This is a multi-part message in MIME format. ---------------0610060234948 Content-Type: text/plain; name="06-281.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-281.keywords" QED, Spectral Analysis, Renormalization Group, Ground State ---------------0610060234948 Content-Type: application/postscript; name="BachKoenenberg 10-06.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BachKoenenberg 10-06.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: F:/eigene Artikel/Endversion/gesendet/BachKoenenberg-06-08-03.dvi %%CreationDate: Fri Oct 06 09:11:00 2006 %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic CMSY10 CMMI10 CMR8 %%+ CMR10 CMMI8 CMSY8 Courier CMMI12 CMR12 MSBM7 CMEX10 MSBM10 EUFM10 %%+ CMBX12 CMSY6 CMMI6 LASY10 CMR6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\texmf\MiKTeX\bin\dvips.exe %+ "F:/eigene Artikel/Endversion/gesendet/BachKoenenberg-06-08-03.dvi" %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.10.06:0910 %%BeginProcSet: texc.pro %! 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put dup 49 /infinity put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 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 104 /h put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 77 /circleplusdisplay put dup 79 /circlemultiplydisplay put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 102 /tildewider put dup 103 /tildewidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 113 /radicalBig 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 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 59 /emptyset put dup 66 /B put dup 70 /F put dup 72 /H put dup 79 /O put dup 83 /S put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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D9ED5F3DDE9F57FF1B9AAE19AE28 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 78 /N put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 10 /Omega put dup 20 /caron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 73 /I put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g 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 115 /s put dup 116 /t put 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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 74 /J put dup 76 /L put dup 80 /P put dup 81 /Q put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 107 /k put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 122 /z put dup 123 /dotlessi put dup 126 /vector put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 80 /P put dup 101 /e put dup 102 /f put dup 104 /h put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 122 /z put dup 123 /dotlessi put dup 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fp(\(II.4\))427 2416 y Fj(\000)p Fm(b)18 b Fl(:=)f(inf)23 b Fm(\033)t Fl([\()p Fj(\000)p Fl(\001)1099 2431 y Ft(x)1144 2416 y Fm(=)p Fl(4\))f(+)g Fm(V)f Fl(\()p Fm(x)p Fl(\)])75 b Fm(>)g Fj(\0001)22 b Fm(:)1158 b Fp(\(II.5\))166 2647 y(The)25 b(operator)996 2780 y Fl(\024)971 2806 y Fm(H)60 b Fl(=)1256 2723 y Fi(X)1241 2907 y Ft(\025)p Fv(=)p Fs(\006)1408 2688 y Fi(Z)1524 2806 y Fm(!)t Fl(\()1622 2779 y Fm(~)1627 2806 y(k)s(;)17 b(\025)p Fl(\))g Fm(a)1888 2764 y Fs(\003)1926 2806 y Fl(\()1959 2779 y Fm(~)1964 2806 y(k)s(;)g(\025)p Fl(\))p Fm(a)p Fl(\()2241 2779 y Fm(~)2246 2806 y(k)s(;)g(\025)p Fl(\))g Fm(d)2507 2764 y Fv(3)2545 2806 y Fm(k)26 b(;)597 b Fp(\(II.6\))166 3068 y(with)31 b Fm(!)t Fl(\()473 3042 y Fm(~)478 3068 y(k)r(;)17 b(\025)p Fl(\))40 b(:=)g Fj(j)876 3042 y Fm(~)881 3068 y(k)s Fj(j)p Fp(,)31 b(is)g(the)h(Hamiltonian)d (of)j(the)f(free)i(quantized)e(radiation)g(\002eld,)g(rep-)166 3198 y(resenting)d(its)g(ener)n(gy)-6 b(.)1017 3173 y Fm(~)999 3198 y(A)29 b 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y(ground)e(state)h(projection)f Fm(P)1173 1475 y Ft(t;s)1279 1460 y Fp(of)h Fm(H)1468 1475 y Ft(t;s)1575 1460 y Fp(is)f(gi)n(v)o(en)f(by)1423 1675 y Fm(P)1486 1690 y Ft(t;s)1620 1647 y Fj(\030)1621 1679 y Fl(=)1774 1642 y Fi(e)1750 1675 y Fm(P)1813 1690 y Ft(t)1865 1675 y Fj(\012)g Fm(P)2028 1690 y Fv(\012)2079 1671 y Fc(t)2079 1706 y Fd(1)2170 1675 y Fm(;)999 b Fp(\(II.34\))166 1904 y(where)463 1871 y Fi(e)440 1904 y Fm(P)503 1919 y Ft(t)563 1904 y Fp(is)30 b(the)g(ground)g(state)g(projection)g(of)1895 1871 y Fi(f)1886 1904 y Fm(H)1967 1919 y Ft(t)1997 1904 y Fp(,)g(and)h Fm(P)2290 1919 y Fv(\012)2341 1900 y Fc(t)2341 1936 y Fd(1)2440 1904 y Fp(is)f(the)h(projection)e(onto)h(the)166 2024 y(v)n(acuum)24 b(in)g Fj(F)10 b Fl([)p Ff(h)767 1988 y Ft(t)767 2049 y Fs(1)842 2024 y Fl(])p Fp(.)24 b(In)h(particular)l(,)1403 2239 y Fm(P)1466 2254 y Ft(t;s)1601 2239 y Fl(=)52 b Fm(P)1792 2254 y Ft(t;t)1919 2239 y Fl(=:)h Fm(P)2138 2254 y Ft(t)2189 2239 y Fm(;)980 b Fp(\(II.35\))166 2454 y(for)32 b(all)f Fm(s)40 b(>)g(t)h Fj(\025)f Fl(0)p Fp(.)32 b(Similarly)-6 b(,)30 b(we)h(de\002ne)h Fm(E)1861 2469 y Ft(t;s)1984 2454 y Fl(:=)40 b(inf)23 b Fm(\033)t Fl([)p Fm(H)2428 2469 y Ft(t;s)2510 2454 y Fl(])32 b Fp(to)f(be)h(the)f(ground)g(state)166 2574 y(ener)n(gy)25 b(of)g Fm(H)649 2589 y Ft(t;s)756 2574 y Fp(and)g(observ)o(e)f(that)1389 2789 y Fm(E)1461 2804 y Ft(t;s)1596 2789 y Fl(=)52 b Fm(E)1796 2804 y Ft(t;t)1924 2789 y Fl(=:)g Fm(E)2151 2804 y Ft(t)2203 2789 y Fm(;)966 b Fp(\(II.36\))166 3004 y(for)25 b(all)g Fm(s)i(>)h(t)g Fj(\025)g Fl(0)p Fp(.)c(W)-8 b(e)26 b(are)f(no)n(w)f(in)h(position)e (to)h(formulate)g(our)h(main)f(result.)166 3224 y Fq(Theor)n(em)j(II.2) 48 b Fh(Assume)20 b(that)h Fm(\013)1343 3188 y Fv(3)1410 3224 y Fj(\024)28 b Fl(\(2500)p Fm(#)1806 3188 y Fv(2)1855 3224 y Fl(+)10 b Fm(b)g Fl(+)g(6)p Fm(\024)2183 3188 y Fv(2)2232 3224 y Fl(+)g Fm(e)2363 3239 y Fv(0)2402 3224 y Fl(\))2440 3188 y Fs(\000)p Fv(1)2535 3224 y Fh(,)21 b(wher)l(e)i Fm(#)28 b Fl(:=)f(\001)p 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4025 y Fh(wher)l(e)32 b Fm(H)526 3984 y Fv(+)518 4045 y Ft(t)624 4025 y Fl(:=)39 b Fm(H)847 4040 y Ft(t)903 4025 y Fj(\000)28 b Fm(E)1080 4040 y Ft(t)1141 4025 y Fh(and)1360 4000 y Fl(_)1321 4025 y Fm(W)1413 4040 y Ft(t)1482 4025 y Fl(:=)39 b Fm(v)t Fl(\()1728 4000 y Fm(~)1713 4025 y(G)1790 4040 y Ft(t)1820 4025 y Fl(\))26 b Fj(\001)h Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)2111 4040 y Ft(t)2155 4000 y Fm(~)2141 4025 y(G)p Fl(\))26 b(+)h Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)2557 4040 y Ft(t)2601 4000 y Fm(~)2587 4025 y(G)p Fl(\))26 b Fj(\001)h Fm(v)t Fl(\()2887 4000 y Fm(~)2872 4025 y(G)2949 4040 y Ft(t)2978 4025 y Fl(\))p Fh(,)k(and)g(ther)l(e)166 4145 y(e)n(xists)24 b(a)h(constant)f Fm(C)941 4155 y Fv(_)918 4172 y Ft(P)1004 4145 y Fm(<)j Fj(1)e Fh(suc)o(h)f(that)1651 4120 y Fl(_)1618 4145 y Fm(P)1681 4160 y Ft(t)1735 4145 y Fh(obe)m(ys)h(the)g(following)e(norm)h(estimate)o(,)1345 4376 y Fj(k)1428 4351 y Fl(_)1395 4376 y Fm(P)1458 4391 y Ft(t)1487 4376 y Fj(k)53 b(\024)g Fm(C)1813 4386 y 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y(struct)28 b(a)i(ground)e(state)h Fm(\036)1071 5395 y Fv(gs)1173 5380 y Fl(=)35 b(lim)1420 5395 y Ft(t)p Fs(!1)1607 5380 y Fm(\036)1665 5395 y Ft(t)1694 5380 y Fp(,)29 b(where)h Fm(P)2084 5395 y Fv(gs)2186 5380 y Fl(=)35 b Fj(j)p Fm(\036)2383 5395 y Fv(gs)2450 5380 y Fj(ih)p Fm(\036)2586 5395 y Fv(gs)2653 5380 y Fj(j)28 b Fp(and)h Fm(P)2945 5395 y Ft(t)3010 5380 y Fl(=)35 b Fj(j)p Fm(\036)3207 5395 y Ft(t)3236 5380 y Fj(ih)p Fm(\036)3372 5395 y Ft(t)3401 5380 y Fj(j)p Fp(,)1773 5712 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 166 83 a Fp(b)n(ut)21 b(we)h(do)g(not)f(sho)n(w)g (its)g(uniqueness.)f(The)i(proof)g(of)f(uniqueness,)g(ho)n(we)n(v)o(er) l(,)f(can)i(be)g(found)166 203 y(in)31 b([5].)h(Another)f(f)o(act)h(we) g(use)g(is)f(the)g(boundedness)g(of)g Fj(k)n Fm(~)-47 b(x)q(\036)2375 218 y Ft(t)2404 203 y Fj(k)31 b Fp(uniformly)g(in)g Fm(t)40 b Fj(\025)h Fl(0)p Fp(.)31 b(W)-8 b(e)166 324 y(refer)26 b(the)f(reader)g(again)g(to)f([5])h(for)g(a)g(proof)g(of)g (this)f(bound.)166 547 y(Eqs.)g(\(II.37\))g(and)g(\(II.38\))h(are)g (stated)f(and)g(pro)o(v)o(ed)f(separately)h(as)g(Theorems)g(V)-13 b(.3)24 b(and)g(VI.1,)166 668 y(belo)n(w)-6 b(.)23 b(The)o(y)i(imply)e (that)1060 924 y Fm(P)1123 939 y Fs(1)1251 924 y Fl(:=)68 b(lim)1406 979 y Ft(t)p Fs(!1)1589 924 y Fm(P)1652 939 y Ft(t)1734 924 y Fl(=)53 b Fm(P)1926 939 y Fv(0)2010 924 y Fl(+)2130 807 y Fi(Z)2213 833 y Fs(1)2176 995 y Fv(0)2337 899 y Fl(_)2304 924 y Fm(P)2367 939 y Ft(t)2424 924 y Fm(dt)22 b(;)637 b Fp(\(II.40\))166 1202 y(e)o(xists)40 b(and)i(is)f(a)h(rank-one)g(projection.)f(\(Note)g(that)g(due)h(to)f (the)h(continuity)e(of)3211 1177 y Fl(_)3178 1202 y Fm(P)3241 1217 y Ft(t)3271 1202 y Fp(,)i(we)166 1322 y(may)e(de\002ne)h(the)e (inte)o(gral)h(in)f(\(II.40\))i(as)f(a)g(Riemann)g(inte)o(gral.\))g(W) -8 b(e)40 b(may)g(choose)g(nor)n(-)166 1443 y(malized)i(v)o(ectors)g Fm(\036)911 1458 y Ft(t)940 1443 y Fm(;)17 b(\036)1042 1458 y Fs(1)1176 1443 y Fj(2)61 b(H)43 b Fp(such)f(that)g Fm(P)1910 1458 y Ft(t)2000 1443 y Fl(=)60 b Fj(j)p Fm(\036)2222 1458 y Ft(t)2251 1443 y Fj(ih)p Fm(\036)2387 1458 y Ft(t)2416 1443 y Fj(j)p Fp(,)42 b Fm(P)2574 1458 y Fs(1)2709 1443 y Fl(=)60 b Fj(j)p Fm(\036)2931 1458 y Fs(1)3005 1443 y Fj(ih)p Fm(\036)3141 1458 y Fs(1)3215 1443 y Fj(j)p Fp(,)42 b(and)166 1563 y Fm(\036)224 1578 y Fs(1)340 1563 y Fl(=)g(lim)594 1578 y Ft(t)p Fs(!1)781 1563 y Fm(\036)839 1578 y Ft(t)869 1563 y Fp(.)32 b(Lemma)g(IV)-13 b(.1)33 b(implies)e(that)h Fm(E)2043 1578 y Fs(1)2160 1563 y Fl(:=)42 b(lim)2440 1578 y Ft(t)p Fs(!1)2627 1563 y Fm(E)2699 1578 y Ft(t)2762 1563 y Fp(e)o(xists,)31 b(and)h(hence)166 1683 y(we)d(also)g(ha)n(v)o(e)g(the)g(con)l(v)o(er)n (gence)g(of)h Fm(H)1585 1698 y Ft(t)1614 1683 y Fm(\036)1672 1698 y Ft(t)1737 1683 y Fl(=)36 b Fm(E)1921 1698 y Ft(t)1951 1683 y Fm(\036)2009 1698 y Ft(t)2074 1683 y Fj(!)f Fm(E)2281 1698 y Fs(1)2356 1683 y Fm(\036)2414 1698 y Fs(1)2489 1683 y Fp(.)29 b(On)g(the)g(other)g(hand,)f(de-)166 1804 y(noting)d Fm(H)528 1819 y Fs(1)632 1804 y Fl(:=)31 b Fm(H)8 b Fl(\()p Fm(\013)q Fl(\))p Fp(,)25 b(it)g(is)h(not)f(dif)n (\002cult)h(to)f(sho)n(w)g(\(in)h(a)h(similar)e(f)o(ashion)g(as)h(in)g (the)g(proof)166 1933 y(of)21 b(Lemma)g(III.3)h(belo)n(w\))f(that)g Fm(W)1351 1948 y Ft(t;)p Fs(1)1498 1933 y Fl(:=)28 b Fm(v)t Fl(\()1733 1908 y Fm(~)1718 1933 y(G)1795 1948 y Ft(t)1824 1933 y Fl(\))9 b Fj(\001)g Fm(A)p Fl(\()2034 1908 y Fm(~)2019 1933 y(G)2096 1948 y Ft(t;)p Fs(1)2216 1933 y Fl(\))g(+)g Fm(A)p Fl(\()2474 1908 y Fm(~)2459 1933 y(G)2536 1948 y Ft(t;)p Fs(1)2656 1933 y Fl(\))g Fj(\001)g Fm(v)t Fl(\()2844 1908 y Fm(~)2829 1933 y(G)2906 1948 y Ft(t)2936 1933 y Fl(\))g(+)g Fm(A)p Fl(\()3194 1908 y Fm(~)3179 1933 y(G)3256 1948 y Ft(t;)p Fs(1)3376 1933 y Fl(\))3414 1897 y Fv(2)166 2054 y Fp(obe)o(ys)24 b(a)h(norm)f(bound)1088 2191 y Fi(\015)1088 2241 y(\015)1088 2291 y(\015)p Fm(W)1226 2306 y Ft(t;)p Fs(1)1363 2291 y Fl(\()p Fm(H)1490 2250 y Fv(+)1482 2316 y Ft(t)1570 2291 y Fl(+)e Fm(\032)1718 2306 y Ft(t)1748 2291 y Fl(\))1786 2250 y Fs(\000)p Fv(1)1881 2191 y Fi(\015)1881 2241 y(\015)1881 2291 y(\015)52 b Fj(\024)h(O)s Fl(\()p Fm(\032)2279 2240 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2279 2311 y Ft(t)2445 2291 y Fl(\))22 b Fm(:)664 b Fp(\(II.41\))166 2531 y(Since)27 b Fm(H)495 2546 y Fs(1)570 2531 y Fm(\036)628 2546 y Ft(t)689 2531 y Fl(=)k Fm(H)877 2546 y Ft(t)907 2531 y Fm(\036)965 2546 y Ft(t)1018 2531 y Fl(+)24 b Fm(W)1210 2546 y Ft(t;)p Fs(1)1329 2531 y Fm(\036)1387 2546 y Ft(t)1417 2531 y Fp(,)j(this)f(norm)g(bound)g(implies)g(that)g Fm(\036)2708 2546 y Ft(t)2769 2531 y Fj(2)32 b Fl(dom\()p Fm(H)3170 2546 y Fs(1)3245 2531 y Fl(\))27 b Fp(and)166 2651 y(that)385 2779 y Fi(\015)385 2828 y(\015)385 2878 y(\015)p Fl(\()p Fm(H)550 2893 y Ft(t)602 2878 y Fj(\000)22 b Fm(H)782 2893 y Fs(1)857 2878 y Fl(\))17 b Fm(\036)970 2893 y Ft(t)999 2779 y Fi(\015)999 2828 y(\015)999 2878 y(\015)52 b Fj(\024)i Fm(\032)1278 2893 y Ft(t)1335 2779 y Fi(\015)1335 2828 y(\015)1335 2878 y(\015)p Fm(W)1473 2893 y Ft(t;)p Fs(1)1610 2878 y Fl(\()p Fm(H)1737 2837 y Fv(+)1729 2903 y Ft(t)1817 2878 y Fl(+)22 b Fm(\032)1965 2893 y Ft(t)1995 2878 y Fl(\))2033 2837 y Fs(\000)p Fv(1)2128 2779 y Fi(\015)2128 2828 y(\015)2128 2878 y(\015)52 b Fj(\024)h(O)s Fl(\()p Fm(\032)2526 2827 y Fv(1)p Ft(=)p Fv(2)2526 2898 y Ft(t)2637 2878 y Fl(\))f Fj(!)g Fl(0)22 b Fm(;)219 b Fp(\(II.42\))166 3118 y(as)36 b Fm(t)49 b Fj(!)f(1)p Fp(.)36 b(Thus,)f(the)h(closedness)g(of)g Fm(H)1759 3133 y Fs(1)1869 3118 y Fp(implies)f(that)h Fm(\036)2443 3133 y Fs(1)2566 3118 y Fj(2)49 b Fl(dom\()p Fm(H)2984 3133 y Fs(1)3058 3118 y Fl(\))p Fp(,)36 b(and)h(we)166 3238 y(obtain)737 3366 y Fm(H)818 3381 y Fs(1)909 3366 y Fm(\036)967 3381 y Fs(1)1094 3366 y Fl(=)68 b(lim)1223 3420 y Ft(t)p Fs(!1)1389 3366 y Fj(f)p Fm(H)1520 3381 y Ft(t)1566 3366 y Fm(\036)1624 3381 y Ft(t)1654 3366 y Fj(g)52 b Fl(=)68 b(lim)1884 3420 y Ft(t)p Fs(!1)2051 3366 y Fj(f)p Fm(E)2173 3381 y Ft(t)2219 3366 y Fm(\036)2277 3381 y Ft(t)2307 3366 y Fj(g)52 b Fl(=)g Fm(E)2609 3381 y Fs(1)2701 3366 y Fm(\036)2759 3381 y Fs(1)2856 3366 y Fm(:)313 b Fp(\(II.43\))166 3543 y(Thus,)21 b Fm(\036)470 3558 y Fs(1)566 3543 y Fp(is)g(a)h(normalized)f(eigen)l(v)o(ector)g(of) h Fm(H)1857 3558 y Fs(1)1931 3543 y Fp(.)g(Finally)-6 b(,)20 b(a)i(simple)e(continuity)g(ar)n(gument)166 3663 y(using)k(that)g Fm(E)653 3678 y Ft(t)711 3663 y Fl(=)j(inf)d Fm(\033)t Fl([)p Fm(H)1116 3678 y Ft(t)1145 3663 y Fl(])h Fp(sho)n(ws)f(that)g Fm(E)1715 3678 y Fs(1)1818 3663 y Fl(=)j(inf)d Fm(\033)t Fl([)p Fm(H)2223 3678 y Fs(1)2298 3663 y Fl(])p Fp(.)490 b Fb(2)166 4208 y Fq(III)99 b(Basic)25 b(Estimates)166 4552 y Fp(In)20 b(this)f(section)g(we)i(collect)e(some) h(basic)g(estimates)f(and)g(apply)h(them)f(to)h(establish)f(a)h(relati) n(v)o(e)166 4673 y(quadratic)25 b(form)f(bound)g Fj(\006)p Fm(W)1223 4688 y Ft(t;s)1333 4673 y Fj(\024)k(O)s Fl(\()p Fm(\013)1621 4637 y Fv(3)p Ft(=)p Fv(2)1731 4673 y Fl(\)\()p Fm(t)22 b Fj(\000)h Fm(s)p Fl(\)\()p Fm(H)2175 4631 y Fv(+)2167 4693 y Ft(t)2256 4673 y Fl(+)f Fm(\032)2404 4688 y Ft(t)2434 4673 y Fl(\))p Fp(,)i(where)361 5023 y Fm(W)453 5038 y Ft(t;s)552 5023 y Fl(:=)16 b Fm(H)752 5038 y Ft(t;s)856 5023 y Fj(\000)23 b Fm(H)1037 5038 y Ft(t;t)1164 5023 y Fl(=)52 b Fm(v)t Fl(\()1396 4998 y Fm(~)1381 5023 y(G)1458 5038 y Ft(s)1495 5023 y Fl(\))1533 4982 y Fv(2)1594 5023 y Fj(\000)23 b Fm(v)t Fl(\()1798 4998 y Fm(~)1783 5023 y(G)1860 5038 y Ft(t)1889 5023 y Fl(\))1927 4982 y Fv(2)3163 5023 y Fp(\(III.44\))565 5201 y Fl(=)671 5104 y Fi(\020)721 5201 y Fm(v)t Fl(\()825 5175 y Fm(~)810 5201 y(G)887 5216 y Ft(t)916 5201 y Fl(\))f(+)g Fm(A)p Fl(\()1200 5175 y Fm(~)1185 5201 y(G)1262 5216 y Ft(t;s)1344 5201 y Fl(\))1382 5104 y Fi(\021)1432 5127 y Fv(2)1515 5201 y Fj(\000)45 b Fm(v)t Fl(\()1741 5175 y Fm(~)1726 5201 y(G)1803 5216 y Ft(t)1832 5201 y Fl(\))1870 5159 y Fv(2)565 5380 y Fl(=)30 b Fm(v)t Fl(\()775 5355 y Fm(~)760 5380 y(G)837 5395 y Ft(t)867 5380 y Fl(\))22 b Fj(\001)f Fm(A)p Fl(\()1102 5355 y Fm(~)1087 5380 y(G)1164 5395 y Ft(t;s)1246 5380 y Fl(\))45 b(+)f Fm(A)p Fl(\()1575 5355 y Fm(~)1560 5380 y(G)1637 5395 y Ft(t;s)1719 5380 y Fl(\))22 b Fj(\001)f Fm(v)t Fl(\()1932 5355 y Fm(~)1917 5380 y(G)1994 5395 y Ft(t)2024 5380 y Fl(\))44 b(+)g Fm(A)p Fl(\()2352 5355 y Fm(~)2337 5380 y(G)2414 5395 y Ft(t;s)2496 5380 y Fl(\))2534 5339 y Fv(2)1773 5712 y Fp(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 166 83 a Fp(and)956 203 y Fm(H)1045 162 y Fv(+)1037 228 y Ft(t;s)1171 203 y Fl(:=)52 b Fm(H)1407 218 y Ft(t;s)1511 203 y Fj(\000)23 b Fm(E)1683 218 y Ft(t;s)1787 203 y Fm(;)111 b(H)2014 162 y Fv(+)2006 228 y Ft(t)2125 203 y Fl(:=)53 b Fm(H)2362 218 y Ft(t)2414 203 y Fj(\000)22 b Fm(E)2585 218 y Ft(t)2637 203 y Fm(;)499 b Fp(\(III.45\))166 342 y(for)25 b Fm(s)j Fj(\025)g Fm(t)g Fj(\025)g Fl(0)p Fp(.)d(W)-8 b(e)25 b(be)o(gin)f(by)g(recalling)h(the)f (follo)n(wing)f(standard)i(estimates.)166 562 y Fq(Lemma)g(III.1)48 b Fh(F)-10 b(or)24 b(all)1065 536 y Fm(~)1051 562 y(f)39 b Fj(2)28 b Ff(h)1284 526 y Fv(3)1323 562 y Fh(,)d(suc)o(h)f(that)g Fm(!)1824 526 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2002 536 y Fm(~)1988 562 y(f)38 b Fj(2)28 b Ff(h)2220 526 y Fv(3)2260 562 y Fh(,)d(and)f(all)g Fm(r)30 b(>)e Fl(0)p Fh(,)d(we)g(have)401 786 y Fi(\015)401 836 y(\015)401 886 y(\015)p Fm(a)p Fl(\()550 860 y Fm(~)536 886 y(f)11 b Fl(\))17 b(\()712 861 y(\024)688 886 y Fm(H)7 b Fl(\()828 860 y Fm(~)814 886 y(f)j Fl(\))22 b(+)g Fm(r)s Fl(\))1115 845 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1280 786 y Fi(\015)1280 836 y(\015)1280 886 y(\015)17 b Fj(\024)1437 786 y Fi(\015)1437 836 y(\015)1437 886 y(\015)p Fm(!)1548 845 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1726 860 y Fm(~)1712 886 y(f)1771 786 y Fi(\015)1771 836 y(\015)1771 886 y(\015)22 b Fm(;)1297 b Fp(\(III.46\))361 969 y Fi(\015)361 1019 y(\015)361 1069 y(\015)p Fm(a)458 1027 y Fs(\003)498 1069 y Fl(\()550 1042 y Fm(~)536 1069 y(f)11 b Fl(\))17 b(\()712 1043 y(\024)688 1069 y Fm(H)7 b Fl(\()828 1042 y Fm(~)814 1069 y(f)j Fl(\))22 b(+)g Fm(r)s Fl(\))1115 1027 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1280 969 y Fi(\015)1280 1019 y(\015)1280 1069 y(\015)17 b Fj(\024)1437 969 y Fi(\015)1437 1019 y(\015)1437 1069 y(\015)p Fl(\()p Fm(!)1586 1027 y Fs(\000)p Fv(1)1701 1069 y Fl(+)22 b Fm(r)1846 1027 y Fs(\000)p Fv(1)1940 1069 y Fl(\))1978 1027 y Fv(1)p Ft(=)p Fv(2)2102 1042 y Fm(~)2088 1069 y(f)2147 969 y Fi(\015)2147 1019 y(\015)2147 1069 y(\015)g Fm(;)921 b Fp(\(III.47\))166 1278 y Fh(and)361 1600 y Fm(A)p Fl(\()486 1574 y Fm(~)472 1600 y(f)11 b Fl(\))569 1559 y Fv(2)625 1600 y Fj(\024)17 b Fl(4)785 1500 y Fi(\015)784 1550 y(\015)784 1600 y(\015)p Fl(\()p Fm(!)933 1559 y Fs(\000)p Fv(1)1049 1600 y Fl(+)22 b Fm(r)1194 1559 y Fs(\000)p Fv(1)1288 1600 y Fl(\))1326 1559 y Fv(1)p Ft(=)p Fv(2)1450 1574 y Fm(~)1436 1600 y(f)1495 1500 y Fi(\015)1495 1550 y(\015)1495 1600 y(\015)1541 1527 y Fv(2)1597 1600 y Fl(\()1660 1575 y(\024)1635 1600 y Fm(H)8 b Fl(\()1776 1574 y Fm(~)1762 1600 y(f)i Fl(\))22 b(+)g Fm(r)s Fl(\))g Fm(;)1051 b Fp(\(III.48\))166 1809 y Fh(wher)l(e)1095 1904 y Fl(\024)1070 1930 y Fm(H)8 b Fl(\()1211 1903 y Fm(~)1197 1930 y(f)i Fl(\))53 b(:=)1501 1812 y Fi(Z)1547 2001 y Fv(supp)1714 1984 y Ft(~)1705 2001 y(f)1767 1930 y Fm(!)t Fl(\()p Fm(k)s Fl(\))17 b Fm(a)2030 1888 y Fs(\003)2068 1930 y Fl(\()p Fm(k)s Fl(\))p Fm(a)p Fl(\()p Fm(k)s Fl(\))g Fm(dk)25 b(:)613 b Fp(\(III.49\))166 2138 y Fh(If)25 b(additionally)d Fj(k)p Fl(\()p Fm(!)911 2102 y Fs(\000)p Fv(1)1027 2138 y Fl(+)g Fm(\024)1181 2102 y Fs(\000)p Fv(1)1275 2138 y Fl(\))1313 2102 y Fv(1)p Ft(=)p Fv(2)1437 2111 y Fm(~)1423 2138 y(f)11 b Fj(k)1532 2102 y Fv(2)1599 2138 y Fj(\024)28 b Fl(1)p Fm(=)p Fl(2)c Fh(holds)g(true)o(,)g(then)1030 2359 y Fm(v)t Fl(\()1133 2333 y Fm(~)1119 2359 y(f)11 b Fl(\))1216 2318 y Fv(2)1308 2359 y Fj(\024)53 b Fl(2)1504 2263 y Fi(\020)1553 2359 y Fm(H)1642 2318 y Fv(+)1700 2359 y Fl(\()1752 2333 y Fm(~)1738 2359 y(f)11 b Fl(\))22 b(+)g Fm(b)h Fl(+)f Fm(\024)g Fl(+)g Fj(k)2357 2333 y Fm(~)2343 2359 y(f)10 b Fj(k)2451 2318 y Fv(2)2491 2263 y Fi(\021)2562 2359 y Fm(;)574 b Fp(\(III.50\))166 2603 y Fh(wher)l(e)33 b Fm(b)43 b Fl(:=)f Fj(\000)17 b Fl(inf)23 b Fm(\033)t Fl([)991 2564 y Fv(1)p 991 2580 36 4 v 991 2638 a(4)1037 2603 y Fl(\()p Fj(\000)p Fl(\001)1233 2618 y Ft(x)1277 2603 y Fl(\))28 b(+)g Fm(V)21 b Fl(\()p Fm(x)p Fl(\)])43 b Fm(<)f Fj(1)p Fh(,)32 b Fm(H)2090 2567 y Fv(+)2148 2603 y Fl(\()2200 2577 y Fm(~)2186 2603 y(f)11 b Fl(\))42 b(:=)g Fm(H)8 b Fl(\()2611 2577 y Fm(~)2597 2603 y(f)i Fl(\))28 b Fj(\000)g Fm(E)6 b Fl(\()2956 2577 y Fm(~)2942 2603 y(f)11 b Fl(\))p Fh(,)32 b Fm(E)6 b Fl(\()3226 2577 y Fm(~)3212 2603 y(f)11 b Fl(\))42 b(:=)166 2743 y(inf)24 b Fm(\033)t Fl([)p Fm(H)8 b Fl(\()528 2717 y Fm(~)514 2743 y(f)i Fl(\)])p Fh(,)25 b(and)1087 2874 y Fm(H)8 b Fl(\()1228 2848 y Fm(~)1214 2874 y(f)j Fl(\))52 b(:=)h Fm(v)t Fl(\()1622 2848 y Fm(~)1608 2874 y(f)10 b Fl(\))1704 2833 y Fv(2)1782 2874 y Fl(+)39 b Fm(V)21 b Fl(\()p Fm(x)p Fl(\))39 b(+)2285 2849 y(\024)2260 2874 y Fm(H)8 b Fl(\()2401 2848 y Fm(~)2387 2874 y(f)i Fl(\))22 b Fm(:)631 b Fp(\(III.51\))166 3112 y Fh(Pr)l(oof:)39 b Fp(The)h(proof)g(of)g (\(III.46\)\226\(III.48\))g(can)h(be)f(found,)f(e.g.,)h(in)g([2,3].)g (W)-8 b(e)40 b(only)f(gi)n(v)o(e)166 3232 y(a)c(proof)g(of)g (\(III.50\))h(which)e(is)h(similar)f(to)g([5,)h(Lemma)g(II.2].)g(W)-8 b(e)35 b(\002rst)g(use)g(\(III.48\))h(and)166 3353 y Fj(\000)p Fl(\001)324 3368 y Ft(x)396 3353 y Fl(=)28 b(\()p Fj(\000)p Fm(i)657 3328 y(~)648 3353 y Fj(r)731 3368 y Ft(x)776 3353 y Fl(\))814 3317 y Fv(2)878 3353 y Fp(to)c(obtain)361 3699 y Fj(\000)p Fl(\001)519 3714 y Ft(x)581 3699 y Fl(=)674 3602 y Fi(\020)724 3699 y Fm(v)t Fl(\()827 3672 y Fm(~)813 3699 y(f)10 b Fl(\))23 b Fj(\000)f Fm(A)p Fl(\()1156 3672 y Fm(~)1142 3699 y(f)11 b Fl(\))1239 3602 y Fi(\021)1288 3625 y Fv(2)3163 3699 y Fp(\(III.52\))581 3879 y Fl(=)17 b Fm(v)t Fl(\()777 3853 y Fm(~)763 3879 y(f)11 b Fl(\))860 3838 y Fv(2)944 3879 y Fj(\000)44 b Fm(v)t Fl(\()1168 3853 y Fm(~)1154 3879 y(f)11 b Fl(\))22 b Fj(\001)g Fm(A)p Fl(\()1448 3853 y Fm(~)1434 3879 y(f)10 b Fl(\))45 b Fj(\000)f Fm(A)p Fl(\()1821 3853 y Fm(~)1807 3879 y(f)11 b Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()2079 3853 y Fm(~)2065 3879 y(f)10 b Fl(\))45 b(+)f Fm(A)p Fl(\()2451 3853 y Fm(~)2437 3879 y(f)11 b Fl(\))2534 3838 y Fv(2)580 4035 y Fj(\024)17 b Fl(2)g Fm(v)t Fl(\()843 4008 y Fm(~)829 4035 y(f)10 b Fl(\))925 3993 y Fv(2)1009 4035 y Fl(+)44 b(2)17 b Fm(A)p Fl(\()1320 4008 y Fm(~)1306 4035 y(f)10 b Fl(\))1402 3993 y Fv(2)580 4209 y Fj(\024)17 b Fl(2)g Fm(v)t Fl(\()843 4182 y Fm(~)829 4209 y(f)10 b Fl(\))925 4168 y Fv(2)1009 4209 y Fl(+)44 b(8)1195 4109 y Fi(\015)1194 4159 y(\015)1194 4209 y(\015)q Fl(\()p Fm(!)1344 4168 y Fs(\000)p Fv(1)1459 4209 y Fl(+)22 b Fm(r)1604 4168 y Fs(\000)p Fv(1)1698 4209 y Fl(\))1736 4168 y Fv(1)p Ft(=)p Fv(2)1860 4182 y Fm(~)1846 4209 y(f)1905 4109 y Fi(\015)1905 4159 y(\015)1905 4209 y(\015)1951 4135 y Fv(2)2007 4209 y Fl(\()2070 4184 y(\024)2045 4209 y Fm(H)8 b Fl(\()2186 4182 y Fm(~)2172 4209 y(f)i Fl(\))22 b(+)g Fm(r)s Fl(\))g Fm(:)166 4438 y Fp(Since)h Fm(V)49 b Fj(2)28 b Fm(L)676 4401 y Fv(2)716 4438 y Fl(\()p Fg(R)826 4401 y Fv(3)865 4438 y Fl(;)17 b Fg(R)981 4396 y Fs(\000)981 4459 y Fv(0)1040 4438 y Fl(\))c(+)g Fm(L)1246 4401 y Fs(1)1321 4438 y Fl(\()p Fg(R)1431 4401 y Fv(3)1470 4438 y Fl(;)k Fg(R)1586 4396 y Fs(\000)1586 4459 y Fv(0)1645 4438 y Fl(\))p Fp(,)22 b(we)h(ha)n(v)o(e)f(that)2258 4398 y Fv(1)p 2258 4414 V 2258 4472 a(4)2303 4438 y Fl(\()p Fj(\000)p Fl(\001)2499 4453 y Ft(x)2544 4438 y Fl(\))13 b(+)g Fm(V)21 b Fl(\()p Fm(x)p Fl(\))28 b Fj(\025)g(\000)p Fm(b)h(>)f Fj(\0001)166 4558 y Fp(and)d(hence)g Fl(0)i Fj(\024)i(\000)p Fm(V)22 b Fl(\()p Fm(x)p Fl(\))28 b Fj(\024)1204 4519 y Fv(1)p 1204 4535 V 1204 4592 a(4)1249 4558 y Fl(\()p Fj(\000)p Fl(\001)1445 4573 y Ft(x)1490 4558 y Fl(\))22 b(+)g Fm(b)p Fp(.)j(This)f(and)h(\(III.52\))g(imply)361 4929 y Fm(v)t Fl(\()464 4903 y Fm(~)450 4929 y(f)11 b Fl(\))547 4888 y Fv(2)603 4929 y Fl(=)18 b Fm(H)8 b Fl(\()838 4903 y Fm(~)824 4929 y(f)i Fl(\))22 b Fj(\000)g Fm(V)g Fl(\()p Fm(x)p Fl(\))g Fj(\000)1398 4904 y Fl(\024)1373 4929 y Fm(H)8 b Fl(\()1514 4903 y Fm(~)1500 4929 y(f)i Fl(\))80 b Fj(\024)h Fm(H)8 b Fl(\()1975 4903 y Fm(~)1961 4929 y(f)i Fl(\))22 b(+)2187 4861 y(1)p 2187 4906 49 4 v 2187 4997 a(4)2246 4929 y(\()p Fj(\000)p Fl(\001)2442 4944 y Ft(x)2487 4929 y Fl(\))g(+)g Fm(b)g Fj(\000)2833 4904 y Fl(\024)2808 4929 y Fm(H)8 b Fl(\()2949 4903 y Fm(~)2935 4929 y(f)i Fl(\))603 5162 y Fj(\024)17 b Fm(H)8 b Fl(\()838 5136 y Fm(~)824 5162 y(f)i Fl(\))22 b(+)g Fm(b)45 b Fl(+)1256 5095 y(1)p 1256 5139 V 1256 5230 a(2)1331 5162 y Fm(v)t Fl(\()1434 5136 y Fm(~)1420 5162 y(f)10 b Fl(\))1516 5121 y Fv(2)3163 5162 y Fp(\(III.53\))697 5363 y Fl(+)795 5267 y Fi(\020)844 5363 y Fl(2)17 b Fj(k)p Fl(\()p Fm(!)1063 5322 y Fs(\000)p Fv(1)1178 5363 y Fl(+)22 b Fm(r)1323 5322 y Fs(\000)p Fv(1)1417 5363 y Fl(\))1455 5322 y Fv(1)p Ft(=)p Fv(2)1579 5337 y Fm(~)1565 5363 y(f)11 b Fj(k)1674 5322 y Fv(2)1735 5363 y Fj(\000)23 b Fl(1)1884 5267 y Fi(\021)1958 5338 y Fl(\024)1933 5363 y Fm(H)8 b Fl(\()2074 5337 y Fm(~)2060 5363 y(f)i Fl(\))45 b(+)f(2)17 b Fj(k)p Fl(\()p Fm(!)2540 5322 y Fs(\000)p Fv(1)2655 5363 y Fl(+)22 b Fm(r)2800 5322 y Fs(\000)p Fv(1)2894 5363 y Fl(\))2932 5322 y Fv(1)p Ft(=)p Fv(2)3055 5337 y Fm(~)3042 5363 y(f)10 b Fj(k)3150 5322 y Fv(2)3206 5363 y Fm(r)25 b(:)1773 5712 y Fp(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 166 97 a Fp(W)-8 b(e)25 b(choose)g Fm(r)30 b Fl(:=)e Fm(\024)d Fp(and)g(use)f(that)h Fl(2)17 b Fj(k)p Fl(\()p Fm(!)1629 61 y Fs(\000)p Fv(1)1744 97 y Fl(+)22 b Fm(r)1889 61 y Fs(\000)p Fv(1)1983 97 y Fl(\))2021 61 y Fv(1)p Ft(=)p Fv(2)2144 71 y Fm(~)2131 97 y(f)10 b Fj(k)2239 61 y Fv(2)2306 97 y Fj(\024)28 b Fl(1)d Fp(to)f(arri)n(v)o (e)h(at)440 334 y Fm(v)t Fl(\()543 308 y Fm(~)529 334 y(f)11 b Fl(\))626 293 y Fv(2)717 334 y Fj(\024)54 b Fl(2)897 238 y Fi(\020)946 334 y Fm(H)8 b Fl(\()1087 308 y Fm(~)1073 334 y(f)i Fl(\))22 b(+)g Fm(b)h Fl(+)f Fm(\024)1507 238 y Fi(\021)1609 334 y Fj(\024)53 b Fl(2)1788 238 y Fi(\020)1838 334 y Fm(H)1927 293 y Fv(+)1985 334 y Fl(\()2037 308 y Fm(~)2023 334 y(f)11 b Fl(\))22 b(+)g Fm(E)6 b Fl(\()2370 308 y Fm(~)2356 334 y(f)11 b Fl(\))22 b(+)g Fm(b)g Fl(+)g Fm(\024)2790 238 y Fi(\021)2862 334 y Fm(:)274 b Fp(\(III.54\))166 558 y(Finally)-6 b(,)25 b(we)h(use)g(the)g(min-max)e(principle)i(with)f(the)h(trial)f(state)h Fm(')2518 573 y Ft(el)2599 558 y Fj(\012)e Fl(\012)p Fp(,)i(where)h Fm(')3155 573 y Ft(el)3239 558 y Fp(is)f(the)166 678 y(normalized)f(ground)h(state)f(of)i Fm(H)1348 693 y Ft(el)1436 678 y Fl(=)j Fj(\000)p Fl(\001)1700 693 y Ft(x)1768 678 y Fl(+)22 b Fm(V)g Fl(\()p Fm(x)p Fl(\))p Fp(,)k(i.e.,)g Fm(H)2381 693 y Ft(el)2440 678 y Fm(')2504 693 y Ft(el)2592 678 y Fl(=)k Fm(e)2743 693 y Fv(0)2783 678 y Fm(')2847 693 y Ft(el)2905 678 y Fp(,)c(with)g Fm(e)3205 693 y Fv(0)3274 678 y Fm(<)k Fl(0)p Fp(.)166 799 y(This)24 b(gi)n(v)o(es)459 1023 y Fm(E)6 b Fl(\()589 997 y Fm(~)575 1023 y(f)11 b Fl(\))52 b Fj(\024)i(h)p Fm(')958 1038 y Ft(el)1038 1023 y Fj(\012)23 b Fl(\012)p Fj(j)k Fm(H)8 b Fl(\()1404 997 y Fm(~)1390 1023 y(f)j Fl(\))17 b Fm(')1568 1038 y Ft(el)1648 1023 y Fj(\012)23 b Fl(\012)p Fj(i)52 b Fl(=)h Fm(e)2083 1038 y Fv(0)2145 1023 y Fl(+)22 b Fj(k)2307 997 y Fm(~)2293 1023 y(f)10 b Fj(k)2401 982 y Fv(2)2493 1023 y Fj(\024)53 b(k)2687 997 y Fm(~)2673 1023 y(f)11 b Fj(k)2782 982 y Fv(2)2843 1023 y Fm(:)293 b Fp(\(III.55\))631 1247 y Fb(2)166 1590 y Fp(In)36 b(order)g(to)f(apply)h(these)f(estimates)g(to)h(the)f (choices)2169 1564 y Fm(~)2155 1590 y(G)2232 1605 y Ft(t)2297 1590 y Fp(and)2492 1564 y Fm(~)2477 1590 y(G)2554 1605 y Ft(t;s)2636 1590 y Fp(,)h(for)2862 1563 y Fm(~)2849 1590 y(f)11 b Fp(,)35 b(we)i(ne)o(xt)d(de-)166 1710 y(ri)n(v)o(e)d (bounds)f(on)i(the)f Fm(L)1019 1674 y Fv(2)1059 1710 y Fp(-norms)g(appearing)g(on)h(the)f(right)g(sides)g(of)h (\(III.46\)\226\(III.48\))g(and)166 1830 y(\(III.50\).)26 b(Suppose)f(that)g Fm(s)k(>)g(t)g Fj(\025)h Fl(0)p Fp(.)25 b(Using)g Fl(0)k Fj(\024)g Fm(\037)2004 1845 y Ft(s)2064 1830 y Fj(\000)23 b Fm(\037)2225 1845 y Ft(t)2284 1830 y Fj(\024)29 b Fl(\()p Fm(s)23 b Fj(\000)g Fm(t)p Fl(\))p Fj(k)p Fm(@)2771 1845 y Ft(t)2801 1830 y Fm(\037)2862 1845 y Ft(t)2892 1830 y Fj(k)2942 1845 y Fs(1)3016 1830 y Fp(,)j(we)g(deri)n(v)o(e)166 1951 y(the)f(follo)n(wing)e(estimates)g (on)i(the)f Fm(L)1455 1915 y Fv(2)1495 1951 y Fl(\()p Fg(R)1605 1915 y Fv(3)1667 1951 y Fj(\002)e Fg(Z)1832 1966 y Fv(2)1872 1951 y Fl(;)17 b Fg(C)1988 1915 y Fv(3)2027 1951 y Fl(\))p Fp(-norm,)25 b(for)g Fm(\014)33 b Fj(\024)28 b Fl(1)p Fm(=)p Fl(2)p Fp(,)361 2315 y Fj(k)p Fm(!)476 2274 y Fs(\000)p Ft(\014)608 2290 y Fm(~)594 2315 y(G)671 2330 y Ft(t;s)753 2315 y Fj(k)803 2274 y Fv(2)859 2315 y Fl(=)953 2198 y Fi(Z)1052 2219 y(h)1092 2315 y Fm(\037)1153 2330 y Ft(s)1189 2315 y Fl(\()p Fm(k)s Fl(\))23 b Fj(\000)f Fm(\037)1502 2330 y Ft(t)1532 2315 y Fl(\()p Fm(k)s Fl(\))1662 2219 y Fi(i)1701 2242 y Fv(2)1757 2315 y Fm(!)t Fl(\()p Fm(k)s Fl(\))1952 2274 y Fs(\000)p Fv(2)p Ft(\014)2105 2315 y Fj(j)2148 2290 y Fm(~)2133 2315 y(G)p Fl(\()p Fm(k)s Fl(\))p Fj(j)2368 2274 y Fv(2)2429 2315 y Fm(dk)859 2533 y Fj(\024)17 b Fl(\()p Fm(s)22 b Fj(\000)g Fm(t)p Fl(\))1231 2491 y Fv(2)1288 2533 y Fj(k)p Fm(@)1389 2548 y Ft(t)1418 2533 y Fm(\037)1479 2548 y Ft(t)1509 2533 y Fj(k)1559 2491 y Fv(2)1559 2557 y Fs(1)1667 2415 y Fi(Z)1713 2604 y Fs(f)p Ft(\032)1784 2612 y Fc(s)1818 2604 y Fs(\024j)p Ft(k)r Fs(j\024)p Fv(2)p Ft(\032)2078 2612 y Fc(t)2105 2604 y Fs(g)2161 2533 y Fm(!)t Fl(\()p Fm(k)s Fl(\))2356 2491 y Fs(\000)p Fv(2)p Ft(\014)2509 2533 y Fj(j)2552 2507 y Fm(~)2537 2533 y(G)o Fl(\()p Fm(k)s Fl(\))p Fj(j)2771 2491 y Fv(2)2833 2533 y Fm(dk)859 2800 y Fj(\024)17 b Fl(\()p Fm(s)22 b Fj(\000)g Fm(t)p Fl(\))1231 2759 y Fv(2)1288 2800 y Fj(k)p Fm(@)1389 2815 y Ft(t)1418 2800 y Fm(\037)1479 2815 y Ft(t)1509 2800 y Fj(k)1559 2759 y Fv(2)1559 2825 y Fs(1)1683 2733 y Fl(8)p Fm(\031)e(\013)1870 2697 y Fv(3)p 1660 2777 272 4 v 1660 2869 a Fl(2\(2)p Fm(\031)t Fl(\))1893 2840 y Fv(3)1975 2683 y Fi(Z)2058 2709 y Fv(2)p Ft(\032)2129 2717 y Fc(t)2021 2872 y Fv(0)2178 2800 y Fm(r)2225 2759 y Fv(1)p Fs(\000)p Fv(2)p Ft(\014)2414 2800 y Fm(dr)859 3073 y Fj(\024)d Fl(\()p Fm(s)22 b Fj(\000)g Fm(t)p Fl(\))1231 3032 y Fv(2)1288 3073 y Fj(k)p Fm(@)1389 3088 y Ft(t)1418 3073 y Fm(\037)1479 3088 y Ft(t)1509 3073 y Fj(k)1559 3032 y Fv(2)1559 3098 y Fs(1)1683 3006 y Fm(\013)1746 2970 y Fv(3)p 1660 3050 148 4 v 1660 3141 a Fl(2)p Fm(\031)1768 3113 y Fv(2)1834 3073 y Fl(2)17 b Fm(\032)1950 3026 y Fv(2)p Fs(\000)p Fv(2)p Ft(\014)1950 3093 y(t)2144 3073 y Fm(:)992 b Fp(\(III.56\))166 3320 y(Hence,)216 3544 y Fj(k)281 3519 y Fm(~)266 3544 y(G)343 3559 y Ft(t;s)424 3544 y Fj(k)52 b(\024)g Fm(\021)20 b(\032)773 3559 y Ft(t)820 3544 y Fm(\013)883 3503 y Fv(3)p Ft(=)p Fv(2)1009 3544 y Fl(\()p Fm(s)f Fj(\000)g Fm(t)p Fl(\))g Fm(;)111 b Fp(and)94 b Fj(k)p Fm(!)1791 3503 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1986 3519 y Fm(~)1972 3544 y(G)2049 3559 y Ft(t;s)2131 3544 y Fj(k)51 b(\024)h Fm(\021)21 b(\032)2480 3493 y Fv(1)p Ft(=)p Fv(2)2480 3564 y Ft(t)2607 3544 y Fm(\013)2670 3503 y Fv(3)p Ft(=)p Fv(2)2796 3544 y Fl(\()p Fm(s)d Fj(\000)i Fm(t)p Fl(\))e Fm(;)50 b Fp(\(III.57\))166 3768 y(using)24 b(that)g Fj(k)p Fm(p)p Fj(k)730 3783 y Fs(1)832 3768 y Fj(\024)k Fm(\031)t(=)p Fl(2)p Fp(,)d(and)f (similarly)870 4002 y Fj(k)935 3976 y Fm(~)920 4002 y(G)997 4017 y Ft(t)1026 4002 y Fj(k)53 b(\024)g Fm(\024)17 b(\013)1395 3960 y Fv(3)p Ft(=)p Fv(2)1599 4002 y Fp(and)94 b Fj(k)p Fm(!)1952 3960 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2131 3976 y Fm(~)2116 4002 y(G)2193 4017 y Ft(t)2223 4002 y Fj(k)52 b(\024)h Fm(\024)17 b(\013)2591 3960 y Fv(3)p Ft(=)p Fv(2)2723 4002 y Fm(:)413 b Fp(\(III.58\))166 4226 y(F)o(or)28 b(the)g(formulation)e(of)i(the)g(ne)o(xt)f(lemma,)g(we)h(introduce)f Fm(\034)45 b Fl(:=)2544 4186 y Fv(1)p 2543 4202 38 4 v 2543 4260 a Ft(\021)2608 4226 y Fl(ln)16 b(2)34 b Fm(>)f Fl(0)28 b Fp(and)f(observ)o(e)166 4346 y(that)d Fm(e)385 4310 y Ft(\021)r(\034)494 4346 y Fl(=)k(2)c Fp(and)h(hence)g Fm(\032)1147 4361 y Ft(t)p Fs(\000)p Ft(\034)1299 4346 y Fl(=)i(2)p Fm(\032)1501 4361 y Ft(t)1531 4346 y Fp(.)e(W)-8 b(e)25 b(recall)g(from)g(\(II.32\))g(that)352 4572 y Fl(\024)326 4597 y Fm(H)407 4612 y Ft(t)p Fs(\000)p Ft(\034)t(;s)631 4597 y Fl(=)760 4480 y Fi(Z)806 4669 y Ft(!)r Fv(\()p Ft(k)r Fv(\))p Ft(<\032)1036 4678 y Fc(t)p Fd(\000)p Fc(\034)1168 4597 y Fm(!)1229 4612 y Ft(s)1266 4597 y Fl(\()p Fm(k)s Fl(\))17 b Fm(a)1464 4556 y Fs(\003)1503 4597 y Fl(\()p Fm(k)s Fl(\))p Fm(a)p Fl(\()p Fm(k)s Fl(\))g Fm(dk)55 b Fl(=)2116 4480 y Fi(Z)2162 4669 y Ft(!)r Fv(\()p Ft(k)r Fv(\))p Ft(<)p Fv(2)p Ft(\032)2427 4677 y Fc(t)2477 4597 y Fm(!)2538 4612 y Ft(s)2575 4597 y Fl(\()p Fm(k)s Fl(\))17 b Fm(a)2773 4556 y Fs(\003)2812 4597 y Fl(\()p Fm(k)s Fl(\))p Fm(a)p Fl(\()p Fm(k)s Fl(\))g Fm(dk)24 b(:)3163 4762 y Fp(\(III.59\))166 4984 y Fq(Lemma)h(III.2)48 b Fh(Let)32 b Fm(\034)51 b Fl(:=)1164 4944 y Fv(1)p 1162 4960 V 1162 5018 a Ft(\021)1227 4984 y Fl(ln)17 b(2)39 b Fm(>)h Fl(0)32 b Fh(and)f Fl(0)39 b Fj(\024)i Fm(t)f Fj(\024)g Fm(s)p Fh(,)32 b(and)f(suppose)f(that)h Fm(\013)3075 4947 y Fv(3)3154 4984 y Fj(\024)40 b Fl(\()p Fm(b)28 b Fl(+)166 5118 y(6)p Fm(\024)271 5082 y Fv(2)333 5118 y Fl(+)22 b Fm(e)476 5133 y Fv(0)537 5118 y Fl(+)g(1\))722 5082 y Fs(\000)p Fv(1)816 5118 y Fh(.)j(Then)1249 5317 y Fl(\024)1224 5342 y Fm(H)1305 5357 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1499 5342 y Fl(+)d Fm(\032)1647 5357 y Ft(t)1729 5342 y Fj(\024)53 b Fl(2\()p Fm(H)2035 5301 y Fv(+)2027 5367 y Ft(t;s)2131 5342 y Fl(+)22 b Fm(\032)2279 5357 y Ft(t)2309 5342 y Fl(\))g Fm(:)767 b Fp(\(III.60\))1773 5712 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 166 83 a Fh(Pr)l(oof:)24 b Fp(W)-8 b(e)25 b(\002rst)g(set)312 282 y Fm(~)297 307 y(F)360 322 y Ft(t)445 307 y Fl(:=)56 b Fe(1)p Fl([)p Fm(!)31 b Fj(\025)d Fl(2)p Fm(\032)983 322 y Ft(t)1013 307 y Fl(])1055 282 y Fm(~)1040 307 y(G)118 b Fp(and)g Fl(\001)1593 282 y Fm(~)1578 307 y(G)1655 322 y Ft(t)1740 307 y Fl(:=)55 b(\()p Fm(!)2001 266 y Fs(\000)p Fv(1)2117 307 y Fl(+)22 b Fm(\032)2265 266 y Fs(\000)p Fv(1)2265 331 y Ft(t)2360 307 y Fl(\))2398 266 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2563 210 y Fi(\020)2627 282 y Fm(~)2612 307 y(G)2689 322 y Ft(t)2741 307 y Fj(\000)2855 282 y Fm(~)2840 307 y(F)2903 322 y Ft(t)2933 210 y Fi(\021)3005 307 y Fm(;)131 b Fp(\(III.61\))166 531 y(and)25 b(note)f(that)274 790 y Fj(k)p Fl(\001)420 765 y Fm(~)405 790 y(G)482 805 y Ft(t)512 790 y Fj(k)562 749 y Fv(2)653 790 y Fj(\024)809 722 y Fl(2)p 794 766 80 4 v 794 858 a Fm(\032)844 873 y Ft(t)900 673 y Fi(Z)946 861 y Ft(\032)982 869 y Fc(t)1010 861 y Fs(\024)p Ft(!)r Fv(\()p Ft(k)r Fv(\))p Fs(\024)p Fv(2)p Ft(\032)1330 869 y Fc(t)1380 790 y Fj(j)1423 765 y Fm(~)1408 790 y(G)o Fl(\()p Fm(k)s Fl(\))p Fj(j)1642 749 y Fv(2)1698 790 y Fm(dk)55 b Fj(\024)2046 722 y Fm(\013)2109 686 y Fv(3)p 1995 766 205 4 v 1995 858 a Fl(2)p Fm(\031)21 b(\032)2170 873 y Ft(t)2243 673 y Fi(Z)2326 699 y Fv(2)p 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y Fj(\000)12 b Fm(v)t Fl(\()3203 1256 y Fm(~)3189 1281 y(F)3252 1296 y Ft(t)3282 1281 y Fl(\))3320 1240 y Fv(2)3359 1185 y Fi(o)3427 1281 y Fm(;)3163 1402 y Fp(\(III.63\))166 1522 y(which)24 b(implies)361 1865 y Fm(H)8 b Fl(\()502 1839 y Fm(~)488 1865 y(F)551 1880 y Ft(t)580 1865 y Fl(\))22 b(+)764 1839 y(\024)738 1865 y Fm(H)819 1880 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1044 1865 y Fj(\024)53 b Fm(H)1255 1880 y Ft(t;s)1375 1865 y Fj(\000)1491 1768 y Fi(n)1547 1865 y Fm(v)t Fl(\()1651 1839 y Fm(~)1636 1865 y(G)1713 1880 y Ft(t)1742 1865 y Fl(\))1780 1824 y Fv(2)1842 1865 y Fj(\000)22 b Fm(v)t Fl(\()2044 1839 y Fm(~)2030 1865 y(F)2093 1880 y Ft(t)2123 1865 y Fl(\))2161 1824 y Fv(2)2200 1768 y Fi(o)3163 1865 y Fp(\(III.64\))378 2044 y Fl(=)16 b Fm(H)551 2059 y Ft(t;s)672 2044 y Fl(+)38 b(2)p Fm(v)t Fl(\()938 2019 y Fm(~)924 2044 y(F)987 2059 y Ft(t)1017 2044 y Fl(\))22 b Fj(\001)f Fm(a)p Fl(\()1230 2019 y Fm(~)1215 2044 y(G)1292 2059 y Ft(t)1344 2044 y Fj(\000)1458 2019 y Fm(~)1444 2044 y(F)1507 2059 y Ft(t)1537 2044 y Fl(\))38 b(+)h(2)p Fm(a)1828 2003 y Fs(\003)1867 2044 y Fl(\()1920 2019 y Fm(~)1905 2044 y(G)1982 2059 y Ft(t)2034 2044 y Fj(\000)2148 2019 y Fm(~)2134 2044 y(F)2197 2059 y Ft(t)2226 2044 y Fl(\))23 b Fj(\001)e Fm(v)t Fl(\()2439 2019 y Fm(~)2425 2044 y(F)2488 2059 y Ft(t)2518 2044 y Fl(\))38 b(+)h Fm(A)p Fl(\()2835 2019 y Fm(~)2820 2044 y(G)2897 2059 y Ft(t)2949 2044 y Fj(\000)3063 2019 y Fm(~)3048 2044 y(F)3111 2059 y Ft(t)3141 2044 y Fl(\))3179 2003 y Fv(2)3241 2044 y Fm(;)166 2280 y Fp(since)21 b Fm(v)t Fl(\()495 2254 y Fm(~)481 2280 y(F)544 2295 y Ft(t)574 2280 y Fl(\))10 b Fj(\001)g Fm(a)p Fl(\()764 2254 y Fm(~)749 2280 y(G)826 2295 y Ft(t)865 2280 y Fj(\000)967 2254 y Fm(~)952 2280 y(F)1015 2295 y Ft(t)1045 2280 y Fl(\))28 b(=)f Fm(a)p Fl(\()1318 2254 y Fm(~)1303 2280 y(G)1380 2295 y Ft(t)1420 2280 y Fj(\000)1522 2254 y Fm(~)1507 2280 y(F)1570 2295 y Ft(t)1600 2280 y Fl(\))10 b Fj(\001)g Fm(v)t Fl(\()1789 2254 y Fm(~)1775 2280 y(F)1838 2295 y Ft(t)1867 2280 y Fl(\))21 b Fp(and)h Fm(v)t Fl(\()2195 2254 y Fm(~)2181 2280 y(F)2244 2295 y Ft(t)2273 2280 y Fl(\))10 b Fj(\001)g Fm(a)2410 2243 y Fs(\003)2449 2280 y Fl(\()2502 2254 y Fm(~)2487 2280 y(G)2564 2295 y Ft(t)2604 2280 y Fj(\000)2705 2254 y Fm(~)2691 2280 y(F)2754 2295 y Ft(t)2784 2280 y Fl(\))27 b(=)h Fm(a)3004 2243 y Fs(\003)3044 2280 y Fl(\()3097 2254 y Fm(~)3082 2280 y(G)3159 2295 y Ft(t)3198 2280 y Fj(\000)3300 2254 y Fm(~)3285 2280 y(F)3348 2295 y Ft(t)3378 2280 y Fl(\))10 b Fj(\001)166 2409 y Fm(v)t Fl(\()269 2384 y Fm(~)255 2409 y(F)318 2424 y Ft(t)347 2409 y Fl(\))p Fp(,)21 b(due)g(to)g(the)f (Coulomb)g(gauge)g(condition)g(and)g(the)h(f)o(act)g(that)g Fl(supp)2764 2384 y Fm(~)2749 2409 y(F)2812 2424 y Ft(t)2850 2409 y Fj(\\)7 b Fl(supp)3156 2384 y Fm(~)3141 2409 y(G)3218 2424 y Ft(t)3275 2409 y Fl(=)28 b Fj(;)p Fp(.)166 2529 y(Applying)23 b(Eqs.)i(\(III.46\),)g(\(III.48\),)g(and)g(\(III.50\),)g (we)g(obtain)f(that,)g(for)h(an)o(y)g Fl(0)i Fm(<)h(")f Fj(\024)h Fl(2)p Fp(,)361 2904 y Fm(H)8 b Fl(\()502 2879 y Fm(~)488 2904 y(F)551 2919 y Ft(t)580 2904 y Fl(\))22 b(+)764 2879 y(\024)738 2904 y Fm(H)819 2919 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1008 2904 y Fj(\024)17 b Fm(H)1183 2919 y Ft(t;s)1303 2904 y Fl(+)39 b(2)p Fm("v)t Fl(\()1616 2879 y Fm(~)1602 2904 y(F)1665 2919 y Ft(t)1694 2904 y Fl(\))1732 2863 y Fv(2)1810 2904 y Fl(+)1934 2837 y(2)p 1934 2881 49 4 v 1936 2972 a Fm(")1993 2904 y Fj(k)p Fl(\001)2139 2879 y Fm(~)2124 2904 y(G)2201 2919 y Ft(t)2231 2904 y Fj(k)2281 2863 y Fv(2)2362 2879 y Fl(\024)2337 2904 y Fm(H)2418 2919 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1118 3102 y Fl(+)17 b Fj(k)p Fl(\001)1357 3077 y Fm(~)1342 3102 y(G)1419 3117 y Ft(t)1449 3102 y Fj(k)1499 3061 y Fv(2)1554 3102 y Fl(\()1618 3077 y(\024)1592 3102 y Fm(H)1673 3117 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1867 3102 y Fl(+)22 b Fm(\032)2015 3117 y Ft(t)2045 3102 y Fl(\))1008 3294 y Fj(\024)17 b Fm(H)1183 3309 y Ft(t;s)1303 3294 y Fl(+)39 b(4)p Fm(")17 b(H)1619 3253 y Fv(+)1676 3294 y Fl(\()1728 3269 y Fm(~)1714 3294 y(F)1777 3309 y Ft(t)1807 3294 y Fl(\))39 b(+)2008 3227 y(4)p 2008 3271 V 2010 3363 a Fm(")2084 3294 y Fj(k)p Fl(\001)2230 3269 y Fm(~)2215 3294 y(G)2292 3309 y Ft(t)2321 3294 y Fj(k)2371 3253 y Fv(2)2453 3269 y Fl(\024)2427 3294 y Fm(H)2508 3309 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1118 3496 y Fl(+)17 b(4)p Fm(")1306 3399 y Fi(\020)1355 3496 y Fm(b)23 b Fl(+)f Fm(\024)g Fl(+)g Fm(e)1738 3511 y Fv(0)1800 3496 y Fl(+)g Fj(k)1963 3470 y Fm(~)1948 3496 y(G)o Fj(k)2074 3454 y Fv(2)2114 3399 y Fi(\021)2202 3496 y Fl(+)39 b Fj(k)p Fl(\001)2463 3470 y Fm(~)2448 3496 y(G)2525 3511 y Ft(t)2554 3496 y Fj(k)2604 3454 y Fv(2)2660 3496 y Fm(\032)2710 3511 y Ft(t)2762 3496 y Fm(:)374 b Fp(\(III.65\))166 3753 y(W)-8 b(e)28 b(choose)f Fm(")32 b Fl(:=)851 3714 y Fv(1)p 851 3730 36 4 v 851 3788 a(8)896 3753 y Fj(k)p Fl(\001)1042 3728 y Fm(~)1027 3753 y(G)1104 3768 y Ft(t)1134 3753 y Fj(k)1184 3717 y Fv(2)1250 3753 y Fp(and)c(observ)o(e)e(that)h Fl(4)p Fm(")32 b Fl(=)2179 3714 y Fv(1)p 2179 3730 V 2179 3788 a(2)2224 3753 y Fj(k)p Fl(\001)2370 3728 y Fm(~)2355 3753 y(G)2432 3768 y Ft(t)2462 3753 y Fj(k)2512 3717 y Fv(2)2583 3753 y Fj(\024)2703 3714 y Fv(1)p 2703 3730 V 2703 3788 a(2)2776 3753 y Fp(and)2957 3714 y Fv(4)p 2957 3730 V 2958 3788 a Ft(")3019 3753 y Fj(k)p Fl(\001)3165 3728 y Fm(~)3150 3753 y(G)3227 3768 y Ft(t)3256 3753 y Fj(k)3306 3717 y Fv(2)3378 3753 y Fl(=)176 3834 y Fv(1)p 176 3850 V 176 3908 a(2)221 3874 y Fp(.)25 b(Inserting)f(this)g(choice) h(into)f(\(III.65\))h(yields)361 4261 y Fm(H)450 4220 y Fv(+)509 4261 y Fl(\()561 4236 y Fm(~)547 4261 y(F)610 4276 y Ft(t)639 4261 y Fl(\))d(+)823 4236 y(\024)798 4261 y Fm(H)879 4276 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1067 4261 y Fj(\024)17 b Fm(H)1250 4220 y Fv(+)1242 4286 y Ft(t;s)1362 4261 y Fl(+)1487 4194 y(1)p 1487 4238 49 4 v 1487 4329 a(2)1546 4165 y Fi(\020)1595 4261 y Fm(H)1684 4220 y Fv(+)1743 4261 y Fl(\()1795 4236 y Fm(~)1781 4261 y(F)1844 4276 y Ft(t)1873 4261 y Fl(\))23 b(+)2057 4236 y(\024)2032 4261 y Fm(H)2113 4276 y Ft(t)p Fs(\000)p Ft(\034)t(;s)2284 4165 y Fi(\021)3163 4261 y Fp(\(III.66\))1177 4494 y Fl(+)1280 4427 y(1)p 1280 4471 V 1280 4563 a(2)1339 4494 y Fj(k)p Fl(\001)1485 4469 y Fm(~)1470 4494 y(G)1547 4509 y Ft(t)1576 4494 y Fj(k)1626 4453 y Fv(2)1666 4398 y Fi(\020)1715 4494 y Fm(b)g Fl(+)f Fm(\024)g Fl(+)g Fm(e)2098 4509 y Fv(0)2160 4494 y Fl(+)g Fj(k)2323 4469 y Fm(~)2308 4494 y(G)p Fj(k)2435 4453 y Fv(2)2496 4494 y Fl(+)g(2)p Fm(\032)2693 4509 y Ft(t)2723 4398 y Fi(\021)2811 4494 y Fl(+)39 b Fm(E)2998 4509 y Ft(t)3050 4494 y Fj(\000)22 b Fm(E)6 b Fl(\()3279 4469 y Fm(~)3265 4494 y(F)3328 4509 y Ft(t)3358 4494 y Fl(\))22 b Fm(;)166 4770 y Fp(which)i(we)i(can) f(solv)o(e)e(for)i Fm(H)1202 4734 y Fv(+)1261 4770 y Fl(\()1313 4745 y Fm(~)1299 4770 y(F)1362 4785 y Ft(t)1392 4770 y Fl(\))d(+)1575 4745 y(\024)1550 4770 y Fm(H)1631 4785 y Ft(t)p Fs(\000)p Ft(\034)t(;s)1827 4770 y Fp(with)i(the)h (result)166 5010 y Fm(H)255 4969 y Fv(+)314 5010 y Fl(\()366 4984 y Fm(~)352 5010 y(F)415 5025 y Ft(t)444 5010 y Fl(\))10 b(+)603 4984 y(\024)578 5010 y Fm(H)659 5025 y Ft(t)p Fs(\000)p Ft(\034)t(;s)879 5010 y Fj(\024)50 b Fl(2)17 b Fm(H)1161 4969 y Fv(+)1153 5034 y Ft(t;s)1260 5010 y Fl(+)26 b Fj(k)p Fl(\001)1508 4984 y Fm(~)1493 5010 y(G)1570 5025 y Ft(t)1600 5010 y Fj(k)1650 4969 y Fv(2)1689 4913 y Fi(\020)1739 5010 y Fm(b)10 b Fl(+)g Fm(\024)g Fl(+)g Fm(e)2073 5025 y Fv(0)2121 5010 y Fl(+)g Fj(k)2272 4984 y Fm(~)2257 5010 y(G)o Fj(k)2383 4969 y Fv(2)2432 5010 y Fl(+)g(2)p Fm(\032)2617 5025 y Ft(t)2646 4913 y Fi(\021)2722 5010 y Fl(+)26 b(2)2873 4913 y Fi(\020)2922 5010 y Fm(E)2994 5025 y Ft(t)3034 5010 y Fj(\000)10 b Fm(E)c Fl(\()3251 4984 y Fm(~)3237 5010 y(F)3300 5025 y Ft(t)3330 5010 y Fl(\))3368 4913 y Fi(\021)3427 5010 y Fm(:)3163 5130 y Fp(\(III.67\))166 5250 y(Denoting)26 b(by)h Fm( )754 5265 y Ft(")819 5250 y Fp(an)h Fm(")p Fp(-approximate)e(ground)h(state)g(of)g Fm(H)8 b Fl(\()2316 5225 y Fm(~)2302 5250 y(F)2365 5265 y Ft(t)2394 5250 y Fl(\))p Fp(,)28 b(i.e.,)f(a)h(normalized)e(v)o(ector)166 5380 y(in)g Fl(dom\()p Fm(H)8 b Fl(\()633 5355 y Fm(~)619 5380 y(F)682 5395 y Ft(t)711 5380 y Fl(\)\))30 b Fj(\023)g Fl(dom\()p Fm(H)1227 5395 y Ft(t;s)1309 5380 y Fl(\))p Fp(,)25 b(for)i(which)e Fj(h)p Fm( )1911 5395 y Ft(")1948 5380 y Fj(j)k Fm(H)8 b Fl(\()2146 5355 y Fm(~)2132 5380 y(F)2195 5395 y Ft(t)2225 5380 y Fl(\))p Fm( )2326 5395 y Ft(")2363 5380 y Fj(i)29 b(\024)h Fm(E)6 b Fl(\()2668 5355 y Fm(~)2654 5380 y(F)2717 5395 y Ft(t)2747 5380 y Fl(\))23 b(+)g Fm(")p Fp(,)i(we)h(observ)o(e)1748 5712 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 166 83 a Fp(that)166 322 y Fm(E)238 337 y Ft(t)288 322 y Fj(\000)21 b Fm(E)6 b Fl(\()516 297 y Fm(~)502 322 y(F)565 337 y Ft(t)595 322 y Fl(\))52 b Fj(\024)814 225 y Fi(D)865 322 y Fm( )928 337 y Ft(")993 222 y Fi(\014)993 272 y(\014)993 322 y(\014)1048 225 y(\020)1098 322 y Fm(H)1179 337 y Ft(t;s)1281 322 y Fj(\000)21 b Fm(H)8 b Fl(\()1520 297 y Fm(~)1506 322 y(F)1569 337 y Ft(t)1598 322 y Fl(\))1636 225 y Fi(\021)1702 322 y Fm(\036)1760 337 y Ft(")1797 225 y Fi(E)1899 322 y Fl(=)2027 222 y Fi(\015)2027 272 y(\015)2027 322 y(\015)2088 297 y Fm(~)2074 322 y(G)2151 337 y Ft(t)2201 322 y Fj(\000)2313 297 y Fm(~)2298 322 y(F)2361 337 y Ft(t)2391 222 y Fi(\015)2391 272 y(\015)2391 322 y(\015)2437 248 y Fv(2)2497 322 y Fl(+)20 b Fm(")52 b Fj(\024)h Fm(\032)2871 337 y Ft(t)2917 322 y Fj(k)p Fl(\001)3063 297 y Fm(~)3048 322 y(G)3125 337 y Ft(t)3155 322 y Fj(k)3205 281 y Fv(2)3264 322 y Fl(+)21 b Fm(")f(:)3163 442 y Fp(\(III.68\))166 562 y(Letting)k Fm(")j Fj(!)g Fl(0)e Fp(and)g(additionally)e(using)h Fm(\032)1714 577 y Ft(t)1771 562 y Fj(\024)k Fm(\024)d Fp(and)g Fj(k)2191 537 y Fm(~)2176 562 y(G)p Fj(k)2303 526 y Fv(2)2370 562 y Fj(\024)j Fm(\024)2531 526 y Fv(2)2571 562 y Fp(,)c(we)h(arri)n(v)o(e)g(at)216 814 y Fm(H)305 772 y Fv(+)363 814 y Fl(\()415 788 y Fm(~)401 814 y(F)464 829 y Ft(t)494 814 y Fl(\))c(+)676 788 y(\024)650 814 y Fm(H)731 829 y Ft(t)p Fs(\000)p Ft(\034)t(;s)955 814 y Fj(\024)53 b Fl(2)17 b Fm(H)1240 772 y Fv(+)1232 838 y Ft(t;s)1351 814 y Fl(+)38 b Fj(k)p Fl(\001)1611 788 y Fm(~)1596 814 y(G)1673 829 y Ft(t)1702 814 y Fj(k)1752 772 y Fv(2)1791 717 y Fi(\020)1841 814 y Fm(b)22 b Fl(+)f(6)p Fm(\024)2106 772 y Fv(2)2166 814 y Fl(+)g Fm(e)2308 829 y Fv(0)2348 717 y Fi(\021)2450 814 y Fj(\024)53 b Fl(2\()p Fm(H)2756 772 y Fv(+)2748 838 y Ft(t;s)2850 814 y Fl(+)21 b Fm(\032)2997 829 y Ft(t)3027 814 y Fl(\))g Fm(;)50 b Fp(\(III.69\))166 1078 y(since)35 b Fm(\013)469 1042 y Fv(3)555 1078 y Fj(\024)47 b Fl(\()p Fm(b)30 b Fl(+)f(6)p Fm(\024)998 1042 y Fv(2)1068 1078 y Fl(+)g Fm(e)1218 1093 y Fv(0)1287 1078 y Fl(+)h(1\))1480 1042 y Fs(\000)p Fv(1)1574 1078 y Fp(.)35 b(This)f(establishes)g(the)h(claim)g(because) 3102 1053 y Fl(\024)3077 1078 y Fm(H)3158 1093 y Ft(t)p Fs(\000)p Ft(\034)t(;s)3376 1078 y Fj(\024)166 1210 y Fm(H)255 1174 y Fv(+)314 1210 y Fl(\()366 1185 y Fm(~)352 1210 y(F)415 1225 y Ft(t)444 1210 y Fl(\))22 b(+)628 1185 y(\024)602 1210 y Fm(H)683 1225 y Ft(t)p Fs(\000)p Ft(\034)t(;s)855 1210 y Fp(.)490 b Fb(2)166 1560 y Fp(The)31 b(desired)g(estimate)g(on)g Fm(W)1258 1575 y Ft(t;s)1371 1560 y Fp(is)g(no)n(w)f(a)i(simple)e(consequence)h(of)h(Lemmata)e (III.1)i(and)166 1680 y(III.2)166 1910 y Fq(Lemma)25 b(III.3)48 b Fh(F)-10 b(or)24 b(all)g Fl(0)k Fj(\024)g Fm(t)g(<)f(s)e Fh(and)f Fm(\013)1707 1874 y Fv(3)1774 1910 y Fj(\024)k Fl(\()p Fm(b)23 b Fl(+)f(6)p Fm(\024)2184 1874 y Fv(2)2245 1910 y Fl(+)g Fm(e)2388 1925 y Fv(0)2450 1910 y Fl(+)g(1\))2635 1874 y Fs(\000)p Fv(1)2729 1910 y Fh(,)495 2061 y Fi(\015)495 2111 y(\015)495 2161 y(\015)p Fl(\()p Fm(H)668 2120 y Fv(+)660 2185 y Ft(t;s)763 2161 y Fl(+)g Fm(\032)911 2176 y Ft(t)941 2161 y Fl(\))979 2120 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1161 2161 y Fm(W)1253 2176 y Ft(t;s)1351 2161 y Fl(\()p Fm(H)1478 2120 y Fv(+)1470 2185 y Ft(t;s)1574 2161 y Fl(+)g Fm(\032)1722 2176 y Ft(t)1752 2161 y Fl(\))1790 2120 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1955 2061 y Fi(\015)1955 2111 y(\015)1955 2161 y(\015)53 b Fj(\024)g Fl(8)17 b Fm(\021)i(\013)2380 2120 y Fv(3)p Ft(=)p Fv(2)2507 2161 y Fl(\()p Fm(s)j Fj(\000)g Fm(t)p Fl(\))h Fm(:)328 b Fp(\(III.70\))166 2508 y Fh(Pr)l(oof:)30 b Fp(W)-8 b(e)32 b(\002rst)g(pick)g Fl(0)40 b Fm(<)g(")g Fj(\024)h Fl(1)31 b Fp(and)h(observ)o(e)f(that,)g(by)g(Lemma)g(III.1,)h (the)g(follo)n(wing)166 2629 y(quadratic)25 b(form)f(estimate)g(holds)g (true,)361 2990 y Fj(\006)p Fm(W)530 3005 y Ft(t;s)630 2990 y Fl(=)17 b Fj(\006)p Fm(v)t Fl(\()904 2965 y Fm(~)889 2990 y(G)966 3005 y Ft(t)996 2990 y Fl(\))22 b Fj(\001)g Fm(A)p Fl(\()1232 2965 y Fm(~)1217 2990 y(G)1294 3005 y Ft(t;s)1376 2990 y Fl(\))44 b Fj(\006)h Fm(A)p Fl(\()1706 2965 y Fm(~)1691 2990 y(G)1768 3005 y Ft(t;s)1850 2990 y Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()2064 2965 y Fm(~)2049 2990 y(G)2126 3005 y Ft(t)2155 2990 y Fl(\))44 b Fj(\006)h Fm(A)p Fl(\()2485 2965 y Fm(~)2470 2990 y(G)2547 3005 y Ft(t;s)2629 2990 y Fl(\))2667 2949 y Fv(2)3163 2990 y Fp(\(III.71\))629 3148 y Fj(\024)17 b Fm(")g(v)t Fl(\()890 3123 y Fm(~)875 3148 y(G)952 3163 y Ft(t)981 3148 y Fl(\))1019 3107 y Fv(2)1102 3148 y Fl(+)45 b(2)p Fm(")1318 3107 y Fs(\000)p Fv(1)1428 3148 y Fm(A)p Fl(\()1554 3123 y Fm(~)1539 3148 y(G)1616 3163 y Ft(t;s)1698 3148 y Fl(\))1736 3107 y Fv(2)629 3325 y Fj(\024)17 b Fl(2)p Fm(")835 3229 y Fi(\020)884 3325 y Fm(H)973 3284 y Fv(+)1031 3325 y Fl(\()1084 3300 y Fm(~)1069 3325 y(G)1146 3340 y Ft(t)1176 3325 y Fl(\))22 b(+)g(1)1383 3229 y Fi(\021)1477 3325 y Fl(+)44 b(8)p Fm(")1692 3284 y Fs(\000)p Fv(1)1802 3225 y Fi(\015)1802 3275 y(\015)1802 3325 y(\015)p Fl(\()p Fm(!)1951 3284 y Fs(\000)p Fv(1)2067 3325 y Fl(+)22 b Fm(\032)2215 3284 y Fs(\000)p Fv(1)2215 3350 y Ft(t)2310 3325 y Fl(\))2348 3284 y Fv(1)p Ft(=)p Fv(2)2472 3300 y Fm(~)2458 3325 y(G)2535 3340 y Ft(t;s)2616 3225 y Fi(\015)2616 3275 y(\015)2616 3325 y(\015)2663 3252 y Fv(2)2719 3229 y Fi(\020)2793 3300 y Fl(\024)2768 3325 y Fm(H)8 b Fl(\()2910 3300 y Fm(~)2895 3325 y(G)2972 3340 y Ft(t;s)3053 3325 y Fl(\))23 b(+)f Fm(\032)3262 3340 y Ft(t)3292 3229 y Fi(\021)3363 3325 y Fm(:)166 3598 y Fp(Using)j Fm(H)519 3561 y Fv(+)578 3598 y Fl(\()631 3572 y Fm(~)616 3598 y(G)693 3613 y Ft(t)722 3598 y Fl(\))31 b Fj(\024)g Fm(H)988 3556 y Fv(+)980 3617 y Ft(t;s)1061 3598 y Fp(,)1138 3572 y Fl(\024)1113 3598 y Fm(H)7 b Fl(\()1254 3572 y Fm(~)1239 3598 y(G)1316 3613 y Ft(t;s)1398 3598 y Fl(\))30 b Fj(\024)1599 3572 y Fl(\024)1574 3598 y Fm(H)1655 3613 y Ft(t;s)1737 3598 y Fp(,)c(and)g(\(III.57\),)h(Eq.)f(\(III.71\))h(and)f(Lemma)f (III.2)166 3718 y(imply)e(that)779 3913 y Fj(\006)p Fm(W)948 3928 y Ft(t;s)1083 3913 y Fj(\024)1213 3792 y Fi(\022)1284 3845 y Fl(2)p Fm(")p 1284 3890 95 4 v 1291 3981 a(\032)1341 3996 y Ft(t)1427 3913 y Fl(+)1552 3845 y(8)p Fm(\021)1653 3809 y Fv(2)1708 3845 y Fm(\013)1771 3809 y Fv(3)1827 3845 y Fm(\032)1877 3860 y Ft(t)1924 3845 y Fl(\()p Fm(s)f Fj(\000)g Fm(t)p Fl(\))2202 3809 y Fv(2)p 1552 3890 691 4 v 1874 3981 a Fm(")2252 3792 y Fi(\023)2330 3817 y(\020)2379 3913 y Fm(H)2468 3872 y Fv(+)2460 3937 y Ft(t;s)2564 3913 y Fl(+)g Fm(\032)2712 3928 y Ft(t)2742 3817 y Fi(\021)2814 3913 y Fm(:)322 b Fp(\(III.72\))166 4143 y(Choosing)24 b Fm(")j Fl(:=)h(2)p Fm(\021)t(\013)941 4106 y Fv(3)p Ft(=)p Fv(2)1050 4143 y Fm(\032)1100 4158 y Ft(t)1130 4143 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fp(,)i(we)g(obtain)f (\(III.70\).)490 b Fb(2)166 4727 y Fq(IV)99 b(Spectral)26 b(Gap)166 5077 y Fp(The)i(goal)f(of)g(this)g(section)g(is)g(to)g(pro)o (v)o(e)g(that,)g(for)g Fm(t)33 b Fj(\025)h Fl(0)p Fp(,)27 b(the)g(ground)g(state)g(ener)n(gies)h Fm(E)3315 5092 y Ft(t)3378 5077 y Fl(=)166 5197 y(inf)c Fm(\033)t Fl([)p Fm(H)468 5212 y Ft(t)497 5197 y Fl(])34 b Fp(are)f(isolated)f(eigen)l (v)n(alues)h(of)g(multiplicity)d(one)j(and)g(to)g(deri)n(v)o(e)f(a)h (quantitati)n(v)o(e)166 5330 y(estimate)24 b(on)h(the)f(spectral)h(gap) f Fl(inf)1432 5234 y Fi(\020)1481 5330 y Fm(\033)t Fl([)p Fm(H)1656 5289 y Fv(+)1648 5350 y Ft(t;s)1730 5330 y Fl(])e Fj(n)g(f)p Fl(0)p Fj(g)2000 5234 y Fi(\021)2074 5330 y Fp(of)j Fm(H)2263 5345 y Ft(t;s)2370 5330 y Fp(abo)o(v)o(e)f Fm(E)2702 5345 y Ft(t;s)2811 5330 y Fl(=)k Fm(E)2987 5345 y Ft(t)3017 5330 y Fp(,)d(for)g Fm(s)i Fj(\025)h Fm(t)p Fp(.)1748 5712 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 166 83 a Fq(Lemma)25 b(IV)-14 b(.1)48 b Fh(F)-10 b(or)31 b(all)g Fl(0)41 b Fj(\024)h Fm(t)f(<)g(s)f Fj(\024)i Fm(t)27 b Fl(+)h Fm(\021)36 b Fh(and)31 b Fm(\013)2146 47 y Fv(3)2226 83 y Fj(\024)42 b Fl(\(64)26 b(+)i Fm(b)g Fl(+)f(6)p Fm(\024)2888 47 y Fv(2)2955 83 y Fl(+)g Fm(e)3103 98 y Fv(0)3143 83 y Fl(\))3181 47 y Fs(\000)p Fv(1)3275 83 y Fh(,)32 b(the)166 203 y(dif)n(fer)l(ence)25 b(of)f(the)h(gr)l(ound)e(state)h(ener)l(gies) h(obe)m(ys)1159 430 y Fj(j)p Fm(E)1259 445 y Ft(t)1311 430 y Fj(\000)d Fm(E)1482 445 y Ft(s)1519 430 y Fj(j)53 b(\024)g Fl(8)p Fm(\021)20 b(\013)1910 389 y Fv(3)p Ft(=)p Fv(2)2036 430 y Fl(\()p Fm(s)i Fj(\000)h Fm(t)p Fl(\))17 b Fm(\032)2382 445 y Ft(t)2434 430 y Fm(:)710 b Fp(\(IV)-13 b(.73\))166 760 y Fh(Pr)l(oof:)24 b Fp(Due)g(to)h(Lemma)f(III.3)h(we)g (ha)n(v)o(e)g(the)g(follo)n(wing)d(lo)n(wer)j(bound)f(for)h Fm(H)2921 775 y Ft(s)361 1089 y Fm(H)442 1104 y Ft(s)496 1089 y Fl(=)17 b Fm(E)661 1104 y Ft(t)713 1089 y Fl(+)22 b Fm(H)900 1048 y Fv(+)892 1114 y Ft(t;s)996 1089 y Fl(+)g Fm(W)1186 1104 y Ft(t;s)3171 1089 y Fp(\(IV)-13 b(.74\))496 1251 y Fl(=)17 b Fm(E)661 1266 y Ft(t)713 1251 y Fj(\000)23 b Fm(\032)863 1266 y Ft(t)915 1251 y Fl(+)f(\()p Fm(H)1140 1210 y Fv(+)1132 1275 y Ft(t;s)1236 1251 y Fl(+)g Fm(\032)1384 1266 y Ft(t)1414 1251 y Fl(\))1452 1210 y Fv(1)p Ft(=)p Fv(2)944 1325 y Fi(\022)1005 1446 y Fe(1)44 b Fl(+)g(\()p Fm(H)1352 1405 y Fv(+)1344 1471 y Ft(t;s)1448 1446 y Fl(+)22 b Fm(\032)1596 1461 y Ft(t)1626 1446 y Fl(\))1664 1405 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1846 1446 y Fm(W)1938 1461 y Ft(t;s)2036 1446 y Fl(\()p Fm(H)2163 1405 y Fv(+)2155 1471 y Ft(t;s)2259 1446 y Fl(+)g Fm(\032)2407 1461 y Ft(t)2437 1446 y Fl(\))2475 1405 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2640 1325 y Fi(\023)2717 1446 y Fl(\()p Fm(H)2844 1405 y Fv(+)2836 1471 y Ft(t;s)2940 1446 y Fl(+)g Fm(\032)3088 1461 y Ft(s)3125 1446 y Fl(\))3163 1405 y Fv(1)p Ft(=)p Fv(2)495 1654 y Fj(\025)17 b Fm(E)661 1669 y Ft(t)713 1654 y Fj(\000)23 b Fm(\032)863 1669 y Ft(t)915 1654 y Fl(+)1013 1557 y Fi(\020)1063 1654 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)1442 1613 y Fv(3)p Ft(=)p Fv(2)1551 1654 y Fl(\()p Fm(s)22 b Fj(\000)g Fm(t)p Fl(\))1829 1557 y Fi(\021)1896 1654 y Fl(\()p Fm(H)2023 1613 y Fv(+)2015 1678 y Ft(t;s)2118 1654 y Fl(+)h Fm(\032)2267 1669 y Ft(t)2296 1654 y Fl(\))496 1836 y(=)17 b Fm(E)661 1851 y Ft(t)730 1836 y Fj(\000)39 b Fl(8)p Fm(\021)t(\013)1010 1795 y Fv(3)p Ft(=)p Fv(2)1119 1836 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1448 1851 y Ft(t)1517 1836 y Fl(+)1632 1740 y Fi(\020)1681 1836 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)2060 1795 y Fv(3)p Ft(=)p Fv(2)2169 1836 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2448 1740 y Fi(\021)2514 1836 y Fm(H)2603 1795 y Fv(+)2595 1861 y Ft(t;s)2699 1836 y Fm(;)166 2087 y Fp(using)j(that)g Fl(1)d Fj(\000)i Fl(8)p Fm(\021)t(\013)923 2051 y Fv(3)p Ft(=)p Fv(2)1032 2087 y Fl(\()p Fm(s)e Fj(\000)h Fm(t)p Fl(\))32 b Fj(\025)g Fl(1)23 b Fj(\000)h Fl(8)p Fm(\013)1739 2051 y Fv(3)p Ft(=)p Fv(2)1880 2087 y Fj(\025)32 b Fl(0)p Fp(.)26 b(Similarly)-6 b(,)25 b(we)i(deri)n(v)o(e)f(the)h(follo)n(wing) 166 2207 y(upper)e(bound)451 2443 y Fm(H)532 2458 y Ft(s)622 2443 y Fj(\024)53 b Fm(E)824 2458 y Ft(t)892 2443 y Fl(+)39 b(8)p Fm(\021)t(\013)1171 2402 y Fv(3)p Ft(=)p Fv(2)1280 2443 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1609 2458 y Ft(t)1678 2443 y Fl(+)1792 2347 y Fi(\020)1842 2443 y Fl(1)44 b(+)g(8)p Fm(\021)t(\013)2219 2402 y Fv(3)p Ft(=)p Fv(2)2328 2443 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2607 2347 y Fi(\021)2673 2443 y Fm(H)2762 2402 y Fv(+)2754 2468 y Ft(t;s)2858 2443 y Fm(:)286 b Fp(\(IV)-13 b(.75\))166 2670 y(By)25 b(\(IV)-13 b(.74\),)25 b(the)g(ground)f(state) g(ener)n(gy)i Fm(E)1668 2685 y Ft(s)1730 2670 y Fp(is)e(bound)g(from)g (belo)n(w)g(by)1170 2897 y Fm(E)1242 2912 y Ft(s)1332 2897 y Fj(\025)53 b Fm(E)1534 2912 y Ft(t)1602 2897 y Fj(\000)40 b Fl(8)p Fm(\021)20 b(\013)1899 2856 y Fv(3)p Ft(=)p Fv(2)2025 2897 y Fl(\()p Fm(s)i Fj(\000)h Fm(t)p Fl(\))17 b Fm(\032)2371 2912 y Ft(t)2423 2897 y Fm(:)721 b Fp(\(IV)-13 b(.76\))166 3124 y(T)-8 b(o)34 b(establish)f(an)h(upper)h (bound)e(on)h Fm(E)1565 3139 y Ft(s)1636 3124 y Fp(we)h(\002x)f Fm(")45 b(>)g Fl(0)34 b Fp(and)g(suppose)f(that)h Fm(\036)2996 3139 y Ft(")3078 3124 y Fj(2)45 b(H)36 b Fp(is)d(a)166 3244 y(normalized)27 b Fm(")p Fp(-approximate)g(ground)h(state)g(of)g Fm(H)1961 3259 y Ft(t;s)2043 3244 y Fp(,)g(i.e.,)g Fj(h)p Fm(\036)2368 3259 y Ft(")2404 3244 y Fj(j)p Fm(H)2521 3203 y Fv(+)2513 3264 y Ft(t;s)2594 3244 y Fm(\036)2652 3259 y Ft(")2689 3244 y Fj(i)33 b(\024)h Fm(")p Fp(.)28 b(By)h(the)f(min-)166 3365 y(max)d(principle)f(and)g(\(IV)-13 b(.75\),)25 b(we)g(ha)n(v)o(e)216 3601 y Fm(E)288 3616 y Ft(s)376 3601 y Fj(\024)52 b(h)p Fm(\036)602 3616 y Ft(")655 3601 y Fj(j)27 b Fm(H)791 3616 y Ft(s)845 3601 y Fm(\036)903 3616 y Ft(")939 3601 y Fj(i)51 b(\024)h Fm(E)1230 3616 y Ft(t)1295 3601 y Fl(+)34 b(8)p Fm(\021)t(\013)1569 3559 y Fv(3)p Ft(=)p Fv(2)1678 3601 y Fl(\()p Fm(s)18 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1999 3616 y Ft(t)2063 3601 y Fl(+)2174 3504 y Fi(\020)2223 3601 y Fl(1)36 b(+)g(8)p Fm(\021)t(\013)2584 3559 y Fv(3)p Ft(=)p Fv(2)2693 3601 y Fl(\()p Fm(s)18 b Fj(\000)h Fm(t)p Fl(\))2964 3504 y Fi(\021)3030 3601 y Fm(")f(;)50 b Fp(\(IV)-13 b(.77\))166 3827 y(and)25 b(the)f(desired)h(bound)f(follo)n(ws)f(from)i Fm(")i Fj(!)h Fl(0)p Fp(.)489 b Fb(2)166 4171 y Fp(F)o(or)25 b Fl(0)i Fj(\024)h Fm(t)g(<)g(s)p Fp(,)d(we)g(introduce)f(three)h (nonne)o(gati)n(v)o(e)d(numbers)361 4500 y Fl(gap)513 4524 y Ft(t;s)611 4500 y Fl(:=)108 b(sup)731 4583 y Ft( )777 4592 y Fa(0)812 4583 y Fs(2Hnf)p Fv(0)p Fs(g)1248 4500 y Fl(inf)1101 4564 y Ft(\036)p Fs(?)p Ft( )1244 4573 y Fa(0)1279 4564 y Ft(;)11 b Fs(k)p Ft(\036)p Fs(k)p Fv(=1)1513 4500 y Fj(h)p Fm(\036)17 b Fj(j)27 b Fm(H)1771 4459 y Fv(+)1763 4525 y Ft(t;s)1861 4500 y Fm(\036)p Fj(i)21 b Fm(;)1165 b Fp(\(IV)-13 b(.78\))413 4723 y Fl(gap)565 4746 y Ft(t)611 4723 y Fl(:=)108 b(sup)731 4806 y Ft( )777 4815 y Fa(0)812 4806 y Fs(2Hnf)p Fv(0)p Fs(g)1248 4723 y Fl(inf)1101 4786 y Ft(\036)p Fs(?)p Ft( )1244 4795 y Fa(0)1279 4786 y Ft(;)11 b Fs(k)p Ft(\036)p Fs(k)p Fv(=1)1513 4723 y Fj(h)p Fm(\036)17 b Fj(j)27 b Fm(H)1771 4681 y Fv(+)1763 4747 y Ft(t)1846 4723 y Fm(\036)p Fj(i)21 b Fm(;)1180 b Fp(\(IV)-13 b(.79\))429 4951 y Fi(g)413 4958 y Fl(gap)565 4982 y Ft(t)611 4958 y Fl(:=)122 b(sup)731 5041 y Ft( )777 5050 y Fa(0)812 5041 y Fs(2H)919 5049 y Fc(t)947 5041 y Fs(nf)p Fv(0)p Fs(g)1276 4958 y Fl(inf)1129 5022 y Ft(\036)p Fs(?)p Ft( )1272 5031 y Fa(0)1307 5022 y Ft(;)11 b Fs(k)p Ft(\036)p Fs(k)p Fv(=1)1541 4958 y Fj(h)p Fm(\036)17 b Fj(j)1718 4926 y Fi(f)1709 4958 y Fm(H)1798 4917 y Fv(+)1790 4983 y Ft(t)1874 4958 y Fm(\036)p Fj(i)k Fm(;)1152 b Fp(\(IV)-13 b(.80\))166 5249 y(which)22 b(are)i(later)e(sho)n(wn)g(to)g(be)h(the)f (spectral)h(gaps)f(abo)o(v)o(e)f(the)i(ground)f(state)g(ener)n(gy)h(of) g Fm(H)3347 5264 y Ft(t;s)3429 5249 y Fp(,)166 5380 y Fm(H)247 5395 y Ft(t)277 5380 y Fp(,)h(and)503 5347 y Fi(f)495 5380 y Fm(H)576 5395 y Ft(t)606 5380 y Fp(,)g(respecti)n(v)o (ely)-6 b(,)23 b(where)i Fj(H)1528 5395 y Ft(t)1586 5380 y Fl(:=)i Fj(H)1800 5395 y Ft(el)1881 5380 y Fj(\012)c(F)10 b Fl([)p Ff(h)2142 5395 y Ft(t)2171 5380 y Fl(])25 b Fp(\(see)h(Eqs.)e(\(II.31\))h(and)g(\(II.33\)\).)1748 5712 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 166 83 a Fq(Lemma)25 b(IV)-14 b(.2)48 b Fh(Assume)24 b Fm(\013)1129 47 y Fv(3)1196 83 y Fj(\024)k Fl(\(64)22 b(+)g Fm(b)h Fl(+)f(6)p Fm(\024)1824 47 y Fv(2)1885 83 y Fl(+)g Fm(e)2028 98 y Fv(0)2068 83 y Fl(\))2106 47 y Fs(\000)p Fv(1)2200 83 y Fh(.)j(Then)857 310 y Fl(gap)1009 334 y Ft(t;s)1143 310 y Fl(=)53 b(min)p Fj(f)1500 303 y Fi(g)1484 310 y Fl(gap)1636 334 y Ft(t)1693 310 y Fm(;)44 b(\032)1814 325 y Ft(s)1851 310 y Fj(g)53 b(\025)g Fl(min)p Fj(f)p Fl(gap)2448 334 y Ft(t)2505 310 y Fm(;)45 b(\032)2627 325 y Ft(s)2664 310 y Fj(g)22 b Fm(;)408 b Fp(\(IV)-13 b(.81\))166 538 y Fh(for)24 b(all)g Fl(0)k Fj(\024)g Fm(t)g(<)f(s)h Fj(\024)g Fm(t)23 b Fl(+)f Fm(\021)t Fh(.)166 762 y(Pr)l(oof:)i Fp(W)-8 b(e)26 b(\002rst)f(remark)g(that)g(the)g(\002rst)g(equality)f Fl(gap)2090 785 y Ft(t;s)2200 762 y Fl(=)29 b(min)p Fj(f)2533 755 y Fi(g)2517 762 y Fl(gap)2669 785 y Ft(t)2698 762 y Fm(;)17 b(\032)2792 777 y Ft(s)2829 762 y Fj(g)25 b Fp(in)g(\(IV)-13 b(.81\))25 b(im-)166 882 y(plies)i(that)g Fl(gap)711 906 y Ft(t)773 882 y Fl(=)33 b(gap)1034 906 y Ft(t;t)1141 882 y Fl(=)g(min)q Fj(f)1479 875 y Fi(g)1463 882 y Fl(gap)1614 906 y Ft(t)1644 882 y Fm(;)17 b(\032)1738 897 y Ft(t)1768 882 y Fj(g)32 b(\024)1977 875 y Fi(g)1961 882 y Fl(gap)2112 906 y Ft(t)2170 882 y Fp(and)27 b(hence)h(the)g (second)f(inequality)166 1002 y(in)k(\(IV)-13 b(.81\).)31 b(It)g(therefore)h(suf)n(\002ces)f(to)f(sho)n(w)g(that)h Fl(gap)2109 1026 y Ft(t;s)2231 1002 y Fl(=)39 b(min)p Fj(f)2574 995 y Fi(g)2558 1002 y Fl(gap)2710 1026 y Ft(t)2739 1002 y Fm(;)17 b(\032)2833 1017 y Ft(s)2870 1002 y Fj(g)p Fp(.)31 b(T)-8 b(o)31 b(this)f(end,)166 1142 y(we)k(recall)h(from)f (\(II.30\))g(that)g Fm(H)1356 1157 y Ft(t;s)1483 1142 y Fl(=)1612 1109 y Fi(f)1603 1142 y Fm(H)1684 1157 y Ft(t)1743 1142 y Fj(\012)29 b Fe(1)1905 1106 y Ft(t)1905 1166 y Fs(1)2009 1142 y Fl(+)g Fe(1)2170 1157 y Ft(t)2229 1142 y Fj(\012)2361 1117 y Fl(\024)2335 1142 y Fm(H)2416 1157 y Ft(t;s)2532 1142 y Fp(on)34 b Fj(H)46 b Fl(=)f Fj(H)3001 1157 y Ft(t)3060 1142 y Fj(\012)29 b(F)10 b Fl([)p Ff(h)3327 1106 y Ft(t)3327 1166 y Fs(1)3402 1142 y Fl(])p Fp(.)166 1262 y(Denoting)42 b(by)g Fm(P)785 1277 y Fv(\012)900 1262 y Fl(:=)61 b Fe(1)1120 1277 y Ft(el)1214 1262 y Fj(\012)36 b Fe(1)1383 1277 y Ft(t)1448 1262 y Fj(\012)g(j)p Fl(\012)1659 1226 y Ft(t)1659 1287 y Fs(1)1734 1262 y Fj(ih)p Fl(\012)1882 1226 y Ft(t)1882 1287 y Fs(1)1956 1262 y Fj(j)43 b Fp(the)f(orthogonal)f(projection)h (onto)g(the)166 1383 y(v)n(acuum)37 b(v)o(ector)h Fl(\012)873 1346 y Ft(t)873 1407 y Fs(1)1000 1383 y Fj(2)53 b(F)10 b Fl([)p Ff(h)1280 1346 y Ft(t)1280 1407 y Fs(1)1354 1383 y Fl(])p Fp(,)38 b(we)h(de\002ne)f Fm(\037)53 b Fl(:)f Fj(H)h(!)f(H)2446 1398 y Ft(t)2514 1383 y Fp(by)38 b Fm(P)2715 1398 y Fv(\012)2770 1383 y Fm(\036)52 b Fl(=:)g Fm(\037)3096 1398 y Ft(\036)3174 1383 y Fj(\012)33 b Fl(\012)3354 1346 y Ft(t)3354 1407 y Fs(1)3429 1383 y Fp(.)166 1503 y(Note)28 b(that)f Fm(H)646 1518 y Ft(t;s)756 1503 y Fp(and)h Fm(P)991 1518 y Fv(\012)1074 1503 y Fp(commute.)f (Hence,)h(for)g(an)o(y)g(normalized)f Fm(\036)h Fp(with)f Fm(P)2950 1518 y Fv(\012)3005 1503 y Fm(\036)33 b Fj(6)p Fl(=)g(0)p Fp(,)28 b(i.e.,)166 1623 y Fl(0)f Fm(<)h Fj(k)p Fm(\037)457 1638 y Ft(\036)503 1623 y Fj(k)g Fl(=)f Fj(k)p Fm(P)797 1638 y Fv(\012)852 1623 y Fm(\036)p Fj(k)g(\024)h Fl(1)p Fp(,)d(we)g(ha)n(v)o(e)361 1978 y Fj(h)p Fm(\036)17 b Fj(j)27 b Fm(H)619 1937 y Fv(+)611 2002 y Ft(t;s)709 1978 y Fm(\036)p Fj(i)17 b Fl(=)916 1882 y Fi(D)967 1978 y Fm(P)1030 1993 y Fv(\012)1085 1978 y Fm(\036)1160 1878 y Fi(\014)1159 1928 y(\014)1159 1978 y(\014)27 b Fl(\()1260 1946 y Fi(f)1252 1978 y Fm(H)1333 1993 y Ft(t)1385 1978 y Fj(\012)c Fe(1)1541 1937 y Ft(t)1541 2002 y Fs(1)1615 1978 y Fl(\))17 b Fm(P)1733 1993 y Fv(\012)1788 1978 y Fm(\036)1846 1882 y Fi(E)1946 1978 y Fl(+)2072 1882 y Fi(D)2123 1978 y Fm(P)2200 1937 y Fs(?)2186 2002 y Fv(\012)2258 1978 y Fm(\036)2333 1878 y Fi(\014)2332 1928 y(\014)2332 1978 y(\014)28 b Fl(\()p Fe(1)2482 1993 y Ft(t)2534 1978 y Fj(\012)2658 1953 y Fl(\024)2633 1978 y Fm(H)2714 1993 y Ft(t;s)2796 1978 y Fl(\))17 b Fm(P)2928 1937 y Fs(?)2914 2002 y Fv(\012)2986 1978 y Fm(\036)p Fj(i)822 2160 y(\025)916 2064 y Fi(D)967 2160 y Fm(P)1030 2175 y Fv(\012)1085 2160 y Fm(\036)1160 2061 y Fi(\014)1159 2111 y(\014)1159 2160 y(\014)27 b Fl(\()1260 2128 y Fi(f)1252 2160 y Fm(H)1333 2175 y Ft(t)1385 2160 y Fj(\012)c Fe(1)1541 2119 y Ft(t)1541 2185 y Fs(1)1615 2160 y Fl(\))17 b Fm(P)1733 2175 y Fv(\012)1788 2160 y Fm(\036)1846 2064 y Fi(E)1946 2160 y Fl(+)50 b Fm(\032)2122 2175 y Ft(s)2176 2160 y Fj(k)p Fm(P)2303 2119 y Fs(?)2289 2185 y Fv(\012)2361 2160 y Fm(\036)p Fj(k)2469 2119 y Fv(2)3171 2160 y Fp(\(IV)-13 b(.82\))823 2368 y Fl(=)17 b Fj(k)p Fm(\037)1027 2383 y Ft(\036)1073 2368 y Fj(k)1123 2327 y Fv(2)1179 2247 y Fi(\034)1301 2301 y Fm(\037)1362 2316 y Ft(\036)p 1251 2345 207 4 v 1251 2436 a Fj(k)p Fm(\037)1362 2451 y Ft(\036)1408 2436 y Fj(k)1485 2244 y Fi(\014)1485 2293 y(\014)1485 2343 y(\014)1485 2393 y(\014)1548 2336 y(f)1540 2368 y Fm(H)1621 2383 y Ft(t)1727 2301 y Fm(\037)1788 2316 y Ft(\036)p 1677 2345 V 1677 2436 a Fj(k)p Fm(\037)1788 2451 y Ft(\036)1834 2436 y Fj(k)1894 2247 y Fi(\035)2006 2368 y Fl(+)50 b Fm(\032)2182 2383 y Ft(s)2235 2272 y Fi(\020)2285 2368 y Fl(1)22 b Fj(\000)g(k)p Fm(\037)2566 2383 y Ft(\036)2612 2368 y Fj(k)2662 2327 y Fv(2)2702 2272 y Fi(\021)2773 2368 y Fm(;)166 2654 y Fp(which)28 b(we)g(may)g(e)o(xtend)f(to)g(the)h(case)h Fj(k)p Fm(\037)1640 2669 y Ft(\036)1686 2654 y Fj(k)k Fl(=)h(0)28 b Fp(by)f(setting)g(the)h (\002rst)g(term)g(of)g(the)g(sum)f(in)166 2775 y(the)e(last)f(line)g (equal)h(to)f(zero.)i(It)e(follo)n(ws)g(for)h(an)o(y)f Fm( )32 b Fj(2)c(H)23 b(n)f(f)p Fl(0)p Fj(g)i Fp(with)g Fm(\037)2734 2790 y Ft( )2815 2775 y Fj(6)p Fl(=)j(0)e Fp(that)492 3111 y Fl(inf)361 3175 y Ft(\036)p Fs(?)p Ft( )r(;)12 b Fs(k)p Ft(\036)p Fs(k)p Fv(=1)741 3111 y Fj(h)p Fm(\036)17 b Fj(j)26 b Fm(H)998 3070 y Fv(+)990 3136 y Ft(t;s)1088 3111 y Fm(\036)p Fj(i)378 3357 y(\025)257 b Fl(inf)472 3421 y Ft(\037)516 3433 y Fc(\036)557 3421 y Fs(?)p Ft(\037)656 3433 y Fc( )702 3421 y Ft(;)12 b Fv(0)p Fs(\024k)p Ft(\037)903 3433 y Fc(\036)944 3421 y Fs(k\024)p Fv(1)1086 3236 y Fi(\032)1148 3357 y Fj(k)p Fm(\037)1259 3372 y Ft(\036)1305 3357 y Fj(k)1355 3316 y Fv(2)1411 3236 y Fi(\034)1533 3290 y Fm(\037)1594 3305 y Ft(\036)p 1483 3334 V 1483 3426 a Fj(k)p Fm(\037)1594 3441 y Ft(\036)1640 3426 y Fj(k)1717 3233 y Fi(\014)1717 3282 y(\014)1717 3332 y(\014)1717 3382 y(\014)1780 3325 y(f)1772 3357 y Fm(H)1861 3316 y Fv(+)1853 3382 y Ft(t)1996 3290 y Fm(\037)2057 3305 y Ft(\036)p 1946 3334 V 1946 3426 a Fj(k)p Fm(\037)2057 3441 y Ft(\036)2103 3426 y Fj(k)2163 3236 y Fi(\035)2275 3357 y Fl(+)50 b Fm(\032)2451 3372 y Ft(s)2505 3261 y Fi(\020)2554 3357 y Fl(1)22 b Fj(\000)h(k)p Fm(\037)2836 3372 y Ft(\036)2882 3357 y Fj(k)2932 3316 y Fv(2)2971 3261 y Fi(\021)3021 3236 y(\033)379 3611 y Fl(=)17 b(min)651 3490 y Fi(\022)882 3611 y Fl(inf)729 3675 y Ft(\037)p Fs(?)p Ft(\037)872 3687 y Fc( )918 3675 y Ft(;)11 b Fs(k)p Ft(\037)p Fs(k)p Fv(=1)1171 3515 y Fi(n)1226 3611 y Fj(h)p Fm(\037)17 b Fj(j)1406 3579 y Fi(f)1398 3611 y Fm(H)1487 3570 y Fv(+)1479 3636 y Ft(t)1562 3611 y Fm(\037)p Fj(i)1662 3515 y Fi(o)1742 3611 y Fm(;)42 b(\032)1861 3626 y Ft(s)1898 3490 y Fi(\023)1981 3611 y Fm(:)1163 b Fp(\(IV)-13 b(.83\))166 3888 y(Therefore)361 4241 y Fl(gap)513 4264 y Ft(t;s)611 4241 y Fj(\025)132 b Fl(sup)705 4320 y Ft( )r Fs(2H)p Fv(:)20 b Ft(\037)944 4332 y Fc( )991 4320 y Fs(6)p Fv(=0)1098 4119 y Fi(\032)1177 4241 y Fl(min)1356 4119 y Fi(\022)1587 4241 y Fl(inf)1433 4304 y Ft(\037)p Fs(?)p Ft(\037)1576 4316 y Fc( )1623 4304 y Ft(;)11 b Fs(k)p Ft(\037)p Fs(k)p Fv(=1)1875 4144 y Fi(n)1931 4241 y Fj(h)p Fm(\037)17 b Fj(j)2111 4208 y Fi(f)2102 4241 y Fm(H)2191 4199 y Fv(+)2183 4265 y Ft(t)2267 4241 y Fm(\037)p Fj(i)2367 4144 y Fi(o)2447 4241 y Fm(;)41 b(\032)2565 4256 y Ft(s)2602 4119 y Fi(\023)q(\033)612 4507 y Fl(=)17 b(min)885 4386 y Fi(\022)1077 4507 y Fl(sup)962 4587 y Ft( )r Fs(2H)p Fv(:)j Ft(\037)1201 4599 y Fc( )1248 4587 y Fs(6)p Fv(=0)1355 4386 y Fi(\024)1569 4507 y Fl(inf)1415 4571 y Ft(\037)p Fs(?)p Ft(\037)1558 4583 y Fc( )1604 4571 y Ft(;)12 b Fs(k)p Ft(\037)p Fs(k)p Fv(=1)1857 4411 y Fi(n)1912 4507 y Fj(h)p Fm(\037)17 b Fj(j)2092 4475 y Fi(f)2084 4507 y Fm(H)2173 4466 y Fv(+)2165 4532 y Ft(t)2248 4507 y Fm(\037)p Fj(i)2348 4411 y Fi(o)2403 4386 y(\025)2472 4507 y Fm(;)42 b(\032)2591 4522 y Ft(s)2628 4386 y Fi(\023)612 4749 y Fl(=)17 b(min)885 4653 y Fi(n)956 4742 y(g)940 4749 y Fl(gap)1092 4773 y Ft(t)1149 4749 y Fm(;)44 b(\032)1270 4764 y Ft(s)1308 4653 y Fi(o)1385 4749 y Fm(;)1759 b Fp(\(IV)-13 b(.84\))166 4977 y(establishing)32 b(the)i(lo)n(wer)g(bound)f(in)h(\(IV)-13 b(.81\).)35 b(F)o(or)f(the)g(upper)g(bound,)f(we)i(\002x)f Fm(")45 b(>)g Fl(0)34 b Fp(and)166 5097 y(choose)i Fl(~)-60 b Fm(\037)27 b Fj(2)h(H)734 5112 y Ft(t)789 5097 y Fp(such)d(that)1101 5317 y Fi(g)1086 5324 y Fl(gap)1237 5348 y Ft(t)1319 5324 y Fj(\024)54 b Fm(")49 b Fl(+)180 b(inf)1671 5388 y Ft(\037)p Fs(?)8 b Fv(~)-43 b Ft(\037)o(;)12 b Fs(k)p Ft(\037)p Fs(k)p Fv(=1)2049 5324 y Fj(h)p Fm(\037)17 b Fj(j)2229 5292 y Fi(f)2221 5324 y Fm(H)2310 5283 y Fv(+)2302 5349 y Ft(t)2385 5324 y Fm(\037)p Fj(i)22 b Fm(:)637 b Fp(\(IV)-13 b(.85\))1748 5712 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 166 83 a Fp(Then)236 296 y Fl(min)415 200 y Fi(n)486 289 y(g)470 296 y Fl(gap)622 320 y Ft(t)679 296 y Fm(;)45 b(\032)801 311 y Ft(s)838 200 y Fi(o)946 296 y Fj(\024)53 b Fm(")c Fl(+)264 b(inf)1297 360 y Ft(\036)p Fs(?)8 b Fv(~)-43 b Ft(\037)p Fs(\012)p Fv(\012)1544 341 y Fc(t)1544 376 y Fd(1)1609 360 y Ft(;)11 b Fs(k)p Ft(\036)p Fs(k)p Fv(=1)1859 200 y Fi(n)1915 296 y Fj(h)p Fm(\036)17 b Fj(j)26 b Fm(H)2172 255 y Fv(+)2164 321 y Ft(t;s)2262 296 y Fm(\036)p Fj(i)2359 200 y Fi(o)2467 296 y Fj(\024)53 b Fm(")c Fl(+)h(gap)2970 320 y Ft(t;s)3074 296 y Fm(;)70 b Fp(\(IV)-13 b(.86\))166 554 y(and)25 b(the)f(upper)h(bound)f(in)h(\(IV)-13 b(.81\))25 b(follo)n(ws)e(by)h (taking)g Fm(")k Fj(!)f Fl(0)p Fp(.)490 b Fb(2)166 894 y Fq(Theor)n(em)27 b(IV)-14 b(.3)47 b Fh(Assume)23 b Fm(\013)1187 858 y Fv(3)1254 894 y Fj(\024)28 b Fl(\(2500)p Fm(#)1650 858 y Fv(2)1708 894 y Fl(+)18 b Fm(b)g Fl(+)g(6)p Fm(\024)2060 858 y Fv(2)2118 894 y Fl(+)g Fm(e)2257 909 y Fv(0)2297 894 y Fl(\))2335 858 y Fs(\000)p Fv(1)2429 894 y Fh(,)24 b(wher)l(e)g Fm(#)k Fl(:=)g(\001)p Fm(e)3083 909 y Ft(el)3142 894 y Fm(=\024)g Fj(\024)g Fl(1)p Fh(.)166 1015 y(Then)d Fm(E)462 1030 y Ft(t)517 1015 y Fh(is)f(an)h(isolated,)e (non-de)l(g)o(ener)o(ate)h(eig)o(en)l(value)g(of)h Fm(H)2377 1030 y Ft(t)2406 1015 y Fh(,)g(and)1067 1228 y Fl(inf)1202 1131 y Fi(\020)1251 1228 y Fm(\033)t Fl([)p Fm(H)1426 1187 y Fv(+)1418 1252 y Ft(t)1485 1228 y Fl(])d Fj(n)g(f)p Fl(0)p Fj(g)1755 1131 y Fi(\021)1857 1228 y Fl(=)52 b(gap)2137 1251 y Ft(t)2219 1228 y Fj(\025)h Fm(#)17 b(\032)2473 1243 y Ft(t)2525 1228 y Fm(:)619 b Fp(\(IV)-13 b(.87\))166 1560 y Fh(Pr)l(oof:)24 b Fp(W)-8 b(e)25 b(\002rst)g(remark)g(that)f Fl(gap)1415 1583 y Ft(t)1472 1560 y Fj(\025)k Fm(#)17 b(\032)1701 1575 y Ft(t)1731 1560 y Fp(,)25 b(for)g(all)g Fm(t)j Fj(\025)g Fl(0)p Fp(,)c(if)n(f)h Fm(J)37 b Fl(=)27 b Fg(R)2696 1518 y Fv(+)2696 1581 y(0)2755 1560 y Fp(,)e(where)931 1784 y Fm(J)61 b Fl(:=)1202 1688 y Fi(n)1257 1784 y Fm(s)28 b Fj(2)g Fg(R)1522 1684 y Fi(\014)1522 1734 y(\014)1522 1784 y(\014)c Fj(8)p Fl(0)k Fj(\024)g Fm(t)g Fj(\024)h Fm(s)e Fl(:)53 b(gap)2285 1807 y Ft(t)2342 1784 y Fj(\025)28 b Fm(#\032)2554 1799 y Ft(t)2584 1688 y Fi(o)2662 1784 y Fm(:)482 b Fp(\(IV)-13 b(.88\))166 1997 y(Ob)o(viously)-6 b(,)22 b Fm(J)34 b Fp(is)25 b(an)g(\(possibly)e(de)o(generate\))i (interv)n(al.)f Fm(J)37 b Fj(3)28 b Fl(0)d Fp(is)g(not)f(empty)g(since) h Fl(gap)3311 2021 y Fv(0)3378 1997 y Fl(=)166 2117 y(min)q Fj(f)p Fl(\001)p Fm(e)505 2132 y Ft(el)564 2117 y Fm(;)17 b(\024)p Fj(g)27 b Fl(=)h(\001)p Fm(e)971 2132 y Ft(el)1057 2117 y Fl(=)g Fm(#\024)g Fl(=)g Fm(#\032)1513 2132 y Fv(0)1553 2117 y Fp(.)d(W)-8 b(e)25 b(no)n(w)f(demonstrate)g(that)g (the)h(assumption)1409 2331 y Fm(s)1455 2346 y Fs(\003)1547 2331 y Fl(:=)53 b(sup)17 b Fm(J)62 b(<)53 b Fj(1)960 b Fp(\(IV)-13 b(.89\))166 2544 y(leads)33 b(to)g(a)g(contradiction.)f (T)-8 b(o)33 b(this)f(end,)h(we)h(\002x)f Fl(0)43 b Fm(<)g(\016)k Fj(\024)d Fl(min)q Fj(f)p Fm(\021)t(;)59 b(\021)2804 2508 y Fs(\000)p Fv(1)2915 2544 y Fl(ln\(1)p Fm(=#)p Fl(\))p Fj(g)33 b Fp(and)166 2664 y(choose)e Fl(0)39 b Fj(\024)h Fm(t)g Fj(\024)g Fm(s)917 2679 y Fs(\003)995 2664 y Fm(<)g(s)31 b Fp(such)g(that)f Fm(t)40 b Fj(2)g Fm(J)9 b Fp(,)31 b Fm(s)51 b(=)-60 b Fj(2)39 b Fm(J)9 b Fp(,)32 b(and)f Fm(s)c Fj(\000)g Fm(t)40 b Fj(\024)g Fm(\016)t Fp(.)31 b(Due)g(to)g(\(IV)-13 b(.73\),)166 2784 y(\(IV)g(.74\),)25 b(and)g(Lemma)f(IV)-13 b(.2,)25 b(we)g(ha)n(v)o(e)320 2998 y Fm(H)409 2957 y Fv(+)401 3022 y Ft(s)520 2998 y Fj(\025)53 b Fm(E)722 3013 y Ft(t)774 2998 y Fj(\000)22 b Fm(E)945 3013 y Ft(s)1021 2998 y Fj(\000)39 b Fl(8)p Fm(\021)t(\013)1301 2957 y Fv(3)p Ft(=)p Fv(2)1410 2998 y Fl(\()p Fm(s)23 b Fj(\000)f Fm(t)p Fl(\))p Fm(\032)1739 3013 y Ft(t)1808 2998 y Fl(+)1923 2901 y Fi(\020)1972 2998 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)2351 2957 y Fv(3)p Ft(=)p Fv(2)2460 2998 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2739 2901 y Fi(\021)2805 2998 y Fm(H)2894 2957 y Fv(+)2886 3022 y Ft(t;s)2990 2998 y Fm(;)154 b Fp(\(IV)-13 b(.90\))166 3211 y(which)24 b(implies)g(that)361 3541 y Fl(gap)513 3565 y Ft(s)566 3541 y Fj(\025)17 b Fm(E)732 3556 y Ft(t)784 3541 y Fj(\000)23 b Fm(E)956 3556 y Ft(s)1032 3541 y Fj(\000)39 b Fl(8)p Fm(\021)t(\013)1312 3500 y Fv(3)p Ft(=)p Fv(2)1421 3541 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1750 3556 y Ft(t)1819 3541 y Fl(+)1933 3445 y Fi(\020)1983 3541 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)2362 3500 y Fv(3)p Ft(=)p Fv(2)2471 3541 y Fl(\()p Fm(s)22 b Fj(\000)g Fm(t)p Fl(\))2749 3445 y Fi(\021)2816 3541 y Fl(gap)2968 3565 y Ft(t;s)3171 3541 y Fp(\(IV)-13 b(.91\))567 3724 y Fl(=)17 b Fj(\000)g Fl(16)p Fm(\021)t(\013)967 3683 y Fv(3)p Ft(=)p Fv(2)1076 3724 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1405 3739 y Ft(t)1474 3724 y Fl(+)1589 3627 y Fi(\020)1638 3724 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)2017 3683 y Fv(3)p Ft(=)p Fv(2)2126 3724 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2405 3627 y Fi(\021)2488 3724 y Fl(min)2667 3627 y Fi(n)2738 3717 y(g)2722 3724 y Fl(gap)2874 3747 y Ft(t)2920 3724 y Fm(;)45 b(\032)3042 3739 y Ft(s)3079 3627 y Fi(o)566 3906 y Fj(\025)17 b(\000)g Fl(16)p Fm(\021)t(\013)967 3865 y Fv(3)p Ft(=)p Fv(2)1076 3906 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))p Fm(\032)1405 3921 y Ft(t)1474 3906 y Fl(+)1589 3810 y Fi(\020)1638 3906 y Fl(1)44 b Fj(\000)h Fl(8)p Fm(\021)t(\013)2017 3865 y Fv(3)p Ft(=)p Fv(2)2126 3906 y Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2405 3810 y Fi(\021)2488 3906 y Fl(min)2667 3810 y Fi(n)2722 3906 y Fl(gap)2874 3930 y Ft(t)2920 3906 y Fm(;)45 b(\032)3042 3921 y Ft(s)3079 3810 y Fi(o)3156 3906 y Fm(:)166 4136 y Fp(Since)25 b Fm(t)j Fj(2)g Fm(J)34 b Fp(and)25 b Fm(s)d Fj(\000)h Fm(t)28 b Fj(\024)g Fm(\016)k Fj(\024)c Fm(\021)1394 4100 y Fs(\000)p Fv(1)1504 4136 y Fl(ln)q(\(1)p Fm(=#)p Fl(\))p Fp(,)894 4362 y Fl(min)1073 4266 y Fi(n)1128 4362 y Fl(gap)1280 4385 y Ft(t)1326 4362 y Fm(;)45 b(\032)1448 4377 y Ft(s)1485 4266 y Fi(o)1593 4362 y Fj(\025)53 b Fl(min)1902 4266 y Fi(n)1957 4362 y Fm(#)17 b(\032)2081 4377 y Ft(t)2128 4362 y Fm(;)44 b(\032)2249 4377 y Ft(s)2286 4266 y Fi(o)2394 4362 y Fl(=)53 b Fm(#)17 b(\032)2647 4377 y Ft(t)2699 4362 y Fm(:)445 b Fp(\(IV)-13 b(.92\))166 4587 y(On)25 b(the)f(other)h(hand,)f Fm(s)40 b(=)-61 b Fj(2)28 b Fm(J)34 b Fp(implies)24 b Fl(gap)1658 4611 y Ft(s)1723 4587 y Fj(\024)k Fm(#\032)1935 4602 y Ft(s)1997 4587 y Fp(and)d(hence)577 4837 y Fl(1)52 b Fj(\025)h Fm(f)11 b Fl([)p Fm(\021)t Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\)])52 b(:=)h Fm(e)1505 4795 y Ft(\021)r Fv(\()p Ft(s)p Fs(\000)p Ft(t)p Fv(\))1715 4715 y Fi(\022)1776 4837 y Fl(1)44 b Fj(\000)h Fl(24)p Fm(\013)2152 4795 y Fv(3)p Ft(=)p Fv(2)2261 4837 y Fm(#\021)t Fl(\()p Fm(s)22 b Fj(\000)h Fm(t)p Fl(\))2649 4715 y Fi(\023)2732 4837 y Fm(;)412 b Fp(\(IV)-13 b(.93\))166 5087 y(additionally)23 b(using)h Fm(#)k Fj(\024)g Fl(1)p Fp(.)d(Note,)f(ho)n(we)n(v)o(er)l(,)g (that)g Fm(f)11 b Fl([0])27 b(=)h(1)c Fp(and)1227 5345 y Fm(f)1286 5304 y Fs(0)1309 5345 y Fl([0])53 b(=)f(1)22 b Fj(\000)h Fl(24)p Fm(\013)1925 5304 y Fv(3)p Ft(=)p Fv(2)2034 5345 y Fm(#)53 b Fj(\025)2284 5277 y Fl(1)p 2284 5321 49 4 v 2284 5413 a(2)2365 5345 y Fm(;)779 b Fp(\(IV)-13 b(.94\))1748 5712 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 166 86 a Fp(since)24 b Fm(\013)458 50 y Fv(3)p Ft(=)p Fv(2)595 86 y Fj(\024)29 b Fl(\(50)p Fm(#)p Fl(\))932 50 y Fs(\000)p Fv(1)1026 86 y Fp(.)24 b(So,)h Fm(f)11 b Fl([)p Fm(")p Fl(])27 b Fm(>)g Fl(1)p Fp(,)d(for)h(suf)n(\002ciently)e(small)g(v)n(alues)h(of)g Fm(")k(>)f Fl(0)p Fp(,)d(yielding)166 206 y(a)29 b(contradiction)e(to)h (\(IV)-13 b(.93\).)29 b(It)f(follo)n(ws)f(that)h Fm(s)1874 221 y Fs(\003)1948 206 y Fl(=)34 b Fj(1)28 b Fp(and)g Fm(J)44 b Fl(=)34 b Fg(R)2638 165 y Fv(+)2638 228 y(0)2697 206 y Fp(,)29 b(i.e.,)f(for)g(all)h Fm(t)34 b Fj(\025)h Fl(0)p Fp(,)166 327 y(we)30 b(ha)n(v)o(e)g Fl(gap)678 350 y Ft(t)745 327 y Fj(\025)37 b Fm(#)17 b(\032)983 342 y Ft(t)1013 327 y Fp(.)30 b(The)g(min-max)e(principle)h(no)n(w)g (immediately)f(implies)g(that)i Fm(E)3328 342 y Ft(t)3387 327 y Fp(is)166 447 y(an)25 b(isolated,)f(non-de)o(generate)g(eigen)l (v)n(alue)g(of)h Fm(H)1921 462 y Ft(t)1951 447 y Fp(.)489 b Fb(2)166 1007 y Fq(V)100 b(Gr)n(ound)25 b(State)h(Pr)n(ojections)166 1353 y Fp(The)d(present)h(section)e(is)h(de)n(v)n(oted)g(to)g(the)g (construction)f(of)i(the)f(spectral)g(projection)g Fm(P)3224 1368 y Ft(t)3277 1353 y Fp(onto)166 1473 y(the)31 b(simple)e(eigen)l(v) n(alue)h Fm(E)1143 1488 y Ft(t)1204 1473 y Fp(of)h Fm(H)1399 1488 y Ft(t)1428 1473 y Fp(,)g(for)g Fm(t)39 b(>)g Fl(0)p Fp(.)31 b(T)-8 b(o)30 b(this)g(end,)h(we)g(choose)f Fl(0)39 b Fm(<)g(\016)j(<)3351 1434 y Ft(#)p 3300 1450 144 4 v 3300 1507 a Fv(128)p Ft(\021)166 1615 y Fp(and)30 b Fm(s)36 b(>)h Fl(0)30 b Fp(such)f(that)g Fm(s)d Fj(\000)g Fm(\016)41 b(<)c(t)g(<)f(s)p Fp(.)30 b(Since)g Fm(\021)t(\016)21 b(e)2074 1578 y Ft(\021)r(\016)2186 1615 y Fj(\024)2342 1575 y Ft(#)p 2310 1591 106 4 v 2310 1649 a Fv(128)2443 1615 y Fm(e)2488 1578 y Ft(#=)p Fv(128)2712 1615 y Fj(\024)2851 1575 y Ft(#)p 2836 1591 71 4 v 2836 1649 a Fv(64)2917 1615 y Fp(,)30 b(we)g(ha)n(v)o(e)f(the)166 1735 y(estimate)336 2009 y Fj(j)p Fm(E)436 2024 y Ft(t)488 2009 y Fj(\000)23 b Fm(E)660 2024 y Ft(s)697 2009 y Fj(j)52 b(\024)h Fl(8)17 b Fm(\021)j(\013)1104 1968 y Fv(3)p Ft(=)p Fv(2)1230 2009 y Fl(\()p Fm(s)i Fj(\000)h Fm(t)p Fl(\))17 b Fm(\032)1576 2024 y Ft(t)1658 2009 y Fj(\024)53 b Fl(16)17 b Fm(\013)1966 1968 y Fv(3)p Ft(=)p Fv(2)2091 2009 y Fm(\021)t(\016)k(e)2252 1968 y Ft(\021)r(\016)2344 2009 y Fm(\032)2394 2024 y Ft(s)2484 2009 y Fj(\024)2624 1941 y Fm(\013)2687 1905 y Fv(3)p Ft(=)p Fv(2)2813 1941 y Fm(#)p 2624 1986 247 4 v 2723 2077 a Fl(8)2897 2009 y Fm(\032)2947 2024 y Ft(s)3006 2009 y Fm(:)171 b Fp(\(V)-13 b(.95\))166 2264 y(W)-8 b(e)25 b(de\002ne)1174 2417 y Fl(\000)1235 2432 y Ft(t)1317 2417 y Fl(:=)1473 2295 y Fi(\032)1535 2417 y Fm(z)32 b Fj(2)c Fg(C)1806 2292 y Fi(\014)1806 2342 y(\014)1806 2392 y(\014)1806 2442 y(\014)1859 2417 y Fj(j)p Fm(z)t Fj(j)f Fl(=)2105 2349 y(1)p 2105 2393 49 4 v 2105 2485 a(4)2180 2417 y Fm(#)17 b(\032)2304 2432 y Ft(t)2334 2295 y Fi(\033)2419 2417 y Fm(;)758 b Fp(\(V)-13 b(.96\))166 2615 y(and)31 b(we)h(observ)o(e)f(that)g Fl(gap)1162 2638 y Ft(t;s)1283 2615 y Fj(\025)41 b Fl(min)p Fj(f)p Fl(gap)1765 2638 y Ft(t)1794 2615 y Fm(;)17 b(\032)1888 2630 y Ft(s)1925 2615 y Fj(g)40 b(\025)g Fm(#\032)2239 2630 y Ft(t)2310 2615 y Fj(\025)g Fm(#\032)2534 2630 y Ft(s)2603 2615 y Fp([see)32 b(Eq.)f(\(IV)-13 b(.92\)])32 b(and)166 2735 y(\(V)-13 b(.95\))25 b(imply)361 3068 y Fj(j)p Fm(\025)d Fj(\000)g Fm(E)639 3083 y Ft(s)699 3068 y Fj(\000)g Fm(z)843 3083 y Ft(s)880 3068 y Fj(j)17 b(\025)g(j)p Fm(\025)k Fj(\000)i Fm(E)1297 3083 y Ft(t)1327 3068 y Fj(j)f(\000)g(j)p Fm(E)1576 3083 y Ft(t)1628 3068 y Fj(\000)h Fm(E)1800 3083 y Ft(s)1837 3068 y Fj(j)e(\000)i(j)p Fm(z)2059 3083 y Ft(s)2096 3068 y Fj(j)925 3262 y(\025)17 b Fl(gap)1170 3286 y Ft(t;s)1274 3262 y Fj(\000)1384 3195 y Fm(#)p 1384 3239 58 4 v 1388 3331 a Fl(8)1468 3262 y Fm(\032)1518 3277 y Ft(s)1577 3262 y Fj(\000)1687 3195 y Fm(#)p 1687 3239 V 1691 3331 a Fl(4)1771 3262 y Fm(\032)1821 3277 y Ft(s)1938 3262 y Fj(\025)2106 3195 y Fl(5)p Fm(#)p 2106 3239 107 4 v 2135 3331 a Fl(8)2239 3262 y Fm(\032)2289 3277 y Ft(s)2348 3262 y Fm(;)829 b Fp(\(V)-13 b(.97\))166 3525 y(for)23 b(e)n(v)o(ery)g Fm(z)591 3540 y Ft(s)656 3525 y Fj(2)28 b Fl(\000)811 3540 y Ft(s)871 3525 y Fp(and)23 b Fm(\025)k Fj(2)h Fm(\033)t Fl([)p Fm(H)1383 3540 y Ft(t;s)1465 3525 y Fl(])16 b Fj(n)g(f)p Fm(E)1696 3540 y Ft(t)1725 3525 y Fj(g)p Fp(.)23 b(Therefore,)h Fm(E)2336 3540 y Ft(t)2389 3525 y Fp(is)f(the)g(only)f (point)g(in)h Fm(\033)t Fl([)p Fm(H)3320 3540 y Ft(t;s)3402 3525 y Fl(])p Fp(,)166 3645 y(which)i(is)f(encircled)h(by)g Fj(f)p Fm(E)1163 3660 y Ft(s)1199 3645 y Fj(g)d(\000)h Fl(\000)1432 3660 y Ft(s)1494 3645 y Fp(and)i Fl(dist\()p Fm(\033)t Fl([)p Fm(H)2025 3660 y Ft(t;s)2107 3645 y Fl(])p Fm(;)34 b Fl(\000)2256 3660 y Ft(s)2292 3645 y Fl(\))28 b Fj(\025)2473 3606 y Ft(#)p 2473 3622 42 4 v 2476 3679 a Fv(8)2525 3645 y Fm(\032)2575 3660 y Ft(s)2612 3645 y Fp(.)d(The)g(same)g(statement)166 3765 y(holds)f(for)h Fm(\033)t Fl([)p Fm(H)715 3780 y Ft(t)744 3765 y Fl(])g Fp(instead)g(of)g Fm(\033)t Fl([)p Fm(H)1379 3780 y Ft(t;s)1460 3765 y Fl(])p Fp(.)g(The)g(Riesz-formula)g(yields)1070 4163 y Fm(P)1133 4178 y Ft(t)1179 4163 y Fl(=)1289 4095 y Fj(\000)p Fl(1)p 1281 4140 141 4 v 1281 4231 a(2)p Fm(\031)t(i)1449 4046 y Fi(Z)1495 4234 y Fv(\000)1539 4242 y Fc(s)1808 4095 y Fm(dz)1904 4110 y Ft(s)p 1603 4140 544 4 v 1603 4231 a Fm(H)1684 4246 y Ft(t)1736 4231 y Fj(\000)d Fm(E)1907 4246 y Ft(s)1966 4231 y Fl(+)g Fm(z)2109 4246 y Ft(s)3204 4163 y Fp(\(V)-13 b(.98\))1179 4416 y Fl(=)16 b Fm(P)1334 4431 y Ft(t;s)1444 4416 y Fl(=)1565 4348 y Fj(\000)p Fl(1)p 1557 4393 141 4 v 1557 4484 a(2)p Fm(\031)t(i)1725 4299 y Fi(Z)1771 4487 y Fv(\000)1815 4495 y Fc(s)2110 4348 y Fm(dz)2206 4363 y Ft(s)p 1879 4393 596 4 v 1879 4484 a Fm(H)1960 4499 y Ft(t;s)2064 4484 y Fj(\000)22 b Fm(E)2235 4499 y Ft(s)2295 4484 y Fl(+)g Fm(z)2438 4499 y Ft(s)2507 4416 y Fm(:)670 b Fp(\(V)-13 b(.99\))166 4710 y Fq(Lemma)25 b(V)-14 b(.1)48 b Fh(Let)23 b Fm(s)28 b(>)g Fl(0)22 b Fh(and)g Fm(\014)33 b Fj(2)c(f)p Fl(0)p Fm(;)17 b Fl(1)p Fj(g)p Fh(,)k(and)h(assume)g(that)g Fm(\013)2469 4674 y Fv(3)2536 4710 y Fj(\024)28 b Fl(\(2500)p Fm(#)2932 4674 y Fv(2)2985 4710 y Fl(+)14 b Fm(b)g Fl(+)g(6)p Fm(\024)3325 4674 y Fv(2)3378 4710 y Fl(+)166 4831 y Fm(e)211 4846 y Fv(0)251 4831 y Fl(\))289 4794 y Fs(\000)p Fv(1)409 4831 y Fh(and)25 b Fl(0)30 b Fm(<)f(\016)34 b(<)29 b Fl(min)q Fj(f)1223 4791 y Ft(#)p 1173 4807 144 4 v 1173 4865 a Fv(128)p Ft(\021)1326 4831 y Fm(;)17 b(s)p Fj(g)p Fh(.)25 b(Then,)i(for)e(all)g Fm(s)e Fj(\000)g Fm(\016)34 b(<)29 b(t)h(<)g(r)i Fj(\024)e Fm(s)c Fh(and)f(all)g Fm(z)3168 4846 y Ft(s)3235 4831 y Fj(2)30 b Fl(\000)3392 4846 y Ft(s)3429 4831 y Fh(,)166 4951 y(we)c(have)166 5104 y Fi(\015)166 5154 y(\015)166 5204 y(\015)212 5107 y(n)268 5204 y Fl(\()p Fm(H)387 5219 y Ft(r)n(;s)488 5204 y Fj(\000)16 b Fm(E)653 5219 y Ft(s)706 5204 y Fl(+)g Fm(z)843 5219 y Ft(s)879 5204 y Fl(\))917 5163 y Fs(\000)p Fv(1)1027 5204 y Fj(\000)g Fl(\()p Fm(H)1239 5219 y Ft(t;s)1336 5204 y Fj(\000)g Fm(E)1501 5219 y Ft(s)1554 5204 y Fl(+)g Fm(z)1691 5219 y Ft(s)1727 5204 y Fl(\))1765 5163 y Fs(\000)p Fv(1)1859 5107 y Fi(o)1931 5204 y Fl(\()p Fm(H)2058 5163 y Fv(+)2050 5228 y Ft(t;s)2148 5204 y Fl(+)g Fm(\032)2290 5219 y Ft(t)2319 5204 y Fl(\))2357 5163 y Ft(\014)s(=)p Fv(2)2475 5104 y Fi(\015)2475 5154 y(\015)2475 5204 y(\015)51 b Fj(\024)2758 5136 y Fl(96)17 b Fm(\021)p 2710 5180 262 4 v 2710 5284 a(#)g(\032)2834 5237 y Fv(1)p Fs(\000)p Ft(\014)2834 5304 y(t)2998 5204 y Fm(\013)3061 5163 y Fv(3)p Ft(=)p Fv(2)3187 5204 y Fl(\()p Fm(r)i Fj(\000)d Fm(t)p Fl(\))3154 5380 y Fp(\(V)-13 b(.100\))1748 5712 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 166 83 a Fh(and)515 230 y Fi(\015)515 279 y(\015)515 329 y(\015)p Fl(\()p Fm(P)662 344 y Ft(s)721 329 y Fj(\000)22 b Fm(P)883 344 y Ft(t)913 329 y Fl(\))17 b(\()p Fm(H)1095 288 y Fv(+)1087 354 y Ft(t;s)1190 329 y Fl(+)22 b Fm(\032)1338 344 y Ft(t)1368 329 y Fl(\))1406 288 y Ft(\014)s(=)p Fv(2)1524 230 y Fi(\015)1524 279 y(\015)1524 329 y(\015)52 b Fj(\024)h Fl(96)17 b Fm(\021)j(#)1992 288 y Fs(\000)p Fv(1)2103 329 y Fm(\013)2166 288 y Fv(3)p Ft(=)p Fv(2)2293 329 y Fm(\032)2343 278 y Ft(\014)s(=)p Fv(2)2343 349 y Ft(t)2477 329 y Fl(\()p Fm(s)i Fj(\000)h Fm(t)p Fl(\))f Fm(:)349 b Fp(\(V)-13 b(.101\))166 688 y Fh(Pr)l(oof:)21 b Fp(W)-8 b(e)22 b(\002rst)g(e)o(xpand)f(the)h(dif)n (ference)g(of)g(the)g(resolv)o(ents)f(in)g(Eq.)h(\(V)-13 b(.100\))22 b(in)f(a)h(Neumann)166 809 y(series)j(as)361 1163 y Fl(\()p Fm(H)480 1178 y Ft(r)n(;s)588 1163 y Fj(\000)e Fm(E)760 1178 y Ft(s)819 1163 y Fl(+)f Fm(z)962 1178 y Ft(s)999 1163 y Fl(\))1037 1122 y Fs(\000)p Fv(1)1154 1163 y Fj(\000)g Fl(\()p Fm(H)1372 1178 y Ft(t;s)1476 1163 y Fj(\000)h Fm(E)1648 1178 y Ft(s)1707 1163 y Fl(+)f Fm(z)1850 1178 y Ft(s)1887 1163 y Fl(\))1925 1122 y Fs(\000)p Fv(1)3154 1163 y Fp(\(V)-13 b(.102\))378 1369 y Fl(=)501 1261 y Fs(1)477 1286 y Fi(X)470 1468 y Ft(n)p Fv(=1)603 1369 y Fl(\()p Fm(H)722 1384 y Ft(t;s)826 1369 y Fj(\000)23 b Fm(E)998 1384 y Ft(s)1057 1369 y Fl(+)f Fm(z)1200 1384 y Ft(s)1237 1369 y Fl(\))1275 1328 y Fs(\000)p Fv(1)1369 1273 y Fi(h)1430 1369 y Fj(\000)h Fm(W)1622 1384 y Ft(t;r)1722 1369 y Fl(\()p Fm(H)1841 1384 y Ft(t;s)1945 1369 y Fj(\000)f Fm(E)2116 1384 y Ft(s)2175 1369 y Fl(+)g Fm(z)2318 1384 y Ft(s)2355 1369 y Fl(\))2393 1328 y Fs(\000)p Fv(1)2488 1273 y Fi(i)2527 1296 y Ft(n)2596 1369 y Fm(:)166 1679 y Fp(W)-8 b(e)25 b(observ)o(e)g(that,)f(for)h Fm(\014)33 b Fl(=)28 b(0)p Fm(;)17 b Fl(1)p Fp(,)361 1956 y Fi(\015)361 2006 y(\015)361 2056 y(\015)p Fl(\()p Fm(H)526 2071 y Ft(t;s)630 2056 y Fj(\000)23 b Fm(E)802 2071 y Ft(s)861 2056 y Fl(+)f Fm(z)1004 2071 y Ft(s)1041 2056 y Fl(\))1079 2015 y Fs(\000)p Fv(1)1173 1959 y Fi(h)1235 2056 y Fj(\000)g Fm(W)1426 2071 y Ft(t;r)1526 2056 y Fl(\()p Fm(H)1645 2071 y Ft(t;s)1749 2056 y Fj(\000)g Fm(E)1920 2071 y Ft(s)1979 2056 y Fl(+)g Fm(z)2122 2071 y Ft(s)2160 2056 y Fl(\))2198 2015 y Fs(\000)p Fv(1)2292 1959 y Fi(i)2331 1982 y Ft(n)2395 2056 y Fl(\()p Fm(H)2514 2071 y Ft(t;s)2617 2056 y Fl(+)g Fm(\032)2765 2071 y Ft(t)2795 2056 y Fl(\))2833 2015 y Ft(\014)s(=)p Fv(2)2951 1956 y Fi(\015)2951 2006 y(\015)2951 2056 y(\015)378 2289 y Fj(\024)472 2190 y Fi(\015)472 2240 y(\015)472 2289 y(\015)p Fl(\()p Fm(H)645 2248 y Fv(+)637 2314 y Ft(t;r)742 2289 y Fl(+)g Fm(\032)890 2304 y Ft(t)920 2289 y Fl(\))958 2248 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1123 2190 y Fi(\015)1123 2240 y(\015)1123 2289 y(\015)1169 2216 y Fv(2)p Fs(\000)p Ft(\014)1328 2165 y Fi(\015)1328 2215 y(\015)1328 2264 y(\015)1328 2314 y(\015)1500 2220 y Fm(H)1589 2179 y Fv(+)1581 2240 y Ft(t;r)1686 2220 y Fl(+)g Fm(\032)1834 2235 y Ft(t)p 1384 2266 596 4 v 1384 2358 a Fm(H)1473 2316 y Fv(+)1465 2378 y Ft(t;s)1569 2358 y Fj(\000)h Fm(E)1741 2373 y Ft(s)1800 2358 y Fl(+)f Fm(z)1943 2373 y Ft(s)1990 2165 y Fi(\015)1990 2215 y(\015)1990 2264 y(\015)1990 2314 y(\015)2036 2191 y Ft(n)472 2526 y Fj(\001)500 2426 y Fi(\015)500 2476 y(\015)500 2526 y(\015)p Fl(\()p Fm(H)673 2484 y Fv(+)665 2550 y Ft(t;r)770 2526 y Fl(+)g Fm(\032)918 2541 y Ft(t)948 2526 y Fl(\))986 2484 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1167 2526 y Fm(W)1259 2541 y Ft(t;r)1359 2526 y Fl(\()p Fm(H)1486 2484 y Fv(+)1478 2550 y Ft(t;r)1583 2526 y Fl(+)g Fm(\032)1731 2541 y Ft(t)1761 2526 y Fl(\))1799 2484 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1963 2426 y Fi(\015)1963 2476 y(\015)1963 2526 y(\015)2010 2452 y Ft(n)378 2711 y Fj(\024)17 b Fm(\032)522 2660 y Fv(\()p Ft(\014)s(=)p Fv(2\))p Fs(\000)p Fv(1)522 2730 y Ft(t)802 2614 y Fi(\020)851 2711 y Fl(4)p Fm(#)957 2669 y Fs(\000)p Fv(1)1074 2711 y Fj(\001)22 b Fl(8)p Fm(\021)t(\013)1288 2669 y Fv(3)p Ft(=)p Fv(2)1397 2711 y Fl(\()p Fm(r)j Fj(\000)d Fm(t)p Fl(\))1676 2614 y Fi(\021)1726 2637 y Ft(n)1795 2711 y Fm(;)1332 b Fp(\(V)-13 b(.103\))166 2957 y(using)24 b(Lemma)g(III.3)h(and)361 3260 y Fi(\015)361 3309 y(\015)361 3359 y(\015)361 3409 y(\015)533 3315 y Fm(H)622 3274 y Fv(+)614 3335 y Ft(t;r)719 3315 y Fl(+)d Fm(\032)867 3330 y Ft(t)p 417 3361 V 417 3453 a Fm(H)506 3411 y Fv(+)498 3472 y Ft(t;s)602 3453 y Fj(\000)h Fm(E)774 3468 y Ft(s)833 3453 y Fl(+)f Fm(z)976 3468 y Ft(s)1023 3260 y Fi(\015)1023 3309 y(\015)1023 3359 y(\015)1023 3409 y(\015)1149 3384 y Fj(\024)1307 3260 y Fi(\015)1307 3309 y(\015)1307 3359 y(\015)1307 3409 y(\015)1480 3315 y Fm(H)1569 3274 y Fv(+)1561 3335 y Ft(t;r)1666 3315 y Fl(+)g Fm(\032)1814 3330 y Ft(t)p 1363 3361 597 4 v 1363 3453 a Fm(H)1452 3411 y Fv(+)1444 3472 y Ft(t;r)1549 3453 y Fj(\000)h Fm(E)1721 3468 y Ft(s)1780 3453 y Fl(+)f Fm(z)1923 3468 y Ft(s)1970 3260 y Fi(\015)1970 3309 y(\015)1970 3359 y(\015)1970 3409 y(\015)3154 3384 y Fp(\(V)-13 b(.104\))378 3650 y Fj(\024)17 b Fl(max)670 3529 y Fi(\032)997 3583 y Fm(\032)1047 3598 y Ft(t)p 742 3627 591 4 v 742 3718 a Fj(j)p Fm(z)815 3733 y Ft(s)874 3718 y Fj(\000)23 b Fm(E)1046 3733 y Ft(s)1105 3718 y Fl(+)f Fm(E)1275 3733 y Ft(t)1305 3718 y Fj(j)1365 3650 y Fm(;)113 b Fl(sup)1436 3728 y Ft(\026)p Fs(\025)p Fv(gap)1643 3745 y Fc(t;r)1737 3526 y Fi(\014)1737 3575 y(\014)1737 3625 y(\014)1737 3675 y(\014)2003 3583 y Fm(\026)22 b Fl(+)g Fm(\032)2232 3598 y Ft(t)p 1775 3627 716 4 v 1775 3718 a Fm(\026)f Fj(\000)i Fm(z)2000 3733 y Ft(s)2059 3718 y Fj(\000)g Fm(E)2231 3733 y Ft(s)2290 3718 y Fl(+)f Fm(E)2460 3733 y Ft(t)2500 3526 y Fi(\014)2500 3575 y(\014)2500 3625 y(\014)2500 3675 y(\014)2544 3529 y(\033)2659 3650 y Fj(\024)2803 3583 y Fl(4)p 2799 3627 58 4 v 2799 3718 a Fm(#)2889 3650 y(:)166 3979 y Fp(Note)g(that)g(here)h Fm(t)28 b(>)f(s)13 b Fj(\000)g Fm(\016)27 b Fp(insures)22 b(that)f Fm(\026)28 b Fj(\025)g Fl(gap)1953 4002 y Ft(t;r)2063 3979 y Fl(=)g Fm(#\032)2274 3994 y Ft(t)2304 3979 y Fp(.)23 b(Hence)g Fm(\026)13 b Fl(+)g Fm(\032)2840 3994 y Ft(t)2896 3979 y Fj(\024)29 b Fl(\(1)13 b(+)3203 3940 y Fv(1)p 3200 3956 42 4 v 3200 4013 a Ft(#)3252 3979 y Fl(\))p Fm(\026)27 b Fj(\024)179 4070 y Fv(2)p 176 4086 V 176 4143 a Ft(#)228 4109 y Fm(\026)e Fp(and)f Fj(j)p Fm(\026)e Fj(\000)g Fm(z)733 4124 y Ft(s)793 4109 y Fj(\000)g Fm(E)964 4124 y Ft(s)1023 4109 y Fl(+)g Fm(E)1193 4124 y Ft(t)1223 4109 y Fj(j)28 b(\025)g Fm(\026)22 b Fj(\000)1574 4070 y Ft(#)p 1574 4086 V 1577 4143 a Fv(4)1626 4109 y Fm(\032)1676 4124 y Ft(t)1728 4109 y Fj(\000)1838 4070 y Ft(#)p 1838 4086 V 1841 4143 a Fv(8)1890 4109 y Fm(\032)1940 4124 y Ft(t)1997 4109 y Fj(\025)2113 4070 y Fv(5)p 2113 4086 36 4 v 2113 4143 a(8)2158 4109 y Fm(\026)p Fp(.)166 4342 y(F)o(or)31 b Fm(s)c Fj(\000)g Fm(t)39 b(>)g Fl(0)31 b Fp(suf)n(\002ciently)f(small,)g Fl(32)p Fm(\021)t(#)1736 4306 y Fs(\000)p Fv(1)1830 4342 y Fm(\013)1893 4306 y Fv(3)p Ft(=)p Fv(2)2003 4342 y Fl(\()p Fm(s)c Fj(\000)i Fm(t)p Fl(\))39 b Fj(\024)2457 4303 y Fv(1)p 2457 4319 V 2457 4377 a(2)2502 4342 y Fp(.)31 b(This)f(implies)g(the)g(norm)166 4463 y(con)l(v)o(er)n(gence)j(of)f(the)g(Neumann)g(series)g(on)g(the)g (right)f(side)h(of)g(\(V)-13 b(.102\))33 b(with)e Fm(r)44 b Fl(:=)e Fm(s)32 b Fp(and)166 4583 y(yields)24 b(Estimate)g(\(V)-13 b(.100\))24 b(by)h(summing)e(up)h(a)h(geometric)g(series.)166 4816 y(Eq.)g(\(V)-13 b(.101\))24 b(is)h(an)g(immediate)e(consequence)i (of)g(\(V)-13 b(.100\))25 b(and)f(\(V)-13 b(.99\))25 b(because)166 5103 y Fm(P)229 5118 y Ft(s)285 5103 y Fj(\000)20 b Fm(P)445 5118 y Ft(t)526 5103 y Fl(=)52 b Fm(P)717 5118 y Ft(s;s)825 5103 y Fj(\000)20 b Fm(P)985 5118 y Ft(t;s)1118 5103 y Fl(=)1263 5036 y Fj(\000)p Fl(1)p 1256 5080 141 4 v 1256 5172 a(2)p Fm(\031)t(i)1423 4986 y Fi(Z)1469 5175 y Fv(\000)1513 5183 y Fc(s)1568 5007 y Fi(n)1623 5103 y Fl(\()p Fm(H)1742 5118 y Ft(s)1798 5103 y Fj(\000)f Fm(E)1966 5118 y Ft(s)2023 5103 y Fl(+)g Fm(z)2163 5118 y Ft(s)2200 5103 y Fl(\))2238 5062 y Fs(\000)p Fv(1)2351 5103 y Fj(\000)h Fl(\()p Fm(H)2567 5118 y Ft(t;s)2668 5103 y Fj(\000)g Fm(E)2837 5118 y Ft(s)2893 5103 y Fl(+)f Fm(z)3033 5118 y Ft(s)3070 5103 y Fl(\))3108 5062 y Fs(\000)p Fv(1)3203 5007 y Fi(o)3275 5103 y Fm(dz)3371 5118 y Ft(s)3427 5103 y Fm(;)3154 5259 y Fp(\(V)-13 b(.105\))166 5380 y(and)25 b(hence)1748 5712 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 361 0 a Fi(\015)361 50 y(\015)361 100 y(\015)p Fl(\()p Fm(P)508 115 y Ft(s)567 100 y Fj(\000)23 b Fm(P)730 115 y Ft(t)759 100 y Fl(\))17 b(\()p Fm(H)941 58 y Fv(+)933 124 y Ft(t;s)1036 100 y Fl(+)22 b Fm(\032)1184 115 y Ft(t)1214 100 y Fl(\))1252 58 y Ft(\014)s(=)p Fv(2)1370 0 y Fi(\015)1370 50 y(\015)1370 100 y(\015)53 b Fj(\024)1608 b Fp(\(V)-13 b(.106\))404 252 y Fm(#)17 b(\032)528 267 y Ft(t)p 404 296 155 4 v 457 388 a Fl(3)629 319 y(sup)607 398 y Ft(z)640 406 y Fc(s)673 398 y Fs(2)p Fv(\000)764 406 y Fc(s)814 220 y Fi(\015)814 269 y(\015)814 319 y(\015)860 223 y(n)916 319 y Fl(\()p Fm(H)1035 334 y Ft(s)1093 319 y Fj(\000)23 b Fm(E)1265 334 y Ft(s)1324 319 y Fl(+)f Fm(z)1467 334 y Ft(s)1504 319 y Fl(\))1542 278 y Fs(\000)p Fv(1)1659 319 y Fj(\000)g Fl(\()p Fm(H)1877 334 y Ft(t;s)1981 319 y Fj(\000)h Fm(E)2153 334 y Ft(s)2212 319 y Fl(+)f Fm(z)2355 334 y Ft(s)2392 319 y Fl(\))2430 278 y Fs(\000)p Fv(1)2524 223 y Fi(o)2596 319 y Fl(\()p Fm(H)2723 278 y Fv(+)2715 344 y Ft(t;s)2819 319 y Fl(+)g Fm(\032)2967 334 y Ft(t)2997 319 y Fl(\))3035 278 y Ft(\014)s(=)p Fv(2)3152 220 y Fi(\015)3152 269 y(\015)3152 319 y(\015)h Fm(:)631 583 y Fb(2)166 924 y Fp(Ne)o(xt,)29 b(we)h(establish)e(the)i (continuous)e(dif)n(ferentiability)g(of)i(the)f(map)h Fm(P)2738 939 y Fv(\()p Fs(\001)p Fv(\))2853 924 y Fl(:)37 b Fg(R)2989 887 y Fv(+)3085 924 y Fj(!)f(B)s Fl([)p Fj(H)q Fl(])p Fp(,)166 1044 y Fm(t)28 b Fj(7!)f Fm(P)419 1059 y Ft(t)449 1044 y Fp(,)e(in)f(operator)h(norm.)f(T)-8 b(o)25 b(this)e(end,)i(we)g(introduce)961 1232 y Fl(_)922 1257 y Fm(W)1014 1272 y Ft(t)1096 1257 y Fl(:=)53 b Fm(v)t Fl(\()1356 1232 y Fm(~)1341 1257 y(G)1418 1272 y Ft(t)1447 1257 y Fl(\))22 b Fj(\001)g Fm(A)p Fl(\()g(_)-49 b Fm(\037)1729 1272 y Ft(t)1773 1232 y Fm(~)1759 1257 y(G)p Fl(\))38 b(+)h Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)2199 1272 y Ft(t)2243 1232 y Fm(~)2229 1257 y(G)p Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()2520 1232 y Fm(~)2505 1257 y(G)2582 1272 y Ft(t)2611 1257 y Fl(\))g Fm(;)456 b Fp(\(V)-13 b(.107\))166 1470 y(with)47 b Fl(_)-50 b Fm(\037)429 1485 y Ft(t)487 1470 y Fl(:=)27 b Fm(@)668 1485 y Ft(t)698 1470 y Fm(\037)759 1485 y Ft(t)789 1470 y Fp(.)e(Our)g(goal)f(is)g(to)h(pro)o(v)o(e)f(the) g(follo)n(wing)f(theorem.)166 1690 y Fq(Theor)n(em)k(V)-14 b(.2)47 b Fh(Assume)33 b(that)f Fm(\013)1346 1653 y Fv(3)1429 1690 y Fj(\024)43 b Fl(\(2500)p Fm(#)1840 1653 y Fv(2)1908 1690 y Fl(+)28 b Fm(b)h Fl(+)f(6)p Fm(\024)2291 1653 y Fv(2)2359 1690 y Fl(+)g Fm(e)2508 1705 y Fv(0)2547 1690 y Fl(\))2585 1653 y Fs(\000)p Fv(1)2680 1690 y Fh(.)33 b(F)-10 b(or)32 b Fl(0)43 b Fm(<)g(s)g(<)g Fj(1)p Fh(,)166 1810 y Fm(z)211 1825 y Ft(s)276 1810 y Fj(2)28 b Fl(\000)431 1825 y Ft(s)468 1810 y Fh(,)c(and)h Fl(0)i Fm(<)h(\016)j Fj(\024)1112 1771 y Ft(#)p 1062 1787 144 4 v 1062 1844 a Fv(128)p Ft(\021)1215 1810 y Fh(,)25 b(the)g(map)548 2074 y Fg(R)620 2033 y Fv(+)701 2074 y Fj(\\)e Fl(\()p Fm(s)f Fj(\000)g Fm(\016)n(;)34 b(s)p Fl(\))55 b Fj(3)h Fm(t)d Fj(7!)1598 1978 y Fi(\020)1648 2074 y Fm(H)1729 2089 y Ft(t;s)1832 2074 y Fj(\000)23 b Fm(E)2004 2089 y Ft(s)2063 2074 y Fl(+)f Fm(z)2206 2089 y Ft(s)2243 1978 y Fi(\021)2293 2001 y Fs(\000)p Fv(1)2443 2074 y Fj(2)55 b(B)s Fl([)p Fj(H)q Fl(])383 b Fp(\(V)-13 b(.108\))166 2287 y Fh(is)24 b(continuously)f(dif)n(fer)l(entiable)o(,)h(and)g(its)g (derivative)g(is)h(given)f(by)243 2477 y Fm(@)p 226 2522 93 4 v 226 2613 a(@)5 b(t)328 2448 y Fi(\020)378 2545 y Fm(H)459 2560 y Ft(t;s)555 2545 y Fj(\000)15 b Fm(E)719 2560 y Ft(s)772 2545 y Fl(+)g Fm(z)908 2560 y Ft(s)945 2448 y Fi(\021)994 2471 y Fs(\000)p Fv(1)1139 2545 y Fl(=)1266 2448 y Fi(\020)1315 2545 y Fm(H)1396 2560 y Ft(t;s)1493 2545 y Fj(\000)g Fm(E)1657 2560 y Ft(s)1709 2545 y Fl(+)g Fm(z)1845 2560 y Ft(s)1882 2448 y Fi(\021)1932 2471 y Fs(\000)p Fv(1)2026 2545 y Fl(\()p Fj(\000)2181 2520 y Fl(_)2141 2545 y Fm(W)2233 2560 y Ft(t)2263 2545 y Fl(\))2301 2448 y Fi(\020)2351 2545 y Fm(H)2432 2560 y Ft(t;s)2529 2545 y Fj(\000)g Fm(E)2693 2560 y Ft(s)2745 2545 y Fl(+)g Fm(z)2881 2560 y Ft(s)2918 2448 y Fi(\021)2968 2471 y Fs(\000)p Fv(1)3077 2545 y Fm(:)50 b Fp(\(V)-13 b(.109\))166 2890 y Fh(Pr)l(oof:)24 b Fp(W)-8 b(e)26 b(compute)f(the)g(dif)n(ferential)g(quotient)f(in)h Fm(t)k Fj(2)h Fg(R)2272 2854 y Fv(+)2353 2890 y Fj(\\)23 b Fl(\()p Fm(s)g Fj(\000)g Fm(\016)n(;)34 b(s)p Fl(\))25 b Fp(and)h(compare)f(it) 166 3010 y(to)546 3131 y Fm(D)627 3146 y Ft(t;s)762 3131 y Fl(:=)52 b(\()p Fm(H)1036 3146 y Ft(t;s)1140 3131 y Fj(\000)23 b Fm(E)1312 3146 y Ft(s)1371 3131 y Fl(+)f Fm(z)1514 3146 y Ft(s)1551 3131 y Fl(\))1589 3089 y Fs(\000)p Fv(1)1683 3131 y Fl(\()p Fj(\000)1838 3105 y Fl(_)1798 3131 y Fm(W)1890 3146 y Ft(t)1920 3131 y Fl(\)\()p Fm(H)2077 3146 y Ft(t;s)2181 3131 y Fj(\000)h Fm(E)2353 3146 y Ft(s)2412 3131 y Fl(+)f Fm(z)2555 3146 y Ft(s)2592 3131 y Fl(\))2630 3089 y Fs(\000)p Fv(1)2746 3131 y Fm(:)381 b Fp(\(V)-13 b(.110\))166 3273 y(From)25 b(the)f(second)h(resolv)o(ent) f(equation,)g(we)h(get,)f(for)h Fm(r)31 b Fj(2)d Fl(\()p Fm(t;)17 b(s)p Fl(\))p Fp(,)24 b(that)303 3463 y Fl(1)p 226 3507 204 4 v 226 3598 a Fm(r)h Fj(\000)d Fm(t)439 3434 y Fi(n)495 3530 y Fl(\()p Fm(H)614 3545 y Ft(r)n(;s)714 3530 y Fj(\000)14 b Fm(E)877 3545 y Ft(s)927 3530 y Fl(+)g Fm(z)1062 3545 y Ft(s)1099 3530 y Fl(\))1137 3489 y Fs(\000)p Fv(1)1245 3530 y Fj(\000)g Fl(\()p Fm(H)1455 3545 y Ft(t;s)1551 3530 y Fj(\000)g Fm(E)1714 3545 y Ft(s)1764 3530 y Fl(+)g Fm(z)1899 3545 y Ft(s)1936 3530 y Fl(\))1974 3489 y Fs(\000)p Fv(1)2068 3434 y Fi(o)2165 3530 y Fj(\000)42 b Fm(D)2365 3545 y Ft(t;s)2497 3530 y Fl(=)50 b Fm(X)2704 3545 y Ft(t;r)2818 3530 y Fl(+)30 b Fm(Y)2981 3545 y Ft(t;r)3077 3530 y Fm(;)50 b Fp(\(V)-13 b(.111\))166 3775 y(where)25 b(the)g(operators)g Fm(X)1058 3790 y Ft(t;r)1166 3775 y Fp(and)f Fm(Y)1391 3790 y Ft(t;r)1499 3775 y Fp(are)h(gi)n(v)o(en)f (by)454 4096 y Fm(X)535 4111 y Ft(t;r)634 4096 y Fl(:=)17 b(\()p Fm(H)873 4111 y Ft(t;s)977 4096 y Fj(\000)22 b Fm(E)1148 4111 y Ft(s)1207 4096 y Fl(+)g Fm(z)1350 4111 y Ft(s)1387 4096 y Fl(\))1425 4055 y Fs(\000)p Fv(1)1536 4096 y Fl(\()p Fj(\000)p Fl(\001)p Fm(W)1824 4111 y Ft(t;r)1908 4096 y Fl(\)\()p Fm(H)2065 4111 y Ft(r)n(;s)2173 4096 y Fj(\000)h Fm(E)2345 4111 y Ft(s)2404 4096 y Fl(+)f Fm(z)2547 4111 y Ft(s)2584 4096 y Fl(\))2622 4055 y Fs(\000)p Fv(1)2738 4096 y Fm(;)389 b Fp(\(V)-13 b(.112\))361 4248 y Fl(\001)p Fm(W)534 4263 y Ft(t;r)634 4248 y Fl(:=)17 b(\()p Fm(r)24 b Fj(\000)f Fm(t)p Fl(\))1033 4207 y Fs(\000)p Fv(1)1128 4248 y Fm(W)1220 4263 y Ft(t;r)1325 4248 y Fj(\000)1464 4223 y Fl(_)1425 4248 y Fm(W)1517 4263 y Ft(t)648 4406 y Fl(=)30 b Fm(v)t Fl(\()858 4380 y Fm(~)843 4406 y(G)920 4421 y Ft(t)949 4406 y Fl(\))22 b Fj(\001)g Fm(A)p Fl(\(\001)1266 4380 y Fm(~)1251 4406 y(G)1328 4421 y Ft(t;r)1411 4406 y Fl(\))39 b(+)g Fm(A)p Fl(\(\001)1810 4380 y Fm(~)1795 4406 y(G)1872 4421 y Ft(t;r)1955 4406 y Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()2169 4380 y Fm(~)2154 4406 y(G)2231 4421 y Ft(t)2260 4406 y Fl(\))39 b(+)f Fm(A)p Fl(\(\001)2658 4380 y Fm(~)2643 4406 y(G)2720 4421 y Ft(t;r)2804 4406 y Fl(\))2842 4364 y Fv(2)2903 4406 y Fm(;)224 b Fp(\(V)-13 b(.113\))166 4630 y(with)24 b Fl(\001)464 4605 y Fm(~)449 4630 y(G)526 4645 y Ft(t;r)637 4630 y Fl(:=)k(\()p Fm(r)c Fj(\000)f Fm(t)p Fl(\))1047 4594 y Fs(\000)p Fv(1)1156 4605 y Fm(~)1142 4630 y(G)1219 4645 y Ft(t;r)1324 4630 y Fj(\000)45 b Fl(_)-50 b Fm(\037)1484 4645 y Ft(t)1529 4605 y Fm(~)1514 4630 y(G)p Fp(,)25 b(and)216 4859 y Fm(Y)273 4874 y Ft(t;r)405 4859 y Fl(:=)49 b(\()p Fm(H)676 4874 y Ft(t;s)767 4859 y Fj(\000)9 b Fm(E)925 4874 y Ft(s)972 4859 y Fl(+)g Fm(z)1102 4874 y Ft(s)1139 4859 y Fl(\))1177 4817 y Fs(\000)p Fv(1)1327 4833 y Fl(_)1288 4859 y Fm(W)1380 4874 y Ft(t)1427 4762 y Fi(n)1482 4859 y Fl(\()p Fm(H)1601 4874 y Ft(t;s)1692 4859 y Fj(\000)g Fm(E)1850 4874 y Ft(s)1897 4859 y Fl(+)g Fm(z)2027 4874 y Ft(s)2065 4859 y Fl(\))2103 4817 y Fs(\000)p Fv(1)2206 4859 y Fj(\000)g Fl(\()p Fm(H)2411 4874 y Ft(r)n(;s)2507 4859 y Fj(\000)g Fm(E)2665 4874 y Ft(s)2713 4859 y Fl(+)g Fm(z)2843 4874 y Ft(s)2880 4859 y Fl(\))2918 4817 y Fs(\000)p Fv(1)3012 4762 y Fi(o)3077 4859 y Fm(:)50 b Fp(\(V)-13 b(.114\))166 5071 y(An)25 b(application)e(of)i(Lemma)f(V)-13 b(.1)25 b(and)g(\(V)-13 b(.104\))25 b(yields)252 5325 y Fj(k)p Fm(Y)359 5340 y Ft(t;r)442 5325 y Fj(k)52 b(\024)684 5258 y Fl(384)17 b Fm(\021)p 684 5302 215 4 v 695 5394 a(#)752 5365 y Fv(2)808 5394 y Fm(\032)858 5409 y Ft(t)925 5325 y Fm(\013)988 5284 y Fv(3)p Ft(=)p Fv(2)1115 5325 y Fl(\()p Fm(r)24 b Fj(\000)f Fm(t)p Fl(\))1411 5226 y Fi(\015)1411 5276 y(\015)1411 5325 y(\015)p Fl(\()p Fm(H)1584 5284 y Fv(+)1576 5350 y Ft(t;s)1680 5325 y Fl(+)f Fm(\032)1828 5340 y Ft(t)1858 5325 y Fl(\))1896 5284 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2077 5325 y Fl(\()p Fj(\000)2232 5300 y Fl(_)2192 5325 y Fm(W)2284 5340 y Ft(t)2314 5325 y Fl(\))17 b(\()p Fm(H)2496 5284 y Fv(+)2488 5350 y Ft(t;s)2592 5325 y Fl(+)22 b Fm(\032)2740 5340 y Ft(t)2770 5325 y Fl(\))2808 5284 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2972 5226 y Fi(\015)2972 5276 y(\015)2972 5325 y(\015)h Fm(;)86 b Fp(\(V)-13 b(.115\))1748 5712 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 166 83 a Fp(where)37 b(the)f(norm)f(on)h(the)g (right)g(side)g(is)f(bounded)h(uniformly)f(in)h Fm(t)49 b Fj(2)g Fl(\()p Fm(s)30 b Fj(\000)h Fm(\016)n(;)17 b(s)p Fl(\))37 b Fp(by)f(an)166 203 y(estimate)24 b(similar)g(to)g (\(III.70\).)166 434 y(F)o(or)h(the)f(estimate)g(on)h Fm(X)1037 449 y Ft(t;r)1145 434 y Fp(we)g(note)f(that)216 723 y Fm(Z)283 738 y Ft(t;r)419 723 y Fl(:=)574 623 y Fi(\015)574 673 y(\015)574 723 y(\015)p Fl(\()p Fm(H)747 682 y Fv(+)739 748 y Ft(t;r)844 723 y Fl(+)e Fm(\032)992 738 y Ft(t)1022 723 y Fl(\))1060 682 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1242 723 y Fl(\()p Fj(\000)p Fl(\001)p Fm(W)1530 738 y Ft(t;r)1613 723 y Fl(\))17 b(\()p Fm(H)1795 682 y Fv(+)1787 748 y Ft(t;r)1892 723 y Fl(+)22 b Fm(\032)2040 738 y Ft(t)2070 723 y Fl(\))2108 682 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)2273 623 y Fi(\015)2273 673 y(\015)2273 723 y(\015)52 b Fj(\024)2543 656 y Fl(32)p 2511 700 161 4 v 2511 810 a Fm(\032)2561 759 y Fv(3)p Ft(=)p Fv(2)2561 818 y Ft(s)2682 623 y Fi(\015)2682 673 y(\015)2682 723 y(\015)p Fl(\001)2824 698 y Fm(~)2809 723 y(G)2886 738 y Ft(t;r)2969 623 y Fi(\015)2969 673 y(\015)2969 723 y(\015)3015 777 y Fv(2)3077 723 y Fm(:)e Fp(\(V)-13 b(.116\))166 1027 y(A)25 b(second-order)g(T)-8 b(aylor)24 b(e)o(xpansion)g(yields) 361 1396 y Fj(j)p Fl(\001)485 1371 y Fm(~)470 1396 y(G)547 1411 y Ft(t;r)630 1396 y Fj(j)16 b(\024)h Fl(\()p Fm(r)25 b Fj(\000)e Fm(t)p Fl(\))1048 1355 y Fs(\000)p Fv(1)1159 1297 y Fi(\014)1159 1346 y(\014)1159 1396 y(\014)p Fm(\037)1248 1411 y Ft(r)1308 1396 y Fj(\000)f Fm(\037)1468 1411 y Ft(t)1520 1396 y Fj(\000)h Fl(\()p Fm(r)i Fj(\000)d Fm(t)p Fl(\))h(_)-50 b Fm(\037)1960 1411 y Ft(t)1990 1297 y Fi(\014)1990 1346 y(\014)1990 1396 y(\014)17 b Fj(j)2078 1371 y Fm(~)2063 1396 y(G)o Fj(j)52 b(\024)h Fl(\()p Fm(r)25 b Fj(\000)e Fm(t)p Fl(\))2646 1297 y Fi(\014)2646 1346 y(\014)2646 1396 y(\014)o Fm(@)2729 1355 y Fv(2)2724 1421 y Ft(t)2770 1396 y Fm(\037)2831 1411 y Ft(t)2860 1297 y Fi(\014)2860 1346 y(\014)2860 1396 y(\014)2888 1450 y Fs(1)2979 1396 y Fj(j)3022 1371 y Fm(~)3007 1396 y(G)p Fj(j)674 1580 y(\024)17 b Fl(6)g Fm(\021)886 1538 y Fv(2)942 1580 y Fj(k)p Fm(p)1041 1538 y Fs(0)1064 1580 y Fj(k)1114 1595 y Fs(1)1205 1580 y Fl(\()p Fm(r)25 b Fj(\000)d Fm(t)p Fl(\))17 b Fj(j)1544 1554 y Fm(~)1529 1580 y(G)p Fj(j)k Fm(;)1472 b Fp(\(V)-13 b(.117\))166 1830 y(pointwise)26 b(a.)i(e.)g(in)f Fm(k)s Fp(.)h(Since)g Fj(j)1279 1805 y Fm(~)1264 1830 y(G)o Fj(j)g Fp(is)f(square-inte)o (grable,)g(Lebesgue')-5 b(s)27 b(dominated)f(con)l(v)o(er)n(-)166 1959 y(gence)21 b(theorem)e(implies)g(that)h Fl(lim)1384 1974 y Ft(r)r Fs(!)p Ft(t)1535 1959 y Fj(k)p Fl(\001)1681 1934 y Fm(~)1666 1959 y(G)1743 1974 y Ft(t;r)1826 1959 y Fj(k)1876 1974 y Fv(2)1943 1959 y Fl(=)28 b(0)20 b Fp(and)g(hence)g(also)g Fl(lim)2849 1974 y Ft(r)r Fs(!)p Ft(t)2999 1959 y Fj(k)p Fm(Z)3116 1974 y Ft(t;r)3199 1959 y Fj(k)27 b Fl(=)h(0)p Fp(.)166 2080 y(W)l(ith)c(this)g(in)g (mind,)g(we)h(apply)f(Lemma)h(V)-13 b(.1)24 b(and)h(\(V)-13 b(.104\))25 b(again)f(and)h(obtain)361 2505 y Fj(k)p Fm(X)492 2520 y Ft(t;r)575 2505 y Fj(k)16 b(\024)751 2437 y Fm(Z)818 2452 y Ft(t;r)p 745 2481 V 745 2592 a Fm(\032)795 2541 y Fv(1)p Ft(=)p Fv(2)795 2612 y Ft(t)932 2380 y Fi(\015)932 2430 y(\015)932 2480 y(\015)932 2529 y(\015)1104 2435 y Fm(H)1193 2394 y Fv(+)1185 2455 y Ft(t;r)1290 2435 y Fl(+)22 b Fm(\032)1438 2450 y Ft(t)p 988 2481 596 4 v 988 2573 a Fm(H)1077 2532 y Fv(+)1069 2593 y Ft(t;s)1173 2573 y Fj(\000)h Fm(E)1345 2588 y Ft(s)1404 2573 y Fl(+)f Fm(z)1547 2588 y Ft(s)1594 2380 y Fi(\015)1594 2430 y(\015)1594 2480 y(\015)1594 2529 y(\015)1662 2505 y Fj(\001)1712 2380 y Fi(\015)1712 2430 y(\015)1712 2480 y(\015)1712 2529 y(\015)1794 2435 y Fl(\()p Fm(H)1921 2394 y Fv(+)1913 2455 y Ft(t;s)2017 2435 y Fl(+)g Fm(\032)2165 2450 y Ft(t)2195 2435 y Fl(\))2233 2399 y Fv(1)p Ft(=)p Fv(2)p 1768 2481 601 4 v 1768 2573 a Fm(H)1857 2544 y Fv(+)1849 2598 y Ft(r)n(;s)1957 2573 y Fj(\000)h Fm(E)2129 2588 y Ft(s)2188 2573 y Fl(+)f Fm(z)2331 2588 y Ft(s)2378 2380 y Fi(\015)2378 2430 y(\015)2378 2480 y(\015)2378 2529 y(\015)3160 2505 y Fp(\(V)-13 b(.118\))641 2800 y Fj(\024)751 2733 y Fm(Z)818 2748 y Ft(t;r)p 745 2777 161 4 v 745 2888 a Fm(\032)795 2837 y Fv(1)p Ft(=)p Fv(2)795 2908 y Ft(t)932 2676 y Fi(\015)932 2726 y(\015)932 2775 y(\015)932 2825 y(\015)1104 2731 y Fm(H)1193 2690 y Fv(+)1185 2751 y Ft(t;r)1290 2731 y Fl(+)22 b Fm(\032)1438 2746 y Ft(t)p 988 2777 596 4 v 988 2869 a Fm(H)1077 2827 y Fv(+)1069 2888 y Ft(t;s)1173 2869 y Fj(\000)h Fm(E)1345 2884 y Ft(s)1404 2869 y Fl(+)f Fm(z)1547 2884 y Ft(s)1594 2676 y Fi(\015)1594 2726 y(\015)1594 2775 y(\015)1594 2825 y(\015)1662 2800 y Fj(\001)1712 2679 y Fi(\032)1774 2676 y(\015)1774 2726 y(\015)1774 2775 y(\015)1774 2825 y(\015)1854 2731 y Fl(\()p Fm(H)1981 2690 y Fv(+)1973 2751 y Ft(t;s)2077 2731 y Fl(+)g Fm(\032)2225 2746 y Ft(t)2255 2731 y Fl(\))2293 2695 y Fv(1)p Ft(=)p Fv(2)p 1830 2777 V 1830 2869 a Fm(H)1919 2827 y Fv(+)1911 2888 y Ft(t;s)2015 2869 y Fj(\000)h Fm(E)2187 2884 y Ft(s)2246 2869 y Fl(+)f Fm(z)2389 2884 y Ft(s)2436 2676 y Fi(\015)2436 2726 y(\015)2436 2775 y(\015)2436 2825 y(\015)2532 2800 y Fl(+)2702 2733 y(96)17 b Fm(\021)p 2668 2777 235 4 v 2668 2888 a(#)g(\032)2792 2837 y Fv(1)p Ft(=)p Fv(2)2792 2908 y Ft(t)2929 2800 y Fm(\013)2992 2759 y Fv(3)p Ft(=)p Fv(2)3118 2800 y Fl(\()p Fm(r)25 b Fj(\000)d Fm(t)p Fl(\))3397 2679 y Fi(\033)641 3076 y Fj(\024)745 3009 y Fl(16)17 b Fm(Z)927 3024 y Ft(t;r)p 745 3053 264 4 v 781 3145 a Fm(#)838 3116 y Fv(2)894 3145 y Fm(\032)944 3160 y Ft(t)1036 2980 y Fi(n)1091 3076 y Fl(1)50 b(+)g(24)p Fm(\021)19 b(\013)1544 3035 y Fv(3)p Ft(=)p Fv(2)1671 3076 y Fm(\016)1718 2980 y Fi(o)1795 3076 y Fm(:)166 3368 y Fp(Therefore,)27 b Fl(lim)745 3383 y Ft(r)r Fs(!)p Ft(t)896 3368 y Fj(k)p Fm(X)1027 3383 y Ft(t;r)1109 3368 y Fj(k)j Fl(=)f(lim)1429 3383 y Ft(r)r Fs(!)p Ft(t)1580 3368 y Fj(k)p Fm(Y)1687 3383 y Ft(t;r)1769 3368 y Fj(k)h Fl(=)f(0)c Fp(which)h(establishes)e(the)i(dif)n(ferentiabil-)166 3488 y(ity)d(of)g Fm(t)28 b Fj(7!)g Fl(\()p Fm(H)711 3503 y Ft(t;s)810 3488 y Fj(\000)17 b Fm(E)976 3503 y Ft(s)1031 3488 y Fj(\000)h Fm(z)1171 3503 y Ft(s)1208 3488 y Fl(\))1246 3452 y Fs(\000)p Fv(1)1364 3488 y Fp(and)23 b(\(V)-13 b(.109\).)23 b(The)h(continuity)e(of)h(the)h(deri)n(v)n(ati)n (v)o(e)d(\(and,)i(in)166 3609 y(f)o(act,)i(e)n(v)o(en)f(smoothness\))f (is)i(sho)n(wn)e(similarly)-6 b(.)488 b Fb(2)166 3960 y Fp(A)25 b(direct)g(consequence)g(of)f(Theorem)h(V)-13 b(.2)25 b(is)166 4191 y Fq(Cor)n(ollary)f(V)-14 b(.3)48 b Fh(Assume)24 b(that)g Fm(\013)1357 4154 y Fv(3)1424 4191 y Fj(\024)k Fl(\(2500)p Fm(#)1820 4154 y Fv(2)1881 4191 y Fl(+)22 b Fm(b)h Fl(+)f(6)p Fm(\024)2246 4154 y Fv(2)2308 4191 y Fl(+)g Fm(e)2451 4206 y Fv(0)2490 4191 y Fl(\))2528 4154 y Fs(\000)p Fv(1)2622 4191 y Fh(.)166 4311 y(Then)k Fm(P)454 4326 y Fv(\()p Fs(\001)p Fv(\))563 4311 y Fj(2)k Fm(C)736 4275 y Fv(1)776 4311 y Fl(\()p Fg(R)886 4275 y Fv(+)945 4311 y Fl(;)17 b Fj(B)s Fl([)p Fj(H)q Fl(]\))26 b Fh(is)g(continuously)e(dif)n(fer)l(entiable)h(in)g (oper)o(ator)g(norm,)g(and)h(its)166 4431 y(derivative)e(is)h(given)f (by)394 4806 y Fl(_)361 4831 y Fm(P)424 4846 y Ft(t)470 4831 y Fl(=)580 4764 y Fj(\000)p Fl(1)p 573 4808 141 4 v 573 4900 a(2)p Fm(\031)t(i)740 4714 y Fi(Z)786 4903 y Fv(\000)830 4911 y Fc(t)879 4735 y Fi(\020)929 4831 y Fm(H)1018 4790 y Fv(+)1010 4856 y Ft(t)1098 4831 y Fl(+)e Fm(z)1241 4846 y Ft(t)1271 4735 y Fi(\021)1321 4758 y Fs(\000)p Fv(1)1415 4831 y Fl(\()p Fj(\000)1570 4806 y Fl(_)1530 4831 y Fm(W)1622 4846 y Ft(t)1652 4831 y Fl(\))1690 4735 y Fi(\020)1740 4831 y Fm(H)1829 4790 y Fv(+)1821 4856 y Ft(t)1910 4831 y Fl(+)g Fm(z)2053 4846 y Ft(t)2083 4735 y Fi(\021)2132 4758 y Fs(\000)p Fv(1)2243 4831 y Fm(dz)2339 4846 y Ft(t)3154 4831 y Fp(\(V)-13 b(.119\))470 5094 y Fl(=)579 5027 y Fm(P)656 4991 y Fs(?)642 5051 y Ft(t)p 573 5071 148 4 v 573 5162 a Fm(H)662 5121 y Fv(+)654 5182 y Ft(t)747 5094 y Fl(\()p Fj(\000)902 5069 y Fl(_)862 5094 y Fm(W)954 5109 y Ft(t)984 5094 y Fl(\))17 b Fm(P)1102 5109 y Ft(t)1175 5094 y Fl(+)44 b Fm(P)1358 5109 y Ft(t)1405 5094 y Fl(\()p Fj(\000)1559 5069 y Fl(_)1520 5094 y Fm(W)1612 5109 y Ft(t)1642 5094 y Fl(\))1712 5027 y Fm(P)1789 4991 y Fs(?)1775 5051 y Ft(t)p 1706 5071 V 1706 5162 a Fm(H)1795 5121 y Fv(+)1787 5182 y Ft(t)1886 5094 y Fm(;)1241 b Fp(\(V)-13 b(.120\))166 5380 y Fh(for)24 b(all)g Fm(t)k(>)g Fl(0)p Fh(.)1748 5712 y Fp(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 166 83 a Fh(Pr)l(oof:)32 b Fp(Let)h Fm(t)43 b(>)g Fl(0)33 b Fp(and)g Fm(s)43 b(>)g Fl(0)33 b Fp(such)g(that)g Fm(t)43 b Fj(2)h Fl(\()p Fm(s)28 b Fj(\000)h Fm(\016)n(;)17 b(s)p Fl(\))p Fp(,)33 b(for)g(a)h(suitable)e Fm(\016)47 b(>)c Fl(0)33 b Fp(as)g(in)166 203 y(Theorem)25 b(V)-13 b(.2.)24 b(W)-8 b(e)25 b(ha)n(v)o(e)1179 460 y Fm(P)1242 475 y Ft(t)1324 460 y Fl(=)1470 393 y Fj(\000)p Fl(1)p 1463 437 141 4 v 1463 529 a(2)p Fm(\031)t(i)1630 343 y Fi(Z)1676 532 y Fv(\000)1720 540 y Fc(s)2016 393 y Fm(dz)2112 408 y Ft(s)p 1784 437 598 4 v 1784 529 a Fm(H)1865 544 y Ft(t;s)1969 529 y Fj(\000)e Fm(E)2141 544 y Ft(s)2200 529 y Fj(\000)f Fm(z)2344 544 y Ft(s)2414 460 y Fm(:)713 b Fp(\(V)-13 b(.121\))166 743 y(Its)25 b(deri)n(v)n(ati)n(v)o(e)d(is)i(gi)n(v)o(en)g(by)369 990 y Fl(_)336 1015 y Fm(P)399 1030 y Ft(t)481 1015 y Fl(=)627 948 y Fj(\000)p Fl(1)p 620 992 141 4 v 620 1084 a(2)p Fm(\031)t(i)787 898 y Fi(Z)833 1087 y Fv(\000)877 1095 y Fc(s)932 919 y Fi(\020)981 1015 y Fm(H)1062 1030 y Ft(t;s)1166 1015 y Fj(\000)f Fm(E)1338 1030 y Ft(s)1397 1015 y Fl(+)f Fm(z)1540 1030 y Ft(s)1577 919 y Fi(\021)1626 942 y Fs(\000)p Fv(1)1721 1015 y Fl(\()p Fj(\000)1875 990 y Fl(_)1836 1015 y Fm(W)1928 1030 y Ft(t)1958 1015 y Fl(\))1996 919 y Fi(\020)2045 1015 y Fm(H)2126 1030 y Ft(t;s)2230 1015 y Fj(\000)h Fm(E)2402 1030 y Ft(s)2461 1015 y Fl(+)f Fm(z)2604 1030 y Ft(s)2641 919 y Fi(\021)2691 942 y Fs(\000)p Fv(1)2802 1015 y Fm(dz)2898 1030 y Ft(s)2957 1015 y Fm(:)170 b Fp(\(V)-13 b(.122\))166 1282 y(W)-8 b(e)29 b(obtain)f(\(V)-13 b(.119\))29 b(in)f(the)h(limit)e Fm(s)35 b Fj(!)g Fm(t)p Fp(.)29 b(Eq.)f(\(V)-13 b(.120\))29 b(follo)n(ws)e(from)i(Cauchy')-5 b(s)28 b(inte)o(gral)166 1402 y(formula,)23 b(observing)g(that)h Fm(z)1166 1417 y Ft(t)1224 1402 y Fj(7!)j Fl(\()p Fm(H)1470 1417 y Ft(t)1518 1402 y Fj(\000)20 b Fm(E)1687 1417 y Ft(t)1736 1402 y Fj(\000)g Fm(z)1878 1417 y Ft(t)1908 1402 y Fl(\))1946 1366 y Fs(\000)p Fv(1)2040 1402 y Fm(P)2117 1366 y Fs(?)2103 1427 y Ft(t)2199 1402 y Fp(is)k(re)o(gular)l(,)g(for)g Fm(z)2801 1417 y Ft(t)2858 1402 y Fj(2)29 b Fl(\000)3014 1417 y Ft(t)3043 1402 y Fp(,)24 b(and)g(from)166 1523 y(the)h(identity)675 1497 y Fl(_)642 1523 y Fm(P)705 1538 y Ft(t)762 1523 y Fl(=)j Fm(P)929 1538 y Ft(t)1008 1497 y Fl(_)975 1523 y Fm(P)1038 1538 y Ft(t)1084 1523 y Fm(P)1161 1486 y Fs(?)1147 1547 y Ft(t)1242 1523 y Fl(+)22 b Fm(P)1417 1486 y Fs(?)1403 1547 y Ft(t)1508 1497 y Fl(_)1475 1523 y Fm(P)1538 1538 y Ft(t)1584 1523 y Fm(P)1647 1538 y Ft(t)1677 1523 y Fp(.)489 b Fb(2)166 2081 y Fq(VI)99 b(Integrability)25 b(of)g(the)h(Deri)o(v)o(ati)o(v)o(e) e(of)h(the)h(Gr)n(ound)g(State)f(Pr)n(ojection)166 2426 y Fp(In)33 b(this)f(last)h(section)f(we)h(pro)o(v)o(e)f(that)h(the)f (norm)h(of)g(the)g(deri)n(v)n(ati)n(v)o(e)2659 2401 y Fl(_)2626 2426 y Fm(P)2689 2441 y Ft(t)2762 2426 y Fl(=)42 b Fm(@)2931 2441 y Ft(t)2961 2426 y Fm(P)3024 2441 y Ft(t)3087 2426 y Fp(of)33 b Fm(P)3266 2442 y Fv(\()p Fs(\001)p Fv(\))3387 2426 y Fj(2)166 2558 y Fm(C)243 2522 y Fv(1)282 2558 y Fl(\()p Fg(R)392 2522 y Fv(+)451 2558 y Fl(;)17 b Fj(B)s Fl([)p Fj(H)q Fl(]\))34 b Fp(decays)f(e)o (xponentially)f(in)g Fm(t)p Fp(.)i(This)e(implies)g(the)h(inte)o (grability)e(of)3217 2533 y Fl(_)3184 2558 y Fm(P)3247 2573 y Ft(t)3310 2558 y Fp(and)166 2679 y(the)h(fundamental)f(theorem)g (of)h(calculus)g(then)f(yields)g(the)h(e)o(xistence)f(of)h(the)g(limit) e Fm(P)3262 2694 y Fs(1)3378 2679 y Fl(=)166 2799 y Fm(P)229 2814 y Fv(0)290 2799 y Fl(+)389 2728 y Fi(R)444 2754 y Fs(1)428 2824 y Fv(0)568 2774 y Fl(_)535 2799 y Fm(P)598 2814 y Ft(t)644 2799 y Fm(dt)p Fp(.)166 3024 y Fq(Theor)n(em)d(VI.1)48 b Fh(Assume)25 b(that)g Fm(\013)1385 2988 y Fv(3)1453 3024 y Fj(\024)30 b Fl(\(2500)p Fm(#)1851 2988 y Fv(2)1912 3024 y Fl(+)23 b Fm(b)g Fl(+)g(6)p Fm(\024)2279 2988 y Fv(2)2341 3024 y Fl(+)f Fm(e)2484 3039 y Fv(0)2524 3024 y Fl(\))2562 2988 y Fs(\000)p Fv(1)2656 3024 y Fh(.)k(Then)g(ther) l(e)g(e)n(xists)f(a)166 3145 y(constant)c Fm(C)619 3155 y Fv(_)596 3172 y Ft(P)682 3145 y Fm(<)27 b Fj(1)22 b Fh(suc)o(h)f(that)g(the)h(derivative)1884 3120 y Fl(_)1851 3145 y Fm(P)1914 3160 y Ft(t)1971 3145 y Fl(=)28 b Fm(@)2126 3160 y Ft(t)2156 3145 y Fm(P)2219 3160 y Ft(t)2270 3145 y Fh(of)22 b Fm(P)2433 3160 y Fv(\()p Fs(\001)p Fv(\))2539 3145 y Fj(2)28 b Fm(C)2710 3109 y Fv(1)2750 3145 y Fl(\()p Fg(R)2860 3109 y Fv(+)2918 3145 y Fl(;)17 b Fj(B)s Fl([)p Fj(H)q Fl(]\))23 b Fh(obe)m(ys)166 3265 y(the)i(following)e(norm)h (estimate)o(,)1345 3505 y Fj(k)1428 3480 y Fl(_)1395 3505 y Fm(P)1458 3520 y Ft(t)1487 3505 y Fj(k)53 b(\024)g Fm(C)1813 3515 y Fv(_)1790 3532 y Ft(P)1870 3505 y Fm(\013)1933 3464 y Fv(3)p Ft(=)p Fv(2)2065 3505 y Fm(\032)2115 3454 y Fv(1)p Ft(=)p Fv(2)2115 3525 y Ft(t)2248 3505 y Fm(;)833 b Fp(\(VI.123\))166 3736 y Fh(for)24 b(all)g Fm(t)k(>)g Fl(0)p Fh(.)166 3962 y(Pr)l(oof:)22 b Fp(First,)g(we)h(observ)o(e)g (that,)f(for)h Fm(\027)34 b Fj(2)28 b(f)p Fl(1)p Fm(;)17 b Fl(2)p Fm(;)g Fl(3)p Fj(g)22 b Fp(and)h(a)g(suf)n(\002ciently)f(re)o (gular)g(and)h(decay-)166 4082 y(ing)h(function)g Fm(F)42 b Fl(:)27 b Fg(R)906 4046 y Fv(3)905 4107 y Ft(~)-34 b(x)972 4082 y Fj(\002)23 b Fl(\()p Fg(R)1182 4046 y Fv(3)1179 4107 y Ft(~)1182 4124 y(k)1247 4082 y Fj(\002)f Fg(Z)1412 4097 y Fv(2)1452 4082 y Fl(\))28 b Fj(!)f Fg(C)p Fp(,)361 4360 y Fi(h)400 4456 y Fm(v)447 4471 y Ft(\027)491 4456 y Fl(\()544 4431 y Fm(~)529 4456 y(G)606 4471 y Ft(t)635 4456 y Fl(\))h Fm(;)44 b(A)p Fl(\()p Fm(F)14 b Fl(\))998 4360 y Fi(i)3108 4456 y Fp(\(VI.124\))378 4639 y Fl(=)470 4542 y Fi(h)532 4639 y Fj(\000)22 b Fm(i@)715 4654 y Ft(x)755 4662 y Fc(\027)826 4639 y Fm(;)44 b(a)948 4598 y Fs(\003)988 4639 y Fl(\()p Fm(F)14 b Fl(\))22 b(+)g Fm(a)p Fl(\()p Fm(F)14 b Fl(\))1465 4542 y Fi(i)1548 4639 y Fl(+)1668 4542 y Fi(h)1707 4639 y Fm(a)1758 4598 y Fs(\003)1798 4639 y Fl(\()p Fm(G)1913 4654 y Ft(t;\027)2001 4639 y Fl(\))22 b(+)g Fm(a)p Fl(\()p Fm(G)2325 4654 y Ft(t;\027)2414 4639 y Fl(\))27 b Fm(;)45 b(a)2602 4598 y Fs(\003)2641 4639 y Fl(\()p Fm(F)14 b Fl(\))22 b(+)g Fm(a)p Fl(\()p Fm(F)14 b Fl(\))3118 4542 y Fi(i)378 4821 y Fl(=)i Fj(\000)p Fm(ia)631 4780 y Fs(\003)672 4821 y Fl(\()p Fm(@)761 4836 y Ft(x)801 4844 y Fc(\027)843 4821 y Fm(F)e Fl(\))22 b Fj(\000)h Fm(ia)p Fl(\()p Fm(@)1253 4836 y Ft(x)1293 4844 y Fc(\027)1336 4821 y Fm(F)14 b Fl(\))44 b(+)g(2)17 b(Im)1797 4725 y Fi(D)1848 4821 y Fm(G)1925 4836 y Ft(t;\027)2013 4722 y Fi(\014)2013 4772 y(\014)2013 4821 y(\014)p Fm(F)2118 4725 y Fi(E)2190 4821 y Fm(;)378 5004 y Fl(=)f Fj(\000)p Fm(iA)p Fl(\()p Fm(@)742 5019 y Ft(x)782 5027 y Fc(\027)826 5004 y Fm(F)e Fl(\))44 b(+)g(2)17 b(Im)1287 4908 y Fi(D)1338 5004 y Fm(G)1415 5019 y Ft(t;\027)1503 4904 y Fi(\014)1503 4954 y(\014)1503 5004 y(\014)o Fm(F)1607 4908 y Fi(E)1680 5004 y Fm(;)166 5247 y Fp(where)30 b Fm(G)516 5262 y Ft(t;\027)641 5247 y Fl(:=)36 b Fm(\037)841 5262 y Ft(t)871 5247 y Fm(G)948 5262 y Ft(\027)991 5247 y Fp(.)30 b(Choosing)e Fm(F)50 b Fl(:=)59 b(_)-49 b Fm(\037)1771 5262 y Ft(t)1801 5247 y Fm(G)1878 5262 y Ft(\027)1950 5247 y Fp(and)30 b(summing)d(o)o(v)o(er)i Fm(\027)43 b Fj(2)37 b(f)p Fl(1)p Fm(;)17 b Fl(2)p Fm(;)g Fl(3)p Fj(g)p Fp(,)28 b(this)166 5380 y(and)d(the)f(Coulomb)g(gauge)h(condition)1550 5355 y Fm(~)1541 5380 y Fj(r)1624 5395 y Ft(x)1690 5380 y Fj(\001)1754 5355 y Fm(~)1740 5380 y(G)i Fl(=)h(0)d Fp(imply)e(that) 1748 5712 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 361 96 a Fm(v)t Fl(\()465 71 y Fm(~)450 96 y(G)527 111 y Ft(t)556 96 y Fl(\))22 b Fj(\001)g Fm(A)p Fl(\()h(_)-50 b Fm(\037)838 111 y Ft(t)883 71 y Fm(~)868 96 y(G)p Fl(\))39 b Fj(\000)g Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)1310 111 y Ft(t)1354 71 y Fm(~)1340 96 y(G)p Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()1631 71 y Fm(~)1616 96 y(G)1693 111 y Ft(t)1722 96 y Fl(\))52 b(=)1220 b Fp(\(VI.125\))394 279 y Fj(\000)p Fm(i)17 b(A)p Fl(\()23 b(_)-50 b Fm(\037)693 294 y Ft(t)733 254 y Fm(~)724 279 y Fj(r)807 294 y Ft(x)873 279 y Fj(\001)937 254 y Fm(~)922 279 y(G)p Fl(\))45 b(+)f(Im)1318 158 y Fi(\032)1397 162 y(Z)1497 179 y(\014)1497 229 y(\014)1497 279 y(\014)1539 254 y Fm(~)1524 279 y(G)p Fl(\()p Fm(k)s Fl(\))1731 179 y Fi(\014)1731 229 y(\014)1731 279 y(\014)1759 206 y Fv(2)1820 279 y Fm(@)1871 294 y Ft(t)1902 183 y Fi(\020)1951 279 y Fm(\037)2012 238 y Fv(2)2012 303 y Ft(t)2052 279 y Fl(\()p Fm(k)s Fl(\))2182 183 y Fi(\021)2253 279 y Fm(dk)2358 158 y Fi(\033)2473 279 y Fl(=)52 b(0)22 b Fm(:)166 559 y Fp(Thus)i Fj(\000)500 534 y Fl(_)467 559 y Fm(P)530 574 y Ft(t)588 559 y Fl(=)j Fm(Q)768 574 y Ft(t)820 559 y Fl(+)22 b Fm(Q)995 523 y Fs(\003)995 583 y Ft(t)1035 559 y Fp(,)j(where)1138 847 y Fm(Q)1215 862 y Ft(t)1297 847 y Fl(:=)53 b(2)1534 779 y Fm(P)1611 743 y Fs(?)1597 804 y Ft(t)p 1528 824 148 4 v 1528 915 a Fm(H)1617 874 y Fv(+)1609 935 y Ft(t)1702 847 y Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)1874 862 y Ft(t)1918 822 y Fm(~)1904 847 y(G)p Fl(\))22 b Fj(\001)g Fm(v)t Fl(\()2195 822 y Fm(~)2180 847 y(G)2257 862 y Ft(t)2286 847 y Fl(\))17 b Fm(P)2404 862 y Ft(t)2455 847 y Fm(:)626 b Fp(\(VI.126\))166 1124 y(Hence,)35 b(it)f(suf)n(\002ces)g(to)g(sho)n(w)g(the)g(e)o (xistence)g(of)g(a)h(contant)f Fm(C)52 b(<)46 b Fj(1)34 b Fp(such)g(that)g Fj(k)p Fm(Q)3251 1139 y Ft(t)3281 1124 y Fj(k)45 b(\024)166 1258 y Fm(C)24 b(\032)310 1207 y Fv(1)p Ft(=)p Fv(2)310 1277 y Ft(t)420 1258 y Fp(.)h(T)-8 b(o)24 b(this)g(end,)h(we)g(use)g Fl(2)p Fm(v)t Fl(\()1412 1232 y Fm(~)1397 1258 y(G)1474 1273 y Ft(t)1503 1258 y Fl(\))i(=)h Fj(\000)p Fm(i)p Fl([)p Fm(H)1898 1216 y Fv(+)1890 1277 y Ft(t)1974 1258 y Fm(;)k(~)-48 b(x)q Fl(])25 b Fp(and)f Fm(H)2399 1216 y Fv(+)2391 1277 y Ft(t)2458 1258 y Fm(P)2521 1273 y Ft(t)2578 1258 y Fl(=)k(0)c Fp(and)h(obtain)361 1666 y Fm(Q)438 1681 y Ft(t)485 1666 y Fl(=)16 b Fj(\000)p Fm(i)721 1599 y(P)798 1563 y Fs(?)784 1623 y Ft(t)p 714 1643 V 714 1735 a Fm(H)803 1693 y Fv(+)795 1754 y Ft(t)889 1666 y Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)1061 1681 y Ft(t)1105 1641 y Fm(~)1090 1666 y(G)p Fl(\))22 b Fj(\001)g Fl([)p Fm(H)1393 1625 y Fv(+)1385 1691 y Ft(t)1468 1666 y Fm(;)32 b(~)-47 b(x)p Fl(])17 b Fm(P)1691 1681 y Ft(t)1801 1666 y Fl(=)80 b Fj(\000)p Fm(i)2131 1558 y Fv(3)2089 1583 y Fi(X)2084 1764 y Ft(\027)t Fv(=1)2246 1599 y Fm(P)2323 1563 y Fs(?)2309 1623 y Ft(t)p 2240 1643 V 2240 1735 a Fm(H)2329 1693 y Fv(+)2321 1754 y Ft(t)2414 1666 y Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)2586 1681 y Ft(t)2616 1666 y Fm(G)2693 1681 y Ft(\027)2736 1666 y Fl(\))17 b Fm(H)2880 1625 y Fv(+)2872 1691 y Ft(t)2955 1666 y Fm(x)3010 1681 y Ft(\027)3070 1666 y Fm(P)3133 1681 y Ft(t)485 1965 y Fl(=)f Fj(\000)p Fm(i)751 1857 y Fv(3)709 1882 y Fi(X)704 2063 y Ft(\027)t Fv(=1)866 1898 y Fm(P)943 1861 y Fs(?)929 1922 y Ft(t)p 860 1942 V 860 2033 a Fm(H)949 1992 y Fv(+)941 2053 y Ft(t)1034 1965 y Fl([)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)1233 1980 y Ft(t)1263 1965 y Fm(G)1340 1980 y Ft(\027)1383 1965 y Fl(\))17 b Fm(;)33 b(H)1579 1980 y Ft(t)1609 1965 y Fl(])17 b Fm(x)1708 1980 y Ft(\027)1768 1965 y Fm(P)1831 1980 y Ft(t)1904 1965 y Fj(\000)45 b Fm(i)17 b(P)2153 1924 y Fs(?)2139 1990 y Ft(t)2228 1965 y Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)2400 1980 y Ft(t)2445 1940 y Fm(~)2430 1965 y(G)p Fl(\))22 b Fj(\001)e Fm(~)-47 b(x)17 b(P)2752 1980 y Ft(t)2803 1965 y Fm(:)278 b Fp(\(VI.127\))166 2253 y(Note)25 b(that)361 2578 y Fj(\000)p Fm(i)p Fl([)p Fm(A)p Fl(\()e(_)-50 b Fm(\037)670 2593 y Ft(t)701 2578 y Fm(G)778 2593 y Ft(\027)821 2578 y Fl(\))17 b Fm(;)33 b(H)1017 2593 y Ft(t)1046 2578 y Fl(])17 b(=)f Fj(\000)p Fm(i)p Fl([)p Fm(A)p Fl(\()24 b(_)-51 b Fm(\037)1491 2593 y Ft(t)1522 2578 y Fm(G)1599 2593 y Ft(\027)1642 2578 y Fl(\))17 b Fm(;)1782 2552 y Fl(\024)1757 2578 y Fm(H)1838 2593 y Ft(t)1868 2578 y Fl(])44 b Fj(\000)h Fm(i)p Fl([)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)2293 2593 y Ft(t)2323 2578 y Fm(G)2400 2593 y Ft(\027)2443 2578 y Fl(\))17 b Fm(;)33 b(v)t Fl(\()2662 2552 y Fm(~)2647 2578 y(G)2724 2593 y Ft(t)2753 2578 y Fl(\))2791 2536 y Fv(2)2830 2578 y Fl(])1090 2795 y(=)16 b Fm(A)p Fl(\()p Fm(i!)27 b Fl(_)-50 b Fm(\037)1452 2810 y Ft(t)1482 2795 y Fm(G)1559 2810 y Ft(\027)1602 2795 y Fl(\))44 b Fj(\000)1855 2687 y Fv(3)1812 2712 y Fi(X)1806 2893 y Ft(\026)p Fv(=1)1955 2795 y Fm(i)p Fl([)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)2187 2810 y Ft(t)2217 2795 y Fm(G)2294 2810 y Ft(\027)2337 2795 y Fl(\))17 b Fm(;)33 b(v)2503 2754 y Fv(2)2499 2819 y Ft(\026)2546 2795 y Fl(\()2599 2769 y Fm(~)2584 2795 y(G)2661 2810 y Ft(t)2691 2795 y Fl(\)])22 b Fm(:)303 b Fp(\(VI.128\))166 3096 y(A)35 b(straightforw)o(ard,)e(b)n(ut)i(some)n (what)e(lengthy)g(computation)g(repeatedly)h(using)g(\(VI.124\))166 3216 y(yields)361 3559 y Fj(\000)p Fm(i)p Fl([)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)670 3574 y Ft(t)701 3559 y Fm(G)778 3574 y Ft(\027)821 3559 y Fl(\))17 b Fm(;)33 b(v)987 3518 y Fv(2)983 3583 y Ft(\026)1030 3559 y Fl(\()1083 3534 y Fm(~)1068 3559 y(G)1145 3574 y Ft(t)1174 3559 y Fl(\)])1869 b Fp(\(VI.129\))378 3726 y Fl(=)16 b Fj(\000)p Fm(i)p Fl([)p Fm(A)p Fl(\()24 b(_)-51 b Fm(\037)779 3741 y Ft(t)810 3726 y Fm(G)887 3741 y Ft(\027)930 3726 y Fl(\))17 b Fm(;)33 b(v)1092 3741 y Ft(\026)1139 3726 y Fl(\()1192 3700 y Fm(~)1177 3726 y(G)1254 3741 y Ft(t)1283 3726 y Fl(\)])17 b Fm(v)1412 3741 y Ft(\026)1459 3726 y Fl(\()1512 3700 y Fm(~)1497 3726 y(G)1574 3741 y Ft(t)1603 3726 y Fl(\))39 b Fj(\000)g Fm(v)1843 3741 y Ft(\026)1890 3726 y Fl(\()1943 3700 y Fm(~)1928 3726 y(G)2005 3741 y Ft(t)2034 3726 y Fl(\))17 b Fm(i)p Fl([)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)2321 3741 y Ft(t)2351 3726 y Fm(G)2428 3741 y Ft(\027)2471 3726 y Fl(\))17 b Fm(;)33 b(v)2633 3741 y Ft(\026)2680 3726 y Fl(\()2733 3700 y Fm(~)2718 3726 y(G)2795 3741 y Ft(t)2824 3726 y Fl(\)])378 3886 y(=)470 3790 y Fi(n)526 3886 y Fm(A)p Fl(\()p Fm(i)22 b Fl(_)-49 b Fm(\037)731 3901 y Ft(t)761 3886 y Fm(@)812 3901 y Ft(x)852 3909 y Fc(\026)898 3886 y Fm(G)975 3901 y Ft(\027)1018 3886 y Fl(\))22 b(+)g(2)p Fm(i)17 b Fl(Im)p Fj(h)p Fm(\037)1491 3901 y Ft(t)1520 3886 y Fm(G)1597 3901 y Ft(\026)1644 3886 y Fj(j)22 b Fl(_)-49 b Fm(\037)1733 3901 y Ft(t)1762 3886 y Fm(G)1839 3901 y Ft(\027)1882 3886 y Fj(i)1921 3790 y Fi(o)1993 3886 y Fm(v)2040 3901 y Ft(\026)2087 3886 y Fl(\()2140 3861 y Fm(~)2125 3886 y(G)2202 3901 y Ft(t)2231 3886 y Fl(\))492 4069 y(+)22 b Fm(v)637 4084 y Ft(\026)684 4069 y Fl(\()737 4044 y Fm(~)722 4069 y(G)799 4084 y Ft(t)829 4069 y Fl(\))884 3973 y Fi(n)939 4069 y Fm(A)p Fl(\()p Fm(i)g Fl(_)-49 b Fm(\037)1144 4084 y Ft(t)1174 4069 y Fm(@)1225 4084 y Ft(x)1265 4092 y Fc(\026)1311 4069 y Fm(G)1388 4084 y Ft(\027)1431 4069 y Fl(\))22 b(+)g(2)p Fm(i)17 b Fl(Im)p Fj(h)p Fm(\037)1904 4084 y Ft(t)1933 4069 y Fm(G)2010 4084 y Ft(\026)2057 4069 y Fj(j)22 b Fl(_)-49 b Fm(\037)2146 4084 y Ft(t)2175 4069 y Fm(G)2252 4084 y Ft(\027)2295 4069 y Fj(i)2334 3973 y Fi(o)378 4236 y Fl(=)16 b Fj(\000)p Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)719 4251 y Ft(t)750 4236 y Fm(k)804 4195 y Fv(2)843 4236 y Fm(G)920 4251 y Ft(\027)963 4236 y Fl(\))22 b(+)g(2)p Fm(i)17 b Fl(Im)p Fj(h)p Fm(k)1426 4251 y Ft(\026)1472 4236 y Fm(G)1549 4251 y Ft(t;\026)1641 4236 y Fj(j)22 b Fl(_)-49 b Fm(\037)1730 4251 y Ft(t)1760 4236 y Fm(G)1837 4251 y Ft(\027)1880 4236 y Fj(i)492 4397 y Fl(+)22 b(2)639 4300 y Fi(n)694 4397 y Fm(A)p Fl(\()h(_)-50 b Fm(\037)866 4412 y Ft(t)896 4397 y Fm(k)947 4412 y Ft(\026)994 4397 y Fm(G)1071 4412 y Ft(\027)1114 4397 y Fl(\))22 b(+)g(2)p Fm(i)17 b Fl(Im)p Fj(h)p Fm(\037)1587 4412 y Ft(t)1616 4397 y Fm(G)1693 4412 y Ft(\026)1740 4397 y Fj(j)22 b Fl(_)-49 b Fm(\037)1829 4412 y Ft(t)1858 4397 y Fm(G)1935 4412 y Ft(\027)1979 4397 y Fj(i)2018 4300 y Fi(o)2089 4397 y Fm(v)2136 4412 y Ft(\026)2183 4397 y Fl(\()2236 4371 y Fm(~)2221 4397 y(G)2298 4412 y Ft(t)2328 4397 y Fl(\))22 b Fm(:)166 4623 y Fp(Summing)h(o)o(v)o(er)g Fm(\026)k Fl(=)h(1)p Fm(;)17 b Fl(2)p Fm(;)g Fl(3)23 b Fp(and)h(taking)f(the)h(Coulomb)f (gauge)g(condition,)g(which)g(implies)166 4743 y(that)335 4717 y Fm(~)340 4743 y(k)j Fj(\001)481 4718 y Fm(~)466 4743 y(G)543 4758 y Ft(t)573 4743 y Fl(\()p Fm(k)s Fl(\))i(=)f(0)p Fp(,)e(we)g(ha)n(v)o(e)f(that)361 5077 y Fj(\000)p Fm(i)p Fl([)p Fm(A)p Fl(\()f(_)-50 b Fm(\037)670 5092 y Ft(t)701 5077 y Fm(G)778 5092 y Ft(\027)821 5077 y Fl(\))17 b Fm(;)33 b(H)1017 5092 y Ft(t)1046 5077 y Fl(])17 b(=)f Fm(A)p Fl(\()p Fm(i!)27 b Fl(_)-50 b Fm(\037)1452 5092 y Ft(t)1482 5077 y Fm(G)1559 5092 y Ft(\027)1602 5077 y Fl(\))39 b Fj(\000)g Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)1967 5092 y Ft(t)1997 5077 y Fm(k)2051 5035 y Fv(2)2090 5077 y Fm(G)2167 5092 y Ft(\027)2210 5077 y Fl(\))860 b Fp(\(VI.130\))1205 5294 y Fl(+)22 b(2)1417 5186 y Fv(3)1374 5211 y Fi(X)1368 5392 y Ft(\026)p Fv(=1)1517 5197 y Fi(n)1573 5294 y Fm(A)p Fl(\()g(_)-49 b Fm(\037)1745 5309 y Ft(t)1774 5294 y Fm(k)1825 5309 y Ft(\026)1872 5294 y Fm(G)1949 5309 y Ft(\027)1992 5294 y Fl(\))22 b(+)g(2)p Fm(i)17 b Fl(Im)p Fj(h)p Fm(\037)2465 5309 y Ft(t)2495 5294 y Fm(G)2572 5309 y Ft(\026)2618 5294 y Fj(j)22 b Fl(_)-49 b Fm(\037)2707 5309 y Ft(t)2737 5294 y Fm(G)2814 5309 y Ft(\027)2857 5294 y Fj(i)2896 5197 y Fi(o)2967 5294 y Fm(v)3014 5309 y Ft(\026)3061 5294 y Fl(\()3114 5268 y Fm(~)3099 5294 y(G)3176 5309 y Ft(t)3206 5294 y Fl(\))22 b Fm(:)1748 5712 y Fp(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 166 101 a Fp(Inserting)24 b(this)g(into)g (\(VI.127\),)g(we)h(obtain)f(that)g Fm(Q)1955 116 y Ft(t)2013 101 y Fl(=)j Fm(Q)2193 50 y Fv(\(1\))2193 121 y Ft(t)2310 101 y Fl(+)21 b Fm(Q)2484 50 y Fv(\(2\))2484 121 y Ft(t)2601 101 y Fl(+)g Fm(Q)2775 50 y Fv(\(3\))2775 121 y Ft(t)2891 101 y Fl(+)h Fm(Q)3066 50 y Fv(\(4\))3066 121 y Ft(t)3161 101 y Fp(,)i(where)361 484 y Fm(Q)438 433 y Fv(\(1\))438 504 y Ft(t)549 484 y Fl(:=)17 b Fj(\000)762 417 y Fm(P)839 381 y Fs(?)825 441 y Ft(t)p 756 461 148 4 v 756 553 a Fm(H)845 511 y Fv(+)837 572 y Ft(t)930 484 y Fm(A)1003 388 y Fi(\020)1053 484 y Fl(\()p Fm(k)1145 443 y Fv(2)1207 484 y Fj(\000)22 b Fm(i!)t Fl(\))g(_)-49 b Fm(\037)1503 499 y Ft(t)1547 459 y Fm(~)1533 484 y(G)1610 499 y Ft(t)1639 388 y Fi(\021)1711 484 y Fj(\001)20 b Fm(~)-47 b(x)17 b(P)1896 499 y Ft(t)1947 484 y Fm(;)1134 b Fp(\(VI.131\))361 706 y Fm(Q)438 655 y Fv(\(2\))438 726 y Ft(t)549 706 y Fl(:=)17 b Fj(\000)p Fm(iP)856 665 y Fs(?)842 731 y Ft(t)932 706 y Fm(A)p Fl(\()22 b(_)-49 b Fm(\037)1104 721 y Ft(t)1148 681 y Fm(~)1134 706 y(G)1211 721 y Ft(t)1240 706 y Fl(\))22 b Fj(\001)e Fm(~)-47 b(x)17 b(P)1485 721 y Ft(t)1537 706 y Fm(;)1544 b Fp(\(VI.132\))361 923 y Fm(Q)438 873 y Fv(\(3\))438 943 y Ft(t)549 923 y Fl(:=)17 b(2)812 816 y Fv(3)770 840 y Fi(X)734 1022 y Ft(\026;\027)t Fv(=1)958 856 y Fm(P)1035 820 y Fs(?)1021 881 y Ft(t)p 952 900 V 952 992 a Fm(H)1041 951 y Fv(+)1033 1012 y Ft(t)1126 923 y Fm(A)p Fl(\()23 b(_)-50 b Fm(\037)1298 938 y Ft(t)1328 923 y Fm(k)1379 938 y Ft(\026)1425 923 y Fm(G)1502 938 y Ft(\027)1546 923 y Fl(\))17 b Fm(v)1648 938 y Ft(\026)1694 923 y Fl(\()1747 898 y Fm(~)1732 923 y(G)1809 938 y Ft(t)1838 923 y Fl(\))g Fm(x)1948 938 y Ft(\027)2008 923 y Fm(P)2071 938 y Ft(t)2123 923 y Fm(;)958 b Fp(\(VI.133\))361 1235 y Fm(Q)438 1184 y Fv(\(4\))438 1255 y Ft(t)549 1235 y Fl(:=)17 b(4)p Fm(i)846 1127 y Fv(3)803 1152 y Fi(X)768 1333 y Ft(\026;\027)t Fv(=1)991 1168 y Fm(P)1068 1132 y Fs(?)1054 1192 y Ft(t)p 985 1212 V 985 1303 a Fm(H)1074 1262 y Fv(+)1066 1323 y Ft(t)1160 1235 y Fl(Im)p Fj(h)p Fm(\037)1376 1250 y Ft(t)1405 1235 y Fm(G)1482 1250 y Ft(\026)1529 1235 y Fj(j)22 b Fl(_)-49 b Fm(\037)1618 1250 y Ft(t)1647 1235 y Fm(G)1724 1250 y Ft(\027)1768 1235 y Fj(i)17 b Fm(v)1871 1250 y Ft(\026)1917 1235 y Fl(\()1970 1210 y Fm(~)1955 1235 y(G)2032 1250 y Ft(t)2061 1235 y Fl(\))g Fm(x)2171 1250 y Ft(\027)2231 1235 y Fm(P)2294 1250 y Ft(t)2346 1235 y Fm(:)735 b Fp(\(VI.134\))166 1527 y(W)-8 b(e)20 b(estimate)f(each)h(of)g(these)f(terms)g(separately) -6 b(.)19 b(Using)g(the)h(pre)n(vious)e(estimates,)g(especially)166 1647 y(Lemma)24 b(III.2,)h(we)g(ha)n(v)o(e)361 2023 y Fj(k)p Fm(Q)488 1972 y Fv(\(1\))488 2043 y Ft(t)583 2023 y Fj(k)16 b(\024)h Fl(8)809 1899 y Fi(\015)808 1948 y(\015)808 1998 y(\015)808 2048 y(\015)871 1956 y Fm(P)948 1920 y Fs(?)934 1980 y Ft(t)p 865 2000 V 865 2091 a Fm(H)954 2050 y Fv(+)946 2111 y Ft(t)1039 2023 y Fl(\()1102 1998 y(\024)1077 2023 y Fm(H)1158 2038 y Ft(t)p Fs(\000)p Ft(\034)t(;t)1344 2023 y Fl(+)22 b Fm(\032)1492 2038 y Ft(t)1522 2023 y Fl(\))1560 1982 y Fv(1)p Ft(=)p Fv(2)1670 1899 y Fi(\015)1670 1948 y(\015)1670 1998 y(\015)1670 2048 y(\015)1733 1923 y(\015)1733 1973 y(\015)1733 2023 y(\015)p Fm(\032)1829 1972 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1829 2043 y Ft(t)1994 2023 y Fl(\()p Fm(k)2086 1982 y Fv(2)2148 2023 y Fl(+)g Fm(i!)t Fl(\))g(_)-49 b Fm(\037)2443 2038 y Ft(t)2487 1998 y Fm(~)2472 2023 y(G)2549 2038 y Ft(t)2579 1923 y Fi(\015)2579 1973 y(\015)2579 2023 y(\015)22 b Fj(k)n Fm(~)-47 b(xP)2815 2038 y Ft(t)2845 2023 y Fj(k)649 2298 y(\024)753 2230 y Fl(128)17 b Fm(\021)i Fj(k)p Fm(p)p Fj(k)1133 2245 y Fs(1)1224 2230 y Fm(\013)1287 2194 y Fv(3)p Ft(=)p Fv(2)p 753 2275 644 4 v 1008 2366 a Fm(\031)i(#)1423 2298 y(\032)1473 2313 y Ft(t)1520 2298 y Fj(k)n Fm(~)-47 b(x\036)1683 2313 y Ft(t)1713 2298 y Fj(k)22 b Fm(;)1319 b Fp(\(VI.135\))361 2500 y Fj(k)p Fm(Q)488 2449 y Fv(\(2\))488 2520 y Ft(t)583 2500 y Fj(k)16 b(\024)h Fl(16)858 2400 y Fi(\015)857 2450 y(\015)857 2500 y(\015)p Fm(\032)953 2449 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)953 2520 y Ft(t)1141 2500 y Fl(_)-49 b Fm(\037)1180 2515 y Ft(t)1224 2475 y Fm(~)1209 2500 y(G)1286 2515 y Ft(t)1316 2400 y Fi(\015)1316 2450 y(\015)1316 2500 y(\015)1379 2400 y(\015)1379 2450 y(\015)1379 2500 y(\015)p Fl(\()p Fm(H)1552 2459 y Fv(+)1544 2525 y Ft(t)1632 2500 y Fl(+)22 b Fm(\032)1780 2515 y Ft(t)1810 2500 y Fl(\))1848 2459 y Fv(1)p Ft(=)p Fv(2)1973 2500 y Fm(~)-47 b(x\036)2088 2515 y Ft(t)2118 2400 y Fi(\015)2118 2450 y(\015)2118 2500 y(\015)649 2736 y Fj(\024)753 2669 y Fl(32)17 b Fm(\021)i Fj(k)p Fm(p)p Fj(k)1084 2684 y Fs(1)1175 2669 y Fm(\013)1238 2633 y Fv(3)p Ft(=)p Fv(2)p 753 2713 596 4 v 1021 2805 a Fm(\031)1375 2736 y(\032)1425 2686 y Fv(1)p Ft(=)p Fv(2)1425 2756 y Ft(t)1552 2637 y Fi(\015)1552 2687 y(\015)1552 2736 y(\015)p Fl(\()p Fm(H)1725 2695 y Fv(+)1717 2761 y Ft(t)1805 2736 y Fl(+)j Fm(\032)1953 2751 y Ft(t)1983 2736 y Fl(\))2021 2695 y Fv(1)p Ft(=)p Fv(2)2146 2736 y Fm(~)-47 b(x\036)2261 2751 y Ft(t)2291 2637 y Fi(\015)2291 2687 y(\015)2291 2736 y(\015)22 b Fm(;)745 b Fp(\(VI.136\))361 2997 y Fj(k)p Fm(Q)488 2946 y Fv(\(3\))488 3017 y Ft(t)583 2997 y Fj(k)16 b(\024)h Fl(8)809 2873 y Fi(\015)808 2922 y(\015)808 2972 y(\015)808 3022 y(\015)871 2930 y Fm(P)948 2894 y Fs(?)934 2954 y Ft(t)p 865 2974 148 4 v 865 3065 a Fm(H)954 3024 y Fv(+)946 3085 y Ft(t)1039 2997 y Fl(\()1102 2972 y(\024)1077 2997 y Fm(H)1158 3012 y Ft(t)p Fs(\000)p Ft(\034)t(;t)1344 2997 y Fl(+)22 b Fm(\032)1492 3012 y Ft(t)1522 2997 y Fl(\))1560 2956 y Fv(1)p Ft(=)p Fv(2)1670 2873 y Fi(\015)1670 2922 y(\015)1670 2972 y(\015)1670 3022 y(\015)1733 2897 y(\015)1733 2947 y(\015)1733 2997 y(\015)p Fm(\032)1829 2946 y Fs(\000)p Fv(1)p Ft(=)p Fv(2)1829 3017 y Ft(t)1994 2997 y Fj(j)2017 2971 y Fm(~)2022 2997 y(k)s Fj(j)39 b Fl(_)-49 b Fm(\037)2182 3012 y Ft(t)2226 2972 y Fm(~)2211 2997 y(G)2288 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454 y Fp(W)-8 b(e)29 b(\002nally)f(use)g(the)g(f)o(act)g (that)g Fl(sup)1420 478 y Ft(t>)p Fv(0)1556 454 y Fj(k)n Fm(~)-47 b(x\036)1719 469 y Ft(t)1749 454 y Fj(k)34 b(\024)g Fm(C)41 b(<)34 b Fj(1)27 b Fp(is)h(uniformly)f(bounded,)g(which)166 575 y(is)34 b(the)g(pro)o(v)o(ed)e(in)i([5,)g(Theorem)g(III.10].)g (Inserting)g(this)f(uniform)g(bound)g(on)h Fj(k)n Fm(~)-47 b(x\036)3196 590 y Ft(t)3226 575 y Fj(k)34 b Fp(and)166 695 y(\(VI.141\))25 b(into)f(\(VI.140\),)g(we)h(arri)n(v)o(e)g(at)f (the)h(asserted)g(claim.)489 b Fb(2)166 1171 y Fq(Refer)n(ences)166 1504 y Fx([1])26 b(A.)h(Arai)h(and)h(M.)e(Hiroka)o(w)o(a.)50 b(On)28 b(the)h(e)o(xistence)h(and)f(uniqueness)j(of)c(ground)i(states) g(of)f(the)f(a)297 1617 y(generalized)f(spin-boson)f(model.)34 b FA(J)n(.)23 b(Funct.)g(Anal.)p Fx(,)g(151\(2\):455\226503,)28 b(1997.)166 1785 y([2])e(V)-12 b(.)37 b(Bach,)h(J.)g(Fr)8 b(\250)-38 b(ohlich,)40 b(and)f(I.)e(M.)h(Sigal.)80 b(Quantum)39 b(electrodynamics)k(of)38 b(con\002ned)i(non-)297 1898 y(relati)n(vistic)26 b(particles.)36 b FA(Adv)-7 b(.)23 b(in)g(Math.)h Fx(,)e(137:299\226395,)27 b(1998.)166 2066 y([3])f(V)-12 b(.)39 b(Bach,)g(J.)h(Fr)8 b(\250)-38 b(ohlich,)40 b(and)h(I.)e(M.)g(Sigal.)85 b(Renormalization)43 b(group)e(analysis)h(of)e(spectral)297 2179 y(problems)25 b(in)e(quantum)i(\002eld)f(theory)-6 b(.)35 b FA(Adv)-7 b(.)23 b(in)g(Math.)g Fx(,)g(137:205\226298,)k(1998.)166 2347 y([4])f(V)-12 b(.)38 b(Bach,)g(J.)g(Fr)8 b(\250)-38 b(ohlich,)40 b(and)f(I.)f(M.)g(Sigal.)82 b(Spectral)40 b(analysis)h(for)e(systems)h(of)e(atoms)h(and)297 2460 y(molecules)24 b(coupled)g(to)e(the)g(quantized)j(radiation)f(\002eld.) 30 b FA(Commun.)21 b(Math.)h(Phys.)p Fx(,)f(207\(2\):249\226)297 2573 y(290,)j(1999.)166 2741 y([5])i(V)-12 b(.)23 b(Bach,)h(J.)f(Fr)8 b(\250)-38 b(ohlich,)25 b(and)g(A.)d(Pizzo.)35 b(Infrared-\002nite)27 b(algorithms)g(in)d(QED:)e(The)i(groundstate)297 2854 y(of)42 b(an)g(atom)g(interacting)j(with)c(the)i(quantized)h(radiation) h(\002eld.)92 b FA(Commun.)41 b(Math.)h(Phys.)p Fx(,)297 2967 y(February)25 b(2006.)34 b(DOI:)23 b(10.1007/s00220-005)q(-14)q (78)q(-3.)166 3135 y([6])j(Ch.)h(Gerard,)48 b(On)27 b(the)h(e)o (xistence)i(of)e(ground)i(states)f(for)f(massless)h(Pauli-Fierz)g (Hamiltonians,)297 3247 y FA(Ann.)23 b(H.)f(P)-7 b(oincar)889 3248 y(\264)884 3247 y(e)p Fx(,1\(3\):)26 b(443\226459,)g(2000.)34 b(DOI:)45 b(10.1007/s00023-00)q(5-)q(000)q(-2)q(.)166 3415 y([7])26 b(Ch.)k(Gerard,)56 b(A)29 b(remark)i(on)g(the)g(paper:)h (\224On)e(the)h(e)o(xistence)h(of)f(ground)h(states)g(for)f(massless) 297 3528 y(Pauli-Fierz)25 b(Hamiltonians\224,)35 b(preprint,)26 b(mp-arc)e(06-146,)h(2006.)166 3696 y([8])h(M.)46 b(Griesemer)l(,)i(E.) e(Lieb,)g(and)i(M.)e(Loss.)109 b(Ground)48 b(states)g(in)f(nonrelati)n (vistic)52 b(quantum)297 3809 y(electrodynamics.)37 b FA(In)l(vent.)25 b(math.)p Fx(,)e(June)i(2001.)34 b(DOI)22 b(10.1007/s00222-01)q(0-0)q(15)q(9.)166 3977 y([9])k(F)-7 b(.)18 b(Hiroshima.)28 b(Functional)22 b(Inte)o(gral)g(Representation)i (of)c(a)g(Model)h(in)g(QED.)j FA(Re)o(v)-7 b(.)19 b(Math.)i(Phys.)p Fx(,)297 4090 y(9\(4\):489\226530,)27 b(1997.)166 4258 y([10])f(F)-7 b(.)20 b(Hiroshima.)29 b(Ground)22 b(states)g(of)f(a)f (model)i(in)f(nonrelati)n(vistic)k(quantum)d(electrodynamics)j(II.)297 4371 y FA(J)n(.)d(Math.)i(Phys.)f Fx(,)f(41\(2\):)j(611\226674)h(,)d (2000.)166 4539 y([11])j(F)-7 b(.)27 b(Hiroshima.)51 b(Self-adjointness)33 b(of)28 b(the)h(P)o(auli-Fierz)g(Hamiltonian)i (for)e(arbitrary)i(coupling)297 4652 y(constants.)d FA(Ann.)19 b(Henri)h(P)-7 b(oincar)1391 4653 y(\264)1386 4652 y(e)p Fx(,)21 b(3\(1\):)g(171\226201,)h(2002.)k(DOI:)19 b(10.1007/s00023-00)q (2-)q(86)q(15-)297 4765 y(8.)166 4933 y([12])26 b(F)-7 b(.)25 b(Hiroshima)i(and)f(H.)f(Spohn.)41 b(Groundstate)29 b(De)o(generac)o(y)e(of)f(the)g(P)o(auli-Fierz)h(Hamiltonian)297 5046 y(with)c(spin.)34 b FA(Adv)-7 b(.)23 b(Theor)-10 b(.)24 b(Math.)f(Phys.)p Fx(,)g(5\(6\):)i(1091\2261104,2001.)166 5214 y([13])h(A.)81 b(Pizzo.)222 b(One-particle)84 b(\(improper\))h (states)e(in)f(Nelson')-5 b(s)84 b(massless)g(model.)297 5327 y FA(Ann.)23 b(H.)f(P)-7 b(oincar)889 5328 y(\264)884 5327 y(e)p Fx(,)24 b(4\(3\):439\226486,)k(2003.)1748 5712 y Fp(22)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0610060234948--