b)-31 b(y)270 b(large)g(disorder.)460 b(If)269 b(w)-31 b(e)271 b(\257x)f(an)g(energy)g Fu(E)371 b(>)308 b(E)45429 54872 y Fx(0)6075 56034 y Fv(and)384 b(a)g(length)h(scale)f Fu(L)g Fv(w)-31 b(e)384 b(ma)-31 b(y)385 b(adjust)f Fu(\270)f Fv(so)h(that)h(for)f(an)-31 b(y)384 b Fu(x)332 b Fr(2)f Ft(R)35270 55633 y Fs(d)36170 56034 y Fv(it)385 b(is)e(o)-31 b(v)g(erwhelmingly)6075 57363 y(lik)g(ely)394 b(that)g Fu(E)457 b Fv(is)393 b(far)g(b)31 b(elo)-31 b(w)394 b(the)f(b)31 b(ottom)394 b(of)g(the)f(sp)31 b(ectrum)392 b(of)h(the)h(\\lo)31 b(cal)394 b(Hamiltonian,")6075 58803 y(i.e.,)363 b Fu(H)9168 58401 y Fx(\()p Fs(B)10221 58067 y Fl(L)10183 58622 y(x)10805 58401 y Fx(\))11206 58803 y Fv(.)489 b(Heuristically)-92 b(,)364 b(this)c(suggests)g(that)g Fu(E)424 b Fv(lies)360 b(in)f(the)h(lo)31 b(calization)364 b(regime,)e(since)6075 60242 y(the)369 b(resolv)-31 b(en)g(t)370 b(\()p Fu(H)14073 59841 y Fx(\()p Fs(B)15126 59507 y Fl(L)15088 60062 y(x)15711 59841 y Fx(\))16358 60242 y Fr(\241)246 b Fu(E)64 b Fv(\))18776 59841 y Fp(\241)p Fx(1)20334 60242 y Fv(is)369 b(t)-31 b(ypically)372 b(b)31 b(ounded)369 b(with)i(small)f(norm.)7403 61571 y(In)493 b(the)h(discrete)f(setting,)526 b(the)494 b(quan)-31 b(titativ)g(e)497 b(b)31 b(ound)493 b Ft(E)p Fv(\()p Fr(j)p Fu(G)p Fv(\()p Fu(x;)184 b(x)p Fv(\))p Fr(j)36403 61169 y Fs(s)36879 61571 y Fv(\))515 b Fh(.)g Fv(1)p Fu(=\270)40952 61169 y Fs(s)41919 61571 y Fv(pro)-31 b(vides)6075 62899 y(the)378 b(basis)g(for)h(the)f(`single-site')j (criterion)e(of)g([4)q(].)520 b(F)-92 b(or)377 b(the)i(op)31 b(erators)378 b(considered)g(here,)j(b)-31 b(y)6075 64227 y(taking)383 b Fu(\270)e Fv(large)h(enough)g(one)f(ma)-31 b(y)382 b(directly)h(v)-31 b(erify)382 b(the)f(lo)31 b(calization)385 b(condition)f(eq.)d(\(1.14\))6075 65556 y(at)523 b(an)-31 b(y)524 b(\257xed)f(\257nite)g(scale)g Fu(L)g Fv(allo)-31 b(w)g(ed)526 b(in)d(Theorem)h(1.2.)955 b(In)522 b(fact,)563 b(eq.)523 b(\(1.14\))i(ma)-31 b(y)525 b(b)31 b(e)6075 66884 y(satis\257ed)452 b(uniformly)h(for)f(all)h (energies)f(in)g(an)g(arbitrary)g(\257nite)h(in)-31 b(terv)-61 b(al.)742 b(F)-92 b(or)451 b(this)h(result,)6075 68213 y(w)-31 b(e)476 b(assume)f(blo)-31 b(w-up)477 b(regularit)-31 b(y)478 b FB(A3)22815 67811 y Fp(0)23601 68213 y Fv(to)f(insure)e(that) h(the)g(constan)-31 b(ts)477 b(whic)-31 b(h)476 b(app)31 b(ear)476 b(in)6075 69541 y(eq.)369 b(\(1.14\))j(remain)e(b)31 b(ounded)369 b(as)g Fu(\270)g Fv(increases.)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 6075 3345 a Fx(44)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y FB(Theorem)568 b(6.2.)628 b Fm(L)-57 b(et)510 b Fu(H)601 b Fm(satisfy)510 b FB(IAD)p Fm(,)h FB(A1)p Fm(,)540 b FB(A2)511 b Fm(and)e FB(A3)33059 5600 y Fp(0)33371 6001 y Fm(.)851 b(Then)511 b(to)e(every)i Fu(E)42863 6167 y Fx(1)43874 6001 y Fr(2)j Ft(R)6075 7330 y Fm(and)439 b Fv(0)387 b Fu(<)f(s)g(<)g Fv(1)p Fu(=)p Fv(3)441 b Fm(ther)-57 b(e)440 b(exists)f Fu(\270)21230 7496 y Fx(0)22112 7330 y Fv(=)386 b Fu(\270)24005 7496 y Fx(0)24502 7330 y Fv(\()p Fu(E)25749 7496 y Fx(1)26246 7330 y Fu(;)184 b(s)p Fv(\))441 b Fm(such)e(that)g(for)g(every)h Fu(\270)386 b(>)h(\270)40703 7496 y Fx(0)41638 7330 y Fm(ther)-57 b(e)440 b(ar)-57 b(e)6075 8658 y(c)g(onstants)395 b Fu(A)308 b(<)g Fr(1)396 b Fm(and)f Fu(\271)308 b(>)f Fv(0)397 b Fm(such)f(that)f(for)h(any)g(op)-57 b(en)397 b(set)f Fv(\255)308 b Fr(\275)f Ft(R)36065 8256 y Fs(d)13209 11476 y Fv(lim)184 b(sup)14247 12414 y Fs(")p Fp(#)p Fx(0)16782 11476 y Ft(E)17704 9915 y Fq(\265)18519 9871 y(\260)18519 10535 y(\260)18519 11199 y(\260)18519 11863 y(\260)19134 11476 y Fu(\302)19827 11642 y Fs(x)23640 10727 y Fv(1)p 20517 11221 6800 45 v 20517 12255 a Fu(H)21527 11935 y Fx(\(\255\))23152 12255 y Fr(\241)246 b Fu(E)310 b Fr(\241)246 b Fv(i)p Fu(")27450 11476 y(\302)28143 11642 y Fs(y)28674 9871 y Fq(\260)28674 10535 y(\260)28674 11199 y(\260)28674 11863 y(\260)29289 10144 y Fs(s)29763 9915 y Fq(\266)30885 11476 y Fr(\267)307 b Fu(Ae)33398 11019 y Fp(\241)p Fs(\271)152 b Fx(dist)36216 11130 y Fk(\255)36802 11019 y Fx(\()p Fs(x;y)28 b Fx(\))43096 11476 y Fv(\(6.14\))6075 14443 y Fm(for)396 b(al)57 b(l)395 b Fu(E)371 b Fr(2)308 b Fv([)p Fu(E)12785 14609 y Fx(0)13282 14443 y Fu(;)184 b(E)14590 14609 y Fx(1)15088 14443 y Fv(])396 b Fm(and)g Fu(x;)184 b(y)348 b Fr(2)307 b Fv(\255)p Fm(.)6075 16684 y FB(Remark:)491 b Fv(The)370 b(pro)31 b(of)369 b(will)i(sho)-31 b(w)370 b(that)g(w)-31 b(e)370 b(can)f(tak)-31 b(e)371 b Fu(\270)30439 16850 y Fx(0)31243 16684 y Fr(/)307 b Fv(\(1)247 b(+)f Fr(j)p Fu(E)35872 16850 y Fx(1)36615 16684 y Fr(\241)g Fu(E)38539 16850 y Fx(0)39036 16684 y Fr(j)p Fv(\))39773 16282 y Fx(2+3)p Fs(d=)p Fx(2)42747 16684 y Fv(.)6075 19194 y Fm(Pr)-57 b(o)g(of.)554 b Fv(As)596 b(men)-31 b(tioned)598 b(in)f(the)f(in)-31 b(tro)31 b(duction,)656 b(w)-31 b(e)597 b(cannot)h(use)d(Lemma)j(1.1)f (to)g(v)-31 b(erify)6075 20522 y(eq.)482 b(\(1.14\),)513 b(since)482 b(the)g(r.h.s.)h(of)f(eq.)h(\(1.7\))h(do)31 b(es)481 b(not)i(approac)-31 b(h)483 b(0)f(as)g Fu(\270)495 b Fr(!)g(1)p Fv(.)831 b(Instead)6075 21850 y(w)-31 b(e)510 b(\257x)g(an)-31 b(y)510 b Fu(L)542 b(>)g Fv(23)p Fu(r)g Fv(and)510 b(use)f(the)h(metho)31 b(d)511 b(discussed)e(in)h(Section)h (3.1)g(to)f(sho)-31 b(w)511 b(that)6075 23290 y Ft(E)p Fv(\()p Fr(k)p Fu(\302)8489 23456 y Fs(\256)9120 23290 y Fv(\()p Fu(H)10560 22889 y Fx(\()p Fs(B)11613 22555 y Fl(L)11575 23110 y(\256)12197 22889 y Fx(\))12857 23290 y Fr(\241)258 b Fu(E)64 b Fv(\))15287 22889 y Fp(\241)p Fx(1)16476 23290 y FB(1)17112 23509 y Fs(\261)28 b(B)18249 23287 y Fl(L)18211 23730 y(\256)18889 23290 y Fr(k)19442 22889 y Fs(s)19916 23290 y Fv(\))388 b(approac)-31 b(hes)388 b(0)g(as)f Fu(\270)338 b Fr(!)g(1)p Fv(.)548 b(This)387 b(metho)31 b(d)389 b(giv)-31 b(es)388 b(v)-31 b(olume)6075 24619 y(dep)31 b(enden)-31 b(t)307 b(b)31 b(ounds)307 b(on)g(fractional)j(momen)-31 b(ts,)322 b(but)307 b(as)g Fu(L)h Fv(is)f(\257xed)g(in)g(the)h(presen)-31 b(t)307 b(argumen)-31 b(t)6075 25947 y(that)370 b(is)f(of)h(no)f(imp)31 b(ortance.)7403 27275 y(By)369 b(the)h(co)-31 b(v)g(ering)370 b(condition)i(in)d FB(A2)h Fv(it)g(su\261ces)f(to)h(sho)-31 b(w)370 b(that)15682 30093 y Ft(E)16604 28532 y Fq(\265)17419 28488 y(\260)17419 29152 y(\260)17419 29817 y(\260)17419 30481 y(\260)18034 30093 y Fu(U)18790 30259 y Fs(\257)21884 29344 y Fv(1)p 19520 29839 5283 45 v 19520 30912 a Fu(H)20530 30571 y Fx(\()p Fs(B)21583 30350 y Fl(L)21545 30792 y(\256)22167 30571 y Fx(\))22814 30912 y Fr(\241)246 b Fu(E)24935 30093 y(U)25691 30259 y Fs(\263)26199 28488 y Fq(\260)26199 29152 y(\260)26199 29817 y(\260)26199 30481 y(\260)26814 28762 y Fs(s)27287 28532 y Fq(\266)28409 30093 y Fr(!)308 b Fv(0)1107 b(as)369 b Fu(\270)308 b Fr(!)f(1)6778 b Fv(\(6.15\))6075 32851 y(for)252 b(all)i Fu(\257)58 b(;)184 b(\263)391 b Fr(2)307 b(I)82 b Fv(.)453 b(Con)-31 b(v)g(ergence)254 b(needs)e(to)h(b)31 b(e)252 b(sho)-31 b(wn)253 b(uniformly)i(with)e (resp)31 b(ect)252 b(to)h Fu(\256)t(;)184 b(\257)58 b(;)184 b(\263)392 b Fr(2)308 b(I)6075 34179 y Fv(and)404 b Fu(E)431 b Fr(2)366 b Fv([)p Fu(E)11738 34345 y Fx(0)12235 34179 y Fu(;)184 b(E)13543 34345 y Fx(1)14041 34179 y Fv(].)599 b(One)404 b(ma)-31 b(y)406 b(then)e(switc)-31 b(h)406 b(to)f(complex)h(energy)f(using)g(Lemma)g(6.1)h(and)6075 35508 y(complete)371 b(the)e(pro)31 b(of)369 b(with)i(Theorem)f(1.2.) 7403 36948 y(T)-92 b(o)473 b(deriv)-31 b(e)472 b(eq.)g(\(6.15\),)501 b(w)-31 b(e)472 b(write)h Fu(H)569 b Fv(=)479 b Fu(H)26496 36546 y Fx(\()p Fs(B)27549 36212 y Fl(L)27511 36767 y(\256)28132 36546 y Fx(\))28534 36948 y Fv(,)498 b(understanding)472 b(henceforth)h(that)g(all)6075 38414 y(op)31 b(erators)538 b(are)f(restricted)h(to)h Fu(L)20537 38013 y Fx(2)21034 38414 y Fv(\()p Fu(B)22360 38013 y Fs(L)22304 38688 y(\256)23022 38414 y Fv(\).)999 b(Let)27169 38135 y Fq(b)26910 38414 y Fu(H)679 b Fv(=)588 b Fu(H)449 b Fr(\241)358 b Fu(\270\264)33742 38580 y Fs(\257)34339 38414 y Fu(U)35095 38580 y Fs(\257)36050 38414 y Fr(\241)h Fu(\270\264)38466 38580 y Fs(\263)38973 38414 y Fu(U)39729 38580 y Fs(\263)40775 38414 y Fv(and)538 b(de\257ne)6075 39777 y Fu(\264)6625 39943 y Fp(\247)7679 39777 y Fv(=)8980 39341 y Fx(1)p 8980 39522 442 45 v 8980 40159 a(2)9554 39777 y Fv(\()p Fu(\264)10534 39943 y Fs(\257)11352 39777 y Fr(\247)221 b Fu(\264)12984 39943 y Fs(\263)13492 39777 y Fv(\),)360 b(so)d(that)h Fu(\264)18872 39943 y Fs(\257)19468 39777 y Fu(U)20224 39943 y Fs(\257)21042 39777 y Fv(+)221 b Fu(\264)22674 39943 y Fs(\263)23182 39777 y Fu(U)23938 39943 y Fs(\263)24754 39777 y Fv(=)307 b Fu(\264)26472 39943 y Fx(+)27207 39777 y Fv(\()p Fu(U)28393 39943 y Fs(\257)29211 39777 y Fv(+)221 b Fu(U)31049 39943 y Fs(\263)31557 39777 y Fv(\))g(+)g Fu(\264)33840 39943 y Fp(\241)34587 39777 y Fv(\()p Fu(U)35773 39943 y Fs(\257)36591 39777 y Fr(\241)g Fu(U)38429 39943 y Fs(\263)38937 39777 y Fv(\).)489 b(Assumption)6075 41105 y FB(A2)370 b Fv(and)f(the)h (Birman-Sc)-31 b(h)g(winger)371 b(argumen)-31 b(t)371 b(yield)12386 43720 y Fr(k)p Fu(U)13695 43886 y Fs(\257)15771 42972 y Fv(1)p 14426 43466 3244 45 v 14426 44480 a Fu(H)336 b Fr(\241)246 b Fu(E)17802 43720 y(U)18558 43886 y Fs(\263)19067 43720 y Fr(k)677 b(\267)369 b Fu(b)22002 43886 y Fx(+)22737 43720 y Fr(k)p Fv(\()p Fu(U)24476 43886 y Fs(\257)25319 43720 y Fv(+)246 b Fu(U)27182 43886 y Fs(\263)27691 43720 y Fv(\))28121 43263 y Fx(1)p Fs(=)p Fx(2)30991 42972 y Fv(1)p 29645 43466 V 29645 44480 a Fu(H)336 b Fr(\241)246 b Fu(E)33022 43720 y Fv(\()p Fu(U)34208 43886 y Fs(\257)35052 43720 y Fv(+)g Fu(U)36915 43886 y Fs(\263)37423 43720 y Fv(\))37853 43263 y Fx(1)p Fs(=)p Fx(2)39245 43720 y Fr(k)20297 46288 y Fv(=)369 b Fu(b)22002 46454 y Fx(+)22921 45015 y Fq(\260)22921 45679 y(\260)22921 46343 y(\260)23536 46288 y Fv([)24107 46008 y Fq(b)23843 46288 y Fu(K)24862 45814 y Fp(\241)p Fx(1)24783 46613 y Fs(E)43 b(;\264)26195 46724 y Fj(\241)27151 46288 y Fv(+)246 b Fu(\270\264)29454 46454 y Fx(+)30189 46288 y Fv(])30496 45831 y Fp(\241)p Fx(1)31685 45015 y Fq(\260)31685 45679 y(\260)31685 46343 y(\260)32792 46288 y Fu(;)43096 44910 y Fv(\(6.16\))6075 49024 y(where)9446 48744 y Fq(b)9183 49024 y Fu(K)10123 49190 y Fs(E)43 b(;\264)11535 49301 y Fj(\241)12552 49024 y Fv(=)307 b(\()p Fu(U)14906 49190 y Fs(\257)15564 49024 y Fv(+)60 b Fu(U)17241 49190 y Fs(\263)17749 49024 y Fv(\))18179 48622 y Fx(1)p Fs(=)p Fx(2)19570 49024 y Fv(\()20291 48744 y(^)20000 49024 y Fu(H)151 b Fv(+)60 b Fu(\270\264)23188 49190 y Fp(\241)23934 49024 y Fv(\()p Fu(U)25120 49190 y Fs(\257)25777 49024 y Fr(\241)g Fu(U)27454 49190 y Fs(\263)27962 49024 y Fv(\))g Fr(\241)g Fu(E)k Fv(\))30684 48622 y Fp(\241)p Fx(1)31873 49024 y Fv(\()p Fu(U)33059 49190 y Fs(\257)33717 49024 y Fv(+)c Fu(U)35394 49190 y Fs(\263)35901 49024 y Fv(\))36331 48622 y Fx(1)p Fs(=)p Fx(2)37999 49024 y Fv(in)277 b Fu(L)39951 48622 y Fx(2)40448 49024 y Fv(\(\(supp)185 b Fu(U)44531 49190 y Fs(\257)45187 49024 y Fr([)6075 50435 y Fv(supp)f Fu(U)9297 50601 y Fs(\263)9805 50435 y Fv(\))246 b Fr(\\)g Fu(B)12361 50033 y Fs(L)12305 50709 y(\256)13023 50435 y Fv(\).)7403 51763 y(No)-31 b(w)529 b Fu(\264)10633 51929 y Fx(+)11940 51763 y Fr(2)573 b Fv([0)p Fu(;)184 b Fr(1)p Fv(\))530 b(and)e(with)i Fu(\264)22211 51929 y Fp(\241)23485 51763 y Fv(\257xed)e(one)h(can)f(use)g(the)g(argumen)-31 b(t)530 b(describ)31 b(ed)527 b(in)6075 53092 y(Section)370 b(3.1)h(to)e(sho) -31 b(w)370 b(that)11930 54387 y Fq(\257)11930 55051 y(\257)11930 55716 y(\257)12299 54431 y(n)13037 55660 y Fu(\264)13587 55826 y Fx(+)14629 55660 y Fr(2)307 b Fv([0)p Fu(;)184 b Fr(1)p Fv(\))618 b(:)d Fr(k)p Fv([)21227 55380 y Fq(b)20962 55660 y Fu(K)21981 55187 y Fp(\241)p Fx(1)21902 55986 y Fs(E)43 b(;\264)23314 56097 y Fj(\241)24271 55660 y Fv(+)246 b Fu(\270\264)26574 55826 y Fx(+)27308 55660 y Fv(])27615 55203 y Fp(\241)p Fx(1)28804 55660 y Fr(k)308 b Fu(>)f(t)31233 54431 y Fq(o)31971 54387 y(\257)31971 55051 y(\257)31971 55716 y(\257)32648 55660 y Fr(\267)33949 54860 y Fv(\(1)247 b(+)f Fu(\273)36770 55026 y Fs(E)43 b(;\264)38182 55137 y Fj(\241)38892 54860 y Fv(\))p 33949 55406 5374 45 v 36113 56420 a Fu(\270t)39763 55660 y(;)3026 b Fv(\(6.17\))6075 58126 y(where)369 b Fu(\273)9760 58292 y Fs(E)43 b(;\264)11172 58403 y Fj(\241)12251 58126 y Fv(ma)-31 b(y)370 b(b)31 b(e)369 b(expressed)f(as)h(a)g(sp)31 b(ectral)370 b(shift)g(function:)7182 61037 y Fu(\273)7666 61203 y Fs(E)43 b(;\264)9078 61314 y Fj(\241)10464 61037 y Fr(\267)677 b Fv(T)-92 b(r)13327 59809 y Fq(h)13850 61037 y Fu(P)154 b Fv(\()15404 60758 y Fq(b)15145 61037 y Fu(H)336 b Fv(+)246 b Fu(\270\264)18704 61203 y Fp(\241)19450 61037 y Fv(\()p Fu(U)20636 61203 y Fs(x)21440 61037 y Fr(\241)g Fu(U)23303 61203 y Fs(y)23835 61037 y Fv(\))308 b Fu(<)f(E)64 b Fv(\))27052 59809 y Fq(i)17696 63473 y Fr(\241)246 b Fv(T)-92 b(r)20128 62244 y Fq(h)20651 63473 y Fu(P)154 b Fv(\()22205 63193 y Fq(b)21946 63473 y Fu(H)336 b Fv(+)246 b Fu(\270\264)25505 63639 y Fp(\241)26251 63473 y Fv(\()p Fu(U)27437 63639 y Fs(x)28241 63473 y Fr(\241)g Fu(U)30104 63639 y Fs(y)30636 63473 y Fv(\))g(+)g Fu(\270)p Fv(\()p Fu(U)34251 63639 y Fs(x)35055 63473 y Fv(+)f Fu(U)36917 63639 y Fs(y)37449 63473 y Fv(\))308 b Fu(<)g(E)64 b Fv(\))40667 62244 y Fq(i)41682 63473 y Fu(:)1107 b Fv(\(6.18\))6075 65960 y(Since)319 b(b)31 b(oth)319 b(Sc)-31 b(hr\304)-553 b(odinger)319 b(op)31 b(erators)319 b(app)31 b(earing)320 b(in)f(eq.)g(\(6.18\))j(are)c(p)31 b(ositiv)-31 b(e)320 b(p)31 b(erturbations)6075 67288 y(of)369 b Fu(H)8255 67454 y Fx(0)8752 67288 y Fv(,)h(w)-31 b(e)369 b(ma)-31 b(y)371 b(b)31 b(ound)369 b(eac)-31 b(h)370 b(trace)f(b)-31 b(y)369 b(a)h(W)-92 b(eyl-t)-31 b(yp)31 b(e)370 b(b)31 b(ound)369 b(\(see)g(eq.)h(\(3.8\)\))i(to)e (yield)12356 69541 y Fu(\273)12840 69707 y Fs(E)43 b(;\264)14252 69818 y Fj(\241)15639 69541 y Fr(\267)676 b Fv(T)-92 b(r)184 b Fu(P)154 b Fv(\()p Fu(H)20716 69707 y Fx(0)21521 69541 y Fu(<)307 b(E)64 b Fv(\))677 b Fr(\267)f Fv(const.)370 b(\(1)247 b(+)f Fr(j)p Fu(E)311 b Fr(\241)246 b Fu(E)35114 69707 y Fx(0)35610 69541 y Fr(j)p Fv(\))36347 69084 y Fs(d=)p Fx(2)37759 69541 y Fu(L)38512 69084 y Fs(d)39336 69541 y Fu(:)3453 b Fv(\(6.19\))p eop end %%Page: 45 45 TeXDict begin 45 44 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(45)7403 6001 y Fv(Av)-31 b(eraging)294 b(o)-31 b(v)g(er)292 b Fu(\264)15430 6167 y Fx(+)16165 6001 y Fu(;)184 b(\264)17206 6167 y Fp(\241)18245 6001 y Fv(as)292 b(in)g(the)g(pro)31 b(of)292 b(of)g(Prop.)g(4.2)h(w)-31 b(e)293 b(obtain)g(the)f(follo)-31 b(wing)296 b(b)31 b(ound:)6628 8699 y(Prob)9168 7138 y Fq(\265)9983 7093 y(\260)9983 7758 y(\260)9983 8422 y(\260)9983 9086 y(\260)10598 8699 y Fu(U)11354 8865 y Fs(\257)14448 7950 y Fv(1)p 12084 8444 5283 45 v 12084 9517 a Fu(H)13094 9176 y Fx(\()p Fs(B)14147 8955 y Fl(L)14109 9398 y(\256)14730 9176 y Fx(\))15378 9517 y Fr(\241)246 b Fu(E)17498 8699 y(U)18254 8865 y Fs(\263)18763 7093 y Fq(\260)18763 7758 y(\260)18763 8422 y(\260)18763 9086 y(\260)19685 8699 y Fu(>)308 b(t)p Fr(jF)22467 8241 y Fs(c)22357 8972 y(\257)40 b(;\263)23670 7138 y Fq(\266)25068 8699 y Fr(\267)583 b Fv(const.)277 b Fu(D)30570 8241 y Fx(2)31343 8699 y Fu(L)32096 8241 y Fs(d)33021 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Fm(er)-57 b(go)g(dic)p Fv(,)480 b(i.e.)756 b(its)457 b(distribution)i(is)d(in)-31 b(v)-61 b(arian)-31 b(t)6075 17453 y(under)391 b(a)h(su\261cien)-31 b(tly)394 b(large)f(group)f(of)h (translations.)563 b(In)391 b(particular)j(this)e(implies)i(that)f(the) 6075 18781 y(sp)31 b(ectrum)357 b(of)h Fu(H)13069 18947 y Fs(!)14067 18781 y Fv(is)g(a)f(non-random)i(set,)h(c.f.)489 b([51)r(,)358 b(64)q(].)489 b(Hence,)360 b(throughout)f(w)-31 b(e)358 b(mak)-31 b(e)359 b(the)6075 20110 y(additional)372 b(assumption)7418 21736 y FB(A4:)554 b Fr(I)448 b Fv(is)366 b(a)g(lattice)i(con)-31 b(taining)369 b(the)e(origin,)h Fu(U)27865 21902 y Fs(\256)28803 21736 y Fv(=)308 b Fu(U)121 b Fv(\()p Fr(\242)240 b(\241)g Fu(\256)t Fv(\),)367 b(and)f Fu(H)37812 21902 y Fs(!)38819 21736 y Fv(is)g(ergo)31 b(dic)367 b(with)9924 23065 y(resp)31 b(ect)369 b(to)h(shifts)f(in)h Fr(I)82 b Fv(.)6075 24691 y(W)-92 b(e)376 b(will)i(drop)f(the)g(suprem) -31 b(um)377 b(o)-31 b(v)g(er)377 b Fu(\256)k Fv(in)c(eq.)h(\(1.14\))h (and)e(w)-31 b(ork)378 b(with)g Fu(\256)325 b Fv(=)320 b(0,)379 b(since)e(if)g FB(A4)6075 26020 y Fv(holds)369 b(eq.)h(\(1.14\))i(is)d(equiv)-61 b(alen)-31 b(t)371 b(to)7938 28669 y Fu(b)8413 28835 y Fs(s)8886 28669 y Fv(\()p Fu(\270;)184 b(E)64 b Fv(\)\(1)249 b(+)d Fu(L)p Fv(\))15286 28211 y Fx(2\()p Fs(d)p Fp(\241)p Fx(1\))18253 28669 y Fv(lim)184 b(sup)19291 29606 y Fs(")p Fp(#)p Fx(0)21826 28669 y Ft(E)22748 27108 y Fq(\265)23564 27063 y(\260)23564 27728 y(\260)23564 28392 y(\260)23564 29056 y(\260)24179 28669 y Fu(\302)24872 28835 y Fx(0)29175 27920 y Fv(1)p 25685 28414 7533 45 v 25685 29551 a Fu(H)26695 29192 y Fx(\()p Fs(B)27748 28925 y Fl(L)27710 29432 y Fk(0)28332 29192 y Fx(\))28979 29551 y Fr(\241)246 b Fu(E)310 b Fr(\241)246 b Fu(i")33535 28669 y FB(1)34171 28932 y Fs(\261)28 b(B)35308 28665 y Fl(L)35270 29173 y Fk(0)35948 27063 y Fq(\260)35948 27728 y(\260)35948 28392 y(\260)35948 29056 y(\260)36563 27337 y Fs(s)37036 27108 y Fq(\266)38527 28669 y Fu(<)677 b Fv(1)308 b Fu(;)1863 b Fv(\(6.21\))6075 31344 y(where)13574 32804 y Fu(b)14049 32970 y Fs(s)14522 32804 y Fv(\()p Fu(\270;)184 b(E)64 b Fv(\))679 b(:=)d Fu(M)20997 32970 y Fs(s;\270)22258 32804 y Fv(\(1)247 b(+)f(1)p Fu(=\270)p Fv(\))26777 32346 y Fx(2)p Fs(s)27693 32804 y Fv(\(1)h(+)f Fr(j)p Fu(E)310 b Fr(\241)246 b Fu(E)33388 32970 y Fx(0)33885 32804 y Fr(j)p Fv(\))34622 32346 y Fx(5)p Fs(s)p Fx(\()p Fs(d)p Fx(+2\))38118 32804 y Fu(:)4671 b Fv(\(6.22\))7403 34535 y(F)-92 b(or)470 b(simplicit)-31 b(y)-92 b(,)499 b(w)-31 b(e)471 b(shall)h(assume)f(that)h(the)f(lattice)h Fr(I)559 b Fv(=)476 b Ft(Z)34459 34133 y Fs(d)35446 34535 y Fv(and)471 b(w)-31 b(ork)471 b(throughout)6075 35863 y(with)325 b(the)e(`)p Fu(`)11179 35461 y Fp(1)12118 35863 y Fv(-norm')h(on)g Ft(R)17933 35461 y Fs(d)18450 35863 y Fv(:)469 b Fr(j)p Fu(x)p Fr(j)308 b Fv(=)g(max)24010 36029 y Fs(j)24661 35863 y Fr(j)p Fu(x)25601 36029 y Fs(j)26068 35863 y Fr(j)p Fv(.)477 b(Th)-31 b(us)324 b(balls)g(are)g(cub)31 b(es)322 b(with)j(sides)e(parallel)6075 37191 y(to)387 b(the)f(co-ordinate)i(axes)e(and)h(unit)g(balls)g(are)f(fundamen)-31 b(tal)388 b(cells)f(for)f(the)h(lattice)h Fr(I)82 b Fv(.)543 b(The)6075 38520 y(reader)303 b(should)i(ha)-31 b(v)g(e)305 b(no)f(problem)h(extending)g(the)g(results)e(b)31 b(elo)-31 b(w)306 b(to)f(more)f(general)h(lattices,)6075 39848 y(in)554 b(whic)-31 b(h)554 b(case)g(it)g(is)g(natural)h(to)f(w)-31 b(ork)555 b(with)g(the)f(`)p Fu(`)30077 39446 y Fp(1)31015 39848 y Fv(-norm')h(induced)f(on)g Ft(R)41596 39446 y Fs(d)42666 39848 y Fv(b)-31 b(y)554 b(the)6075 41176 y(decomp)31 b(osition)371 b(of)f(a)g(v)-31 b(ector)370 b(in)-31 b(to)370 b(comp)31 b(onen)-31 b(ts)371 b(parallel)g(to)f (lattice)h(generators.)7403 42505 y(The)400 b(densit)-31 b(y)401 b(of)f(states)h(measure)f(for)g(an)g(ergo)31 b(dic)400 b(random)h(Sc)-31 b(hr\304)-553 b(odinger)400 b(op)31 b(erator)401 b(is)f(a)6075 43833 y(Borel)327 b(measure)g Fu(\267)f Fv(on)i(the)f(real)g(line)h(de\257ned)e(through)i (a)f(limiting)j(pro)31 b(cedure)326 b(via)i(its)f(action)6075 45162 y(on)369 b(compactly)j(supp)31 b(orted)369 b(con)-31 b(tin)g(uous)370 b(functions:)16522 46084 y Fq(Z)17813 47591 y Fu(f)119 b Fv(\()p Fu(t)p Fv(\))p Fu(\267)p Fv(\()p Fu(dt)p Fv(\))309 b(:=)725 b(lim)23993 48305 y Fs(L)p Fp(!1)27748 46842 y Fv(1)p 26683 47336 2684 45 v 26683 48350 a(\(2)p Fu(L)p Fv(\))28849 48030 y Fs(d)29684 47591 y Fv(T)-92 b(r)184 b Fu(f)119 b Fv(\()p Fu(H)33110 47134 y Fx(\(\244)34062 47245 y Fl(L)34647 47134 y Fx(\))35048 47591 y Fv(\))7618 b(\(6.23\))6075 50274 y(where)438 b(\244)10114 50440 y Fs(L)11199 50274 y Fv(=)423 b Fu(B)13379 49872 y Fs(L)13323 50548 y Fx(0)14463 50274 y Fv(=)g([)p Fr(\241)p Fu(L;)184 b(L)p Fv(])19219 49872 y Fs(d)19738 50274 y Fv(.)700 b(F)-92 b(or)437 b(the)i(op)31 b(erators)438 b(considered)h(here,)455 b FB(A4)439 b Fv(implies)h(that)6075 51602 y(this)369 b(limit)j(exists,)e(is)f(non-random)h([51)r(],)g(and)f (is)g(equal)h(to)8053 52586 y Fq(Z)9345 54093 y Fu(f)119 b Fv(\()p Fu(t)p Fv(\))p Fu(\267)p Fv(\(d)p Fu(t)p Fv(\))678 b(=)1093 b(lim)15994 54807 y Fs(L)p Fp(!1)19749 53344 y Fv(1)p 18684 53838 V 18684 54852 a(\(2)p Fu(L)p Fv(\))20850 54532 y Fs(d)21685 54093 y Fv(T)-92 b(r)184 b Ft(E)23932 52864 y Fq(\263)24594 54093 y Fu(f)119 b Fv(\()p Fu(H)26695 53636 y Fx(\(\244)27647 53747 y Fl(L)28231 53636 y Fx(\))28633 54093 y Fv(\))29063 52864 y Fq(\264)30401 54093 y Fv(=)32425 53344 y(1)p 32071 53838 1263 45 v 32071 54852 a Fr(jC)65 b(j)33466 54093 y Ft(E)184 b Fv(\()q(T)-92 b(r)184 b FB(1)36780 54259 y Fp(C)37356 54093 y Fu(f)119 b Fv(\()p Fu(H)90 b Fv(\)\))493 b Fu(;)1979 b Fv(\(6.24\))6075 56776 y(where)520 b Fr(C)585 b Fv(is)521 b(an)-31 b(y)521 b(unit)h(cell)f(of)h(the)f(lattice)i Fr(I)82 b Fv(,)558 b(e.g.,)i([0)p Fu(;)184 b Fv(1])32002 56374 y Fs(d)32522 56776 y Fv(.)947 b(In)520 b(addition,)561 b(the)521 b(random)6075 58104 y(op)31 b(erators)410 b(studied)h(here)e(satisfy)i(a)g(W)-92 b(egner)409 b(estimate)j(whic)-31 b(h)411 b(sho)-31 b(ws)411 b(that)g(the)g(densit)-31 b(y)411 b(of)6075 59432 y(states)344 b(measure)g(is)g(absolutely)i(con)-31 b(tin)g(uous)346 b(with)g(a)e(densit)-31 b(y)345 b(whic)-31 b(h)345 b(is)f(in)g Fu(L)38767 58900 y Fs(p)38767 59766 y(l)t(oc)40284 59432 y Fv(for)g(all)h(\257nite)6075 60761 y Fu(p)369 b Fv([17)q(].)7403 62089 y(W)-92 b(e)369 b(also)h(in)-31 b(tro)31 b(duce)369 b(the)h(\257nite)g(v)-31 b(olume)371 b(measures)17740 62796 y Fq(Z)18355 65311 y Fs(A)19261 64303 y Fu(\267)19899 64469 y Fs(L)20561 64303 y Fv(\(d)p Fu(t)p Fv(\))677 b(=)g Ft(E)25573 63074 y Fq(\263)26234 64303 y Fv(T)-92 b(r)184 b Fu(P)28270 64469 y Fs(A)28991 64303 y Fv(\()p Fu(H)30431 63846 y Fx(\(\244)31383 63957 y Fl(L)31968 63846 y Fx(\))32369 64303 y Fv(\))32799 63074 y Fq(\264)33952 64303 y Fu(:)8837 b Fv(\(6.25\))6075 66884 y(Again)272 b Fu(\267)9843 67050 y Fs(L)10777 66884 y Fv(is)f(absolutely)i(con)-31 b(tin)g(uous)273 b(with)g(a)e(densit)-31 b(y)272 b(in)g Fu(L)31317 66352 y Fs(p)31317 67218 y(l)t(oc)32760 66884 y Fv([17)r(].)460 b(Note)272 b(that)h(lim)41664 67050 y Fs(L)42511 66884 y Fu(\267)43149 67050 y Fs(L)44119 66884 y Fv(=)307 b Fu(\267)6075 68213 y Fv(in)502 b(the)g(sense)e(of)i (w)-31 b(eak)503 b(con)-31 b(v)g(ergence)503 b(in)f(the)g(dual)g(space) f(of)i(the)e(compactly)j(supp)31 b(orted)6075 69541 y(con)-31 b(tin)g(uous)371 b(functions.)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 6075 3345 a Fx(46)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)7403 6001 y Fv(If)338 b Fu(E)403 b Fv(is)338 b(a)h(band)g(edge)g(of)g(the)g (almost)h(sure)e(sp)31 b(ectrum)338 b(of)h Fu(H)90 b Fv(,)345 b(e.g.)483 b(if)339 b Fu(E)372 b Fv(=)307 b(inf)272 b Fu(\276)40 b Fv(\()p Fu(H)90 b Fv(\),)345 b(then)6075 7330 y(it)329 b(is)g(generally)h(exp)31 b(ected)329 b(that)g Fu(\267)p Fv([)p Fu(E)64 b(;)184 b(E)230 b Fv(+)165 b(\242)p Fu(E)64 b Fv(])330 b(v)-61 b(anishes)328 b(to)h(high)h(v)-31 b(ery)328 b(order)g(in)h(\242)p Fu(E)64 b Fv(.)480 b(This)6075 8658 y(phenomenon)438 b(w)-31 b(as)439 b(\257rst)e(observ)-31 b(ed)437 b(b)-31 b(y)438 b(I.)f(M.)h(Lifshitz)g([46)r(])g(and)f(the)h (regions)g(where)f(suc)-31 b(h)6075 9986 y(estimates)370 b(hold)g(are)f(called)i(`Lifshitz)f(tails'.)7403 11315 y(In)406 b(the)g(Lifshitz)i(tail)g(regime)f(one)f(exp)31 b(ects)406 b(it)h(is)g(v)-31 b(ery)406 b(rare)g(to)h(\257nd)f(an)g (eigen)-31 b(v)-61 b(alue)408 b(of)f(a)6075 12643 y(\257nite)266 b(v)-31 b(olume)267 b(op)31 b(erator)266 b(near)f(the)h(band)f(edge)h Fu(E)64 b Fv(.)458 b(With)266 b(fa)-31 b(v)g(orable)267 b(estimates)g(this)f(suggests)6075 13971 y(lo)31 b(calization)455 b(should)c(hold)g(near)g Fu(E)515 b Fv(since)451 b(the)g(Com)-31 b(b)31 b(es-Thomas)453 b(estimate)g([18)q(,)f(11)q(])f(ma)-31 b(y)6075 15300 y(t)g(ypically)372 b(b)31 b(e)369 b(used)f(to)i(obtain)h (resolv)-31 b(en)g(t)370 b(deca)-31 b(y)370 b(across)f(\257nite)h(v)-31 b(olumes.)7403 16628 y(One)402 b(result)g(in)h(this)g(v)-31 b(ein)403 b(sho)-31 b(ws)402 b(that)i(smallness)f(b)31 b(ounds)401 b(for)i(the)f(\257nite)h(v)-31 b(olume)404 b(mea-)6075 17957 y(sures)307 b Fu(\267)9435 18123 y Fs(L)10405 17957 y Fv(ma)-31 b(y)310 b(b)31 b(e)308 b(used)f(to)i(v)-31 b(erify)309 b(the)g(lo)31 b(calization)312 b(criterion)d(in)g(Theorem)g (1.2)g(in)g(a)g(similar)6075 19285 y(w)-31 b(a)g(y)322 b(as)e(done)h(for)f(initial)j(length)f(scale)f(estimates)h(in)e(pro)31 b(ofs)321 b(of)g(band)f(edge)h(lo)31 b(calization)324 b(via)6075 20613 y(m)-31 b(ultiscale)371 b(analysis.)6075 22627 y FB(Theorem)619 b(6.3.)655 b Fm(L)-57 b(et)552 b Fu(H)642 b Fm(satisfy)551 b FB(IAD)p Fm(,)i FB(A1-4)p Fm(.)977 b(Supp)-57 b(ose)551 b(that)f(for)i(some)f Fu(\257)647 b Fr(2)589 b Fv(\(0)p Fu(;)184 b Fv(2\))p Fm(,)6075 23955 y Fu(\273)358 b(>)308 b Fv(2\()p Fu(d)247 b Fr(\241)f Fv(1\))p Fm(,)397 b Fu(C)13509 24121 y Fx(1)14314 23955 y Fu(>)307 b Fv(0)p Fm(,)397 b Fu(C)17562 24121 y Fx(2)18366 23955 y Fu(>)308 b Fv(0)p Fm(,)397 b(and)f Fu(E)371 b Fr(2)308 b Ft(R)396 b Fm(it)g(holds)f(for)h(su\261ciently)h(lar)-57 b(ge)397 b Fu(L)f Fm(that)15037 25840 y Fu(\267)15675 26006 y Fs(L)16522 24943 y Fq(\241)17029 25840 y Fv([)p Fu(E)311 b Fr(\241)246 b Fu(C)20362 26006 y Fx(1)20858 25840 y Fu(L)21611 25383 y Fp(\241)p Fs(\257)22901 25840 y Fu(;)184 b(E)311 b Fv(+)245 b Fu(C)26417 26006 y Fx(1)26914 25840 y Fu(L)27667 25383 y Fp(\241)p Fs(\257)28957 25840 y Fv(])29264 24943 y Fq(\242)30475 25840 y Fu(<)703 b(C)32830 26006 y Fx(2)33327 25840 y Fu(L)34080 25383 y Fp(\241)p Fs(\273)26 b Fp(\241)p Fs(d)36656 25840 y Fu(:)6133 b Fv(\(6.26\))6075 27670 y Fm(Then)396 b(the)g(c)-57 b(onclusion)397 b(of)f(The)-57 b(or)g(em)397 b(6.1)f(holds.)6075 29684 y(Pr)-57 b(o)g(of.)554 b Fv(The)471 b(pro)31 b(of)470 b(is)g(v)-31 b(ery)470 b(similar)i(to)f(the)g(pro)31 b(of)470 b(of)h(Theorem)g(6.1.)797 b(This)470 b(time,)497 b(de\257ne)6075 31012 y(\255)6874 31178 y Fs(G)7926 31012 y Fv(:=)308 b Fr(f)p Fu(!)348 b Fv(:)492 b(dist)q(\()p Fu(\276)40 b Fv(\()p 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Fs(\277)45495 60323 y Fv(\))6075 61658 y(for)369 b Fu(h)307 b Fr(!)h Fv(0,)370 b(see)e([39)r(,)h(Prop.)h(4.1].)494 b(Th)-31 b(us)370 b(Theorem)g(6.3)g(is)f(applicable)j(if)d Fu(\277)433 b(>)307 b Fv(\(3)p Fu(d)247 b Fr(\241)f Fv(2\))p Fu(=)p Fv(2.)7403 62986 y(T)-92 b(o)311 b(complemen)-31 b(t)313 b(the)e(ab)31 b(o)-31 b(v)g(e)312 b(discussion,)323 b(it)311 b(is)f(in)-31 b(teresting)313 b(to)e(note)g(that)h(one)f(ma) -31 b(y)312 b(pro)-31 b(v)g(e)6075 64314 y(lo)31 b(calization)373 b(directly)d(from)g(an)f(estimate)i(on)f(the)f(in\257nite)h(v)-31 b(olume)371 b(densit)-31 b(y)370 b(of)g(states.)6075 66328 y FB(Theorem)436 b(6.4.)560 b Fm(L)-57 b(et)405 b Fu(H)495 b Fm(satisfy)405 b FB(IAD)p Fm(,)g FB(A1-4)p Fm(.)538 b(F)-85 b(or)404 b(e)-57 b(ach)405 b Fu(\273)374 b(>)323 b Fv(3)p Fu(d)253 b Fr(\241)f Fv(1)405 b Fm(and)g Fu(E)387 b Fr(2)323 b Ft(R)p Fm(,)408 b(ther)-57 b(e)6075 67656 y(exists)395 b Fu(C)387 b Fv(=)307 b Fu(C)79 b Fv(\()p Fu(E)64 b(;)184 b(\273)51 b(;)184 b(\270)p 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y(`)36482 44635 y Fv(,)h(is)f(de\257ned)g(to)h(b)31 b(e)17299 47224 y Fu(\267)17937 46767 y Fs(P)17937 47498 y(`)18675 47224 y Fv(\()p Fu(A)p Fv(\))678 b(:=)23232 46475 y(1)p 23020 46969 978 45 v 23020 47983 a Fu(`)23481 47664 y Fs(d)24131 47224 y Ft(E)25053 46327 y Fq(\241)25560 47224 y Fv(T)-92 b(r)185 b FB(1)27522 47390 y Fx(\244)28128 47538 y Fl(`=)p Fk(2)29353 47224 y Fu(P)30064 47390 y Fs(A)30785 47224 y Fv(\()p Fu(H)32225 46767 y Fs(P)32135 47498 y(`)32964 47224 y Fv(\))33394 46327 y Fq(\242)34393 47224 y Fu(:)8396 b Fv(\(6.33\))6075 49728 y(By)369 b(translation)j(in) -31 b(v)-61 b(ariance)370 b(of)g(the)f(distribution)i(of)f Fu(H)30122 49327 y Fs(P)30032 50042 y(`)30860 49728 y Fv(,)g(this)f(is)g(also)h(giv)-31 b(en)371 b(b)-31 b(y)18703 51842 y Fu(\267)19341 51385 y Fs(P)19341 52116 y(`)20078 51842 y Fv(\()p Fu(A)p Fv(\))678 b(=)f Ft(E)24906 50946 y Fq(\241)25413 51842 y Fv(T)-92 b(r)185 b FB(1)27375 52008 y Fp(C)27950 51842 y Fu(P)28661 52008 y Fs(A)29381 51842 y Fv(\()p 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a Fx(48)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(Using)370 b(the)f(Hel\256er-Sj\304)-553 b(ostrand)370 b(form)-31 b(ula)371 b([32)q(,)f(20)q(],)g(write)7182 8250 y FB(1)7818 8416 y Fp(C)8393 8250 y Fv(\()p Fu(f)119 b Fv(\()p Fu(H)10924 7793 y Fs(P)10834 8524 y(`)11663 8250 y Fv(\))247 b Fr(\241)e Fu(f)119 b Fv(\()p Fu(H)90 b Fv(\)\))p FB(1)17043 8416 y Fp(C)14027 10752 y Fv(=)15564 9245 y Fq(Z)16179 11761 y Fp(J)16785 11878 y Fl(\261)17221 11761 y Fp(\243)p Fx([)p Fp(\241)p Fx(1)p Fs(;)p Fx(1])20491 9523 y Fq(\263)21152 10752 y Fu(@)21794 10918 y Fx(\271)-495 b Fs(z)22457 10460 y Fq(e)22250 10752 y Fu(f)22792 10918 y Fs(n)23394 10752 y Fv(\()p Fu(z)49 b Fv(\))24818 9523 y Fq(\264)25849 10752 y FB(1)26485 10918 y Fp(C)27244 9191 y Fq(\265)29747 10003 y Fv(1)p 28192 10497 3665 45 v 28192 11541 a Fu(z)294 b Fr(\241)246 b Fu(H)31118 11160 y Fs(P)31028 11875 y(`)32235 10752 y Fr(\241)34662 10003 y 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b(supp)31 b(orted)6075 33238 y(inside)391 b(\244)10008 33438 y Fs(`=)p Fx(2)p Fp(\241)p Fs(r)12377 33549 y Fl(o)12876 33238 y Fv(.)558 b(W)-92 b(e)390 b(c)-31 b(ho)31 b(ose)392 b(\243)f(with)h Fg(r)p Fv(\243)f(supp)31 b(orted)391 b(in)g(a)g(strip)h(of)f(width)i(one)e(near)g(the)6075 34567 y(b)31 b(oundary)369 b(of)h(\244)13089 34767 y Fs(`=)p Fx(2)p Fp(\241)p Fs(r)15458 34878 y Fl(o)15956 34567 y Fv(.)7403 35895 y(Using)330 b(a)g(\\comm)-31 b(utator)333 b(argumen)-31 b(t")332 b(similar)f(to)g(that)f(in)g(the)g (pro)31 b(of)330 b(of)h(Lemmas)f(4.3)h(and)6075 37223 y(4.4,)371 b(w)-31 b(e)369 b(ma)-31 b(y)371 b(sho)-31 b(w)370 b(that)g(for)f Fu(z)356 b Fr(2)308 b(J)21696 37389 y Fs(\261)22428 37223 y Fr(\243)246 b(f)p Fv([)p Fr(\241)p Fv(1)p Fu(;)184 b Fv(0\))249 b Fr([)d Fv(\(0)p Fu(;)184 b Fv(1])p Fr(g)s Fv(,)17801 39616 y([)p Fu(H)28 b(;)184 b Fv(\243])22221 38867 y(1)p 21035 39361 V 21035 40375 a Fu(z)294 b Fr(\241)246 b Fu(H)24094 39616 y FB(1)24730 39782 y Fp(C)25982 39616 y Fv(=)676 b Fu(T)154 b Fv(\252)30500 38867 y(1)p 29314 39361 V 29314 40375 a Fu(z)294 b Fr(\241)246 b Fu(H)32373 39616 y FB(1)33009 39782 y Fp(C)33892 39616 y Fu(;)8897 b Fv(\(6.40\))6075 41890 y(where)369 b Fu(T)522 b Fv(is)369 b(a)h(trace)f(class)h(op)31 b(erator)369 b(with)20977 43761 y(T)-92 b(r)184 b Fr(j)p Fu(T)154 b Fr(j)677 b(\267)g Fu(C)79 b Fv(\()p Fu(E)64 b Fv(\))370 b Fu(`)29374 43304 y Fs(d)p Fp(\241)p Fx(1)43096 43761 y Fv(\(6.41\))6075 45632 y(and)252 b(\252)h(is)f(a)h(function)h(whic) -31 b(h)253 b(is)g(iden)-31 b(tically)256 b(one)c(on)h(the)g(supp)31 b(ort)252 b(of)h Fg(r)p Fv(\243)f(and)h(also)g(supp)31 b(orted)6075 46960 y(in)369 b(a)h(strip)f(of)g(width)i(order)d(one)i (near)f(the)g(b)31 b(oundary)370 b(of)f(\244)31587 47160 y Fs(`=)p Fx(2)p Fp(\241)p Fs(r)33956 47271 y Fl(o)34455 46960 y Fv(.)492 b(Therefore)8182 48007 y Fq(\257)8182 48672 y(\257)8182 49336 y(\257)8182 50000 y(\257)8551 49613 y Fv(T)-92 b(r)184 b(1)10429 49779 y Fp(C)12692 48864 y Fv(1)p 11137 49358 3665 45 v 11137 50402 a Fu(z)294 b Fr(\241)246 b Fu(H)14063 50020 y Fs(P)13973 50735 y(`)15119 49613 y Fv([)p Fu(H)28 b(;)184 b Fv(\243])19539 48864 y(1)p 18353 49358 2927 45 v 18353 50372 a Fu(z)294 b Fr(\241)246 b Fu(H)21412 49613 y FB(1)22048 49779 y Fp(C)22623 48007 y Fq(\257)22623 48672 y(\257)22623 49336 y(\257)22623 50000 y(\257)23300 49613 y Fr(\267)369 b Fu(C)79 b Fv(\()p Fu(E)64 b Fv(\))p Fu(`)27602 49155 y Fs(d)p Fp(\241)p Fx(1)29806 48007 y Fq(\260)29806 48672 y(\260)29806 49336 y(\260)29806 50000 y(\260)30421 49613 y Fv(1)30974 49779 y Fp(C)33237 48864 y Fv(1)p 31682 49358 3665 45 v 31682 50402 a Fu(z)294 b Fr(\241)246 b Fu(H)34608 50020 y Fs(P)34518 50735 y(`)35479 49613 y Fv(\252)36340 48007 y Fq(\260)36340 48672 y(\260)36340 49336 y(\260)36340 50000 y(\260)37509 48007 y(\260)37509 48672 y(\260)37509 49336 y(\260)37509 50000 y(\260)38124 49613 y Fv(\252)40304 48864 y(1)p 39118 49358 2927 45 v 39118 50372 a Fu(z)294 b Fr(\241)246 b Fu(H)42177 49613 y FB(1)42813 49779 y Fp(C)43388 48007 y Fq(\260)43388 48672 y(\260)43388 49336 y(\260)43388 50000 y(\260)23300 52641 y Fr(\267)369 b Fu(C)79 b Fv(\()p Fu(E)64 b Fv(\))370 b Fu(`)27972 52183 y Fs(d)p Fp(\241)p Fx(1)30386 51892 y Fv(1)p 30123 52386 1080 45 v 30123 53400 a Fu(y)30706 53080 y Fx(2)31335 52641 y Fv(e)31827 52183 y Fp(\241)p Fs(\271)p Fx(\()p Fs(E)43 b Fx(\))p Fs(y)28 b(`)35651 52641 y Fu(;)43096 54556 y Fv(\(6.42\))6075 55884 y(where)396 b(w)-31 b(e)398 b(ha)-31 b(v)g(e)398 b(used)e(the)h(Com)-31 b(b)31 b(es-Thomas)399 b(b)31 b(ound)397 b([18)q(,)g(11)q(])h(to)f(estimate)i(the)e(resolv)-31 b(en)g(t)6075 57213 y(norms.)7403 58541 y(Putting)371 b(this)f(in)-31 b(to)371 b(eq.)e(\(6.37\))j(w)-31 b(e)370 b(\257nd)f(that)7119 60037 y Fq(\257)7119 60701 y(\257)7488 60978 y Fv(T)-92 b(r)184 b FB(1)9449 61144 y Fp(C)10025 60978 y Fv(\()p Fu(f)119 b Fv(\()p Fu(H)12556 60521 y Fs(P)12466 61252 y(`)13295 60978 y Fv(\))246 b Fr(\241)g Fu(f)119 b Fv(\()p Fu(H)90 b Fv(\)\))p FB(1)18675 61144 y Fp(C)19252 60037 y Fq(\257)19252 60701 y(\257)20298 60978 y Fr(\267)368 b Fu(C)79 b Fv(\()p Fu(E)64 b Fv(\))24322 59471 y Fq(Z)24939 61987 y Fp(J)25545 62104 y Fl(\261)25981 61987 y Fp(\243)p Fx([)p Fp(\241)p Fx(1)p Fs(;)p Fx(1])29920 60229 y Fu(y)30503 59827 y Fs(n)p 29384 60723 2258 45 v 29384 61737 a Fu(\261)29918 61418 y Fs(n)p Fx(+1)31774 60978 y Fu(`)32235 60521 y Fs(d)p Fp(\241)p Fx(1)34649 60229 y Fv(1)p 34386 60723 1080 45 v 34386 61737 a Fu(y)34969 61418 y Fx(2)35598 60978 y Fv(e)36090 60521 y Fp(\241)p Fs(\271)p Fx(\()p Fs(E)43 b Fx(\))p Fs(y)28 b(`)39791 60978 y Fv(d)p Fu(x)p Fv(d)p Fu(y)20298 63618 y Fr(\267)368 b Fu(C)79 b Fv(\()p Fu(E)64 b(;)184 b(n)p Fv(\))g Fu(`)25938 63161 y Fs(d)p Fp(\241)p Fs(n)27697 63618 y Fu(\261)28231 63161 y Fp(\241)p Fs(n)29833 63618 y Fu(;)43096 62048 y Fv(\(6.43\))6075 65556 y(whic)-31 b(h)370 b(completes)g(the)g(pro)31 b(of)369 b(of)h(eq.)g(\(6.35\).)7403 66884 y(W)-92 b(e)440 b(no)-31 b(w)442 b(recall)g(some)g(standard)f(facts)h(from)f(the)g(Blo) 31 b(c)-31 b(h/Flo)31 b(quet)444 b(theory)d(of)h(p)31 b(erio)g(dic)6075 68213 y(op)g(erators,)415 b(c.f.)601 b([53)q(].)g(F)-92 b(or)404 b(eac)-31 b(h)406 b FB(k)367 b Fr(2)g Fu(B)56 b(Z)483 b Fv(with)407 b Fu(B)56 b(Z)445 b Fv(=)368 b([0)p Fu(;)184 b Fv(2)p Fu(\274)40 b Fv(])33597 67811 y Fs(d)34520 68213 y Fv(de\257ne)405 b(the)g(restriction)i(of) 6075 69541 y Fu(H)7085 69139 y Fs(P)6995 69854 y(`)8286 69541 y Fv(to)464 b(\244)10502 69741 y Fs(`=)p Fx(2)12285 69541 y Fv(with)g(quasi-p)31 b(erio)g(dic)464 b(b)31 b(oundary)464 b(conditions)h(at)f(quasi-momen)-31 b(tum)466 b FB(k)p Fu(=`)d Fv(to)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(49)6075 6001 y Fv(b)31 b(e)391 b(the)i(self-adjoin)-31 b(t)395 b(op)31 b(erator)392 b Fu(H)20659 6167 y Fs(`)p Fx(;)p Ff(k)22257 6001 y Fv(on)g Fu(L)24570 5600 y 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Fs(P)17251 14377 y(`)17989 14104 y Fv(\()p Fr(J)19169 14304 y Fs(\261)28 b(=)p Fx(2)20551 14104 y Fv(\))677 b(=)24352 13355 y(1)p 23328 13849 2602 45 v 23328 14863 a(\(2)p Fu(\274)40 b Fv(\))25412 14543 y Fs(d)26246 12597 y Fq(Z)26861 15112 y Fs(B)e(Z)28465 14104 y Fu(\267)29103 14270 y Fs(`)p Fx(;)p Ff(k)30309 14104 y Fv(\()p Fr(J)31489 14304 y Fs(\261)28 b(=)p Fx(2)32870 14104 y Fv(\))184 b(d)p FB(k)308 b Fu(;)7710 b Fv(\(6.44\))6075 16608 y(where)369 b(the)g(quasi-p)31 b(erio)g(dic)370 b(densities)g Fu(\267)23434 16774 y Fs(`)p Fx(;)p Ff(k)25008 16608 y Fv(are)f(de\257ned)g(b)-31 b(y)18230 19087 y Fu(\267)18868 19253 y Fs(`)p Fx(;)p Ff(k)20074 19087 y Fv(\()p Fu(A)p Fv(\))677 b(:=)24631 18338 y(1)p 24419 18832 978 45 v 24419 19846 a Fu(`)24880 19526 y Fs(d)25529 19087 y Ft(E)184 b Fv(\()q(T)-92 b(r)184 b Fu(P)28918 19253 y Fs(A)29639 19087 y Fv(\()p Fu(H)30989 19253 y Fs(l)t Fx(;)p Ff(k)32109 19087 y Fv(\)\))493 b Fu(:)9327 b Fv(\(6.45\))7403 21378 y(No)-31 b(w)575 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Fp(0)16325 43374 y Fu(\261)42 b Fv(\244)17628 43540 y Fs(`)18054 41769 y Fq(\260)18054 42433 y(\260)18054 43097 y(\260)18054 43762 y(\260)18669 42043 y Fs(s)19142 41813 y Fq(\266)20633 43374 y Fr(\267)677 b Fu(A)23001 42917 y Fs(s)23659 43374 y Fu(\261)24193 42917 y Fp(\241)p Fs(s)25542 43374 y Fv(e)26034 42917 y Fp(\241)p Fs(s\271\261)28112 42583 y Fk(1)p Fl(=)p Fk(2)29328 42917 y Fs(`)13502 46196 y Fv(+)615 b Fu(C)79 b Fv(\()p Fu(t;)184 b(\270;)g(E)64 b Fv(\))19617 45739 y Fs(s=t)20881 46196 y Fu(`)21342 45739 y Fx(\(2+)p Fs(s)p Fx(\)\()p Fs(d)p Fp(\241)p Fx(1\))26098 45299 y Fq(\241)26606 46196 y Fu(`)27067 45739 y Fs(d)27583 46196 y Fu(\267)p Fv(\()p Fr(J)29401 46362 y Fs(\261)29887 46196 y Fv(\))616 b(+)f Fu(C)33200 46362 y Fs(n)33802 46196 y Fu(`)34263 45739 y Fx(2)p Fs(d)p Fp(\241)p Fs(n)36460 46196 y Fu(\261)36994 45739 y Fp(\241)p Fs(n)38288 45299 y Fq(\242)38796 45474 y Fx(1)p Fp(\241)p Fs(s=t)41682 46196 y Fu(;)1107 b Fv(\(6.48\))6075 48118 y(for)369 b(an)-31 b(y)370 b Fu(s)307 b(<)h(t)f(<)g Fv(1.)493 b(Th)-31 b(us)12321 50698 y Fu(`)12782 50241 y Fx(2\()p Fs(d)p Fp(\241)p Fx(1\))15565 50698 y Ft(E)16303 50241 y Fp(0)16798 49137 y Fq(\265)17613 49093 y(\260)17613 49757 y(\260)17613 50421 y(\260)17613 51085 y(\260)18228 50698 y Fu(\302)18921 50864 y Fx(0)21786 49949 y Fv(1)p 19550 50443 V 19550 51457 a Fu(H)20470 51623 y Fs(`)p Fx(;)p Fs(!)21678 51402 y Fj(0)22277 51457 y Fr(\241)246 b Fu(E)24265 51137 y Fp(0)24709 50698 y Fu(\261)42 b Fv(\244)26012 50864 y Fs(`)26437 49093 y Fq(\260)26437 49757 y(\260)26437 50421 y(\260)26437 51085 y(\260)27052 49366 y Fs(s)27525 49137 y Fq(\266)29016 50698 y Fr(\267)677 b Fu(C)448 b Fv(max)q(\()p Fu(A)35113 50864 y Fx(1)35611 50698 y Fu(;)184 b(A)36932 50864 y Fx(2)37430 50698 y Fu(;)g(A)38751 50864 y Fx(3)39248 50698 y Fv(\))3418 b(\(6.49\))6075 53242 y(with)15951 54683 y Fu(A)16781 54849 y Fx(1)17954 54683 y Fv(=)369 b Fu(\261)19718 54226 y Fp(\241)p Fs(s)20883 54683 y Fu(`)21344 54226 y Fx(2\()p Fs(d)p Fp(\241)p 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61169 y Fx(\))9249 61571 y Fv(is)426 b(replaced)g(b)-31 b(y)426 b(the)g(random)h(quasi-p)31 b(erio)g(dic)427 b(op)31 b(erator)426 b Fu(H)35131 61737 y Fs(`)35557 61571 y Fv(,)441 b(w)-31 b(e)426 b(see)f(that)i(there)f (is)6075 62899 y(\257xed)306 b(quan)-31 b(tit)g(y)308 b Fu(B)363 b Fv(=)307 b Fu(B)56 b Fv(\()p Fu(s;)184 b(\270;)g(E)64 b Fv(\))308 b(suc)-31 b(h)306 b(that)h(if)f(max)28295 63065 y Fs(j)28761 62899 y Fv(\()p Fu(A)30021 63065 y Fs(j)30488 62899 y Fv(\))i Fu(<)g(B)361 b Fv(then)306 b(the)g(conclusion)h(of)g(the)6075 64227 y(presen)-31 b(t)307 b(theorem)h(holds,)321 b(i.e.,)g(the)308 b(in)-31 b(terv)-61 b(al)308 b Fr(J)25841 64427 y Fs(\261)28 b(=)p Fx(4)27530 64227 y Fv(is)307 b(con)-31 b(tained)309 b(in)f(the)g(lo)31 b(calization)311 b(regime.)6075 65556 y(Note)441 b(that)g(the)g (in\257nite)g(v)-31 b(olume)442 b(op)31 b(erator)441 b(do)31 b(es)440 b(not)g(dep)31 b(end)440 b(on)g(the)h(quasi-momen)-31 b(tum)6075 66884 y FB(k)p Fv(.)806 b(Nonetheless,)501 b(the)474 b(\257nite)g(v)-31 b(olume)476 b(quasi-p)31 b(erio)g(dic)474 b(op)31 b(erators)474 b(ma)-31 b(y)475 b(b)31 b(e)473 b(used)h(b)31 b(ecause)6075 68213 y(lo)g(cally)-92 b(,)377 b(i.e.,)g(for)d(functions)h(supp)31 b(orted)373 b(in)h(the)h(in)-31 b(terior)375 b(of)f(the)g(cub)31 b(e)373 b(\244)37271 68379 y Fs(`)37697 68213 y Fv(,)i(they)g(agree)f (with)6075 69541 y(the)369 b(in\257nite)i(v)-31 b(olume)370 b(op)31 b(erator.)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 6075 3345 a Fx(50)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)7403 6001 y Fv(T)-92 b(o)407 b(obtain)i(a)e(concrete)g(result,)416 b(w)-31 b(e)407 b(let)h Fu(`)370 b Fr(\274)g Fu(\261)27257 5600 y Fp(\241)p Fs(r)28845 6001 y Fv(for)407 b(some)g Fu(r)401 b(>)370 b Fv(1.)606 b(Then)406 b Fu(A)40742 6167 y Fx(1)41609 6001 y Fu(<)370 b(B)462 b Fv(and)6075 7330 y Fu(A)6905 7496 y Fx(3)7811 7330 y Fu(<)409 b(B)486 b Fv(for)430 b(all)i(su\261cien)-31 b(tly)432 b(small)f Fu(\261)472 b Fv(pro)-31 b(vided)431 b(w)-31 b(e)431 b(c)-31 b(ho)31 b(ose)431 b Fu(n)f Fv(su\261cien)-31 b(tly)433 b(large.)676 b(Th)-31 b(us,)6075 8658 y(there)415 b(is)g Fu(\261)10604 8824 y Fx(0)11485 8658 y Fu(>)385 b Fv(0)416 b(suc)-31 b(h)415 b(that)h(for)g Fu(\261)426 b(<)385 b(\261)23084 8824 y Fx(0)23996 8658 y Fv(the)415 b(condition)j(max)32913 8824 y Fs(j)33380 8658 y Fv(\()p Fu(A)34640 8824 y Fs(j)35107 8658 y Fv(\))385 b Fu(<)f(B)471 b Fv(is)415 b(equiv)-61 b(alen)-31 b(t)417 b(to)6075 9986 y(requiring)369 b(that)17802 11351 y Fu(A)18632 11517 y Fx(2)19436 11351 y Fv(=)307 b Fu(\261)21138 10894 y Fp(\241)p Fs(\273)26 b Fx(\(1)p Fp(\241)p Fs(s=t)p Fx(\))25345 11351 y Fu(\267)p Fv(\()p Fr(J)27163 11517 y Fs(\261)27650 11351 y Fv(\))28080 10894 y Fx(1)p Fp(\241)p Fs(s=t)31151 11351 y Fu(<)676 b(B)363 b(;)8898 b Fv(\(6.51\))6075 12911 y(where)494 b Fu(\273)567 b Fv(=)515 b Fu(r)31 b(d)330 b Fv(+)f Fu(r)31 b Fv(\()p Fu(d)330 b Fr(\241)g Fv(1\)\(2)h(+)e Fu(s)p Fv(\))p Fu(=)p Fv(\(1)i Fr(\241)f Fu(s=t)p Fv(\).)868 b(With)494 b(the)h(appropriate)g(c)-31 b(hoice)495 b(of)g(0)516 b Fu(<)6075 14240 y(s)400 b(<)h(t)g(<)g Fv(1)425 b(and)h Fu(r)432 b(>)401 b Fv(1)425 b(w)-31 b(e)426 b(can)f(obtain)i(an)-31 b(y)426 b Fu(\273)452 b(>)401 b Fv(3)p Fu(d)284 b Fr(\241)f Fv(1.)661 b(This)426 b(completes)h(the)e(pro)31 b(of)426 b(of)6075 15568 y(Theorem)370 b(6.4.)32592 b Fh(\244)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(51)16566 6001 y Fw(Appendix)419 b Fv(A.)554 b Fw(Technical)419 b(comments)7403 7994 y Fv(F)-92 b(ollo)-31 b(wing)342 b(are)d(some)g(commen)-31 b(ts)341 b(of)e(tec)-31 b(hnical)341 b(nature,)346 b(whic)-31 b(h)340 b(are)f(in)-31 b(tended)339 b(to)h(supple-)6075 9322 y(men)-31 b(t)370 b(the)f(discussion)h(of)f(the)h(assumptions)g (and)f(results)g(stated)h(in)g(the)f(in)-31 b(tro)31 b(duction.)6075 10651 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17922 y Fi(R)p Fp(n)p Fx([)p Fs(x)p Fp(\241)p Fs(";x)p Fx(+)p Fs(")p Fx(])30786 16164 y Fu(f)119 b Fv(\()p Fu(y)40 b Fv(\))p 30554 16659 2569 45 v 30554 17673 a Fu(x)246 b Fr(\241)g Fu(y)33440 16913 y(dy)40 b(;)8513 b Fv(\(B.7\))6075 19679 y(whenev)-31 b(er)445 b(this)g(limit)j(exists.)720 b(The)445 b(same)h(de\257nition)g(applies) g(when)g Fu(f)563 b Fv(tak)-31 b(es)446 b(v)-61 b(alues)446 b(in)f(a)6075 21008 y(Hilb)31 b(ert)350 b(space,)355 b(in)350 b(whic)-31 b(h)351 b(case)f(the)g(r.h.s.)h(of)f(\(B.7\))i(is)e (in)-31 b(terpreted)350 b(as)g(a)h(Bo)31 b(c)-31 b(hner)350 b(in)-31 b(tegral.)6075 23258 y FB(Prop)35 b(osition)426 b(B.1.)553 b Fm(Supp)-57 b(ose)396 b(that)f Fv(\251)308 b Fr(2)f Fu(H)25400 22857 y Fx(2)25897 23258 y Fv(\()p Ft(C)27126 22857 y Fp(\241)27874 23258 y Fv(\))p Fm(,)397 b(i.e.)h Fv(\251)308 b(:)g Ft(C)33485 22857 y Fp(\241)34540 23258 y Fr(!)g Ft(C)396 b Fm(is)g(analytic)h(and)19365 26069 y Fv(sup)19393 26968 y Fs(y)28 b(>)p Fx(0)21216 24563 y Fq(Z)21831 27078 y Fi(R)22630 26069 y Fr(j)p Fv(\251\()p Fu(x)247 b Fr(\241)f Fu(iy)40 b Fv(\))p Fr(j)27854 25612 y Fx(2)28536 26069 y Fu(dx)307 b(<)h Fr(1)p Fu(:)10784 b Fv(\(B.8\))6075 29087 y Fm(Then)463 b(the)h(b)-57 b(oundary)464 b(value)f Fv(\251\()p Fu(x)p Fv(\))430 b(=)g(lim)24242 29253 y Fs(y)28 b Fp(#)p Fx(0)25854 29087 y Fv(\251\()p Fu(x)295 b Fr(\241)g Fu(iy)40 b Fv(\))463 b Fm(exists)g(for)g(almost)g (every)h Fu(x)429 b Fr(2)g Ft(R)p Fm(,)6075 30416 y Fv(\251)308 b Fr(2)f Fu(L)8980 30014 y Fx(2)9477 30416 y Fv(\()p Ft(R)p Fv(\))p Fm(,)398 b(and)e(its)g(r)-57 b(e)g(al)396 b(and)g(imaginary)i(p)-57 b(arts)395 b(ar)-57 b(e)396 b(c)-57 b(onjugate,)398 b(i.e.)16463 32661 y Fv(Re)185 b(\251\()p Fu(x)p Fv(\))308 b(=)g Fu(H)90 b Fv(\(Im)185 b(\251\)\()p Fu(x)p Fv(\))1133 b Fm(for)396 b(a.e.)h Fu(x)307 b Fr(2)h Ft(R)p Fu(:)7883 b Fv(\(B.9\))6075 34912 y FB(Prop)35 b(osition)559 b(B.2.)622 b Fm(L)-57 b(et)503 b Fr(H)513 b Fm(b)-57 b(e)504 b(a)e(sep)-57 b(ar)g(able)502 b(Hilb)-57 b(ert)503 b(sp)-57 b(ac)g(e)503 b(and)f Fu(f)620 b Fr(2)500 b Fu(L)38921 34510 y Fx(1)39418 34912 y Fv(\()p Ft(R)p Fu(;)184 b Fr(H)11 b Fv(\))504 b Fm(in)g(the)6075 36240 y(sense)340 b(of)g(Bo)-57 b(chner)341 b(inte)-57 b(gr)g(ation.)493 b(Then)340 b(the)g(Hilb)-57 b(ert)341 b(tr)-57 b(ansform)340 b Fu(H)90 b(f)119 b Fv(\()p Fu(y)40 b Fv(\))308 b Fr(2)f(H)351 b Fm(exists)339 b(for)h(a.e.)6075 37568 y Fu(y)347 b Fr(2)307 b Ft(R)p Fm(,)366 b(and)357 b(ther)-57 b(e)357 b(exists)g(a)f(c)-57 b(onstant)358 b Fu(C)386 b(<)308 b Fr(1)p Fm(,)365 b(indep)-57 b(endent)359 b(of)e Fu(f)119 b Fm(,)366 b(such)357 b(that)f(for)h(al)57 b(l)356 b Fu(t)307 b(>)h Fv(0)14738 40318 y Fr(jf)p Fu(y)348 b Fr(2)308 b Ft(R)g Fv(:)554 b Fr(k)p Fu(H)90 b(f)119 b Fv(\()p Fu(y)40 b Fv(\))p Fr(k)23723 40484 y Fp(H)24845 40318 y Fu(>)308 b(t)p Fr(gj)g(\267)28883 39569 y Fu(C)p 28883 40063 871 45 v 29119 41077 a(t)30071 38811 y Fq(Z)30686 41326 y Fi(R)31485 40318 y Fr(k)p Fu(f)119 b Fv(\()p Fu(x)p Fv(\))p Fr(k)34745 40484 y Fp(H)35745 40318 y Fu(dx:)5605 b Fv(\(B.10\))7403 43190 y(A)301 b(mo)31 b(dern)302 b(pro)31 b(of)301 b(of)h(Prop)31 b(osition)303 b(B.1)f(can)g(b)31 b(e)301 b(found)g(in)h([27)q(].)471 b(Prop)31 b(osition)303 b(B.2,)316 b(i.e.)303 b(the)6075 44518 y(w)-31 b(eak-)p Fu(L)9594 44116 y Fx(1)10092 44518 y Fv(-prop)31 b(ert)-31 b(y)421 b(of)g(the)g(Hilb)31 b(ert)421 b(transform,)435 b(is)420 b(w)-31 b(ell)422 b(kno)-31 b(wn)422 b(for)f(the)g(case)f Fr(H)404 b Fv(=)393 b Ft(C)p Fv(.)647 b(A)6075 45846 y(pro)31 b(of)390 b(in)g(the)g(con)-31 b(text)392 b(of)e(more)g(general)h(Calderon-Zygm)-31 b(und)393 b(inequalities)f(can)e(b)31 b(e)390 b(found)6075 47175 y(in)454 b([63)r(,)476 b(Chapter)455 b(2],)477 b(where)454 b(it)h(is)f(remark)-31 b(ed)455 b(that)g(the)g(result)f (extends)g(to)i(the)e Fr(H)11 b Fv(-v)-61 b(alued)6075 48503 y(case.)928 b(Detailed)516 b(pro)31 b(ofs)515 b(of)g(suc)-31 b(h)514 b(results)g(for)g(v)-31 b(ector-v)-61 b(alued)516 b(functions,)553 b(whic)-31 b(h)515 b(con)-31 b(tain)6075 49832 y(Prop)31 b(osition)371 b(B.2)f(as)f(a)g(sp)31 b(ecial)370 b(case,)g(can)f(b)31 b(e)369 b(found)h(in)f([55)r(].)7403 51160 y(W)-92 b(e)490 b(no)-31 b(w)491 b(apply)g(these)f(facts)h(to)g Fu(T)22841 51326 y Fs(\261)23327 51160 y Fv(\()p Fu(v)40 b Fv(\))509 b(:=)h Fu(M)28146 50758 y Fp(\244)28654 51160 y Fv(\()p Fu(A)328 b Fr(\241)e Fu(v)366 b Fv(+)327 b Fu(i\261)42 b Fv(\))34865 50758 y Fp(\241)p Fx(1)36054 51160 y Fu(M)121 b Fv(.)855 b(The)491 b(trace)f(class)6075 52488 y(norm)437 b(will)i(b)31 b(e)436 b(denoted)i Fr(k)292 b(\242)g(k)19016 52654 y Fx(1)19513 52488 y Fv(.)696 b(The)437 b(real)g(and)h(imaginary)h(parts)e(of)h(b)31 b(ounded)437 b(op)31 b(erators)6075 53817 y(are)369 b(de\257ned)g(as)g (usual)g(b)-31 b(y)370 b(Re)184 b Fu(C)387 b Fv(=)21499 53381 y Fx(1)p 21499 53562 442 45 v 21499 54198 a(2)22073 53817 y Fv(\()p Fu(C)326 b Fv(+)246 b Fu(C)25597 53415 y Fp(\244)26106 53817 y Fv(\))370 b(and)f(Im)184 b Fu(C)388 b Fv(=)33201 53381 y Fx(1)p 33044 53562 755 45 v 33044 54198 a(2)p Fs(i)33931 53817 y Fv(\()p Fu(C)326 b Fr(\241)246 b Fu(C)37455 53415 y Fp(\244)37964 53817 y Fv(\).)6075 56067 y FB(Lemma)424 b(B.3.)552 b Fm(F)-85 b(or)396 b(every)h Fu(\261)349 b(>)307 b Fv(0)397 b Fm(it)f(holds)f(that)13368 57310 y Fq(Z)13983 59825 y Fi(R)14782 58817 y Fr(k)p Fv(Im)185 b Fu(T)17489 58983 y Fs(\261)17975 58817 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)19965 58983 y Fs(H)59 b(S)21576 58817 y Fu(dv)347 b Fr(\267)24204 57310 y Fq(Z)24819 59825 y Fi(R)25619 58817 y Fr(k)p Fv(Im)p Fu(T)28141 58983 y Fs(\261)28627 58817 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)30617 58983 y Fx(1)31299 58817 y Fu(dv)347 b Fv(=)307 b Fu(\274)40 b Fr(k)p Fu(M)121 b Fr(k)36899 58360 y Fx(2)36899 59090 y Fs(H)59 b(S)38325 58817 y Fu(:)4234 b Fv(\(B.11\))6075 61689 y Fm(Pr)-57 b(o)g(of.)554 b Fv(The)306 b(\257rst)f(part)h(of)g (\(B.11\))i(follo)-31 b(ws)308 b(from)e Fr(k)119 b(\242)g(k)28651 61855 y Fs(H)59 b(S)30386 61689 y Fr(\267)307 b(k)119 b(\242)g(k)33205 61855 y Fx(1)33703 61689 y Fv(.)471 b(Let)306 b Fu(E)64 b Fv(\()p Fu(t)p Fv(\))306 b(b)31 b(e)305 b(the)h(sp)31 b(ectral)6075 63017 y(resolution)286 b(of)g(the)g(selfadjoin)-31 b(t)288 b(dilation)g Fu(L)d Fv(of)h Fu(A)g Fv(and)f Fu(\301)307 b Fr(2)h(H)31668 63183 y Fx(1)32164 63017 y Fv(.)465 b(Then)285 b(b)-31 b(y)286 b(\(B.1\),)304 b(the)286 b(sp)31 b(ectral)6075 64345 y(theorem,)370 b(and)g(F)-92 b(ubini)9858 65599 y Fq(Z)10965 67106 y Fr(h)p Fv(Im)185 b Fu(T)13549 67272 y Fs(\261)14034 67106 y Fv(\()p Fu(v)40 b Fv(\))p Fu(\301;)184 b(\301)p Fr(i)17712 67272 y Fp(H)18470 67383 y Fk(1)19143 67106 y Fu(dv)1147 b Fv(=)23371 65599 y Fq(Z)676 b(Z)28971 66357 y Fu(\261)p 26086 66851 6303 45 v 26086 67865 a Fv(\()p Fu(x)247 b Fr(\241)f Fu(v)40 b Fv(\))29510 67545 y Fx(2)30252 67865 y Fv(+)246 b Fu(\261)31893 67545 y Fx(2)32707 67106 y Fu(d)p Fr(k)p Fu(E)64 b Fv(\()p Fu(x)p Fv(\))p Fu(P)37075 66649 y Fp(\244)37584 67106 y Fu(M)121 b(\301)p Fr(k)39992 66649 y Fx(2)40246 67347 y(~)39992 67543 y Fp(H)40990 67106 y Fu(dv)21403 69541 y Fv(=)1107 b Fu(\274)40 b Fr(k)p Fu(M)121 b(\301)p Fr(k)27003 69084 y Fx(2)27003 69815 y Fp(H)27761 69926 y Fk(1)28248 69541 y Fu(:)14311 b Fv(\(B.12\))p eop end %%Page: 56 56 TeXDict begin 56 55 bop 6075 3345 a Fx(56)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(Let)341 b(\()p Fu(\301)9120 6167 y Fs(n)9723 6001 y Fv(\))h(b)31 b(e)341 b(an)h(orthonormal)i(basis)d(in)h Fr(H)24686 6167 y Fx(1)25182 6001 y Fv(.)484 b(W)-92 b(e)340 b(ha)-31 b(v)g(e)343 b(Im)184 b Fu(T)32529 6167 y Fs(\261)33015 6001 y Fv(\()p Fu(v)40 b Fv(\))308 b Fr(\270)g Fv(0)341 b(and)h(th)-31 b(us)342 b(b)-31 b(y)342 b(\(B.12\))13595 6854 y Fq(Z)14886 8360 y Fr(k)p Fv(Im)185 b Fu(T)17593 8526 y Fs(\261)18079 8360 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)20069 8526 y Fx(1)20751 8360 y Fu(dv)1146 b Fv(=)24978 6854 y Fq(Z)26270 8360 y Fv(T)-92 b(r\(Im)185 b Fu(T)29995 8526 y Fs(\261)30480 8360 y Fv(\()p Fu(v)40 b Fv(\)\))184 b Fu(dv)23010 11263 y Fv(=)24978 9756 y Fq(Z)26270 10212 y(X)26796 12536 y Fs(n)27869 11263 y Fr(h)p Fv(Im)h Fu(T)30453 11429 y Fs(\261)30939 11263 y Fv(\()p Fu(v)40 b Fv(\))p Fu(\301)33036 11429 y Fs(n)33638 11263 y Fu(;)184 b(\301)34789 11429 y Fs(n)35392 11263 y Fr(i)35822 11429 y Fp(H)36580 11540 y Fk(1)37253 11263 y Fu(dv)23010 14046 y Fv(=)1107 b Fu(\274)40 b Fr(k)p Fu(M)121 b Fr(k)27950 13589 y Fx(2)27950 14320 y Fs(H)59 b(S)29376 14046 y Fu(:)13183 b Fv(\(B.13\))45064 15839 y Fh(\244)6075 17832 y FB(Lemma)522 b(B.4.)604 b Fm(L)-57 b(et)476 b Fr(H)16466 17998 y Fs(H)59 b(S)18366 17832 y Fm(denote)475 b(the)g(sep)-57 b(ar)g(able)475 b(Hilb)-57 b(ert)476 b(sp)-57 b(ac)g(e)475 b(of)g(al)57 b(l)474 b(Hilb)-57 b(ert-Schmidt)6075 19160 y(op)g(er)g(ators)395 b(on)h Fr(H)13403 19326 y Fx(1)13899 19160 y Fm(.)7403 20489 y(\(a\))f(If)h Fu(\301)307 b Fr(2)h(H)13380 20655 y Fx(1)13876 20489 y Fm(,)397 b(then)f Fr(h)p Fu(T)18150 20655 y Fs(\261)18636 20489 y Fv(\()p Fr(\242)p Fv(\))p Fu(\301;)184 b(\301)p Fr(i)310 b(2)d Fu(H)24409 20087 y Fx(2)24906 20489 y Fv(\()p Ft(C)26135 20087 y Fp(\241)26883 20489 y Fv(\))397 b Fm(for)f(e)-57 b(ach)396 b(\257xe)-57 b(d)396 b Fu(\261)350 b(>)307 b Fv(0)p Fm(.)7403 21817 y(\(b\))396 b(F)-85 b(or)395 b(almost)g(every)i Fu(v)347 b Fr(2)307 b Ft(R)397 b Fm(one)f(has)20258 23610 y Fv(Re)184 b Fu(T)22396 23776 y Fs(\261)22882 23610 y Fv(\()p Fu(v)40 b Fv(\))307 b(=)h Fu(H)90 b Fv(\(Im)185 b Fu(T)29389 23776 y Fs(\261)29875 23610 y Fv(\)\()p Fu(v)40 b Fv(\))11124 b(\(B.14\))6075 25403 y Fm(in)397 b(the)e(sense)h(of)h(Hilb)-57 b(ert)396 b(tr)-57 b(ansforms)396 b(of)g Fr(H)24933 25569 y Fs(H)59 b(S)26358 25403 y Fm(-value)-57 b(d)397 b(functions.)6075 27396 y(Pr)-57 b(o)g(of.)554 b Fv(Consider)412 b(arbitrary)g Fu(\301)e Fv(and)i Fu(\303)450 b Fv(in)412 b Fr(H)25668 27562 y Fx(1)26575 27396 y Fv(and)g(use)e(\(B.1\))j(and)f(the)f(sp)31 b(ectral)411 b(theorem)6075 28724 y(to)370 b(estimate)6358 29300 y Fq(Z)7649 30807 y Fr(jh)p Fu(T)9033 30973 y Fs(\261)9520 30807 y Fv(\()p Fu(v)285 b Fr(\241)246 b Fu(iy)40 b Fv(\))p Fu(\301;)184 b(\303)40 b Fr(ij)15922 30350 y Fx(2)16605 30807 y Fu(dv)1147 b Fr(\267)20832 29300 y Fq(Z)22124 30807 y Fr(k)p Fv(\()p Fu(L)247 b Fr(\241)f Fu(v)285 b Fr(\241)246 b Fu(i)p Fv(\()p Fu(y)287 b Fv(+)246 b Fu(\261)42 b Fv(\)\))31285 30350 y Fp(\241)p Fx(1)32474 30807 y Fu(P)33339 30350 y Fp(\244)33847 30807 y Fu(M)121 b(\301)p Fr(k)36255 30350 y Fx(2)36936 30807 y Fu(dv)224 b Fr(k)p Fu(M)121 b(\303)40 b Fr(k)41335 30350 y Fx(2)18865 33709 y Fv(=)20832 32203 y Fq(Z)677 b(Z)27821 32961 y Fv(1)p 23548 33455 9100 45 v 23548 34469 a(\()p Fu(x)246 b Fr(\241)g Fu(v)40 b Fv(\))26971 34149 y Fx(2)27714 34469 y Fv(+)246 b(\()p Fu(y)285 b Fv(+)246 b Fu(\261)42 b Fv(\))32150 34149 y Fx(2)32965 33709 y Fu(dv)224 b(d)p Fr(k)p Fu(E)64 b Fv(\()p Fu(x)p Fv(\))p Fu(P)38670 33252 y Fp(\244)39179 33709 y Fu(M)121 b(\301)p Fr(k)41587 33252 y Fx(2)42083 33709 y Fr(k)p Fu(M)g(\303)40 b Fr(k)45145 33252 y Fx(2)18865 36414 y Fv(=)21865 35665 y Fu(\274)p 20965 36159 2470 45 v 20965 37173 a(y)286 b Fv(+)246 b Fu(\261)23567 36414 y Fr(k)p Fu(M)121 b(\301)p Fr(k)26528 35957 y Fx(2)27025 36414 y Fr(k)p Fu(M)g(\303)40 b Fr(k)30087 35957 y Fx(2)30892 36414 y Fr(\267)32193 35665 y Fu(\274)p 32193 36159 671 45 v 32261 37173 a(\261)32996 36414 y Fr(k)p Fu(M)121 b(\301)p Fr(k)35957 35957 y Fx(2)36454 36414 y Fr(k)p Fu(M)g(\303)40 b Fr(k)39516 35957 y Fx(2)40013 36414 y Fu(:)2546 b Fv(\(B.15\))7403 38733 y(Summing)308 b(this)f(o)-31 b(v)g(er)308 b(an)e(orthonormal)j(basis)e(of)g(v)-31 b(ectors)307 b Fu(\303)346 b Fv(and)307 b(then)g(o)-31 b(v)g(er)307 b(an)g(orthonor-)6075 40061 y(mal)370 b(basis)f(of)h(v)-31 b(ectors)370 b Fu(\301)e Fv(leads)i(to)17750 40852 y Fq(Z)19041 42359 y Fr(k)p Fu(T)20241 42525 y Fs(\261)20727 42359 y Fv(\()p Fu(v)286 b Fr(\241)246 b Fu(iy)40 b Fv(\))p Fr(k)25034 41901 y Fx(2)25034 42632 y Fs(H)59 b(S)26645 42359 y Fu(dy)347 b Fr(\267)29412 41610 y Fu(\274)p 29412 42104 V 29480 43118 a(\261)30216 42359 y Fr(k)p Fu(M)121 b Fr(k)32517 41901 y Fx(4)32517 42632 y Fs(H)59 b(S)33943 42359 y Fu(:)8616 b Fv(\(B.16\))7403 44718 y(This)513 b(implies)g(\(a\).)923 b(In)512 b(fact,)549 b(w)-31 b(e)513 b(ha)-31 b(v)g(e)513 b(pro)-31 b(v)g(en)513 b(the)f(stronger)g(result)g (that)i Fu(T)41950 44884 y Fs(\261)42435 44718 y Fv(\()p Fr(\242)p Fv(\))g(is)e(a)6075 46046 y(Hilb)31 b(ert-Sc)-31 b(hmidt-v)-61 b(alued)372 b Fu(H)18711 45644 y Fx(2)19208 46046 y Fv(-function)e(in)g(the)f(lo)-31 b(w)g(er)371 b(half)f(plane.)7403 47374 y(By)477 b(Lemma)h(B.3)f(and)g(Prop)31 b(osition)479 b(B.2,)504 b Fu(H)90 b Fv(\(Im)186 b Fu(T)29805 47540 y Fs(\261)30290 47374 y Fv(\))477 b(exists)g(almost)i(ev)-31 b(erywhere)476 b(as)h(a)6075 48703 y(Hilb)31 b(ert-Sc)-31 b(hmidt)453 b(op)31 b(erator.)739 b(Since)452 b(the)f(strong)h(top)31 b(ology)454 b(is)d(w)-31 b(eak)g(er)452 b(than)g(the)f(Hilb)31 b(ert-)6075 50031 y(Sc)-31 b(hmidt)317 b(top)31 b(ology)-92 b(,)330 b(this)316 b(implies)h(the)f(existence)h(of)f Fu(H)90 b Fr(h)p Fv(Im)186 b Fu(T)32203 50197 y Fs(\261)32688 50031 y Fv(\()p Fr(\242)p Fv(\))p Fu(\301;)e(\301)p Fr(i)310 b Fv(=)d Fr(h)p Fu(H)90 b Fv(\(Im)186 b Fu(T)41599 50197 y Fs(\261)42085 50031 y Fv(\)\()p Fr(\242)p Fv(\))p Fu(\301;)e(\301)p Fr(i)6075 51359 y Fv(for)383 b(ev)-31 b(ery)384 b Fu(\301)330 b Fr(2)h(H)13716 51525 y Fx(1)14213 51359 y Fv(.)535 b(By)383 b(\(a\))i(and)f(Prop)31 b(osition)385 b(B.1)f(the)g(latter)g (is)g(equal)g(to)g Fr(h)p Fv(Re)185 b Fu(T)41722 51525 y Fs(\261)42208 51359 y Fv(\()p Fr(\242)p Fv(\))p Fu(\301;)f(\301)p Fr(i)p Fv(.)6075 52688 y(W)-92 b(e)511 b(conclude)i(\(B.14\))h(since)e (b)31 b(ounded)513 b(op)31 b(erators)512 b(are)g(determined)h(b)-31 b(y)512 b(their)h(quadratic)6075 54016 y(form.)36435 b Fh(\244)7403 56009 y Fv(W)-92 b(e)246 b(are)g(no)-31 b(w)247 b(prepared)e(to)i(pro)-31 b(v)g(e)247 b(\(B.6\))h(and)e(thereb) -31 b(y)247 b(complete)h(the)e(pro)31 b(of)247 b(of)f(Lemma)h(4.1.)6075 57337 y(W)-92 b(e)368 b(ha)-31 b(v)g(e,)371 b(using)e(Lemma)i (B.4\(b\),)6075 59619 y Fr(jf)p Fu(v)347 b Fv(:)493 b Fr(k)p Fu(T)9819 59785 y Fs(\261)10305 59619 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)12295 59785 y Fs(H)59 b(S)14029 59619 y Fu(>)307 b(t)p Fr(gj)1108 b(\267)f(jf)p Fu(v)348 b Fv(:)492 b Fr(k)p Fv(Re)185 b Fu(T)24769 59785 y Fs(\261)25255 59619 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)27245 59785 y Fs(H)59 b(S)28978 59619 y Fu(>)30356 58870 y(t)p 30280 59364 554 45 v 30280 60378 a Fv(2)30966 59619 y Fr(gj)247 b Fv(+)f Fr(jf)p Fu(v)348 b Fv(:)308 b Fr(k)p Fv(Im)185 b Fu(T)38247 59785 y Fs(\261)38732 59619 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)40722 59785 y Fs(H)59 b(S)42456 59619 y Fu(>)43834 58870 y(t)p 43757 59364 V 43757 60378 a Fv(2)44444 59619 y Fr(gj)17565 62251 y Fv(=)1107 b Fr(jf)p Fu(v)348 b Fv(:)492 b Fr(k)p Fu(H)90 b Fv(\(Im)186 b Fu(T)26225 62417 y Fs(\261)26710 62251 y Fv(\)\()p Fu(v)40 b Fv(\))p Fr(k)29130 62417 y Fs(H)59 b(S)30865 62251 y Fu(>)32243 61502 y(t)p 32166 61996 V 32166 63010 a Fv(2)32852 62251 y Fr(gj)247 b Fv(+)f Fr(jf)p Fu(v)348 b Fv(:)308 b Fr(k)p Fv(Im)185 b Fu(T)40133 62417 y Fs(\261)40619 62251 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)42609 62417 y Fs(H)59 b(S)44343 62251 y Fu(>)45721 61502 y(t)p 45644 61996 V 45644 63010 a Fv(2)46330 62251 y Fr(gj)17565 65032 y(\267)19666 64283 y Fv(2\()p Fu(C)326 b Fv(+)246 b(1\))p 19666 64777 4192 45 v 21561 65791 a Fu(t)24174 63525 y Fq(Z)25466 65032 y Fr(k)p Fv(Im)185 b Fu(T)28173 65198 y Fs(\261)28659 65032 y Fv(\()p Fu(v)40 b Fv(\))p Fr(k)30649 65198 y Fs(H)59 b(S)32259 65032 y Fu(dv)40 b(;)10413 b Fv(\(B.17\))6075 67330 y(where)508 b(in)h(the)g(end)f(Prop) 31 b(osition)510 b(B.2)g(and)e(Cheb)-31 b(yc)g(hev's)510 b(inequalit)-31 b(y)511 b(w)-31 b(ere)509 b(used.)910 b(Th)-31 b(us)6075 68658 y(\(B.6\))371 b(is)e(a)g(consequence)h(of)f (Lemma)h(B.3.)p eop end %%Page: 57 57 TeXDict begin 57 56 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(57)10286 6001 y Fw(Appendix)419 b Fv(C.)555 b Fw(A)418 b(disorder-a)-98 b(vera)-25 b(ged)417 b(spectral)j(shift)d(bound)7403 7994 y Fv(Here)483 b(w)-31 b(e)483 b(presen)-31 b(t)483 b(a)h(new)f(result,)512 b(whic)-31 b(h)484 b(amoun)-31 b(ts)485 b(to)f(the)f(b)31 b(oundedness)482 b(-under)h(the)6075 9322 y(a)-31 b(v)g(erage)366 b(o)-31 b(v)g(er)366 b(lo)31 b(cal)367 b(disorder)e(\(but)h(at)g(\257xed)f(energy)g(!\)-)h(of)g(the) f(fractional)j(momen)-31 b(t)367 b(of)f(the)6075 10651 y(sp)31 b(ectral)472 b(shift)g(asso)31 b(ciated)473 b(with)g(the)f (addition,)500 b(to)473 b(a)f(Sc)-31 b(hr\304)-553 b(odinger)472 b(op)31 b(erator,)498 b(of)473 b(a)f(lo)31 b(cal)6075 11979 y(\\p)g(oten)-31 b(tial)372 b(bump".)7403 13307 y(As)383 b(w)-31 b(as)383 b(explained)h(in)g(the)f(in)-31 b(tro)31 b(duction,)389 b(some)383 b(of)g(the)g(di\261culties)i(whic) -31 b(h)384 b(ha)-31 b(v)g(e)384 b(in)f(the)6075 14636 y(past)317 b(imp)31 b(eded)317 b(the)g(exten)-31 b(tion)319 b(of)e(fractional)i(momen)-31 b(t)318 b(metho)31 b(ds)318 b(to)f(the)g(con)-31 b(tin)g(uum)319 b(can)e(b)31 b(e)6075 15964 y(traced)370 b(to)g(the)g(lac)-31 b(k)371 b(of)f(uniform)g(b)31 b(ounds)369 b(on)h(suc)-31 b(h)370 b(sp)31 b(ectral)369 b(shifts.)494 b(The)370 b(follo)-31 b(wing)374 b(results)6075 17292 y(is)419 b(enabled)i(b)-31 b(y)420 b(the)g(metho)31 b(ds)420 b(of)h(Section)g(4.)645 b(While)420 b(the)g(analysis)h(presen) -31 b(ted)419 b(ab)31 b(o)-31 b(v)g(e)421 b(do)31 b(es)6075 18621 y(not)435 b(pro)31 b(ceede)434 b(through)i(this)f(b)31 b(ound,)451 b(the)435 b(issues)f(in)-31 b(v)g(olv)g(ed)437 b(are)d(closely)i(related,)452 b(and)435 b(the)6075 19949 y(result)369 b(ma)-31 b(y)370 b(pro)-31 b(vide)370 b(a)g(useful)f(to)31 b(ol.)6075 22117 y FB(Theorem)464 b(C.1.)574 b Fm(L)-57 b(et)428 b Fu(H)17218 22283 y Fs(t)17971 22117 y Fv(=)19455 21837 y Fq(b)19196 22117 y Fu(H)358 b Fv(+)269 b Fu(tV)672 b Fm(wher)-57 b(e)26738 21837 y Fq(b)26479 22117 y Fu(H)517 b Fm(satis\257es)427 b FB(A1)h Fm(and)f Fu(V)672 b Fm(is)427 b(a)g(non-ne)-57 b(gative)6075 23445 y(b)g(ounde)g(d)403 b(function)g(with)e(c)-57 b(omp)g(act)403 b(supp)-57 b(ort.)526 b(L)-57 b(et)402 b Fu(U)523 b Fm(b)-57 b(e)402 b(a)g(non-ne)-57 b(gative)404 b(b)-57 b(ounde)g(d)403 b(function)6075 24774 y(such)490 b(that)g Fu(V)737 b Fm(is)491 b(strictly)g(p)-57 b(ositive)492 b(thr)-57 b(oughout)490 b(the)h(set)g Fu(Q)479 b Fv(=)h Fr(f)p Fu(q)348 b Fv(:)308 b(dist\()p Fu(q)40 b(;)184 b Fv(supp)q(\()p Fu(U)121 b Fv(\)\))308 b Fu(<)g(\261)42 b Fr(g)6075 26102 y Fm(with)423 b(some)i Fu(\261)400 b(>)359 b Fv(0)424 b Fm(and)g(set)g Fu(v)18923 26268 y Fp(\241)20029 26102 y Fv(=)358 b(inf)22595 26268 y Fs(x)p Fp(2)p Fs(Q)24630 26102 y Fu(V)246 b Fv(\()p Fu(x)p Fv(\))p Fm(.)595 b(Then,)432 b(for)424 b(any)h Fv(0)359 b Fu(<)g(s)f(<)h Fv(min)q(\(2)p Fu(=d;)184 b Fv(1)p Fu(=)p Fv(2\))6075 27430 y Fm(ther)-57 b(e)396 b(is)g Fu(C)10812 27596 y Fs(s;\261)12286 27430 y Fu(<)308 b Fr(1)396 b Fm(such)f(that)g(the)h(sp)-57 b(e)g(ctr)g(al)396 b(shift)h(function,)h(de\257ne)-57 b(d)397 b(as)14544 29396 y Fu(\273)51 b Fv(\()p Fu(t;)184 b(E)64 b Fv(\))705 b(=)f(T)-92 b(r)184 b([)p Fu(P)338 b Fv(\()q Fu(H)24013 29562 y Fs(t)24710 29396 y Fu(<)308 b(E)64 b Fv(\))246 b Fr(\241)g Fu(P)338 b Fv(\()q Fu(H)30943 29562 y Fs(t)31578 29396 y Fv(+)246 b Fu(U)428 b(<)308 b(E)64 b Fv(\)])493 b Fu(;)5948 b Fv(\(C.1\))6075 31500 y Fm(satis\257es,)396 b(for)g(any)g Fu(E)372 b Fr(\270)307 b Fv(inf)272 b Fu(\276)40 b Fv(\()19712 31220 y Fq(b)19454 31500 y Fu(H)89 b Fv(\))p Fm(:)9850 32750 y Fq(Z)10957 33024 y Fx(1)10465 35265 y(0)11638 34257 y Fr(j)q Fu(\273)51 b Fv(\()p Fu(t;)184 b(E)64 b Fv(\))p Fr(j)15422 33700 y Fs(s)16080 34257 y Fv(d)p Fu(t)703 b Fr(\267)g Fu(C)20153 34423 y Fs(s;\261)21505 34257 y Fr(k)p Fu(U)121 b Fr(k)23488 34589 y Fp(1)24610 34257 y Fv(\(1)247 b(+)f Fr(j)p Fu(E)311 b Fr(\241)246 b Fu(E)30306 34423 y Fs(o)30798 34257 y Fr(j)h Fv(+)f Fr(k)p Fu(V)g Fr(k)34457 34589 y Fp(1)35395 34257 y Fv(\))35825 33800 y Fs(s)p Fx(\(2)p 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44669 y Fu(\273)28912 44835 y Fs(L)29575 44669 y Fv(\()p Fu(t;)g(E)64 b Fv(\))11197 b(\(C.3\))6075 47166 y(where)411 b Fu(\273)9802 47332 y Fs(L)10465 47166 y Fv(\()p Fu(t;)184 b(E)64 b Fv(\))413 b(is)f(computed)g(with)h Fu(H)23386 46591 y Fx(\(\244)24338 46702 y Fl(L)24922 46591 y Fx(\))25268 46702 y Fl(o)23296 47438 y Fs(t)26179 47166 y Fv(in)f(place)h(of)f Fu(H)32608 47332 y Fs(t)33409 47166 y Fv(with)h(\244)36742 47332 y Fs(L)37783 47166 y Fv(=)378 b([)p Fr(\241)p Fu(L;)184 b(L)p Fv(])42494 46764 y Fs(d)43013 47166 y Fv(.)619 b(It)412 b(is)6075 48494 y(use\260)277 b(to)h(note)g(that,)297 b(b)31 b(ecause)277 b Fu(U)398 b Fv(is)278 b(non-negativ)-31 b(e,)298 b(the)278 b(di\256erence)f(of)h(pro)61 b(jections)279 b(app)31 b(earing)6075 49822 y(in)285 b(eq.)g(\(C.1\))i(is)e(a)g(p)31 b(ositiv)-31 b(e)286 b(semi-de\257nite)g(op)31 b(erator)286 b(so)f(its)g(trace)g(is)g(equal)g(to)h(its)f(trace)g(norm.)6075 51366 y(An)263 b(application)j(of)e(F)-92 b(atou's)263 b(lemma)i(yields)f(eq.)g(\(C.2\))h(for)e Fu(H)31746 51532 y Fs(t)32399 51366 y Fv(pro)-31 b(vided)264 b(it)g(holds)f(for)g Fu(H)43236 50791 y Fx(\(\244)44188 50902 y Fl(L)44772 50791 y Fx(\))45118 50902 y Fl(o)43146 51638 y Fs(t)45618 51366 y Fv(.)7403 52833 y(Throughout)505 b(the)f(rest)f(of)h(the)g(pro) 31 b(of)504 b(w)-31 b(e)504 b(\257x)g Fu(L)532 b(>)f Fv(0)504 b(and)g(write)g Fu(H)37865 52999 y Fs(t)38759 52833 y Fv(and)41305 52553 y Fq(b)41046 52833 y Fu(H)593 b Fv(for)504 b(the)6075 54161 y(restrictions)375 b(of)g(these)f(op)31 b(erators)375 b(to)h([)p Fr(\241)p Fu(L;)184 b(L)p Fv(])25662 53759 y Fs(d)26555 54161 y Fv(with)376 b(Diric)-31 b(hlet)375 b(b)31 b(oundary)375 b(conditions.)511 b(W)-92 b(e)6075 55489 y(b)31 b(egin)369 b(with)i(the)e(observ)-61 b(ation)370 b(that)21788 57455 y Fu(\273)51 b Fv(\()p Fu(t;)184 b(E)64 b Fv(\))679 b(=)d(T)-92 b(r)184 b Fu(P)29207 57621 y Fs(t)29904 57455 y Fu(;)13193 b Fv(\(C.4\))6075 59420 y(where)252 b Fu(P)9870 59586 y Fs(t)10512 59420 y 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67575 y(Note)g(that)h(for)e(an)-31 b(y)370 b Fu(n)308 b Fr(\270)f Fv(1,)19382 68600 y Fq(\260)19382 69264 y(\260)19997 69541 y Fu(K)21016 69067 y Fp(\241)p Fs(n)20937 69813 y(t)22311 69541 y Fu(P)23022 69707 y Fs(t)23412 69541 y Fv(\(1)246 b(+)g Fu(K)26767 69067 y Fp(\241)p Fx(1)26688 69813 y Fs(t)27956 69541 y Fv(\))28386 69084 y Fs(n)28989 68600 y Fq(\260)28989 69264 y(\260)30281 69541 y Fr(\267)307 b Fv(1)h Fu(:)10787 b Fv(\(C.6\))p eop end %%Page: 58 58 TeXDict begin 58 57 bop 6075 3345 a Fx(58)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y Fv(Th)-31 b(us)369 b(b)-31 b(y)369 b(the)h(H\304)-553 b(older)370 b(inequalit)-31 b(y)372 b(for)d(trace)g(norms)16185 7795 y Fu(\273)51 b Fv(\()p Fu(t;)184 b(E)64 b Fv(\))678 b Fr(\267)e(k)q Fu(K)23140 7338 y Fs(n)23061 8068 y(t)23743 7795 y Fr(k)24296 8127 y Fs(H)59 b(S)26091 6854 y Fq(\260)26091 7518 y(\260)26706 7795 y Fv(\(1)247 b(+)f Fu(K)30062 7321 y Fp(\241)p Fx(1)29983 8067 y Fs(t)31250 7795 y Fv(\))31680 7338 y Fp(\241)p Fs(n)32975 6854 y Fq(\260)32975 7518 y(\260)33590 8237 y Fs(H)59 b(S)35508 7795 y Fu(:)7589 b Fv(\(C.7\))6075 9588 y(Noting)371 b(that)16216 11376 y(\(1)247 b(+)f Fu(K)19572 10903 y Fp(\241)p Fx(1)19493 11649 y Fs(t)20761 11376 y Fv(\))21191 10919 y Fp(\241)p Fx(1)23056 11376 y Fv(=)677 b Fu(U)25471 10919 y Fx(1)p Fs(=)p Fx(2)29605 10627 y Fv(1)p 26995 11122 5774 45 v 26995 12135 a Fu(H)27915 12301 y Fs(t)28551 12135 y Fv(+)246 b Fu(U)366 b Fr(\241)246 b Fu(E)32901 11376 y(U)33778 10919 y Fx(1)p Fs(=)p Fx(2)35476 11376 y Fu(;)7621 b Fv(\(C.8\))6075 13413 y(w)-31 b(e)369 b(\257nd)7366 15059 y Fq(Z)8473 15333 y Fx(1)7981 17574 y(0)9154 16566 y Fv(d)p Fu(t)184 b Fr(j)q Fu(\273)51 b Fv(\()p Fu(t;)184 b(E)64 b Fv(\))p Fr(j)14137 16009 y Fs(s)15287 16566 y Fr(\267)16824 14673 y Fq(\303)17701 15059 y(Z)18808 15333 y Fx(1)18316 17574 y(0)19489 14961 y Fq(\260)19489 15625 y(\260)19489 16289 y(\260)19489 16953 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Fu(t)39230 18744 y Fq(!)40106 18973 y Fx(1)p Fs(=)p Fx(2)41989 20637 y Fu(;)1108 b Fv(\(C.9\))6075 23579 y(where)369 b Fr(k)p Fu(A)p Fr(k)11213 23911 y Fs(m)12361 23579 y Fv(=)308 b(\(T)-92 b(r)184 b Fr(j)q Fu(A)p Fr(j)16731 23023 y Fs(m)17571 23579 y Fv(\))18001 23177 y Fx(1)p Fs(=m)19738 23579 y Fv(.)7403 24907 y(Lemma)291 b(4.1)h(and)f(the)f(represen)-31 b(tation)292 b(eq.)e(\(4.39\))k(used) 289 b(in)i(the)f(pro)31 b(of)291 b(of)g(Lemma)g(4.3)h(ma)-31 b(y)6075 26236 y(b)31 b(e)364 b(used)h(to)h(sho)-31 b(w)366 b(that)g(the)f(in)-31 b(tegrals)367 b(on)f(the)f(righ)-31 b(t)366 b(hand)f(side)h(of)f(eq.)h(\(C.9\))h(are)e(b)31 b(ounded)6075 27564 y(if)423 b(\(1\))g(2)p Fu(ns)397 b(<)f Fv(1)423 b(and)f(\(2\))i(4)p Fu(n)397 b(>)f(d)423 b Fv(\()p Fu(d)f Fv(is)h(the)f(dimension\).)654 b(W)-92 b(e)422 b(no)-31 b(w)423 b(outline)h(the)f(pro)31 b(of)423 b(of)6075 28893 y(this)369 b(assertion.)7403 30221 y(The)g(argumen)-31 b(ts)371 b(used)d(in)i(Lemma)g(4.3)g(can)g(b)31 b(e)369 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b(in)g Fr(I)34543 36409 y Fs(p)35446 36243 y Fv(for)g(an)-31 b(y)392 b Fu(p)346 b(>)f(d=)p Fv(2,)399 b(with)6075 37571 y(uniform)370 b(b)31 b(ounds)369 b(on)g(its)h Fr(I)17763 37737 y Fs(p)18643 37571 y Fv(norm.)493 b(Since)369 b Fr(k)q(\242k)26272 37904 y Fx(2)p Fs(n)27623 37571 y Fr(\267)308 b(k\242k)30206 37904 y Fs(H)59 b(S)32001 37571 y Fv(w)-31 b(e)369 b(\257nd)g(that)7086 38803 y Fq(Z)8193 39077 y Fx(1)7701 41319 y(0)8874 38705 y Fq(\260)8874 39369 y(\260)8874 40033 y(\260)8874 40697 y(\260)9489 40310 y Fu(U)10366 39853 y Fx(1)p Fs(=)p Fx(2)13385 39561 y Fv(1)p 11889 40055 V 11889 41069 a Fu(H)12809 41235 y Fs(t)13445 41069 y Fr(\241)246 b Fu(E)15566 40310 y(U)16443 39853 y Fx(1)p Fs(=)p Fx(2)17834 38705 y Fq(\260)17834 39369 y(\260)17834 40033 y(\260)17834 40697 y(\260)18449 38979 y Fx(2)p Fs(ns)18449 41417 y Fx(2)p Fs(n)20095 40310 y Fv(d)p Fu(t)676 b Fr(\267)23324 38803 y Fq(Z)24431 39077 y Fx(1)23939 41319 y(0)25112 38705 y Fq(\260)25112 39369 y(\260)25112 40033 y(\260)25112 40697 y(\260)25727 40310 y Fu(U)26604 39853 y Fx(1)p Fs(=)p Fx(2)27995 40310 y Fu(T)154 b Fv(\243)31284 39561 y(1)p 29790 40055 V 29790 41069 a Fu(H)30710 41235 y Fs(t)31345 41069 y Fr(\241)246 b Fu(E)33466 40310 y Fv(\243)p Fu(T)35128 39853 y Fp(y)35588 40310 y Fu(U)36465 39853 y Fx(1)p Fs(=)p Fx(2)37856 38705 y Fq(\260)37856 39369 y(\260)37856 40033 y(\260)37856 40697 y(\260)38471 38979 y Fx(2)p Fs(ns)38471 41417 y(H)59 b(S)40548 40310 y Fv(+)615 b(O\(1\))309 b Fu(;)42850 42358 y Fv(\(C.11\))6075 43686 y(with)347 b(a)f(similar)i(expression)e (for)g(the)g(other)h(factor)g(in)f(eq.)h(\(C.9\).)487 b(The)347 b(w)-31 b(eak)347 b Fu(L)40032 43285 y Fx(1)40875 43686 y Fv(b)31 b(ound)346 b(can)6075 45015 y(b)31 b(e)368 b(used)h(to)h(b)31 b(ound)369 b(this)h(\257nal)g(in)-31 b(tegral)371 b(since)15871 47268 y(\243)18360 46519 y(1)p 16865 47013 V 16865 48027 a Fu(H)17785 48193 y Fs(t)18421 48027 y Fr(\241)246 b Fu(E)20541 47268 y Fv(\243)677 b(=)f(\243)p Fu(V)25369 46811 y Fp(\241)p Fx(1)p 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37442 y(\267)9601 37666 y(S.)e(Birman.)g(On) f(the)g(n)-26 b(um)g(b)26 b(er)469 b(of)g(eigen)-26 b(v)-52 b(alues)471 b(in)f(a)g(quan)-26 b(tum)468 b(scattering)j(problem.)f Fy(V)-73 b(estnik)8009 38773 y(L)-49 b(eningr)g(ad.)340 b(Univ.)p FA(,)314 b(16\(13\):163{166,)320 b(1961.)6075 39885 y([15])472 b(M.)9700 39662 y(\267)9674 39885 y(S.)542 b(Birman)g(and)g(S.)g(B.)g(En)-26 b(tina.)543 b(A)e(stationary)i (approac)-26 b(h)542 b(to)g(abstract)h(scattering)g(theory)-78 b(.)8009 40992 y Fy(Math.)339 b(USSR)i(Izv.)p FA(,)314 b(1,)g(1967.)6075 42099 y([16])472 b(J.-M.)341 b(Com)-26 b(b)26 b(es)341 b(and)g(P)-78 b(.)341 b(D.)g(Hislop.)i(Lo)26 b(calization)343 b(for)e(some)h(con)-26 b(tin)g(uous,)348 b(random)341 b(Hamiltonians)i(in)8009 43206 y Fe(d)p FA(-dimensions.)315 b Fy(J.)340 b(F)-73 b(unct.)340 b(A)-24 b(nal.)p FA(,)314 b(124:149{180,)319 b(1994.)6075 44313 y([17])472 b(J.-M.)257 b(Com)-26 b(b)26 b(es,)269 b(P)-78 b(.)257 b(D.)g(Hislop,)270 b(and)257 b(Sh)-26 b(u)256 b(Nak)-52 b(am)-26 b(ura.)257 b(The)g Fe(L)30477 44001 y Fd(p)30959 44313 y FA(-theory)f(of)h(the)g(sp)26 b(ectral)258 b(shift)e(function,)8009 45420 y(the)250 b(Wegner)h(estimate,)265 b(and)251 b(the)f(in)-26 b(tegrated)251 b(densit)-26 b(y)250 b(of)h(states)g(for)f(some)i(random)e(op)26 b(erators.)252 b Fy(Comm.)8009 46527 y(Math.)339 b(Phys.)p FA(,)313 b(218\(1\):113{130,)319 b(2001.)6075 47634 y([18])472 b(J.-M.)478 b(Com)-26 b(b)26 b(es)478 b(and)g(L.)g(Thomas.)h (Asymptotic)f(b)26 b(eha)-26 b(viour)478 b(of)g(eigenfunctions)h(for)f (m)-26 b(ultiparticle)8009 48741 y(Sc)g(hr\304)-470 b(odinger)313 b(op)26 b(erators.)315 b Fy(Comm.)341 b(Math.)e(Phys.)p FA(,)312 b(34:251{270,)319 b(1973.)6075 49848 y([19])472 b(H.)285 b(L.)g(Cycon,)291 b(R.)285 b(G.)h(F)-78 b(ro)26 b(ese,)291 b(W.)285 b(Kirsc)-26 b(h,)291 b(and)285 b(B.)h(Simon.)g Fy(Schr\304)-486 b(odinger)313 b(op)-49 b(er)g(ators)313 b(with)f(applic)-49 b(ation)8009 50955 y(to)453 b(quantum)h(me)-49 b(chanics)454 b(and)f(glob)-49 b(al)453 b(ge)-49 b(ometry)p FA(.)437 b(T)-78 b(exts)434 b(and)h(Monographs)h(in)f(Ph)-26 b(ysics.)436 b(Springer-)8009 52062 y(V)-78 b(erlag,)314 b(Berlin,)h(study)e(edition,)h(1987.)6075 53169 y([20])472 b(E.)336 b(B.)g(Da)-26 b(vies.)337 b Fy(Sp)-49 b(e)g(ctr)g(al)362 b(the)-49 b(ory)360 b(and)h(di\256er)-49 b(ential)360 b(op)-49 b(er)g(ators)p FA(,)342 b(v)-26 b(olume)336 b(42)g(of)g Fy(Cambridge)361 b(Studies)f(in)8009 54276 y(A)-49 b(dvanc)g(e)g(d)342 b(Mathematics)p FA(.)313 b(Cam)-26 b(bridge)314 b(Univ)-26 b(ersit)g(y)313 b(Press,)h(Cam)-26 b(bridge,)314 b(1995.)6075 55383 y([21])472 b(S.)422 b(De)g(Bi)-26 b(\265)-444 b(evre)421 b(and)h(J.)g(V.)g(Pul)-26 b(\266)-444 b(e.)422 b(Propagating)h(edge)g(states)f(for)f(a)h (magnetic)h(hamiltonian.)h Fy(Math.)8009 56490 y(Phys.)339 b(Ele)-49 b(c.)340 b(Jour.)p FA(,)314 b(5,)g(1999.)6075 57597 y([22])472 b(L.)404 b(de)f(Branges.)i(P)-26 b(erturbations)404 b(of)f(self-adjoin)-26 b(t)404 b(transformations.)h Fy(A)-24 b(mer.)424 b(J.)g(Math.)p 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b(101\(1\):21{46,)319 b(1985.)6075 66452 y([26])472 b(J.)404 b(F)-78 b(r\304)-470 b(ohlic)-26 b(h)403 b(and)g(T.)g(Sp)26 b(encer.)404 b(Absence)f(of)h(di\256usion)f(in)h(the)f(Anderson)g(tigh) -26 b(t)404 b(binding)f(mo)26 b(del)405 b(for)8009 67559 y(large)315 b(disorder)e(or)h(lo)-26 b(w)314 b(energy)-78 b(.)314 b Fy(Comm.)340 b(Math.)f(Phys.)p FA(,)313 b(88\(2\):151{184,) 319 b(1983.)p eop end %%Page: 60 60 TeXDict begin 60 59 bop 6075 3345 a Fx(60)3146 b(M.)298 b(AIZENMAN,)h(A.)g(ELGAR)-71 b(T,)297 b(S.)h(NABOK)-24 b(O,)298 b(J.)g(H.)g(SCHENKER,)h(AND)g(G.)f(STOLZ)6075 6001 y FA([27])472 b(J.)304 b(B.)g(Garnett.)g Fy(Bounde)-49 b(d)332 b(analytic)e(functions)p FA(.)304 b(Academic)g(Press)g(Inc.)g ([Harcourt)g(Brace)g(Jo)-26 b(v)-52 b(ano)-26 b(vic)g(h)8009 7108 y(Publishers],)314 b(New)f(Y)-78 b(ork,)314 b(1981.)6075 8215 y([28])472 b(F.)330 b(Germinet)h(and)f(A.)h(Klein.)g(Bo)26 b(otstrap)331 b(m)-26 b(ultiscale)332 b(analysis)g(and)e(lo)26 b(calization)334 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b(Math.)f(Phys.)p FA(,)346 b(77\(2\):127{)8009 52508 y(135,)315 b(1980.)6075 53615 y([46])472 b(I.)346 b(M.)g(Lifshitz.)g (Energy)f(sp)26 b(ectrum)346 b(structure)f(and)h(quan)-26 b(tum)345 b(states)i(of)f(disordered)g(condensed)g(sys-)8009 54722 y(tems.)314 b Fy(Soviet)340 b(Physics)f(Usp)-49 b(ekhi)p FA(,)314 b(7:549{573,)k(1965.)6075 55829 y([47])472 b(F.)352 b(Martinelli)h(and)e(E.)h(Scopp)26 b(ola.)354 b(In)-26 b(tro)26 b(duction)351 b(to)i(the)e(mathematical)j(theory)e (of)f(Anderson)h(lo)26 b(cal-)8009 56936 y(ization.)315 b Fy(R)-24 b(iv.)339 b(Nuovo)i(Cimento)f(\(3\))p FA(,)313 b(10\(10\):1{90,)318 b(1987.)6075 58043 y([48])472 b(N.)311 b(Mott)f(and)h(W.)g(Tw)-26 b(ose.)311 b(The)f(theory)g(of)h(impurit)-26 b(y)310 b(conduction.)h Fy(A)-49 b(dvanc)g(es)339 b(in)f(Physics)p FA(,)310 b(page)i(107,)8009 59150 y(1961.)6075 60257 y([49])472 b(S.)323 b(N.)g(Nab)26 b(ok)-26 b(o.)324 b(The)e(structure)g (of)h(singularities)i(of)d(op)26 b(erator)324 b(functions)f(with)f(a)h (p)26 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b(del)433 b([MR)f(93c:47057].)k(In)c Fy(Partial)449 b(di\256er)-49 b(ential)449 b(e)-49 b(quations,)478 b(VIII)p FA(,)433 b(pages)g(87{153.)8009 69113 y(Springer,)313 b(Berlin,)i(1996.)p eop end %%Page: 61 61 TeXDict begin 61 60 bop 13203 3345 a Fx(LOCALIZA)-71 b(TION)387 b(IN)h(RANDOM)h(SCHR)28750 3149 y(\304)28631 3345 y(ODINGER)e(OPERA)-71 b(TORS)6244 b(61)6075 6001 y FA([53])472 b(M.)275 b(Reed)h(and)f(B.)h(Simon.)h Fy(Metho)-49 b(ds)304 b(of)g(mo)-49 b(dern)305 b(mathematic)-49 b(al)305 b(physics.)d(IV.)j(Analysis)e(of)h(op)-49 b(er)g(ators)p FA(.)8009 7108 y(Academic)315 b(Press)e([Harcourt)h(Brace)g(Jo)-26 b(v)-52 b(ano)-26 b(vic)g(h)314 b(Publishers],)g(New)g(Y)-78 b(ork,)313 b(1978.)6075 8215 y([54])472 b(W.)255 b(Rudin.)g Fy(R)-49 b(e)g(al)285 b(and)h(c)-49 b(omplex)286 b(analysis)p FA(.)253 b(McGra)-26 b(w-Hill)255 b(Bo)26 b(ok)255 b(Co.,)267 b(New)255 b(Y)-78 b(ork,)265 b(third)254 b(edition,)268 b(1987.)6075 9322 y([55])472 b(J.)268 b(Sc)-26 b(h)g(w)g(artz.)267 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b(Thouless.)f(Electrons)h(in)f(disordered)f(systems)h(and)g(the)g (theory)f(of)h(lo)26 b(calization.)396 b Fy(Phys.)411 b(R)-49 b(ep.)p FA(,)8009 31461 y(13C:93,)316 b(1974.)6075 32568 y([67])472 b(H.)245 b(v)-26 b(on)244 b(Dreifus)h(and)g(A.)g (Klein.)h(A)e(new)h(pro)26 b(of)245 b(of)f(lo)26 b(calization)249 b(in)c(the)f(Anderson)h(tigh)-26 b(t)245 b(binding)g(mo)26 b(del.)8009 33675 y Fy(Comm.)340 b(Math.)f(Phys.)p FA(,)313 b(124\(2\):285{299,)319 b(1989.)6075 34782 y([68])472 b(T.)534 b(H.)g(W)-78 b(ol\256.)534 b(Unique)g(con)-26 b(tin)g(uation)535 b(for)e Fc(j)p FA(\242)p Fe(u)p Fc(j)629 b(\267)f Fe(V)209 b Fc(jr)p Fe(u)p Fc(j)535 b FA(and)f(related)g (problems.)h Fy(R)-49 b(ev.)545 b(Mat.)8009 35889 y(Ib)-49 b(er)g(o)g(americ)g(ana)p FA(,)316 b(6\(3-4\):155{200,)j(1990.)6075 36996 y([69])472 b(T.)253 b(H.)h(W)-78 b(ol\256.)254 b(A)f(prop)26 b(ert)-26 b(y)252 b(of)h(measures)h(in)g Fb(R)24683 36684 y Fd(N)25705 36996 y FA(and)f(an)h(application)h(to)f (unique)e(con)-26 b(tin)g(uation.)255 b Fy(Ge)-49 b(om.)8009 38103 y(F)-73 b(unct.)340 b(A)-24 b(nal.)p FA(,)314 b(2\(2\):225{284,)k (1992.)7403 40179 y Fz(M.)334 b(Aizenman,)g(Princeton)i(University)7403 41286 y Fy(E-mail)k(addr)-49 b(ess)71 b FA(:)418 b Fa (aizenman@princeton.edu)7403 43168 y Fz(A.)334 b(Elgar)-59 b(t,)335 b(St)-59 b(anf)-20 b(ord)336 b(University)7403 44275 y Fy(E-mail)k(addr)-49 b(ess)71 b FA(:)418 b Fa (elgart@CIMS.nyu.edu)7403 46156 y Fz(S.)335 b(Nabok)-20 b(o,)333 b(St.)439 b(Petersbur)-20 b(g)335 b(St)-59 b(a)g(te)336 b(University)7403 47263 y Fy(E-mail)k(addr)-49 b(ess)71 b FA(:)418 b Fa(naboko@math.su.se)7403 49145 y Fz(J.)334 b(H.)g(Schenker,)i(ETH)e(Z)17741 49071 y(\304)17714 49145 y(urich)7403 50252 y Fy(E-mail)340 b(addr)-49 b(ess)71 b FA(:)418 b Fa(schenker@alumni.princeton.edu)7403 52134 y Fz(G.)334 b(Stolz,)h(University)g(of)f(Alabama)h(Birmingham)7403 53241 y Fy(E-mail)340 b(addr)-49 b(ess)71 b FA(:)418 b Fa(stolz@math.uab.edu)p eop end %%Trailer userdict /end-hook 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