Content-Type: multipart/mixed; boundary="-------------0703161146217" This is a multi-part message in MIME format. ---------------0703161146217 Content-Type: text/plain; name="07-62.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-62.keywords" quantum graph, vertex coupling, approximation, norm resolvent convergence ---------------0703161146217 Content-Type: application/postscript; name="vertapp.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="vertapp.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: vertapp.dvi %%CreationDate: Fri Mar 16 17:18:57 2007 %%Pages: 33 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR10 CMR8 CMSY8 CMBX10 CMTI8 CMMI8 CMMI6 CMR6 CMMI10 %%+ CMMI7 CMSY10 CMSY7 CMTI10 CMR7 CMEX10 CMBXTI10 MSBM10 CMMIB10 CMMI5 %%+ CMTI7 CMR5 MSBM7 CMSY5 CMR9 CMTI9 CMBX9 CMTT9 CMMI9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 vertapp %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2007.03.16:1718 %%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: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % 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 (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 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 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 69 /E put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 20 /caron put dup 28 /oslash put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3C89F3D8DBAC85E06561646557A B16BAE67D1C3D5F6EC51F05178B6CBC34F9F491056E5ADE1256277DB904A7F75 6105D774449D6CBB70F2F539D739D011BDBE27A9674A5CA3D5BD42B7FBFBBA35 89966597639DE60F2A4812AFAEFBBBB26C9E24F72439F3AF2A7D5C0D2AFB5146 58112FC60E9307B9FE29090D81441A1C3B7E371656A3321A0031038D54BB6670 0594C9C72FDDE3DE50EE1CFBDEEE5B6DDBA17B1D6328651FE1EFD1E2B8376949 5353E35A988064FDB6CE37ED4570B49903AF26304DE40164903476EC2F4830D6 4995CEE9B37B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 49 /one put dup 50 /two put dup 51 /three put dup 105 /i put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI7 %!PS-AdobeFont-1.1: CMTI7 1.0 %%CreationDate: 1991 Aug 18 21:07:18 % 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 (CMTI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI7 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 dup 99 /c put dup 110 /n put dup 116 /t put readonly def /FontBBox{-27 -250 1268 750}readonly def /UniqueID 5000825 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 20 /kappa put dup 33 /omega put dup 59 /comma put dup 62 /greater put dup 65 /A put dup 66 /B put dup 96 /lscript put dup 100 /d put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 117 /u put dup 118 /v put dup 119 /w put dup 126 /vector put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 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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 14 /delta put dup 110 /n put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 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 78 /N put dup 82 /R put dup 126 /planckover2pi1 put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 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 45 /hyphen put dup 65 /A put dup 81 /Q put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-29 -250 1274 754}readonly def /UniqueID 5000771 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 76 /circleplustext put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 59 /semicolon put dup 67 /C put dup 70 /F put dup 76 /L put dup 77 /M put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 71 /G put dup 102 /braceleft put dup 103 /braceright put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 15 /bullet put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 32 /arrowleft put dup 33 /arrowright put dup 39 /similarequal put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 71 /G put dup 72 /H put dup 74 /J put dup 79 /O put dup 82 /R put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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6F3959B0265FD36F9B66838C2BE122181FEC6025BE75F9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 28 /tau put dup 33 /omega 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 72 /H put dup 76 /L put dup 78 /N put dup 83 /S put dup 84 /T put dup 85 /U put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n 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 126 /vector put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 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 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 96 /lscript 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 114 /r 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 readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 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 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 106 /j put dup 107 /k put dup 110 /n put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 20 /caron put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 55 /seven put dup 57 /nine 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 73 /I put dup 75 /K 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 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{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 67 /C put dup 73 /I put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T 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 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /circlecopyrt put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 13 /fl put dup 20 /caron put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 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 readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 3 /Lambda put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 21 /breve put dup 22 /macron put dup 34 /quotedblright put dup 35 /numbersign put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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b Fl(R)21682 17380 y Fo(+)22418 17782 y Ft(;)431 b(j)370 b FC(=)308 b(1)p Ft(;)184 b(:)g(:)g(:)k(;)c(n)p FC(;)371 b(w)-31 b(e)370 b(denote)g(them)g(as)f(\000)f(or)h(\000)41998 17948 y Fs(n)42601 17782 y FC(.)5535 21239 y Fz(2.2.)655 b Fm(V)-98 b(ertex)459 b(c)-65 b(ouplings)5535 23231 y FC(Since)423 b(the)f(Hamiltonian)k(men)-31 b(tioned)424 b(ab)31 b(o)-31 b(v)g(e)423 b(is)f(a)g(second-order)g(op)31 b(erator,)423 b(the)f(matc)-31 b(hing)5535 24670 y(conditions)378 b(in)-31 b(v)g(olv)g(e)378 b(b)31 b(oundary)376 b(v)-61 b(alues)376 b(of)g(the)g(functions)h(in)f(the)g(v)-31 b(ertex)377 b(and)f(of)g(their)g(\014rst)5535 26109 y(deriv)-61 b(ativ)-31 b(es.)553 b(Both)e(regarded)g(as)g(one-sided)g(limits,)i(the)e(deriv) -61 b(ativ)-31 b(es)552 b(are)f(tak)-31 b(en)552 b(in)f(the)5535 27548 y(out)-31 b(w)g(ard)348 b(direction.)g(W)-92 b(e)345 b(arrange)i(them)g(in)-31 b(to)347 b(column)h(v)-31 b(ectors)346 b(\011\(0\))h(and)g(\011)38812 27147 y Fq(0)39122 27548 y FC(\(0\).)h(The)e(self-)5535 28988 y(adjoin)-31 b(tness)337 b(of)f Ft(H)90 b FC(,)336 b(whic)-31 b(h)337 b(in)e(the)h(ph)-31 b(ysical)337 b(language)h(means)d(conserv)-61 b(ation)337 b(of)f(probabilit)-31 b(y)5535 30427 y(curren)g(t)364 b(at)i(the)f(v)-31 b(ertex,)365 b(is)g(expressed)e(through)i(a)g (linear)g(relation)i(b)31 b(et)-31 b(w)g(een)365 b(these)g(v)-31 b(ectors,)20230 32563 y Ft(A)p FC(\011\(0\))248 b(+)e Ft(B)56 b FC(\011)26446 32106 y Fq(0)26755 32563 y FC(\(0\))309 b(=)f(0)184 b Ft(;)12420 b FC(\(2.1\))5535 34700 y(b)-31 b(y)271 b([22)q(])g(the)f(op)31 b(erator)271 b Ft(H)360 b FC(is)270 b(self-adjoin)-31 b(t)272 b(if)f(and)f(only)h(if)g Ft(A;)184 b(B)364 b Fr(2)307 b Fl(C)33612 34299 y Fs(n;n)35295 34700 y FC(satisfy)271 b(the)g(conditions)16069 36837 y(rank\()p Ft(A;)184 b(B)56 b FC(\))309 b(=)e Ft(n)184 b(;)1293 b(AB)26983 36380 y Fq(\003)28106 36837 y FC(is)369 b(self-adjoin)-31 b(t)s Ft(;)8258 b FC(\(2.2\))5535 38974 y(where)336 b(\()p Ft(A;)184 b(B)56 b FC(\))337 b(denotes)f(the)g Ft(n)179 b Fr(\002)g FC(2)p Ft(n)338 b FC(matrix)f(with)g Ft(A;)184 b(B)392 b FC(forming)337 b(the)f(\014rst)g(and)g(the)g (second)5535 40413 y Ft(n)410 b FC(columns,)h(resp)31 b(ectiv)-31 b(ely)-92 b(.)410 b(This)g(parametrization)j(is)c(ob)-31 b(viously)411 b(non-unique,)g(since)e Ft(A;)184 b(B)5535 41852 y FC(can)408 b(b)31 b(e)407 b(replaced)h(b)-31 b(y)408 b Ft(C)79 b(A;)184 b(C)79 b(B)464 b FC(with)409 b(an)-31 b(y)409 b(regular)e Ft(n)273 b Fr(\002)e Ft(n)408 b FC(matrix)h Ft(C)79 b FC(.)408 b(This)g(defect)g(can)g(b)31 b(e)5535 43291 y(corrected)369 b(b)-31 b(y)370 b(c)-31 b(ho)31 b(osing)370 b(the)g(matrices)g(in)f(the)h(standard)f(form)h ([23)q(,24)r(],)17275 45428 y(\()p Ft(U)d Fr(\000)246 b Ft(I)87 b FC(\)\011\(0\))247 b(+)f(i\()p Ft(U)368 b FC(+)246 b Ft(I)87 b FC(\)\011)29400 44971 y Fq(0)29710 45428 y FC(\(0\))309 b(=)f(0)184 b Ft(;)9465 b FC(\(2.3\))5535 47565 y(where)275 b Ft(U)396 b FC(is)275 b(an)h Ft(n)58 b Fr(\002)g Ft(n)277 b FC(unitary)f(matrix;)h(the)f(Hamiltonian)j 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FC(with)h(this)e(prop)31 b(ert)-31 b(y)445 b(is)h(c)-31 b(haracterized)6614 35559 y(b)g(y)528 b(t)-31 b(w)g(o)529 b(parameters,)g Ft(U)692 b FC(=)571 b Ft(aI)439 b FC(+)351 b Ft(bJ)634 b FC(with)529 b Fr(j)p Ft(a)p Fr(j)572 b FC(=)g(1)528 b(and)g Fr(j)p Ft(a)352 b FC(+)f Ft(nb)p Fr(j)573 b FC(=)f(1,)528 b Fp(cf.)g FC([20)r(],)g(the)6614 36998 y(corresp)31 b(onding)369 b(b)31 b(oundary)370 b(conditions)h(b)31 b(eing)10541 39397 y(\()p Ft(a)247 b Fr(\000)f FC(1\))184 b(\()q Ft( )15229 39563 y Fs(j)15696 39397 y FC(\(0\))247 b Fr(\000)f Ft( )19184 39563 y Fs(k)19729 39397 y FC(\(0\)\))i(+)e(i\()p Ft(a)h FC(+)f(1\))26770 38500 y Fn(\000)27279 39397 y Ft( )28040 38939 y Fq(0)28000 39670 y Fs(j)28466 39397 y FC(\(0\))i Fr(\000)e Ft( )31995 38939 y Fq(0)31955 39670 y Fs(k)32499 39397 y FC(\(0\))33912 38500 y Fn(\001)34753 39397 y FC(=)332 b(0)184 b Ft(;)1293 b(j)r(;)184 b(k)344 b Fr(2)363 b FC(^)-609 b Ft(n)185 b(;)44189 40282 y FC(\(2.7\))11752 41944 y(\()p Ft(a)247 b Fr(\000)f FC(1)g(+)g 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Fr(J)514 b(\000)309 b Ft(I)551 b FC(b)31 b(e)464 b(the)h(corresp)31 b(onding)464 b(matrix)i(of)f(the)g (condition)i(\(2.3\).)6614 52789 y(Giv)-31 b(en)379 b(a)g(family)i(of)e (real-v)-61 b(alued)380 b(functions)g Ft(W)477 b FC(=)324 b Fr(f)p Ft(W)30150 52955 y Fs(j)30940 52789 y FC(:)577 b Ft(j)386 b Fr(2)379 b FC(^)-609 b Ft(n)q Fr(g)p FC(,)379 b(for)g(simplicit)-31 b(y)382 b(assumed)6614 54228 y(to)370 b(b)31 b(e)369 b(compactly)i(supp)31 b(orted,)370 b(w)-31 b(e)369 b(de\014ne)g(scaled)h(p)31 b(oten)-31 b(tials)371 b(at)f(graph)g(edges)f(b)-31 b(y)19433 56975 y Ft(W)20478 57141 y Fs(\017;j)22069 56975 y FC(:=)23862 56226 y(1)p 23862 56721 554 45 v 23914 57735 a Ft(\017)24733 56975 y(W)25778 57141 y Fs(j)26429 55747 y Fn(\020)27223 56226 y Ft(x)p 27223 56721 633 45 v 27315 57735 a(\017)27988 55747 y Fn(\021)29019 56975 y Ft(;)1291 b(j)371 b Fr(2)363 b FC(^)-608 b Ft(n)185 b(:)10543 b FC(\(2.8\))6614 59635 y(Starting)433 b(from)f(the)f(free)g(b)31 b(oundary)431 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68930 y(j)45 b Fo(=1)29420 67707 y Fn(R)30158 67980 y Fo(1)29943 68992 y(0)30839 68598 y Ft(W)31884 68764 y Fs(j)32351 68598 y FC(\()p Ft(x)p FC(\))184 b Ft(dx)g Fp(.)6614 70923 y FC(F)-92 b(or)305 b Fz(pro)35 b(of)427 b FC(see)305 b([15)q(])h(where)f(a)h(more)f (general)h(result)f(of)h(this)g(t)-31 b(yp)31 b(e)306 b(is)f(deriv)-31 b(ed,)306 b(together)g(with)6614 72362 y(other)370 b(extensions)g(of)f(the)h(standard)f(Sturm-Liouville)j (theory)e(to)g(star)f(graphs.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(6)972 b Fy(P.)340 b(Exner)h(and)f(O.)g(T)-73 b(ur)-49 b(ek)5535 11900 y Fz(2.4.)655 b Fm(Appr)-65 b(oximation)459 b(of)g(singular)g(p)-65 b(ermutation-invariant)459 b(c)-65 b(ouplings)5535 13892 y 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33647 y(true)454 b(in)g(the)g(two)f(exclude)-57 b(d)453 b(c)-57 b(ases,)454 b(pr)-57 b(ovide)g(d)455 b(we)e(r)-57 b(eplac)g(e)454 b(the)g(ab)-57 b(ove)454 b Ft(u)p FC(\()p Ft(d)p FC(\))h Fp(by)f Fr(\000)p Ft(nd)41989 33246 y Fq(\000)p Fo(1)43632 33647 y Fp(and)5535 35086 y Ft(\020)82 b(d)6677 34685 y Fq(\000)p Fs(\027)8316 35086 y Fp(with)396 b Fl(R)308 b Fr(3)f Ft(\020)389 b Fr(6)p FC(=)308 b(0)396 b Fp(and)g Ft(\027)378 b(>)308 b FC(2)p Fp(,)397 b(r)-57 b(esp)g(e)g(ctively.)5535 37293 y Fz(Pro)35 b(of)439 b FC(can)317 b(b)31 b(e)316 b(found)i(in)e([20)r(],)h(the)g(particular)h(case)f(of)g Ft(\016)30078 36891 y Fq(0)30036 37567 y Fs(s)30826 37293 y FC(coupling)h(\(2.5\))h(in)e(whic)-31 b(h)318 b Ft(u)p FC(\()p Ft(d)p FC(\))308 b(=)5535 38732 y Fr(\000)p Ft(\014)58 b(d)7656 38330 y Fq(\000)p Fo(2)9214 38732 y FC(and)370 b Ft(v)40 b FC(\()p Ft(d)p FC(\))308 b(=)f Fr(\000)p Ft(d)16293 38330 y Fq(\000)p Fo(1)17851 38732 y FC(w)-31 b(as)370 b(discussed)e(in)h([19)r(].)5535 42267 y Fz(3.)637 b(CS-t)-35 b(yp)35 b(e)426 b(appro)-35 b(ximation)424 b(of)h(singular)g(couplings)5535 44260 y FC(After)443 b(the)f(preliminaries)h(let)g(us)e(turn)h(to)h(our)e(prop)31 b(er)441 b(task,)i(namely)h(appro)-31 b(ximations)445 b(of)5535 45699 y(singular)432 b(couplings)h Fp(\022)-566 b(a)453 b(la)516 b FC(Cheon)432 b(and)f(Shigehara,)i(i.e.)g(b)-31 b(y)431 b(means)h(of)g(additional)i Ft(\016)473 b FC(in)-31 b(ter-)5535 47138 y(actions,)480 b(prop)31 b(erly)478 b(scaled,)h(on)f(edges)g(of)h(our)f(star)g(graph,)h(without)h(the)e (requiremen)-31 b(t)479 b(of)5535 48577 y(p)31 b(erm)-31 b(utation)372 b(in)-31 b(v)-61 b(ariance.)5535 52112 y Fz(3.1.)655 b Fm(The)459 b(class)h(of)f(appr)-65 b(oximable)461 b(c)-65 b(ouplings)5535 54104 y FC(The)293 b(\014rst)f(question)h(is)g (ho)-31 b(w)293 b(large)g(is)g(the)f(class)h(of)g(op)31 b(erators)292 b Ft(H)32479 54270 y Fs(U)33513 54104 y FC(whic)-31 b(h)294 b(can)e(can)h(b)31 b(e)292 b(treated)5535 55543 y(in)378 b(this)f(w)-31 b(a)g(y)-92 b(.)379 b(W)-92 b(e)377 b(are)g(going)i(to)f(answ)-31 b(er)377 b(it)h(using)g(the)g (tec)-31 b(hnique)378 b(of)g([16)q(],)g(i.e.)h(lo)31 b(oking)379 b(in)-31 b(to)5535 56982 y(con)g(v)g(ergence)370 b(of)g(the)g(corresp)31 b(onding)369 b(b)31 b(oundary)369 b(conditions.)5535 59410 y Fz(Prop)35 b(osition)468 b(3.1.)566 b Fp(L)-57 b(et)430 b FC(\000)f Fp(b)-57 b(e)430 b(a)f(star)f(gr)-57 b(aph)429 b(with)f Ft(n)i Fp(semi-in\014nite)h(e)-57 b(dges)430 b(and)f FC(\000\()p Ft(d)p FC(\))h Fp(b)-57 b(e)430 b(a)5535 60849 y(gr)-57 b(aph)379 b(obtaine)-57 b(d)380 b(fr)-57 b(om)380 b FC(\000)g Fp(by)g(adding)f(a)h(\014nite)g (numb)-57 b(er)381 b(of)e(vertic)-57 b(es)381 b(at)d(e)-57 b(ach)380 b(e)-57 b(dge.)381 b(Consider)5535 62288 y(a)j(family)i Fr(f)p FC(\000\()p Ft(d)p FC(\))309 b(:)554 b Ft(d)307 b Fr(2)h Fl(R)16419 61887 y Fo(+)17154 62288 y Fr(g)385 b Fp(of)f(such)g(gr)-57 b(aphs)384 b(with)g(the)g(pr)-57 b(op)g(erties)384 b(that)f(the)h(numb)-57 b(er)386 b(of)e(the)5535 63727 y(adde)-57 b(d)399 b(vertic)-57 b(es)399 b(at)g(e)-57 b(ach)399 b(e)-57 b(dge)399 b(is)g(indep)-57 b(endent)400 b(of)f Ft(d)g Fp(and)g(their)g(distanc)-57 b(es)399 b(fr)-57 b(om)399 b(the)g(c)-57 b(entr)g(e)5535 65167 y(ar)g(e)344 b Fr(O)31 b FC(\()p Ft(d)p FC(\))346 b Fp(as)d Ft(d)308 b Fr(!)f FC(0)14271 65333 y Fo(+)15007 65167 y Fp(.)344 b(Supp)-57 b(ose)344 b(that)f(a)h(family)h(of)g(functions)g FC(\011)33156 65333 y Fs(d)33979 65167 y Fr(2)308 b Ft(W)36224 64765 y Fo(2)p Fs(;)p Fo(2)37609 65167 y FC(\()q(\000)246 b Fr(n)g FC(\()p Fr(f)p Ft(c)p Fr(g)i([)e Ft(V)43670 65333 y Fs(d)44185 65167 y FC(\)\))q Fp(,)5535 66606 y(wher)-57 b(e)341 b Ft(c)g Fp(is)f(the)h(c)-57 b(entr)g(e)342 b(of)f FC(\000)f Fp(and)h Ft(V)20573 66772 y Fs(d)21429 66606 y Fp(is)g(the)g(set)f(of)h(adde)-57 b(d)340 b(vertic)-57 b(es,)342 b(satis\014es)e(the)h(c)-57 b(onditions)5535 68045 y(\(2.4\))535 b(with)g Ft(d)p Fp(-dep)-57 b(endent)537 b(p)-57 b(ar)g(ameters,)536 b(and)f(that)g(it)h(c)-57 b(onver)g(ges)537 b(to)e FC(\011)561 b Fr(2)f Ft(W)39816 67643 y Fo(2)p Fs(;)p Fo(2)41202 68045 y FC(\(\000)246 b Fr(n)g(f)p Ft(c)p Fr(g)p FC(\))5535 69484 y Fp(which)432 b(ob)-57 b(eys)432 b(the)f(c)-57 b(ondition)433 b(\(2.1\))e(with)g (some)g Ft(A;)370 b(B)486 b Fp(satisfying)433 b(the)e(r)-57 b(e)g(quir)g(ements)434 b(\(2.2\).)5535 70923 y(The)362 b(family)h(of)e(the)g(c)-57 b(onditions)363 b(\(2.1\))d(which)i(c)-57 b(an)362 b(b)-57 b(e)362 b(obtaine)-57 b(d)362 b(in)h(this)e(way)g(dep) -57 b(ends)361 b(on)h FC(2)p Ft(n)5535 72362 y Fp(p)-57 b(ar)g(ameters)396 b(if)h Ft(n)308 b(>)f FC(2)p Fp(,)397 b(and)f(on)g(thr)-57 b(e)g(e)397 b(p)-57 b(ar)g(ameters)396 b(for)g Ft(n)308 b FC(=)f(2)p Fp(.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 22416 9083 a Fy(Appr)-49 b(oximations)340 b(of)f(vertex)i(c) -49 b(ouplings)340 b(in)g(quantum)h(gr)-49 b(aphs)971 b FB(7)6614 11955 y Fz(Pro)35 b(of.)638 b FC(The)553 b Ft(\016)594 b FC(coupling)555 b(in)e(the)g(cen)-31 b(tre)552 b(of)i(\000)e(is)h(expressed)e(b)-31 b(y)553 b(the)g(condition)i(\(2.4\).)6614 13394 y(Consider)321 b(\014rst)f Ft(\016)362 b FC(in)-31 b(teractions)323 b(on)e(a)g(hal\015ine)g(and)g(lo)31 b(ok)322 b(ho)-31 b(w)321 b(the)g(b)31 b(oundary)321 b(v)-61 b(alues)321 b(c)-31 b(hange)6614 14833 y(when)410 b(w)-31 b(e)410 b(pass)f(b)31 b(et)-31 b(w)g(een)410 b(di\013eren)-31 b(t)410 b(sites.)g(Supp)31 b(ose)409 b(that)h(at)h(a)e(p)31 b(oin)-31 b(t)411 b Ft(x)e FC(the)g(function)i(and)6614 16272 y(its)329 b(deriv)-61 b(ativ)-31 b(e)329 b(ha)-31 b(v)g(e)330 b(the)e(righ)-31 b(t)329 b(limits,)i(and)d(that)i Ft(x)164 b FC(+)g Ft(\017)328 b FC(is)g(the)g(site)h(of)g(a)f Ft(\016)370 b FC(in)-31 b(teraction,)331 b(then)6614 17711 y(the)370 b(T)-92 b(a)-31 b(ylor)370 b(expansion)h(giv)-31 b(es)9447 20085 y Ft( )40 b FC(\()p Ft(x)246 b FC(+)g Ft(\017)13073 20251 y Fq(\000)13821 20085 y FC(\))308 b(=)f Ft( )40 b FC(\()p Ft(x)17551 20251 y Fo(+)18286 20085 y FC(\))247 b(+)f Ft(\017 )21280 19628 y Fq(0)21590 20085 y FC(\()p Ft(x)22653 20251 y Fo(+)23388 20085 y FC(\))h(+)f Fr(O)31 b FC(\()p Ft(\017)26963 19628 y Fo(2)27460 20085 y FC(\))184 b Ft(;)1293 b( )30435 19628 y Fq(0)30745 20085 y FC(\()p Ft(x)246 b FC(+)g Ft(\017)33610 20251 y Fq(\000)34358 20085 y FC(\))308 b(=)f Ft( )37025 19628 y Fq(0)37336 20085 y FC(\()p Ft(x)38399 20251 y Fo(+)39134 20085 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(\017)p FC(\))184 b Ft(;)6614 22459 y FC(and)370 b(the)f Ft(\016)411 b FC(in)-31 b(teraction)372 b(is)d(according)h(to)g(\(2.4\))h(describ)31 b(ed)369 b(b)-31 b(y)6830 24832 y Ft( )40 b FC(\()p Ft(x)246 b FC(+)g Ft(\017)10456 24998 y Fo(+)11191 24832 y FC(\))308 b(=)g Ft( )40 b FC(\()p Ft(x)246 b FC(+)g Ft(\017)16724 24998 y Fq(\000)17471 24832 y FC(\))308 b(=:)g Ft( )40 b FC(\()p Ft(x)246 b FC(+)g Ft(\017)p FC(\))184 b Ft(;)2400 b( )27393 24375 y Fq(0)27703 24832 y FC(\()p Ft(x)247 b FC(+)f Ft(\017)30569 24998 y Fo(+)31304 24832 y FC(\))g Fr(\000)g Ft( )33848 24375 y Fq(0)34159 24832 y FC(\()p Ft(x)g FC(+)g Ft(\017)37024 24998 y Fq(\000)37771 24832 y FC(\))308 b(=)g Ft(\013)t FC(\()p Ft(\017)p FC(\))p Ft( )40 b FC(\()p Ft(x)247 b FC(+)f Ft(\017)p FC(\))184 b Ft(;)6614 27206 y FC(where)486 b Ft(\013)t FC(\()p Ft(\017)p FC(\))i(is)e(the)h(coupling)h(parameter.)f(The)g(ma)-31 b(y)487 b(b)31 b(e)486 b Ft(\017)p FC(-dep)31 b(enden)-31 b(t)487 b(but)f(w)-31 b(e)487 b(supp)31 b(ose)6614 28645 y(suc)-31 b(h)369 b(a)g(dep)31 b(endence)369 b(that)h(the)g(error)e (terms)h(can)h(b)31 b(e)368 b(neglected)j(as)e Ft(\017)307 b Fr(!)h FC(0)38550 28811 y Fo(+)39285 28645 y FC(;)370 b(then)f(w)-31 b(e)370 b(ha)-31 b(v)g(e)11763 31019 y Ft( )40 b FC(\()p Ft(x)246 b FC(+)g Ft(\017)p FC(\))308 b(=)g Ft( )40 b FC(\()p Ft(x)19120 31185 y Fo(+)19855 31019 y FC(\))246 b(+)g Ft(\017 )22848 30562 y Fq(0)23159 31019 y FC(\()p Ft(x)24222 31185 y Fo(+)24957 31019 y FC(\))g(+)g Fr(O)31 b FC(\()p Ft(\017)28531 30562 y Fo(2)29029 31019 y FC(\))184 b Ft(;)10718 32790 y( )11479 32333 y Fq(0)11789 32790 y FC(\()p Ft(x)246 b FC(+)g Ft(\017)14654 32956 y Fo(+)15389 32790 y FC(\))308 b(=)g Ft( )18057 32333 y Fq(0)18367 32790 y FC(\()p Ft(x)19430 32956 y Fo(+)20165 32790 y FC(\))247 b(+)f Fr(O)31 b FC(\()p Ft(\017)p FC(\))247 b(+)f Ft(\013)t FC(\()p Ft(\017)p FC(\)\()p Ft( )40 b FC(\()p Ft(x)29799 32956 y Fo(+)30536 32790 y FC(\))246 b(+)g Ft(\017 )33529 32333 y Fq(0)33840 32790 y FC(\()p Ft(x)34903 32956 y Fo(+)35638 32790 y FC(\))g(+)g Fr(O)31 b FC(\()p Ft(\017)39212 32333 y Fo(2)39710 32790 y FC(\)\))308 b(=)16127 34561 y(=)g(\(1)247 b(+)f Ft(\013)t FC(\()p Ft(\017)p FC(\))p Ft(\017)p FC(\))p Ft( )23294 34104 y Fq(0)23606 34561 y FC(\()p Ft(x)24669 34727 y Fo(+)25404 34561 y FC(\))h(+)f Ft(\013)t FC(\()p Ft(\017)p FC(\))p Ft( )40 b FC(\()p Ft(x)31033 34727 y Fo(+)31769 34561 y FC(\))247 b(+)f Fr(O)31 b FC(\()p Ft(\017)p FC(\))247 b(+)f Ft(\013)t FC(\()p Ft(\017)p FC(\))p Fr(O)31 b FC(\()p Ft(\017)40940 34104 y Fo(2)41439 34561 y FC(\))184 b Ft(;)6614 36935 y FC(so)383 b(that)h Ft( )40 b FC(\()p Ft(x)255 b FC(+)g Ft(\017)p FC(\))384 b(and)f Ft( )17784 36533 y Fq(0)18094 36935 y FC(\()p Ft(x)256 b FC(+)f Ft(\017)20978 37101 y Fo(+)21713 36935 y FC(\))383 b(dep)31 b(end)383 b(on)g Ft( )40 b FC(\()p Ft(x)29759 37101 y Fo(+)30494 36935 y FC(\))383 b(and)g Ft( )34234 36533 y Fq(0)34544 36935 y FC(\()p Ft(x)35607 37101 y Fo(+)36343 36935 y FC(\))g(linearly)h(up)f(to)h(error)6614 38374 y(terms.)401 b(In)g(case)g(of)g(a)g(\014nite)h(n)-31 b(um)g(b)31 b(er)401 b(of)g Ft(\016)443 b FC(in)-31 b(teractions)403 b(on)e(a)g(hal\015ine)h(one)f(can)h(sho)-31 b(w)401 b(in)g(a)6614 39813 y(similar)347 b(w)-31 b(a)g(y)348 b(recursiv)-31 b(ely)346 b(that)h(the)f(function)h(v)-61 b(alue)346 b(and)g(the)g(righ)-31 b(t)347 b(limit)h(of)e(the)g(deriv)-61 b(ativ)-31 b(e)6614 41252 y(at)402 b(the)e(site)h(of)g(the)g(last)h Ft(\016)442 b FC(dep)31 b(ends,)400 b(up)g(to)i(error)d(terms,)i (linearly)h(on)f(the)g(function)h(v)-61 b(alue)6614 42691 y(and)370 b(the)f(righ)-31 b(t)370 b(limit)i(of)d(the)h(deriv)-61 b(ativ)-31 b(e)370 b(for)g(the)f(\014rst)g Ft(\016)411 b FC(in)-31 b(teraction.)8275 44130 y(Let)439 b(us)g(apply)i(this)e (conclusion)i(to)f(the)g(edges)g(of)g(our)f(star)g(graph.)i(W)-92 b(e)438 b(denote)i(b)-31 b(y)440 b Ft(d)45998 44296 y Fs(j)6614 45569 y FC(the)333 b(distance)g(of)g(the)f(last)h Ft(\016)374 b FC(in)-31 b(teraction)335 b(on)e(the)f Ft(j)63 b FC(-th)333 b(hal\015ine)h(family)g(of)f(edges)f(in)h(\000\()p Ft(d)p FC(\);)g(b)-31 b(y)6614 47008 y(assumption)371 b(w)-31 b(e)369 b(ha)-31 b(v)g(e)371 b Ft(d)17227 47174 y Fs(j)18001 47008 y FC(=)307 b Fr(O)31 b FC(\()p Ft(d)p FC(\).)371 b(Then)e(w)-31 b(e)370 b(ha)-31 b(v)g(e)14308 49235 y(~)14070 49527 y Ft(f)14731 48952 y Fo(\(1\))14612 49835 y Fs(j)15919 49527 y FC(\()p Ft(d)p FC(\))p Ft( )18076 49693 y Fs(j)18544 49527 y FC(\()p Ft(d)19550 49693 y Fs(j)20017 49527 y FC(\))332 b(=)370 b(~)-591 b Ft(g)22540 48952 y Fo(\(1\))22500 49835 y Fs(j)23729 49527 y FC(\()p Ft(d)p FC(\))p Ft( )40 b FC(\(0\))248 b(+)28705 49235 y(~)28694 49527 y Ft(h)29332 48952 y Fo(\(1\))29332 49835 y Fs(j)30520 49527 y FC(\()p Ft(d)p FC(\))p Ft( )32717 49070 y Fq(0)32677 49800 y Fs(j)33145 49527 y FC(\(0\))f(+)296 b(~)-603 b Ft(r)36442 48952 y Fo(\(1\))36411 49835 y Fs(j)37631 49527 y FC(\()p Ft(d)p FC(\))184 b Ft(;)13573 51293 y FC(~)13335 51585 y Ft(f)13996 51010 y Fo(\(2\))13877 51893 y Fs(j)15184 51585 y FC(\()p Ft(d)p FC(\))p Ft( )17381 51128 y Fq(0)17341 51859 y Fs(j)17809 51585 y FC(\()p Ft(d)18815 51751 y Fs(j)19282 51957 y Fo(+)20017 51585 y FC(\))332 b(=)370 b(~)-591 b Ft(g)22540 51010 y Fo(\(2\))22500 51893 y Fs(j)23729 51585 y FC(\()p Ft(d)p FC(\))p Ft( )40 b FC(\(0\))248 b(+)28705 51293 y(~)28694 51585 y Ft(h)29332 51010 y Fo(\(2\))29332 51893 y Fs(j)30520 51585 y FC(\()p Ft(d)p FC(\))p Ft( )32717 51128 y Fq(0)32677 51859 y Fs(j)33145 51585 y FC(\(0\))f(+)296 b(~)-603 b Ft(r)36442 51010 y Fo(\(2\))36411 51893 y Fs(j)37631 51585 y FC(\()p Ft(d)p FC(\))6614 54251 y(for)513 b(some)11607 53959 y(~)11369 54251 y Ft(f)12030 53676 y Fo(\(1\))11911 54559 y Fs(j)13219 54251 y Ft(;)222 b FC(~)-591 b Ft(g)14278 53676 y Fo(\(1\))14238 54559 y Fs(j)15467 54251 y Ft(;)15970 53959 y FC(~)15958 54251 y Ft(h)16596 53676 y Fo(\(1\))16596 54559 y Fs(j)17785 54251 y Ft(;)18515 53959 y FC(~)18276 54251 y Ft(f)18937 53676 y Fo(\(2\))18818 54559 y Fs(j)20126 54251 y Ft(;)223 b FC(~)-592 b Ft(g)21185 53676 y Fo(\(2\))21145 54559 y Fs(j)22375 54251 y Ft(;)22878 53959 y FC(~)22866 54251 y Ft(h)23504 53676 y Fo(\(2\))23504 54559 y Fs(j)25239 54251 y FC(:)547 b Fl(R)26892 53849 y Fo(+)28174 54251 y Fr(!)g Fl(R)p FC(.)513 b(The)g(functions)564 b(~)-603 b Ft(r)39312 53676 y Fo(\(1\))39281 54559 y Fs(j)41013 54251 y FC(and)563 b(~)-603 b Ft(r)43839 53676 y Fo(\(2\))43808 54559 y Fs(j)45028 54251 y FC(\()p Ft(d)p FC(\))6614 55690 y(are)465 b(error)f(terms)h(and)h(w)-31 b(e)465 b(supp)31 b(ose)465 b(that)h(they)f(can)h(b)31 b(e)464 b(neglected)j(in)e(the)h(limit.)h(W)-92 b(e)465 b(are)6614 57129 y(in)-31 b(terested)382 b(in)f(the)g(situation)i(when)e(the)g (last)h(relations)g(can)f(b)31 b(e)380 b(in)-31 b(v)g(erted)382 b(and)f Ft( )40 b FC(\(0\))p Ft(;)371 b( )44624 56727 y Fq(0)44584 57417 y Fs(j)45050 57129 y FC(\(0\))6614 58568 y(can)f(b)31 b(e)368 b(expressed)g(b)-31 b(y)370 b(means)f(of)h Ft( )22024 58734 y Fs(j)22490 58568 y FC(\()p Ft(d)23496 58734 y Fs(j)23963 58568 y FC(\))g(and)f Ft( )27676 58166 y Fq(0)27636 58856 y Fs(j)28103 58568 y FC(\()p Ft(d)29109 58734 y Fs(j)29576 58940 y Fo(+)30311 58568 y FC(\),)13012 61234 y Ft( )40 b FC(\(0\))333 b(=)f Ft(f)17373 60659 y Fo(\(1\))17254 61542 y Fs(j)18562 61234 y FC(\()p Ft(d)p FC(\))p Ft( )20719 61400 y Fs(j)21186 61234 y FC(\()p Ft(d)22192 61400 y Fs(j)22659 61234 y FC(\))247 b(+)f Ft(g)25011 60659 y Fo(\(1\))24971 61542 y Fs(j)26199 61234 y FC(\()p Ft(d)p FC(\))p Ft( )28396 60777 y Fq(0)28356 61507 y Fs(j)28823 61234 y FC(\()p Ft(d)29829 61400 y Fs(j)30296 61606 y Fo(+)31031 61234 y FC(\))h(+)f Fr(R)p FC(\()p Ft(d)p FC(\))184 b Ft(;)1293 b(j)371 b Fr(2)363 b FC(^)-608 b Ft(n)185 b(;)4187 b FC(\(3.1\))12585 63297 y Ft( )13346 62840 y Fq(0)13306 63571 y Fs(j)13772 63297 y FC(\(0\))334 b(=)16950 63005 y(~)16712 63297 y Ft(f)17373 62722 y Fo(\(2\))17254 63606 y Fs(j)18562 63297 y FC(\()p Ft(d)p FC(\))p Ft( )20719 63463 y Fs(j)21186 63297 y FC(\()p Ft(d)22192 63463 y Fs(j)22659 63297 y FC(\))247 b(+)283 b(~)-590 b Ft(g)25011 62722 y Fo(\(2\))24971 63606 y Fs(j)26199 63297 y FC(\()p Ft(d)p FC(\))p Ft( )28396 62840 y Fq(0)28356 63571 y Fs(j)28823 63297 y FC(\()p Ft(d)29829 63463 y Fs(j)30296 63669 y Fo(+)31031 63297 y FC(\))247 b(+)f Fr(R)p FC(\()p Ft(d)p FC(\))184 b Ft(;)1785 b(j)371 b Fr(2)363 b FC(^)-608 b Ft(n)185 b(;)3695 b FC(\(3.2\))6614 65671 y(where)508 b(w)-31 b(e)508 b(ha)-31 b(v)g(e)508 b(in)-31 b(tro)31 b(duced)509 b Fr(R)p FC(\()p Ft(d)p FC(\))g(as)e(the)h(sym)-31 b(b)31 b(ol)509 b(for)f(a)g(generic)g(remainder;)g(w)-31 b(e)508 b(still)6614 67110 y(assume)430 b(that)h(it)f(can)g(b)31 b(e)429 b(neglected)i(with)g(resp)31 b(ect)428 b(to)j(the)f(other)f (terms)h(as)g Ft(d)408 b Fr(!)g FC(0)43086 67276 y Fo(+)43821 67110 y FC(.)430 b(The)6614 68549 y(equations)371 b(\(3.1\))g(yield)f (for)g Ft(j)r(;)184 b(k)343 b Fr(2)363 b FC(^)-608 b Ft(n)369 b FC(the)g(conditions)6614 70923 y Ft(f)7275 70348 y Fo(\(1\))7156 71231 y Fs(j)8464 70923 y FC(\()p Ft(d)p FC(\))p Ft( )10621 71089 y Fs(j)11088 70923 y FC(\()p Ft(d)12094 71089 y Fs(j)12561 70923 y FC(\))158 b Fr(\000)g Ft(f)14829 70348 y Fo(\(1\))14710 71257 y Fs(k)16018 70923 y FC(\()p Ft(d)p FC(\))p Ft( )18175 71089 y Fs(k)18721 70923 y FC(\()p Ft(d)19727 71089 y Fs(k)20273 70923 y FC(\))g(+)g Ft(g)22448 70348 y Fo(\(1\))22408 71231 y Fs(j)23636 70923 y FC(\()p Ft(d)p FC(\))p Ft( )25833 70466 y Fq(0)25793 71196 y Fs(j)26261 70923 y FC(\()p Ft(d)27267 71089 y Fs(j)27734 71295 y Fo(+)28469 70923 y FC(\))g Fr(\000)g Ft(g)30644 70348 y Fo(\(1\))30604 71257 y Fs(k)31832 70923 y FC(\()p Ft(d)p FC(\))p Ft( )34029 70466 y Fq(0)33989 71196 y Fs(k)34535 70923 y FC(\()p Ft(d)35541 71089 y Fs(k)36086 71144 y Fo(+)36821 70923 y FC(\))308 b(=)g Fr(R)p FC(\()p Ft(d)p FC(\))184 b Ft(;)1294 b(j)r(;)184 b(k)343 b Fr(2)363 b FC(^)-609 b Ft(n)44189 72362 y FC(\(3.3\))p eop end %%Page: 8 8 TeXDict begin 8 7 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(8)972 b Fy(P.)340 b(Exner)h(and)f(O.)g(T)-73 b(ur)-49 b(ek)5535 11955 y FC(and)370 b(from)f(\(3.2\))j(together)e (with)h(the)e(second)g(one)h(of)f(the)h(conditions)h(\(2.4\))g(w)-31 b(e)370 b(get)12079 14825 y Ft(\013)t( )40 b FC(\(0\))309 b(=)16974 13441 y Fs(n)16448 13773 y Fn(X)16442 16156 y Fs(k)19 b Fo(=1)18237 13596 y Fn(\020)18898 14825 y Ft(f)19559 14250 y Fo(\(2\))19440 15159 y Fs(k)20748 14825 y FC(\()p Ft(d)p FC(\))p Ft( )22905 14991 y Fs(k)23450 14825 y FC(\()p Ft(d)24456 14991 y Fs(k)25002 14825 y FC(\))246 b(+)g Ft(g)27353 14250 y Fo(\(2\))27313 15159 y Fs(k)28541 14825 y FC(\()p Ft(d)p FC(\))p Ft( )30738 14368 y Fq(0)30698 15098 y Fs(k)31244 14825 y FC(\()p Ft(d)32250 14991 y Fs(k)32795 15046 y Fo(+)33530 14825 y FC(\))33960 13596 y Fn(\021)34868 14825 y FC(+)g Fr(R)p FC(\()p Ft(d)p FC(\))184 b Ft(:)4270 b FC(\(3.4\))5535 17862 y(W)-92 b(e)350 b(substitute)i(for)f Ft( )40 b FC(\(0\))352 b(from)f(\(3.1\))j(and)d(p)31 b(erform)350 b(a)h(rep)31 b(eated)351 b(summation)i(of)438 b(\(3.4\))353 b(o)-31 b(v)g(er)5535 19301 y Ft(j)63 b FC(.)370 b(After)f(an)h(easy)f (rearrangemen)-31 b(t)370 b(w)-31 b(e)370 b(get)7180 20725 y Fs(n)6654 21057 y Fn(X)6688 23415 y Fs(j)45 b Fo(=1)8438 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26056 y FC(in)392 b(the)f(equations)i (\(3.3\))h(and)d(\(3.5\))s(.)g(Before)h(that)g(w)-31 b(e)5535 27495 y(m)g(ultiply)472 b(b)31 b(oth)471 b(sides)e(b)-31 b(y)469 b(a)h(p)31 b(o)-31 b(w)g(er)470 b(of)g Ft(d)g FC(suc)-31 b(h)469 b(that)i(the)f(righ)-31 b(t-hand)471 b(side)e(tends)h(to)g(zero)5535 28934 y(as)402 b Ft(d)362 b Fr(!)g FC(0)9887 29100 y Fo(+)10622 28934 y FC(,)403 b(while)g(at)f(least)h(one)f(co)31 b(e\016cien)-31 b(t)404 b(at)e(the)g(left-hand)i(side)d(remains)i(nonzero,)f(in)5535 30373 y(other)319 b(w)-31 b(ords,)319 b(w)-31 b(e)319 b(use)e(the)i(assumed)f(existence)h(of)g(the)g(limit)h(in)f(whic)-31 b(h)319 b(the)g(error)e(terms)h(can)5535 31813 y(b)31 b(e)369 b(neglected)h(w.r.t.)h(the)f(leading)h(ones.)e(The)h(equation)h (\(3.3\))g(acquires)e(then)h(the)f(form)11829 33906 y Ft(c)12308 34072 y Fs(j)12774 33906 y Ft( )13495 34072 y Fs(j)13962 33906 y FC(\(0\))247 b Fr(\000)f Ft(c)17208 34072 y Fs(k)17753 33906 y Ft( )18474 34072 y Fs(k)19019 33906 y FC(\(0\))h(+)f Ft(t)22186 34072 y Fs(j)22652 33906 y Ft( )23413 33449 y Fq(0)23373 34179 y Fs(j)23840 33906 y FC(\(0)24823 34072 y Fo(+)25558 33906 y FC(\))h Fr(\000)f Ft(t)27742 34072 y Fs(k)28286 33906 y Ft( )29047 33449 y Fq(0)29007 34179 y Fs(k)29552 33906 y FC(\(0)30535 34072 y Fo(+)31271 33906 y FC(\))308 b(=)f(0)184 b Ft(;)1293 b(j)r(;)184 b(k)343 b Fr(2)363 b FC(^)-608 b Ft(n)4018 b FC(\(3.6\))5535 35999 y(while)371 b(\(3.5\))g(giv)-31 b(es)18301 37485 y Fs(n)17776 37817 y Fn(X)17809 40175 y Fs(j)45 b Fo(=1)19559 38869 y Ft(\015)20132 39035 y Fs(j)20599 38869 y Ft( )21320 39035 y Fs(j)21786 38869 y FC(\(0\))248 b(+)25079 37485 y Fs(n)24553 37817 y Fn(X)24587 40175 y Fs(j)45 b Fo(=1)26337 38869 y Ft(\034)26821 39035 y Fs(j)27287 38869 y Ft( )28048 38412 y Fq(0)28008 39142 y Fs(j)28475 38869 y FC(\(0)29458 39035 y Fo(+)30194 38869 y FC(\))308 b(=)f(0)184 b Ft(;)9966 b FC(\(3.7\))5535 41970 y(where)479 b Ft(c)9325 42136 y Fs(j)9791 41970 y Ft(;)370 b(t)10868 42136 y Fs(j)11334 41970 y Ft(;)f(\015)12583 42136 y Fs(j)13050 41970 y Ft(;)g(\034)14210 42136 y Fs(j)15155 41970 y FC(are)478 b(the)h(appropriate)h(limiting)i(v)-61 b(alues)478 b(of)h(the)g(functions)h(in)-31 b(v)g(olv)g(ed.)5535 43409 y(The)370 b(obtained)h(conditions)g(can)e(also)h(b)31 b(e)369 b(written)h(in)g(a)f(matrix)i(form,)8175 44707 y Fn(0)8175 46656 y(B)8175 47320 y(B)8175 47984 y(B)8175 48648 y(B)8175 49312 y(B)8175 50021 y(@)9357 45396 y Ft(c)9836 45562 y Fo(1)10711 45396 y Fr(\000)p Ft(c)12051 45562 y Fo(2)13522 45396 y FC(0)974 b Fr(\001)184 b(\001)g(\001)1213 b FC(0)9357 46835 y Ft(c)9836 47001 y Fo(1)11353 46835 y FC(0)974 b Fr(\000)p Ft(c)14220 47001 y Fo(3)15049 46835 y Fr(\001)184 b(\001)g(\001)1213 b FC(0)9691 48048 y(.)9691 48491 y(.)9691 48933 y(.)15203 48159 y(.)15633 48491 y(.)16064 48822 y(.)9357 50372 y Ft(c)9836 50538 y Fo(1)11353 50372 y FC(0)1616 b(0)974 b Fr(\001)184 b(\001)g(\001)519 b(\000)p Ft(c)18197 50538 y Fs(n)9310 51812 y Ft(\015)9883 51978 y Fo(1)11095 51812 y Ft(\015)11668 51978 y Fo(2)13264 51812 y Ft(\015)13837 51978 y Fo(3)15049 51812 y Fr(\001)184 b(\001)g(\001)902 b Ft(\015)17813 51978 y Fs(n)18965 44707 y Fn(1)18965 46656 y(C)18965 47320 y(C)18965 47984 y(C)18965 48648 y(C)18965 49312 y(C)18965 50021 y(A)8175 52708 y(|)p 8673 52708 4884 133 v 4884 w({z)p 14553 52708 V 4884 w(})13722 53754 y Fs(A)20119 48593 y FC(\011\(0\))247 b(+)23747 44707 y Fn(0)23747 46656 y(B)23747 47320 y(B)23747 47984 y(B)23747 48648 y(B)23747 49312 y(B)23747 50021 y(@)24924 45396 y Ft(t)25324 45562 y Fo(1)26194 45396 y Fr(\000)p Ft(t)27455 45562 y Fo(2)28885 45396 y FC(0)935 b Fr(\001)184 b(\001)g(\001)1174 b FC(0)24924 46835 y Ft(t)25324 47001 y Fo(1)26796 46835 y FC(0)935 b Fr(\000)p Ft(t)29545 47001 y Fo(3)30373 46835 y Fr(\001)184 b(\001)g(\001)1174 b FC(0)25218 48048 y(.)25218 48491 y(.)25218 48933 y(.)30527 48159 y(.)30957 48491 y(.)31388 48822 y(.)24924 50372 y Ft(t)25324 50538 y Fo(1)26796 50372 y FC(0)1536 b(0)935 b Fr(\001)184 b(\001)g(\001)519 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y(A)45836 27082 y Ft(:)6614 32240 y FC(Supp)31 b(ose)381 b(\014rst)g(that)i Ft(c)16155 32406 y Fs(j)16876 32240 y FC(+)254 b(i)p Ft(t)18698 32406 y Fs(j)19493 32240 y Fr(6)p FC(=)328 b(0)382 b(for)g(all)h Ft(j)391 b Fr(2)384 b FC(^)-608 b Ft(n)p FC(,)382 b(then)g(b)-31 b(y)382 b(equiv)-61 b(alen)-31 b(t)383 b(ro)-31 b(w)382 b(manipulations)6614 33679 y(w)-31 b(e)370 b(pass)f(to)h(the)f(matrix)i (\()p Ft(D)31 b Fr(j)p Ft(V)247 b FC(\),)370 b(where)6614 39175 y Ft(D)339 b FC(=)9037 34957 y Fn(0)9037 36906 y(B)9037 37570 y(B)9037 38234 y(B)9037 38898 y(B)9037 39562 y(B)9037 40227 y(B)9037 40935 y(@)10172 35967 y Fr(\000)11217 34738 y Fn(\020)11879 35967 y Ft(\015)12452 36133 y Fo(1)13194 35967 y FC(+)246 b(i)p Ft(\034)15092 36133 y Fo(1)15835 35967 y FC(+)g(\()p Ft(c)17851 36133 y Fo(1)18348 35967 y FC(+)19209 36133 y Fi(1)19834 35967 y FC(\))20448 35137 y Fn(P)21617 35411 y Fi(n)21617 36299 y Fs(`)p Fo(=)p Fi(1)23610 35473 y Fs(\015)24075 35590 y Fj(`)24461 35473 y Fo(+i)p Fs(\034)25800 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46202 y(B)9037 46866 y(B)9037 47530 y(B)9037 48238 y(@)10172 44574 y FC(\()p Ft(c)11081 44740 y Fo(1)11824 44574 y Fr(\000)246 b FC(i)p Ft(t)13638 44740 y Fo(1)14135 44574 y FC(\))p Ft(S)310 b Fr(\000)246 b FC(2i)17654 44081 y Fs(c)18050 44192 y Fh(1)18482 44081 y Fs(\034)18890 44192 y Fh(1)19323 44081 y Fq(\000)p Fs(\015)20480 44192 y Fh(1)20911 44081 y Fs(t)21245 44192 y Fh(1)p 17655 44320 4024 45 v 18404 44956 a Fs(c)18800 45067 y Fh(1)19232 44956 y Fo(+i)p Fs(t)20497 45067 y Fh(1)24634 44574 y Fr(\000)p FC(2i)26488 44081 y Fs(c)26884 44192 y Fh(2)27316 44081 y Fs(\034)27724 44192 y Fh(2)28156 44081 y Fq(\000)p Fs(\015)29313 44192 y Fh(2)29745 44081 y Fs(t)30079 44192 y Fh(2)p 26488 44320 V 27238 44956 a Fs(c)27634 45067 y Fh(2)28066 44956 y Fo(+i)p Fs(t)29331 45067 y Fh(2)33468 44574 y Fr(\001)184 b(\001)g(\001)3115 b(\000)p FC(2i)39726 44081 y Fs(c)40122 44192 y Fj(n)40666 44081 y Fs(\034)41074 44192 y Fj(n)41617 44081 y Fq(\000)p Fs(\015)42774 44192 y Fj(n)43316 44081 y Fs(t)43650 44192 y Fj(n)p 39727 44320 4467 45 v 40588 44956 a Fs(c)40984 45067 y Fj(n)41526 44956 y Fo(+i)p Fs(t)42791 45067 y Fj(n)13501 45638 y Fo(2i)p 13501 45819 692 45 v 13554 46456 a Fs(S)14458 45580 y(c)14854 45691 y Fh(1)15286 45580 y Fs(\034)15694 45691 y Fh(1)16126 45580 y Fq(\000)p Fs(\015)17283 45691 y Fh(1)17715 45580 y Fs(t)18049 45691 y Fh(1)p 14458 45820 4024 45 v 15208 46456 a Fs(c)15604 46567 y Fh(1)16036 46456 y Fo(+i)p Fs(t)17301 46567 y Fh(1)22143 46074 y Fr(\000)p Ft(c)23483 46240 y Fo(2)24225 46074 y FC(+)246 b(i)p Ft(t)26039 46240 y Fo(2)26782 46074 y FC(+)28022 45638 y Fo(2i)p 28022 45819 692 45 v 28075 46456 a Fs(S)28979 45580 y(c)29375 45691 y Fh(2)29807 45580 y Fs(\034)30215 45691 y Fh(2)30647 45580 y Fq(\000)p Fs(\015)31804 45691 y Fh(2)32236 45580 y Fs(t)32570 45691 y Fh(2)p 28979 45820 4024 45 v 29729 46456 a Fs(c)30125 46567 y Fh(2)30557 46456 y Fo(+i)p Fs(t)31822 46567 y Fh(2)33468 46074 y Fr(\001)184 b(\001)g(\001)38388 45638 y Fo(2i)p 38388 45819 692 45 v 38441 46456 a Fs(S)39345 45580 y(c)39741 45691 y Fj(n)40283 45580 y Fs(\034)40691 45691 y Fj(n)41235 45580 y Fq(\000)p Fs(\015)42392 45691 y Fj(n)42934 45580 y Fs(t)43268 45691 y Fj(n)p 39345 45820 4467 45 v 40205 46456 a Fs(c)40601 46567 y Fj(n)41144 46456 y Fo(+i)p Fs(t)42409 46567 y Fj(n)15838 47348 y FC(.)15838 47790 y(.)15838 48233 y(.)33621 47458 y(.)34052 47790 y(.)34482 48122 y(.)13501 49236 y Fo(2i)p 13501 49417 692 45 v 13554 50054 a Fs(S)14458 49178 y(c)14854 49289 y Fh(1)15286 49178 y Fs(\034)15694 49289 y Fh(1)16126 49178 y Fq(\000)p Fs(\015)17283 49289 y Fh(1)17715 49178 y Fs(t)18049 49289 y Fh(1)p 14458 49418 4024 45 v 15208 50054 a Fs(c)15604 50165 y Fh(1)16036 50054 y Fo(+i)p Fs(t)17301 50165 y Fh(1)25149 49236 y Fo(2i)p 25149 49417 692 45 v 25202 50054 a Fs(S)26106 49178 y(c)26502 49289 y Fh(2)26934 49178 y Fs(\034)27342 49289 y Fh(2)27774 49178 y Fq(\000)p Fs(\015)28931 49289 y Fh(2)29363 49178 y Fs(t)29697 49289 y Fh(2)p 26106 49418 4024 45 v 26856 50054 a Fs(c)27252 50165 y Fh(2)27683 50054 y Fo(+i)p Fs(t)28948 50165 y Fh(2)33468 49672 y Fr(\001)g(\001)g(\001)519 b(\000)p Ft(c)36616 49838 y Fs(n)37464 49672 y FC(+)246 b(i)p Ft(t)39278 49838 y Fs(n)40127 49672 y FC(+)41367 49236 y Fo(2i)p 41367 49417 692 45 v 41420 50054 a Fs(S)42324 49178 y(c)42720 49289 y Fj(n)43263 49178 y Fs(\034)43671 49289 y Fj(n)44214 49178 y Fq(\000)p Fs(\015)45371 49289 y Fj(n)45913 49178 y Fs(t)46247 49289 y Fj(n)p 42324 49418 4467 45 v 43184 50054 a Fs(c)43580 50165 y Fj(n)44123 50054 y Fo(+i)p Fs(t)45388 50165 y Fj(n)47089 43589 y Fn(1)47089 45537 y(C)47089 46202 y(C)47089 46866 y(C)47089 47530 y(C)47089 48238 y(A)48243 47143 y Ft(;)6614 51941 y FC(where)373 b(w)-31 b(e)374 b(ha)-31 b(v)g(e)375 b(denoted)f Ft(S)k FC(=)20429 51110 y Fn(P)21597 51384 y Fs(n)21597 52273 y(`)p Fo(=1)23461 51447 y Fs(\015)23926 51564 y Fj(`)24312 51447 y Fo(+i)p Fs(\034)25651 51564 y Fj(`)p 23461 51686 2576 45 v 23533 52322 a Fs(c)23929 52439 y Fj(`)24314 52322 y Fo(+i)p Fs(t)25579 52439 y Fj(`)26169 51941 y FC(.)c(Since)g(w)-31 b(e)374 b(used)f(only)h(equiv)-61 b(alen)-31 b(t)375 b(manipula-)6614 53380 y(tions,)307 b(the)g(diagonal)h(matrix)f Ft(D)337 b FC(should)307 b(ha)-31 b(v)g(e)306 b(the)h(same)f(rank)g(as)g Ft(A)119 b FC(+)g(i)p Ft(B)56 b FC(,)306 b(hence)g(it)h(m)-31 b(ust)306 b(b)31 b(e)6614 54819 y(regular)384 b(b)31 b(ecause)384 b(none)g(of)g(its)g(diagonal)j(elemen)-31 b(ts)384 b(is)g(zero.)g(Consequen)-31 b(tly)-92 b(,)386 b(w)-31 b(e)385 b(can)f(divide)6614 56258 y(eac)-31 b(h)375 b(ro)-31 b(w)376 b(of)f(\()p Ft(D)31 b Fr(j)p Ft(V)247 b FC(\))375 b(b)-31 b(y)375 b(the)g(corresp)31 b(onding)374 b(diagobal)k(elemen)-31 b(t)376 b(of)f Ft(D)31 b FC(.)375 b(This)h(yields)f(\()p Ft(I)87 b Fr(j)p Ft(U)121 b FC(\),)6614 57697 y(where)444 b Ft(U)565 b FC(is)445 b(the)f(sough)-31 b(t)446 b(unitary)g(matrix)g(and)e(its)h(diagonal)i(and)e (o\013-diagonal)i(elemen)-31 b(ts)6614 59136 y(are)369 b(giv)-31 b(en)371 b(b)-31 b(y)14434 61669 y Ft(U)15190 61835 y Fs(j)45 b(j)16375 61669 y FC(=)19615 60920 y(2i\()p Ft(c)21384 61086 y Fs(j)21852 60920 y Ft(\034)22336 61086 y Fs(j)23048 60920 y Fr(\000)246 b Ft(t)24555 61086 y Fs(j)25021 60920 y Ft(\015)25594 61086 y Fs(j)26061 60920 y FC(\))p 17676 61414 10755 45 v 17676 62562 a(\()p Ft(c)18585 62728 y Fs(j)19298 62562 y FC(+)g(i)p Ft(t)21112 62728 y Fs(j)21578 62562 y FC(\))22008 62242 y Fo(2)22690 61732 y Fn(P)23859 62006 y Fs(n)23859 62894 y(`)p Fo(=1)25723 62068 y Fs(\015)26188 62185 y Fj(`)26573 62068 y Fo(+i)p Fs(\034)27912 62185 y Fj(`)p 25723 62308 2576 45 v 25795 62944 a Fs(c)26191 63061 y Fj(`)26576 62944 y Fo(+i)p Fs(t)27841 63061 y Fj(`)28809 61669 y Fr(\000)30049 60920 y Ft(c)30528 61086 y Fs(j)31241 60920 y Fr(\000)g FC(i)p Ft(t)33055 61086 y Fs(j)p 30049 61414 3473 45 v 30049 62428 a Ft(c)30528 62594 y Fs(j)31241 62428 y FC(+)g(i)p Ft(t)33055 62594 y Fs(j)33839 61669 y Ft(;)14356 65083 y(U)15112 65249 y Fs(j)45 b(k)16375 65083 y FC(=)21418 64334 y(2)p Ft(i)p FC(\()p 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b(of)g(all)g(the)g(expressions)e Ft(c)34075 69650 y Fs(j)34788 69484 y FC(+)246 b Ft(it)36676 69650 y Fs(j)37142 69484 y FC(.)8275 70923 y(So)386 b(far)h(w)-31 b(e)387 b(ha)-31 b(v)g(e)387 b(not)g(emplo)-31 b(y)g(ed)389 b(the)d(second)h(one)f(of)h(the)g(requiremen)-31 b(ts)387 b(\(2.2\))r(,)g(namely)6614 72362 y(the)272 b(self-adjoin)-31 b(tness)272 b(of)g(the)f(matrix)i Ft(AB)24199 71960 y Fq(\003)24707 72362 y FC(.)f(This)f(is)g(equiv)-61 b(alen)-31 b(t)273 b(to)f(unitarit)-31 b(y)273 b(of)f Ft(U)121 b FC(,)271 b(ho)-31 b(w)g(ev)g(er,)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(10)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 11900 y FC(it)372 b(is)f(easier)g(to)h(c)-31 b(hec)g(k)372 b(it)g(in)f(its)h(original)h(v)-31 b(ersion.)372 b(By)g(a)f(straigh)-31 b(tforw)g(ard)374 b(computation)g(w)-31 b(e)5535 13339 y(\014nd)369 b(that)i(the)e(pro)31 b(duct)369 b Ft(AB)17934 12937 y Fq(\003)18750 13339 y FC(=)307 b Ft(AB)21644 12937 y Fs(T)22710 13339 y FC(equals)6305 15062 y Fn(0)6305 17010 y(B)6305 17674 y(B)6305 18338 y(B)6305 19003 y(B)6305 19667 y(B)6305 20331 y(B)6305 20995 y(B)6305 21659 y(B)6305 22368 y(@)7534 16027 y Ft(c)8013 16193 y Fo(1)8510 16027 y Ft(t)8910 16193 y Fo(1)9652 16027 y FC(+)246 b Ft(c)11238 16193 y Fo(2)11735 16027 y Ft(t)12135 16193 y Fo(2)14764 16027 y Ft(c)15243 16193 y Fo(1)15739 16027 y Ft(t)16139 16193 y Fo(1)20381 16027 y Ft(c)20860 16193 y Fo(1)21356 16027 y Ft(t)21756 16193 y Fo(1)24291 16027 y Fr(\001)184 b(\001)g(\001)2664 b Ft(c)28723 16193 y Fo(1)29219 16027 y Ft(t)29619 16193 y Fo(1)34900 16027 y Ft(c)35379 16193 y Fo(1)35876 16027 y Ft(\034)36360 16193 y Fo(1)37102 16027 y Fr(\000)246 b Ft(c)38688 16193 y Fo(2)39185 16027 y 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Fo(1)29219 18905 y Ft(t)29619 19071 y Fo(1)34900 18905 y Ft(c)35379 19071 y Fo(1)35876 18905 y Ft(\034)36360 19071 y Fo(1)37102 18905 y Fr(\000)246 b Ft(c)38688 19071 y Fo(4)39185 18905 y Ft(\034)39669 19071 y Fo(4)9929 20118 y FC(.)9929 20561 y(.)9929 21004 y(.)24445 20229 y(.)24875 20561 y(.)25306 20893 y(.)29026 20118 y(.)29026 20561 y(.)29026 21004 y(.)9146 22443 y Ft(c)9625 22609 y Fo(1)10122 22443 y Ft(t)10522 22609 y Fo(1)14764 22443 y Ft(c)15243 22609 y Fo(1)15739 22443 y Ft(t)16139 22609 y Fo(1)20381 22443 y Ft(c)20860 22609 y Fo(1)21356 22443 y Ft(t)21756 22609 y Fo(1)24291 22443 y Fr(\001)184 b(\001)g(\001)945 b Ft(c)27004 22609 y Fo(1)27501 22443 y Ft(t)27901 22609 y Fo(1)28643 22443 y FC(+)246 b Ft(c)30229 22609 y Fs(n)30832 22443 y Ft(t)31232 22609 y Fs(n)34794 22443 y Ft(c)35273 22609 y Fo(1)35770 22443 y Ft(\034)36254 22609 y Fo(1)36996 22443 y Fr(\000)g Ft(c)38582 22609 y Fs(n)39185 22443 y Ft(\034)39669 22609 y Fs(n)7440 23882 y Ft(\015)8013 24048 y Fo(1)8510 23882 y Ft(t)8910 24048 y Fo(1)9652 23882 y Fr(\000)g Ft(\015)11332 24048 y Fo(2)11829 23882 y Ft(t)12229 24048 y Fo(2)13057 23882 y Ft(\015)13630 24048 y Fo(1)14127 23882 y Ft(t)14527 24048 y Fo(1)15269 23882 y Fr(\000)g Ft(\015)16949 24048 y Fo(3)17446 23882 y Ft(t)17846 24048 y Fo(3)18674 23882 y Ft(\015)19247 24048 y Fo(1)19744 23882 y Ft(t)20144 24048 y Fo(1)20886 23882 y Fr(\000)g Ft(\015)22566 24048 y Fo(4)23063 23882 y Ft(t)23463 24048 y Fo(4)24291 23882 y Fr(\001)184 b(\001)g(\001)851 b Ft(\015)27004 24048 y Fo(1)27501 23882 y Ft(t)27901 24048 y Fo(1)28643 23882 y Fr(\000)246 b Ft(\015)30323 24048 y Fs(n)30926 23882 y Ft(t)31326 24048 y Fs(n)32260 23882 y Ft(\015)32833 24048 y Fo(1)33330 23882 y Ft(\034)33814 24048 y Fo(1)34557 23882 y FC(+)f Ft(\015)36236 24048 y Fo(2)36733 23882 y Ft(\034)37217 24048 y Fo(2)37960 23882 y FC(+)h Fr(\001)184 b(\001)g(\001)j Ft(\015)41116 24048 y Fs(n)41718 23882 y Ft(\034)42202 24048 y Fs(n)42971 15062 y Fn(1)42971 17010 y(C)42971 17674 y(C)42971 18338 y(C)42971 19003 y(C)42971 19667 y(C)42971 20331 y(C)42971 20995 y(C)42971 21659 y(C)42971 22368 y(A)44308 19943 y Ft(;)5535 26548 y FC(hence)529 b Ft(AB)10496 26147 y Fq(\003)11533 26548 y FC(is)h(self-adjoin)-31 b(t)531 b(if)f(and)f(only)h(if)g Ft(c)26391 26714 y Fo(1)26888 26548 y Ft(\034)27372 26714 y Fo(1)28221 26548 y Fr(\000)353 b Ft(c)29914 26714 y Fs(j)30380 26548 y Ft(\034)30864 26714 y Fs(j)31905 26548 y FC(=)574 b Ft(\015)33913 26714 y Fo(1)34409 26548 y Ft(t)34809 26714 y Fo(1)35658 26548 y Fr(\000)353 b Ft(\015)37445 26714 y Fs(j)37911 26548 y Ft(t)38311 26714 y Fs(j)39307 26548 y FC(holds)529 b(for)g(all)5535 27987 y Ft(j)371 b FC(=)307 b(2)p Ft(;)184 b(:)g(:)g(:)189 b(;)184 b(n)p FC(,)371 b(and)e(therefore)9440 30779 y Ft(c)9919 30945 y Fo(1)10416 30779 y Ft(\034)10900 30945 y Fo(1)11642 30779 y Fr(\000)246 b Ft(\015)13322 30945 y Fo(1)13819 30779 y Ft(t)14219 30945 y Fo(1)15023 30779 y FC(=)307 b Ft(c)16670 30945 y Fo(2)17167 30779 y Ft(\034)17651 30945 y Fo(2)18394 30779 y Fr(\000)245 b Ft(\015)20073 30945 y Fo(2)20570 30779 y Ft(t)20970 30945 y Fo(2)21774 30779 y FC(=)307 b Ft(c)23421 30945 y Fo(3)23918 30779 y Ft(\034)24402 30945 y Fo(3)25145 30779 y Fr(\000)246 b Ft(\015)26825 30945 y Fo(3)27321 30779 y Ft(t)27721 30945 y Fo(3)28525 30779 y FC(=)308 b Fr(\001)184 b(\001)g(\001)309 b FC(=)f Ft(c)32940 30945 y Fs(n)33543 30779 y Ft(\034)34027 30945 y Fs(n)34875 30779 y Fr(\000)246 b Ft(\015)36555 30945 y Fs(n)37158 30779 y Ft(t)37558 30945 y Fs(n)38344 30779 y Ft(:)3906 b FC(\(3.10\))5535 33571 y(W)-92 b(e)494 b(denote)h(the)f(common)i(v)-61 b(alue)495 b Ft(c)21307 33737 y Fs(j)21774 33571 y Ft(\034)22258 33737 y Fs(j)23053 33571 y Fr(\000)330 b Ft(\015)24817 33737 y Fs(j)25283 33571 y Ft(t)25683 33737 y Fs(j)26644 33571 y FC(as)494 b Ft(\024)g FC(and)g(recall)h(that)h(w)-31 b(e)495 b(ha)-31 b(v)g(e)495 b(denoted)5535 35010 y Ft(S)371 b FC(=)7754 34180 y Fn(P)8922 34454 y Fs(n)8922 35342 y(`)p Fo(=1)10786 34516 y Fs(\015)11251 34633 y Fj(`)11637 34516 y Fo(+i)p Fs(\034)12976 34633 y Fj(`)p 10786 34756 2576 45 v 10858 35392 a Fs(c)11254 35509 y Fj(`)11639 35392 y Fo(+i)p Fs(t)12904 35509 y Fj(`)13494 35010 y FC(,)f(then)f(the)h(matrix)g Ft(U)490 b FC(giv)-31 b(en)370 b(b)-31 b(y)385 b(\(3.9\))371 b(can)f(b)31 b(e)369 b(simpli\014ed,)6869 40673 y Ft(U)428 b FC(=)9221 37120 y Fn(0)9221 39068 y(B)9221 39732 y(B)9221 40396 y(B)9221 41060 y(B)9221 41769 y(@)12001 37544 y Fo(2i)p Fs(\024)p 10489 37725 4235 45 v 10489 38362 a Fo(\()p Fs(c)11231 38473 y Fh(1)11662 38362 y Fo(+i)p Fs(t)12927 38473 y Fh(1)13359 38362 y Fo(\))13705 38141 y Fh(2)14137 38362 y Fs(S)15101 37980 y Fr(\000)16341 37526 y Fs(c)16737 37637 y Fh(1)17169 37526 y Fq(\000)p Fo(i)p Fs(t)18446 37637 y Fh(1)p 16341 37725 2537 45 v 16347 38362 a Fs(c)16743 38473 y Fh(1)17175 38362 y Fo(+i)p Fs(t)18440 38473 y Fh(1)23065 37544 y Fo(2i)p Fs(\024)p 20161 37725 7019 45 v 20161 38362 a Fo(\()p Fs(c)20903 38473 y Fh(1)21335 38362 y Fo(+i)p Fs(t)22600 38473 y Fh(1)23031 38362 y Fo(\)\()p Fs(c)24119 38473 y Fh(2)24551 38362 y Fo(+i)p Fs(t)25816 38473 y Fh(2)26247 38362 y Fo(\))p Fs(S)28330 37980 y Fr(\001)184 b(\001)g(\001)35584 37544 y Fo(2i)p Fs(\024)p 32570 37725 7240 45 v 32570 38362 a Fo(\()p Fs(c)33312 38473 y Fh(1)33743 38362 y Fo(+i)p Fs(t)35008 38473 y Fh(1)35440 38362 y Fo(\)\()p Fs(c)36528 38473 y Fj(n)37070 38362 y Fo(+i)p Fs(t)38335 38473 y Fj(n)38877 38362 y Fo(\))p Fs(S)14078 39127 y Fo(2i)p Fs(\024)p 11174 39308 7019 45 v 11174 39945 a Fo(\()p Fs(c)11916 40056 y Fh(2)12348 39945 y Fo(+i)p Fs(t)13613 40056 y Fh(2)14044 39945 y Fo(\)\()p Fs(c)15132 40056 y Fh(1)15564 39945 y Fo(+i)p Fs(t)16829 40056 y Fh(1)17260 39945 y Fo(\))p Fs(S)20987 39127 y Fo(2i)p Fs(\024)p 19475 39308 4235 45 v 19475 39945 a Fo(\()p Fs(c)20217 40056 y Fh(2)20649 39945 y Fo(+i)p Fs(t)21914 40056 y Fh(2)22346 39945 y Fo(\))22692 39723 y Fh(2)23124 39945 y Fs(S)24088 39563 y Fr(\000)25328 39109 y Fs(c)25724 39220 y Fh(2)26156 39109 y Fq(\000)p Fo(i)p Fs(t)27433 39220 y Fh(2)p 25328 39308 2537 45 v 25334 39945 a Fs(c)25730 40056 y Fh(2)26162 39945 y Fo(+i)p Fs(t)27427 40056 y Fh(2)28330 39563 y Fr(\001)g(\001)g(\001)35584 39127 y Fo(2i)p Fs(\024)p 32570 39308 7240 45 v 32570 39945 a Fo(\()p Fs(c)33312 40056 y Fh(2)33743 39945 y Fo(+i)p Fs(t)35008 40056 y Fh(2)35440 39945 y Fo(\)\()p Fs(c)36528 40056 y Fj(n)37070 39945 y Fo(+i)p Fs(t)38335 40056 y Fj(n)38877 39945 y Fo(\))p Fs(S)14529 40919 y FC(.)14529 41362 y(.)14529 41805 y(.)28483 41030 y(.)28914 41362 y(.)29344 41694 y(.)14078 42808 y Fo(2i)p Fs(\024)p 11063 42989 V 11063 43626 a Fo(\()p Fs(c)11805 43737 y Fj(n)12348 43626 y Fo(+i)p Fs(t)13613 43737 y Fj(n)14155 43626 y Fo(\)\()p Fs(c)15243 43737 y Fh(1)15675 43626 y Fo(+i)p Fs(t)16940 43737 y Fh(1)17371 43626 y Fo(\))p Fs(S)23065 42808 y Fo(2i)p Fs(\024)p 20050 42989 V 20050 43626 a Fo(\()p Fs(c)20792 43737 y Fj(n)21335 43626 y Fo(+i)p Fs(t)22600 43737 y Fj(n)23142 43626 y Fo(\)\()p Fs(c)24230 43737 y Fh(2)24661 43626 y Fo(+i)p Fs(t)25926 43737 y Fh(2)26358 43626 y Fo(\))p Fs(S)28330 43244 y Fr(\001)g(\001)g(\001) 33396 42808 y Fo(2i)p Fs(\024)p 30270 42989 7462 45 v 30270 43626 a Fo(\()p Fs(c)31012 43737 y Fj(n)31555 43626 y Fo(+i)p Fs(t)32820 43737 y Fj(n)33362 43626 y Fo(\)\()p Fs(c)34450 43737 y Fj(n)34992 43626 y Fo(+i)p Fs(t)36257 43737 y Fj(n)36800 43626 y Fo(\))p Fs(S)38111 43244 y Fr(\000)39350 42790 y Fs(c)39746 42901 y Fj(n)40289 42790 y Fq(\000)p Fo(i)p Fs(t)41566 42901 y Fj(n)p 39350 42989 2759 45 v 39356 43626 a Fs(c)39752 43737 y Fj(n)40295 43626 y Fo(+i)p Fs(t)41560 43737 y Fj(n)42407 37120 y Fn(1)42407 39068 y(C)42407 39732 y(C)42407 40396 y(C)42407 41060 y(C)42407 41769 y(A)43745 40673 y Ft(:)5535 46113 y FC(Let)433 b(us)e(sho)-31 b(w)433 b(that)h(the)e(matrix)866 b(\(3.9\))435 b(can)d(b)31 b(e)432 b(parametrized)i(b)-31 b(y)432 b(2)p Ft(n)h FC(real)g(n)-31 b(um)g(b)31 b(ers.)432 b(W)-92 b(e)5535 47552 y(rewrite)370 b(the)f(quan)-31 b(tit)g(y)372 b Ft(S)433 b FC(in)-31 b(tro)31 b(duced)370 b(ab)31 b(o)-31 b(v)g(e)370 b(in)f(the)h(follo)-31 b(wing)373 b(w)-31 b(a)g(y)-92 b(,)10603 51059 y Ft(S)371 b FC(=)13347 49675 y Fs(n)12821 50008 y Fn(X)12875 52391 y Fs(`)p Fo(=1)14737 50310 y FC(\()p Ft(\015)15740 50476 y Fs(`)16413 50310 y FC(+)246 b(i)p Ft(\034)18311 50476 y Fs(`)18737 50310 y FC(\)\()p Ft(c)20076 50476 y Fs(`)20749 50310 y Fr(\000)g FC(i)p Ft(t)22563 50476 y Fs(`)22989 50310 y FC(\))p 14737 50805 8683 45 v 17466 51818 a Ft(c)17945 51437 y Fo(2)17945 52152 y Fs(`)18688 51818 y FC(+)g Ft(t)20195 51437 y Fo(2)20195 52152 y Fs(`)23860 51059 y FC(=)25554 49675 y Fs(n)25028 50008 y Fn(X)25082 52391 y Fs(`)p Fo(=1)26945 50310 y Ft(c)27424 50476 y Fs(`)27850 50310 y Ft(\015)28423 50476 y Fs(`)29095 50310 y FC(+)f Ft(t)30601 50476 y Fs(`)31027 50310 y Ft(\034)31511 50476 y Fs(`)p 26945 50805 4993 45 v 27828 51818 a Ft(c)28307 51437 y Fo(2)28307 52152 y Fs(`)29050 51818 y FC(+)h Ft(t)30557 51437 y Fo(2)30557 52152 y Fs(`)32316 51059 y FC(+)g(i)p Ft(\024)35078 49675 y Fs(n)34552 50008 y Fn(X)34606 52391 y Fs(`)p Fo(=1)37804 50310 y FC(1)p 36469 50805 3225 45 v 36469 51818 a Ft(c)36948 51437 y Fo(2)36948 52152 y Fs(`)37690 51818 y FC(+)g Ft(t)39197 51437 y Fo(2)39197 52152 y Fs(`)40011 51059 y Ft(;)5535 54795 y FC(and)387 b(mak)-31 b(e)389 b(\014rst)d(sev)-31 b(eral)387 b(observ)-61 b(ations:)388 b(\(i\))h(regarding)403 b(\(3.8\))389 b(as)e(a)g(system)g(of)g(linear)h(equa-)5535 56234 y(tions)316 b(its)g(solv)-61 b(abilit)-31 b(y)318 b(is)d(not)h(a\013ected)f(if)h(the)f(last)h(one)f(is)g(m)-31 b(ultiplied)318 b(b)-31 b(y)315 b(a)h(nonzero)f(n)-31 b(um)g(b)31 b(er.)5535 57673 y(A)-31 b(t)300 b(the)f(same)g(time,)h (the)f(v)-61 b(alue)300 b(of)f Ft(\024)f FC(is)h(directly)h(prop)31 b(ortional)300 b(to)g Ft(\015)34273 57839 y Fs(j)34739 57673 y FC(,)g Ft(\034)35830 57839 y Fs(j)36296 57673 y FC(,)f(and)g(consequen)-31 b(tly)-92 b(,)5535 59112 y(one)437 b(can)f(supp)31 b(ose)436 b(without)i(loss)f(of)f(generalit) -31 b(y)439 b(that)e Ft(\024)420 b 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b(can)g(divide)g(all)g(but) g(the)f(last)i(of)e(the)h(equations)h(in)e(the)h(system)f(\(3.8\))5535 66307 y(b)-31 b(y)370 b Fr(j)p Ft(c)7859 66473 y Fo(1)8602 66307 y FC(+)246 b(i)p Ft(t)10416 66473 y Fo(1)10913 66307 y Fr(j)369 b FC(whic)-31 b(h)370 b(is)f(nonzero)h(b)-31 b(y)369 b(assumption.)7196 67746 y(With)g(the)h(ab)31 b(o)-31 b(v)g(e)370 b(con)-31 b(v)g(en)g(tion)372 b(w)-31 b(e)370 b(can)f(denote)h Ft(c)28338 67912 y Fo(1)29081 67746 y FC(+)246 b(i)p Ft(t)30895 67912 y Fo(1)31699 67746 y FC(=:)308 b(e)33667 67345 y Fo(i)p Fs(\022)34787 67746 y FC(and)370 b(Re)184 b Ft(S)371 b FC(=:)308 b Ft(\032)369 b FC(so)g(that)18676 71362 y Ft(S)i FC(=)308 b Ft(\032)246 b FC(+)g(i)23311 69470 y Fn( )24188 71362 y FC(1)h(+)26620 69979 y Fs(n)26095 70311 y Fn(X)26148 72694 y Fs(`)p Fo(=2)29347 70614 y FC(1)p 28011 71108 V 28011 72122 a Ft(c)28490 71740 y Fo(2)28490 72456 y Fs(`)29233 72122 y FC(+)e Ft(t)30739 71740 y Fo(2)30739 72456 y Fs(`)31369 69470 y Fn(!)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(11)6614 11900 y FC(and)370 b Ft(U)489 b FC(can)370 b(b)31 b(e)368 b(written)j(explicitly)g(as)8559 16709 y Ft(U)428 b FC(=)10911 13155 y Fn(0)10911 15104 y(B)10911 15768 y(B)10911 16432 y(B)10911 17096 y(B)10911 17805 y(@)12179 13580 y Fo(2i)p 12179 13761 692 45 v 12232 14398 a Fs(S)13372 14016 y FC(e)13864 13614 y Fq(\000)p Fo(2i)p Fs(\022)15995 14016 y Fr(\000)246 b FC(e)17594 13614 y Fq(\000)p Fo(2i)p Fs(\022)22640 13580 y Fo(2i)p 21085 13761 3802 45 v 21085 14398 a(\()p Fs(c)21827 14509 y Fh(2)22258 14398 y Fo(+i)p Fs(t)23523 14509 y Fh(2)23955 14398 y Fo(\))p Fs(S)25388 14016 y FC(e)25880 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JcC<$JcC<$JcD>A!tYG4n:CUjs+13$s2t@l~> JcC<$JcC<$JcD>A"$chd`.@r>s+13$s2t@l~> JcC<$JcC<$JcD>A"$chd`.@r>s+13$s2t@l~> JcC<$JcC<$JcD>A"$chd`.@r>s+13$s2t@l~> JcC<$JcC<$JcD>A")n5?OFdF_s+13$s2t@l~> JcC<$JcC<$JcD>A")n5?OFdF_s+13$s2t@l~> JcC<$JcC<$JcD>A")n5?OFdF_s+13$s2t@l~> JcC<$JcC<$JcD>A"/#VoA:ac3s+13$s2t@l~> JcC<$JcC<$JcD>A"/#VoA:ac3s+13$s2t@l~> JcC<$JcC<$JcD>A"/#VoA:ac3s+13$s2t@l~> JcC<$JcC<$JcD>A"4.#J0S07Ts+13$s2t@l~> JcC<$JcC<$JcD>A"4.#J0S07Ts+13$s2t@l~> JcC<$JcC<$JcD>A"4.#J0S07Ts+13$s2t@l~> JcC<$JcC<$JcD>A"T/B&#lK8RJcC<$JcEpnJ,~> JcC<$JcC<$JcD>A"T/B&#lK8RJcC<$JcEpnJ,~> JcC<$JcC<$JcD>A"T/B&#lK8RJcC<$JcEpnJ,~> JcC<$JcC<$JcD;@"#']Td"24Js+13$s3(Fm~> JcC<$JcC<$JcD;@"#']Td"24Js+13$s3(Fm~> JcC<$JcC<$JcD;@"#']Td"24Js+13$s3(Fm~> JcC<$JcC<$JcD;@"(2*/Uk/Pss+13$s3(Fm~> JcC<$JcC<$JcD;@"(2*/Uk/Pss+13$s3(Fm~> JcC<$JcC<$JcD;@"(2*/Uk/Pss+13$s3(Fm~> JcC<$JcC<$JcD;@"- JcC<$JcC<$JcD;@"- JcC<$JcC<$JcD;@"- JcC<$JcC<$JcD;@"2Fm:7"PAhs+13$s3(Fm~> JcC<$JcC<$JcD;@"2Fm:7"PAhs+13$s3(Fm~> JcC<$JcC<$JcD;@"2Fm:7"PAhs+13$s3(Fm~> JcC<$JcC<$JcD;@"7QEn0S07Ts+13$s3(Fm~> JcC<$JcC<$JcD;@"7QEn0S07Ts+13$s3(Fm~> JcC<$JcC<$JcD;@"7QEn0S07Ts+13$s3(Fm~> JcC<$JcC<$JcD8?!fRtpJcC<$JcC<$bQ!(~> JcC<$JcC<$JcD8?!fRtpJcC<$JcC<$bQ!(~> JcC<$JcC<$JcD8?!fRtpJcC<$JcC<$bQ!(~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> JcC<$JcC<$JcC<$JcC<$JcC<$j8XW~> %%EndData showpage %%Trailer end %%EOF %%EndDocument @endspecial 69021 41670 a currentpoint grestore moveto 69021 41670 a 16577 41670 a currentpoint currentpoint translate 1 0.37993 div 1 0.37993 div scale neg exch neg exch translate 16577 41670 a 16537 43884 a FB(Fig.)314 b(1.)h(Sc)-26 b(heme)313 b(of)h(a)g(2)p Fx(n)p FB(-parameter)g(appro)-26 b(ximation)6614 48633 y FC(F)-92 b(or)470 b(the)h(sak)-31 b(e)471 b(of)h(brevit)-31 b(y)472 b(w)-31 b(e)471 b(will)h(not)g (indicate)g(the)f Ft(d)p FC(-dep)31 b(endence)471 b(of)g(the)g (parameters)6614 50073 y Ft(u;)369 b(v)8461 50239 y Fs(j)8927 50073 y Ft(;)h(w)10396 50239 y Fs(j)11222 50073 y FC(and)360 b(the)g(distance)h Ft(D)391 b FC(unless)359 b(necessary)-92 b(.)359 b(The)h(b)31 b(oundary)360 b(conditions)i(whic)-31 b(h)360 b(the)6614 51512 y(functions)370 b Ft( )12107 51678 y Fo(1)12604 51512 y Ft(;)184 b(:)g(:)g(:)k(;)c( )15784 51678 y Fs(n)16757 51512 y FC(on)369 b(\000)g(ha)-31 b(v)g(e)370 b(to)g(satisfy)g(are)8957 54931 y Ft( )9678 55097 y Fo(1)10174 54931 y FC(\(0\))309 b(=)f Ft( )13786 55097 y Fo(2)14282 54931 y FC(\(0\))h(=)f Fr(\001)184 b(\001)g(\001)310 b FC(=)d Ft( )20661 55097 y Fs(n)21264 54931 y FC(\(0\))i(=:)e Ft( )40 b FC(\(0\))184 b Ft(;)30053 53547 y Fs(n)29527 53879 y Fn(X)29561 56237 y Fs(j)45 b Fo(=1)31311 54931 y Ft( )32072 54474 y Fq(0)32032 55204 y Fs(j)32498 54931 y FC(\(0)33481 55097 y Fo(+)34217 54931 y FC(\))308 b(=)f Ft(u )40 b FC(\(0\))4705 b(\(3.14\))8957 57829 y Ft( )9678 57995 y Fs(j)10144 57829 y FC(\()p Ft(D)11490 57995 y Fo(+)12226 57829 y FC(\))308 b(=)f Ft( )14853 57995 y Fs(j)15320 57829 y FC(\()p Ft(D)16666 57995 y Fq(\000)17414 57829 y FC(\))h(=:)g Ft( )20349 57995 y Fs(j)20815 57829 y FC(\()p Ft(D)31 b FC(\))184 b Ft(;)1294 b( )25168 57371 y Fq(0)25128 58102 y Fs(j)25594 57829 y FC(\()p Ft(D)26940 57995 y Fo(+)27676 57829 y FC(\))247 b Fr(\000)f Ft( )30221 57371 y Fq(0)30181 58102 y Fs(j)30647 57829 y FC(\()p Ft(D)31993 57995 y Fq(\000)32741 57829 y FC(\))308 b(=)g Ft(v)35185 57995 y Fs(j)35651 57829 y Ft( )36372 57995 y Fs(j)36838 57829 y FC(\()p Ft(D)31 b FC(\))4991 b(\(3.15\))8957 59600 y Ft( )9678 59766 y Fs(j)10144 59600 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)13451 59766 y Fq(\006)14198 59600 y FC(\))308 b(=:)g Ft( )17133 59766 y Fs(j)17599 59600 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\))184 b Ft(;)1293 b( )23881 59143 y Fq(0)23841 59873 y Fs(j)24308 59600 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)27614 59766 y Fo(+)28349 59600 y FC(\))h Fr(\000)f Ft( )30894 59143 y Fq(0)30854 59873 y Fs(j)31320 59600 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)34627 59766 y Fq(\000)35374 59600 y FC(\))308 b(=)g Ft(w)38073 59766 y Fs(j)38540 59600 y Ft( )39261 59766 y Fs(j)39727 59600 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\))184 b Ft(:)-319 b FC(\(3.16\))6614 62304 y(F)-92 b(urther)387 b(relations)j(whic)-31 b(h)389 b(will)h(in)e(the)g(follo)-31 b(wing)392 b(serv)-31 b(e)388 b(to)h(determine)f(the)h(parameter)g(de-) 6614 63743 y(p)31 b(endence)300 b(on)h Ft(d)g FC(are)f(obtained)j(from) e(T)-92 b(a)-31 b(ylor)302 b(expansion)g(of)f(the)g(resp)31 b(ectiv)-31 b(e)301 b(w)-31 b(a)g(v)g(e)302 b(functions,)8662 66447 y Ft( )9383 66613 y Fs(j)9850 66447 y FC(\()p Ft(D)31 b FC(\))309 b(=)e Ft( )13855 66613 y Fs(j)14322 66447 y FC(\(0\))247 b(+)f Ft(D)31 b( )18797 65990 y Fq(0)18757 66721 y Fs(j)19224 66447 y FC(\(0)20207 66613 y Fo(+)20943 66447 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(D)25015 65990 y Fo(2)25512 66447 y FC(\))184 b Ft(;)2400 b( )29594 65990 y Fq(0)29554 66721 y Fs(j)30021 66447 y FC(\()p Ft(D)31367 66613 y Fq(\000)32115 66447 y FC(\))308 b(=)f Ft( )34782 65990 y Fq(0)34742 66721 y Fs(j)35209 66447 y FC(\(0)36192 66613 y Fo(+)36928 66447 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(D)g FC(\))184 b Ft(;)1715 b FC(\(3.17\))8662 68218 y Ft( )9383 68384 y Fs(j)9850 68218 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\))308 b(=)f Ft( )15784 68384 y Fs(j)16251 68218 y FC(\()p Ft(D)31 b FC(\))247 b(+)f Ft(d )20749 67761 y Fq(0)20709 68492 y Fs(j)21175 68218 y FC(\()p Ft(D)22521 68384 y Fo(+)23257 68218 y FC(\))h(+)f Fr(O)31 b FC(\()p Ft(d)26959 67761 y Fo(2)27456 68218 y FC(\))184 b Ft(;)493 b( )29631 67761 y Fq(0)29591 68492 y Fs(j)30058 68218 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)33365 68384 y Fq(\000)34112 68218 y FC(\))308 b(=)f Ft( )36779 67761 y Fq(0)36739 68492 y Fs(j)37206 68218 y FC(\()p Ft(D)38552 68384 y Fo(+)39288 68218 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))217 b(\(3.18\))6614 70923 y(for)397 b Ft(j)416 b Fr(2)408 b FC(^)-608 b Ft(n)p FC(.)397 b(W)-92 b(e)395 b(need)i(to)g(\014nd)f(relations)i(b)31 b(et)-31 b(w)g(een)397 b(the)g(v)-61 b(alues)396 b Ft( )34637 71089 y Fo(1)35134 70923 y FC(\()p Ft(D)296 b FC(+)264 b Ft(d)p FC(\))p Ft(;)184 b(:)g(:)g(:)189 b(;)184 b( )42088 71089 y Fs(n)42691 70923 y FC(\()p Ft(D)296 b FC(+)264 b Ft(d)p FC(\))6614 72362 y(and)462 b Ft( )9620 71960 y Fq(0)9580 72637 y Fo(1)10077 72362 y FC(\()p Ft(D)339 b FC(+)308 b Ft(d)13507 72528 y Fo(+)14242 72362 y FC(\))p Ft(;)184 b(:)g(:)g(:)k(;)c( )17892 71960 y Fq(0)17852 72635 y Fs(n)18456 72362 y FC(\()p Ft(D)340 b FC(+)307 b Ft(d)21886 72528 y Fo(+)22621 72362 y FC(\).)463 b(T)-92 b(o)462 b(this)g(aim)h(w)-31 b(e)463 b(express)d(them)j(\014rst)e(in)h (terms)g(of)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(14)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 11957 y Ft( )40 b FC(\(0\))370 b(and)g Ft( )10993 11555 y Fq(0)10953 12245 y Fs(j)11419 11957 y FC(\(0)12402 12123 y Fo(+)13138 11957 y FC(\).)g(Using)g(the)f (relations)i(\(3.15\))h(and)d(\(3.17\))j(w)-31 b(e)370 b(get)10571 14275 y Ft( )11332 13818 y Fq(0)11292 14549 y Fs(j)11758 14275 y FC(\()p Ft(D)13104 14441 y Fo(+)13840 14275 y FC(\))308 b(=)f Ft( )16507 13818 y Fq(0)16467 14549 y Fs(j)16934 14275 y FC(\(0)17917 14441 y Fo(+)18653 14275 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)25046 14441 y Fs(j)25512 14275 y FC(\()p Ft( )26663 14441 y Fs(j)27130 14275 y FC(\(0\))i(+)e Ft(D)31 b( )31606 13818 y Fq(0)31566 14549 y Fs(j)32032 14275 y FC(\(0)33015 14441 y Fo(+)33751 14275 y FC(\))247 b(+)f Fr(O)31 b FC(\()p Ft(D)37824 13818 y Fo(2)38321 14275 y FC(\)\))308 b(=)14578 16047 y(=)f Ft(v)16283 16213 y Fs(j)16749 16047 y Ft( )40 b FC(\(0\))248 b(+)e(\(1)h(+)e Ft(v)23151 16213 y Fs(j)23617 16047 y Ft(D)31 b FC(\))p Ft( )25755 15590 y Fq(0)25715 16320 y Fs(j)26183 16047 y FC(\(0)27166 16213 y Fo(+)27902 16047 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)34295 16213 y Fs(j)34761 16047 y Fr(O)31 b FC(\()p Ft(D)37050 15590 y Fo(2)37547 16047 y FC(\))184 b Ft(:)5535 18247 y FC(Substituting)314 b(in)-31 b(to)314 b(the)e(\014rst)f(one)h(of)g(the)g(relations)h (\(3.18\))i(and)d(using)624 b(\(3.15\))314 b(again)g(w)-31 b(e)312 b(\014nd)6642 20780 y Ft( )7363 20946 y Fs(j)7830 20780 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\))308 b(=)g(\(1)247 b(+)f Ft(dv)16493 20946 y Fs(j)16959 20780 y FC(\))p Ft( )40 b FC(\(0\))248 b(+)e(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)27098 20946 y Fs(j)27564 20780 y Ft(D)31 b FC(\)\))186 b Ft( )30318 20323 y Fq(0)30278 21054 y Fs(j)30745 20780 y FC(\(0)31728 20946 y Fo(+)32463 20780 y FC(\))21637 22551 y(+)246 b Fr(O)31 b FC(\()p Ft(D)25033 22094 y Fo(2)25531 22551 y FC(\))246 b(+)g Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(dv)33076 22717 y Fs(j)33542 22551 y Fr(O)31 b FC(\()p Ft(D)35831 22094 y Fo(2)36329 22551 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)40030 22094 y Fo(2)40527 22551 y FC(\))184 b Ft(:)1109 b FC(\(3.19\))5535 24752 y(The)438 b(already)h(obtained)g (expression)e(for)h Ft( )24402 24350 y Fq(0)24362 25040 y Fs(j)24828 24752 y FC(\()p Ft(D)26174 24918 y Fo(+)26910 24752 y FC(\))g(together)h(with)f(the)g(second)g(one)f(of)h(the)5535 26191 y(relations)740 b(\(3.18\))372 b(giv)-31 b(e)8674 28392 y Ft( )9435 27935 y Fq(0)9395 28666 y Fs(j)9861 28392 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)13168 28558 y Fq(\000)13915 28392 y FC(\))308 b(=)g Ft(v)16359 28558 y Fs(j)16825 28392 y Ft( )40 b FC(\(0\))247 b(+)f(\(1)h(+)f Ft(v)23227 28558 y Fs(j)23693 28392 y Ft(D)31 b FC(\))p Ft( )25831 27935 y Fq(0)25791 28666 y Fs(j)26258 28392 y FC(\(0)27241 28558 y Fo(+)27977 28392 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)34370 28558 y Fs(j)34836 28392 y Fr(O)31 b FC(\()p Ft(D)37125 27935 y Fo(2)37623 28392 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))184 b Ft(:)5535 30593 y FC(Substituting)330 b(from)e(here)f(and)h(\(3.19\))j(in)-31 b(to)329 b(the)f(second)f(one)h (of)g(the)g(relations)h(\(3.16\))i(w)-31 b(e)328 b(get)5535 32032 y(after)370 b(a)f(simple)i(rearrangemen)-31 b(t)6642 34565 y Ft( )7403 34108 y Fq(0)7363 34838 y Fs(j)7830 34565 y FC(\()p Ft(D)31 b FC(+)p Ft(d)10644 34731 y Fo(+)11379 34565 y FC(\))308 b(=)g(\()p Ft(v)14253 34731 y Fs(j)14965 34565 y FC(+)246 b Ft(w)16864 34731 y Fs(j)17331 34565 y FC(\(1)h(+)f Ft(dv)20781 34731 y Fs(j)21247 34565 y FC(\)\))186 b Ft( )40 b FC(\(0\)+\()q(1)247 b(+)f Ft(v)28203 34731 y Fs(j)28669 34565 y Ft(D)277 b FC(+)246 b Ft(w)31761 34731 y Fs(j)32412 34565 y FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)38593 34731 y Fs(j)39059 34565 y Ft(D)31 b FC(\)\))q(\))185 b Ft( )42243 34108 y Fq(0)42203 34838 y Fs(j)42670 34565 y FC(\(0)43653 34731 y Fo(+)44389 34565 y FC(\))5831 36336 y(+)246 b Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)11548 36502 y Fs(j)12014 36336 y Fr(O)31 b FC(\()p Ft(D)14303 35879 y Fo(2)14801 36336 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))247 b(+)f Ft(w)21078 36502 y Fs(j)21730 35439 y Fn(\000)22237 36336 y Fr(O)31 b FC(\()p Ft(D)24526 35879 y Fo(2)25023 36336 y FC(\))247 b(+)f Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(dv)32569 36502 y Fs(j)33035 36336 y Fr(O)31 b FC(\()p Ft(D)35324 35879 y Fo(2)35821 36336 y FC(\))247 b(+)f Fr(O)31 b FC(\()p Ft(d)39523 35879 y Fo(2)40020 36336 y FC(\))40450 35439 y Fn(\001)41142 36336 y Ft(:)1108 b FC(\(3.20\))5535 38537 y(Next)432 b(w)-31 b(e)432 b(eliminate)i Ft( )15615 38135 y Fq(0)15575 38825 y Fs(j)16042 38537 y FC(\(0)17025 38703 y Fo(+)17761 38537 y FC(\))d(\(for)h(simplicit)-31 b(y)434 b(w)-31 b(e)432 b(write)g Ft( )31267 38135 y Fq(0)31227 38825 y Fs(j)31694 38537 y FC(\(0\)\))h(from)f(the)f(obtained)i(rela-)5535 39976 y(tions)441 b(\(3.19\))h(and)e(\(3.20\))s(,)g(m)-31 b(ultiplying)443 b(them)e(b)-31 b(y)440 b(1)293 b(+)g Ft(v)30404 40142 y Fs(j)30870 39976 y Ft(D)325 b FC(+)293 b Ft(w)34057 40142 y Fs(j)34708 39976 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)40889 40142 y Fs(j)41355 39976 y Ft(D)31 b FC(\)\))440 b(and)5535 41415 y Ft(D)236 b FC(+)205 b Ft(d)p FC(\(1)h(+)f Ft(v)11121 41581 y Fs(j)11587 41415 y Ft(D)31 b FC(\),)350 b(resp)31 b(ectiv)-31 b(ely)-92 b(,)349 b(and)g(subtracting.)i(In)d(the)h(resulting)g(expression)g(the) f(co)31 b(ef-)5535 42854 y(\014cien)-31 b(t)370 b(at)g Ft( )40 b FC(\(0\))371 b(equals)e(one,)6827 45387 y(\(1)247 b(+)f Ft(v)9701 45553 y Fs(j)10167 45387 y Ft(D)277 b FC(+)246 b Ft(w)13259 45553 y Fs(j)13910 45387 y FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)20091 45553 y Fs(j)20557 45387 y Ft(D)31 b FC(\)\))q(\))185 b Ft( )23701 45553 y Fs(j)24168 45387 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\))19205 47158 y(=)307 b Ft( )40 b FC(\(0\))247 b(+)f(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)30082 47324 y Fs(j)30548 47158 y Ft(D)31 b FC(\)\))186 b Ft( )33302 46701 y Fq(0)33262 47431 y Fs(j)33728 47158 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)37035 47324 y Fo(+)37770 47158 y FC(\))g(+)g Fr(R)40491 47324 y Fs(j)41142 47158 y Ft(;)1108 b FC(\(3.21\))5535 49358 y(with)371 b(the)e(remainder)h(term)6642 51891 y Fr(R)7580 52057 y Fs(j)8354 51891 y FC(:=)308 b(\()q(1)246 b(+)g Ft(v)12704 52057 y Fs(j)13170 51891 y Ft(D)277 b FC(+)246 b Ft(w)16262 52057 y Fs(j)16914 51891 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)23094 52057 y Fs(j)23560 51891 y Ft(D)31 b FC(\)\))r(\))185 b(\()p Fr(O)31 b FC(\()p Ft(D)28703 51434 y Fo(2)29200 51891 y FC(\))247 b(+)f Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(dv)36746 52057 y Fs(j)37212 51891 y Fr(O)31 b FC(\()p Ft(D)39501 51434 y Fo(2)39999 51891 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)43700 51434 y Fo(2)44197 51891 y FC(\)\))14244 53662 y Fr(\000)246 b FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)21532 53828 y Fs(j)21998 53662 y Ft(D)31 b FC(\)\))23990 52766 y Fn(\000)24498 53662 y FC(\()p Fr(O)g FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)29538 53828 y Fs(j)30004 53662 y Fr(O)31 b FC(\()p Ft(D)32293 53205 y Fo(2)32791 53662 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))21316 55434 y(+)246 b Ft(w)23215 55600 y Fs(j)23867 54537 y Fn(\000)24374 55434 y Fr(O)31 b FC(\()p Ft(D)26663 54977 y Fo(2)27160 55434 y FC(\))247 b(+)f Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(dv)34706 55600 y Fs(j)35172 55434 y Fr(O)31 b FC(\()p Ft(D)37461 54977 y Fo(2)37959 55434 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)41660 54977 y Fo(2)42157 55434 y FC(\))42587 54537 y Fn(\001)186 b(\001)43971 55434 y Ft(:)5535 57634 y FC(So)311 b(far)g(the)g(edge)g(index)g(has)g(b)31 b(een)310 b(k)-31 b(ept)312 b(\014xed.)f(Subtracting)h(m)-31 b(utually)314 b(the)d(relations)h(\(3.21\))5535 59073 y(for)370 b(di\013eren)-31 b(t)369 b(v)-61 b(alues)369 b(of)h Ft(j)r(;)184 b(k)343 b Fr(2)308 b Fl(N)p FC(,)370 b(w)-31 b(e)369 b(can)h(eliminate)i Ft( )40 b FC(\(0\),)6827 61606 y(\(1)247 b(+)f Ft(v)9701 61772 y Fs(j)10167 61606 y Ft(D)277 b FC(+)246 b Ft(w)13259 61772 y Fs(j)13910 61606 y FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)20091 61772 y Fs(j)20557 61606 y Ft(D)31 b FC(\)\))q(\))185 b Ft( )23701 61772 y Fs(j)24168 61606 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\))14335 63377 y Fr(\000)g FC(\(1)g(+)g Ft(v)18315 63543 y Fs(k)18860 63377 y Ft(D)277 b FC(+)246 b Ft(w)21952 63543 y Fs(k)22681 63377 y FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)28862 63543 y Fs(k)29406 63377 y Ft(D)31 b FC(\)\))r(\))185 b Ft( )32551 63543 y Fs(k)33095 63377 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)p FC(\))7410 65149 y(=)307 b(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)14759 65315 y Fs(j)15225 65149 y Ft(D)31 b FC(\)\))185 b Ft( )17978 64692 y Fq(0)17938 65422 y Fs(j)18405 65149 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)21712 65315 y Fo(+)22447 65149 y FC(\))g Fr(\000)g FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)30411 65315 y Fs(k)30955 65149 y Ft(D)31 b FC(\)\))186 b Ft( )33709 64692 y Fq(0)33669 65422 y Fs(k)34213 65149 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)37520 65315 y Fo(+)38255 65149 y FC(\))h(+)f Fr(R)40977 65315 y Fs(j)41689 65149 y Fr(\000)g(R)43734 65315 y Fs(k)42557 66588 y FC(\(3.22\))5535 68722 y(Returning)371 b(to)f(the)g(relations)h(\(3.19\))i(and)d(\(3.20\))i(w)-31 b(e)370 b(can)g(eliminate)j(from)d(them)h Ft( )40 b FC(\(0\))370 b(in)g(a)5535 70161 y(similar)h(w)-31 b(a)g(y)371 b(as)e(ab)31 b(o)-31 b(v)g(e)370 b(arriving)g(at)g(the)g(relation)7983 72362 y(\(1)247 b(+)f Ft(dv)11433 72528 y Fs(j)11899 72362 y FC(\))p Ft( )13090 71905 y Fq(0)13050 72635 y Fs(j)13517 72362 y FC(\()p Ft(D)278 b FC(+)246 b Ft(d)16824 72528 y Fo(+)17559 72362 y FC(\))g Fr(\000)g FC(\()q Ft(v)20310 72528 y Fs(j)21022 72362 y FC(+)g Ft(w)22921 72528 y Fs(j)23387 72362 y FC(\(1)h(+)f Ft(dv)26837 72528 y Fs(j)27304 72362 y FC(\)\))185 b Ft( )29070 72528 y Fs(j)29537 72362 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\))308 b(=)g Ft( )35511 71905 y Fq(0)35471 72635 y Fs(j)35937 72362 y FC(\(0\))248 b Fr(\000)38989 72082 y FC(~)38705 72362 y Fr(R)39643 72528 y Fs(j)42557 72362 y FC(\(3.23\))p eop end %%Page: 15 15 TeXDict begin 15 14 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(15)6614 11900 y FC(with)371 b(the)e(remainder)g(term)8006 13878 y(~)7721 14158 y Fr(R)8659 14324 y Fs(j)9433 14158 y FC(:=)308 b(\()q Ft(v)11877 14324 y Fs(j)12589 14158 y FC(+)246 b Ft(w)14488 14324 y Fs(j)14954 14158 y FC(\(1)h(+)f Ft(dv)18404 14324 y Fs(j)18871 14158 y FC(\)\))185 b(\()p Fr(O)31 b FC(\()p Ft(D)22635 13701 y Fo(2)23133 14158 y FC(\))246 b(+)g Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))248 b(+)e Ft(dv)30679 14324 y Fs(j)31145 14158 y Fr(O)31 b FC(\()p Ft(D)33434 13701 y Fo(2)33931 14158 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)37632 13701 y Fo(2)38130 14158 y FC(\)\))p Fr(\000)6614 15929 y(\000)p FC(\(1+)p Ft(dv)10432 16095 y Fs(j)10899 15929 y FC(\))11513 15033 y Fn(\000)12022 15929 y FC(\()p Fr(O)g FC(\()p Ft(D)g FC(\))247 b(+)f Ft(v)17062 16095 y Fs(j)17528 15929 y Fr(O)31 b FC(\()p Ft(D)19817 15472 y Fo(2)20315 15929 y FC(\))246 b(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))247 b(+)f Ft(w)26592 16095 y Fs(j)27243 15033 y Fn(\000)27751 15929 y Fr(O)31 b FC(\()p Ft(D)30040 15472 y Fo(2)30537 15929 y FC(\))247 b(+)f Ft(d)p Fr(O)31 b FC(\()p Ft(D)g FC(\))247 b(+)f Ft(dv)38083 16095 y Fs(j)38549 15929 y Fr(O)31 b FC(\()p Ft(D)40838 15472 y Fo(2)41335 15929 y FC(\))247 b(+)f Fr(O)31 b FC(\()p Ft(d)45037 15472 y Fo(2)45534 15929 y FC(\))45964 15033 y Fn(\001)q(\001)47348 15929 y Ft(:)6614 17856 y FC(Summing)371 b(the)e(ab)31 b(o)-31 b(v)g(e)371 b(relations)f(o)-31 b(v)g(er)370 b Ft(j)h Fr(2)307 b Fl(N)370 b FC(and)f(using)739 b(\(3.14\))372 b(w)-31 b(e)369 b(get)7694 19175 y Fs(n)7168 19507 y Fn(X)7201 21864 y Fs(j)45 b Fo(=1)8767 20558 y FC(\(1)38 b(+)g Ft(dv)11800 20724 y Fs(j)12267 20558 y FC(\))p Ft( )13458 20101 y Fq(0)13418 20832 y Fs(j)13885 20558 y FC(\()p Ft(D)69 b FC(+)38 b Ft(d)16775 20724 y Fo(+)17512 20558 y FC(\))g Fr(\000)19406 19175 y Fs(n)18879 19507 y Fn(X)18913 21864 y Fs(j)45 b Fo(=1)20663 20558 y FC(\()q Ft(v)21631 20724 y Fs(j)22343 20558 y FC(+)246 b Ft(w)24242 20724 y Fs(j)24709 20558 y FC(\(1)h(+)f Ft(dv)28159 20724 y Fs(j)28625 20558 y FC(\)\))185 b Ft( )30391 20724 y Fs(j)30858 20558 y FC(\()p Ft(D)69 b FC(+)38 b Ft(d)p FC(\))309 b(=)f Ft(u )40 b FC(\(0\))e Fr(\000)39928 19175 y Fs(n)39401 19507 y Fn(X)39435 21864 y Fs(j)45 b Fo(=1)41470 20279 y FC(~)41186 20558 y Fr(R)42124 20724 y Fs(j)42775 20558 y Ft(:)554 b FC(\(3.24\))6614 23554 y(The)412 b(righ)-31 b(t-hand)414 b(side)e(can)g(rewritten)h(using)f(the)g(con)-31 b(tin)g(uit)g(y)415 b(condition)f(\(3.14\))h(in)d(com)-31 b(bi-)6614 24993 y(nation)371 b(with)f(the)g(relation)h(\(3.21\))s(,) 7721 28028 y Ft(u )40 b FC(\(0\))309 b(=)12154 27279 y Ft(u)p 12139 27773 665 45 v 12139 28787 a(n)13646 26644 y Fs(n)13121 26976 y Fn(X)13154 29334 y Fs(j)45 b Fo(=1)14904 28028 y Ft( )15625 28194 y Fs(j)16092 28028 y FC(\(0\))308 b(=)19130 27279 y Ft(u)p 19115 27773 V 19115 28787 a(n)20622 26644 y Fs(n)20096 26976 y Fn(X)20130 29334 y Fs(j)45 b Fo(=1)21880 27131 y Fn(\000)22572 28028 y FC(\(1)247 b(+)f Ft(v)25446 28194 y Fs(j)25912 28028 y Ft(D)277 b FC(+)246 b Ft(w)29004 28194 y Fs(j)29655 28028 y FC(\()q Ft(D)277 b FC(+)246 b Ft(d)p FC(\(1)h(+)f Ft(v)35836 28194 y Fs(j)36302 28028 y Ft(D)31 b FC(\)\))q(\))185 b Ft( )39446 28194 y Fs(j)39913 28028 y FC(\()p Ft(D)277 b FC(+)246 b Ft(d)p FC(\))26195 31046 y Fr(\000)g FC(\()p Ft(D)278 b FC(+)245 b Ft(d)p FC(\(1)j(+)e Ft(v)33483 31212 y Fs(j)33949 31046 y Ft(D)31 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Fs(k)28743 70602 y Ft(\014)29369 70768 y Fs(k)30159 70602 y Fr(\000)h FC(1)h(+)f Ft(\014)33799 70768 y Fs(k)34344 70602 y Ft(d)p FC(\))p Ft( )36071 70768 y Fs(k)36616 70602 y FC(\()p Ft(d)37622 70145 y Fo(3)38365 70602 y FC(+)g Ft(d)p FC(\))14107 72373 y(=)308 b(\()p Ft(\013)16414 72539 y Fs(j)17127 72373 y FC(+)246 b Ft(d)p FC(\))p Ft( )20001 71916 y Fq(0)19961 72647 y Fs(j)20428 72373 y FC(\()p Ft(d)21434 71916 y Fo(3)22177 72373 y FC(+)g Ft(d)23860 72539 y Fo(+)24595 72373 y FC(\))h Fr(\000)e FC(\()p Ft(\013)27516 72539 y Fs(k)28308 72373 y FC(+)h Ft(d)p FC(\))p Ft( )31182 71916 y Fq(0)31142 72647 y Fs(k)31687 72373 y FC(\()p Ft(d)32693 71916 y Fo(3)33436 72373 y FC(+)g Ft(d)35119 72539 y Fo(+)35854 72373 y FC(\))h(+)f Fr(O)31 b FC(\()p Ft(d)p FC(\))184 b Ft(:)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 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Fr(j)p Ft(\013)21604 72528 y Fo(1)22101 72362 y Ft(\014)22727 72528 y Fo(1)23470 72362 y Fr(\000)246 b FC(1)g Fr(\000)g FC(i)p Ft(\013)27498 72528 y Fo(1)27995 72362 y Fr(j)308 b FC(=)g(1)12225 b(\(3.31\))p eop end %%Page: 17 17 TeXDict begin 17 16 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(17)6614 11900 y FC(and)370 b(to)g(set)19896 13264 y Fs(n)19370 13596 y Fn(X)19467 15979 y Fs(l)t Fo(=1)21286 13899 y Ft(\014)21912 14065 y Fs(l)22252 13899 y FC(\()p Ft(\013)23390 14065 y Fs(l)23730 13899 y Ft(\014)24356 14065 y Fs(l)24942 13899 y Fr(\000)246 b FC(1\))h(+)f Ft(\013)29094 14065 y Fs(l)p 21286 14393 8148 45 v 21440 15407 a FC(\()p Ft(\013)22578 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b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(18)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)7196 12418 y FC(Let)512 b(us)f(construct)h(\014rst)g Fr(G)19047 11886 y Fs(!)28 b(;)d(~)-479 b(\013;)20790 11682 y(~)20734 11886 y(\014)18981 12752 y(k)21843 12418 y FC(for)512 b(the)g(star-graph)g(Hamiltonian)k(referring)c(to)g(the) 5535 13857 y(condition)355 b(\(2.1\))g(in)d(the)h(cen)-31 b(tre.)353 b(W)-92 b(e)352 b(b)31 b(egin)353 b(with)h Ft(n)e FC(indep)31 b(enden)-31 b(t)353 b(hal\015ines)h(with)g(Diric)-31 b(hlet)5535 15296 y(condition)372 b(at)e(its)f(endp)31 b(oin)-31 b(ts;)371 b(Green's)d(function)i(for)g(eac)-31 b(h)369 b(of)h(them)g(is)f(w)-31 b(ell-kno)g(wn)372 b(to)e(b)31 b(e)18650 17823 y Fr(G)19308 17989 y Fo(i)p Fs(\024)20133 17823 y FC(\()p Ft(x;)184 b(y)40 b 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58188 y Fn(Z)9884 58461 y Fo(+)p Fq(1)9392 60703 y Fo(0)11686 57802 y Fn( )13242 58311 y Fs(n)12716 58643 y Fn(X)12562 60987 y Fs(m)p Fo(=1)14653 59694 y Ft(A)15483 59860 y Fs(j)45 b(m)16735 59694 y Ft(\025)17381 59860 y Fs(ml)18506 59694 y FC(\()p Ft(k)19547 59237 y Fo(2)20045 59694 y FC(\))246 b(+)g Ft(B)22668 59860 y Fs(j)45 b(l)23664 59694 y Fr(\000)246 b Ft(\024)26273 58311 y Fs(n)25747 58643 y Fn(X)25593 60987 y Fs(m)p Fo(=1)27684 59694 y Ft(B)28524 59860 y Fs(j)45 b(m)29776 59694 y Ft(\025)30422 59860 y Fs(ml)31547 59694 y FC(\()p Ft(k)32588 59237 y Fo(2)33085 59694 y FC(\))33515 57802 y Fn(!)34577 59694 y FC(e)35069 59237 y Fq(\000)p Fs(\024y)36728 59354 y Fj(l)37104 59694 y Ft( )37825 59860 y Fs(l)38165 59694 y FC(\()p Ft(y)39138 59860 y Fs(l)39478 59694 y FC(\)d)p Ft(y)41066 59860 y Fs(l)41713 59694 y FC(=)308 b(0)184 b Ft(;)5535 62587 y FC(with)366 b Ft(j)371 b Fr(2)363 b FC(^)-609 b Ft(n)q FC(.)364 b(W)-92 b(e)364 b(require)g(that)i(the)e(left-hand)i(side)f(v)-61 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67474 y Ft(:)5535 69484 y FC(Notice)439 b(that)g(the)f (matrix)h Ft(A)292 b Fr(\000)g Ft(\024B)492 b FC(is)438 b(regular)g(in)g(view)g(of)h(the)e(\014rst)h(conditions)h(in)f(\(2.2\)) r(;)5535 70923 y(since)375 b Ft(A;)370 b(B)430 b FC(are)376 b(real)f(and)h(Im)184 b Ft(\024)318 b Fr(6)p FC(=)g(0,)376 b(the)f(requiremen)-31 b(t)376 b(rank)q(\()p Ft(A;)184 b(B)56 b FC(\))319 b(=)e Ft(n)376 b FC(implies)h(that)f(w)-31 b(e)5535 72362 y(ha)g(v)g(e)370 b(also)h(rank\()p Ft(A)246 b Fr(\000)g Ft(\024B)56 b FC(\))307 b(=)h Ft(n)p FC(.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(19)8275 11900 y FC(Let)325 b(us)g(no)-31 b(w)327 b(concen)-31 b(trate)326 b(on)g(the)g(class)f(of)h(couplings)h(for)f (whic)-31 b(h)326 b(w)-31 b(e)326 b(established)h(in)e(the)6614 13339 y(previous)369 b(section)i(the)e(b)31 b(oundary)369 b(condition)j(con)-31 b(v)g(ergence.)370 b(In)f(this)h(case)f Ft(A)246 b Fr(\000)g Ft(\024B)424 b FC(equals)6807 14758 y Fn(0)6807 16706 y(B)6807 17371 y(B)6807 18035 y(B)6807 18743 y(@)10069 15502 y Ft(\013)10777 15668 y Fo(1)11273 15502 y FC(\()p Ft(\014)12329 15668 y Fo(1)13073 15502 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1)3726 b Fr(\000)p FC(\()p Ft(\013)22879 15668 y Fo(2)23376 15502 y FC(\()p Ft(\014)24432 15668 y Fo(2)25175 15502 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1\))1600 b Fr(\001)184 b(\001)g(\001)6018 b FC(0)13458 16715 y(.)13458 17158 y(.)13458 17601 y(.)31439 16826 y(.)31870 17158 y(.)32300 17490 y(.)10069 19040 y Ft(\013)10777 19206 y Fo(1)11273 19040 y FC(\()p Ft(\014)12329 19206 y Fo(1)13073 19040 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1)7853 b(0)5726 b Fr(\001)184 b(\001)g(\001)1785 b(\000)p FC(\()p Ft(\013)36359 19206 y Fs(n)36962 19040 y FC(\()p Ft(\014)38018 19206 y Fs(n)38867 19040 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1\))7941 20479 y(\()p Ft(\014)8997 20645 y Fo(1)9741 20479 y FC(+)g Ft(\024)p FC(\))12100 19582 y Fn(\000)12741 20043 y Fs(!)p 12741 20224 586 45 v 12760 20860 a(n)13459 20479 y Ft(\013)14167 20645 y Fo(1)14910 20479 y Fr(\000)g FC(1)16570 19582 y Fn(\001)17323 20479 y Fr(\000)18563 20043 y Fs(!)p 18563 20224 V 18582 20860 a(n)19614 20479 y FC(\()p Ft(\014)20670 20645 y Fo(2)21413 20479 y FC(+)g Ft(\024)p FC(\))23772 19582 y Fn(\000)24413 20043 y Fs(!)p 24413 20224 V 24432 20860 a(n)25131 20479 y Ft(\013)25839 20645 y Fo(2)26582 20479 y Fr(\000)g FC(1)28242 19582 y Fn(\001)28995 20479 y Fr(\000)30235 20043 y Fs(!)p 30235 20224 V 30254 20860 a(n)31286 20479 y Fr(\001)184 b(\001)g(\001)519 b FC(\()p Ft(\014)34150 20645 y Fs(n)34999 20479 y FC(+)246 b Ft(\024)p FC(\))37358 19582 y Fn(\000)37999 20043 y Fs(!)p 37999 20224 V 38018 20860 a(n)38717 20479 y Ft(\013)39425 20645 y Fs(n)40274 20479 y Fr(\000)g FC(1)41934 19582 y Fn(\001)42687 20479 y Fr(\000)43927 20043 y Fs(!)p 43927 20224 V 43946 20860 a(n)44812 14758 y Fn(1)44812 16706 y(C)44812 17371 y(C)44812 18035 y(C)44812 18743 y(A)45965 17979 y Ft(;)6614 22620 y FC(and)336 b(a)g(tedious)g(b)-31 b(y)336 b(straigh)-31 b(tforw)g(ard)339 b(computation)f(yields)e(an)g(explicit)i(form)e(of)g(the)g(matrix)6614 24059 y Fr(\000)p FC(\()p Ft(A)247 b Fr(\000)f Ft(\024B)56 b FC(\))12053 23658 y Fq(\000)p Fo(1)13241 24059 y Ft(B)g FC(,)368 b(namely)6741 25843 y Fn(\002)7202 26739 y Fr(\000)p FC(\()p Ft(A)247 b Fr(\000)f Ft(\024B)56 b FC(\))12641 26282 y Fq(\000)p Fo(1)13829 26739 y Ft(B)14725 25843 y Fn(\003)15185 27182 y Fs(j)45 b(l)16243 26739 y FC(=)22576 25991 y(1)p 17237 26485 11233 45 v 17237 27650 a Ft(!)286 b Fr(\000)19319 26819 y Fn(P)20487 27093 y Fs(n)20487 27982 y(m)p Fo(=1)24334 27156 y Fs(\014)24836 27267 y Fj(m)25571 27156 y Fo(+)p Fs(\024)p 22766 27395 5571 45 v 22766 28031 a(\013)23341 28142 y Fj(m)24077 28031 y Fo(\()p Fs(\014)24925 28142 y Fj(m)25660 28031 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)28848 26739 y Fr(\001)37017 25991 y FC(1)p 29534 26485 15520 45 v 29534 27499 a(\()p Ft(\013)30672 27665 y Fs(j)31140 27499 y FC(\()p Ft(\014)32196 27665 y Fs(j)32909 27499 y FC(+)245 b Ft(\024)p FC(\))i Fr(\000)f FC(1\)\()p Ft(\013)38558 27665 y Fs(l)38899 27499 y FC(\()p Ft(\014)39955 27665 y Fs(l)40541 27499 y FC(+)g Ft(\024)p FC(\))h Fr(\000)f FC(1\))39521 29675 y(for)369 b Ft(j)h Fr(6)p FC(=)308 b Ft(l)207 b(;)6614 31115 y Fn(\002)7075 32012 y Fr(\000)p FC(\()p Ft(A)247 b Fr(\000)f Ft(\024B)56 b FC(\))12514 31555 y Fq(\000)p Fo(1)13702 32012 y Ft(B)14598 31115 y Fn(\003)15058 32455 y Fs(j)45 b(j)16243 32012 y FC(=)22576 31263 y(1)p 17237 31757 11233 45 v 17237 32922 a Ft(!)286 b Fr(\000)19319 32092 y Fn(P)20487 32365 y Fs(n)20487 33254 y(m)p Fo(=1)24334 32428 y Fs(\014)24836 32539 y Fj(m)25571 32428 y Fo(+)p Fs(\024)p 22766 32667 5571 45 v 22766 33304 a(\013)23341 33415 y Fj(m)24077 33304 y Fo(\()p Fs(\014)24925 33415 y Fj(m)25660 33304 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)28848 32012 y Fr(\001)33449 31263 y FC(1)p 29534 31757 8383 45 v 29534 32771 a(\()p Ft(\013)30672 32937 y Fs(j)31140 32771 y FC(\()p Ft(\014)32196 32937 y Fs(j)32909 32771 y FC(+)245 b Ft(\024)p FC(\))i Fr(\000)f FC(1\))37420 32451 y Fo(2)38296 32012 y FC(+)42461 31263 y Ft(\013)43169 31429 y Fs(j)p 39536 31757 7026 45 v 39536 32771 a Ft(\013)40244 32937 y Fs(j)40710 32771 y FC(\()p Ft(\014)41766 32937 y Fs(j)42479 32771 y FC(+)g Ft(\024)p FC(\))h Fr(\000)f FC(1)46878 32012 y Ft(:)6614 35732 y FC(In)329 b(this)h(w)-31 b(a)g(y)330 b(w)-31 b(e)330 b(get)g(the)g(Green)e(function)j Fr(G)25793 35200 y Fs(!)28 b(;)d(~)-479 b(\013)o(;)27535 34995 y(~)27479 35200 y(\014)25727 36049 y Fo(i)p Fs(\024)28077 35732 y FC(.)329 b(As)g(w)-31 b(e)330 b(ha)-31 b(v)g(e)330 b(men)-31 b(tioned)331 b(ab)31 b(o)-31 b(v)g(e,)331 b(it)f(is)f(an)6614 37171 y Ft(n)247 b Fr(\002)f Ft(n)369 b FC(matrix-v)-61 b(alued)371 b(function)g(the)e(\()p Ft(j)r(;)184 b(l)22 b FC(\)-th)372 b(elemen)-31 b(t)370 b(of)g(whic)-31 b(h)370 b(is)f(giv)-31 b(en)370 b(b)-31 b(y)6614 40019 y Fr(G)7338 39487 y Fs(!)28 b(;)d(~)-479 b(\013;)9081 39283 y(~)9025 39487 y(\014)7272 40353 y Fo(i)p Fs(\024;j)45 b(l)9622 40019 y FC(\()p Ft(x;)184 b(y)40 b FC(\))309 b(=)p Ft(\016)13851 40185 y Fs(j)45 b(l)14786 38458 y Fn(\022)15734 39270 y FC(sinh)184 b Ft(\024x)19163 39436 y Fs(<)20156 39270 y FC(e)20648 38868 y Fq(\000)p Fs(\024x)22360 38979 y Fj(>)p 15734 39764 7337 45 v 19083 40778 a Ft(\024)23449 40019 y FC(+)246 b(e)25048 39562 y Fq(\000)p Fs(\024)p Fo(\()p Fs(x)p Fo(+)p Fs(y)28 b Fo(\))31722 39270 y Ft(\013)32430 39436 y Fs(j)p 28796 39764 7026 45 v 28796 40778 a Ft(\013)29504 40944 y Fs(j)29971 40778 y FC(\()p Ft(\014)31027 40944 y Fs(j)31740 40778 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1)35954 38458 y Fn(\023)36954 40019 y FC(+)12498 43000 y(+)18831 42251 y(1)p 13492 42746 11233 45 v 13492 43910 a Ft(!)285 b Fr(\000)15573 43080 y Fn(P)16742 43354 y Fs(n)16742 44242 y(m)p Fo(=1)20588 43416 y Fs(\014)21090 43527 y Fj(m)21825 43416 y Fo(+)p Fs(\024)p 19021 43656 5571 45 v 19021 44292 a(\013)19596 44403 y Fj(m)20331 44292 y Fo(\()p Fs(\014)21179 44403 y Fj(m)21914 44292 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)25103 43000 y Fr(\001)33272 42251 y FC(1)p 25789 42746 15520 45 v 25789 43759 a(\()p Ft(\013)26927 43925 y Fs(j)27394 43759 y FC(\()p Ft(\014)28450 43925 y Fs(j)29163 43759 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1\)\()p Ft(\013)34812 43925 y Fs(l)35153 43759 y FC(\()p Ft(\014)36209 43925 y Fs(l)36796 43759 y FC(+)g Ft(\024)p FC(\))g Fr(\000)g FC(1\))41810 43000 y(e)42302 42543 y Fq(\000)p Fs(\024x)44069 43000 y FC(e)44561 42543 y Fq(\000)p Fs(\024y)46488 43000 y FC(;)6614 46196 y(w)-31 b(e)370 b(use)e(the)i(con)-31 b(v)g(en)g(tion)372 b(that)e Ft(x)f FC(is)g(from)h(the)f Ft(j)63 b FC(-th)370 b(hal\015ine)g(and)f Ft(y)409 b FC(from)370 b(the)f Ft(l)22 b FC(-th)370 b(one.)8275 47635 y(Next)528 b(w)-31 b(e)529 b(will)g(pass)f(to)h(resolv)-31 b(en)g(t)529 b(construction)g(for)f(the)g(appro)-31 b(ximating)532 b(family)e(of)6614 49074 y(op)31 b(erators)397 b Ft(H)12553 48672 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)14789 49074 y FC(\()p Ft(d)p FC(\).)398 b(As)e(a)g(starting)i(p)31 b(oin)-31 b(t)397 b(w)-31 b(e)397 b(consider)f Ft(n)g FC(indep)31 b(enden)-31 b(t)397 b(hal\015ines)g(with)6614 50513 y(Diric)-31 b(hlet)409 b(endp)31 b(oin)-31 b(ts;)408 b(w)-31 b(e)408 b(kno)-31 b(w)409 b(that)g(the)e(appropriate)i(Green's) e(function)h(is)g Fr(G)41840 50679 y Fo(i)p Fs(\024)42664 50513 y FC(\()p Ft(x;)184 b(y)40 b FC(\))373 b(=)6614 51952 y Ft(\024)7252 51550 y Fq(\000)p Fo(1)8625 51952 y FC(sinh)184 b Ft(\024x)12054 52118 y Fs(<)12801 51952 y FC(e)13293 51550 y Fq(\000)p Fs(\024x)15005 51661 y Fj(>)15715 51952 y FC(.)322 b(The)f(sough)-31 b(t)322 b(resolv)-31 b(en)g(t)322 b(k)-31 b(ernel)321 b(will)h(b)31 b(e)321 b(then)g(found)g(in)g(sev)-31 b(eral)322 b(steps.)6614 53391 y(Eac)-31 b(h)537 b(of)f(them)g(represen)-31 b(ts)534 b(an)i(application)j(of)d(Krein's)f(form)-31 b(ula.)538 b(First)d(w)-31 b(e)536 b(add)g(the)g Ft(\016)6614 54830 y FC(in)-31 b(teraction)291 b(with)e(the)f(parameter)h Ft(v)327 b FC(at)289 b(the)f(distance)h Ft(d)30348 54428 y Fo(3)31132 54830 y FC(from)g(the)f(endp)31 b(oin)-31 b(t,)290 b(then)e(another)6614 56269 y(one)369 b(with)h(the)f (parameter)h Ft(w)398 b FC(at)370 b(the)f(distance)g Ft(d)246 b FC(+)f Ft(d)29628 55867 y Fo(3)30124 56269 y FC(,)369 b(again)i(from)e(the)g(endp)31 b(oin)-31 b(t.)371 b(This)e(is)6614 57708 y(done)391 b(on)g(eac)-31 b(h)392 b(hal\015ine)g(separately)-92 b(.)392 b(In)e(the)h(\014nal)h(step)e(w) -31 b(e)392 b(\014nd)e(Green's)g(function)i(for)f(the)6614 59147 y(star)356 b(in)h(whic)-31 b(h)357 b(the)g(Diric)-31 b(hlet)357 b(ends)f(are)g(replaced)h(b)-31 b(y)356 b(the)h Ft(\016)398 b FC(coupling)358 b(with)f(the)g(parameter)6614 60586 y Ft(u)p FC(.)369 b(That)i(will)g(require,)e(of)h(course,)f(to)h (distinguish)g(the)g(hal\015ines)g(b)-31 b(y)369 b(their)h(indices.) 8275 62025 y(The)f(\014rst)g(step)g(is)g(rather)g(standard)g([19)r(])g (and)h(resulting)g(Green)e(function)j(is)11679 64468 y Fr(G)12403 64011 y Fs(v)12337 64742 y Fo(i)p Fs(\024)13162 64468 y FC(\()p Ft(x;)184 b(y)40 b FC(\))308 b(=)g Fr(G)17864 64634 y Fo(i)p Fs(\024)18688 64468 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)26586 63719 y Ft(v)p 22742 64214 8264 45 v 22742 65228 a FC(1)g(+)e Ft(v)286 b Fr(\001)246 b(G)26682 65394 y Fo(i)p Fs(\024)27507 65228 y FC(\()p Ft(d)28513 64908 y Fo(3)29010 65228 y Ft(;)184 b(d)30077 64908 y Fo(3)30575 65228 y FC(\))31507 64468 y Fr(G)32165 64634 y Fo(i)p Fs(\024)32990 64468 y FC(\()p Ft(y)40 b(;)184 b(d)35070 64011 y Fo(3)35568 64468 y FC(\))369 b Fr(G)37025 64634 y Fo(i)p Fs(\024)37850 64468 y FC(\()p Ft(x;)184 b(d)39980 64011 y Fo(3)40478 64468 y FC(\))g Ft(:)2790 b FC(\(4.5\))6614 67214 y(Adding)399 b(another)f Ft(\016)439 b FC(in)-31 b(teraction)400 b(at)e(the)g(distance)g Ft(d)g FC(from)g(the)g(previous)f(one)h(w)-31 b(e)398 b(seek)f(the)6614 68653 y(k)-31 b(ernel)418 b(in)f(the)h(form)f Ft(R)16733 68252 y Fs(v)28 b(;w)18183 68653 y FC(\()p Ft(k)19224 68252 y Fo(2)19721 68653 y FC(\))388 b(=)f Ft(R)22637 68252 y Fs(v)23162 68653 y FC(\()p Ft(k)24203 68252 y Fo(2)24700 68653 y FC(\))279 b(+)f Ft(\025)p FC(\()p Ft(k)28235 68252 y Fo(2)28732 68653 y FC(\))29162 67757 y Fn(\000)29670 68653 y Ft(\036)p FC(\()p 30760 67701 1108 45 v Ft(k)31371 68333 y Fo(2)31868 68653 y FC(\))p Ft(;)184 b Fr(\001)33096 67757 y Fn(\001)33605 68653 y Ft(\036)p FC(\()p Ft(k)35306 68252 y Fo(2)35803 68653 y FC(\))417 b(where)g(the)h(\014rst)e(term)6614 70092 y(is)369 b Ft(R)8577 69691 y Fs(v)9102 70092 y FC(\()p Ft(k)10143 69691 y Fo(2)10640 70092 y FC(\))308 b(:=)g Fr(G)13578 69691 y Fs(v)13512 70389 y Fo(i)p Fs(\024)14706 70092 y FC(and)370 b(the)f(de\014ciency-subspace)g(elemen)-31 b(t)371 b Ft(\036)p FC(\()p Ft(k)34254 69691 y Fo(2)34751 70092 y FC(\))e(is)g(c)-31 b(hosen)369 b(as)20106 72362 y Ft(\036)p FC(\()p Ft(k)21807 71905 y Fo(2)22304 72362 y FC(\)\()p Ft(x)p FC(\))308 b(:=)g Fr(G)26735 71905 y Fs(v)26669 72635 y Fo(i)p Fs(\024)27494 72362 y FC(\()p Ft(x;)184 b(d)247 b FC(+)f Ft(d)31554 71905 y Fo(3)32051 72362 y FC(\))184 b Ft(:)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(20)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 12026 y FC(W)-92 b(e)358 b(apply)i(this)f(Ansatz)h (to)f(an)-31 b(y)360 b Ft( )347 b Fr(2)308 b Ft(L)22620 11624 y Fo(2)23117 12026 y FC(\()p Fl(R)24346 11624 y Fo(+)25082 12026 y FC(\))359 b(and)g(denote)g Ft(h)308 b FC(:=)g Ft(R)34841 11624 y Fs(v)28 b(;w)36290 12026 y FC(\()p Ft(k)37331 11624 y Fo(2)37828 12026 y FC(\))p Ft( )40 b FC(.)359 b(It)g(is)g(easy)g(to)5535 13465 y(c)-31 b(hec)g(k)370 b(that)p 10916 12502 4051 45 v 370 w Fr(G)11640 13084 y Fs(v)11574 13782 y Fo(i)39 b(\026)-480 b Fs(\024)12399 13465 y FC(\()p Ft(x;)184 b(y)40 b FC(\))309 b(=)e Fr(G)17167 13063 y Fs(v)17101 13761 y Fo(i)p Fs(\024)17926 13465 y FC(\()p Ft(x;)184 b(y)40 b FC(\),)370 b(hence)f(w)-31 b(e)370 b(can)f(write)h Ft(h)f FC(explicitly)j(as)5970 16627 y Ft(h)p FC(\()p Ft(x)p FC(\))308 b(=)9577 15120 y Fn(Z)10684 15394 y Fo(+)p Fq(1)10192 17635 y Fo(0)12486 16627 y Fr(G)13210 16170 y Fs(v)13144 16900 y Fo(i)p Fs(\024)13969 16627 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p Ft( )g FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)287 b FC(+)246 b Ft(\025)p FC(\()p Ft(k)23163 16170 y Fo(2)23660 16627 y FC(\))24274 15120 y Fn(Z)25382 15394 y Fo(+)p Fq(1)24890 17635 y Fo(0)27184 16627 y Fr(G)27908 16170 y Fs(v)27842 16900 y Fo(i)p Fs(\024)28666 16627 y FC(\()p Ft(y)40 b(;)184 b(d)247 b FC(+)f Ft(d)32676 16170 y Fo(3)33173 16627 y FC(\))p Ft( )40 b FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)287 b Fr(\001)246 b(G)38713 16170 y Fs(v)38647 16900 y Fo(i)p Fs(\024)39472 16627 y FC(\()p Ft(x;)184 b(d)247 b FC(+)f Ft(d)43532 16170 y Fo(3)44029 16627 y FC(\))184 b Ft(:)5535 19536 y FC(By)312 b(de\014nition)h(this)g(function)g(this)f (function)h(b)31 b(elongs)312 b(to)h(the)f(domain)h(of)f(the)g(op)31 b(erator)312 b(with)5535 20975 y(t)-31 b(w)g(o)371 b Ft(\016)411 b FC(in)-31 b(teractions,)372 b(in)d(particular,)i(it)f (has)f(to)h(satisfy)g(the)g(b)31 b(oundary)369 b(conditions)16093 23436 y Ft(h)p FC(\()p Ft(d)246 b FC(+)g Ft(d)19666 22979 y Fo(3)20163 23602 y(+)20898 23436 y FC(\))308 b(=)f Ft(h)p FC(\()p Ft(d)247 b FC(+)f Ft(d)26378 22979 y Fo(3)26874 23602 y Fq(\000)27622 23436 y FC(\))308 b(=:)g Ft(h)p FC(\()p Ft(d)246 b FC(+)g Ft(d)33409 22979 y Fo(3)33906 23436 y FC(\))184 b Ft(;)8283 b FC(\(4.6\))15187 25207 y Ft(h)15825 24750 y Fq(0)16135 25207 y FC(\()p Ft(d)247 b FC(+)f Ft(d)19071 24750 y Fo(3)19567 25373 y(+)20302 25207 y FC(\))h Fr(\000)f Ft(h)22724 24750 y Fq(0)23034 25207 y FC(\()p Ft(d)h FC(+)f Ft(d)25970 24750 y Fo(3)26466 25373 y Fq(\000)27214 25207 y FC(\))308 b(=)f Ft(w)276 b Fr(\001)247 b Ft(h)p FC(\()p Ft(d)f FC(+)g Ft(d)34315 24750 y Fo(3)34812 25207 y FC(\))184 b Ft(:)7377 b FC(\(4.7\))5535 27667 y(Green's)369 b(function)h(con)-31 b(tin)g(uit)g(y)372 b(implies)f(\(4.6\))r(.)f(F)-92 b(urthermore,)369 b(w)-31 b(e)369 b(ha)-31 b(v)g(e)5535 30789 y Ft(h)6173 30332 y Fq(0)6484 30789 y FC(\()p Ft(x)p FC(\))308 b(=)9453 29282 y Fn(Z)10560 29556 y Fo(+)p Fq(1)10068 31798 y Fo(0)12495 30040 y Ft(@)61 b Fr(G)13868 29638 y Fs(v)13802 30337 y Fo(i)p Fs(\024)14627 30040 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 12495 30534 4700 45 v 14204 31548 a Ft(@)61 b(x)17328 30789 y( )40 b FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)74 b FC(+)34 b Ft(\025)p FC(\()p Ft(k)23530 30332 y Fo(2)24027 30789 y FC(\))24641 29282 y Fn(Z)25749 29556 y Fo(+)p Fq(1)25257 31798 y Fo(0)27551 30789 y Fr(G)28275 30332 y Fs(v)28209 31063 y Fo(i)p Fs(\024)29033 30789 y FC(\()p Ft(y)40 b(;)184 b(d)34 b FC(+)g Ft(d)32618 30332 y Fo(3)33116 30789 y FC(\))p Ft( )40 b FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)74 b Fr(\001)37640 30040 y Ft(@)61 b Fr(G)39013 29638 y Fs(v)38947 30337 y Fo(i)p Fs(\024)39774 30040 y FC(\()p Ft(x;)184 b(d)247 b FC(+)f Ft(d)43834 29638 y Fo(3)44331 30040 y FC(\))p 37640 30534 7120 45 v 40560 31548 a Ft(@)61 b(x)45078 30789 y(;)5535 33831 y FC(whic)-31 b(h)398 b(allo)-31 b(ws)398 b(us)e(to)i(express)d Ft(h)19406 33429 y Fq(0)19716 33831 y FC(\()p Ft(d)265 b FC(+)g Ft(d)22689 33429 y Fo(3)22689 34105 y(+)23424 33831 y FC(\))f Fr(\000)h Ft(h)25882 33429 y Fq(0)26192 33831 y FC(\()p Ft(d)g FC(+)f Ft(d)29164 33429 y Fo(3)29164 34105 y Fq(\000)29911 33831 y FC(\).)398 b(The)e(\014rst)h(term)f(ob) -31 b(viously)399 b(do)31 b(es)5535 35270 y(not)347 b(con)-31 b(tribute)348 b(to)f(the)g(di\013erence,)f(while)i(the)f(con)-31 b(tribution)348 b(of)f(the)g(second)f(one)h(simpli\014es)5535 36869 y(in)370 b(view)g(of)10905 36332 y Fs(@)48 b Fq(G)c Fo(\()p Fs(x;y)28 b Fo(\))p 10905 36614 3032 45 v 11910 37251 a Fs(@)48 b(x)14069 35596 y Fn(\014)14069 36260 y(\014)14069 36924 y(\014)14438 37644 y Fs(y)14887 37755 y Fh(+)15813 36869 y Fr(\000)17185 36332 y Fs(@)g Fq(G)c Fo(\()p Fs(x;y)28 b Fo(\))p 17185 36614 V 18190 37251 a Fs(@)48 b(x)20349 35596 y Fn(\014)20349 36260 y(\014)20349 36924 y(\014)20718 37644 y Fs(y)21167 37755 y Ff(\000)22185 36869 y FC(=)307 b Fr(\000)p FC(1)370 b(to)g(the)f(form)10457 40692 y Ft(h)11095 40235 y Fq(0)11405 40692 y FC(\()p Ft(d)247 b FC(+)f Ft(d)14341 40235 y Fo(3)14341 40965 y(+)15076 40692 y FC(\))h Fr(\000)f Ft(h)17498 40235 y Fq(0)17808 40692 y FC(\()p Ft(d)h FC(+)e Ft(d)20743 40235 y Fo(3)20743 40965 y Fq(\000)21491 40692 y FC(\))308 b(=)f Fr(\000)p Ft(\025)p FC(\()p Ft(k)25945 40235 y Fo(2)26442 40692 y FC(\))27056 39185 y Fn(Z)28164 39459 y Fo(+)p Fq(1)27672 41700 y Fo(0)29966 40692 y Fr(G)30690 40235 y Fs(v)30624 40965 y Fo(i)p Fs(\024)31449 40692 y FC(\()p Ft(y)40 b(;)184 b(d)247 b FC(+)f Ft(d)35459 40235 y Fo(3)35955 40692 y FC(\))p Ft( )40 b FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)225 b(:)5535 43733 y FC(T)-92 b(o)370 b(satisfy)386 b(\(4.7\))371 b(the)e(co)31 b(e\016cien)-31 b(t)371 b Ft(\025)p FC(\()p Ft(k)22040 43332 y Fo(2)22537 43733 y FC(\))f(m)-31 b(ust)369 b(ob)31 b(ey)370 b(the)f(condition)8226 45348 y Fn(Z)9333 45622 y Fo(+)p Fq(1)8841 47864 y Fo(0)11134 45958 y Fn(\002)11596 46855 y Ft(\025)p FC(\()p Ft(k)13283 46398 y Fo(2)13780 46855 y FC(\))246 b(+)g Ft(w)276 b FC(+)246 b Ft(w)30 b(\025)p FC(\()p Ft(k)20247 46398 y Fo(2)20744 46855 y FC(\))p Fr(G)21898 46398 y Fs(v)21832 47129 y Fo(i)p Fs(\024)22658 46855 y FC(\()p Ft(d)246 b FC(+)g Ft(d)25593 46398 y Fo(3)26090 46855 y Ft(;)184 b(d)247 b FC(+)f Ft(d)29087 46398 y Fo(3)29584 46855 y FC(\))30014 45958 y Fn(\003)30660 46855 y Fr(G)31384 46398 y Fs(v)31318 47129 y Fo(i)p Fs(\024)32143 46855 y FC(\()p Ft(y)40 b(;)184 b(d)247 b FC(+)f Ft(d)36153 46398 y Fo(3)36650 46855 y FC(\))p Ft( )40 b FC(\()p Ft(y)g FC(\))184 b(d)p Ft(y)348 b FC(=)307 b(0)5535 49897 y(for)348 b(an)-31 b(y)349 b Ft( )e Fr(2)307 b Ft(L)12144 49495 y Fo(2)12641 49897 y FC(\()p Fl(R)13870 49495 y Fo(+)14606 49897 y FC(\),)349 b(where)e(w)-31 b(e)349 b(ha)-31 b(v)g(e)349 b(tak)-31 b(en)349 b(Green's)e(function)i (symmetry)f(with)h(resp)31 b(ect)5535 51336 y(to)348 b(the)f(argumen)-31 b(t)348 b(in)-31 b(terc)g(hange)349 b(in)-31 b(to)348 b(accoun)-31 b(t.)349 b(Consequen)-31 b(tly)-92 b(,)349 b(the)e(square)f(brac)-31 b(k)g(et)348 b(has)f(to)5535 52775 y(v)-61 b(anish)370 b(and)f(w)-31 b(e)370 b(get)g(the)f(form)-31 b(ula)371 b(for)e(the)h(k)-31 b(ernel)369 b(with)i(t)-31 b(w)g(o)370 b Ft(\016)411 b FC(in)-31 b(teractions,)6273 55348 y Fr(G)6997 54816 y Fs(v)28 b(;w)6931 55665 y Fo(i)p Fs(\024)8447 55348 y FC(\()p Ft(x;)184 b(y)40 b FC(\))309 b(=)e Fr(G)13215 54891 y Fs(v)13149 55621 y Fo(i)p Fs(\024)13974 55348 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)23800 54599 y Ft(w)p 18028 55093 12368 45 v 18028 56107 a FC(1)f(+)g Ft(w)276 b Fr(\001)247 b(G)22280 55726 y Fs(v)22214 56424 y Fo(i)p Fs(\024)23039 56107 y FC(\()p Ft(d)f FC(+)g Ft(d)25974 55787 y Fo(3)26471 56107 y Ft(;)184 b(d)247 b FC(+)f Ft(d)29468 55787 y Fo(3)29965 56107 y FC(\))30897 55348 y Fr(G)31621 54891 y Fs(v)31555 55621 y Fo(i)p Fs(\024)32380 55348 y FC(\()p Ft(y)40 b(;)184 b(d)247 b FC(+)f Ft(d)36390 54891 y Fo(3)36886 55348 y FC(\))370 b Fr(G)38410 54891 y Fs(v)38344 55621 y Fo(i)p Fs(\024)39169 55348 y FC(\()p Ft(x;)184 b(d)247 b FC(+)f Ft(d)43229 54891 y Fo(3)43726 55348 y FC(\))184 b Ft(:)43110 57365 y FC(\(4.8\))7196 58804 y(The)324 b(remaining)h(step)e(will)j(b)31 b(e)323 b(more)h(complicated)i(b)31 b(ecause)323 b(w)-31 b(e)324 b(are)f(going)j(to)e(in)-31 b(tro)31 b(duce)5535 60243 y(a)469 b(coupling)h(b)31 b(et)-31 b(w)g(een)470 b(di\013eren)-31 b(t)469 b(hal\015ines)g(w)-31 b(orking)471 b(this)e(with)g(matrix-v)-61 b(alued)471 b(functions.)5535 61682 y(Our)369 b(to)31 b(ol)370 b(will)h(b)31 b(e)369 b(again)i(Krein's)e(form)-31 b(ula)371 b(whic)-31 b(h)370 b(no)-31 b(w)370 b(tak)-31 b(es)370 b(the)g(form)9011 64858 y Ft(R)9852 65098 y Fs(H)10636 64876 y Fj(u;)-17 b(~)-390 b(v)19 b(;)98 b(~)-505 b(w)12715 64858 y FC(\()p Ft(k)13756 64401 y Fo(2)14253 64858 y FC(\))308 b(=)g Ft(R)17001 65098 y Fs(H)17768 64876 y Fj(~)-390 b(v)19 b(;)98 b(~)-505 b(w)19142 64858 y FC(\()p Ft(k)20183 64401 y Fo(2)20680 64858 y FC(\))246 b(+)23206 63474 y Fs(n)22680 63806 y Fn(X)22463 66189 y Fs(j)m(;l)t Fo(=1)24679 64858 y Ft(\025)25325 65024 y Fs(j)45 b(l)26076 64858 y FC(\()p Ft(k)27117 64401 y Fo(2)27614 64858 y FC(\))28228 63629 y Fn(\020)28890 64858 y Ft(\036)29550 65024 y Fs(l)29890 64858 y FC(\()p 30320 63906 1108 45 v Ft(k)30931 64538 y Fo(2)31428 64858 y FC(\))p Ft(;)184 b Fr(\001)32656 63629 y Fn(\021)33319 65632 y Fs(L)33926 65411 y Fh(2)34358 65632 y Fo(\(\()p Fg(R)35610 65411 y Fh(+)36234 65632 y Fo(\))36580 65411 y Fj(n)37123 65632 y Fo(\))37770 64858 y Fr(\001)246 b Ft(\036)38983 65024 y Fs(j)39449 64858 y FC(\()p Ft(k)40490 64401 y Fo(2)40988 64858 y FC(\))184 b Ft(;)5535 68484 y FC(where)369 b(the)h(functions)g Ft(\036)16075 68650 y Fs(j)16541 68484 y FC(\()p Ft(k)17582 68082 y Fo(2)18079 68484 y FC(\))g(will)h(b)31 b(e)368 b(c)-31 b(hosen)369 b(as)15274 70650 y Fn(\000)15781 71547 y Ft(\036)16441 71713 y Fs(j)16907 71547 y FC(\()p Ft(k)17948 71090 y Fo(2)18446 71547 y FC(\)\()p Ft(x)p FC(\))20369 70650 y Fn(\001)20877 71990 y Fs(m)22025 71547 y FC(=)308 b Ft(\016)23686 71713 y Fs(j)45 b(m)25184 71547 y Fr(\001)26003 70798 y Ft(@)61 b Fr(G)27376 70266 y Fs(v)27819 70377 y Fj(m)28554 70266 y Fs(;w)29457 70377 y Fj(m)27310 71115 y Fo(i)p Fs(\024)30248 70798 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 26003 71292 6813 45 v 28794 72306 a Ft(@)61 b(y)32949 69942 y Fn(\014)32949 70606 y(\014)32949 71270 y(\014)32949 71934 y(\014)33318 72654 y Fs(y)28 b Fo(=0)35340 71547 y Ft(:)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(21)6614 12056 y FC(W)-92 b(e)377 b(apply)h(this)g(Ansatz)f(to)i (an)e(arbitrary)h(\011)321 b(=)g Fr(f)p Ft( )28749 12222 y Fo(1)29246 12056 y Ft(;)184 b(:)g(:)g(:)k(;)c( )32426 12222 y Fs(n)33030 12056 y Fr(g)33583 11654 y Fs(T)34657 12056 y FC(and)378 b(denote)g(the)f(elemen)-31 b(ts)6614 13495 y(of)370 b(the)f(resulting)h(v)-31 b(ector)370 b(as)f Ft(h)19662 13661 y Fs(j)20129 13495 y FC(,)g(explicitly)8022 16537 y Ft(h)8660 16703 y Fs(j)9126 16537 y FC(\()p Ft(x)p FC(\))308 b(=)11972 15030 y Fn(Z)13079 15304 y Fo(+)p Fq(1)12587 17545 y Fo(0)14881 16537 y Fr(G)15605 15937 y Fs(v)16048 16048 y Fj(j)16462 15937 y Fs(;w)17365 16048 y Fj(j)15539 16854 y Fo(i)p Fs(\024)17834 16537 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p Ft( )21122 16703 y Fs(j)21589 16537 y FC(\()p Ft(y)g FC(\)d)p Ft(y)409 b FC(+)10927 19816 y(+)12498 18432 y Fs(n)11972 18764 y Fn(X)12069 21147 y Fs(l)t Fo(=1)13756 19816 y Ft(\025)14402 19982 y Fs(j)45 b(l)15152 19816 y FC(\()p Ft(k)16193 19359 y Fo(2)16691 19816 y FC(\))17305 18309 y Fn(Z)18412 18583 y Fo(+)p Fq(1)17920 20825 y Fo(0)20480 19067 y Ft(@)61 b Fr(G)21853 18535 y Fs(v)22296 18652 y Fj(l)22617 18535 y Fs(;w)23520 18652 y Fj(l)21787 19384 y Fo(i)p Fs(\024)23897 19067 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 20480 19561 5986 45 v 22832 20575 a Ft(@)61 b(x)26598 18211 y Fn(\014)26598 18875 y(\014)26598 19539 y(\014)26598 20203 y(\014)26967 20923 y Fs(x)p Fo(=0)28830 19816 y Ft( )29551 19982 y Fs(l)29890 19816 y FC(\()p Ft(y)40 b FC(\))184 b(d)p Ft(y)287 b Fr(\001)33781 19067 y Ft(@)61 b Fr(G)35154 18467 y Fs(v)35597 18578 y Fj(j)36011 18467 y Fs(;w)36914 18578 y Fj(j)35088 19384 y Fo(i)p Fs(\024)37383 19067 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 33781 19561 6170 45 v 36250 20575 a Ft(@)61 b(y)40084 18211 y Fn(\014)40084 18875 y(\014)40084 19539 y(\014)40084 20203 y(\014)40453 20923 y Fs(y)28 b Fo(=0)42474 19816 y Ft(:)44189 18318 y FC(\(4.9\))6614 23039 y(where)286 b(w)-31 b(e)286 b(ha)-31 b(v)g(e)286 b(used)f(Green's)g(function)i(symmetry)f(and)g(the)g(fact)g (that)h(its)f(complex)h(conju-)6614 24478 y(gation)330 b(is)d(equiv)-61 b(alen)-31 b(t)330 b(to)e(switc)-31 b(hing)330 b(from)e Ft(\024)f FC(to)370 b(\026)-595 b Ft(\024)p FC(.)328 b(As)f(b)31 b(efore)327 b(the)h(functions)h Ft(h)40642 24644 y Fo(1)41138 24478 y Ft(;)184 b(h)42267 24644 y Fo(2)42765 24478 y Ft(;)g(:)g(:)g(:)j(;)d(h)45861 24644 y Fs(n)6614 25917 y FC(ha)-31 b(v)g(e)299 b(to)h(satisfy)f(the)f (b)31 b(oundary)299 b(conditions)h(expressing)e(the)h Ft(\016)340 b FC(coupling)300 b(in)e(the)h(star)f(cen)-31 b(tre,)17640 28257 y Ft(h)18278 28423 y Fo(1)18774 28257 y FC(\(0\))309 b(=)f Ft(h)22303 28423 y Fo(2)22799 28257 y FC(\(0\))h(=)e Fr(\001)184 b(\001)g(\001)310 b FC(=)e Ft(h)29095 28423 y Fs(n)29697 28257 y FC(\(0\))h(=:)f Ft(h)p FC(\(0\))184 b Ft(;)8199 b FC(\(4.10\))17261 30029 y Ft(h)17899 29572 y Fq(0)17899 30302 y Fo(1)18396 30029 y FC(\(0\))247 b(+)f Ft(h)21801 29572 y Fq(0)21801 30302 y Fo(2)22298 30029 y FC(\(0\))h(+)f Fr(\001)184 b(\001)g(\001)248 b FC(+)e Ft(h)28347 29572 y Fq(0)28347 30302 y Fs(n)28950 30029 y FC(\(0\))309 b(=)e Ft(u)246 b Fr(\001)g Ft(h)p FC(\(0\))184 b Ft(;)7821 b FC(\(4.11\))6614 32369 y(for)329 b(an)-31 b(y)329 b Ft( )11039 32535 y Fo(1)11536 32369 y Ft(;)184 b(:)g(:)g(:)k(;)c( )14716 32535 y Fs(n)15627 32369 y Fr(2)308 b Ft(L)17426 31967 y Fo(2)17922 32369 y FC(\()p Fl(R)19151 31967 y Fo(+)19887 32369 y FC(\).)330 b(Let)f(us)f(\014rst)g(express)g Ft(h)30978 32535 y Fs(j)31444 32369 y 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45 v 28778 72122 a Ft(@)61 b(x)32545 69757 y Fn(\014)32545 70422 y(\014)32545 71086 y(\014)32545 71750 y(\014)32914 72469 y Fs(x)p Fo(=0)34776 71362 y Ft( )35497 71528 y Fs(l)35837 71362 y FC(\()p Ft(y)40 b FC(\))184 b(d)p Ft(y)225 b(:)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(22)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 11955 y FC(No)-31 b(w)343 b(w)-31 b(e)342 b(can)g(\014nd)g(what)h(is)f(required)f(to)h(ful\014ll)i(the)e (conditions)h(\(4.10\))s(,)f(i.e.)h Ft(h)39136 12121 y Fs(j)39602 11955 y FC(\(0\))309 b(=)f Ft(h)43131 12121 y Fs(m)43971 11955 y FC(\(0\))5535 13394 y(for)370 b(all)g Ft(j)r(;)184 b(m)308 b Fr(2)363 b FC(^)-608 b Ft(n)p FC(.)369 b(This)h(is)f(true)g(pro)-31 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a Ft(@)61 b(y)38805 33882 y Fn(\014)38805 34546 y(\014)38805 35210 y(\014)38805 35874 y(\014)39174 36594 y Fs(y)28 b Fo(=0)40827 33594 y Fn(!)41703 33550 y(\014)41703 34214 y(\014)41703 34878 y(\014)41703 35542 y(\014)41703 36206 y(\014)42072 36926 y Fs(x)p Fo(=0)44119 35487 y Ft(;)5535 38803 y FC(The)370 b(b)31 b(oundary)369 b(condition)i(\(4.11\))h(then)e(requires)e(that)j(the)e(expression) 11246 41008 y Fs(n)10720 41340 y Fn(X)10817 43723 y Fs(l)t Fo(=1)12503 40885 y Fn(Z)13610 41159 y Fo(+)p Fq(1)13118 43400 y Fo(0)15412 40167 y Fn(0)15412 42159 y(@)16381 42392 y FC(1)246 b(+)g Ft(\025)18933 42558 y Fs(l)19983 41008 y(n)19457 41340 y Fn(X)19491 43698 y Fs(j)45 b Fo(=1)21822 41643 y FC(d)p 21506 42137 1248 45 v 21506 43151 a(d)p Ft(x)23071 40499 y Fn( )24213 41643 y Ft(@)61 b Fr(G)25586 41043 y Fs(v)26029 41154 y Fj(j)26443 41043 y Fs(;w)27346 41154 y Fj(j)25520 41960 y Fo(i)p Fs(\024)27815 41643 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 24213 42137 6170 45 v 26682 43151 a Ft(@)61 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b(and)g(this)h(in)f(turn)g(yields) 10271 52837 y Ft(\025)10917 53003 y Fs(l)11564 52837 y FC(=)12733 50612 y Fn(2)12733 52605 y(4)13471 52837 y Ft(u)246 b Fr(\000)15983 51454 y Fs(n)15457 51786 y Fn(X)15491 54143 y Fs(j)45 b Fo(=1)17823 52088 y FC(d)p 17507 52583 1248 45 v 17507 53597 a(d)p Ft(x)19072 50944 y Fn( )20214 52088 y Ft(@)61 b Fr(G)21587 51489 y Fs(v)22030 51600 y Fj(j)22443 51489 y Fs(;w)23346 51600 y Fj(j)21521 52405 y Fo(i)p Fs(\024)23815 52088 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 20214 52583 6170 45 v 22683 53597 a Ft(@)61 b(y)26516 51232 y Fn(\014)26516 51896 y(\014)26516 52561 y(\014)26516 53225 y(\014)26885 53944 y Fs(y)28 b Fo(=0)28538 50944 y Fn(!)29415 50900 y(\014)29415 51564 y(\014)29415 52228 y(\014)29415 52893 y(\014)29415 53557 y(\014)29783 54276 y Fs(x)p Fo(=0)31462 50612 y Fn(3)31462 52605 y(5)32200 50842 y Fq(\000)p Fo(1)35048 52837 y FC(for)370 b(all)g Ft(l)330 b Fr(2)363 b FC(^)-609 b Ft(n)5535 56365 y FC(sho)-31 b(wing,)511 b(in)f(particular,)h(that)f 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b(G)16107 65859 y Fs(v)16550 65970 y Fj(j)16964 65859 y Fs(;w)17867 65970 y Fj(j)16041 66776 y Fo(i)p Fs(\024)18336 66459 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b(+)24160 64167 y Fs(@)48 b Fq(G)25259 63614 y Fj(v)25659 63792 y(j)26071 63614 y(;w)26883 63792 y(j)25215 64426 y Fh(i)p Fj(\024)27352 64167 y Fo(\()p Fs(x;y)28 b Fo(\))p 24160 64515 5126 45 v 26224 65151 a Fs(@)48 b(y)29418 63165 y Fn(\014)29418 63829 y(\014)29418 64493 y(\014)29418 65157 y(\014)29787 65877 y Fs(y)28 b Fo(=0)31686 64770 y Fr(\001)32505 64167 y Fs(@)48 b Fq(G)33604 63706 y Fj(v)34004 63900 y(l)34324 63706 y(;w)35136 63900 y(l)33560 64426 y Fh(i)p Fj(\024)35513 64167 y Fo(\()p Fs(x;y)28 b Fo(\))p 32505 64515 4941 45 v 34465 65151 a Fs(@)48 b(x)37579 63497 y Fn(\014)37579 64161 y(\014)37579 64825 y(\014)37948 65544 y Fs(x)p Fo(=0)p 22390 66204 18741 45 v 22390 67942 a Ft(u)246 b Fr(\000)24377 67112 y Fn(P)25545 67386 y Fs(n)25545 68274 y(m)p Fo(=1)28208 67506 y(d)p 27957 67687 991 45 v 27957 68324 a(d)p Fs(x)29265 66381 y Fn(\022)30346 67339 y Fs(@)48 b Fq(G)31445 66962 y Fj(v)31845 67073 y(m)32578 66962 y(;w)33390 67073 y(m)31401 67599 y Fh(i)p Fj(\024)34181 67339 y Fo(\()p Fs(x;y)28 b Fo(\))p 30346 67688 5769 45 v 32731 68324 a Fs(@)48 b(y)36247 66669 y Fn(\014)36247 67333 y(\014)36247 67998 y(\014)36616 68717 y Fs(y)28 b Fo(=0)38269 66381 y Fn(\023)39084 66337 y(\014)39084 67001 y(\014)39084 67666 y(\014)39084 68330 y(\014)39453 69049 y Fs(x)p Fo(=0)41448 66459 y Ft(:)802 b FC(\(4.13\))5535 70923 y(The)388 b(\014rst)f(deriv)-61 b(ativ)-31 b(e)388 b(in)g(the)f(n)-31 b(umerator)389 b(w)-31 b(as)388 b(found)f(in)h(\(4.12\))i(and)d(b)-31 b(y)388 b(Green's)e(function)5535 72362 y(symmetry)489 b(the)f(other)h(one)f(is)g(giv)-31 b(en)489 b(b)-31 b(y)488 b(the)g(same)h(expression,)f(with)h Ft(y)527 b FC(replaced)488 b(b)-31 b(y)489 b Ft(x)p FC(.)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(23)6614 12730 y FC(The)425 b(same)f(relation)i(allo)-31 b(ws)427 b(us)c(to)j(compute)27054 12294 y Fo(d)p 26803 12475 991 45 v 26803 13112 a(d)p Fs(x)28111 11169 y Fn(\022)29192 12127 y Fs(@)48 b Fq(G)30291 11749 y Fj(v)30691 11860 y(m)31424 11749 y(;w)32236 11860 y(m)30247 12387 y Fh(i)p Fj(\024)33027 12127 y Fo(\()p Fs(x;y)28 b Fo(\))p 29192 12475 5769 45 v 31577 13112 a Fs(@)48 b(y)35093 11457 y Fn(\014)35093 12121 y(\014)35093 12785 y(\014)35462 13505 y Fs(y)28 b Fo(=0)37115 11169 y Fn(\023)37930 12730 y FC(,)424 b(in)h(particular,)h(to)6614 14661 y(ev)-61 b(aluate)371 b(the)e(quan)-31 b(tit)g(y)372 b(app)31 b(earing)370 b(in)g(the)f(square)g(brac)-31 b(k)g(et)370 b(ab)31 b(o)-31 b(v)g(e,)8488 17184 y(d)p 8171 17678 1248 45 v 8171 18692 a(d)p Ft(x)9736 16040 y Fn( )10878 17184 y Ft(@)61 b Fr(G)12251 16652 y Fs(v)12694 16763 y Fj(m)13429 16652 y Fs(;w)14332 16763 y Fj(m)12185 17501 y Fo(i)p Fs(\024)15123 17184 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 10878 17678 6813 45 v 13669 18692 a Ft(@)61 b(y)17824 16327 y Fn(\014)17824 16992 y(\014)17824 17656 y(\014)17824 18320 y(\014)18193 19039 y Fs(y)28 b Fo(=0)19846 16040 y Fn(!)20722 15995 y(\014)20722 16659 y(\014)20722 17324 y(\014)20722 17988 y(\014)20722 18652 y(\014)21091 19372 y Fs(x)p Fo(=0)23077 17932 y FC(=)307 b Fr(\000)p Ft(\024)246 b Fr(\000)31073 17184 y Ft(v)31610 17350 y Fs(m)p 27230 17678 9065 45 v 27230 18692 a FC(1)g(+)g Ft(v)29673 18858 y Fs(m)30760 18692 y Fr(\001)g(G)31971 18858 y Fo(i)p Fs(\024)32796 18692 y FC(\()p Ft(d)33802 18372 y Fo(3)34299 18692 y Ft(;)184 b(d)35366 18372 y Fo(3)35864 18692 y FC(\))36796 17932 y(e)37288 17475 y Fq(\000)p Fs(\024d)38959 17141 y Fh(3)40061 17932 y Fr(\001)247 b FC(e)41107 17475 y Fq(\000)p Fs(\024d)42778 17141 y Fh(3)16178 21087 y Fr(\000)23428 20338 y Ft(w)24220 20504 y Fs(m)p 17418 20833 13654 45 v 17418 21847 a FC(1)f(+)g Ft(w)20116 22013 y Fs(m)21204 21847 y Fr(\001)g(G)22481 21355 y Fs(v)22924 21466 y Fj(m)22415 22164 y Fo(i)p Fs(\024)23714 21847 y FC(\()p Ft(d)h FC(+)f Ft(d)26650 21527 y Fo(3)27147 21847 y Ft(;)184 b(d)247 b FC(+)f Ft(d)30144 21527 y Fo(3)30641 21847 y FC(\))31450 21087 y Fr(\001)32003 19859 y Fn(\020)32664 21087 y FC(e)33156 20630 y Fq(\000)p Fs(\024)p Fo(\()p Fs(d)p Fo(+)p Fs(d)36314 20296 y Fh(3)36746 20630 y Fo(\))20144 24105 y Fr(\000)25227 23356 y Ft(v)25764 23522 y Fs(m)p 21383 23851 9065 45 v 21383 24864 a FC(1)h(+)f Ft(v)23827 25030 y Fs(m)24913 24864 y Fr(\001)h(G)26125 25030 y Fo(i)p Fs(\024)26949 24864 y FC(\()p Ft(d)27955 24545 y Fo(3)28453 24864 y Ft(;)184 b(d)29520 24545 y Fo(3)30017 24864 y FC(\))30950 24105 y(e)31442 23648 y Fq(\000)p Fs(\024d)33113 23314 y Fh(3)33969 24105 y Fr(G)34627 24271 y Fo(i)p Fs(\024)35452 24105 y FC(\()p Ft(d)246 b FC(+)g Ft(d)38387 23648 y Fo(3)38884 24105 y Ft(;)184 b(d)39951 23648 y Fo(3)40449 24105 y FC(\))40879 22876 y Fn(\021)41540 23106 y Fo(2)42221 24105 y Ft(:)1108 b FC(\(4.14\))6614 26758 y(The)520 b(relations)h(\(4.13\))i(and)d(\(4.14\))j(together)e(with)g(\(4.12\))h (and)e(its)h(mirror)f(coun)-31 b(terpart)6614 28198 y(describ)31 b(e)368 b(completely)k(Green's)c(function)j Fr(G)25537 27666 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)25471 28515 y Fo(i)p Fs(\024)28142 28198 y FC(of)370 b(the)g(appro)-31 b(ximating)372 b(op)31 b(erators.)8275 29637 y(After)413 b(deriving)g(explicit)i(expressions)d(for)h(the)g(resolv)-31 b(en)g(t)414 b(w)-31 b(e)413 b(can)g(pass)f(to)i(our)e(prop)31 b(er)6614 31256 y(goal)390 b(whic)-31 b(h)389 b(is)e(to)i(pro)-31 b(v)g(e)388 b(that)h(the)g(matrix-v)-61 b(alued)390 b(k)-31 b(ernel)388 b Fr(G)33105 30724 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)33039 31573 y Fo(i)p Fs(\024)35729 31256 y FC(con)-31 b(v)g(erges)388 b(to)h Fr(G)42803 30724 y Fs(!)28 b(;)d(~)-479 b(\013;)44546 30519 y(~)44490 30724 y(\014)42737 31573 y Fo(i)p Fs(\024)45475 31256 y FC(as)6614 32695 y Ft(d)308 b Fr(!)f FC(0)9465 32861 y Fo(+)10570 32695 y FC(whic)-31 b(h)370 b(in)f(terms)g(of)h(their)g (comp)31 b(onen)-31 b(ts)370 b(can)f(b)31 b(e)369 b(written)i(as)17719 35034 y(lim)17256 35756 y Fs(d)p Fq(!)p Fo(0+)19904 33761 y Fn(\015)19904 34425 y(\015)19904 35089 y(\015)20519 35034 y Fr(G)21243 34502 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)21177 35368 y Fo(i)p Fs(\024;j)45 b(l)23726 35034 y Fr(\000)246 b(G)25557 34502 y Fs(!)28 b(;)d(~)-479 b(\013;)27300 34297 y(~)27244 34502 y(\014)25491 35368 y Fo(i)p Fs(\024;j)45 b(l)27841 33761 y Fn(\015)27841 34425 y(\015)27841 35089 y(\015)28456 35808 y Fs(L)29063 35587 y Fh(2)29495 35808 y Fo(\()p Fg(R)30401 35587 y Fh(+)31025 35808 y Fq(\002)p Fg(R)32277 35587 y Fh(+)32900 35808 y Fo(\))33609 35034 y FC(=)308 b(0)184 b Ft(:)6614 38083 y FC(Dep)31 b(ending)275 b(on)h(the)f(v)-61 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Ft(x;)184 b(y)40 b FC(\).)344 b(Let)e(us)6614 53930 y(start)370 b(with)g(expressions)f(whic)-31 b(h)370 b(app)31 b(ear)369 b(in)h(the)f(form)-31 b(ulae)371 b(rep)31 b(eatedly)-92 b(.)370 b(The)f(\014rst)g(one)h(is)16222 55932 y Ft(v)16759 56098 y Fs(m)p 12378 56427 V 12378 57441 a FC(1)247 b(+)f Ft(v)14822 57607 y Fs(m)15908 57441 y Fr(\001)h(G)17120 57607 y Fo(i)p Fs(\024)17944 57441 y FC(\()p Ft(d)18950 57121 y Fo(3)19448 57441 y Ft(;)184 b(d)20515 57121 y Fo(3)21012 57441 y FC(\))21883 56681 y(=)28023 55868 y Fr(\000)29243 55432 y Fo(1)p 29017 55613 893 45 v 29017 56250 a Fs(d)29478 56028 y Fh(3)30288 55868 y FC(+)31528 55414 y Fs(\013)32103 55525 y Fj(m)p 31528 55613 1311 45 v 31737 56250 a Fs(d)32198 56028 y Fh(2)p 23184 56427 14626 45 v 23184 57736 a FC(1)247 b(+)25091 56840 y Fn(\000)25598 57736 y Fr(\000)26818 57301 y Fo(1)p 26592 57482 893 45 v 26592 58118 a Fs(d)27053 57897 y Fh(3)27864 57736 y FC(+)29103 57283 y Fs(\013)29678 57394 y Fj(m)p 29103 57482 1311 45 v 29312 58118 a Fs(d)29773 57897 y Fh(2)30547 56840 y Fn(\001)31300 57736 y Fr(\001)31986 57301 y Fo(sinh)151 b Fs(\024d)34696 57079 y Fh(3)35329 57301 y Fo(e)35723 57079 y Ff(\000)p Fj(\024d)37190 56916 y Fh(3)p 31986 57482 5691 45 v 34573 58118 a Fs(\024)38251 56681 y FC(=)307 b(\()p Fr(\003)p FC(\))6614 59812 y(Using)370 b(sinh\()p Ft(x)p FC(\))308 b(=)g Ft(x)245 b FC(+)h Fr(O)31 b FC(\()p Ft(x)18630 59410 y Fo(2)19127 59812 y FC(\))369 b(and)h(e)22571 59410 y Fs(x)23436 59812 y FC(=)307 b(1)247 b(+)f Fr(O)31 b FC(\()p Ft(x)p FC(\))369 b(w)-31 b(e)370 b(get)11767 61968 y(sinh)185 b Ft(\024d)15140 61566 y Fo(3)15882 61968 y FC(e)16374 61566 y Fq(\000)p Fs(\024d)18045 61232 y Fh(3)p 11767 62462 6766 45 v 14831 63476 a Ft(\024)18973 62717 y FC(=)20274 61968 y(\()p Ft(\024d)21918 61566 y Fo(3)22661 61968 y FC(+)246 b Fr(O)31 b FC(\()p Ft(d)25686 61566 y Fo(6)26184 61968 y FC(\)\)\(1)248 b(+)d Fr(O)31 b FC(\()p Ft(d)31299 61566 y Fo(3)31797 61968 y FC(\)\))p 20274 62462 12384 45 v 26147 63476 a Ft(\024)33098 62717 y FC(=)307 b Ft(d)34842 62260 y Fo(3)35339 62717 y FC(\(1)247 b(+)f Fr(O)31 b FC(\()p Ft(d)39594 62260 y Fo(3)40091 62717 y FC(\)\))184 b Ft(;)6614 65094 y FC(and)370 b(this)f(in)h(turn)f (allo)-31 b(ws)371 b(us)e(to)g(express)f(\(*\))j(as)e(follo)-31 b(ws,)9205 67762 y(\()p Fr(\003)p FC(\))309 b(=)e Fr(\000)13349 67014 y FC(1)p 13089 67508 1073 45 v 13089 68522 a Ft(d)13665 68202 y Fo(3)14541 67762 y Fr(\001)21319 67014 y FC(1)247 b Fr(\000)f Ft(\013)23934 67180 y Fs(m)24775 67014 y Ft(d)p 15227 67508 16217 45 v 15227 68582 a FC(1)h(+)17134 67685 y Fn(\000)17641 68582 y Fr(\000)18861 68146 y Fo(1)p 18635 68327 893 45 v 18635 68963 a Fs(d)19096 68742 y Fh(3)19906 68582 y FC(+)21146 68128 y Fs(\013)21721 68239 y Fj(m)p 21146 68327 1311 45 v 21355 68963 a Fs(d)21816 68742 y Fh(2)22590 67685 y Fn(\001)23343 68582 y Fr(\001)f FC(\()p Ft(d)24902 68262 y Fo(3)25400 68582 y FC(\(1)h(+)f Fr(O)31 b FC(\()p Ft(d)29655 68262 y Fo(3)30152 68582 y FC(\)\)\))31884 67762 y(=)307 b Fr(\000)34305 67014 y FC(1)p 34046 67508 1073 45 v 34046 68522 a Ft(d)34622 68202 y Fo(4)35497 67762 y Fr(\001)36051 66202 y Fn(\022)37496 67014 y FC(1)p 36999 67508 1549 45 v 36999 68522 a Ft(\013)37707 68688 y Fs(m)38926 67762 y FC(+)246 b Fr(O)31 b FC(\()p Ft(d)p FC(\))42381 66202 y Fn(\023)43566 67762 y Ft(:)6614 70923 y FC(The)455 b(next)g(frequen)-31 b(t)455 b(expression)f(is)g Ft(w)23385 71089 y Fs(m)24411 70026 y Fn(\000)24918 70923 y FC(1)247 b(+)f Ft(w)27617 71089 y Fs(m)28704 70923 y Fr(\001)h(G)29982 70431 y Fs(v)30425 70542 y Fj(m)29916 71240 y Fo(i)p Fs(\024)31215 70923 y FC(\()p Ft(d)g FC(+)f Ft(d)34151 70521 y Fo(3)34647 70923 y Ft(;)184 b(d)247 b FC(+)f Ft(d)37644 70521 y Fo(3)38141 70923 y FC(\))38571 70026 y Fn(\001)39079 70255 y Fq(\000)p Fo(1)40267 70923 y FC(.)455 b(W)-92 b(e)454 b(emplo)-31 b(y)6614 72362 y(relation)344 b(\(4.5\))g(with)f Ft(v)k FC(=)307 b Ft(v)18349 72528 y Fs(m)19531 72362 y FC(and)342 b(the)g(expansion)h(e)29141 71960 y Fs(x)30006 72362 y FC(=)307 b(1)192 b(+)f Ft(x)g FC(+)g Fr(O)31 b FC(\()p Ft(x)36822 71960 y Fo(2)37319 72362 y FC(\))342 b(together)h(with)g(the)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(24)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 11900 y FC(explicit)371 b(form)f(of)g Fr(G)13976 12066 y Fo(i)p Fs(\024)14801 11900 y FC(;)f(this)h(yields)f (after)h(a)g(straigh)-31 b(tforw)g(ard)371 b(computation)5535 15113 y Fr(G)6259 14621 y Fs(v)6702 14732 y Fj(m)6193 15430 y Fo(i)p Fs(\024)7493 15113 y FC(\()p Ft(d)80 b FC(+)g Ft(d)10096 14656 y Fo(3)10593 15113 y Ft(;)184 b(d)80 b FC(+)g Ft(d)13257 14656 y Fo(3)13755 15113 y FC(\))308 b(=)g Ft(d)16422 13552 y Fn(\022)17237 15113 y FC(1)247 b Fr(\000)f Ft(\024d)g Fr(\000)22330 14364 y Ft(d)p 21844 14858 1549 45 v 21844 15872 a(\013)22552 16038 y Fs(m)23771 15113 y FC(+)g Fr(O)31 b FC(\()p Ft(d)26796 14656 y Fo(2)27294 15113 y FC(\))27724 13552 y Fn(\023)28846 15113 y FC(=)308 b Fr(G)30673 15279 y Fo(i)p Fs(\024)31497 15113 y FC(\()p Ft(d)80 b FC(+)g Ft(d)34100 14656 y Fo(3)34598 15113 y Ft(;)184 b(d)80 b FC(+)g Ft(d)37262 14656 y Fo(3)37760 15113 y FC(\))g Fr(\000)39583 14364 y Ft(d)40159 13962 y Fo(2)p 39345 14858 V 39345 15872 a Ft(\013)40053 16038 y Fs(m)41107 15113 y FC(+)g Fr(O)31 b FC(\()p Ft(d)43966 14656 y Fo(3)44463 15113 y FC(\))184 b Ft(;)5535 18281 y FC(and)370 b(therefore)16322 20818 y Ft(w)17114 20984 y Fs(m)p 10312 21312 13654 45 v 10312 22326 a FC(1)247 b(+)f Ft(w)13011 22492 y Fs(m)14098 22326 y Fr(\001)g(G)15375 21834 y Fs(v)15818 21945 y Fj(m)15309 22643 y Fo(i)p Fs(\024)16609 22326 y FC(\()p Ft(d)g FC(+)g Ft(d)19544 22006 y Fo(3)20041 22326 y Ft(;)184 b(d)247 b FC(+)f Ft(d)23038 22006 y Fo(3)23535 22326 y FC(\))24406 21567 y(=)307 b Fr(\000)26828 20818 y FC(1)p 26568 21312 1073 45 v 26568 22326 a Ft(d)27144 22006 y Fo(2)27958 19674 y Fn( )31884 20818 y FC(1)p 28967 21312 6387 45 v 28967 22380 a Ft(\014)29593 22546 y Fs(m)30680 22380 y FC(+)246 b Ft(\024)g Fr(\000)34345 21944 y Fo(1)p 33911 22125 1311 45 v 33911 22762 a Fs(\013)34486 22873 y Fj(m)35733 21567 y FC(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))39188 19674 y Fn(!)40434 21567 y Ft(:)5535 25387 y FC(No)-31 b(w)430 b(w)-31 b(e)429 b(can)g(expand)g(the)g(\014rst)f(term)h(in)f Fr(G)24888 24855 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)24822 25721 y Fo(i)p Fs(\024;j)45 b(l)27125 25387 y FC(\()p Ft(x;)184 b(y)40 b FC(\).)430 b(Using)f(\(4.8\))i(for)d (the)h(parameters)5535 26826 y Ft(v)347 b FC(=)307 b Ft(v)8124 26992 y Fs(j)8590 26826 y Ft(;)370 b(w)338 b FC(=)307 b Ft(w)12357 26992 y Fs(j)13193 26826 y FC(together)370 b(with)h(the)e(previous)h(result)f(w)-31 b(e)369 b(get)8152 30122 y Fr(G)8876 29522 y Fs(v)9319 29633 y Fj(j)9732 29522 y Fs(;w)10635 29633 y Fj(j)8810 30439 y Fo(i)p Fs(\024)11104 30122 y FC(\()p Ft(x;)184 b(y)40 b FC(\))309 b(=)15281 29373 y(sinh)185 b Ft(\024x)245 b FC(e)19448 28971 y Fq(\000)p Fs(\024y)p 15281 29867 5910 45 v 17917 30881 a Ft(\024)8398 33261 y FC(+)9897 32512 y(1)p 9637 33006 1073 45 v 9637 34020 a Ft(d)10213 33700 y Fo(2)11027 31368 y Fn( )14953 32512 y FC(1)p 12037 33006 6387 45 v 12037 34074 a Ft(\014)12663 34240 y Fs(m)13750 34074 y FC(+)h Ft(\024)f Fr(\000)17415 33638 y Fo(1)p 16980 33819 1311 45 v 16980 34456 a Fs(\013)17555 34567 y Fj(m)18802 33261 y FC(+)h Fr(O)31 b FC(\()p Ft(d)p FC(\))22257 31368 y Fn(!)23821 32512 y FC(sinh)185 b Ft(\024)p FC(\()p Ft(d)246 b FC(+)g Ft(d)29553 32110 y Fo(3)30050 32512 y FC(\))184 b(e)31156 32110 y Fq(\000)p Fs(\024y)p 23821 33006 9078 45 v 28041 34020 a Ft(\024)33534 32512 y FC(sinh)g Ft(\024)p FC(\()p Ft(d)247 b FC(+)f Ft(d)39266 32110 y Fo(3)39762 32512 y FC(\))184 b(e)40868 32110 y Fq(\000)p Fs(\024x)p 33534 33006 9104 45 v 37766 34020 a Ft(\024)8459 36718 y FC(=)9760 35969 y(sinh)h Ft(\024x)245 b FC(e)13927 35567 y Fq(\000)p Fs(\024y)p 9760 36463 5910 45 v 12396 37477 a Ft(\024)16048 36718 y FC(+)20205 35969 y(1)p 17288 36463 6387 45 v 17288 37531 a Ft(\014)17914 37697 y Fs(m)19001 37531 y FC(+)h Ft(\024)g Fr(\000)22666 37095 y Fo(1)p 22232 37276 1311 45 v 22232 37913 a Fs(\013)22807 38024 y Fj(m)24177 36718 y FC(e)24669 36261 y Fq(\000)p Fs(\024x)26805 36718 y FC(e)27297 36261 y Fq(\000)p Fs(\024y)29285 35821 y Fn(\000)29793 36718 y FC(1)g(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))34047 35821 y Fn(\001)5535 40202 y FC(As)277 b(for)f(the)h(second)g(term)g(in)g(\(\(4.13\))s(\),)h(w)-31 b(e)277 b(\014rst)f(expand)h(the)g(deriv)-61 b(ativ)-31 b(e)279 b(in)d(the)h(denominator)5535 41641 y(using)370 b Fr(G)9090 41807 y Fo(i)p Fs(\024)9915 41641 y FC(\()p Ft(d)246 b FC(+)g Ft(d)12850 41239 y Fo(3)13347 41641 y Ft(;)184 b(d)14414 41239 y Fo(3)14912 41641 y FC(\))308 b(=)f Ft(d)17394 41239 y Fo(3)17891 41641 y FC(\(1)247 b(+)f Fr(O)31 b FC(\()p Ft(d)p FC(\)\))370 b(and)g(\(4.14\).)i(A)d (direct)h(computation)i(yields)9780 44454 y(d)p 9463 44948 1248 45 v 9463 45962 a(d)p Ft(x)11028 43310 y Fn( )12170 44454 y Ft(@)61 b Fr(G)13543 43922 y Fs(v)13986 44033 y Fj(m)14722 43922 y Fs(;w)15625 44033 y Fj(m)13477 44771 y Fo(i)p Fs(\024)16415 44454 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 12170 44948 6813 45 v 14961 45962 a Ft(@)61 b(y)19116 43598 y Fn(\014)19116 44262 y(\014)19116 44926 y(\014)19116 45590 y(\014)19485 46310 y Fs(y)28 b Fo(=0)21138 43310 y Fn(!)22014 43266 y(\014)22014 43930 y(\014)22014 44594 y(\014)22014 45258 y(\014)22014 45923 y(\014)22383 46642 y Fs(x)p Fo(=0)24369 45203 y FC(=)25930 44454 y(1)p 25670 44948 1073 45 v 25670 45962 a Ft(d)26246 45643 y Fo(4)27122 45203 y Fr(\001)27675 43642 y Fn(\022)30781 44454 y Ft(\014)31407 44620 y Fs(m)32494 44454 y FC(+)246 b Ft(\024)p 28623 44948 7775 45 v 28623 45962 a(\013)29331 46128 y Fs(m)30172 45962 y FC(\()p Ft(\014)31228 46128 y Fs(m)32315 45962 y FC(+)g Ft(\024)p FC(\))h Fr(\000)f FC(1)36776 45203 y(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))40231 43642 y Fn(\023)41416 45203 y Ft(;)5535 48759 y FC(and)370 b(therefore)5535 50377 y Fn( )6412 52270 y Ft(u)245 b Fr(\000)9078 50886 y Fs(n)8552 51218 y Fn(X)8398 53563 y Fs(m)p Fo(=1)11071 51521 y FC(d)p 10755 52015 1248 45 v 10755 53029 a(d)p Ft(x)12320 50377 y Fn( )13462 51521 y Ft(@)61 b Fr(G)14835 50989 y Fs(v)15278 51100 y Fj(m)16013 50989 y Fs(;w)16916 51100 y Fj(m)14769 51838 y Fo(i)p Fs(\024)17707 51521 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 13462 52015 6813 45 v 16252 53029 a Ft(@)61 b(y)20407 50665 y Fn(\014)20407 51329 y(\014)20407 51993 y(\014)20407 52657 y(\014)20776 53377 y Fs(y)28 b Fo(=0)22429 50377 y Fn(!)23306 50333 y(\014)23306 50997 y(\014)23306 51661 y(\014)23306 52325 y(\014)23306 52989 y(\014)23675 53709 y Fs(x)p Fo(=0)25353 50377 y Fn(!)26229 50606 y Fq(\000)p Fo(1)27725 52270 y FC(=)307 b Ft(d)29469 51813 y Fo(4)30151 50377 y Fn( )36499 51521 y FC(1)p 31160 52015 11233 45 v 31160 53180 a Ft(!)286 b Fr(\000)33242 52350 y Fn(P)34410 52624 y Fs(n)34410 53512 y(m)p Fo(=1)38257 52686 y Fs(\014)38759 52797 y Fj(m)39494 52686 y Fo(+)p Fs(\024)p 36689 52925 5571 45 v 36689 53562 a(\013)37264 53673 y Fj(m)38000 53562 y Fo(\()p Fs(\014)38848 53673 y Fj(m)39582 53562 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)42771 52270 y FC(+)246 b Fr(O)31 b FC(\()p Ft(d)p FC(\))46226 50377 y Fn(!)47472 52270 y Ft(:)5535 55872 y FC(Next)399 b(w)-31 b(e)398 b(expand)g(the)g(deriv)-61 b(ativ)-31 b(es)398 b(whic)-31 b(h)399 b(app)31 b(ear)398 b(in)f(the)h(n)-31 b(umerator)399 b(using)f(the)g(relation)5535 57311 y Fr(G)6259 56819 y Fs(v)6702 56930 y Fj(m)6193 57628 y Fo(i)p Fs(\024)7493 57311 y FC(\()p Ft(x;)184 b(d)247 b FC(+)f Ft(d)11553 56910 y Fo(3)12050 57311 y FC(\))308 b(=)f Ft(d)p FC(\(1)247 b(+)f Fr(O)31 b FC(\()p Ft(d)p FC(\)\))184 b(e)20323 56910 y Fq(\000)p Fs(\024x)22093 57311 y FC(;)369 b(it)h(giv)-31 b(es)7092 60174 y Ft(@)61 b Fr(G)8465 59642 y Fs(v)8908 59753 y Fj(m)9644 59642 y Fs(;w)10547 59753 y Fj(m)8399 60491 y Fo(i)p Fs(\024)11337 60174 y FC(\()p Ft(x;)184 b(y)40 b FC(\))p 7092 60668 6813 45 v 9883 61682 a Ft(@)61 b(y)14038 59318 y Fn(\014)14038 59982 y(\014)14038 60646 y(\014)14038 61311 y(\014)14407 62030 y Fs(y)28 b Fo(=0)16367 60923 y FC(=)308 b(e)18028 60466 y Fq(\000)p Fs(\024x)20041 60923 y Fr(\000)25125 60174 y Ft(v)25662 60340 y Fs(m)p 21281 60668 9065 45 v 21281 61682 a FC(1)247 b(+)f Ft(v)23725 61848 y Fs(m)24811 61682 y Fr(\001)g(G)26022 61848 y Fo(i)p Fs(\024)26847 61682 y FC(\()p Ft(d)27853 61363 y Fo(3)28350 61682 y Ft(;)184 b(d)29417 61363 y Fo(3)29915 61682 y FC(\))30847 60923 y(e)31339 60466 y Fq(\000)p Fs(\024d)33010 60132 y Fh(3)33867 60923 y Fr(G)34525 61089 y Fo(i)p Fs(\024)35349 60923 y FC(\()p Ft(x;)g(d)37479 60466 y Fo(3)37977 60923 y FC(\))p Fr(\000)17732 63849 y(\000)24982 63100 y Ft(w)25774 63266 y Fs(m)p 18972 63594 13654 45 v 18972 64608 a FC(1)247 b(+)f Ft(w)21671 64774 y Fs(m)22758 64608 y Fr(\001)g(G)24035 64116 y Fs(v)24478 64227 y Fj(m)23969 64925 y Fo(i)p Fs(\024)25269 64608 y FC(\()p Ft(d)g FC(+)g Ft(d)28204 64289 y Fo(3)28701 64608 y Ft(;)184 b(d)247 b FC(+)f Ft(d)31698 64289 y Fo(3)32195 64608 y FC(\))33127 63849 y Fr(\001)25280 66973 y FC(=)26841 66224 y(1)p 26581 66719 1073 45 v 26581 67733 a Ft(d)27157 67413 y Fo(2)27971 65412 y Fn(\022)32529 66224 y FC(1)p 28919 66719 7775 45 v 28919 67733 a Ft(\013)29627 67899 y Fs(m)30468 67733 y FC(\()p Ft(\014)31524 67899 y Fs(m)32612 67733 y FC(+)g Ft(\024)p FC(\))g Fr(\000)g FC(1)37072 66973 y(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))40527 65412 y Fn(\023)41527 66973 y FC(e)42019 66516 y Fq(\000)p Fs(\024x)43971 66973 y Ft(:)5535 70646 y FC(and)401 b(the)g(analogous)i(expression)d(for)22156 70043 y Fs(@)48 b Fq(G)23255 69665 y Fj(v)23655 69776 y(m)24389 69665 y(;w)25201 69776 y(m)23211 70303 y Fh(i)p Fj(\024)25991 70043 y Fo(\()p Fs(x;y)28 b Fo(\))p 22156 70391 5769 45 v 24529 71028 a Fs(@)48 b(x)28058 69373 y Fn(\014)28058 70037 y(\014)28058 70701 y(\014)28427 71421 y Fs(x)p Fo(=0)30505 70646 y FC(with)402 b Ft(x)e FC(replaced)h(b)-31 b(y)401 b Ft(y)40 b FC(.)400 b(This)h(de-)5535 72362 y(termines)499 b(the)g(b)31 b(eha)-31 b(viour)500 b(of)g(the)f(second)f(term)h(at)h(the)f(righ)-31 b(t-hand)500 b(side)f(of)g(\(\(4.13\))t(\))g(as)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(25)6614 12214 y Ft(d)308 b Fr(!)f FC(0)9465 12380 y Fo(+)10201 12214 y FC(,)369 b(and)h(for)f(the)g(full)h(k)-31 b(ernel)370 b Fr(G)22554 11682 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)22488 12548 y Fo(i)p Fs(\024;j)45 b(l)24790 12214 y FC(\()p Ft(x;)184 b(y)40 b FC(\))370 b(w)-31 b(e)370 b(consequen)-31 b(tly)371 b(ha)-31 b(v)g(e)7906 19418 y Fr(G)8630 18886 y Fs(u;)k(~)-427 b(v)28 b(;)116 b(~)-570 b(w)8564 19752 y Fo(i)p Fs(\024;j)45 b(l)10866 19418 y FC(\()p Ft(x;)184 b(y)40 b FC(\))309 b(=)e Ft(\016)15402 19584 y Fs(j)45 b(l)16337 17525 y Fn( )17347 18669 y FC(sinh)184 b Ft(\024x)369 b FC(e)21637 18267 y Fq(\000)p Fs(\024y)p 17347 19163 6033 45 v 20044 20177 a Ft(\024)23758 19418 y FC(+)25715 18669 y(1)247 b(+)e Fr(O)31 b FC(\()p Ft(d)p FC(\))p 24997 19163 5691 45 v 24997 20231 a Ft(\014)25623 20397 y Fs(j)26336 20231 y FC(+)246 b Ft(\024)g Fr(\000)29840 19795 y Fo(1)p 29566 19976 990 45 v 29566 20613 a Fs(\013)30141 20724 y Fj(j)31067 19418 y FC(e)31559 18961 y Fq(\000)p Fs(\024x)33696 19418 y FC(e)34188 18961 y Fq(\000)p Fs(\024y)35930 17525 y Fn(!)8152 23211 y FC(+)9259 21318 y Fn( )15607 22462 y FC(1)p 10268 22957 11233 45 v 10268 24121 a Ft(!)285 b Fr(\000)12349 23291 y Fn(P)13518 23565 y Fs(n)13518 24453 y(m)p Fo(=1)17365 23627 y Fs(\014)17867 23738 y Fj(m)18602 23627 y Fo(+)p Fs(\024)p 15797 23867 5571 45 v 15797 24503 a(\013)16372 24614 y Fj(m)17107 24503 y Fo(\()p Fs(\014)17955 24614 y Fj(m)18690 24503 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)21879 23211 y Fr(\001)25801 22462 y FC(1)p 22565 22957 7026 45 v 22565 23970 a Ft(\013)23273 24136 y Fs(j)23740 23970 y FC(\()p Ft(\014)24796 24136 y Fs(j)25509 23970 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1)29969 23211 y Fr(\001)33765 22462 y FC(1)p 30655 22957 6772 45 v 30655 23970 a Ft(\013)31363 24136 y Fs(l)31703 23970 y FC(\()p Ft(\014)32759 24136 y Fs(l)33346 23970 y FC(+)g Ft(\024)p FC(\))g Fr(\000)g FC(1)37806 23211 y(+)g Fr(O)31 b FC(\()p Ft(d)p FC(\))41261 21318 y Fn(!)42323 23211 y FC(e)42815 22754 y Fq(\000)p Fs(\024x)44583 23211 y FC(e)45075 22754 y Fq(\000)p Fs(\024y)47001 23211 y Ft(:)6614 29916 y FC(On)369 b(the)h(other)f(hand,)h(for)f Ft(x)307 b Fr(\024)h Ft(y)408 b FC(w)-31 b(e)370 b(ha)-31 b(v)g(e)6614 36211 y Fr(G)7338 35679 y Fs(!)28 b(;)d(~)-479 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Fj(m)21914 40484 y Fo(+)p Fs(\024)p Fo(\))p Fq(\000)p Fo(1)25103 39192 y Fr(\001)33272 38444 y FC(1)p 25789 38938 15520 45 v 25789 39952 a(\()p Ft(\013)26927 40118 y Fs(j)27394 39952 y FC(\()p Ft(\014)28450 40118 y Fs(j)29163 39952 y FC(+)246 b Ft(\024)p FC(\))g Fr(\000)g FC(1\)\()p Ft(\013)34812 40118 y Fs(l)35153 39952 y FC(\()p Ft(\014)36209 40118 y Fs(l)36796 39952 y FC(+)g Ft(\024)p FC(\))g Fr(\000)g FC(1\))41810 39192 y(e)42302 38735 y Fq(\000)p Fs(\024x)44069 39192 y FC(e)44561 38735 y Fq(\000)p Fs(\024y)46488 39192 y Ft(;)6614 45897 y FC(hence)369 b(the)g(Green)g(function)h (di\013erence)f(satis\014es)12633 51676 y Fr(G)13357 51144 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)13291 52009 y Fo(i)p Fs(\024;j)45 b(l)15594 51676 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)20239 51144 y Fs(!)28 b(;)d(~)-479 b(\013;)21982 50939 y(~)21926 51144 y(\014)20173 52009 y Fo(i)p Fs(\024;j)45 b(l)22523 51676 y FC(\()p Ft(x;)184 b(y)40 b FC(\))308 b(=)g Fr(O)31 b FC(\()p Ft(d)p FC(\))247 b(e)29654 51218 y Fq(\000)p Fs(\024x)31421 51676 y FC(e)31913 51218 y Fq(\000)p Fs(\024y)34762 51676 y FC(as)615 b Ft(d)308 b Fr(!)f FC(0)39218 51842 y Fo(+)40138 51676 y Ft(:)6614 57454 y FC(The)385 b(same)h(estimate)h (is)e(ob)-31 b(viously)387 b(v)-61 b(alid)386 b(also)g(for)f Ft(d)334 b(<)g(y)373 b(<)334 b(x)p FC(,)385 b(hence)g(there)g(is)g(a)g (constan)-31 b(t)6614 58893 y Ft(K)448 b FC(indep)31 b(enden)-31 b(t)370 b(of)g Ft(d;)g(x)e FC(and)i Ft(y)408 b FC(suc)-31 b(h)369 b(that)15944 63579 y Fn(\014)15944 64243 y(\014)15944 64907 y(\014)16313 64852 y Fr(G)17037 64320 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)16971 65186 y Fo(i)p Fs(\024;j)45 b(l)19273 64852 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)23918 64320 y Fs(!)28 b(;)d(~)-479 b(\013;)25661 64115 y(~)25605 64320 y(\014)23852 65186 y Fo(i)p Fs(\024;j)45 b(l)26202 64852 y FC(\()p Ft(x;)184 b(y)40 b FC(\))28769 63579 y Fn(\014)28769 64243 y(\014)28769 64907 y(\014)29447 64852 y Ft(<)307 b(K)264 b(d)246 b FC(e)33133 64395 y Fq(\000)p Fs(\024x)34901 64852 y FC(e)35393 64395 y Fq(\000)p Fs(\024y)43636 64852 y FC(\(4.15\))6614 70923 y(holds)373 b(for)g(all)g Ft(d)313 b(<)g FC(1)p Ft(;)434 b(x)312 b Fr(\025)h Ft(d)249 b FC(+)f Ft(d)20738 70521 y Fo(3)21607 70923 y FC(and)373 b Ft(y)352 b Fr(\025)313 b Ft(d)248 b FC(+)g Ft(d)28341 70521 y Fo(3)28838 70923 y FC(.)373 b(No)-31 b(w)373 b(w)-31 b(e)373 b(are)g(in)f(p)31 b(osition)374 b(to)g(estimate)6614 72362 y(the)386 b(Hilb)31 b(ert-Sc)-31 b(hmidt)387 b(norm)f(of)g(the)f(resolv)-31 b(en)g(t)387 b(di\013erence)e(for)g(the)h(op)31 b(erators)385 b Ft(H)42012 71960 y Fs(!)28 b(;)d(~)-479 b(\013;)43755 71756 y(~)43699 71960 y(\014)44681 72362 y FC(and)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 5535 9083 a FB(26)973 b Fy(P.)340 b(Exner)h(and)f(O.)f(T)-73 b(ur)-49 b(ek)5535 12083 y Ft(H)6545 11682 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)8782 12083 y FC(\()p Ft(d)p FC(\))370 b(whic)-31 b(h)370 b(can)f(b)31 b(e)369 b(written)i(explicitly)g(as)e(follo)-31 b(ws,)5659 14296 y Fn(\015)5659 14961 y(\015)6274 15237 y Ft(R)7115 15477 y Fs(H)7899 15256 y Fj(u;)-17 b(~)-390 b(v)19 b(;)98 b(~)-505 b(w)9922 15477 y Fo(\()p Fs(d)p Fo(\))11130 15237 y FC(\()p Ft(k)12171 14780 y Fo(2)12669 15237 y FC(\))p Fr(\000)450 b Ft(R)15251 15623 y Fs(H)16035 15402 y Fj(!)20 b(;)26 b(~)-433 b(\013)o(;)17603 15256 y(~)17559 15402 y(\014)18135 14296 y Fn(\015)18135 14961 y(\015)18750 14570 y Fo(2)18750 15680 y(2)19554 15237 y FC(=)21465 13854 y Fs(n)20939 14186 y Fn(X)20723 16569 y Fs(j)m(;l)t Fo(=1)22939 13731 y Fn(Z)24046 14004 y Fo(+)p Fq(1)23554 16246 y Fo(0)25847 13731 y Fn(Z)26954 14004 y Fo(+)p Fq(1)26462 16246 y Fo(0)28756 13964 y Fn(\014)28756 14629 y(\014)28756 15293 y(\014)29125 15237 y Fr(G)29849 14705 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)29783 15571 y Fo(i)p Fs(\024;j)45 b(l)32086 15237 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)36731 14705 y Fs(!)28 b(;)d(~)-479 b(\013;)38474 14501 y(~)38418 14705 y(\014)36665 15571 y Fo(i)p Fs(\024;j)45 b(l)39015 15237 y FC(\()p Ft(x;)184 b(y)40 b FC(\))41582 13964 y Fn(\014)41582 14629 y(\014)41582 15293 y(\014)41952 14218 y Fo(2)42817 15237 y FC(d)p Ft(x)p FC(d)p Ft(y)10893 19101 y FC(=)12804 17717 y Fs(n)12278 18049 y Fn(X)12061 20432 y Fs(j)m(;l)t Fo(=1)14277 17540 y Fn(\022)15092 17594 y(Z)16199 17868 y Fo(+)p Fq(1)15707 20110 y Fs(d)p Fo(+)p Fs(d)17309 19888 y Fh(3)18001 17594 y Fn(Z)19108 17868 y Fo(+)p Fq(1)18616 20110 y Fs(d)p Fo(+)p Fs(d)20218 19888 y Fh(3)20910 17828 y Fn(\014)20910 18492 y(\014)20910 19156 y(\014)21279 19101 y Fr(G)22003 18569 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)21937 19435 y Fo(i)p Fs(\024;j)45 b(l)24239 19101 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)28884 18569 y Fs(!)28 b(;)d(~)-479 b(\013;)30627 18365 y(~)30571 18569 y(\014)28818 19435 y Fo(i)p Fs(\024;j)45 b(l)31168 19101 y FC(\()p Ft(x;)184 b(y)40 b FC(\))33735 17828 y Fn(\014)33735 18492 y(\014)33735 19156 y(\014)34105 18081 y Fo(2)34971 19101 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 23060 y FC(+)15446 21554 y Fn(Z)16553 21827 y Fs(d)p Fo(+)p Fs(d)18155 21493 y Fh(3)16061 24069 y Fs(d)16522 23848 y Fh(3)18826 21554 y Fn(Z)19933 21827 y Fo(+)p Fq(1)19441 24069 y Fs(d)p Fo(+)p Fs(d)21043 23848 y Fh(3)21735 21787 y Fn(\014)21735 22451 y(\014)21735 23116 y(\014)22104 23060 y Fr(G)22828 22528 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)22762 23394 y Fo(i)p Fs(\024;j)45 b(l)25064 23060 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)29709 22528 y Fs(!)28 b(;)d(~)-479 b(\013;)31452 22324 y(~)31396 22528 y(\014)29643 23394 y Fo(i)p Fs(\024;j)45 b(l)31993 23060 y FC(\()p Ft(x;)184 b(y)40 b FC(\))34560 21787 y Fn(\014)34560 22451 y(\014)34560 23116 y(\014)34930 22041 y Fo(2)35796 23060 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 26528 y FC(+)15446 25021 y Fn(Z)16553 25295 y Fo(+)p Fq(1)16061 27536 y Fs(d)p Fo(+)p Fs(d)17663 27315 y Fh(3)18355 25021 y Fn(Z)19462 25295 y Fs(d)p Fo(+)p Fs(d)21064 24961 y Fh(3)18970 27536 y Fs(d)19431 27315 y Fh(3)21735 25255 y Fn(\014)21735 25919 y(\014)21735 26583 y(\014)22104 26528 y Fr(G)22828 25996 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)22762 26862 y Fo(i)p Fs(\024;j)45 b(l)25064 26528 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)29709 25996 y Fs(!)28 b(;)d(~)-479 b(\013;)31452 25791 y(~)31396 25996 y(\014)29643 26862 y Fo(i)p Fs(\024;j)45 b(l)31993 26528 y FC(\()p Ft(x;)184 b(y)40 b FC(\))34560 25255 y Fn(\014)34560 25919 y(\014)34560 26583 y(\014)34930 25508 y Fo(2)35796 26528 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 29995 y FC(+)15446 28488 y Fn(Z)16553 28762 y Fs(d)17014 28428 y Fh(3)16061 31004 y Fo(0)17685 28488 y Fn(Z)18792 28762 y Fo(+)p Fq(1)18300 31004 y Fs(d)p Fo(+)p Fs(d)19902 30782 y Fh(3)20594 28722 y Fn(\014)20594 29386 y(\014)20594 30050 y(\014)20963 29995 y Fr(G)21687 29463 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)21621 30329 y Fo(i)p Fs(\024;j)45 b(l)23924 29995 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)28569 29463 y Fs(!)28 b(;)d(~)-479 b(\013;)30312 29259 y(~)30256 29463 y(\014)28503 30329 y Fo(i)p Fs(\024;j)45 b(l)30853 29995 y FC(\()p Ft(x;)184 b(y)40 b FC(\))33420 28722 y Fn(\014)33420 29386 y(\014)33420 30050 y(\014)33790 28975 y Fo(2)34655 29995 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 33463 y FC(+)15446 31956 y Fn(Z)16553 32229 y Fo(+)p Fq(1)16061 34471 y Fs(d)p Fo(+)p Fs(d)17663 34250 y Fh(3)18355 31956 y Fn(Z)19462 32229 y Fs(d)19923 31895 y Fh(3)18970 34471 y Fo(0)20594 32189 y Fn(\014)20594 32854 y(\014)20594 33518 y(\014)20963 33463 y Fr(G)21687 32931 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)21621 33796 y Fo(i)p Fs(\024;j)45 b(l)23924 33463 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)28569 32931 y Fs(!)28 b(;)d(~)-479 b(\013;)30312 32726 y(~)30256 32931 y(\014)28503 33796 y Fo(i)p Fs(\024;j)45 b(l)30853 33463 y FC(\()p Ft(x;)184 b(y)40 b FC(\))33420 32189 y Fn(\014)33420 32854 y(\014)33420 33518 y(\014)33790 32443 y Fo(2)34655 33463 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 36930 y FC(+)15446 35423 y Fn(Z)16553 35697 y Fs(d)p Fo(+)p Fs(d)18155 35363 y Fh(3)16061 37938 y Fs(d)16522 37717 y Fh(3)18826 35423 y Fn(Z)19933 35697 y Fs(d)p Fo(+)p Fs(d)21535 35363 y Fh(3)19441 37938 y Fs(d)19902 37717 y Fh(3)22206 35657 y Fn(\014)22206 36321 y(\014)22206 36985 y(\014)22575 36930 y Fr(G)23299 36398 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)23233 37264 y Fo(i)p Fs(\024;j)45 b(l)25535 36930 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)30180 36398 y Fs(!)28 b(;)d(~)-479 b(\013;)31923 36193 y(~)31867 36398 y(\014)30114 37264 y Fo(i)p Fs(\024;j)45 b(l)32464 36930 y FC(\()p Ft(x;)184 b(y)40 b FC(\))35031 35657 y Fn(\014)35031 36321 y(\014)35031 36985 y(\014)35401 35910 y Fo(2)36267 36930 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 40305 y FC(+)15446 38798 y Fn(Z)16553 39072 y Fs(d)17014 38738 y Fh(3)16061 41314 y Fo(0)17685 38798 y Fn(Z)18792 39072 y Fs(d)p Fo(+)p Fs(d)20394 38738 y Fh(3)18300 41314 y Fs(d)18761 41092 y Fh(3)21065 39032 y Fn(\014)21065 39696 y(\014)21065 40360 y(\014)21434 40305 y Fr(G)22158 39773 y Fs(u;)-27 b(~)-427 b(v)28 b(;)116 b(~)-570 b(w)22092 40639 y Fo(i)p Fs(\024;j)45 b(l)24395 40305 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)29040 39773 y Fs(!)28 b(;)d(~)-479 b(\013;)30783 39569 y(~)30727 39773 y(\014)28974 40639 y Fo(i)p Fs(\024;j)45 b(l)31324 40305 y FC(\()p Ft(x;)184 b(y)40 b FC(\))33891 39032 y Fn(\014)33891 39696 y(\014)33891 40360 y(\014)34261 39285 y Fo(2)35127 40305 y FC(d)p Ft(x)p FC(d)p Ft(y)14339 43680 y FC(+)15446 42173 y Fn(Z)16553 42447 y Fs(d)p Fo(+)p Fs(d)18155 42113 y Fh(3)16061 44689 y Fs(d)16522 44467 y Fh(3)18826 42173 y Fn(Z)19933 42447 y Fs(d)20394 42113 y Fh(3)19441 44689 y Fo(0)21065 42407 y Fn(\014)21065 43071 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Fo(i)p Fs(\024;j)45 b(l)30193 47069 y FC(\()p Ft(x;)184 b(y)40 b FC(\))32760 45796 y Fn(\014)32760 46460 y(\014)32760 47124 y(\014)33130 46049 y Fo(2)33996 47069 y FC(d)p Ft(x)p FC(d)p Ft(y)36442 45176 y Fn(!)37686 47069 y Ft(:)5535 50332 y FC(The)370 b(inequalit)-31 b(y)387 b(\(4.15\))372 b(mak)-31 b(es)370 b(it)g(p)31 b(ossible)369 b(to)h(estimate)h(the)f(\014rst)e(one)i(of)g (the)f(in)-31 b(tegrals,)6827 52422 y Fn(Z)7934 52695 y Fo(+)p Fq(1)7442 54937 y Fs(d)p Fo(+)p Fs(d)9044 54716 y Fh(3)9736 52422 y Fn(Z)10843 52695 y Fo(+)p Fq(1)10351 54937 y Fs(d)p Fo(+)p Fs(d)11953 54716 y Fh(3)12644 52655 y Fn(\014)12644 53320 y(\014)12644 53984 y(\014)13013 53928 y Fr(G)13737 53396 y Fs(u;)k(~)-427 b(v)28 b(;)116 b(~)-570 b(w)13671 54262 y Fo(i)p Fs(\024;j)45 b(l)15974 53928 y FC(\()p Ft(x;)184 b(y)40 b FC(\))247 b Fr(\000)f(G)20619 53396 y Fs(!)28 b(;)d(~)-479 b(\013;)22362 53192 y(~)22306 53396 y(\014)20553 54262 y Fo(i)p Fs(\024;j)45 b(l)22903 53928 y FC(\()p Ft(x;)184 b(y)40 b 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44744 72362 643 45 v eop end %%Page: 27 27 TeXDict begin 27 26 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 b(17:18)i(WSPC/INSTR)-31 b(UCTION)371 b(FILE)2214 b(v)-31 b(ertapp)p 0 155 34 2657 v -3421 952 2657 34 v 52764 952 V 51967 155 34 2657 v 0 82406 V -3421 78952 2657 34 v 51967 82406 34 2657 v 52797 78952 2657 34 v 21946 9083 a Fy(Appr)-49 b(oximations)339 b(of)h(vertex)g(c) -49 b(ouplings)341 b(in)f(quantum)g(gr)-49 b(aphs)972 b FB(27)6614 11900 y Fz(5.)637 b(Appro)-35 b(ximations)425 b(with)g(added)g(edges)6614 13892 y FC(W)-92 b(e)396 b(ha)-31 b(v)g(e)397 b(seen)f(that)h(a)g(CS{t)-31 b(yp)31 b(e)398 b(sc)-31 b(heme)396 b(can)h(pro)31 b(duce)396 b(a)g(2)p Ft(n)p FC(-parameter)i(family)h(of)e(\(self-)6614 15331 y(adjoin)-31 b(t\))476 b(couplings)f(out)f(of)g(the)g(whole)g (set)f(dep)31 b(ending)474 b(on)g Ft(n)34119 14929 y Fo(2)35089 15331 y FC(real)g(n)-31 b(um)g(b)31 b(ers.)473 b(T)-92 b(o)474 b(get)g(a)6614 16770 y(wider)366 b(class)f(w)-31 b(e)366 b(ha)-31 b(v)g(e)366 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FC(\(\000)246 b Fr(n)g FC(\()p Fr(f)p Ft(c)p Fr(g)i([)e Ft(V)34659 33146 y Fs(d)35175 32980 y FC(\)\))q Fp(,)355 b(wher)-57 b(e)355 b Ft(c)g Fp(is)g(the)f(c)-57 b(entr)g(e)6614 34419 y(of)357 b FC(\000)p Fp(,)g(and)f Ft(V)12020 34585 y Fs(d)12893 34419 y Fp(is)g(the)h(set)f(of)h(the)f (vertic)-57 b(es)357 b(adde)-57 b(d)357 b(on)f(the)h(hal\015ines,)f (satis\014es)g(the)h(c)-57 b(onditions)6614 35858 y(\(2.4\))535 b(with)g Ft(d)p Fp(-dep)-57 b(endent)537 b(p)-57 b(ar)g(ameters,)536 b(and)f(that)g(it)h(c)-57 b(onver)g(ges)537 b(to)e FC(\011)561 b Fr(2)f Ft(W)40895 35456 y Fo(2)p Fs(;)p Fo(2)42280 35858 y FC(\()q(\000)246 b Fr(n)g(f)p Ft(c)p Fr(g)p FC(\))6614 37297 y Fp(which)431 b(ob)-57 b(eys)433 b(the)e(c)-57 b(ondition)433 b(\(2.1\))e(with)g(some)g Ft(A;)370 b(B)486 b Fp(satisfying)433 b(the)e(r)-57 b(e)g(quir)g(ements)434 b(\(2.2\).)6614 38736 y(The)350 b(family)i(of)e(the)f(c)-57 b(onditions)351 b(\(2.1\))f(which)f(c)-57 b(an)351 b(b)-57 b(e)351 b(obtaine)-57 b(d)351 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b(orts)552 b(either)h(a)f Ft(\016)594 b FC(in)-31 b(teraction)555 b(or)d(a)h Ft(\016)6614 47075 y FC(coupling)328 b(at)f(the)f(endp)31 b(oin)-31 b(t)328 b(of)e(an)h(added)f(edge.)h(W)-92 b(e)325 b(arrange)i(the)f(function)i(v)-61 b(alues)326 b(at)h(these)6614 48514 y(p)31 b(oin)-31 b(ts)283 b(in)-31 b(to)284 b(the)e Ft(n)p FC(-tuple)i(\011\()p Ft(d)p FC(\),)f(and)f(similarly)j(\011) 27863 48112 y Fq(0)28173 48514 y FC(\()p Ft(d)29179 48680 y Fo(+)29915 48514 y FC(\))d(is)g(the)h Ft(n)p FC(-tuple)g(of)g(righ) -31 b(t)283 b(deriv)-61 b(ativ)-31 b(es.)6614 49953 y(Let)419 b(us)e(stress)g(that)j(this)e(a)h(sym)-31 b(b)31 b(olic)420 b(notation;)h(the)e(elemen)-31 b(ts)419 b(are)f Ft( )37397 50119 y Fs(j)37864 49953 y FC(\()p Ft(d)38870 50119 y Fs(j)39337 49953 y FC(\))g(and)h Ft( )43148 49551 y Fq(0)43108 50241 y Fs(j)43574 49953 y FC(\()p Ft(d)44580 50119 y Fs(j)45 b Fo(+)45727 49953 y FC(\),)6614 51392 y(resp)31 b(ectiv)-31 b(ely)-92 b(.)8275 52831 y(As)401 b(in)h(the)g(pro)31 b(of)402 b(of)g(Prop)31 b(osition)404 b(3.1)f(w)-31 b(e)402 b(can)g(use)f(\(2.4\))j(to)e(express)f(these)g(quan)-31 b(tities)6614 54270 y(through)370 b(the)g(common)g(v)-61 b(alue)370 b Ft( )40 b FC(\(0\))370 b(and)g(the)f(righ)-31 b(t)370 b(deriv)-61 b(ativ)-31 b(es)371 b(\011)35649 53868 y Fq(0)35959 54270 y FC(\(0)36942 54436 y Fo(+)37678 54270 y FC(\))f(at)f(the)h(origin)14939 56250 y Ft(M)16013 56416 y Fo(1)16509 56250 y FC(\()p Ft(d)p FC(\)\011\()p Ft(d)p FC(\))335 b(=)d Ft( )40 b FC(\(0\))247 b Fr(\001)f Ft(m)25716 56416 y Fo(2)26213 56250 y FC(\()p Ft(d)p FC(\))h(+)f Ft(M)30077 56416 y Fo(3)30573 56250 y FC(\()p Ft(d)p FC(\)\011)32870 55793 y Fq(0)33182 56250 y FC(\(0\))h(+)f Fr(R)p FC(\()p Ft(d)p FC(\))184 b Ft(;)14078 58021 y(N)14967 58187 y Fo(1)15464 58021 y FC(\()p Ft(d)p FC(\)\011)17761 57564 y Fq(0)18072 58021 y FC(\()p Ft(d)19078 58187 y Fo(+)19814 58021 y FC(\))333 b(=)f Ft( )40 b FC(\(0\))247 b Fr(\001)f Ft(n)25408 58187 y Fo(2)25905 58021 y FC(\()p Ft(d)p FC(\))h(+)f Ft(N)29584 58187 y Fo(3)30081 58021 y FC(\()p Ft(d)p FC(\)\011)32378 57564 y Fq(0)32690 58021 y FC(\(0\))h(+)35742 57741 y(~)35457 58021 y Fr(R)q FC(\()p Ft(d)p FC(\))6614 60001 y(for)399 b(some)g Ft(M)12215 60167 y Fo(1)12712 60001 y Ft(;)184 b(M)14277 60167 y Fo(3)14774 60001 y Ft(;)g(N)16154 60167 y Fo(1)16652 60001 y Ft(;)g(N)18032 60167 y Fo(3)18887 60001 y FC(:)358 b Fl(R)20351 59600 y Fo(+)21443 60001 y Fr(!)f Fl(R)23706 59600 y Fs(n;n)25120 60001 y FC(,)665 b Ft(m)27064 60167 y Fo(2)27560 60001 y Ft(;)184 b(n)28715 60167 y Fo(2)29570 60001 y FC(:)358 b Fl(R)31034 59600 y Fo(+)32126 60001 y Fr(!)f Fl(R)34389 59600 y Fs(n)35391 60001 y FC(and)399 b(error)f(terms)h Fr(R)p Ft(;)45147 59721 y FC(~)44861 60001 y Fr(R)358 b FC(:)6614 61440 y Fl(R)7413 61039 y Fo(+)8464 61440 y Fr(!)316 b Fl(R)10686 61039 y Fs(n)11663 61440 y FC(supp)31 b(osed)373 b(to)i(b)31 b(e)374 b(negligible)j(as)d Ft(d)316 b Fr(!)g FC(0)28473 61606 y Fo(+)29208 61440 y FC(;)374 b(w)-31 b(e)375 b(ma)-31 b(y)375 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y Fq(0)31947 66193 y FC(\()p Ft(d)32953 66359 y Fo(+)33689 66193 y FC(\))31360 67632 y Ft( )40 b FC(\(0\))31154 69071 y(\011)32015 68669 y Fq(0)32326 69071 y FC(\(0\))34285 64012 y Fn(1)34285 65960 y(C)34285 66624 y(C)34285 67333 y(A)35561 66901 y FC(=)36730 64676 y Fn(0)36730 66669 y(@)37864 65473 y Ft(o)p FC(\(1\))37864 66912 y Ft(o)p FC(\(1\))38563 68351 y(0)39981 64676 y Fn(1)39981 66669 y(A)6614 70923 y FC(T)-92 b(o)348 b(\014nd)e(an)i(appro)-31 b(ximation)350 b(in)e(the)f(describ)31 b(ed)346 b(sense)g(one)h(has)g(to)h(\014nd)f(a) g(relation)i(b)31 b(et)-31 b(w)g(een)6614 72362 y(\011\()p Ft(d)p FC(\))405 b(and)e(\011)12363 71960 y Fq(0)12674 72362 y FC(\()p Ft(d)13680 72528 y Fo(+)14415 72362 y FC(\))h(eliminating)j Ft( )40 b FC(\(0\))p Ft(;)371 b FC(\011)24777 71960 y Fq(0)25088 72362 y FC(\(0\).)405 b(Since)f(the)f(former)h(are)f(determined)i(b)-31 b(y)403 b(the)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 664 -122 a FC(Marc)-31 b(h)369 b(16,)i(2007)1108 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b(ely)-92 b(.)7196 46344 y(T)g(o)416 b(construct)g(appro)-31 b(ximation)420 b(of)c(the)g(corresp)31 b(onding)416 b(op)31 b(erator)416 b Ft(H)37159 45942 y Fs(A;B)39266 46344 y FC(w)-31 b(e)416 b(ha)-31 b(v)g(e)417 b(\014nd)5535 47783 y(suitable)j(family)h(of)f(graphs)18148 47503 y(~)18079 47783 y(\000\()p Ft(d)p FC(\).)g(The)f(decomp)31 b(osition)422 b(of)d(the)g(matrix)h(in)f(\(5.1\))i(in)-31 b(to)420 b(the)5535 49222 y(diagonal)372 b(and)e(o\013-diagonal)i(part) d(inspires)g(the)g(follo)-31 b(wing)373 b(sc)-31 b(heme:)8413 51438 y Fr(\017)554 b FC(the)390 b(cen)-31 b(tre)389 b(of)g(\000)g(supp)31 b(orts)388 b(a)i Ft(\016)430 b FC(coupling)391 b(with)f(the)f(parameter)h Ft(u)p FC(\()p Ft(d)p FC(\))g(the)f(dep)31 b(en-)9520 52877 y(dence)369 b(of)h(whic)-31 b(h)370 b(on)g Ft(d)f FC(will)i(b)31 b(e)368 b(sp)31 b(eci\014ed)369 b(b)31 b(elo)-31 b(w)8413 54316 y Fr(\017)554 b FC(at)383 b(eac)-31 b(h)382 b(edge)f(of)h(\000)f (w)-31 b(e)382 b(place)h(a)e Ft(\016)423 b FC(coupling)384 b(at)e(the)g(distance)g Ft(d)f FC(from)h(the)g(cen)-31 b(tre;)9520 55755 y(the)331 b(corresp)31 b(onding)331 b(parameter)g Ft(v)24258 55921 y Fs(j)24724 55755 y FC(\()p Ft(d)p FC(\),)h(to)f(b)31 b(e)330 b(again)i(sp)31 b(eci\014ed,)331 b(will)h(b)31 b(e)330 b(related)h(to)9520 57194 y(the)370 b(diagonal)i(elemen)-31 b(t)370 b Ft(D)20860 57360 y Fs(j)45 b(j)22107 57194 y FC(of)370 b(the)f(matrix)i Ft(D)8413 58633 y Fr(\017)554 b FC(the)521 b(pairs)f(of)h(edges)f (whose)g(indices)h Ft(j)r(;)184 b(k)556 b FC(corresp)31 b(ond)519 b(to)i(nonzero)g(elemen)-31 b(ts)521 b(of)9520 60073 y(the)501 b(matrix)g Ft(S)563 b FC(w)-31 b(e)501 b(join)g(b)-31 b(y)500 b(an)g(additional)j(edge,)e(whose)f(endp)31 b(oin)-31 b(ts)501 b(are)f(the)g Ft(\016)9520 61512 y FC(coupling)312 b(sites)d(men)-31 b(tioned)312 b(ab)31 b(o)-31 b(v)g(e,)311 b(and)f(in)g(the)f(middle)i(of)f(this)g(edge)g(w) -31 b(e)310 b(place)g(the)9520 62951 y Ft(\016)411 b FC(in)-31 b(teraction)372 b(with)e(a)g(parameter)g Ft(w)25523 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