Content-Type: multipart/mixed; boundary="-------------0810311755241" This is a multi-part message in MIME format. ---------------0810311755241 Content-Type: text/plain; name="08-207.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-207.keywords" Linearized NLS, endpoint bifurcations, Birman-Schwinger principle, Feshbach map ---------------0810311755241 Content-Type: application/postscript; name="bif.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bif.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: bif.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR8 CMBX12 CMR12 CMMI12 CMMI8 CMSY10 MSBM10 CMSY8 CMR7 %%+ CMR10 CMEX10 CMTI12 CMMI6 CMR6 CMMIB10 CMSY6 MSBM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o bif.ps bif.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.10.31:1724 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) 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 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-4 -948 1329 786}readonly 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 30 /phi put dup 32 /psi put readonly def /FontBBox{-15 -250 1216 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly 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 33 /omega put dup 115 /s put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine 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 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W 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 readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /parenleftBig put dup 17 /parenrightBig put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 100 /hatwidest put dup 115 /radicalBigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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E36966A334BD53D376350BBF4B6E8ED3F33190955995D0F470650D4ED7A74D3D 4B63FA3AE08BFE7456DE55 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 48 /zero put dup 49 /one put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 74 /J put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 8E8DCC80DD0C49E137CDC9AC08BAE39282E26A7A4D8C159B95F227BDA2A281AF A9DAEBF31F504380B20812A211CF9FEB112EC29A3FB3BD3E81809FC6293487A7 455EB3B879D2B4BD46942BB1243896264722CB59146C3F65BD59B96A74B12BB2 9A1354AF174932210C6E19FE584B1B14C00E746089CBB17E68845D7B3EA05105 EEE461E3697FCF835CBE6D46C75523478E766832751CF6D96EC338BDAD57D53B 52F5340FAC9FE0456AD13101824234B262AC0CABA43B62EBDA39795BAE6CFE97 563A50AAE1F195888739F2676086A9811E5C9A4A7E0BF34F3E25568930ADF80F 0BDDAC3B634AD4BA6A59720EA4749236CF0F79ABA4716C340F98517F6F06D9AB 7ED8F46FC1868B5F3D3678DF71AA772CF1F7DD222C6BF19D8EF0CFB7A76FC6D1 0AD323C176134907AB375F20CFCD667AB094E2C7CB2179C4283329C9E435E7A4 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A4DE3E3F6222F 7A252B780C86447859579C68E52691E144F836C1C62F19A12EFB710343D33262 1F7955FE5C37074CE5F9C7ABF1A241078519A4D7913A0AD861E0E357B50FB730 E757C0D26390E6028FAC61EB0E9414716AC8406A6E35DC70A7C1AA524804FC8E 985CC3604A2BE0A8235CC895B2B33CB7EE85FE4F2CD817BAC3D27ADD295D0A0E BC0E8D849952BCA7325DC261A785CD2305BC377AC61AC5E5B2CD3164CFF033CB 5436B8000673A4D763ED26273130702447C75A774C7799FB8C3E54A2E34D1710 CF7883A9B05285C7DF30F314455A4428A5369D92C0348D45BF4AEC5E16611D16 1E5EF015900F4DF63A58DC233BEE88417B204DBD110AACD1DE3D750F9C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % 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 (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 91 /hatwide put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 D10F04E2C2822D3E73F25B81C424627D3D9A158EAB554233A25D3C6849ABA86F 1F25C1667CB57D2E79B7803083CB7CC0616467F68450D9A3FEAB534EB9721003 DBFEEFD050F3AC3492F5C74162A9A531ECEC0F47610B4940E946D21CAA771D30 A6C27ECBA11708CC46C62396BF9D1990D579D0C394899D24FE7A4382EA18E7E1 160E7283AF5BE17254790628E79FCC206F28B5566075B3A5697D5209062544FF D85FD89D6F43D6588B242AB2666B5D2861CD38A8CE676503EDFAE84D12A71E77 8405E468FE391F4F3F50D2C57ED55512036B0DB8E76A7EF413ED08673E56DE2C 16A3B65CD478433C0D2F9FEC4E662D54DAA43CFA6957D2A9AF8979BE06F70B68 ED4C8C493D6DAC4971A3F1D010A7726D084EC1074FECD7D12D72AE16C26194AF 21AF5774D9B860EEE8608D34F150092F09C19959BAA670022B9A9F263CD391E3 74DD1D1B4CD4D75273CAA4E37F68C631723E08FA35AD34C0AFB4621AE6689861 854D16CE1C375FD159A337E221A6FF1CFFB5693A0623E7EBB58C2969F590D081 AD92DD9E5322E26D6A15023664AC73A355998BCC48ADD0E7A4BC79790519606F A1FEF6075033BCD422EE8233B83D1E7C20043280D531223D5AD4D5B41669F884 95CE4D6DDE819B588742B930C579EDF743F2C74C95F717FAA6154FADC3FE2975 F59CFB1C1A29059487E75C48505BAEAD7145667D4E18E46E610C868A257173ED 0D30EAA4C090854DD8378E92D0A376226EA7DA63798F247BAC770FE26D70E72F 90CCFAADF118304646955DE0B9D25C4B419FBF62FB42ED17CD96B73429AC9385 DA5E26A4CA9E2B940EF216ED17A21FE1B2A9DC7F3D1C294E1653E337DE641754 ADE4310F47E3F275DDD5E92E372CB87A19EEA30B57EC893FD86E12AEB60D5F64 3B5360EC8A5ECE7F6842A2B55B8B62AC167F9700432CD2629E716997619E97CE 9401FB641D587B29A8E80F20E33E72E79500E13B5DA9F4C46117F325C0253627 74D6286E6F1DAF626F6B2BA6A1686898D216A8876E4EF7AF362E573156AD2A1A E85299286E99272E0506AA5D870782106D83B7399D8B67B24C4969B047B04CEE EC23D656B62F94395B21999DFA8DE3B02EA8DE07042A33374793E485602236CC 488305D11B10E9CDD4B04FD2036F1801CE011190E406D7695CF14BD7DC860857 275FEC8941D8C23839C35FDBFF547A8A70950F8995F7E15F0341508A32ED8049 D60D861623790DC321BC66A0D031EC75F0D52FC13BD36896CDF77C35945B7FF1 5872B58DBACFF9E423AD32C0EEA6E9E95B39AC34DCE4B5FD4D77254EADE019C9 BBBAEB05DEC0A64E42906BFDCDE8F49E67BDA885ECA7783A2A9C697332B1F488 2C185BCF0A760B59E32B06EBE5DB6509D4E96A5864BCC95C17A09CB3406AD543 95031E3E608C6BCD643F61E6E1ED7FC4E027A72E82ADA61D48AC1E7B9E5E4EB4 67510B114664F6C7B931DDF5D85D1C48D548A1512B8B4A53451F9B12F943D80C 3A1947964ED8AEA1E218138F647263E9D96BFBE8B58558DDE011B478B74C1449 48008521E4D9A3C4683283A6DAC3D1AE86829F3ECE5EA2A58A41B1923019D279 08248D3FC131F2A82E286631FCDDCC721ADDC9D4DCED202CBE647658479AE22B A38C5D0BDF0BB27C1F349DB3B2D250099CE3164CC7A216C22E2067D1935E4B06 7E353D18043E36D07DB5C7F5A545AF18B42F01AD2E71FAF101412FB69A3633C8 11232A7C78CDE8AD78B4CDC852BD8BB52C51B77F52D1EF4A8827C07ECD878659 47F3A929E1D3A66DC624FB8308D897DB2EB4717395FABC39E6298E022556B51F 2BB65F29BC1A8C0134CE40DCEDC6CB7E78CEC0228ACD527885C37033A67F853F AA8FBDAE73056BDE1F03C510E26588835B2FE9D1D78BF11BCBF32E7AC77C8F 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 72 /H put dup 75 /K put dup 76 /L 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 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 17 /eta put dup 25 /pi put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 59 /comma put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 72 /H put dup 76 /L put dup 78 /N put dup 84 /T put dup 101 /e put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 25 /pi put dup 27 /sigma put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 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 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67 3C15C2153C061EB541F66C11E7EE77D5D77C0B11E1AC55101DA976CCACAB6993 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 22 /macron put dup 34 /quotedblright 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 64 /at 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 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 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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Fp(Vitali)33 b(V)-8 b(ougalter)111 612 y(Univ)m(ersit)m(y)35 b(of)d(T)-8 b(oron)m(to,)33 b(Departmen)m(t)g(of)f(Mathematics,)i(T)-8 b(oron)m(to,)33 b(ON,)g(M5S)f(2E4,)h(Canada)1237 780 y(e-mail:)44 b(vitali@math.toron)m(to.edu)1706 1397 y Fq(Abstract)86 1565 y Fp(W)-8 b(e)44 b(pro)m(v)m(e)h(the)f(instabilit)m (y)h(of)e(threshold)h(resonances)h(and)f(eigen)m(v)-5 b(alues)45 b(of)e(the)h(linearized)h(NLS)-60 1734 y(op)s(erator.)f(W)-8 b(e)33 b(compute)h(the)g(asymptotic)g(appro)m(ximations)g(of)f(the)g (eigen)m(v)-5 b(alues)35 b(app)s(earing)e(from)g(the)-60 1902 y(endp)s(oin)m(t)28 b(singularities)g(in)f(terms)h(of)e(the)i(p)s (erturbations)f(applied)h(to)f(the)g(original)g(NLS)g(equation.)42 b(Our)-60 2071 y(metho)s(d)33 b(in)m(v)m(olv)m(es)i(suc)m(h)f(tec)m (hniques)i(as)d(the)g(Birman-Sc)m(h)m(winger)h(principle)g(and)f(the)g (F)-8 b(esh)m(bac)m(h)34 b(map.)1543 2598 y Fq(1.)50 b(In)m(tro)s(duction)86 2856 y Fp(In)33 b(this)g(w)m(ork)h(w)m(e)g 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Fo(\036)1635 3878 y Fr(0)1674 3863 y Fp(\()p Fo(x)p Fp(\))p Fo(e)1850 3822 y Fn(i!)r(t)1951 3863 y Fo(;)212 b(!)31 b(>)c Fp(0)p Fo(;)1314 b Fp(\(2\))-60 4078 y(where)34 b Fo(\036)280 4093 y Fr(0)319 4078 y Fp(\()p Fo(x)p Fp(\))f(is)g(an)f (exp)s(onen)m(tially)j(decreasing)f(solution)f(of)f(the)h(elliptic)h (problem,)1054 4329 y Fm(\000)p Fp(\001)p Fo(\036)1270 4344 y Fr(0)1332 4329 y Fp(+)22 b Fo(!)t(\036)1553 4344 y Fr(0)1614 4329 y Fp(+)g Fo(U)1778 4344 y Fr(0)1818 4329 y Fp(\()p Fo(x)p Fp(\))p Fo(\036)2007 4344 y Fr(0)2068 4329 y Fp(+)g Fo(F)2229 4344 y Fr(0)2269 4329 y Fp(\()p Fo(\036)2365 4344 y Fr(0)2404 4286 y(2)2443 4329 y Fp(\))p Fo(\036)2539 4344 y Fr(0)2606 4329 y Fp(=)28 b(0)p Fo(;)989 b Fp(\(3\))-60 4579 y(suc)m(h)42 b(that)f Fo(\036)446 4594 y Fr(0)526 4579 y Fp(:)h Fl(R)667 4543 y Fr(3)747 4579 y Fm(!)f Fl(R)p Fo(;)58 b(\036)1103 4594 y Fr(0)1183 4579 y Fm(2)42 b Fl(C)1363 4543 y Fk(1)1438 4579 y Fp(.)67 b(The)42 b(existence)h(of)d(suc)m(h)i(solutions)g(w)m(as)g(pro)m(v)m (en)g(for)e(a)g(large)-60 4748 y(class)f(of)f(nonlinearities)i(\(see)g (e.g.)99 b([A)33 b(02)o(],)79 b([BL)32 b(83I],)78 b([BL)33 b(83I)s(I],)78 b([BLP)33 b(81],)78 b([M)33 b(93)o(],)79 b([Str)32 b(77]\).)p -60 4836 1584 4 v 52 4897 a Fj(1)89 4927 y Fi(AMS)d(sub)5 b(ject)27 b(classi\014cation:)36 b(Primary:)f(35Q55;)25 b(Secondary:)36 b(47J15,)25 b(81Q05)1896 5176 y Fp(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop -60 -73 a Fp(These)31 b(solutions)e(are)g(called) g(the)h(ground)e(states)i(if)f(they)g(are)g(p)s(ositiv)m(e)h(in)f(the)g (whole)h(space)g Fl(R)3511 -109 y Fr(3)3579 -73 y Fp(and)f(the)-60 95 y(excited)34 b(states)g(if)e(they)h(are)g(sign-inde\014nite.)45 b(Linearization)33 b(of)f(the)h(NLS)g(equation)g(near)g(the)f(solution) -60 264 y(\(2\))g(leads)i(to)e(the)h(sp)s(ectral)g(problem)h(for)e(the) h(op)s(erator)e Fm(L)2153 279 y Fr(0)2220 264 y Fp(:=)d Fo(\033)2406 279 y Fr(1)2446 264 y Fp(H)2519 279 y Fr(0)2591 264 y Fp(considered)34 b(on)f Fo(L)3270 228 y Fr(2)3310 264 y Fp(\()p Fl(R)3420 228 y Fr(3)3459 264 y Fo(;)17 b Fl(C)3575 228 y Fr(2)3614 264 y Fp(\):)1700 519 y Fm(L)1769 534 y Fr(0)1808 519 y Fq(u)28 b Fp(=)f Fo(z)t Fq(u)p Fo(;)1636 b Fp(\(4\))-60 774 y(where)34 b Fq(u)c Fp(=)e(\()p Fo(u;)17 b(w)s Fp(\))667 738 y Fn(T)721 774 y Fo(;)g(\033)820 789 y Fr(1)893 774 y Fp(is)33 b(the)h(standard)g(P)m(auli)g(matrix)f (and)h(the)g(energy)g(op)s(erator)e(H)3278 789 y Fr(0)3350 774 y Fp(is)i(the)g(second)-60 942 y(v)-5 b(ariation)32 b(of)g(the)h(Hamiltonian)g(functional)g(for)f(the)h(NLS)g(equation)g (\(1\):)955 1266 y Fo(\033)1010 1281 y Fr(1)1078 1266 y Fp(=)1181 1066 y Fh(0)1181 1245 y(@)1310 1191 y Fp(0)83 b(1)1310 1359 y(1)g(0)1532 1066 y Fh(1)1532 1245 y(A)1636 1266 y Fo(;)212 b Fp(H)1948 1281 y Fr(0)2015 1266 y Fp(=)2119 1066 y Fh(0)2119 1245 y(@)2247 1191 y Fo(L)2313 1206 y Fr(+)p Fn(;)33 b Fr(0)2625 1191 y Fp(0)2329 1359 y(0)165 b Fo(L)2609 1374 y Fk(\000)p Fn(;)33 b Fr(0)2797 1066 y Fh(1)2797 1245 y(A)3775 1266 y Fp(\(5\))-60 1585 y(where)k Fo(L)291 1600 y Fk(\006)p Fn(;)32 b Fr(0)470 1585 y Fp(=)g Fm(\000)p Fp(\001)25 b(+)f Fo(!)j Fp(+)d Fo(V)1106 1600 y Fk(\006)p Fn(;)32 b Fr(0)1288 1585 y Fp(is)k(the)g(pair)f(of)g(Sc)m (hr\177)-49 b(odinger)37 b(op)s(erators)e(with)h(the)g(p)s(oten)m (tials)g Fo(V)3754 1600 y Fk(\006)p Fn(;)c Fr(0)-60 1753 y Fp(giv)m(en)i(b)m(y)497 1922 y Fo(V)554 1937 y Fr(+)p Fn(;)e Fr(0)700 1922 y Fp(\()p Fo(x)p Fp(\))c(=)g Fo(U)1029 1937 y Fr(0)1068 1922 y Fp(\()p Fo(x)p Fp(\))23 b(+)f Fo(F)1383 1937 y Fr(0)1422 1922 y Fp(\()p Fo(\036)1518 1881 y Fr(2)1518 1946 y(0)1557 1922 y Fp(\))h(+)f(2)p Fo(F)1842 1881 y Fk(0)1828 1946 y Fr(0)1867 1922 y Fp(\()p Fo(\036)1963 1881 y Fr(2)1963 1946 y(0)2002 1922 y Fp(\))p Fo(\036)2098 1881 y Fr(2)2098 1946 y(0)2137 1922 y Fo(;)212 b(V)2433 1937 y Fk(\000)p Fn(;)32 b Fr(0)2607 1922 y Fp(=)27 b Fo(U)2776 1937 y Fr(0)2816 1922 y Fp(\()p Fo(x)p Fp(\))c(+)f Fo(F)3131 1937 y Fr(0)3170 1922 y Fp(\()p 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b(05)o(],)79 b([V],)g([VP)33 b(06],)79 b([CGNT)33 b(07]\).)62 b(Its)39 b(sp)s(ectrum)h Fo(\033)t Fp(\()p Fm(L)3823 2828 y Fr(0)3862 2813 y Fp(\))-60 2981 y(is)33 b(symmetric)i(with)e(resp)s(ect)h(to)e(the)i(real)e(and)h (imaginary)g(axes.)45 b(The)33 b(essen)m(tial)i(sp)s(ectrum)f(consists) g(of)-60 3150 y(t)m(w)m(o)i(branc)m(hes)i(\()p Fm(\0001)p Fo(;)17 b Fm(\000)p Fo(!)t Fp(])35 b(and)h([)p Fo(!)t(;)17 b Fm(1)p Fp(\))34 b(suc)m(h)j(that)e(the)h(p)s(oin)m(ts)h Fm(\006)p Fo(!)i Fp(are)d(called)g(the)g Fg(thr)-5 b(esholds)p Fp(,)36 b(while)-60 3318 y(the)44 b(eigen)m(v)-5 b(alues)46 b(could)f(b)s(e)f(lo)s(cated)g(an)m(ywhere)h(in)f(the)h(complex)g (plane.)78 b(F)-8 b(or)43 b(the)h(op)s(erator)g Fm(L)3714 3333 y Fr(0)3796 3318 y Fp(its)-60 3487 y(p)s(oin)m(t)36 b(sp)s(ectrum)i(is)f(denoted)g(as)f Fo(\033)1280 3502 y Fn(p)1321 3487 y Fp(\()p Fm(L)1428 3502 y Fr(0)1467 3487 y Fp(\).)54 b(The)37 b(notations)g Fm(h)p Fq(f)10 b Fo(;)17 b Fq(g)r Fm(i)35 b Fp(are)h(used)i(for)d(the)i(inner)g(pro)s (duct)f(of)-60 3655 y Fq(f)10 b Fo(;)17 b Fq(g)34 b Fm(2)e Fo(L)282 3619 y Fr(2)322 3655 y Fp(\()p Fl(R)432 3619 y Fr(3)471 3655 y Fo(;)17 b Fl(C)587 3619 y Fr(2)626 3655 y Fp(\),)134 b(\()p Fo(f)5 b(;)17 b(g)t Fp(\))34 b(for)h Fo(f)5 b(;)17 b(g)35 b Fm(2)d Fo(L)1578 3619 y Fr(2)1618 3655 y Fp(\()p Fl(R)1728 3619 y Fr(3)1768 3655 y Fp(\),)133 b Fm(k)p Fo(f)11 b Fm(k)2125 3670 y Fr(2)2199 3655 y Fp(for)34 b(the)i(norm)f(of)g Fo(f)43 b Fm(2)32 b Fo(L)3147 3619 y Fr(2)3187 3655 y Fp(\()p Fl(R)3297 3619 y Fr(3)3336 3655 y Fp(\),)k Fm(h)p Fq(u)p Fo(;)17 b Fq(v)r Fm(i)3682 3670 y Fn(N)3783 3655 y Fp(for)-60 3824 y(the)33 b(scalar)g(pro)s(duct)g(of)f Fq(u)p Fo(;)17 b Fq(v)29 b Fm(2)f Fl(C)1222 3788 y Fn(N)1322 3824 y Fp(and)33 b Fm(j)p Fq(u)p Fm(j)f Fp(for)g(the)h(length)g(of)f Fq(u)c Fm(2)g Fl(C)2644 3788 y Fr(3)2683 3824 y Fp(.)86 4085 y(A)22 b(solution)g Fq(u)601 4100 y Fr(0)663 4085 y Fp(of)f(the)h(sp)s(ectral)h(problem)f(\(4\))g(at)f Fo(z)32 b 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b(on)h(the)f(bifurcations)i(from)e(a)g(zero)g (resonance)i(or)e(a)-60 4590 y(simple)d(eigen)m(v)-5 b(alue)39 b(at)d Fo(z)k Fp(=)34 b Fo(!)40 b Fp(in)d(three)h(dimensions) h(b)m(y)e(estimating)h(these)g(bifurcations)g(in)f(terms)g(of)-60 4759 y(the)44 b(p)s(erturbation)g(applied)h(to)f(the)g(original)g(NLS)g (equation.The)i(main)e(result)h(of)e(the)h(w)m(ork)h(is)g(the)-60 4927 y(pro)s(of)31 b(of)g(the)h(structural)h(instabilit)m(y)g(of)e(the) i(singularities)g(of)e(b)s(oth)g(kinds)j(at)d(the)h(end)h(of)e(the)h (essen)m(tial)1896 5176 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop -60 -73 a Fp(sp)s(ectrum)37 b Fo(z)h Fp(=)32 b Fo(!)39 b Fp(under)d(generic)h(p)s(erturbations)f(and)g(the)g (computation)g(of)f(the)h(asymptotics)h(of)f(the)-60 95 y(eigen)m(v)-5 b(alues)28 b(whic)m(h)g(can)e(app)s(ear)g(in)h(the)f (neigh)m(b)s(orho)s(o)s(d)g(of)g(the)h(p)s(oin)m(t)f Fo(z)32 b Fp(=)c Fo(!)h Fp(when)f(suc)m(h)f(p)s(erturbations)-60 264 y(are)33 b(applied.)44 b(W)-8 b(e)33 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y(p)1435 1898 y Fo(;)1502 1872 y Fp(^)1479 1898 y Fo(B)t Fq(v)1618 1869 y Fn(i)1616 1922 y(p)1656 1898 y Fm(i)1727 1814 y Fp(+)g Fo(O)s Fp(\()p Fo(")1997 1746 y Fe(3)p 1996 1758 31 3 v 1996 1799 a(2)2041 1814 y Fp(\))p Fo(;)121 b(i)28 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)2697 1773 y Fr(+)2687 1847 y Fn(B)2743 1828 y Fc(\000)2822 1814 y Fp(+)22 b Fo(N)3008 1773 y Fk(\000)2998 1847 y Fn(B)3054 1828 y Fe(+)86 2122 y Fg(\(ii\))34 b(When)g Fm(h)p Fq(v)641 2086 y Fn(i)639 2147 y(p)679 2122 y Fo(;)746 2097 y Fp(^)723 2122 y Fo(B)5 b Fq(v)863 2086 y Fn(i)861 2147 y(p)900 2122 y Fm(i)28 b Fo(<)f Fp(0)34 b Fg(and)g Fo(\025)1399 2086 y Fn(i)1399 2147 y(p)1466 2122 y Fo(<)28 b Fp(0)34 b Fg(the)g(eigenvalues)f(move)h(in)g(the)g(c)-5 b(omplex)33 b(plane,)h(such)g(that)-60 2290 y(they)h(b)-5 b(e)g(c)g(ome)34 b(c)-5 b(omplex)34 b(c)-5 b(onjugate)34 b(p)-5 b(airs)34 b(and)h(their)g(asymptotics)f(ar)-5 b(e)35 b(given)f(by)240 2579 y Fo(z)285 2594 y Fn(i)313 2579 y Fp(\()p Fo(")p Fp(\))28 b(=)f Fo(!)f Fp(+)c 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y(p)-5 b(erturb)g(ation)34 b(for)f Fo(")g Fg(smal)5 b(l)33 b(enough,)g(only)g(the)g(bifur)-5 b(c)g(ations)33 b(of)g(the)h(\(i\))f(and)g(the)g(\(iii\))g(kind)g(in)g (The)-5 b(or)g(em)-60 4759 y(1.4)36 b(ar)-5 b(e)36 b(p)-5 b(ossible,)36 b(sinc)-5 b(e)36 b(the)g(eigenvalues)f(of)h(the)h(line)-5 b(arize)g(d)35 b(op)-5 b(er)g(ator)36 b(ar)-5 b(e)37 b(r)-5 b(estricte)g(d)36 b(to)h(the)f(r)-5 b(e)g(al)37 b(and)-60 4927 y(imaginary)d(axes)g(in)h(this)g(c)-5 b(ase.)1896 5176 y Fp(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 86 -73 a Fp(Bifurcations)24 b(of)e(resonances)i (at)f(the)g(endp)s(oin)m(ts)h(w)m(ere)g(studied)g(in)f(one)g(dimension) h(using)f(the)h(metho)s(d)-60 95 y(of)k(Ev)-5 b(ans)30 b(functions)f(\(see)59 b([PKA)33 b(98)o(],)59 b([KS)32 b(02],)58 b([KS)33 b(04)o(]\),)d(n)m(umerically)h(in)e(one)g(and)f(t)m (w)m(o)i(dimensions)-60 264 y(in)g(\()g([CGNT)j(07)o(]\).)43 b(The)31 b(primary)f(goal)f(of)g(the)i(study)f(of)g(suc)m(h)h 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y(here)37 b(is)h(whether)f(the)g(endp)s(oin)m(t)h(bifurcations)f(lead)g(to)g(the) g(app)s(earance)g(of)f(the)h(eigen)m(v)-5 b(alues)39 b(lying)e(in)-60 1612 y(the)c(upp)s(er)h(half-plane)f(whic)m(h)h(leads) g(to)f(instabilitary)h(of)e(solitary)i(w)m(a)m(v)m(es.)47 b(Theorems)35 b(1.3)d(and)h(1.4)g(are)-60 1781 y(the)28 b(statemen)m(ts)h(ab)s(out)e(the)g(fate)h(of)e(the)i(threshold)g (resonance)h(and)f(the)f(N-fold)g(degenerate)h(eigen)m(v)-5 b(alue)-60 1949 y(when)33 b(p)s(erturbation)g(is)g(applied)g(to)f(the)h (NLS)f(equation,)h(whic)m(h)h(is)f(the)g(generalization)g(of)e(the)i(w) m(ork)g(of)-60 2118 y(\()38 b([CP)c(05)o(]\),)40 b(pro)m(v)m(en)g(via)e (tec)m(hniques)k(di\013eren)m(t)d(than)f(the)h(resolv)m(en)m(t)i (expansions)f(and)e(the)h(mappings)-60 2286 y(b)s(et)m(w)m(een)c(the)e (w)m(eigh)m(ted)h(Sob)s(olev)f(spaces.)86 2554 y(In)h(the)g(next)g (section)g(w)m(e)h(in)m(v)m(estigate)g(ho)m(w)f(the)g(p)s(erturbations) g(to)f(the)g(equation)i(\(1\))d(en)m(tail)i(mo)s(di-)-60 2723 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4416 y Fr(+)p Fn(;)35 b Fr(0)2382 4401 y Fp(\))2420 4360 y Fk(\000)p Fr(1)2515 4401 y Fp(\()p Fo(U)2619 4416 y Fn(p)2658 4401 y Fo(\036)2716 4416 y Fr(0)2756 4401 y Fp(\))-60 4623 y Fg(in)g(the)f(\014rst)h(c)-5 b(ase)35 b(and)280 4892 y Fo(V)337 4907 y Fr(+)p Fn(;p)479 4892 y Fp(=)27 b Fm(\000)p Fp(2)p Fo(\036)766 4907 y Fr(0)806 4892 y Fm(f)p Fp(3)p Fo(F)982 4850 y Fk(0)968 4916 y Fr(0)1007 4892 y Fp(\()p Fo(\036)1103 4850 y Fr(2)1103 4916 y(0)1142 4892 y Fp(\))22 b(+)g(2)p Fo(\036)1407 4850 y Fr(2)1407 4916 y(0)1446 4892 y Fo(F)1523 4850 y Fk(00)1509 4916 y Fr(0)1565 4892 y Fp(\()p Fo(\036)1661 4850 y Fr(2)1661 4916 y(0)1700 4892 y Fp(\))p Fm(g)p Fp(\()p Fo(L)1892 4907 y Fr(+)p Fn(;)35 b Fr(0)2041 4892 y Fp(\))2079 4850 y Fk(\000)p Fr(1)2173 4892 y Fp(\()p Fo(F)2274 4907 y Fn(p)2314 4892 y Fp(\()p Fo(\036)2410 4850 y Fr(2)2410 4916 y(0)2449 4892 y Fp(\))p Fo(\036)2545 4907 y Fr(0)2584 4892 y Fp(\))22 b(+)g Fo(F)2805 4907 y Fn(p)2845 4892 y Fp(\()p Fo(\036)2941 4850 y Fr(2)2941 4916 y(0)2980 4892 y Fp(\))g(+)g(2)p Fo(F)3264 4850 y Fk(0)3250 4916 y Fn(p)3290 4892 y Fp(\()p Fo(\036)3386 4850 y Fr(2)3386 4916 y(0)3425 4892 y Fp(\))p Fo(\036)3521 4850 y Fr(2)3521 4916 y(0)1896 5176 y Fp(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 904 -73 a Fo(V)961 -58 y Fk(\000)p Fn(;p)1103 -73 y Fp(=)27 b Fm(\000)p Fp(2)p Fo(\036)1390 -58 y Fr(0)1430 -73 y Fo(F)1507 -114 y Fk(0)1493 -49 y Fr(0)1532 -73 y Fp(\()p Fo(\036)1628 -114 y Fr(2)1628 -49 y(0)1668 -73 y Fp(\)\()p Fo(L)1810 -58 y Fr(+)p Fn(;)34 b Fr(0)1959 -73 y Fp(\))1997 -114 y Fk(\000)p Fr(1)2091 -73 y Fp(\()p Fo(F)2192 -58 y Fn(p)2232 -73 y Fp(\()p Fo(\036)2328 -114 y Fr(2)2328 -49 y(0)2367 -73 y Fp(\))p Fo(\036)2463 -58 y Fr(0)2502 -73 y Fp(\))22 b(+)g Fo(F)2723 -58 y Fn(p)2763 -73 y Fp(\()p Fo(\036)2859 -114 y Fr(2)2859 -49 y(0)2898 -73 y Fp(\))-60 143 y Fg(in)35 b(the)f(se)-5 b(c)g(ond)34 b(c)-5 b(ase.)86 404 y Fp(T)d(o)33 b(ev)-5 b(aluate)32 b(the)h(corrections)g(to)f(the)h(standing)f(solitary)h(w)m 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Fr(0)2491 1673 y Fp(\()p Fo(\036)2587 1632 y Fr(2)2626 1673 y Fp(\))p Fo(\036)28 b Fp(=)f(0)-60 1927 y(A)32 b(straigh)m(tforw)m(ard)h(computation)g(using)g(\(3\))f(leads)h(to)f (the)h(equation)g(for)f(the)h(correction)g(term)g(to)f(the)-60 2096 y(solitary)h(w)m(a)m(v)m(e)i(solution)1191 2264 y Fo(\036)1249 2279 y Fn(p)1316 2264 y Fp(=)27 b Fo(L)1485 2223 y Fk(\000)p Fr(1)1485 2289 y(+)p Fn(;)33 b Fr(0)1632 2264 y Fp([)p Fm(\000)p Fo(")p Fm(K)1858 2279 y Fr(1)1898 2264 y Fp(\()p Fo(\036)1994 2279 y Fr(0)2033 2264 y Fp(\))22 b Fm(\000)h(K)2269 2279 y Fr(2)2308 2264 y Fp(\()p Fo(\036)2404 2279 y Fr(0)2443 2264 y Fo(;)17 b(\036)2545 2279 y Fn(p)2584 2264 y Fp(\)])1078 b(\(13\))-60 2481 y(with)32 b Fm(K)237 2496 y Fr(1)276 2481 y Fp(\()p Fo(\036)372 2496 y Fr(0)411 2481 y Fp(\))c(=)f Fo(U)646 2496 y Fn(p)686 2481 y Fo(\036)744 2496 y Fr(0)815 2481 y Fp(and)k Fm(K)1079 2496 y Fr(2)1118 2481 y Fp(\()p Fo(\036)1214 2496 y Fr(0)1253 2481 y Fo(;)17 b(\036)1355 2496 y Fn(p)1394 2481 y Fp(\))28 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3170 y Fq(F)-9 b(act)38 b(1)32 b Fg(If)j Fo(\036)582 3185 y Fn(p)649 3170 y Fm(2)28 b Fo(H)832 3134 y Fr(2)871 3170 y Fp(\()p Fl(R)981 3134 y Fr(3)1020 3170 y Fp(\))35 b Fg(then)f Fo(L)1375 3129 y Fk(\000)p Fr(1)1375 3195 y(+)p Fn(;)h Fr(0)1525 3170 y Fp([)p Fo(")p Fm(K)1674 3185 y Fr(1)1713 3170 y Fp(\()p Fo(\036)1809 3185 y Fr(0)1848 3170 y Fp(\))22 b(+)g Fm(K)2082 3185 y Fr(2)2121 3170 y Fp(\()p Fo(\036)2217 3185 y Fr(0)2256 3170 y Fo(;)17 b(\036)2358 3185 y Fn(p)2398 3170 y Fp(\)])27 b Fm(2)h Fo(H)2673 3134 y Fr(2)2712 3170 y Fp(\()p Fl(R)2822 3134 y Fr(3)2862 3170 y Fp(\))p Fg(.)86 3431 y Fq(F)-9 b(act)38 b(2)32 b Fg(If)j Fo(";)51 b Fm(k)p Fo(\036)756 3380 y Fr(\(1\))756 3441 y Fn(p)850 3431 y Fm(k)900 3447 y Fn(H)963 3428 y Fe(2)1001 3431 y Fo(;)h Fm(k)p Fo(\036)1188 3380 y Fr(\(1\))1188 3441 y Fn(p)1282 3431 y Fm(k)1332 3447 y Fn(H)1395 3428 y Fe(2)1468 3431 y Fg(ar)-5 b(e)35 b(smal)5 b(l)34 b(enough,)g(then)593 3685 y Fm(k)p Fo(L)709 3644 y Fk(\000)p Fr(1)709 3710 y(+)p Fn(;)e Fr(0)855 3685 y Fp([)p Fm(K)958 3700 y Fr(2)998 3685 y Fp(\()p Fo(\036)1094 3700 y Fr(0)1133 3685 y Fo(;)17 b(\036)1235 3644 y Fr(\(1\))1235 3710 y Fn(p)1329 3685 y Fp(\)])22 b Fm(\000)h Fo(L)1582 3644 y Fk(\000)p Fr(1)1582 3710 y(+)p Fn(;)32 b Fr(0)1728 3685 y Fp([)p Fm(K)1831 3700 y Fr(2)1871 3685 y Fp(\()p Fo(\036)1967 3700 y Fr(0)2006 3685 y Fo(;)17 b(\036)2108 3644 y Fr(\(2\))2108 3710 y Fn(p)2202 3685 y Fp(\)])p Fm(k)2317 3702 y Fn(H)2380 3683 y Fe(2)2446 3685 y Fm(\024)2561 3618 y Fp(1)p 2561 3662 49 4 v 2561 3753 a(2)2620 3685 y Fm(k)p Fo(\036)2728 3644 y Fr(\(1\))2728 3710 y Fn(p)2844 3685 y Fm(\000)22 b Fo(\036)3001 3644 y Fr(\(2\))3001 3710 y Fn(p)3096 3685 y Fm(k)3146 3702 y Fn(H)3209 3683 y Fe(2)86 4031 y Fg(Pr)-5 b(o)g(of)47 b(of)g(the)f(F)-7 b(act)46 b(1.)83 b Fp(Since)46 b(the)h(p)s(oten)m(tial)e(p)s(erturbation)h Fm(j)p Fo(U)2605 4046 y Fn(p)2645 4031 y Fp(\()p Fo(x)p Fp(\))p Fm(j)k(\024)g Fo(C)r(;)62 b(x)50 b Fm(2)h Fl(R)3436 3995 y Fr(3)3475 4031 y Fp(,)e(w)m(e)e(ha)m(v)m(e)-60 4200 y Fm(kK)66 4215 y Fr(1)105 4200 y Fp(\()p Fo(\036)201 4215 y Fr(0)240 4200 y Fp(\))p Fm(k)328 4215 y Fr(2)402 4200 y Fm(\024)35 b Fo(C)7 b Fm(k)p Fo(\036)699 4215 y Fr(0)738 4200 y Fm(k)788 4215 y Fr(2)861 4200 y Fo(<)34 b Fm(1)p Fp(.)55 b(Here)38 b(and)e(further)h(with)g(a)f(sligh)m(t)i (abuse)f(of)f(notation)g Fo(C)44 b Fp(stands)37 b(for)f(a)-60 4369 y(\014nite,)d(p)s(ositiv)m(e)h(constan)m(t.)45 b(Th)m(us)34 b Fo(L)1322 4327 y Fk(\000)p Fr(1)1322 4393 y(+)p Fn(;)e Fr(0)1469 4369 y Fp([)p Fo(")p Fm(K)1618 4384 y Fr(1)1657 4369 y Fp(\()p Fo(\036)1753 4384 y Fr(0)1792 4369 y Fp(\)])c Fm(2)g Fo(H)2068 4332 y Fr(2)2107 4369 y Fp(\()p Fl(R)2217 4332 y Fr(3)2256 4369 y Fp(\).)86 4537 y(Clearly)37 b Fm(j)p Fo("U)568 4552 y Fn(p)608 4537 y Fo(\036)666 4552 y Fn(p)705 4537 y Fm(j)c(\024)g Fo("C)7 b Fm(j)p Fo(\036)1085 4552 y Fn(p)1124 4537 y Fm(j)p Fp(.)53 b(Let)36 b Fo(\021)t Fp(\()p Fo(\036)p Fp(\))c(:=)h Fo(F)1827 4552 y Fr(0)1867 4537 y Fp(\()p Fo(\036)1963 4501 y Fr(2)2002 4537 y Fp(\))p Fo(\036)p Fp(.)53 b(F)-8 b(or)35 b(this)h(smo)s(oth)g(function)g Fo(\021)t Fp(\()p Fo(\036)p Fp(\))d Fm(2)h Fo(C)3678 4501 y Fk(1)3752 4537 y Fp(\()p Fl(R)p Fp(\))-60 4706 y(w)m(e)g(ha)m(v)m(e)796 4874 y Fo(\021)t Fp(\()p Fo(\036)p Fp(\))22 b Fm(\000)g Fo(\021)t Fp(\()p Fo(\036)1251 4889 y Fr(0)1290 4874 y Fp(\))g Fm(\000)h Fo(\021)1502 4833 y Fk(0)1525 4874 y Fp(\()p Fo(\036)1621 4889 y Fr(0)1660 4874 y Fp(\)\()p Fo(\036)f Fm(\000)g Fo(\036)1973 4889 y Fr(0)2013 4874 y Fp(\))27 b(=)2182 4739 y Fh(Z)2281 4765 y Fn(\036)2237 4964 y(\036)2279 4973 y Fe(0)2344 4764 y Fh(\020)2420 4739 y(Z)2520 4765 y Fn(s)2476 4964 y(\036)2518 4973 y Fe(0)2573 4874 y Fo(\021)2625 4833 y Fk(00)2667 4874 y Fp(\()p Fo(q)t Fp(\))p Fo(dq)2888 4764 y Fh(\021)2947 4874 y Fo(ds)1896 5176 y Fp(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop -60 -73 a Fp(and)33 b Fo(\036)188 -58 y Fn(p)255 -73 y Fm(2)28 b Fo(L)415 -109 y Fk(1)490 -73 y Fp(\()p Fl(R)600 -109 y Fr(3)639 -73 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Fn(p)1698 720 y Fp(\))p Fm(j)27 b(\024)i Fo("C)7 b Fm(j)p Fo(\036)2106 735 y Fn(p)2144 720 y Fm(j)22 b Fp(+)g Fo(C)7 b Fm(j)p Fo(\036)2455 735 y Fn(p)2494 720 y Fm(j)2522 679 y Fr(2)-60 985 y Fp(Since)35 b Fo(\036)254 1000 y Fn(p)323 985 y Fm(2)30 b Fo(L)485 948 y Fr(2)525 985 y Fp(\()p Fl(R)635 948 y Fr(3)674 985 y Fp(\))23 b Fm(\\)g Fo(L)890 948 y Fk(1)965 985 y Fp(\()p Fl(R)1075 948 y Fr(3)1115 985 y Fp(\),)34 b(w)m(e)h(ha)m(v)m(e)g Fo(\036)1643 1000 y Fn(p)1712 985 y Fm(2)30 b Fo(L)1874 948 y Fr(4)1914 985 y Fp(\()p Fl(R)2024 948 y Fr(3)2063 985 y Fp(\).)47 b(Th)m(us)35 b Fm(K)2499 1000 y Fr(2)2538 985 y Fp(\()p Fo(\036)2634 1000 y Fr(0)2674 985 y Fo(;)17 b(\036)2776 1000 y Fn(p)2815 985 y Fp(\))29 b Fm(2)h Fo(L)3044 948 y Fr(2)3084 985 y Fp(\()p Fl(R)3194 948 y Fr(3)3233 985 y Fp(\))k(and,)g(therefore)-60 1153 y Fo(L)6 1112 y Fk(\000)p Fr(1)6 1178 y(+)p Fn(;)e Fr(0)153 1153 y Fp([)p Fm(K)256 1168 y Fr(2)295 1153 y Fp(\()p Fo(\036)391 1168 y Fr(0)430 1153 y Fo(;)17 b(\036)532 1168 y Fn(p)571 1153 y Fp(\)])28 b Fm(2)g Fo(H)847 1117 y Fr(2)886 1153 y Fp(\()p Fl(R)996 1117 y Fr(3)1036 1153 y Fp(\).)p 3824 1324 60 62 v 86 1588 a Fg(Pr)-5 b(o)g(of)32 b(of)f(the)h(F)-7 b(act)30 b(2.)42 b Fp(Let)29 b Fo(\036)1180 1603 y Fr(1)1247 1588 y Fp(:=)f Fo(\036)1436 1603 y Fr(0)1490 1588 y Fp(+)15 b Fo(\036)1639 1537 y Fr(\(1\))1639 1598 y Fn(p)1762 1588 y Fp(and)29 b Fo(\036)2006 1603 y Fr(2)2073 1588 y Fp(:=)e Fo(\036)2261 1603 y Fr(0)2315 1588 y Fp(+)15 b Fo(\036)2464 1537 y Fr(\(2\))2464 1598 y Fn(p)2558 1588 y Fp(.)42 b(The)30 b(norms)g Fm(k)p Fo(\036)3222 1537 y Fr(\(1)p Fn(;)p Fr(2\))3222 1598 y Fn(p)3371 1588 y Fm(k)3421 1603 y Fk(1)3524 1588 y Fp(are)f(\014nite)-60 1756 y(and)40 b(su\016cien)m(tly)j(small)e(b)m(y)h(means)f(of)f(the)g (Sob)s(olev)h(em)m(b)s(edding)h(theorem.)67 b(A)41 b(trivial)f (computation)-60 1925 y(yields)450 2093 y Fm(K)526 2108 y Fr(2)566 2093 y Fp(\()p Fo(\036)662 2108 y Fr(0)701 2093 y Fo(;)17 b(\036)803 2052 y Fr(\(1\))803 2118 y Fn(p)896 2093 y Fp(\))22 b Fm(\000)h(K)1132 2108 y Fr(2)1171 2093 y Fp(\()p Fo(\036)1267 2108 y Fr(0)1307 2093 y Fo(;)17 b(\036)1409 2052 y Fr(\(2\))1409 2118 y Fn(p)1502 2093 y Fp(\))28 b(=)f Fo(F)1734 2108 y Fr(0)1774 2093 y Fp(\()p Fo(\036)1870 2052 y Fr(2)1870 2118 y(1)1909 2093 y Fp(\))p Fo(\036)2005 2108 y Fr(1)2066 2093 y Fm(\000)c Fo(F)2229 2108 y Fr(0)2269 2093 y Fp(\()p Fo(\036)2365 2052 y Fr(2)2365 2118 y(2)2404 2093 y Fp(\))p Fo(\036)2500 2108 y Fr(2)2561 2093 y Fp(+)f Fo("U)2771 2108 y Fn(p)2811 2093 y Fp(\()p Fo(\036)2907 2052 y Fr(\(1\))2907 2118 y Fn(p)3023 2093 y Fm(\000)g Fo(\036)3180 2052 y Fr(\(2\))3180 2118 y Fn(p)3274 2093 y Fp(\))p Fm(\000)970 2314 y(\000)p Fp(2)p Fo(F)1173 2273 y Fk(0)1159 2339 y Fr(0)1199 2314 y Fp(\()p Fo(\036)1295 2273 y Fr(2)1295 2339 y(0)1334 2314 y Fp(\))p Fo(\036)1430 2273 y Fr(2)1430 2339 y(0)1469 2314 y Fp(\()p Fo(\036)1565 2273 y Fr(\(1\))1565 2339 y Fn(p)1681 2314 y Fm(\000)h Fo(\036)1839 2273 y Fr(\(2\))1839 2339 y Fn(p)1933 2314 y Fp(\))f Fm(\000)h Fo(F)2156 2329 y Fr(0)2195 2314 y Fp(\()p Fo(\036)2291 2273 y Fr(2)2291 2339 y(0)2330 2314 y Fp(\)\()p Fo(\036)2464 2273 y Fr(\(1\))2464 2339 y Fn(p)2580 2314 y Fm(\000)g Fo(\036)2738 2273 y Fr(\(2\))2738 2339 y Fn(p)2832 2314 y Fp(\))-60 2535 y(Due)39 b(to)f(the)h(b)s(ound)g Fm(j)p Fo(U)851 2550 y Fn(p)891 2535 y Fp(\()p Fo(x)p Fp(\))p Fm(j)f(\024)h Fo(C)r(;)55 b(x)38 b Fm(2)h Fl(R)1628 2499 y Fr(3)1706 2535 y Fp(w)m(e)h(ha)m(v)m(e)g Fm(k)p Fo("U)2249 2550 y Fn(p)2288 2535 y Fp(\()p Fo(\036)2384 2484 y Fr(\(1\))2384 2545 y Fn(p)2505 2535 y Fm(\000)27 b Fo(\036)2667 2484 y Fr(\(2\))2667 2545 y Fn(p)2761 2535 y Fm(k)2811 2550 y Fr(2)2888 2535 y Fm(\024)39 b Fo("C)7 b Fm(k)p Fo(\036)3235 2484 y Fr(\(1\))3235 2545 y Fn(p)3355 2535 y Fm(\000)27 b Fo(\036)3517 2484 y Fr(\(2\))3517 2545 y Fn(p)3611 2535 y Fm(k)3661 2550 y Fr(2)3700 2535 y Fp(.)62 b(T)-8 b(o)-60 2704 y(estimate)34 b(the)f(remaining)g(terms)g (in)g(the)g(righ)m(t)g(side)h(of)e(the)h(iden)m(tit)m(y)h(ab)s(o)m(v)m (e)g(w)m(e)f(use)h(the)f(form)m(ula)726 2847 y Fh(Z)826 2873 y Fn(\036)868 2882 y Fe(1)781 3073 y Fn(\036)823 3082 y Fe(2)923 2872 y Fh(\020)999 2847 y(Z)1099 2873 y Fn(s)1054 3073 y(\036)1096 3082 y Fe(0)1152 2983 y Fo(\021)1204 2942 y Fk(00)1246 2983 y Fp(\()p Fo(q)t Fp(\))p Fo(dq)1467 2872 y Fh(\021)1526 2983 y Fo(ds)27 b Fp(=)h Fo(\021)t Fp(\()p Fo(\036)1902 2998 y Fr(1)1941 2983 y Fp(\))22 b Fm(\000)g Fo(\021)t Fp(\()p Fo(\036)2248 2998 y Fr(2)2287 2983 y Fp(\))g Fm(\000)h Fo(\021)2499 2942 y Fk(0)2522 2983 y Fp(\()p Fo(\036)2618 2998 y Fr(0)2657 2983 y Fp(\)\()p Fo(\036)2791 2998 y Fr(1)2852 2983 y Fm(\000)g Fo(\036)3010 2998 y Fr(2)3049 2983 y Fp(\))p Fo(;)-60 3259 y Fp(whic)m(h)34 b(implies)392 3523 y Fm(k)p Fo(F)505 3538 y Fr(0)544 3523 y Fp(\()p Fo(\036)640 3482 y Fr(2)640 3548 y(1)680 3523 y Fp(\))p Fo(\036)776 3538 y Fr(1)837 3523 y Fm(\000)22 b Fo(F)999 3538 y Fr(0)1039 3523 y Fp(\()p Fo(\036)1135 3482 y Fr(2)1135 3548 y(2)1174 3523 y Fp(\))p Fo(\036)1270 3538 y Fr(2)1331 3523 y Fm(\000)h Fp(2)p Fo(F)1557 3482 y Fk(0)1543 3548 y Fr(0)1582 3523 y Fp(\()p Fo(\036)1678 3482 y Fr(2)1678 3548 y(0)1717 3523 y Fp(\))p Fo(\036)1813 3482 y Fr(2)1813 3548 y(0)1853 3523 y Fp(\()p Fo(\036)1949 3482 y Fr(\(1\))1949 3548 y Fn(p)2065 3523 y Fm(\000)f Fo(\036)2222 3482 y Fr(\(2\))2222 3548 y Fn(p)2316 3523 y Fp(\))g Fm(\000)h Fo(F)2539 3538 y Fr(0)2579 3523 y Fp(\()p Fo(\036)2675 3482 y Fr(2)2675 3548 y(0)2714 3523 y Fp(\)\()p Fo(\036)2848 3482 y Fr(\(1\))2848 3548 y Fn(p)2964 3523 y Fm(\000)g Fo(\036)3122 3482 y Fr(\(2\))3122 3548 y Fn(p)3216 3523 y Fp(\))p Fm(k)3304 3538 y Fr(2)3370 3523 y Fm(\024)280 b Fp(\(15\))1109 3788 y Fm(\024)28 b Fo(C)7 b Fp(\()p Fm(k)p Fo(\036)1437 3747 y Fr(\(1\))1437 3813 y Fn(p)1530 3788 y Fm(k)1580 3803 y Fk(1)1677 3788 y Fp(+)22 b Fm(k)p Fo(\036)1883 3747 y Fr(\(2\))1883 3813 y Fn(p)1977 3788 y Fm(k)2027 3803 y Fk(1)2102 3788 y Fp(\))p Fm(k)p Fo(\036)2248 3747 y Fr(\(1\))2248 3813 y Fn(p)2363 3788 y Fm(\000)h Fo(\036)2521 3747 y Fr(\(2\))2521 3813 y Fn(p)2615 3788 y Fm(k)2665 3803 y Fr(2)2704 3788 y Fo(;)-60 4009 y Fp(and)33 b(therefore)441 4273 y Fm(kK)567 4288 y Fr(2)606 4273 y Fp(\()p Fo(\036)702 4288 y Fr(0)741 4273 y Fo(;)17 b(\036)843 4232 y Fr(\(1\))843 4298 y Fn(p)937 4273 y Fp(\))22 b Fm(\000)h(K)1173 4288 y Fr(2)1212 4273 y Fp(\()p Fo(\036)1308 4288 y Fr(0)1347 4273 y Fo(;)17 b(\036)1449 4232 y Fr(\(2\))1449 4298 y Fn(p)1543 4273 y Fm(k)1593 4288 y Fr(2)1660 4273 y Fm(\024)28 b Fo(C)7 b Fp(\()p Fo(")21 b Fp(+)i Fm(k)p Fo(\036)2154 4232 y Fr(\(1\))2154 4298 y Fn(p)2247 4273 y Fm(k)2297 4288 y Fk(1)2394 4273 y Fp(+)f Fm(j)p Fo(\036)2578 4232 y Fr(\(2\))2578 4298 y Fn(p)2672 4273 y Fm(k)2722 4288 y Fk(1)2796 4273 y Fp(\))p Fm(k)p Fo(\036)2942 4232 y Fr(\(1\))2942 4298 y Fn(p)3058 4273 y Fm(\000)h Fo(\036)3216 4232 y Fr(\(2\))3216 4298 y Fn(p)3310 4273 y Fm(k)3360 4288 y Fr(2)-60 4538 y Fp(Hence)34 b(w)m(e)g(arriv)m(e)f(at)955 4706 y Fm(k)p Fo(L)1071 4665 y Fk(\000)p Fr(1)1071 4731 y(+)p Fn(;)f Fr(0)1217 4706 y Fp([)p Fm(K)1320 4721 y Fr(2)1360 4706 y Fp(\()p Fo(\036)1456 4721 y Fr(0)1495 4706 y Fo(;)17 b(\036)1597 4665 y Fr(\(1\))1597 4731 y Fn(p)1691 4706 y Fp(\)])22 b Fm(\000)g Fo(L)1943 4665 y Fk(\000)p Fr(1)1943 4731 y(+)p Fn(;)33 b Fr(0)2090 4706 y Fp([)p Fm(K)2193 4721 y Fr(2)2233 4706 y Fp(\()p Fo(\036)2329 4721 y Fr(0)2368 4706 y Fo(;)17 b(\036)2470 4665 y Fr(\(2\))2470 4731 y Fn(p)2564 4706 y Fp(\)])p Fm(k)2679 4723 y Fn(H)2742 4704 y Fe(2)2808 4706 y Fm(\024)711 4927 y(\024)28 b(k)p Fo(L)932 4886 y Fk(\000)p Fr(1)932 4952 y(+)p Fn(;)k Fr(0)1078 4927 y Fm(k)1128 4944 y Fn(L)1176 4925 y Fe(2)1211 4944 y Fk(!)p Fn(H)1345 4925 y Fe(2)1383 4927 y Fo(C)7 b Fp(\()p Fo(")22 b Fp(+)g Fm(k)p Fo(\036)1772 4886 y Fr(\(1\))1772 4952 y Fn(p)1866 4927 y Fm(k)1916 4942 y Fk(1)2013 4927 y Fp(+)g Fm(k)p Fo(\036)2219 4886 y Fr(\(1\))2219 4952 y Fn(p)2312 4927 y Fm(k)2362 4942 y Fk(1)2437 4927 y Fp(\))p Fm(k)p Fo(\036)2583 4886 y Fr(\(1\))2583 4952 y Fn(p)2699 4927 y Fm(\000)g Fo(\036)2856 4886 y Fr(\(2\))2856 4952 y Fn(p)2951 4927 y Fm(k)3001 4944 y Fn(H)3064 4925 y Fe(2)3102 4927 y Fo(;)1896 5176 y Fp(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop -60 -72 a Fp(whic)m(h)34 b(completes)g(the)f(pro) s(of)f(of)g(the)h(F)-8 b(act)32 b(2)h(due)g(to)f(the)h(smallness)i(of)d Fo(")g Fp(and)h(the)g(norms)g Fm(k)p Fo(\036)3520 -123 y Fr(\(1)p Fn(;)p Fr(2\))3520 -62 y Fn(p)3669 -72 y Fm(k)3719 -57 y Fk(1)3793 -72 y Fp(.)p 3824 98 60 62 v 86 349 a(No)m(w)d(w)m(e)h (turn)f(to)f(the)h(studies)h(of)e(the)h(situation)f(when)i(the)f (nonlinear)g(term)g(is)g(b)s(eing)g(p)s(erturb)s(ed)g(in)-60 517 y(the)j(NLS)g(equation.)44 b(Hence)34 b(the)f(standing)g(w)m(a)m(v) m(e)i(solution)e(will)g(satisfy)g(the)g(elliptic)h(equation)919 758 y Fm(\000)p Fp(\001)p Fo(\036)22 b Fp(+)g Fo(!)t(\036)f Fp(+)h Fo(U)1563 773 y Fr(0)1603 758 y Fp(\()p Fo(x)p Fp(\))p Fo(\036)g Fp(+)g([)p Fo(F)2002 773 y Fr(0)2042 758 y Fp(\()p Fo(\036)2138 717 y Fr(2)2177 758 y Fp(\))g(+)g Fo("F)2444 773 y Fn(p)2484 758 y Fp(\()p Fo(\036)2580 717 y Fr(2)2619 758 y Fp(\)])p Fo(\036)27 b Fp(=)h(0)-60 999 y(W)-8 b(e)33 b(easily)h(obtain)e(the)h(equation)h(for)e(the)h (correction)g(term)g(to)f(the)h(solitary)g(w)m(a)m(v)m(e)i(solution)e (via)g(\(3\).)1177 1239 y Fo(\036)1235 1254 y Fn(p)1302 1239 y Fp(=)28 b Fo(L)1472 1198 y Fk(\000)p Fr(1)1472 1264 y(+)p Fn(;)k Fr(0)1619 1239 y Fp([)p Fm(\000)p Fo(")p Fm(K)1845 1254 y Fr(3)1884 1239 y Fp(\()p Fo(\036)1980 1254 y Fr(0)2019 1239 y Fp(\))22 b Fm(\000)h(K)2255 1254 y Fr(4)2294 1239 y Fp(\()p Fo(\036)2390 1254 y Fr(0)2430 1239 y Fo(;)17 b(\036)2532 1254 y Fn(p)2571 1239 y Fp(\)])p Fo(;)1064 b Fp(\(16\))-60 1480 y(where)46 b Fm(K)310 1495 y Fr(3)350 1480 y Fp(\()p Fo(\036)446 1495 y Fr(0)485 1480 y Fp(\))i(=)h Fo(F)759 1495 y Fn(p)799 1480 y Fp(\()p Fo(\036)895 1444 y Fr(2)895 1505 y(0)934 1480 y Fp(\))p Fo(\036)1030 1495 y Fr(0)1114 1480 y Fp(and)c Fm(K)1392 1495 y Fr(4)1431 1480 y Fp(\()p Fo(\036)1527 1495 y Fr(0)1567 1480 y Fo(;)17 b(\036)1669 1495 y Fn(p)1708 1480 y Fp(\))48 b(=)h Fo(F)1982 1495 y Fr(0)2022 1480 y Fp(\()p Fo(\036)2118 1444 y Fr(2)2157 1480 y Fp(\))p Fo(\036)30 b Fp(+)h Fo("F)2499 1495 y Fn(p)2538 1480 y Fp(\()p Fo(\036)2634 1444 y Fr(2)2673 1480 y Fp(\))p Fo(\036)f Fm(\000)h Fo(F)2970 1495 y Fr(0)3010 1480 y Fp(\()p Fo(\036)3106 1444 y Fr(2)3106 1505 y(0)3145 1480 y Fp(\))p Fo(\036)3241 1495 y Fr(0)3311 1480 y Fm(\000)g Fo(F)3482 1495 y Fr(0)3521 1480 y Fp(\()p Fo(\036)3617 1444 y Fr(2)3617 1505 y(0)3657 1480 y Fp(\))p Fo(\036)3753 1495 y Fn(p)3823 1480 y Fm(\000)-60 1649 y Fo(F)17 1613 y Fk(0)3 1673 y Fr(0)43 1649 y Fp(\()p Fo(\036)139 1613 y Fr(2)139 1673 y(0)178 1649 y Fp(\)2)p Fo(\036)323 1613 y Fr(2)323 1673 y(0)362 1649 y Fo(\036)420 1664 y Fn(p)486 1649 y Fm(\000)c Fo("F)699 1664 y Fn(p)738 1649 y Fp(\()p Fo(\036)834 1613 y Fr(2)834 1673 y(0)873 1649 y Fp(\))p Fo(\036)969 1664 y Fr(0)1009 1649 y Fp(.)63 b(Analogously)40 b(to)f(the)h(\014rst)g(case)g(when)g(the)g(p)s(oten)m(tial)g(term)f(w)m (as)i(b)s(eing)-60 1817 y(p)s(erturb)s(ed)33 b(w)m(e)h(pro)m(v)m(e)g (the)f(t)m(w)m(o)g(crucial)g(facts.)86 2070 y Fq(F)-9 b(act)38 b(3)32 b Fg(If)j Fo(\036)582 2085 y Fn(p)649 2070 y Fm(2)28 b Fo(H)832 2033 y Fr(2)871 2070 y Fp(\()p Fl(R)981 2033 y Fr(3)1020 2070 y Fp(\))35 b Fg(then)f Fo(L)1375 2028 y Fk(\000)p Fr(1)1375 2094 y(+)p Fn(;)h Fr(0)1525 2070 y Fp([)p Fo(")p Fm(K)1674 2085 y Fr(3)1713 2070 y Fp(\()p Fo(\036)1809 2085 y Fr(0)1848 2070 y Fp(\))22 b(+)g Fm(K)2082 2085 y Fr(4)2121 2070 y Fp(\()p Fo(\036)2217 2085 y Fr(0)2256 2070 y Fo(;)17 b(\036)2358 2085 y Fn(p)2398 2070 y Fp(\)])27 b Fm(2)h Fo(H)2673 2033 y Fr(2)2712 2070 y Fp(\()p Fl(R)2822 2033 y Fr(3)2862 2070 y Fp(\))p Fg(.)86 2322 y Fq(F)-9 b(act)38 b(4)32 b Fg(If)j Fo(";)17 b Fm(k)p Fo(\036)722 2271 y Fr(\(1\))722 2332 y Fn(p)815 2322 y Fm(k)865 2339 y Fn(H)928 2320 y Fe(2)966 2322 y Fo(;)g Fm(k)p Fo(\036)1118 2271 y Fr(\(2\))1118 2332 y Fn(p)1212 2322 y Fm(k)1262 2339 y Fn(H)1325 2320 y Fe(2)1398 2322 y Fg(ar)-5 b(e)35 b(smal)5 b(l)34 b(enough,)g(then)593 2563 y Fm(k)p Fo(L)709 2522 y Fk(\000)p Fr(1)709 2587 y(+)p Fn(;)e Fr(0)855 2563 y Fp([)p Fm(K)958 2578 y Fr(4)998 2563 y Fp(\()p Fo(\036)1094 2578 y Fr(0)1133 2563 y Fo(;)17 b(\036)1235 2522 y Fr(\(1\))1235 2587 y Fn(p)1329 2563 y Fp(\)])22 b Fm(\000)h Fo(L)1582 2522 y Fk(\000)p Fr(1)1582 2587 y(+)p Fn(;)32 b Fr(0)1728 2563 y Fp([)p Fm(K)1831 2578 y Fr(4)1871 2563 y Fp(\()p Fo(\036)1967 2578 y Fr(0)2006 2563 y Fo(;)17 b(\036)2108 2522 y Fr(\(2\))2108 2587 y Fn(p)2202 2563 y Fp(\)])p Fm(k)2317 2579 y Fn(H)2380 2561 y Fe(2)2446 2563 y Fm(\024)2561 2495 y Fp(1)p 2561 2540 49 4 v 2561 2631 a(2)2620 2563 y Fm(k)p Fo(\036)2728 2522 y Fr(\(1\))2728 2587 y Fn(p)2844 2563 y Fm(\000)22 b Fo(\036)3001 2522 y Fr(\(2\))3001 2587 y Fn(p)3096 2563 y Fm(k)3146 2579 y Fn(H)3209 2561 y Fe(2)86 2887 y Fg(Pr)-5 b(o)g(of)40 b(of)g(the)g(F)-7 b(act)39 b(3.)59 b Fp(Since)40 b Fm(j)p Fo(F)1354 2902 y Fn(p)1393 2887 y Fp(\()p Fo(\036)1489 2851 y Fr(2)1489 2912 y(0)1528 2887 y Fp(\))p Fm(j)d(\024)h Fo(C)r(;)54 b(x)37 b Fm(2)h Fl(R)2167 2851 y Fr(3)2206 2887 y Fp(,)i(w)m(e)f(obtain)f Fm(kK)2857 2902 y Fr(3)2896 2887 y Fp(\()p Fo(\036)2992 2902 y Fr(0)3031 2887 y Fp(\))p Fm(k)3119 2902 y Fr(2)3195 2887 y Fm(\024)g Fo(C)7 b Fm(k)p Fo(\036)3495 2902 y Fr(0)3534 2887 y Fm(k)3584 2902 y Fr(2)3660 2887 y Fo(<)37 b Fm(1)p Fp(.)-60 3056 y(Hence)d Fo(L)296 3015 y Fk(\000)p Fr(1)296 3080 y(+)p Fn(;)f Fr(0)443 3056 y Fp([)p Fo(")p Fm(K)592 3071 y Fr(3)631 3056 y Fp(\()p Fo(\036)727 3071 y Fr(0)766 3056 y Fp(\)])28 b Fm(2)h Fo(H)1043 3020 y Fr(2)1082 3056 y Fp(\()p Fl(R)1192 3020 y Fr(3)1231 3056 y Fp(\).)44 b(Let)33 b Fo(\021)1563 3071 y Fn(p)1603 3056 y Fp(\()p Fo(\036)p Fp(\))28 b(:=)g Fo(F)1959 3071 y Fn(p)1999 3056 y Fp(\()p Fo(\036)2095 3020 y Fr(2)2134 3056 y Fp(\))p Fo(\036)p Fp(.)44 b(F)-8 b(or)32 b(this)i(smo)s(oth)f (function)g Fo(\021)3442 3071 y Fn(p)3510 3056 y Fm(2)28 b Fo(C)3681 3020 y Fk(1)3789 3056 y Fp(w)m(e)-60 3257 y(ha)m(v)m(e)41 b Fo(\021)220 3272 y Fn(p)260 3257 y Fp(\()p Fo(\036)p Fp(\))22 b Fm(\000)h Fo(\021)564 3272 y Fn(p)603 3257 y Fp(\()p Fo(\036)699 3272 y Fr(0)738 3257 y Fp(\))28 b(=)908 3121 y Fh(Z)1007 3148 y Fn(\036)963 3347 y(\036)1005 3356 y Fe(0)1070 3257 y Fo(\021)1122 3216 y Fk(0)1118 3282 y Fn(p)1158 3257 y Fp(\()p Fo(s)p Fp(\))p Fo(ds)o Fp(.)66 b(By)40 b(means)h(of)f(the)g(Sob)s(olev)h(em)m (b)s(edding)h(theorem)e Fo(\036)3545 3272 y Fn(p)3625 3257 y Fm(2)h Fo(L)3798 3221 y Fk(1)3873 3257 y Fp(.)-60 3447 y(Th)m(us)34 b Fm(j)p Fo("F)324 3462 y Fn(p)363 3447 y Fp(\()p Fo(\036)459 3411 y Fr(2)498 3447 y Fp(\))p Fo(\036)22 b Fm(\000)h Fo("F)825 3462 y Fn(p)864 3447 y Fp(\()p Fo(\036)960 3411 y Fr(2)960 3472 y(0)999 3447 y Fp(\))p Fo(\036)1095 3462 y Fr(0)1134 3447 y Fm(j)28 b(\024)g Fo("C)7 b Fm(j)p Fo(\036)1504 3462 y Fn(p)1543 3447 y Fm(j)p Fp(,)32 b(whic)m(h)i(along)e(with)h(the)g(estimate)h (\(14\))e(yields)1279 3688 y Fm(jK)1383 3703 y Fr(4)1422 3688 y Fp(\()p Fo(\036)1518 3703 y Fr(0)1557 3688 y Fo(;)17 b(\036)1659 3703 y Fn(p)1698 3688 y Fp(\))p Fm(j)27 b(\024)i Fo("C)7 b Fm(j)p Fo(\036)2106 3703 y Fn(p)2144 3688 y Fm(j)22 b Fp(+)g Fo(C)7 b Fm(j)p Fo(\036)2455 3703 y Fn(p)2494 3688 y Fm(j)2522 3647 y Fr(2)-60 3928 y Fp(Since)28 b(the)g(correction)g(to)e(the)i(solitary)f(w)m(a)m(v)m(e)i(solution)f Fo(\036)2077 3943 y Fn(p)2144 3928 y Fm(2)g Fo(L)2304 3892 y Fr(2)2344 3928 y Fp(\()p Fl(R)2454 3892 y Fr(3)2493 3928 y Fp(\))11 b Fm(\\)g Fo(L)2685 3892 y Fk(1)2761 3928 y Fp(\()p Fl(R)2871 3892 y Fr(3)2910 3928 y Fp(\),)28 b(w)m(e)g(ha)m(v)m(e)h Fo(\036)3419 3943 y Fn(p)3486 3928 y Fm(2)f Fo(L)3646 3892 y Fr(4)3686 3928 y Fp(\()p Fl(R)3796 3892 y Fr(3)3835 3928 y Fp(\).)-60 4097 y(Therefore,)34 b Fm(K)485 4112 y Fr(4)524 4097 y Fp(\()p Fo(\036)620 4112 y Fr(0)659 4097 y Fo(;)17 b(\036)761 4112 y Fn(p)800 4097 y Fp(\))28 b Fm(2)g Fo(L)1026 4061 y Fr(2)1066 4097 y Fp(\()p Fl(R)1176 4061 y Fr(3)1215 4097 y Fp(\))33 b(and)f Fo(L)1541 4056 y Fk(\000)p Fr(1)1541 4121 y(+)p Fn(;)h Fr(0)1688 4097 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Fr(2)1717 4952 y(1)1756 4927 y Fp(\))p Fo(\036)1852 4942 y Fr(1)1914 4927 y Fm(\000)22 b Fo(F)2076 4942 y Fr(0)2116 4927 y Fp(\()p Fo(\036)2212 4886 y Fr(2)2212 4952 y(2)2251 4927 y Fp(\))p Fo(\036)2347 4942 y Fr(2)2408 4927 y Fp(+)g Fo("F)2615 4942 y Fn(p)2655 4927 y Fp(\()p Fo(\036)2751 4886 y Fr(2)2751 4952 y(1)2790 4927 y Fp(\))p Fo(\036)2886 4942 y Fr(1)2947 4927 y Fm(\000)h Fo("F)3156 4942 y Fn(p)3195 4927 y Fp(\()p Fo(\036)3291 4886 y Fr(2)3291 4952 y(2)3330 4927 y Fp(\))p Fo(\036)3426 4942 y Fr(2)3465 4927 y Fm(\000)1896 5176 y Fp(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 970 -73 a Fm(\000)p Fo(F)1110 -58 y Fr(0)1150 -73 y Fp(\()p Fo(\036)1246 -114 y Fr(2)1246 -49 y(0)1285 -73 y Fp(\)\()p Fo(\036)1419 -114 y Fr(\(1\))1419 -49 y Fn(p)1535 -73 y Fm(\000)23 b Fo(\036)1693 -114 y Fr(\(2\))1693 -49 y Fn(p)1787 -73 y Fp(\))f Fm(\000)h Fo(F)2024 -114 y Fk(0)2010 -49 y Fr(0)2049 -73 y Fp(\()p Fo(\036)2145 -114 y Fr(2)2145 -49 y(0)2184 -73 y Fp(\)2)p Fo(\036)2329 -114 y Fr(2)2329 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Fp(When)i(the)g(p)s(oten)m(tial)f(term)g(in) h(the)f(NLS)g(equation)h(is)g(b)s(eing)-60 2719 y(p)s(erturb)s(ed,)40 b(b)m(y)f(means)h(of)d(the)i(F)-8 b(acts)38 b(1)g(and)g(2)g(along)g (with)h(the)f(con)m(taction)h(lemma)g(w)m(e)g(obtain)f(that)-60 2887 y(the)33 b(solution)g Fo(\036)535 2902 y Fn(p)607 2887 y Fp(to)f(the)h(equation)h(\(13\))d(exists)k(and)e(is)g(unique)h (and)e(has)h(the)g(asymptotics)918 3153 y Fo(\036)976 3168 y Fn(p)1016 3153 y Fp(\()p Fo(x)p Fp(\))28 b(=)f Fo(\036)p Fp(\()p Fo(x)p Fp(\))22 b Fm(\000)h Fo(\036)1647 3168 y Fr(0)1686 3153 y Fp(\()p Fo(x)p Fp(\))28 b(=)g Fm(\000)p Fo("L)2138 3112 y Fk(\000)p Fr(1)2138 3178 y(+)p Fn(;)k Fr(0)2285 3153 y Fp(\()p Fo(U)2389 3168 y Fn(p)2428 3153 y Fo(\036)2486 3168 y Fr(0)2526 3153 y Fp(\))22 b(+)g Fo(O)s Fp(\()p Fo(")2846 3112 y Fr(2)2884 3153 y Fp(\))-60 3419 y(in)27 b(the)h Fo(H)300 3383 y Fr(2)339 3419 y Fp(\()p Fl(R)449 3383 y Fr(3)488 3419 y Fp(\))g Fm(\032)g Fo(L)725 3383 y Fk(1)800 3419 y Fp(\()p Fl(R)910 3383 y 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Fp(\()p Fo(R)295 4891 y Fr(3)334 4927 y Fp(\))28 b Fm(\032)g Fo(L)571 4891 y Fk(1)646 4927 y Fp(\()p Fl(R)756 4891 y Fr(3)795 4927 y Fp(\),)33 b(whic)m(h)h(completes)g(the)f(pro)s(of)f (in)h(the)g(\014rst)g(case.)1871 5176 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 86 -73 a Fg(Part)28 b(II.)c Fp(When)h(the)g(p)s (erturbation)g(is)g(b)s(eing)g(applied)h(to)e(the)h(nonlinear)g(term)g (in)g(the)g(NLS)g(equation,)-60 95 y(via)30 b(the)g(F)-8 b(acts)29 b(3,)h(4)f(and)g(the)h(con)m(traction)g(lemma)g(w)m(e)h (arriv)m(e)f(at)f(the)h(existence)i(of)d(the)h(unique)g(solution)-60 264 y(to)i(the)h(equation)h(\(16\))d(ha)m(ving)j(the)f(asymptotics)748 531 y Fo(\036)806 546 y Fn(p)845 531 y Fp(\()p Fo(x)p Fp(\))28 b(=)g Fo(\036)p Fp(\()p Fo(x)p Fp(\))22 b Fm(\000)h Fo(\036)1477 546 y Fr(0)1516 531 y Fp(\()p Fo(x)p Fp(\))28 b(=)g Fm(\000)p Fo(")p Fp(\()p Fo(L)2006 546 y Fr(+)p Fn(;)k Fr(0)2152 531 y Fp(\))2190 490 y Fk(\000)p Fr(1)2285 531 y Fp(\()p Fo(F)2386 546 y Fn(p)2425 531 y Fp(\()p 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Fo(O)s Fp(\()p Fo(")1627 1512 y Fr(2)1665 1554 y Fp(\))p Fo(;)212 b("F)2065 1512 y Fk(0)2051 1578 y Fn(p)2091 1554 y Fp(\()p Fo(\036)2187 1512 y Fr(2)2226 1554 y Fp(\))p Fo(\036)2322 1512 y Fr(2)2389 1554 y Fp(=)27 b Fo("F)2615 1512 y Fk(0)2601 1578 y Fn(p)2641 1554 y Fp(\()p Fo(\036)2737 1512 y Fr(2)2737 1578 y(0)2776 1554 y Fp(\))p Fo(\036)2872 1512 y Fr(2)2872 1578 y(0)2933 1554 y Fp(+)22 b Fo(O)s Fp(\()p Fo(")3193 1512 y Fr(2)3231 1554 y Fp(\))-60 1776 y(A)33 b(trivial)g(calculation)g(using)g(\(11\))f (and)h(\(12\))f(giv)m(es)-60 2042 y Fo(L)6 2057 y Fr(+)93 2042 y Fp(=)c Fo(L)263 2057 y Fr(+)p Fn(;)k Fr(0)429 2042 y Fm(\000)20 b Fp(2)p Fo("\036)679 2057 y Fr(0)718 2042 y Fm(f)p Fp(3)p Fo(F)894 2001 y Fk(0)880 2067 y Fr(0)919 2042 y Fp(\()p Fo(\036)1015 2001 y Fr(2)1015 2067 y(0)1054 2042 y Fp(\))g(+)f(2)p Fo(\036)1314 2001 y Fr(2)1314 2067 y(0)1353 2042 y Fo(F)1430 2001 y Fk(00)1416 2067 y Fr(0)1472 2042 y Fp(\()p Fo(\036)1568 2001 y Fr(2)1568 2067 y(0)1607 2042 y Fp(\))p Fm(g)p Fp(\()p Fo(L)1799 2057 y Fr(+)p Fn(;)33 b Fr(0)1946 2042 y Fp(\))1984 2001 y Fk(\000)p Fr(1)2078 2042 y Fp(\()p Fo(F)2179 2057 y Fn(p)2219 2042 y Fp(\()p Fo(\036)2315 2001 y Fr(2)2315 2067 y(0)2354 2042 y Fp(\))p Fo(\036)2450 2057 y Fr(0)2489 2042 y Fp(\))20 b(+)f Fo("F)2751 2057 y Fn(p)2791 2042 y Fp(\()p Fo(\036)2887 2001 y Fr(2)2887 2067 y(0)2926 2042 y Fp(\))h(+)f(2)p Fo("F)3251 2001 y Fk(0)3237 2067 y Fn(p)3276 2042 y Fp(\()p Fo(\036)3372 2001 y Fr(2)3372 2067 y(0)3411 2042 y Fp(\))p Fo(\036)3507 2001 y Fr(2)3507 2067 y(0)3566 2042 y Fp(+)h Fo(O)s Fp(\()p Fo(")3824 2001 y Fr(2)3862 2042 y Fp(\))575 2309 y Fo(L)641 2324 y Fk(\000)728 2309 y Fp(=)27 b Fo(L)897 2324 y Fk(\000)p Fn(;)33 b Fr(0)1066 2309 y Fm(\000)23 b Fp(2)p Fo("\036)1319 2324 y Fr(0)1358 2309 y Fo(F)1435 2268 y Fk(0)1421 2334 y Fr(0)1460 2309 y Fp(\()p Fo(\036)1556 2268 y Fr(2)1556 2334 y(0)1595 2309 y Fp(\)\()p Fo(L)1737 2324 y Fr(+)p Fn(;)33 b Fr(0)1884 2309 y Fp(\))1922 2268 y Fk(\000)p Fr(1)2016 2309 y Fp(\()p 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Fp(\))2046 4916 y Fo(g)t Fp(\()p Fo(x)p Fp(\))f Fm(\000)p Fp(\001)23 b(+)f Fo(!)k Fp(+)c Fo(f)11 b Fp(\()p Fo(x)p Fp(\))3241 4622 y Fh(1)3241 4802 y(A)3345 4823 y Fo(;)355 b Fp(\(18\))1871 5176 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop -60 -73 a Fp(suc)m(h)27 b(that)f Fo(V)415 -58 y Fk(\006)474 -73 y Fp(\()p Fo(x)p Fp(\))i(=)f Fo(f)11 b Fp(\()p Fo(x)p Fp(\))e Fm(\006)g Fo(g)t Fp(\()p Fo(x)p Fp(\).)41 b(The)27 b(solution)f(of)g(the)g(sp)s(ectral)h (problem)f(ab)s(o)m(v)m(e)h Fd( )3174 -50 y Fr(0)3241 -73 y Fp(=)h(\()p Fo( )3446 -58 y Fr(0)p Fn(;)33 b Fr(1)3573 -73 y Fo(;)17 b( )3680 -58 y Fr(0)p Fn(;)33 b Fr(2)3807 -73 y Fp(\))3845 -109 y Fn(T)-60 95 y Fp(at)f(the)h(end)g(of)g(the)g (essen)m(tial)h(sp)s(ectrum)g Fo(z)f Fp(=)27 b Fo(!)36 b Fp(satis\014es)e(the)f(system)1173 223 y Fh(\()1270 305 y Fp(\001)p Fo( )1414 320 y Fr(0)p Fn(;)g Fr(1)1569 305 y Fp(=)27 b Fo(f)1720 320 y Fr(0)1760 305 y Fo( )1823 320 y Fr(0)p Fn(;)33 b Fr(1)1972 305 y Fp(+)22 b Fo(g)2117 320 y Fr(0)2156 305 y Fo( )2219 320 y Fr(0)p Fn(;)33 b Fr(2)1270 473 y Fp(\(\001)22 b Fm(\000)h Fp(2)p Fo(!)t Fp(\))p Fo( )1726 488 y Fr(0)p Fn(;)31 b Fr(2)1879 473 y Fp(=)d Fo(g)2030 488 y Fr(0)2069 473 y Fo( )2132 488 y Fr(0)p Fn(;)33 b Fr(1)2282 473 y Fp(+)22 b Fo(f)2428 488 y Fr(0)2467 473 y Fo( )2530 488 y Fr(0)p Fn(;)33 b Fr(2)3727 393 y Fp(\(19\))-60 708 y(W)-8 b(e)43 b(consider)g(the)g (solutions)g(of)f(the)g(system)i(\(19\),)h(suc)m(h)e(that)f(\(1)29 b(+)f Fm(j)p Fo(x)p Fm(j)2760 672 y Fr(2)2799 708 y Fp(\))2837 671 y Fk(\000)2903 644 y Ff(s)p 2902 656 31 3 v 2902 697 a Fe(2)2947 708 y Fd( )3026 731 y Fr(0)3110 708 y Fm(2)44 b Fo(L)3286 672 y Fr(2)3326 708 y Fp(\()p Fl(R)3436 672 y Fr(3)3475 708 y Fo(;)17 b Fl(C)3591 672 y Fr(2)3630 708 y Fp(\))42 b(with)-60 876 y Fo(s)31 b(>)134 837 y Fr(1)p 134 853 36 4 v 134 911 a(2)214 876 y Fp(or)k Fd( )415 900 y Fr(0)486 876 y Fm(2)c Fo(L)649 840 y Fr(2)689 876 y Fp(\()p Fl(R)799 840 y Fr(3)838 876 y Fo(;)17 b Fl(C)954 840 y Fr(2)994 876 y Fp(\).)49 b(By)36 b(in)m(v)m(erting)g(the)f (Laplacian)g(the)g(\014rst)g(comp)s(onen)m(t)h(of)e(the)h(solution)g (of)-60 1045 y(this)e(system)i(can)e(b)s(e)f(written)i(as)1497 1213 y Fo( )1560 1228 y Fr(0)p Fn(;)f Fr(1)1715 1213 y Fp(=)28 b Fm(\000)1961 1146 y Fo(C)2031 1161 y Fr(0)p 1906 1190 219 4 v 1906 1282 a Fp(4)p Fo(\031)t Fm(j)p Fo(x)p Fm(j)2157 1213 y Fp(+)22 b Fo(\037;)1384 b Fp(\(20\))-60 1434 y(where)1036 1603 y Fo(C)1106 1618 y Fr(0)1173 1603 y Fp(=)1277 1467 y Fh(Z)1332 1693 y Fb(R)1380 1674 y Fe(3)1419 1603 y Fp(\()p Fo(f)1505 1618 y Fr(0)1544 1603 y Fp(\()p Fo(x)p Fp(\))p Fo( )1738 1618 y Fr(0)p Fn(;)33 b Fr(1)1866 1603 y Fp(\()p Fo(x)p Fp(\))22 b(+)g Fo(g)2164 1618 y Fr(0)2203 1603 y Fp(\()p Fo(x)p Fp(\))p Fo( )2397 1618 y Fr(0)p Fn(;)33 b Fr(2)2525 1603 y Fp(\()p Fo(x)p Fp(\)\)d)p Fo(x)-60 1889 y Fp(and)48 b Fo(\037)p Fp(\()p Fo(x)p Fp(\))28 b(=)508 1822 y(1)p 479 1866 108 4 v 479 1958 a(4)p Fo(\031)613 1754 y Fh(Z)668 1979 y Fb(R)716 1960 y Fe(3)772 1779 y Fh(\020)872 1822 y Fp(1)p 841 1866 111 4 v 841 1958 a Fm(j)p Fo(x)p Fm(j)984 1889 y(\000)1211 1822 y Fp(1)p 1093 1866 284 4 v 1093 1958 a Fm(j)p Fo(x)23 b Fm(\000)f Fo(y)t Fm(j)1387 1779 y Fh(\021)1447 1889 y Fp(\()p Fo(f)1533 1904 y Fr(0)1572 1889 y Fp(\()p Fo(y)t Fp(\))p Fo( )1763 1904 y Fr(0)p Fn(;)p Fr(1)1857 1889 y Fp(\()p Fo(y)t Fp(\))f(+)h Fo(g)2151 1904 y Fr(0)2190 1889 y Fp(\()p Fo(y)t Fp(\))p Fo( )2381 1904 y Fr(0)p Fn(;)p Fr(2)2475 1889 y Fp(\()p Fo(y)t Fp(\)\)d)o Fo(y)s Fp(.)90 b(Clearly)49 b Fo( )3279 1904 y Fr(0)p Fn(;)32 b Fr(2)3460 1889 y Fm(2)54 b Fo(L)3646 1853 y Fr(2)3686 1889 y Fp(\()p Fl(R)3796 1853 y Fr(3)3835 1889 y Fp(\).)-60 2058 y(Let)33 b(us)g(sho)m(w)h(that)e Fo(\037)h Fp(is)g(square)g(in)m (tegrable)h(as)f(w)m(ell.)86 2324 y Fq(F)-9 b(act)38 b(5)32 b Fg(F)-7 b(or)34 b(the)h(se)-5 b(c)g(ond)34 b(term)h(in)f(the)h (right)g(side)f(of)h(the)g(identity)g(\(20\))f(we)h(have)f Fo(\037)p Fp(\()p Fo(x)p Fp(\))28 b Fm(2)g Fo(L)3616 2288 y Fr(2)3656 2324 y Fp(\()p Fl(R)3766 2288 y Fr(3)3805 2324 y Fp(\))86 2590 y Fg(Pr)-5 b(o)g(of.)43 b Fp(Let)33 b(us)g(in)m(tro)s(duce)h(the)f(auxiliary)g(function)1155 2855 y Fo(s)p Fp(\()p Fo(x)p Fp(\))28 b(:=)g Fo(f)1539 2870 y Fr(0)1578 2855 y Fp(\()p Fo(x)p Fp(\))p Fo( )1772 2870 y Fr(0)p Fn(;)p Fr(1)1867 2855 y Fp(\()p Fo(x)p Fp(\))23 b(+)f Fo(g)2166 2870 y Fr(0)2205 2855 y Fp(\()p Fo(x)p Fp(\))p Fo( )2399 2870 y Fr(0)p Fn(;)p Fr(2)2494 2855 y Fp(\()p Fo(x)p Fp(\))33 b Fo(;)-60 3119 y Fp(whic)m(h)44 b(is)e(square)i(in)m(tegrable)f(since)h Fo( )1427 3134 y Fr(0)p Fn(;)p Fr(1)1563 3119 y Fp(and)f Fo( )1826 3134 y Fr(0)p Fn(;)p Fr(2)1963 3119 y Fp(b)s(elong)f(to)g(w)m(eigh)m(ted)i Fo(L)2895 3083 y Fr(2)2977 3119 y Fp(spaces)g(and)e Fo(f)3533 3134 y Fr(0)3615 3119 y Fp(and)g Fo(g)3861 3134 y Fr(0)-60 3288 y Fp(deca)m(y)34 b(exp)s(onen)m(tially)-8 b(.)45 b(Hence)34 b Fo(\037)p Fp(\()p Fo(x)p Fp(\))28 b(=)1495 3220 y(1)p 1466 3265 108 4 v 1466 3356 a(4)p Fo(\031)1600 3152 y Fh(Z)1655 3378 y Fb(R)1703 3359 y Fe(3)1759 3177 y Fh(\020)1859 3220 y Fp(1)p 1828 3265 111 4 v 1828 3356 a Fm(j)p Fo(x)p Fm(j)1971 3288 y(\000)2198 3220 y Fp(1)p 2081 3265 284 4 v 2081 3356 a Fm(j)p Fo(x)22 b Fm(\000)h Fo(y)t Fm(j)2374 3177 y Fh(\021)2434 3288 y Fo(s)p Fp(\()p Fo(y)t Fp(\))p Fo(dy)34 b Fp(and)f(therefore)1528 3628 y Fh(b)-63 b Fo(\037)p Fp(\()p Fo(p)p Fp(\))27 b(=)1848 3560 y Fh(b)-56 b Fo(s)p Fp(\(0\))22 b Fm(\000)h Fh(b)-56 b Fo(s)p Fp(\()p Fo(p)p Fp(\))p 1847 3605 464 4 v 2034 3696 a Fo(p)2083 3667 y Fr(2)-60 3892 y Fp(Here)44 b(and)f(further)g (do)m(wn)h(the)f(hat)g(sym)m(b)s(ol)i(stands)f(for)e(the)i(F)-8 b(ourier)42 b(transform.)75 b(F)-8 b(or)42 b(pro)m(ving)i(our)-60 4061 y(statemen)m(t)34 b(w)m(e)g(use)f(the)g(decomp)s(osition)1484 4325 y Fh(b)-63 b Fo(\037)p Fp(\()p Fo(p)p Fp(\))28 b(=)35 b Fh(b)-63 b Fo(\037)1854 4340 y Fr(1)1894 4325 y Fp(\()p Fo(p)p Fp(\))22 b(+)30 b Fh(b)-63 b Fo(\037)2200 4340 y Fr(2)2239 4325 y Fp(\()p Fo(p)p Fp(\))-60 4590 y(with)53 b Fh(b)-63 b Fo(\037)235 4605 y Fr(1)274 4590 y Fp(\()p Fo(p)p Fp(\))28 b(:=)568 4523 y Fh(b)-55 b Fo(s)p Fp(\(0\))21 b Fm(\000)j Fh(b)-56 b Fo(s)p Fp(\()p Fo(p)p Fp(\))p 568 4567 V 755 4658 a Fo(p)804 4630 y Fr(2)1041 4590 y Fo(\037)1102 4606 y Fk(fj)p Fn(p)p Fk(j\024)p Fr(1)p Fk(g)1386 4590 y Fp(and)52 b Fh(b)-63 b Fo(\037)1648 4605 y Fr(2)1688 4590 y Fp(\()p Fo(p)p Fp(\))27 b(:=)1981 4523 y Fh(b)-55 b Fo(s)p Fp(\(0\))21 b Fm(\000)j Fh(b)-56 b Fo(s)p Fp(\()p Fo(p)p Fp(\))p 1981 4567 V 2168 4658 a Fo(p)2217 4630 y Fr(2)2454 4590 y Fo(\037)2515 4606 y Fk(fj)p Fn(p)p Fk(j)p Fn(>)p Fr(1)p Fk(g)2754 4590 y Fp(,)48 b(where)d Fo(\037)3183 4606 y Fk(fj)p Fn(p)p Fk(j\024)p Fr(1)p Fk(g)3467 4590 y Fp(stands)g(for)-60 4759 y(the)e(c)m(haracteristic)h(function)f(of)f(the)h(unit)g(ball)f (in)h(the)g(F)-8 b(ourier)42 b(space)h(cen)m(tered)i(at)d(the)h(origin) f(and)-60 4927 y Fo(\037)1 4943 y Fk(fj)p Fn(p)p Fk(j)p Fn(>)p Fr(1)p Fk(g)270 4927 y Fp(for)30 b(the)g(c)m(haracteristic)h (function)g(of)e(its)h(complemen)m(t.)45 b(The)30 b(functions)h Fo(s)p Fp(\()p Fo(x)p Fp(\))f(and)g Fm(j)p Fo(x)p Fm(j)p Fo(s)p Fp(\()p Fo(x)p Fp(\))g(ha)m(v)m(e)1871 5176 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop -60 -73 a Fp(\014nite)35 b(norms)g(in)g Fo(L)668 -109 y Fr(1)708 -73 y Fp(\()p Fl(R)818 -109 y Fr(3)857 -73 y Fp(\),)g(whic)m(h)i(can)d(b)s(e)h(easily)h(sho)m(wn)g (b)m(y)f(means)h(of)e(the)h(Sc)m(h)m(w)m(arz)i(inequalit)m(y)f(using) -60 95 y(that)g Fo(f)203 110 y Fr(0)243 95 y Fp(\()p Fo(x)p Fp(\))g(and)g Fo(g)650 110 y Fr(0)690 95 y Fp(\()p Fo(x)p Fp(\))g(deca)m(y)i(exp)s(onen)m(tially)g(and)e Fo( )1993 110 y Fr(0)p Fn(;)p Fr(1)2088 95 y Fp(\()p Fo(x)p Fp(\))h(and)f Fo( )2512 110 y Fr(0)p Fn(;)p Fr(2)2607 95 y Fp(\()p Fo(x)p Fp(\))g(b)s(elong)g(to)g(the)h(w)m(eigh)m(ted)h Fo(L)3860 59 y Fr(2)-60 264 y Fp(space.)45 b(Th)m(us)34 b(w)m(e)f(ha)m(v)m(e)900 523 y Fm(j)8 b Fh(b)-63 b Fo(\037)989 538 y Fr(2)1028 523 y Fp(\()p Fo(p)p Fp(\))p Fm(j)27 b(\024)1314 412 y Fh(\020)1478 456 y Fp(2)p 1383 500 239 4 v 1383 608 a(\(2)p Fo(\031)t Fp(\))1577 544 y Fe(3)p 1577 556 31 3 v 1577 597 a(2)1648 387 y Fh(Z)1703 613 y Fb(R)1751 594 y Fe(3)1806 523 y Fm(j)p Fo(s)p Fp(\()p Fo(x)p Fp(\))p Fm(j)p Fo(dx)2145 412 y Fh(\021)2214 452 y Fo(\037)2275 468 y Fk(fj)p Fn(p)p Fk(j)p Fn(>)p Fr(1)p Fk(g)p 2214 500 301 4 v 2321 591 a Fo(p)2370 563 y Fr(2)2553 523 y Fm(2)h Fo(L)2713 482 y Fr(2)2753 523 y Fp(\()p Fl(R)2863 482 y Fr(3)2902 523 y Fp(\))-60 801 y(In)47 b(order)h(to)e(sho)m(w)j(the)e(square)h(in)m(tegrabilit)m(y)h(of)e(the) g(remaining)h(term)g(w)m(e)g(will)g(mak)m(e)g(use)g(of)f(the)-60 970 y(follo)m(wing)33 b(represen)m(tation)1155 1156 y Fh(b)-56 b Fo(s)p Fp(\()p Fo(p)p Fp(\))27 b(=)i Fh(b)-56 b Fo(s)p Fp(\(0\))22 b(+)1747 1021 y Fh(Z)1846 1047 y Fk(j)p Fn(p)p Fk(j)1802 1246 y Fr(0)2005 1089 y Fo(@)p 1952 1133 164 4 v 1952 1225 a(@)5 b Fm(j)p Fo(\021)t Fm(j)2126 1156 y Fh(b)-55 b Fo(s)p Fp(\()p Fm(j)p Fo(\021)t Fm(j)p Fo(;)17 b Fp(\012\))p Fo(d)p Fm(j)p Fo(\021)t Fm(j)30 b Fo(;)-60 1375 y Fp(where)k(\012)f(stands)g(for)f(the)h(angle) g(v)-5 b(ariables)33 b(on)g(the)g(sphere.)45 b(This)34 b(implies)1151 1687 y Fh(b)-64 b Fo(\037)1203 1702 y Fr(1)1243 1687 y Fp(\()p Fo(p)p Fp(\))27 b(=)h Fm(\000)1586 1517 y Fh(R)1653 1543 y Fk(j)p Fn(p)p Fk(j)1633 1632 y Fr(0)1797 1558 y Fn(@)p 1758 1574 119 4 v 1758 1631 a(@)t Fk(j)p Fn(\021)r Fk(j)1887 1597 y Fh(b)-56 b Fo(s)p Fp(\()p Fm(j)p Fo(\021)t Fm(j)p Fo(;)17 b Fp(\012\))p Fo(d)p Fm(j)p Fo(\021)t Fm(j)p 1586 1664 801 4 v 1942 1755 a Fo(p)1991 1726 y Fr(2)2397 1687 y Fo(\037)2458 1702 y Fk(fj)p Fn(p)p Fk(j\024)p Fr(1)p Fk(g)-60 2009 y Fp(Clearly)34 b Fm(jr)390 2024 y Fn(p)430 2009 y Fh(b)-56 b Fo(s)p Fp(\()p Fo(p)p Fp(\))p Fm(j)27 b(\024)865 1942 y Fp(1)p 770 1986 239 4 v 770 2094 a(\(2)p Fo(\031)t Fp(\))964 2030 y Fe(3)p 964 2042 31 3 v 964 2083 a(2)1035 1873 y Fh(Z)1090 2099 y Fb(R)1138 2080 y Fe(3)1194 2009 y Fm(j)p Fo(s)p Fp(\()p Fo(x)p Fp(\))p Fm(jj)p Fo(x)p Fm(j)p Fo(dx)o Fp(,)33 b(whic)m(h)h(yields)792 2353 y Fm(j)8 b Fh(b)-63 b Fo(\037)881 2368 y Fr(1)921 2353 y Fp(\()p Fo(p)p Fp(\))p Fm(j)27 b(\024)1206 2243 y Fh(\020)1370 2286 y Fp(1)p 1275 2331 239 4 v 1275 2439 a(\(2)p Fo(\031)t Fp(\))1469 2375 y Fe(3)p 1469 2387 31 3 v 1469 2428 a(2)1540 2218 y Fh(Z)1595 2443 y Fb(R)1643 2425 y Fe(3)1699 2353 y Fm(j)p Fo(s)p Fp(\()p Fo(x)p Fp(\))p Fm(jj)p Fo(x)p Fm(j)p Fo(dx)2149 2243 y Fh(\021)2245 2286 y Fp(1)p 2218 2331 105 4 v 2218 2422 a Fm(j)p Fo(p)p Fm(j)2332 2353 y Fo(\037)2393 2369 y Fk(fj)p Fn(p)p Fk(j\024)p Fr(1)p Fk(g)2660 2353 y Fm(2)i Fo(L)2821 2312 y Fr(2)2860 2353 y Fp(\()p Fl(R)2970 2312 y Fr(3)3010 2353 y Fp(\))p 3824 2623 60 62 v 86 2884 a(Hence)k(there)f(are)f(t)m(w)m(o)h(distinct)h(t)m (yp)s(es)f(of)f(solutions)i(for)d(the)i(sp)s(ectral)g(problem)g(\(17\)) f(at)g(the)g(end)h(of)-60 3053 y(the)h(essen)m(tial)i(sp)s(ectrum)f (\(see)f(e.g.)76 b([CP)34 b(05)o(]\).)86 3316 y Fq(I.)k(Threshold)g (resonance)86 3579 y Fp(This)k(situation)f(o)s(ccurs)g(when)g Fo(C)1373 3594 y Fr(0)1453 3579 y Fm(6)p Fp(=)g(0.)66 b(Therefore)42 b(the)e(solution)h Fd( )2793 3602 y Fr(0)2873 3579 y Fp(of)f(the)h(system)h(\(19\))d(suc)m(h)-60 3747 y(that)j(\(1)29 b(+)g Fm(j)p Fo(x)p Fm(j)493 3711 y Fr(2)532 3747 y Fp(\))570 3711 y Fk(\000)636 3684 y Ff(s)p 635 3696 31 3 v 635 3737 a Fe(2)680 3747 y Fd( )759 3771 y Fr(0)843 3747 y Fm(2)45 b Fo(L)1020 3711 y Fr(2)1060 3747 y Fp(\()p Fl(R)1170 3711 y Fr(3)1209 3747 y Fo(;)17 b Fl(C)1325 3711 y Fr(2)1364 3747 y Fp(\))43 b(with)g Fo(s)i(>)1898 3708 y Fr(1)p 1898 3725 36 4 v 1898 3782 a(2)1986 3747 y Fp(deca)m(ys)g(as)2467 3708 y Fr(1)p 2446 3725 79 4 v 2446 3782 a Fk(j)p Fn(x)p Fk(j)2534 3747 y Fp(.)74 b(F)-8 b(rom)42 b(the)h(represen)m(tation)i(\(20\))-60 3916 y(of)36 b(the)h(\014rst)h(comp)s(onen)m(t)f(of)f(the)i(solution)f Fo( )1659 3931 y Fr(0)p Fn(;)c Fr(1)1823 3916 y Fp(it)j(follo)m(ws)i (that)e(the)h(threshold)h(resonance)g(is)f(alw)m(a)m(ys)-60 4085 y(simple.)43 b(Indeed,)30 b(if)e(there)g(w)m(ere)h(t)m(w)m(o)g(of) e(them:)42 b Fd( )1819 4042 y Fr(1)1819 4109 y(0)1886 4085 y Fp(and)28 b Fd( )2150 4042 y Fr(2)2150 4109 y(0)2189 4085 y Fp(,)h(w)m(e)g(could)f(assume)h(them)g(ha)m(ving)f(the)g(same) -60 4253 y(constan)m(t)37 b Fo(C)407 4268 y Fr(0)483 4253 y Fp(in)g(the)g(form)m(ula)g(\(20\),)g(whic)m(h)h(can)f(b)s(e)g (ac)m(hiev)m(ed)i(b)m(y)e(m)m(ultiplying)i(the)e(equations)g(of)g(the) -60 4422 y(system)29 b(\(19\))e(b)m(y)h(the)g(appropriate)f(constan)m (t.)43 b(Then)28 b(their)g(di\013erence)h Fd( )2689 4379 y Fr(1)2689 4446 y(0)2740 4422 y Fm(\000)11 b Fd( )2908 4379 y Fr(2)2908 4446 y(0)2975 4422 y Fm(2)28 b Fo(L)3135 4385 y Fr(2)3175 4422 y Fp(\()p Fl(R)3285 4385 y Fr(3)3324 4422 y Fo(;)17 b Fl(C)3440 4385 y Fr(2)3480 4422 y Fp(\))27 b(will)h(b)s(e)f(a)-60 4590 y(solution)32 b(of)f(the)h(system)h (\(19\).)43 b(But)32 b(the)f(co)s(existence)j(of)d(a)h(zero)f (resonance)i(and)f(an)f(eigen)m(v)-5 b(alue)33 b(at)f(the)-60 4759 y(threshold)j(is)f(b)s(ey)m(ond)h(the)f(scop)s(e)h(of)e(our)g(w)m (ork,)i(whic)m(h)h(is)e(assumed)h(in)f(the)g(Theorem)h(1.3.)47 b(Therefore,)-60 4927 y Fd( )19 4884 y Fr(1)19 4952 y(0)91 4927 y Fp(and)33 b Fd( )360 4884 y Fr(2)360 4952 y(0)432 4927 y Fp(coincide.)1871 5176 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 86 -73 a Fp(W)-8 b(e)26 b(normalize)f Fd( )762 -50 y Fr(0)827 -73 y Fp(suc)m(h)h(that)e Fo(C)1312 -58 y Fr(0)1379 -73 y Fp(=)1483 -156 y Fm(p)p 1566 -156 108 4 v 83 x Fp(4)p Fo(\031)k Fp(and)d(in)g(the)h(original)e(v)-5 b(ariables)26 b(for)e(our)h(sp)s(ectral)g(problem)-60 95 y(the)33 b(solution)g(is)g Fq(u)637 110 y Fr(0)705 95 y Fp(=)27 b(\()p Fo(u)902 110 y Fr(0)941 95 y Fo(;)17 b(w)1055 110 y Fr(0)1094 95 y Fp(\))1132 59 y Fn(T)1187 95 y Fp(.)86 354 y Fq(I)s(I.)37 b(Eigen)m(v)-6 b(alue)86 612 y Fp(This)34 b(is)f(the)g(case)h(when)f Fo(C)1106 627 y Fr(0)1173 612 y Fp(=)28 b(0)k(and,)h(therefore)g Fd( )2063 636 y Fr(0)2130 612 y Fm(2)28 b Fo(L)2290 576 y Fr(2)2330 612 y Fp(\()p Fl(R)2440 576 y Fr(3)2480 612 y Fo(;)17 b Fl(C)2596 576 y Fr(2)2635 612 y Fp(\).)86 871 y(The)23 b(eigen)m(v)-5 b(alue)24 b(at)d(the)i(end)g(of)e(the)i (essen)m(tial)h(sp)s(ectrum)f(no)m(w)g(is)f(assumed)i(to)e(b)s(e)g Fo(N)10 b Fp(-fold)22 b(degenerate,)-60 1039 y(suc)m(h)36 b(that)f(the)h(corresp)s(onding)f(eigen)m(v)m(ectors)j(are)d Fd( )1967 996 y Fn(i)1967 1064 y Fr(0)2038 1039 y Fp(=)d(\()p Fo( )2251 1003 y Fn(i)2247 1064 y Fr(0)p Fn(;)h Fr(1)2374 1039 y Fo(;)17 b( )2485 1003 y Fn(i)2481 1064 y Fr(0)p Fn(;)33 b Fr(2)2608 1039 y Fp(\))2646 1003 y Fn(T)2701 1039 y Fo(;)87 b(i)32 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)10 b Fp(,)36 b(for)e(whic)m(h)j(w)m(e)-60 1208 y(ha)m(v)m(e)1320 1241 y Fh(Z)1375 1466 y Fb(R)1423 1447 y Fe(3)1478 1376 y Fo(s)1524 1335 y Fn(i)1553 1376 y Fp(\()p Fo(x)p Fp(\)d)p Fo(x)28 b Fp(=)g(0)p Fo(;)49 b(i)28 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)1217 b Fp(\(21\))86 1681 y(F)-8 b(or)24 b(the)g(sp)s(ectral) h(problem)g(\(17\))f(according)h(to)f(the)g(Birman-Sc)m(h)m(winger)i (principle)g(\(see)f(e.g.)65 b([S)33 b(79],)-60 1850 y([LSW)g(76]\))c(eac)m(h)i(eigen)m(v)-5 b(alue)31 b Fo(z)j Fp(is)c(corresp)s(onden)m(t)h(to)e(the)h(eigen)m(v)-5 b(alue)31 b Fm(\000)p Fp(1)f(of)f(the)h(same)g(m)m(ultiplicit)m(y)i(of) -60 2018 y(the)h(problem)1596 2187 y Fo(K)1679 2202 y Fn(\013;)f(")1813 2187 y Fq(w)d Fp(=)e Fm(\000)p Fq(w)33 b Fo(;)1483 b Fp(\(22\))-60 2402 y(where)34 b(the)f(Birman-Sc)m(h)m (winger)h(op)s(erator)e(is)h(de\014ned)h(as)1378 2653 y Fo(K)1461 2668 y Fn(\013;)e(")1623 2653 y Fp(:=)27 b Fm(j)p Fo(Q)1858 2668 y Fr(0)1898 2653 y Fm(j)1936 2585 y Fe(1)p 1935 2597 31 3 v 1935 2638 a(2)1980 2653 y Fo(R)2054 2668 y Fn(\013)2104 2653 y Fo(Q)p Fm(j)p Fo(Q)2286 2668 y Fr(0)2325 2653 y Fm(j)2353 2612 y Fk(\000)2418 2585 y Fe(1)p 2418 2597 V 2418 2638 a(2)3727 2653 y Fp(\(23\))-60 2905 y(on)32 b Fo(L)141 2868 y Fr(2)181 2905 y Fp(\()p Fl(R)291 2868 y Fr(3)331 2905 y Fo(;)17 b Fl(C)447 2868 y Fr(2)486 2905 y Fp(\))32 b(with)h(the)g(op)s(erator-v)-5 b(alued)32 b(terms)169 3225 y Fo(R)243 3240 y Fn(\013)320 3225 y Fp(=)424 3024 y Fh(0)424 3204 y(@)552 3149 y Fo(G)629 3164 y Fn(\013)916 3149 y Fp(0)591 3318 y(0)122 b Fo(G)839 3289 y Fk(p)p 898 3289 217 3 v 56 x Fr(2)p Fn(!)r Fk(\000)p Fn(\013)1079 3326 y Fe(2)1160 3024 y Fh(1)1160 3204 y(A)1263 3225 y Fo(;)50 b(G)1417 3240 y Fn(\013)1494 3225 y Fp(=)27 b(\()p Fm(\000)p Fp(\001)c(+)f Fo(\013)1977 3183 y Fr(2)2016 3225 y Fp(\))2054 3183 y Fk(\000)p Fr(1)2181 3225 y Fp(and)33 b Fo(G)2448 3196 y Fk(p)p 2506 3196 V 2506 3252 a Fr(2)p Fn(!)r Fk(\000)p Fn(\013)2687 3233 y Fe(2)2755 3225 y Fp(=)27 b(\()p Fm(\000)p Fp(\001)c(+)f(2)p Fo(!)j Fm(\000)e Fo(\013)3473 3183 y Fr(2)3512 3225 y Fp(\))3550 3183 y Fk(\000)p Fr(1)3644 3225 y Fo(;)-60 3539 y Fp(the)33 b(family)g(of)f(p)s(oten)m(tials)944 3854 y Fo(Q)27 b Fp(=)h Fo(Q)1229 3869 y Fr(0)1291 3854 y Fp(+)22 b Fo("Q)1512 3869 y Fn(p)1579 3854 y Fp(=)1683 3653 y Fh(0)1683 3833 y(@)1811 3778 y Fo(f)1859 3793 y Fr(0)1983 3778 y Fo(g)2030 3793 y Fr(0)1812 3947 y Fo(g)1859 3962 y Fr(0)1982 3947 y Fo(f)2030 3962 y Fr(0)2111 3653 y Fh(1)2111 3833 y(A)2220 3854 y Fp(+)g Fo(")2381 3653 y Fh(0)2381 3833 y(@)2509 3778 y Fo(f)2557 3793 y Fn(p)2681 3778 y Fo(g)2728 3793 y Fn(p)2510 3947 y Fo(g)2557 3962 y Fn(p)2680 3947 y Fo(f)2728 3962 y Fn(p)2809 3653 y Fh(1)2809 3833 y(A)-60 4174 y Fp(and)33 b(the)g(solution)g(of)f(the)h(sp)s(ectral)g(problem)h (\(22\))1660 4425 y Fq(w)29 b Fp(=)f Fm(j)p Fo(Q)1979 4440 y Fr(0)2018 4425 y Fm(j)2056 4357 y Fe(1)p 2056 4369 31 3 v 2056 4410 a(2)2101 4425 y Fd( )-60 4676 y Fp(The)34 b(sp)s(ectral)f(parameter)g Fo(\013)g Fp(is)g(de\014ned)h (relativ)m(ely)h(to)d(the)h(sp)s(ectrum)h(of)e(the)h(problem)g(\(17\))f (via)1672 4927 y Fo(z)g Fp(=)c Fo(!)d Fm(\000)e Fo(\013)2102 4886 y Fr(2)2141 4927 y Fo(;)1871 5176 y Fp(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop -60 -73 a Fp(suc)m(h)39 b(that)e Fo(\013)h Fp(b)s(elonging)g(to)f(the)g(\014rst)h(sheet)h(of)e(the)h (Riemann)g(surface)g Fm(f)p Fo(\013)e Fp(:)g(Re)p Fo(\013)h(>)f Fp(0)p Fm(g)h Fp(corresp)s(onds)-60 95 y(to)c(an)g(eigen)m(v)-5 b(alue)35 b(near)e Fo(!)j Fp(and)e Fo(\033)1223 110 y Fn(p)1263 95 y Fp(\()p Fm(L)p Fp(\))e(is)i(empt)m(y)h(in)e(the)h(neigh) m(b)s(orho)s(o)s(d)f(of)f Fo(!)37 b Fp(when)d Fo(\013)g Fp(b)s(elongs)g(to)f(the)-60 264 y(second)39 b(sheet)g(of)e(the)h (Riemann)h(surface)f Fm(f)p Fo(\013)f Fp(:)f(Re)q Fo(\013)h(<)f Fp(0)p Fm(g)p Fp(.)58 b(F)-8 b(or)37 b(the)h(matrix)g(v)-5 b(alued)39 b(p)s(oten)m(tials)f(here)-60 432 y(and)33 b(further)g(w)m(e)g(use)h(the)f(notations)f(in)h(the)g(sense)i(of)d (the)h(sp)s(ectral)g(calculus)1090 677 y Fo(Q)1167 692 y Fr(0)1235 677 y Fp(=)27 b Fo(sig)t(nQ)1603 692 y Fr(0)1643 677 y Fm(j)p Fo(Q)1748 692 y Fr(0)1787 677 y Fm(j)p Fo(;)114 b(Q)2043 590 y Fe(1)p 2043 602 31 3 v 2043 643 a(2)2033 702 y Fr(0)2115 677 y Fp(=)28 b Fo(sig)t(nQ)2484 692 y Fr(0)2523 677 y Fm(j)p Fo(Q)2628 692 y Fr(0)2668 677 y Fm(j)2706 609 y Fe(1)p 2705 621 V 2705 662 a(2)-60 922 y Fp(The)33 b(Birman-Sc)m(h)m(winger)i(op)s(erator)d Fo(K)1428 937 y Fn(\013;)g(")1594 922 y Fp(is)h(compact)g(due)h(to)e (the)h(exp)s(onen)m(tial)h(deca)m(y)f(of)f(the)h(p)s(oten-)-60 1091 y(tials)h(and)g(of)f(the)h(k)m(ernel)h(of)e(the)i(op)s(erator)e Fo(R)1663 1106 y Fn(\013)1746 1091 y Fp(in)h(three)g(dimension.)48 b(The)35 b(resolv)m(en)m(t)h(op)s(erator)d(can)h(b)s(e)-60 1259 y(expanded)g(as)1050 1428 y Fo(R)1124 1443 y Fn(\013)1202 1428 y Fp(=)27 b Fo(R)1379 1443 y Fr(0)1441 1428 y Fp(+)22 b Fo(\013)q(R)1676 1443 y Fr(1)1738 1428 y Fp(+)g Fo(\013)1899 1387 y Fr(2)1938 1428 y Fo(R)2012 1443 y Fr(2)2073 1428 y Fp(+)g Fo(\013)2234 1387 y Fr(3)2274 1428 y Fo(R)2348 1443 y Fr(3)2409 1428 y Fp(+)g Fo(O)s Fp(\()p Fo(\013)2686 1387 y Fr(4)2725 1428 y Fp(\))p Fo(;)-60 1641 y Fp(where)34 b(the)f(k)m(ernels)i(of)d(the)h(op)s(erators)f(are)h(giv)m(en)g(b)m(y) 571 1949 y Fo(R)645 1964 y Fr(0)685 1949 y Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b(=)1042 1749 y Fh(0)1042 1928 y(@)1288 1834 y Fr(1)p 1181 1850 250 4 v 1181 1907 a(4)p Fn(\031)r Fk(j)p Fn(x)p Fk(\000)p Fn(y)r Fk(j)1689 1873 y Fp(0)1281 2042 y(0)1533 2003 y Fn(e)1566 1979 y Fc(\000)1614 1938 y(p)p 1664 1938 72 3 v 41 x Fe(2)p Ff(!)r Fc(j)p Ff(x)p Fc(\000)p Ff(y)r Fc(j)p 1533 2019 360 4 v 1588 2076 a Fr(4)p Fn(\031)r Fk(j)p Fn(x)p Fk(\000)p Fn(y)r Fk(j)1944 1749 y Fh(1)1944 1928 y(A)2048 1949 y Fo(;)114 b(R)2263 1964 y Fr(1)2303 1949 y Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b(=)2660 1749 y Fh(0)2660 1928 y(@)2789 1873 y Fm(\000)2898 1834 y Fr(1)p 2876 1851 79 4 v 2876 1908 a(4)p Fn(\031)3048 1873 y Fp(0)2852 2042 y(0)147 b(0)3138 1749 y Fh(1)3138 1928 y(A)3242 1949 y Fo(;)568 2392 y(R)642 2407 y Fr(2)682 2392 y Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b(=)1039 2192 y Fh(0)1039 2371 y(@)1178 2269 y Fk(j)p Fn(x)p Fk(\000)p Fn(y)r Fk(j)p 1178 2294 172 4 v 1225 2351 a Fr(8)p Fn(\031)1608 2317 y Fp(0)1239 2485 y(0)1452 2446 y Fn(e)1485 2423 y Fc(\000)1533 2382 y(p)p 1583 2382 72 3 v 41 x Fe(2)p Ff(!)r Fc(j)p Ff(x)p Fc(\000)p Ff(y)r Fc(j)p 1452 2463 360 4 v 1523 2527 a Fr(8)p Fn(\031)1601 2472 y Fk(p)p 1660 2472 82 3 v 55 x Fr(2)p Fn(!)1863 2192 y Fh(1)1863 2371 y(A)1967 2392 y Fo(;)114 b(R)2182 2407 y Fr(3)2222 2392 y Fp(\()p Fo(x;)17 b(y)t Fp(\))27 b(=)2579 2192 y Fh(0)2579 2371 y(@)2708 2317 y Fm(\000)2795 2269 y Fk(j)p Fn(x)p Fk(\000)p Fn(y)r Fk(j)2967 2246 y Fe(2)p 2795 2294 206 4 v 2842 2351 a Fr(24)p Fn(\031)3094 2317 y Fp(0)2835 2485 y(0)210 b(0)3184 2192 y Fh(1)3184 2371 y(A)-60 2678 y Fp(and)35 b Fo(O)s Fp(\()p Fo(\013)311 2642 y Fr(4)350 2678 y Fp(\))g(here)h(and)f(b)s(elo)m(w)h(stands)g(for) f(the)h(terms)g(of)f(the)g(order)h(four)e(and)i(higher)g(and)f (analytic)h(in)-60 2847 y Fo(\013)f Fp(for)f Fo(\013)h Fp(small)g(enough.)49 b(The)35 b(expansion)h(ab)s(o)m(v)m(e)f(yields)h (the)f(con)m(v)m(ergen)m(t)i(expansion)f(for)d Fo(K)3480 2862 y Fn(\013;)g(")3649 2847 y Fp(in)i(the)-60 3015 y(op)s(erator)25 b(norm)h(\()p Fo(L)678 2979 y Fr(2)745 3015 y Fm(!)i Fo(L)939 2979 y Fr(2)978 3015 y Fp(\))e(due)g(to)g(the)g (exp)s(onen)m(tial)h(deca)m(y)g(of)e(the)h(p)s(oten)m(tials.)42 b(First)26 b(w)m(e)h(consider)g(the)-60 3184 y(case)k(of)f(a)g(m)m (ultiple)i(eigen)m(v)-5 b(alue)31 b(at)f Fo(z)j Fp(=)27 b Fo(!)34 b Fp(with)d(corresp)s(onding)g(eigenfunctions)h Fd( )3126 3141 y Fn(i)3126 3208 y Fr(0)3165 3184 y Fo(;)47 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:::;)g(N)10 b Fp(.)42 b(By)-60 3352 y(letting)24 b Fo(\013)q(;)17 b(")27 b Fm(!)g Fp(0)d(w)m(e)h (obtain)e(the)i(Birman-Sc)m(h)m(winger)g(problem)g(at)e(the)i(edge)f (of)f(the)i(essen)m(tial)h(sp)s(ectrum)-60 3521 y(of)39 b(the)h(linearized)g(NLS)g(op)s(erator)f Fo(K)1389 3536 y Fr(0)p Fn(;)32 b Fr(0)1516 3521 y Fq(w)1599 3485 y Fn(i)1597 3545 y Fr(0)1675 3521 y Fp(=)39 b Fm(\000)p Fq(w)1950 3485 y Fn(i)1948 3545 y Fr(0)1988 3521 y Fo(;)56 b(i)40 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)49 b Fp(with)40 b Fq(w)2916 3485 y Fn(i)2914 3545 y Fr(0)2993 3521 y Fp(=)f Fm(j)p Fo(Q)3213 3536 y Fr(0)3252 3521 y Fm(j)3290 3457 y Fe(1)p 3290 3469 31 3 v 3290 3510 a(2)3334 3521 y Fd( )3413 3478 y Fn(i)3413 3545 y Fr(0)3492 3521 y Fm(2)h Fo(L)3664 3485 y Fr(2)3743 3521 y Fp(and)-60 3689 y Fo(K)23 3704 y Fr(0)p Fn(;)32 b Fr(0)178 3689 y Fp(=)27 b Fm(j)p Fo(Q)386 3704 y Fr(0)425 3689 y Fm(j)463 3625 y Fe(1)p 463 3637 V 463 3679 a(2)508 3689 y Fo(R)582 3704 y Fr(0)621 3689 y Fo(Q)708 3602 y Fe(1)p 709 3614 V 709 3655 a(2)698 3714 y Fr(0)753 3689 y Fp(.)44 b(This)33 b(is)g(equiv)-5 b(alen)m(t)35 b(to)1308 3934 y Fd( )1388 3891 y Fn(i)1388 3959 y Fr(0)1455 3934 y Fp(=)27 b Fm(\000)p Fo(R)1709 3949 y Fr(0)1749 3934 y Fo(Q)1826 3949 y Fr(0)1866 3934 y Fd( )1945 3891 y Fn(i)1945 3959 y Fr(0)1985 3934 y Fo(;)49 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:::;)g(N)1205 b Fp(\(24\))-60 4179 y(Let)33 b Fq(q)174 4143 y Fn(i)174 4204 y Fr(0)241 4179 y Fp(:=)28 b(\()p Fo(q)457 4143 y Fn(i)453 4204 y Fr(0)p Fn(;)k Fr(1)580 4179 y Fo(;)17 b(q)671 4143 y Fn(i)667 4204 y Fr(0)p Fn(;)32 b Fr(2)793 4179 y Fp(\))831 4143 y Fn(T)914 4179 y Fp(=)c Fo(Q)1095 4194 y Fr(0)1134 4179 y Fd( )1214 4136 y Fn(i)1214 4204 y Fr(0)1253 4179 y Fo(;)49 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:::;)g(N)5 b(:)32 b Fp(A)h(trivial)g(calculation)g(using)g(\(21\))f (yields)670 4437 y Fo(R)744 4452 y Fr(1)784 4437 y Fq(q)843 4396 y Fn(i)843 4462 y Fr(0)910 4437 y Fp(=)c(0)p Fo(;)1204 4302 y Fh(Z)1259 4527 y Fb(R)1307 4508 y Fe(3)1362 4437 y Fo(q)1409 4396 y Fn(i)1405 4462 y Fr(0)p Fn(;)33 b Fr(1)1532 4437 y Fp(\()p Fo(x)p Fp(\)d)q Fo(x)28 b Fp(=)1904 4302 y Fh(Z)1960 4527 y Fb(R)2008 4508 y Fe(3)2063 4437 y Fo(s)2109 4396 y Fn(i)2137 4437 y Fp(\()p Fo(x)p Fp(\)d)q Fo(x)g Fp(=)f(0)p Fo(;)114 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:::;)g(N)567 b Fp(\(25\))-60 4682 y(The)34 b(sp)s(ectral)f(problem)g (adjoin)m(t)g(to)f(the)h(unp)s(erturb)s(ed)h(Birman-Sc)m(h)m(winger)g (principle)g(is)1259 4927 y Fo(K)1349 4886 y Fk(\003)1342 4952 y Fr(0)p Fn(;)e Fr(0)1469 4927 y Fq(w)1552 4886 y Fn(i)g Fk(\003)1550 4952 y Fr(0)1675 4927 y Fp(=)27 b Fm(\000)p Fq(w)1938 4886 y Fn(i)33 b Fk(\003)1936 4952 y Fr(0)2034 4927 y Fo(;)50 b(i)28 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)1871 5176 y Fp(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop -60 -73 a Fp(with)1513 95 y Fo(K)1603 54 y Fk(\003)1596 120 y Fr(0)p Fn(;)33 b Fr(0)1751 95 y Fp(=)28 b Fo(Q)1942 8 y Fe(1)p 1942 20 31 3 v 1942 61 a(2)1932 120 y Fr(0)1986 95 y Fo(R)2060 110 y Fr(0)2100 95 y Fm(j)p Fo(Q)2205 110 y Fr(0)2244 95 y Fm(j)2282 27 y Fe(1)p 2282 39 V 2282 80 a(2)-60 318 y Fp(and)33 b(the)g(solutions)g(of)f(the)h(adjoin)m(t)g(problem)g(are)g(related)g (to)f(the)h(original)g(ones)g(as)1226 586 y Fq(w)1309 545 y Fn(i)f Fk(\003)1307 611 y Fr(0)1432 586 y Fp(=)c Fo(s)1582 545 y Fn(i)1582 611 y Fr(0)1621 586 y Fo(sig)t(nQ)1886 601 y Fr(0)1925 586 y Fq(w)2008 545 y Fn(i)2006 611 y Fr(0)2046 586 y Fo(;)49 b(i)28 b Fp(=)f(1)p Fo(;)17 b(:::;)g(N)5 b(;)-60 854 y Fp(where)41 b Fo(s)275 818 y Fn(i)275 879 y Fr(0)354 854 y Fp(=)f Fo(sig)t(n)p Fm(h)p Fo(sig)t(nQ)962 869 y Fr(0)1001 854 y Fq(w)1084 818 y Fn(i)1082 879 y Fr(0)1122 854 y Fo(;)17 b Fq(w)1249 818 y Fn(i)1247 879 y 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y Fp(.)86 1969 y Fq(De\014nition)k(3.1)33 b Fg(F)-7 b(or)34 b(an)h(arbitr)-5 b(ary)35 b(function)f Fq(f)k Fm(2)29 b Fo(L)2130 1933 y Fr(2)2169 1969 y Fp(\()p Fl(R)2279 1933 y Fr(3)2319 1969 y Fo(;)17 b Fl(C)2435 1933 y Fr(2)2474 1969 y Fp(\))1462 2282 y Fo(P)1525 2297 y Fr(0)1564 2282 y Fq(f)39 b Fp(:=)1808 2157 y Fn(N)1767 2187 y Fh(X)1782 2397 y Fn(i)p Fr(=1)1911 2282 y Fm(h)p Fq(w)2033 2241 y Fn(i)34 b Fk(\003)2031 2306 y Fr(0)2131 2282 y Fo(;)17 b Fq(f)10 b Fm(i)p Fq(w)2341 2241 y Fn(i)2339 2306 y Fr(0)-60 2594 y Fg(The)34 b(op)-5 b(er)g(ator)35 b(de\014ne)-5 b(d)34 b(ab)-5 b(ove)34 b(satis\014es)g Fo(P)1559 2558 y Fr(2)1545 2619 y(0)1625 2594 y Fp(=)28 b Fo(P)1792 2609 y Fr(0)1866 2594 y Fg(and)34 b(its)h(adjoint)g(is)f (given)g(by)1456 2906 y Fo(P)1533 2865 y Fk(\003)1519 2931 y Fr(0)1572 2906 y Fq(g)29 b Fp(=)1801 2782 y Fn(N)1760 2812 y Fh(X)1775 3022 y Fn(i)p Fr(=1)1904 2906 y Fm(h)p Fq(w)2026 2865 y Fn(i)2024 2931 y Fr(0)2063 2906 y Fo(;)17 b Fq(g)r Fm(i)p Fq(w)2287 2865 y Fn(i)34 b Fk(\003)2285 2931 y Fr(0)-60 3219 y Fg(for)h(any)f Fq(g)c Fm(2)e Fo(L)528 3182 y Fr(2)568 3219 y Fp(\()p Fl(R)678 3182 y Fr(3)717 3219 y Fo(;)17 b Fl(C)833 3182 y Fr(2)872 3219 y Fp(\))p Fg(.)86 3437 y Fp(This)32 b(enables)g(us)g(to)e(restrict)i(the)f (problem)h(to)e Fo(R)q(anP)2131 3452 y Fr(0)2171 3437 y Fp(,)h(the)h(range)e(of)h(the)g(pro)5 b(jection)32 b Fo(P)3460 3452 y Fr(0)3499 3437 y Fp(,)f(whic)m(h)h(is)-60 3605 y(the)h(space)h(of)e Fo(N)43 b Fp(dimensions.)86 3824 y Fq(De\014nition)32 b(3.2)27 b Fg(The)i(F)-7 b(eshb)i(ach)28 b(map)h(\(se)-5 b(e)30 b(p.207)e(of)60 b([GS)35 b(03)o(],)c(also)59 b([BFS)34 b(98)o(]\))c(for)f(the)h(Birman-)-60 3992 y(Schwinger)k(op)-5 b(er)g(ator)34 b Fo(K)871 4007 y Fn(\013;)h(")1042 3992 y Fg(is)g(the)g(op)-5 b(er)g(ator)35 b Fo(F)41 b Fp(:)28 b Fo(R)q(anP)2099 4007 y Fr(0)2166 3992 y Fm(!)f Fo(R)q(anP)2540 4007 y Fr(0)2580 3992 y Fg(,)35 b(such)g(that)859 4261 y Fo(F)42 b Fp(:=)27 b Fo(P)1157 4276 y Fr(0)1197 4261 y Fp(\()p Fo(K)1318 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b(of)d(the)h(Hilb)s(ert-Sc)m(hmidt)h(op)s (erator)e Fo(K)3560 4762 y Fn(\013;)33 b(")3695 4747 y Fp(,)43 b(the)-60 4916 y(sp)s(ectrum)34 b(of)504 4890 y(\026)478 4916 y Fo(K)561 4931 y Fn(\013;)e(")717 4916 y Fp(+)22 b(1)33 b(is)g(empt)m(y)i(in)e(the)g(neigh)m(b)s(orho)s(o)s(d) g(of)f(the)h(origin)g(and)g(therefore,)g(the)h(op)s(erator)1871 5176 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop -60 -73 a Fp(\()4 -98 y(\026)-22 -73 y Fo(K)61 -58 y Fn(\013;)32 b(")215 -73 y Fp(+)19 b(1\))397 -109 y Fk(\000)p Fr(1)523 -73 y Fp(is)32 b(w)m(ell)g (de\014ned.)45 b(W)-8 b(e)32 b(denote)g(the)g(\014rst)g(term)f(in)h (the)g(de\014nition)g(of)f(the)h(F)-8 b(esc)m(h)m(bac)m(h)34 b(map)-60 95 y(as)1592 264 y Fo(U)k Fp(:=)28 b Fo(P)1890 279 y Fr(0)1929 264 y Fo(K)2012 279 y Fn(\013;)k(")2146 264 y Fo(P)2209 279 y Fr(0)-60 480 y Fp(and)h(the)g(second)h(one)e(as) 1099 649 y Fo(D)e Fp(:=)e Fo(P)1404 664 y Fr(0)1443 649 y Fo(K)1526 664 y Fn(\013;)33 b(")1683 624 y Fp(\026)1661 649 y Fo(P)1724 664 y Fr(0)1763 649 y Fp(\()1827 624 y(\026)1801 649 y Fo(K)1884 664 y Fn(\013;)f(")2040 649 y Fp(+)22 b(1\))2225 608 y Fk(\000)p Fr(1)2341 624 y Fp(\026)2319 649 y Fo(P)2382 664 y Fr(0)2421 649 y Fo(K)2504 664 y Fn(\013;)33 b(")2639 649 y Fo(P)2702 664 y Fr(0)-60 865 y Fp(Since)39 b(the)e(F)-8 b(esh)m(bac)m(h)39 b(op)s(erator)e(has)h (the)g(isolated)g(eigen)m(v)-5 b(alue)39 b Fm(\000)p Fp(1)e(of)g(m)m(ultiplicit)m(y)j Fo(N)48 b Fp(as)37 b(w)m(ell)i(as)f (the)-60 1034 y(Birman-Sc)m(h)m(winger)d(op)s(erator)c(\(see)67 b([GS)32 b(03],)65 b([BFS)32 b(98]\))g(w)m(e)i(ha)m(v)m(e)g(the)f(sp)s (ectral)g(problem)1703 1288 y Fo(F)14 b Fq(v)29 b Fp(=)f Fm(\000)p Fq(v)r Fo(;)-60 1542 y Fp(where)34 b Fq(v)29 b Fp(=)414 1467 y Fh(P)519 1493 y Fn(N)519 1571 y(k)r Fr(=1)668 1542 y Fo(c)710 1557 y Fn(k)753 1542 y Fq(w)836 1506 y Fn(k)834 1566 y Fr(0)878 1542 y Fp(.)44 b(Th)m(us)1603 1710 y Fo(det)p Fp(\()p Fo(F)36 b Fp(+)22 b Fo(I)8 b Fp(\))27 b(=)h(0)1489 b(\(27\))-60 1927 y(Hence)33 b(the)f(implicit)h (dep)s(endence)h(of)e Fo(\013)g Fp(on)g Fo(")f Fp(is)h(giv)m(en)h(b)m (y)g(this)f(equation)h(and)f(the)g(goal)f(is)h(to)g(estimate)-60 2095 y(the)37 b(matrix)h(elemen)m(ts)h(of)d(the)i(F)-8 b(esh)m(bac)m(h)38 b(op)s(erator.)56 b(Due)37 b(to)f(the)h (orthogonalit)m(y)g(relations)h(\(26\))e(they)-60 2264 y(are)d(giv)m(en)g(b)m(y)1201 2432 y Fo(F)1264 2447 y Fn(i;j)1372 2432 y Fp(=)27 b Fm(h)p Fq(w)1597 2391 y Fn(i)32 b Fk(\003)1595 2457 y Fr(0)1693 2432 y Fo(;)17 b(F)d Fq(w)1897 2385 y Fn(j)1895 2457 y Fr(0)1933 2432 y Fm(i)p Fo(;)114 b Fp(1)28 b Fm(\024)g Fo(i;)17 b(j)34 b Fm(\024)28 b Fo(N)-60 2649 y Fp(Let)33 b(us)g(\014rst)g(ev)-5 b(aluate)33 b(the)g(matrix)g(elemen)m(ts)i(of)d(the)h(op)s(erator)f Fo(U)10 b Fp(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)g (lemma.)86 2863 y Fq(Lemma)39 b(3.3)33 b Fg(F)-7 b(or)34 b(al)5 b(l)35 b Fo(i;)17 b(j)33 b Fp(=)28 b(1)p Fo(;)17 b(:::;)g(N)45 b Fg(we)34 b(have)648 3117 y Fo(U)714 3132 y Fn(i;j)822 3117 y Fp(=)28 b Fm(\000)p Fo(\016)1046 3132 y Fn(i;j)1149 3117 y Fm(\000)23 b Fp(2)p Fo("s)1390 3076 y Fn(i)1390 3142 y Fr(0)1428 3117 y Fp(\()1481 3092 y(^)1466 3117 y Fo(V)1523 3132 y Fn(p)1563 3117 y Fp(\))1601 3132 y Fn(i;j)1703 3117 y Fm(\000)g Fp(2)p Fo(\013)1915 3076 y Fr(2)1954 3117 y Fo(s)2000 3076 y Fn(i)2000 3142 y Fr(0)2062 3092 y Fp(^)2039 3117 y Fo(B)2113 3132 y Fn(i;j)2215 3117 y Fp(+)f Fo(\013)2376 3076 y Fr(2)2416 3117 y Fo("s)2508 3076 y Fn(i)2508 3142 y Fr(0)2547 3117 y Fm(h)p Fq(q)2645 3076 y Fn(i)2645 3142 y Fr(0)2684 3117 y Fo(;)17 b(R)2802 3132 y Fr(2)2841 3117 y Fo(Q)2918 3132 y Fn(p)2958 3117 y Fd( )3038 3070 y Fn(j)3038 3141 y Fr(0)3077 3117 y Fm(i)p Fp(+)952 3386 y(+)1038 3319 y Fo(\013)1101 3283 y Fr(3)1140 3319 y Fo(s)1186 3283 y Fn(i)1186 3343 y Fr(0)p 1038 3363 188 4 v 1054 3454 a Fp(12)p Fo(\031)1236 3386 y Fp(\()1290 3361 y(^)1274 3386 y Fo(S)1340 3345 y Fk(\003)1395 3361 y Fp(^)1379 3386 y Fo(S)6 b Fp(\))1483 3401 y Fn(i;j)1585 3386 y Fp(+)22 b Fo(\013)1746 3345 y Fr(3)1785 3386 y Fo("s)1877 3345 y Fn(i)1877 3411 y Fr(0)1916 3386 y Fm(h)p Fq(q)2014 3345 y Fn(i)2014 3411 y Fr(0)2053 3386 y Fo(;)17 b(R)2171 3401 y Fr(3)2211 3386 y Fo(Q)2288 3401 y Fn(p)2328 3386 y Fd( )2407 3339 y Fn(j)2407 3411 y Fr(0)2446 3386 y Fm(i)22 b Fp(+)g Fo(O)s Fp(\()p Fo(\013)2784 3345 y Fr(4)2823 3386 y Fp(\))p Fo(;)-60 3603 y Fg(wher)-5 b(e)34 b Fo(O)s Fp(\()p Fo(\013)394 3566 y Fr(4)433 3603 y Fp(\))g Fg(is)h(analytic)g (in)f Fo(\013)i Fg(for)e Fo(\013)i Fg(smal)5 b(l)34 b(enough.)86 3817 y(Pr)-5 b(o)g(of.)43 b(Step)35 b(I.)d Fp(F)-8 b(or)32 b(the)h(\014rst)g(term)g(of)f(the)h(F)-8 b(esh)m(bac)m(h)34 b(op)s(erator)e(w)m(e)i(ha)m(v)m(e)g(the)f(expansion)1417 4107 y Fo(U)38 b Fp(=)1679 3983 y Fr(3)1625 4013 y Fh(X)1635 4222 y Fn(s)p Fr(=0)1785 4107 y Fo(\013)1848 4066 y Fn(s)1885 4107 y Fo(U)1951 4122 y Fn(s)2010 4107 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)2287 4066 y Fr(4)2325 4107 y Fp(\))33 b Fo(;)-60 4422 y Fp(where)38 b Fo(U)292 4437 y Fn(s)363 4422 y Fp(=)c Fo(P)536 4437 y Fr(0)576 4422 y Fo(K)659 4437 y Fn(s;)d(")780 4422 y Fo(P)843 4437 y Fr(0)919 4422 y Fp(and)36 b Fo(K)1195 4437 y Fn(s;)c(")1351 4422 y Fp(=)i Fm(j)p Fo(Q)1566 4437 y Fr(0)1606 4422 y Fm(j)1644 4358 y Fe(1)p 1643 4370 31 3 v 1643 4411 a(2)1688 4422 y Fo(R)1762 4437 y Fn(s)1799 4422 y Fo(Q)p Fm(j)p Fo(Q)1981 4437 y Fr(0)2020 4422 y Fm(j)2048 4385 y Fk(\000)2113 4358 y Fe(1)p 2113 4370 V 2113 4411 a(2)2158 4422 y Fo(;)53 b(s)34 b Fp(=)g(0)p Fo(;)17 b(:::;)g Fp(3,)37 b(the)g(term)g Fo(O)s Fp(\()p Fo(\013)3347 4385 y Fr(4)3385 4422 y Fp(\))f(is)h (analytic)-60 4590 y(in)32 b Fo(\013)h Fp(for)f Fo(\013)g Fp(small)h(enough)g(and)f(the)g(expansion)i(con)m(v)m(erges)g(in)f(the) f(op)s(erator)g(norm)g(\()p Fo(L)3259 4554 y Fr(2)3326 4590 y Fm(!)c Fo(L)3520 4554 y Fr(2)3560 4590 y Fp(\))j(due)i(to)-60 4759 y(the)j(exp)s(onen)m(tial)h(deca)m(y)g(of)e(the)h(p)s(oten)m (tials.)53 b(Let)35 b(us)h(ev)-5 b(aluate)36 b(the)g(matrix)g(elemen)m (ts)i(for)d(eac)m(h)h(of)g(the)-60 4927 y(op)s(erators)c Fo(U)437 4942 y Fn(s)474 4927 y Fp(.)1871 5176 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 86 -73 a Fg(Step)41 b(II.)c Fp(Using)i(the)g (expression)i(for)d(the)h(pro)5 b(jection)40 b(op)s(erator)e(from)g (Lemma)h(3.1,)h(the)f(relations)-60 95 y(\(26\),)31 b(\(24\),)g(the)g (v)-5 b(ariable)32 b(c)m(hange)g(for)e(the)i(eigenfunctions)g(of)f(the) h(sp)s(ectral)f(problem)h(\(17\))f(and)g(for)g(the)-60 264 y(elemen)m(ts)k(of)d(the)h(energy)h(op)s(erator)d(w)m(e)j(obtain) 873 525 y(\()p Fo(U)977 540 y Fr(0)1017 525 y Fp(\))1055 540 y Fn(i;j)1162 525 y Fp(=)28 b Fm(h)p Fq(w)1388 484 y Fn(i)k Fk(\003)1386 550 y Fr(0)1483 525 y Fo(;)17 b(P)1590 540 y Fr(0)1629 525 y Fo(K)1712 540 y Fr(0)p Fn(;)32 b(")1836 525 y Fo(P)1899 540 y Fr(0)1938 525 y Fq(w)2021 478 y Fn(j)2019 550 y Fr(0)2059 525 y Fm(i)27 b Fp(=)h Fm(h)p Fq(w)2351 484 y Fn(i)k Fk(\003)2349 550 y Fr(0)2446 525 y Fo(;)17 b(K)2573 540 y Fr(0)p Fn(;)32 b Fr(0)2700 525 y Fq(w)2783 478 y Fn(j)2781 550 y Fr(0)2820 525 y Fm(i)g Fp(+)471 787 y(+)g Fo(")p Fm(h)p Fq(w)747 746 y Fn(i)g Fk(\003)745 811 y Fr(0)842 787 y Fo(;)17 b Fm(j)p Fo(Q)991 802 y Fr(0)1030 787 y Fm(j)1068 718 y Fe(1)p 1068 730 31 3 v 1068 772 a(2)1112 787 y Fo(R)1186 802 y Fr(0)1226 787 y Fo(Q)1303 802 y Fn(p)1343 787 y Fm(j)p Fo(Q)1448 802 y Fr(0)1487 787 y Fm(j)1515 746 y Fk(\000)1580 718 y Fe(1)p 1580 730 V 1580 772 a(2)1624 787 y Fq(w)1707 740 y Fn(j)1705 811 y Fr(0)1745 787 y Fm(i)27 b Fp(=)h Fm(\000)p Fo(\016)2035 802 y Fn(i;)k(j)2170 787 y Fm(\000)23 b Fp(2)p Fo("s)2411 746 y Fn(i)2411 811 y Fr(0)2450 787 y Fp(\()2503 762 y(^)2488 787 y Fo(V)2545 802 y Fn(p)2584 787 y Fp(\))2622 802 y Fn(i;j)2702 787 y Fo(;)114 b Fp(1)28 b Fm(\024)g Fo(i;)17 b(j)34 b Fm(\024)28 b Fo(N)86 1102 y Fg(Step)34 b(III.)d Fp(F)-8 b(or)31 b(an)m(y)h(op)s(erator)f Fo(A)h Fp(suc)m(h)i(that)d(the)h(quadratic)h(form)f(b)s(elo)m(w)g(is)h (de\014ned,)g(using)g(\(25\))e(w)m(e)-60 1271 y(ha)m(v)m(e)576 1439 y Fm(h)p Fq(w)698 1398 y Fn(i)h Fk(\003)696 1464 y Fr(0)794 1439 y Fo(;)17 b(P)901 1454 y Fr(0)940 1439 y Fo(K)1023 1454 y Fr(1)p Fn(;)32 b(")1147 1439 y Fo(A)p Fq(w)1303 1392 y Fn(j)1301 1464 y Fr(0)1340 1439 y Fm(i)c Fp(=)f Fo(s)1556 1398 y Fn(i)1556 1464 y Fr(0)1596 1439 y Fm(h)p Fo(R)1709 1454 y Fr(1)1748 1439 y Fq(q)1807 1398 y Fn(i)1807 1464 y Fr(0)1847 1439 y Fo(;)17 b(Q)p Fm(j)p Fo(Q)2073 1454 y Fr(0)2112 1439 y Fm(j)2140 1398 y Fk(\000)2205 1371 y Fe(1)p 2205 1383 V 2205 1424 a(2)2249 1439 y Fo(A)p Fq(w)2405 1392 y Fn(j)2403 1464 y Fr(0)2443 1439 y Fm(i)27 b Fp(=)h(0)p Fo(;)49 b Fp(1)27 b Fm(\024)h Fo(i;)17 b(j)34 b Fm(\024)28 b Fo(N)-60 1659 y Fp(Th)m(us)34 b(c)m(ho)s(osing)f Fo(A)28 b Fp(=)g Fo(P)851 1674 y Fr(0)922 1659 y Fp(yields)917 1920 y(\()p Fo(U)1021 1935 y Fr(1)1060 1920 y Fp(\))1098 1935 y Fn(i;j)1206 1920 y Fp(=)f Fm(h)p Fq(w)1431 1879 y Fn(i)32 b Fk(\003)1429 1945 y Fr(0)1527 1920 y Fo(;)17 b(P)1634 1935 y Fr(0)1673 1920 y Fo(K)1756 1935 y Fr(1)p Fn(;)32 b(")1880 1920 y Fo(P)1943 1935 y Fr(0)1982 1920 y Fq(w)2065 1873 y Fn(j)2063 1945 y 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Fr(2)2649 2708 y Fo(Q)2726 2723 y Fn(p)2766 2708 y Fd( )2845 2660 y Fn(j)2845 2732 y Fr(0)2884 2708 y Fm(ig)p Fo(;)49 b Fp(1)28 b Fm(\024)g Fo(i;)17 b(j)33 b Fm(\024)28 b Fo(N)-60 2969 y Fp(Th)m(us)34 b(it)f(remains)g(to)g (estimate)572 3230 y Fm(h)p Fq(q)670 3189 y Fn(i)670 3255 y Fr(0)709 3230 y Fo(;)17 b(R)827 3245 y Fr(2)867 3230 y Fq(q)926 3183 y Fn(j)926 3255 y Fr(0)965 3230 y Fm(i)28 b Fp(=)1175 3163 y(1)p 1145 3208 108 4 v 1145 3299 a(8)p Fo(\031)1263 3230 y Fm(f)p Fp(\()p Fo(q)1398 3189 y Fn(i)1394 3255 y Fr(0)p Fn(;)k Fr(1)1521 3230 y Fo(;)17 b Fm(j)p Fo(x)p Fm(j)k(\003)h Fo(q)1816 3183 y Fn(j)1812 3255 y Fr(0)p Fn(;)33 b Fr(1)1939 3230 y Fp(\))23 b(+)f(\()p Fo(q)2183 3189 y Fn(i)2179 3255 y Fr(0)p Fn(;)32 b Fr(2)2306 3230 y Fo(;)2433 3163 y Fp(1)p 2359 3208 197 4 v 2359 3227 a Fm(p)p 2442 3227 114 4 v 83 x Fp(2)p Fo(!)2565 3230 y(e)2610 3189 y Fk(\000)2665 3134 y(p)p 2724 3134 82 3 v 55 x Fr(2)p Fn(!)s Fk(j)p Fn(x)p Fk(j)2911 3230 y Fm(\003)22 b Fo(q)3030 3183 y Fn(j)3026 3255 y Fr(0)p Fn(;)33 b Fr(2)3153 3230 y Fp(\))p Fm(g)p Fo(;)-60 3492 y Fp(where)k Fm(\003)f Fp(stands)h(for)e(the)h (con)m(v)m(olution.)55 b(W)-8 b(e)36 b(write)h(the)f(\014rst)h(term)f (in)g(the)h(righ)m(t)f(side)h(of)e(the)h(equalit)m(y)-60 3660 y(ab)s(o)m(v)m(e)d(as)859 3762 y(1)p 829 3806 108 4 v 829 3897 a(8)p Fo(\031)947 3829 y Fp(\()p Fo(q)1032 3788 y Fn(i)1028 3854 y Fr(0)p Fn(;)g Fr(1)1155 3829 y Fo(;)17 b Fm(j)p Fo(x)p Fm(j)22 b(\003)g Fo(q)1451 3782 y Fn(j)1447 3853 y Fr(0)p Fn(;)32 b Fr(1)1574 3829 y Fp(\))27 b(=)h Fm(\000)1860 3762 y Fp(1)p 1830 3806 V 1830 3897 a(8)p Fo(\031)1975 3762 y(@)p 1958 3806 93 4 v 1958 3897 a(@)5 b(t)2060 3829 y Fm(j)2088 3845 y Fn(t)p Fr(=0)2203 3826 y Fe(+)2258 3829 y Fp(\()p Fo(q)2343 3788 y Fn(i)2339 3854 y Fr(0)p Fn(;)33 b Fr(1)2466 3829 y Fo(;)17 b(e)2555 3788 y Fk(\000)p Fn(t)p Fk(j)p Fn(x)p Fk(j)2741 3829 y Fm(\003)22 b Fo(q)2860 3782 y Fn(j)2856 3853 y Fr(0)p Fn(;)32 b Fr(1)2983 3829 y Fp(\))-60 4049 y(Ha)m(ving)k(the)f(form)m(ula)g(for)f(the)i(heat)f(k)m(ernel)i(of)d (the)h(ro)s(ot)f(of)h(the)g(Laplacian)g(handy)h(\(see)g(e.g.)51 b([LL)32 b(97]\))-60 4217 y(w)m(e)i(easily)f(obtain)981 4376 y Fl([)970 4408 y Fo(e)1015 4379 y Fk(\000)p Fn(t)p Fk(j)p Fn(x)p Fk(j)1179 4408 y Fp(\()p Fo(p)p Fp(\))28 b(=)1445 4341 y(\(2)p Fo(\031)t Fp(\))1639 4277 y Fe(3)p 1639 4289 31 3 v 1639 4330 a(2)p 1445 4385 239 4 v 1515 4477 a Fo(\031)1574 4448 y Fr(2)1874 4341 y Fo(t)p 1703 4385 377 4 v 1703 4477 a Fp([)p Fo(t)1765 4448 y Fr(2)1827 4477 y Fp(+)22 b Fo(p)1974 4448 y Fr(2)2013 4477 y Fp(])2040 4448 y Fr(2)2090 4408 y Fo(;)114 b(p)28 b Fm(2)g Fl(R)2474 4367 y Fr(3)2513 4408 y Fo(;)114 b(t)28 b(>)g Fp(0)-60 4628 y(Th)m(us)387 4822 y(1)p 357 4866 108 4 v 357 4958 a(8)p Fo(\031)475 4889 y Fp(\()p Fo(q)560 4848 y Fn(i)556 4914 y Fr(0)p Fn(;)33 b Fr(1)683 4889 y Fo(;)17 b Fm(j)p Fo(x)p Fm(j)22 b(\003)g Fo(q)979 4842 y Fn(j)975 4914 y Fr(0)p Fn(;)32 b Fr(1)1102 4889 y Fp(\))27 b(=)h Fm(\000)p Fp(\()1399 4854 y Fh(d)1386 4889 y Fo(q)1433 4855 y Fn(i)1429 4914 y Fr(0)p Fn(;)33 b Fr(1)1556 4889 y Fp(\()p Fo(p)p Fp(\))p Fo(;)1755 4822 y Fp(1)p 1735 4866 89 4 v 1735 4958 a Fo(p)1784 4929 y Fr(4)1846 4841 y Fh(d)1833 4889 y Fo(q)1880 4842 y Fn(j)1876 4914 y Fr(0)p Fn(;)g Fr(1)2003 4889 y Fp(\()p Fo(p)p Fp(\)\))28 b(=)f Fm(\000)p Fp(\(\()p Fm(\000)p Fp(\001\))2646 4848 y Fk(\000)p Fr(1)2742 4889 y Fo(q)2789 4848 y Fn(i)2785 4914 y Fr(0)p Fn(;)32 b Fr(1)2912 4889 y Fo(;)17 b Fp(\()p Fm(\000)p Fp(\001\))3190 4848 y Fk(\000)p Fr(1)3285 4889 y Fo(q)3332 4842 y Fn(j)3328 4914 y Fr(0)p Fn(;)32 b Fr(1)3455 4889 y Fp(\))1871 5176 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop -60 -73 a Fp(F)-8 b(or)34 b(the)i(quadratic)g (form)f(remaining)h(to)f(estimate)h(via)g(the)g(ab)s(o)m(v)m(e)g(men)m (tioned)g(heat)g(k)m(ernel)h(with)f Fo(t)c Fp(=)-60 13 y Fm(p)p 23 13 114 4 v 82 x Fp(2)p Fo(!)k Fp(w)m(e)d(deriv)m(e)648 296 y(1)p 619 341 108 4 v 619 432 a(8)p Fo(\031)737 364 y Fp(\()p Fo(q)822 322 y Fn(i)818 388 y Fr(0)p Fn(;)f Fr(2)945 364 y Fo(;)1072 296 y Fp(1)p 998 341 197 4 v 998 361 a Fm(p)p 1081 361 114 4 v 82 x Fp(2)p Fo(!)1204 364 y(e)1249 322 y Fk(\000)1304 267 y(p)p 1363 267 82 3 v 55 x Fr(2)p Fn(!)s Fk(j)p Fn(x)p Fk(j)1550 364 y Fm(\003)22 b Fo(q)1669 316 y Fn(j)1665 388 y Fr(0)p Fn(;)33 b Fr(2)1792 364 y Fp(\))28 b(=)f(\()2012 328 y Fh(d)1999 364 y Fo(q)2046 329 y Fn(i)2042 388 y Fr(0)p Fn(;)33 b Fr(2)2169 364 y Fp(\()p Fo(p)p Fp(\))p Fo(;)2542 296 y Fp(1)p 2348 341 437 4 v 2348 432 a(\(2)p Fo(!)25 b Fp(+)d Fo(p)2668 403 y Fr(2)2707 432 y Fp(\))2745 403 y Fr(2)2808 315 y Fh(d)2795 364 y Fo(q)2842 316 y Fn(j)2838 388 y Fr(0)p Fn(;)32 b Fr(2)2965 364 y Fp(\()p Fo(p)p Fp(\)\))27 b(=)1076 656 y(=)h(\(\()p Fm(\000)p Fp(\001)22 b(+)g(2)p Fo(!)t Fp(\))1686 615 y Fk(\000)p Fr(1)1780 656 y Fo(q)1827 615 y Fn(i)1823 680 y Fr(0)p Fn(;)32 b Fr(2)1950 656 y Fo(;)17 b Fp(\()p Fm(\000)p Fp(\001)22 b(+)g(2)p Fo(!)t Fp(\))2462 615 y Fk(\000)p Fr(1)2556 656 y Fo(q)2603 609 y Fn(j)2599 680 y Fr(0)p Fn(;)32 b Fr(2)2726 656 y Fp(\))-60 878 y(Therefore)189 1146 y Fm(h)p Fq(q)287 1105 y Fn(i)287 1171 y Fr(0)326 1146 y Fo(;)17 b(R)444 1161 y Fr(2)483 1146 y Fq(q)542 1099 y Fn(j)542 1171 y Fr(0)582 1146 y Fm(i)27 b Fp(=)h Fm(\000)p Fp(\(\()p Fm(\000)p Fp(\001\))1101 1105 y Fk(\000)p Fr(1)1196 1146 y Fo(q)1243 1105 y Fn(i)1239 1171 y Fr(0)p Fn(;)33 b Fr(1)1366 1146 y Fo(;)17 b Fp(\()p Fm(\000)p Fp(\001\))1644 1105 y Fk(\000)p Fr(1)1739 1146 y Fo(q)1786 1099 y Fn(j)1782 1171 y Fr(0)p Fn(;)33 b Fr(1)1909 1146 y Fp(\))22 b(+)g(\(\()p Fm(\000)p Fp(\001)h(+)f(2)p Fo(!)t Fp(\))2574 1105 y Fk(\000)p Fr(1)2667 1146 y Fo(q)2714 1105 y Fn(i)2710 1171 y Fr(0)p Fn(;)33 b Fr(2)2837 1146 y Fo(;)17 b Fp(\()p Fm(\000)p Fp(\001)23 b(+)f(2)p Fo(!)t Fp(\))3350 1105 y Fk(\000)p Fr(1)3443 1146 y Fo(q)3490 1099 y Fn(j)3486 1171 y Fr(0)p Fn(;)33 b Fr(2)3613 1146 y Fp(\))-60 1415 y(By)45 b(means)h(of)e(the)g(Birman-Sc)m(h)m(winger)j(principle)f(at)e 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Fr(0)1977 2020 y Fo(;)17 b(R)2095 2035 y Fr(2)2134 2020 y Fo(Q)2211 2035 y Fn(p)2251 2020 y Fd( )2330 1973 y Fn(j)2330 2044 y Fr(0)2370 2020 y Fm(i)p Fo(;)49 b Fp(1)27 b Fm(\024)h Fo(i;)17 b(j)34 b Fm(\024)28 b Fo(N)86 2388 y Fg(Step)34 b(V.)d Fp(F)-8 b(or)30 b(the)i(cubic)g(term)f(in)g(the)h(expansion)g (of)f(the)g(op)s(erator)g Fo(U)41 b Fp(using)32 b(the)f(de\014nitions)i (of)e(the)-60 2556 y(op)s(erators)f(and)g(the)g(v)m(ector-functions)i (in)m(v)m(olv)m(ed)h(in)d(it)g(along)f(with)i(the)f(iden)m(tit)m(y)i (\(25\))e(w)m(e)h(easily)g(obtain)521 2824 y(\()p Fo(U)625 2839 y Fr(3)665 2824 y Fp(\))703 2839 y Fn(i;j)811 2824 y Fp(=)c Fm(h)p Fq(w)1036 2783 y Fn(i)32 b Fk(\003)1034 2849 y Fr(0)1132 2824 y Fo(;)17 b(P)1239 2839 y Fr(0)1278 2824 y Fo(K)1361 2839 y Fr(3)p Fn(;)32 b(")1485 2824 y Fo(P)1548 2839 y Fr(0)1587 2824 y Fq(w)1670 2777 y Fn(j)1668 2849 y Fr(0)1707 2824 y Fm(i)c Fp(=)f Fo(s)1923 2783 y Fn(i)1923 2849 y Fr(0)1963 2824 y Fm(h)p Fq(q)2061 2783 y Fn(i)2061 2849 y Fr(0)2100 2824 y Fo(;)17 b(R)2218 2839 y Fr(3)2257 2824 y Fq(q)2316 2777 y Fn(j)2316 2849 y Fr(0)2356 2824 y Fm(i)22 b Fp(+)g Fo("s)2607 2783 y Fn(i)2607 2849 y Fr(0)2646 2824 y Fm(h)p Fq(q)2744 2783 y Fn(i)2744 2849 y Fr(0)2783 2824 y Fo(;)17 b(R)2901 2839 y Fr(3)2941 2824 y Fo(Q)3018 2839 y Fn(p)3058 2824 y Fd( )3137 2777 y Fn(j)3137 2849 y Fr(0)3176 2824 y Fm(i)28 b Fp(=)905 3108 y(=)1054 3040 y Fo(s)1100 3004 y Fn(i)1100 3065 y Fr(0)p 1019 3085 157 4 v 1019 3176 a Fp(12)p Fo(\031)1185 3108 y Fp(\()1240 3082 y(^)1223 3108 y Fo(S)1289 3066 y Fk(\003)1345 3082 y Fp(^)1328 3108 y Fo(S)6 b Fp(\))1432 3123 y Fn(i;j)1534 3108 y Fp(+)22 b Fo("s)1724 3066 y Fn(i)1724 3132 y Fr(0)1763 3108 y Fm(h)p Fq(q)1861 3066 y Fn(i)1861 3132 y Fr(0)1901 3108 y Fo(;)17 b(R)2019 3123 y Fr(3)2058 3108 y Fo(Q)2135 3123 y Fn(p)2175 3108 y Fd( )2254 3060 y Fn(j)2254 3132 y Fr(0)2294 3108 y Fm(i)p Fo(;)49 b Fp(1)27 b Fm(\024)h Fo(i;)17 b(j)34 b Fm(\024)28 b Fo(N)-60 3330 y Fp(Collecting)34 b(all)e(the)h(estimates)h(deriv)m(ed)h(in)e(the)g Fo(S)6 b(teps)32 b(I)e Fm(\000)23 b Fo(V)54 b Fp(w)m(e)34 b(complete)f(the)g (pro)s(of)f(of)g(the)h(lemma.)p 3824 3501 60 62 v 86 3767 a Fq(Lemma)39 b(3.4)33 b Fg(F)-7 b(or)34 b(al)5 b(l)35 b Fo(i;)17 b(j)33 b Fp(=)28 b(1)p Fo(;)17 b(:::;)g(N)45 b Fg(we)34 b(have)575 4035 y Fo(D)656 4050 y Fn(i;j)764 4035 y Fp(=)28 b Fo(")914 3994 y Fr(2)953 4035 y Fp(\()p Fo(D)1072 4050 y Fr(0)1111 4035 y Fp(\))1149 4050 y Fn(i;j)1251 4035 y Fp(+)22 b Fo(\013)q(")1458 3994 y Fr(2)1497 4035 y Fp(\()p Fo(D)1616 4050 y Fr(1)1655 4035 y Fp(\))1693 4050 y Fn(i;j)1796 4035 y Fp(+)g Fo(\013)1957 3994 y Fr(2)1996 4035 y Fo(")p Fp(\()p Fo(D)2161 4050 y Fr(2)2200 4035 y Fp(\))2238 4050 y Fn(i;j)2340 4035 y Fp(+)g Fo(\013)2501 3994 y Fr(3)2540 4035 y Fo(")p Fp(\()p Fo(D)2705 4050 y Fr(3)2744 4035 y Fp(\))2782 4050 y Fn(i;j)2884 4035 y Fp(+)g Fo(O)s Fp(\()p Fo(\013)3161 3994 y Fr(4)3200 4035 y Fp(\))p Fo(;)-60 4303 y Fg(wher)-5 b(e)30 b(her)-5 b(e)30 b(and)g(further)h(with)f(a)h(slight)f(abuse)g(of)g(notations)g Fp(\()p Fo(D)2379 4318 y Fn(k)2422 4303 y Fp(\))2460 4318 y Fn(i;j)2540 4303 y Fo(;)47 b(k)31 b Fp(=)c(0)p Fo(;)17 b(:::;)g Fp(3)30 b Fg(stand)g(for)h(the)f(terms)-60 4472 y(analytic)38 b(in)f Fo(")p Fg(,)i(such)f(that)g(their)g(exact)g (values)f(ar)-5 b(e)38 b(unimp)-5 b(ortant)38 b(and)f Fo(O)s Fp(\()p Fo(\013)2861 4435 y Fr(4)2900 4472 y Fp(\))h Fg(is)f(analytic)h(in)g Fo(\013)h Fg(for)e Fo(\013)-60 4640 y Fg(smal)5 b(l)34 b(enough.)1871 5176 y Fp(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 86 -73 a Fg(Pr)-5 b(o)g(of.)42 b Fp(Let)28 b(us)g(expand)h(eac)m(h)g(of)e(the)h(three)h(op)s(erator)e (v)-5 b(alued)28 b(factors)g(in)m(v)m(olv)m(ed)i(in)e(the)g(second)h (term)-60 95 y Fo(D)38 b Fp(of)d(the)h(F)-8 b(esh)m(bac)m(h)38 b(map)d(in)h(terms)h(of)e(the)h(parameter)g Fo(\013)g Fp(up)g(to)f(the)h(third)g(order)g(inclusiv)m(ely)-8 b(.)55 b(Th)m(us)-60 264 y(the)33 b(\014rst)g(one)g(can)g(b)s(e)g (written)g(as)1121 523 y Fo(P)1184 538 y Fr(0)1224 523 y Fo(K)1307 538 y Fn(\013;)f(")1463 498 y Fp(\026)1441 523 y Fo(P)1504 538 y Fr(0)1571 523 y Fp(=)1729 399 y Fr(3)1674 429 y Fh(X)1685 638 y Fn(s)p Fr(=0)1835 523 y Fo(\013)1898 482 y Fn(s)1934 523 y Fo(P)1997 538 y Fr(0)2037 523 y Fo(K)2120 538 y Fn(s;)f(")2263 498 y Fp(\026)2241 523 y Fo(P)2304 538 y Fr(0)2365 523 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)2642 482 y Fr(4)2681 523 y Fp(\))-60 799 y(and)33 b(the)g(third)g(one)f(analogously)h(as)1143 1052 y(\026)1121 1078 y Fo(P)1184 1093 y Fr(0)1224 1078 y Fo(K)1307 1093 y Fn(\013;)f(")1441 1078 y Fo(P)1504 1093 y Fr(0)1571 1078 y Fp(=)1729 953 y Fr(3)1674 983 y Fh(X)1685 1192 y Fn(s)p Fr(=0)1835 1078 y Fo(\013)1898 1036 y Fn(s)1956 1052 y Fp(\026)1934 1078 y Fo(P)1997 1093 y Fr(0)2037 1078 y Fo(K)2120 1093 y Fn(s;)f(")2241 1078 y Fo(P)2304 1093 y Fr(0)2365 1078 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)2642 1036 y Fr(4)2681 1078 y Fp(\))-60 1365 y(with)32 b(the)g Fo(K)411 1380 y Fn(s;)g(")564 1365 y Fp(op)s(erators)g(giv)m(en)h(in)e(the)i(Lemma)f(3.3.)43 b(T)-8 b(o)31 b(expand)i(the)f(middle)h(term)f(w)m(e)h(use)3605 1340 y(\026)3579 1365 y Fo(K)3662 1380 y Fn(\013;)g(")3824 1365 y Fp(=)-60 1459 y Fh(P)45 1486 y Fr(3)45 1563 y Fn(s)p Fr(=0)189 1534 y Fo(\013)252 1498 y Fn(s)314 1509 y Fp(\026)288 1534 y Fo(K)371 1549 y Fn(s;")482 1534 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)759 1498 y Fr(4)798 1534 y Fp(\))32 b(and)h(therefore,)842 1822 y(\()906 1797 y(\026)880 1822 y Fo(K)963 1837 y Fn(\013;)f(")1119 1822 y Fp(+)22 b(1\))1304 1781 y Fk(\000)p Fr(1)1426 1822 y Fp(=)28 b(\(1)21 b(+)1791 1698 y Fr(3)1736 1728 y Fh(X)1747 1937 y Fn(s)p Fr(=1)1897 1822 y Fo(\013)1960 1781 y Fn(s)1996 1822 y Fo(M)2090 1837 y Fn(s;)32 b(")2234 1822 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)2511 1781 y Fr(4)2550 1822 y Fp(\)\))2626 1781 y Fk(\000)p Fr(1)2720 1822 y Fo(M)2814 1837 y Fr(0)p Fn(;)33 b(")2971 1822 y Fo(;)-60 2098 y Fp(where)h(the)f(op)s(erators)1166 2300 y Fo(M)1260 2315 y Fn(s;)f(")1409 2300 y 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b(")2927 2787 y Fp(\))p Fo(M)3059 2802 y Fr(0)p Fn(;)h(")3184 2787 y Fp(+)659 3029 y(+)p Fo(\013)798 2988 y Fr(3)837 3029 y Fp(\()p Fo(M)969 3044 y Fr(1)p Fn(;)f(")1093 3029 y Fo(M)1187 3044 y Fr(2)p Fn(;)h(")1334 3029 y Fp(+)22 b Fo(M)1526 3044 y Fr(2)p Fn(;)33 b(")1650 3029 y Fo(M)1744 3044 y Fr(1)p Fn(;)g(")1891 3029 y Fm(\000)23 b Fo(M)2085 3044 y Fr(3)p Fn(;)32 b(")2231 3029 y Fm(\000)23 b Fo(M)2435 2988 y Fr(3)2425 3054 y(1)p Fn(;)33 b(")2549 3029 y Fp(\))p Fo(M)2681 3044 y Fr(0)p Fn(;)g(")2828 3029 y Fp(+)22 b Fo(O)s Fp(\()p Fo(\013)3105 2988 y Fr(4)3143 3029 y Fp(\))-60 3241 y(W)-8 b(e)28 b(m)m(ultiply)h(out)f(the)g (expansions)i(deriv)m(ed)f(ab)s(o)m(v)m(e)g(for)e(eac)m(h)h(of)f(the)i (three)f(factors)g(con)m(tributing)g(to)g(the)-60 3409 y(op)s(erator)35 b Fo(D)i Fp(and)f(estimate)g(the)g(matrix)g(elemen)m (ts)h Fo(D)2013 3424 y Fn(i;j)2125 3409 y Fp(=)c Fm(h)p Fq(w)2356 3373 y Fn(i)e Fk(\003)2354 3434 y Fr(0)2451 3409 y Fo(;)17 b(D)s Fq(w)2662 3362 y Fn(j)2660 3434 y 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b(1.4.)73 b Fp(By)43 b(means)g(of)g(the)g(lemmas)g(men)m(tioned)i(ab)s(o)m(v)m(e)e(the)g (equation)g(\(27\))f(is)-60 4759 y(equiv)-5 b(alen)m(t)34 b(to)1584 4927 y Fo(detW)14 b Fp(\()p Fo(";)j(\013)q Fp(\))26 b(=)i(0)p Fo(;)1871 5176 y Fp(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop -60 -73 a Fp(where)726 95 y Fo(W)818 110 y Fn(i;j)898 95 y Fp(\()p Fo(";)17 b(\013)q Fp(\))26 b(=)i Fo(")p Fp(\()1356 70 y(^)1341 95 y Fo(V)1398 110 y Fn(p)1437 95 y Fp(\))1475 110 y Fn(i;j)1577 95 y Fp(+)22 b Fo(\013)1738 54 y Fr(2)1801 70 y Fp(^)1777 95 y Fo(B)1851 110 y Fn(i;j)1954 95 y Fm(\000)2091 28 y Fo(\013)2154 -8 y Fr(3)p 2063 72 157 4 v 2063 164 a Fp(24)p Fo(\031)2230 95 y Fp(\()2284 70 y(^)2268 95 y Fo(S)2334 54 y Fk(\003)2390 70 y Fp(^)2373 95 y Fo(S)6 b Fp(\))2477 110 y Fn(i;j)2579 95 y Fp(+)22 b Fo(")2723 54 y Fr(2)2762 95 y Fp(\()p Fo(D)2881 110 y Fr(0)2920 95 y Fp(\))2958 110 y Fn(i;j)3039 95 y Fp(+)620 313 y(+)p Fo(\013)q(")805 272 y Fr(2)843 313 y Fp(\()p Fo(D)962 328 y Fr(1)1002 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b(w)m(e)g(ha)m(v)m(e)g(the) f(follo)m(wing)g(p)s(ossibilities.)86 2194 y(\(i\))27 b Fm(h)p Fq(v)316 2157 y Fn(i)314 2218 y(p)354 2194 y Fo(;)421 2168 y Fp(^)398 2194 y Fo(B)5 b Fq(v)538 2157 y Fn(i)536 2218 y(p)576 2194 y Fm(i)27 b Fo(<)h Fp(0)e(and)i Fo(\025)1063 2157 y Fn(i)1063 2218 y(p)1130 2194 y Fo(>)f Fp(0)g(or)g Fm(h)p Fq(v)1523 2157 y Fn(i)1521 2218 y(p)1561 2194 y Fo(;)1628 2168 y Fp(^)1605 2194 y Fo(B)5 b Fq(v)1745 2157 y Fn(i)1743 2218 y(p)1782 2194 y Fm(i)28 b Fo(>)f Fp(0)g(and)g Fo(\025)2269 2157 y Fn(i)2269 2218 y(p)2337 2194 y Fo(<)g Fp(0.)42 b(The)28 b(situation)f(when)i(the)e(sp)s(ectral) -60 2362 y(parameter)38 b(has)h(a)e(p)s(ositiv)m(e)i(real)f(part)g (corresp)s(onds)h(to)f(the)g(eigen)m(v)-5 b(alues)40 b(bifurcating)e(along)g(the)g(real)-60 2531 y(line)33 b(ha)m(ving)h(the)f(asymptotics)795 2803 y Fo(z)840 2818 y Fn(i)896 2803 y Fp(=)28 b Fo(!)d Fp(+)d Fo(")1391 2727 y(\025)1448 2691 y Fn(i)1448 2752 y(p)p 1240 2780 398 4 v 1240 2887 a Fm(h)p Fq(v)1340 2858 y Fn(i)1338 2911 y(p)1377 2887 y Fo(;)1444 2862 y Fp(^)1421 2887 y Fo(B)5 b Fq(v)1561 2858 y Fn(i)1559 2911 y(p)1599 2887 y Fm(i)1670 2803 y Fp(+)22 b Fo(O)s Fp(\()p Fo(")1940 2735 y Fe(3)p 1939 2747 31 3 v 1939 2788 a(2)1983 2803 y Fp(\))p Fo(;)114 b(i)28 b Fp(=)g(1)p Fo(;)17 b(:::;)g(N)2633 2762 y Fr(+)2623 2836 y Fn(B)2679 2817 y Fc(\000)2757 2803 y Fp(+)22 b Fo(N)2943 2762 y Fk(\000)2933 2836 y Fn(B)2989 2817 y Fe(+)-60 3090 y Fp(When)38 b(the)g(sp)s(ectral)g(parameter)g(mo)m(v)m (es)h(to)e(the)h(second)g(sheet)h(of)e(the)h(Riemann)g(surface,)h(the)f (eigen-)-60 3259 y(v)-5 b(alues)38 b(near)g(the)f(edge)h(of)f(the)h (essen)m(tial)h(sp)s(ectrum)g(of)e(the)g(linearized)i(NLS)f(op)s (erator)e(are)i(not)f(b)s(eing)-60 3427 y(pro)s(duced.)86 3643 y(\(ii\))45 b Fm(h)p Fq(v)361 3607 y Fn(i)359 3667 y(p)398 3643 y Fo(;)465 3617 y Fp(^)442 3643 y Fo(B)5 b Fq(v)582 3607 y Fn(i)580 3667 y(p)620 3643 y Fm(i)48 b Fo(<)f Fp(0)e(and)f Fo(\025)1182 3607 y Fn(i)1182 3667 y(p)1270 3643 y Fo(<)j 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begin 23 22 bop -60 -73 a Fp(assymptotics)35 b(of)d(the)i(eigen)m(v)-5 b(alues)34 b(bifurcating)f(from)g(the)g(edge) h(of)e(the)h(essen)m(tial)i(sp)s(ectrum)f(are)f(giv)m(en)-60 95 y(b)m(y)1214 264 y Fo(z)t Fp(\()p Fo(")p Fp(\))28 b(=)g Fo(!)d Fm(\000)e Fp(4)p Fo(")1798 223 y Fr(2)1837 264 y Fm(h)p Fq(u)1938 279 y Fr(0)1977 264 y Fo(;)17 b(V)2078 279 y Fn(p)2118 264 y Fq(u)2180 279 y Fr(0)2219 264 y Fm(i)2258 223 y Fr(2)2320 264 y Fp(+)22 b Fo(o)p Fp(\()p Fo(")2549 223 y Fr(2)2588 264 y Fp(\))p 3824 488 60 62 v 86 854 a Fq(Ac)m(kno)m(wledgemen)m(t)82 b Fg(The)42 b(authors)g(thank)g(I.M.Sigal)g(for)g(suggesting)g(the)g(pr) -5 b(oblem,)43 b(for)f(valu-)-60 1023 y(able)c(discussions,)h(for)f(p) -5 b(artial)39 b(supp)-5 b(ort)39 b(via)f(NSER)n(C)g(gr)-5 b(ant)39 b(No.)57 b(7901,)39 b(Gang)f(Zhou,)i(M.Merkli)f(and)-60 1191 y(D.Pelinovsky)33 b(for)i(stimulating)g(discussions.)-60 1597 y Fa(References)-60 1914 y Fp([A)e(02])235 b(S.)29 b(Adac)m(hi:)42 b(\\A)29 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Fq(58,)34 b(1)p Fp(,)29 b(1{29,)g(\(2005\))-60 3722 y([G)j(90])232 b(M.Grillakis:)57 b(\\Analysis)40 b(of)e(the)h(linearization)g(around)g (a)f(critical)h(p)s(oin)m(t)g(of)f(an)g(in\014nite-)433 3890 y(dimensional)25 b(Hamiltonian)f(system",)j(Comm.Pure)e (Appl.Math.)f Fq(43,)29 b(3)p Fp(,)c(299{333,)f(\(1990\))-60 4192 y([GS)32 b(03])178 b(S.)64 b(Gustafson,)71 b(I.M.)63 b(Sigal:)105 b(\\Mathematical)64 b(concepts)h(of)d(quan)m(tum)j(mec)m (hanics",)433 4360 y(Springer{V)-8 b(erlag,)33 b(Berlin,)h(249)d(pp.,)i (\(2003\))-60 4662 y([GS)f(05])178 b(Z.)24 b(Gang,)g(I.M.)h(Sigal:)39 b(\\Asymptotic)25 b(stabilit)m(y)g(of)e(nonlinear)h(Sc)m(hr\177)-49 b(odinger)25 b(equations)g(with)433 4830 y(p)s(oten)m(tial",)33 b(Rev.Math.Ph)m(ys.)i Fq(17,)j(10)p Fp(,)33 b(1143{1207,)d(\(2005\)) 1871 5176 y(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop -60 -73 a Fp([HS)33 b(96])181 b(P)-8 b(.D.)48 b(Hislop,)53 b(I.M.)48 b(Sigal:)74 b(\\In)m(tro)s (duction)49 b(to)f(sp)s(ectral)g(theory)h(with)f(applications)h(to)433 95 y(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erators",)e(Springer,)i (\(1996\))-60 397 y([KS)f(02])178 b(T.)46 b(Kapitula,)i(B.)d (Sandstede:)70 b(\\Edge)45 b(bifurcations)h(for)e(near)h(in)m(tegrable) h(systems)i(via)433 565 y(Ev)-5 b(ans)34 b(functions)f(tec)m(hniques",) i(SIAM)f(J.)f(Math.Anal.)g Fq(33,)k(5)p Fp(,)c(1117{1143,)e(\(2002\)) -60 867 y([KS)i(04])178 b(T.)42 b(Kapitula,)i(B.)e(Sandstede:)63 b(\\Eigen)m(v)-5 b(alues)44 b(and)e(resonances)h(using)f(the)g(Ev)-5 b(ans)43 b(func-)433 1035 y(tions",)33 b(Discrete)h(Con)m (tin.Dyn.Syst.)h Fq(10,)i(4)p Fp(,)c(857{869,)e(\(2004\))-60 1337 y([KS)i(80])178 b(M.)42 b(Klaus,)i(B.)e(Simon:)62 b(\\Coupling)42 b(constan)m(t)g(thresholds)i(in)d(nonrelativistic)j (quan)m(tum)433 1505 y(mec)m(hanics.)35 b(I.)e(Short-range)f(t)m(w)m (o-b)s(o)s(dy)h(case",)g(Ann.Ph)m(ys.)i Fq(130,)j(2)p Fp(,)33 b(251{281,)e(\(1980\))-60 1806 y([LL)i(97])186 b(E.)43 b(Lieb,)j(M.)e(Loss:)64 b(\\Analysis",)47 b(Graduate)42 b(studies)j(in)e(Mathematics,)k(14,)e(American)433 1975 y(Mathematical)34 b(So)s(ciet)m(y)-8 b(,)33 b(Pro)m(vidence,)i(RI,)e (\(1997\))-60 2276 y([LSW)g(76])93 b(E.)39 b(Lieb,)i(B.)e(Simon,)i(A.)e (Wigh)m(tman:)57 b(\\Studies)39 b(in)g(Mathematical)h(Ph)m(ysics:)59 b(Essa)m(ys)41 b(in)433 2445 y(Honor)32 b(of)h(V)-8 b(alen)m(tine)33 b(Bargmann",)g(Princeton)h(Univ)m(ersit)m(y)h(Press,)f(\(1976\))-60 2746 y([M)f(93])219 b(K.)32 b(McLeo)s(d:)44 b(\\Uniqueness)35 b(of)c(p)s(ositiv)m(e)i(radial)f(solutions)h(of)e(\001)p Fo(u)21 b Fp(+)g Fo(f)11 b Fp(\()p Fo(u)p Fp(\))27 b(=)g(0)32 b(in)g Fl(R)3645 2710 y Fn(n)3692 2746 y Fo(:)g Fp(I)s(I.")433 2915 y(T)-8 b(rans.Amer.Math.So)s(c.)35 b Fq(339,)i(2)p Fp(,)c(495{505,)e(\(1993\))-60 3216 y([PKA)i(98])93 b(D.)37 b(P)m(elino)m(vsky)-8 b(,)42 b(Y.)c(Kivshar,)i(V.)e(Afanasjev:)54 b(\\In)m(ternal)38 b(mo)s(des)h(of)e(en)m(v)m(elop)s(e)i(solitons",)433 3385 y(Ph)m(ys.D)34 b Fq(116,)k(1{2)p Fp(,)32 b(121{142,)f(\(1998\))-60 3686 y([P)i(04])242 b(G.)62 b(P)m(erelman:)105 b(\\Asymptotic)64 b(stabilit)m(y)g(of)e(m)m(ulti-soliton)i(solutions)f(for)f(nonlinear) 433 3855 y(Sc)m(hr\177)-49 b(odinger)49 b(equations",)k(Comm.)48 b(P)m(artial)g(Di\013eren)m(tial)g(Equations)h Fq(29,)59 b(7{8)p Fp(,)52 b(1051{)433 4023 y(1095,)32 b(\(2004\))-60 4324 y([Str)h(77])178 b(W.)23 b(Strauss:)39 b(\\Existence)24 b(of)e(solitary)g(w)m(a)m(v)m(es)j(in)d(higher)h(dimensions",)j (Comm.Math.Ph)m(ys.)433 4493 y Fq(55,)38 b(2)p Fp(,)32 b(149{162,)f(\(1977\))-60 4794 y([S)i(79])254 b(B.)28 b(Simon:)41 b(\\F)-8 b(unctional)27 b(in)m(tegration)h(and)f(quan)m (tum)i(ph)m(ysics",)h(Academic)f(Press,)h(\(1979\))1871 5176 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop -60 -73 a Fp([S])385 b(W.)36 b(Sc)m(hlag:)51 b(\\Stable)37 b(manifolds)f(for)g(an)g(orbitally)g (unstable)h(NLS",)g(to)e(app)s(ear)h(in)g(Ann.)433 95 y(of)c(Math.)-60 397 y([V])366 b(V.)37 b(V)-8 b(ougalter:)51 b(\\On)36 b(the)h(negativ)m(e)g(index)h(theorem)f(for)f(the)h (linearized)h(NLS)e(problem",)433 565 y(to)c(app)s(ear)h(in)g (Canad.Math.Bull.)-60 867 y([VP)g(06])169 b(V.)40 b(V)-8 b(ougalter,)42 b(D.)d(P)m(elino)m(vsky:)62 b(\\Eigen)m(v)-5 b(alues)41 b(of)f(zero)g(energy)h(in)f(the)g(linearized)i(NLS)433 1035 y(problem",)33 b(J.)g(Math.)g(Ph)m(ys.)h Fq(47,)k(6)p Fp(,)33 b(13)f(pp.,)h(\(2006\))-60 1337 y([W)g(85])208 b(M.I.)39 b(W)-8 b(einstein:)56 b(\\Mo)s(dulation)37 b(stabilit)m(y)j(of)d(ground)h(states)h(of)e(nonlinear)i(Sc)m(hr\177) -49 b(odinger)433 1505 y(equations",)34 b(SIAM)f(J.Math.Anal.")g Fq(16,)38 b(3)p Fp(,)33 b(472{491,)e(\(1985\))1871 5176 y(26)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0810311755241--