Content-Type: multipart/mixed; boundary="-------------0604301720547" This is a multi-part message in MIME format. ---------------0604301720547 Content-Type: text/plain; name="06-141.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-141.keywords" QED, Infrared Problem, Rayleigh Scattering, Expansion, Bohr's Frequency condition, Algorithm ---------------0604301720547 Content-Type: application/postscript; name="BFP3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BFP3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: BFP3.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMSY10 CMBX10 CMSY7 CMSL12 CMBX12 %%+ CMMI12 CMR8 CMMI8 MSBM10 CMTI12 CMSY8 CMEX10 EUFM10 CMSY6 CMMI6 CMR6 %%+ MSAM10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o BFP3.ps BFP3.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 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: 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 32 /parenleftBigg put dup 33 /parenrightBigg put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 101 /tildewide put dup 102 /tildewider put dup 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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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBB2A7C1B5D8E7E8AA0 5B10EA43D6A8ED61AF5B23D49920D8F79DAB6A59062134D84AC0100187A6CD1F 80F5DDD9D222ACB1C23326A7656A635C4A241CCD32CBFDF8363206B8AA36E107 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 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 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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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 (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 /.notdef put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 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y(\024)1003 3818 y Fp(H)32 b Fr(is)26 b(the)f(Hamiltonian)g(of)g(the)g(quan)m (tized,)j(free)e(electromagnetic)257 3938 y(\014eld.)45 b(This)33 b(op)s(erator)f(is)h(giv)m(en)h(b)m(y)1129 4157 y(\024)1103 4183 y Fp(H)68 b Fr(:=)1431 4100 y Fi(X)1415 4284 y Fm(\025)p Fn(=)p Fj(\006)1583 4065 y Fi(Z)1682 4183 y Fp(d)1733 4141 y Fn(3)1773 4183 y Fp(k)30 b(a)1905 4141 y Fj(\003)1945 4183 y Fr(\()1978 4156 y Fp(~)1983 4183 y(k)s(;)17 b(\025)p Fr(\))g Fo(j)2216 4156 y Fp(~)2221 4183 y(k)r Fo(j)g Fp(a)p Fr(\()2403 4156 y Fp(~)2408 4183 y(k)r(;)g(\025)p Fr(\))22 b Fp(;)611 b Fr(\(I.11\))257 4505 y(where)42 b Fp(a)598 4468 y Fj(\003)637 4505 y Fr(\()670 4478 y Fp(~)675 4505 y(k)s(;)17 b(\025)p Fr(\))40 b(and)g Fp(a)p Fr(\()1189 4478 y Fp(~)1194 4505 y(k)t(;)17 b(\025)p Fr(\))39 b(are)i(the)f(usual)h(photon)f(creation-)g(and)h (annihilation)257 4625 y(op)s(erators)33 b(ob)s(eying)g(the)g (canonical)g(comm)m(utation)g(relations)860 4845 y([)p Fp(a)938 4809 y Fj(\003)978 4845 y Fr(\()1011 4819 y 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Fr(\),)j(where)257 2981 y Fo(D)s Fr(\()p Fp(H)456 2996 y Fn(0)495 2981 y Fr(\))33 b(is)g(the)g(domain)g(of)f(the)h(selfadjoin)m(t)g(op)s(erator) 1253 3192 y Fp(H)1334 3207 y Fn(0)1434 3192 y Fr(:=)60 b Fo(\000)p Fr(\001)1754 3207 y Fm(~)-34 b(x)1849 3192 y Fo(\000)50 b Fp(V)22 b Fr(\()p Fp(x)p Fr(\))50 b(+)2387 3166 y(\024)2362 3192 y Fp(H)29 b(:)761 b Fr(\(I.17\))257 3402 y(The)33 b(op)s(erator)d Fp(H)39 b Fr(is)32 b(b)s(ounded)g(from)e (b)s(elo)m(w,)j(and)e(the)h(in\014m)m(um)h(of)d(the)i(sp)s(ectrum)h(is) 257 3523 y(a)27 b(non-degenerate)g(eigen)m(v)-5 b(alue,)30 b(the)d(groundstate)g(energy)-8 b(,)29 b Fp(E)2584 3538 y Fm(g)r(s)2657 3523 y Fr(,)f(corresp)s(onding)g(to)e(a)257 3643 y(unique)35 b(eigen)m(v)m(ector,)h Fp(\036)1168 3658 y Fm(g)r(s)1240 3643 y Fr(.)46 b(There)35 b(is)e(an)h(ionization)f (threshold)i(\006,)f(\006)29 b Fp(>)g(E)3118 3658 y Fm(g)r(s)3191 3643 y Fr(,)k(ab)s(o)m(v)m(e)257 3763 y(whic)m(h)39 b(the)e(sp)s 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Fn(3)p Fm(=)p Fn(2)2431 1589 y Fi(q)p 2514 1589 159 4 v 100 x Fr(2)p Fo(j)p Fp(k)s Fo(j)3270 1596 y Fr(\(I)s(I.2\))257 1940 y(with)501 1914 y(^)480 1940 y Fp(f)10 b Fr(\()571 1914 y Fp(~)576 1940 y(k)s Fr(\))28 b Fo(2)g Fp(C)867 1904 y Fj(1)860 1965 y Fn(0)942 1940 y Fr(\()p Fl(R)1052 1904 y Fn(3)1091 1940 y Fo(nf)p Fr(0)p Fo(g)p Fr(\).)43 b(W)-8 b(e)33 b(construct)g(a)g(v)m(ector)g(of)f(test)i(functions)820 2193 y Fp(~)806 2220 y(f)854 2235 y Fm(t)884 2220 y Fr(\()o Fp(~)-48 b(y)s Fr(\))28 b(:=)1185 2137 y Fi(X)1169 2321 y Fm(\025)p Fn(=)p Fj(\006)1337 2102 y Fi(Z)1435 2220 y Fp(~)-48 b(")p Fr(\()1515 2193 y Fp(~)1520 2220 y(k)s(;)17 b(\025)p Fr(\))1713 2178 y Fj(\003)1773 2193 y Fr(^)1752 2220 y Fp(f)1811 2178 y Fm(\025)1856 2220 y Fr(\()1889 2193 y Fp(~)1894 2220 y(k)s Fr(\))p Fp(e)2031 2178 y Fj(\000)p Fm(i)p Fj(j)p Fm(k)r Fj(j)p Fm(t)p Fn(+)p Fm(i)2290 2161 y(~)2293 2178 y(k)q Fj(\001)p Fm(~)-35 b(y)2597 2152 y Fp(d)2648 2116 y Fn(3)2687 2152 y Fp(k)p 2402 2196 535 4 v 2402 2313 a Fr(\(2)p Fp(\031)t Fr(\))2586 2284 y Fn(3)p Fm(=)p Fn(2)2695 2213 y Fi(q)p 2778 2213 159 4 v 100 x Fr(2)p Fo(j)p Fp(k)s Fo(j)3270 2220 y Fr(\(I)s(I.3\))257 2542 y(satisfying)34 b(the)f(w)m(a)m(v)m(e)h(equation)g(\(I)s(I.1\),)e (with)939 2715 y Fi(X)924 2900 y Fm(\025)p Fn(=)p Fj(\006)1090 2798 y Fp(~)-48 b(")p Fr(\()1170 2772 y Fp(~)1175 2798 y(k)s(;)17 b(\025)p Fr(\))1368 2757 y Fj(\003)1428 2772 y Fr(^)1407 2798 y Fp(f)1466 2757 y Fm(\025)1511 2798 y Fr(\()1544 2772 y Fp(~)1549 2798 y(k)s Fr(\))28 b(=:)1820 2744 y(^)1813 2772 y Fp(~)1799 2798 y(f)11 b Fr(\()1891 2772 y Fp(~)1896 2798 y(k)s Fr(\))28 b Fo(2)g Fp(C)2187 2757 y Fj(1)2180 2823 y Fn(0)2261 2798 y Fr(\()p Fl(R)2371 2757 y Fn(3)2411 2798 y Fo(nf)p Fr(0)p Fo(g)p Fr(;)17 b Fl(C)2726 2757 y Fn(3)2764 2798 y Fr(\))p Fp(:)257 3098 y Fr(An)33 b(asymptotic)h(v)m(ector)f(p)s(oten)m(tial)g(is)g (constructed)h(as)e(an)g(LSZ)g(\()p Fp(t)c Fo(!)f(\0061)p Fr(\))33 b(limit)g(of)257 3218 y(in)m(terp)s(olating)g(\014eld)h(op)s (erators)765 3485 y Fp(~)747 3510 y(A)t Fr([)865 3484 y Fp(~)851 3510 y(f)899 3525 y Fm(t)929 3510 y Fp(;)17 b(t)p Fr(])28 b(:=)f Fp(i)1243 3393 y Fi(Z)1343 3414 y(\020)1411 3485 y Fp(~)1392 3510 y(A)5 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))22 b Fo(\001)1758 3442 y Fp(@)1828 3416 y(~)1814 3442 y(f)1862 3457 y Fm(t)1892 3442 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p 1758 3487 262 4 v 1842 3578 a Fp(@)5 b(t)2051 3510 y Fo(\000)2161 3442 y Fp(@)2236 3417 y(~)2217 3442 y(A)g Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))p 2161 3487 341 4 v 2285 3578 a Fp(@)5 b(t)2533 3510 y Fo(\001)2597 3484 y Fp(~)2583 3510 y(f)2631 3525 y Fm(t)2661 3510 y Fr(\()o Fp(~)-48 b(y)s Fr(\))2788 3414 y Fi(\021)2837 3510 y Fp(d)2888 3469 y Fn(3)2927 3510 y Fp(y)t(;)264 b Fr(\(I)s(I.4\))257 3783 y(with)493 3757 y Fp(~)480 3783 y(f)528 3798 y Fm(t)590 3783 y Fr(as)33 b(in)f(\(I)s(I.3\))h(and)1368 3987 y Fp(~)1350 4012 y(A)5 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))27 b(:=)h Fp(e)1837 3971 y Fm(iH)5 b(t)1972 3987 y Fp(~)1954 4012 y(A)t Fr(\()o Fp(~)-48 b(y)s Fr(\))p Fp(e)2203 3971 y Fj(\000)p Fm(iH)5 b(t)2375 4012 y Fp(:)868 b Fr(\(I)s(I.5\))257 4233 y(If)27 b(the)h(photon)f(w)m(ere)i(a)e(massiv)m(e)i(particle,)g(the)e(smeared)i (\014eld)f(op)s(erator)e(\(I)s(I.4\))h(w)m(ould)257 4353 y(con)m(v)m(erge)32 b(strongly)-8 b(,)31 b(as)f Fp(t)e Fo(!)f(1)p Fr(,)k(on)e(a)h(dense)i(linear)e(subspace)i(of)d(the)i(Hilb) s(ert)f(space)257 4473 y(of)38 b(the)g(system.)62 b(F)-8 b(or)38 b(massless)i(photons,)g(con)m(v)m(ergence)h(of)c(\(I)s(I.4\))h (only)h(holds)f(on)g(a)257 4594 y(subspace)j(of)d(v)m(ectors)i(in)f (the)g(Hilb)s(ert)g(space)g(whose)h(maximal)f(energy)h(is)f(so)f(small) 257 4714 y(that)47 b(the)h(propagation)f(sp)s(eed)h(of)f(the)h (electron)g(is)g(strictly)g(b)s(elo)m(w)g(the)g(sp)s(eed)h(of)257 4834 y(ligh)m(t,)41 b(e.g.,)f(on)e(the)h(space)g Fo(H)1395 4849 y Fn(\006)1450 4834 y Fr(;)i(see)f([7],)g([11)o(].)61 b(The)39 b(existence)i(of)d(strong)g(limits)i(of)257 4955 y(the)c(smeared)h(\014eld)f(op)s(erators)g(in)f(Eq.)i(\(I)s (I.4\),)f(as)f Fp(t)e Fo(!)f(\0061)p Fr(,)37 b(implies)g(the)f (existence)257 5075 y(of)c(asymptotic)i(creation-)f(and)f(annihilation) h(op)s(erators)1285 5295 y Fo(f)p Fp(a)1386 5254 y Fj(\003)1386 5320 y Fm(out=in)1593 5295 y Fr(\()1626 5269 y Fp(~)1631 5295 y(k)s(;)17 b(\025)p Fr(\))g Fp(;)32 b(a)1951 5311 y Fm(out=in)2158 5295 y Fr(\()2191 5269 y Fp(~)2196 5295 y(k)s(;)17 b(\026)p Fr(\))p Fo(g)p Fp(;)802 b Fr(\(I)s(I.6\))p eop end %%Page: 8 9 TeXDict begin 8 8 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1900 b Fr(8)257 573 y(de\014ned)44 b(on)d(a)h(dense)h (subspace)h(of)d Fo(H)1734 588 y Fn(\006)1831 573 y Fr(\(see)i (\(I.18\)\))e(and)h(ob)s(eying)g(the)h(canonical)257 693 y(comm)m(utation)36 b(relations.)51 b(In)36 b(fact,)f(the)g(limits) h(of)f(the)g(op)s(erators)g(in)g(\(I)s(I.4\),)h(as)f Fp(t)d Fo(!)257 814 y(\0061)p Fr(,)h(corresp)s(ond)h(to)790 1033 y Fp(~)772 1058 y(A)845 1017 y Fm(out=in)1052 1058 y Fr([)1093 1032 y Fp(~)1079 1058 y(f)11 b Fr(])27 b(=)1311 975 y Fi(X)1296 1160 y Fm(\025)p Fn(=)p Fj(\006)1463 941 y Fi(Z)1563 1058 y Fp(a)1614 1017 y Fj(\003)1614 1083 y Fm(out=in)1821 1058 y Fr(\()1854 1032 y Fp(~)1859 1058 y(k)s(;)17 b(\025)p Fr(\))2073 1032 y(^)2052 1058 y Fp(f)2111 1017 y Fm(\025)2156 1058 y Fr(\()2189 1032 y Fp(~)2194 1058 y(k)s Fr(\))p Fp(d)2337 1017 y Fn(3)2376 1058 y Fp(k)30 b Fr(=:)e Fp(a)2639 1017 y Fj(\003)2639 1083 y Fm(out=in)2846 1058 y Fr(\()p Fp(f)11 b Fr(\))289 b(\(I)s(I.7\))257 1352 y(and,)33 b(similarly)-8 b(,)738 1571 y Fp(~)720 1596 y(A)793 1555 y Fm(out=in)1000 1596 y Fr([)p Fo(\000)1125 1542 y Fr(\026)1118 1570 y Fp(~)1104 1596 y(f)11 b Fr(])28 b(=)1337 1513 y Fi(X)1321 1698 y Fm(\025)p Fn(=)p Fj(\006)1489 1479 y Fi(Z)1588 1596 y Fp(a)1639 1612 y Fm(out=in)1846 1596 y Fr(\()1879 1570 y Fp(~)1884 1596 y(k)s(;)17 b(\025)p Fr(\))p 2077 1489 234 4 v 2098 1570 a(^)2077 1596 y Fp(f)2136 1568 y Fm(\025)2181 1596 y Fr(\()2214 1570 y Fp(~)2219 1596 y(k)s Fr(\))p Fp(d)2362 1555 y Fn(3)2401 1596 y Fp(k)31 b Fr(=:)c Fp(a)2664 1612 y Fm(out=in)2871 1596 y Fr(\()p Fp(f)11 b Fr(\))p Fp(:)237 b Fr(\(I)s(I.8\))257 1896 y(Using)32 b(estimates)g(pro)m(v)m (en)h(in)e([7)o(])g(and)g(in)g(Lemma)h(I)s(I.1,)f(b)s(elo)m(w,)h(one)f (can)g(pro)m(v)m(e)h(that,)257 2029 y(for)428 2002 y(^)407 2029 y Fp(f)466 1992 y Fm(\025)511 2029 y Fr(\()544 2002 y Fp(~)549 2029 y(k)s Fr(\))p Fp(;)685 2002 y Fr(^)685 2029 y Fp(h)741 1992 y Fm(\025)786 2029 y Fr(\()819 2002 y Fp(~)824 2029 y(k)s Fr(\))c Fo(2)h Fp(L)1105 2044 y Fn(2)1145 1932 y Fi(\020)1194 2029 y Fl(R)1266 1992 y Fn(3)1306 2029 y Fr(;)17 b(\(1)k(+)1588 1989 y Fn(1)p 1567 2005 78 4 v 1567 2074 a Fj(j)1584 2056 y Fm(~)1587 2074 y(k)q Fj(j)1654 2029 y Fr(\))1692 1992 y Fj(\000)p Fn(1)1787 2029 y Fr(\))p Fp(d)1876 1992 y Fn(3)1915 2029 y Fp(k)1969 1932 y Fi(\021)2051 2029 y Fr(the)34 b(follo)m(wing)f (relations)g(hold)h(in)f(the)257 2168 y(sense)i(of)d(quadratic)h (forms,)g(on)f Fo(H)1570 2183 y Fn(\006)1626 2168 y Fr(:)257 2288 y(i\))835 2538 y([)p Fp(a)913 2554 y Fm(out=in)1120 2538 y Fr(\()p Fp(f)11 b Fr(\))p Fp(;)17 b(a)1350 2497 y Fj(\003)1350 2563 y Fm(out=in)1556 2538 y Fr(\()p Fp(h)p Fr(\)])28 b(=)g(\()p Fp(f)5 b(;)17 b(h)p Fr(\))27 b(=)2222 2455 y Fi(X)2207 2640 y Fm(\025)p Fn(=)p Fj(\006)2374 2421 y Fi(Z)p 2474 2431 234 4 v 2495 2512 a Fr(^)2474 2538 y Fp(f)2533 2509 y Fm(\025)2578 2538 y Fr(\()2611 2512 y Fp(~)2616 2538 y(k)s Fr(\))2709 2512 y(^)2708 2538 y Fp(h)2764 2497 y Fm(\025)2809 2538 y Fr(\()2842 2512 y Fp(~)2847 2538 y(k)t Fr(\))p Fp(d)2991 2497 y Fn(3)3030 2538 y Fp(k)189 b Fr(\(I)s(I.9\))835 2763 y([)p Fp(a)913 2778 y Fm(out=in)1120 2763 y Fr(\()p Fp(f)11 b Fr(\))p Fp(;)17 b(a)1350 2778 y Fm(out=in)1556 2763 y Fr(\()p Fp(h)p Fr(\)])28 b(=)g([)p Fp(a)1925 2722 y Fj(\003)1925 2787 y Fm(out=in)2132 2763 y Fr(\()p Fp(f)11 b Fr(\))p Fp(;)17 b(a)2362 2722 y Fj(\003)2362 2787 y Fm(out=in)2568 2763 y Fr(\()p Fp(h)p Fr(\)])28 b(=)f(0)257 2983 y(ii\))294 3203 y Fp(e)339 3162 y Fm(itH)456 3203 y Fp(a)507 3162 y Fj(\003)507 3227 y Fm(out=in)714 3203 y Fr(\()p Fp(h)p Fr(\))p Fp(e)891 3162 y Fj(\000)p Fm(itH)1091 3203 y Fr(=)g Fp(a)1245 3162 y Fj(\003)1245 3227 y Fm(out=in)1452 3203 y Fr(\()p Fp(e)1535 3162 y Fm(it!)1635 3203 y Fp(h)p Fr(\))98 b Fp(;)114 b(e)2013 3162 y Fm(itH)2130 3203 y Fp(a)2181 3218 y Fm(out=in)2388 3203 y Fr(\()p Fp(h)p Fr(\))p Fp(e)2565 3162 y Fj(\000)p Fm(itH)2765 3203 y Fr(=)27 b Fp(a)2919 3218 y Fm(out=in)3126 3203 y Fr(\()p Fp(e)3209 3162 y Fj(\000)p Fm(it!)3364 3203 y Fp(h)p Fr(\))3222 3323 y(\(I)s(I.10\))257 3444 y(where)k(the)f(F)-8 b(ourier)30 b(transform)f(of)g(eac)m(h)i(comp)s(onen)m(t)f(of)f Fp(e)2453 3407 y Fm(it!)2554 3444 y Fp(h)g Fr(is)h(obtained)g(from)g (the)257 3573 y(ones)k(of)e Fp(h)g Fr(b)m(y)i(m)m(ultiplying)g(b)m(y)g Fp(e)1511 3536 y Fm(it!)r Fn(\()1630 3519 y Fm(~)1633 3536 y(k)s Fn(\))1704 3573 y Fr(,)f Fp(!)t Fr(\()1862 3546 y Fp(~)1867 3573 y(k)r Fr(\))28 b(:=)f Fo(j)2139 3546 y Fp(~)2144 3573 y(k)s Fo(j)p Fr(.)404 3693 y(Since,)35 b(in)e(this)h(pap)s(er,)g(w)m(e)h(are)e(in)m(terested)i(in)f(Ra)m (yleigh)g(scattering,)h(w)m(e)f(restrict)257 3813 y(our)42 b(atten)m(tion)h(to)e(the)i(construction)g(of)f(asymptotic)h(states)g (describing)h(an)d(atom)257 3934 y(b)s(elo)m(w)26 b(the)f(ionization)g (threshold,)i(\006.)41 b(\(They)26 b(ha)m(v)m(e)g(sp)s(ectral)g(supp)s (ort)e(strictly)i(b)s(elo)m(w)257 4054 y(the)d(ionization)g (threshold\).)41 b(F)-8 b(or)22 b(suc)m(h)i(states,)h(the)e(p)s (osition)g(of)f(the)h(electron)g(remains)257 4174 y(close)j(to)f(the)g (one)g(of)g(the)g(proton)g(for)f(all)h(times.)42 b(More)25 b(precisely)-8 b(,)29 b(suc)m(h)d(states)g(exhibit)257 4295 y(exp)s(onen)m(tial)i(deca)m(y)g(in)f(the)g(distance)h(b)s(et)m(w) m(een)h(the)e(electron)h(and)e(the)h(n)m(ucleus.)44 b(This)257 4415 y(implies)39 b(that)f(the)f(con)m(v)m(ergence)k(of)c(\(I)s(I.4\),) h(as)g(time)g Fp(t)e Fo(!)g(\0061)p Fr(,)i(is)g(faster)g(than)f(an)m(y) 257 4536 y(in)m(v)m(erse)h(p)s(o)m(w)m(er)e(of)f(time)g Fp(t)p Fr(.)52 b(This)36 b(implies)h(that)d(when)j(applied)f(to)e(v)m (ectors)j(in)e Fo(H)3351 4551 y Fn(\006)p Fj(\000)p Fm(\016)257 4656 y Fr(the)29 b(op)s(erators)e(de\014ned)j(in)e(\(I)s(I.4\))g(con)m (v)m(erge)h(faster)f(than)g(an)m(y)h(in)m(v)m(erse)h(p)s(o)m(w)m(er)f (of)f(time)257 4776 y Fp(t)p Fr(,)i(as)g Fp(t)e Fo(!)f(1)p Fr(.)42 b(This)30 b(has)g(imp)s(ortan)m(t)f(consequences)k(for)28 b(the)i(expansion)h(of)d(Ra)m(yleigh)257 4897 y(scattering)34 b(amplitudes)g(in)f(the)g(\014nestructure)h(constan)m(t.)257 5017 y(In)f(the)g(follo)m(wing,)g(w)m(e)h(mak)m(e)g(use)f(of)f(the)h (\014eld)h(equation)1039 5283 y Fp(~)1029 5308 y Fo(r)1112 5323 y Fm(~)-35 b(y)1176 5308 y Fo(\001)1235 5283 y Fp(~)1226 5308 y Fo(r)1309 5323 y Fm(~)g(y)1368 5283 y Fp(~)1350 5308 y(A)5 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))22 b Fo(\000)1765 5241 y Fp(@)1821 5204 y Fn(2)1880 5215 y Fp(~)1862 5241 y(A)t Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))p 1765 5285 380 4 v 1889 5376 a Fp(@)5 b(t)1980 5348 y Fn(2)2182 5308 y Fr(=)28 b Fo(\000)2379 5283 y Fp(~)2363 5308 y(J)2426 5267 y Fm(tr)2490 5308 y Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))p Fp(;)499 b Fr(\(I)s(I.11\))p eop end %%Page: 9 10 TeXDict begin 9 9 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1900 b Fr(9)257 573 y(where)283 806 y Fp(~)267 832 y(J)330 790 y Fm(tr)393 832 y Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))28 b(:=)g Fo(\000)906 764 y Fp(\013)969 728 y Fn(3)p Fm(=)p Fn(2)p 845 808 294 4 v 845 900 a Fr(\(2)p Fp(\031)t Fr(\))1029 871 y Fn(3)p Fm(=)p Fn(2)1181 749 y Fi(X)1165 933 y Fm(\025)p Fn(=)p Fj(\006)1333 714 y Fi(Z)1416 832 y Fr(\()l Fp(~)-45 b(v)s Fr(\()p Fp(t)p Fr(\))23 b Fo(\001)d Fp(~)-48 b(")p Fr(\()1766 805 y Fp(~)1771 832 y(k)s(;)17 b(\025)p Fr(\))1964 790 y Fj(\003)2003 832 y Fr(\))o Fp(~)-48 b(")p Fr(\()2120 805 y Fp(~)2125 832 y(k)s(;)17 b(\025)p Fr(\)\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\))p Fp(e)2617 790 y Fj(\000)p Fm(i)2693 773 y(~)2696 790 y(k)p Fj(\001)p Fn(\()p Fm(~)-35 b(y)r Fj(\000)p Fm(\013)o(~)h(x)p Fn(\()p Fm(t)p Fn(\)\))3069 832 y Fp(d)3120 790 y Fn(3)3159 832 y Fp(k)25 b Fr(+)d Fp(h:c:)3222 1031 y Fr(\(I)s(I.12\))257 1161 y(with)27 b Fp(~)-45 b(v)t Fr(\()p Fp(t)p Fr(\))28 b(=)f Fp(e)815 1125 y Fm(iH)5 b(t)928 1161 y Fp(~)-45 b(v)t(e)1028 1125 y Fj(\000)p Fm(iH)5 b(t)1227 1161 y Fr(=)28 b Fp(e)1376 1125 y Fm(iH)5 b(t)1493 1161 y Fr(\()p Fo(\000)p Fp(i)1650 1136 y(~)1641 1161 y Fo(r)1723 1176 y Fm(~)-34 b(x)1785 1161 y Fo(\000)18 b Fp(\013)1953 1097 y Fd(3)p 1953 1109 31 4 v 1953 1150 a(2)2015 1136 y Fp(~)1997 1161 y(A)p Fr(\()p Fp(\013)o(~)-47 b(x)p Fr(\)\))p Fp(e)2347 1125 y Fj(\000)p Fm(iH)5 b(t)2549 1161 y Fr(and)29 b Fp(~)-48 b(x)q Fr(\()p Fp(t)p Fr(\))27 b(=)h Fp(e)3079 1125 y Fm(iH)5 b(t)3194 1161 y Fp(~)-47 b(xe)3296 1125 y Fj(\000)p Fm(iH)5 b(t)3468 1161 y Fr(.)257 1281 y(Equation)23 b(\(I)s(I.11\))e(is)h(meaningful)h(as)f(an)f (equation)i(b)s(et)m(w)m(een)h(densely)g(de\014ned)f(op)s(erator-)257 1401 y(v)-5 b(alued)34 b(distributions.)257 1522 y(By)f(Co)s(ok's)h (argumen)m(t,)f(the)g(existence)i(of)d(the)h(limits)1609 1739 y(lim)1566 1794 y Fm(t)p Fj(!\0061)1822 1714 y Fp(~)1804 1739 y(A)5 b Fr([)1923 1713 y Fp(~)1909 1739 y(f)1957 1754 y Fm(t)1986 1739 y Fp(;)17 b(t)p Fr(])p Fp( )t(;)1036 b Fr(\(I)s(I.13\))257 1984 y(for)32 b Fp( )g Fo(2)c(H)679 1999 y Fn(\006)734 1984 y Fr(,)33 b(follo)m(ws)g(from)g(the)g (existence)i(of)d(the)h(in)m(tegral)616 2128 y Fi(Z)699 2154 y Fj(\0061)662 2317 y Fn(0)872 2178 y Fp(@)p 855 2222 93 4 v 855 2313 a(@)5 b(t)975 2220 y(~)957 2245 y(A)t Fr([)1075 2219 y Fp(~)1061 2245 y(f)1109 2260 y Fm(t)1139 2245 y Fp(;)17 b(t)p Fr(])p Fp( )t(dt)28 b Fr(=)699 2514 y(=)858 2397 y Fi(Z)941 2424 y Fj(\0061)904 2586 y Fn(0)1087 2514 y Fp(i)1137 2397 y Fi(Z)1264 2447 y Fp(@)p 1246 2491 V 1246 2583 a(@)5 b(t)1348 2418 y Fi(\020)1416 2489 y Fp(~)1398 2514 y(A)t Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))22 b Fo(\001)1763 2447 y Fp(@)1834 2421 y(~)1819 2447 y(f)1867 2462 y Fm(t)1898 2447 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p 1763 2491 262 4 v 1848 2583 a Fp(@)5 b(t)2057 2514 y Fo(\000)2166 2447 y Fp(@)2241 2422 y(~)2222 2447 y(A)h Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))p 2166 2491 341 4 v 2290 2583 a Fp(@)5 b(t)2539 2514 y Fo(\001)2602 2488 y Fp(~)2588 2514 y(f)2636 2529 y Fm(t)2666 2514 y Fr(\()o Fp(~)-48 b(y)s Fr(\))2793 2418 y Fi(\021)2843 2514 y Fp(d)2894 2473 y Fn(3)2933 2514 y Fp(y)t( )t(dt)84 b Fr(\(I)s(I.14\))699 2783 y(=)858 2666 y Fi(Z)941 2692 y Fj(\0061)904 2854 y Fn(0)1087 2783 y Fp(i)1137 2666 y Fi(Z)1236 2686 y(h)q(\020)1335 2758 y Fp(~)1325 2783 y Fo(r)1408 2798 y Fm(~)-35 b(y)1472 2783 y Fo(\001)1531 2758 y Fp(~)1521 2783 y Fo(r)1604 2798 y Fm(~)g(y)1664 2758 y Fp(~)1646 2783 y(A)t Fr(\()o Fp(~)-48 b(y)s(;)17 b(t)p Fr(\))22 b Fo(\000)2061 2715 y Fp(@)2117 2679 y Fn(2)2175 2690 y Fp(~)2157 2715 y(A)t Fr(\()o Fp(~)-48 b(y)t(;)17 b(t)p Fr(\))p 2061 2759 380 4 v 2185 2851 a Fp(@)5 b(t)2276 2822 y Fn(2)2450 2686 y Fi(\021)2522 2783 y Fo(\001)2585 2756 y Fp(~)2572 2783 y(f)2620 2798 y Fm(t)2650 2783 y Fr(\()o Fp(~)-48 b(y)s Fr(\))2777 2686 y Fi(i)2816 2783 y Fp(d)2867 2742 y Fn(3)2906 2783 y Fp(y)t( )t(dt)111 b Fr(\(I)s(I.15\))257 3034 y(where,)46 b(in)c(passing)g(from)g(\(I)s(I.14\))f(to)h(\(I)s(I.15\),)h(w)m(e)g (use)g(the)f(w)m(a)m(v)m(e)i(equation)e(\(I)s(I.1\),)257 3165 y(and,)37 b(taking)e(in)m(to)h(accoun)m(t)g(the)g(rapid)g(spatial) f(deca)m(y)i(of)2493 3138 y Fp(~)2479 3165 y(f)2527 3180 y Fm(t)2557 3165 y Fr(\()o Fp(~)-48 b(y)s Fr(\),)36 b(w)m(e)h(then)f (in)m(tegrate)257 3285 y(b)m(y)e(parts)f(t)m(wice)h(in)d Fp(~)-47 b(y)s Fr(.)257 3406 y(In)m(tro)s(ducing)34 b(the)f(notation) 1225 3622 y Fp(~)1210 3647 y(J)1273 3606 y Fm(tr)1336 3647 y Fr([)1377 3621 y Fp(~)1363 3647 y(f)1411 3662 y Fm(t)1441 3647 y Fp(;)17 b(t)p Fr(])28 b(:=)1705 3530 y Fi(Z)1821 3622 y Fp(~)1805 3647 y(J)1868 3606 y Fm(tr)1931 3647 y Fr(\()o Fp(~)-48 b(y)t(;)17 b(t)p Fr(\))22 b Fo(\001)2223 3621 y Fp(~)2209 3647 y(f)2257 3662 y Fm(t)2287 3647 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p Fp(dy)t(;)678 b Fr(\(I)s(I.16\))257 3895 y(and)33 b(using)g(the)g(\014eld)h(equation)f(\(I)s(I.11\),)f(w)m (e)i(ma)m(y)f(rewrite)h(\(I)s(I.14\))e(as)h(follo)m(ws:)988 4045 y Fi(Z)1071 4071 y Fj(\0061)1034 4233 y Fn(0)1239 4094 y Fp(@)p 1227 4138 82 4 v 1227 4230 a(@)1278 4245 y Fm(t)1336 4137 y Fp(~)1318 4162 y(A)5 b Fr([)1437 4135 y Fp(~)1423 4162 y(f)1471 4177 y Fm(t)1500 4162 y Fp(;)17 b(t)p Fr(])p Fp( )t(dt)28 b Fr(=)g Fo(\000)p Fp(i)2018 4045 y Fi(Z)2101 4071 y Fj(\0061)2064 4233 y Fn(0)2263 4137 y Fp(~)2247 4162 y(J)2310 4121 y Fm(tr)2374 4162 y Fr([)2415 4135 y Fp(~)2401 4162 y(f)2449 4177 y Fm(t)2479 4162 y Fp(;)17 b(t)p Fr(])p Fp( )t(dt:)457 b Fr(\(I)s(I.17\))257 4420 y(The)36 b(follo)m(wing)e(lemma)h(guaran)m(tees)g(the)g(con)m(v)m (ergence)i(of)d(the)g(in)m(tegral)h(o)m(v)m(er)g(time)g Fp(t)257 4540 y Fr(on)e(the)g(righ)m(t)g(side)g(of)f(Eq.)i(\(I)s (I.17\).)257 4741 y Fg(Lemma)46 b(I)s(I.1.)e Fk(L)-5 b(et)40 b Fp( )k Fk(b)-5 b(e)39 b(a)h(ve)-5 b(ctor)39 b(b)-5 b(elonging)39 b(to)h(the)f(subsp)-5 b(ac)g(e)39 b Fo(H)2893 4756 y Fn(\006)p Fj(\000)p Fm(\016)3077 4741 y Fk(de\014ne)-5 b(d)39 b(in)257 4895 y(\(I.18\).)86 b(F)-7 b(or)817 4828 y Fi(P)905 4916 y Fm(\025)p Fn(=)p Fj(\006)1075 4895 y Fp(~)-47 b(")o Fr(\()1155 4869 y Fp(~)1160 4895 y(k)s(;)17 b(\025)p Fr(\))1353 4859 y Fj(\003)1413 4869 y Fr(^)1392 4895 y Fp(f)1451 4859 y Fm(\025)1496 4895 y Fr(\()1529 4869 y Fp(~)1534 4895 y(k)s Fr(\))54 b(=:)1858 4840 y(^)1850 4869 y Fp(~)1837 4895 y(f)11 b Fr(\()1929 4869 y Fp(~)1934 4895 y(k)r Fr(\))54 b Fo(2)g Fp(C)2276 4859 y Fj(1)2269 4919 y Fn(0)2351 4895 y Fr(\()p Fl(R)2461 4859 y Fn(3)2500 4895 y Fo(nf)p Fr(0)p Fo(g)p Fr(;)17 b Fl(C)2815 4859 y Fn(3)2854 4895 y Fr(\))49 b Fk(the)g(fol)5 b(lowing)257 5015 y(estimate)35 b(holds:)1371 5159 y Fo(k)1437 5134 y Fp(~)1421 5159 y(J)1484 5118 y Fm(tr)1548 5159 y Fr([)1589 5133 y Fp(~)1575 5159 y(f)1623 5174 y Fm(t)1653 5159 y Fp(;)17 b(t)p Fr(])p Fp( )t Fo(k)27 b(\024)2113 5092 y Fp(C)2183 5107 y Fm(m)p 2018 5136 327 4 v 2018 5227 a Fr(1)22 b(+)g Fo(j)p Fp(t)p Fo(j)2278 5199 y Fm(m)2354 5159 y Fp(;)841 b Fr(\(I)s(I.18\))257 5388 y Fk(for)35 b(any)g Fp(m)28 b Fo(2)g Fl(N)p Fk(,)35 b(wher)-5 b(e)34 b Fp(C)1289 5403 y Fm(m)1390 5388 y Fk(is)h(a)f(\014nite)h(c)-5 b(onstant)34 b(\(dep)-5 b(ending)34 b(on)g Fp(m)p Fk(\).)p eop end %%Page: 10 11 TeXDict begin 10 10 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(10)257 573 y Fk(Pr)-5 b(o)g(of.)257 693 y Fr(W)d(e)33 b(write)h(the)f(iden)m(tit)m(y)h(op)s(erator)e(in)h (electron)g(p)s(osition)g(space)h(as)520 913 y Fg(1)27 b Fr(=)h Fp(\037)p Fr(\()p Fo(h)n Fp(~)-47 b(x)p Fo(i)22 b(\000)g Fp(\014)6 b Fo(j)p Fp(t)p Fo(j)p Fr(\))22 b(+)g Fp(\037)1431 872 y Fm(c)1466 913 y Fr(\()p Fo(h)n Fp(~)-47 b(x)p Fo(i)22 b(\000)g Fp(\014)6 b Fo(j)p Fp(t)p Fo(j)p Fr(\))97 b Fp(;)114 b Fo(h)n Fp(~)-47 b(x)p Fo(i)28 b Fr(:=)f(\()n Fp(~)-47 b(x)23 b Fo(\001)d Fp(~)-47 b(x)22 b Fr(+)g(1\))2915 845 y Fd(1)p 2915 857 31 4 v 2915 898 a(2)3222 913 y Fr(\(I)s(I.19\))257 1133 y(for)i(0)k Fp(<)f(\014)34 b(<)27 b Fr(1,)f(where)g(the)f(function)g Fp(\037)p Fr(\()p Fp(y)t Fr(\)\(=)i(1)6 b Fo(\000)g Fp(\037)2209 1097 y Fm(c)2244 1133 y Fr(\()p Fp(y)t Fr(\)\))23 b(is)i(a)g(non-negativ)m(e)g Fp(C)3240 1097 y Fj(1)3315 1133 y Fr(\()p Fl(R)p Fr(\)-)257 1254 y(function)41 b(equal)g(to)e(1,)j(for)d Fp(y)44 b(<)c Fo(\000)p Fr(1,)i(and)e(equal)h(to)f(0,)h(for)f Fp(y)j(>)d Fr(1.)66 b(The)41 b(op)s(erator)273 1349 y Fp(~)257 1374 y(J)320 1338 y Fm(tr)384 1374 y Fr([)425 1348 y Fp(~)411 1374 y(f)459 1389 y Fm(t)489 1374 y Fp(;)17 b(t)p Fr(])32 b(is)h(giv)m(en)h(b)m(y)1336 1541 y Fp(~)1320 1566 y(J)1383 1525 y Fm(tr)1447 1566 y Fr([)1488 1540 y Fp(~)1474 1566 y(f)1522 1581 y Fm(t)1552 1566 y Fp(;)17 b(t)p Fr(])28 b(=)1831 1458 y Fn(3)1789 1483 y Fi(X)1793 1668 y Fm(l)q Fn(=1)1926 1566 y Fp(g)1977 1525 y Fm(l)1973 1591 y(t)2002 1566 y Fr(\()n Fp(~)-47 b(x)p Fr(\()p Fp(t)p Fr(\)\))p Fp(v)2295 1525 y Fm(l)2321 1566 y Fr(\()p 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3406 y Fe(g)r(s)2269 3383 y Fp(;)936 b Fr(\(I)s(I)s(I.9\))257 3603 y(where)40 b Fo(h\001i)651 3618 y Fm(\036)693 3626 y Fe(g)r(s)800 3603 y Fr(denotes)g(the)f(exp)s (ectation)h(v)-5 b(alue)39 b(with)g(resp)s(ect)h(to)f(the)g (groundstate)257 3723 y Fp(\036)315 3738 y Fm(g)r(s)388 3723 y Fr(.)257 3844 y(Exploiting)34 b(Eq.)f(\(I)s(I.17\),)g(w)m(e)g (\014nd)g(that)624 4087 y Fo(h)681 4062 y Fp(~)663 4087 y(A)736 4046 y Fm(out)841 4087 y Fr([)p Fo(\000)966 4033 y Fr(\026)959 4061 y Fp(~)945 4087 y(f)11 b Fr(])1049 4062 y Fp(~)1031 4087 y(A)1104 4046 y Fm(in)1175 4087 y Fr([)1195 4061 y Fp(~)1202 4087 y(h)p Fr(])p Fo(i)1324 4102 y Fm(\036)1366 4110 y Fe(g)r(s)1463 4087 y Fr(=)1645 b(\(I)s(I)s(I.10\))707 4264 y(=)83 b Fo(h)923 4239 y Fp(~)905 4264 y(A)978 4223 y Fm(out)1082 4264 y Fr([)p Fo(\000)1208 4209 y Fr(\026)1200 4238 y Fp(~)1186 4264 y(f)11 b Fr(])1291 4239 y Fp(~)1272 4264 y(A)1345 4223 y Fm(out)1451 4264 y Fr([)1471 4238 y Fp(~)1478 4264 y(h)p Fr(])p Fo(i)1600 4279 y Fm(\036)1642 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Fr(\(I)s(I)s(I.12\))866 4704 y Fo(\000)p Fp(i)993 4586 y Fi(Z)1076 4613 y Fn(+)p Fj(1)1039 4775 y(\0001)1206 4704 y Fo(h)1261 4678 y Fp(~)1245 4704 y(J)1308 4662 y Fm(tr)1371 4704 y Fr([)1391 4677 y Fp(~)1398 4704 y(h)1454 4719 y Fm(s)1491 4704 y Fp(;)17 b(s)p Fr(])1626 4678 y Fp(~)1608 4704 y(A)1681 4662 y Fm(in)1752 4704 y Fr([)p Fo(\000)1878 4649 y Fr(\026)1870 4677 y Fp(~)1856 4704 y(f)11 b Fr(])p Fo(i)1981 4719 y Fm(\036)2023 4727 y Fe(g)r(s)2093 4704 y Fp(ds;)257 4965 y Fr(where)41 b(the)f(third)g(term)h(in)e(\(I)s(I)s(I.12\))h(can)g (b)s(e)f(added)i(for)e(free,)j(b)s(ecause)f(it)f(actually)257 5085 y(v)-5 b(anishes.)52 b(This)36 b(is)g(seen)g(b)m(y)f(noticing)h (that)e(the)i(groundstate)f(is)g(a)g(v)-5 b(acuum)35 b(for)g(the)257 5205 y(asymptotic)26 b(annihilation)e(op)s(erators;)j (see,)g(e.g.,)f([8].)41 b(No)m(w,)26 b(using)f(again)f(Eq.)h(\(I)s (I.17\),)p eop end %%Page: 13 14 TeXDict begin 13 13 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(13)257 573 y(w)m(e)34 b(get)317 796 y Fo(h)374 770 y Fp(~)356 796 y(A)429 754 y Fm(out)534 796 y Fr([)p Fo(\000)659 741 y Fr(\026)652 769 y Fp(~)638 796 y(f)11 b Fr(])742 770 y Fp(~)724 796 y(A)797 754 y Fm(in)868 796 y Fr([)888 769 y Fp(~)895 796 y(h)p Fr(])p Fo(i)1017 811 y Fm(\036)1059 819 y Fe(g)r(s)1156 796 y Fr(=)1952 b(\(I)s(I)s(I.13\))400 993 y(=)83 b(\()p Fp(f)5 b(;)17 b(h)p Fr(\))22 b Fo(\000)910 876 y Fi(Z)993 902 y Fn(+)p Fj(1)956 1064 y(\0001)1139 876 y Fi(Z)1222 902 y Fn(+)p Fj(1)1185 1064 y(\0001)1368 876 y Fi(Z)54 b(Z)p 1568 908 204 4 v 1568 993 a Fp(f)1616 1008 y Fm(t)1645 993 y Fr(\()n Fp(~)-47 b(z)5 b Fr(\))p Fo(hT)25 b Fr(\()1943 968 y Fp(~)1927 993 y(J)1990 952 y Fm(tr)2054 993 y Fr(\()n Fp(~)-47 b(z)t(;)17 b(t)p Fr(\))2274 968 y Fp(~)2258 993 y(J)2321 952 y Fm(tr)2385 993 y Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))p Fo(i)2641 1008 y Fm(\036)2683 1016 y Fe(g)r(s)2745 966 y Fp(~)2752 993 y(h)2808 1008 y Fm(s)2845 993 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p Fp(d)3023 952 y Fn(3)3062 993 y Fp(y)t(d)3165 952 y Fn(3)3203 993 y Fp(z)t(dsdt)559 1235 y Fr(+)p Fp(i)685 1118 y Fi(Z)768 1144 y Fn(+)p Fj(1)731 1307 y(\0001)897 1235 y Fo(h)p Fr([)981 1210 y Fp(~)963 1235 y(A)5 b Fr([)p Fo(\000)1166 1181 y Fr(\026)1159 1209 y Fp(~)1145 1235 y(f)1193 1250 y Fm(s)1230 1235 y Fp(;)17 b(s)p Fr(])p Fp(;)1406 1210 y(~)1391 1235 y(J)1454 1194 y Fm(tr)1517 1235 y Fr([)1537 1209 y Fp(~)1544 1235 y(h)1600 1250 y Fm(s)1637 1235 y Fp(;)g(s)p Fr(]])p Fo(i)1820 1250 y Fm(\036)1862 1258 y Fe(g)r(s)1931 1235 y Fp(ds)1156 b Fr(\(I)s(I)s(I.14\))257 1476 y(where)29 b Fo(T)53 b Fr(is)28 b(the)f(time)h(ordered)g(pro)s (duct.)42 b(A)28 b(scalar)g(pro)s(duct)f(b)s(et)m(w)m(een)j(v)m(ector)e (quan-)257 1596 y(tities)41 b(dep)s(ending)g(on)f(the)h(same)f(p)s (osition)h(v)-5 b(ariables)40 b(is)h(understo)s(o)s(d,)h(here)f(and)f (in)257 1716 y(the)33 b(follo)m(wing.)257 1837 y(The)h(equal)f(time)h (comm)m(utator)f(in)f(Eq.)i(\(I)s(I)s(I.14\))e(corresp)s(onds)i(to)952 2118 y Fo(\000)1046 2001 y Fi(Z)54 b(Z)p 1278 2011 211 4 v 1292 2092 a Fp(~)1278 2118 y(f)1326 2133 y Fm(s)1363 2118 y Fr(\()n Fp(~)-47 b(z)5 b Fr(\)[)1526 2050 y Fp(@)1601 2025 y(~)1582 2050 y(A)g Fr(\()n Fp(~)-47 b(z)5 b(;)17 b(s)p Fr(\))p 1526 2095 350 4 v 1649 2186 a Fp(@)5 b(s)1885 2118 y(;)1945 2093 y(~)1929 2118 y(J)1992 2077 y Fm(tr)2055 2118 y Fr(\()o Fp(~)-48 b(y)t(;)17 b(s)p Fr(\)])2293 2092 y Fp(~)2300 2118 y(h)2356 2133 y Fm(s)2392 2118 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p Fp(d)2570 2077 y Fn(3)2609 2118 y Fp(y)t(d)2712 2077 y Fn(3)2751 2118 y Fp(z)388 b Fr(\(I)s(I)s(I.15\))257 2348 y(that)33 b(is,)g(in)g(general,)g(a)f (non)h(v)-5 b(anishing)33 b(function)g(of)f(the)h(electron)h(p)s (osition.)257 2468 y(In)d(order)f(to)g(get)g(a)g(more)g(symmetrical)i (expression)h(that)d(can)g(b)s(e)g(easily)i(generalized)257 2589 y(to)c(scattering)h(amplitudes)g(with)g(an)e(arbitrary)h(\014nite) h(n)m(um)m(b)s(er)g(of)f(photons,)h(starting)257 2709 y(from)e(the)g(expression)i(in)e(Eq.)h(\(I)s(I)s(I.12\))e(w)m(e)i 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3634 y Fj(\000)p Fm(\017)1959 3675 y Fo(\000)2068 3608 y Fp(R)p 2068 3652 V 2081 3744 a Fr(2)2251 3675 y(and)97 b Fp(s)28 b Fo(\025)g Fp(\013)2747 3634 y Fj(\000)p Fm(\017)2857 3675 y Fr(+)2965 3608 y Fp(R)p 2965 3652 V 2978 3744 a Fr(2)3050 3675 y Fp(;)107 b Fr(\(I)s(I)s(I.17\))257 3905 y(where)34 b(0)28 b Fp(<)f(\017)h(<<)g Fr(1)k(and)h Fp(R)28 b(>)g Fr(0)k(is)h(an)g Fp(\013)q Fr(-indep)s(enden)m(t)h(n)m (um)m(b)s(er.)257 4026 y Fg(Remark)257 4146 y Fr(This)49 b(is)g(a)f(crucial)g(step)h(for)e(the)i(expansion)g(in)f Fp(\013)h Fr(w)m(e)g(are)f(going)f(to)h(carry)g(out.)257 4267 y(Because)i(of)e(the)g(fast)g(con)m(v)m(ergence)j(pro)m(v)m(ed)f (in)e(Lemma)h(I)s(I.1,)j(w)m(e)d(can)f(c)m(ho)s(ose)h(a)257 4387 y(\\short")41 b(time)g(scale,)j Fp(\013)1177 4351 y Fj(\000)p Fm(\017)1264 4387 y Fr(,)f(as)e(a)f(time)i(cuto\013,)g(pro) m(vided)h(the)e(cuto\013-function)g Fp(\030)k Fr(is)257 4507 y(smo)s(oth,)33 b(uniformly)h(in)e Fp(\013)q Fr(.)404 4628 y(It)g(follo)m(ws)i(from)e(Lemma)h(I)s(I.1,)g(Inequalit)m(y)h(\(I) s(I.18\),)f(that)1237 4876 y Fp(i)1287 4759 y Fi(Z)1369 4785 y Fn(+)p Fj(1)1333 4948 y(\0001)1499 4876 y Fo(h)1556 4851 y Fp(~)1538 4876 y(A)1611 4835 y Fm(out)1716 4876 y Fr([)p Fo(\000)1841 4822 y Fr(\026)1834 4850 y Fp(~)1820 4876 y(f)11 b Fr(])1922 4851 y Fp(~)1906 4876 y(J)1969 4835 y Fm(tr)2032 4876 y Fr([)2052 4850 y Fp(~)2059 4876 y(h)2115 4891 y Fm(s)2153 4876 y Fp(;)17 b(s)p Fr(])p Fo(i)2309 4891 y Fm(\036)2351 4899 y Fe(g)r(s)2419 4876 y Fp(ds)668 b Fr(\(I)s(I)s(I.18\))257 5117 y(can)33 b(b)s(e)g(appro)m (ximated)h(b)m(y)1128 5360 y Fp(i)1178 5242 y Fi(Z)1260 5269 y Fn(+)p Fj(1)1224 5431 y(\0001)1390 5360 y Fo(h)1447 5334 y Fp(~)1429 5360 y(A)1502 5318 y Fm(out)1606 5360 y Fr([)p Fo(\000)1732 5305 y Fr(\026)1724 5333 y Fp(~)1710 5360 y(f)11 b Fr(])1812 5334 y Fp(~)1796 5360 y(J)1859 5318 y Fm(tr)1923 5360 y Fr(\()1954 5333 y Fp(~)1961 5360 y(h)2017 5375 y Fm(s)2054 5360 y Fp(;)17 b(s)p Fr(\))p Fo(i)2221 5375 y Fm(\036)2263 5383 y Fe(g)r(s)2332 5360 y Fp(\030)5 b Fr(\()p Fp(s)p Fr(\))p Fp(ds;)558 b Fr(\(I)s(I)s(I.19\))p eop end %%Page: 14 15 TeXDict begin 14 14 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(14)257 573 y(up)34 b(to)e(an)h(error)g(term)g (of)f(order)h Fp(o)p Fr(\()p Fp(\013)1640 537 y Fm(N)1707 573 y Fr(\),)g(for)f(an)m(y)i Fp(N)39 b Fo(2)28 b Fl(N)p Fr(.)45 b(After)33 b(some)g(in)m(tegrations)257 693 y(b)m(y)h(parts,)f (w)m(e)g(end)h(up)f(with)674 952 y Fo(\000)p Fp(i)801 834 y Fi(Z)884 861 y Fn(+)p Fj(1)847 1023 y(\0001)1030 834 y Fi(Z)1113 952 y Fo(h)1170 926 y Fp(~)1152 952 y(A)1225 910 y Fm(out)1330 952 y Fr([)p Fo(\000)1455 897 y Fr(\026)1448 925 y Fp(~)1434 952 y(f)11 b Fr(])1538 926 y Fp(~)1520 952 y(A)p Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))p Fo(i)1849 967 y Fm(\036)1891 975 y Fe(g)r(s)1960 952 y Fc(\003)p Fr(\()2068 925 y Fp(~)2075 952 y(h)2131 967 y Fm(s)2168 952 y Fr(\()o Fp(~)-48 b(y)s Fr(\))p Fp(\030)5 b Fr(\()p Fp(s)p Fr(\)\))p Fp(d)2554 910 y Fn(3)2593 952 y Fp(y)t(ds:)415 b Fr(\(I)s(I)s(I.20\))257 1207 y(Similarly)1215 1366 y Fo(\000)p Fp(i)1342 1249 y Fi(Z)1425 1275 y Fn(+)p Fj(1)1388 1437 y(\0001)1555 1366 y Fo(h)1610 1341 y Fp(~)1594 1366 y(J)1657 1325 y Fm(tr)1720 1366 y Fr([)1740 1339 y Fp(~)1747 1366 y(h)1803 1381 y Fm(s)1840 1366 y Fp(;)17 b(s)p Fr(])1975 1341 y Fp(~)1957 1366 y(A)2030 1325 y Fm(in)2101 1366 y Fr([)p Fo(\000)2227 1311 y Fr(\026)2219 1339 y Fp(~)2205 1366 y(f)11 b Fr(])p Fo(i)2330 1381 y Fm(\036)2372 1389 y Fe(g)r(s)2441 1366 y Fp(ds)646 b Fr(\(I)s(I)s(I.21\))257 1576 y(is)33 b(appro)m(ximated)h(b)m(y)1005 1834 y Fp(i)1055 1717 y Fi(Z)1138 1743 y Fn(+)p Fj(1)1101 1905 y(\0001)1268 1834 y Fo(h)1325 1809 y Fp(~)1307 1834 y(A)p Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))1614 1809 y Fp(~)1597 1834 y(A)1670 1793 y Fm(in)1740 1834 y Fr([)p Fo(\000)1866 1779 y Fr(\026)1858 1808 y Fp(~)1844 1834 y(f)11 b Fr(])p Fo(i)1969 1849 y Fm(\036)2011 1857 y Fe(g)r(s)2081 1834 y Fc(\003)p Fr(\()2189 1808 y Fp(~)2196 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y(totic)k(photons,)g(w)m(e)h(ha)m(v)m(e)g(to)e(con)m(trol)h(the)g (norm)f(of)h(v)m(ectors)h(of)e(the)h(form)667 3491 y Fi(Z)750 3517 y Fn(+)p Fj(1)713 3679 y(\0001)896 3608 y Fo(\001)22 b(\001)996 3491 y Fi(Z)1079 3517 y Fn(+)p Fj(1)1042 3679 y(\0001)1225 3491 y Fi(Z)1324 3608 y Fo(\001)g(\001)1424 3491 y Fi(Z)1556 3608 y Fo(T)1636 3556 y Fi(b)1636 3566 y Fm(\022)1675 3608 y Fr(\()1731 3583 y Fp(~)1713 3608 y(A)p Fr(\()o Fp(~)-48 b(y)1872 3623 y Fn(1)1911 3608 y Fp(;)17 b(s)2001 3623 y Fn(1)2040 3608 y Fr(\))22 b Fo(\001)g(\001)g(\001)2268 3583 y Fp(~)2249 3608 y(A)q Fr(\()o Fp(~)-48 b(y)2409 3623 y Fm(n)2455 3608 y Fp(;)17 b(s)2545 3623 y Fm(n)2592 3608 y Fr(\)\))p Fp(\036)2726 3623 y Fm(g)r(s)2820 3608 y Fo(\002)287 b Fr(\(I)s(I)s(I.30\))1564 3864 y Fo(\002)1694 3756 y Fm(n)1663 3781 y Fi(Y)1659 3963 y Fm(i)p Fn(=1)1789 3864 y Fc(\003)p Fr(\()1897 3838 y Fp(~)1904 3864 y(h)1960 3879 y Fm(i;s)2037 3889 y Fe(i)2068 3864 y Fr(\()o Fp(~)-48 b(y)2154 3879 y Fm(i)2181 3864 y Fr(\))p 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begin 16 16 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(16)257 573 y Fk(uniformly)35 b(in)g Fp(\013)q Fk(:)682 708 y Fi(Z)765 734 y Fn(+)p Fj(1)728 897 y(\0001)911 825 y Fo(\001)17 b(\001)g(\001)1044 708 y Fi(Z)1127 734 y Fn(+)p Fj(1)1090 897 y(\0001)1273 708 y Fi(Z)1373 825 y Fo(\001)g(\001)g(\001)1506 708 y Fi(Z)1640 825 y Fo(T)1720 774 y Fi(b)1720 784 y Fm(\022)1759 825 y Fr(\()1817 800 y Fp(~)1797 825 y(X)8 b Fr(\()o Fp(~)-48 b(y)1972 840 y Fn(1)2010 825 y Fp(;)17 b(s)2100 840 y Fn(1)2140 825 y Fr(\))22 b Fo(\001)f(\001)h(\001)2370 800 y Fp(~)2349 825 y(X)8 b Fr(\()o Fp(~)-48 b(y)2524 840 y Fm(n)2570 825 y Fp(;)17 b(s)2660 840 y Fm(n)2707 825 y Fr(\)\))22 b Fo(\002)302 b Fr(\(I)s(I)s(I.31\))1209 1081 y Fo(\002)1338 974 y Fm(n)1306 998 y Fi(Y)1303 1183 y Fm(l)q Fn(=1)1432 1081 y Fc(\003)p Fr(\()1540 1055 y Fp(~)1547 1081 y(h)1603 1096 y Fm(l)q(;s)1678 1108 y Fe(l)1706 1081 y Fr(\()o Fp(~)-48 b(y)1792 1096 y Fm(l)1817 1081 y Fr(\))p Fp(\030)5 b Fr(\()p Fp(s)1987 1096 y Fm(l)2012 1081 y Fr(\)\))2140 974 y Fm(n)2108 998 y Fi(Y)2105 1183 y Fm(l)q Fn(=1)2234 1081 y Fp(ds)2331 1096 y Fm(l)2356 1081 y Fp(d)2407 1040 y Fn(3)2446 1081 y Fp(y)2494 1096 y Fm(l)2688 1014 y Fr(1)p 2530 1058 365 4 v 2530 1150 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))2847 1121 y Fm(n)257 1412 y Fk(wher)-5 b(e)553 1387 y Fp(~)533 1412 y(X)7 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))35 b Fk(is)g(either)1273 1387 y Fp(~)1255 1412 y(A)p Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))35 b Fk(or)1742 1359 y Fr(_)1724 1387 y Fp(~)1706 1412 y(A)p Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))34 b Fk(and)1110 1651 y Fo(T)1189 1599 y Fi(b)1189 1609 y Fm(\022)1228 1651 y Fr(\()1287 1626 y Fp(~)1266 1651 y(X)8 b Fr(\()o Fp(~)-48 b(y)1441 1666 y Fn(1)1480 1651 y Fp(;)17 b(s)1570 1666 y Fn(1)1609 1651 y Fr(\))22 b Fo(\001)g(\001)g(\001)1839 1626 y Fp(~)1818 1651 y(X)8 b Fr(\()o Fp(~)-48 b(y)1993 1666 y Fm(n)2040 1651 y Fp(;)17 b(s)2130 1666 y Fm(n)2176 1651 y Fr(\)\))28 b(:=)801 b(\(I)s(I)s(I.32\))951 1805 y(=)1118 1722 y Fi(X)1110 1906 y Fm(p)p Fj(2P)1284 1780 y Fp(~)1264 1805 y(X)7 b Fr(\()o Fp(~)-48 b(y)1438 1820 y Fm(p)p Fn(\(1\))1568 1805 y Fp(;)17 b(s)1658 1820 y Fm(p)p Fn(\(1\))1787 1805 y Fr(\))22 b Fo(\001)g(\001)g(\001)2017 1780 y Fp(~)1997 1805 y(X)8 b Fr(\()o Fp(~)-48 b(y)2172 1820 y Fm(p)p Fn(\()p Fm(n)p Fn(\))2308 1805 y Fp(;)17 b(s)2398 1820 y Fm(p)p Fn(\()p Fm(n)p Fn(\))2535 1805 y Fr(\))23 b Fo(\002)1309 2054 y(\002)1395 2020 y Fi(b)1386 2054 y Fp(\022)s Fr(\()p Fp(s)1518 2069 y Fm(p)p Fn(\(1\))1670 2054 y Fo(\000)g Fp(s)1816 2069 y Fm(p)p Fn(\(2\))1946 2054 y Fr(\))f Fo(\001)g(\001)f(\001)2164 2020 y Fi(b)2155 2054 y Fp(\022)s Fr(\()p Fp(s)2287 2069 y Fm(p)p Fn(\()p Fm(n)p Fj(\000)p Fn(1\))2537 2054 y Fo(\000)i Fp(s)2683 2069 y Fm(p)p Fn(\()p Fm(n)p Fn(\))2820 2054 y Fr(\))p Fp(;)257 2267 y Fk(and)35 b Fo(P)43 b Fk(b)-5 b(eing)34 b(the)h(gr)-5 b(oup)35 b(of)f(p)-5 b(ermutations)35 b(of)f Fp(n)i Fk(elements.)257 2466 y(Pr)-5 b(o)g(of.)43 b Fr(The)34 b(pro)s(of)e(is)h(b)m(y)g(induction)h(in)f Fp(n)p Fr(.)257 2586 y(Giv)m(en)i(the)f(p)s(erm)m(utation)h(\()p Fp(p)p Fr(\(1\))p Fp(;)17 b Fo(\001)22 b(\001)g(\001\001)p Fp(;)17 b(p)p Fr(\()p Fp(j)6 b Fr(\)\),)33 b(w)m(e)i(assume)g(that,)g(for)e(1)c Fo(\024)i Fp(j)k Fo(\024)c Fp(n)23 b Fo(\000)g Fr(1,)257 2706 y(the)33 b(follo)m(wing)g(statemen)m(ts)i(are)d(true:)403 2905 y Fo(\017)48 b Fr(H1\))33 b(The)g(op)s(erator)501 3045 y Fi(Z)584 3071 y Fn(+)p Fj(1)547 3234 y(\0001)730 3162 y Fo(\001)22 b(\001)830 3045 y Fi(Z)913 3071 y Fn(+)p Fj(1)876 3234 y(\0001)1059 3045 y Fi(Z)1159 3162 y Fo(\001)g(\001)1259 3045 y Fi(Z)1411 3137 y Fp(~)1391 3162 y(X)8 b Fr(\()o Fp(~)-48 b(y)1566 3178 y Fm(p)p Fn(\(1\))1695 3162 y Fp(;)17 b(s)1785 3178 y Fm(p)p Fn(\(1\))1914 3162 y Fr(\))22 b Fo(\001)g(\001)g(\001)2144 3137 y Fp(~)2124 3162 y(X)8 b Fr(\()o Fp(~)-48 b(y)2299 3178 y Fm(p)p Fn(\()p Fm(j)t Fn(\))2425 3162 y Fp(;)17 b(s)2515 3178 y Fm(p)p Fn(\()p Fm(j)t Fn(\))2642 3162 y Fr(\))p Fo(\002)730 b Fr(\(I)s(I)s(I.33\))765 3440 y Fo(\002)859 3326 y Fm(j)t Fj(\000)p Fn(1)867 3357 y Fi(Y)864 3541 y Fm(l)q Fn(=1)1007 3406 y Fi(b)998 3440 y Fp(\022)s Fr(\()p Fp(s)1130 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))1269 3440 y Fo(\000)22 b Fp(s)1414 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(+1\))1621 3440 y Fr(\))1715 3326 y Fm(j)1678 3357 y Fi(Y)1675 3541 y Fm(l)q Fn(=1)1804 3440 y Fc(\003)p Fr(\()1912 3413 y Fp(~)1919 3440 y(h)1975 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))p Fm(;s)2140 3469 y Fe(p)p Fd(\()p Fe(l)p Fd(\))2248 3440 y Fr(\()o Fp(~)-48 b(y)2334 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))2450 3440 y Fr(\))p Fp(\030)5 b Fr(\()p Fp(s)2620 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))2736 3440 y Fr(\)\))2868 3326 y Fm(j)2831 3357 y Fi(Y)2828 3541 y Fm(l)q Fn(=1)2957 3440 y Fp(ds)3054 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))3170 3440 y Fp(d)3221 3398 y Fn(3)3260 3440 y Fp(y)3308 3455 y Fm(p)p Fn(\()p Fm(l)q Fn(\))3587 3372 y Fr(1)p 3434 3416 355 4 v 3434 3508 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))3752 3479 y Fm(j)501 3720 y Fr(is)33 b(b)s(ounded)h(uniformly)f (in)g Fp(\013)q Fr(;)403 3934 y Fo(\017)48 b Fr(H2\))22 b(F)-8 b(or)21 b Fp(u)27 b Fo(2)h Fl(R)p Fr(,)c(and)e(for)f(functions)i Fp(\020)8 b Fr(,)18 b Fp(~)-44 b(g)25 b Fr(b)s(elonging)c(to)h(the)g (families)h Fo(f)3148 3887 y Fm(d)3184 3864 y Fe(k)3222 3887 y Fm(\030)s Fn(\()p Fm(s)3316 3899 y Fe(l)3340 3887 y Fn(\))p 3148 3911 220 4 v 3204 3971 a Fm(ds)3273 3948 y Fe(k)3273 3996 y(l)3377 3934 y Fo(j)p Fp(l)30 b Fr(=)501 4116 y(1)p Fp(;)17 b(::;)g(n)g Fr(;)33 b Fp(k)39 b Fr(=)d(1)p Fp(;)17 b Fr(2)p Fp(;)g Fr(3)p Fp(;)g Fo(\001)23 b(\001)i(\001g)37 b Fr(and)g Fo(f)1781 4070 y Fm(d)1817 4046 y Fe(k)1851 4052 y Fm(~)1856 4070 y(h)1897 4082 y Fe(l)1921 4070 y Fn(\()p Fm(~)-35 b(y)1983 4082 y Fe(l)2007 4070 y Fm(;s)2060 4082 y Fe(l)2083 4070 y Fn(\))p 1781 4093 330 4 v 1893 4154 a Fm(ds)1962 4131 y Fe(k)1962 4179 y(l)2121 4116 y Fo(j)p Fp(l)38 b Fr(=)d(1)p Fp(;)17 b(::;)g(n)g Fr(;)33 b Fp(k)39 b Fr(=)d(1)p Fp(;)17 b Fr(2)p Fp(;)g Fr(3)p Fp(;)g Fo(\001)23 b(\001)i(\001g)p Fr(,)38 b(re-)501 4256 y(sp)s(ectiv)m(ely)-8 b(,)36 b(the)d(op)s(erator)602 4392 y Fi(Z)686 4418 y Fn(+)p Fj(1)649 4581 y(\0001)832 4392 y Fi(Z)915 4509 y Fr(\()p Fp(H)c Fr(+)22 b Fp(i)p Fr(\))1232 4468 y Fm(j)t Fj(\000)p Fn(1)1380 4484 y Fp(~)1359 4509 y(X)8 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\)\))1712 4476 y Fi(b)1703 4509 y Fp(\022)s Fr(\()p Fp(u)k Fo(\000)i Fp(s)p Fr(\))l Fp(~)-45 b(g)2097 4524 y Fm(s)2133 4509 y Fr(\()o Fp(~)d(y)s Fr(\))p Fp(\020)8 b Fr(\()p Fp(s)p Fr(\)\))p Fp(d)2522 4468 y Fn(3)2560 4509 y Fp(y)t(ds)2871 4442 y Fr(1)p 2719 4486 355 4 v 2719 4577 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))3036 4549 y Fm(j)3184 4509 y Fr(\(I)s(I)s(I.34\))501 4773 y(is)33 b(b)s(ounded,)h(uniformly)f(in)g Fp(u)f Fr(and)h(in)f Fp(\013)q Fr(.)257 4972 y(W)-8 b(e)34 b(\014rst)g(pro)m(v)m(e)g(that)f(H2\))g(holds)h(when)g Fp(j)40 b Fr(is)33 b(replaced)i(b)m(y)f Fp(j)28 b Fr(+)23 b(1.)45 b(F)-8 b(or)33 b(this)g(purp)s(ose)257 5092 y(w)m(e)h(consider) g(the)f(scalar)g(pro)s(duct)358 5345 y(\()p Fp( )t(;)507 5228 y Fi(Z)590 5254 y Fn(+)p Fj(1)553 5416 y(\0001)736 5228 y Fi(Z)819 5345 y Fr(\()p Fp(H)d Fr(+)22 b Fp(i)p Fr(\))1137 5304 y Fm(j)1194 5320 y Fp(~)1174 5345 y(X)7 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\)\))1526 5311 y Fi(b)1517 5345 y Fp(\022)s Fr(\()p Fp(u)22 b Fo(\000)g Fp(s)p Fr(\))l Fp(~)-45 b(g)1911 5360 y Fm(s)1948 5345 y Fr(\()o Fp(~)d(y)s Fr(\))p Fp(\020)8 b Fr(\()p Fp(s)p Fr(\)\))p Fp(d)2337 5304 y Fn(3)2375 5345 y Fp(y)t(ds)2730 5277 y Fr(1)p 2534 5321 445 4 v 2534 5413 a(\()p Fp(H)28 b Fr(+)22 b Fp(i)p Fr(\))2850 5384 y Fm(j)t Fn(+1)2987 5345 y Fp(\036)p Fr(\))101 b(\(I)s(I)s(I.35\))p eop end %%Page: 17 18 TeXDict begin 17 17 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(17)257 573 y(where)49 b Fp( )57 b Fo(2)c(D)s Fr(\()p Fp(H)1000 537 y Fm(m)1065 573 y Fr(\),)e(for)c(an)m(y)h Fp(m)53 b Fo(2)g Fl(N)p Fr(,)e(and)c Fp(\036)g Fr(is)h(an)f(arbitrary)h(v)m(ector.)88 b(The)257 693 y(expression)35 b(in)e(Eq.)h(\(I)s(I)s(I.35\))e(can)g(b)s(e)h (written)h(as)e(\(recall)2443 660 y Fi(b)2433 693 y Fp(\022)2478 708 y Fm(u;s)2604 693 y Fr(=)2716 660 y Fi(b)2707 693 y Fp(\022)t Fr(\()p Fp(t)22 b Fo(\000)g Fp(s)p Fr(\)\))501 843 y Fi(Z)584 870 y Fn(+)p Fj(1)547 1032 y(\0001)730 843 y Fi(Z)813 961 y Fr(\()p Fp( )t(;)17 b Fr(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))1279 919 y Fm(j)t Fj(\000)p Fn(1)1427 935 y Fp(~)1406 961 y(X)8 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\)\))1759 927 y Fi(b)1750 961 y Fp(\022)1795 976 y Fm(u;s)1888 961 y Fp(~)-45 b(g)1939 976 y Fm(s)1976 961 y Fr(\()o Fp(~)d(y)s Fr(\))p Fp(\020)8 b Fr(\()p Fp(s)p Fr(\)\))p Fp(d)2365 919 y Fn(3)2403 961 y Fp(y)t(ds)2713 893 y Fr(1)p 2562 937 355 4 v 2562 1029 a(\()p Fp(H)28 b Fr(+)22 b Fp(i)p Fr(\))2878 1000 y Fm(j)2925 961 y Fp(\036)p Fr(\))187 b(\(I)s(I)s(I.36\))340 1251 y Fo(\000)84 b Fp(i)551 1134 y Fi(Z)634 1160 y Fn(+)p Fj(1)597 1322 y(\0001)780 1134 y Fi(Z)863 1251 y Fr(\()p Fp( )t(;)17 b Fr(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))1329 1210 y Fm(j)t Fj(\000)p Fn(1)1466 1183 y Fp(@)1544 1158 y(~)1522 1183 y(X)9 b Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))p 1466 1227 363 4 v 1596 1319 a Fp(@)5 b(s)1848 1217 y Fi(b)1838 1251 y Fp(\022)1883 1266 y Fm(u;s)1977 1251 y Fp(~)-45 b(g)2028 1266 y Fm(s)2065 1251 y Fr(\()o Fp(~)d(y)s Fr(\))p Fp(\020)8 b Fr(\()p Fp(s)p Fr(\)\))p Fp(d)2454 1210 y Fn(3)2492 1251 y Fp(y)t(ds)2847 1183 y Fr(1)p 2651 1227 445 4 v 2651 1319 a(\()p Fp(H)28 b Fr(+)22 b Fp(i)p Fr(\))2967 1290 y Fm(j)t Fn(+1)3104 1251 y Fp(\036)p Fr(\))292 b Fp(:)-311 b Fr(\(I)s(I)s(I.37\))257 1527 y(In)m(tegrating)36 b(b)m(y)g(parts)f(in)h(the)f(time)h(v)-5 b(ariable)36 b Fp(s)p Fr(,)f(the)h(expression)i(in)d(Eq.)h(\(I)s(I)s (I.37\))f(is)257 1647 y(seen)f(to)e(b)s(e)h(giv)m(en)h(b)m(y)307 1911 y Fp(i)357 1794 y Fi(Z)440 1820 y Fn(+)p Fj(1)403 1982 y(\0001)586 1794 y Fi(Z)669 1911 y Fr(\()p Fp( )t(;)17 b Fr(\()p Fp(H)8 b Fr(+)p Fp(i)p Fr(\))1092 1870 y Fm(j)t Fj(\000)p Fn(1)1239 1886 y Fp(~)1218 1911 y(X)g Fr(\()o Fp(~)-48 b(y)s(;)17 b(s)p Fr(\))1557 1843 y Fp(@)p 1534 1888 103 4 v 1534 1979 a(@)5 b(s)1647 1815 y Fi(\020)1705 1877 y(b)1696 1911 y Fp(\022)1741 1926 y Fm(u;s)1835 1911 y Fp(~)-45 b(g)1886 1926 y Fm(s)1922 1911 y Fr(\()o Fp(~)d(y)s Fr(\))p Fp(\020)8 b Fr(\()p Fp(s)p Fr(\)\))2259 1815 y Fi(\021)2517 1843 y Fr(1)p 2319 1888 445 4 v 2319 1979 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))2636 1950 y Fm(j)t Fn(+1)2773 1911 y Fp(\036)p Fr(\))p Fp(d)2920 1870 y Fn(3)2959 1911 y Fp(y)t(ds:)49 b Fr(\(I)s(I)s(I.38\))257 2187 y(By)31 b(the)g(induction)h(h)m(yp)s(othesis)g(H2\),)f(w)m(e)h (conclude)f(that)g(the)g(absolute)g(v)-5 b(alue)31 b(of)f(the)257 2307 y(scalar)j(pro)s(duct)g(\(I)s(I)s(I.37\))f(is)h(b)s(ounded)h(b)m (y)1662 2527 y Fp(C)7 b Fo(k)p Fp( )t Fo(kk)p Fp(\036)p Fo(k)p Fp(;)1093 b Fr(\(I)s(I)s(I.39\))257 2747 y(where)52 b(C)e(is)h(a)f(p)s(ositiv)m(e)h(constan)m(t)g(indep)s(enden)m(t)h(of)e Fp( )k Fr(and)c Fp(\036)p Fr(.)96 b(Due)50 b(to)g(Riesz')257 2868 y(lemma,)h(the)46 b(op)s(erator)f(in)h(\(I)s(I)s(I.35\))g(is)g(b)s (ounded.)85 b(Because)47 b(of)f(our)f(assumptions)257 2988 y(on)397 2954 y Fi(b)388 2988 y Fp(\022)433 3003 y Fm(u)p Fj(\000)p Fm(s)566 2988 y Fp(;)12 b(~)-44 b(g)657 3003 y Fm(s)693 2988 y Fr(\()o Fp(~)c(y)s Fr(\))p Fp(;)17 b(\020)8 b Fr(\()p Fp(s)p Fr(\),)27 b(the)g(constan)m(t)i Fp(C)34 b Fr(can)27 b(b)s(e)h(c)m(hosen)h(to)d(b)s(e)i(indep)s(enden)m (t)h(of)e Fp(\013)h Fr(and)257 3108 y Fp(u)p Fr(.)404 3229 y(T)-8 b(urning)50 b(to)f(H1\),)k(w)m(e)e(\014rst)f(in)m(tro)s (duce)g(a)f(shorthand)h(notation:)77 b(Expression)257 3349 y(\(I)s(I)s(I.33\))31 b(is)h(abbreviated)h(b)m(y)f Fo(X)15 b Fr(\()p Fp(p)p Fr(\(1\))p Fp(;)i Fo(\001\001)p Fp(;)g(p)p Fr(\()p Fp(j)6 b Fr(\)\),)29 b(and)j(the)f(expression)j(in)d (Eq.)i(\(I)s(I)s(I.34\))257 3469 y(b)m(y)g Fo(A)472 3484 y Fm(u)516 3469 y Fr(\()l Fp(~)-45 b(g)t(;)17 b(\020)8 b Fr(\).)41 b(Because)34 b(of)d(prop)s(ert)m(y)h(H2\),)f(for)g Fp(j)26 b Fr(+)20 b(1,)32 b(and)f(assuming)i(H1\))f(holds)g(for)257 3590 y Fp(j)6 b Fr(,)33 b(the)g(follo)m(wing)g(inequalit)m(y)503 3810 y Fo(kX)15 b Fr(\()p Fp(p)p Fr(\(1\))p Fp(;)i Fo(\001\001)p Fp(;)g(p)p Fr(\()p Fp(j)26 b Fr(+)c(1\)\))p Fo(k)1763 b Fr(\(I)s(I)s(I.40\))343 4034 y Fo(\024)83 b Fr(2)p Fo(kX)15 b Fr(\()p Fp(p)p Fr(\(1\))p Fp(;)i Fo(\001\001)p Fp(;)g(p)p Fr(\()p Fp(j)6 b Fr(\)\))p Fo(k)p Fr(sup)1447 4057 y Fm(s)1480 4071 y Fe(p)p Fd(\()p Fe(j)s Fd(\))1597 4034 y Fo(kA)1727 4049 y Fm(s)1760 4063 y Fe(p)p Fd(\()p Fe(j)s Fd(\))1875 4034 y Fr(\()1923 3965 y Fp(d)1967 3939 y(~)1974 3965 y(h)2030 3981 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2247 3965 y Fr(\()o Fp(~)-48 b(y)2333 3981 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2550 3965 y Fp(;)17 b(s)2640 3981 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2856 3965 y Fr(\))p 1923 4011 972 4 v 2252 4102 a Fp(ds)2349 4118 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2904 4034 y Fp(;)2958 3965 y(d\030)5 b Fr(\()p Fp(s)3141 3981 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))3357 3965 y Fr(\))p 2958 4011 438 4 v 3020 4102 a Fp(ds)3117 4118 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))3405 4034 y Fr(\))p Fo(k)503 4315 y Fr(+)p Fo(kX)15 b Fr(\()p Fp(p)p Fr(\(1\))p Fp(;)i Fo(\001\001)p Fp(;)g(p)p Fr(\()p Fp(j)6 b Fr(\)\))p Fo(k)p Fr(sup)1474 4339 y Fm(s)1507 4353 y Fe(p)p Fd(\()p Fe(j)s Fd(\))1624 4315 y Fo(kA)1754 4330 y Fm(s)1787 4344 y Fe(p)p Fd(\()p Fe(j)s Fd(\))1902 4315 y Fr(\()1933 4289 y Fp(~)1940 4315 y(h)1996 4331 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2214 4315 y Fp(;)2267 4246 y(d)2318 4210 y Fn(2)2357 4246 y Fp(\030)f Fr(\()p Fp(s)2489 4262 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2706 4246 y Fr(\))p 2267 4292 477 4 v 2349 4383 a Fp(ds)2446 4349 y Fn(2)2446 4412 y Fm(p)p Fn(\()p Fm(j)t Fn(+1\))2754 4315 y Fr(\))p Fo(k)257 4611 y Fr(implies)36 b(that)d(H1\))h(holds)g(for)f Fp(j)c Fr(+)23 b(1,)34 b(to)s(o.)46 b(Since)35 b(H1\))e(and)h(H2\))g (are)f(ob)m(viously)j(true)257 4732 y(for)e Fp(j)j Fr(=)30 b(1,)35 b(they)g(hold)g(for)f(an)m(y)h Fp(j)h Fo(\024)31 b Fp(n)p Fr(.)49 b(Lemma)35 b(I)s(I)s(I.1)g(follo)m(ws)g(from)f(H1\))g (and)g(H2\).)p 3425 4852 4 66 v 3429 4790 59 4 v 3429 4852 V 3487 4852 4 66 v 257 4973 a(In)41 b(our)f(deriv)-5 b(ation)40 b(of)f Fk(r)-5 b(e)g(duction)42 b(formulae)47 b Fr(w)m(e)41 b(neglect)g(forw)m(ard)f(scattering,)i(i.e.,)257 5093 y(w)m(e)31 b(assume)h(that)d(\()p Fp(f)1027 5108 y Fm(i)1055 5093 y Fp(;)17 b(h)1155 5108 y Fm(j)1192 5093 y Fr(\))27 b(=)h(0,)i(for)g(arbitrary)f Fp(i;)17 b(j)6 b Fr(.)43 b(The)31 b(general)f(case)h(can)f(easily)h(b)s(e)257 5213 y(deriv)m(ed)k(from)d(the)g(result)i(b)s(elo)m(w,)f(at)f(the)h (price)g(of)f(more)h(complicated)g(expressions.)p eop end %%Page: 18 19 TeXDict begin 18 18 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(18)257 573 y Fg(Prop)s(osition)52 b(I)s(I)s(I.2.)c Fk(Under)e(the)g(assumptions)g(in)g(Eq.)g(\(III.2\))e (and)i(of)g(L)-5 b(emma)257 703 y(III.1)36 b(on)h Fo(f)672 677 y Fp(~)679 703 y(h)735 718 y Fm(j)772 703 y Fo(g)p Fk(,)h Fo(f)954 677 y Fp(~)940 703 y(f)988 718 y Fm(p)1027 703 y Fo(g)g Fk(and)f Fp(\030)5 b Fk(,)37 b(and)g(if)h Fr(\()p Fp(f)1798 718 y Fm(p)1838 703 y Fp(;)17 b(h)1938 718 y Fm(j)1974 703 y Fr(\))33 b(=)f(0)p Fk(,)38 b(for)f(al)5 b(l)38 b Fp(p)f Fk(and)g Fp(j)6 b Fk(,)38 b(the)g(S-matrix)257 824 y(element)1220 903 y Fi(\020)1307 891 y Fm(m)1369 868 y Ff(0)1296 916 y Fi(Y)1286 1098 y Fm(p)p Fn(=1)1447 974 y Fp(~)1429 999 y(A)1502 958 y Fm(out)1606 999 y Fr([)1647 973 y Fp(~)1633 999 y(f)1681 1014 y Fm(p)1721 999 y Fr(])p Fp(\036)1806 1014 y Fm(g)r(s)1879 999 y Fp(;)1952 891 y Fm(m)1931 916 y Fi(Y)1922 1099 y Fm(j)t Fn(=1)2080 974 y Fp(~)2062 999 y(A)2135 958 y Fm(in)2206 999 y Fr([)2226 973 y Fp(~)2233 999 y(h)2289 1014 y Fm(j)2325 999 y Fr(])p Fp(\036)2410 1014 y Fm(g)r(s)2483 903 y Fi(\021)3184 999 y Fr(\(I)s(I)s(I.41\))257 1236 y Fk(is)d(given)f(by) 257 1494 y Fp(i)290 1453 y Fm(m)p Fn(+)p Fm(m)469 1430 y Ff(0)514 1377 y Fi(Z)597 1403 y Fn(+)p Fj(1)560 1566 y(\0001)743 1494 y Fo(\001)22 b(\001)842 1377 y Fi(Z)925 1403 y Fn(+)p 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Fm(m)1765 1732 y Fi(Y)1756 1915 y Fm(j)t Fn(=1)1895 1815 y Fc(\003)p Fr(\()2003 1789 y Fp(~)2010 1815 y(h)2066 1830 y Fm(j;s)2148 1840 y Fe(j)2184 1815 y Fr(\()o Fp(~)-48 b(y)2270 1830 y Fm(j)2306 1815 y Fr(\))p Fp(\030)5 b Fr(\()p Fp(s)2476 1830 y Fm(j)2512 1815 y Fr(\)\))2625 1707 y Fm(m)2687 1684 y Ff(0)2614 1732 y Fi(Y)2605 1914 y Fm(p)p Fn(=1)2747 1815 y Fp(dt)2833 1830 y Fm(p)2873 1815 y Fp(d)2924 1774 y Fn(3)2963 1815 y Fp(z)3008 1830 y Fm(p)3094 1707 y(m)3072 1732 y Fi(Y)3064 1915 y Fm(j)t Fn(=1)3203 1815 y Fp(ds)3300 1830 y Fm(j)3337 1815 y Fp(d)3388 1774 y Fn(3)3427 1815 y Fp(y)3475 1830 y Fm(j)3511 1815 y Fp(;)257 2077 y Fk(up)35 b(to)g(an)g(err)-5 b(or)35 b(term)g(of)f(arbitr)-5 b(arily)35 b(high)f(or)-5 b(der)35 b(in)g Fp(\013)q Fk(.)257 2249 y(Pr)-5 b(o)g(of.)65 b Fr(The)41 b(pro)s(of)d(is)j(b)m(y)f(induction.) 66 b(If)40 b(there)h(are)e(only)i(one)f(incoming)g(and)g(one)257 2370 y(outgoing)32 b(photon,)h(w)m(e)g(ha)m(v)m(e)h(pro)m(v)m(en)g (this)f(result)g(at)f(the)h(b)s(eginning)g(of)f(the)h(section.)257 2490 y(Notice)j(that,)f(using)g(the)h(same)f(argumen)m(ts,)i(w)m(e)f (can)f(eliminate)h(one)f(incoming)g(and)257 2610 y(one)23 b(outgoing)f(photon)g(from)g(the)h(asymptotic)g(states)g(in)g(the)f (scalar)h(pro)s(duct)g(\(I)s(I)s(I.41\).)257 2731 y(The)37 b(inductiv)m(e)g(assumption)g(is)f(that)f Fp(l)27 b Fr(+)d Fp(l)1907 2695 y Fj(0)1963 2731 y Fr(=)32 b Fp(n)24 b Fo(\000)h Fr(1\()p Fp(<)32 b(m)24 b Fr(+)g Fp(m)2744 2695 y Fj(0)2768 2731 y Fr(\))35 b(photons)h(can)g(b)s(e)257 2851 y(eliminated)e(yielding)g(the)f(expression)392 2993 y Fi(Z)492 3013 y(\020)558 3002 y Fm(m)620 2978 y Ff(0)643 3002 y Fj(\000)p Fm(l)720 2978 y Ff(0)597 3027 y Fi(Y)587 3208 y Fm(p)p Fn(=1)777 3084 y Fp(~)759 3110 y(A)832 3069 y Fm(out)936 3110 y Fr([)977 3083 y Fp(~)963 3110 y(f)1011 3125 y Fm(p)1051 3110 y Fr(])p Fp(\036)1136 3125 y Fm(g)r(s)1209 3110 y Fp(;)17 b Fo(T)1332 3058 y Fi(b)1332 3068 y Fm(\022)1371 3110 y Fr(\()1450 3002 y Fm(l)1472 2978 y Ff(0)1419 3027 y Fi(Y)1409 3208 y Fm(p)p Fn(=1)1569 3084 y Fp(~)1551 3110 y(A)p Fr(\()n Fp(~)-47 b(z)1707 3125 y Fm(p)1747 3110 y Fp(;)17 b(t)1826 3125 y Fm(p)1866 3110 y Fr(\))1971 3002 y Fm(l)1929 3027 y Fi(Y)1921 3209 y Fm(j)t Fn(=1)2078 3084 y Fp(~)2060 3110 y(A)p Fr(\()o Fp(~)-48 b(y)2219 3125 y Fm(j)2255 3110 y Fp(;)17 b(s)2345 3125 y Fm(j)2381 3110 y Fr(\)\))2474 3002 y Fm(m)p Fj(\000)p Fm(l)2490 3027 y Fi(Y)2482 3209 y Fm(j)t Fn(=1)2647 3084 y Fp(~)2629 3110 y(A)2702 3069 y Fm(in)2773 3110 y Fr([)2793 3083 y Fp(~)2800 3110 y(h)2856 3125 y Fm(j)2893 3110 y Fr(])p Fp(\036)2978 3125 y Fm(g)r(s)3051 3013 y Fi(\021)3100 3110 y Fo(\002)7 b Fr(\(I)s(I)s(I.43\))656 3431 y Fo(\002)791 3323 y Fm(l)813 3299 y Ff(0)760 3348 y Fi(Y)750 3529 y Fm(p)p Fn(=1)892 3431 y Fc(\003)p Fr(\()p 1007 3323 V 1021 3404 a Fp(~)1007 3431 y(f)1055 3446 y Fm(p;t)1136 3454 y Fe(p)1177 3431 y Fr(\()n Fp(~)-47 b(z)1260 3446 y Fm(i)1288 3431 y Fr(\))p Fp(\030)5 b Fr(\()p Fp(t)1447 3446 y Fm(p)1486 3431 y Fr(\)\))1629 3323 y Fm(l)1587 3348 y Fi(Y)1579 3530 y Fm(j)t Fn(=1)1718 3431 y Fc(\003)p Fr(\()1826 3404 y Fp(~)1833 3431 y(h)1889 3446 y Fm(j;s)1971 3456 y Fe(j)2007 3431 y Fr(\()o Fp(~)-48 b(y)2093 3446 y Fm(j)2129 3431 y Fr(\))p Fp(\030)5 b Fr(\()p Fp(s)2299 3446 y Fm(j)2335 3431 y Fr(\)\))2468 3323 y Fm(l)2490 3299 y Ff(0)2437 3348 y Fi(Y)2427 3529 y Fm(p)p Fn(=1)2569 3431 y Fp(dt)2655 3446 y Fm(p)2695 3431 y Fp(d)2746 3390 y Fn(3)2785 3431 y Fp(z)2830 3446 y Fm(p)2937 3323 y(l)2895 3348 y Fi(Y)2887 3530 y Fm(j)t Fn(=1)3026 3431 y Fp(ds)3123 3446 y Fm(j)3159 3431 y Fp(d)3210 3390 y Fn(3)3249 3431 y Fp(y)3297 3446 y Fm(j)3333 3431 y Fp(;)257 3698 y Fr(up)46 b(to)f(an)h(error)f(term)h(of)f (arbitrarily)g(high)h(order)f(in)h Fp(\013)q Fr(.)82 b(T)-8 b(o)45 b(pro)m(v)m(e)i(\(I)s(I)s(I.41\))e(for)257 3818 y Fp(l)30 b Fr(+)e Fp(l)451 3782 y Fj(0)502 3818 y Fr(+)g(1)41 b(=)h Fp(n)p Fr(,)h(another)e(photon,)i(for)e(example)h (an)f(outgoing)f(photon,)j(m)m(ust)f(b)s(e)257 3938 y(eliminated)30 b(from)e(the)h(outgoing)f(state)h(in)g(Expression)h(\(I)s(I)s(I.41\),)f (with)g Fp(l)16 b Fr(+)e Fp(l)3102 3902 y Fj(0)3153 3938 y Fr(=)27 b Fp(n)14 b Fo(\000)g Fr(1.)257 4059 y(This)30 b(can)f(b)s(e)g(accomplished)h(b)m(y)g(rep)s(eating)f(op)s(erations)g (i\))f(and)h(ii\))f(at)h(the)g(b)s(eginning)257 4179 y(of)j(this)i(section,)f(with)h(the)f(follo)m(wing)g(mo)s (di\014cations:)257 4300 y(a\))g(The)g(coun)m(terpart)h(of)e(the)h(exp) s(ectation)g(v)-5 b(alue)33 b(in)g(\(I)s(I)s(I.23\))f(is)h(giv)m(en)h (b)m(y:)314 4414 y Fi(\020)363 4510 y Fp( )430 4469 y Fm(out)426 4535 y(m)488 4516 y Ff(0)511 4535 y Fj(\000)p Fm(l)588 4516 y Ff(0)610 4535 y Fj(\000)p Fn(1)705 4510 y Fp(;)17 b(T)820 4459 y Fi(b)820 4469 y Fm(\022)858 4510 y Fr(\()914 4485 y Fp(~)896 4510 y(A)p Fr(\()n Fp(~)-47 b(z)1052 4525 y Fn(1)1092 4510 y Fp(;)17 b(t)1171 4525 y Fn(1)1210 4510 y Fr(\))22 b Fo(\001)g(\001)1366 4485 y Fp(~)1348 4510 y(A)p Fr(\()n Fp(~)-47 b(z)1504 4525 y Fm(l)1526 4506 y Ff(0)1549 4525 y Fn(+1)1643 4510 y Fp(;)17 b(t)1722 4525 y Fm(l)1744 4506 y Ff(0)1766 4525 y Fn(+1)1860 4510 y Fr(\))22 b Fo(\001)1970 4441 y( )-77 b(!)2003 4510 y Fp(@)p 2092 4403 615 4 v 2106 4484 a(~)2092 4510 y(f)2140 4525 y Fm(l)2162 4506 y Ff(0)2184 4525 y Fn(+1)p Fm(;t)2319 4544 y Fe(l)2339 4530 y Ff(0)2362 4544 y Fd(+1)2447 4510 y Fr(\()n Fp(~)-47 b(z)2530 4525 y Fm(l)2552 4506 y Ff(0)2575 4525 y Fn(+1)2669 4510 y Fr(\))22 b Fo(\001)g(\001)2825 4485 y Fp(~)2807 4510 y(A)p Fr(\()o Fp(~)-48 b(y)2966 4525 y Fm(l)2991 4510 y Fp(;)17 b(s)3081 4525 y Fm(l)3107 4510 y Fr(\)\))p Fp( )3250 4469 y Fm(in)3246 4535 y(m)p Fj(\000)p Fm(l)3389 4414 y Fi(\021)3184 4643 y Fr(\(I)s(I)s(I.44\))257 4763 y(where)1201 5003 y Fp( )1268 4961 y Fm(out)1264 5027 y(m)1326 5008 y Ff(0)1349 5027 y Fj(\000)p Fm(l)1426 5008 y Ff(0)1448 5027 y Fj(\000)p Fn(1)1625 5003 y Fr(:=)1811 4895 y Fm(m)1873 4871 y Ff(0)1896 4895 y Fj(\000)p Fm(l)1973 4871 y Ff(0)1995 4895 y Fj(\000)p Fn(1)1895 4920 y Fi(Y)1885 5101 y Fm(p)p Fn(=1)2120 4977 y Fp(~)2102 5003 y(A)2175 4961 y Fm(out)2280 5003 y Fr([)2321 4976 y Fp(~)2307 5003 y(f)2355 5018 y Fm(p)2394 5003 y Fr(])p Fp(\036)2479 5018 y Fm(g)r(s)3184 5003 y Fr(\(I)s(I)s(I.45\))1335 5317 y Fp( )1402 5276 y Fm(in)1398 5342 y(m)p Fj(\000)p Fm(l)1625 5317 y Fr(:=)1811 5210 y Fm(m)p Fj(\000)p Fm(l)1828 5234 y Fi(Y)1820 5417 y Fm(j)t Fn(=1)1985 5292 y Fp(~)1967 5317 y(A)2040 5276 y Fm(in)2111 5317 y Fr([)2131 5291 y Fp(~)2138 5317 y(h)2194 5332 y Fm(j)2231 5317 y Fr(])p Fp(\036)2316 5332 y Fm(g)r(s)2388 5317 y Fr(;)769 b(\(I)s(I)s(I.46\))p eop end %%Page: 19 20 TeXDict begin 19 19 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(19)257 573 y(b\))32 b(As)g(b)s(efore,)g (using)h(Lemma)f(I)s(I)s(I.1,)g(w)m(e)h(ma)m(y)f(cuto\013)g(the)g(in)m (tegration)g(o)m(v)m(er)h Fp(t)3222 588 y Fm(l)3244 569 y Ff(0)3266 588 y Fn(+1)3392 573 y Fr(b)m(y)257 693 y(in)m(tro)s (ducing)40 b(the)f(smo)s(oth)g(function)g(of)g(compact)g(supp)s(ort)g 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2150 y(will)35 b(pro)m(v)m(e,)h(can)e(b)s(e)g(c)m(hosen)h(to)f(b)s(e)g (of)f(arbitrarily)i(high)f(order)g(in)g(the)g(\014nestructure)257 2270 y(constan)m(t)g Fp(\013)q Fr(.)257 2391 y(The)28 b Fk(r)-5 b(e)g(duction)29 b(formulae)k Fr(deriv)m(ed)c(in)e(the)g (previous)h(Section)f(are)g(not)f(a)g(particularly)257 2511 y(con)m(v)m(enien)m(t)44 b(starting)c(p)s(oin)m(t)h(to)f(dev)m (elop)j(an)d(algorithm)h(for)f(calculating)h(S-matrix)257 2632 y(elemen)m(ts.)62 b(Ho)m(w)m(ev)m(er,)41 b(with)d(a)g(sligh)m(t)g (mo)s(di\014cation)g(that)g(dep)s(ends)h(on)e(the)i(desired)257 2752 y(order,)31 b Fp(o)p Fr(\()p Fp(\013)685 2716 y Fm(N)752 2752 y Fr(\),)f Fp(N)38 b Fr(=)28 b(1)p Fp(;)17 b Fr(2)p Fp(;)g Fr(3)p Fp(;)g(::)p Fr(,)29 b(of)g(accuracy)i(of)e(the)i (algorithm,)f(w)m(e)h(can,)g(in)f(essence,)257 2872 y(rep)s(eat)25 b(the)g(reduction)h(pro)s(cedure)f(in)g(Prop)s(osition)g(I)s(I)s(I.2.) 41 b(This)26 b(pro)s(cedure)f(giv)m(es)h(rise)257 2993 y(to)36 b(time-ordered)h(pro)s(ducts.)55 b(Our)36 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Fd(3)p 2797 4145 31 4 v 2797 4187 a(2)2838 4161 y Fn(\()p Fm(m)p Fn(+)p Fm(m)3044 4137 y Ff(0)3067 4161 y Fn(\))3099 4197 y Fr(,)g(w)m(e)g(ma)m(y)257 4318 y(assume)g(that)f Fp(N)48 b(>)1068 4279 y Fn(3)p 1068 4295 36 4 v 1068 4352 a(2)1113 4318 y Fr(\()p Fp(m)26 b Fr(+)h Fp(m)1450 4282 y Fj(0)1473 4318 y Fr(\).)61 b(The)40 b Fp(N)10 b Fo(\000)p Fr(dep)s(enden)m(t,)43 b(mo)s(di\014ed)c(reduction)h(pro-) 257 4438 y(cedure)35 b(di\013ers)g(from)e(the)h(usual)h(one)e(in)h(the) g(follo)m(wing)g(w)m(a)m(y:)47 b(Before)34 b(eliminating)g(a)257 4559 y(photon)f(from)f(one)h(of)f(the)h(asymptotic)h(states,)g(e.g.)43 b(from)33 b(the)g(incoming)g(state)1448 4831 y Fp( )1515 4790 y Fm(in)1511 4855 y(m)1613 4831 y Fr(=)1747 4723 y Fm(m)1725 4748 y Fi(Y)1717 4930 y Fm(j)t Fn(=1)1874 4805 y Fp(~)1856 4831 y(A)1929 4790 y Fm(in)2000 4831 y Fr([)2020 4804 y Fp(~)2027 4831 y(h)2083 4846 y Fm(j)2120 4831 y Fr(])p Fp(\036)2205 4846 y Fm(g)r(s)2278 4831 y Fp(;)930 b Fr(\(IV.1\))p eop end %%Page: 20 21 TeXDict begin 20 20 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(20)257 573 y(w)m(e)38 b(apply)f(the)g(op)s (erator)f(\()p Fp(H)c Fr(+)25 b Fp(i)p Fr(\))1569 537 y Fm(n)1616 573 y Fr(,)38 b Fp(n)d Fr(=)f([)p Fp(N)i Fo(\000)2137 534 y Fn(3)p 2137 550 36 4 v 2137 607 a(2)2182 573 y Fr(\()p Fp(m)25 b Fr(+)g Fp(m)2516 537 y Fj(0)2540 573 y Fr(\))f(+)h(1])37 b(\(where)g([)p Fo(\001)p Fr(])g(is)g(the)257 693 y(in)m(teger)d(part\))e(to)g(obtain)867 868 y Fm(n)829 892 y Fi(X)826 1074 y Fm(p)p Fn(=1)968 829 y Fi( )1034 908 y Fp(n)1039 1044 y(p)1092 829 y Fi(!)1158 976 y Fr(\()p Fp(E)1268 991 y Fm(g)r(s)1363 976 y Fr(+)22 b Fp(i)p Fr(\))1532 934 y Fm(n)p Fj(\000)p Fm(p)1728 868 y(m)1699 892 y Fi(X)1686 1077 y Fm(l)1707 1086 y Fd(1)1742 1077 y Fn(=1)1890 868 y Fm(m)1862 892 y Fi(X)1849 1077 y Fm(l)1870 1086 y Fd(2)1904 1077 y Fn(=1)2011 976 y Fo(\001)g(\001)g(\001)2197 868 y Fm(m)2169 892 y Fi(X)2155 1077 y Fm(l)2176 1085 y Fe(p)2212 1077 y Fn(=1)2349 868 y Fm(m)2327 892 y 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y(=)83 b(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))980 2536 y Fj(\000)p Fm(n)1140 2469 y(n)1102 2494 y Fi(X)1099 2675 y Fm(p)p Fn(=1)1241 2431 y Fi( )1307 2510 y Fp(n)1311 2645 y(p)1365 2431 y Fi(!)1431 2577 y Fr(\()p Fp(E)1541 2592 y Fm(g)r(s)1636 2577 y Fr(+)g Fp(i)p Fr(\))1805 2536 y Fm(n)p Fj(\000)p Fm(p)2001 2469 y(m)1972 2494 y Fi(X)1959 2679 y Fm(l)1980 2688 y Fd(1)2015 2679 y Fn(=1)2121 2577 y Fo(\001)g(\001)2263 2469 y Fm(m)2235 2494 y Fi(X)2221 2679 y Fm(l)2242 2687 y Fe(p)2278 2679 y Fn(=1)2415 2469 y Fm(m)2393 2494 y Fi(Y)2385 2676 y Fm(j)t Fn(=1)2542 2552 y Fp(~)2524 2577 y(A)2597 2536 y Fm(in)2668 2577 y Fr([)2688 2551 y Fp(~)2695 2577 y(h)2751 2525 y Fn(\()p Fm(l)2799 2534 y Fd(1)2835 2525 y Fm(;l)2876 2534 y Fd(2)2910 2525 y Fm(;)p Fj(\001\001)p Fm(;l)3011 2533 y Fe(p)3045 2525 y Fn(\))2751 2600 y Fm(j)3077 2577 y Fr(])p Fp(\036)3162 2592 y Fm(g)r(s)3235 2577 y Fr(\(IV.4\))662 2883 y(+\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))1056 2842 y Fj(\000)p Fm(n)1158 2883 y Fr(\()p 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b(time-ordered)h(pro)s(duct)f(is)g(giv)m(en)g(b) m(y)672 4337 y Fo(T)752 4286 y Fi(b)752 4296 y Fm(\022)726 4362 y(N)794 4241 y Fi(\020)873 4229 y Fm(m)935 4206 y Ff(0)863 4254 y Fi(Y)860 4439 y Fm(l)q Fn(=1)1007 4312 y Fp(~)988 4337 y(A)q Fr(\()n Fp(~)-47 b(z)1145 4352 y Fm(l)1171 4337 y Fp(;)17 b(t)1250 4352 y Fm(l)1276 4337 y Fr(\))1360 4229 y Fm(m)1339 4254 y Fi(Y)1330 4436 y Fm(j)t Fn(=1)1488 4312 y Fp(~)1469 4337 y(A)q Fr(\()o Fp(~)-48 b(y)1629 4352 y Fm(i)1656 4337 y Fp(;)17 b(s)1746 4352 y Fm(i)1774 4337 y Fr(\))1812 4241 y Fi(\021)1889 4337 y Fr(:=)1243 b(\(IV.6\))838 4658 y(=)28 b Fo(T)1022 4607 y Fi(b)1021 4617 y Fm(\022)1060 4562 y Fi(\020)1140 4550 y Fm(m)1202 4527 y Ff(0)1129 4575 y Fi(Y)1127 4760 y Fm(l)q Fn(=1)1239 4658 y Fo(f)1306 4633 y Fp(~)1289 4658 y(A)o Fr(\()n Fp(~)-47 b(z)1444 4673 y Fm(l)1471 4658 y Fp(;)17 b(t)1550 4673 y Fm(l)1576 4658 y Fr(\))1782 4591 y(1)p 1624 4635 365 4 v 1624 4726 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))1941 4698 y Fm(n)1998 4658 y 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Fm(\034)2243 1059 y Fd(1)2427 1024 y Fr(1)p 2292 1068 318 4 v 2292 1159 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))2620 1091 y Fp(H)2701 1106 y Fm(I)2741 1091 y Fp(e)2786 1050 y Fj(\000)p Fm(iH)2923 1059 y Fd(0)2957 1050 y Fm(\034)2988 1059 y Fd(1)3027 1091 y Fp(d\034)3120 1106 y Fn(1)3160 1091 y Fp(e)3205 1050 y Fm(iH)3287 1059 y Fd(0)3322 1050 y Fm(t)643 1262 y Fo(\001)g(\001)g(\001)g(\001)f(\001)h(\001)g(\001)g(\001)g(\001)f (\001)h(\001)g(\001)g(\001)g(\001)f(\001)h(\001)g(\001)g(\001)g(\001)g (\001)f(\001)h(\001)643 1454 y Fr(+)803 1386 y Fp(i)836 1350 y Fm(n)p Fj(\000)p Fn(1)p 729 1431 V 729 1522 a Fr(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))1073 1337 y Fi(Z)1156 1363 y Fm(t)1119 1525 y Fn(0)1186 1454 y Fr(\()1224 1346 y Fm(n)p Fj(\000)p Fn(2)1237 1371 y Fi(Y)1234 1555 y Fm(l)q Fn(=1)1374 1337 y Fi(Z)1457 1363 y Fm(\034)1488 1375 y Fe(l)1420 1525 y Fn(0)1516 1454 y Fr(\))1571 1346 y Fm(n)p Fj(\000)p Fn(1)1584 1371 y Fi(Y)1576 1553 y Fm(j)t Fn(=1)1720 1358 y Fi(\020)1770 1454 y Fp(e)1815 1413 y Fm(iH)1897 1422 y Fd(0)1932 1413 y Fm(\034)1963 1423 y Fe(n)p Ff(\000)p Fe(j)2230 1386 y Fr(1)p 2096 1431 V 2096 1522 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))2424 1454 y Fp(H)2505 1469 y Fm(I)2545 1454 y Fp(e)2590 1413 y Fj(\000)p Fm(iH)2727 1422 y Fd(0)2761 1413 y Fm(\034)2792 1423 y Fe(n)p Ff(\000)p Fe(j)2915 1358 y Fi(\021)2987 1454 y Fo(\002)1033 1694 y(\002)p Fp(d\034)1203 1709 y Fm(n)p Fj(\000)p Fn(1)1363 1694 y Fo(\001)g(\001)g(\001)g Fp(d\034)1606 1709 y Fn(1)1645 1694 y Fp(e)1690 1653 y Fm(iH)1772 1662 y Fd(0)1807 1653 y Fm(t)643 1908 y Fr(+)p Fp(i)752 1867 y Fm(n)816 1791 y Fi(Z)899 1817 y Fm(t)862 1980 y Fn(0)928 1908 y Fr(\()966 1800 y Fm(n)p Fj(\000)p Fn(1)980 1825 y Fi(Y)976 2010 y Fm(l)q Fn(=1)1116 1791 y Fi(Z)1199 1817 y Fm(\034)1230 1829 y Fe(l)1162 1980 y Fn(0)1259 1908 y Fr(\))p Fp(e)1342 1867 y Fm(iH)5 b(\034)1460 1875 y Fe(n)1651 1841 y Fr(1)p 1517 1885 V 1517 1977 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))1845 1908 y Fp(H)1926 1923 y Fm(I)1965 1908 y Fp(e)2010 1867 y Fj(\000)p Fm(iH)2147 1876 y Fd(0)2182 1867 y Fm(\034)2213 1875 y Fe(n)2282 1908 y Fo(\002)876 b Fr(\(IV.9\))838 2210 y Fo(\002)932 2102 y Fm(n)p Fj(\000)p Fn(1)946 2127 y Fi(Y)938 2309 y Fm(j)t Fn(=1)1082 2114 y Fi(\020)1131 2210 y Fp(e)1176 2169 y Fm(iH)1258 2178 y Fd(0)1293 2169 y Fm(\034)1324 2179 y Fe(n)p Ff(\000)p Fe(j)1592 2143 y Fr(1)p 1458 2187 V 1458 2279 a(\()p Fp(H)29 b Fr(+)22 b Fp(i)p Fr(\))1785 2210 y Fp(H)1866 2225 y Fm(I)1906 2210 y Fp(e)1951 2169 y Fj(\000)p Fm(iH)2088 2178 y Fd(0)2123 2169 y Fm(\034)2154 2179 y Fe(n)p Ff(\000)p Fe(j)2277 2114 y Fi(\021)2327 2210 y Fp(d\034)2420 2225 y Fm(n)2489 2210 y Fo(\001)g(\001)f(\001)h Fp(d\034)2731 2225 y Fn(1)2771 2210 y Fp(e)2816 2169 y Fm(iH)2898 2178 y Fd(0)2933 2169 y Fm(t)2962 2210 y Fp(;)257 2513 y Fr(where)33 b Fp(H)619 2528 y Fn(0)689 2513 y Fr(has)f(b)s(een)g(de\014ned)h(in)e(Eq.)i (\(I.17\),)e(and)g Fp(H)2278 2528 y Fm(I)2346 2513 y Fr(:=)c Fp(H)h Fo(\000)20 b Fp(H)2763 2528 y Fn(0)2802 2513 y Fr(.)43 b(W)-8 b(e)32 b(then)f(apply)257 2634 y(the)i(Neumann)h(series)g(expansion)g(of)e(the)h(resolv)m(en)m(t)1058 2853 y(1)p 924 2897 V 924 2988 a(\()p Fp(H)c Fr(+)22 b Fp(i)p Fr(\))1335 2920 y(=)1494 2812 y Fm(n)p Fj(\000)p Fn(1)1500 2837 y Fi(X)1499 3019 y Fm(j)t Fn(=0)1643 2824 y Fi(h)1843 2853 y Fr(1)p 1692 2897 350 4 v 1692 2988 a(\()p Fp(H)1811 3003 y Fn(0)1873 2988 y Fr(+)g Fp(i)p Fr(\))2052 2824 y Fi(\020)2124 2920 y Fo(\000)g Fp(H)2304 2935 y Fm(I)2504 2853 y Fr(1)p 2354 2897 V 2354 2988 a(\()p Fp(H)2473 3003 y Fn(0)2534 2988 y Fr(+)g Fp(i)p Fr(\))2713 2824 y Fi(\021)2763 2847 y Fm(j)2800 2824 y Fi(i)3186 2920 y Fr(\(IV.10\))1494 3205 y(+)1730 3138 y(1)p 1580 3182 V 1580 3274 a(\()p Fp(H)1699 3289 y Fn(0)1760 3274 y Fr(+)g Fp(i)p Fr(\))1939 3109 y Fi(\020)2011 3205 y Fo(\000)g Fp(H)2191 3220 y Fm(I)2375 3138 y Fr(1)p 2241 3182 318 4 v 2241 3274 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))2569 3109 y Fi(\021)2618 3132 y Fm(n)2665 3205 y Fp(;)494 b Fr(\(IV.11\))257 3476 y(and)33 b(w)m(e)h(exploit)f (the)g Fp(\013)q Fr(-indep)s(endence)i(on)d(the)h(norm)g(b)s(ounds)g (of)f(the)h(op)s(erators)526 3735 y Fp(\013)589 3693 y Fj(\000)654 3666 y Fd(3)p 654 3678 31 4 v 654 3720 a(2)886 3667 y Fr(1)p 708 3711 405 4 v 708 3817 a(\()p Fp(H)827 3832 y Fn(0)889 3817 y Fr(+)22 b Fp(i)p Fr(\))1068 3753 y Fd(1)p 1068 3765 31 4 v 1068 3807 a(2)1122 3735 y Fp(H)1203 3750 y Fm(I)1431 3667 y Fr(1)p 1253 3711 405 4 v 1253 3817 a(\()p Fp(H)1372 3832 y Fn(0)1433 3817 y Fr(+)g Fp(i)p Fr(\))1612 3753 y Fd(1)p 1613 3765 31 4 v 1613 3807 a(2)1667 3735 y Fp(;)114 b(\013)1871 3693 y Fj(\000)1936 3666 y Fd(3)p 1936 3678 V 1936 3720 a(2)2152 3667 y Fr(1)p 1990 3711 373 4 v 1990 3817 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))2318 3753 y Fd(1)p 2318 3765 31 4 v 2318 3807 a(2)2373 3735 y Fp(H)2454 3750 y Fm(I)2681 3667 y Fr(1)p 2503 3711 405 4 v 2503 3817 a(\()p Fp(H)2622 3832 y Fn(0)2684 3817 y Fr(+)g Fp(i)p Fr(\))2863 3753 y Fd(1)p 2863 3765 31 4 v 2863 3807 a(2)3186 3735 y Fr(\(IV.12\))342 3965 y Fi(Z)460 4057 y Fp(~)442 4082 y(A)p Fr(\()n Fp(~)-47 b(z)598 4097 y Fm(l)624 4082 y Fp(;)17 b Fr(0\))p 755 3975 314 4 v 769 4055 a Fp(~)755 4082 y(f)803 4097 y Fm(l)828 4082 y Fr(\()n Fp(~)-47 b(z)911 4097 y Fm(l)938 4082 y Fp(;)17 b Fr(0\))o Fp(d)1119 4041 y Fn(3)1158 4082 y Fp(z)1203 4097 y Fm(l)1417 4014 y Fr(1)p 1240 4059 405 4 v 1240 4165 a(\()p Fp(H)1359 4180 y Fn(0)1420 4165 y Fr(+)22 b Fp(i)p Fr(\))1599 4101 y Fd(1)p 1599 4113 31 4 v 1599 4154 a(2)1654 4082 y Fp(;)1795 3965 y Fi(Z)1913 4057 y Fp(~)1894 4082 y(A)q Fr(\()n Fp(~)-47 b(z)2051 4097 y Fm(l)2077 4082 y Fp(;)17 b Fr(0\))p 2208 3975 314 4 v 2222 4055 a Fp(~)2208 4082 y(f)2256 4097 y Fm(l)2281 4082 y Fr(\()n Fp(~)-47 b(z)2364 4097 y Fm(l)2391 4082 y Fp(;)17 b Fr(0\))o Fp(d)2572 4041 y Fn(3)2611 4082 y Fp(z)2656 4097 y Fm(l)2854 4014 y Fr(1)p 2692 4059 373 4 v 2692 4165 a(\()p Fp(H)30 b Fr(+)22 b Fp(i)p Fr(\))3020 4101 y Fd(1)p 3020 4113 31 4 v 3020 4154 a(2)3075 4082 y Fp(:)84 b Fr(\(IV.13\))257 4321 y(Using)31 b(the)f(fact)f(that)h (the)g(time-in)m(tegrations)h(extend)g(o)m(v)m(er)g(in)m(terv)-5 b(als)30 b(of)g(length)g(pro-)257 4441 y(p)s(ortional)25 b(to)g Fp(\013)845 4405 y Fj(\000)p Fm(\017)932 4441 y Fr(,)i(w)m(e)f(conclude)h(that)e(the)g(remainder)i(term)e(\(IV.9\))h (in)f(the)h(Duhamel)257 4570 y(expansion)i(is)e(b)s(ounded)h(b)m(y)f Fp(const)8 b Fo(\001)g Fp(\013)1650 4534 y Fm(n)p Fn(\()1730 4507 y Fd(3)p 1731 4519 V 1731 4560 a(2)1771 4534 y Fj(\000)p Fm(\017)p Fn(\))1886 4570 y Fr(.)42 b(Therefore)27 b(the)f(op)s(erator) f(\(IV.6\))h(can)g(b)s(e)257 4691 y(appro)m(ximated,)g(up)d(to)f(an)h (error)f(term)h(of)f Fp(o)p Fr(\()p Fp(\013)1960 4655 y Fm(N)2027 4691 y Fr(\),)j(b)m(y)e(\014nitely)h(man)m(y)f(expressions) j(only)257 4811 y(in)m(v)m(olving)41 b(the)e(propagator)f(and)h(the)h (resolv)m(en)m(t)h(of)d(the)h(Hamiltonian)h Fp(H)3087 4826 y Fn(0)3126 4811 y Fr(,)g(b)s(esides)257 4932 y(the)31 b(groundstate)h Fp(\036)1017 4947 y Fm(g)r(s)1120 4932 y Fr(and)f(the)g(groundstate)g(energy)h Fp(E)2391 4947 y Fm(g)r(s)2463 4932 y Fr(.)43 b(These)33 b(latter)d(quan)m(tities)257 5052 y(can)35 b(b)s(e)g(calculated)h(using)g(the)f(algorithm)f(dev)m (elop)s(ed)j(in)e([2])g(and)g(outlined)g(in)g(next)257 5172 y(section.)45 b(W)-8 b(e)33 b(can)g(therefore)g(state)g(the)g (main)g(result)g(of)f(the)h(pap)s(er:)p eop end %%Page: 22 23 TeXDict begin 22 22 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(22)257 573 y Fg(Theorem)43 b(IV.1.)g Fk(F)-7 b(or)38 b Fp(\013)d Fo(\024)45 b Fr(\026)-58 b Fp(\013)p Fk(,)40 b(with)48 b Fr(\026)-59 b Fp(\013)35 b Fo(\021)45 b Fr(\026)-58 b Fp(\013)2040 588 y Fm(N)2145 573 y Fk(smal)5 b(l)38 b(enough,)h(the)g(S-matrix)f(ele-)257 693 y(ments)d Fp(S)610 657 y Fm(m)672 634 y Ff(0)694 657 y Fm(;m)604 718 y(\013)781 693 y Fr(\()p Fo(f)883 667 y Fp(~)869 693 y(f)917 708 y Fm(i)945 693 y Fo(g)p Fp(;)17 b Fo(f)1082 667 y Fp(~)1089 693 y(h)1145 708 y Fm(j)1181 693 y Fo(g)p Fr(\))p Fk(,)34 b(wher)-5 b(e)34 b Fr(\()p Fp(f)1694 708 y Fm(i)1722 693 y Fp(;)17 b(h)1822 708 y Fm(j)1859 693 y Fr(\))27 b(=)h(0)p Fk(,)35 b(have)f(exp)-5 b(ansions)33 b(of)h(the)h(form)460 962 y Fp(S)526 921 y Fm(m)588 898 y Ff(0)611 921 y Fm(;m)520 987 y(\013)697 962 y Fr(\()p Fo(f)799 936 y Fp(~)785 962 y(f)833 977 y Fm(i)861 962 y Fo(g)p Fp(;)17 b Fo(f)998 936 y Fp(~)1005 962 y(h)1061 977 y Fm(j)1097 962 y Fo(g)p Fr(\))1185 921 y Fm(conn)1367 962 y Fr(=)1481 854 y Fn(2)p Fm(N)1471 879 y Fi(X)1471 1064 y Fm(`)p Fn(=6)1607 962 y Fp(S)1673 915 y Fm(m)1735 892 y Ff(0)1758 915 y Fm(;m)1667 987 y(l)1844 962 y Fr(\()p Fo(f)1946 936 y Fp(~)1932 962 y(f)1980 977 y Fm(i)2008 962 y Fo(g)p Fp(;)g Fo(f)2145 936 y Fp(~)2152 962 y(h)2208 977 y Fm(j)2244 962 y Fo(g)p Fr(;)g Fp(\013)q Fr(\))p Fp(\013)2514 894 y Fe(`)p 2511 906 31 4 v 2511 947 a Fd(2)2605 962 y Fr(+)50 b Fp(o)p Fr(\()p Fp(\013)2879 921 y Fm(N)2945 962 y Fr(\))203 b(\(IV.14\))257 1221 y Fk(with)464 1413 y Fr(lim)456 1469 y Fm(\013)p Fj(!)p Fn(0)624 1413 y Fp(\013)687 1372 y Fm(\016)725 1413 y Fo(j)p Fp(S)819 1365 y Fm(m)881 1342 y Ff(0)903 1365 y Fm(;m)813 1438 y(l)989 1413 y Fr(\()p Fo(f)1091 1386 y Fp(~)1077 1413 y(f)1125 1428 y Fm(i)1153 1413 y Fo(g)p Fp(;)17 b Fo(f)1290 1386 y Fp(~)1297 1413 y(h)1353 1428 y Fm(j)1389 1413 y Fo(g)p Fr(;)g Fp(\013)q Fr(\))27 b(=)g(0)199 b Fp(;)117 b Fk(for)34 b(arbitr)-5 b(ary)100 b Fp(\016)32 b(>)c Fr(0)o Fp(;)199 b Fr(\(IV.15\))257 1656 y Fk(for)41 b Fp(N)50 b Fr(=)40 b(3)p Fp(;)17 b Fr(4)p Fp(;)g Fr(5)p Fp(;)g(:)g(:)g(:)m Fk(.)64 b(The)41 b(c)-5 b(o)g(e\016cients)40 b Fp(S)1923 1608 y Fm(m)1985 1585 y Ff(0)2008 1608 y Fm(;m)1917 1681 y(l)2094 1656 y Fr(\()p Fo(f)2196 1629 y Fp(~)2182 1656 y(f)2230 1671 y Fm(i)2258 1656 y Fo(g)p Fp(;)17 b Fo(f)2395 1629 y Fp(~)2402 1656 y(h)2458 1671 y Fm(j)2494 1656 y Fo(g)p Fr(;)g Fp(\013)q Fr(\))40 b Fk(ar)-5 b(e)41 b(c)-5 b(omputable)41 b(in)257 1776 y(terms)32 b(of)f(\014nitely)g (many)h(c)-5 b(onver)g(gent)30 b(inte)-5 b(gr)g(als,)32 b(for)f(arbitr)-5 b(ary)32 b Fp(l)e(<)d Fo(1)k Fk(\(with)h Fp(l)e Fo(\025)e Fr(6)p Fk(\).)257 2061 y Fa(IV.2)135 b(Expansion)65 b(of)f(the)g(groundstate)i(and)d(the)i(ground-)648 2210 y(state)46 b(energy)257 2395 y Fr(The)33 b(\014nal)e(and)g(tec)m (hnically)j(most)e(subtle)g(step)g(in)g(the)g(calculation)f(of)g(the)h (S-matrix)257 2515 y(elemen)m(ts)47 b(concerns)e(the)g(calculation)f (of)g(the)h(groundstate)f(and)g(the)h(groundstate)257 2636 y(energy)-8 b(.)58 b(Because)39 b(of)d(infrared)h(div)m(ergences,) 42 b(whic)m(h)c(in)m(v)-5 b(alidate)38 b(a)f(straigh)m(tforw)m(ard)257 2756 y(T)-8 b(a)m(ylor)49 b(expansion,)55 b(an)48 b(iterativ)m(e)i (construction)f(m)m(ust)h(b)s(e)f(emplo)m(y)m(ed)h(to)e(remo)m(v)m(e) 257 2876 y(an)43 b(infrared)g(cuto\013)f(in)h(photon)g(momen)m(tum)h (space)f(and)g(to)f(devise)j(a)d(con)m(v)m(ergen)m(t)257 2997 y(algorithm.)76 b(Suc)m(h)45 b(a)e(construction)i(has)e(b)s(een)i (dev)m(elop)s(ed)g(in)f([2])f(on)g(the)h(basis)h(of)257 3117 y(results)31 b(in)f([10])g(and)f([1].)43 b(In)30 b(the)g(follo)m(wing,)h(w)m(e)f(describ)s(e)i(the)e(results)h(of)e([2]) h(without)257 3238 y(pro)m(viding)43 b(pro)s(ofs;)i(but)c(see)i([1],)g ([2])e(and)h([10].)69 b(The)42 b(main)g(ideas)g(underlying)h(the)257 3358 y(construction)h(of)e(the)h(groundstate)g(will)h(b)s(e)f (discussed,)48 b(and)42 b(the)h(strategy)h(of)e(the)257 3478 y(re-expansion)34 b(will)g(b)s(e)e(outlined.)257 3734 y Fg(IV.2.1)112 b(Notation)257 3919 y Fr(In)33 b(the)g(follo)m (wing)g(of)f(this)h(Section,)h(w)m(e)g(simplify)g(our)e(notation)g(b)m (y)i(setting)257 4135 y Fp(k)52 b Fr(:=)c(\()544 4109 y Fp(~)549 4135 y(k)s(;)17 b(\025)p Fr(\))k Fp(;)111 b(!)t Fr(\()p Fp(k)s Fr(\))48 b Fo(\021)h(j)p Fp(k)s Fo(j)f Fr(:=)g Fo(j)1602 4109 y Fp(~)1607 4135 y(k)s Fo(j)21 b Fp(;)111 b Fr(and)2116 4018 y Fi(Z)2216 4135 y Fp(f)11 b Fr(\()p Fp(k)s Fr(\))20 b Fp(dk)52 b Fr(:=)2746 4052 y Fi(X)2730 4236 y Fm(\025)p Fn(=)p Fj(\006)2898 4018 y Fi(Z)2997 4135 y Fp(f)11 b Fr(\()3089 4109 y Fp(~)3094 4135 y(k)s(;)17 b(\025)p Fr(\))g Fp(d)3355 4094 y Fn(3)3393 4135 y Fp(k)24 b(;)257 4406 y Fr(for)g(an)m(y)h(in)m(tegrable)g (functions)g Fp(f)11 b Fr(\()p Fo(\001)p Fp(;)17 b(\025)p Fr(\),)p Fp(\025)27 b Fr(=)g Fo(\006)p Fr(.)42 b(Giv)m(en)25 b(an)f(op)s(erator-v)-5 b(alued)24 b(function)257 4526 y Fp(F)42 b Fr(:)28 b Fl(R)489 4490 y Fn(3)550 4526 y Fo(\002)23 b Fl(Z)716 4541 y Fn(2)783 4526 y Fo(!)k(B)s Fr(\()p Fo(H)1100 4541 y Fm(el)1160 4526 y Fr(\),)32 b(w)m(e)i(write)1230 4748 y Fp(a)1281 4707 y Fj(\003)1321 4748 y Fr(\()p Fp(F)14 b Fr(\))82 b(:=)1742 4631 y Fi(Z)1842 4748 y Fp(F)14 b Fr(\()p Fp(k)s Fr(\))22 b Fo(\012)g Fp(a)2221 4707 y Fj(\003)2261 4748 y Fr(\()p Fp(k)s Fr(\))p Fp(dk)693 b Fr(\(IV.16\))1269 4966 y Fp(a)p Fr(\()p Fp(F)14 b Fr(\))83 b(:=)1742 4849 y Fi(Z)1842 4966 y Fp(F)14 b Fr(\()p Fp(k)s Fr(\))2049 4925 y Fj(\003)2110 4966 y Fo(\012)23 b Fp(a)p Fr(\()p Fp(k)s Fr(\))p Fp(dk)s(:)663 b Fr(\(IV.17\))257 5187 y(This)34 b(allo)m(ws)g(us)f(to)f(write)h(the)g (v)m(elo)s(cit)m(y)i(op)s(erator)28 b Fp(~)-45 b(v)36 b Fr(\(rescaled)e(b)m(y)f(2\))g(as)1215 5388 y Fp(~)-45 b(v)64 b Fr(:=)c Fo(\000)p Fp(i)1612 5363 y(~)1603 5388 y Fo(r)1686 5403 y Fm(x)1769 5388 y Fr(+)39 b Fp(a)1935 5347 y Fj(\003)1974 5388 y Fr(\()2027 5363 y Fp(~)2012 5388 y(G)p Fr(\))g(+)g Fp(a)p Fr(\()2385 5363 y Fp(~)2370 5388 y(G)p Fr(\))22 b Fp(;)652 b Fr(\(IV.18\))p eop end %%Page: 23 24 TeXDict begin 23 23 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(23)257 573 y(where)554 548 y Fp(~)539 573 y(G)28 b Fr(:)g Fl(R)771 537 y Fn(3)832 573 y Fo(\002)23 b Fl(Z)998 588 y Fn(2)1065 573 y Fo(!)28 b(B)s Fr(\()p Fo(H)1383 588 y Fm(el)1442 573 y Fr(\))1480 537 y Fn(3)1552 573 y Fr(are)k(the)h(m)m(ultiplication)h(op)s(erators)f (de\014ned)h(b)m(y)1082 823 y Fp(~)1068 848 y(G)p Fr(\()p Fp(k)s Fr(\))60 b(:=)1568 780 y Fp(\013)1631 744 y Fn(3)p Fm(=)p Fn(2)p 1508 825 294 4 v 1508 916 a Fr(\(2)p Fp(\031)t Fr(\))1692 887 y Fn(3)p Fm(=)p Fn(2)1887 780 y Fr(\003\()p Fp(k)s Fr(\))p 1849 825 275 4 v 1849 841 a Fi(q)p 1932 841 192 4 v 100 x Fr(2)17 b Fo(j)p Fp(k)s Fo(j)2161 848 y Fp(e)2206 807 y Fj(\000)p Fm(i\013)2327 789 y(~)2330 807 y(k)r Fj(\001)o Fm(~)-34 b(x)2459 848 y Fp(~)-48 b(")p Fr(\()p Fp(k)s Fr(\))22 b Fp(:)501 b Fr(\(IV.19\))257 1153 y(In)33 b(terms)h(of)e(the)h(v)m(elo)s(cit)m(y)h(op)s(erator,)e (the)h(Hamiltonian)g(assumes)i(the)e(simple)h(form)1366 1362 y Fp(H)68 b Fr(=)56 b Fp(~)-45 b(v)1713 1321 y Fn(2)1791 1362 y Fo(\000)40 b Fp(V)21 b Fr(\()p Fp(x)p Fr(\))39 b(+)2296 1337 y(\024)2271 1362 y Fp(H)7 b(:)800 b Fr(\(IV.20\))257 1571 y(W)-8 b(e)33 b(de\014ne)h(a)e(decreasing)i(sequence,)i(\()p Fp(\033)1785 1586 y Fm(n)1832 1571 y Fr(\))1870 1534 y Fj(1)1870 1595 y Fm(n)p Fn(=0)2007 1571 y Fr(,)d(of)f(energy)i (scales)g(b)m(y)f(setting)1595 1779 y Fp(\033)1650 1794 y Fm(n)1757 1779 y Fr(:=)61 b Fp(\024)22 b(\013)2062 1738 y Fm(n)2131 1779 y Fp(:)1028 b Fr(\(IV.21\))257 1988 y(Note,)36 b(in)g(passing,)g(that,)g(for)f(all)g Fp(n)d Fo(2)g Fl(N)1804 2003 y Fn(0)1844 1988 y Fr(,)k(w)m(e)g(ha)m(v)m (e)g(that)f Fp(\033)2549 2003 y Fm(n)p Fn(+1)2719 1988 y Fo(\024)d Fp(\024\013)h(<)f Fr(1.)51 b(T)-8 b(o)35 b(cut)257 2108 y(the)43 b(in)m(teraction)g(Hamiltonian)f(in)m(to)g (slices)i(corresp)s(onding)f(to)f(ev)m(er)h(lo)m(w)m(er)g(energy)257 2229 y(scales,)35 b(w)m(e)e(mak)m(e)h(use)f(of)f(the)h(op)s(erators)318 2412 y Fp(~)303 2438 y(G)380 2453 y Fm(n)427 2438 y Fr(\()p Fp(k)s Fr(\))60 b(:=)g Fg(1)836 2341 y Fi(\020)886 2438 y Fp(\033)941 2453 y Fm(n)1016 2438 y Fo(\024)28 b(j)p Fp(k)s Fo(j)1231 2341 y Fi(\021)1317 2412 y Fp(~)1302 2438 y(G)p Fr(\()p Fp(k)s Fr(\))22 b Fp(;)112 b Fr(and)1936 2412 y Fp(~)1921 2438 y(G)1998 2396 y Fm(m)1998 2462 y(n)2065 2438 y Fr(\()p Fp(k)s Fr(\))60 b(:=)g Fg(1)2474 2341 y Fi(\020)2524 2438 y Fp(\033)2579 2453 y Fm(n)2654 2438 y Fo(\024)28 b(j)p Fp(k)s Fo(j)f Fp(<)h(\033)3055 2453 y Fm(m)3122 2341 y Fi(\021)3208 2412 y Fp(~)3193 2438 y(G)p Fr(\()p Fp(k)s Fr(\))22 b Fp(;)3186 2570 y Fr(\(IV.22\))257 2700 y(for)35 b(all)f Fp(k)h Fo(2)d Fl(R)802 2664 y Fn(3)865 2700 y Fo(\002)24 b Fl(Z)1032 2715 y Fn(2)1106 2700 y Fr(and)35 b Fp(m;)17 b(n)32 b Fo(2)g Fl(N)1687 2715 y Fn(0)1726 2700 y Fr(,)k(with)f Fp(m)d(<)f(n)p Fr(.)51 b(Note)34 b(that)2839 2675 y Fp(~)2824 2700 y(G)d Fr(=)3040 2633 y Fi(P)3127 2660 y Fj(1)3127 2724 y Fm(n)p Fn(=0)3296 2675 y Fp(~)3281 2700 y(G)3358 2664 y Fm(n)3358 2724 y(n)p Fn(+1)257 2836 y Fr(and)e(that)666 2811 y Fp(~)651 2836 y(G)728 2851 y Fm(n)804 2836 y Fr(is)g(the)h (coupling)f(function)h(of)e(the)i(in)m(teraction)g(Hamiltonian)f (cuto\013)g(in)257 2956 y(the)34 b(infrared)e(region,)h Fo(j)p Fp(k)s Fo(j)28 b(\024)g Fp(\033)1409 2971 y Fm(n)1457 2956 y Fr(.)44 b(W)-8 b(e)33 b(factorize)g(F)-8 b(o)s(c)m(k)33 b(space)g Fo(F)38 b Fr(=)28 b Fo(F)10 b Fr(\()p Fh(h)p Fr(\))32 b(in)m(to)h(tensor)257 3077 y(pro)s(ducts)e(b)m(y)g(in)m(tro)s (ducing)f(suitable)h(subspaces)h(corresp)s(onding)f(to)e(the)h (one-photon)257 3197 y(Hilb)s(ert)j(space)h(corresp)s(onding)g(to)e (di\013eren)m(t)i(energy)f(scales:)1029 3406 y Fh(h)1081 3421 y Fm(n)1188 3406 y Fr(:=)60 b Fp(L)1417 3365 y Fn(2)1457 3309 y Fi(h)1496 3406 y Fo(K)1572 3421 y Fm(n)1619 3309 y Fi(i)1753 3406 y Fr(and)94 b Fh(h)2056 3365 y Fm(m)2056 3430 y(n)2183 3406 y Fr(:=)60 b Fp(L)2412 3365 y Fn(2)2452 3309 y Fi(h)2491 3406 y Fo(K)2568 3365 y Fm(m)2567 3430 y(n)2635 3309 y Fi(i)2697 3406 y Fp(;)462 b Fr(\(IV.23\))257 3614 y(where,)498 3823 y(for)32 b(0)c Fo(\024)g Fp(n)g Fo(\024)g(1)p Fr(,)93 b Fo(K)1316 3838 y Fm(n)1457 3823 y Fr(:=)1636 3727 y Fi(n)1692 3823 y Fr(\()1725 3797 y Fp(~)1730 3823 y(k)s(;)17 b(\034)11 b Fr(\))28 b Fo(2)g Fl(R)2113 3782 y Fn(3)2174 3823 y Fo(\002)23 b Fl(Z)2340 3838 y Fn(2)2407 3724 y Fi(\014)2407 3773 y(\014)2407 3823 y(\014)33 b Fp(\033)2523 3838 y Fm(n)2598 3823 y Fo(\024)28 b Fp(!)t Fr(\()2801 3797 y Fp(~)2806 3823 y(k)r Fr(\))2897 3727 y Fi(o)2975 3823 y Fp(;)184 b Fr(\(IV.24\))282 4006 y(for)32 b(0)27 b Fo(\024)h Fp(m)g(<)g(n)g Fo(\024)g(1)p Fr(,)83 b Fo(K)1307 3970 y Fm(m)1306 4030 y(n)1457 4006 y Fr(:=)1636 3910 y Fi(n)1692 4006 y Fr(\()1725 3980 y Fp(~)1730 4006 y(k)s(;)17 b(\034)11 b Fr(\))28 b Fo(2)g Fl(R)2113 3965 y Fn(3)2174 4006 y Fo(\002)23 b Fl(Z)2340 4021 y Fn(2)2407 3906 y Fi(\014)2407 3956 y(\014)2407 4006 y(\014)33 b Fp(\033)2523 4021 y Fm(n)2598 4006 y Fo(\024)28 b Fp(!)t Fr(\()2801 3980 y Fp(~)2806 4006 y(k)r Fr(\))g Fp(<)f(\033)3083 4021 y Fm(m)3150 3910 y Fi(o)3206 4006 y Fp(:)22 b(:)-96 b Fr(\(IV.25\))257 4231 y(Note)31 b(that)g Fo(K)778 4195 y Fn(0)777 4256 y Fm(n)852 4231 y Fo(\032)d(K)1033 4246 y Fm(n)1110 4231 y Fr(is)k(a)e(prop)s(er)h(subset.)45 b(F)-8 b(or)30 b(in)m(tegers)i(1) 27 b Fo(\024)i Fp(m)f(<)f(n)h(<)g(`)f Fo(\024)h(1)p Fr(,)j(w)m(e)257 4352 y(ha)m(v)m(e)38 b(the)e(disjoin)m(t)h(decomp)s(osition)g Fo(K)1730 4367 y Fm(`)1797 4352 y Fr(=)c Fo(K)1982 4367 y Fm(m)2074 4352 y Fo([)24 b(K)2241 4315 y Fm(m)2240 4376 y(n)2333 4352 y Fo([)h(K)2501 4315 y Fm(n)2500 4376 y(`)2548 4352 y Fr(,)37 b(and)f(hence)i(the)e(direct)257 4472 y(sum)1421 4592 y Fh(h)1473 4607 y Fm(`)1566 4565 y Fo(\030)1566 4597 y Fr(=)1703 4592 y Fh(h)1755 4607 y Fm(m)1844 4592 y Fo(\010)23 b Fh(h)1996 4551 y Fm(m)1996 4617 y(n)2084 4592 y Fo(\010)g Fh(h)2236 4551 y Fm(n)2236 4617 y(`)2305 4592 y Fp(;)854 b Fr(\(IV.26\))257 4762 y(whic)m(h)34 b(giv)m(es)g(rise)g(to)e(the)h(isomorphism)1371 4971 y Fo(F)1443 4986 y Fm(`)1536 4943 y Fo(\030)1537 4975 y Fr(=)1674 4971 y Fo(F)1746 4986 y Fm(m)1834 4971 y Fo(\012)23 b(F)2016 4930 y Fm(m)2006 4995 y(n)2104 4971 y Fo(\012)g(F)2286 4930 y Fm(n)2276 4995 y(`)2354 4971 y Fp(;)805 b Fr(\(IV.27\))257 5179 y(with)34 b Fo(F)552 5194 y Fm(n)626 5179 y Fr(:=)27 b Fo(F)10 b Fr(\()p Fh(h)928 5194 y Fm(n)975 5179 y Fr(\))32 b(and)h Fo(F)1317 5143 y Fm(m)1307 5204 y(n)1411 5179 y Fr(:=)27 b Fo(F)10 b Fr(\()p Fh(h)1713 5143 y Fm(m)1713 5204 y(n)1779 5179 y Fr(\).)44 b(In)33 b(particular,)f(for)g(an)m(y)i Fp(n)28 b Fo(2)g Fl(N)p Fr(,)1146 5388 y Fo(F)70 b Fr(=)60 b Fo(F)1496 5403 y Fj(1)1630 5360 y Fo(\030)1631 5392 y Fr(=)1768 5388 y Fo(F)1840 5403 y Fm(n)1909 5388 y Fo(\012)22 b(F)2090 5347 y Fm(n)2080 5413 y(n)p Fn(+1)2239 5388 y Fo(\012)h(F)2421 5347 y Fm(n)p Fn(+1)2411 5413 y Fj(1)2580 5388 y Fp(:)579 b Fr(\(IV.28\))p eop end %%Page: 24 25 TeXDict begin 24 24 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(24)257 573 y(W)-8 b(e)33 b(set)927 693 y Fo(H)1011 708 y Fm(n)1118 693 y Fr(:=)60 b Fo(H)1365 708 y Fm(el)1446 693 y Fo(\012)23 b(F)1618 708 y Fm(n)1759 693 y Fr(and)95 b Fo(H)2096 652 y Fm(m)2095 718 y(n)2223 693 y Fr(:=)60 b Fo(H)2470 708 y Fm(el)2551 693 y Fo(\012)22 b(F)2732 652 y Fm(m)2722 718 y(n)2799 693 y Fp(:)360 b Fr(\(IV.29\))257 856 y(F)-8 b(or)47 b(energy-scale)i(indices)g Fp(m;)17 b(n)53 b Fo(2)g Fl(N)1773 871 y Fn(0)1812 856 y Fr(,)e(with)d Fp(m)53 b(<)g(n)p Fr(,)e(w)m(e)d(de\014ne)h(the)e(v)m (elo)s(cit)m(y)257 977 y(op)s(erator)27 b Fp(~)-45 b(v)696 992 y Fm(n)743 977 y Fr(,)31 b(the)h(\014eld-energy)g(op)s(erators)1945 951 y(\024)1919 977 y Fp(H)2000 992 y Fm(n)2047 977 y Fr(,)2131 951 y(\024)2106 977 y Fp(H)2195 940 y Fm(m)2187 1001 y(n)2261 977 y Fr(,)f(and)g(the)h(Hamiltonian)f Fp(H)3314 992 y Fm(n)3392 977 y Fr(b)m(y)890 1179 y Fp(~)-45 b(v)941 1194 y Fm(n)1071 1179 y Fr(:=)83 b Fo(\000)p Fp(i)1376 1154 y(~)1367 1179 y Fo(r)1450 1194 y Fm(x)1534 1179 y Fr(+)38 b Fp(a)1699 1138 y Fj(\003)1739 1179 y Fr(\()1792 1154 y Fp(~)1777 1179 y(G)1854 1194 y Fm(n)1901 1179 y Fr(\))g(+)h Fp(a)p Fr(\()2196 1154 y Fp(~)2181 1179 y(G)2258 1194 y Fm(n)2305 1179 y Fr(\))22 b Fp(;)794 b Fr(\(IV.30\))886 1329 y(\024)860 1354 y Fp(H)941 1369 y Fm(n)1071 1354 y Fr(:=)1257 1237 y Fi(Z)1357 1354 y Fg(1)p Fr(\()p Fp(\033)1506 1369 y Fm(n)1581 1354 y Fo(\024)28 b(j)p Fp(k)s Fo(j)p Fr(\))21 b Fp(!)t Fr(\()p Fp(k)s Fr(\))h Fp(a)2123 1313 y Fj(\003)2163 1354 y Fr(\()p Fp(k)s Fr(\))p Fp(a)p Fr(\()p Fp(k)s Fr(\))g Fp(dk)j(;)536 b Fr(\(IV.31\))858 1547 y(\024)833 1572 y Fp(H)922 1531 y Fm(m)914 1597 y(n)1071 1572 y Fr(:=)1257 1455 y Fi(Z)1357 1572 y Fg(1)p Fr(\()p Fp(\033)1506 1587 y Fm(n)1581 1572 y Fo(\024)28 b(j)p Fp(k)s Fo(j)f Fp(<)h(\033)1982 1587 y Fm(m)2049 1572 y Fr(\))22 b Fp(!)t Fr(\()p Fp(k)s Fr(\))f Fp(a)2376 1531 y Fj(\003)2416 1572 y Fr(\()p Fp(k)s Fr(\))p Fp(a)p Fr(\()p Fp(k)s Fr(\))h Fp(dk)j(;)283 b Fr(\(IV.32\))860 1760 y Fp(H)941 1775 y Fm(n)1071 1760 y Fr(:=)79 b Fp(~)-45 b(v)1319 1719 y Fn(2)1304 1785 y Fm(n)1398 1760 y Fo(\000)39 b Fp(V)22 b Fr(\()p Fp(x)p Fr(\))39 b(+)1903 1735 y(\024)1877 1760 y Fp(H)1958 1775 y Fm(n)2027 1760 y Fp(;)1132 b Fr(\(IV.33\))257 1954 y(as)45 b(op)s(erators)g(on)g Fo(H)1065 1969 y Fm(n)1157 1954 y Fr(and)g Fo(H)1444 1918 y Fm(m)1443 1978 y(n)1511 1954 y Fr(,)j(resp)s(ectiv)m(ely)-8 b(.)83 b(W)-8 b(e)45 b(in)m(tro)s(duce)h(the)g(groundstate)257 2074 y(energy)34 b(at)e(scale)i Fp(n)f Fr(and)f(groundstate)h(energy)h (di\013erences)935 2268 y Fp(E)1007 2283 y Fm(n)1114 2268 y Fr(:=)60 b(inf)24 b Fp(\033)t Fr(\()p Fp(H)1590 2283 y Fm(n)1637 2268 y Fr(\))94 b(and)h Fp(E)2099 2227 y Fm(m)2093 2292 y(n)2226 2268 y Fr(:=)60 b Fp(E)2461 2283 y Fm(m)2550 2268 y Fo(\000)22 b Fp(E)2721 2283 y Fm(n)2791 2268 y Fp(:)368 b Fr(\(IV.34\))257 2461 y(T)-8 b(o)42 b(compare)g(Hamiltonians,)j Fp(H)1528 2476 y Fm(n)1616 2461 y Fr(and)d Fp(H)1896 2476 y Fm(n)p Fn(+1)2033 2461 y Fr(,)i(at)d(successiv)m(e)k(energy)e(scales,)i(it)d(is)257 2582 y(con)m(v)m(enien)m(t)31 b(to)d(de\014ne)h(p)s(ositiv)m(e)h(op)s (erators)d Fp(H)1999 2546 y Fn(+)1991 2606 y Fm(n)2086 2582 y Fr(and)2279 2549 y Fi(f)2271 2582 y Fp(H)2360 2546 y Fn(+)2352 2606 y Fm(n)2447 2582 y Fr(on)h Fo(H)2662 2597 y Fm(n)2737 2582 y Fr(and)g Fo(H)3006 2597 y Fm(n)p Fn(+1)3143 2582 y Fr(,)h(resp)s(ec-)257 2702 y(tiv)m(ely)-8 b(,)35 b(b)m(y)1109 2896 y Fp(H)1198 2855 y Fn(+)1190 2920 y Fm(n)1339 2896 y Fr(:=)83 b Fp(H)1606 2911 y Fm(n)1692 2896 y Fo(\000)39 b Fp(E)1880 2911 y Fm(n)1949 2896 y Fp(;)1210 b Fr(\(IV.35\))1117 3009 y Fi(f)1109 3041 y Fp(H)1198 3000 y Fn(+)1190 3066 y Fm(n)1339 3041 y Fr(:=)83 b Fp(H)1614 3000 y Fn(+)1606 3066 y Fm(n)1695 3041 y Fo(\012)22 b Fg(1)1850 3000 y Fm(n)1850 3066 y(n)p Fn(+1)2032 3041 y Fr(+)44 b Fg(1)2208 3056 y Fm(n)2277 3041 y Fo(\012)2402 3016 y Fr(\024)2377 3041 y Fp(H)2466 3000 y Fm(n)2458 3066 y(n)p Fn(+1)2617 3041 y Fp(;)542 b Fr(\(IV.36\))257 3235 y(where)46 b(w)m(e)f(denote)g(the)f(iden)m(tit)m(y)i(op)s(erator)e (on)g Fo(H)2219 3250 y Fm(n)2310 3235 y Fr(and)g(on)g Fo(F)2740 3198 y Fm(m)2730 3259 y(n)2850 3235 y Fr(b)m(y)h Fg(1)3053 3250 y Fm(n)3144 3235 y Fr(and)g Fg(1)3402 3198 y Fm(m)3402 3259 y(n)3468 3235 y Fr(,)257 3355 y(resp)s(ectiv)m (ely)-8 b(.)56 b(W)-8 b(e)36 b(iden)m(tify)e Fp(~)-45 b(v)1405 3370 y Fm(n)1487 3355 y Fr(with)33 b Fp(~)-45 b(v)1760 3370 y Fm(n)1831 3355 y Fo(\012)25 b Fg(1)1989 3319 y Fm(n)1989 3380 y(n)p Fn(+1)2162 3355 y Fr(acting)36 b(on)f Fo(H)2680 3370 y Fm(n)p Fn(+1)2818 3355 y Fr(.)52 b(Note)36 b(that,)h(for)257 3475 y Fp(n)28 b Fo(2)g Fl(N)509 3490 y Fn(0)549 3475 y Fr(,)1096 3596 y Fp(~)-45 b(v)1147 3611 y Fm(n)p Fn(+1)1345 3596 y Fr(=)56 b Fp(~)-45 b(v)1528 3611 y Fm(n)1614 3596 y Fr(+)38 b Fp(a)1779 3555 y Fj(\003)1819 3596 y Fr(\()1872 3571 y Fp(~)1857 3596 y(G)1934 3555 y Fm(n)1934 3620 y(n)p Fn(+1)2071 3596 y Fr(\))g(+)h Fp(a)p Fr(\()2366 3571 y Fp(~)2351 3596 y(G)2428 3555 y Fm(n)2428 3620 y(n)p Fn(+1)2565 3596 y Fr(\))22 b Fp(:)534 b Fr(\(IV.37\))257 3759 y(Similarly)-8 b(,)34 b(giv)m(en)g Fp( )1009 3774 y Fm(n)1084 3759 y Fo(2)28 b(H)1262 3774 y Fm(n)1309 3759 y Fr(,)33 b(w)m(e)g(de\014ne)h(a)e(v)m(ector)1286 3926 y(~)1266 3952 y Fp( )1329 3967 y Fm(n)1437 3952 y Fr(:=)60 b Fp( )1663 3967 y Fm(n)1732 3952 y Fo(\012)23 b Fr(\012)1902 3911 y Fm(n)1902 3977 y(n)p Fn(+1)2100 3952 y Fo(2)50 b(H)2300 3967 y Fm(n)p Fn(+1)2459 3952 y Fp(;)700 b Fr(\(IV.38\))257 4146 y(where)42 b(\012)617 4161 y Fm(n)704 4146 y Fr(and)e(\012)971 4110 y Fm(m)971 4171 y(n)1079 4146 y Fr(denote)g(the)h(v)-5 b(acuum)41 b(v)m(ectors)h(in)e Fo(F)2476 4161 y Fm(n)2563 4146 y Fr(and)g Fo(F)2842 4110 y Fm(m)2832 4171 y(n)2908 4146 y Fr(,)i(resp)s(ectiv)m(ely)-8 b(.)257 4266 y(With)33 b(these)h(notations,)f(w)m(e)h(ha)m(v)m(e)f(that)1231 4460 y Fp(H)1320 4419 y Fn(+)1312 4485 y Fm(n)p Fn(+1)1476 4460 y Fr(=)1588 4428 y Fi(f)1580 4460 y Fp(H)1669 4419 y Fn(+)1661 4485 y Fm(n)1766 4460 y Fr(+)39 b Fp(W)1987 4419 y Fm(n)1973 4485 y(n)p Fn(+1)2149 4460 y Fr(+)f Fp(E)2341 4419 y Fm(n)2335 4485 y(n)p Fn(+1)2495 4460 y Fp(;)664 b Fr(\(IV.39\))257 4654 y(where)369 4847 y Fp(W)475 4806 y Fm(n)461 4872 y(n)p Fn(+1)681 4847 y Fr(:=)83 b(\()l Fp(~)-45 b(v)952 4862 y Fm(n)p Fn(+1)1090 4847 y Fr(\))1128 4806 y Fn(2)1206 4847 y Fo(\000)39 b Fr(\()l Fp(~)-45 b(v)1407 4862 y Fm(n)1454 4847 y Fr(\))1492 4806 y Fn(2)3186 4847 y Fr(\(IV.40\))695 5028 y(=)96 b(2)17 b Fp(a)984 4987 y Fj(\003)1023 5028 y Fr(\()1076 5003 y Fp(~)1061 5028 y(G)1138 4987 y Fm(n)1138 5053 y(n)p Fn(+1)1275 5028 y Fr(\))22 b Fo(\001)c Fp(~)-45 b(v)1432 5043 y Fm(n)1529 5028 y Fr(+)50 b(2)13 b Fp(~)-45 b(v)1768 5043 y Fm(n)1837 5028 y Fo(\001)21 b Fp(a)p Fr(\()1990 5003 y Fp(~)1975 5028 y(G)2052 4987 y Fm(n)2052 5053 y(n)p Fn(+1)2190 5028 y Fr(\))49 b(+)2403 4932 y Fi(\020)2453 5028 y Fp(a)2504 4987 y Fj(\003)2543 5028 y Fr(\()2596 5003 y Fp(~)2581 5028 y(G)2658 4987 y Fm(n)2658 5053 y(n)p Fn(+1)2795 5028 y Fr(\))22 b(+)g Fp(a)p Fr(\()3057 5003 y Fp(~)3042 5028 y(G)3119 4987 y Fm(n)3119 5053 y(n)p Fn(+1)3257 5028 y Fr(\))3295 4932 y Fi(\021)3344 4955 y Fn(2)695 5207 y Fr(=)96 b(2)17 b Fp(a)984 5166 y Fj(\003)1023 5207 y Fr(\()1076 5182 y Fp(~)1061 5207 y(G)1138 5166 y Fm(n)1138 5232 y(n)p Fn(+1)1275 5207 y Fr(\))22 b Fo(\001)c Fp(~)-45 b(v)1432 5222 y Fm(n)1529 5207 y Fr(+)50 b(2)13 b Fp(~)-45 b(v)1768 5222 y Fm(n)1837 5207 y Fo(\001)21 b Fp(a)p Fr(\()1990 5182 y Fp(~)1975 5207 y(G)2052 5166 y Fm(n)2052 5232 y(n)p Fn(+1)2190 5207 y Fr(\))49 b(+)h Fp(a)2454 5166 y Fj(\003)2494 5207 y Fr(\()2547 5182 y Fp(~)2532 5207 y(G)2609 5166 y Fm(n)2609 5232 y(n)p Fn(+1)2746 5207 y Fr(\))22 b Fo(\001)g Fp(a)2907 5166 y Fj(\003)2946 5207 y Fr(\()2999 5182 y Fp(~)2984 5207 y(G)3061 5166 y Fm(n)3061 5232 y(n)p Fn(+1)3198 5207 y Fr(\))895 5388 y(+)28 b Fp(a)p Fr(\()1103 5363 y Fp(~)1088 5388 y(G)1165 5347 y Fm(n)1165 5413 y(n)p Fn(+1)1302 5388 y Fr(\))22 b Fo(\001)g Fp(a)p Fr(\()1516 5363 y Fp(~)1501 5388 y(G)1578 5347 y Fm(n)1578 5413 y(n)p Fn(+1)1715 5388 y Fr(\))50 b(+)f(2)17 b Fp(a)2045 5347 y Fj(\003)2084 5388 y Fr(\()2137 5363 y Fp(~)2122 5388 y(G)2199 5347 y Fm(n)2199 5413 y(n)p Fn(+1)2336 5388 y Fr(\))22 b Fo(\001)g Fp(a)p Fr(\()2550 5363 y Fp(~)2535 5388 y(G)2612 5347 y Fm(n)2612 5413 y(n)p Fn(+1)2749 5388 y Fr(\))50 b(+)2963 5289 y Fi(\015)2963 5338 y(\015)2963 5388 y(\015)3024 5363 y Fp(~)3009 5388 y(G)3086 5347 y Fm(n)3086 5413 y(n)p Fn(+1)3223 5289 y Fi(\015)3223 5338 y(\015)3223 5388 y(\015)3269 5315 y Fn(2)3331 5388 y Fp(;)p eop end %%Page: 25 26 TeXDict begin 25 25 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(25)257 589 y(with)470 490 y Fi(\015)470 540 y(\015)470 589 y(\015)531 564 y Fp(~)516 589 y(G)593 553 y Fm(n)593 614 y(n)p Fn(+1)730 490 y Fi(\015)730 540 y(\015)730 589 y(\015)776 516 y Fn(2)843 589 y Fr(:=)974 518 y Fi(R)1046 589 y Fo(j)1089 564 y Fp(~)1074 589 y(G)1151 553 y Fm(n)1151 614 y(n)p Fn(+1)1288 589 y Fr(\()p Fp(k)s Fr(\))p Fo(j)1446 553 y Fn(2)1502 589 y Fp(dk)s Fr(.)40 b(In)23 b(Eq.)h(\(IV.39\),)h(w)m(e)f(mak)m(e)g (use)g(of)f(the)g(Coulom)m(b)257 744 y(gauge)33 b(condition)971 718 y Fp(~)962 744 y Fo(r)22 b(\001)1135 718 y Fp(~)1117 744 y(A)p Fr(\()p Fp(x)p Fr(\))28 b(=)g(0,)k(whic)m(h)i(implies)g(that) 307 950 y Fp(a)358 909 y Fj(\003)398 950 y Fr(\()451 925 y Fp(~)436 950 y(G)513 909 y Fm(n)513 974 y(n)p Fn(+1)650 950 y Fr(\))16 b Fo(\001)c Fp(~)-45 b(v)795 965 y Fm(n)899 950 y Fr(=)53 b Fp(~)-45 b(v)1079 965 y Fm(n)1142 950 y Fo(\001)16 b Fp(a)1237 909 y Fj(\003)1277 950 y Fr(\()1330 925 y Fp(~)1315 950 y(G)1392 909 y Fm(n)1392 974 y(n)p Fn(+1)1529 950 y Fr(\))94 b(and)h Fp(a)p Fr(\()2017 925 y Fp(~)2002 950 y(G)2079 909 y Fm(n)2079 974 y(n)p Fn(+1)2216 950 y Fr(\))16 b Fo(\001)c Fp(~)-45 b(v)2361 965 y Fm(n)2465 950 y Fr(=)53 b Fp(~)-45 b(v)2645 965 y Fm(n)2708 950 y Fo(\001)16 b Fp(a)p Fr(\()2856 925 y Fp(~)2841 950 y(G)2918 909 y Fm(n)2918 974 y(n)p Fn(+1)3055 950 y Fr(\))g Fp(:)50 b Fr(\(IV.41\))257 1206 y Fg(IV.2.2)112 b(Preliminary)38 b(results)g(and)h(outline)e(of)h(strategy)257 1391 y Fr(Here)e(w)m(e)f(describ)s(e)h(some)g(results)f(deriv)m(ed)i(in)d ([1],)h(concerning)h(the)f(construction)h(of)257 1511 y(the)d(groundstate)g(of)f Fp(H)8 b Fr(,)32 b(that)g(will)h(b)s(e)f (used)i(again)d(and)i(again)f(in)g(the)h(re-expansion)257 1632 y(pro)s(cedure.)55 b(A)36 b(k)m(ey)i(ingredien)m(t)f(used)g(in)f (our)g(construction)h(of)f(the)g(groundstate)h(of)257 1752 y Fp(H)j Fr(is)33 b(the)h(simple)g(iden)m(tit)m(y)1544 1872 y(2)13 b Fp(~)-45 b(v)63 b Fr(=)d Fp(i)17 b Fr([)p Fp(H)25 b(;)31 b(~)-47 b(x)p Fr(])p Fp(;)978 b Fr(\(IV.42\))257 2037 y(whic)m(h)24 b(implies)f(that)f(the)h(in)m(teraction)g(term)f(in) g Fp(H)8 b Fr(,)24 b(whic)m(h)g(is)e Fk(mar)-5 b(ginal)31 b Fr(in)23 b(the)f Fk(infr)-5 b(ar)g(e)g(d)257 2157 y Fr(region)30 b(\(in)g(the)g(sense)i(of)d(p)s(o)m(w)m(er)i(coun)m (ting\),)g(is)f(actually)g Fk(infr)-5 b(ar)g(e)g(d-irr)g(elevant)38 b Fr(on)30 b(the)257 2278 y(subspace)35 b(of)d(all)g(those)i(states)f (where)h(the)f(electron)h(is)f(b)s(ound)f(to)h(the)g(n)m(ucleus.)257 2461 y Fg(Prop)s(osition)45 b(IV.2.)f Fk(Assume)c(Hyp)-5 b(othesis)40 b(1.)61 b(Then)39 b(ther)-5 b(e)40 b(exist)g(c)-5 b(onstants)40 b Fr(0)d Fp(<)257 2582 y(C)334 2546 y Fj(0)385 2582 y Fo(\024)28 b Fp(C)7 b Fk(,)35 b Fp(C)f Fo(\025)28 b Fr(4)34 b Fk(such)h(that,)f(for)g(al)5 b(l)34 b Fp(\013)29 b(<)1900 2543 y Fn(1)p 1872 2559 91 4 v 1872 2616 a(2)p Fm(C)1972 2582 y Fk(,)35 b Fp(E)2109 2597 y Fm(n)2190 2582 y Fk(is)f(an)g(eigenvalue)f(of)h(multiplicity)257 2702 y(one.)45 b(Mor)-5 b(e)g(over)836 2919 y Fr(inf)970 2822 y Fi(h)1009 2919 y Fp(\033)t Fr(\()p Fp(H)1195 2878 y Fn(+)1187 2943 y Fm(n)1254 2919 y Fr(\))22 b Fo(n)g(f)p Fr(0)p Fo(g)1535 2822 y Fi(i)1657 2919 y Fr(=:)83 b(gap)1994 2942 y Fm(n)2131 2919 y Fo(\025)91 b Fr([1)22 b Fo(\000)2507 2851 y Fr(3)p 2507 2895 49 4 v 2507 2987 a(4)2566 2919 y Fp(C)7 b(\013)q Fr(])17 b Fp(\033)2805 2934 y Fm(n)2873 2919 y Fp(;)286 b Fr(\(IV.43\))836 3120 y(inf)970 3024 y Fi(h)1009 3120 y Fp(\033)t Fr(\()1114 3088 y Fi(f)1106 3120 y Fp(H)1195 3079 y Fn(+)1187 3145 y Fm(n)1254 3120 y Fr(\))22 b Fo(n)g(f)p Fr(0)p Fo(g)1535 3024 y Fi(i)1657 3120 y Fr(=:)1858 3113 y Fi(g)1843 3120 y Fr(gap)1994 3143 y Fm(n)2131 3120 y Fr(=)91 b Fp(\033)2353 3135 y Fm(n)p Fn(+1)2512 3120 y Fp(;)647 b Fr(\(IV.44\))257 3317 y Fk(and)700 3550 y Fr(sup)650 3629 y Fm(z)s Fj(2)p Fn(\000)777 3638 y Fe(n)p Fd(+1)913 3425 y Fi(\015)913 3475 y(\015)913 3525 y(\015)913 3575 y(\015)987 3429 y(\022)1193 3483 y Fr(1)p 1058 3527 320 4 v 1066 3599 a Fi(f)1058 3631 y Fp(H)1147 3603 y Fn(+)1139 3656 y Fm(n)1228 3631 y Fo(\000)22 b Fp(z)1387 3429 y Fi(\023)1448 3452 y Fn(1)p Fm(=)p Fn(2)1558 3454 y Fi(\020)1630 3550 y Fo(\000)g Fp(W)1835 3509 y Fm(n)1821 3575 y(n)p Fn(+1)1959 3454 y Fi(\021)2008 3429 y(\022)2215 3483 y Fr(1)p 2079 3527 V 2087 3599 a Fi(f)2079 3631 y Fp(H)2168 3603 y Fn(+)2160 3656 y Fm(n)2249 3631 y Fo(\000)h Fp(z)2408 3429 y Fi(\023)2470 3452 y Fn(1)p Fm(=)p Fn(2)2607 3425 y Fi(\015)2607 3475 y(\015)2607 3525 y(\015)2607 3575 y(\015)2736 3550 y Fo(\024)84 b Fp(C)23 b(\013)g(;)84 b Fr(\(IV.45\))257 3833 y Fk(wher)-5 b(e)41 b Fr(\000)600 3848 y Fm(n)p Fn(+1)776 3833 y Fr(:=)e Fo(f)978 3794 y Fn(1)p 978 3810 36 4 v 978 3868 a(4)1023 3833 y Fp(\033)1078 3848 y Fm(n)p Fn(+1)1216 3833 y Fp(e)1261 3797 y Fm(i#)1370 3833 y Fo(2)h Fl(C)p Fo(j)p Fp(#)f Fo(2)h Fr([0)p Fp(;)17 b Fr(2)p Fp(\031)t Fr(\))p Fo(g)p Fk(.)62 b(The)40 b(sp)-5 b(e)g(ctr)g(al)41 b(pr)-5 b(oje)g(ctions)3267 3801 y Fi(e)3243 3833 y Fp(P)3306 3848 y Fm(n)3392 3833 y Fr(:=)257 3968 y Fo(j)298 3942 y Fr(~)285 3968 y Fp(\036)343 3983 y Fm(n)390 3968 y Fo(ih)481 3942 y Fr(~)468 3968 y Fp(\036)526 3983 y Fm(n)572 3968 y Fo(k)39 b Fk(and)h Fp(P)919 3983 y Fm(n)p Fn(+1)1092 3968 y Fr(:=)d Fo(j)p Fp(\036)1318 3983 y Fm(n)p Fn(+1)1454 3968 y Fo(ih)p Fp(\036)1590 3983 y Fm(n)p Fn(+1)1727 3968 y Fo(j)p Fk(,)j(wher)-5 b(e)2118 3942 y Fr(~)2105 3968 y Fp(\036)2163 3983 y Fm(n)2250 3968 y Fk(and)39 b Fp(\036)2502 3983 y Fm(n)p Fn(+1)2679 3968 y Fk(ar)-5 b(e)39 b(gr)-5 b(oundstates)40 b(of)266 4056 y Fi(f)257 4089 y Fp(H)338 4104 y Fm(n)420 4089 y Fk(and)34 b Fp(H)690 4104 y Fm(n)p Fn(+1)827 4089 y Fk(,)h(r)-5 b(esp)g(e)g(ctively,)34 b(c)-5 b(orr)g(esp)g(ond)34 b(to)1280 4300 y Fi(e)1256 4332 y Fp(P)1319 4347 y Fm(n)1429 4332 y Fr(=)1585 4265 y Fo(\000)p Fr(1)p 1577 4309 141 4 v 1577 4400 a(2)p Fp(\031)t(i)1745 4215 y Fi(Z)1791 4404 y Fn(\000)1835 4413 y Fe(n)p Fd(+1)2094 4265 y Fp(dz)2190 4280 y Fm(n)p Fn(+1)p 1985 4309 452 4 v 1993 4381 a Fi(f)1985 4413 y Fp(H)2074 4385 y Fn(+)2066 4438 y Fm(n)2155 4413 y Fo(\000)22 b Fp(z)2299 4428 y Fm(n)p Fn(+1)2469 4332 y Fp(;)690 b Fr(\(IV.46\))257 4592 y Fk(and)1011 4737 y Fp(P)1074 4752 y Fm(n)p Fn(+1)1273 4737 y Fr(=)1429 4670 y Fo(\000)p Fr(1)p 1422 4714 141 4 v 1422 4805 a(2)p Fp(\031)t(i)1589 4620 y Fi(Z)1635 4809 y Fn(\000)1679 4818 y Fe(n)p Fd(+1)2140 4670 y Fp(dz)2236 4685 y Fm(n)p Fn(+1)p 1829 4714 854 4 v 1829 4805 a Fp(H)1910 4820 y Fm(n)p Fn(+1)2070 4805 y Fo(\000)22 b Fp(E)2241 4820 y Fm(n)p Fn(+1)2401 4805 y Fo(\000)g Fp(z)2545 4820 y Fm(n)p Fn(+1)2715 4737 y Fp(;)444 b Fr(\(IV.47\))257 4962 y Fk(r)-5 b(esp)g(e)g(ctively,)46 b(with)d Fr(\000)1089 4977 y Fm(n)1181 4962 y Fr(:=)h Fo(f)1388 4923 y Fn(1)p 1388 4939 36 4 v 1388 4996 a(4)1433 4962 y Fp(\033)1488 4977 y Fm(n)1535 4962 y Fp(e)1580 4926 y Fm(i#)1695 4962 y Fo(2)h Fl(C)p Fo(j)p Fp(#)f Fo(2)h Fr([0)p Fp(;)17 b Fr(2)p Fp(\031)t Fr(\))p Fo(g)p Fk(.)70 b(Their)44 b(di\013er)-5 b(enc)g(e)42 b(has)h(a)257 5082 y(norm)35 b(c)-5 b(onver)g(gent)34 b(series)g(exp)-5 b(ansion)625 5331 y Fp(P)688 5346 y Fm(n)p Fn(+1)847 5331 y Fo(\000)970 5299 y Fi(e)947 5331 y Fp(P)1010 5346 y Fm(n)1119 5331 y Fr(=)1275 5264 y Fo(\000)p Fr(1)p 1268 5308 141 4 v 1268 5400 a(2)p Fp(\031)t(i)1464 5224 y Fj(1)1440 5248 y Fi(X)1435 5430 y Fm(\027)t Fn(=1)1581 5214 y Fi(Z)1627 5403 y Fn(\000)1671 5412 y Fe(n)p Fd(+1)1795 5331 y Fr(\()p Fo(\000)p Fr(1\))1997 5290 y Fm(\027)2062 5331 y Fp(Y)2140 5281 y Fn(\()p Fm(\027)t Fn(\))2119 5353 y Fm(n)p Fn(+1)2256 5331 y Fr(\()p Fp(z)2339 5346 y Fm(n)p Fn(+1)2476 5331 y Fr(\))22 b Fp(dz)2632 5346 y Fm(n)p Fn(+1)2792 5331 y Fp(;)367 b Fr(\(IV.48\))p eop end %%Page: 26 27 TeXDict begin 26 26 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(26)257 573 y Fk(wher)-5 b(e)1068 713 y Fp(Y)1146 662 y Fn(\()p Fm(\027)t Fn(\))1125 735 y Fm(n)p Fn(+1)1262 713 y Fr(\()p Fp(z)t Fr(\))28 b(:=)1691 645 y(1)p 1556 690 320 4 v 1564 762 a Fi(f)1556 794 y Fp(H)1645 765 y Fn(+)1637 819 y Fm(n)1726 794 y Fo(\000)22 b Fp(z)1885 592 y Fi(\022)1946 713 y Fp(W)2052 672 y Fm(n)2038 737 y(n)p Fn(+1)2337 645 y Fr(1)p 2202 690 V 2210 762 a Fi(f)2202 794 y Fp(H)2291 765 y Fn(+)2283 819 y Fm(n)2372 794 y Fo(\000)g Fp(z)2531 592 y Fi(\023)2592 615 y Fm(\027)2657 713 y Fp(;)502 b Fr(\(IV.49\))257 950 y Fk(and)1264 1071 y Fo(k)p Fp(P)1377 1086 y Fm(n)p Fn(+1)1536 1071 y Fo(\000)1659 1038 y Fi(e)1635 1071 y Fp(P)1698 1086 y Fm(n)1745 1071 y Fo(k)63 b(\024)g Fp(C)2075 1029 y Fj(0)2114 1071 y Fp(\013)2177 1029 y Fn(\()p Fm(n)p Fn(+2\))p Fm(=)p Fn(2)2462 1071 y Fp(:)697 b Fr(\(IV.50\))257 1245 y Fk(The)35 b(eigenvalue)e Fp(E)999 1260 y Fj(1)1102 1245 y Fo(\021)28 b Fp(E)1279 1260 y Fm(g)r(s)1387 1245 y Fk(is)35 b(non)f(de)-5 b(gener)g(ate.)404 1448 y Fr(If)46 b(w)m(e)h(are)g(in)m(terested)h(in)f(deriving)g(an)g (explicit)g(expression)i(for)d Fp(\036)3040 1463 y Fm(g)r(s)3159 1448 y Fr(up)g(to)g(a)257 1569 y(remainder)40 b(term)f(of)g(order)f Fp(o)p Fr(\()p Fp(\013)1489 1533 y Fm(N)1556 1569 y Fr(\),)i(w)m(e)g (ma)m(y)g(as)f(w)m(ell)h(consider)g(the)f(v)m(ector)h Fp(\036)3276 1584 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)3468 1569 y Fr(,)257 1689 y(b)s(ecause)1369 1809 y Fo(k)p Fp(\036)1477 1824 y Fn(2)p Fm(N)g Fj(\000)p Fn(1)1691 1809 y Fo(\000)23 b Fp(\036)1849 1824 y Fm(g)r(s)1921 1809 y Fo(k)28 b(\024)g Fp(o)p Fr(\()p Fp(\013)2252 1768 y Fm(N)2319 1809 y Fr(\))p Fp(:)802 b Fr(\(IV.51\))257 1984 y(The)34 b(b)s(ound)g(\(IV.51\))e(follo)m(ws)i(from)f(Prop)s (osition)g(IV.2.)45 b(Up)34 b(to)e(normalization,)i(the)257 2104 y(v)m(ector)40 b Fp(\036)614 2119 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)845 2104 y Fr(is)39 b(giv)m(en)g(b)m(y)g(the)g(pro)s (duct)g Fp(P)1959 2119 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2151 2104 y Fp(P)2214 2119 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)2433 2104 y Fo(\001)26 b(\001)g(\001)f Fp(P)2657 2119 y Fn(1)2735 2104 y Fr(of)38 b(the)h(pro)5 b(jections)257 2225 y Fo(f)p Fp(P)370 2240 y Fn(2)p Fm(N)i Fj(\000)p Fn(1)563 2225 y Fp(;)17 b(::::;)g(P)822 2240 y Fn(1)861 2225 y Fo(g)29 b Fr(applied)i(to)e(the)h(groundstate,)h Fp(\036)2180 2240 y Fn(0)2219 2225 y Fr(,)f(of)f(the)h(bare)g(Hamiltonian)g Fp(H)3410 2183 y Fn(+)3402 2246 y(0)3468 2225 y Fr(,)257 2345 y(i.e.)1344 2465 y Fp(\036)1402 2480 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1622 2465 y Fo(/)28 b Fp(P)1790 2480 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2005 2465 y Fo(\001)22 b(\001)g(\001)f(\001)p Fp(P)2245 2480 y Fn(1)2284 2465 y Fp(\036)2342 2480 y Fn(0)2382 2465 y Fp(:)777 b Fr(\(IV.52\))257 2640 y(F)-8 b(or)26 b(a)g(more)g(precise)i(v)m(ersion)g(of)d(F)-8 b(orm)m(ula)26 b(\(IV.52\))g(see)i([2)o(].)42 b(F)-8 b(rom)25 b(Prop)s(osition)i(IV.2,)257 2760 y(more)f(precisely)h(from)d (estimate)i(\(IV.45\),)h(w)m(e)f(infer)f(that)g(the)g(expansion)h (\(IV.48\))f(can)257 2880 y(b)s(e)h(truncated)g(at)f(a)g(\014nite)h (order)g(in)g(suc)m(h)g(a)g(w)m(a)m(y)g(that)f(the)h(remainder)h(term)f (is)f Fp(o)p Fr(\()p Fp(\013)3363 2844 y Fm(N)3430 2880 y Fr(\).)257 3001 y(Th)m(us,)35 b(the)e(v)m(ector)1472 3121 y Fp(P)1549 3080 y Fm(T)1535 3146 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1750 3121 y Fo(\001)22 b(\001)g(\001)f(\001)p Fp(P)2004 3080 y Fm(T)1990 3146 y Fn(1)2059 3121 y Fp(\036)2117 3136 y Fn(0)2253 3121 y Fp(;)906 b Fr(\(IV.53\))257 3296 y(where)371 3565 y(\()p Fp(P)472 3580 y Fm(m)538 3565 y Fr(\))576 3524 y Fm(T)659 3565 y Fr(:=)28 b Fo(\000)923 3498 y Fr(1)p 877 3542 141 4 v 877 3633 a(2)p Fp(\031)t(i)1074 3457 y Fm(N)1046 3482 y Fi(X)1045 3664 y Fm(j)t Fn(=0)1184 3448 y Fi(Z)1230 3637 y Fn(\000)1274 3645 y Fe(m)1353 3565 y Fp(dz)1449 3580 y Fm(m)1737 3498 y Fr(1)p 1526 3542 472 4 v 1534 3614 a Fi(f)1526 3646 y Fp(H)1615 3605 y Fn(+)1607 3668 y Fm(m)p Fj(\000)p Fn(1)1786 3646 y Fo(\000)22 b Fp(z)1930 3661 y Fm(m)2007 3469 y Fi(h)2046 3565 y Fr(\()p Fo(\000)p Fp(W)2267 3524 y Fm(m)p Fj(\000)p Fn(1)2253 3590 y Fm(m)2424 3565 y Fr(\))2700 3498 y(1)p 2489 3542 V 2497 3614 a Fi(f)2489 3646 y Fp(H)2578 3605 y Fn(+)2570 3668 y Fm(m)p Fj(\000)p Fn(1)2748 3646 y Fo(\000)h Fp(z)2893 3661 y Fm(m)2970 3469 y Fi(i)3009 3492 y Fm(j)3046 3565 y Fp(;)113 b Fr(\(IV.54\))257 3875 y(is)26 b(an)f(appro)m(ximate)h(expression)h(for)d Fp(\036)1693 3890 y Fm(g)r(s)1766 3875 y Fr(,)i(up)f(to)g(a)f(remainder)i(term)g Fp(o)p Fr(\()p Fp(\013)2964 3839 y Fm(N)3031 3875 y Fr(\))e(and)h(up)h (to)257 3995 y(a)32 b(normalization)h(factor.)42 b(The)33 b(truncation)g(in)f(the)h(de\014nition)g(of)e Fp(P)2850 3959 y Fm(T)2836 4020 y(m)2937 3995 y Fr(only)i(dep)s(ends)257 4116 y(on)g Fp(N)10 b Fr(.)257 4236 y(F)-8 b(rom)41 b(no)m(w)i(on,)g(w) m(e)g(fo)s(cus)f(our)f(atten)m(tion)h(on)g(the)f(analysis)i(of)e(the)h (\014nitely)h(man)m(y)257 4357 y(resolv)m(en)m(ts)36 b(app)s(earing)c(on)h(the)g(righ)m(t)f(side)i(of)e(\(IV.54\),)h(for)f Fp(m)c Fr(=)f(2)p Fp(N)33 b Fo(\000)22 b Fr(1.)404 4477 y(Our)37 b(deriv)-5 b(ation)39 b(of)e(explicit)i(\014nite)g (expressions)h(for)d(the)i(v)m(ector)f(\(IV.53\))g(relies)257 4597 y(on)33 b(three)g(op)s(erations)g(to)f(b)s(e)h(iterated)g(a)f (\014nite)i(n)m(um)m(b)s(er)g(of)e(times:)387 4801 y(i\))49 b(The)k(photon)e(creation-)g(and)h(annihilation)g(op)s(erators)f(are)g (Wic)m(k-ordered,)501 4921 y(shell)34 b(b)m(y)f(shell;)360 5124 y(ii\))49 b(the)33 b(iden)m(tit)m(y)h(op)s(erator,)d Fg(1)1504 5139 y Fm(n)1551 5124 y Fr(,)h(in)h(the)f(space)h Fo(H)2235 5139 y Fm(n)2314 5124 y Fr(is)g(decomp)s(osed)h(in)m(to)e (the)g(sum)501 5245 y Fp(P)564 5260 y Fm(n)633 5245 y Fr(+)22 b Fp(P)808 5209 y Fj(?)794 5270 y Fm(n)867 5245 y Fr(;)p eop end %%Page: 27 28 TeXDict begin 27 27 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(27)333 573 y(iii\))49 b(t)m(w)m(o)43 b(sligh)m(tly)h(di\013eren)m(t,)i(truncated)d(Neumann)g(expansions,)k (whic)m(h)d(w)m(e)f(call)501 693 y Fo(A)34 b Fr(and)h Fo(B)s Fr(,)h(are)e(used)i(to)e(re-expand)i(the)f(con)m(tributions)h (asso)s(ciated)g(with)f Fp(P)3437 657 y Fj(?)3423 718 y Fm(n)501 814 y Fr(coming)e(from)g(op)s(eration)f(ii\).)257 989 y Fk(R)-5 b(e-exp)g(ansion)33 b(pr)-5 b(o)g(c)g(e)g(dur)g(e)257 1109 y Fr(As)33 b(a)g(\014rst)g(step,)g(w)m(e)h(expand)g(the)f(resolv)m (en)m(ts)1852 1267 y(1)p 1560 1312 634 4 v 1568 1384 a Fi(f)1560 1416 y Fp(H)1649 1375 y Fn(+)1641 1440 y(2)p Fm(N)7 b Fj(\000)p Fn(2)1855 1416 y Fo(\000)23 b Fp(z)2000 1431 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)3186 1335 y Fr(\(IV.55\))257 1604 y(app)s(earing)28 b(in)f(the)h(truncated)g(pro)5 b(jection)28 b Fp(P)1950 1568 y Fm(T)1936 1628 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2156 1604 y Fr(in)28 b(Expression)h(\(IV.53\))f (un)m(til)g(only)257 1724 y(resolv)m(en)m(ts)40 b(of)d(the)g (Hamiltonian)h Fp(H)1653 1683 y Fn(+)1645 1748 y(2)p Fm(N)7 b Fj(\000)p Fn(3)1874 1724 y Fr(are)38 b(left.)57 b(W)-8 b(e)38 b(then)f(put)h(the)f(expressions)257 1844 y(obtained)c(in)g(a)g(form)f(that)h(enables)h(us)f(to)f(iterate)h(the)g (op)s(eration,)g(un)m(til)g(only)g(resol-)257 1965 y(v)m(en)m(ts)i(of)d (the)h(bare)g(Hamiltonian)g Fp(H)1656 1924 y Fn(+)1648 1987 y(0)1747 1965 y Fr(are)f(left.)257 2085 y(This)23 b(can)f(b)s(e)g(accomplished)i(if)e(w)m(e)g(tak)m(e)h(in)m(to)f(accoun) m(t)g(that)g(the)g(op)s(erator)f(in)h(Eq.)h(\(IV.55\))257 2206 y(is)g(applied)f(to)g(a)f(v)m(ector)i(con)m(taining)f(only)g(a)g (\014nite,)j Fp(N)10 b Fr(-dep)s(enden)m(t,)26 b(but)c Fp(\013)q Fr(-indep)s(enden)m(t,)257 2326 y(n)m(um)m(b)s(er)42 b(of)f(photons)g(with)g(momen)m(ta)h(in)f(the)g(shell)h Fo(K)2387 2285 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2386 2350 y(2)p Fm(N)g Fj(\000)p Fn(1)2580 2326 y Fr(,)42 b(thanks)g(to)e(the)h(trun-)257 2446 y(cation)g(in)f Fp(P)756 2410 y Fm(T)742 2471 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)935 2446 y Fr(.)67 b(After)40 b(Wic)m(k-ordering)i(of)d(the)i (photon)g(op)s(erators)f(in)g(the)h(shell)257 2567 y Fo(K)334 2526 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)333 2591 y(2)p Fm(N)g Fj(\000)p Fn(1)527 2567 y Fr(,)45 b(the)e(original)e (resolv)m(en)m(t)k(\(IV.55\))d(is)g(replaced)i(b)m(y)f(a)f(\014nite,)j Fp(N)10 b Fr(-dep)s(enden)m(t)257 2687 y(n)m(um)m(b)s(er)34 b(of)e(resolv)m(en)m(ts)k(of)c(the)h(form)1852 2826 y(1)p 1358 2870 1037 4 v 1358 2962 a Fp(H)1447 2921 y Fn(+)1439 2986 y(2)p Fm(N)7 b Fj(\000)p Fn(2)1654 2962 y Fo(\000)22 b Fp(z)1798 2977 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2013 2962 y Fr(+)2111 2895 y Fi(P)2199 2983 y Fm(j)2252 2962 y Fo(j)p Fp(k)2331 2977 y Fm(j)2367 2962 y Fo(j)3186 2894 y Fr(\(IV.56\))257 3145 y(applied)37 b(to)f(a)f(v)m(ector)i(in)f Fo(H)1306 3160 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1499 3145 y Fr(,)37 b(where)g(the)g(sum,)2257 3079 y Fi(P)2345 3166 y Fm(j)2398 3145 y Fo(j)p Fp(k)2477 3160 y Fm(j)2513 3145 y Fo(j)p Fr(,)f(of)g(energies)h Fo(j)p Fp(k)3170 3160 y Fm(j)3206 3145 y Fo(j)p Fr(,)g Fo(f)p Fp(k)3399 3160 y Fm(j)3468 3145 y Fr(:)257 3265 y Fp(\033)312 3280 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)558 3265 y Fo(\024)54 b(j)p Fp(k)768 3280 y Fm(j)804 3265 y Fo(j)e(\024)h Fp(\033)1069 3280 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1263 3265 y Fo(g)47 b Fr(is)h(\014nite)g(and)f(dep)s(ends)i(only)f(on)f Fp(N)10 b Fr(.)89 b(The)48 b(k)m(ey)h(idea)257 3386 y(underlying)39 b(the)f(re-expansion)h(of)e Fp(\036)1666 3401 y Fm(g)r(s)1775 3386 y Fr(is)h(to)f(split)h(the)g(op)s(erator)f(\(IV.56\))g(in)m(to)g (t)m(w)m(o)257 3506 y(pieces,)878 3665 y(1)p 384 3709 V 384 3800 a Fp(H)473 3759 y Fn(+)465 3825 y(2)p Fm(N)7 b Fj(\000)p Fn(2)680 3800 y Fo(\000)23 b Fp(z)825 3815 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1040 3800 y Fr(+)1138 3734 y Fi(P)1225 3821 y Fm(j)1278 3800 y Fo(j)p Fp(k)1357 3815 y Fm(j)1393 3800 y Fo(j)1431 3732 y Fp(P)1494 3747 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)1784 3732 y Fr(and)2543 3665 y(1)p 2049 3709 V 2049 3800 a Fp(H)2138 3759 y Fn(+)2130 3825 y(2)p Fm(N)g Fj(\000)p Fn(2)2345 3800 y Fo(\000)22 b Fp(z)2489 3815 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2704 3800 y Fr(+)2802 3734 y Fi(P)2890 3821 y Fm(j)2943 3800 y Fo(j)p Fp(k)3022 3815 y Fm(j)3058 3800 y Fo(j)3096 3732 y Fp(P)3173 3691 y Fj(?)3159 3757 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)3351 3732 y Fp(:)3186 3917 y Fr(\(IV.57\))257 4037 y(The)25 b(\014rst)e(one)h(is)f(prop)s(ortional)g(to)g(the)g(pro)5 b(jection)24 b Fp(P)2235 4052 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2428 4037 y Fr(,)25 b(the)f(factor)e(of)h(prop)s(ortion-)257 4158 y(alit)m(y)i(b)s(eing)f(an)f(explicit)j(n)m(um)m(b)s(er.)42 b(Up)24 b(to)f(a)g(remainder)i(term)f(of)g(order)f Fp(o)p Fr(\()p Fp(\013)3082 4122 y Fm(N)3149 4158 y Fr(\),)j Fp(P)3303 4173 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)257 4278 y Fr(can)31 b(b)s(e)g(expanded)i(b)m(y)f(using)f(Eq.)h(\(IV.48\))f(and) f(then)i(truncated)g(as)f(in)g(Eq.)g(\(IV.54\).)257 4399 y(The)44 b(second)g(term)f(in)g(\(IV.57\))g(is)g(analyzed)h(b)m(y)f (using)h(the)f(Neumann)h(expansion)257 4519 y(b)s(elo)m(w,)34 b(whic)m(h)g(w)m(e)g(call)e(of)h(t)m(yp)s(e)g Fo(A)p Fr(:)499 4745 y Fp(P)576 4704 y Fj(?)562 4769 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1259 4677 y Fr(1)p 765 4721 V 773 4794 a Fi(f)765 4826 y Fp(H)854 4785 y Fn(+)846 4850 y(2)p Fm(N)g Fj(\000)p Fn(3)1060 4826 y Fo(\000)23 b Fp(z)1205 4841 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1420 4826 y Fr(+)1518 4760 y Fi(P)1606 4847 y Fm(j)1659 4826 y Fo(j)p Fp(k)1738 4841 y Fm(j)1774 4826 y Fo(j)1812 4745 y Fp(P)1889 4704 y Fj(?)1875 4769 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)3568 4745 y Fr(\(IV.58\))340 5024 y(+)83 b Fp(P)576 4983 y Fj(?)562 5049 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1259 4957 y Fr(1)p 765 5001 V 773 5073 a Fi(f)765 5106 y Fp(H)854 5065 y Fn(+)846 5130 y(2)p Fm(N)g Fj(\000)p Fn(3)1060 5106 y Fo(\000)23 b Fp(z)1205 5121 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1420 5106 y Fr(+)1518 5039 y Fi(P)1606 5127 y Fm(j)1659 5106 y Fo(j)p Fp(k)1738 5121 y Fm(j)1774 5106 y Fo(j)1812 5024 y Fp(P)1889 4983 y Fj(?)1875 5049 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)2089 5024 y Fr(+)22 b Fp(P)2264 4983 y Fj(?)2250 5049 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2947 4957 y Fr(1)p 2453 5001 V 2461 5073 a Fi(f)2453 5106 y Fp(H)2542 5065 y Fn(+)2534 5130 y(2)p Fm(N)g Fj(\000)p Fn(3)2748 5106 y Fo(\000)23 b Fp(z)2893 5121 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)3108 5106 y Fr(+)3206 5039 y Fi(P)3294 5127 y Fm(j)3347 5106 y Fo(j)p Fp(k)3426 5121 y Fm(j)3462 5106 y Fo(j)3500 5024 y Fp(P)3577 4983 y Fj(?)3563 5049 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)3777 5024 y Fo(\002)499 5317 y(\002)619 5210 y Fj(1)595 5234 y Fi(X)593 5417 y Fm(j)t Fn(=1)733 5221 y Fi(h\020)843 5317 y Fo(\000)23 b Fp(W)1049 5276 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1035 5342 y(2)p Fm(N)g Fj(\000)p Fn(2)1264 5317 y Fo(\000)22 b Fr(\()p Fp(E)1479 5276 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1473 5342 y(2)p Fm(N)g Fj(\000)p Fn(2)1672 5317 y Fr(\))1710 5221 y Fi(\021)2263 5250 y Fr(1)p 1769 5294 V 1777 5366 a Fi(f)1769 5399 y Fp(H)1858 5358 y Fn(+)1850 5423 y(2)p Fm(N)g Fj(\000)p Fn(3)2065 5399 y Fo(\000)23 b Fp(z)2210 5414 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2425 5399 y Fr(+)2523 5332 y Fi(P)2610 5420 y Fm(j)2664 5399 y Fo(j)p Fp(k)2743 5414 y Fm(j)2778 5399 y Fo(j)2816 5317 y Fp(P)2893 5276 y Fj(?)2879 5342 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)3072 5221 y Fi(i)3111 5244 y Fm(j)3147 5317 y Fp(:)p eop end %%Page: 28 29 TeXDict begin 28 28 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(28)257 573 y(This)49 b(expansion)g(con)m(v)m (erges,)54 b(b)s(ecause)49 b(the)f(absolute)g(v)-5 b(alue)48 b(of)f(the)h(energy)h(shift)257 693 y Fp(E)335 652 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)329 718 y(2)p Fm(N)g Fj(\000)p Fn(2)561 693 y Fr(is)33 b(v)m(ery)h(small,)f(namely)h(of)e(order)h Fo(O)s Fr(\()p Fp(\013)2043 657 y Fn(2)p Fm(N)2145 693 y Fr(\);)f(and)257 814 y(the)h(expression)1724 905 y(1)p 1230 950 1037 4 v 1238 1022 a Fi(f)1230 1054 y Fp(H)1319 1013 y Fn(+)1311 1079 y(2)p Fm(N)7 b Fj(\000)p Fn(3)1526 1054 y Fo(\000)22 b Fp(z)1670 1069 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1886 1054 y Fr(+)1984 988 y Fi(P)2071 1075 y Fm(j)2124 1054 y Fo(j)p Fp(k)2203 1069 y Fm(j)2239 1054 y Fo(j)2277 973 y Fp(P)2354 932 y Fj(?)2340 998 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)3186 973 y Fr(\(IV.59\))257 1234 y(is)33 b(b)s(ounded)h(in)f(norm)f(b)m(y)i Fp(\033)1317 1193 y Fj(\000)p Fn(1)1313 1259 y(2)p Fm(N)7 b Fj(\000)p Fn(2)1506 1234 y Fr(,)33 b(b)s(ecause)g(of)g(the)g(orthogonal)e(pro)5 b(jection)34 b Fp(P)3234 1198 y Fj(?)3220 1259 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)3412 1234 y Fr(.)257 1355 y(The)33 b(expansion)g(in)f(Eq.)g(\(IV.58\))f(yields)j(additional)d(p)s(o)m(w)m (ers)i(of)e Fp(\013)q Fr(.)43 b(It)32 b(can)g(therefore)257 1475 y(b)s(e)h(truncated)h(at)e(some)i Fp(N)10 b Fo(\000)p Fr(dep)s(enden)m(t)36 b(order.)44 b(Finally)-8 b(,)33 b Fp(P)2558 1439 y Fj(?)2544 1500 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2765 1475 y Fr(=)28 b Fg(1)2925 1490 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)3140 1475 y Fo(\000)23 b Fp(P)3303 1490 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)257 1595 y Fr(has)44 b(to)e(b)s(e)h(expanded)i(and)e(truncated)g(similarly)h(to)f Fp(P)2417 1610 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2609 1595 y Fr(.)75 b(As)43 b(a)g(result)g(of)g(the)257 1716 y(previous)25 b(op)s(erations)e(\(decomp)s(osition)i(\(IV.57\),)g(and)e (Neumann)h(expansion)h Fo(A)p Fr(\),)g(the)257 1836 y(resolv)m(en)m(t) 35 b(\(IV.56\))d(is)i(represen)m(ted)h(b)m(y)e(a)f(p)s(olynomial)h(in)g (the)g(follo)m(wing)g(op)s(erators:)403 2039 y Fo(\017)48 b Fr(the)33 b(resolv)m(en)m(ts)1360 2212 y(1)p 866 2256 V 874 2328 a Fi(f)866 2360 y Fp(H)955 2319 y Fn(+)947 2385 y(2)p Fm(N)7 b Fj(\000)p Fn(3)1162 2360 y Fo(\000)22 b Fp(z)1306 2375 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1521 2360 y Fr(+)1619 2294 y Fi(P)1707 2381 y Fm(j)1760 2360 y Fo(j)p Fp(k)1839 2375 y Fm(j)1875 2360 y Fo(j)2010 2279 y Fp(;)2454 2212 y Fr(1)p 2161 2256 634 4 v 2169 2328 a Fi(f)2161 2360 y Fp(H)2250 2319 y Fn(+)2242 2385 y(2)p Fm(N)g Fj(\000)p Fn(3)2457 2360 y Fo(\000)23 b Fp(z)2602 2375 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2805 2279 y Fr(;)354 b(\(IV.60\))403 2630 y Fo(\017)48 b Fr(the)33 b(slice)h(in)m(teraction)g Fp(W)1478 2589 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1464 2654 y(2)p Fm(N)g Fj(\000)p Fn(2)1670 2630 y Fr(;)403 2833 y Fo(\017)48 b Fr(and)33 b(the)g(energy)h(shift)f Fp(E)1469 2792 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1463 2858 y(2)p Fm(N)g Fj(\000)p Fn(2)1661 2833 y Fr(.)257 3037 y(Returning)34 b(to)g(the)g(expression)i(\(IV.53\),)e(w)m(e)g (then)g(Wic)m(k-order)h(the)f(photon)g(op)s(era-)257 3157 y(tors)24 b(corresp)s(onding)g(to)f(photon)g(momen)m(ta)h(in)f (the)h(shell)g Fo(K)2461 3116 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)2460 3182 y(2)p Fm(N)g Fj(\000)p Fn(2)2654 3157 y Fr(.)41 b(This)24 b(yields)h(\014nitely)257 3278 y(man)m(y)36 b(terms,)g(and)f(the)g(resolv)m(en)m(ts)i(in)e(Eq.)h(\(IV.60\))e(are)h (replaced)h(b)m(y)f(resolv)m(en)m(ts)j(of)257 3398 y(the)33 b(form)1062 3570 y(1)p 378 3614 1417 4 v 378 3706 a Fp(H)467 3665 y Fn(+)459 3730 y(2)p Fm(N)7 b Fj(\000)p Fn(3)674 3706 y Fo(\000)22 b Fp(z)818 3721 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1034 3706 y Fr(+)1132 3639 y Fi(P)1219 3727 y Fm(j)1272 3706 y Fo(j)p Fp(k)1351 3721 y Fm(j)1387 3706 y Fo(j)22 b Fr(+)1535 3639 y Fi(P)1623 3727 y Fm(i)1667 3706 y Fo(j)p Fp(q)1738 3721 y Fm(i)1767 3706 y Fo(j)1902 3638 y Fp(;)2535 3570 y Fr(1)p 2053 3614 1013 4 v 2053 3706 a Fp(H)2142 3665 y Fn(+)2134 3730 y(2)p Fm(N)7 b Fj(\000)p Fn(3)2349 3706 y Fo(\000)22 b Fp(z)2493 3721 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2708 3706 y Fr(+)2806 3639 y Fi(P)2894 3727 y Fm(i)2939 3706 y Fo(j)p Fp(q)3010 3721 y Fm(i)3038 3706 y Fo(j)3186 3638 y Fr(\(IV.61\))257 3934 y(where)31 b(the)f(photon)g(momen)m(ta)g Fp(q)1494 3949 y Fm(i)1552 3934 y Fr(all)f(b)s(elong)h(to)f(the)h(shell)h Fo(K)2571 3893 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)2570 3958 y(2)p Fm(N)g Fj(\000)p Fn(2)2764 3934 y Fr(,)30 b(and)g(the)g(n)m(um)m(b)s(er)257 4054 y(of)j(terms)h(in)f(the)g(sum) 1130 3988 y Fi(P)1217 4075 y Fm(i)1262 4054 y Fo(j)p Fp(q)1333 4069 y Fm(i)1361 4054 y Fo(j)g Fr(is)g(b)s(ounded)h(b)m(y)f (an)g Fp(N)10 b Fo(\000)p Fr(dep)s(enden)m(t)36 b(n)m(um)m(b)s(er,)e (thanks)257 4175 y(to)c(the)g(truncation)g(in)g Fp(P)1201 4139 y Fm(T)1187 4200 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1380 4175 y Fr(.)43 b(As)30 b(b)s(efore,)g(w)m(e)h(decomp)s(ose)h(the)e (resolv)m(en)m(ts)i(in)m(to)e(t)m(w)m(o)257 4295 y(pieces,)c(using)d (the)f(pro)5 b(jections)23 b Fp(P)1514 4310 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1729 4295 y Fr(and)22 b Fp(P)1985 4259 y Fj(?)1971 4320 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)2163 4295 y Fr(.)40 b(This)23 b(yields)h(terms)f(prop)s(ortional)257 4416 y(to)29 b(the)g(pro)5 b(jection)29 b Fp(P)1059 4431 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1252 4416 y Fr(,)29 b(with)g(an)g(explicit)h(factor)e(of)g(prop)s(ortionalit)m(y)-8 b(,)30 b(and)e(terms)257 4536 y(of)k(the)h(form)749 4708 y(1)p 267 4752 V 267 4844 a Fp(H)356 4803 y Fn(+)348 4868 y(2)p Fm(N)7 b Fj(\000)p Fn(3)563 4844 y Fo(\000)23 b Fp(z)708 4859 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)923 4844 y Fr(+)1021 4777 y Fi(P)1108 4865 y Fm(i)1153 4844 y Fo(j)p Fp(q)1224 4859 y Fm(i)1252 4844 y Fo(j)1290 4776 y Fp(P)1367 4734 y Fj(?)1353 4800 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)1643 4776 y Fp(;)2478 4708 y Fr(1)p 1794 4752 1417 4 v 1794 4844 a Fp(H)1883 4803 y Fn(+)1875 4868 y(2)p Fm(N)g Fj(\000)p Fn(3)2090 4844 y Fo(\000)23 b Fp(z)2235 4859 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2450 4844 y Fr(+)2548 4777 y Fi(P)2635 4865 y Fm(j)2688 4844 y Fo(j)p Fp(k)2767 4859 y Fm(j)2803 4844 y Fo(j)22 b Fr(+)2951 4777 y Fi(P)3039 4865 y Fm(i)3084 4844 y Fo(j)p Fp(q)3155 4859 y Fm(i)3183 4844 y Fo(j)3220 4776 y Fp(P)3297 4734 y Fj(?)3283 4800 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)3476 4776 y Fp(:)3186 4961 y Fr(\(IV.62\))257 5081 y(The)38 b(\014rst)f(term)g(in)g(Eq.)g(\(IV.62\))g(is)g(analyzed)g(b)m(y)h (using)f(the)g(Neumann)h(expansion)257 5201 y(of)30 b(t)m(yp)s(e)h Fo(A)p Fr(,)f(see)i(\(IV.58\),)e(with)h(2)p Fp(N)c Fo(\000)18 b Fr(2)30 b(replaced)h(b)m(y)g(2)p Fp(N)d Fo(\000)17 b Fr(3.)43 b(The)31 b(second)g(term)g(in)p eop end %%Page: 29 30 TeXDict begin 29 29 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(29)257 573 y(Eq.)35 b(\(IV.62\))e(is)h (treated)g(b)m(y)g(applying)g(the)g(follo)m(wing)g(expansion,)h(whic)m (h)g(w)m(e)g(call)f(of)257 693 y(t)m(yp)s(e)g Fo(B)s Fr(:)571 938 y Fp(P)648 897 y Fj(?)634 963 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1319 871 y Fr(1)p 837 915 1013 4 v 845 987 a Fi(f)837 1019 y Fp(H)926 978 y Fn(+)918 1044 y(2)p Fm(N)g Fj(\000)p Fn(4)1132 1019 y Fo(\000)23 b Fp(z)1277 1034 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1492 1019 y Fr(+)1590 953 y Fi(P)1678 1040 y Fm(i)1723 1019 y Fo(j)p Fp(q)1794 1034 y Fm(i)1822 1019 y Fo(j)1859 938 y Fp(P)1936 897 y Fj(?)1922 963 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)3186 938 y Fr(\(IV.63\))412 1215 y(+)83 b Fp(P)648 1174 y Fj(?)634 1240 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1319 1148 y Fr(1)p 837 1192 V 845 1264 a Fi(f)837 1297 y Fp(H)926 1255 y Fn(+)918 1321 y(2)p Fm(N)g Fj(\000)p Fn(4)1132 1297 y Fo(\000)23 b Fp(z)1277 1312 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1492 1297 y Fr(+)1590 1230 y Fi(P)1678 1317 y Fm(i)1723 1297 y Fo(j)p Fp(q)1794 1312 y Fm(i)1822 1297 y Fo(j)1859 1215 y Fp(P)1936 1174 y Fj(?)1922 1240 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)2137 1215 y Fo(\002)571 1523 y(\002)677 1415 y Fn(2)p Fm(N)667 1440 y Fi(X)665 1622 y Fm(j)t Fn(=1)805 1427 y Fi(h\020)915 1523 y Fo(\000)23 b Fp(W)1121 1482 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)1107 1547 y(2)p Fm(N)g Fj(\000)p Fn(3)1336 1523 y Fo(\000)22 b Fp(E)1513 1482 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)1507 1547 y(2)p Fm(N)g Fj(\000)p Fn(3)1706 1523 y Fr(\))2236 1455 y(1)p 1754 1500 V 1762 1572 a Fi(f)1754 1604 y Fp(H)1843 1563 y Fn(+)1835 1628 y(2)p Fm(N)g Fj(\000)p Fn(4)2049 1604 y Fo(\000)23 b Fp(z)2194 1619 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2409 1604 y Fr(+)2507 1538 y Fi(P)2595 1625 y Fm(i)2640 1604 y Fo(j)p Fp(q)2711 1619 y Fm(i)2739 1604 y Fo(j)2776 1523 y Fp(P)2853 1482 y Fj(?)2839 1547 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)766 1808 y Fr(+)842 1712 y Fi(\020)892 1808 y Fp(z)937 1823 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)1152 1808 y Fo(\000)23 b Fp(z)1297 1823 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1512 1808 y Fr(+)1610 1725 y Fi(X)1654 1907 y Fm(j)1746 1808 y Fo(j)p Fp(k)1825 1823 y Fm(j)1861 1808 y Fo(j)1889 1712 y Fi(\021)2430 1741 y Fr(1)p 1948 1785 V 1956 1857 a Fi(f)1948 1889 y Fp(H)2037 1848 y Fn(+)2029 1914 y(2)p Fm(N)g Fj(\000)p Fn(4)2244 1889 y Fo(\000)23 b Fp(z)2389 1904 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2604 1889 y Fr(+)2702 1823 y Fi(P)2790 1910 y Fm(i)2834 1889 y Fo(j)p Fp(q)2905 1904 y Fm(i)2933 1889 y Fo(j)2971 1808 y Fp(P)3048 1767 y Fj(?)3034 1833 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)3227 1712 y Fi(i)3266 1735 y Fm(j)412 2093 y Fr(+)1063 2026 y(1)p 581 2070 V 581 2162 a Fp(H)670 2120 y Fn(+)662 2186 y(2)p Fm(N)g Fj(\000)p Fn(3)877 2162 y Fo(\000)23 b Fp(z)1022 2177 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1237 2162 y Fr(+)1335 2095 y Fi(P)1422 2182 y Fm(i)1467 2162 y Fo(j)p Fp(q)1538 2177 y Fm(i)1566 2162 y Fo(j)1604 2093 y Fp(P)1681 2052 y Fj(?)1667 2118 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)1882 2093 y Fo(\002)1227 b Fr(\(IV.64\))571 2358 y Fo(\002)648 2262 y Fi(h)q(\020)760 2358 y Fo(\000)22 b Fp(W)965 2317 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)951 2382 y(2)p Fm(N)g Fj(\000)p Fn(3)1180 2358 y Fo(\000)22 b Fp(E)1357 2317 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)1351 2382 y(2)p Fm(N)g Fj(\000)p Fn(3)1550 2358 y Fr(\))2080 2290 y(1)p 1598 2335 V 1606 2407 a Fi(f)1598 2439 y Fp(H)1687 2398 y Fn(+)1679 2463 y(2)p Fm(N)g Fj(\000)p Fn(4)1894 2439 y Fo(\000)22 b Fp(z)2038 2454 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2254 2439 y Fr(+)2352 2373 y Fi(P)2439 2460 y Fm(i)2484 2439 y Fo(j)p Fp(q)2555 2454 y Fm(i)2583 2439 y Fo(j)2621 2358 y Fp(P)2698 2317 y Fj(?)2684 2382 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)766 2635 y Fr(+)842 2539 y Fi(\020)892 2635 y Fp(z)937 2650 y Fn(2)p Fm(N)g Fj(\000)p Fn(2)1152 2635 y Fo(\000)23 b Fp(z)1297 2650 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1512 2635 y Fr(+)1610 2552 y Fi(X)1654 2734 y Fm(j)1746 2635 y Fo(j)p Fp(k)1825 2650 y Fm(j)1861 2635 y Fo(j)1889 2539 y Fi(\021)2430 2568 y Fr(1)p 1948 2612 V 1956 2684 a Fi(f)1948 2717 y Fp(H)2037 2675 y Fn(+)2029 2741 y(2)p Fm(N)g Fj(\000)p Fn(4)2244 2717 y Fo(\000)23 b Fp(z)2389 2732 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2604 2717 y Fr(+)2702 2650 y Fi(P)2790 2737 y Fm(i)2834 2717 y Fo(j)p Fp(q)2905 2732 y Fm(i)2933 2717 y Fo(j)2971 2635 y Fp(P)3048 2594 y Fj(?)3034 2660 y Fn(2)p Fm(N)g Fj(\000)p Fn(3)3227 2539 y Fi(i)3266 2562 y Fm(N)g Fn(+1)257 2924 y Fr(In)38 b(this)f(truncated)h(expansion,)h(the)f(remainder)f(term)h(\(IV.64\))e (is)i(pro)m(v)m(en)g(to)f(b)s(e)g(of)257 3045 y(order)g Fp(o)p Fr(\()p Fp(\013)664 3008 y Fm(N)731 3045 y Fr(\))f(with)h(resp)s (ect)h(to)e(the)g(original)g(expression)j(in)e(\(IV.62\).)54 b(T)-8 b(o)37 b(see)g(this,)257 3165 y(w)m(e)42 b(use)g(the)f(b)s (ounds)g(on)g(the)g(shift)g(of)f(the)h(in)m(tegration)g(v)-5 b(ariable,)43 b Fp(z)2904 3180 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)3125 3165 y Fo(\000)28 b Fp(z)3275 3180 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)3468 3165 y Fr(,)257 3285 y(and)33 b(on)g(the)g(sum)958 3219 y Fi(P)1046 3306 y Fm(j)1099 3285 y Fo(j)p Fp(k)1178 3300 y Fm(j)1214 3285 y Fo(j)p Fr(:)44 b(Both)32 b(quan)m(tities)j(are)e(of)f(order)h Fp(\033)2590 3300 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2783 3285 y Fr(,)33 b(the)h(\014rst)f(one)g(b)m(y)257 3406 y(de\014nition)39 b(of)f Fp(z)858 3421 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1089 3406 y Fr(and)38 b Fp(z)1329 3421 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1522 3406 y Fr(,)40 b(and)e(the)g(second)i(one)e(b)s(ecause)h(of)f(the)h(fact)e(that)257 3526 y(the)c(n)m(um)m(b)s(er)h(of)e(terms)i(in)1277 3460 y Fi(P)1364 3547 y Fm(j)1418 3526 y Fo(j)p Fp(k)1497 3541 y Fm(j)1532 3526 y Fo(j)f Fr(is)g(\014nite)g(and)g Fp(\013)q Fr(-indep)s(enden)m(t.)257 3647 y(The)k(new)f(features)g(of)e (our)h(expansion)i(of)e(t)m(yp)s(e)h Fo(B)s Fr(,)g(as)g(compared)g(to)f (an)g(expansion)257 3767 y(of)d(t)m(yp)s(e)i Fo(A)p Fr(,)e(are)h(as)f (follo)m(ws:)257 3887 y(The)41 b(replacemen)m(t)g(of)e(the)h(in)m (tegration)g(v)-5 b(ariable)40 b Fp(z)2230 3902 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2462 3887 y Fr(b)m(y)41 b(the)f(v)-5 b(ariable)39 b Fp(z)3197 3902 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)3430 3887 y Fr(is)257 4008 y(p)s(ossible)32 b(b)s(ecause)g(of)e (Estimate)i(\(IV.44\))e(on)h(the)g(sp)s(ectral)g(gap:)42 b(One)31 b(then)h(expands)257 4128 y(in)h(the)g(energies)h Fo(j)p Fp(k)987 4143 y Fm(j)1023 4128 y Fo(j)p Fr(,)e(whic)m(h)j(are)d (b)s(ounded)h(b)m(y)h Fp(const)22 b Fo(\001)g Fp(\033)2441 4143 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)2634 4128 y Fr(.)257 4248 y(The)38 b(norm)e(of)g(the)h(remaining)g(resolv)m(en)m(ts)i(is)d (b)s(ounded)h(b)m(y)h Fp(const)25 b Fo(\001)f Fp(\033)2925 4207 y Fj(\000)p Fn(1)2921 4273 y(2)p Fm(N)7 b Fj(\000)p Fn(3)3114 4248 y Fr(.)55 b(Finally)257 4369 y(w)m(e)34 b(truncate)f Fp(P)868 4333 y Fj(?)854 4393 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)1047 4369 y Fr(.)257 4489 y(Hence,)38 b(starting)e(with)g(a)f(resolv)m(en)m(t)j(of)d(the)h(form)g(giv)m(en)g (in)g(Eq.)h(\(IV.56\),)f(and)g(then)257 4610 y(p)s(erforming)29 b(the)f(op)s(erations)g(just)h(describ)s(ed)h(w)m(e)f(end)g(up)f(with)h (a)f(p)s(olynomial)g(in)h(the)257 4730 y(follo)m(wing)k(op)s(erators:) 257 4850 y(*)g(Resolv)m(en)m(ts,)1441 5028 y(1)p 959 5072 V 967 5144 a Fi(f)959 5176 y Fp(H)1048 5135 y Fn(+)1040 5201 y(2)p Fm(N)7 b Fj(\000)p Fn(4)1255 5176 y Fo(\000)22 b Fp(z)1399 5191 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1615 5176 y Fr(+)1713 5110 y Fi(P)1800 5197 y Fm(i)1845 5176 y Fo(j)p Fp(q)1916 5191 y Fm(i)1944 5176 y Fo(j)2079 5095 y Fp(;)2425 5028 y Fr(1)p 2133 5072 634 4 v 2141 5144 a Fi(f)2133 5176 y Fp(H)2222 5135 y Fn(+)2214 5201 y(2)p Fm(N)g Fj(\000)p Fn(4)2429 5176 y Fo(\000)22 b Fp(z)2573 5191 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(3)2776 5095 y Fr(;)383 b(\(IV.65\))257 5388 y(*)33 b(slice)h(in)m(teractions)f Fp(W)1185 5347 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)1171 5412 y(2)p Fm(N)g Fj(\000)p Fn(3)1378 5388 y Fr(;)p eop end %%Page: 30 31 TeXDict begin 30 30 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(30)257 573 y(*)30 b(energy)h(shifts)g Fp(E)979 532 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)973 597 y(2)p Fm(N)g Fj(\000)p Fn(3)1172 573 y Fr(,)30 b(di\013erence)i(of)d (in)m(tegration)h(v)-5 b(ariables,)32 b Fp(z)2739 588 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)2948 573 y Fo(\000)18 b Fp(z)3088 588 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)3281 573 y Fr(,)30 b(and)257 693 y(energy)k(sums,)841 627 y Fi(P)929 714 y Fm(j)982 693 y Fo(j)p Fp(k)1061 708 y Fm(j)1097 693 y Fo(j)o Fr(.)257 814 y(W)-8 b(e)42 b(pro)s(ceed)h(b)m (y)f(Wic)m(k-ordering)h(the)f(photon)g(creation-)f(and)h(annihilation)g (op)s(er-)257 934 y(ators)j(in)f(the)h(shell)g Fo(K)1130 893 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)1129 958 y(2)p Fm(N)g Fj(\000)p Fn(3)1368 934 y Fr(in)44 b(ev)m(ery)i(term)f(obtained)g(so)f (far,)j(starting)e(from)f(the)257 1054 y(v)m(ector)39 b(in)f(Eq.)h(\(IV.53\),)g(using)f(\(IV.54\),)h(for)e Fp(m)g Fr(=)g(2)p Fp(N)f Fo(\000)26 b Fr(1,)39 b(and)f(then)g (expanding)257 1175 y(the)33 b(resolv)m(en)m(ts)j(as)c(describ)s(ed)j (ab)s(o)m(v)m(e.)44 b(Tw)m(o)34 b(t)m(yp)s(es)f(of)g(resolv)m(en)m(ts)i (result:)1075 1366 y(1)p 381 1410 1437 4 v 381 1502 a Fp(H)470 1461 y Fn(+)462 1526 y(2)p Fm(N)7 b Fj(\000)p Fn(4)676 1502 y Fo(\000)23 b Fp(z)821 1517 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(2)1036 1502 y Fr(+)1134 1436 y Fi(P)1222 1523 y Fm(i)1267 1502 y Fo(j)p Fp(q)1338 1517 y Fm(i)1366 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Fr(,)38 b(and)f(the)h(n)m(um)m(b)s(er) g(of)f(terms)h(is)g(b)s(ounded)g(b)m(y)g(an)f Fp(N)10 b Fr(-dep)s(enden)m(t,)40 b(but)257 1959 y Fp(\013)q Fr(-indep)s(enden)m(t)34 b(in)m(teger.)404 2079 y(After)g(inserting)h (the)f(partition)g(of)f(unit)m(y)-8 b(,)36 b Fg(1)2084 2094 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)2306 2079 y Fr(=)30 b Fp(P)2475 2094 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)2691 2079 y Fr(+)23 b Fp(P)2867 2043 y Fj(?)2853 2104 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(4)3045 2079 y Fr(,)34 b(w)m(e)h(arriv)m(e)257 2200 y(at)26 b(the)h(expressions)i(in)d(\(IV.62\),)i(but)e(with)h(2)p Fp(N)20 b Fo(\000)9 b Fr(3)26 b(replaced)i(b)m(y)f(2)p Fp(N)19 b Fo(\000)9 b Fr(4.)42 b(Th)m(us)28 b(,)f(our)257 2320 y(re-expansion)34 b(pro)s(cedure,)f(based)g(on)f(truncated)h (Neumann)h(expansions)g(of)e(t)m(yp)s(e)h Fo(A)257 2441 y Fr(and)g Fo(B)j Fr(and)d(on)f(Wic)m(k-ordering,)i(can)f(b)s(e)g (iterated.)257 2561 y(By)h(applying)f(the)h(re-expansion)g(pro)s (cedure,)g(scale)g(b)m(y)g(scale,)g(to)e(all)h(the)h(resolv)m(en)m(ts) 257 2681 y(app)s(earing)g(in)f(the)h(truncated)g(pro)5 b(jections)35 b(of)e(Expression)j(\(IV.53\),)d(w)m(e)i(ev)m(en)m (tually)257 2802 y(end)40 b(up)g(with)g(an)g(expansion)g(of)f (\(IV.53\))h(in)m(v)m(olving)h(only)f(\\bare")f(resolv)m(en)m(ts,)k (i.e.,)257 2922 y(resolv)m(en)m(ts)36 b(of)c(the)h(form)1214 3115 y(1)p 660 3159 1158 4 v 660 3251 a Fp(H)749 3209 y Fn(+)741 3272 y(0)830 3251 y Fo(\000)22 b Fp(z)974 3266 y Fn(2)1036 3251 y Fr(+)1134 3184 y Fi(P)1222 3271 y Fm(i)1267 3251 y Fo(j)p Fp(q)1338 3266 y Fm(i)1366 3251 y Fo(j)f Fr(+)1513 3184 y Fi(P)1601 3271 y Fm(i)1625 3253 y Ff(0)1668 3251 y Fo(j)p Fp(q)1743 3216 y Fj(0)1739 3276 y Fm(i)1763 3257 y Ff(0)1790 3251 y Fo(j)1925 3182 y Fp(;)2343 3115 y Fr(1)p 1979 3159 779 4 v 1979 3251 a Fp(H)2068 3209 y Fn(+)2060 3272 y(0)2148 3251 y Fo(\000)i Fp(z)2293 3266 y Fn(1)2355 3251 y Fr(+)2453 3184 y Fi(P)2540 3271 y Fm(i)2564 3253 y Ff(0)2608 3251 y Fo(j)p Fp(q)2683 3216 y Fj(0)2679 3276 y Fm(i)2703 3257 y Ff(0)2729 3251 y Fo(j)2767 3182 y Fp(;)392 b Fr(\(IV.67\))257 3417 y(with)34 b(momen)m(ta)f Fo(f)p Fp(q)998 3432 y Fm(i)1026 3417 y Fo(g)p Fr(,)g Fo(f)p Fp(q)1233 3381 y Fj(0)1229 3442 y Fm(i)1253 3423 y Ff(0)1279 3417 y Fo(g)g Fr(b)s(elonging)f(to)h(the)g (shells)h Fo(K)2429 3381 y Fn(1)2428 3442 y(2)2501 3417 y Fr(and)f Fo(K)2768 3381 y Fn(0)2767 3442 y(1)2808 3417 y Fr(,)f(resp)s(ectiv)m(ely)-8 b(.)404 3538 y(The)32 b(argumen)m(ts)g(describ)s(ed)h(ab)s(o)m(v)m(e)e(represen)m(t)i(the)f (essen)m(tial)h(ingredien)m(ts)g(in)e(the)257 3658 y(re-expansion)i (pro)s(cedure)f(dev)m(elop)s(ed)h(in)f([2)o(],)g(where)g(the)g (mathematical)g(details)g(are)257 3779 y(presen)m(ted.)404 3899 y(The)22 b(main)g(di\016cult)m(y)i(in)e(the)g(inductiv)m(e)i(pro)s (of)d(of)g(con)m(v)m(ergence)k(of)c(the)h(re-expansion)257 4019 y(pro)s(cedure)34 b(is)g(related)f(to)g(determining)i(the)e (energy)h(shifts)g Fp(E)2614 3983 y Fm(n)2608 4044 y(m)2708 4019 y Fr(more)f(explicitly)-8 b(.)47 b(In)257 4140 y(re-expanding)38 b Fp(\036)902 4155 y Fn(2)p Fm(N)7 b Fj(\000)p Fn(1)1095 4140 y Fr(,)38 b(all)f(the)g(energy)h(shifts)g(up)g(to)e(scale)i(2)p Fp(N)e Fo(\000)25 b Fr(1)37 b(app)s(ear)g(in)g(our)257 4260 y(form)m(ulas.)71 b(The)42 b(energy)h(shifts)f(can)g(b)s(e)g (expressed)i(in)d(terms)i(of)e(the)h(groundstate)257 4381 y(v)m(ectors)32 b(on)e(scales)h(up)f(to)g(2)p Fp(N)d Fo(\000)17 b Fr(2.)43 b(Explicit)31 b(expressions)i(for)c(the)i(energy) g(shifts)f Fp(E)3434 4344 y Fm(n)3428 4405 y(m)257 4501 y Fr(are)c(obtained)g(from)f(the)h(re-expansions)h(of)e Fo(f)p Fp(\036)2001 4516 y Fm(i)2029 4501 y Fo(j)p Fp(i)j Fr(=)f(1)p Fp(;)17 b(:::;)g Fr(2)p Fp(N)g Fo(\000)8 b Fr(2)p Fo(g)p Fr(,)27 b(up)f(to)f(remainder)257 4621 y(terms)39 b(of)e Fp(o)p Fr(\()p Fp(\013)798 4585 y Fm(N)865 4621 y Fr(\).)59 b(The)39 b(result)g(of)e(the)h(re-expansion)h(of)e (the)i(groundstate)f(and)g(the)257 4742 y(groundstate)33 b(energy)h(presen)m(ted)h(in)e([2])f(is:)257 4862 y Fk(F)-7 b(or)34 b Fp(\013)29 b Fo(\024)37 b Fr(\026)-58 b Fp(\013)p Fk(,)35 b(with)45 b Fr(\026)-59 b Fp(\013)28 b Fo(\021)38 b Fr(\026)-59 b Fp(\013)1231 4877 y Fm(N)1334 4862 y Fk(smal)5 b(l)34 b(enough,)g(the)h(gr)-5 b(oundstate)34 b(ener)-5 b(gy)35 b Fp(E)3017 4877 y Fn(gs)3112 4862 y Fo(\021)28 b Fp(E)3289 4877 y Fn(gs)3357 4862 y Fr(\()p Fp(\013)q Fr(\))p eop end %%Page: 31 32 TeXDict begin 31 31 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(31)257 573 y Fk(and)35 b(the)g(gr)-5 b(oundstate)34 b Fp(\036)1188 588 y Fn(gs)1283 573 y Fo(\021)28 b Fp(\036)1446 588 y Fn(gs)1513 573 y Fr(\()p Fp(\013)1624 509 y Fd(1)p 1624 521 31 4 v 1624 562 a(2)1668 573 y Fr(\))35 b Fk(have)f(exp)-5 b(ansions)33 b(of)i(the)g(form)1036 845 y Fp(E)1108 860 y Fn(gs)1176 845 y Fr(\()p Fp(\013)q Fr(\))82 b(=)h Fp(E)1628 860 y Fn(0)1690 845 y Fr(+)1816 737 y Fm(N)1788 762 y Fi(X)1788 947 y Fm(`)p Fn(=3)1924 845 y Fp(")1970 860 y Fm(`)2020 845 y Fr(\()p Fp(\013)q Fr(\))17 b 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b(w)m(e)g(ha)m(v)m(e)g(used)g(the)f(standard)g(de\014nition) h(for)e Fp(T)14 b Fr(,)24 b(the)g(T-matrix)f(op)s(erator.)39 b(The)257 5388 y(leading)30 b(order)f(term)g(is)g(of)g(order)g Fp(\013)1586 5352 y Fn(3)1654 5388 y Fr(and)g(arises)g(from)g(the)g (expression)i(in)f(Eq.)f(\(V.2\).)p eop end %%Page: 32 33 TeXDict begin 32 32 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(32)257 573 y(In)33 b(fact,)f(due)g(to)g(one)g (p)s(o)m(w)m(er)h(of)f Fp(\013)g Fr(m)m(ultiplying)i(the)e(electron)h (p)s(osition)f(op)s(erator,)e Fp(~)-47 b(x)p Fr(,)257 693 y(in)31 b(the)g(transv)m(erse)h(curren)m(t,)g(the)f(term)g(in)f (Eq.)i(\(V.3\))e(is)h(of)e(order)i Fp(\013)2817 657 y Fn(4)2856 693 y Fr(.)43 b(In)30 b(computing)257 814 y(the)j(leading)g (term,)h(w)m(e)f(\014rst)g(rewrite)h(the)f(in)m(tegrand)g(in)g (Expression)i(\(V.2\))d(as)810 1060 y Fp(e)855 1019 y Fm(iE)931 1027 y Fe(g)r(s)996 1019 y Fn(\()p Fm(t)p Fj(\000)p Fm(s)p Fn(\))p 1168 953 204 4 v 1182 1034 a Fp(~)1168 1060 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Fm(~)934 3910 y(k)q Fj(j)p Fn(\))p Fm(t)1049 3951 y Fp(e)1094 3910 y Fj(\000)p Fm(i)p Fn(\()p Fj(\000E)1292 3918 y Fe(n)1335 3910 y Fn(+)p Fj(E)1427 3919 y Fd(0)1462 3910 y Fn(+)p Fj(j)p Fm(~)-35 b(q)q Fj(j)p Fn(\))p Fm(s)1654 3951 y Fp(\022)s Fr(\()p Fp(t)22 b Fo(\000)h Fp(s)p Fr(\))p 1991 3778 198 4 v 2012 3857 a(^)1991 3884 y Fp(f)2050 3855 y Fm(\025)p Fn(\()2115 3837 y Fm(~)2118 3855 y(k)r Fn(\))p 1991 3928 V 1993 3944 a Fi(q)p 2076 3944 110 4 v 112 x Fo(j)2099 4029 y Fp(~)2104 4056 y(k)s Fo(j)2209 3857 y Fr(^)2208 3884 y Fp(h)2264 3848 y Fm(\025)5 b Fj(0)2333 3884 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 2208 3928 248 4 v 2239 3944 a Fi(q)p 2322 3944 103 4 v 100 x Fo(j)o Fp(~)g(q)t Fo(j)2475 3884 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2475 3928 499 4 v 2700 4019 a(8)2984 3951 y Fp(d)3035 3910 y Fn(3)3074 3951 y Fp(k)s(d)3179 3910 y Fn(3)3218 3951 y Fp(q)t(:)257 4268 y Fr(where)31 b(\001)617 4283 y Fm(n)664 4268 y Fo(E)36 b Fr(:=)28 b Fo(E)937 4283 y Fm(n)999 4268 y Fo(\000)15 b(E)1144 4283 y Fn(0)1212 4268 y Fr(and)30 b Fo(E)1452 4283 y Fm(n)1527 4268 y Fr(is)g(the)g(energy)g(of)f(the)g(eigenstate)i Fp( )2883 4283 y Fm(n)2930 4268 y Fr(.)42 b(In)m(tro)s(ducing)257 4388 y(the)33 b(v)-5 b(ariable)33 b Fp(u)27 b Fr(:=)h Fp(t)22 b Fo(\000)h Fp(s)28 b(>)f Fr(0,)33 b(the)g(in)m(tegral)g(in)f (Eq.)i(\(V.8\))e(can)h(b)s(e)g(written)g(as)329 4652 y Fp(i)379 4535 y Fi(Z)462 4561 y Fn(+)p Fj(1)425 4723 y(\0001)608 4652 y Fp(dt)711 4535 y Fi(Z)793 4561 y Fn(+)p Fj(1)757 4723 y Fn(0)940 4652 y Fp(du)1064 4535 y Fi(Z)52 b(Z)1245 4652 y Fr(\(\001)1364 4667 y Fm(n)1411 4652 y Fo(E)9 b Fr(\))1511 4611 y Fn(2)1569 4569 y Fi(X)1567 4754 y Fm(\025;\025)1669 4735 y Ff(0)1691 4652 y Fr(\()p Fp( )1792 4667 y Fn(0)1832 4652 y Fp(;)15 b(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")p Fr(\()2082 4626 y Fp(~)2087 4652 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2343 4667 y Fm(n)2390 4652 y Fr(\)\()p Fp( )2529 4667 y Fm(n)2576 4652 y Fp(;)e(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")p Fr(\()o Fp(~)g(q)s(;)17 b(\025)2978 4611 y Fj(0)3001 4652 y Fr(\))3039 4611 y Fj(\003)3079 4652 y Fp( )3142 4667 y Fn(0)3181 4652 y Fr(\))p Fo(\002)-26 b Fr(\(V.9\))1080 4990 y Fo(\002)p Fp(e)1202 4948 y Fm(i)p Fn(\()p Fj(j)1270 4931 y Fm(~)1273 4948 y(k)r Fj(j\000j)p Fm(~)-35 b(q)q Fj(j)p Fn(\))p Fm(t)1517 4990 y Fp(e)1562 4948 y Fm(i)p Fn(\()p Fj(\000E)1705 4956 y Fe(n)1748 4948 y Fn(+)p Fj(E)1840 4957 y Fd(0)1875 4948 y Fn(+)p Fj(j)p Fm(~)g(q)q Fj(j)p Fn(\))p Fm(u)p 2086 4815 234 4 v 2107 4896 a Fr(^)2086 4922 y Fp(f)2145 4893 y Fm(\025)2190 4922 y Fr(\()2223 4896 y Fp(~)2228 4922 y(k)s Fr(\))p 2086 4966 V 2106 4983 a Fi(q)p 2189 4983 110 4 v 111 x Fo(j)2212 5068 y Fp(~)2217 5094 y(k)s Fo(j)2340 4896 y Fr(^)2339 4922 y Fp(h)2395 4886 y Fm(\025)5 b Fj(0)2464 4922 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 2339 4966 248 4 v 2371 4983 a Fi(q)p 2454 4983 103 4 v 100 x Fo(j)o Fp(~)g(q)s Fo(j)2607 4922 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2607 4966 499 4 v 2832 5058 a(8)3116 4990 y Fp(d)3167 4948 y Fn(3)3206 4990 y Fp(k)s(d)3311 4948 y Fn(3)3350 4990 y Fp(q)t(:)257 5323 y Fr(W)-8 b(e)36 b(insert)g(regularizing)g (factors)f Fp(e)1601 5287 y Fj(\000)p Fm(\017)p Fj(j)p Fm(t)p Fj(j)1753 5323 y Fr(,)h Fp(\017)c(>)g Fr(0,)k(and)f Fp(e)2344 5287 y Fj(\000)p Fm(\026)p Fj(j)p Fm(u)p Fj(j)2526 5323 y Fr(,)g Fp(\026)d(>)g Fr(0,)j(and)h(then)f(pass)p eop end %%Page: 33 34 TeXDict begin 33 33 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(33)257 573 y(to)33 b(the)g(limits)g Fp(\017)28 b Fo(!)f Fr(0)33 b(and)f Fp(\026)c Fo(!)f Fr(0)32 b(of)423 817 y Fp(i)473 700 y Fi(Z)556 726 y Fn(+)p Fj(1)519 889 y(\0001)703 817 y Fp(dt)806 700 y Fi(Z)888 726 y Fn(+)p Fj(1)851 889 y Fn(0)1034 817 y Fp(du)1158 700 y Fi(Z)53 b(Z)1340 817 y Fr(\(\001)1459 832 y Fm(n)1506 817 y Fo(E)9 b Fr(\))1606 776 y Fn(2)1664 734 y Fi(X)1661 919 y Fm(\025;\025)1763 900 y Ff(0)1786 817 y Fr(\()p Fp( )1887 832 y Fn(0)1927 817 y Fp(;)15 b(~)-47 b(x)22 b Fo(\001)e Fp(~)-47 b(")o Fr(\()2176 791 y Fp(~)2181 817 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2437 832 y Fm(n)2484 817 y Fr(\)\()p Fp( )2623 832 y Fm(n)2670 817 y Fp(;)e(~)-47 b(x)23 b Fo(\001)d Fp(~)-47 b(")o Fr(\()o Fp(~)f(q)t(;)17 b(\025)3073 776 y Fj(0)3096 817 y Fr(\))3134 776 y Fj(\003)3173 817 y Fp( )3236 832 y Fn(0)3276 817 y Fr(\))i(\(V.10\))423 1155 y Fo(\002)p Fp(e)545 1114 y Fm(i)p Fn(\()p Fj(j)613 1096 y Fm(~)616 1114 y(k)s Fj(j\000j)p Fm(~)-35 b(q)q Fj(j)p Fn(\))p Fm(t)860 1155 y Fp(e)905 1114 y Fj(\000)p Fm(\017)p Fj(j)p Fm(t)p Fj(j)1058 1155 y Fp(e)1103 1114 y Fm(i)p Fn(\()p Fj(\000E)1246 1122 y Fe(n)1289 1114 y Fn(+)p Fj(E)1381 1123 y Fd(0)1416 1114 y Fn(+)p Fj(j)p Fm(~)g(q)q Fj(j)p Fn(\))p Fm(u)1616 1155 y Fp(e)1661 1114 y Fj(\000)p Fm(\026u)p 1814 980 234 4 v 1835 1061 a Fr(^)1814 1087 y Fp(f)1873 1059 y Fm(\025)1918 1087 y Fr(\()1951 1061 y Fp(~)1956 1087 y(k)s Fr(\))p 1814 1131 V 1835 1148 a Fi(q)p 1918 1148 110 4 v 111 x Fo(j)1941 1233 y Fp(~)1946 1259 y(k)r Fo(j)2069 1061 y Fr(^)2068 1087 y Fp(h)2124 1051 y Fm(\025)t Fj(0)2192 1087 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 2068 1131 248 4 v 2099 1148 a Fi(q)p 2182 1148 103 4 v 100 x Fo(j)o Fp(~)g(q)s Fo(j)2335 1087 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2335 1131 499 4 v 2560 1223 a(8)2844 1155 y Fp(d)2895 1114 y Fn(3)2934 1155 y Fp(k)s(d)3039 1114 y Fn(3)3078 1155 y Fp(q)t(:)257 1457 y Fr(An)33 b(explicit)h(calculation)g(giv)m (es)257 1688 y Fp(l)r(im)406 1703 y Fm(\017)p Fj(!)p Fn(0)546 1688 y Fp(l)r(im)695 1703 y Fm(\026)p Fj(!)p Fn(0)848 1688 y Fp(i)898 1571 y Fi(Z)998 1688 y Fp(d)1049 1647 y Fn(3)1088 1688 y Fp(q)1151 1571 y Fi(Z)1251 1688 y Fp(d)1302 1647 y Fn(3)1341 1688 y Fp(k)s Fr(\(\001)p Fo(E)1567 1703 y Fm(n)1614 1688 y Fr(\))1652 1647 y Fn(2)1710 1605 y Fi(X)1708 1790 y Fm(\025;\025)1810 1771 y Ff(0)1832 1688 y Fr(\()p Fp( )1933 1703 y Fn(0)1973 1688 y Fp(;)15 b(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")p Fr(\()2223 1661 y Fp(~)2228 1688 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2484 1703 y Fm(n)2530 1688 y Fr(\)\()p Fp( )2669 1703 y Fm(n)2717 1688 y Fp(;)e(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")o Fr(\()o Fp(~)g(q)t(;)17 b(\025)3119 1647 y Fj(0)3142 1688 y Fr(\))3180 1647 y Fj(\003)3220 1688 y Fp( )3283 1703 y Fn(0)3322 1688 y Fr(\))p Fo(\002)474 b Fr(\(V.11\))694 2025 y Fo(\002)1062 1958 y Fr(2)p Fp(\017)p 781 2002 648 4 v 781 2109 a Fr(\()p Fo(j)842 2083 y Fp(~)847 2109 y(k)t Fo(j)21 b(\000)i(j)o Fp(~)-48 b(q)s Fo(j)p Fr(\))1191 2081 y Fn(2)1253 2109 y Fr(+)22 b Fp(\017)1390 2081 y Fn(2)1462 2025 y Fo(\001)1879 1958 y Fp(i)p 1521 2002 748 4 v 1521 2094 a Fo(E)1574 2109 y Fn(0)1636 2094 y Fo(\000)g(E)1788 2109 y Fm(n)1857 2094 y Fr(+)g Fo(j)o Fp(~)-48 b(q)s Fo(j)22 b Fr(+)g Fp(i\026)p 2289 1851 234 4 v 2310 1932 a Fr(^)2289 1958 y Fp(f)2348 1929 y Fm(\025)2393 1958 y Fr(\()2426 1932 y Fp(~)2431 1958 y(k)s Fr(\))p 2289 2002 V 2310 2019 a Fi(q)p 2393 2019 110 4 v 111 x Fo(j)2416 2104 y Fp(~)2421 2130 y(k)s Fo(j)2544 1932 y Fr(^)2543 1958 y Fp(h)2599 1922 y Fm(\025)t Fj(0)2668 1958 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 2543 2002 248 4 v 2574 2019 a Fi(q)p 2657 2019 103 4 v 100 x Fo(j)o Fp(~)g(q)t Fo(j)2810 1958 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2810 2002 499 4 v 3035 2094 a(8)535 2315 y(=)83 b Fp(i)744 2198 y Fi(Z)54 b(Z)944 2315 y Fp(d)995 2274 y Fn(3)1034 2315 y Fp(q)t(d)1132 2274 y Fn(3)1171 2315 y Fp(k)s Fr(\(\001)1344 2330 y Fm(n)1391 2315 y Fo(E)9 b Fr(\))1491 2274 y Fn(2)1549 2232 y Fi(X)1546 2417 y Fm(\025;\025)1648 2398 y Ff(0)1671 2315 y Fr(\()p Fp( )1772 2330 y Fn(0)1811 2315 y Fp(;)15 b(~)-47 b(x)23 b Fo(\001)d Fp(~)-47 b(")o Fr(\()2061 2289 y Fp(~)2066 2315 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2322 2330 y Fm(n)2369 2315 y Fr(\)\()p Fp( )2508 2330 y Fm(n)2555 2315 y Fp(;)e(~)-47 b(x)23 b Fo(\001)d Fp(~)-48 b(")p Fr(\()o Fp(~)g(q)t(;)17 b(\025)2958 2274 y Fj(0)2981 2315 y Fr(\))3019 2274 y Fj(\003)3058 2315 y Fp( )3121 2330 y Fn(0)3161 2315 y Fr(\))22 b Fo(\002)613 b Fr(\(V.12\))792 2653 y Fo(\002)p Fr(2)p Fp(\031)977 2612 y Fn(2)1017 2653 y Fp(\016)t Fr(\()p Fo(j)1125 2627 y Fp(~)1130 2653 y(k)r Fo(j)22 b(\000)h(j)o Fp(~)-48 b(q)s Fo(j)p Fr(\))p Fp(\016)t Fr(\()p Fo(E)1611 2668 y Fn(0)1672 2653 y Fo(\000)22 b(E)1824 2668 y Fm(n)1893 2653 y Fr(+)g Fo(j)o Fp(~)-48 b(q)t Fo(j)p Fr(\))p 2142 2478 234 4 v 2162 2559 a(^)2142 2585 y Fp(f)2201 2557 y Fm(\025)2245 2585 y Fr(\()2278 2559 y Fp(~)2283 2585 y(k)s Fr(\))p 2142 2630 V 2162 2646 a Fi(q)p 2245 2646 110 4 v 112 x Fo(j)2268 2731 y Fp(~)2273 2758 y(k)s Fo(j)2396 2559 y Fr(^)2395 2585 y Fp(h)2451 2549 y Fm(\025)5 b Fj(0)2520 2585 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 2395 2630 248 4 v 2426 2646 a Fi(q)p 2509 2646 103 4 v 100 x Fo(j)o Fp(~)g(q)t Fo(j)2662 2585 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2662 2630 499 4 v 2887 2721 a(8)694 2957 y Fo(\000)p Fp(P)865 2840 y Fi(Z)965 2957 y Fp(d)1016 2916 y Fn(3)1055 2957 y Fp(q)1355 2890 y Fr(1)p 1112 2934 536 4 v 1112 3025 a Fo(E)1165 3040 y Fn(0)1226 3025 y Fo(\000)22 b(E)1378 3040 y Fm(n)1447 3025 y Fr(+)g Fo(j)o Fp(~)-48 b(q)s Fo(j)1674 2840 y Fi(Z)1773 2957 y Fp(d)1824 2916 y Fn(3)1863 2957 y Fp(k)s Fr(\(\001)2036 2972 y Fm(n)2084 2957 y Fo(E)9 b Fr(\))2184 2916 y Fn(2)2241 2874 y Fi(X)2239 3059 y Fm(\025;\025)2341 3040 y Ff(0)2363 2957 y Fr(\()p Fp( )2464 2972 y Fn(0)2504 2957 y Fp(;)15 b(~)-47 b(x)23 b Fo(\001)d Fp(~)-48 b(")p Fr(\()2754 2931 y Fp(~)2759 2957 y(k)s(;)17 b(\025)p Fr(\))p Fp( )3015 2972 y Fm(n)3062 2957 y Fr(\)\()p Fp( )3201 2972 y Fm(n)3248 2957 y Fp(;)e(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")p Fr(\()o Fp(~)g(q)t(;)17 b(\025)3651 2916 y Fj(0)3674 2957 y Fr(\))3712 2916 y Fj(\003)3751 2957 y Fp( )3814 2972 y Fn(0)3854 2957 y Fr(\))i(\(V.13\))792 3295 y Fo(\002)p Fr(2)p Fp(\031)t(\016)t Fr(\()p Fo(j)1085 3268 y Fp(~)1090 3295 y(k)s Fo(j)j(\000)g(j)o Fp(~)-48 b(q)t Fo(j)p Fr(\))p 1444 3120 234 4 v 1464 3201 a(^)1444 3227 y Fp(f)1503 3199 y Fm(\025)1547 3227 y Fr(\()1580 3201 y Fp(~)1585 3227 y(k)s Fr(\))p 1444 3272 V 1464 3288 a Fi(q)p 1547 3288 110 4 v 111 x Fo(j)1570 3373 y Fp(~)1575 3399 y(k)s Fo(j)1698 3201 y Fr(^)1697 3227 y Fp(h)1753 3191 y Fm(\025)5 b Fj(0)1822 3227 y Fr(\()o Fp(~)-48 b(q)t Fr(\))p 1697 3272 248 4 v 1728 3288 a Fi(q)p 1811 3288 103 4 v 100 x Fo(j)o Fp(~)g(q)t Fo(j)1965 3227 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 1965 3272 499 4 v 2190 3363 a(8)257 3610 y(where)30 b Fp(P)42 b Fr(stands)30 b(for)e(the)h (principal)h(part)e(of)2127 3571 y Fn(1)p 1977 3587 335 4 v 1977 3645 a Fj(E)2014 3654 y Fd(0)2049 3645 y Fj(\000E)2141 3653 y Fe(n)2184 3645 y Fn(+)p Fj(j)p Fm(~)-35 b(q)q Fj(j)2322 3610 y Fr(.)42 b(By)29 b(similar)g(calculations)h(w)m(e)257 3744 y(see)k(that)e(con)m(tribution)i(in)f(Eq.)g(\(V.5\))g(giv)m(es:) 423 3980 y Fp(i)473 3863 y Fi(Z)54 b(Z)673 3980 y Fp(d)724 3939 y Fn(3)763 3980 y Fp(q)t(d)861 3939 y Fn(3)900 3980 y Fp(k)s Fr(\(\001)1073 3995 y Fm(n)1120 3980 y Fo(E)9 b Fr(\))1220 3939 y Fn(2)1278 3897 y Fi(X)1275 4082 y Fm(\025;\025)1377 4063 y Ff(0)1400 3980 y Fr(\()p Fp( )1501 3995 y Fn(0)1540 3980 y Fp(;)15 b(~)-47 b(x)23 b Fo(\001)d Fp(~)-47 b(")o Fr(\()1790 3954 y Fp(~)1795 3980 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2051 3995 y Fm(n)2098 3980 y Fr(\)\()p Fp( )2237 3995 y Fm(n)2284 3980 y Fp(;)e(~)-47 b(x)23 b Fo(\001)d Fp(~)-48 b(")p Fr(\()o Fp(~)g(q)t(;)17 b(\025)2687 3939 y Fj(0)2710 3980 y Fr(\))2748 3939 y Fj(\003)2787 3980 y Fp( )2850 3995 y Fn(0)2890 3980 y Fr(\))22 b Fo(\002)612 b Fr(\(V.14\))521 4318 y Fo(\002)p Fr(2)p Fp(\031)706 4277 y Fn(2)746 4318 y Fp(\016)t Fr(\()p Fo(j)854 4291 y Fp(~)859 4318 y(k)r Fo(j)22 b(\000)h(j)o Fp(~)-48 b(q)s Fo(j)p Fr(\))p Fp(\016)t Fr(\()p Fo(\000E)1417 4333 y Fn(0)1478 4318 y Fr(+)22 b Fo(E)1629 4333 y Fm(n)1698 4318 y Fr(+)g Fo(j)o Fp(~)-48 b(q)s Fo(j)p Fr(\))p 1946 4143 234 4 v 1967 4224 a(^)1946 4250 y Fp(f)2005 4222 y Fm(\025)2050 4250 y Fr(\()2083 4224 y Fp(~)2088 4250 y(k)s Fr(\))p 1946 4294 V 1967 4311 a Fi(q)p 2050 4311 110 4 v 111 x Fo(j)2073 4396 y Fp(~)2078 4422 y(k)s Fo(j)2201 4224 y Fr(^)2200 4250 y Fp(h)2256 4214 y Fm(\025)t Fj(0)2325 4250 y Fr(\()o Fp(~)g(q)s Fr(\))p 2200 4294 248 4 v 2231 4311 a Fi(q)p 2314 4311 103 4 v 100 x Fo(j)o Fp(~)g(q)t Fo(j)2467 4250 y Fr(\003\()p Fo(j)p Fp(k)s Fo(j)p Fr(\)\003\()p Fo(j)p Fp(q)t Fo(j)p Fr(\))p 2467 4294 499 4 v 2692 4386 a(8)423 4622 y Fo(\000)517 4505 y Fi(Z)617 4622 y Fp(d)668 4581 y Fn(3)707 4622 y Fp(q)1045 4555 y Fr(1)p 764 4599 612 4 v 764 4690 a Fo(\000E)894 4705 y Fn(0)956 4690 y Fr(+)22 b Fo(E)1107 4705 y Fm(n)1175 4690 y Fr(+)g Fo(j)o Fp(~)-48 b(q)t Fo(j)1402 4505 y Fi(Z)1502 4622 y Fp(d)1553 4581 y Fn(3)1592 4622 y Fp(k)s Fr(\(\001)1765 4637 y Fm(n)1812 4622 y Fo(E)9 b Fr(\))1912 4581 y Fn(2)1970 4539 y Fi(X)1968 4724 y Fm(\025;\025)2070 4705 y Ff(0)2092 4622 y Fr(\()p Fp( )2193 4637 y Fn(0)2233 4622 y Fp(;)15 b(~)-47 b(x)22 b Fo(\001)f Fp(~)-48 b(")o Fr(\()2482 4596 y Fp(~)2487 4622 y(k)s(;)17 b(\025)p Fr(\))p Fp( )2743 4637 y Fm(n)2790 4622 y Fr(\)\()p Fp( )2929 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(a)h(\(connected\))i(scattering)f(amplitude)257 5388 y(of)38 b(the)g(t)m(yp)s(e)g(calculated)h(ab)s(o)m(v)m(e,)h(with)f Fp(f)47 b Fr(=)36 b Fp(h)p Fr(,)j(is)g(di\013eren)m(t)g(from)e(zero,)j (in)e(leading)p eop end %%Page: 34 35 TeXDict begin 34 34 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(34)257 573 y(order,)35 b(only)g(if)g(the)f (photon)h(w)m(a)m(v)m(e)h(function)f(do)s(es)g(not)f(v)-5 b(anish)35 b(for)f(photon)h(energies)257 693 y(corresp)s(onding)d(to)e (a)h(di\013erence,)i Fo(E)1594 708 y Fm(n)1659 693 y Fo(\000)19 b(E)1808 708 y Fn(0)1847 693 y Fr(,)31 b(of)f(the)h(energy)h Fo(E)2543 708 y Fm(n)2620 693 y Fr(of)e(an)h(excited)h(b)s(ound-)257 814 y(state)39 b(and)g(the)f(groundstate)h(energy)h Fo(E)1785 829 y Fn(0)1824 814 y Fr(.)60 b(In)39 b(fact,)h(it)e(corresp)s(onds)i (to)e(Expression)257 934 y(\(V.12\).)43 b(Assuming)33 b(the)f(v)-5 b(alidit)m(y)33 b(of)e(the)g(optical)h(theorem,)h(the)f (total)f(cross)h(section)257 1054 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4830 y([7])49 b(J.)29 b(F)-8 b(r\177)-49 b(ohlic)m(h,)30 b(M.Griesemer,)i(B.)d(Sc)m(hlein.)40 b(Asymptotic)30 b(electromagnetic)h(\014elds)458 4951 y(in)d(mo)s(dels)g(of)f(quan)m(tum-mec)m(hanical)j(matter)d(in)m (teracting)i(with)f(the)f(quan)m(tized)458 5071 y(radiation)32 b(\014eld.)44 b Fk(A)-5 b(dv.)35 b(Math.)e Fr(164)f(\(2\):)43 b(349{398,)31 b(2001)306 5268 y([8])49 b(J.)j(F)-8 b(r\177)-49 b(ohlic)m(h,)58 b(M.Griesemer,)i(B.)52 b(Sc)m(hlein.)104 b(Asymptotic)54 b(completeness)i(for)458 5388 y(Ra)m(yleigh)33 b(scattering.)44 b Fk(A)n(nn.)35 b(H.)g(Poinc)-5 b(ar)n(\023)-47 b(e)p Fr(,3:)41 b(107{170,)31 b(2002)p eop end %%Page: 35 36 TeXDict begin 35 35 bop 257 266 a Fs(BFP-Ra)m(ylScatt,)33 b(24-April-2006)1852 b Fr(35)306 573 y([9])49 b(M.Griesemer.)122 b(Exp)s(onen)m(tial)60 b(deca)m(y)g(and)f(ionization)f(threshold)i(in)f (non-)458 693 y(relativistic)30 b(quan)m(tum)g(electro)s(dynamics.)39 b Fk(J.)31 b(F)-7 b(unct.)30 b(A)n(nal.)p Fr(,210)e(\(3\):)41 b(321{340,)458 814 y(2004)257 1017 y([10])49 b(A.Pizzo.)90 b(One-particle)49 b(\(improp)s(er\))f(states)h(in)f(Nelson's)h (massless)i(mo)s(del.)458 1137 y Fk(A)n(nn.)34 b(H.)h(Poinc)-5 b(ar)n(\023)-47 b(e)p Fr(,)31 b(4\(3\):439{486,)g(2003.)257 1341 y([11])49 b(A.Pizzo.)d(Scattering)33 b(of)g(an)g(Infraparticle:)45 b(the)33 b(One-P)m(article)h(Sector)g(in)f(Nel-)458 1461 y(son's)g(massless)i(mo)s(del.)44 b Fk(A)n(nn.)34 b(H.)i(Poinc)-5 b(ar)n(\023)-47 b(e)p Fr(,)30 b(6)j(:553{606,)e(2005.)257 1665 y([12])49 b(Reed)33 b(Simon.)44 b(Scattering)33 b(Theory)-8 b(.)44 b Fk(A)-5 b(c)g(ademic)35 b(Pr)-5 b(ess)p Fr(.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0604301720547--