Content-Type: multipart/mixed; boundary="-------------0408270802292" This is a multi-part message in MIME format. ---------------0408270802292 Content-Type: text/plain; name="04-262.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-262.comments" revised version ---------------0408270802292 Content-Type: text/plain; name="04-262.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-262.keywords" scattering states, infraparticle, infrared problem, Q.E.D. ---------------0408270802292 Content-Type: application/postscript; name="segundo2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="segundo2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: segundo2.dvi %%Pages: 53 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMMI10 CMMI7 CMR5 CMR10 CMMI9 CMMI6 CMR6 CMR9 MSBM10 %%+ CMBX9 CMSY9 TeX-cmex9 CMSY6 CMBX6 MSBM7 CMSY8 CMR8 TeX-cmex8 CMMI8 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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: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 D10F04E2CB7C8461B10646CD63AFEB7608468CA0FCFC4D3458FB43D22879B515 27DD9CCF44C2BFCD95A4DE911E4915FBC02335E9999FD9B546134081D6DA3792 EC4A76DEBA77635BE52E09986268A919CB48B5EFB1A1301EE0683CB5709BC8CE D819D799020CBA673BA39C911075501395B1FD20EAD392C9D5A8C9FD1198C737 D1A614CF0C0432F29DDEB4BF9DB026908DBE89EE522B7D55DE9BF64AFBE6248E 2E10466655EB9083E7D23E3F0EE26154F191BEBC9987930CD4B4CABE1275BDF9 8755EF3D531FDF91D54954FC53F15A38D1E8F8D1E36447484FA2C09D34813615 838B6330FEAE536D08376E4A0FDDF58CDF5647C9F1FF3A7D1ACAD376DB3CADB6 9459F7A5D4F1864863B79E9F93A1EDE8B99C3138D26227C01F6FE0AAC800F2E5 94DD81CF7B1355B642CE45CB532FC5B535D66EDFFEA076C009E87406D9772D71 848C3C53B7496A5D6B58679EF11E114C5F457C6A0D3CDE50278E4A89D5393B1C F877CF4E2142A4D045C4AA9138105D748903BACC28FD43DFEDB341E1FCDBE2EA D412498FBB5374D6836CFBEB13D4C2B7B9625C25B037FDA9DCC42F5679C4B3C1 6340E341F73A921013790936801297855EDE27C68907C6C87633FE64CBDCFEC9 DD8EC8784857A09FE395D7FE095A87CC113B930AD09643AC6F9F5A44A1154C89 7951C1CEF380FE489886299A1C5A2E1FB2D475FD3478BEDD0EC21A4CBAD46924 9AF698FCD47B9891629762AAD9E00C0C31B4F54AA5E59A575DB50A8961F05D7B D54237D5CE59FDB5C3E83AFEB532CC4F900F7E85358A528D56A6FBE9F3002E56 EA59E9C2459EEA543F906E74846306E52E324E00D300A78B984ADD86E21B5C08 4E19F5F28704CCA9BC6F5B8E11C74674881613A81A1763E4217B4FC14BD6FF18 B37A14375DF5F7CCD561DC7ABC388F4539424239C618F1D92CD37F507FF94133 927748C386DF1C1EA2F61DC341C76A4E72E64AA9723292B83079A9C3FCEA7355 51666A804736DB26A32A5E44562BF0EEC98870D116148D9E17AFD856254CF45F 25F15F274DE5DBC4B3CCF7C6FF153BF8242EF5EDC3FCCD9B43F935521B1FF4EE B28366EAFBD200118C925D66D52481AB06AE5A5625621A8AF272644946C20207 24365BD70A52C4528DCE46B91DC303AE2B893D2092BAF0549F41212112F76EAE BF473258D66078DA79B12C2A33BCDC15B23546FAD43454AE82431C133C810075 2B59E52FF11F7CF77CA45C8A17AD59D4847D0CFF1209F7DC1A856D12F1D1EE4C 83CBC71DD3D37DC5639AF38B8D7B05594D3707D6D1E097610A0E16BD6F1882EA 553860902939460F9170C1D2F9EB293EB3EF0909F9C27E946921C9AAA772DC77 B1CB264C4C73BD4963D0DDB3C184889BD8AC3EAB60609684D051312506B99A06 C819A6CFFEB6AB715BD14A5E5BC4666DEE9234385BF7DD9679EAA06626FE3801 938D6AA08EF6A0CBB97ECD97B1C2AAF4971535E3BF61CAD947B712EDFC99C18E 601BDCAF8FD8FAFDDED104A1D9FF1260AA5E00607519C5BDAAB857B63C902E6C 3A8AF90A6CB7AD7463232B6A2BEAC5FD75A6F69DF254B3F859BD06A632B32AF2 27BA2914514732E74A332F6587E1352B64E0E5053B6646A1265653041F9B446F 7FA484367B729C0C726E190F204E4F94103BD8F8E103D7D8B204D3BF9BCAD2F0 76803AD0C81922885ED79DF4E0103D152A1DFED6DA7FA2213407D5E172862184 2A2FCBB49016627BA7866BAD6AB0B9E02A36EAECB500F9BB2BC0C8C0FC378AA5 E54618BE5B0780A8364329CFAA166390CAA72D77254470E8849F9735924D2385 E4BE9BC60FC412481B5C05277917D5C576D44C1E0F10CA72206B0FC0E5290D0C E5DBE8DEF2F080C5647547478024F22307F44E1C9688DA9F8E8DA71804C1ED19 5C67E96717D02D74ED95A855B7ECB495D60F149FFD1E86986A6FF994BBD1877C E37DE7F2A6EA2BC93E7B49819BAC5FB8929C616BDB5C793613A677DD11C0B652 B1319E98F8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 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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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: TeX-cmex7 %!PS-AdobeFont-1.0: TeX-cmex7 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex7) def /FullName (TeX\040cmex7\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 16 /quotedblleft put dup 17 /quotedblright put dup 18 /quotedblbase put dup 19 /guillemotleft put dup 32 /space put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 82 /R put dup 98 /b put readonly def /FontBBox {-14 -2954 1627 771} readonly def /UniqueID 4314415 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BAE4F093966CD2192CE95EB0F323A6BABFDACCFCF27D91F7869A0E46CA5 9AAF6905783E8AC1F3F9875A76F97187738432F8D14E61574CB292FFB9740871 66839799D8CAF6E0DFE00012EE6D46A2B3655F29705BE37FD5EDA1C765AA2CF5 C5AD37207ED1EE9DB82FF31A33307FFA16911406557336AF92F50B603C7BD336 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1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX5 %!PS-AdobeFont-1.1: CMBX5 1.0 %%CreationDate: 1991 Aug 20 16:35:12 % 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 (CMBX5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX5 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{-40 -250 1483 756}readonly def /UniqueID 5000763 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F068F48FD40DA38A480ECD24C95AA52D1EB8D 502E7D1E35B648C712F04A3F32986A7D18D3C6D4AC7333184C6D44BDF95915A2 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 126 /planckover2pi1 put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 F992DEA5FC3897E2CC8B7A90238E61E41622DE80F438DD994C73275CC52249D9 F6686F87F394FB7BB668138B210BEC9E46415A1B58C990B81E7D7DD301143517 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX6 %!PS-AdobeFont-1.1: CMBX6 1.0 %%CreationDate: 1991 Aug 20 16:35:30 % 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 (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 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{-49 -250 1367 753}readonly def /UniqueID 5000764 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F05C11F9A72F5DA508C30BC4BF52C8B1EC5FB 7F9DDDD0964A6D591DFD9EADB67720E5596FA8FBC900296430DED80E17C9C632 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF4CA109080392ED99D52AE696FCA 304F241590CAE3A062DB22B67184E9FB3C7B36E2F99F499BF767DFB7D98AAB94 8483AF926E6DCE81FB32D957D1387D29287F2C7E70B8AE55410C0197602106C1 F9694FC68415AE7C623CC00EC7E52B520E4D6E9C3D8E715EA5A4AACAC4937996 01D6C379BDAE7DABD95664F7356F346E97522B39855C248831F37BE0278AEF17 0D958F943A90AC251C7DBABD81DA205B5EC1BC432F6FEA6DFF7EC7A3A382421F FA72EDA1F73AA3B9AF53A7B795D34B4DD2944A5B9F5680F7A05D8BEB26055E5D A4B628639AB25D0728C1614429B54901C14D1B0C49DA42C2A8A1C3FDBB45A64D A5034E82C0428BCE5D51E1404E12367E9AF7CF9CFB7571E69D904C390A8B40B9 BDA6A8ADFE3B26CD653F451092EE86ECC5682C88D04A3C9D5F0FFD3033CE7293 2E1563EF1C022CD7BA9D64E39A8230585DD496AC68623EC0F3432B99DFCC870A 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16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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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. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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y(the)f(main)f(gap)g(w)n(e)g(w)n(an)n(t)g(to)h(\034ll)f(in)h(the)g (presen)n(t)f(pap)r(er.)515 2167 y(The)40 b(\034rst)f(systematic)g (analysis)g(of)g(scattering)g(in)h(the)g(translationally)e(in)n(v)-5 b(arian)n(t)39 b(Nel-)515 2258 y(son's)29 b(mo)r(del)g(has)g(b)r(een)h (done)f(b)n(y)h(F)-7 b(r\366hlic)n(h)28 b(in)i(t)n(w)n(o)f(pap)r(ers)g ([F)-7 b(r.1])29 b(and)g([F)-7 b(r.2],)29 b(where)g(the)515 2350 y(second)g(one)h(pro)n(vides)f(sp)r(ectral)h(results)g(exploited)g (in)g([F)-7 b(r.1].)30 b(In)h(this)f(study)-7 b(,)31 b(indications)515 2441 y(coming)h(from)h(solv)-5 b(able)33 b(mo)r(dels)g(are)g(mastered,)f(t)n(w)n(o)h(di\033eren)n(t)g(approac)n (hes)e(to)j(collision)515 2532 y(theory)26 b(are)h(dev)n(elop)r(ed)g (and)g(man)n(y)g(useful)h(tec)n(hnical)f(to)r(ols)g(are)g(pro)n(vided.) f(Starting)h(from)515 2623 y(the)37 b(in)n(tuitiv)n(e)f(picture,)h(a)f (recip)r(e)g(is)h(attempted)g(for)f(the)h(v)n(ector)e(in)i(the)g(Hilb)r (ert)g(space,)515 2715 y FH( )568 2683 y FG(out)p 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b(is)i(carried)f(on)h(in)g(a)g(subsequen)n(t)g(paragraph.) d(Indeed)j(it)h(is)f(surely)f(crucial)h(to)f(outline)515 3354 y(\034rst)33 b(the)g(ph)n(ysical)f(and)h(mathematical)g(asp)r (ects)g(in)n(v)n(olv)n(ed,)e(then)j(w)n(e)f(can)f(b)r(etter)i(justify) 515 3445 y(new)25 b(conceptual)f(and)h(tec)n(hnical)f(steps)h (su\036cien)n(t)g(for)f(a)h(consisten)n(t)f(collision)g(theory)g(b)r (oth)515 3537 y(for)j(the)h(infraparticle)e(and)i(the)g(b)r(osons.)515 3719 y(In)18 b(the)h(framew)n(ork)e(of)i(general)e(quan)n(tum)h (\034eld)h(theory)-7 b(,)18 b(analogous)e(issues)i(has)g(b)r(een)h (treated)515 3811 y(b)n(y)i(Buc)n(hholz)g([Bu.1])h([Bu.2],)f(who)g (established)h(the)g(asymptotic)f(con)n(v)n(ergence)e(of)j(massless)515 3902 y(b)r(oson)h(\034elds)i(applying)e(Huyghens')h(principle.)h(Also)f (the)g(problem)g(of)g(the)h(asymptotic)f(de-)515 3993 y(scription)j(of)h(an)g(infraparticle)f(has)g(b)r(een)i(addressed)e(b)n (y)h(Buc)n(hholz,)f(P)n(ormann)g(and)h(Stein)515 4085 y([B.P)-7 b(.S.])25 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bop 515 523 a FO(heren)n(t)23 b(sectors.)g(In)h(this) g(resp)r(ect,)g(it)g(is)g(w)n(orth)n(while)f(to)h(p)r(oin)n(t)g(out)g (that)g(the)h(main)f(ph)n(ysical)515 614 y(feature)g(in)h(the)g(mo)r (del)g(discussed)f(in)h(our)f(pap)r(er)g(is)g(nev)n(er)g(exp)r(ected)h (in)g(the)g(con\034ned)f(case,)515 706 y(namely)e(the)i(co)r(existence) e(of)h(inequiv)-5 b(alen)n(t)23 b(non-F)-7 b(o)r(c)n(k)22 b(represen)n(tations)f(of)i(the)g(asymptotic)515 797 y(b)r(oson)f(\034eld)h(lab)r(elled)g(b)n(y)g(the)g(asymptotic)g (\(mean\))g(v)n(elo)r(cit)n(y)f(of)h(the)g(electron.)f(The)h(asymp-)515 888 y(totic)f(completeness)g(in)g(the)h(con\034ned)f(and)g(infrared)g (regularized)e(case)i(has)f(b)r(een)i(discussed)515 980 y(b)n(y)k(Gerard)f(in)i([Ge.].)515 1071 y(The)33 b(asymptotic)f(con)n (v)n(ergence)f(of)i(the)g(radiation)f(\034eld)h(in)h(non-relativistic)d (Q.E.D.)j(has)515 1162 y(b)r(een)j(established)f(in)i([F.G.S.])f(for)g (small)f(energy)g(con\034gurations)f(of)i(the)g(system.)g(The)515 1254 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FE(\))12 b FH(;)h(a)g FE(\()p FA(k)p FE(\))3426 4563 y Fy(\011)3471 4623 y FO(:)1577 4803 y Fz(F)29 b FE(=)21 b Fz(\010)1802 4767 y Fx(1)1802 4816 y FG(j)s FF(=0)1911 4803 y FH(S)1958 4811 y FG(j)1990 4803 y FH(L)2042 4767 y FF(2)2090 4720 y Fy(h)2126 4743 y(\000)2162 4803 y FB(R)2216 4767 y FF(3)2250 4743 y Fy(\001)2285 4756 y FG(j)2318 4720 y Fy(i)2384 4803 y FH(:)905 b FI(\(1.5\))1970 5059 y FO(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 515 523 a FO(An)28 b(elemen)n(t)f(of)h Fz(H)g FO(is)g(a)f(sequence)g Fz(f)p FH( )1730 531 y FG(n)1773 523 y Fz(g)h FO(of)f(functions)h(on)f FB(R)2460 491 y FF(3\()q FG(n)p FF(+1\))2686 523 y FO(with)h Fz(k)p FH( )s Fz(k)22 b FH(<)f Fz(1)p FO(,)28 b(where)1003 742 y Fz(k)p FH( )s Fz(k)1132 706 y FF(2)1188 742 y FE(=)1297 649 y Fx(1)1272 671 y Fy(X)1269 829 y FG(n)p FF(=0)1398 641 y Fy(Z)p 1487 677 496 4 v 1487 742 a FH( )1540 721 y FG(n)1595 742 y FE(\()p FA(x)p FH(;)13 b FA(k)1753 750 y FF(1)1787 742 y FH(;)g(::)p FA(k)1910 750 y FG(n)1953 742 y FE(\))p FH( )2036 706 y FG(n)2091 742 y FE(\()p FA(x)p FH(;)g FA(k)2249 750 y FF(1)2283 742 y FH(;)g(::)p FA(k)2406 750 y FG(n)2449 742 y FE(\))g FH(d)2532 706 y FF(3)2566 742 y FH(k)2606 750 y FF(1)2641 742 y FH(:::d)2744 706 y FF(3)2779 742 y FH(k)2819 750 y FG(n)2861 742 y FH(d)2901 706 y FF(3)2936 742 y FH(x)515 971 y FO(and)37 b(eac)n(h)f FH( )935 940 y FG(n)990 971 y FE(\()p FA(x)p FH(;)13 b FA(k)1148 979 y FF(1)1182 971 y FH(;)g(::)p FA(k)1305 979 y FG(n)1348 971 y FE(\))37 b FO(is)g(symmetric)g(in)g FA(k)2075 979 y FF(1)2110 971 y FH(;)13 b(:::;)h FA(k)2289 979 y FG(n)2331 971 y FO(.)37 b(The)h FH(n)e FE(=)g(0)i FO(comp)r(onen)n(t)f(corre-)515 1063 y(sp)r(onds)21 b(to)g(the)h (tensor)f(pro)r(duct)h(of)f(the)h(v)-5 b(acuum)22 b(subspace)e(of)i Fz(F)29 b FO(with)22 b(the)g(non-relativistic)515 1154 y(particle)27 b(space)f FH(L)1088 1122 y FF(2)1136 1094 y Fy(\000)1171 1154 y FB(R)1225 1122 y FF(3)1260 1094 y Fy(\001)1295 1154 y FI(.)515 1337 y FT(Standar)-5 b(d)34 b(r)-5 b(esults)35 b(ab)-5 b(out)37 b FH(H)c FT(and)38 b FH(P)11 b FT(:)515 1519 y FO(i\))28 b(The)f(op)r(erators)1370 1650 y FA(P)22 b FE(=)f FA(p)c Fz(\012)g FH(I)23 b FE(+)16 b FH(I)23 b Fz(\012)1943 1548 y Fy(Z)2032 1650 y FA(k)p FH(a)2120 1614 y Fx(y)2165 1650 y FE(\()p FA(k)p FE(\))12 b FH(a)h FE(\()p FA(k)p FE(\))f FH(d)2497 1614 y FF(3)2531 1650 y FH(k)20 b(;)515 1818 y FO(where)27 b FN(I)34 b FO(is)28 b(the)g(iden)n(tit)n(y)f(op)r(erator,)f(are)h(essen)n(tially)f (self-adjoin)n(t)h(\(e.s.a.\))h(in)1696 1994 y FH(D)c FE(:=)1901 1923 y Fy(_)1886 2082 y FG(n)p Fx(2)p Fv(N)2014 1994 y FH(h)17 b Fz(\012)g FH( )2205 1958 y FG(n)2265 1994 y FH(:)533 2229 y FN(D)k FO(is)d(the)h(set)f(of)g(the)h(\034nite)g (linear)e(com)n(binations)h(of)g(v)n(ectors)f(of)h(w)n(a)n(v)n(e)f (function)i FH(h)13 b FE(\()o FA(x)p FE(\))g FH( )3273 2198 y FG(n)3328 2229 y FE(\()p FA(k)3405 2237 y FF(1)3439 2229 y FH(;)g(:::)p FA(k)3583 2237 y FG(n)3626 2229 y FE(\))p FO(,)515 2321 y(where)25 b FH(h)13 b FE(\()p FA(x)p FE(\))21 b Fz(2)g FH(S)1074 2260 y Fy(\000)1110 2321 y FB(R)1164 2289 y FF(3)1198 2260 y Fy(\001)1233 2321 y FO(\(the)27 b(space)e(of)h(Sc)n(h)n(w)n(artz)e(test)i (functions\),)h FH( )2696 2289 y FG(n)2751 2321 y FE(\()p FA(k)2828 2329 y FF(1)2862 2321 y FH(;)13 b(:::)p FA(k)3006 2329 y FG(n)3049 2321 y FE(\))21 b Fz(2)h FH(S)3224 2289 y FG(s)3270 2260 y Fy(\000)3305 2321 y FB(R)3359 2289 y FF(3)p FG(n)3432 2260 y Fy(\001)515 2421 y FO(\(symmetric)28 b(Sc)n(h)n(w)n(artz)e(test)i(functions\))h(and)f FH(n)22 b Fz(2)h FB(N)p FO(.)28 b(Since)g FA(p)g FO(and)2723 2361 y Fy(R)2787 2421 y FA(k)p FH(a)2875 2390 y Fx(y)2920 2421 y FE(\()p FA(k)p FE(\))12 b FH(a)h FE(\()p FA(k)p FE(\))f FH(d)3252 2390 y FF(3)3286 2421 y FH(k)31 b FO(are)515 2513 y(e.s.a.)26 b(in)g FH(S)881 2452 y Fy(\000)916 2513 y FB(R)970 2481 y FF(3)1005 2452 y Fy(\001)1067 2513 y FO(and)1227 2457 y Fy(W)1291 2536 y FG(n)p Fx(2)p Fv(N)1424 2513 y FH( )1477 2481 y FG(n)1546 2513 y FO(resp)r(ectiv)n(ely)-7 b(,)25 b(the)i(result)f(easily)g(follo)n(ws)f(for)h(the)h FA(P)g FO(op)r(era-)515 2604 y(tors.)g(The)i(sp)r(ectrum)f(of)g(eac)n (h)g(comp)r(onen)n(t)g(of)g FA(P)h FO(is)f(the)h(real)f(axis,)f(the)i (sp)r(ectral)f(measure)515 2695 y(is)f(absolutely)g(con)n(tin)n(uous)f (with)i(resp)r(ect)g(to)f(the)h(Leb)r(esgue)f(measure.)515 2878 y(ii\))38 b(The)g(in)n(teraction)f(term)g(in)h(the)g(Hamiltonian)g (is)g(an)f(in\034nitesimal)h(small)f(p)r(erturba-)515 2969 y(tion)27 b(\(in)h(the)g(sense)f(of)h(Kato\))f(with)h(resp)r(ect)f (to)1682 3180 y FH(H)1746 3188 y FF(0)1802 3180 y FE(:=)1925 3131 y FA(p)1974 3100 y FF(2)p 1914 3163 107 4 v 1914 3230 a FE(2)p FH(m)2047 3180 y FE(+)17 b FH(H)2194 3144 y FG(ph)2279 3180 y FH(:)1010 b FI(\(1.6\))515 3365 y FO(Hence)31 b FH(H)36 b FO(is)31 b(b)r(ounded)h(from)e(b)r(elo)n(w,)h (it)g(is)g(e.s.a.)f(in)i FH(D)h FO(and)e(its)g(self-adjoin)n(tness)f (domain)515 3456 y(\(s.a.d.\),)e FH(D)15 b FE(\()p FH(H)6 b FE(\))o FO(,)28 b(coincides)f(with)h FH(D)15 b FE(\()p FH(H)1790 3464 y FF(0)1824 3456 y FE(\))27 b FO(\(s.a.d.)h(of)f FH(H)2289 3464 y FF(0)2324 3456 y FO(\).)515 3639 y(iii\))h(The)f (groups)g FH(e)1119 3607 y FG(i)p Fw(a)p Fx(\001)p Fw(P)1244 3639 y FO(and)g FH(e)1441 3607 y FG(i\034)6 b(H)1556 3639 y FO(,)27 b FH(\034)j Fz(2)22 b FB(R)p FH(;)30 b FA(a)21 b Fz(2)h FB(R)2039 3607 y FF(3)2101 3639 y FO(,)27 b(comm)n(ute.)515 3822 y(iv\))34 b(The)f(join)n(t)h(sp)r(ectral)f (decomp)r(osition)g(of)h(the)g(Hilb)r(ert)g(space)f(with)h(resp)r(ect)g (to)f(the)h FA(P)515 3913 y FO(op)r(erators)25 b(is)j Fu(H)19 b Ft(=)1120 3859 y Fs(R)1167 3880 y Fx(\010)1231 3913 y Fu(H)1291 3924 y Fw(P)1342 3913 y Fr(d)1378 3890 y FF(3)1413 3913 y Fr(P)37 b FO(where)27 b Fz(H)1800 3921 y Fw(P)1878 3913 y FO(is)h(a)f(cop)n(y)g(of)g Fz(F)7 b FO(.)515 4004 y(Indeed)27 b(to)h(the)g(improp)r(er)f(eigen)n(v)n (ectors)e(of)i(the)h FA(P)g FO(op)r(erators,)e FE(\010)2622 3973 y FG(n)2622 4019 y Fw(P)2673 4004 y FI(,)i FO(where)544 4180 y FE(\010)599 4144 y FG(n)599 4193 y Fw(P)663 4180 y FE(\()p FA(x)p FH(;)13 b FA(k)821 4188 y FF(1)855 4180 y FH(;)g(::)p FA(k)978 4188 y FG(n)1021 4180 y FE(\))21 b(:=)h(\()o(2)p FH(\031)s FE(\))1319 4144 y Fx(\000)1377 4125 y FL(3)p 1377 4133 29 3 v 1377 4166 a(2)1432 4180 y FH(e)1468 4144 y FG(i)p FF(\()p Fw(P)p Fx(\000)p Fw(k)1645 4154 y FL(1)1678 4144 y Fx(\000)p FG(:::)p Fx(\000)p Fw(k)1867 4152 y Fq(n)1907 4144 y FF(\))p Fx(\001)p Fw(x)1991 4180 y FH(')2041 4144 y FG(n)2041 4193 y Fw(P)2105 4180 y FE(\()p FA(k)2182 4188 y FF(1)2216 4180 y FH(;)13 b(::)p FA(k)2339 4188 y FG(n)2382 4180 y FE(\))166 b FH(')2628 4144 y FG(n)2628 4193 y Fw(P)2692 4180 y FE(\()p FA(k)2769 4188 y FF(1)2803 4180 y FH(;)13 b(::)p FA(k)2926 4188 y FG(n)2969 4180 y FE(\))21 b Fz(2)g FH(S)3143 4144 y FG(s)3190 4120 y Fy(\000)3225 4180 y FB(R)3279 4144 y FF(3)p FG(n)3352 4120 y Fy(\001)3417 4180 y FH(;)515 4340 y FO(w)n(e)27 b(can)g(relate)g(a)g(natural)g(scalar)f(pro)r(duct:) 1000 4481 y Fy(\000)1036 4542 y FE(\010)1091 4506 y Fx(0)p FG(n)1091 4555 y Fw(P)1152 4542 y FH(;)13 b FE(\010)1241 4506 y FG(m)1241 4555 y Fw(P)1300 4481 y Fy(\001)1356 4542 y FE(=)22 b FH(\016)1472 4550 y FG(n;m)1600 4440 y Fy(Z)p 1690 4476 431 4 v 1690 4542 a FH(')1740 4516 y Fx(0)p FG(n)1740 4563 y Fw(P)1814 4542 y FE(\()o FA(k)1890 4550 y FF(1)1925 4542 y FH(;)13 b(::)p FA(k)2048 4550 y FG(n)2091 4542 y FE(\))o FH(')2170 4506 y FG(m)2170 4555 y Fw(P)2242 4542 y FE(\()p FA(k)2319 4550 y FF(1)2353 4542 y FH(;)g(::)p FA(k)2476 4550 y FG(m)2535 4542 y FE(\))g FH(d)2618 4506 y FF(3)2652 4542 y FH(k)2692 4550 y FF(1)2727 4542 y FH(:::d)2830 4506 y FF(3)2865 4542 y FH(k)2905 4550 y FG(n)2961 4542 y FH(:)328 b FI(\(1.7\))515 4719 y FO(The)21 b(v)n(ector)e(space)p 1136 4653 321 4 v 1136 4663 a Fy(W)1200 4742 y FG(n)p Fx(2)p FG(N)1351 4719 y FE(\010)1406 4693 y FG(n)1406 4740 y Fw(P)1478 4719 y FO(is)i(de\034ned)g(as)f(the)h(closure)f(of)h(the)g(\034nite)h (linear)e(com)n(binations)515 4810 y(of)40 b(the)g(w)n(a)n(v)n(e)f (functions)h FE(\010)1418 4778 y FG(n)1418 4825 y Fw(P)1482 4810 y FE(\()p FA(x)p FH(;)13 b FA(k)1640 4818 y FF(1)1674 4810 y FH(;)h(::)p FA(k)1798 4818 y FG(n)1840 4810 y FE(\))40 b FO(in)h(the)g(norm)e(whic)n(h)h(arises)f(from)h(the)g (scalar)1970 5059 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 515 523 a FO(pro)r(duct)27 b(\(1.7\).)h(Starting) f(from)g(this)h(space,)e(w)n(e)i(uniquely)f(de\034ne)h(the)g(linear)f (application)1684 712 y FA(I)1717 720 y Fw(P)1790 712 y FE(:)p 1832 624 253 4 v 1856 641 a Fy(_)1832 800 y FG(n)p Fx(2)p FG(N)1978 712 y FE(\010)2033 686 y FG(n)2033 733 y Fw(P)2106 712 y Fz(!)21 b(F)2266 676 y FG(b)3310 712 y FI(\(1.8\))542 944 y FO(b)n(y)28 b(the)g(prescription:)662 1136 y FA(I)695 1144 y Fw(P)759 1136 y FE(\(\010)844 1100 y FG(n)844 1149 y Fw(P)908 1136 y FE(\()p FA(x)p FH(;)13 b FA(k)1066 1144 y FF(1)1100 1136 y FH(;)g(::)p FA(k)1223 1144 y FG(n)1266 1136 y FE(\)\))21 b(=)1485 1087 y(1)p 1438 1119 132 4 v 1438 1134 a Fz(p)p 1502 1134 68 4 v 64 x FH(n)p FE(!)1593 1034 y Fy(Z)1682 1136 y FH(b)1715 1100 y Fx(y)1761 1136 y FE(\()p FA(k)1838 1144 y FF(1)1872 1136 y FE(\))12 b FH(::::b)2031 1100 y Fx(y)2078 1136 y FE(\()p FA(k)2155 1144 y FG(n)2197 1136 y FE(\))h FH(')2290 1100 y FG(n)2290 1149 y Fw(P)2354 1136 y FE(\()p FA(k)2431 1144 y FF(1)2465 1136 y FH(;)g(::)p FA(k)2588 1144 y FG(n)2631 1136 y FE(\))f FH(d)2713 1100 y FF(3)2748 1136 y FH(k)2788 1144 y FF(1)2822 1136 y FH(:::d)2925 1100 y FF(3)2961 1136 y FH(k)3001 1144 y FG(n)3043 1136 y FH( )3093 1144 y FF(0)3141 1136 y FH(;)148 b FI(\(1.9\))515 1346 y FO(where)33 b FH(b)13 b FE(\()o FA(k)p FE(\))g FH(;)g(b)993 1315 y Fx(y)1038 1346 y FE(\()p FA(k)p FE(\))p FI(,)34 b FO(whic)n(h)f(formally)g(corresp)r(ond)f(to)i FH(a)12 b FE(\()p FA(k)p FE(\))g FH(e)2522 1315 y FG(i)p Fw(k)p Fx(\001)p Fw(x)2640 1346 y FH(;)h(a)2715 1315 y Fx(y)2761 1346 y FE(\()o FA(k)p FE(\))g FH(e)2916 1315 y Fx(\000)p FG(i)p Fw(k)p Fx(\001)p Fw(x)3082 1346 y FO(,)33 b(are)g(anni-)515 1438 y(hilation)k(and)g(creation)g(op)r (erator-v)-5 b(alued)35 b(temp)r(ered)j(distributions)f(in)h(the)g(F)-7 b(o)r(c)n(k)37 b(space)515 1529 y Fz(F)577 1497 y FG(b)630 1509 y Fz(\030)630 1533 y FE(=)711 1529 y Fz(F)7 b FO(,)20 b(and)f FH( )1019 1537 y FF(0)1073 1529 y FO(is)g(the)h(related)e(v)-5 b(acuum.)19 b(The)h(giv)n(en)e(norm)h(for)f FE(\010)2625 1497 y FG(n)2625 1544 y Fw(P)2696 1529 y FO(is)h(equal)g(to)g Fz(k)p FA(I)3146 1537 y Fw(P)3210 1529 y FE(\(\010)3295 1497 y FG(n)3295 1544 y Fw(P)3346 1529 y FE(\))p Fz(k)3414 1552 y Fx(F)515 1620 y FO(\()p Fz(k\001k)645 1643 y Fx(F)726 1620 y FO(is)27 b(the)h(F)-7 b(o)r(c)n(k)27 b(norm\).)h(The)f (application)g FA(I)2054 1628 y Fw(P)2133 1620 y FO(is)g(on)n(to)g(and) h(isometric.)515 1803 y(v\))i(Since)g FE([)p FH(H)r(;)12 b FA(P)p FE(])26 b(=)e(0)p FO(,)31 b(w)n(e)e(ha)n(v)n(e)g(that)h FH(H)g FE(=)1922 1743 y Fy(R)1986 1803 y FH(H)2050 1811 y Fw(P)2100 1803 y FH(d)2140 1771 y FF(3)2174 1803 y FH(P)11 b FO(,)30 b(where)f FH(H)2593 1811 y Fw(P)2668 1803 y FE(:)c Fz(H)2779 1811 y Fw(P)2855 1803 y Fz(!)g(H)3022 1811 y Fw(P)3102 1803 y FO(is)30 b(e.s.a.)f(in)515 1894 y FH(D)580 1862 y FG(b)633 1894 y FE(:=)735 1838 y Fy(W)799 1917 y FG(n)p Fx(2)p Fv(N)932 1894 y FE(\010)987 1862 y FG(n)987 1909 y Fw(P)1043 1894 y FO(.)23 b(In)g(terms)g(of)g(the)g(v) -5 b(ariables)22 b FA(P)p FO(,)h FH(b)13 b FE(\()p FA(k)p FE(\))f FH(;)h(b)2321 1862 y Fx(y)2367 1894 y FE(\()p FA(k)p FE(\))o FO(,)23 b(the)h(op)r(erator)d FH(H)3052 1902 y Fw(P)3125 1894 y FO(is)i(written)515 1986 y(as)k(follo)n(ws:) 1016 2193 y FH(H)1080 2201 y Fw(P)1151 2193 y FE(=)1242 2077 y Fy(\000)1277 2138 y FA(P)1337 2106 y FG(ph)1427 2138 y Fz(\000)17 b FA(P)1564 2077 y Fy(\001)1600 2092 y FF(2)p 1242 2176 392 4 v 1385 2244 a FE(2)p FH(m)1661 2193 y FE(+)g FH(g)1790 2092 y Fy(Z)1867 2112 y FG(\024)1833 2262 y FF(0)1920 2111 y Fy(\020)1966 2193 y FH(b)c FE(\()o FA(k)p FE(\))k(+)g FH(b)2245 2158 y Fx(y)2290 2193 y FE(\()p FA(k)p FE(\))2397 2111 y Fy(\021)2516 2145 y FH(d)2556 2113 y FF(3)2590 2145 y FH(k)p 2465 2176 218 4 v 2465 2192 a Fy(p)p 2542 2192 141 4 v 64 x FE(2)c Fz(j)q FA(k)p Fz(j)2710 2193 y FE(+)j FH(H)2856 2158 y FG(ph)2945 2193 y FH(:)305 b FI(\(1.10\))515 2419 y FO(b)r(eing)30 b FH(H)811 2387 y FG(ph)908 2419 y Fz(\021)993 2359 y Fy(R)1057 2419 y Fz(j)q FA(k)p Fz(j)25 b FH(b)1205 2387 y Fx(y)1251 2419 y FE(\()p FA(k)p FE(\))12 b FH(b)h FE(\()o FA(k)p FE(\))g FH(d)1575 2387 y FF(3)1609 2419 y FH(k)33 b FO(and)d FA(P)1906 2387 y FG(ph)2004 2419 y Fz(\021)2089 2359 y Fy(R)2153 2419 y FA(k)12 b FH(b)2245 2387 y Fx(y)2291 2419 y FE(\()p FA(k)p FE(\))g FH(b)h FE(\()p FA(k)p FE(\))f FH(d)2615 2387 y FF(3)2649 2419 y FH(k)33 b FO(when)d(applied)g(on)g(the)515 2519 y(\034b)r(er)d (spaces)g Fz(H)1026 2527 y Fw(P)1077 2519 y FO(.)515 2718 y Fp(Notations)515 2917 y FO(W)-7 b(e)33 b(collect)f(standard)g (notations)g(and)g(some)h(con)n(v)n(en)n(tions)d(whic)n(h)j(are)f(used) g(throughout)515 3017 y(the)c(pap)r(er:)515 3216 y(1\))35 b FH(C)684 3184 y FG(n)739 3156 y Fy(\000)774 3216 y FB(R)828 3184 y FF(3)863 3156 y Fy(\001)933 3216 y FO(denotes)g(the)g (space)g(of)g(functions)g(on)g FB(R)2265 3184 y FF(3)2335 3216 y FO(with)g(con)n(tin)n(uous)f(deriv)-5 b(ativ)n(es)34 b(up)515 3316 y(to)e(degree)f FN(n)p FO(,)i FH(C)1050 3284 y FG(n)1045 3329 y FF(0)1105 3255 y Fy(\000)1140 3316 y FB(R)1194 3284 y FF(3)1229 3255 y Fy(\001)1296 3316 y FO(denotes)f(the)h(subset)f(of)h(compact)f(supp)r(ort)g (functions)g(con)n(tained)515 3415 y(in)c FH(C)672 3384 y FG(n)727 3355 y Fy(\000)762 3415 y FB(R)816 3384 y FF(3)851 3355 y Fy(\001)886 3415 y FO(.)515 3515 y(2\))j(The)g(sym)n(b) r(ol)g Fo(j)p FN(:)p Fo(j)h FO(will)g(denote)f(the)h(absolute)e(v)-5 b(ale)31 b(for)g Fn(C)h FO(n)n(um)n(b)r(ers)e(as)h(w)n(ell)g(as)g(the)h (eu-)515 3615 y(clidean)g(norm)g(for)g(v)n(ectors)f(in)i FB(R)1598 3583 y FG(n)1640 3615 y FO(,)g FH(n)d(>)f FE(1)p FO(.)k(Multiplication)g(b)r(et)n(w)n(een)f(real)g(n)n(um)n(b)r(ers)g (and)515 3714 y(scalar)25 b(pro)r(ducts)i(b)r(et)n(w)n(een)f(v)n (ectors)g(in)h FB(R)1845 3682 y FG(n)1887 3714 y FO(,)g FH(n)22 b(>)f FE(1)p FO(,)28 b(are)e(b)r(oth)h(denoted)g(b)n(y)g(the)g (m)n(ultiplica-)515 3814 y(tion)g(sign)g(\020)7 b Fo(\001)p FO(\021)g FJ(.)515 3913 y FO(3\))27 b(Giv)n(en)h(a)f(function)h FH(\037)12 b FE(\()p FH(u)p FE(\))p FJ(,)29 b FH(supp\037)d FO(is)i(the)g(supp)r(ort)f(of)g(the)h(function.)515 4013 y(4\))d(F)-7 b(or)24 b FE(3)p FO(-dimensional)g(in)n(tegrals)g(w)n(e)g (use)h(only)g(one)f(in)n(tegration)g(sym)n(b)r(ol)g(and)h(the)g (explicit)515 4113 y(in)n(tegration)33 b(b)r(ounds)i(are)f(referred)f (to)i(the)g(radial)f(part)g(of)h(the)g(in)n(tegration)e(v)-5 b(ariable.)34 b(If)515 4212 y(necessary)25 b(the)j(notations)f(are)g (less)g(compressed)p FJ(.)515 4312 y FO(5\))g(The)h(notation)f FH(s)17 b Fz(\000)f FH(l)q(im)28 b FO(means)f(strong)g(limit.)515 4412 y(6\))h(Giv)n(en)g(a)g(self-adjoin)n(t)g(op)r(erator)f FH(A)p FO(,)h FH(D)15 b FE(\()p FH(A)p FE(\))28 b FO(is)h(the)f (corresp)r(onding)f(domain.)h(The)g(nota-)515 4511 y(tion)f FJ(h.c.)j FO(means)d(hermitian)g(conjugate.)515 4611 y(7\))32 b(The)g(op)r(erators,)f Fz(r)p FH(E)1317 4579 y FG(\033)1371 4611 y FE(\()p FA(P)p FE(\))26 b FH(;)g(W)1637 4619 y FG(\033)1691 4611 y FE(\()o Fz(r)p FH(E)1845 4579 y FG(\033)1900 4611 y FE(\()o FA(P)p FE(\))q(\))p FI(,)32 b FO(are)g(functions)g(of)h(the)f(total)g(momen)n(tum)515 4710 y(op)r(erator)d FA(P)p FO(.)j(F)-7 b(or)30 b(brevit)n(y)h(the)g (dep)r(endence)h(on)f FA(P)g FO(is)g(some)g(times)g(di\033eren)n(tly)g (indicated)515 4810 y(\(e.g.)c FH(E)760 4778 y FG(\033)756 4825 y Fw(P)806 4810 y FO(\).)1970 5059 y(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 515 523 a FO(8\))30 b(In)g(the)h(estimates)f (that)h(w)n(e)f(pro)r(duce)g(throughout)f(the)i(pap)r(er,)f(w)n(e)g (generically)e(call)i FN(C)515 623 y FO(all)35 b(the)g(m)n (ultiplicativ)n(e)g(constan)n(ts)f(whic)n(h)h(are)f(are)g(time)i(indep) r(enden)n(t,)g(uniform)f(in)g(the)515 722 y(infrared)26 b(cut-o\033)i(and)f(in)h(the)g(cell)g(partition.)515 955 y FC(1.2)112 b(F)-9 b(r\366hlic)m(h's)36 b(construction)515 1100 y FO(The)i(issues)g(and)g(the)g(results)g(w)n(e)g(are)f(going)g (to)h(discuss)g(concern)f(the)i(mo)r(del)f(with)h(an)515 1191 y(ultra)n(violet)33 b(cut-o\033)h(and)g(are)g(connected)g(to)g (the)g(infrared)g(di\036culties)h(whic)n(h)f(a\033ect)g(the)515 1282 y(form)n(ulation)26 b(of)i(scattering)f(theory)-7 b(.)27 b(F)-7 b(o)r(cusing)27 b(our)g(atten)n(tion)h(on)f(suc)n(h)h (asp)r(ect,)f(w)n(e)h(recall)515 1374 y(that)g(in)f(F)-7 b(r\366hlic)n(h's)27 b(pap)r(er)g([F)-7 b(r.1])27 b(the)h(follo)n(wing) f(cases)f(are)h(in)n(v)n(estigated)f(and)h(compared:)515 1465 y(1\))g(The)h(massiv)n(e)e(and)h(the)h(massless)f(cases)f(as)h (far)g(as)g(the)h(b)r(oson)f(\034eld)h(is)f(concerned;)515 1556 y(2\))c(Both)h(the)g(non-relativistic)e(and)i(the)g(relativistic)f (disp)r(ersion)g(la)n(w)f(for)i(the)g(c)n(harged)e(par-)515 1648 y(ticle)28 b(kinetic)f(energy)g(in)h(the)f(Hamiltonian.)515 1830 y(The)f(scattering)g(problem)g(is)g(studied)h(in)f(a)h(time-dep)r (enden)n(t)g(approac)n(h,)d(b)n(y)i(adapting)g(the)515 1922 y(\020Haag-Ruelle\021)31 b(framew)n(ork)24 b([Ha.])j(to)f(the)h (mixed)f(c)n(haracter)e(of)j(the)g(mo)r(del,)f(in)h(fact)f(quan-)515 2013 y(tum)j(mec)n(hanical)e(non-relativistic)g(matter)h(co)r(exists)f (with)i(a)f(quan)n(tum)g(relativistic)f(\034eld.)515 2104 y(The)35 b(adopted)f(pro)r(cedure)g(is)h(successful)f(as)h(far)f (as)g(one)h(particle)f(states)h(for)f(the)h(c)n(harge)515 2195 y(are)28 b(a)n(v)-5 b(ailable.)29 b(It)g(is)h(alw)n(a)n(ys)e(the)h (case)g(in)h(presence)f(of)g(massiv)n(e)f(b)r(osons;)h(in)h(the)g (massless)515 2287 y(case)k(only)h(if)h(an)f(infrared)f (regularization,)f(for)i(instance)g(a)g(cut-o\033,)g(is)g(imp)r(osed)g (on)g(the)515 2378 y(in)n(teraction.)28 b(Starting)h(from)h(the)g (one-particle)e(states)h(and)g(the)h(asymptotic)f(limit)i(of)e(the)515 2469 y FJ(L.S.Z.)39 b FO(smeared)e(\034eld,)h(the)h(asymptotic)e (picture)h(is)g(simply)g(giv)n(en)f(b)n(y)h(a)g(free)f(electron)515 2561 y(with)24 b(a)f(renormalized)e(disp)r(ersion)i(la)n(w)g(and)g (free)g(b)r(osons)g(in)h(the)g(F)-7 b(o)r(c)n(k)23 b(represen)n (tation.)e(W)-7 b(e)515 2652 y(recall)26 b(that)h(in)g(the)g(massless)e (case)h(the)h(asymptotic)g(con)n(v)n(ergence)d(of)j(the)g FJ(L.S.Z.)h FO(smeared)515 2743 y(\034eld)37 b(requires,)e(as)h (additional)g(condition,)h(that)g(the)g(asymptotic)f(v)n(elo)r(cit)n(y) g(of)h(the)g(non-)515 2835 y(relativistic)22 b(particle)h(is)h (strictly)f(smaller)f(with)i(resp)r(ect)g(to)f(the)h(b)r(oson)f(v)n (elo)r(cit)n(y)f(\(the)i(sp)r(eed)515 2926 y(of)j(the)h(ligh)n(t\).)515 3017 y(Suc)n(h)i(a)h(ph)n(ysical)f(description)g(fails)g(in)h(the)g (true)g(\(no)g(infrared)f(regularization\))e(Nelson's)515 3109 y(massless)33 b(mo)r(del)h(and)g(t)n(w)n(o)g(alternativ)n(e)f (scattering)g(descriptions)h(are)f(therefore)g(consid-)515 3200 y(ered.)515 3291 y(The)20 b(\034rst)f(one)h(is)g(an)g(attempt)g (to)g(generalize)e(Haag-Ruelle)h(theory)g(b)n(y)h(a)f(limiting)i (construc-)515 3383 y(tion)f(starting)f(from)g(the)i(mo)r(del)f(with)g (an)g(infrared)f(cut-o\033.)h(This)g(approac)n(h)d(is)j(reconsidered) 515 3474 y(and)27 b(dev)n(elop)r(ed)g(in)h(this)g(pap)r(er,)f(where)g (it)h(is)f(pro)n(v)n(ed)f(to)i(b)r(e)g(consisten)n(t.)515 3565 y(The)k(second)g(one)f(assumes)g(the)i(existence)f(of)g(the)h (asymptotic)e(b)r(oson)h(\(free\))g(algebra)f(to)515 3657 y(de\034ne)c(the)g(time-space)f(translation)g(generators)f(for)h (the)h(asymptotic)g(c)n(harge)e(as)h(a)h(di\033er-)515 3748 y(ence:)d(these)g(are)f(obtained)h(b)n(y)g(subtracting)g(from)g (the)g(full)h(generators)d(the)j(corresp)r(onding)515 3839 y(ones)h(for)g(the)h(asymptotic)f(b)r(osons)g(\(for)g(details)g (see)h([F)-7 b(r.1]\).)26 b(Similar)g(concepts)h(w)n(ere)e(later)515 3931 y(exploited)31 b(in)h(Q.E.D.)g(\(see)f([F.M.S.]\))i(in)f(the)g (Wigh)n(tman)f(framew)n(ork)e(of)j(quan)n(tum)f(\034eld)515 4022 y(theory)-7 b(.)29 b(In)i(that)g(con)n(text)f(a)g(ten)n(tativ)n(e) g(recip)r(e)g(has)g(b)r(een)h(pro)n(vided)e(for)h(the)h(construction) 515 4113 y(of)c(the)h(asymptotic)f(c)n(harged)f(\034elds.)515 4296 y(The)35 b(\034rst)g(approac)n(h)e(\(the)j(only)e(one)h(w)n(e)g (are)f(in)n(terested)g(in\))i(requires)e(a)g(careful)h(analy-)515 4387 y(sis)23 b(of)i(the)f(one-particle)f(improp)r(er)g(states)h(or,)f (in)h(other)g(w)n(ords,)f(of)h(the)g(one-particle)f(states)515 4479 y(corresp)r(onding)31 b(to)h(Hamiltonians)h(with)g(smaller)f(and)h (smaller)f(infrared-cuto\033)f FH(\033)36 b FO(in)d(the)515 4570 y(in)n(teraction)f(term.)i(The)f(underlying)g(conjecture)g(is)g (that)h(a)f(su\036cien)n(tly)h(re\034ned)f(con)n(trol)515 4661 y(on)28 b(the)g(one-particle)f(states)h(\(whic)n(h)h(disapp)r(ear) e(from)h(the)g(Hilb)r(ert)h(space)f Fz(H)h FO(in)f(the)h(limit)515 4753 y FH(\033)24 b Fz(!)d FE(0)p FO(\))j(should)g(predict)g(the)g(lo)n (w)f(energy)g(b)r(eha)n(vior)f(of)i(the)h(b)r(oson)e(cloud)g(app)r (earing)g(in)h(the)1970 5059 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 515 523 a FO(scattering)22 b(states.)h(This)h (asp)r(ect)f(is)g(clearly)g(crucial)f(in)i(order)e(to)i(de\034ne)f(an)h (appro)n(ximating)515 614 y(v)n(ector)32 b FH( )16 b FE(\()p FH(t)p FE(\))34 b FO(of)g FH( )1112 583 y FG(out)1207 614 y FO(.)g(T)-7 b(o)33 b(motiv)-5 b(ate)34 b(wh)n(y)-7 b(,)34 b(in)h(our)e(opinion,)h(the)g(w)n(a)n(y)f(follo)n(w)n(ed)g(in)h ([F)-7 b(r.1])515 706 y(is)29 b(the)h(correct)f(one)g(to)h(understand)f (the)h(scattering)e(b)r(eha)n(vior,)h(w)n(e)g(review)g(that)h(analysis) 515 797 y(b)r(efore)d(\034lling)h(some)f(conceptual)h(steps)f(to)n(w)n (ards)f(a)i(mo)r(di\034ed)g(de\034nition)g(of)g FH( )15 b FE(\()p FH(t)p FE(\))28 b FO(and)f(the)515 888 y(pro)r(of)g(of)g(its) h(con)n(v)n(ergence)d(in)j(time.)515 1071 y(The)39 b(sp)r(ectral)g (results)g(b)r(ehind)h(the)g(de\034nition)g(of)f FH( )16 b FE(\()o FH(t)p FE(\))40 b FO(in)f([F)-7 b(r.1])39 b(are)g(concerned)f (with)515 1162 y(the)31 b(ground)f(states)g(of)h(the)g(Hamiltonians)f FH(H)2006 1170 y Fw(P)2056 1162 y FO(,)h(and)g(they)g(are)f(ac)n(hiev)n (ed)f(through)h(a)h(non)515 1254 y(constructiv)n(e)j(metho)r(d)i (already)e(used)h(b)n(y)g(Glimm)h(and)f(Ja\033e)g([G.J.].)h(The)f(main) g(results)515 1345 y(are:)515 1436 y(The)e(absence,)g(in)g(the)h(not)f (\(infrared\))g(regularized)f(case,)g(of)i(a)e(ground)h(state)g(for)g FH(H)3315 1444 y Fw(P)3398 1436 y FO(in)515 1528 y(the)f(Hilb)r(ert)h (space)e Fz(H)1244 1536 y Fw(P)1316 1508 y Fz(\030)1316 1532 y FE(=)1398 1528 y Fz(F)39 b FO(and)32 b(its)g(existence)g(and)g (uniqueness)g(in)g(the)h FA(P)p Fz(\000)p FO(dep)r(enden)n(t)515 1619 y(coheren)n(t)c(represen)n(tation)f(of)1494 1559 y Fy(\010)1539 1619 y FH(b)13 b FE(\()p FA(k)p FE(\))f FH(;)h(b)1771 1587 y Fx(y)1817 1619 y FE(\()p FA(k)p FE(\))1923 1559 y Fy(\011)1998 1619 y FO(with)31 b(coheren)n(t)e (factor)g FH(c)13 b FE(\()p FA(k)p FE(\))30 b FO(singled)g(out)g(b)n(y) 515 1710 y(the)e(infrared)e(b)r(eha)n(vior:)1301 1872 y FH(c)13 b FE(\()p FA(k)p FE(\))21 b Fz(!)1552 1881 y Fw(k)p Fx(!)p FF(0)1705 1872 y Fz(\000)2182 1824 y FH(g)p 1775 1855 854 4 v 1775 1889 a Fz(p)p 1839 1889 39 4 v 61 x FE(2)13 b Fz(j)p FA(k)p Fz(j)1989 1895 y FL(3)p 1989 1903 29 3 v 1989 1936 a(2)2044 1868 y Fy(\020)2090 1950 y FE(1)18 b Fz(\000)2225 1931 y Fy(b)2223 1950 y FA(k)e Fz(\001)i(r)p FH(E)e FE(\()p FA(P)p FE(\))2582 1868 y Fy(\021)2660 1872 y FE(;)590 b FI(\(1.11\))515 2135 y FO(The)27 b(follo)n(wing)g(prop)r(erties)f(whic)n(h)i(hold)f (for)g(the)h(ground)f(state)g(energy)g FH(E)16 b FE(\()p FA(P)p FE(\))p FO(:)515 2235 y(The)27 b(energy)f FH(E)16 b FE(\()p FA(P)p FE(\))22 b(=)f FH(E)c FE(\()o Fz(j)q FA(P)p Fz(j)p FE(\))28 b FO(is)f(absolutely)f(con)n(tin)n(uous,)g (therefore)2783 2199 y FG(@)s(E)s FF(\()p Fw(P)p FF(\))p 2783 2218 179 4 v 2812 2261 a FG(@)s Fx(j)p Fw(P)p Fx(j)2998 2235 y FO(exists)h(almost)515 2339 y(ev)n(erywhere;)515 2430 y(Concerning)f(the)i(deriv)-5 b(ativ)n(e)866 2525 y Fy(\014)866 2570 y(\014)866 2615 y(\014)902 2575 y FG(@)s(E)s FF(\()o Fw(P)p FF(\))p 902 2594 V 931 2636 a FG(@)s Fx(j)p Fw(P)p Fx(j)1090 2525 y Fy(\014)1090 2570 y(\014)1090 2615 y(\014)1137 2611 y FH(<)21 b FE(1)154 b(for)h FA(P)22 b FE(:)f Fz(j)q FA(P)p Fz(j)h FH(<)f(M)91 b FE(if)235 b FH(H)2494 2619 y FF(0)2549 2611 y FE(=)2648 2578 y Fw(p)2686 2557 y FL(2)p 2640 2594 85 4 v 2640 2636 a FF(2)p FG(m)2752 2611 y FE(+)17 b FH(H)2899 2579 y FG(ph)866 2751 y Fy(\014)866 2796 y(\014)866 2840 y(\014)902 2801 y FG(@)s(E)s FF(\()o Fw(P)p FF(\))p 902 2820 179 4 v 931 2862 a FG(@)s Fx(j)p Fw(P)p Fx(j)1090 2751 y Fy(\014)1090 2796 y(\014)1090 2840 y(\014)1137 2837 y FH(<)k FE(1)83 b(for)23 b(an)n(y)166 b FA(P)21 b Fz(2)h FB(R)1947 2805 y Fw(3)2151 2837 y FE(if)90 b FH(H)2349 2845 y FF(0)2404 2837 y FE(=)2485 2772 y Fy(p)p 2562 2772 280 4 v 65 x FA(p)2611 2815 y FF(2)2663 2837 y FE(+)17 b FH(m)2808 2815 y FF(2)2859 2837 y FE(+)g FH(H)3006 2805 y FG(ph)3095 2837 y FH(:)515 3021 y FO(A)34 b(crucial)f(tec)n (hnical)g(to)r(ol)g(is)h(in)n(v)n(olv)n(ed)e(in)i(the)h(ab)r(o)n(v)n(e) d(results.)h(It)h(is)g(an)f(almost)g(explicit)515 3113 y(expression)24 b(for)i(the)g(action)g(of)g FH(b)13 b FE(\()o FA(k)p FE(\))26 b FO(on)g(the)g(ground)f(state)h FH( )2489 3081 y FG(\033)2486 3128 y Fw(P)2563 3113 y FO(of)g(the)h(Hamiltonian)f FH(H)3338 3121 y Fw(P)p FG(;\033)3444 3113 y FO(,)515 3204 y(i.e.)h(with)i(an)e(infrared)g(cut-o\033)g FH(\033)j FO(in)e(the)g(in)n(teraction)f(term)p FI(.)g FO(The)h(to)r(ol)f(is)801 3391 y FH(b)13 b FE(\()p FA(k)p FE(\))f FH( )1019 3355 y FG(\033)1016 3404 y Fw(P)1088 3391 y FE(=)1268 3343 y FH(g)p 1179 3374 218 4 v 1179 3390 a Fy(p)p 1256 3390 141 4 v 64 x FE(2)h Fz(j)q FA(k)p Fz(j)1419 3286 y Fy(\022)1850 3343 y FE(1)p 1486 3374 768 4 v 1486 3442 a FH(E)1547 3420 y FG(\033)1601 3442 y FE(\()p FA(P)p FE(\))k Fz(\000)g(j)p FA(k)p Fz(j)g(\000)g FH(H)2062 3451 y Fw(P)p Fx(\000)p Fw(k)p FG(;\033)2263 3286 y Fy(\023)2333 3391 y FH( )2386 3355 y FG(\033)2383 3404 y Fw(P)2587 3391 y FH(\033)24 b Fz(\024)d(j)p FA(k)p Fz(j)h(\024)f FH(\024)299 b FI(\(1.12\))515 3612 y FO(where)28 b FH(E)817 3581 y FG(\033)871 3612 y FE(\()p FA(P)p FE(\))g FO(is)h(the)f(eigen)n(v)-5 b(alue)28 b(of)g FH( )1793 3581 y FG(\033)1790 3627 y Fw(P)1841 3612 y FO(,)h FH(H)1957 3620 y Fw(P)p FG(;\033)2063 3612 y FH( )2116 3581 y FG(\033)2113 3627 y Fw(P)2187 3612 y FE(=)22 b FH(E)2330 3581 y FG(\033)2384 3612 y FE(\()p FA(P)p FE(\))13 b FH( )2570 3581 y FG(\033)2567 3627 y Fw(P)2618 3612 y FO(.)29 b(The)f(resolv)n(en)n(t)f(form)n(ula) 515 3704 y(\(1.12\))34 b(clearly)g(pla)n(ys)f(an)i(imp)r(ortan)n(t)g (role)f(also)f(in)j(the)f(presen)n(t)f(pap)r(er)h(b)r(ecause)f(it)h (con-)515 3795 y(tains)g(a)f(structural)g(information)h(ab)r(out)g(the) g(logarithmic)f(div)n(ergence,)g(in)h(the)g(infrared)515 3886 y(limit)26 b FH(\033)e Fz(!)d FE(0)p FO(,)26 b(of)f(the)g(b)r (oson)g(n)n(um)n(b)r(er)g(op)r(erator,)e FH(N)30 b FE(:=)2281 3826 y Fy(R)2345 3886 y FH(b)2378 3855 y Fx(y)2423 3886 y FE(\()p FA(k)p FE(\))12 b FH(b)h FE(\()p FA(k)p FE(\))f FH(d)2747 3855 y FF(3)2782 3886 y FH(k)r FI(,)25 b FO(ev)-5 b(aluated)25 b(on)g(the)515 3978 y(ground)h(state)i(of)f FH(H)1158 3986 y Fw(P)p FG(;\033)1265 3978 y FO(.)515 4160 y(Once)36 b(the)h(previous)e(sp)r(ectral)h(information)g(is)h(kno) n(wn,)f(the)h(main)f(issue)h(is)f(ho)n(w)g(to)h(de-)515 4252 y(\034ne)23 b(a)g(v)n(ector)f FH( )1030 4220 y FG(\033)1027 4267 y(h;\024)1118 4277 y FL(1)1167 4252 y FE(\()p FH(t)p FE(\))f Fz(2)g(H)j FO(with)g(the)f(prop)r(ert)n(y)f(that)i FE(lim)2377 4260 y FG(t)p Fx(!1)p FG(;\033)r Fx(!)p FF(0)2687 4252 y FH( )2740 4220 y FG(\033)2737 4267 y(h;\024)2828 4277 y FL(1)2878 4252 y FE(\()p FH(t)p FE(\))f FO(represen)n(ts,)e(in) 515 4343 y(Heisen)n(b)r(erg)g(picture,)h(an)g(asymptotic)g(electron)f (with)i(w)n(a)n(v)n(e)e(function)h FH(h)h FO(in)g(the)f(asymptotic)515 4434 y(momen)n(tum)j(of)h(the)g(c)n(harged)e(particle,)h(with)h(the)f (exp)r(ected)h(freely)f(mo)n(ving)g(\(soft\))h(photon)515 4526 y(cloud)33 b(surrounding)g(it,)h(where)f(the)h(b)r(oson)g (frequency)f(is)h(up)g(to)f(the)i(threshold)e FH(\024)3233 4534 y FF(1)3268 4526 y FO(.)h(The)515 4617 y(w)n(a)n(v)n(e)25 b(function)j(of)f(the)g(asymptotic)g(\(soft\))g(photons)g(whic)n(h)g (form)f(the)i(cloud)f(is)g(suggested)515 4708 y(b)n(y)33 b(the)i(sp)r(ectral)e(analysis)g(of)g(one-particle)g(states.)g(More)g (precisely)g(it)i(is)f(link)n(ed)f(to)h(the)515 4800 y(coheren)n(t)25 b(represen)n(tations)f(\(1.11\))h(singled)g(out)h(in)g (the)h(limit)f FH(\033)e Fz(!)d FE(0)27 b FO(for)e(di\033eren)n(t)h FA(P)p FO(.)h(This)1970 5059 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 515 523 a FO(in)n(terpretation)22 b(of)h(the)g(limiting)g(v)n(ectors)f(requires)g(an)g FJ(a)k(p)l(osteriori)33 b FO(justi\034cation)24 b(from)e(the)515 614 y(action)27 b(of)g(the)h(asymptotic)f(observ)-5 b(ables)26 b(\(to)i(b)r(e)g(constructed\))f(on)g(them.)515 706 y(In)35 b(order)f(to)i(construct)f(the)h(generic)e(v)n(ector)g FH( )2084 674 y FG(\033)2081 721 y(h;\024)2172 731 y FL(1)2222 706 y FE(\()p FH(t)p FE(\))o FO(,)i(F)-7 b(r\366hlic)n(h)35 b(starts)f(from)h(the)h(w)n(a)n(v)n(e)515 797 y(function,)e(in)f(terms) g(of)g(the)h(c)n(harged)d(particle)i(p)r(osition)g(op)r(erator)e(and)i (the)h(b)r(osons)e(mo-)515 888 y(men)n(ta)23 b(v)-5 b(ariables,)22 b(of)h(a)g(one-particle)f(state)h(corresp)r(onding)e(to)i(the)h(mo)r (del)f(with)h(\(infrared\))515 980 y(cut-o\033)30 b FH(\033)i FO(in)e(the)g(in)n(teraction)f(term)h(of)g(the)g(Hamiltonian)g (\(1.3\).)f(Let)h(it)g(b)r(e)h(giv)n(en)e(b)n(y)g(the)515 1071 y(sequence)619 1169 y Fy(\()680 1297 y FH( )733 1261 y FG(\033)r FF(\()p FG(n)p FF(\))874 1297 y FE(\()p FA(x)p FH(;)13 b FA(k)1032 1305 y FF(1)1066 1297 y FH(;)g(:::;)h FA(k)1245 1305 y FG(n)1287 1297 y FE(\))22 b(:)43 b FH(n)21 b Fz(2)h FB(N)1686 1226 y Fy(X)1723 1383 y FG(n)1810 1195 y Fy(Z)1900 1211 y(\014)1900 1256 y(\014)1900 1300 y(\014)1925 1297 y FH( )1978 1261 y FG(\033)r FF(\()o FG(n)p FF(\))2119 1297 y FE(\()p FA(x)p FH(;)12 b FA(k)2276 1305 y FF(1)2311 1297 y FH(;)h(:::;)h FA(k)2490 1305 y FG(n)2532 1297 y FE(\))2562 1211 y Fy(\014)2562 1256 y(\014)2562 1300 y(\014)2587 1231 y FF(2)2635 1297 y FH(d)2675 1261 y FF(3)2709 1297 y FH(xd)2793 1261 y FF(3)2827 1297 y FH(k)2867 1305 y FF(1)2902 1297 y FH(:::d)3005 1261 y FF(3)3040 1297 y FH(k)3080 1305 y FG(n)3144 1297 y FH(<)21 b Fz(1)3302 1169 y Fy(\))542 1527 y FO(where)1029 1637 y FH( )1082 1602 y FG(\033)r FF(\()o FG(n)p FF(\))1223 1637 y FE(\()p FA(x)p FH(;)12 b FA(k)1380 1645 y FF(1)1415 1637 y FH(;)h(:::;)h FA(k)1594 1645 y FG(n)1636 1637 y FE(\))21 b(=)g(\(2)p FH(\031)s FE(\))1913 1602 y Fx(\000)1971 1582 y FL(3)p 1971 1590 29 3 v 1971 1623 a(2)2026 1536 y Fy(Z)2116 1637 y FH(e)2152 1602 y FG(i)p Fw(p)p Fx(\001)p Fw(x)2285 1618 y Fy(e)2271 1637 y FH( )2324 1602 y FG(\033)r FF(\()p FG(n)p FF(\))2321 1651 y Fw(p)2465 1637 y FE(\()p FA(k)2542 1645 y FF(1)2576 1637 y FH(;)13 b(:::;)h FA(k)2755 1645 y FG(n)2797 1637 y FE(\))f FH(d)2880 1602 y FF(3)2914 1637 y FH(p)318 b FI(\(1.13\))515 1800 y FO(In)25 b FH( )669 1769 y FG(\033)666 1816 y(h;\024)757 1826 y FL(1)807 1800 y FE(\()o FH(t)p FE(\))g FO(the)h(subscript)e FH(h)i FO(is)f(referred)e(to)i(the)h(w)n(a)n(v)n(e)d(function)j(in)f FA(P)h FO(of)f(the)g(one-particle)515 1892 y(state.)i(The)g(supp)r(ort) f(of)h FN(h)g FO(has)g(to)g(b)r(e)g(restricted)f(to)h(a)g(neigh)n(b)r (orho)r(o)r(d)e(of)i FA(P)22 b FE(=)f(0)28 b FO(suc)n(h)e(that)515 1983 y Fz(jr)p FH(E)661 1951 y FG(\033)715 1983 y FE(\()p FA(P)p FE(\))p Fz(j)i FH(<)g FE(1)p FO(.)k(In)g(the)g(expression)e (\(1.13\))h(the)h FA(P)p FI(-)p FO(dep)r(endence)g(is)g(hidden)g(in)g (the)g(\(sym-)515 2088 y(metric\))27 b(function)1144 2068 y Fy(e)1131 2088 y FH( )1184 2050 y FG(\033)r FF(\()o FG(n)p FF(\))1181 2097 y Fw(p)1324 2088 y FE(\()p FA(k)1401 2096 y FF(1)1435 2088 y FH(;)13 b(:::;)h FA(k)1614 2096 y FG(n)1657 2088 y FE(\))27 b FO(where)g FA(p)h FO(is)f(the)h(c)n (harged)e(particle)h(momen)n(tum.)515 2270 y(By)g(the)h(substitution)g FA(p)21 b FE(=)h FA(P)17 b Fz(\000)g FA(k)1603 2278 y FF(1)1654 2270 y Fz(\000)g FH(:::)h Fz(\000)f FA(k)1936 2278 y FG(n)1978 2270 y FO(,)28 b(w)n(e)f(ha)n(v)n(e)633 2447 y FH( )686 2415 y FG(\033)r FF(\()o FG(n)p FF(\))826 2447 y FE(\()p FA(x)p FH(;)13 b FA(k)984 2455 y FF(1)1018 2447 y FH(;)g(:::;)i FA(k)1198 2455 y FG(n)1240 2447 y FE(\))83 b(=)f(\(2)p FH(\031)s FE(\))1640 2412 y Fx(\000)1698 2392 y FL(3)p 1698 2400 V 1698 2433 a(2)1753 2387 y Fy(R)1817 2447 y FH(e)1853 2415 y FG(i)p Fw(P)p Fx(\001)p Fw(x)1981 2447 y FH(e)2017 2415 y Fx(\000)p FG(i)p FF(\()p Fw(k)2147 2425 y FL(1)2180 2415 y FF(+)p FG(::)p FF(+)p Fw(k)2348 2423 y Fq(n)2388 2415 y FF(\))p Fx(\001)p Fw(x)2485 2427 y Fy(e)2471 2447 y FH( )2524 2409 y FG(\033)r FF(\()p FG(n)p FF(\))2521 2456 y Fw(p)2665 2447 y FE(\()p FA(k)2742 2455 y FF(1)2776 2447 y FH(;)13 b(:::;)h FA(k)2955 2455 y FG(n)2998 2447 y FE(\))e FH(d)3080 2415 y FF(3)3115 2447 y FH(p)1353 2556 y FE(=)1527 2496 y Fy(R)1591 2556 y FH(e)1627 2524 y FG(i)p Fw(P)p Fx(\001)p Fw(x)1755 2556 y FH(e)1791 2524 y Fx(\000)p FG(i)p Fw(P)1908 2503 y Fq(ph)1977 2524 y Fx(\001)p Fw(x)2036 2556 y FH(h)i FE(\()o FA(P)p FE(\))f FH( )2279 2518 y FG(\033)r FF(\()p FG(n)p FF(\))2276 2577 y Fw(P)2420 2556 y FE(\()p FA(k)2497 2564 y FF(1)2531 2556 y FH(;)g(:::;)h FA(k)2710 2564 y FG(n)2752 2556 y FE(\))f FH(d)2835 2524 y FF(3)2870 2556 y FH(P)3271 2493 y FI(\(1.14\))542 2719 y FO(where)994 2811 y FE(\(2)p FH(\031)s FE(\))1139 2775 y Fx(\000)1196 2755 y FL(3)p 1196 2763 V 1196 2796 a(2)1265 2791 y Fy(e)1252 2811 y FH( )1305 2773 y FG(\033)r FF(\()o FG(n)p FF(\))1302 2832 y Fw(p)p FF(=)p Fw(P)p Fx(\000)p Fw(k)1518 2842 y FL(1)1550 2832 y Fx(\000)p FG(::)p Fx(\000)p Fw(k)1720 2840 y Fq(n)1778 2811 y FE(\()p FA(k)1855 2819 y FF(1)1889 2811 y FH(;)g(:::;)h FA(k)2068 2819 y FG(n)2110 2811 y FE(\))22 b(=)f FH(h)13 b FE(\()p FA(P)p FE(\))g FH( )2486 2773 y FG(\033)r FF(\()o FG(n)p FF(\))2483 2832 y Fw(P)2626 2811 y FE(\()p FA(k)2703 2819 y FF(1)2737 2811 y FH(;)g(:::;)h FA(k)2916 2819 y FG(n)2959 2811 y FE(\))282 b FI(\(1.15\))515 2935 y FO(with)28 b(the)g(normalization)1274 3045 y Fy(X)1310 3202 y FG(n)1398 3014 y Fy(Z)1487 3030 y(\014)1487 3075 y(\014)1487 3120 y(\014)1513 3116 y FH( )1566 3078 y FG(\033)r FF(\()o FG(n)p FF(\))1563 3137 y Fw(P)1707 3116 y FE(\()o FA(k)1783 3124 y FF(1)1818 3116 y FH(;)13 b(:::;)h FA(k)1997 3124 y FG(n)2039 3116 y FE(\))2069 3030 y Fy(\014)2069 3075 y(\014)2069 3120 y(\014)2095 3051 y FF(2)2142 3116 y FH(d)2182 3080 y FF(3)2216 3116 y FH(k)2256 3124 y FF(1)2291 3116 y FH(::::d)2415 3080 y FF(3)2451 3116 y FH(k)2491 3124 y FG(n)2554 3116 y FE(=)22 b(1)13 b FH(:)515 3347 y FO(In)23 b(order)e(to)i(prop)r(erly)e (con)n(trol)h(the)h(action,)f(on)h(the)g(one-particle)e(state,)i(of)f (the)i(W)-7 b(eyl)23 b(op)r(er-)515 3438 y(ator)h(\020carrying\021)29 b(the)c(b)r(oson)g(cloud,)g(one)f(\034rst)h(de\034nes)g(the)h(op)r (erator)d(v)-5 b(alued)25 b(distributions)515 3469 y Fy(\010)560 3529 y FH(a)12 b FE(\()p FA(k)p FE(\))g FH(;)h(a)807 3497 y Fx(y)853 3529 y FE(\()o FA(k)p FE(\))959 3469 y Fy(\011)1024 3529 y FO(smeared)19 b(out)h(with)h(functions)f FH(f)h FE(\()p FA(k)p FH(;)13 b FA(P)p FE(\))g FH(;)21 b FO(where)e FA(P)i FO(is)f(the)h(the)f(total)g(momen-)515 3639 y(tum,)j(and)f(then)g(applied)h(to)f(the)g(v)n(ector)1824 3619 y Fy(e)1810 3639 y FH( )1863 3601 y FG(\033)r FF(\()o FG(n)p FF(\))1860 3648 y Fw(p)2004 3639 y FE(\()o FA(k)2080 3647 y FF(1)2115 3639 y FH(;)13 b(:::;)h FA(k)2294 3647 y FG(n)2336 3639 y FE(\))p FO(.)22 b(Exploiting)h(the)f(decomp)r (osition)515 3731 y(of)27 b Fz(H)h FO(on)g(the)g(sp)r(ectrum)f(of)h FA(P)p FO(,)g(the)g(de\034nition)g(is:)667 3826 y Fy(Z)757 3927 y FH(a)12 b FE(\()p FA(k)p FE(\))g FH(f)21 b FE(\()p FA(k)p FH(;)13 b FA(P)p FE(\))g FH(d)1241 3891 y FF(3)1275 3927 y FH(k)r( )1370 3891 y FG(\033)r FF(\()p FG(n)p FF(\))1511 3927 y FE(\()p FA(x)p FH(;)g FA(k)1669 3935 y FF(1)1703 3927 y FH(;)g(:::;)h FA(k)1882 3935 y FG(n)1925 3927 y FE(\))21 b(:=)1214 b FI(\(1.16\))586 4179 y FE(=)736 4119 y Fy(R)28 b(R)864 4179 y FH(a)13 b FE(\()o FA(k)p FE(\))g FH(f)21 b FE(\()p FA(k)p FH(;)13 b FA(P)p FE(\))g FH(e)1345 4147 y FG(i)p Fw(P)p Fx(\001)p Fw(x)p Fx(\000)p FG(i)p Fw(P)1586 4126 y Fq(ph)1655 4147 y Fx(\001)p Fw(x)1714 4179 y FH(h)g FE(\()p FA(P)p FE(\))g FH( )1957 4141 y FG(\033)r FF(\()o FG(n)p FF(\))1954 4200 y Fw(P)2097 4179 y FE(\()p FA(k)2174 4187 y FF(1)2209 4179 y FH(;)g(:::;)h FA(k)2388 4187 y FG(n)2430 4179 y FE(\))f FH(d)2513 4147 y FF(3)2547 4179 y FH(P)e(d)2647 4147 y FF(3)2681 4179 y FH(k)586 4379 y FE(=)83 b(\()o(2)p FH(\031)s FE(\))873 4344 y Fx(\000)931 4324 y FL(3)p 931 4332 V 931 4365 a(2)986 4326 y Fz(p)p 1050 4326 47 4 v 53 x FH(n)1110 4319 y Fy(R)1174 4379 y FH(e)1210 4348 y Fx(\000)p FG(i)p Fw(P)1327 4327 y Fq(ph)1395 4348 y Fx(\001)p Fw(x)p FF(+)p FG(i)p Fw(P)p Fx(\001)p Fw(x)1638 4319 y Fy(R)1702 4379 y FH(f)22 b FE(\()o FA(k)p FH(;)13 b FA(P)p FE(\))g FH(e)2010 4348 y Fx(\000)p FG(i)p Fw(k)p Fx(\001)p Fw(x)2190 4360 y Fy(e)2176 4379 y FH( )2229 4341 y FG(\033)r FF(\()o FG(n)p FF(\))2226 4401 y Fw(P)p Fx(\000)p Fw(k)p Fx(\000)p Fw(k)2441 4411 y FL(2)2474 4401 y FG(:::)p Fx(\000)p Fw(k)2615 4409 y Fq(n)2673 4379 y FE(\()p FA(k)p FH(;)f FA(k)2830 4387 y FF(2)2865 4379 y FH(;)h(::;)g FA(k)3022 4387 y FG(n)3065 4379 y FE(\))f FH(d)3147 4348 y FF(3)3182 4379 y FH(k)r(d)3264 4348 y FF(3)3298 4379 y FH(P)28 b(:)515 4543 y FO(A)g(similar)e(pro)r(cedure)h(is)g(used)h(for)f(the)h (action)f(of)g(the)h(op)r(erator)2627 4483 y Fy(R)2691 4543 y FH(a)2732 4512 y Fx(y)2777 4543 y FE(\()p FA(k)p FE(\))13 b FH(f)21 b FE(\()p FA(k)p FH(;)12 b FA(P)p FE(\))h FH(d)3208 4512 y FF(3)3243 4543 y FH(k)30 b FO(.)515 4726 y(On)38 b(the)h(basis)f(of)h(the)g(previous)e(de\034nitions,)i (the)g(\034nal)g(expression)e(w)n(e)h(are)g(in)n(terested)1970 5059 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 515 523 a FO(in)24 b(can)f(b)r(e)i(handled)f (after)f(ha)n(ving)g(expanded,)h(in)g(terms)g(of)g(the)g(generator,)e (the)i(formal)f(ex-)515 614 y(pression)j(for)h(the)h FJ(L.S.Z.)g FO(W)-7 b(eyl)28 b(op)r(erator)d(\020carrying\021)32 b(the)c(b)r(oson)e(cloud.)i(In)f(other)g(w)n(ords,)515 706 y(it)f(means)f(that)g(the)h(appro)n(ximating)e(v)n(ector)g FH( )2018 674 y FG(\033)2015 721 y(h;\024)2106 731 y FL(1)2155 706 y FE(\()p FH(t)p FE(\))h FO(is)h(de\034ned)f(starting)g (from)g(eac)n(h)g(pro-)515 797 y(jection)i(on)h(the)g FH(n)p Fz(\000)p FO(particle)f(subspace)p FI(,)g FO(namely)p FI(:)706 917 y Fy(\020)752 999 y FH(e)788 964 y Fx(\000)p FG(iH)908 972 y Fq(\033)947 964 y FG(t)975 999 y FH( )1028 964 y FG(\033)1025 1013 y(h;k)1113 1023 y FL(1)1163 999 y FE(\()p FH(t)p FE(\))1250 917 y Fy(\021)1296 934 y FF(\()p FG(n)p FF(\))1399 999 y FE(\()p FA(x)p FH(;)13 b FA(k)1557 1007 y FF(1)1591 999 y FH(;)g(::)p FA(k)1714 1007 y FG(n)1757 999 y FE(\))21 b(:=)1382 b FI(\(1.17\))556 1273 y FE(=)699 1123 y Fy(0)699 1258 y(@)766 1213 y(R)830 1273 y FH(e)866 1242 y Fx(\000)p FG(iH)990 1221 y Fq(ph)1059 1242 y FG(t)1087 1273 y FH(e)1123 1178 y Fx(\000)p FG(g)1213 1138 y Fm(R)1251 1157 y Fq(\024)1285 1169 y FL(1)1241 1197 y Fq(\033)1495 1151 y(a)p FL(\()p Fl(k)p FL(\))q Fk(\000)p Fq(a)1682 1134 y Fk(y)1714 1151 y FL(\()p Fl(k)p FL(\))p 1343 1166 601 3 v 1343 1204 a Fk(p)p 1388 1204 29 3 v 1388 1238 a FL(2)q Fk(j)p Fl(k)p Fk(j)1498 1196 y FL(3)p 1498 1203 V 1498 1234 a(2)1540 1246 y FE(\()1570 1238 y FL(1)p Fk(\000)1646 1227 y Fj(b)1643 1238 y Fl(k)p Fk(\001r)p Fq(E)1787 1226 y(\033)1827 1238 y FL(\()p Fl(P)p FL(\))1914 1246 y FE(\))1954 1178 y FG(d)1986 1157 y FL(3)2018 1178 y FG(k)2056 1273 y FH(e)2092 1242 y FG(iH)2168 1221 y Fq(ph)2237 1242 y FG(t)2265 1273 y FH(e)2301 1242 y Fx(\000)p FG(iE)2419 1221 y Fq(\033)2458 1242 y FF(\()p Fw(P)p FF(\))o FG(t)2580 1273 y 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35 x FL(2)p Fk(j)p Fl(k)p Fk(j)1710 1504 y FL(3)p 1710 1511 V 1710 1542 a(2)1752 1555 y FE(\()1782 1547 y FL(1)p Fk(\000)1858 1536 y Fj(b)1855 1547 y Fl(k)q Fk(\001r)p Fq(E)2000 1535 y(\033)2039 1547 y FL(\()p Fl(P)p FL(\))2126 1555 y FE(\))2166 1486 y FG(d)2198 1465 y FL(3)2230 1486 y FG(k)2268 1582 y FH(e)2304 1550 y Fx(\000)p FG(iE)2422 1529 y Fq(\033)2461 1550 y FF(\()p Fw(P)p FF(\))p FG(t)2584 1582 y FH(h)g FE(\()p FA(P)p FE(\))g FH( )2827 1550 y FG(\033)2824 1597 y Fw(P)2875 1582 y FH(d)2915 1550 y FF(3)2949 1582 y FH(P)3009 1409 y Fy(1)3009 1540 y(C)3009 1588 y(A)3076 1426 y FF(\()p FG(n)p FF(\))3192 1582 y FE(\()p FA(k)3269 1590 y FF(1)3303 1582 y FH(;)g(::)p FA(k)3426 1590 y FG(n)3469 1582 y FE(\))25 b FH(:)515 1911 y FO(F)-7 b(or)37 b(a)h(\034xed)f(infrared)g (cut-o\033)h FH(\033)s FO(,)g(the)g(time)h(limit)f(of)g(the)g(v)n (ector)f FH( )2797 1879 y FG(\033)2794 1926 y(h;\024)2885 1936 y FL(1)2935 1911 y FE(\()p FH(t)p FE(\))g FO(is)h(obtained)515 2002 y(exploiting)27 b(Hepp's)g(metho)r(d)h([He.].)g(The)g(pro)r(of)f (basically)f(relies)h(on)g(the)h(estimates)f(of)g(the)515 2093 y(exp)r(ectation)e(v)-5 b(alues)25 b(of)h(p)r(olynomials)f(in)1852 2033 y Fy(\010)1897 2093 y FH(b)13 b FE(\()p FA(k)p FE(\))f FH(;)h(b)2129 2062 y Fx(y)2175 2093 y FE(\()p FA(k)p FE(\))2281 2033 y Fy(\011)2352 2093 y FO(on)25 b(the)h(ground)f(state)g FH( )3140 2062 y FG(\033)3137 2108 y Fw(P)3214 2093 y FO(\(gener-)515 2185 y(alized)30 b(resolv)n(en)n(t)f(form)n(ulas,)g (see)i([F)-7 b(r.1]\))30 b(and)h(on)f(an)h(implicit)g(propagation)e (estimate)h(for)515 2276 y(the)f(electron)f(con)n(tained)g(in)h(the)g (constrain)n(t)f Fz(jr)p FH(E)2131 2244 y FG(\033)2185 2276 y FE(\()p FA(P)p FE(\))q Fz(j)23 b FH(<)g FE(1)29 b FO(whic)n(h)g(holds)g(in)g(a)f(neigh)n(b)r(or-)515 2367 y(ho)r(o)r(d)f(of)h FA(P)21 b FE(=)h(0)p FO(.)515 2550 y(The)40 b(ultimate)i(motiv)-5 b(ation)40 b(for)g(the)h(previous)f (construction)g(is)g(ho)n(w)n(ev)n(er)f(the)i(limit)g(in)515 2641 y(time)33 b(of)g FH( )859 2650 y FG(h;\024)950 2660 y FL(1)1000 2641 y FE(\()p FH(t)p FE(\))g FO(with)g(no)g(infrared)f (cut-o\033)i FH(\033)s FO(.)e(In)i(the)f(ph)n(ysical)f(situation)h (without)h(in-)515 2733 y(frared)c(cut-o\033,)h(it)g(indeed)g(represen) n(ts)f(a)g(minimal)h(\(with)h(resp)r(ect)f(to)g(the)g(photon)g(cloud\)) 515 2824 y(description)c(of)h(an)g(infraparticle)f(in)h(a)g(scattering) e(state.)i(It)h(means)e(that)h(a)g(photon)g(cloud)515 2915 y(of)j(soft)h(photons)f(is)g(una)n(v)n(oidable,)f(i.e.)i FH(\024)1850 2923 y FF(1)1916 2915 y FO(can)g(b)r(e)g(arbitrarily)d (small)i(but)h(not)g(zero.)f(The)515 3007 y(subspace)c(generated)h(b)n (y)g(suc)n(h)g(v)n(ectors)f(can)h(b)r(e)g(seen)h(as)e(a)h(one-particle) f(subspace,)h(up)h(to)515 3098 y(an)e(observ)-5 b(abilit)n(y)26 b(threshold)h(in)h(the)g(energy)e(of)i(the)g(asymptotic)f(photons.)515 3513 y FC(1.3)112 b(Minimal)35 b(asymptotic)h(electron)515 3658 y FO(Let)30 b(us)f(inquire)h(ab)r(out)g(the)g(features)f(and)h (the)g(problems)f(of)h(the)g(previous)e(construction.)515 3749 y(As)35 b(already)f(p)r(oin)n(ted)i(out,)g(no)f(problem)g(arises)f (in)i(the)g(norm)f(con)n(trol)f(and)i(in)f(the)h(con-)515 3840 y(v)n(ergence)28 b(in)i(time)h(of)e FH( )1294 3809 y FG(\033)1291 3856 y(h;\024)1382 3866 y FL(1)1432 3840 y FE(\()p FH(t)p FE(\))h FO(as)f(long)g(as)g FH(\033)f Fz(6)p FE(=)c(0)p FO(,)31 b(b)r(ecause)e(the)i(series)d(expansion)h(of) h(the)515 3932 y(W)-7 b(eyl)30 b(op)r(erators)f(in)i(terms)f(of)g(the)h (generator)d(can)j(b)r(e)f(con)n(trolled,)g(basically)-7 b(,)29 b(due)i(to)f(the)515 4023 y(regularit)n(y)21 b(prop)r(erties)i (in)g FA(P)h FO(of)f FH( )1595 3991 y FG(\033)1592 4038 y Fw(P)1667 4023 y FO(in)g(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)i FA(P)e FE(=)f(0)p FO(.)j(The)f(situation)g(c)n(hanges)515 4114 y(drastically)k(for)h FH(\033)d FE(=)d(0)p FO(.)29 b(If)g(w)n(e)f(remo)n(v)n(e)f(the)h(cut-o\033)h FH(\033)i FO(in)e(the)f(expression)f(\(1.17\),)h(the)h(de\034-)515 4206 y(nition)e(of)f(the)i(v)n(ector)d(at)i(\034nite)g(times)g(b)r (ecomes)f(a)h(delicate)f(issue.)h(The)g(previous)e(metho)r(d)515 4297 y(fails)e(b)r(ecause)g(of)h(div)n(ergences)d(app)r(earing)i(in)g (the)h(series)f(expansion)f(of)i(the)g(W)-7 b(eyl)23 b(op)r(erator)1487 4558 y FH(e)1523 4463 y Fx(\000)p FG(g)1613 4422 y Fm(R)1652 4442 y Fq(\024)1686 4454 y FL(1)1641 4482 y Fq(\033)1895 4436 y(a)p FL(\()q Fl(k)p FL(\))p Fk(\000)p Fq(a)2082 4419 y Fk(y)2114 4436 y FL(\()q Fl(k)p FL(\))p 1743 4451 601 3 v 1743 4489 a Fk(p)p 1789 4489 29 3 v 34 x FL(2)p Fk(j)p Fl(k)p Fk(j)1898 4481 y FL(3)p 1898 4488 V 1898 4519 a(2)1941 4531 y FE(\()1971 4523 y FL(1)p Fk(\000)2047 4512 y Fj(b)2044 4523 y Fl(k)p Fk(\001r)p Fq(E)2188 4511 y(\033)2227 4523 y FL(\()q Fl(P)p FL(\))2314 4531 y FE(\))2354 4463 y FG(d)2386 4442 y FL(3)2419 4463 y FG(k)2474 4558 y FH(:)776 b FI(\(1.18\))515 4719 y FO(The)33 b(expansion)e(is)i(tec)n(hnically)f(forced)g(b)r (ecause)g(the)i FA(P)p Fz(\000)p FO(\034b)r(er)e(spaces)g Fz(H)2944 4727 y Fw(P)3028 4719 y FO(are)g(not)g(pre-)515 4810 y(serv)n(ed)h(under)i(the)g(action)f(of)h(the)g(op)r(erator)e (\(1.18\).)h(The)h(de\034nition)h(at)e(\034nite)i(times)f(of)1950 5059 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 515 523 a FO(eac)n(h)37 b FH(n)p Fz(\000)p FO(comp)r(onen)n(t)i(in)f(\(1.17\))g(and)g(summabilit)n(y)g (in)h FN(n)f FO(is)h(still)g(w)n(ell)f(founded)g(b)n(y)h(as-)515 614 y(suming)e(some)g(regularit)n(y)e(prop)r(erties)h(whic)n(h)i(can)f (b)r(e)h(ev)n(en)n(tually)e(reconciled)g(with)i(the)515 706 y(existence)24 b(of)g(the)h(second)f(deriv)-5 b(ativ)n(e)23 b(of)i(the)g(ground)e(state)h(energy)f FH(E)17 b FE(\()p FA(P)p FE(\))p FO(.)25 b(Ho)n(w)n(ev)n(er,)d(ev)n(en)515 797 y(in)28 b(these)f(assumptions,)g(the)h(time)g(asymptotic)f(b)r(eha) n(vior)f(is)i(practically)e(out)h(of)h(con)n(trol.)515 980 y(The)e(di\036culties)h(coming)e(from)h(the)h(expression)d(without) j(the)g(cut-o\033)f FH(\033)j FO(are)c(at)h(the)h(origin)515 1071 y(of)32 b(an)g(alternativ)n(e)g(and,)g(for)g(some)g(asp)r(ects,)g (conceptually)g(di\033eren)n(t)g(recip)r(e)g(for)g(the)h(ap-)515 1162 y(pro)n(ximating)24 b(v)n(ector)g(that)i(w)n(e)g(denote)f(b)n(y)h FH( )1954 1171 y FG(h;\024)2045 1181 y FL(1)2094 1162 y FE(\()p FH(t)p FE(\))p FO(.)g(The)g(new)f(main)h(ingredien)n(ts)f (that)h(w)n(e)515 1254 y(in)n(tro)r(duce)h(are:)515 1345 y(A)f(con)n(v)n(ergence)c(sc)n(heme)j(based)g(on)g(a)g(diagonal)f (limiting)i(pro)r(cedure)e(to)i(b)r(etter)g(follo)n(w)e(the)515 1436 y(slo)n(w)i(asymptotic)h(decoupling)g(due)h(to)f(the)h(in)n (teraction)e(with)i(infrared)f(b)r(osons.)f(It)i(means)515 1528 y(that)f(the)h(infrared)e(cut-o\033)i(in)f(the)h(appro)n(ximating) d(v)n(ectors)h(is)h(remo)n(v)n(ed)f(only)g(asymptoti-)515 1619 y(cally)h(in)h(time;)515 1710 y(A)21 b(constructiv)n(e)e(c)n (haracterization)f(of)j(one)f(particle)g(states)g(pro)n(vided)g(in)h ([Pi.])g(whic)n(h)g(enable)515 1802 y(us)27 b(to)h(hea)n(vily)e(use)i (strong)e(regularit)n(y)g(prop)r(erties)g(\(for)h(details)h(see)f (Section)g(3)h(in)f([Pi.]\);)515 1893 y(The)32 b(propagation)e (estimate)i(pro)n(vided)f(b)n(y)h(the)h(non-relativistic)d(lo)r(calit)n (y)i(of)g(the)g(mo)r(del,)515 1984 y(namely)26 b(the)i(decoupling)e (mec)n(hanism)h(used)g(in)g(Haag-Ruelle)e(theory)i(can)f(b)r(e)i(repro) r(duced)515 2076 y(in)h(terms)g(of)g(cluster)f(prop)r(erties)g(at)h (\034xed)g(time)h(of)f(the)g(photon)g(\034eld)g(and)g(of)g(the)g (curren)n(t)515 2167 y(densit)n(y)e(\034eld)h(asso)r(ciated)e(to)i(the) g(electron.)515 2350 y(Going)d(to)h(tec)n(hnical)g(details,)g(our)g (prop)r(osal)e(for)i(the)h(generic)e(v)n(ector)g FH( )2816 2359 y FG(h;\024)2907 2369 y FL(1)2956 2350 y FE(\()p FH(t)p FE(\))h FO(in)n(tro)r(duces:)515 2441 y(A)i(time)g(dep)r(enden)n (t)g(cut-o\033)f FH(\033)1499 2449 y FG(t)1527 2441 y FO(,)h(whic)n(h)f(is)h(remo)n(v)n(ed)e(at)h(a)g(rate)g(faster)g(than) 3003 2410 y FF(1)p 3003 2424 31 4 v 3006 2466 a FG(t)3043 2441 y FO(;)515 2532 y(The)e(transformation)f(of)i(the)g(in)n(tegral)e (in)i FH(d)1916 2500 y FF(3)1951 2532 y FH(P)36 b FO(to)26 b(a)f(Riemann)g(sum)h(b)n(y)g(a)f(time)h(dep)r(enden)n(t)515 2623 y(cell-partition)g(of)i(the)g FA(P)p Fz(\000)p FO(momen)n(tum)g (space;)515 2715 y(A)c(phase)g(factor,)f(already)g(someho)n(w)g(presen) n(t)g(in)i(the)f(ten)n(tativ)n(e)g(construction)f(b)n(y)h(F)-7 b(r\366hlic)n(h)515 2806 y(for)36 b(the)g(case)g FH(\033)h FE(=)e(0)p FO(,)i(whic)n(h)f(is)g(here)g(exploited,)g(in)h(applying)f (Co)r(ok's)f(argumen)n(t,)g(as)h(a)515 2897 y(function)31 b(of)f(the)h(v)-5 b(ariable)1410 2862 y Fw(x)p FF(\()p FG(t)p FF(\))p 1410 2880 109 4 v 1452 2923 a FG(t)1554 2897 y FE(:=)25 b FH(e)1696 2866 y FG(iH)t(t)1810 2867 y Fw(x)p 1810 2881 37 4 v 1816 2923 a FG(t)1856 2897 y FH(e)1892 2866 y Fx(\000)p FG(iH)t(t)2043 2897 y FO(,)31 b(that)f(is)h(the)f(electron)g(mean)g(v)n(elo)r(cit)n(y)f(\(at)515 2989 y(time)f FH(t)p FO(\))f(up)h(to)g(a)f(correction)f(of)h(order)f FH(t)1809 2957 y Fx(\000)p FF(1)1892 2989 y FO(.)515 3171 y(The)36 b(t)n(w)n(o)f(main)h(di\033erences)f(with)i(resp)r(ect)e (to)h(F)-7 b(r\366hlic)n(h's)35 b(prop)r(osal,)g(namely)g(the)i(diag-) 515 3263 y(onal)29 b(limit)i(and)f(the)g(cell)g(partition)f(of)h(the)h FA(P)p Fz(\000)p FO(space,)e(represen)n(t)g(the)h(building)g(blo)r(c)n (ks)g(of)515 3354 y(a)i(strategy)f(con)n(trolling,)h(sim)n(ultaneously) -7 b(,)31 b(the)i(logarithmic)f(div)n(ergences)f(arising)g(in)i(the)515 3445 y(t)n(w)n(o)27 b(limits)i FH(\033)c Fz(!)d FE(0)28 b FO(and)g FH(t)22 b Fz(!)g(1)p FO(.)28 b(T)-7 b(o)28 b(implemen)n(t)h(our)e(strategy)-7 b(,)27 b(the)i(use)f(of)g (di\033eren)n(t)g(time)515 3537 y(scales)e(is)i(crucial.)e(They)i(are)e (basically:)515 3628 y(The)g(rate)g FH(\033)898 3636 y FG(t)953 3628 y FO(of)h(the)g(remo)n(v)-5 b(al)25 b(of)h(the)h (infrared)f(cut-o\033,)h(b)n(y)f(whic)n(h)h(w)n(e)f(approac)n(h)f(the)i (limit)515 3719 y FH(\033)d FE(=)d(0)i FO(of)g(no)g(infrared)f (cut-o\033)h(cutting)h(a)n(w)n(a)n(y)d(the)i(frozen)g(degrees)e(of)i (freedom)g(at)g(the)g(giv)n(en)515 3811 y(time)28 b(scale)f FH(t)p FO(;)515 3902 y(The)35 b(slo)n(w)n(er)e(rate)h(of)h(the)g (partition)g(go)n(v)n(erned)d(b)n(y)j(an)g(exp)r(onen)n(t)f FH(\017)p FO(,)i(determined)f(b)n(y)f(the)515 3993 y(\(estimated\))28 b(time)g(scale)f(of)g(the)h(decoupling.)515 4176 y(Let)k(us)g(an)n (ticipate)g(the)g(exp)r(ected)h(adv)-5 b(an)n(tages)30 b(of)i(these)h(constructiv)n(e)e(mo)r(di\034cations)g(in)515 4267 y(con)n(trolling)26 b(the)i(t)n(w)n(o)e(quan)n(tities)1281 4429 y Fz(k)p FH( )1369 4438 y FG(h;\024)1460 4448 y FL(1)1510 4429 y FE(\()p FH(t)p FE(\))o Fz(k)1636 4393 y FF(2)1760 4429 y FH(;)90 b Fz(k)p FH( )1959 4438 y FG(h;\024)2050 4448 y FL(1)2100 4429 y FE(\()p FH(t)2158 4437 y FF(2)2192 4429 y FE(\))17 b Fz(\000)g FH( )2366 4438 y FG(h;\024)2457 4448 y FL(1)2506 4429 y FE(\()p FH(t)2564 4437 y FF(1)2599 4429 y FE(\))o Fz(k)2667 4393 y FF(2)3271 4429 y FI(\(1.19\))542 4591 y FO(that)28 b(w)n(e)f(will)h(study)g(in)g(the)g(pap)r(er.)515 4774 y FJ(1\))j FO(By)c(the)h(transformation)e(of)h(the)h(in)n(tegral)f(to)g (a)g(Riemann)h(sum:)1950 5059 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 515 523 a FO(W)-7 b(e)33 b(can)f(replace)f(the) i(series)e(expansion)g(of)i(the)g(W)-7 b(eyl)32 b(op)r(erators)f(b)n(y) h(a)g(\020cell-expansion\021)515 614 y(in)c(the)g FA(P)p Fz(\000)p FO(space)g(whic)n(h)g(w)n(e)f(can)h(easily)f(con)n(trol)g (exploiting)g(the)i(cluster)e(prop)r(ert)n(y)g(of)h(the)515 706 y(system.)d(In)h(this)g(resp)r(ect,)g(w)n(e)f(an)n(ticipate)h(here) f(that)h(di\033eren)n(t)g(v)-5 b(alues)25 b(of)h FA(P)g FO(in)g(the)g(expan-)515 797 y(sion)h(corresp)r(ond)f(to)h(di\033eren)n (t)h(asymptotic)f(v)n(elo)r(cities)g(of)g(the)h(c)n(harged)e(particle;) 515 888 y(F)-7 b(or)29 b(all)g(\034nite)i(times,)f(w)n(e)f(deal)h(with) g(an)g(expression)e(in)i(terms)g(of)f(b)r(ounded)i(op)r(erators)c(in) 515 980 y(the)c(Hilb)r(ert)g(space,)f(that)h(w)n(e)f(can)g(actually)g (handle)g(without)h(considering,)e(in)i(general,)e(an)n(y)515 1071 y(particular)26 b(w)n(a)n(v)n(e)g(function)i(represen)n(tation)e (but)i(simply)f(abstract)g(calculus.)515 1254 y FJ(2\))k FO(By)c(in)n(tro)r(ducing)g(a)g(time-dep)r(enden)n(t)h(cut-o\033)g FH(\033)2159 1262 y FG(t)2214 1254 y FO(:)515 1345 y(W)-7 b(e)26 b(can)f(exploit)g(the)h(unitarit)n(y)f(prop)r(ert)n(y)g(of)g (the)h(W)-7 b(eyl)26 b(op)r(erators)d(as)i(long)g(as)g FH(\033)3129 1353 y FG(t)3178 1345 y FH(>)c FE(0)p FO(.)26 b(F)-7 b(or)515 1436 y(eac)n(h)35 b(cell,)h(it)g(pro)n(vides)e (a-priori)g(estimates)h(without)i(resumming)e(con)n(tributions)g(whic)n (h)515 1528 y(are)23 b(logarithmically)g(div)n(ergen)n(t)g(in)i(the)f (infrared)g(limit.)h(Moreo)n(v)n(er)d(the)j(a-priori)d(estimates)515 1619 y(matc)n(h)g(easily)f(with)i(the)g(p)r(o)n(w)n(er)e(la)n(w)h(deca) n(y)f(of)h(the)h(v)-5 b(anishing)22 b(quan)n(tities)g(whic)n(h)g (neutralize)515 1710 y(the)28 b(div)n(ergen)n(t)e(terms;)515 1802 y(W)-7 b(e)30 b(can)f(extend)g(or)g(simply)g(push)h(to)f(the)h (limit)g(of)g(no)f(infrared)f(cut-o\033)i(some)f(prop)r(erties)515 1893 y(whic)n(h)e(hold)h(for)f(the)h(mo)r(del)f(with)h(a)g(\034xed)f (infrared)g(cut-o\033.)g(The)h(prop)r(erties)f(are:)515 1984 y FJ(2i\))k FO(The)d(propagation)d(estimate)1089 2151 y FH(e)1125 2115 y FG(iH)t(t)1229 2151 y FH(f)1287 2068 y Fy(\020)1343 2103 y FA(x)p 1343 2134 47 4 v 1352 2202 a FH(t)1399 2068 y Fy(\021)1458 2151 y FH(e)1494 2115 y Fx(\000)p FG(iH)t(t)1667 2151 y Fz(!)1744 2159 y FG(t)p Fx(!1)1915 2151 y FH(f)c FE(\()p Fz(r)p FH(E)16 b FE(\()p FA(P)p FE(\)\))167 b FH(f)29 b Fz(2)22 b FH(C)2656 2115 y Fx(1)2651 2164 y FF(0)2734 2091 y Fy(\000)2769 2151 y FB(R)2823 2115 y FF(3)2858 2091 y Fy(\001)515 2319 y FO(whic)n(h)33 b(morally)e(holds)i(on)g(one)f(particle)g(improp) r(er)h(states,)f(as)g(it)i(can)e(b)r(e)i(deduced)f(from)515 2419 y(Theorem)21 b(4.2.)g(This)h(extrap)r(olated)f(prop)r(ert)n(y)g (is)g(nothing)h(but)h(the)f(limiting)g(case)g(for)f FH(\033)j Fz(!)d FE(0)515 2519 y FO(of)36 b(the)g(analogous)e(con)n(v)n(ergence)g (whic)n(h)i(can)f(b)r(e)i(easily)e(pro)n(v)n(ed)g(in)h(the)h(case)e(of) h(a)g(\034xed)515 2618 y FH(\033)s Fz(\000)p FO(cuto\033)27 b(dynamics;)515 2718 y FJ(2ii\))i FO(The)24 b(fact)h(that,)g(for)e(a)i (\034xed)f FH(\033)s Fz(\000)p FO(cuto\033)g(dynamics,)g(the)h(one)f (particle)g(states)g(are)f(v)-5 b(acua)515 2818 y(for)20 b(the)i(annihilation)f(part)g(of)g(the)h(asymptotic)e(b)r(oson)h (\034eld.)g(It)h(turns)f(out)g(to)h(b)r(e)f(extremely)515 2917 y(useful)h(in)g(treating)g(the)g(o\033-diagonal)e(terms,)i(with)g (resp)r(ect)g(to)g(the)g(partition,)g(of)g(the)g(quan-)515 3017 y(tities)28 b(\(1.19\);)515 3216 y FJ(3\))37 b FO(The)d(phase)f (factor)f(is)i(emplo)n(y)n(ed)f(in)h(Co)r(ok's)e(argumen)n(t,)h(in)h (analogy)d(with)k(Dollard's)515 3316 y(treatmen)n(t)26 b(of)h(Coulom)n(b)f(scattering)g([Do.])h(\(see)f(also)g([K.F.]\),)h (though)g(the)g(presen)n(t)f(phase)515 3415 y(factor)k(is)g(only)g(a)h (tec)n(hnical)f(to)r(ol)g(in)h(the)g(follo)n(wing)f(sense.)g(In)h(con)n (trast)e(to)h(the)h(Coulom)n(b)515 3515 y(phase,)20 b(it)i(is)f(in)g (fact)h(con)n(v)n(ergen)n(t)c(for)j FH(t)g Fz(!)g(1)p FO(.)g(It)h(is)f(seemingly)f(a)n(v)n(oidable,)f(nev)n(ertheless)h(it)i (is)515 3615 y(helpful)28 b(in)h(our)e(framew)n(ork)e(b)r(ecause)j(pro) n(vides)e(a)i(useful)g(subtraction)f(in)h(the)g(application)515 3714 y(of)f(Co)r(ok's)g(argumen)n(t.)515 4013 y(The)39 b(explicit)g(construction)f(of)h(the)h(generic)e(asymptotic)g(v)n (ector)g(will)h(clarify)g(the)g(mo-)515 4113 y(tiv)-5 b(ations)26 b(for)g(the)i(strategy)d(in)n(v)n(ok)n(ed)g(so)h(far.)g(In) h(the)g(new)g(recip)r(e)g(for)f(the)h(v)n(ector)e FH( )3216 4122 y FG(h;\024)3307 4132 y FL(1)3357 4113 y FE(\()p FH(t)p FE(\))o FO(,)515 4212 y(the)j(k)n(ey)f(di\033eren)n(t)g(p)r(oin) n(ts)h(of)f(view)h(to)f(b)r(e)h(stressed)f(are:)515 4312 y(The)j(construction)f(of)h(the)g(v)n(ector)f(is)h(analyzed)f(in)h (terms)f(of)h(a)g(\020regular\021)k(blo)r(c)n(k)c(giv)n(en)f(b)n(y)515 4412 y(the)21 b(one-particle)e(states)i(of)g(transformed)e (Hamiltonians)h FH(H)2444 4380 y FG(w)2438 4427 y Fw(P)p FG(;\033)2566 4412 y FO(\(see)h(the)g(expression)e(\(1.21\))515 4511 y(written)28 b(later\))g(and)g(a)f(\020dressing\021)34 b(blo)r(c)n(k)27 b(whic)n(h)h(is)g(di\033eren)n(t)g(from)g(the)h(ph)n (ysical)e(dressing)515 4611 y(photon)g(cloud;)515 4710 y(The)19 b(infrared)g(cut-o\033)h(remo)n(v)-5 b(al)18 b(is)i(an)f FJ(a)k(p)l(osteriori)30 b FO(result)20 b(and)f(a)g(b)n (ypro)r(duct)h(of)f(the)h(asymp-)515 4810 y(totic)27 b(decoupling.)1950 5059 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 515 523 a FO(Our)20 b(construction)g(should)h (b)r(e)h(simpli\034ed)f(in)g(order)f(to)h(treat)g(generalizations,)d (for)j(instance)515 623 y(more)g(than)h(one)g(electron.)f(Hop)r(efully) i(some)e(constructiv)n(e)g(device)h(is)g(not)g(necessary)e(or)h(can)515 722 y(b)r(e)j(made)g(less)f(cum)n(b)r(ersome)g(in)i(a)e(mo)r(di\034ed)i (and)e(impro)n(v)n(ed)g(recip)r(e.)h(Ho)n(w)n(ev)n(er)e(the)i(presen)n (t)515 822 y(construction)g(represen)n(ts)g(a)h(starting)f(p)r(oin)n(t) i(for)f(simpler)g(descriptions)f(of)h(the)h(asymptotic)515 922 y(decoupling)h(and)g(for)g(a)g(precise)g(analysis)f(of)i(the)g(in)n (v)n(olv)n(ed)e(time)i(scales.)515 1021 y(The)22 b(en)n(tire)f (construction)g(is)h(self-con)n(tained)f(assuming)g(the)i(results)e(in) h([Pi.])h(together)e(with)515 1121 y(the)27 b(resolv)n(en)n(t)f(form)n (ula)g(\(1.12\))g(\(for)h(details,)g([F)-7 b(r.1]\).)27 b(The)g(only)g(crucial)f(constructiv)n(e)g(h)n(y-)515 1220 y(p)r(othesis)21 b(not)g(pro)n(v)n(en)e(y)n(et)i(\(but)h(ph)n (ysically)e(reasonable\))f(concerns)h(a)h(p)r(ositiv)n(e)g(b)r(ound)g (from)515 1320 y(b)r(elo)n(w)i(for)h(the)g(second)f(deriv)-5 b(ativ)n(e)23 b(of)h(the)g(ground)f(state)h(energy)f FH(E)2681 1288 y FG(\033)2735 1320 y FE(\()p FA(P)p FE(\))h FO(uniform)g(in)g FH(\033)g(>)d FE(0)515 1420 y FO(and)27 b(in)h(the)g(region)e(of)i FA(P)g FO(w)n(e)f(are)f(in)n(terested)i(in.) 515 1734 y FP(1.3.1)93 b(Assumptions)29 b(for)j(the)g(construction)515 1978 y Fp(Sp)-5 b(e)g(ctr)g(al)34 b(pr)-5 b(op)g(erties)515 2161 y FO(W)e(e)28 b(recall)e(some)h(sp)r(ectral)g(results)g(stated)h (in)f([Pi.])i(whic)n(h)e(hold)h(for)f FA(P)22 b Fz(2)f FE(\006)1605 2339 y(\006)h(:=)1784 2256 y Fy(n)1835 2339 y FA(P)g FE(:)35 b Fz(j)p FA(P)p Fz(j)22 b FH(<)2193 2291 y(m)p 2188 2322 77 4 v 2188 2390 a FE(20)2275 2256 y Fy(o)2356 2339 y FH(;)515 2523 y FO(when)g(the)g(coupling)g(constan)n (t)f FH(g)k FO(and)d(the)g(ratio)2087 2496 y FG(k)p 2077 2508 55 3 v 2077 2549 a(m)2164 2523 y FO(are)f(su\036cien)n(tly)h (small)p FI(.)g FO(The)g(constrain)n(t)515 2615 y(on)f FK(\006)h FO(re\035ects)e(the)i(mixed)g(c)n(haracter)d(of)j(the)g(mo)r (del,)g(whic)n(h)f(forces)f(to)i(restrict)f(the)h(ph)n(ysical)515 2706 y(region)g(to)h(the)h(set)f Fz(f)q FA(P)e FE(:)35 b Fz(j)p FA(P)p Fz(j)22 b FH(<)f(m)p Fz(g)p FO(;)j(the)f(adopted)g (more)g(restrictiv)n(e)f(constrain)n(t)g(is)h(only)g(due)515 2797 y(to)k(tec)n(hnical)g(reasons.)515 2980 y(Giv)n(en)g(the)h(W)-7 b(eyl)28 b(op)r(erator)1361 3214 y FH(W)1434 3222 y FG(\033)1488 3214 y FE(\()p Fz(r)p FH(E)1643 3178 y FG(\033)1639 3227 y Fw(P)1689 3214 y FE(\))22 b(=)f FH(e)1858 3140 y Fx(\000)p FG(g)1948 3100 y Fm(R)1986 3119 y Fq(\024)1976 3159 y(\033)2099 3114 y(b)p FL(\()p Fl(k)p FL(\))q Fk(\000)p Fq(b)2274 3097 y Fk(y)2306 3114 y FL(\()q Fl(k)p FL(\))p 2045 3129 393 3 v 2045 3179 a Fk(j)p Fl(k)p Fk(j)2117 3187 y FE(\()2146 3179 y FL(1)p Fk(\000)2222 3168 y Fj(b)2219 3179 y Fl(k)q Fk(\001r)p Fq(E)2364 3165 y(\033)2362 3201 y Fl(P)2408 3187 y FE(\))2485 3121 y Fq(d)2516 3104 y FL(3)2548 3121 y Fq(k)p 2458 3129 150 3 v 2458 3136 a Fx(p)p 2507 3136 100 3 v 2507 3172 a FL(2)p Fk(j)q Fl(k)p Fk(j)3271 3214 y FI(\(1.20\))515 3369 y FO(and)27 b(the)h(transformed)e(Hamiltonian) 1368 3524 y FH(H)1438 3488 y FG(w)1432 3537 y Fw(P)p FG(;\033)1560 3524 y FE(:=)21 b FH(W)1735 3532 y FG(\033)1789 3524 y FE(\()p Fz(r)p FH(E)1944 3488 y FG(\033)1940 3537 y Fw(P)1990 3524 y FE(\))13 b FH(H)2097 3532 y Fw(P)p FG(;\033)2204 3524 y FH(W)2288 3488 y Fx(y)2277 3537 y FG(\033)2332 3524 y FE(\()p Fz(r)p FH(E)2487 3488 y FG(\033)2483 3537 y Fw(P)2533 3524 y FE(\))30 b FH(;)657 b FI(\(1.21\))515 3678 y FO(the)28 b(corresp)r(onding)d(non)j (degenerate)e(ground)g(eigen)n(v)n(ector)g FI(in)f Fz(H)2635 3686 y Fw(P)677 3913 y FH(\036)723 3877 y FG(\033)723 3926 y Fw(P)795 3913 y FE(:=)950 3829 y Fz(\000)1050 3798 y FF(1)p 1020 3812 91 4 v 1020 3854 a(2)p FG(\031)r(i)1133 3769 y Fy(H)1169 3856 y FG(\015)1340 3798 y FF(1)p 1232 3812 246 4 v 1232 3854 a FG(H)1286 3837 y Fq(w)1282 3875 y Fl(P)p Fq(;\033)1383 3854 y Fx(\000)p FG(E)1488 3829 y FH(dE)c( )1656 3837 y FF(0)p 907 3896 826 4 v 907 3906 a Fy(\015)907 3950 y(\015)907 3995 y(\015)950 3992 y Fz(\000)1050 3961 y FF(1)p 1020 3975 91 4 v 1020 4017 a(2)p FG(\031)r(i)1133 3931 y Fy(H)1169 4019 y FG(\015)1340 3961 y FF(1)p 1232 3975 246 4 v 1232 4017 a FG(H)1286 4000 y Fq(w)1282 4038 y Fl(P)p Fq(;\033)1383 4017 y Fx(\000)p FG(E)1488 3992 y FH(dE)g( )1656 4000 y FF(0)1690 3906 y Fy(\015)1690 3950 y(\015)1690 3995 y(\015)2063 3830 y(n)2114 3913 y FH(\015)k FE(:)43 b FH(E)25 b Fz(2)d FB(C)f FH(;)90 b Fz(j)p FH(E)21 b Fz(\000)c FH(E)2821 3877 y FG(\033)2817 3926 y Fw(P)2867 3913 y Fz(j)22 b FE(=)3001 3865 y FH(\033)p 3001 3896 47 4 v 3005 3964 a FE(4)3058 3830 y Fy(o)3271 3913 y FI(\(1.22\))515 4176 y FO(is)g(regular)f(as)g(function)i(of)g FH(\033)i FO(and)d FA(P)h FO(in)f(the)h(space)f Fu(F)2187 4152 y FG(b)2218 4176 y Fi(,)g FO(according)f(to)h(the)h(follo)n(wing)e(results:)515 4334 y FJ(The)l(or)l(em)30 b(3.2)41 b FO([Pi.])515 4491 y FJ(F)-6 b(or)37 b Fh(P)20 b Fu(2)f Ft(\006)p FJ(,)30 b(the)g(limit)36 b Fr(s)16 b Fu(\000)f Ft(lim)1482 4500 y FG(\033)r Fx(!)p FF(0)1627 4491 y Fr(\036)1669 4468 y FG(\033)1669 4514 y Fw(P)1739 4491 y Ft(=:)k Fr(\036)1875 4502 y Fw(P)1954 4491 y FJ(exists.)515 4580 y FO(Moreo)n(v)n(er)32 b(the)j(con)n(v)n(ergence)d(of)j Fr(\036)1647 4557 y FG(\033)1647 4603 y Fw(P)1733 4580 y FO(to)f(the)h(non)g(zero)f(v)n (ector)f FH(\036)2645 4588 y Fw(P)2731 4580 y FO(in)i Fu(H)2895 4591 y Fw(P)2975 4563 y Fu(\030)2975 4584 y Ft(=)3060 4580 y Fu(F)3117 4557 y FG(b)3184 4580 y FO(and)f(the)515 4694 y(con)n(v)n(ergence)27 b Fz(r)p FH(E)1101 4662 y FG(\033)1097 4709 y Fw(P)1172 4694 y Fz(!)e(r)p FH(E)1395 4702 y Fw(P)1475 4694 y FO(are)k(estimated)g(with)i(errors)c(at)j(most) f(of)h(order)3052 4640 y Fs(\000)3094 4666 y FG(\033)p 3094 4678 38 3 v 3095 4720 a(\024)3141 4640 y Fs(\001)3184 4638 y FL(1)p 3184 4646 29 3 v 3184 4679 a(4)3222 4658 y Fx(\000)p FG(\016)3334 4694 y FO(and)515 4757 y Fs(\000)557 4783 y FG(\033)p 557 4795 38 3 v 558 4836 a(\024)604 4757 y Fs(\001)647 4755 y FL(1)p 647 4763 29 3 v 647 4796 a(4)717 4810 y FO(resp)r(ectiv)n(ely)-7 b(,)27 b(where)g FH(\016)d(>)d FE(0)28 b FO(is)f(arbitrarily)f(small)p FI(;)1950 5059 y FO(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 515 602 a FJ(L)l(emma)30 b(3.3)40 b FO([Pi.])515 760 y FJ(The)30 b(fol)t(lowing)j(H\366lder)d(estimate)g (holds:)1333 880 y Fu(jr)p Fr(E)1468 852 y FG(\033)1520 880 y Ft(\()q Fh(P)p Ft(\))16 b Fu(\000)g(r)p Fr(E)1832 852 y FG(\033)1885 880 y Ft(\()p Fh(P)g Ft(+)g(\001)p Fh(P)p Ft(\))p Fu(j)k(\024)f Fr(C)i Fu(\001)15 b(j)p Ft(\001)p Fh(P)p Fu(j)2593 831 y FL(1)p 2579 839 57 3 v 2579 872 a(16)515 991 y FJ(wher)l(e)27 b(the)h(c)l(onstant)j Fr(C)f FJ(is)d(uniform)h(in)j Ft(0)20 b Fr(<)f(\033)k(<)c(\024\017)24 b FJ(,)j(in)32 b Fh(P)p Fr(;)11 b Fh(P)g Ft(+)g(\001)p Fh(P)20 b Fu(2)g Ft(\006)k FJ(and)34 b Ft(\001)p Fh(P)20 b Fu(2)3181 974 y Fs(b)3175 991 y Fr(I)t FJ(,)28 b(wher)l(e)521 1101 y Fs(b)515 1117 y Fr(I)h Ft(:=)675 1024 y Fs(\032)728 1117 y Ft(\001)p Fh(P)20 b Ft(:)911 1084 y Fx(j)p FF(\001)p Fw(P)p Fx(j)p 911 1102 135 3 v 951 1143 a FG(m)1075 1117 y Fu(\024)1149 1044 y Fs(\020)1240 1090 y FF(1)p 1201 1102 108 3 v 1201 1143 a(3)p FG(C)1282 1156 y Fj(b)1275 1166 y Fq(I)1319 1044 y Fs(\021)1371 1042 y FL(8)p 1371 1050 29 3 v 1371 1083 a(3)1425 1117 y Fr(;)j(m)1540 1069 y FL(3)p 1541 1077 V 1541 1110 a(4)1595 1117 y Fu(j)p Ft(\001)p Fh(P)p Fu(j)1758 1069 y FL(1)p 1758 1077 V 1758 1109 a(4)1820 1117 y Fu(\024)d Fr(\024\017)1965 1024 y Fs(\033)2048 1117 y FJ(and)41 b Fr(C)2276 1130 y Fm(b)2270 1143 y FG(I)2337 1117 y FJ(is)33 b(a)h(c)l(onstant)e (su\036ciently)i(lar)l(ger)515 1246 y(than)g Ft(1)p FJ(.)515 1429 y(The)l(or)l(em)c(3.4)41 b FO([Pi.])515 1611 y FJ(Under)33 b(the)h(c)l(onstructive)f(hyp)l(otheses,)i(for)1979 1584 y FG(k)p 1969 1596 55 3 v 1969 1637 a(m)2065 1611 y FJ(and)41 b Fr(g)34 b FJ(su\036ciently)g(smal)t(l,)g(the)g(norm)f(dif-)515 1703 y(fer)l(enc)l(e)e(b)l(etwe)l(en)j Fr(\036)1147 1679 y FG(\033)1147 1725 y Fw(P)1227 1703 y FJ(and)j Fr(\036)1437 1679 y FG(\033)1437 1725 y Fw(P)p FF(+\001)p Fw(P)1660 1703 y FJ(is)31 b(H\366lder)g(in)k Fu(j)p Ft(\001)p Fh(P)p Fu(j)28 b FJ(with)k(c)l(o)l(e\036cient)2913 1675 y FF(1)p 2898 1687 61 3 v 2898 1729 a(16)2985 1703 y Fu(\000)16 b Fr(\016)r FR(,)29 b Fr(\016)23 b(>)e Ft(0)29 b FJ(and)515 1794 y(arbitr)l(arily)41 b(smal)t(l.)g(The)g(multiplic)l(ative)g(c)l (onstant,)d Fr(C)2290 1806 y FG(\016)2325 1794 y FJ(,)i(is)g(uniform)g (in)45 b Ft(0)35 b Fu(\024)g Fr(\033)j(<)d(\024\017)p FJ(,)k(in)515 1885 y Fh(P)p Fr(;)12 b Fh(P)k Ft(+)f(\001)p Fh(P)20 b Fu(2)g Ft(\006)28 b FJ(and)36 b Ft(\001)p Fh(P)20 b Fu(2)g Fr(I)5 b FJ(,)28 b Fr(I)k FJ(a)e(su\036ciently)g(smal)t(l)h (\034xe)l(d)e(b)l(al)t(l)i(ar)l(ound)36 b Ft(\001)p Fh(P)20 b Ft(=)g(0)p FJ(.)515 2068 y FO(In)30 b(our)f(construction)g(w)n(e)g (will)i(assume)e(the)h(results)f(ab)r(o)n(v)n(e)g(\()p FJ(The)l(or)l(em)k(3.2)43 b FO([Pi.],)30 b FJ(L)l(emma)515 2159 y(3.3)45 b FO([Pi.])33 b(and)f FJ(The)l(or)l(em)j(3.4)45 b FO([Pi.]\))33 b(with)g(co)r(e\036cien)n(ts)2361 2129 y FF(1)p 2361 2143 31 4 v 2361 2185 a(4)2433 2159 y FO(and)2625 2129 y FF(1)p 2609 2143 61 4 v 2609 2185 a(16)2713 2159 y FO(resp)r(ectiv)n(ely)-7 b(,)31 b(with)i(no)515 2251 y(substan)n(tial)27 b(di\033erence)g(for)g(our)g(pro)r(cedure)f(and)i (the)g(con)n(ten)n(t)f(of)g(the)h(\034nal)g(results.)515 2433 y(F)-7 b(rom)34 b(the)h(analysis)f(in)h([F)-7 b(r.1])34 b(and)g([Pi.])i(w)n(e)e(get)h(that)g(for)f FA(P)f Fz(2)f FE(\006)k FO(w)n(e)e(ha)n(v)n(e)g(small)g(elec-)515 2525 y(tron)27 b(\020v)n(elo)r(cities\021)34 b(that)28 b(means)f Fz(jr)p FH(E)1696 2493 y FG(\033)1751 2525 y FE(\()o FA(P)p FE(\))q Fz(j)22 b FH(<)f FE(1)78 b Fz(8)p FH(\033)30 b FO(.)e(In)g(later)f(constructions)g(w)n(e)g(assume)515 2616 y(that)i(the)h(upp)r(er)g(frequency)-7 b(,)29 b FH(\024)1519 2624 y FF(1)1554 2616 y FI(,)h FO(in)g(the)f(b)r(oson)g (cloud)h(of)f FH( )2458 2625 y FG(h;\024)2549 2635 y FL(1)2599 2616 y FE(\()o FH(t)p FE(\))h FO(is)f(small)g(enough)g(suc)n (h)515 2707 y(that)f(for)f FA(P)21 b Fz(2)h FE(\006)1109 2839 y Fz(jr)p FH(E)1255 2803 y FG(\033)1310 2839 y FE(\()o FA(P)c FE(+)f FA(k)p FE(\))o Fz(j)22 b FH(<)f(v)1734 2803 y FG(max)1882 2839 y FH(<)g FE(1)154 b Fz(8)p FH(\033)n(;)30 b Fz(8)p FA(k)38 b FE(:)22 b(0)g FH(<)f Fz(j)p FA(k)p Fz(j)g(\024)g FH(\024)2838 2847 y FF(1)3271 2839 y FI(\(1.23\))515 2971 y FO(for)27 b(a)g(giv)n(en)g(and)g(\034xed)h FH(v)1330 2939 y FG(max)1478 2971 y FH(>)21 b FE(0)p FI(.)515 3154 y FP(Remark)31 b(1.1)515 3245 y FO(W)-7 b(e)33 b(will)h(treat)e(the)i (con)n(v)n(ergence)c(in)k Fz(H)g FO(of)f(a)f(v)n(ector)g(giv)n(en)g(as) h(a)g(direct)g(in)n(tegral)e(on)i(the)515 3336 y(\034b)r(er)25 b(spaces)g Fz(H)1022 3344 y Fw(P)1073 3336 y FO(.)g(In)h(order)e(to)h (a)n(v)n(oid)f(an)n(y)h(confusion)g(in)h(dealing)e(with)i(v)n(ectors)e (b)r(elonging)515 3428 y(to)f(di\033eren)n(t)h(\034b)r(er)g(spaces,)f (w)n(e)g(will)h(use)g(explicitly)g(the)g(isomorphism)e FA(I)2818 3436 y Fw(P)2893 3428 y FO(in)i(our)f(notations)515 3519 y(di\033eren)n(tly)j(from)g([Pi.].)h(Therefore,)e(for)h(instance,) g(the)h(prop)r(ert)n(y)e(in)h FJ(The)l(or)l(em)k(3.4)39 b FO([Pi.])27 b(is)515 3610 y(rewritten)g(as)g(follo)n(ws:)1168 3742 y Fz(k)p FA(I)1239 3750 y Fw(P)p FF(+)p Fw(\001P)1453 3742 y FE(\()p FH(\036)1529 3706 y FG(\033)1529 3755 y Fw(P)p FF(+\001)p Fw(P)1723 3742 y FE(\))17 b Fz(\000)f FA(I)1879 3750 y Fw(P)1943 3742 y FE(\()p FH(\036)2019 3706 y FG(\033)2019 3755 y Fw(P)2070 3742 y FE(\))p Fz(k)2138 3767 y Fx(F)2212 3742 y Fz(\024)21 b FH(C)2348 3751 y FG(\016)2399 3742 y Fz(\001)d(j)p FE(\001)p FA(P)p Fz(j)2629 3687 y FL(1)p 2615 3695 57 3 v 2615 3728 a(16)2681 3706 y Fx(\000)p FG(\016)2793 3742 y FH(:)515 3966 y Fp(Sp)-5 b(e)g(ctr)g(al)34 b(hyp)-5 b(othesis)515 4148 y FO(W)e(e)22 b(also)f(assume)g(the)h(follo)n(wing)f(not)h(pro)n(v)n(en)e(h)n(yp)r (othesis,)i(whic)n(h)g(allo)n(ws)e(the)j(construction)515 4240 y(of)k(a)g(\(time-dep)r(enden)n(t\))i(cell)f(partition)f(with)h (the)g(desired)f(prop)r(erties:)515 4422 y FP(Hyp)s(othesis)j(H)p Fg(0)515 4514 y FJ(F)-6 b(or)34 b FA(P)22 b Fz(2)f FE(\006)26 b FJ(,)i(ther)l(e)f(exists)g(a)h(p)l(ositive)h(c)l(onstant)2131 4483 y FF(1)p 2102 4497 88 4 v 2102 4539 a FG(m)2156 4547 y Fq(r)2225 4514 y FJ(such)f(that)f(the)h(fol)t(lowing)h(ine)l (qualities)515 4605 y(hold)i(uniformly)g(in)j FH(\033)24 b(>)d FE(0)p FJ(:)1195 4734 y FH(@)t(E)1301 4702 y FG(\033)1355 4734 y FE(\()p FA(P)p FE(\))p 1195 4765 281 4 v 1255 4833 a FH(@)16 b Fz(j)q FA(P)p Fz(j)1506 4782 y(\025)1597 4734 y(j)q FA(P)p Fz(j)p 1597 4765 104 4 v 1597 4833 a FH(m)1665 4841 y FG(r)1892 4782 y FH(and)2210 4734 y(@)2255 4702 y FF(2)2289 4734 y FH(E)2350 4702 y FG(\033)2404 4734 y FE(\()p FA(P)p FE(\))p 2210 4765 315 4 v 2269 4833 a FH(@)2314 4811 y FF(2)2362 4833 y Fz(j)p FA(P)p Fz(j)2556 4782 y(\025)2679 4734 y FE(1)p 2647 4765 103 4 v 2647 4833 a FH(m)2715 4841 y FG(r)2776 4782 y FH(:)1950 5059 y FO(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 515 523 a FO(Assuming)27 b(this)h(h)n(yp)r (othesis,)f(the)h(application)1377 685 y FA(J)1423 693 y FG(\033)1486 685 y FE(:)21 b FA(P)h Fz(!)f(r)p FH(E)1833 649 y FG(\033)1887 685 y FE(\()p FA(P)p FE(\))167 b FA(P)22 b Fz(2)f FE(\006)p FH(;)35 b(\033)24 b(>)p FE(0)666 b FI(\(1.24\))515 847 y FO(is)27 b(a)g(bijection)h(and)g(the)g (determinan)n(t)f(of)g(the)h(Jacobian)e(satis\034es)h(the)h(inequalit)n (y)1178 1061 y FH(det)12 b FA(dJ)1389 1069 y FG(\033)1452 1061 y FE(=)1592 1013 y(1)p 1543 1044 138 4 v 1543 1121 a Fz(j)p FA(P)p Fz(j)1646 1086 y FF(2)1703 956 y Fy(\022)1770 1013 y FH(@)t(E)1876 981 y FG(\033)1930 1013 y FE(\()p FA(P)p FE(\))p 1770 1044 281 4 v 1829 1112 a FH(@)17 b Fz(j)p FA(P)p Fz(j)2060 956 y Fy(\023)2116 974 y FF(2)2168 1061 y Fz(\001)2216 1013 y FH(@)2261 981 y FF(2)2296 1013 y FH(E)2357 981 y FG(\033)2411 1013 y FE(\()p FA(P)p FE(\))p 2216 1044 315 4 v 2276 1112 a FH(@)2321 1091 y FF(2)2368 1112 y Fz(j)q FA(P)p Fz(j)2562 1061 y(\025)2685 1013 y FE(1)p 2654 1044 103 4 v 2654 1112 a FH(m)2722 1091 y FF(3)2722 1125 y FG(r)2783 1061 y FE(;)515 1276 y FO(concerning)28 b(the)h(calculation)g(of)g(the)h(determinan)n(t,)f (w)n(e)g(recall)f(that)i(the)f(function)h FH(E)3293 1244 y FG(\033)3347 1276 y FE(\()p FA(P)p FE(\))515 1367 y FO(is)d(in)n(v)-5 b(arian)n(t)27 b(under)g(rotations)f(and)i(b)r (elongs)e(to)i FH(C)2154 1336 y Fx(1)2232 1307 y Fy(\000)2267 1367 y FB(R)2321 1336 y FF(3)2356 1307 y Fy(\001)2418 1367 y FO(\(see)g([F)-7 b(r.1]\).)515 1459 y(Under)30 b(this)g(assumption,)f(giv)n(en)h FH(O)1675 1467 y Fw(P)1751 1459 y Fz(\032)25 b FE(\006)30 b FO(and)g(the)g(corresp)r(onding)e (region)h FH(O)3076 1468 y Fx(r)p FG(E)3175 1451 y Fq(\033)3172 1489 y Fl(P)3253 1459 y FO(in)h(the)515 1561 y Fz(r)p FH(E)640 1529 y FG(\033)636 1576 y Fw(P)686 1561 y Fz(\000)p FO(space,)20 b FH(O)1042 1569 y Fw(P)1115 1561 y FE(=)h FA(J)1242 1529 y Fx(\000)p FF(1)1242 1574 y FG(\033)1337 1501 y Fy(\000)1372 1561 y FH(O)1430 1570 y Fx(r)p FG(E)1529 1553 y Fq(\033)1526 1591 y Fl(P)1577 1501 y Fy(\001)1612 1561 y FO(,)h(the)g(follo)n(wing)e(relation)g(holds)h(b)r(et)n(w)n(een) g(their)g(v)n(olumes:)1655 1738 y FH(V)1700 1746 y FG(O)1747 1757 y Fl(P)1818 1738 y Fz(\024)g FH(m)1967 1702 y FF(3)1967 1751 y FG(r)2019 1738 y Fz(\001)c FH(V)2102 1746 y FG(O)2149 1761 y Fk(r)p Fq(E)2241 1747 y(\033)2239 1784 y Fl(P)2306 1738 y FH(:)944 b FI(\(1.25\))515 1920 y Fp(Cel)5 b(l)33 b(p)-5 b(artition)515 2102 y FO(Let)26 b(us)g(consider)f(a)h(region)e (con)n(tained)i(in)g FE(\006)p FI(,)h FO(for)f(con)n(v)n(enience)e(a)i (cub)r(e)h(of)f(v)n(olume)f FH(V)38 b FE(=)21 b FH(L)3411 2071 y FF(3)3446 2102 y FI(.)515 2194 y FO(W)-7 b(e)35 b(no)n(w)f(construct)g(a)g(time-dep)r(enden)n(t,)h FH(t)d Fz(\035)g FE(1)p FO(,)k(cell-partition)d FE(\000)2727 2162 y FF(\()p FG(t)p FF(\))2838 2194 y FO(of)i(the)g(v)n(olume)f FH(V)16 b FO(,)515 2285 y(according)25 b(to)j(the)g(follo)n(wing)e (recip)r(e:)515 2376 y(A)n(t)j(time)h FH(t)23 b Fz(\035)h FE(1)p FO(,)29 b(the)h(linear)e(dimension)h(of)g(eac)n(h)f(cell)h(is) 2383 2346 y FG(L)p 2368 2360 72 4 v 2368 2402 a FF(2)2398 2388 y Fq(n)2479 2376 y FO(where)f FH(n)c Fz(2)g FB(N)p FH(;)39 b(n)24 b Fz(\025)f FE(1)p FO(,)30 b(is)f(suc)n(h)515 2468 y(that)1469 2570 y FE(\(2)1537 2534 y FG(n)1580 2570 y FE(\))1620 2513 y FL(1)p 1620 2521 29 3 v 1621 2554 a Fq(\017)1683 2570 y Fz(\024)21 b FH(t)g(<)1894 2509 y Fy(\000)1930 2570 y FE(2)1968 2534 y FG(n)p FF(+1)2088 2509 y Fy(\001)2133 2503 y FL(1)p 2133 2511 V 2134 2544 a Fq(\017)2341 2570 y FH(\017)h(>)f FE(0)515 2703 y FO(and)27 b(the)h(small)f(exp)r(onen)n(t)h FH(\017)21 b Fz(\034)g FE(1)28 b FO(is)g(\034xed)f(only)g(a)h(p)r(osteriori.)515 2802 y(This)34 b(de\034nition)h(implies)g(that)g(the)g(total)g(n)n(um)n (b)r(er)f(of)h(cells)f(at)h(time)g FH(t)g FO(is)f FH(N)21 b FE(\()p FH(t)p FE(\))32 b(=)g(\(2)3337 2770 y FG(n)3380 2802 y FE(\))3410 2767 y FF(3)3444 2802 y FO(,)515 2902 y(where)d FH(n)24 b FE(=)g([)q(log)1032 2921 y FF(2)1079 2902 y FH(t)1107 2870 y FG(\017)1137 2902 y FE(])q FO(,)29 b FK([)p FN(:)p FK(])h FO(is)f(the)h(in)n(teger)f(part.)g(W)-7 b(e)30 b(call)f FE(\000)2369 2864 y FF(\()p FG(t)p FF(\))2369 2922 y FG(j)2474 2902 y FO(the)h FH(j)2654 2870 y FG(th)2748 2902 y FO(cell,)g(cen)n(tered)f(in)p 3352 2841 93 4 v 30 w FA(P)3412 2910 y FG(j)3444 2902 y FO(,)515 3001 y(b)r(elonging)e(to)g(the)h(partition,)f FE(\000)1552 2970 y FF(\()p FG(t)p FF(\))1628 3001 y FI(,)h FO(at)f(time)h FH(t)p FO(.)515 3217 y FP(1.3.2)93 b(De\034nition)31 b(of)h(the)g(v)m(ector)g FN( )1870 3229 y FM(h;\024)1968 3237 y FL(1)2018 3217 y FK(\()q FN(t)p FK(\))515 3362 y FO(The)22 b(generic)g(v)n(ector)f FH( )1251 3371 y FG(h;\024)1342 3381 y FL(1)1392 3362 y FE(\()p FH(t)p FE(\))h FO(is)h(constructed)f(starting)g(from)g(a)g(one-particle)g (state)g(for)g(the)515 3453 y(Hamiltonian)27 b FH(H)1056 3461 y FG(\033)1092 3469 y Fq(t)1123 3453 y FO(,)h(of)f(w)n(a)n(v)n(e)f (function)i FH(h)g FO(in)g FA(P)p Fz(\000)p FO(v)-5 b(ariables.)515 3545 y(A)25 b FA(P)p Fz(\000)p FO(dep)r(enden)n(t)h FJ(L.S.Z.)g FO(W)-7 b(eyl)25 b(op)r(erator,)e(in)j(prop)r(erly)d(ev)n(olv)n(ed)h (photon)h(v)-5 b(ariables)23 b(is)i(ap-)515 3636 y(plied,)31 b(cell)f(b)n(y)g(cell,)h(on)f(the)g(considered)g(one-particle)f(state.) h(The)g(smearing)f(function)i(in)515 3727 y(the)d(generator)e(of)j(the) f(W)-7 b(eyl)29 b(op)r(erator)d(has)i(frequency)g(supp)r(ort)g(in)g (the)h(set)f FH(\033)3062 3735 y FG(t)3113 3727 y Fz(\024)22 b(j)p FA(k)p Fz(j)g(\024)h FH(\024)3433 3735 y FF(1)515 3819 y FO(where)31 b FH(\033)803 3827 y FG(t)858 3819 y Fz(!)c FE(0)32 b FO(for)f Fz(j)p FH(t)p Fz(j)d(!)f FE(+)p Fz(1)k FO(and)h FH(\024)1743 3827 y FF(1)1809 3819 y FO(is)g(an)f(arbitrarily)e(small)i(p)r(ositiv)n(e)g(n)n(um)n(b)r (er)h(whic)n(h)515 3910 y(satis\034es)c(the)i(constrain)n(t)e(\(1.23\)) g(.)i(The)f(b)r(eha)n(vior)f(of)h(the)h(smearing)e(function)h(at)h FA(k)23 b FE(=)h(0)30 b FO(is)515 4001 y(lab)r(elled)d(b)n(y)h(the)g (sp)r(ectral)f(v)-5 b(alues)27 b(of)g(the)h(op)r(erator)e Fz(r)p FH(E)2336 3970 y FG(\033)2390 4001 y FE(\()p FA(P)p FE(\))p FO(.)515 4184 y(1\))h(W)-7 b(e)28 b(start)f(from)g(the)h(v)n (ector)1542 4385 y FH( )1595 4347 y FF(\()o FG(t)p FF(\))1592 4405 y FG(j;\033)1672 4413 y Fq(t)1724 4385 y FE(:=)1827 4284 y Fy(Z)1869 4454 y FF(\000)1907 4426 y FL(\()q Fq(t)p FL(\))1907 4474 y Fq(j)1996 4385 y FH(h)14 b FE(\()o FA(P)p FE(\))f FH( )2239 4349 y FG(\033)2275 4357 y Fq(t)2236 4405 y Fw(P)2307 4385 y FH(d)2347 4349 y FF(3)2381 4385 y FH(P)841 b FI(\(1.26\))515 4618 y FO(where:)515 4709 y FH(h)13 b FE(\()p FA(P)p FE(\))21 b Fz(2)h FH(C)846 4678 y FF(1)841 4722 y(0)893 4649 y Fy(\000)928 4709 y FB(R)982 4678 y FF(3)1034 4709 y Fz(n)c FE(0)1128 4649 y Fy(\001)1191 4709 y FO(has)27 b(supp)r(ort)g(inside)h(the)g(cub)r(e)g FH(V)44 b FO(;)515 4810 y FH(\033)559 4818 y FG(t)608 4810 y FE(=)21 b FH(t)717 4778 y Fx(\000)p FG(\014)834 4810 y FO(,)28 b(where)f FH(\014)e Fz(\035)c FE(1)28 b FO(is)g(\034xed)f(only)g(a)h(p)r(osteriori)p FI(;)1950 5059 y FO(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 515 523 a FH( )568 489 y FG(\033)604 497 y Fq(t)565 544 y Fw(P)656 523 y FE(:=)22 b FH(W)843 491 y Fx(y)832 536 y FG(\033)868 544 y Fq(t)911 523 y FE(\()p Fz(r)p FH(E)1066 489 y FG(\033)1102 497 y Fq(t)1062 544 y Fw(P)1133 523 y FE(\))13 b FH(\036)1222 489 y FG(\033)1258 497 y Fq(t)1222 544 y Fw(P)1317 523 y FO(is)27 b(the)h(unique)g(ground) e(state)i(of)f FH(H)2456 531 y Fw(P)p FG(;\033)2558 539 y Fq(t)2589 523 y FO(.)515 614 y(Notice)g(that)1444 693 y Fy(\015)1444 738 y(\015)1444 783 y(\015)1487 779 y FH( )1540 741 y FF(\()o FG(t)p FF(\))1537 799 y FG(j;\033)1617 807 y Fq(t)1648 693 y Fy(\015)1648 738 y(\015)1648 783 y(\015)1712 779 y FE(=)1793 652 y Fy( )1854 677 y(Z)1897 848 y FF(\000)1935 820 y FL(\()p Fq(t)p FL(\))1935 868 y Fq(j)2024 779 y Fz(j)p FH(h)13 b FE(\()p FA(P)p FE(\))p Fz(j)2244 743 y FF(2)2291 779 y FH(d)2331 743 y FF(3)2365 779 y FH(P)2425 652 y Fy(!)2496 649 y FL(1)p 2496 657 29 3 v 2496 690 a(2)515 999 y FO(is)27 b(of)h(order)e FE(\()p FH(N)21 b FE(\()p FH(t)p 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FL(1)1975 1837 y Fq(\033)2009 1848 y(t)2088 1792 y(a)p FL(\()q Fl(k)p FL(\))q Fk(\000)p Fq(a)2276 1775 y Fk(y)2308 1792 y FL(\()p Fl(k)p FL(\))p 2077 1807 321 3 v 2077 1857 a Fk(j)p Fl(k)p Fk(j)2148 1865 y FE(\()2178 1857 y FL(1)p Fk(\000)2254 1846 y Fj(b)2251 1857 y Fl(k)p Fk(\001)p Fl(v)2336 1870 y Fq(j)2367 1865 y FE(\))2444 1798 y Fq(d)2475 1782 y FL(3)2507 1798 y Fq(k)p 2417 1806 150 3 v 2417 1814 a Fx(p)p 2467 1814 100 3 v 36 x FL(2)p Fk(j)p Fl(k)p Fk(j)3271 1892 y FI(\(1.27\))515 2017 y FO(where)j FA(v)800 2025 y FG(j)854 2017 y Fz(\021)c(r)p FH(E)1060 1985 y FG(\033)1096 1993 y Fq(t)1140 1956 y Fy(\000)p 1175 1956 93 4 v 61 x FA(P)1235 2025 y FG(j)1268 1956 y Fy(\001)1329 2017 y FO(is)26 b(the)g(\020v)n(elo)r(cit)n(y\021) 32 b(at)26 b(time)g FN(t)g FO(corresp)r(onding)e(to)i(the)g(cen)n(ter)p 3282 1956 V 26 w FA(P)3342 2025 y FG(j)3401 2017 y FO(of)515 2127 y(the)i(cell)g FE(\000)854 2088 y FF(\()p FG(t)p FF(\))854 2146 y FG(j)930 2127 y FO(.)g(In)g(order)e(not)i(to)g(o)n(v)n 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FG(h;k)728 2885 y FL(1)777 2866 y FE(\()p FH(t)p FE(\))83 b(:=)105 b FH(e)1170 2834 y FG(iH)t(t)1274 2866 y FH(e)1310 2834 y Fx(\000)p FG(iH)1434 2813 y Fq(ph)1502 2834 y FG(t)1543 2809 y Fy(P)1624 2830 y FG(N)5 b FF(\()p FG(t)p FF(\))1624 2888 y FG(j)s FF(=1)1766 2866 y Fz(W)1842 2874 y FG(\033)1878 2882 y Fq(t)1922 2866 y FE(\()p FA(v)1999 2874 y FG(j)2031 2866 y FE(\))13 b FH(e)2110 2834 y FG(iH)2186 2813 y Fq(ph)2255 2834 y FG(t)2283 2866 y FH(e)2319 2833 y FG(i\015)2373 2841 y Fq(\033)2407 2852 y(t)2439 2839 y FE(\()2468 2833 y Fw(v)2504 2844 y Fq(j)2536 2833 y FG(;)p Fx(r)p FG(E)2654 2804 y Fq(\033)2688 2815 y(t)2651 2854 y Fl(P)2719 2833 y FG(;t)2762 2839 y FE(\))2796 2866 y FH(e)2832 2834 y Fx(\000)p FG(iE)2950 2805 y Fq(\033)2984 2816 y(t)2947 2855 y Fl(P)3015 2834 y FG(t)3042 2866 y FH( )3095 2827 y FF(\()p FG(t)p FF(\))3092 2885 y FG(j;\033)3172 2893 y Fq(t)958 3072 y FE(=)94 b FH(e)1148 3040 y FG(iH)t(t)1264 3016 y Fy(P)1345 3037 y FG(N)5 b FF(\()q FG(t)p FF(\))1345 3095 y 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y(The)c(de\034nitions)h(\(1.29\),)f(\(1.30\))f(require)h(some)f (commen)n(ts)h(con)n(tained)g(in)h(the)g(remarks)e(b)r(e-)515 4110 y(lo)n(w,)27 b(whic)n(h)h(giv)n(e)g(us)g(the)h(opp)r(ortunit)n(y)e (to)h(come)g(bac)n(k)g(to)g(the)g(motiv)-5 b(ations)28 b(of)g(the)h(recip)r(e)515 4201 y(here)e(presen)n(ted.)515 4384 y FP(Remark)k(1.2)515 4475 y FO(As)c(far)g(as)g(the)h(b)r(oson)f (cloud)h(and)f(the)h(coheren)n(t)e(factor)h FI(\(1.11\))1474 4669 y Fz(\000)p FH(g)2021 4621 y FE(1)p 1583 4652 917 4 v 1583 4748 a Fz(j)p FA(k)p Fz(j)1685 4684 y Fy(p)p 1762 4684 141 4 v 64 x FE(2)13 b Fz(j)p FA(k)p Fz(j)1915 4665 y Fy(\020)1961 4748 y FE(1)k Fz(\000)2095 4728 y Fy(b)2093 4748 y FA(k)g Fz(\001)g(r)p FH(E)g FE(\()p FA(P)p FE(\))2453 4665 y Fy(\021)1950 5059 y FO(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 515 523 a FO(are)31 b(concerned,)g(b)n(y)g(the) i(in)n(tro)r(duction)e(of)h(the)h(c-n)n(um)n(b)r(er)e FA(v)2484 531 y FG(j)2548 523 y FO(w)n(e)h(implemen)n(t)g(in)g(our)f (for-)515 623 y(malism)d(a)f(crucial)h(ph)n(ysical)f(feature)h(whic)n (h)g(is)g(not)g(exploited)g(in)g(all)g(its)h(consequences)d(in)515 722 y(F)-7 b(r\366hlic)n(h's)29 b(formalism.)g(More)g(precisely)-7 b(,)29 b(the)i(op)r(erator)d(that)i(actually)g(lab)r(els)g(the)g (coher-)515 822 y(en)n(t)c(factor)g(is)h(the)g(asymptotic)f(\(mean\))g (v)n(elo)r(cit)n(y)g(of)h(the)g(electron,)f(that,)g(di\033eren)n(tly)h (from)515 922 y Fz(r)p FH(E)16 b FE(\()p FA(P)p FE(\))p FO(,)34 b(has)e(to)h(comm)n(ute)g(with)g(the)g(asymptotic)g(b)r(oson)f (algebra.)f(The)i(t)n(w)n(o)g(op)r(erators)515 1021 y(w)n(ould)j (coincide)h(on)g(the)g(one-particle)f(states)g(if)i(the)f(latter)g (ones)f(existed,)h(as)g(happ)r(ens)515 1121 y(when)f(a)g(\034xed)h (infrared)e(cut-o\033)i(in)f(the)h(in)n(teraction)f(is)g(considered.)f (W)-7 b(e)37 b(w)n(an)n(t)f(to)g(k)n(eep)515 1220 y(trac)n(k)24 b(of)h(this)g(concept)g(in)g(a)g(limiting)g(construction,)f(in)n(v)n (olving)g(the)h(time)h(dep)r(enden)n(t)g(par-)515 1320 y(tition)h(and)h(the)f(discretized)g(v)n(elo)r(cities)g Fz(r)p FH(E)1938 1288 y FG(\033)1974 1296 y Fq(t)2018 1260 y Fy(\000)p 2053 1259 93 4 v 60 x FA(P)2113 1328 y FG(j)2146 1260 y Fy(\001)2181 1320 y FO(.)h(W)-7 b(e)27 b(will)h(see)f(that)h(the)f(c)n(hosen)g(recip)r(e)515 1420 y(for)j(the)i(dressing)e(cloud)g(is)h(in)h(the)f(end)g(a)g(tec)n (hnically)g(con)n(v)n(enien)n(t)f(w)n(a)n(y)f(to)i(appro)n(ximate)515 1519 y(the)f(op)r(erator)e(asymptotic)h(\(mean\))i(v)n(elo)r(cit)n(y)e (of)g(the)i(electron)e(inside)h(the)g(w)n(a)n(v)n(e)e(function)515 1619 y(of)f(the)h(photon)f(cloud.)g(The)g(reason)f(is)h(that)h(w)n(e)f (can)g(easily)f(exploit)h(the)h(cluster)f(prop)r(ert)n(y)515 1719 y(whic)n(h)e(implemen)n(ts)g(the)h(asymptotic)f(orthogonalit)n(y)d (b)r(et)n(w)n(een)k(o\033-diagonal)d(terms)i(in)g(the)515 1818 y(partition)34 b(of)g(the)h(v)n(ector.)f(W)-7 b(e)35 b(stress)e(that)i(the)g(op)r(erator)e(asymptotic)h(\(mean\))h(v)n(elo)r (cit)n(y)515 1918 y(of)30 b(the)g(electron)g(is)g(not)g(constructed)f (y)n(et.)h(W)-7 b(e)31 b(only)e(use)h(the)h(v)-5 b(alues)29 b(that)i(it)f(is)g(exp)r(ected)515 2017 y(to)i(tak)n(e)f(on)h(the)g (minimal)g(asymptotic)g(electron)f(states)g(for)h FH(k)2561 2025 y FF(1)2627 2017 y FO(smaller)f(and)h(smaller,)f(in)515 2117 y(the)26 b(region)e(of)i(momen)n(ta)f FA(P)h FO(whic)n(h)g(is)g (ph)n(ysically)e(meaningful.)i(Moreo)n(v)n(er)d(the)j(expression)515 2217 y(giv)n(en)33 b(for)i(the)g(smearing)e(function)i(in)g(the)g(W)-7 b(eyl)34 b(op)r(erators)f(\(1.29\))h(enco)r(des)g(someho)n(w,)515 2316 y(already)26 b(at)i(\034nite)h(times,)g(the)f(comm)n(utation)g (prop)r(ert)n(y)f(w)n(e)g(exp)r(ect)i(at)f(asymptotic)f(times)515 2416 y(b)r(et)n(w)n(een)g(the)h(asymptotic)f(b)r(oson)g(algebra)f(and)h (the)h(asymptotic)f(\(mean\))h(v)n(elo)r(cit)n(y)f(of)g(the)515 2516 y(electron,)g(up)h(to)f(an)h(error)d(whic)n(h)j(b)r(ecomes)f (smaller)g(and)h(smaller)e(as)h(time)i(increases)d(and)515 2615 y(the)i(partition)f(gets)g(\034ner)g(and)h(\034ner.)515 2814 y FP(Remark)j(1.3)515 2914 y FO(The)36 b(in)n(tro)r(duction)g(of)g (the)h(phase)f(factor)f FI(\(1.30\))j FO(is)e(related)g(to)g(Co)r(ok's) f(argumen)n(t.)h(The)515 3014 y(\020fast\021)g(cut-o\033)31 b FH(\033)1047 3022 y FG(\034)1116 3014 y FO(is)g(of)f(order)g FH(\034)1564 2982 y Fx(\000)p FG(\014)1652 3014 y FO(,)h(where)f FH(\014)35 b FO(is)30 b(larger)f(than)h FE(1)p FO(.)i(The)e(in)n (tegration)f(b)r(ound)515 3113 y FH(\033)562 3082 y FG(S)559 3126 y(\034)634 3113 y FO(is)g(a)f(\020slo)n(w\021)34 b(infrared)28 b(cut-o\033,)g FH(\033)1679 3082 y FG(S)1676 3126 y(\034)1723 3113 y FE(=)23 b FH(\034)1849 3082 y Fx(\000)p FG(\013)1969 3113 y FO(where)28 b FH(\013)h FO(is)g(a)f(p)r(ositiv)n(e)g(n)n(um)n(b)r(er)g(less)g(than)h(1.)515 3213 y(The)h(in\034nitesimal)h(upp)r(er)f(b)r(ound)h FH(\033)1708 3181 y FG(S)1705 3226 y(\034)1783 3213 y FO(for)f(the)h(in)n(tegral)e(in)h(\(1.30\))g(enables)g(us)g(to)h (replace)515 3313 y(the)g(argumen)n(t)g Fz(r)p FH(E)1162 3278 y FG(\033)1198 3286 y Fq(t)1158 3334 y Fw(P)1260 3313 y FO(with)1463 3277 y Fw(x)p FF(\()p FG(t)p FF(\))p 1463 3295 109 4 v 1505 3338 a FG(t)1612 3313 y FO(\(Corollary)e(A3\))j (for)f(asymptotic)f(times.)i(Therefore)e(w)n(e)515 3412 y(get)f(that)h(the)g(time)g(deriv)-5 b(ativ)n(e)29 b(of)g(the)h(phase)g (factor)e(kills)i(an)f(infrared)g(tail)g(term)h(arising)515 3512 y(from)d(the)i(application)e(of)h(Co)r(ok's)f(argumen)n(t,)h(whic) n(h)g(is)g(not)g(\(absolutely\))f(con)n(v)n(ergen)n(t)f(as)515 3611 y(function)i(of)g FN(t)p FO(.)h(On)e(the)i(basis)e(of)h(partial)f (estimates,)h FH(\013)h FO(is)f(ev)n(en)n(tually)f(c)n(hosen)g (su\036cien)n(tly)515 3711 y(close)36 b(to)g FE(1)i FO(and)e FH(\014)41 b FO(large)36 b(enough)g(with)h(resp)r(ect)f(to)h FE(1)g FO(in)g(order)f(to)g(ac)n(hiev)n(e)g(the)h(strong)515 3811 y(con)n(v)n(ergence)25 b(of)i(the)h(v)n(ector)e FH( )1510 3820 y FG(h;\024)1601 3830 y FL(1)1651 3811 y FE(\()p FH(t)p FE(\))p FO(.)515 4140 y FC(1.4)112 b(Surv)m(ey)38 b(of)f(results)g(and)h(plan)g(of)f(the)h(pap)s(er)515 4284 y FO(After)25 b(ha)n(ving)e(constructed)i(the)g(generic)e(v)n (ector)h FH( )2154 4293 y FG(h;\024)2245 4303 y FL(1)2294 4284 y FE(\()p FH(t)p FE(\))p FO(,)h(w)n(e)f(pro)n(v)n(e)f(the)i (existence)g(of)f(the)515 4376 y(strong)i(limit)1477 4476 y FH(s)17 b Fz(\000)60 b FE(lim)1607 4521 y FG(t)p Fx(!\0061)1813 4476 y FH( )1863 4485 y FG(h;\024)1954 4495 y FL(1)2004 4476 y FE(\()o FH(t)p FE(\))21 b(=:)h FH( )2268 4438 y FG(out)p FF(\()n FG(in)p FF(\))2265 4497 y FG(h;\024)2356 4507 y FL(1)2484 4476 y FH(:)515 4627 y FO(The)34 b(construction)g(is)g(explicitly)g(p)r(erformed)g(in)h (the)g(case)e(\020out\021,)h(the)h(case)e(\020in\021)41 b(is)34 b(com-)515 4719 y(pletely)25 b(analogous.)d(In)k(our)e (notations)g(w)n(e)g(use)h(the)g(sup)r(erscript)2620 4689 y Ff(\()o FM(in)p Ff(\))2765 4719 y FO(to)g(mean)g(either)f(that) 515 4810 y(an)j(analogous)e(structure)i(holds)g(for)g(the)h(ingoing)e (case)h(or)f(to)i(denote)f(b)r(oth)h(the)g(t)n(w)n(o)e(ones,)1950 5059 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 515 523 a FO(for)20 b(instance)g(b)r(oth)i(the) f(t)n(w)n(o)f(asymptotic)g(subspaces.)g(Ho)n(w)n(ev)n(er)e(w)n(e)j(do)f (not)h(claim)f(an)n(ything)515 614 y(ab)r(out)27 b(their)h(relations.) 515 706 y(By)k(analogy)f(with)i(the)g(regularized)e(case,)h(w)n(e)g (de\034ne)h(the)g(in)n(v)-5 b(arian)n(t)32 b(\(under)g(space-time)515 797 y(translation\))26 b(subspaces)806 997 y Fz(H)871 961 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))871 1010 y FG(\024)907 1020 y FL(1)1127 997 y FE(:=)p 1230 901 1930 4 v 1230 914 a Fy(n)1281 926 y(_)1379 997 y FH( )1432 958 y FG(out)p FF(\()o FG(in)p FF(\))1429 1018 y FG(h;\024)1520 1028 y FL(1)1648 997 y FE(\()p FH(\034)t(;)13 b FA(a)p FE(\))38 b(:)99 b FH(h)13 b FE(\()p FA(P)p FE(\))21 b Fz(2)h FH(C)2312 971 y FF(1)2307 1016 y(0)2359 997 y FE(\(\006)c Fz(n)f FE(0\))c FH(;)26 b(\034)k Fz(2)21 b FB(R)13 b FH(;)g FA(a)22 b Fz(2)f FB(R)3073 975 y FF(3)3108 914 y Fy(o)515 1180 y FO(where)32 b(the)i(subscript)f FH(\024)1315 1188 y FF(1)1383 1180 y FO(denotes)g(the)h(upp)r(er)f (frequency)g(in)h(the)f(b)r(oson)g(cloud)g(and)g(the)515 1280 y(v)n(ector)27 b FH( )818 1241 y FG(out)p FF(\()o FG(in)p FF(\))815 1301 y FG(h;\024)906 1311 y FL(1)1034 1280 y FE(\()p FH(\034)t(;)14 b FA(a)p FE(\))28 b FO(corresp)r(onds)f (to)i(the)g(ev)n(olution)f FH(\034)38 b FO(in)29 b(time)g(and)g(to)g(a) f(displacemen)n(t)515 1395 y FA(a)e FO(in)g(space)e(of)i(the)g(state)g (asso)r(ciated)e(to)i(the)g(v)n(ector)e FH( )2268 1356 y FG(out)p FF(\()o FG(in)p FF(\))2265 1416 y FG(h;\024)2356 1426 y FL(1)2472 1395 y FO(.)h(Because)g(of)h(the)g(presence)f(of)515 1496 y(the)d(b)r(oson)e(cloud,)i(the)f(electronic)g(w)n(a)n(v)n(e)f (functions)h Fo(f)p FN(h)p Fo(g)g FO(cannot)g(fully)h(c)n(haracterize)d (the)j(set)515 1588 y(of)27 b(states)g(w)n(e)g(are)f(in)n(terested)h (in.)h(The)f(next)h(step)f(consists)g(in)g(adding)g(\020hard\021)33 b(asymptotic)515 1679 y(b)r(osons)26 b(as)h(result)h(of)f(the)h(limits) 796 1839 y FH(s)17 b Fz(\000)60 b FE(lim)926 1884 y FG(t)p Fx(!\0061)1132 1839 y FH(e)1168 1803 y FG(iH)t(t)1271 1839 y FH(e)1307 1803 y Fx(\000)p FG(iH)1431 1782 y Fq(ph)1500 1803 y FG(t)1528 1839 y FH(e)1564 1803 y FG(i)1586 1809 y FE(\()1616 1803 y FG(a)p FF(\()o FG(\026)p FF(\))q(+)p FG(a)1816 1782 y Fk(y)1847 1803 y FF(\()p FG(\026)p FF(\))1932 1809 y FE(\))1966 1839 y FH(e)2002 1803 y FG(iH)2078 1782 y Fq(ph)2147 1803 y FG(t)2175 1839 y FH(e)2211 1803 y Fx(\000)p FG(iH)t(t)2362 1839 y FH( )2415 1801 y FG(out)p FF(\()o FG(in)p FF(\))2412 1861 y FG(h;\024)2503 1871 y FL(1)2631 1839 y FE(\()p FH(\034)t(;)13 b FA(a)p FE(\))21 b(=:)h FH( )2983 1801 y FG(out)p FF(\()n FG(in)p FF(\))2980 1861 y FG(h;\026)515 2055 y FO(where)c FH(a)787 2023 y Fx(y)832 2055 y FE(\()p FH(\026)p FE(\))j(:=)g(\()p FH(a)13 b FE(\()p FH(\026)p FE(\)\))1281 2019 y Fx(y)1335 2055 y FE(=)1416 1994 y Fy(\000)1451 1995 y(R)1515 2055 y FH(a)g FE(\()o FA(k)p FE(\))k Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))f FH(d)1906 2023 y FF(3)1940 2055 y FH(k)1982 1994 y Fy(\001)2018 2011 y Fx(y)2050 2055 y FI(,)18 b FH(\026)13 b FE(\()p FA(y)q FE(\))21 b(=)2358 1995 y Fy(R)2422 2055 y FH(e)2458 2023 y Fx(\000)p FG(i)p Fw(ky)2610 2055 y Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))f FH(d)2824 2023 y FF(3)2858 2055 y FH(k)r(;)23 b Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))21 b Fz(2)h FH(C)3260 2023 y Fx(1)3255 2068 y FF(0)3338 1994 y Fy(\000)3373 2055 y FB(R)3427 2023 y FF(3)3478 2055 y Fz(n)c FE(0)3572 1994 y Fy(\001)3607 2055 y FO(,)515 2146 y(and)27 b(the)h(dep)r (endence)g(on)f FH(\024)1423 2154 y FF(1)1458 2146 y FH(;\034)22 b(;)13 b FA(a)27 b FO(is)h(omitted)g(in)g(the)g(\034nal)f (expression.)515 2237 y(Finally)-7 b(,)27 b(the)h(prop)r(osed)f (scattering)f(subspaces)h(are)882 2433 y Fz(H)947 2397 y FG(out)p FF(\()p FG(in)p FF(\))1173 2433 y FE(:=)p 1275 2337 1804 4 v 1275 2350 a Fy(n)1326 2362 y(_)1424 2433 y FH( )1477 2394 y FG(out)p FF(\()o FG(in)p FF(\))1474 2454 y FG(h;\026)1719 2433 y FE(:)99 b FH(h)13 b FE(\()p FA(P)p FE(\))21 b Fz(2)h FH(C)2170 2407 y FF(1)2165 2451 y(0)2217 2433 y FE(\(\006)c Fz(n)f FE(0\))c FH(;)k Fy(e)-47 b FH(\026)22 b Fz(2)f FH(C)2690 2407 y Fx(1)2685 2451 y FF(0)2768 2433 y FE(\()p FB(R)2852 2411 y FF(3)2904 2433 y Fz(n)c FE(0\))3028 2350 y Fy(o)3079 2433 y FH(:)515 2624 y FO(On)28 b(these)h(subspaces)e(the)i FH(C)1452 2592 y Fx(1)1447 2637 y FF(0)1546 2624 y FO(functions)g FH(f)37 b FO(of)28 b(the)h(v)-5 b(ariable)28 b FH(e)2567 2592 y FG(iH)t(t)2680 2593 y Fw(x)p 2680 2607 37 4 v 2686 2649 a FG(t)2726 2624 y FH(e)2762 2592 y Fx(\000)p FG(iH)t(t)2942 2624 y FO(con)n(v)n(erge.)f(This)515 2715 y(means)g(that)h(the)g(limits)1539 2807 y FH(s)17 b Fz(\000)60 b FE(lim)1668 2852 y FG(t)p Fx(!\0061)1875 2807 y FH(e)1911 2771 y FG(iH)t(t)2014 2807 y FH(f)2072 2724 y Fy(\020)2128 2758 y FA(x)p 2128 2790 47 4 v 2137 2857 a FH(t)2185 2724 y Fy(\021)2243 2807 y FH(e)2279 2771 y Fx(\000)p FG(iH)t(t)515 2983 y FO(exist)35 b(and)g(generate)e(the)j(comm)n (utativ)n(e)e(algebra)f Fz(A)2240 2945 y FG(out)p FF(\()o FG(in)p FF(\))2240 2992 y Fw(v)2276 3004 y Fq(el)2444 2983 y FO(.)i(Analogously)e(the)j(canonical)515 3088 y(W)-7 b(eyl)29 b(algebra)f Fz(A)1080 3050 y FG(out)p FF(\()o FG(in)p FF(\))1080 3109 y FG(ph)1284 3088 y FO(asso)r(ciated)g (to)h(a)h(free)f(massless)f(b)r(oson)h(\034eld)g(is)h(generated)e(b)n (y)h(the)515 3196 y(strong)d(time)i(limits)g(of)g(the)g FJ(L.S.Z.)g FO(W)-7 b(eyl)28 b(op)r(erators)e(acting)h(on)g(the)h (space)f Fz(H)3052 3165 y FG(out)p FF(\()o FG(in)p FF(\))3256 3196 y FO(:)779 3376 y Fz(W)861 3340 y FG(out)p FF(\()o FG(in)p FF(\))1077 3376 y FE(\()p FH(\020)5 b FE(\))21 b(:=)h FH(s)17 b Fz(\000)60 b FE(lim)1429 3421 y FG(t)p Fx(!\0061)1636 3376 y FH(e)1672 3340 y FG(iH)t(t)1775 3376 y FH(e)1811 3340 y Fx(\000)p FG(iH)1935 3319 y Fq(ph)2004 3340 y FG(t)2032 3376 y FH(e)2068 3340 y FG(i)2090 3346 y FE(\()2119 3340 y FG(a)p FF(\()p FG(\020)s 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FO(A)23 b(w)n(arning)f(is)g(necessary)g(at)g(this)i(p)r(oin)n(t.)f (Previous)f(de\034nitions)h(are)f(arbitrary)f(to)i(some)f(ex-)515 4153 y(ten)n(t,)h(due)g(to)g(the)g(coheren)n(t)f(function)h(in)g(the)h (de\034nition)f(of)g(the)g(minimal)g(asymptotic)f(elec-)515 4245 y(tron)28 b(states,)h(whic)n(h)g(is)h(arbitrary)d(except)i(in)g (the)h(infrared)e(limit.)i(Nev)n(ertheless,)e(through)515 4345 y(the)40 b(\(arti\034cial\))g(distinction)g(b)r(et)n(w)n(een)g Fz(H)1896 4307 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))1896 4353 y FG(\024)1932 4363 y FL(1)2171 4345 y FO(and)g Fz(H)2410 4313 y FG(out)p FF(\()o FG(in)p FF(\))2654 4345 y FO(w)n(e)g(w)n(an)n(t)f(to)h(p)r(oin)n(t)h(out)515 4436 y(that:)515 4527 y(F)-7 b(rom)36 b(a)h(tec)n(hnical)f(p)r(oin)n(t) h(of)g(view,)g(our)f(construction)g(of)h(the)g(scattering)f(subspaces)g (is)515 4627 y(based)27 b(on)g(some,)g(not)h(unique,)f Fz(H)1597 4589 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))1597 4636 y FG(\024)1633 4646 y FL(1)1860 4627 y FO(just)h(to)f(fo)r(cus)h (on)f(the)h(infrared)f(dressing;)515 4719 y(F)-7 b(rom)27 b(a)h(ph)n(ysical)e(p)r(oin)n(t)i(of)g(view,)g(whether)f(the)i (\020hard\021)k(photon)28 b(cloud)f(describ)r(ed)h(b)n(y)f(the)515 4810 y(smearing)21 b(functions)i Fz(f)p FH(\026)p Fz(g)h FO(is)e(totally)g(remo)n(v)-5 b(able,)22 b(the)h(photon)f(cloud)g(link) n(ed)h(to)f(the)h(v)n(ectors)1950 5059 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 515 523 a FO(in)23 b Fz(H)672 485 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))672 531 y FG(\024)708 541 y FL(1)929 523 y FO(is)g(not)f(completely)h(remo)n(v)-5 b(able.)21 b(It)i(means)f(that)h(all)g(scattering)e(states)h(con)n (tain)515 614 y(asymptotic)27 b(photons.)515 706 y(The)35 b(ph)n(ysical)f(quan)n(tities)h(m)n(ust)g(b)r(e)h(indep)r(enden)n(t)f (of)g(the)h(c)n(hoice)e(of)h(the)h(\020one-particle\021)515 806 y(subspace)28 b Fz(H)928 767 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))928 814 y FG(\024)964 824 y FL(1)1163 806 y FH(;)h FO(in)g(particular)f(of)h(the)g(c)n(hoice)g(of)g FH(\024)2325 814 y FF(1)2359 806 y FO(.)h(Indeed,)f(once)f FK(\006)h FO(is)g(\034xed,)g(if)h(w)n(e)515 907 y(considered)f(a)h (di\033eren)n(t)h Fz(H)1392 869 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))1392 935 y FG(\024)1428 917 y Fk(0)1428 954 y FL(1)1627 907 y FO(,)f(i.e.)h(with)g(a)f(di\033eren)n(t)h(upp)r (er)g(frequency)p FI(,)f FH(\024)3097 876 y Fx(0)3120 915 y FF(1)3180 907 y FH(>)c(\024)3310 915 y FF(1)3345 907 y FO(,)k(in)515 1038 y(the)37 b(photon)g(cloud)g(of)h(the)f (generic)g(v)n(ector)f FH( )2050 999 y FG(out)p FF(\()p FG(in)p FF(\))2047 1062 y FG(h;\024)2138 1048 y Fk(0)2160 1072 y FL(1)2253 1038 y FO(,)i(it)f(is)g(not)h(di\036cult)g(to)f(c)n (hec)n(k)f(that)515 1155 y FH( )568 1117 y FG(out)p FF(\()p FG(in)p FF(\))565 1180 y FG(h;\024)656 1166 y Fk(0)679 1190 y FL(1)798 1155 y FO(can)26 b(b)r(e)g(expressed)g(as)f(a)h (dressing)g(of)g FH( )2065 1117 y FG(out)p FF(\()p FG(in)p FF(\))2062 1177 y FG(h;\024)2153 1187 y FL(1)2295 1155 y FO(b)n(y)g(means)g(of)g(asymptotic)g(photons)515 1273 y(and,)f(vice)g(v)n(ersa,)f(that)h FH( )1327 1235 y FG(out)p FF(\()p FG(in)p FF(\))1324 1295 y FG(h;\024)1415 1305 y FL(1)1556 1273 y FO(can)g(b)r(e)h(obtained)f(from)g FH( )2403 1235 y FG(out)p FF(\()p FG(in)p FF(\))2400 1297 y FG(h;\024)2491 1283 y Fk(0)2514 1307 y FL(1)2632 1273 y FO(subtracting)f(asymptotic)515 1365 y(photons.)f(F)-7 b(ollo)n(wing)23 b(the)h(pro)r(cedures)f(dev)n(elop)r(ed)g(in)h(the)h (next)f(t)n(w)n(o)f(sections,)g(it)h(is)g(enough)515 1456 y(to)36 b(prop)r(erly)f(c)n(hoice)h(the)g(partition)g(rates)g(for) f(com)n(bining)h(estimates)g(and)g(to)g(tak)n(e)g(in)n(to)515 1547 y(accoun)n(t)f(the)h(mec)n(hanism)f(used)h(in)g(Subsection)g (3.1.1.)f(This)h(is)f(imp)r(ortan)n(t)h(in)g(order)f(to)515 1639 y(iden)n(tify)i(limits)f(obtained)h(with)f(di\033eren)n(t)h (partition)f(rates.)f(In)i(this)g(analysis)e(it)h(is)h(also)515 1730 y(necessary)25 b(to)j(assume)f(the)h(follo)n(wing)e(result)h(pro)n 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FI(,)g Fz(A)2558 2265 y FG(out)p FF(\()p FG(in)p FF(\))2558 2324 y FG(ph)2790 2303 y FO(are)i(indep)r(enden)n(t)h(of)515 2418 y(the)f(construction)e (of)i(the)g(\020one-particle\021)k(space)27 b Fz(H)2172 2380 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))2172 2426 y FG(\024)2208 2436 y FL(1)2434 2418 y FO(but)h(dep)r(end)h(only)e(on)g FK(\006)p FO(.)515 2617 y(Ab)r(out)h(the)g(structure)f(of)g(the)h(pap)r (er:)515 2717 y(In)g(Section)f(2)g(w)n(e)g(study)h(the)g(time)g(b)r (eha)n(vior)e(of)i(the)g(norm)f(of)g(the)h(appro)n(ximating)e(v)n (ector)515 2816 y FH( )565 2825 y FG(h;\024)656 2835 y FL(1)705 2816 y FE(\()p FH(t)p FE(\))p FO(;)515 2916 y(In)h(Section)h(3)f(w)n(e)g(pro)n(v)n(e)f(the)i(strong)e(con)n(v)n (ergence)g(of)h FH( )2325 2925 y FG(h;\024)2416 2935 y FL(1)2466 2916 y FE(\()p FH(t)p FE(\))g FO(for)g FH(t)21 b Fz(!)g(1)p FO(;)515 3015 y(Section)27 b(4)h(con)n(tains)e(the)j (construction)d(of)i(the)g(scattering)f(subspace)g Fz(H)2853 2984 y FG(out)p FF(\()o FG(in)p FF(\))3057 3015 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FG(j;\033)3315 4282 y Fq(t)3347 4171 y Fy(\021)3418 4254 y FH(:)515 4496 y FO(The)24 b(diagonal)f(terms)g (are)h(easily)f(under)h(con)n(trol)f(b)r(ecause)h(their)g(sum)g(is)g (constan)n(t)g(in)g(time:)972 4637 y FG(N)5 b FF(\()p FG(t)p FF(\))979 4665 y Fy(X)982 4822 y FG(j)s FF(=1)1110 4653 y Fy(\020)1156 4736 y FH( )1209 4698 y FF(\()o FG(t)p FF(\))1206 4756 y FG(j;\033)1286 4764 y Fq(t)1317 4736 y FH(;)13 b( )1404 4698 y FF(\()p FG(t)p FF(\))1401 4756 y FG(j;\033)1481 4764 y Fq(t)1513 4653 y Fy(\021)1580 4736 y FE(=)1661 4637 y FG(N)5 b FF(\()q FG(t)p FF(\))1668 4665 y Fy(X)1671 4822 y FG(j)s FF(=1)1799 4634 y Fy(Z)1842 4804 y FF(\000)1880 4777 y FL(\()p Fq(t)p FL(\))1880 4825 y Fq(j)1969 4736 y Fz(j)p FH(h)13 b FE(\()p FA(P)p FE(\))p Fz(j)2189 4700 y FF(2)2236 4736 y FH(d)2276 4700 y FF(3)2311 4736 y FH(P)32 b FE(=)2473 4634 y Fy(Z)2562 4736 y Fz(j)q FH(h)13 b FE(\()o FA(P)p FE(\))q Fz(j)2782 4700 y FF(2)2829 4736 y FH(d)2869 4700 y FF(3)2904 4736 y FH(P)36 b(:)300 b FI(\(2.1\))1950 5059 y FO(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 515 523 a FO(The)29 b(non)g(trivial)g(step)h (consists)e(in)i(pro)n(ving)e(that)i(it)g(is)f(indeed)h(the)f(limit)i (of)e(the)h(squared)515 623 y(norm)25 b(pro)n(vided)g(the)i(partition)f (rate)f(is)h(prop)r(erly)f(c)n(hosen.)h(F)-7 b(or)25 b(this)i(purp)r(ose)f(it)g(is)g(enough)515 722 y(to)20 b(sho)n(w)f(that)i(eac)n(h)e(o\033-diagonal)g(term)h(in)g(the)h(sum) 2186 666 y Fy(P)2267 684 y FG(N)5 b FF(\()q FG(t)p FF(\))2267 744 y FG(l;j)s FF(=1)2436 722 y FO(asymptotically)19 b(v)-5 b(anishes)20 b(with)515 822 y(an)26 b(order)g(in)h FH(t)f FO(substan)n(tially)g(not)h(related)f(to)h(the)g(dimension)g(of) g(the)g(cell.)g(In)g(the)g(end,)g(w)n(e)515 922 y(obtain)34 b(that)h(the)g(sum)g(of)g(the)g(o\033-diagonal)e(terms,)2270 866 y Fy(P)2351 883 y FG(N)5 b FF(\()q FG(t)p FF(\))2351 943 y FG(l;j)s FF(=1)p FG(;l)p Fx(6)p FF(=)p FG(j)2613 922 y FO(,)35 b(v)-5 b(anishes)34 b(for)h FH(t)d Fz(!)g FE(+)p Fz(1)p FO(,)515 1021 y(pro)n(vided)24 b(the)i(exp)r(onen)n(t)f FH(\017)p FO(,)h(whic)n(h)g(determines)f(the)h(gro)n(wth)e(rate)h(of)g (the)h(total)f(n)n(um)n(b)r(er)h(of)515 1121 y(cells,)h FH(N)21 b FE(\()p FH(t)p FE(\))g Fz(\024)g FH(t)1018 1089 y FF(3)p FG(\017)1079 1121 y FO(,)27 b(is)h(su\036cien)n(tly)f (small.)515 1453 y FC(2.1)112 b(Con)m(trol)36 b(of)i(the)f (o\033-diagonal)h(terms)515 1714 y FO(The)27 b(generic)g (o\033-diagonal)e(term)j(is)f(\()p FH(l)c Fz(6)p FE(=)e FH(j)t FO(\))610 1899 y FH(M)684 1908 y FG(l;j)769 1899 y FE(\()o FH(t)p FE(\))g(=)958 1817 y Fy(\020)1004 1899 y FH(e)1040 1863 y FG(i\015)1094 1871 y Fq(\033)1128 1882 y(t)1160 1869 y FE(\()1190 1863 y Fw(v)1226 1875 y Fq(l)1251 1863 y FG(;)p Fx(r)p FG(E)1369 1835 y Fq(\033)1403 1846 y(t)1366 1884 y Fl(P)1434 1863 y FG(;t)1477 1869 y FE(\))1510 1899 y FH( )1563 1861 y FF(\()p FG(t)p FF(\))1560 1921 y FG(l;\033)1635 1929 y Fq(t)1667 1899 y FH(;)13 b(e)1737 1863 y FG(iH)1809 1871 y Fq(\033)1843 1882 y(t)1874 1863 y FG(t)1902 1899 y Fz(W)1978 1908 y FG(\033)2014 1916 y Fq(t)2041 1908 y FG(;l;j)2144 1899 y FE(\()p FH(t)p FE(\))g FH(e)2281 1863 y Fx(\000)p FG(iH)2401 1871 y Fq(\033)2435 1882 y(t)2466 1863 y FG(t)2494 1899 y FH(e)2530 1863 y FG(i\015)2584 1871 y Fq(\033)2618 1882 y(t)2650 1869 y FE(\()2680 1863 y Fw(v)2716 1874 y Fq(j)2747 1863 y FG(;)p Fx(r)p FG(E)2865 1835 y Fq(\033)2899 1846 y(t)2862 1884 y Fl(P)2930 1863 y FG(;t)2973 1869 y FE(\))3007 1899 y FH( )3060 1861 y FF(\()p FG(t)p FF(\))3057 1919 y FG(j;\033)3137 1927 y Fq(t)3168 1817 y Fy(\021)3310 1899 y FI(\(2.2\))515 2084 y FO(where)1103 2218 y Fz(W)1179 2227 y FG(\033)1215 2235 y Fq(t)1242 2227 y FG(;l;j)1345 2218 y FE(\()p FH(t)p FE(\))21 b(:=)h FH(e)1593 2167 y Fx(\000)1651 2126 y Fm(R)1689 2146 y Fq(\024)1723 2158 y FL(1)1678 2186 y Fq(\033)1712 2197 y(t)1781 2140 y(a)p FL(\()p Fl(k)p FL(\))q Fq(e)1919 2123 y(i)p Fk(j)p Fl(k)p Fk(j)p Fq(t)2040 2140 y Fk(\000)p Fq(a)2117 2123 y Fk(y)2149 2140 y FL(\()p Fl(k)p FL(\))q Fq(e)2255 2123 y Fk(\000)p Fq(i)p Fk(j)p Fl(k)p Fk(j)p Fq(t)p 1781 2155 641 3 v 2065 2189 a Fk(j)p Fl(k)p Fk(j)2431 2167 y FG(\021)2463 2179 y Fq(l;j)2533 2173 y FE(\()2565 2154 y Fm(b)2563 2167 y Fw(k)2599 2173 y FE(\))2666 2147 y Fq(d)2697 2130 y FL(3)2729 2147 y Fq(k)p 2639 2155 150 3 v 2639 2163 a Fx(p)p 2689 2163 100 3 v 35 x FL(2)p Fk(j)p Fl(k)p Fk(j)3310 2218 y FI(\(2.3\))515 2343 y FO(and)1343 2473 y FH(\021)1381 2482 y FG(l;j)1465 2390 y Fy(\020)1513 2453 y(b)1511 2473 y FA(k)1558 2390 y Fy(\021)1625 2473 y FE(:=)1944 2425 y FH(g)1986 2405 y Fy(b)1984 2425 y FA(k)16 b Fz(\001)h FE(\()p FA(v)2162 2433 y FG(j)2211 2425 y Fz(\000)g FA(v)2335 2434 y FG(l)2359 2425 y FE(\))p 1738 2456 858 4 v 1738 2469 a Fy(\020)1783 2551 y FE(1)h Fz(\000)1918 2532 y Fy(b)1916 2551 y FA(k)e Fz(\001)i FA(v)2065 2559 y FG(j)2097 2469 y Fy(\021)2160 2551 y Fz(\001)2198 2469 y Fy(\020)2244 2551 y FE(1)g Fz(\000)2378 2532 y Fy(b)2376 2551 y FA(k)f Fz(\001)h FA(v)2526 2560 y FG(l)2549 2469 y Fy(\021)2618 2473 y FH(:)671 b FI(\(2.4\))515 2699 y FO(No)n(w,)25 b(let)h(us)f(consider)g FH(M)1343 2708 y FG(l;j)1427 2699 y FE(\()p FH(t)p FE(\))g FO(as)g(a)h(t)n(w)n (o-v)-5 b(ariable)23 b(function,)j(b)n(y)f(distinguishing)h(the)g(v)-5 b(ari-)515 2790 y(able)26 b FH(t)p FO(,)g(whic)n(h)g(parameterizes)f (the)h(partition)g FE(\000)2062 2758 y FF(\()p FG(t)p FF(\))2164 2790 y FO(and)g(the)h(infrared)f(cut-o\033)g FH(\033)3080 2798 y FG(t)3108 2790 y FO(,)g(from)g(the)515 2881 y(v)-5 b(ariable,)26 b FH(s)p FO(,)i(of)f(the)h(dynamical)f(ev)n (olution.)g(Then,)h(for)f FH(s)21 b Fz(\025)g FH(t)27 b FO(w)n(e)g(de\034ne:)565 3043 y Fy(c)556 3062 y FH(M)630 3071 y FG(l;j)715 3062 y FE(\()p FH(t;)12 b(s)p FE(\))21 b(:=)996 2979 y Fy(\020)1042 3062 y FH(e)1078 3029 y FG(i\015)1132 3037 y Fq(\033)1166 3048 y(t)1198 3035 y FE(\()1227 3029 y Fw(v)1263 3041 y Fq(l)1288 3029 y FG(;)p Fx(r)p FG(E)1406 3001 y Fq(\033)1440 3012 y(t)1403 3050 y Fl(P)1471 3029 y FG(;s)1519 3035 y FE(\))1553 3062 y FH( )1606 3024 y FF(\()p FG(t)p FF(\))1603 3083 y FG(l;\033)1678 3091 y Fq(t)1722 3062 y FH(;)26 b(e)1805 3030 y FG(iH)1877 3038 y Fq(\033)1911 3049 y(t)1942 3030 y FG(s)1976 3062 y Fz(W)2052 3071 y FG(\033)2088 3079 y Fq(t)2115 3071 y FG(;l;j)2218 3062 y FE(\()p FH(s)p FE(\))12 b FH(e)2362 3030 y Fx(\000)p FG(iH)2482 3038 y Fq(\033)2516 3049 y(t)2548 3030 y FG(s)2581 3062 y FH(e)2617 3029 y FG(i\015)2671 3037 y Fq(\033)2705 3048 y(t)2737 3035 y FE(\()2767 3029 y Fw(v)2803 3040 y Fq(j)2834 3029 y FG(;)p Fx(r)p FG(E)2952 3001 y Fq(\033)2986 3012 y(t)2949 3050 y Fl(P)3017 3029 y FG(;s)3065 3035 y FE(\))3099 3062 y FH( )3152 3024 y FF(\()p FG(t)p FF(\))3149 3082 y FG(j;\033)3229 3090 y Fq(t)3261 2979 y Fy(\021)3356 3062 y FI(\(2.5\))542 3247 y FO(where)790 3408 y FH(\015)830 3416 y FG(\033)866 3424 y Fq(t)910 3408 y FE(\()p FA(v)987 3417 y FG(l)1010 3408 y FH(;)i Fz(r)p FH(E)1170 3372 y FG(\033)1206 3380 y Fq(t)1166 3428 y Fw(P)1237 3408 y FH(;)f(s)p FE(\))21 b(:=)937 3687 y Fz(\000)1010 3627 y Fy(R)1061 3647 y FG(s)1046 3714 y FF(1)1107 3582 y Fy(\032)1164 3687 y FH(g)1204 3655 y FF(2)1251 3627 y Fy(R)1302 3647 y FG(\034)6 b Fx(\001)p FG(\034)1391 3626 y Fk(\000)p Fq(\013)1287 3714 y FG(\034)g Fx(\001)p FG(\033)1377 3722 y Fq(t)1505 3638 y FF(cos)1587 3644 y FE(\()1617 3638 y Fw(q)p Fx(\001r)p FG(E)1771 3610 y Fq(\033)1805 3621 y(t)1768 3659 y Fl(P)1836 3638 y Fx(\000j)p Fw(q)p Fx(j)1958 3644 y FE(\))p 1505 3670 484 4 v 1625 3712 a FF(\(1)p Fx(\000)1729 3711 y Fm(b)1727 3712 y Fw(q)q Fx(\001)p Fw(v)1819 3724 y Fq(l)1844 3712 y FF(\))2008 3651 y FG(d)p FF(\012)p FG(d)p Fx(j)p Fw(q)p Fx(j)p 2008 3670 182 4 v 2082 3712 a FG(\034)2200 3582 y Fy(\033)2270 3687 y FH(d\034)149 b FE(for)18 b FH(s)j FE(:)98 b FH(s)2814 3655 y Fx(\000)p FG(\013)2928 3687 y Fz(\025)21 b FH(\033)3053 3695 y FG(t)667 3822 y FE(=)g Fz(f)879 3971 y(\000)952 3911 y Fy(R)1003 3931 y FG(\033)1041 3892 y Fk(\000)1100 3876 y FL(1)p 1096 3883 38 3 v 1096 3914 a Fq(\013)1039 3949 y(t)988 3998 y FF(1)1165 3866 y Fy(\032)1222 3971 y FH(g)1262 3939 y FF(2)1309 3911 y Fy(R)1360 3931 y FG(\034)6 b Fx(\001)p FG(\034)1449 3910 y Fk(\000)p Fq(\013)1345 3998 y FG(\034)g Fx(\001)p FG(\033)1435 4006 y Fq(t)1562 3922 y FF(cos)1644 3928 y FE(\()1674 3922 y Fw(q)p Fx(\001r)p FG(E)1828 3894 y Fq(\033)1862 3905 y(t)1825 3943 y Fl(P)1894 3922 y Fx(\000j)p Fw(q)p Fx(j)2016 3928 y FE(\))p 1562 3954 484 4 v 1683 3996 a FF(\(1)p Fx(\000)1787 3995 y Fm(b)1785 3996 y Fw(q)q Fx(\001)p Fw(v)1877 4008 y Fq(l)1901 3996 y FF(\))2066 3935 y FG(d)p FF(\012)p FG(d)p Fx(j)p Fw(q)p Fx(j)p 2066 3954 182 4 v 2139 3996 a FG(\034)2257 3866 y Fy(\033)2328 3971 y FH(d\034)110 b FE(for)18 b FH(s)j FE(:)99 b FH(s)2834 3939 y Fx(\000)p FG(\013)2947 3971 y FH(<)21 b(\033)3072 3979 y FG(t)3117 3971 y FH(:)3310 3829 y FI(\(2.6\))515 4158 y FO(The)27 b(prop)r(ert)n(y)1034 4139 y Fy(c)1025 4158 y FH(M)1099 4167 y FG(l;j)1183 4158 y FE(\()p FH(t;)13 b(t)p FE(\))21 b Fz(\021)g FH(M)1509 4167 y FG(l;j)1593 4158 y FE(\()p FH(t)p FE(\))27 b FO(follo)n(ws)g(b)n (y)g(de\034nition.)515 4341 y FP(Theorem)j(2.1)515 4432 y FO(Under)d(the)i(constructiv)n(e)d(assumptions)h(and)h(for)f FH(\013)p FO(\()p FH(<)22 b FE(1)p FO(\))28 b(su\036cien)n(tly)g(close) f(to)g(1,)h(the)g(fol-)515 4536 y(lo)n(wing)e(prop)r(erties)h(hold)g (for)g(the)h(o\033-diagonal)e(matrix)h(elemen)n(ts)2688 4517 y Fy(c)2679 4536 y FH(M)2753 4545 y FG(l;j)2838 4536 y FE(\()p FH(t;)12 b(s)p FE(\))p FO(:)515 4719 y(I\))614 4700 y Fy(c)605 4719 y FH(M)679 4728 y FG(l;j)763 4719 y FE(\()p FH(t;)h FE(+)p Fz(1)p FE(\))21 b(:=)g(lim)1252 4727 y FG(s)p Fx(!)p FF(+)p Fx(1)1475 4700 y Fy(c)1466 4719 y FH(M)1540 4728 y FG(l;j)1624 4719 y FE(\()p FH(t;)13 b(s)p FE(\))21 b(=)g(0)1950 5059 y FO(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 515 526 a FO(I)r(I\))28 b Fz(j)p FH(M)732 535 y FG(l;j)817 526 y FE(\()p FH(t)p FE(\))o Fz(j)22 b FE(=)1028 440 y Fy(\014)1028 485 y(\014)1028 530 y(\014)1063 507 y(c)1053 526 y FH(M)1127 535 y FG(l;j)1212 526 y FE(\()p FH(t;)13 b(t)p FE(\))j Fz(\000)1464 507 y Fy(c)1455 526 y FH(M)1529 535 y FG(l;j)1614 526 y FE(\()o FH(t;)d FE(+)p Fz(1)p FE(\))1872 440 y Fy(\014)1872 485 y(\014)1872 530 y(\014)1919 526 y Fz(\024)21 b FH(C)h Fz(\001)17 b FH(t)2143 494 y Fx(\000)p FF(7)p FG(\017)515 709 y FO(pro)n(vided)26 b FE(4)p FH(\017)c(<)f(\021)31 b FO(where)c FH(\021)k FO(is)c(a)g(p)r(ositiv)n(e)g(exp)r(onen)n(t)h FH(\013)p FO(-dep)r(enden)n(t.)515 891 y FJ(Pr)l(o)l(of)515 983 y(analysis)j(of)f(I\))515 1074 y FO(F)-7 b(or)27 b FH(s)21 b Fz(\025)g FH(t)p FO(,)27 b(let)h(us)g(consider)524 1236 y Fy(c)515 1255 y FH(M)597 1219 y FG(\025)589 1268 y(l;j)673 1255 y FE(\()p FH(t;)13 b(s)p FE(\))21 b(:=)1016 1172 y Fy(\020)1062 1255 y FH(e)1098 1222 y FG(i\015)1152 1230 y Fq(\033)1186 1241 y(t)1218 1228 y FE(\()1248 1222 y Fw(v)1284 1234 y Fq(l)1308 1222 y FG(;)p Fx(r)p FG(E)1426 1193 y Fq(\033)1460 1204 y(t)1423 1243 y Fl(P)1491 1222 y FG(;s)1539 1228 y FE(\))1573 1255 y FH( )1626 1217 y FF(\()p FG(t)p FF(\))1623 1276 y FG(l;\033)1698 1284 y Fq(t)1747 1255 y FH(;)26 b(e)1830 1223 y FG(iH)1902 1231 y Fq(\033)1936 1242 y(t)1967 1223 y FG(s)2000 1255 y Fz(W)2082 1223 y FG(\025)2076 1270 y(\033)2112 1278 y Fq(t)2139 1270 y FG(;l;j)2243 1255 y FE(\()o FH(s)p FE(\))13 b FH(e)2387 1223 y Fx(\000)p FG(iH)2507 1231 y Fq(\033)2541 1242 y(t)2572 1223 y FG(s)2605 1255 y FH(e)2641 1222 y FG(i\015)2695 1230 y Fq(\033)2729 1241 y(t)2761 1228 y FE(\()2791 1222 y Fw(v)2827 1233 y Fq(j)2859 1222 y FG(;)p Fx(r)p FG(E)2977 1193 y Fq(\033)3011 1204 y(t)2974 1243 y Fl(P)3042 1222 y FG(;s)3090 1228 y FE(\))3124 1255 y FH( )3177 1217 y FF(\()o FG(t)p FF(\))3174 1275 y FG(j;\033)3254 1283 y Fq(t)3285 1172 y Fy(\021)3333 1255 y FI(\(2.7\))515 1439 y FO(where)1115 1574 y Fz(W)1197 1538 y FG(\025)1191 1587 y(\033)1227 1595 y Fq(t)1254 1587 y FG(;l;j)1357 1574 y FE(\()p FH(s)p FE(\))21 b(:=)g FH(e)1612 1522 y Fx(\000)p FG(\025)1707 1482 y Fm(R)1746 1501 y Fq(\024)1780 1513 y FL(1)1735 1541 y Fq(\033)1769 1552 y(t)1837 1496 y(a)p FL(\()q Fl(k)p FL(\))p Fq(e)1975 1479 y(i)p Fk(j)q Fl(k)p Fk(j)p Fq(s)2101 1496 y Fk(\000)p Fq(a)2178 1479 y Fk(y)2210 1496 y FL(\()p Fl(k)p FL(\))q Fq(e)2316 1479 y Fk(\000)p Fq(i)p Fk(j)p Fl(k)p Fk(j)p Fq(s)p 1837 1511 650 3 v 2126 1544 a Fk(j)p Fl(k)p Fk(j)2496 1522 y FG(\021)2528 1534 y Fq(l;j)2598 1528 y FE(\()2630 1509 y Fm(b)2628 1522 y Fw(k)2664 1528 y FE(\))2731 1503 y Fq(d)2762 1486 y FL(3)2794 1503 y Fq(k)p 2704 1511 150 3 v 2704 1518 a Fx(p)p 2754 1518 100 3 v 36 x FL(2)p Fk(j)p Fl(k)p Fk(j)3310 1574 y FI(\(2.8\))515 1698 y Fr(\025)28 b FO(b)r(eing)f(a)g(real)g(parameter.)515 1790 y(F)-7 b(rom)19 b(the)h(deriv)-5 b(ativ)n(e)1214 1759 y Fe(1)1270 1790 y FO(of)1366 1771 y Fy(c)1357 1790 y FH(M)1439 1758 y FG(\025)1431 1805 y(l;j)1515 1790 y FE(\()p FH(t;)13 b(s)p FE(\))20 b FO(with)g(resp)r(ect)f(to)h(the)g (real)f(parameter)f FH(\025)p FO(,)i(the)g(follo)n(wing)515 1881 y(di\033eren)n(tial)27 b(equation)g(is)g(determined:)1239 2051 y FH(d)1288 2032 y Fy(c)1279 2051 y FH(M)1361 2019 y FG(\025)1353 2066 y(l;j)1437 2051 y FE(\()p FH(t;)13 b(s)p FE(\))p 1239 2089 356 4 v 1374 2156 a FH(d\025)1625 2106 y FE(=)22 b Fz(\000)p FH(\025C)1867 2115 y FG(l;j;\033)1986 2123 y Fq(t)2033 2106 y Fz(\001)2081 2087 y Fy(c)2072 2106 y FH(M)2154 2070 y FG(\025)2146 2119 y(l;j)2230 2106 y FE(\()p FH(t;)13 b(s)p FE(\))k(+)f FH(r)2518 2070 y FG(\025)2516 2119 y(\033)2552 2127 y Fq(t)2596 2106 y FE(\()p FH(t;)d(s)p FE(\))556 b FI(\(2.9\))515 2292 y FO(where)1479 2413 y FH(C)1534 2422 y FG(l;j;\033)1653 2430 y Fq(t)1705 2413 y FE(=)1786 2311 y Fy(Z)1863 2332 y FG(\024)1899 2342 y FL(1)1829 2481 y FG(\033)1865 2489 y Fq(t)1949 2327 y Fy(\014)1949 2372 y(\014)1949 2417 y(\014)1974 2413 y FH(\021)2012 2422 y FG(l;j)2097 2330 y Fy(\020)2145 2393 y(b)2143 2413 y FA(k)2189 2330 y Fy(\021)2235 2327 y(\014)2235 2372 y(\014)2235 2417 y(\014)2261 2347 y FF(2)2348 2365 y FH(d)2388 2333 y FF(3)2422 2365 y FH(k)p 2318 2396 176 4 v 2318 2473 a FE(2)13 b Fz(j)q FA(k)p Fz(j)2459 2437 y FF(3)3271 2413 y FI(\(2.10\))542 2599 y FH(r)579 2568 y FG(\025)577 2613 y(\033)613 2621 y Fq(t)657 2599 y FE(\()p FH(t;)g(s)p FE(\))21 b(=)629 2785 y(=)786 2702 y Fy(\020)832 2785 y Fz(\000W)974 2750 y FG(\025)p Fx(y)968 2806 y FG(\033)1004 2814 y Fq(t)1031 2806 y FG(;l;j)1134 2785 y FE(\()p FH(s)p FE(\))12 b FH(e)1278 2752 y FG(i\015)1332 2760 y Fq(\033)1366 2771 y(t)1398 2758 y FE(\()1428 2752 y Fw(v)1464 2764 y Fq(l)1489 2752 y FG(;)p Fx(r)p FG(E)1607 2724 y Fq(\033)1641 2735 y(t)1604 2773 y Fl(P)1672 2752 y FG(;s)1720 2758 y FE(\))1754 2785 y FH(e)1790 2753 y Fx(\000)p FG(iH)1910 2761 y Fq(\033)1944 2772 y(t)1975 2753 y FG(s)2008 2785 y FH( )2061 2747 y FF(\()p FG(t)p FF(\))2058 2806 y FG(l;\033)2133 2814 y Fq(t)2177 2785 y FH(;)1046 3033 y(;)1093 2973 y Fy(R)1144 2994 y FG(\024)1180 3004 y FL(1)1129 3060 y FG(\033)1165 3068 y Fq(t)1239 2985 y FG(a)p FF(\()p Fw(k)p FF(\))p FG(e)1386 2964 y Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(s)1512 2985 y FG(\021)1544 2997 y Fq(l;j)1614 2991 y FE(\()1646 2972 y Fm(b)1644 2985 y Fw(k)1680 2991 y FE(\))p 1239 3016 471 4 v 1438 3059 a Fx(j)p Fw(k)p Fx(j)1765 3003 y FG(d)1797 2982 y FL(3)1830 3003 y FG(k)p 1730 3017 169 4 v 1730 3026 a Fz(p)p 1794 3026 105 4 v 49 x FF(2)p Fx(j)q Fw(k)p Fx(j)1909 3033 y FH(e)1945 3000 y FG(i\015)1999 3008 y Fq(\033)2033 3019 y(t)2065 3006 y FE(\()2095 3000 y Fw(v)2131 3012 y Fq(l)2155 3000 y FG(;)p Fx(r)p FG(E)2273 2972 y Fq(\033)2307 2983 y(t)2270 3021 y Fl(P)2338 3000 y FG(;s)2386 3006 y FE(\))2420 3033 y FH(e)2456 3001 y Fx(\000)p FG(iH)2576 3009 y Fq(\033)2610 3020 y(t)2642 3001 y FG(s)2675 3033 y FH( )2728 2995 y FF(\()p FG(t)p FF(\))2725 3053 y FG(j;\033)2805 3061 y Fq(t)2836 2928 y Fy(\023)772 3304 y FE(+)844 3199 y Fy(\022)901 3244 y(R)952 3264 y FG(\024)988 3274 y FL(1)937 3331 y FG(\033)973 3339 y Fq(t)1048 3256 y FG(a)p FF(\()o Fw(k)p FF(\))q FG(e)1195 3235 y Fq(i)p Fk(j)p Fl(k)p Fk(j)p Fq(s)1320 3256 y FG(\021)1352 3268 y Fq(l;j)1422 3262 y FE(\()1454 3243 y Fm(b)1452 3256 y Fw(k)1489 3262 y FE(\))p 1048 3287 471 4 v 1246 3329 a Fx(j)p Fw(k)p Fx(j)1574 3273 y FG(d)1606 3252 y FL(3)1638 3273 y FG(k)p 1538 3287 169 4 v 1538 3296 a Fz(p)p 1602 3296 105 4 v 50 x FF(2)p Fx(j)q Fw(k)p Fx(j)1717 3304 y FH(e)1753 3271 y FG(i\015)1807 3279 y Fq(\033)1841 3290 y(t)1873 3277 y FE(\()1903 3271 y Fw(v)1939 3283 y Fq(l)1963 3271 y FG(;)p Fx(r)p FG(E)2081 3243 y Fq(\033)2115 3254 y(t)2078 3292 y Fl(P)2146 3271 y FG(;s)2194 3277 y FE(\))2228 3304 y FH(e)2264 3272 y Fx(\000)p FG(iH)2384 3280 y Fq(\033)2418 3291 y(t)2450 3272 y FG(s)2483 3304 y FH( )2536 3266 y FF(\()p FG(t)p FF(\))2533 3325 y FG(l;\033)2608 3333 y Fq(t)2652 3304 y FH(;)1284 3552 y(;)26 b Fz(W)1413 3520 y FG(\025)1407 3568 y(\033)1443 3576 y Fq(t)1470 3568 y FG(;l;j)1573 3552 y FE(\()p FH(s)p FE(\))13 b FH(e)1718 3519 y FG(i\015)1772 3527 y Fq(\033)1806 3538 y(t)1837 3525 y FE(\()1867 3519 y Fw(v)1903 3531 y Fq(l)1928 3519 y FG(;)p Fx(r)p FG(E)2046 3491 y Fq(\033)2080 3502 y(t)2043 3540 y Fl(P)2111 3519 y FG(;s)2159 3525 y FE(\))2193 3552 y FH(e)2229 3520 y Fx(\000)p FG(iH)2349 3528 y Fq(\033)2383 3539 y(t)2414 3520 y FG(s)2447 3552 y FH( )2500 3514 y FF(\()p FG(t)p FF(\))2497 3572 y FG(j;\033)2577 3580 y Fq(t)2609 3469 y Fy(\021)3271 3169 y FI(\(2.11\))515 3704 y FO(The)27 b(solution)g(of)h(the)g(di\033eren)n(tial)f(equation)g(\(2.9\))g(at)h FH(\025)21 b FE(=)g(1)28 b FO(is)755 3898 y Fy(c)746 3917 y FH(M)820 3926 y FG(l;j)904 3917 y FE(\()p FH(t;)13 b(s)p FE(\))21 b(=)g FH(e)1200 3881 y Fx(\000)1258 3840 y Fq(C)1299 3855 y(l;j;\033)1415 3866 y(t)p 1258 3870 189 3 v 1338 3903 a FL(2)1470 3898 y Fy(c)1461 3917 y FH(M)1543 3881 y FG(\025)p FF(=0)1535 3930 y FG(l;j)1674 3917 y FE(\()p FH(t;)12 b(s)p FE(\))17 b(+)1925 3816 y Fy(Z)2002 3836 y FF(1)1968 3986 y(0)2049 3917 y FH(r)2086 3881 y FG(\025)2123 3861 y Fk(0)2084 3930 y FG(\033)2120 3938 y Fq(t)2164 3917 y FE(\()p FH(t;)c(s)p FE(\))j Fz(\001)i FH(e)2413 3881 y Fx(\000)2470 3840 y Fq(C)2511 3855 y(l;j;\033)2627 3866 y(t)p 2470 3870 V 2551 3903 a FL(2)2669 3887 y FE(\()2699 3881 y FF(1)p Fx(\000)p FG(\025)2814 3861 y Fk(0)p FL(2)2865 3887 y FE(\))2899 3917 y FH(d\025)2984 3881 y Fx(0)3019 3917 y FH(:)231 b FI(\(2.12\))515 4122 y FO(No)n(w,)27 b(notice)g(the)h(follo)n(wing)f(facts:)524 4286 y Fy(c)515 4305 y FH(M)597 4273 y FG(\025)p FF(=0)589 4320 y FG(l;j)728 4305 y FE(\()p FH(t;)12 b(s)p FE(\))37 b(=)f(0)51 b Fz(8)p FH(t;)13 b(s)p FO(,)37 b(b)r(ecause)g(the)h FA(P)p Fz(\000)p FO(supp)r(orts)f(of)h FH( )2406 4266 y FF(\()o FG(t)p FF(\))2403 4324 y FG(j;\033)2483 4332 y Fq(t)2552 4305 y FO(and)f FH( )2776 4266 y FF(\()p FG(t)p FF(\))2773 4326 y FG(l;\033)2848 4334 y Fq(t)2880 4305 y FI(,)e FH(l)j Fz(6)p FE(=)e FH(j)t FO(,)i(are)f(dis-)515 4396 y(join)n(t;)p 515 4465 1182 4 v 607 4536 a Fd(1)642 4559 y Fr( )690 4524 y FF(\()p FG(t)p FF(\))688 4581 y FG(j;\033)768 4589 y Fq(t)828 4559 y Fu(2)28 b Fr(D)14 b Ft(\()p Fr(H)1060 4567 y FG(\033)1096 4575 y Fq(t)1128 4559 y Ft(\))29 b Fi(implies)e(that)j(it)f(b)r(elongs)g(to)h(the)f(domains)f(of)h(the)h (op)r(erators)h Fr(a)12 b Ft(\()q Fr(f)7 b Ft(\))29 b Fi(and)h Fr(a)3307 4536 y Fx(y)3352 4559 y Ft(\()p Fr(f)7 b Ft(\))q Fi(,)515 4656 y Fr(f)27 b Fu(2)19 b Fr(L)690 4632 y FF(2)737 4602 y Fs(\000)769 4656 y Fc(R)817 4632 y FF(3)867 4656 y Fu(n)d Fr(B)971 4664 y FG(\033)1007 4672 y Fq(t)1039 4602 y Fs(\001)1094 4656 y Fi(with)23 b Fr(B)1307 4664 y FG(\033)1343 4672 y Fq(t)1395 4656 y Ft(:=)1489 4602 y Fs(\010)1530 4656 y Fh(k)c Fu(2)q Fc(R)1688 4632 y FF(3)1742 4656 y Ft(:)39 b Fu(j)o Fh(k)p Fu(j)19 b(\024)h Fr(\033)2017 4664 y FG(t)2045 4602 y Fs(\011)2086 4656 y Fi(;)j(therefore)i(the)g(deriv)l(ativ)n(e)g(with)e (resp)r(ect)i(to)f Fr(\025)f Fi(is)515 4735 y(w)n(ell)g(de\034ned.)1950 5059 y FO(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 515 523 a FO(Thanks)27 b(to)g(Theorem)g(A5)g (\(in)h(App)r(endix\),)h(the)f(v)n(ector)735 786 y FH(s)17 b Fz(\000)62 b FE(lim)865 831 y FG(s)p Fx(!)p FF(+)p Fx(1)1075 786 y FH(e)1111 750 y FG(iH)1183 758 y Fq(\033)1217 769 y(t)1248 750 y FG(s)1294 684 y Fy(Z)1371 705 y FG(\024)1407 715 y FL(1)1337 854 y FG(\033)1373 862 y Fq(t)1467 708 y FH(a)12 b FE(\()p FA(k)p FE(\))g FH(e)1675 676 y FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(s)1805 708 y FH(\021)1843 717 y FG(l;j)1927 625 y Fy(\020)1975 688 y(b)1973 708 y FA(k)2020 625 y Fy(\021)p 1467 769 600 4 v 1721 837 a Fz(j)q FA(k)p Fz(j)2136 738 y FH(d)2176 706 y FF(3)2210 738 y FH(k)p 2085 769 218 4 v 2085 785 a Fy(p)p 2162 785 141 4 v 63 x FE(2)h Fz(j)q FA(k)p Fz(j)2313 786 y FH(e)2349 750 y Fx(\000)p FG(iH)2469 758 y Fq(\033)2503 769 y(t)2534 750 y FG(s)2567 786 y FH(e)2603 750 y FG(i\015)2657 758 y Fq(\033)2691 769 y(t)2723 756 y FE(\()2753 750 y Fw(v)2789 761 y Fq(j)2820 750 y FG(;)p Fx(r)p FG(E)2938 722 y Fq(\033)2972 733 y(t)2935 771 y Fl(P)3004 750 y FG(;s)3052 756 y FE(\))3086 786 y FH( )3139 748 y FF(\()o FG(t)p FF(\))3136 806 y FG(j;\033)3216 814 y Fq(t)515 1006 y FO(is)27 b(w)n(ell)h(de\034ned)f (and)h(can)f(b)r(e)h(written)g(as)1085 1146 y Fy(Z)1162 1167 y FG(\024)1198 1177 y FL(1)1127 1317 y FG(\033)1163 1325 y Fq(t)1257 1170 y FH(a)1298 1138 y FG(out)1298 1183 y(\033)1334 1191 y Fq(t)1406 1170 y FE(\()p FA(k)p FE(\))12 b FH(\021)1563 1179 y FG(l;j)1647 1087 y Fy(\020)1695 1150 y(b)1693 1170 y FA(k)1740 1087 y Fy(\021)p 1257 1231 529 4 v 1477 1299 a Fz(j)p FA(k)p Fz(j)1856 1200 y FH(d)1896 1168 y FF(3)1931 1200 y FH(k)p 1806 1231 218 4 v 1806 1247 a Fy(p)p 1883 1247 141 4 v 64 x FE(2)h Fz(j)p FA(k)p Fz(j)2033 1248 y FH(e)2069 1178 y FG(i\015)2123 1186 y Fq(\033)2157 1197 y(t)2189 1093 y Fm( )2234 1178 y Fw(v)2270 1189 y Fq(j)2302 1178 y FG(;)p Fx(r)p FG(E)2420 1149 y Fq(\033)2454 1160 y(t)2417 1199 y Fl(P)2485 1178 y FG(;\033)2542 1139 y Fk(\000)2601 1123 y FL(1)p 2596 1130 38 3 v 2596 1161 a Fq(\013)2540 1196 y(t)2648 1093 y Fm(!)2698 1248 y FH( )2751 1210 y FF(\()o FG(t)p FF(\))2748 1268 y FG(j;\033)2828 1276 y Fq(t)2876 1248 y FH(;)515 1474 y FO(where)34 b FH(a)803 1442 y FG(out)803 1487 y(\033)839 1495 y Fq(t)911 1474 y FE(\()p FA(k)p FE(\))g FO(is)h(the)h(asymptotic)e(annihilation)h(op)r(erator-v)-5 b(alued)32 b(distribution)k(corre-)515 1565 y(sp)r(onding)27 b(to)g(the)h(dynamics)f(go)n(v)n(erned)f(b)n(y)h(the)h(Hamiltonian)f FH(H)2627 1573 y FG(\033)2663 1581 y Fq(t)2694 1565 y FO(;)515 1665 y(Since)g(the)h(v)n(ector)f FH( )1177 1627 y FF(\()o FG(t)p FF(\))1174 1685 y FG(j;\033)1254 1693 y Fq(t)1313 1665 y FO(is)g(a)g(v)-5 b(acuum)28 b(v)n(ector)e(for)2148 1605 y Fy(\010)2193 1665 y FH(a)2234 1634 y FG(out)2234 1679 y(\033)2270 1687 y Fq(t)2342 1665 y FE(\()p FA(k)p FE(\))2448 1605 y Fy(\011)2521 1665 y FO(\(see)h(Theorem)g(A5\),)h(w)n (e)f(get)1706 1843 y FE(lim)1660 1889 y FG(s)p Fx(!)p FF(+)p Fx(1)1871 1843 y FH(r)1908 1807 y FG(\025)1906 1856 y(\033)1942 1864 y Fq(t)1985 1843 y FE(\()p FH(t;)13 b(s)p FE(\))21 b(=)g(0)d FH(:)949 b FI(\(2.13\))515 2096 y FO(Starting)26 b(from)h(the)g(solution)f(\(2.12\))g(and)h(exploiting) g(the)g(dominated)g(con)n(v)n(ergence)d(theo-)515 2187 y(rem,)j(w)n(e)g(ha)n(v)n(e)1008 2375 y Fy(c)999 2394 y FH(M)1073 2403 y FG(l;j)1157 2394 y FE(\()p FH(t;)13 b FE(+)p Fz(1)p FE(\))20 b(=)67 b(lim)1518 2439 y FG(s)p Fx(!)p FF(+)p Fx(1)1728 2292 y Fy(Z)1805 2313 y FF(1)1770 2463 y(0)1852 2394 y FH(r)1889 2358 y FG(\025)1926 2337 y Fk(0)1887 2407 y FG(\033)1923 2415 y Fq(t)1967 2394 y FE(\()p FH(t;)13 b(s)p FE(\))j Fz(\001)i FH(e)2216 2358 y Fx(\000)2273 2316 y Fq(C)2314 2331 y(l;j;\033)2430 2342 y(t)p 2273 2347 189 3 v 2354 2380 a FL(2)2472 2364 y FE(\()2502 2358 y FF(1)p Fx(\000)p FG(\025)2617 2337 y Fk(0)p FL(2)2668 2364 y FE(\))2702 2394 y FH(d\025)2787 2358 y Fx(0)2830 2394 y FE(=)j(0)13 b FH(:)288 b FI(\(2.14\))515 2603 y FJ(analysis)31 b(of)f(II\))515 2694 y FO(Let)d(us)h(consider:) 515 2872 y FH(e)551 2836 y Fx(\000)p FG(iH)671 2844 y Fq(\033)705 2855 y(t)736 2836 y FG(s)797 2824 y FH(d)p 779 2855 76 4 v 779 2923 a(ds)877 2789 y Fy(\020)923 2872 y FH(e)959 2836 y FG(iH)1031 2844 y Fq(\033)1065 2855 y(t)1097 2836 y FG(s)1130 2872 y Fz(W)1206 2880 y FG(\033)1242 2888 y Fq(t)1286 2872 y FE(\()p FA(v)1363 2881 y FG(l)1387 2872 y FH(;)13 b(s)p FE(\))f FH(e)1535 2836 y FG(i\015)1589 2844 y Fq(\033)1623 2855 y(t)1655 2842 y FE(\()1685 2836 y Fw(v)1721 2848 y Fq(l)1746 2836 y FG(;)p Fx(r)p FG(E)1864 2808 y Fq(\033)1898 2819 y(t)1861 2857 y Fl(P)1929 2836 y FG(;s)1977 2842 y FE(\))2011 2872 y FH(e)2047 2836 y Fx(\000)p FG(iH)2167 2844 y Fq(\033)2201 2855 y(t)2232 2836 y FG(s)2265 2789 y Fy(\021)2324 2872 y FH( )2377 2834 y FF(\()p FG(t)p FF(\))2374 2894 y FG(l;\033)2449 2902 y Fq(t)2502 2872 y FE(=)3423 3012 y FI(\(2.15\))842 3172 y FE(=)21 b FH(i)p Fz(W)1025 3180 y FG(\033)1061 3188 y Fq(t)1105 3172 y FE(\()p FA(v)1182 3181 y FG(l)1206 3172 y FH(;)13 b(s)p FE(\))1319 3067 y Fy(\022)1375 3172 y FH(')1425 3180 y FG(\033)1461 3188 y Fq(t)1489 3180 y FG(;)p Fw(v)1544 3192 y Fq(l)1585 3172 y FE(\()p FA(x)p FH(;)g(s)p FE(\))j(+)1865 3121 y FH(d\015)1945 3129 y FG(\033)1981 3137 y Fq(t)2025 3121 y FE(\()p FA(v)2102 3130 y FG(l)2125 3121 y FH(;)d Fz(r)p FH(E)2284 3087 y FG(\033)2320 3095 y Fq(t)2280 3143 y Fw(P)2352 3121 y FH(;)g(s)p FE(\))p 1865 3155 587 4 v 2120 3223 a FH(ds)2461 3067 y Fy(\023)2531 3172 y FH(e)2567 3136 y FG(i\015)2621 3144 y Fq(\033)2655 3155 y(t)2687 3142 y FE(\()2716 3136 y Fw(v)2752 3148 y Fq(l)2777 3136 y FG(;)p Fx(r)p FG(E)2895 3108 y Fq(\033)2929 3119 y(t)2892 3157 y Fl(P)2960 3136 y FG(;s)3008 3142 y FE(\))3042 3172 y FH(e)3078 3136 y Fx(\000)p FG(iH)3198 3144 y Fq(\033)3232 3155 y(t)3263 3136 y FG(s)3297 3172 y FH( )3350 3134 y FF(\()o FG(t)p FF(\))3347 3194 y FG(l;\033)3422 3202 y Fq(t)542 3378 y FO(where)1180 3491 y FH(')1230 3499 y FG(\033)1266 3507 y Fq(t)1293 3499 y FG(;)p Fw(v)1348 3511 y Fq(l)1389 3491 y FE(\()p FA(x)p FH(;)g(s)p FE(\))21 b(:=)g FH(g)1729 3455 y FF(2)1776 3389 y Fy(Z)1853 3410 y FG(\024)1889 3420 y FL(1)1819 3559 y FG(\033)1855 3567 y Fq(t)1948 3443 y FE(cos)14 b(\()p FA(k)j Fz(\001)g FA(x)g Fz(\000)f(j)q FA(k)p Fz(j)d FH(s)p FE(\))p 1948 3474 556 4 v 2028 3487 a Fy(\020)2074 3569 y FE(1)k Fz(\000)2208 3550 y Fy(b)2206 3569 y FA(k)g Fz(\001)g FA(v)2355 3578 y FG(l)2379 3487 y Fy(\021)2514 3491 y FH(d)p FE(\012)p FH(d)c Fz(j)p FA(k)p Fz(j)30 b FH(:)469 b FI(\(2.16\))515 3734 y FO(The)18 b(formal)g(deriv)-5 b(ativ)n(e)17 b(in)i(\(2.15\))e(is)i(op)r (eratorially)d(w)n(ell)i(de\034ned)g(b)r(ecause)g FH( )2958 3696 y FF(\()p FG(t)p FF(\))2955 3754 y FG(j;\033)3035 3762 y Fq(t)3088 3734 y Fz(2)k FH(D)15 b FE(\()p FH(H)3333 3742 y FG(\033)3369 3750 y Fq(t)3400 3734 y FE(\))21 b Fz(\021)515 3834 y FH(D)15 b FE(\()p FH(H)6 b FE(\))750 3804 y Fe(2)787 3834 y FO(.)515 3934 y(First)33 b(w)n(e)g(discuss)g (some)g(preliminary)g(quan)n(tities)g(useful)h(to)f(estimate)h(the)g (norm)f(of)g(the)p 515 4002 1182 4 v 607 4056 a Fd(2)642 4079 y Fi(More)24 b(precisely)-6 b(,)24 b(the)h(result)f(follo)n(ws)f (b)r(ecause:)515 4179 y(The)j(op)r(erators)i Fr(H)1034 4187 y FG(\033)1070 4195 y Fq(t)1102 4179 y Fi(,)d Fr(H)1210 4156 y FG(ph)1282 4179 y Fi(,)h Fr(H)1386 4188 y FF(0)1444 4179 y Ft(=)1539 4148 y Fw(p)1577 4127 y FL(2)p 1532 4164 85 3 v 1532 4205 a FF(2)p FG(m)1644 4179 y Ft(+)17 b Fr(H)1779 4156 y FG(ph)1877 4179 y Fi(and)26 b Fr(g)2064 4126 y Fs(R)2111 4146 y FG(\024)2147 4156 y FL(1)2097 4203 y FG(\033)2133 4211 y Fq(t)2195 4126 y Fs(\000)2228 4179 y Fr(a)12 b Ft(\()q Fh(k)p Ft(\))g Fr(e)2420 4156 y FG(i)p Fw(k)p Fx(\001)p Fw(x)2553 4179 y Ft(+)k Fr(a)2661 4156 y Fx(y)2706 4179 y Ft(\()p Fh(k)p Ft(\))c Fr(e)2848 4156 y Fx(\000)p FG(i)p Fw(k)p Fx(\001)p Fw(x)3014 4126 y Fs(\001)3101 4152 y FG(d)3133 4131 y FL(3)3165 4152 y FG(k)p 3068 4164 164 3 v 3068 4173 a Fu(p)p 3127 4173 105 3 v 46 x FF(2)p Fx(j)p Fw(k)p Fx(j)3267 4179 y Fi(ha)n(v)n(e)27 b(a)515 4290 y(common)22 b(e.s.d.)h Fr(D)r Fi(;)515 4369 y(The)h(deriv)l(ativ)n(e)1694 4454 y Fr(d)1742 4381 y Fs(\020)1784 4454 y Fr(e)1817 4430 y FG(iH)1893 4410 y Fq(ph)1962 4430 y FG(s)1995 4454 y Fr(e)2028 4427 y Fx(\000)p FG(iH)2148 4435 y Fq(\033)2182 4446 y(t)2213 4427 y FG(s)2246 4381 y Fs(\021)p 1694 4509 595 3 v 1957 4574 a Fr(ds)515 4652 y Fi(is)f(an)h(op)r(erator)i(whic)n(h)e(has)g(a)f (closure;)515 4731 y(Appro)n(ximating)18 b(the)g(v)n(ectors)i(in)d Fr(D)d Ft(\()p Fr(H)1579 4739 y FG(\033)1615 4747 y Fq(t)1647 4731 y Ft(\))k Fi(with)f(v)n(ectors)j(in)d Fr(D)j FI(\()p Fi(in)d(the)h(norm)g Fu(k)p Fr(H)2727 4740 y FF(0)2762 4731 y Fr( )r Fu(k)t Ft(+)t Fu(k)p Fr( )r Fu(k)q Fi(\))g(and)g (applying)515 4810 y(the)24 b(formal)f(calculus)h(w)n(e)g(get)g(con)n (v)n(ergen)n(t)j(sequences.)1950 5059 y FO(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 515 523 a FO(expression)26 b(\(2.15\).)515 722 y(i\))i(F)-7 b(rom)27 b(the)h(de\034nition)g(\(2.6\))f(w)n(e)g(ha)n (v)n(e)1430 991 y Fz(\000)p FH(g)1530 959 y FF(2)1576 930 y Fy(R)1627 951 y FG(s)1656 930 y FL(1)p Fk(\000)p Fq(\013)1613 1018 y FG(\033)1649 1026 y Fq(t)1676 1018 y Fx(\001)p FG(s)1799 942 y FF(cos)1881 948 y FE(\()1911 942 y Fw(q)p Fx(\001r)p FG(E)2065 914 y Fq(\033)2099 925 y(t)2062 963 y Fl(P)2130 942 y Fx(\000j)p Fw(q)p Fx(j)2252 948 y FE(\))p 1799 974 484 4 v 1919 1016 a FF(\(1)p Fx(\000)2023 1015 y Fm(b)2021 1016 y Fw(q)q Fx(\001)p Fw(v)2113 1028 y Fq(l)2138 1016 y FF(\))2302 955 y FG(d)p FF(\012)p FG(d)p Fx(j)p Fw(q)p Fx(j)p 2302 974 182 4 v 2378 1016 a FG(s)2673 991 y FE(for)13 b FH(s)21 b(<)h(\033)2963 943 y Fx(\000)3025 924 y FL(1)p 3020 932 38 3 v 3020 965 a Fq(\013)2960 1008 y FG(t)705 1078 y(d\015)769 1086 y Fq(\033)803 1097 y(t)835 1084 y FE(\()865 1078 y Fw(v)901 1090 y Fq(l)925 1078 y FG(;)p Fx(r)p FG(E)1043 1050 y Fq(\033)1077 1061 y(t)1040 1099 y Fl(P)1108 1078 y FG(;s)1156 1084 y FE(\))p 705 1109 482 4 v 915 1152 a FG(ds)1217 1127 y FE(=)f Fz(f)1943 1247 y FE(0)672 b(for)14 b FH(s)21 b Fz(\025)g FH(\033)2943 1200 y Fx(\000)3006 1181 y FL(1)p 3001 1189 38 3 v 3001 1222 a Fq(\013)2940 1265 y FG(t)3070 1247 y FH(:)3271 1102 y FI(\(2.17\))542 1411 y FO(By)28 b(analogy)d(w)n(e)j(de\034ne)1440 1604 y Fz(\000)p FH(g)1540 1572 y FF(2)1587 1544 y Fy(R)1638 1564 y FG(s)1667 1543 y FL(1)p Fk(\000)p Fq(\013)1623 1631 y FG(\033)1659 1639 y Fq(t)1686 1631 y Fx(\001)p FG(s)1809 1556 y FF(cos)1891 1562 y FE(\()1921 1556 y Fw(q)p Fx(\001)1987 1536 y Fl(x)p 1987 1544 34 3 v 1990 1577 a Fq(s)2030 1556 y Fx(\000j)p Fw(q)p Fx(j)2152 1562 y FE(\))p 1809 1587 373 4 v 1874 1629 a FF(\(1)p Fx(\000)1978 1628 y Fm(b)1976 1629 y Fw(q)q Fx(\001)p Fw(v)2068 1641 y Fq(l)2093 1629 y FF(\))2202 1568 y FG(d)p FF(\012)p FG(d)p Fx(j)p Fw(q)p Fx(j)p 2202 1587 182 4 v 2278 1629 a FG(s)2572 1604 y FE(for)14 b FH(s)21 b(<)g(\033)2862 1557 y Fx(\000)2924 1537 y FL(1)p 2920 1545 38 3 v 2920 1578 a Fq(\013)2859 1621 y FG(t)805 1692 y(d\015)869 1700 y Fq(\033)903 1711 y(t)935 1698 y FE(\()965 1692 y Fw(v)1001 1704 y Fq(l)1026 1692 y FG(;)1054 1672 y Fl(x)p 1054 1680 34 3 v 1057 1713 a Fq(s)1098 1692 y FG(;s)1146 1698 y FE(\))p 805 1723 371 4 v 960 1765 a FG(ds)1207 1740 y FE(:=)g Fz(f)1898 1861 y FE(0)617 b(for)14 b FH(s)21 b Fz(\025)g FH(\033)2843 1814 y Fx(\000)2905 1794 y FL(1)p 2900 1802 38 3 v 2900 1835 a Fq(\013)2840 1878 y FG(t)2969 1861 y FH(:)3271 1716 y FI(\(2.18\))542 2029 y FO(ii\))28 b(The)g(function)g Fr(')1190 2037 y FG(\033)1226 2045 y Fq(t)1253 2037 y FG(;)p Fw(v)1308 2049 y Fq(l)1349 2029 y Ft(\()p Fh(x)p Fr(;)11 b(s)p Ft(\))28 b FO(can)f(b)r(e)h(decomp)r(osed)f(as)1314 2191 y FH(')1364 2199 y FG(\033)1400 2207 y Fq(t)1427 2199 y FG(;)p Fw(v)1482 2211 y Fq(l)1524 2191 y FE(\()p FA(x)p FH(;)12 b(s)p FE(\))21 b(=)g FH(')1852 2155 y Fx(\000)1852 2204 y FG(\033)1888 2212 y Fq(t)1916 2204 y FG(;)p Fw(v)1971 2216 y Fq(l)2012 2191 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))j(+)h FH(')2332 2155 y FF(+)2332 2204 y FG(\033)2368 2212 y Fq(t)2396 2204 y FG(;)p Fw(v)2451 2216 y Fq(l)2492 2191 y FE(\()p FA(x)p FH(;)c(s)p FE(\))602 b FI(\(2.19\))542 2353 y FO(where)27 b(the)h(t)n(w)n(o)f(terms)g(on)h(the)g(righ)n(t)e (hand)i(side)f(of)h(the)g(equation)f(\(2.19\))g(are)f(de\034ned)i(as) 1531 2546 y FH(g)1571 2514 y FF(2)1618 2486 y Fy(R)1669 2506 y FG(s)1698 2485 y Fk(\000)p Fq(\013)1654 2573 y FG(\033)1690 2581 y Fq(t)1812 2510 y FF(cos)q(\()p Fw(k)p Fx(\001)p Fw(x)p Fx(\000j)q Fw(k)p Fx(j)p FG(s)p FF(\))p 1812 2529 373 4 v 1872 2588 a FE(\()1902 2582 y FF(1)p Fx(\000)1982 2569 y Fm(b)1980 2582 y Fw(k)q Fx(\001)p Fw(v)2072 2594 y Fq(l)2096 2588 y FE(\))2195 2546 y FH(d)p FE(\012)p FH(d)13 b Fz(j)p FA(k)p Fz(j)83 b FE(for)78 b FH(s)21 b(<)g(\033)2869 2499 y Fx(\000)2932 2479 y FL(1)p 2927 2487 38 3 v 2927 2520 a Fq(\013)2866 2564 y FG(t)890 2671 y FH(')940 2639 y Fx(\000)940 2684 y FG(\033)976 2692 y Fq(t)1004 2684 y FG(;)p Fw(v)1059 2696 y Fq(l)1100 2671 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))21 b(:=)g Fz(f)1963 2792 y FE(0)514 b(for)78 b FH(s)21 b Fz(\025)g FH(\033)2869 2744 y Fx(\000)2932 2725 y FL(1)p 2927 2733 V 2927 2766 a Fq(\013)2866 2809 y FG(t)3271 2654 y FI(\(2.20\))1625 2991 y FH(g)1665 2960 y FF(2)1712 2931 y Fy(R)1763 2952 y FG(\024)1799 2962 y FL(1)1748 3018 y FG(s)1777 3004 y Fk(\000)p Fq(\013)1891 2955 y FF(cos)q(\()p Fw(k)p Fx(\001)p Fw(x)p Fx(\000j)q Fw(k)p Fx(j)p FG(s)p FF(\))p 1891 2974 373 4 v 1951 3033 a FE(\()1981 3027 y FF(1)p Fx(\000)2061 3014 y Fm(b)2059 3027 y Fw(k)p Fx(\001)p Fw(v)2150 3039 y Fq(l)2175 3033 y FE(\))2274 2991 y FH(d)p FE(\012)p FH(d)13 b Fz(j)p FA(k)p Fz(j)83 b FE(for)18 b FH(s)k(<)f(\033)2889 2944 y Fx(\000)2951 2925 y FL(1)p 2946 2933 38 3 v 2946 2966 a Fq(\013)2886 3009 y FG(t)984 3116 y FH(')1034 3084 y FF(+)1034 3129 y FG(\033)1070 3137 y Fq(t)1098 3129 y FG(;)p Fw(v)1153 3141 y Fq(l)1194 3116 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))21 b(:=)g Fz(f)1875 3237 y FH(')1925 3245 y FG(\033)1961 3253 y Fq(t)1989 3245 y FG(;)p Fw(v)2044 3257 y Fq(l)2085 3237 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))332 b(for)18 b FH(s)k Fz(\025)f FH(\033)2889 3190 y Fx(\000)2951 3170 y FL(1)p 2946 3178 V 2946 3211 a Fq(\013)2886 3254 y FG(t)3271 3100 y FI(\(2.21\))515 3368 y FO(iii\))41 b(F)-7 b(or)41 b(implemen)n(ting)g(the)g (propagation)e(estimate)i(concerning)f(the)h(p)r(osition)g(of)g(the)515 3459 y(electron,)35 b(taking)g(in)n(to)g(accoun)n(t)g(Hyp)r(othesis)h (H0)f(\()p Fp(Sp)-5 b(e)g(ctr)g(al)44 b(hyp)-5 b(othesis)7 b FO(,)36 b(Subsection)515 3551 y(1.3.1\))26 b(w)n(e)h(can)h(consider)e (a)h FH(C)1472 3519 y Fx(1)1467 3564 y FF(0)1550 3490 y Fy(\000)1585 3551 y FB(R)1639 3519 y FF(3)1691 3551 y Fz(n)18 b FE(0)1785 3490 y Fy(\001)1848 3551 y FO(function)28 b FH(\037)2221 3560 y FG(h)2289 3551 y FO(with)g(the)g(follo)n(wing)e (prop)r(ert)n(y:)1369 3712 y FH(\037)1417 3721 y FG(h)1470 3712 y FE(\()p Fz(r)p FH(E)1625 3677 y FG(\033)1661 3685 y Fq(t)1621 3732 y Fw(P)1692 3712 y FE(\))21 b Fz(\021)g FE(1)154 b FI(for)h FA(P)21 b Fz(2)q FH(supph)674 b FI(\(2.22\))515 3874 y FO(uniformly)27 b(in)h FN(t)p FO(.)515 4057 y(What)37 b(w)n(e)f(w)n(an)n(t)h(no)n(w)f(to)h(c)n(hec)n(k)f(is)g(that)i(the)f (norm)f(of)h(the)g(expression)f(\(2.15\))g(go)r(es)g(to)515 4148 y(zero)31 b(with)j(an)e(in)n(tegrable)g(rate)g(substan)n(tially)g (indep)r(enden)n(t)i(of)e(the)i(partition)e(rate.)g(F)-7 b(or)515 4240 y(this)31 b(purp)r(ose)g(w)n(e)g(exploit)h(the)f(decomp)r (osition)g(\(2.19\))g(of)g FH(')2490 4248 y FG(\033)2526 4256 y Fq(t)2554 4248 y FG(;)p Fw(v)2609 4260 y Fq(l)2650 4240 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))31 b FO(and)g(the)h(function) 515 4331 y FH(\037)563 4340 y FG(h)616 4331 y FE(\()p Fz(r)p FH(E)771 4297 y FG(\033)807 4305 y Fq(t)767 4352 y Fw(P)838 4331 y FE(\))p Fi(.)27 b FO(W)-7 b(e)28 b(break)f(the)h (expression)e(\(2.15\))h(in)h(separate)e(con)n(tributions)h(and)h (estimate)515 4422 y(the)g(norm)f(for)g(eac)n(h)f(of)i(them)g(to)g(con) n(trol)e(the)i(norm)f(of)g(the)h(original)e(v)n(ector:)1022 4517 y Fy(\015)1022 4562 y(\015)1022 4607 y(\015)1022 4652 y(\015)1092 4577 y FH(d)p 1074 4609 76 4 v 1074 4676 a(ds)1173 4543 y Fy(\020)1219 4626 y FH(e)1255 4590 y FG(iH)1327 4598 y Fq(\033)1361 4609 y(t)1392 4590 y FG(s)1425 4626 y Fz(W)1501 4634 y FG(\033)1537 4642 y Fq(t)1581 4626 y FE(\()p FH(s)p FE(\))13 b FH(e)1726 4590 y FG(i\015)1780 4598 y Fq(\033)1814 4609 y(t)1846 4596 y FE(\()1875 4590 y Fw(v)1911 4602 y Fq(l)1936 4590 y FG(;)p Fx(r)p FG(E)2054 4561 y Fq(\033)2088 4572 y(t)2051 4611 y Fl(P)2119 4590 y FG(;s)2167 4596 y FE(\))2201 4626 y FH(e)2237 4590 y Fx(\000)p FG(iH)2357 4598 y Fq(\033)2391 4609 y(t)2422 4590 y FG(s)2456 4543 y Fy(\021)2514 4626 y FH( )2567 4587 y FF(\()p FG(t)p FF(\))2564 4647 y FG(l;\033)2639 4655 y Fq(t)2671 4517 y Fy(\015)2671 4562 y(\015)2671 4607 y(\015)2671 4652 y(\015)2735 4626 y Fz(\024)476 b FI(\(2.23\))1950 5059 y FO(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 611 637 a Fu(\024)749 561 y Fs(\015)749 600 y(\015)749 640 y(\015)788 637 y Fr(')834 645 y FG(\033)870 653 y Fq(t)898 645 y FG(;)p Fw(v)953 657 y Fq(l)993 637 y Ft(\()p Fh(x)p Fr(;)12 b(s)p Ft(\))1167 564 y Fs(\020)1209 637 y Fr(\037)1253 649 y FG(h)1305 583 y Fs(\000)1337 637 y Fu(r)p Fr(E)1452 606 y FG(\033)1488 614 y Fq(t)1448 660 y Fw(P)1519 583 y Fs(\001)1567 637 y Fu(\000)j Fr(\037)1681 649 y FG(h)1733 564 y Fs(\020)1785 591 y Fh(x)p 1785 622 43 3 v 1790 687 a Fr(s)1838 564 y Fs(\021\021)1934 637 y Fr(e)1967 609 y Fx(\000)p FG(iE)2085 580 y Fq(\033)2119 591 y(t)2082 630 y Fl(P)2150 609 y FG(s)2183 637 y Fr(e)2216 609 y FG(i\015)2270 617 y Fq(\033)2304 628 y(t)2336 613 y Ft(\()2364 609 y Fw(v)2400 621 y Fq(l)2424 609 y FG(;)p Fx(r)p FG(E)2542 580 y Fq(\033)2576 591 y(t)2539 630 y Fl(P)2607 609 y FG(;s)2655 613 y Ft(\))2687 637 y Fr( )2735 602 y FF(\()p FG(t)p FF(\))2733 661 y FG(l;\033)2808 669 y Fq(t)2839 561 y Fs(\015)2839 600 y(\015)2839 640 y(\015)749 844 y Ft(+)816 768 y Fs(\015)816 808 y(\015)816 848 y(\015)855 844 y Fr(')901 817 y FF(+)901 861 y FG(\033)937 869 y Fq(t)964 861 y FG(;)p Fw(v)1019 873 y Fq(l)1060 844 y Ft(\()p Fh(x)p Fr(;)c(s)p Ft(\))h Fr(\037)1277 856 y FG(h)1329 771 y Fs(\020)1381 798 y Fh(x)p 1381 829 V 1386 894 a Fr(s)1434 771 y Fs(\021)1488 844 y Fr(e)1521 816 y Fx(\000)p FG(iE)1639 788 y Fq(\033)1673 799 y(t)1636 837 y Fl(P)1704 816 y FG(s)1737 844 y Fr(e)1770 816 y FG(i\015)1824 824 y Fq(\033)1858 835 y(t)1890 820 y Ft(\()1918 816 y Fw(v)1954 828 y Fq(l)1978 816 y FG(;)p Fx(r)p FG(E)2096 788 y Fq(\033)2130 799 y(t)2093 837 y Fl(P)2161 816 y FG(;s)2209 820 y Ft(\))2241 844 y Fr( )2289 809 y FF(\()p FG(t)p FF(\))2287 868 y FG(l;\033)2362 876 y Fq(t)2393 768 y Fs(\015)2393 808 y(\015)2393 848 y(\015)749 1098 y Ft(+)816 981 y Fs(\015)816 1021 y(\015)816 1061 y(\015)816 1101 y(\015)816 1141 y(\015)855 984 y( )911 1098 y Fr(')957 1070 y Fx(\000)957 1114 y FG(\033)993 1122 y Fq(t)1020 1114 y FG(;)p Fw(v)1075 1126 y Fq(l)1116 1098 y Ft(\()p Fh(x)p Fr(;)f(s)p Ft(\))16 b(+)1374 1045 y Fr(d\015)1446 1053 y FG(\033)1482 1061 y Fq(t)1525 991 y Fs(\000)1558 1045 y Fh(v)1601 1057 y FG(l)1625 1045 y Fr(;)1666 1017 y Fw(x)p 1666 1029 37 3 v 1670 1071 a FG(s)1713 1045 y Fr(;)11 b(s)1777 991 y Fs(\001)p 1374 1083 436 3 v 1557 1147 a Fr(ds)1819 984 y Fs(!)1886 1098 y Fr(\037)1930 1110 y FG(h)1982 1024 y Fs(\020)2034 1051 y Fh(x)p 2034 1083 43 3 v 2039 1147 a Fr(s)2087 1024 y Fs(\021)2141 1098 y Fr(e)2174 1070 y Fx(\000)p FG(iE)2292 1041 y Fq(\033)2326 1052 y(t)2289 1091 y Fl(P)2357 1070 y FG(s)2390 1098 y Fr(e)2423 1070 y FG(i\015)2477 1078 y Fq(\033)2511 1089 y(t)2543 1074 y Ft(\()2571 1070 y Fw(v)2607 1082 y Fq(l)2631 1070 y FG(;)p Fx(r)p FG(E)2749 1041 y Fq(\033)2783 1052 y(t)2746 1091 y Fl(P)2814 1070 y FG(;s)2862 1074 y Ft(\))2894 1098 y Fr( )2942 1062 y FF(\()q FG(t)p FF(\))2940 1121 y FG(l;\033)3015 1129 y Fq(t)3047 981 y Fs(\015)3047 1021 y(\015)3047 1061 y(\015)3047 1101 y(\015)3047 1141 y(\015)749 1363 y Ft(+)816 1247 y Fs(\015)816 1287 y(\015)816 1327 y(\015)816 1367 y(\015)816 1407 y(\015)865 1310 y Fr(d\015)937 1318 y FG(\033)973 1326 y Fq(t)1017 1257 y Fs(\000)1049 1310 y Fh(v)1092 1322 y FG(l)1116 1310 y Fr(;)1158 1283 y Fw(x)p 1158 1295 37 3 v 1162 1336 a FG(s)1204 1310 y Fr(;)g(s)1268 1257 y Fs(\001)p 865 1348 436 3 v 1048 1413 a Fr(ds)1322 1290 y Fs(\020)1364 1363 y Fr(\037)1408 1375 y FG(h)1460 1310 y Fs(\000)1492 1363 y Fu(r)p Fr(E)1607 1332 y FG(\033)1643 1340 y Fq(t)1603 1387 y Fw(P)1674 1310 y Fs(\001)1722 1363 y Fu(\000)16 b Fr(\037)1837 1375 y FG(h)1889 1290 y Fs(\020)1941 1317 y Fh(x)p 1941 1348 43 3 v 1946 1413 a Fr(s)1994 1290 y Fs(\021\021)2090 1363 y Fr(e)2123 1335 y Fx(\000)p FG(iE)2241 1307 y Fq(\033)2275 1318 y(t)2238 1356 y Fl(P)2306 1335 y FG(s)2339 1363 y Fr(e)2372 1335 y FG(i\015)2426 1343 y Fq(\033)2460 1354 y(t)2492 1339 y Ft(\()2519 1335 y Fw(v)2555 1347 y Fq(l)2580 1335 y FG(;)p Fx(r)p FG(E)2698 1307 y Fq(\033)2732 1318 y(t)2695 1356 y Fl(P)2763 1335 y FG(;s)2811 1339 y Ft(\))2842 1363 y Fr( )2890 1328 y FF(\()q FG(t)p FF(\))2888 1387 y FG(l;\033)2963 1395 y Fq(t)2995 1247 y Fs(\015)2995 1287 y(\015)2995 1327 y(\015)2995 1367 y(\015)2995 1407 y(\015)749 1629 y Ft(+)816 1513 y Fs(\015)816 1553 y(\015)816 1593 y(\015)816 1632 y(\015)816 1672 y(\015)855 1516 y( )911 1629 y Fu(\000)976 1576 y Fr(d\015)1048 1584 y FG(\033)1084 1592 y Fq(t)1128 1522 y Fs(\000)1160 1576 y Fh(v)1203 1588 y FG(l)1227 1576 y Fr(;)1268 1549 y Fw(x)p 1268 1561 37 3 v 1272 1602 a FG(s)1315 1576 y Fr(;)11 b(s)1379 1522 y Fs(\001)p 976 1614 436 3 v 1159 1679 a Fr(ds)1437 1629 y Ft(+)1517 1579 y Fr(d\015)1589 1587 y FG(\033)1625 1595 y Fq(t)1669 1525 y Fs(\000)1701 1579 y Fh(v)1744 1591 y FG(l)1769 1579 y Fr(;)g Fu(r)p Fr(E)1915 1547 y FG(\033)1951 1555 y Fq(t)1911 1602 y Fw(P)1982 1579 y Fr(;)g(s)2046 1525 y Fs(\001)p 1517 1614 561 3 v 1763 1679 a Fr(ds)2088 1516 y Fs(!)2155 1629 y Fr(\037)2199 1641 y FG(h)2251 1576 y Fs(\000)2284 1629 y Fu(r)p Fr(E)2399 1598 y FG(\033)2435 1606 y Fq(t)2395 1652 y Fw(P)2465 1576 y Fs(\001)2509 1629 y Fr(e)2542 1601 y Fx(\000)p FG(iE)2660 1573 y Fq(\033)2694 1584 y(t)2657 1622 y Fl(P)2725 1601 y FG(s)2759 1629 y Fr(e)2792 1601 y FG(i\015)2846 1609 y Fq(\033)2880 1620 y(t)2912 1605 y Ft(\()2939 1601 y Fw(v)2975 1613 y Fq(l)3000 1601 y FG(;)p Fx(r)p FG(E)3118 1573 y Fq(\033)3152 1584 y(t)3115 1622 y Fl(P)3183 1601 y FG(;s)3231 1605 y Ft(\))3262 1629 y Fr( )3310 1594 y FF(\()q FG(t)p FF(\))3308 1653 y FG(l;\033)3383 1661 y Fq(t)3415 1513 y Fs(\015)3415 1553 y(\015)3415 1593 y(\015)3415 1632 y(\015)3415 1672 y(\015)515 1813 y FO(No)n(w)28 b(w)n(e)h(explain)g(ho)n(w)f(in)h(the)h (expression)d(ab)r(o)n(v)n(e)h(eac)n(h)g(term)h(is)g(con)n(trolled)e (and)i(wh)n(y)g(w)n(e)515 1915 y(can)e(\034x)g FH(\021)e(>)c FE(0)28 b FO(suc)n(h)f(that)h(a)f(leading)g(order)f(is)i FH(s)2053 1883 y Fx(\000)p FF(1)2152 1915 y Fz(\001)18 b FH(s)2227 1883 y Fx(\000)p FG(\021)2329 1915 y Fz(\001)g(j)p FE(ln)13 b FH(\033)2510 1923 y FG(t)2538 1915 y Fz(j)2559 1880 y FF(2)2611 1915 y Fz(\001)k FH(t)2677 1883 y Fx(\000)2735 1864 y FL(3)p Fq(\017)p 2735 1872 54 3 v 2747 1905 a FL(2)2803 1915 y FO(.)515 2006 y(F)-7 b(rom)27 b(Lemma)g(A4,)h(Theorem) e(A2)i(and)f(Corollary)e(A3)j(w)n(e)f(deduce)g(that:)515 2098 y(After)34 b(the)g(subtraction)f(of)h(the)g(infrared)f(tail)g FH(')2103 2066 y Fx(\000)2103 2111 y FG(\033)2139 2119 y Fq(t)2167 2111 y FG(;)p Fw(v)2222 2123 y Fq(l)2263 2098 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))33 b FO(and)h(exploiting)f (the)h(electron)515 2189 y(disp)r(ersion,)27 b(the)h(decoupling)f(is)g (estimated)h(from)f(ab)r(o)n(v)n(e)f(b)n(y)1319 2355 y FE(sup)1359 2415 y Fw(x)1448 2269 y Fy(\014)1448 2314 y(\014)1448 2359 y(\014)1473 2355 y FH(')1523 2319 y FF(+)1523 2368 y FG(\033)1559 2376 y Fq(t)1587 2368 y FG(;)p Fw(v)1642 2380 y Fq(l)1683 2355 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))f FH(\037)1920 2364 y FG(h)1973 2272 y Fy(\020)2029 2307 y FA(x)p 2029 2338 47 4 v 2034 2406 a FH(s)2086 2272 y Fy(\021)2131 2269 y(\014)2131 2314 y(\014)2131 2359 y(\014)2178 2355 y Fz(\024)21 b FH(C)i Fz(\001)17 b FH(s)2411 2319 y Fx(\000)p FF(2)2510 2355 y Fz(\001)h FH(s)2585 2319 y FG(\013)2642 2355 y FE(;)515 2544 y FO(The)27 b(remainder)g(is)g(con)n(trolled)f(b)n(y)i(com)n (bining)f(the)h(b)r(ounds)1445 2733 y FE(sup)1485 2793 y Fw(x)1574 2733 y Fz(j)p FH(')1645 2741 y FG(\033)1681 2749 y Fq(t)1709 2741 y FG(;)p Fw(v)1764 2753 y Fq(l)1805 2733 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))o Fz(j)84 b(\024)e FH(C)23 b Fz(\001)2344 2625 y Fy(\014)2344 2669 y(\014)2344 2714 y(\014)2344 2759 y(\014)2379 2685 y FE(ln)13 b(\()p FH(\033)2530 2693 y FG(t)2558 2685 y FE(\))p 2379 2716 209 4 v 2466 2784 a FH(s)2598 2625 y Fy(\014)2598 2669 y(\014)2598 2714 y(\014)2598 2759 y(\014)3271 2893 y FI(\(2.24\))1342 3076 y FE(sup)1381 3136 y Fw(x)1470 2945 y Fy(\014)1470 2990 y(\014)1470 3035 y(\014)1470 3080 y(\014)1470 3124 y(\014)1506 3020 y FH(d\015)1586 3028 y FG(\033)1622 3036 y Fq(t)1665 2960 y Fy(\000)1701 3020 y FA(v)1748 3029 y FG(l)1771 3020 y FH(;)1816 2989 y Fw(x)p 1816 3003 37 4 v 1820 3045 a FG(s)1862 3020 y FH(;)g(s)1932 2960 y Fy(\001)p 1506 3059 462 4 v 1699 3127 a FH(ds)1977 2945 y Fy(\014)1977 2990 y(\014)1977 3035 y(\014)1977 3080 y(\014)1977 3124 y(\014)2086 3076 y Fz(\024)82 b FH(C)23 b Fz(\001)2344 2968 y Fy(\014)2344 3012 y(\014)2344 3057 y(\014)2344 3102 y(\014)2379 3027 y FE(ln)13 b(\()p FH(\033)2530 3035 y FG(t)2558 3027 y FE(\))p 2379 3059 209 4 v 2466 3127 a FH(s)2598 2968 y Fy(\014)2598 3012 y(\014)2598 3057 y(\014)2598 3102 y(\014)542 3291 y FO(with)28 b(the)g(propagation)e(estimates)950 3371 y Fy(\015)950 3416 y(\015)950 3460 y(\015)992 3374 y(\020)1038 3457 y FH(\037)1086 3466 y FG(h)1139 3457 y FE(\()p Fz(r)p FH(E)1294 3421 y FG(\033)1330 3429 y Fq(t)1290 3476 y Fw(P)1361 3457 y FE(\))17 b Fz(\000)g FH(\037)1533 3466 y FG(h)1586 3374 y Fy(\020)1642 3408 y FA(x)p 1642 3440 47 4 v 1647 3507 a FH(s)1699 3374 y Fy(\021)o(\021)1803 3457 y FH(e)1839 3421 y Fx(\000)p FG(iE)1957 3392 y Fq(\033)1991 3403 y(t)1954 3442 y Fl(P)2022 3421 y FG(s)2055 3457 y FH(e)2091 3421 y FG(i\015)2145 3429 y Fq(\033)2179 3440 y(t)2211 3427 y FE(\()2241 3421 y Fw(v)2277 3433 y Fq(l)2301 3421 y FG(;)p Fx(r)p FG(E)2419 3392 y Fq(\033)2453 3403 y(t)2416 3442 y Fl(P)2485 3421 y FG(;s)2533 3427 y FE(\))2567 3457 y FH( )2620 3418 y FF(\()o FG(t)p FF(\))2617 3478 y FG(l;\033)2692 3486 y Fq(t)2723 3371 y Fy(\015)2723 3416 y(\015)2723 3460 y(\015)2787 3457 y Fz(\024)2476 3623 y(\024)k FH(C)h Fz(\001)17 b FH(s)2708 3587 y Fx(\000)p FG(\035)2813 3623 y Fz(\001)h(j)p FE(ln)13 b FH(\033)2994 3631 y FG(t)3022 3623 y Fz(j)k(\001)h FH(t)3127 3587 y Fx(\000)3184 3567 y FL(3)p Fq(\017)p 3184 3575 54 3 v 3197 3608 a FL(2)3271 3739 y FI(\(2.25\))564 3791 y Fy(\015)564 3836 y(\015)564 3881 y(\015)564 3925 y(\015)564 3970 y(\015)607 3794 y( )668 3922 y Fz(\000)738 3866 y FH(d\015)818 3874 y FG(\033)854 3882 y Fq(t)897 3805 y Fy(\000)932 3866 y FA(v)979 3875 y FG(l)1003 3866 y FH(;)1047 3835 y Fw(x)p 1047 3849 37 4 v 1051 3891 a FG(s)1093 3866 y FH(;)c(s)1164 3805 y Fy(\001)p 738 3904 462 4 v 930 3972 a FH(ds)1226 3922 y FE(+)1312 3871 y FH(d\015)1392 3879 y FG(\033)1428 3887 y Fq(t)1472 3871 y FE(\()p FA(v)1549 3880 y FG(l)1573 3871 y FH(;)f Fz(r)p FH(E)1732 3837 y FG(\033)1768 3845 y Fq(t)1728 3892 y Fw(P)1799 3871 y FH(;)g(s)p FE(\))p 1312 3904 587 4 v 1568 3972 a FH(ds)1909 3794 y Fy(!)1982 3922 y FH(e)2018 3886 y Fx(\000)p FG(iE)2136 3857 y Fq(\033)2170 3868 y(t)2133 3907 y Fl(P)2201 3886 y FG(s)2234 3922 y FH(e)2270 3886 y FG(i\015)2324 3894 y Fq(\033)2358 3905 y(t)2390 3892 y FE(\()2420 3886 y Fw(v)2456 3898 y Fq(l)2481 3886 y FG(;)p Fx(r)p FG(E)2599 3857 y Fq(\033)2633 3868 y(t)2596 3907 y Fl(P)2664 3886 y FG(;s)2712 3892 y FE(\))2746 3922 y FH( )2799 3883 y FF(\()p FG(t)p FF(\))2796 3943 y FG(l;\033)2871 3951 y Fq(t)2902 3791 y Fy(\015)2902 3836 y(\015)2902 3881 y(\015)2902 3925 y(\015)2902 3970 y(\015)2966 3922 y Fz(\024)2285 4130 y(\024)21 b FH(C)h Fz(\001)17 b FH(s)2517 4095 y Fx(\000)p FF(1)2617 4130 y Fz(\001)g FH(s)2691 4095 y Fx(\000)p FG(\035)2796 4130 y Fz(\001)h(j)p FE(ln)13 b FH(\033)2977 4138 y FG(t)3005 4130 y Fz(j)k(\001)h FH(t)3110 4095 y Fx(\000)3167 4075 y FL(3)p Fq(\017)p 3167 4083 54 3 v 3180 4116 a FL(2)515 4277 y FO(where)j FH(\035)i(>)e FE(0)h FO(for)f FH(\013)h FO(su\036cien)n(tly)f(close)g(to)g FE(1)h FO(and)f FH(\017)h FO(small)f(enough)g(\(Theorem)g(A2,)g(Corollary)515 4369 y(A3\).)27 b(Moreo)n(v)n(er,)e(assuming)i(the)h(follo)n(wing)e (constrain)n(t)1865 4516 y FE(4)p FH(\017)c(<)f(\021)f(;)1155 b FI(\(2.26\))515 4663 y FO(w)n(e)27 b(easily)g(obtain)646 4810 y Fz(j)p FH(M)741 4819 y FG(l;j)826 4810 y FE(\()o FH(t)p FE(\))p Fz(j)22 b(\024)2255 b FI(\(2.27\))1950 5059 y FO(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 574 526 a Fz(\024)717 440 y Fy(\014)717 485 y(\014)717 530 y(\014)743 466 y(R)794 486 y FF(+)p Fx(1)779 553 y FG(t)943 495 y(d)p 928 509 62 4 v 928 551 a(ds)1012 443 y Fy(\020)1058 526 y FH(e)1094 493 y FG(i\015)1148 501 y Fq(\033)1182 512 y(t)1214 499 y FE(\()1244 493 y Fw(v)1280 505 y Fq(l)1304 493 y FG(;)p Fx(r)p FG(E)1422 465 y Fq(\033)1456 476 y(t)1419 514 y Fl(P)1487 493 y FG(;s)1535 499 y FE(\))1569 526 y FH( )1622 488 y FF(\()p FG(t)p FF(\))1619 547 y FG(l;\033)1694 555 y Fq(t)1739 526 y FH(;)26 b(e)1822 494 y FG(iH)1894 502 y Fq(\033)1928 513 y(t)1959 494 y FG(s)1992 526 y Fz(W)2068 535 y FG(\033)o(;l;j)2205 526 y FE(\()p FH(s)p FE(\))13 b FH(e)2350 494 y Fx(\000)p FG(iH)2470 502 y Fq(\033)2504 513 y(t)2535 494 y FG(s)2568 526 y FH(e)2604 493 y FG(i\015)2658 501 y Fq(\033)2692 512 y(t)2724 499 y FE(\()2754 493 y Fw(v)2790 504 y Fq(j)2821 493 y FG(;)p Fx(r)p FG(E)2939 465 y Fq(\033)2973 476 y(t)2936 514 y Fl(P)3005 493 y FG(;s)3053 499 y FE(\))3087 526 y FH( )3140 488 y FF(\()o FG(t)p FF(\))3137 546 y FG(j;\033)3217 554 y Fq(t)3248 443 y Fy(\021)3307 526 y FH(ds)3383 440 y Fy(\014)3383 485 y(\014)3383 530 y(\014)574 752 y Fz(\024)727 692 y Fy(R)779 712 y FF(+)p Fx(1)764 779 y FG(t)903 666 y Fy(\015)903 711 y(\015)903 756 y(\015)970 721 y FG(d)p 956 735 V 956 777 a(ds)1039 669 y Fy(n)1091 752 y FH(e)1127 720 y FG(iH)1199 728 y Fq(\033)1233 739 y(t)1264 720 y FG(s)1297 752 y Fz(W)1373 760 y FG(\033)1409 768 y Fq(t)1453 752 y FE(\()p FA(v)1530 761 y FG(l)1554 752 y FH(;)g(s)p FE(\))g FH(e)1703 719 y FG(i\015)1757 727 y Fq(\033)1791 738 y(t)1823 725 y FE(\()1852 719 y Fw(v)1888 731 y Fq(l)1913 719 y FG(;)p Fx(r)p FG(E)2031 690 y Fq(\033)2065 701 y(t)2028 740 y Fl(P)2096 719 y FG(;s)2144 725 y FE(\))2178 752 y FH(e)2214 720 y Fx(\000)p FG(iH)2334 728 y Fq(\033)2368 739 y(t)2399 720 y FG(s)2433 752 y FH( )2486 714 y FF(\()o FG(t)p FF(\))2483 773 y FG(l;\033)2558 781 y Fq(t)2589 669 y Fy(o)2640 666 y(\015)2640 711 y(\015)2640 756 y(\015)2700 752 y Fz(\001)2738 666 y Fy(\015)2738 711 y(\015)2738 756 y(\015)2781 752 y FH( )2834 714 y FF(\()p FG(t)p FF(\))2831 772 y FG(j;\033)2911 780 y Fq(t)2942 666 y Fy(\015)2942 711 y(\015)2942 756 y(\015)2998 752 y FH(ds)728 978 y FE(+)801 917 y Fy(R)852 938 y FF(+)p Fx(1)837 1005 y FG(t)976 892 y Fy(\015)976 936 y(\015)976 981 y(\015)1044 947 y FG(d)p 1029 961 V 1029 1003 a(ds)1113 895 y Fy(n)1164 978 y FH(e)1200 946 y FG(iH)1272 954 y Fq(\033)1306 965 y(t)1337 946 y FG(s)1371 978 y Fz(W)1447 986 y FG(\033)1483 994 y Fq(t)1527 978 y FE(\()p FA(v)1604 986 y FG(j)1636 978 y FH(;)g(s)p FE(\))f FH(e)1784 945 y FG(i\015)1838 953 y Fq(\033)1872 964 y(t)1904 951 y FE(\()1934 945 y Fw(v)1970 956 y Fq(j)2002 945 y FG(;)p Fx(r)p FG(E)2120 916 y Fq(\033)2154 927 y(t)2117 966 y Fl(P)2185 945 y FG(;s)2233 951 y FE(\))2267 978 y FH(e)2303 946 y Fx(\000)p FG(iH)2423 954 y Fq(\033)2457 965 y(t)2488 946 y FG(s)2521 978 y FH( )2574 939 y FF(\()p FG(t)p FF(\))2571 997 y FG(j;\033)2651 1005 y Fq(t)2683 895 y Fy(o)2734 892 y(\015)2734 936 y(\015)2734 981 y(\015)2794 978 y Fz(\001)2832 892 y Fy(\015)2832 936 y(\015)2832 981 y(\015)2875 978 y FH( )2928 939 y FF(\()p FG(t)p FF(\))2925 999 y FG(l;\033)3000 1007 y Fq(t)3031 892 y Fy(\015)3031 936 y(\015)3031 981 y(\015)3087 978 y FH(ds)574 1181 y Fz(\024)106 b FH(C)22 b Fz(\001)c FH(t)884 1150 y Fx(\000)p FF(7)p FG(\017)515 1314 y FO(Hence)27 b(the)h(sum)g(of)g(the)g(o\033-diagonal)d(terms)i (is)h(b)r(ounded)g(b)n(y)f FH(C)22 b Fz(\001)c FH(t)2680 1282 y Fx(\000)p FG(\017)2785 1314 y FO(.)595 b Fb(\003)515 1588 y FD(3)131 b(Strong)41 b(con)l(v)l(ergence)h(of)f(the)g(appro)l (ximating)h(v)l(ector)515 1911 y FO(In)35 b(order)f(to)h(pro)n(v)n(e)e (the)j(strong)e(con)n(v)n(ergence)e(of)k FH( )2223 1920 y FG(h;\024)2314 1930 y FL(1)2363 1911 y FE(\()p FH(t)p FE(\))p FO(,)f(w)n(e)g(study)g(the)h(norm)e(of)h(the)515 2003 y(v)n(ector)1381 2094 y FE(\001)1445 2102 y FG(t)1469 2112 y FL(2)1501 2102 y FG(;t)1544 2112 y FL(1)1580 2094 y FH( )1630 2103 y FG(h;\024)1721 2113 y FL(1)1780 2094 y FE(:=)21 b FH( )1932 2103 y FG(h;\024)2023 2113 y FL(1)2073 2094 y FE(\()o FH(t)2130 2102 y FF(2)2165 2094 y FE(\))c Fz(\000)g FH( )2339 2103 y FG(h;\024)2430 2113 y FL(1)2479 2094 y FE(\()p FH(t)2537 2102 y FF(1)2571 2094 y FE(\))709 b FI(\(3.1\))515 2219 y FO(for)27 b(arbitrary)e(times)j FH(t)1245 2227 y FF(2)1301 2219 y FH(>)21 b(t)1410 2227 y FF(1)1465 2219 y Fz(\035)h FE(1)p FO(.)515 2310 y(F)-7 b(or)35 b(a)h(time)h(di\033erence,)f FH(t)1379 2318 y FF(2)1435 2310 y Fz(\000)22 b FH(t)1545 2318 y FF(1)1580 2310 y FO(,)36 b(su\036cien)n(tly)g(large)f(w)n(e)g(ha)n(v)n(e)g (di\033eren)n(t)i(partitions)e(cor-)515 2401 y(resp)r(onding)g(to)h FH(t)1084 2409 y FF(2)1154 2401 y FO(and)g FH(t)1352 2409 y FF(1)1423 2401 y FO(resp)r(ectiv)n(ely)f(and)h(then)g FH(N)22 b FE(\()o FH(t)2391 2409 y FF(2)2426 2401 y FE(\))34 b Fz(6)p FE(=)g FH(N)22 b FE(\()o FH(t)2724 2409 y FF(1)2759 2401 y FE(\))p FO(.)36 b(The)g FH(t)3055 2409 y FF(2)3089 2401 y Fz(\000)p FO(partition)515 2501 y(sum,)719 2445 y Fy(P)800 2466 y FG(N)5 b FF(\()p FG(t)901 2476 y FL(2)933 2466 y FF(\))800 2524 y FG(j)s FF(=1)961 2501 y FO(,)33 b(is)g(therefore)e(generally)g(written)i(as)2220 2445 y Fy(P)2301 2466 y FG(N)5 b FF(\()q FG(t)2403 2476 y FL(1)2435 2466 y FF(\))2301 2524 y FG(j)s FF(=1)2476 2445 y Fy(P)2557 2524 y FG(l)p FF(\()p FG(j)s FF(\))2658 2501 y FO(,)32 b(where)g(the)h(index)g FH(l)14 b FE(\()p FH(j)t FE(\))515 2611 y FO(coun)n(ts)35 b(the)h(sub)r(cells,)f(relativ) n(e)g(to)g(the)h FH(t)1865 2619 y FF(2)1899 2611 y Fz(\000)p FO(partition,)f(whic)n(h)h(are)e(con)n(tained)h(in)h(the)g FH(j)3403 2579 y FG(th)515 2702 y FO(cell)e(of)g FE(\000)818 2671 y FF(\()p FG(t)866 2681 y FL(1)898 2671 y FF(\))927 2702 y FO(,)g(with)h FE(1)d Fz(\024)g FH(l)14 b FE(\()o FH(j)t FE(\))33 b Fz(\024)1607 2666 y FG(N)5 b FF(\()q FG(t)1709 2676 y FL(2)1741 2666 y FF(\))p 1607 2685 158 4 v 1607 2728 a FG(N)g FF(\()q FG(t)1709 2738 y FL(1)1741 2728 y FF(\))1775 2702 y FO(.)34 b(In)h(accordance)d(to)i(these)h (notations,)e(the)i(v)n(ector)515 2806 y FE(\001)579 2814 y FG(t)603 2824 y FL(2)635 2814 y FG(;t)678 2824 y FL(1)714 2806 y FH( )764 2815 y FG(h;\024)855 2825 y FL(1)920 2806 y FO(corresp)r(onds)25 b(to)515 3048 y FE(\001)579 3056 y FG(t)603 3066 y FL(2)635 3056 y FG(;t)678 3066 y FL(1)714 3048 y FH( )764 3057 y FG(h;\024)855 3067 y FL(1)913 3048 y FE(=)c FH(e)1030 3012 y FG(iH)t(t)1130 3022 y FL(2)1179 2949 y FG(N)5 b FF(\()p FG(t)1280 2959 y FL(1)1313 2949 y FF(\))1202 2977 y Fy(X)1205 3134 y FG(j)s FF(=1)1349 2977 y Fy(X)1357 3139 y FG(l)p FF(\()p FG(j)s FF(\))1473 3048 y Fz(W)1549 3056 y FG(\033)1585 3064 y Fq(t)1608 3076 y FL(2)1662 2988 y Fy(\000)1697 3048 y FA(v)1744 3060 y FG(l)p FF(\()p FG(j)s FF(\))1844 3048 y FH(;)13 b(t)1906 3056 y FF(2)1941 2988 y Fy(\001)1989 3048 y FH(e)2025 3008 y FG(i\015)2079 3016 y Fq(\033)2113 3027 y(t)2136 3039 y FL(2)2177 3014 y FE(\()2207 3008 y Fw(v)2243 3022 y Fq(l)p FL(\()q Fq(j)r FL(\))2339 3008 y FG(;)p Fx(r)p FG(E)2457 2979 y Fq(\033)2491 2990 y(t)2514 3002 y FL(2)2555 3008 y FF(\()p Fw(P)p FF(\))o FG(;t)2692 3018 y FL(2)2724 3014 y FE(\))2758 3048 y FH(e)2794 3008 y Fx(\000)p FG(iH)2914 3016 y Fq(\033)2948 3027 y(t)2971 3039 y FL(2)3012 3008 y FG(t)3036 3018 y FL(2)3072 3048 y FH( )3125 3010 y FF(\()p FG(t)3173 3020 y FL(2)3205 3010 y FF(\))3122 3072 y FG(l)p FF(\()p FG(j)s FF(\))q FG(;\033)3274 3080 y Fq(t)3297 3092 y FL(2)3346 3249 y FI(\(3.2\))1026 3446 y Fz(\000)p FH(e)1122 3410 y FG(iH)t(t)1222 3420 y FL(1)1270 3346 y FG(N)5 b FF(\()q FG(t)1372 3356 y FL(1)1404 3346 y FF(\))1293 3375 y Fy(X)1296 3532 y FG(j)s FF(=1)1440 3446 y Fz(W)1516 3454 y FG(\033)1552 3462 y Fq(t)1575 3474 y FL(1)1629 3446 y FE(\()p FA(v)1706 3454 y FG(j)1738 3446 y FH(;)13 b(t)1800 3454 y FF(1)1835 3446 y FE(\))f FH(e)1913 3405 y FG(i\015)1967 3413 y Fq(\033)2001 3424 y(t)2024 3436 y FL(1)2066 3411 y FE(\()2095 3405 y Fw(v)2131 3416 y Fq(j)2163 3405 y FG(;)p Fx(r)p FG(E)2281 3376 y Fq(\033)2315 3387 y(t)2338 3399 y FL(1)2378 3405 y FF(\()p Fw(P)p FF(\))p FG(;t)2516 3415 y FL(1)2548 3411 y FE(\))2582 3446 y FH(e)2618 3405 y Fx(\000)p FG(iH)2738 3413 y Fq(\033)2772 3424 y(t)2795 3436 y FL(1)2836 3405 y FG(t)2860 3415 y FL(1)2896 3446 y FH( )2949 3407 y FF(\()p FG(t)2997 3417 y FL(1)3029 3407 y FF(\))2946 3465 y FG(j;\033)3026 3473 y Fq(t)3049 3485 y FL(1)3107 3446 y FH(:)542 3686 y FO(Our)27 b(\034nal)h(goal)e(is)h(to)h(obtain)f (the)h(follo)n(wing)e(estimate)940 3897 y Fz(k)p FE(\001)1042 3905 y FG(t)1066 3915 y FL(2)1099 3905 y FG(;t)1142 3915 y FL(1)1178 3897 y FH( )1228 3906 y FG(h;\024)1319 3916 y FL(1)1356 3897 y Fz(k)21 b FE(=)g Fz(k)q FH( )1585 3906 y FG(h;\024)1676 3916 y FL(1)1725 3897 y FE(\()p FH(t)1783 3905 y FF(2)1818 3897 y FE(\))c Fz(\000)f FH( )1991 3906 y FG(h;\024)2082 3916 y FL(1)2132 3897 y FE(\()p FH(t)2190 3905 y FF(1)2224 3897 y FE(\))p Fz(k)22 b(\024)f FH(C)h Fz(\001)2520 3849 y(j)q FE(ln)13 b FH(t)2647 3857 y FF(2)2681 3849 y Fz(j)2702 3813 y FF(2)p 2520 3880 217 4 v 2582 3957 a FH(t)2610 3922 y FF(2)p FG(\032)2610 3976 y FF(1)2824 3897 y FH(\032)20 b(>)h FE(0)d FH(:)268 b FI(\(3.3\))515 4112 y FO(This)34 b(estimate)f(is)h(su\036cien)n(t)g (to)g(pro)n(v)n(e)e(the)i(strong)f(Cauc)n(h)n(y)g(prop)r(ert)n(y)f(of)i FH( )3042 4121 y FG(h;\024)3133 4131 y FL(1)3183 4112 y FE(\()p FH(t)p FE(\))f FO(b)n(y)h(a)515 4212 y(telescopic)27 b(argumen)n(t)f(\(Theorem)h(3.1\).)515 4444 y FC(3.1)112 b(Outline)36 b(of)i(the)f(pro)s(of)515 4589 y FO(Due)g(to)g(the)h (constructiv)n(e)e(recip)r(e,)h(the)h(time)f(v)-5 b(ariation,)36 b FH(t)2473 4597 y FF(2)2544 4589 y Fz(!)g FH(t)2685 4597 y FF(1)2719 4589 y FO(,)h(yields)g(man)n(y)g(mo)r(di-)515 4680 y(\034cations)e(in)h(the)h(v)n(ector)d FH( )1415 4689 y FG(h;k)1503 4699 y FL(1)1553 4680 y FE(\()p FH(t)p FE(\))o FO(.)j(In)f(addition)g(to)g(the)g(time)g(ev)n(olution,)g(the)g (partition)515 4772 y FE(\000)563 4740 y FF(\()p FG(t)p FF(\))673 4772 y FO(and)d(the)i(infrared)e(cut-o\033)h FH(\033)1619 4780 y FG(t)1680 4772 y FO(are)f(time)i(dep)r(enden)n(t.)f (Before)f(going)g(in)n(to)g(details,)h(it)1950 5059 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 515 523 a FO(is)31 b(w)n(orth)n(while)g(to)g (explain)g(the)h(general)e(mec)n(hanisms)h(whic)n(h)g(prev)n(en)n(t)g (the)h(rising)e(of)i(not)515 614 y(con)n(v)n(ergen)n(t)25 b(terms)i(and)h(then)g(imply)g(the)f(Cauc)n(h)n(y)g(prop)r(ert)n(y)-7 b(.)515 706 y(The)31 b(increase)e(in)j(the)f(n)n(um)n(b)r(er)g(of)g (cells)f(is)h(a)g(p)r(oten)n(tial)g(source)f(of)h(problems)f(in)h(the)g (con-)515 797 y(trol)c(of)g(the)i(di\033erence)e(\(3.3\).)h(Concerning) e(the)i(norm)f(of)h(the)g(piece)f(of)h(v)n(ector)e FE(\001)3154 805 y FG(t)3178 815 y FL(2)3210 805 y FG(;t)3253 815 y FL(1)3290 797 y FH( )3340 806 y FG(h;\024)3431 816 y FL(1)515 888 y FO(corresp)r(onding)f(to)h(eac)n(h)g(cell)h(in)g FE(\000)1625 857 y FF(\()p FG(t)1673 867 y FL(2)1705 857 y FF(\))1733 888 y FO(,)g(if)h(w)n(e)e(only)g(used)h(the)g(b)r (ound)h(coming)e(from)g(the)h(re-)515 980 y(striction)e(of)i(the)f (supp)r(ort)g(in)g(the)h FA(P)p Fz(\000)p FO(v)-5 b(ariable,)25 b(the)i(estimate)f(for)f(the)i(norm)e(of)i(the)f(en)n(tire)515 1086 y(v)n(ector)k(w)n(ould)h(div)n(erge)f(lik)n(e)h FH(N)21 b FE(\()p FH(t)1600 1094 y FF(2)1634 1086 y FE(\))1674 1031 y FL(1)p 1674 1039 29 3 v 1674 1072 a(2)1717 1086 y FO(.)31 b(On)g(the)h(other)f(hand,)h(the)f(\(non-relativistic\))g (clus-)515 1177 y(ter)25 b(prop)r(ert)n(y)g(of)h(the)h(system)e (implies)h(that)h(the)f(comp)r(onen)n(ts)f(with)i(di\033eren)n(t)f (\(electronic\))515 1269 y(\020v)n(elo)r(cities\021)42 b(in)37 b(the)g(cell-partition)f(are)f(asymptotically)h(orthogonal)e (as)i(v)n(ectors)f(in)i(the)515 1360 y(Hilb)r(ert)28 b(space.)g(In)g(order)f(to)h(exploit)g(this)g(mec)n(hanism)g(in)g (Theorem)g(2.1,)f(the)i(rate)e(of)h(the)515 1451 y(partition)f(is)g(c)n (hosen)g(slo)n(w)n(er)f(than)h(the)h(decoupling)f(rate)g(\(constrain)n (t)g(\(2.26\)\).)515 1543 y(Besides)k(the)h(increase)f(of)h(n)n(um)n(b) r(er)g(of)g(cells)f(in)i(time,)f(w)n(e)g(m)n(ust)g(handle)g(t)n(w)n(o)f (delicate)h(as-)515 1634 y(p)r(ects)c(concerning)e(the)i(con)n(v)n (ergence)d(in)j(eac)n(h)e(single)h(cell,)h(namely:)515 1725 y(A)23 b(correction)d(to)j(the)f(asymptotic)g(dynamics,)g(b)n(y)g (means)g(of)h(the)g(phase)f(factor,)f(is)i(required)515 1817 y(in)28 b(the)g(application)e(of)i(Co)r(ok's)e(argumen)n(t;)515 1908 y(The)e(regularit)n(y)e(prop)r(erties)h(concerning)g(the)h(v)n (ector)f FH(\036)2287 1874 y FG(\033)2323 1882 y Fq(t)2287 1929 y Fw(P)2378 1908 y FO(\(Subsection)h(1.3.1\))f(whic)n(h)h(come)515 1999 y(in)n(to)31 b(game)f(and)h(m)n(ust)h(b)r(e)g(exploited)f(in)g (order)f(to)h(c)n(hec)n(k)g(that)g(the)h(dressing)e(cloud)h(com-)515 2090 y(bined)j(with)h(the)f(one)g(particle)g(state)g(giv)n(es)f(rise)g (to)h(a)g(w)n(ell)g(de\034ned)g(v)n(ector)f(in)h(the)h(limit)515 2182 y FH(\033)559 2190 y FG(t)608 2182 y Fz(!)21 b FE(0)p FJ(.)515 2364 y(V)-6 b(ariation)30 b(of)h(the)f(p)l(artition)515 2547 y FO(As)g(preliminary)f(step)h(in)g(the)h(analysis)d(of)i FE(\001)1986 2555 y FG(t)2010 2565 y FL(2)2042 2555 y FG(;t)2085 2565 y FL(1)2122 2547 y FH( )2172 2556 y FG(h;\024)2263 2566 y FL(1)2299 2547 y FO(,)h(w)n(e)e(con)n(trol)g(the)h(v)-5 b(ariation)29 b(of)h(the)515 2638 y(appro)n(ximating)h(v)n(ector)h (when)i(the)g(cell)f(partition)g(c)n(hanges)f(from)h FE(\000)2769 2607 y FF(\()p FG(t)2817 2617 y FL(2)2849 2607 y FF(\))2911 2638 y FO(to)g FE(\000)3066 2607 y FF(\()p FG(t)3114 2617 y FL(1)3146 2607 y FF(\))3174 2638 y FO(,)h(all)f(the)515 2730 y(other)23 b(v)-5 b(ariables)23 b(remaining)h(\034xed)g(at)g(time)h FH(t)1963 2738 y FF(2)1997 2730 y FO(.)g(This)f(means)g(that)g(w)n(e)g(p)r(erform)g(the) h(follo)n(w-)515 2821 y(ing)i(replacemen)n(ts)773 2984 y FH(W)846 2992 y FG(\033)882 3000 y Fq(t)905 3012 y FL(2)958 2924 y Fy(\000)994 2984 y FA(v)1041 2996 y FG(l)p FF(\()o FG(j)s FF(\))1141 2984 y FH(;)13 b(t)1203 2992 y FF(2)1237 2924 y Fy(\001)1448 2984 y Fz(!)182 b FH(W)1780 2992 y FG(\033)1816 3000 y Fq(t)1839 3012 y FL(2)1893 2984 y FE(\()o FA(v)1969 2992 y FG(j)2002 2984 y FH(;)13 b(t)2064 2992 y FF(2)2098 2984 y FE(\))182 b FA(v)2357 2996 y FG(l)p FF(\()p FG(j)s FF(\))2479 2984 y Fz(\021)21 b(r)p FH(E)2685 2938 y FG(\033)2721 2946 y Fq(t)2744 2958 y FL(2)p 2681 2976 147 3 v 2681 3017 a Fw(P)2728 3031 y Fq(l)p FL(\()p Fq(j)r FL(\))2844 2984 y FH(;)30 b FA(v)2942 2992 y FG(j)2996 2984 y Fz(\021)21 b(r)p FH(E)3202 2938 y FG(\033)3238 2946 y Fq(t)3261 2958 y FL(1)p 3198 2976 82 3 v 3198 3017 a Fw(P)3245 3028 y Fq(j)681 3141 y FH(e)717 3100 y FG(i\015)771 3108 y Fq(\033)805 3119 y(t)828 3131 y FL(2)869 3045 y Fm(\020)903 3100 y Fw(v)939 3114 y Fq(l)p FL(\()q Fq(j)r FL(\))1036 3100 y FG(;)p Fx(r)p FG(E)1154 3059 y Fq(\033)1188 3070 y(t)1211 3082 y FL(2)1151 3121 y Fl(P)1251 3100 y FG(;t)1294 3110 y FL(2)1326 3045 y Fm(\021)1448 3141 y Fz(!)83 b FH(e)1644 3100 y FG(i\015)1698 3108 y Fq(\033)1732 3119 y(t)1755 3131 y FL(2)1796 3045 y Fm(\020)1831 3100 y Fw(v)1867 3111 y Fq(j)1898 3100 y FG(;)p Fx(r)p FG(E)2016 3059 y Fq(\033)2050 3070 y(t)2073 3082 y FL(2)2013 3121 y Fl(P)2114 3100 y FG(;t)2157 3110 y FL(2)2189 3045 y Fm(\021)890 3244 y FH( )943 3206 y FF(\()p FG(t)991 3216 y FL(2)1023 3206 y FF(\))940 3268 y FG(l)p FF(\()p FG(j)s FF(\))q FG(;\033)1092 3276 y Fq(t)1115 3288 y FL(2)1448 3244 y Fz(!)296 b FH( )1874 3206 y FF(\()o FG(t)1921 3216 y FL(1)1954 3206 y FF(\))1871 3264 y FG(j;\033)1951 3272 y Fq(t)1974 3284 y FL(2)515 3425 y FO(so)27 b(that)g(the)h(corresp)r (onding)e(mo)r(di\034cation)h(of)h(the)g(appro)n(ximating)d(v)n(ector)h (is)515 3516 y FJ(D0\))751 3761 y FH(e)787 3725 y FG(iH)t(t)887 3735 y FL(2)935 3661 y FG(N)5 b FF(\()q FG(t)1037 3671 y FL(1)1069 3661 y FF(\))959 3690 y Fy(X)961 3847 y FG(j)s FF(=1)1106 3690 y Fy(X)1113 3852 y FG(l)p FF(\()p FG(j)s FF(\))1230 3761 y Fz(W)1306 3769 y FG(\033)1342 3777 y Fq(t)1365 3789 y FL(2)1418 3700 y Fy(\000)1453 3761 y FA(v)1500 3773 y FG(l)p FF(\()p FG(j)s FF(\))1600 3761 y FH(;)14 b(t)1663 3769 y FF(2)1697 3700 y Fy(\001)1745 3761 y FH(e)1781 3720 y FG(i\015)1835 3728 y Fq(\033)1869 3739 y(t)1892 3751 y FL(2)1933 3665 y Fm(\020)1968 3720 y Fw(v)2004 3734 y Fq(l)p FL(\()q Fq(j)r FL(\))2100 3720 y FG(;)p Fx(r)p FG(E)2218 3679 y Fq(\033)2252 3690 y(t)2275 3702 y FL(2)2215 3741 y Fl(P)2316 3720 y FG(;t)2359 3730 y FL(2)2391 3665 y Fm(\021)2429 3761 y FH(e)2465 3721 y Fx(\000)p FG(iH)2585 3729 y Fq(\033)2619 3740 y(t)2642 3752 y FL(2)2683 3721 y FG(t)2707 3731 y FL(2)2743 3761 y FH( )2796 3722 y FF(\()p FG(t)2844 3732 y FL(2)2876 3722 y FF(\))2793 3785 y FG(l)p FF(\()p FG(j)s FF(\))q FG(;\033)2945 3793 y Fq(t)2968 3805 y FL(2)3030 3761 y Fz(!)688 4050 y(!)98 b FH(e)899 4014 y FG(iH)t(t)999 4024 y FL(2)1048 3951 y FG(N)5 b FF(\()p FG(t)1149 3961 y FL(1)1181 3951 y FF(\))1071 3979 y Fy(X)1074 4137 y FG(j)s FF(=1)1218 4050 y Fz(W)1294 4058 y FG(\033)1330 4066 y Fq(t)1353 4078 y FL(2)1407 4050 y FE(\()p FA(v)1484 4058 y FG(j)1516 4050 y FH(;)13 b(t)1578 4058 y FF(2)1612 4050 y FE(\))g FH(e)1691 4010 y FG(i\015)1745 4018 y Fq(\033)1779 4029 y(t)1802 4041 y FL(2)1843 3955 y Fm(\020)1878 4010 y Fw(v)1914 4021 y Fq(j)1945 4010 y FG(;)p Fx(r)p FG(E)2063 3969 y Fq(\033)2097 3980 y(t)2120 3992 y FL(2)2060 4031 y Fl(P)2161 4010 y FG(;t)2204 4020 y FL(2)2236 3955 y Fm(\021)2275 4050 y FH(e)2311 4010 y Fx(\000)p FG(iH)2431 4018 y Fq(\033)2465 4029 y(t)2488 4041 y FL(2)2528 4010 y FG(t)2552 4020 y FL(2)2588 4050 y FH( )2641 4012 y FF(\()p FG(t)2689 4022 y FL(1)2721 4012 y FF(\))2638 4070 y FG(j;\033)2718 4078 y Fq(t)2741 4090 y FL(2)2799 4050 y FH(:)515 4416 y FO(Once)27 b(the)h(partition)f(is)g FE(\000)1347 4385 y FF(\()p FG(t)1395 4395 y FL(1)1427 4385 y FF(\))1456 4416 y FO(,)g(w)n(e)h(can)f(study)g(the)h(v)-5 b(ariation)27 b(of)g(the)h(v)n(ector)980 4656 y FH( )1033 4618 y FF(\()p FG(t)1081 4628 y FL(1)1113 4618 y FF(\))1030 4678 y FG(h;\024)1121 4688 y FL(1)1171 4656 y FE(\()p FH(t)p FE(\))21 b(:=)1382 4557 y FG(N)5 b FF(\()q FG(t)1484 4567 y FL(1)1516 4557 y FF(\))1405 4585 y Fy(X)1447 4743 y FG(j)1553 4656 y FH(e)1589 4620 y FG(iH)t(t)1692 4656 y Fz(W)1768 4664 y FG(\033)1804 4672 y Fq(t)1848 4656 y FE(\()p FA(v)1925 4664 y FG(j)1957 4656 y FH(;)13 b(t)p FE(\))g FH(e)2098 4620 y FG(i\015)2152 4628 y Fq(\033)2186 4639 y(t)2218 4626 y FE(\()2247 4620 y Fw(v)2283 4631 y Fq(j)2315 4620 y FG(;)p Fx(r)p FG(E)2433 4592 y Fq(\033)2467 4603 y(t)2430 4641 y Fl(P)2498 4620 y FG(;t)2541 4626 y FE(\))2575 4656 y FH(e)2611 4620 y Fx(\000)p FG(iE)2729 4592 y Fq(\033)2763 4603 y(t)2726 4641 y Fl(P)2794 4620 y Fx(\001)p FG(t)2840 4656 y FH( )2893 4618 y FF(\()p FG(t)2941 4628 y FL(1)2973 4618 y FF(\))2890 4676 y FG(j;\033)2970 4684 y Fq(t)3310 4656 y FI(\(3.4\))1950 5059 y FO(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 515 523 a FO(b)r(et)n(w)n(een)27 b FH(t)21 b FE(=)g FH(t)993 531 y FF(2)1055 523 y FO(and)28 b FH(t)21 b FE(=)g FH(t)1375 531 y FF(1)1409 523 y FO(.)28 b(The)f(initial)h(constructiv)n(e)e(h)n(yp)r(otheses)h(are)g(not)g (su\036cien)n(t)h(to)515 614 y(p)r(erform)f(the)h(time)g(deriv)-5 b(ativ)n(e:)1725 691 y FH(d)p 1707 722 76 4 v 1707 790 a(ds)1793 739 y( )1846 701 y FF(\()p FG(t)1894 711 y FL(1)1926 701 y FF(\))1843 760 y FG(h;\024)1934 770 y FL(1)1983 739 y FE(\()p FH(s)p FE(\))21 b Fz(j)2121 747 y FG(s)p FF(=)p FG(t)2264 739 y FH(:)515 881 y FO(In)27 b(fact)h(the)g(\(strong\))f(con)n(v)n(ergence)e(pro)n(v)n(ed)h(in)i ([Pi.])g(for)f(the)h(v)n(ector)1615 1018 y FH(\036)1661 982 y FG(\033)1697 990 y Fq(t)1661 1037 y Fw(P)1749 1018 y FE(=)21 b FH(W)1903 1026 y FG(\033)1939 1034 y Fq(t)1983 1018 y FE(\()p Fz(r)p FH(E)2138 982 y FG(\033)2174 990 y Fq(t)2134 1037 y Fw(P)2205 1018 y FE(\))12 b FH( )2300 982 y FG(\033)2336 990 y Fq(t)2297 1037 y Fw(P)515 1155 y FO(do)r(es)27 b(not)h(imply)f(that)h FH(\036)1308 1121 y FG(\033)1344 1129 y Fq(t)1308 1176 y Fw(P)1403 1155 y FO(is)f(\(strongly\))g(Lipsc)n(hitz)h(in)f FH(\033)2367 1163 y FG(t)2423 1155 y FO(in)h(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)i FH(\033)3249 1163 y FG(t)3298 1155 y FE(=)21 b(0)p FO(.)515 1247 y(Assuming)34 b(the)h(Hyp)r(othesis)g(H0)f(and)g(exploiting)g (only)h(the)g(H\366lder)f(prop)r(ert)n(y)f(of)i FH(\036)3297 1212 y FG(\033)3333 1220 y Fq(t)3297 1268 y Fw(P)3398 1247 y FO(in)515 1338 y(neigh)n(b)r(orho)r(o)r(ds)30 b(of)h FH(\033)1215 1346 y FG(t)1271 1338 y FE(=)c(0)32 b FO(and)g FA(P)c FE(=)f(0)32 b FO(\(see)g Fp(Sp)-5 b(e)g(ctr)g(al)39 b(pr)-5 b(op)g(erties)7 b FO(,)31 b(Subsection)h(1.3.1\))515 1442 y(w)n(e)e(p)r(erform)g(some)h(in)n(termediate)f(steps)h(from)f FH( )2118 1404 y FF(\()p FG(t)2166 1414 y FL(1)2198 1404 y FF(\))2115 1464 y FG(h;\024)2206 1474 y FL(1)2256 1442 y FE(\()p FH(t)2314 1450 y FF(2)2348 1442 y FE(\))h FO(to)f FH( )2566 1404 y FF(\()p FG(t)2614 1414 y FL(1)2646 1404 y FF(\))2563 1464 y FG(h;\024)2654 1474 y FL(1)2704 1442 y FE(\()p FH(t)2762 1450 y FF(1)2796 1442 y FE(\))h FO(corresp)r (onding)e(to)515 1534 y(\034nite)k(di\033erences)g(and)f(w)n(e)h(study) g(the)g(norm)f(of)h(eac)n(h)f(con)n(tribution.)g(In)h(the)h(next)f (lines,)515 1625 y(w)n(e)23 b(carefully)f(single)h(out)g(the)h(in)n (termediate)f(v)-5 b(ariation)22 b FJ(D1\))p FO(,)i(in)n(v)n(olving)d (Co)r(ok's)i(argumen)n(t,)515 1716 y(and)31 b(then)h(the)f(v)-5 b(ariations)30 b FJ(D2)11 b Fp(\))s FO(,)32 b FJ(D3.1\))p FO(,)g FJ(D3.2\))j FO(and)c FJ(D3.3)11 b FO(\),)33 b(related)e(to)g (the)h(remo)n(v)-5 b(al)30 b(of)515 1808 y(infrared)d(cut-o\033.)h (Indeed)g(the)g(order)e(in)j(the)f(subsequen)n(t)f(mo)r(di\034cations)g (is)h(imp)r(ortan)n(t)g(to)515 1899 y(get)f(the)h(desired)f(estimate.) 515 2082 y FJ(Co)l(ok's)k(ar)l(gument)515 2264 y FO(Th)p FP(e)39 b FO(bac)n(kw)n(ards)d(time)k(ev)n(olution,)e(at)h(\034xed)g (cut-o\033)g FH(\033)2355 2272 y FG(t)2379 2282 y FL(2)2415 2264 y FO(,)g(corresp)r(onds)e(to)i(the)h(mo)r(di\034-)515 2356 y(cation)515 2447 y FJ(D1\))807 2567 y FG(N)5 b FF(\()p FG(t)908 2577 y FL(1)940 2567 y FF(\))830 2596 y Fy(X)833 2753 y FG(j)s FF(=1)977 2667 y FH(e)1013 2631 y FG(iH)t(t)1113 2641 y FL(2)1175 2667 y Fz(W)1251 2675 y FG(\033)1287 2683 y Fq(t)1310 2695 y FL(2)1363 2667 y FE(\()p FA(v)1440 2675 y FG(j)1472 2667 y FH(;)13 b(t)1534 2675 y FF(2)1569 2667 y FE(\))f FH(e)1647 2626 y FG(i\015)1701 2634 y Fq(\033)1735 2645 y(t)1758 2657 y FL(2)1800 2571 y Fm(\020)1834 2626 y Fw(v)1870 2637 y Fq(j)1902 2626 y FG(;)p Fx(r)p FG(E)2020 2585 y Fq(\033)2054 2596 y(t)2077 2608 y FL(2)2017 2647 y Fl(P)2117 2626 y FG(;t)2160 2636 y FL(2)2192 2571 y Fm(\021)2231 2667 y FH(e)2267 2631 y Fx(\000)p FG(iE)2385 2590 y Fq(\033)2419 2601 y(t)2442 2613 y FL(2)2382 2652 y Fl(P)2482 2631 y FG(t)2506 2641 y FL(2)2542 2667 y FH( )2595 2628 y FF(\()p FG(t)2643 2638 y FL(1)2675 2628 y FF(\))2592 2687 y FG(j;\033)2672 2695 y Fq(t)2695 2707 y FL(2)2911 2667 y Fz(!)854 2949 y(!)1105 2850 y FG(N)5 b FF(\()q FG(t)1207 2860 y FL(1)1239 2850 y FF(\))1129 2878 y Fy(X)1131 3036 y FG(j)s FF(=1)1276 2949 y FH(e)1312 2913 y FG(iH)t(t)1412 2923 y FL(1)1448 2949 y Fz(W)1524 2957 y FG(\033)1560 2965 y Fq(t)1583 2977 y FL(2)1636 2949 y FE(\()p FA(v)1713 2957 y FG(j)1745 2949 y FH(;)13 b(t)1807 2957 y FF(1)1842 2949 y FE(\))f FH(e)1920 2909 y FG(i\015)1974 2917 y Fq(\033)2008 2928 y(t)2031 2940 y FL(2)2073 2854 y Fm(\020)2107 2909 y Fw(v)2143 2920 y Fq(j)2175 2909 y FG(;)p Fx(r)p FG(E)2293 2868 y Fq(\033)2327 2879 y(t)2350 2891 y FL(2)2290 2930 y Fl(P)2390 2909 y FG(;t)2433 2919 y FL(1)2465 2854 y Fm(\021)2504 2949 y FH(e)2540 2913 y Fx(\000)p FG(iE)2658 2873 y Fq(\033)2692 2884 y(t)2715 2896 y FL(2)2655 2934 y Fl(P)2755 2913 y FG(t)2779 2923 y FL(1)2815 2949 y FH( )2868 2911 y FF(\()p FG(t)2916 2921 y FL(1)2948 2911 y FF(\))2865 2969 y FG(j;\033)2945 2977 y Fq(t)2968 2989 y FL(2)3310 2949 y FI(\(3.5\))515 3165 y FO(and)31 b(the)g(study)g(of)g (the)g(di\033erence)g(consists)f(in)h(a)g(standard)f(Co)r(ok's)g (argumen)n(t)g(with)h(the)515 3256 y(subtraction)26 b(of)i(the)g (infrared)f(tail)g(as)g(in)h(Theorem)f(2.1.)515 3439 y FJ(V)-6 b(ariation)30 b(of)h(the)f(infr)l(ar)l(e)l(d)g(cut-o\033)515 3622 y FO(Under)37 b(the)g(v)-5 b(ariation)36 b(of)h(the)g(infrared)f (cut-o\033,)h FH(\033)2200 3630 y FG(t)2224 3640 y FL(2)2296 3622 y Fz(!)f FH(\033)2453 3630 y FG(t)2477 3640 y FL(1)2513 3622 y Fi(,)g FO(the)i(v)n(ector)d(\(3.5\))i(c)n(hanges)515 3713 y(as)27 b(follo)n(ws)824 3813 y FG(N)5 b FF(\()p FG(t)925 3823 y FL(1)957 3813 y FF(\))847 3841 y Fy(X)850 3998 y FG(j)s FF(=1)994 3912 y FH(e)1030 3876 y FG(iH)t(t)1130 3886 y FL(1)1166 3912 y Fz(W)1242 3920 y FG(\033)1278 3928 y Fq(t)1301 3940 y FL(2)1354 3912 y FE(\()p FA(v)1431 3920 y FG(j)1464 3912 y 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4830 y FG(j;\033)2653 4838 y Fq(t)2676 4850 y FL(2)3310 4810 y FI(\(3.8\))1950 5059 y FO(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 515 610 a FO(where)27 b(w)n(e)g(recall)g(that)g Fr(W)1345 618 y FG(\033)1381 626 y Fq(t)1404 638 y FL(2)1456 537 y Fs(\020)1499 610 y Fu(r)p Fr(E)1614 566 y FG(\033)1650 574 y Fq(t)1673 586 y FL(2)1610 633 y Fw(P)1713 537 y Fs(\021)1774 610 y Ft(=)20 b Fr(e)1882 512 y Fx(\000)p FG(g)1972 472 y Fm(R)2011 491 y Fq(\024)2000 531 y(\033)2034 542 y(t)2057 554 y FL(2)2204 486 y Fq(b)p FL(\()q Fl(k)p FL(\))p Fk(\000)p Fq(b)2379 469 y Fk(y)2411 486 y FL(\()q Fl(k)p FL(\))p 2118 501 457 3 v 2118 563 a Fk(j)p Fl(k)p Fk(j)2190 505 y Fj(\022)2225 563 y FL(1)p Fk(\000)2301 552 y Fj(b)2298 563 y Fl(k)p Fk(\001r)p Fq(E)2442 525 y(\033)2476 536 y(t)2499 548 y FL(2)2440 586 y Fl(P)2540 505 y Fj(\023)2622 493 y Fq(d)2653 476 y FL(3)2685 493 y Fq(k)p 2595 501 150 3 v 2595 508 a Fx(p)p 2645 508 100 3 v 36 x FL(2)p Fk(j)p Fl(k)p Fk(j)2758 610 y FO(.)515 724 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FL(1)1165 4672 y FE(\()p FA(v)1242 4680 y FG(j)1274 4672 y FH(;)g(t)1336 4680 y FF(1)1370 4672 y FE(\))g FH(W)1497 4637 y Fx(y)1486 4686 y FG(\033)1522 4694 y Fq(t)1545 4706 y FL(1)1598 4672 y FE(\()p FA(v)1675 4680 y FG(j)1707 4672 y FE(\))g FH(W)1823 4680 y FG(\033)1859 4688 y Fq(t)1882 4700 y FL(2)1935 4672 y FE(\()p FA(v)2012 4680 y FG(j)2044 4672 y FE(\))g FH(W)2171 4637 y Fx(y)2160 4686 y FG(\033)2196 4694 y Fq(t)2219 4706 y FL(2)2272 4612 y Fy(\000)2307 4672 y Fz(r)p FH(E)2432 4626 y FG(\033)2468 4634 y Fq(t)2491 4646 y FL(1)2428 4693 y Fw(P)2532 4612 y Fy(\001)2742 4672 y Fz(!)998 4810 y(!)174 b(W)1325 4818 y FG(\033)1361 4826 y Fq(t)1384 4838 y FL(1)1438 4810 y FE(\()p FA(v)1515 4818 y FG(j)1547 4810 y FH(;)13 b(t)1609 4818 y FF(1)1643 4810 y FE(\))g FH(W)1770 4774 y Fx(y)1759 4823 y FG(\033)1795 4831 y Fq(t)1818 4843 y FL(1)1871 4810 y FE(\()p FA(v)1948 4818 y FG(j)1980 4810 y FE(\))g FH(W)2096 4818 y FG(\033)2132 4826 y Fq(t)2155 4838 y FL(1)2208 4810 y FE(\()p FA(v)2285 4818 y FG(j)2317 4810 y FE(\))g FH(W)2444 4774 y Fx(y)2433 4823 y FG(\033)2469 4831 y Fq(t)2492 4843 y FL(1)2545 4750 y Fy(\000)2580 4810 y Fz(r)p FH(E)2705 4764 y FG(\033)2741 4772 y Fq(t)2764 4784 y FL(1)2701 4831 y Fw(P)2805 4750 y Fy(\001)1950 5059 y FO(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 515 523 a FO(the)36 b(last)g(t)n(w)n(o)f(steps) h(accoun)n(t)f(resp)r(ectiv)n(ely)g(for)h(the)g(di\033erence)g(b)r(et)n (w)n(een)g(the)g(gradien)n(ts)515 614 y Fz(r)p FH(E)640 583 y FG(\033)676 591 y Fq(t)699 603 y FL(2)752 614 y FE(\()p FA(P)p FE(\))p FI(,)p Fz(r)p FH(E)1018 583 y FG(\033)1054 591 y Fq(t)1077 603 y FL(1)1131 614 y FE(\()p FA(P)p FE(\))28 b FO(and)g(for)g(the)h(shift)g FH(\033)1945 622 y FG(t)1969 632 y FL(2)2027 614 y Fz(!)22 b FH(\033)2170 622 y FG(t)2194 632 y FL(1)2259 614 y FO(in)28 b(the)h(t)n(w)n(o)f(W)-7 b(eyl)28 b(op)r(erators)e(on)i(the)515 706 y(v)n(ery)e(righ)n(t.)515 888 y(The)39 b(analysis)f(of)i(eac)n(h)e(di\033erence)i FJ(D0\))p FO(,)g FJ(D1\))p FO(,)f FJ(D2\))p FO(,)h FJ(D3.1\))p FO(,)g FJ(D3.2\))k FO(and)39 b FJ(D3.3\))44 b FO(is)39 b(the)515 980 y(con)n(ten)n(t)25 b(of)g(the)h(remaining)f(part)g(of)g (the)h(Section.)g(The)f(discussion)g(is)g(carried)f(out)i(in)f(Sub-)515 1071 y(sections)20 b(3.1.1,)f(3.1.2,)h(3.1.3)f(and)i(3.1.4,)e(where)h (w)n(e)h(describ)r(e)f(the)h(ph)n(ysical)f(ingredien)n(ts)g(and)515 1162 y(the)31 b(tec)n(hnical)f(steps)g(used)h(to)f(con)n(trol)f(them.)i (It)g(is)f(rather)g(detailed)g(and)h(complete)f(with)515 1254 y(the)25 b(use)g(of)g(some)g(results)f(pro)n(v)n(ed)g(in)h(the)h (App)r(endix.)g(Though)f(it)g(is)g(not)g(alw)n(a)n(ys)e(explicitly)515 1345 y(written,)33 b(w)n(e)g(assume)f(that)h FH(\013)p FE(\()p FH(<)d FE(1\))j FO(is)g(su\036cien)n(tly)g(close)f(to)h FK(1)f FO(and)h(that)g(the)h(constrain)n(t)515 1436 y(\(2.26\))27 b(is)g(satis\034ed.)515 1528 y(W)-7 b(e)28 b(here)f(an)n(ticipate)g (the)h(result:)515 1619 y(Let)j(us)f(assume)g FH(\014)h(>)26 b FE(1)31 b FO(and)g(large)e(enough)h(\(whic)n(h)h(means)f(that)h(the)g (remo)n(v)-5 b(al)30 b(of)h(the)g(in-)515 1710 y(frared)25 b(cut-o\033)h FH(\033)1059 1718 y FG(t)1108 1710 y FE(=)21 b FH(t)1217 1679 y Fx(\000)p FG(\014)1333 1710 y FO(is)26 b(su\036cien)n(tly)g(fast)h(in)f(time\).)h(Then)g(the)f(b)r(ounds)h (\(3.23\),)e(\(3.32\),)515 1802 y(\(3.38\),)32 b(\(3.47\),)h(\(3.49\))f (and)i(\(3.54\))e(-)h(whic)n(h)g(are)g(obtained)g(resp)r(ectiv)n(ely)f (for)h(the)g(norms)515 1893 y(of)h(the)g(v)n(ectors)e(corresp)r(onding) g(to)i(the)g(v)-5 b(ariations)33 b FJ(D0\))p FO(,)h FJ(D1\))p FO(,)g FJ(D2\))p FO(,)g FJ(D3.1\))p FO(,)h FJ(D3.2\))j FO(and)515 1984 y FJ(D3.3\))31 b FO(-)d(are)e(suc)n(h)h(that)h(a)f (leading)g(order)g(term)g(is)1782 2099 y Fy(\022)1849 2155 y FE(ln)13 b(\()o FH(t)1983 2163 y FF(2)2018 2155 y FE(\))p 1849 2187 199 4 v 1916 2254 a FH(t)1944 2219 y FG(\032)1944 2273 y FF(1)2058 2099 y Fy(\023)2114 2116 y FF(2)2179 2204 y FH(:)515 2411 y FO(with)28 b FH(\032)21 b(>)g FE(0)p FO(.)28 b(W)-7 b(e)28 b(can)f(no)n(w)g(state)g(the)h(main) g(theorem)f(of)g(the)h(pap)r(er.)515 2593 y FP(Theorem)i(3.1)515 2685 y FO(F)-7 b(or)38 b FH(\014)k(>)c FE(1)h FO(large)e(enough)h(and)g FH(\013)h FO(su\036cien)n(tly)f(close)g(to)g FE(1)p FO(,)h(the)g(v)n (ector)e FH( )3018 2694 y FG(h;\024)3109 2704 y FL(1)3159 2685 y FE(\()p FH(t)p FE(\))o FI(,)i FO(with)515 2716 y Fy(R)579 2776 y Fz(j)p FH(h)13 b FE(\()p FA(P)p FE(\))p Fz(j)799 2741 y FF(2)846 2776 y FH(d)886 2744 y FF(3)920 2776 y FH(P)34 b(>)23 b FE(0)p FI(,)29 b FO(con)n(v)n(erges)d(strongly) i(for)g FH(t)23 b Fz(!)g FE(+)p Fz(1)28 b FO(to)h(a)f(non)g(zero)g(v)n (ector)f FH( )3129 2744 y FG(out)3126 2792 y(h;\024)3217 2802 y FL(1)3254 2776 y FO(,)i(with)515 2867 y(an)e(error)f(of)h(order) 1169 2837 y FF(1)p 1155 2851 59 4 v 1155 2893 a FG(t)1179 2879 y Fq(\032)1251 2867 y FO(at)g(most,)h(where)f FH(\032)21 b(>)g FE(0)28 b FO(is)f(a)g(prop)r(er)g(small)g(co)r(e\036cien)n(t.)515 3050 y FJ(Pr)l(o)l(of)515 3141 y FO(Starting)c(from)g(Theorem)f(2.1,)h (the)h(time)g(scale)e(related)h(to)g(the)h(partition)f(is)g(tuned)h (accord-)515 3233 y(ing)31 b(to)f(the)i(constrain)p FI(t)27 b FE(4)p FH(\017)g(<)g(\021)s FI(.)k FO(Therefore,)f(for)h FH(\014)g(>)26 b FE(1)31 b FO(large)f(enough)g(and)h FH(\013)h FO(su\036cien)n(tly)515 3324 y(close)26 b(to)i FE(1)p FO(,)g(w)n(e)f(can)g(estimate)1301 3539 y Fz(k)p FH( )1389 3548 y FG(h;\024)1480 3558 y FL(1)1530 3539 y FE(\()p FH(t)1588 3547 y FF(2)1622 3539 y FE(\))17 b Fz(\000)g FH( )1796 3548 y FG(h;\024)1887 3558 y FL(1)1937 3539 y FE(\()o FH(t)1994 3547 y FF(1)2029 3539 y FE(\))p Fz(k)k FH(<)g(C)i Fz(\001)2315 3434 y Fy(\022)2381 3491 y FE(ln)13 b(\()p FH(t)2516 3499 y FF(2)2550 3491 y FE(\))p 2381 3522 199 4 v 2449 3590 a FH(t)2477 3554 y FG(\032)2477 3609 y FF(1)2590 3434 y Fy(\023)2647 3451 y FF(2)3310 3539 y FI(\(3.9\))515 3751 y FO(where)27 b FH(\032)21 b(>)g FE(0)28 b FO(e)f FH(C)g(>)21 b FE(0)28 b FO(are)e(indep)r(enden)n (t)j(of)e FH(t)1987 3759 y FF(1)2049 3751 y FO(and)h FH(t)2239 3759 y FF(2)2301 3751 y FO(\()p FH(t)2361 3759 y FF(2)2417 3751 y Fz(\025)21 b FH(t)2526 3759 y FF(1)2581 3751 y FH(>)p 2662 3696 28 4 v 21 w(t)h Fz(\035)f FE(1)p FO(\).)515 3934 y(No)n(w)38 b(let)g(us)h(consider)e(the)i(sequence)1806 3873 y Fy(\010)1850 3934 y FH(t)1878 3942 y FF(1)1913 3934 y FH(;)13 b(t)1975 3902 y FF(2)1975 3947 y(1)2009 3934 y FH(;)g(:::::t)2176 3902 y FG(n)2176 3947 y FF(1)2220 3934 y FH(;)g(::::)2338 3873 y Fy(\011)2423 3934 y FO(and)38 b(assume)g FH(t)2921 3902 y FG(n)2921 3947 y FF(1)3001 3934 y Fz(\024)g FH(t)3127 3942 y FF(2)3199 3934 y FH(<)g(t)3325 3901 y FG(n)p FF(+1)3325 3953 y(1)3444 3934 y FO(.)515 4025 y(Due)28 b(to)f(the)h(norm)f(prop)r(erties,)g(it)h(follo)n(ws)e (that:)701 4187 y Fz(k)q FH( )790 4196 y FG(h;\024)881 4206 y FL(1)930 4187 y FE(\()p FH(t)988 4195 y FF(2)1023 4187 y FE(\))17 b Fz(\000)f FH( )1196 4196 y FG(h;\024)1287 4206 y FL(1)1337 4187 y FE(\()p FH(t)1395 4195 y FF(1)1429 4187 y FE(\))p Fz(k)22 b(\024)851 4419 y(\024)932 4356 y Fy(\015)932 4401 y(\015)974 4419 y FH( )1024 4428 y FG(h;\024)1115 4438 y FL(1)1165 4359 y Fy(\000)1200 4419 y FH(t)1228 4384 y FF(2)1228 4433 y(1)1263 4359 y Fy(\001)1315 4419 y Fz(\000)17 b FH( )1442 4428 y FG(h;\024)1533 4438 y FL(1)1582 4419 y FE(\()p FH(t)1640 4427 y FF(1)1675 4419 y FE(\))1704 4356 y Fy(\015)1704 4401 y(\015)1764 4419 y FE(+)g FH(:::::)i FE(+)d Fz(k)q FH( )2130 4428 y FG(h;\024)2221 4438 y FL(1)2270 4419 y FE(\()p FH(t)2328 4427 y FF(2)2363 4419 y FE(\))h Fz(\000)f FH( )2536 4428 y FG(h;\024)2627 4438 y FL(1)2677 4419 y FE(\()p FH(t)2735 4384 y FG(n)2735 4433 y FF(1)2777 4419 y FE(\))p Fz(k)426 b FI(\(3.10\))851 4711 y Fz(\024)995 4663 y FH(C)p 993 4694 64 4 v 993 4762 a(t)1021 4726 y FG(\032)1021 4781 y FF(1)1084 4711 y Fz(\001)1122 4584 y Fy(\( )1276 4663 y FE(2)p 1255 4694 82 4 v 1255 4793 a FH(t)1293 4721 y Fq(\032)p 1292 4734 31 3 v 1293 4767 a FL(2)1283 4812 y FF(1)1364 4711 y Fz(\001)17 b FE(ln)c(\()p FH(t)1537 4719 y FF(1)1571 4711 y FE(\))1601 4584 y Fy(!)1662 4601 y FF(2)1713 4711 y FE(+)k FH(:::::::)j FE(+)2033 4584 y Fy( )2104 4663 y FH(n)d FE(+)g(1)p 2104 4694 179 4 v 2134 4793 a FH(t)2172 4721 y Fq(n\032)p 2172 4734 67 3 v 2191 4767 a FL(2)2162 4812 y FF(1)2310 4711 y Fz(\001)g FE(ln)c(\()p FH(t)2483 4719 y FF(1)2517 4711 y FE(\))2547 4584 y Fy(!)2608 4601 y FF(2)2642 4584 y Fy(\))3271 4711 y FI(\(3.11\))1950 5059 y FO(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 515 523 a FO(F)-7 b(or)29 b FH(t)694 531 y FF(1)758 523 y FO(su\036cien)n(tly)g(large,)g FH(t)1432 531 y FF(1)1491 523 y Fz(\025)1585 509 y Fy(b)1575 523 y FH(t)1603 531 y FF(1)1662 523 y FH(>)p 1746 468 28 4 v 24 w(t)24 b Fz(\035)h FE(1)p FO(,)30 b(the)g(series)e(inside)i(the) g(brac)n(k)n(ets)e(in)i(\(3.11\))f(is)515 614 y(b)r(ounded)f(b)n(y)f(a) g(constan)n(t)g(less)g(than)h FK(1)p FO(.)515 706 y(W)-7 b(e)28 b(can)f(conclude)g(that)h Fz(8)p FH(t)1402 714 y FF(1)1436 706 y FH(;)13 b(t)1498 714 y FF(2)1532 706 y FO(,)28 b(where)f FH(t)1851 714 y FF(2)1907 706 y Fz(\025)21 b FH(t)2016 714 y FF(1)2071 706 y Fz(\025)2162 692 y Fy(b)2153 706 y FH(t)2181 714 y FF(1)2215 706 y FO(,)1483 900 y Fz(k)p FH( )1571 909 y FG(h;\024)1662 919 y FL(1)1712 900 y FE(\()p FH(t)1770 908 y FF(2)1804 900 y FE(\))c Fz(\000)g FH( )1978 909 y FG(h;\024)2069 919 y FL(1)2119 900 y FE(\()o FH(t)2176 908 y FF(1)2211 900 y FE(\))p Fz(k)k(\024)2393 852 y FH(C)p 2391 883 64 4 v 2391 951 a(t)2419 915 y FG(\032)2419 970 y FF(1)2478 900 y FH(:)772 b FI(\(3.12\))515 1116 y FO(Because)26 b(of)i(Theorem)e(2.1,)h(the)h (limiting)g(v)n(ector)e(is)i(non)f(zero)g(if)2632 1056 y Fy(R)2696 1116 y Fz(j)p FH(h)13 b FE(\()p FA(P)p FE(\))p Fz(j)2916 1081 y FF(2)2963 1116 y FH(d)3003 1084 y FF(3)3037 1116 y FH(P)32 b(>)21 b FE(0)p FI(.)111 b Fb(\003)515 1332 y FP(3.1.1)93 b(V)-8 b(ariation)32 b(of)g(the)g(partition)515 1477 y FO(The)27 b(squared)g(norm)g(of)g(the)h(di\033erence)g FJ(D0\))i FO(is)515 1577 y Fy(\015)515 1621 y(\015)515 1666 y(\015)515 1711 y(\015)515 1756 y(\015)515 1801 y(\015)557 1631 y FG(N)5 b FF(\()q FG(t)659 1641 y FL(1)691 1631 y FF(\))581 1659 y Fy(X)584 1816 y FG(j)s FF(=1)728 1659 y Fy(X)735 1821 y FG(l)p FF(\()p FG(j)s FF(\))852 1730 y Fz(W)928 1738 y FG(\033)964 1746 y Fq(t)987 1758 y FL(2)1040 1669 y Fy(\000)1075 1730 y FA(v)1122 1742 y FG(l)p FF(\()p FG(j)s FF(\))1223 1730 y FH(;)13 b(t)1285 1738 y FF(2)1319 1669 y Fy(\001)1367 1730 y FH(e)1403 1689 y FG(i\015)1457 1697 y Fq(\033)1491 1708 y(t)1514 1720 y FL(2)1555 1634 y Fm(\020)1590 1689 y Fw(v)1626 1703 y Fq(l)p FL(\()q Fq(j)r FL(\))1722 1689 y FG(;)p Fx(r)p FG(E)1840 1648 y Fq(\033)1874 1659 y(t)1897 1671 y FL(2)1837 1710 y Fl(P)1938 1689 y FG(;t)1981 1699 y FL(2)2013 1634 y Fm(\021)2051 1730 y FH(e)2087 1690 y Fx(\000)p FG(iH)2207 1698 y Fq(\033)2241 1709 y(t)2264 1721 y FL(2)2305 1690 y FG(t)2329 1700 y FL(2)2365 1730 y FH( )2418 1692 y FF(\()p FG(t)2466 1702 y FL(2)2498 1692 y FF(\))2415 1754 y FG(l)p FF(\()p FG(j)s FF(\))q FG(;\033)2567 1762 y Fq(t)2590 1774 y FL(2)2631 1730 y Fz(\000)3297 1935 y FI(\(3.13\))1248 2151 y Fz(\000)1320 2052 y FG(N)5 b FF(\()q FG(t)1422 2062 y FL(1)1454 2052 y FF(\))1344 2080 y Fy(X)1347 2238 y FG(j)s FF(=1)1491 2080 y Fy(X)1498 2242 y FG(l)p FF(\()p FG(j)s FF(\))1615 2151 y Fz(W)1691 2159 y FG(\033)1727 2167 y Fq(t)1750 2179 y FL(2)1803 2151 y FE(\()p FA(v)1880 2159 y FG(j)1912 2151 y FH(;)13 b(t)1974 2159 y FF(2)2009 2151 y FE(\))f FH(e)2087 2111 y FG(i\015)2141 2119 y Fq(\033)2175 2130 y(t)2198 2142 y FL(2)2240 2056 y Fm(\020)2274 2111 y Fw(v)2310 2122 y Fq(j)2342 2111 y FG(;)p Fx(r)p FG(E)2460 2070 y Fq(\033)2494 2081 y(t)2517 2093 y FL(2)2457 2132 y Fl(P)2557 2111 y FG(;t)2600 2121 y FL(2)2632 2056 y Fm(\021)2671 2151 y FH(e)2707 2111 y Fx(\000)p FG(iH)2827 2119 y Fq(\033)2861 2130 y(t)2884 2142 y FL(2)2925 2111 y FG(t)2949 2121 y FL(2)2985 2151 y FH( )3038 2113 y FF(\()p FG(t)3086 2123 y FL(2)3118 2113 y FF(\))3035 2176 y FG(l)p FF(\()p FG(j)s FF(\))p FG(;\033)3186 2184 y Fq(t)3209 2196 y FL(2)3250 1998 y Fy(\015)3250 2043 y(\015)3250 2088 y(\015)3250 2133 y(\015)3250 2178 y(\015)3250 2222 y(\015)3293 2019 y FF(2)542 2403 y FO(where)27 b(w)n(e)h(ha)n(v)n (e)e(used)h(that)786 2584 y FH( )839 2546 y FF(\()p FG(t)887 2556 y FL(1)919 2546 y FF(\))836 2604 y FG(j;\033)916 2612 y Fq(t)939 2624 y FL(2)1002 2584 y FE(=)1083 2482 y Fy(Z)1125 2652 y FF(\000)1163 2623 y FL(\()p Fq(t)1208 2635 y FL(1)1241 2623 y(\))1163 2673 y Fq(j)1285 2584 y FH(h)13 b FE(\()p FA(P)p FE(\))g FH( )1528 2537 y FG(\033)1564 2545 y Fq(t)1587 2557 y FL(2)1525 2605 y Fw(P)1627 2584 y FH(d)1667 2548 y FF(3)1702 2584 y FH(P)32 b FE(=)1864 2513 y Fy(X)1871 2675 y FG(l)p FF(\()p FG(j)s FF(\))1988 2482 y Fy(Z)2030 2652 y FF(\000)2068 2623 y FL(\()p Fq(t)2113 2635 y FL(2)2146 2623 y(\))2068 2676 y Fq(l)p FL(\()p Fq(j)r FL(\))2190 2584 y FH(h)13 b FE(\()p FA(P)p FE(\))g FH( )2433 2537 y FG(\033)2469 2545 y Fq(t)2492 2557 y FL(2)2430 2605 y Fw(P)2532 2584 y FH(d)2572 2548 y FF(3)2607 2584 y FH(P)32 b FE(=)2769 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FA(v)1559 3887 y FG(l)1579 3873 y Fk(0)1601 3887 y FF(\()p FG(j)1653 3873 y Fk(0)1676 3887 y FF(\))1704 3875 y FH(;)g(t)1766 3883 y FF(2)1800 3814 y Fy(\001)1853 3875 y Fz(\000)1938 3856 y Fy(c)1929 3875 y Fz(W)2005 3883 y FG(\033)2041 3891 y Fq(t)2064 3903 y FL(2)2118 3875 y FE(\()p FA(v)2195 3887 y FG(j)2223 3873 y Fk(0)2249 3875 y FH(;)h(t)2312 3883 y FF(2)2346 3875 y FE(\))2376 3792 y Fy(\021)2434 3875 y FH(e)2470 3835 y Fx(\000)p FG(iH)2590 3843 y Fq(\033)2624 3854 y(t)2647 3866 y FL(2)2688 3835 y FG(t)2712 3845 y FL(2)2748 3875 y FH( )2801 3836 y FF(\()p FG(t)2849 3846 y FL(2)2881 3836 y FF(\))2798 3899 y FG(l)p Fx(0)p FF(\()q FG(j)2889 3885 y Fk(0)2911 3899 y FF(\))p FG(;\033)2990 3907 y Fq(t)3013 3919 y FL(2)3054 3792 y Fy(\021)3271 3762 y FI(\(3.15\))515 4061 y FO(The)35 b(sum)g(of)h(the)f(terms)g(where)g FH(j)1651 4030 y Fx(0)1707 4061 y Fz(6)p FE(=)e FH(j)40 b FO(and)35 b FH(l)2064 4030 y Fx(0)2099 4061 y FE(\()p FH(j)t FE(\))f Fz(6)p FE(=)f FH(l)13 b FE(\()p FH(j)t FE(\))36 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FE(2)18 b Fz(\000)1369 4595 y Fy(c)1359 4614 y Fz(W)1441 4582 y Fx(y)1435 4627 y FG(\033)1471 4635 y Fq(t)1494 4647 y FL(2)1548 4614 y FE(\()p FA(v)1625 4622 y FG(j)1657 4614 y FH(;)13 b(t)1719 4622 y FF(2)1754 4614 y FE(\))1805 4595 y Fy(c)1796 4614 y Fz(W)1872 4622 y FG(\033)1908 4630 y Fq(t)1931 4642 y FL(2)1985 4554 y Fy(\000)2020 4614 y FA(v)2067 4626 y FG(l)p FF(\()p FG(j)s FF(\))2167 4614 y FH(;)g(t)2229 4622 y FF(2)2264 4554 y Fy(\001)1263 4723 y Fz(\000)1332 4704 y Fy(c)1323 4723 y Fz(W)1405 4691 y Fx(y)1399 4736 y FG(\033)1435 4744 y Fq(t)1458 4756 y FL(2)1512 4663 y Fy(\000)1547 4723 y FA(v)1594 4735 y FG(l)p FF(\()p FG(j)s FF(\))1694 4723 y FH(;)g(t)1756 4731 y FF(2)1791 4663 y Fy(\001)1848 4704 y(c)1839 4723 y Fz(W)1915 4731 y FG(\033)1951 4739 y Fq(t)1974 4751 y FL(2)2027 4723 y FE(\()p FA(v)2104 4731 y FG(j)2136 4723 y FH(;)g(t)2198 4731 y FF(2)2233 4723 y FE(\))2340 4539 y Fy(!+)2463 4764 y FG(e)2492 4727 y Fk(\000)p Fq(iH)2605 4735 y(\033)2639 4746 y(t)2662 4758 y FL(2)2704 4727 y Fq(t)2727 4739 y FL(2)2763 4764 y FG( )2806 4735 y FL(\()q Fq(t)2852 4747 y FL(2)2884 4735 y(\))2804 4788 y Fq(l)p FL(\()q Fq(j)r FL(\))q Fq(;\033)2949 4799 y(t)2972 4811 y FL(2)3271 4667 y FI(\(3.16\))1950 5059 y FO(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 515 523 a FO(where,)27 b(for)g(a)g(giv)n(en)g (op)r(erator)f FH(O)k FO(and)d(a)g(v)n(ector)g FH(')p FI(,)f Fz(h)o FH(O)r Fz(i)2311 546 y FG(')2383 523 y FO(denotes)h FE(\()p FH(';)13 b(O)r(')p FE(\))q Fi(.)515 706 y FO(No)n(w)27 b(from)g(the)h(sum)g(\(3.16\))e(w)n(e)i(extract)e (the)i(leading)f(con)n(tribution.)515 797 y(Let)g(us)h(start)f (analyzing)515 895 y Fy(\034)572 1000 y FH(e)608 960 y Fx(\000)p FG(i\015)710 968 y Fq(\033)744 979 y(t)767 991 y FL(2)809 905 y Fm(\020)843 960 y Fw(v)879 971 y Fq(j)911 960 y FG(;)p Fx(r)p FG(E)1029 919 y Fq(\033)1063 930 y(t)1086 942 y FL(2)1026 981 y Fl(P)1126 960 y FG(;t)1169 970 y FL(2)1201 905 y Fm(\021)1240 1000 y Fz(W)1322 964 y Fx(y)1316 1014 y FG(\033)1352 1022 y Fq(t)1375 1034 y FL(2)1428 1000 y FE(\()p FA(v)1505 1008 y FG(j)1538 1000 y FH(;)13 b(t)1600 1008 y FF(2)1634 1000 y FE(\))g Fz(W)1753 1008 y FG(\033)1789 1016 y Fq(t)1812 1028 y FL(2)1865 940 y Fy(\000)1900 1000 y FA(v)1947 1012 y FG(l)p FF(\()p FG(j)s FF(\))2048 1000 y FH(;)g(t)2110 1008 y FF(2)2144 940 y Fy(\001)2192 1000 y FH(e)2228 960 y FG(i\015)2282 968 y Fq(\033)2316 979 y(t)2339 991 y FL(2)2380 905 y Fm(\020)2415 960 y Fw(v)2451 974 y Fq(l)p FL(\()q Fq(j)r FL(\))2547 960 y FG(;)p Fx(r)p FG(E)2665 919 y Fq(\033)2699 930 y(t)2722 942 y FL(2)2662 981 y Fl(P)2763 960 y FG(;t)2806 970 y FL(2)2838 905 y Fm(\021)2876 895 y Fy(\035)2934 1076 y FG(e)2963 1038 y Fk(\000)p Fq(iH)3076 1046 y(\033)3110 1057 y(t)3133 1069 y FL(2)3175 1038 y Fq(t)3198 1050 y FL(2)3235 1076 y FG( )3278 1046 y FL(\()p Fq(t)3323 1058 y FL(2)3355 1046 y(\))3276 1099 y Fq(l)p FL(\()p Fq(j)r FL(\))q Fq(;\033)3420 1110 y(t)3443 1122 y FL(2)3471 1000 y FI(\(3.17\))515 1259 y FO(and)27 b(for)g Fr(s)19 b Fu(\025)h Fr(t)955 1268 y FF(2)515 1352 y Fy(\034)572 1457 y FH(e)608 1417 y FG(iH)680 1425 y Fq(\033)714 1436 y(t)737 1448 y FL(2)778 1417 y FG(s)811 1457 y FH(e)847 1417 y Fx(\000)p FG(i\015)949 1425 y Fq(\033)983 1436 y(t)1006 1448 y FL(2)1048 1362 y Fm(\020)1082 1417 y Fw(v)1118 1428 y Fq(j)1149 1417 y FG(;)p Fx(r)p FG(E)1267 1376 y Fq(\033)1301 1387 y(t)1324 1399 y FL(2)1264 1438 y Fl(P)1365 1417 y FG(;s)1413 1362 y Fm(\021)1452 1457 y Fz(W)1534 1421 y Fx(y)1528 1470 y FG(\033)1564 1478 y Fq(t)1587 1490 y FL(2)1640 1457 y FE(\()p FA(v)1717 1465 y FG(j)1749 1457 y FH(;)13 b(s)p FE(\))g Fz(W)1938 1465 y FG(\033)1974 1473 y Fq(t)1997 1485 y FL(2)2051 1397 y Fy(\000)2086 1457 y FA(v)2133 1469 y FG(l)p FF(\()o FG(j)s FF(\))2233 1457 y FH(;)g(s)2303 1397 y Fy(\001)2351 1457 y FH(e)2387 1417 y FG(i\015)2441 1425 y Fq(\033)2475 1436 y(t)2498 1448 y FL(2)2539 1362 y Fm(\020)2574 1417 y Fw(v)2610 1431 y Fq(l)p FL(\()p Fq(j)r FL(\))2706 1417 y FG(;)p Fx(r)p FG(E)2824 1376 y Fq(\033)2858 1387 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1999 y Fw(v)1406 2010 y Fq(j)1438 1999 y FG(;)p Fx(r)p FG(E)1556 1958 y Fq(\033)1590 1969 y(t)1613 1981 y FL(2)1553 2020 y Fl(P)1653 1999 y FG(;\033)1710 1960 y Fk(\000)1770 1944 y FL(1)p 1765 1951 38 3 v 1765 1982 a Fq(\013)1708 2017 y(t)1731 2029 y FL(2)1817 1914 y Fm(!)1866 2069 y FH( )1919 2031 y FF(\()p FG(t)1967 2041 y FL(2)1999 2031 y FF(\))1916 2094 y FG(l)p FF(\()p FG(j)s FF(\))q FG(;\033)2068 2102 y Fq(t)2091 2114 y FL(2)2131 2069 y FH(;)13 b(e)2201 1999 y FG(i\015)2255 2007 y Fq(\033)2289 2018 y(t)2312 2030 y FL(2)2354 1914 y Fm( )2399 1999 y Fw(v)2435 2013 y Fq(l)p FL(\()q Fq(j)r FL(\))2531 1999 y FG(;)p Fx(r)p FG(E)2649 1958 y Fq(\033)2683 1969 y(t)2706 1981 y FL(2)2646 2020 y Fl(P)2747 1999 y FG(;\033)2804 1960 y Fk(\000)2863 1944 y FL(1)p 2858 1951 V 2858 1982 a Fq(\013)2802 2017 y(t)2825 2029 y FL(2)2910 1914 y Fm(!)2960 2069 y FH( )3013 2031 y FF(\()o FG(t)3060 2041 y FL(2)3093 2031 y FF(\))3010 2094 y FG(l)p FF(\()p FG(j)s FF(\))p FG(;\033)3161 2102 y Fq(t)3184 2114 y FL(2)3225 1919 y Fy(1)3225 2054 y(A)3271 2249 y FI(\(3.19\))515 2375 y FO(where)27 b Fr(C)805 2389 y FG(j;l)p FF(\()p FG(j)s FF(\))p FG(;\033)1000 2397 y Fq(t)1023 2409 y FL(2)1084 2375 y Ft(=)1158 2321 y Fs(R)1205 2342 y FG(\024)1241 2352 y FL(1)1192 2399 y FG(\033)1228 2407 y Fq(t)1251 2419 y FL(2)1303 2298 y Fs(\014)1303 2338 y(\014)1303 2378 y(\014)1327 2375 y Fr(\021)1362 2389 y FG(j;l)p FF(\()p FG(j)s FF(\))1518 2301 y Fs(\020)1562 2357 y(b)1561 2375 y Fh(k)1603 2301 y Fs(\021)1646 2298 y(\014)1646 2338 y(\014)1646 2378 y(\014)1669 2319 y FF(2)1745 2347 y FG(d)1777 2327 y FL(3)1809 2347 y FG(k)p 1725 2359 138 3 v 1725 2406 a FF(2)p Fx(j)q Fw(k)p Fx(j)1830 2389 y FL(3)1900 2375 y FO(and)h FH(\021)2100 2387 y FG(l)p FF(\()p FG(j)s FF(\))p FG(;j)2260 2292 y Fy(\020)2308 2355 y(b)2306 2375 y FA(k)2353 2292 y Fy(\021)2420 2375 y FE(:=)2638 2326 y FG(g)2672 2313 y Fm(b)2670 2326 y Fw(k)p Fx(\001)2725 2332 y FE(\()2755 2326 y Fw(v)2791 2337 y Fq(j)2823 2326 y Fx(\000)p Fw(v)2907 2340 y Fq(l)p FL(\()p Fq(j)r FL(\))3003 2332 y FE(\))p 2533 2357 606 4 v 2533 2416 a(\()2562 2410 y FF(1)p Fx(\000)2642 2397 y Fm(b)2640 2410 y Fw(k)q Fx(\001)p Fw(v)2732 2421 y Fq(j)2764 2416 y FE(\))2793 2410 y Fx(\001)2812 2416 y FE(\()2842 2410 y FF(1)p Fx(\000)2922 2397 y Fm(b)2920 2410 y Fw(k)q Fx(\001)p Fw(v)3012 2424 y Fq(l)p FL(\()q Fq(j)r FL(\))3108 2416 y FE(\))3161 2375 y FO(.)515 2557 y(Then)39 b(w)n(e)g(rewrite)g(the)g(exp)r (ectation)g(v)-5 b(alue)39 b(\(3.17\))g(as)f(the)i(limit)g(for)f FH(s)g Fz(!)g FE(+)p Fz(1)g FO(of)g(the)515 2649 y(corresp)r(onding)26 b(quan)n(tit)n(y)i(\(3.18\))g(plus)g(a)g(remainder.)g(The)g(limit)h (corresp)r(onds)e(to)h(the)h(ex-)515 2740 y(pression)g(\(3.19\).)i(The) f(sum)h(of)g(the)h(remainders)d(o)n(v)n(er)g(the)i(cells)g(in)g (\(3.16\))f(amoun)n(ts)g(to)h(a)515 2831 y(quan)n(tit)n(y)c(whic)n(h)g (is)h(at)f(most)g(of)h(order)e FH(t)1812 2799 y Fx(\000)p FF(4)p FG(\017)1812 2850 y FF(2)1921 2831 y FO(.)515 3014 y(Hence)h(the)h(discussion)f(is)h(no)n(w)e(restricted)h(to)h(the)g (follo)n(wing)e(sum)515 3155 y FG(N)5 b FF(\()p FG(t)616 3165 y FL(1)648 3155 y FF(\))538 3183 y Fy(X)541 3340 y FG(j)s FF(=1)685 3183 y Fy(X)692 3345 y FG(l)p FF(\()p FG(j)s FF(\))809 3126 y Fy(*)871 3254 y FE(2)17 b Fz(\000)g FH(e)1039 3213 y FG(i)p FF(\001)p FG(\015)1143 3221 y Fq(\033)1177 3232 y(t)1200 3244 y FL(2)1241 3158 y Fm(\020)1276 3213 y Fw(v)1312 3227 y Fq(l)p FL(\()q Fq(j)r FL(\))1408 3213 y Fx(\000)p Fw(v)1492 3224 y Fq(j)1523 3213 y FG(;)p Fx(r)p FG(E)1641 3172 y Fq(\033)1675 3183 y(t)1698 3195 y FL(2)1638 3234 y Fl(P)1739 3213 y FG(;t)1782 3223 y FL(2)1814 3158 y Fm(\021)1853 3254 y FH(e)1889 3218 y Fx(\000)1946 3161 y Fq(C)1987 3178 y(l)p FL(\()q Fq(j)r FL(\))q Fq(;j;\033)2175 3189 y(t)2198 3201 y FL(2)p 1946 3207 294 3 v 2079 3239 a(2)2271 3254 y Fz(\000)f FH(e)2383 3213 y FG(i)p FF(\001)p FG(\015)2487 3221 y Fq(\033)2521 3232 y(t)2544 3244 y FL(2)2586 3158 y Fm(\020)2620 3213 y Fw(v)2656 3224 y Fq(j)2687 3213 y Fx(\000)p Fw(v)2771 3227 y Fq(l)p FL(\()q Fq(j)r FL(\))2868 3213 y FG(;)p Fx(r)p FG(E)2986 3172 y Fq(\033)3020 3183 y(t)3043 3195 y FL(2)2983 3234 y Fl(P)3083 3213 y FG(;t)3126 3223 y FL(2)3158 3158 y Fm(\021)3197 3254 y FH(e)3233 3218 y Fx(\000)3291 3161 y Fq(C)3332 3178 y(j;l)p FL(\()p Fq(j)r FL(\))r Fq(;\033)3520 3189 y(t)3543 3201 y FL(2)p 3291 3207 V 3423 3239 a(2)3598 3126 y Fy(+)3660 3351 y FG( )3703 3322 y FL(\()p Fq(t)3748 3334 y FL(2)3781 3322 y(\))3701 3375 y Fq(l)p FL(\()p Fq(j)r FL(\))q Fq(;\033)3845 3386 y(t)3868 3398 y FL(2)3883 3254 y FI(\(3.20\))515 3535 y FO(where)601 3771 y FH(e)637 3731 y FG(i)p FF(\001)p FG(\015)741 3739 y Fq(\033)775 3750 y(t)798 3762 y FL(2)840 3676 y Fm(\020)874 3731 y Fw(v)910 3742 y Fq(j)941 3731 y Fx(\000)p Fw(v)1025 3745 y Fq(l)p FL(\()q Fq(j)r FL(\))1122 3731 y FG(;)p Fx(r)p FG(E)1240 3690 y Fq(\033)1274 3701 y(t)1297 3713 y FL(2)1237 3752 y Fl(P)1337 3731 y FG(;t)1380 3741 y FL(2)1412 3676 y Fm(\021)1472 3771 y FE(:=)22 b FH(e)1611 3701 y Fx(\000)p FG(i\015)1713 3709 y Fq(\033)1747 3720 y(t)1770 3732 y FL(2)1811 3616 y Fm( )1856 3701 y Fw(v)1892 3712 y Fq(j)1924 3701 y FG(;)p Fx(r)p FG(E)2042 3660 y Fq(\033)2076 3671 y(t)2099 3683 y FL(2)2039 3722 y Fl(P)2139 3701 y FG(;\033)2196 3661 y Fk(\000)2255 3645 y FL(1)p 2251 3652 38 3 v 2251 3684 a Fq(\013)2194 3719 y(t)2217 3731 y FL(2)2303 3616 y Fm(!)2352 3771 y FH(e)2388 3701 y FG(i\015)2442 3709 y Fq(\033)2476 3720 y(t)2499 3732 y FL(2)2540 3616 y Fm( )2586 3701 y Fw(v)2622 3715 y Fq(l)p FL(\()p Fq(j)r FL(\))2718 3701 y FG(;)p Fx(r)p FG(E)2836 3660 y Fq(\033)2870 3671 y(t)2893 3683 y FL(2)2833 3722 y Fl(P)2933 3701 y FG(;\033)2990 3661 y Fk(\000)3050 3645 y FL(1)p 3045 3652 V 3045 3684 a Fq(\013)2988 3719 y(t)3011 3731 y FL(2)3097 3616 y Fm(!)3163 3771 y FH(:)87 b FI(\(3.21\))515 3933 y FO(An)28 b(estimate)f(from)g(ab)r(o)n(v)n(e)g(of)g(the)h(sum)g(\(3.20\))f(is)g (giv)n(en)g(b)n(y)807 4078 y FG(N)5 b FF(\()q FG(t)909 4088 y FL(1)941 4078 y FF(\))831 4106 y Fy(X)833 4264 y FG(j)s FF(=1)978 4106 y Fy(X)985 4268 y FG(l)p FF(\()p FG(j)s FF(\))1101 4072 y Fy(\022)1158 4091 y(\015)1158 4136 y(\015)1158 4181 y(\015)1201 4177 y FH( )1254 4139 y FF(\()o FG(t)1301 4149 y FL(2)1333 4139 y FF(\))1251 4202 y FG(l)p FF(\()p FG(j)s FF(\))p FG(;\033)1402 4210 y Fq(t)1425 4222 y FL(2)1466 4091 y Fy(\015)1466 4136 y(\015)1466 4181 y(\015)1508 4112 y FF(2)1560 4177 y Fz(\001)17 b FH(C)23 b Fz(\001)17 b FH(t)1742 4130 y Fx(\000)1815 4111 y Fq(\017)p 1800 4119 57 3 v 1800 4152 a FL(16)1742 4196 y FF(1)1887 4177 y Fz(\001)h(j)p FE(ln)13 b(\()p FH(\033)2098 4185 y FG(t)2122 4195 y FL(2)2158 4177 y FE(\))p Fz(j)2209 4072 y Fy(\023)2287 4177 y Fz(\024)21 b FH(C)h Fz(\001)c FH(t)2512 4130 y Fx(\000)2585 4111 y Fq(\017)p 2569 4119 V 2569 4152 a FL(16)2512 4196 y FF(1)2657 4177 y Fz(\001)f(j)q FE(ln)c(\()o FH(\033)2867 4185 y FG(t)2891 4195 y FL(2)2928 4177 y FE(\))p Fz(j)292 b FI(\(3.22\))515 4425 y FO(taking)27 b(in)n(to)g(accoun)n(t)g(that) 1376 4632 y FH(C)1431 4644 y FG(l)p FF(\()o FG(j)s FF(\))q FG(;j;\033)1626 4652 y Fq(t)1649 4664 y FL(2)1711 4632 y FE(:=)1813 4531 y Fy(Z)1890 4551 y FG(\024)1926 4561 y FL(1)1856 4701 y FG(\033)1892 4709 y Fq(t)1915 4721 y FL(2)1976 4546 y Fy(\014)1976 4591 y(\014)1976 4636 y(\014)2001 4632 y FH(\021)2039 4644 y FG(l)p FF(\()q FG(j)s FF(\))p FG(;j)2200 4550 y Fy(\020)2248 4613 y(b)2246 4632 y FA(k)2293 4550 y Fy(\021)2339 4546 y(\014)2339 4591 y(\014)2339 4636 y(\014)2364 4567 y FF(2)2451 4584 y FH(d)2491 4552 y FF(3)2525 4584 y FH(k)p 2421 4615 176 4 v 2421 4692 a FE(2)14 b Fz(j)p FA(k)p Fz(j)2562 4657 y FF(3)1950 5059 y FO(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 515 523 a FO(and)1541 531 y Fy(\014)1541 576 y(\014)1541 621 y(\014)1541 666 y(\014)1566 640 y FH(e)1602 599 y FG(i)p FF(\001)p FG(\015)1706 607 y Fq(\033)1740 618 y(t)1763 630 y FL(2)1804 544 y Fm(\020)1839 599 y Fw(v)1875 613 y Fq(l)p FL(\()q Fq(j)r FL(\))1971 599 y Fx(\000)p Fw(v)2055 610 y Fq(j)2087 599 y FG(;)p Fx(r)p FG(E)2205 558 y Fq(\033)2239 569 y(t)2262 581 y FL(2)2202 620 y Fl(P)2302 599 y FG(;t)2345 609 y FL(2)2377 544 y Fm(\021)2416 531 y Fy(\014)2416 576 y(\014)2416 621 y(\014)2416 666 y(\014)515 809 y FO(are)26 b(b)r(oth)i(b)r(ounded)g(b)n (y)1686 918 y FH(C)22 b Fz(\001)17 b FH(t)1829 871 y Fx(\000)1903 852 y Fq(\017)p 1887 860 57 3 v 1887 893 a FL(16)1829 937 y FF(1)1975 918 y Fz(\001)g(j)p FE(ln)c(\()p FH(\033)2185 926 y FG(t)2209 936 y FL(2)2245 918 y FE(\))p Fz(j)515 1043 y FO(b)r(ecause)26 b(of)h(the)g(di\033erence)f FA(v)1474 1051 y FG(j)1522 1043 y Fz(\000)15 b FA(v)1644 1055 y FG(l)p FF(\()o FG(j)s FF(\))1771 1043 y FO(and)26 b(the)h(regularit)n(y)e(prop)r(erties)h(of)g(the)h(gradien)n(t)f(of)515 1134 y(the)i(ground)e(state)i(energy)e(\(Subsection)i(1.3.1)e(and)h (Lemma)h(A1\).)515 1226 y(In)h(the)g(end,)h(collecting)e(all)h(the)g (partial)g(estimates,)f(w)n(e)h(can)g(conclude)f(that)i(the)f(norm)g (of)515 1317 y(the)f(di\033erence)f FJ(D0\))k FO(is)c(surely)g(b)r (ounded)h(from)f(ab)r(o)n(v)n(e)f(b)n(y)i(a)f(quan)n(tit)n(y)g(of)g (order)1718 1501 y FH(t)1746 1454 y Fx(\000)1819 1434 y Fq(\017)p 1804 1442 V 1804 1475 a FL(16)1746 1520 y FF(1)1891 1501 y Fz(\001)18 b(j)p FE(ln)13 b(\()p FH(\033)2102 1509 y FG(t)2126 1519 y FL(2)2162 1501 y FE(\))p Fz(j)30 b FH(:)1007 b FI(\(3.23\))515 1717 y FP(3.1.2)93 b(Co)s(ok's)31 b(argumen)m(t)515 1862 y FO(The)c(di\033erence)h(corresp)r(onding)d(to) j(the)g(v)-5 b(ariation)26 b FJ(D1\))31 b FO(is)592 2007 y FG(N)5 b FF(\()p FG(t)693 2017 y FL(1)725 2007 y FF(\))615 2035 y Fy(X)618 2193 y FG(j)s FF(=1)762 2001 y Fy(\032)820 2106 y FH(e)856 2070 y FG(iH)t(t)956 2080 y FL(2)991 2106 y Fz(W)1067 2114 y FG(\033)1103 2122 y Fq(t)1126 2134 y FL(2)1180 2106 y FE(\()p FA(v)1257 2114 y FG(j)1289 2106 y FH(;)13 b(t)1351 2114 y FF(2)1385 2106 y FE(\))g FH(e)1464 2066 y FG(i\015)1518 2074 y Fq(\033)1552 2085 y(t)1575 2097 y FL(2)1616 2011 y Fm(\020)1651 2066 y Fw(v)1687 2077 y Fq(j)1718 2066 y FG(;)p Fx(r)p FG(E)1836 2025 y Fq(\033)1870 2036 y(t)1893 2048 y FL(2)1833 2087 y Fl(P)1934 2066 y FG(;t)1977 2076 y FL(2)2009 2011 y Fm(\021)2048 2106 y FH(e)2084 2070 y Fx(\000)p FG(iE)2202 2029 y Fq(\033)2236 2040 y(t)2259 2052 y FL(2)2199 2091 y Fl(P)2299 2070 y FG(t)2323 2080 y FL(2)2359 2106 y FH( )2412 2068 y FF(\()p FG(t)2460 2078 y FL(1)2492 2068 y FF(\))2409 2126 y FG(j;\033)2489 2134 y Fq(t)2512 2146 y FL(2)2553 2106 y Fz(\000)1199 2398 y(\000)p FH(e)1295 2362 y FG(iH)t(t)1395 2372 y FL(1)1430 2398 y Fz(W)1506 2406 y FG(\033)1542 2414 y Fq(t)1565 2426 y FL(2)1619 2398 y FE(\()p FA(v)1696 2406 y FG(j)1728 2398 y FH(;)g(t)1790 2406 y FF(1)1824 2398 y FE(\))g FH(e)1903 2358 y FG(i\015)1957 2366 y Fq(\033)1991 2377 y(t)2014 2389 y FL(2)2055 2303 y Fm(\020)2090 2358 y Fw(v)2126 2369 y Fq(j)2157 2358 y FG(;)p Fx(r)p FG(E)2275 2317 y Fq(\033)2309 2328 y(t)2332 2340 y FL(2)2272 2379 y Fl(P)2373 2358 y FG(;t)2416 2368 y FL(1)2448 2303 y Fm(\021)2486 2398 y FH(e)2522 2362 y Fx(\000)p FG(iE)2640 2321 y Fq(\033)2674 2332 y(t)2697 2344 y FL(2)2637 2383 y Fl(P)2738 2362 y FG(t)2762 2372 y FL(1)2798 2398 y FH( )2851 2360 y FF(\()p FG(t)2899 2370 y FL(1)2931 2360 y FF(\))2848 2418 y FG(j;\033)2928 2426 y Fq(t)2951 2438 y FL(2)2992 2293 y Fy(\033)3079 2398 y FH(:)515 2568 y FO(W)-7 b(e)28 b(estimate)g(the)g(con)n (tribution)f(for)h(eac)n(h)f(cell)g(b)n(y)h(expressing)e(the)j (di\033erence)e(of)h(the)g(t)n(w)n(o)515 2667 y(related)f(v)n(ectors)f (as)h(the)g(follo)n(wing)g(in)n(tegral)f(from)h Fr(t)2192 2676 y FF(1)2255 2667 y FO(to)h Fr(t)2382 2676 y FF(2)2417 2667 y FO(:)928 2777 y Fy(Z)1005 2798 y FG(t)1029 2808 y FL(2)971 2947 y FG(t)995 2957 y FL(1)1106 2830 y FH(d)p 1088 2862 76 4 v 1088 2929 a(ds)1186 2774 y Fy(\032)1244 2879 y FH(e)1280 2843 y FG(iH)t(s)1389 2879 y Fz(W)1465 2887 y FG(\033)1501 2895 y Fq(t)1524 2907 y FL(2)1577 2879 y FE(\()p FA(v)1654 2887 y FG(j)1686 2879 y FH(;)13 b(s)p FE(\))g FH(e)1835 2838 y FG(i\015)1889 2846 y Fq(\033)1923 2857 y(t)1946 2869 y FL(2)1987 2783 y Fm(\020)2022 2838 y Fw(v)2058 2849 y Fq(j)2089 2838 y FG(;)p Fx(r)p FG(E)2207 2797 y Fq(\033)2241 2808 y(t)2264 2820 y FL(2)2204 2859 y Fl(P)2305 2838 y FG(;s)2353 2783 y Fm(\021)2391 2879 y FH(e)2427 2843 y Fx(\000)p FG(iE)2545 2802 y Fq(\033)2579 2813 y(t)2602 2825 y FL(2)2542 2864 y Fl(P)2643 2843 y FG(s)2676 2879 y FH( )2729 2841 y FF(\()p FG(t)2777 2851 y FL(1)2809 2841 y FF(\))2726 2899 y FG(j;\033)2806 2907 y Fq(t)2829 2919 y FL(2)2870 2774 y Fy(\033)2940 2879 y FH(ds)k(:)217 b FI(\(3.24\))515 3055 y FO(Moreo)n(v)n(er)24 b(w)n(e)k(use)f(the)h(inequalit)n(y)613 3150 y Fy(\015)613 3195 y(\015)613 3240 y(\015)613 3285 y(\015)656 3199 y(R)707 3219 y FG(t)731 3229 y FL(2)692 3286 y FG(t)716 3296 y FL(1)805 3228 y FG(d)p 790 3242 62 4 v 790 3284 a(ds)874 3154 y Fy(\032)932 3259 y FH(e)968 3227 y FG(iH)t(s)1076 3259 y Fz(W)1152 3267 y FG(\033)1188 3275 y Fq(t)1211 3287 y FL(2)1265 3259 y FE(\()p FA(v)1342 3267 y FG(j)1374 3259 y FH(;)13 b(s)p FE(\))g FH(e)1523 3218 y FG(i\015)1577 3226 y Fq(\033)1611 3237 y(t)1634 3249 y FL(2)1675 3164 y Fm(\020)1709 3218 y Fw(v)1745 3229 y Fq(j)1777 3218 y FG(;)p Fx(r)p FG(E)1895 3178 y Fq(\033)1929 3189 y(t)1952 3201 y FL(2)1892 3239 y Fl(P)1992 3218 y FG(;s)2040 3164 y Fm(\021)2079 3259 y FH(e)2115 3227 y Fx(\000)p FG(iE)2233 3186 y Fq(\033)2267 3197 y(t)2290 3209 y FL(2)2230 3248 y Fl(P)2330 3227 y FG(s)2364 3259 y FH( )2417 3221 y FF(\()o FG(t)2464 3231 y FL(1)2497 3221 y FF(\))2414 3279 y FG(j;\033)2494 3287 y Fq(t)2517 3299 y FL(2)2558 3154 y Fy(\033)2628 3259 y FH(ds)2704 3150 y Fy(\015)2704 3195 y(\015)2704 3240 y(\015)2704 3285 y(\015)2768 3259 y Fz(\024)767 3529 y(\024)848 3469 y Fy(R)899 3490 y FG(t)923 3500 y FL(2)884 3556 y FG(t)908 3566 y FL(1)972 3421 y Fy(\015)972 3466 y(\015)972 3511 y(\015)972 3556 y(\015)1039 3499 y FG(d)p 1025 3513 V 1025 3555 a(ds)1109 3424 y Fy(\032)1166 3529 y FH(e)1202 3498 y FG(iH)t(s)1311 3529 y Fz(W)1387 3537 y FG(\033)1423 3545 y Fq(t)1446 3557 y FL(2)1500 3529 y FE(\()o FA(v)1576 3537 y FG(j)1609 3529 y FH(;)g(s)p FE(\))f FH(e)1757 3489 y FG(i\015)1811 3497 y Fq(\033)1845 3508 y(t)1868 3520 y FL(2)1910 3434 y Fm(\020)1944 3489 y Fw(v)1980 3500 y Fq(j)2011 3489 y FG(;)p Fx(r)p FG(E)2129 3448 y Fq(\033)2163 3459 y(t)2186 3471 y FL(2)2126 3510 y Fl(P)2227 3489 y FG(;s)2275 3434 y Fm(\021)2314 3529 y FH(e)2350 3498 y Fx(\000)p FG(iE)2468 3457 y Fq(\033)2502 3468 y(t)2525 3480 y FL(2)2465 3519 y Fl(P)2565 3498 y FG(s)2598 3529 y FH( )2651 3491 y FF(\()p FG(t)2699 3501 y FL(1)2731 3491 y FF(\))2648 3549 y FG(j;\033)2728 3557 y Fq(t)2751 3569 y FL(2)2792 3424 y Fy(\033)2850 3421 y(\015)2850 3466 y(\015)2850 3511 y(\015)2850 3556 y(\015)2905 3529 y FH(ds)17 b(:)3271 3394 y FI(\(3.25\))515 3742 y FO(The)27 b(deriv)-5 b(ativ)n(e)27 b(in)h(the)g(expression)e(\(3.24\))h(can)g(b)r(e)h(split)g(as)f(follo)n (ws:)662 3956 y FH(ie)724 3924 y FG(iH)t(s)833 3956 y Fz(W)909 3964 y FG(\033)945 3972 y Fq(t)968 3984 y FL(2)1022 3956 y FE(\()p FA(v)1099 3964 y FG(j)1131 3956 y FH(;)13 b(s)p FE(\))1244 3851 y Fy(\022)1300 3956 y FH(')1350 3964 y FG(\033)1386 3972 y Fq(t)1409 3984 y FL(2)1446 3964 y FG(;)p Fw(v)1501 3975 y Fq(j)1549 3956 y FE(\()p FA(x)p FH(;)g(s)p FE(\))k(+)1829 3900 y FG(d\015)1893 3908 y Fq(\033)1927 3919 y(t)1950 3931 y FL(2)1992 3845 y Fm(\020)2026 3900 y Fw(v)2062 3911 y Fq(j)2094 3900 y FG(;)p Fx(r)p FG(E)2212 3859 y Fq(\033)2246 3870 y(t)2269 3882 y FL(2)2209 3921 y Fl(P)2309 3900 y FG(;s)2357 3845 y Fm(\021)p 1829 3939 563 4 v 2080 3981 a FG(ds)2402 3851 y Fy(\023)2471 3956 y FH(e)2507 3915 y FG(i\015)2561 3923 y Fq(\033)2595 3934 y(t)2618 3946 y FL(2)2660 3860 y Fm(\020)2694 3915 y Fw(v)2730 3926 y Fq(j)2761 3915 y FG(;)p Fx(r)p FG(E)2879 3874 y Fq(\033)2913 3885 y(t)2936 3897 y FL(2)2876 3936 y Fl(P)2977 3915 y FG(;s)3025 3860 y Fm(\021)3064 3956 y FH(e)3100 3924 y Fx(\000)p FG(iE)3218 3883 y Fq(\033)3252 3894 y(t)3275 3906 y FL(2)3215 3945 y Fl(P)3315 3924 y FG(s)3348 3956 y FH( )3401 3918 y FF(\()p FG(t)3449 3928 y FL(1)3481 3918 y FF(\))3398 3976 y FG(j;\033)3478 3984 y Fq(t)3501 3996 y FL(2)3588 3956 y FI(\(3.26\))515 4204 y FE(+)82 b FH(ie)719 4173 y FG(iH)t(s)828 4204 y Fz(W)904 4212 y FG(\033)940 4220 y Fq(t)963 4232 y FL(2)1017 4204 y FE(\()p FA(v)1094 4212 y FG(j)1126 4204 y FH(;)13 b(s)p FE(\))1239 4144 y Fy(\000)1274 4204 y FH(H)22 b Fz(\000)17 b FH(H)1501 4212 y FG(\033)1537 4220 y Fq(t)1560 4232 y FL(2)1601 4144 y Fy(\001)1649 4204 y FH(e)1685 4164 y FG(i\015)1739 4172 y Fq(\033)1773 4183 y(t)1796 4195 y FL(2)1837 4109 y Fm(\020)1871 4164 y Fw(v)1907 4175 y Fq(j)1939 4164 y FG(;)p Fx(r)p FG(E)2057 4123 y Fq(\033)2091 4134 y(t)2114 4146 y FL(2)2054 4185 y Fl(P)2154 4164 y FG(;s)2202 4109 y Fm(\021)2241 4204 y FH(e)2277 4173 y Fx(\000)p FG(iE)2395 4132 y Fq(\033)2429 4143 y(t)2452 4155 y FL(2)2392 4194 y Fl(P)2492 4173 y FG(s)2526 4204 y FH( )2579 4166 y FF(\()o FG(t)2626 4176 y FL(1)2659 4166 y FF(\))2576 4224 y FG(j;\033)2656 4232 y Fq(t)2679 4244 y FL(2)3588 4204 y FI(\(3.27\))515 4386 y FO(where)25 b(the)i(term)e(\(3.26\))h(is) g(analogous)d(to)j(the)g(expression)f(\(2.15\))g(in)h(Theorem)f(2.1)h (a)f(part)515 4486 y(from)i(the)h(ev)n(olution)f(op)r(erator)e(on)j (the)g(v)n(ery)e(left.)515 4685 y FJ(T)-6 b(erm)30 b(\(3.26\))515 4785 y FO(As)d(in)h(Theorem)f(2.1,)g(Section)g(2,)g(w)n(e)h(decomp)r (ose)e FH(')2214 4793 y FG(\033)2250 4801 y Fq(t)2273 4813 y FL(2)2310 4793 y FG(;)p Fw(v)2365 4804 y Fq(j)2413 4785 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))27 b FO(as)1950 5059 y(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 1519 614 a FH(')1569 578 y Fx(\000)1569 628 y FG(\033)1605 636 y Fq(t)1628 648 y FL(2)1665 628 y FG(;)p Fw(v)1720 639 y Fq(j)1768 614 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))j(+)h FH(')2088 578 y FF(+)2088 628 y FG(\033)2124 636 y Fq(t)2147 648 y FL(2)2184 628 y FG(;)p Fw(v)2239 639 y Fq(j)2287 614 y FE(\()p FA(x)p FH(;)c(s)p FE(\))807 b FI(\(3.28\))515 734 y FO(where:)1081 933 y FH(')1131 897 y Fx(\000)1131 946 y FG(\033)1167 954 y Fq(t)1190 966 y FL(2)1227 946 y FG(;)p Fw(v)1282 957 y Fq(j)1330 933 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))21 b(:)83 b(=)g FH(g)1815 897 y FF(2)1861 831 y Fy(Z)1938 851 y FG(s)1967 830 y Fk(\000)p Fq(\013)1904 1001 y FG(\033)1940 1009 y Fq(t)1963 1021 y FL(2)2081 884 y FE(cos)14 b(\()p FA(k)i Fz(\001)i FA(x)f Fz(\000)f(j)q FA(k)p Fz(j)d FH(s)p FE(\))p 2081 915 556 4 v 2157 928 a Fy(\020)2202 1011 y FE(1)18 b Fz(\000)2337 991 y Fy(b)2335 1011 y FA(k)e Fz(\001)i FA(v)2484 1019 y FG(j)2516 928 y Fy(\021)2647 933 y FH(d)p FE(\012)p FH(d)13 b Fz(j)p FA(k)p Fz(j)3271 1149 y FI(\(3.29\))1081 1305 y FH(')1131 1269 y FF(+)1131 1318 y FG(\033)1167 1326 y Fq(t)1190 1338 y FL(2)1227 1318 y FG(;)p Fw(v)1282 1329 y Fq(j)1330 1305 y FE(\()p FA(x)p FH(;)g(s)p FE(\))21 b(:)83 b(=)g FH(g)1815 1269 y FF(2)1861 1203 y Fy(Z)1938 1224 y FG(\024)1974 1234 y FL(1)1904 1373 y FG(s)1933 1359 y Fk(\000)p Fq(\013)2047 1257 y FE(cos)14 b(\()p FA(k)i Fz(\001)i FA(x)e Fz(\000)h(j)q FA(k)p Fz(j)12 b FH(s)p FE(\))p 2047 1288 V 2122 1300 a Fy(\020)2168 1383 y FE(1)18 b Fz(\000)2303 1364 y Fy(b)2301 1383 y FA(k)e Fz(\001)i FA(v)2450 1391 y FG(j)2482 1300 y Fy(\021)2613 1305 y FH(d)p FE(\012)p FH(d)13 b Fz(j)p FA(k)p Fz(j)30 b FH(:)515 1564 y FO(By)25 b(the)g(same)g(pro)r(cedure,) f(exploiting)h(the)g(subtraction)g(of)g(the)h(infrared)e(tail)h FH(')3092 1533 y Fx(\000)3092 1577 y FG(\033)3128 1585 y Fq(t)3151 1597 y FL(2)3188 1577 y FG(;)p Fw(v)3243 1588 y Fq(j)3291 1564 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))515 1706 y FO(b)n(y)23 b(means)h(of)g(the)g(deriv)-5 b(ativ)n(e)23 b(of)h FH(e)1611 1665 y FG(i\015)1665 1673 y Fq(\033)1699 1684 y(t)1722 1696 y FL(2)1763 1610 y Fm(\020)1798 1665 y Fw(v)1834 1676 y Fq(j)1865 1665 y FG(;)p Fx(r)p FG(E)1983 1624 y Fq(\033)2017 1635 y(t)2040 1647 y FL(2)1980 1686 y Fl(P)2081 1665 y FG(;s)2129 1610 y Fm(\021)2192 1706 y FO(and)f(assuming)g(the)i(constrain)n(t)d (\(2.26\),)515 1797 y(w)n(e)i(obtain)f(the)i(follo)n(wing)e(estimate)h (from)g(ab)r(o)n(v)n(e)e(for)i(the)h(norm)e(of)h(the)h(expression)d (\(3.26\):)1477 1980 y FH(C)g Fz(\001)17 b FH(s)1628 1944 y Fx(\000)p FF(1)1728 1980 y Fz(\001)g FH(s)1802 1944 y Fx(\000)p FF(4)p FG(\017)1927 1980 y Fz(\001)h FH(t)1994 1933 y Fx(\000)2052 1913 y FL(3)p Fq(\017)p 2052 1921 54 3 v 2064 1954 a FL(2)1994 1999 y FF(1)2136 1980 y Fz(\001)g FE(\(ln)12 b FH(\033)2325 1988 y FG(t)2349 1998 y FL(2)2386 1980 y FE(\))2416 1944 y FF(2)2484 1980 y FH(:)515 2132 y FJ(T)-6 b(erm)30 b(\(3.27\))515 2223 y FO(As)d(far)g(as)g(the)h(norm)f(is)h(concerned,)e(the)i(v)n(ector)f (\(3.27\))f(is)i(equiv)-5 b(alen)n(t)27 b(to)g(the)h(v)n(ector)798 2352 y Fy(\000)833 2412 y FH(H)23 b Fz(\000)17 b FH(H)1061 2420 y FG(\033)1097 2428 y Fq(t)1120 2440 y FL(2)1160 2352 y Fy(\001)1208 2412 y FH(e)1244 2372 y FG(i\015)1298 2380 y Fq(\033)1332 2391 y(t)1355 2403 y FL(2)1396 2317 y Fm(\020)1431 2372 y Fw(v)1467 2383 y Fq(j)1498 2372 y FG(;)p Fx(r)p FG(E)1616 2331 y Fq(\033)1650 2342 y(t)1673 2354 y FL(2)1613 2393 y Fl(P)1714 2372 y FG(;s)1762 2317 y Fm(\021)1800 2412 y FH( )1853 2374 y FF(\()p FG(t)1901 2384 y FL(1)1933 2374 y FF(\))1850 2432 y FG(j;\033)1930 2440 y Fq(t)1953 2452 y FL(2)2016 2412 y FE(=)1195 b FI(\(3.30\))1137 2674 y FE(=)21 b FH(e)1254 2633 y FG(i\015)1308 2641 y Fq(\033)1342 2652 y(t)1365 2664 y FL(2)1406 2578 y Fm(\020)1440 2633 y Fw(v)1476 2644 y Fq(j)1508 2633 y FG(;)p Fx(r)p FG(E)1626 2592 y Fq(\033)1660 2603 y(t)1683 2615 y FL(2)1623 2654 y Fl(P)1723 2633 y FG(;s)1771 2578 y Fm(\021)1810 2674 y FH(g)1862 2572 y Fy(Z)1939 2592 y FG(\033)1975 2600 y Fq(t)1998 2612 y FL(2)1905 2742 y FF(0)2052 2591 y Fy(\020)2097 2674 y FH(b)13 b FE(\()p FA(k)p FE(\))k(+)f FH(b)2376 2638 y Fx(y)2422 2674 y FE(\()p FA(k)p FE(\))2528 2591 y Fy(\021)2647 2625 y FH(d)2687 2594 y FF(3)2722 2625 y FH(k)p 2597 2657 218 4 v 2597 2672 a Fy(p)p 2674 2672 141 4 v 64 x FE(2)d Fz(j)p FA(k)p Fz(j)2824 2674 y FH( )2877 2635 y FF(\()p FG(t)2925 2645 y FL(1)2957 2635 y FF(\))2874 2694 y FG(j;\033)2954 2702 y Fq(t)2977 2714 y FL(2)515 2885 y FO(whose)27 b(norm)g(can)g(b)r (e)h(estimated)f(starting)g(from)1090 2970 y Fy(\015)1090 3015 y(\015)1090 3060 y(\015)1090 3104 y(\015)1090 3149 y(\015)1132 3101 y FH(g)1185 2999 y Fy(Z)1261 3019 y FG(\033)1297 3027 y Fq(t)1320 3039 y FL(2)1227 3169 y FF(0)1374 3018 y Fy(\020)1420 3101 y FH(b)13 b FE(\()o FA(k)p FE(\))k(+)g FH(b)1699 3065 y Fx(y)1744 3101 y FE(\()p FA(k)p FE(\))1851 3018 y Fy(\021)1970 3052 y FH(d)2010 3021 y FF(3)2044 3052 y FH(k)p 1919 3084 218 4 v 1919 3099 a Fy(p)p 1996 3099 141 4 v 64 x FE(2)c Fz(j)q FA(k)p Fz(j)2147 3101 y FH( )2200 3062 y FF(\()o FG(t)2247 3072 y FL(1)2279 3062 y FF(\))2197 3121 y FG(j;\033)2277 3129 y Fq(t)2300 3141 y FL(2)2340 2970 y Fy(\015)2340 3015 y(\015)2340 3060 y(\015)2340 3104 y(\015)2340 3149 y(\015)2404 3101 y Fz(\024)807 b FI(\(3.31\))1423 3391 y Fz(\024)1504 3260 y Fy(\015)1504 3305 y(\015)1504 3350 y(\015)1504 3395 y(\015)1504 3440 y(\015)1546 3391 y FH(g)1598 3290 y Fy(Z)1675 3310 y FG(\033)1711 3318 y Fq(t)1734 3330 y FL(2)1641 3460 y FF(0)1788 3391 y FH(b)1821 3355 y Fx(y)1866 3391 y FE(\()p FA(k)p FE(\))2046 3343 y FH(d)2086 3311 y FF(3)2120 3343 y FH(k)p 1995 3374 218 4 v 1995 3390 a Fy(p)p 2072 3390 141 4 v 64 x FE(2)13 b Fz(j)q FA(k)p Fz(j)2223 3391 y FH( )2276 3353 y FF(\()o FG(t)2323 3363 y FL(1)2356 3353 y FF(\))2273 3411 y FG(j;\033)2353 3419 y Fq(t)2376 3431 y FL(2)2417 3260 y Fy(\015)2417 3305 y(\015)2417 3350 y(\015)2417 3395 y(\015)2417 3440 y(\015)1423 3644 y Fz(\024)21 b FH(C)h Fz(\001)17 b FH(\033)1663 3652 y FG(t)1687 3662 y FL(2)1740 3644 y Fz(\001)h FH(t)1807 3597 y Fx(\000)1865 3578 y FL(3)p Fq(\017)p 1865 3586 54 3 v 1877 3619 a FL(2)1807 3663 y FF(1)515 3808 y FO(b)r(ecause)27 b FH(b)13 b FE(\()o FA(k)p FE(\))g FH( )1040 3770 y FF(\()p FG(t)1088 3780 y FL(1)1120 3770 y FF(\))1037 3828 y FG(j;\033)1117 3836 y Fq(t)1140 3848 y FL(2)1202 3808 y FE(=)21 b(0)28 b FO(for)f FA(k)21 b Fz(2)h(f)p FA(k)f FE(:)43 b Fz(j)p FA(k)p Fz(j)22 b(\024)o FH(\033)2001 3816 y FG(t)2025 3826 y FL(2)2062 3808 y Fz(g)p FO(.)515 3919 y(In)d(conclusion)f(the)h (norm)f(of)h(the)g(v)n(ector)f(corresp)r(onding)f(to)h(the)i (di\033erence)e FJ(D1\))k FO(is)d(b)r(ounded)515 4010 y(b)n(y)27 b(a)g(quan)n(tit)n(y)g(of)h(order)1491 4128 y FH(t)1519 4081 y Fx(\000)1577 4062 y FL(5)p Fq(\017)p 1577 4070 V 1590 4103 a FL(2)1519 4147 y FF(1)1662 4128 y Fz(\001)17 b FE(\(ln)c FH(\033)1851 4136 y FG(t)1875 4146 y FL(2)1911 4128 y FE(\))1941 4092 y FF(2)1993 4128 y FE(+)j FH(t)2097 4136 y FF(2)2149 4128 y Fz(\001)h FH(\033)2231 4136 y FG(t)2255 4146 y FL(2)2309 4128 y Fz(\001)g FH(t)2385 4062 y FL(3)p Fq(\017)p 2385 4070 V 2397 4103 a FL(2)2375 4147 y FF(1)2470 4128 y FH(:)780 b FI(\(3.32\))515 4342 y FP(3.1.3)93 b(V)-8 b(ariation)32 b(of)g(the)g(infrared)g(cut-o\033:)f Fp(r)-5 b(e)g(gular)34 b(blo)-5 b(ck)515 4487 y FO(Let)27 b(us)h(study)g(the)g(di\033erence) 515 4631 y FG(N)5 b FF(\()p FG(t)616 4641 y FL(1)648 4631 y FF(\))538 4659 y Fy(X)541 4817 y FG(j)s FF(=1)685 4730 y FH(e)721 4695 y FG(iH)t(t)821 4705 y FL(1)857 4730 y Fz(W)933 4738 y FG(\033)969 4746 y Fq(t)992 4758 y FL(2)1046 4730 y FE(\()o FA(v)1122 4738 y FG(j)1155 4730 y FH(;)13 b(t)1217 4738 y FF(1)1251 4730 y FE(\))g FH(W)1378 4695 y Fx(y)1367 4744 y FG(\033)1403 4752 y Fq(t)1426 4764 y FL(2)1479 4670 y Fy(\000)1514 4730 y Fz(r)p FH(E)1639 4684 y FG(\033)1675 4692 y Fq(t)1698 4704 y FL(2)1635 4751 y Fw(P)1739 4670 y Fy(\001)1787 4580 y(2)1787 4715 y(4)1879 4671 y FH(e)1915 4631 y FG(i\015)1969 4639 y Fq(\033)2003 4650 y(t)2026 4662 y FL(2)2068 4576 y Fm(\020)2102 4631 y Fw(v)2138 4642 y Fq(j)2170 4631 y FG(;)p Fx(r)p FG(E)2288 4590 y Fq(\033)2322 4601 y(t)2345 4613 y FL(2)2285 4652 y Fl(P)2385 4631 y FG(;t)2428 4641 y FL(1)2460 4576 y Fm(\021)2499 4671 y FH(W)2572 4679 y FG(\033)2608 4687 y Fq(t)2631 4699 y FL(2)2684 4611 y Fy(\000)2719 4671 y Fz(r)p FH(E)2844 4624 y FG(\033)2880 4632 y Fq(t)2903 4644 y FL(2)2840 4692 y Fw(P)2944 4611 y Fy(\001)2992 4671 y FH(e)3028 4639 y Fx(\000)p FG(iE)3146 4598 y Fq(\033)3180 4609 y(t)3203 4621 y FL(2)3143 4660 y Fl(P)3243 4639 y FG(t)3267 4649 y FL(1)3303 4671 y FH( )3356 4633 y FF(\()p FG(t)3404 4643 y FL(1)3436 4633 y FF(\))3353 4691 y FG(j;\033)3433 4699 y Fq(t)3456 4711 y FL(2)3497 4671 y Fz(\000)1879 4811 y(\000)p FH(e)1975 4770 y FG(i\015)2029 4778 y Fq(\033)2063 4789 y(t)2086 4801 y FL(1)2127 4716 y Fm(\020)2162 4770 y Fw(v)2198 4781 y Fq(j)2229 4770 y FG(;)p Fx(r)p FG(E)2347 4730 y Fq(\033)2381 4741 y(t)2404 4753 y FL(1)2344 4791 y Fl(P)2445 4770 y FG(;t)2488 4780 y FL(1)2520 4716 y Fm(\021)2559 4811 y FH(W)2632 4819 y FG(\033)2668 4827 y Fq(t)2691 4839 y FL(1)2744 4751 y Fy(\000)2779 4811 y Fz(r)p FH(E)2904 4764 y FG(\033)2940 4772 y Fq(t)2963 4784 y FL(1)2900 4832 y Fw(P)3003 4751 y Fy(\001)3051 4811 y FH(e)3087 4779 y Fx(\000)p FG(iE)3205 4738 y Fq(\033)3239 4749 y(t)3262 4761 y FL(1)3202 4800 y Fl(P)3303 4779 y FG(t)3327 4789 y FL(1)3363 4811 y FH( )3416 4773 y FF(\()p FG(t)3464 4783 y FL(1)3496 4773 y FF(\))3413 4831 y FG(j;\033)3493 4839 y Fq(t)3516 4851 y FL(1)3599 4580 y Fy(3)3599 4715 y(5)1950 5059 y FO(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 542 523 a FO(for)27 b(eac)n(h)g(single)g(cell)h (in)g FE(\000)1380 491 y FF(\()p FG(t)1428 501 y FL(1)1460 491 y FF(\))1488 523 y FO(.)g(The)f(norm)g(of)h(the)g(cell-v)n(ector)d (is)j(con)n(trolled)e(as)h(follo)n(ws)625 598 y Fs(\015)625 638 y(\015)625 678 y(\015)625 718 y(\015)664 694 y Fr(e)697 655 y FG(i\015)751 663 y Fq(\033)785 674 y(t)808 686 y FL(2)850 600 y Fm(\020)884 655 y Fw(v)920 666 y Fq(j)952 655 y FG(;)p Fx(r)p FG(E)1070 614 y Fq(\033)1104 625 y(t)1127 637 y FL(2)1067 676 y Fl(P)1167 655 y FG(;t)1210 665 y FL(1)1242 600 y Fm(\021)1281 694 y Fr(e)1314 666 y Fx(\000)p FG(iE)1432 625 y Fq(\033)1466 636 y(t)1489 648 y FL(2)1429 687 y Fl(P)1529 666 y FG(t)1553 676 y FL(1)1590 694 y Fr(W)1657 702 y FG(\033)1693 710 y Fq(t)1716 722 y FL(2)1768 621 y Fs(\020)1810 694 y Fu(r)p Fr(E)1925 651 y FG(\033)1961 659 y Fq(t)1984 671 y FL(2)1921 718 y Fw(P)2024 621 y Fs(\021)2078 694 y Fr( )2126 659 y FF(\()p FG(t)2174 669 y FL(1)2206 659 y FF(\))2124 716 y FG(j;\033)2204 724 y Fq(t)2227 736 y FL(2)2268 694 y Fu(\000)1419 889 y(\000)p Fr(e)1507 849 y FG(i\015)1561 857 y Fq(\033)1595 868 y(t)1618 880 y FL(1)1659 794 y Fm(\020)1694 849 y Fw(v)1730 860 y Fq(j)1761 849 y FG(;)p Fx(r)p FG(E)1879 808 y Fq(\033)1913 819 y(t)1936 831 y FL(1)1876 870 y Fl(P)1977 849 y FG(;t)2020 859 y FL(1)2052 794 y Fm(\021)2090 889 y Fr(e)2123 861 y Fx(\000)p FG(iE)2241 820 y Fq(\033)2275 831 y(t)2298 843 y FL(1)2238 882 y Fl(P)2339 861 y FG(t)2363 871 y FL(1)2399 889 y Fr(W)2466 897 y FG(\033)2502 905 y Fq(t)2525 917 y FL(1)2577 815 y Fs(\020)2619 889 y Fu(r)p Fr(E)2734 845 y FG(\033)2770 853 y Fq(t)2793 865 y FL(1)2730 912 y Fw(P)2833 815 y Fs(\021)2887 889 y Fr( )2935 854 y FF(\()q FG(t)2984 864 y FL(1)3016 854 y FF(\))2933 911 y FG(j;\033)3013 919 y Fq(t)3036 931 y FL(1)3077 793 y Fs(\015)3077 832 y(\015)3077 872 y(\015)3077 912 y(\015)3136 889 y Fu(\024)515 1186 y Fz(\024)683 1078 y Fy(\015)683 1123 y(\015)683 1168 y(\015)683 1213 y(\015)725 1081 y(\024)766 1186 y FH(e)802 1146 y FG(i\015)856 1154 y Fq(\033)890 1165 y(t)913 1177 y FL(2)954 1091 y Fm(\020)989 1146 y Fw(v)1025 1157 y Fq(j)1056 1146 y FG(;)p Fx(r)p FG(E)1174 1105 y Fq(\033)1208 1116 y(t)1231 1128 y FL(2)1171 1167 y Fl(P)1272 1146 y FG(;t)1315 1156 y FL(1)1347 1091 y Fm(\021)1402 1186 y Fz(\000)17 b FH(e)1515 1146 y FG(i\015)1569 1154 y Fq(\033)1603 1165 y(t)1626 1177 y FL(1)1668 1091 y Fm(\020)1702 1146 y Fw(v)1738 1157 y Fq(j)1770 1146 y FG(;)p Fx(r)p FG(E)1888 1105 y Fq(\033)1922 1116 y(t)1945 1128 y FL(1)1885 1167 y Fl(P)1985 1146 y FG(;t)2028 1156 y FL(1)2060 1091 y Fm(\021)2099 1081 y Fy(\025)2152 1186 y FH(e)2188 1155 y Fx(\000)p FG(iE)2306 1114 y Fq(\033)2340 1125 y(t)2363 1137 y FL(2)2303 1176 y Fl(P)2403 1155 y FG(t)2427 1165 y FL(1)2464 1186 y FH(W)2537 1194 y FG(\033)2573 1202 y Fq(t)2596 1214 y FL(2)2649 1126 y Fy(\000)2684 1186 y Fz(r)p FH(E)2809 1140 y FG(\033)2845 1148 y Fq(t)2868 1160 y FL(2)2805 1208 y Fw(P)2909 1126 y Fy(\001)2957 1186 y FH( )3010 1148 y FF(\()o FG(t)3057 1158 y FL(1)3090 1148 y FF(\))3007 1206 y FG(j;\033)3087 1214 y Fq(t)3110 1226 y FL(2)3151 1078 y Fy(\015)3151 1123 y(\015)3151 1168 y(\015)3151 1213 y(\015)3323 1186 y FI(\(3.33\))657 1444 y FE(+)730 1336 y Fy(\015)730 1381 y(\015)730 1426 y(\015)730 1470 y(\015)773 1444 y FH(e)809 1404 y FG(i\015)863 1412 y Fq(\033)897 1423 y(t)920 1435 y FL(1)961 1349 y Fm(\020)996 1404 y Fw(v)1032 1415 y Fq(j)1063 1404 y FG(;)p Fx(r)p FG(E)1181 1363 y Fq(\033)1215 1374 y(t)1238 1386 y FL(1)1178 1425 y Fl(P)1278 1404 y FG(;t)1321 1414 y FL(1)1354 1349 y Fm(\021)1405 1361 y Fy(h)1441 1444 y FH(e)1477 1412 y Fx(\000)p FG(iE)1595 1372 y Fq(\033)1629 1383 y(t)1652 1395 y FL(2)1592 1433 y Fl(P)1693 1412 y FG(t)1717 1422 y FL(1)1770 1444 y Fz(\000)g FH(e)1883 1412 y Fx(\000)p FG(iE)2001 1372 y Fq(\033)2035 1383 y(t)2058 1395 y FL(1)1998 1433 y Fl(P)2098 1412 y FG(t)2122 1422 y FL(1)2158 1361 y Fy(i)2207 1444 y FH(W)2280 1452 y FG(\033)2316 1460 y Fq(t)2339 1472 y FL(2)2393 1384 y Fy(\000)2428 1444 y Fz(r)p FH(E)2553 1398 y FG(\033)2589 1406 y Fq(t)2612 1418 y FL(2)2549 1465 y Fw(P)2652 1384 y Fy(\001)2700 1444 y FH( )2753 1406 y FF(\()p FG(t)2801 1416 y FL(1)2833 1406 y FF(\))2750 1464 y FG(j;\033)2830 1472 y Fq(t)2853 1484 y FL(2)2894 1336 y Fy(\015)2894 1381 y(\015)2894 1426 y(\015)2894 1470 y(\015)3323 1444 y FI(\(3.34\))675 1702 y FE(+)747 1594 y Fy(\015)747 1639 y(\015)747 1683 y(\015)747 1728 y(\015)790 1702 y FH(e)826 1662 y FG(i\015)880 1670 y Fq(\033)914 1681 y(t)937 1693 y FL(1)978 1607 y Fm(\020)1013 1662 y Fw(v)1049 1673 y Fq(j)1080 1662 y FG(;)p Fx(r)p FG(E)1198 1621 y Fq(\033)1232 1632 y(t)1255 1644 y FL(1)1195 1683 y Fl(P)1296 1662 y FG(;t)1339 1672 y FL(1)1371 1607 y Fm(\021)1409 1702 y FH(e)1445 1670 y Fx(\000)p FG(iE)1563 1629 y Fq(\033)1597 1640 y(t)1620 1652 y FL(1)1560 1691 y Fl(P)1661 1670 y FG(t)1685 1680 y FL(1)1734 1619 y Fy(h)1770 1702 y FH(W)1843 1710 y FG(\033)1879 1718 y Fq(t)1902 1730 y FL(2)1955 1642 y Fy(\000)1990 1702 y Fz(r)p FH(E)2115 1655 y FG(\033)2151 1663 y Fq(t)2174 1675 y FL(2)2111 1723 y Fw(P)2215 1642 y Fy(\001)2263 1702 y FH( )2316 1664 y FF(\()p FG(t)2364 1674 y FL(1)2396 1664 y FF(\))2313 1722 y FG(j;\033)2393 1730 y Fq(t)2416 1742 y FL(2)2474 1702 y Fz(\000)g FH(W)2624 1710 y FG(\033)2660 1718 y Fq(t)2683 1730 y FL(1)2736 1642 y Fy(\000)2771 1702 y Fz(r)p FH(E)2896 1655 y FG(\033)2932 1663 y Fq(t)2955 1675 y FL(1)2892 1723 y Fw(P)2996 1642 y Fy(\001)3044 1702 y FH( )3097 1664 y FF(\()o FG(t)3144 1674 y FL(1)3176 1664 y FF(\))3094 1722 y FG(j;\033)3174 1730 y Fq(t)3197 1742 y FL(1)3237 1619 y Fy(i)3274 1594 y(\015)3274 1639 y(\015)3274 1683 y(\015)3274 1728 y(\015)3323 1702 y FI(\(3.35\))515 1913 y FO(and)32 b(eac)n(h)g(term)h(\(3.33\),)f (\(3.34\),)h(\(3.35\))f(is)g(in\034nitesimal)i(in)f(the)g(limit)g(in)h (whic)n(h)e(the)i(in-)515 2004 y(frared)26 b(cut-o\033)i(is)f(remo)n(v) n(ed.)f(The)i(argumen)n(ts)e(for)h(this)h(are)f(the)g(follo)n(wing)g (ones.)515 2187 y FJ(T)-6 b(erm)30 b(\(3.33\))515 2278 y FO(W)-7 b(e)28 b(can)f(easily)g(estimate)612 2372 y Fy(\015)612 2417 y(\015)612 2462 y(\015)612 2506 y(\015)655 2375 y(\024)695 2480 y FH(e)731 2440 y FG(i\015)785 2448 y Fq(\033)819 2459 y(t)842 2471 y FL(2)884 2385 y Fm(\020)918 2440 y Fw(v)954 2451 y Fq(j)986 2440 y FG(;)p Fx(r)p FG(E)1104 2399 y Fq(\033)1138 2410 y(t)1161 2422 y FL(2)1101 2461 y Fl(P)1201 2440 y FG(;t)1244 2450 y FL(1)1276 2385 y Fm(\021)1332 2480 y Fz(\000)17 b FH(e)1445 2440 y FG(i\015)1499 2448 y Fq(\033)1533 2459 y(t)1556 2471 y FL(1)1597 2385 y Fm(\020)1632 2440 y Fw(v)1668 2451 y Fq(j)1699 2440 y FG(;)p Fx(r)p FG(E)1817 2399 y Fq(\033)1851 2410 y(t)1874 2422 y FL(1)1814 2461 y Fl(P)1915 2440 y FG(;t)1958 2450 y FL(1)1990 2385 y Fm(\021)2028 2375 y Fy(\025)2082 2480 y FH(e)2118 2444 y Fx(\000)p FG(iE)2236 2404 y Fq(\033)2270 2415 y(t)2293 2427 y FL(2)2233 2465 y Fl(P)2333 2444 y FG(t)2357 2454 y FL(1)2393 2480 y FH(W)2466 2488 y FG(\033)2502 2496 y Fq(t)2525 2508 y FL(2)2578 2420 y Fy(\000)2614 2480 y Fz(r)p FH(E)2739 2434 y FG(\033)2775 2442 y Fq(t)2798 2454 y FL(2)2735 2501 y Fw(P)2838 2420 y Fy(\001)2886 2480 y FH( )2939 2442 y FF(\()p FG(t)2987 2452 y FL(1)3019 2442 y FF(\))2936 2500 y FG(j;\033)3016 2508 y Fq(t)3039 2520 y FL(2)3080 2372 y Fy(\015)3080 2417 y(\015)3080 2462 y(\015)3080 2506 y(\015)3206 2480 y Fz(\024)1179 2738 y(\024)29 b FE(sup)1260 2799 y Fw(P)p Fx(2)p FF(\006)1404 2630 y Fy(\014)1404 2675 y(\014)1404 2719 y(\014)1404 2764 y(\014)1430 2738 y FH(e)1466 2698 y FG(i\015)1520 2706 y Fq(\033)1554 2717 y(t)1577 2729 y FL(2)1618 2643 y Fm(\020)1652 2698 y Fw(v)1688 2709 y Fq(j)1720 2698 y FG(;)p Fx(r)p FG(E)1838 2657 y Fq(\033)1872 2668 y(t)1895 2680 y FL(2)1835 2719 y Fl(P)1935 2698 y FG(;t)1978 2708 y FL(1)2010 2643 y Fm(\021)2066 2738 y Fz(\000)17 b FH(e)2179 2698 y FG(i\015)2233 2706 y Fq(\033)2267 2717 y(t)2290 2729 y FL(1)2331 2643 y Fm(\020)2366 2698 y Fw(v)2402 2709 y Fq(j)2433 2698 y FG(;)p Fx(r)p FG(E)2551 2657 y Fq(\033)2585 2668 y(t)2608 2680 y FL(1)2548 2719 y Fl(P)2649 2698 y FG(;t)2692 2708 y FL(1)2724 2643 y Fm(\021)2762 2630 y Fy(\014)2762 2675 y(\014)2762 2719 y(\014)2762 2764 y(\014)2805 2738 y Fz(\001)2844 2652 y Fy(\015)2844 2697 y(\015)2844 2742 y(\015)2886 2738 y FH( )2939 2700 y FF(\()p FG(t)2987 2710 y FL(1)3019 2700 y FF(\))2936 2758 y FG(j;\033)3016 2766 y Fq(t)3039 2778 y FL(2)3080 2652 y Fy(\015)3080 2697 y(\015)3080 2742 y(\015)515 2952 y FO(and)27 b(the)h(sup)g(is)f(b)r(ounded)h(in)g (terms)f(of)h(\(see)f(Lemma)g(A1\):)589 3069 y Fy(\014)589 3114 y(\014)614 3132 y FH(\015)654 3140 y FG(\033)690 3148 y Fq(t)713 3160 y FL(2)766 3072 y Fy(\000)802 3132 y FA(v)849 3140 y FG(j)881 3132 y FH(;)13 b Fz(r)p FH(E)1040 3086 y FG(\033)1076 3094 y Fq(t)1099 3106 y FL(2)1036 3153 y Fw(P)1139 3132 y FH(;)h(t)1202 3140 y FF(1)1236 3072 y Fy(\001)1288 3132 y Fz(\000)j FH(\015)1405 3140 y FG(\033)1441 3148 y Fq(t)1464 3160 y FL(1)1517 3072 y Fy(\000)1552 3132 y FA(v)1599 3140 y FG(j)1632 3132 y FH(;)c Fz(r)p FH(E)1791 3086 y FG(\033)1827 3094 y Fq(t)1850 3106 y FL(1)1787 3153 y Fw(P)1890 3132 y FH(;)g(t)1952 3140 y FF(1)1987 3072 y Fy(\001)2022 3069 y(\014)2022 3114 y(\014)2069 3132 y Fz(\024)21 b FH(C)h Fz(\001)c FE(\()o FH(\033)2339 3140 y FG(t)2363 3150 y FL(1)2400 3132 y FE(\))2439 3077 y FL(1)p 2439 3085 29 3 v 2439 3118 a(4)2499 3132 y Fz(\001)f FH(t)2565 3094 y FF(2\(1)p Fx(\000)p FG(\013)p FF(\))2565 3151 y(1)2783 3132 y FE(+)g FH(C)22 b Fz(\001)c FH(t)3004 3140 y FF(1)3055 3132 y Fz(\001)f FH(\033)3137 3140 y FG(t)3161 3150 y FL(1)3271 3132 y FI(\(3.36\))515 3330 y FO(so)25 b(that)h(for)g FH(\013)g FO(su\036cien)n(tly)g(close)f(to)h FE(1)h FO(w)n(e)e(can)h (surely)f(pro)n(vide)g(a)h(b)r(ound)g(of)g(order)f Fr(t)3232 3285 y Fx(\000)3290 3266 y FL(3)p Fq(\017)p 3290 3274 54 3 v 3303 3307 a FL(2)3354 3285 y FF(+1)3232 3351 y(1)3448 3330 y Fu(\001)515 3435 y Ft(\()p Fr(\033)582 3443 y FG(t)606 3453 y FL(1)643 3435 y Ft(\))680 3386 y FL(1)p 680 3394 29 3 v 680 3427 a(4)750 3435 y FO(for)i(the)h(expression)e (\(3.33\);)515 3526 y FJ(T)-6 b(erm)34 b(\(3.34\))515 3617 y FO(Because)26 b(of)i(the)g(regularit)n(y)d(prop)r(erties)i(of)g (the)h(energy)f FH(E)2403 3586 y FG(\033)2399 3632 y Fw(P)2449 3617 y FH(;)13 b FA(P)22 b Fz(2)f FE(\006)p FO(,)29 b(w)n(e)e(easily)f(get)1370 3711 y Fy(\014)1370 3756 y(\014)1370 3801 y(\014)1370 3846 y(\014)1396 3819 y FH(e)1432 3784 y Fx(\000)p FG(iE)1550 3743 y Fq(\033)1584 3754 y(t)1607 3766 y FL(2)1547 3805 y Fl(P)1647 3784 y FG(t)1671 3794 y FL(1)1724 3819 y Fz(\000)17 b FH(e)1837 3784 y Fx(\000)p FG(iE)1955 3743 y Fq(\033)1989 3754 y(t)2012 3766 y FL(1)1952 3805 y Fl(P)2052 3784 y FG(t)2076 3794 y FL(1)2113 3711 y Fy(\014)2113 3756 y(\014)2113 3801 y(\014)2113 3846 y(\014)2160 3819 y Fz(\024)k FH(C)h Fz(\001)17 b FH(\033)2413 3753 y FL(1)p 2413 3761 V 2413 3794 a(4)2400 3837 y FG(t)2424 3847 y FL(1)2477 3819 y Fz(\001)h FH(t)2544 3827 y FF(1)2591 3819 y FH(:)659 b FI(\(3.37\))515 4061 y FO(whic)n(h)27 b(implies)h(a)f(b)r(ound)h (from)f(ab)r(o)n(v)n(e)f(of)i(order)e FH(\033)2159 3995 y FL(1)p 2159 4003 V 2159 4036 a(4)2146 4079 y FG(t)2170 4089 y FL(1)2223 4061 y Fz(\001)18 b FH(t)2290 4014 y Fx(\000)2347 3995 y FL(3)p Fq(\017)p 2347 4003 54 3 v 2360 4036 a FL(2)2411 4014 y FF(+1)2290 4080 y(1)2520 4061 y FO(for)27 b(the)h(norm)f(\(3.34\))p FI(.)515 4153 y FJ(T)-6 b(erm)34 b(\(3.35\))515 4244 y FO(Exploiting)27 b(the)h Fp(Sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(op)g(erties)7 b FO(,)27 b(Subsection)h(1.3.1,)e(w)n(e)h(can)g(estimate:)777 4341 y Fy(\015)777 4386 y(\015)777 4430 y(\015)820 4427 y FH(W)893 4435 y FG(\033)929 4443 y Fq(t)952 4455 y FL(2)1005 4366 y Fy(\000)1040 4427 y Fz(r)p FH(E)1165 4380 y FG(\033)1201 4388 y Fq(t)1224 4400 y FL(2)1161 4448 y Fw(P)1265 4366 y Fy(\001)1313 4427 y FH( )1366 4389 y FF(\()p FG(t)1414 4399 y FL(1)1446 4389 y FF(\))1363 4447 y FG(j;\033)1443 4455 y Fq(t)1466 4467 y FL(2)1524 4427 y Fz(\000)17 b FH(W)1674 4435 y FG(\033)1710 4443 y Fq(t)1733 4455 y FL(1)1786 4366 y Fy(\000)1821 4427 y Fz(r)p FH(E)1946 4380 y FG(\033)1982 4388 y Fq(t)2005 4400 y FL(1)1942 4448 y Fw(P)2046 4366 y Fy(\001)2094 4427 y FH( )2147 4389 y FF(\()o FG(t)2194 4399 y FL(1)2227 4389 y FF(\))2144 4447 y FG(j;\033)2224 4455 y Fq(t)2247 4467 y FL(1)2288 4341 y Fy(\015)2288 4386 y(\015)2288 4430 y(\015)2352 4427 y FE(=)890 4694 y(=)971 4586 y Fy(\015)971 4630 y(\015)971 4675 y(\015)971 4720 y(\015)1014 4634 y(R)1050 4729 y FF(\000)1088 4699 y FL(\()p Fq(t)1133 4711 y FL(1)1166 4699 y(\))1088 4749 y Fq(j)1210 4694 y FH(h)c FE(\()p FA(P)p FE(\))g(\()22 b FH(W)1525 4702 y FG(\033)1561 4710 y Fq(t)1584 4722 y FL(2)1637 4634 y Fy(\000)1673 4694 y Fz(r)p FH(E)1798 4647 y FG(\033)1834 4655 y Fq(t)1857 4667 y FL(2)1794 4715 y Fw(P)1897 4634 y Fy(\001)1945 4694 y FH( )1998 4647 y FG(\033)2034 4655 y Fq(t)2057 4667 y FL(2)1995 4715 y Fw(P)2115 4694 y Fz(\000)17 b FH(W)2265 4702 y FG(\033)2301 4710 y Fq(t)2324 4722 y FL(1)2377 4634 y Fy(\000)2412 4694 y Fz(r)p FH(E)2537 4647 y FG(\033)2573 4655 y Fq(t)2596 4667 y FL(1)2533 4715 y Fw(P)2636 4634 y Fy(\001)2684 4694 y FH( )2737 4647 y FG(\033)2773 4655 y Fq(t)2796 4667 y FL(1)2734 4715 y Fw(P)2860 4694 y FE(\))12 b FH(d)2942 4662 y FF(3)2977 4694 y FH(P)3037 4586 y Fy(\015)3037 4630 y(\015)3037 4675 y(\015)3037 4720 y(\015)1950 5059 y FO(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 905 574 a Fz(\024)986 469 y Fy(\032)1044 514 y(R)1080 609 y FF(\000)1118 580 y FL(\()p Fq(t)1163 592 y FL(1)1196 580 y(\))1118 629 y Fq(j)1240 574 y Fz(j)p FH(h)1305 582 y Fw(P)1356 574 y Fz(j)1377 539 y FF(2)1425 511 y Fy(\015)1425 555 y(\015)1467 574 y FA(I)1500 582 y Fw(P)1564 514 y Fy(\000)1599 574 y FH(\036)1645 528 y FG(\033)1681 536 y Fq(t)1704 548 y FL(2)1645 595 y Fw(P)1745 514 y Fy(\001)1797 574 y Fz(\000)17 b FA(I)1907 582 y Fw(P)1971 514 y Fy(\000)2006 574 y FH(\036)2052 528 y FG(\033)2088 536 y Fq(t)2111 548 y FL(1)2052 595 y Fw(P)2152 514 y Fy(\001)2187 511 y(\015)2187 555 y(\015)2230 527 y FF(2)2230 604 y Fx(F)2295 574 y FH(d)2335 542 y FF(3)2370 574 y FH(P)2430 469 y Fy(\033)2497 467 y FL(1)p 2497 475 29 3 v 2497 508 a(2)818 817 y Fz(\024)k FH(C)i Fz(\001)17 b FE(\()p FH(\033)1089 825 y FG(t)1113 835 y FL(1)1149 817 y FE(\))1189 762 y FL(1)p 1189 770 V 1189 803 a(4)1248 817 y Fz(\001)h FH(t)1315 770 y Fx(\000)1372 750 y FL(3)p Fq(\017)p 1372 758 54 3 v 1385 791 a FL(2)1315 836 y FF(1)1453 817 y FH(:)542 969 y FO(In)28 b(conclusion)f(the)h (norm)f(of)g(the)h(di\033erence)g FJ(D2\))i FO(is)e(surely)f(b)r (ounded)h(b)n(y)1681 1153 y FH(C)22 b Fz(\001)17 b FH(t)1834 1086 y FL(3)p Fq(\017)p 1834 1094 V 1847 1127 a FL(2)1898 1106 y FF(+1)1824 1172 y(1)1996 1153 y Fz(\001)g FE(\()p FH(\033)2108 1161 y FG(t)2132 1171 y FL(1)2168 1153 y FE(\))2208 1097 y FL(1)p 2208 1105 29 3 v 2208 1138 a(4)2280 1153 y FH(:)970 b FI(\(3.38\))515 1367 y FP(3.1.4)93 b(V)-8 b(ariation)32 b(of)g(the)g(infrared)g(cut-o\033:)f Fp(dr)-5 b(essing)35 b(blo)-5 b(ck)515 1611 y(analysis)34 b(of)g(D3.1\))515 1794 y FO(W)-7 b(e)28 b(\034rst)f(de\034ne)h FH(')1119 1756 y FF(\()p FG(t)1167 1766 y FL(1)1199 1756 y FF(\))1119 1814 y FG(j;\033)1199 1822 y Fq(t)1222 1834 y FL(1)1290 1794 y FO(the)g FJ(r)l(e)l(gular)37 b(blo)l(ck)h FO(corresp)r(onding)25 b(to)j(the)f(time)i FN(t)2924 1806 y Ff(1)1004 2003 y FH(')1054 1965 y FF(\()p FG(t)1102 1975 y FL(1)1134 1965 y FF(\))1054 2023 y FG(j;\033)1134 2031 y Fq(t)1157 2043 y FL(1)1219 2003 y FE(:=)22 b FH(e)1358 1963 y FG(i\015)1412 1971 y Fq(\033)1446 1982 y(t)1469 1994 y FL(1)1510 1908 y Fm(\020)1545 1963 y Fw(v)1581 1974 y Fq(j)1612 1963 y FG(;)p Fx(r)p FG(E)1730 1922 y Fq(\033)1764 1933 y(t)1787 1945 y FL(1)1727 1984 y Fl(P)1828 1963 y FG(;t)1871 1973 y FL(1)1903 1908 y Fm(\021)1941 2003 y FH(e)1977 1967 y Fx(\000)p FG(iE)2095 1926 y Fq(\033)2129 1937 y(t)2152 1949 y FL(1)2092 1988 y Fl(P)2193 1967 y FG(t)2217 1977 y FL(1)2253 2003 y FH(W)2326 2011 y FG(\033)2362 2019 y Fq(t)2385 2031 y FL(1)2438 1943 y Fy(\000)2473 2003 y Fz(r)p FH(E)2598 1956 y FG(\033)2634 1964 y Fq(t)2657 1976 y FL(1)2594 2024 y Fw(P)2698 1943 y Fy(\001)2746 2003 y FH( )2799 1965 y FF(\()p FG(t)2847 1975 y FL(1)2879 1965 y FF(\))2796 2023 y FG(j;\033)2876 2031 y Fq(t)2899 2043 y FL(1)2957 2003 y FH(:)293 b FI(\(3.39\))515 2175 y FO(Then)27 b(the)h(di\033erence)g(in)n(v)n(olv)n(ed)e(in)i(the)g (step)f FJ(D3.1\))32 b FO(can)27 b(b)r(e)h(written)g(as:)515 2311 y FG(N)5 b FF(\()p FG(t)616 2321 y FL(1)648 2311 y FF(\))538 2339 y Fy(X)541 2497 y FG(j)s FF(=1)685 2410 y FH(e)721 2374 y FG(iH)t(t)821 2384 y FL(1)870 2327 y Fy(\020)916 2410 y Fz(W)992 2418 y FG(\033)1028 2426 y Fq(t)1051 2438 y FL(2)1104 2410 y FE(\()p FA(v)1181 2418 y FG(j)1213 2410 y FH(;)13 b(t)1275 2418 y FF(1)1310 2410 y FE(\))f FH(W)1436 2374 y Fx(y)1425 2423 y FG(\033)1461 2431 y Fq(t)1484 2443 y FL(2)1538 2410 y FE(\()o FA(v)1614 2418 y FG(j)1647 2410 y FE(\))17 b Fz(\000)f(W)1846 2418 y FG(\033)1882 2426 y Fq(t)1905 2438 y FL(1)1959 2410 y FE(\()p FA(v)2036 2418 y FG(j)2068 2410 y FH(;)d(t)2130 2418 y FF(1)2164 2410 y FE(\))g FH(W)2291 2374 y Fx(y)2280 2423 y FG(\033)2316 2431 y Fq(t)2339 2443 y FL(1)2392 2410 y FE(\()p FA(v)2469 2418 y FG(j)2501 2410 y FE(\))2531 2327 y Fy(\021)2590 2410 y FH(W)2663 2418 y FG(\033)2699 2426 y Fq(t)2722 2438 y FL(2)2775 2410 y FE(\()p FA(v)2852 2418 y FG(j)2884 2410 y FE(\))g FH(W)3011 2374 y Fx(y)3000 2423 y FG(\033)3036 2431 y Fq(t)3059 2443 y FL(2)3112 2350 y Fy(\000)3147 2410 y Fz(r)p FH(E)3272 2364 y FG(\033)3308 2372 y Fq(t)3331 2384 y FL(2)3268 2431 y Fw(P)3371 2350 y Fy(\001)3419 2410 y FH(')3469 2372 y FF(\()q FG(t)3518 2382 y FL(1)3550 2372 y FF(\))3469 2430 y FG(j;\033)3549 2438 y Fq(t)3572 2450 y FL(1)3626 2410 y FH(:)22 b FI(\(3.40\))515 2641 y FO(W)-7 b(e)35 b(can)g(restrict)f(the)h(analysis)e(to)i(eac)n(h) f(single)h(cell.)g(By)f(standard)g(algebraic)f(steps)i(w)n(e)515 2733 y(ha)n(v)n(e)560 2885 y Fz(W)636 2893 y FG(\033)672 2901 y Fq(t)695 2913 y FL(2)749 2885 y FE(\()p FA(v)826 2893 y FG(j)858 2885 y FH(;)13 b(t)920 2893 y FF(1)955 2885 y FE(\))f FH(W)1081 2849 y Fx(y)1070 2898 y FG(\033)1106 2906 y Fq(t)1129 2918 y FL(2)1182 2885 y FE(\()p FA(v)1259 2893 y FG(j)1291 2885 y FE(\))17 b Fz(\000)g(W)1491 2893 y FG(\033)1527 2901 y Fq(t)1550 2913 y FL(1)1604 2885 y FE(\()p FA(v)1681 2893 y FG(j)1713 2885 y FH(;)c(t)1775 2893 y FF(1)1809 2885 y FE(\))g FH(W)1936 2849 y Fx(y)1925 2898 y FG(\033)1961 2906 y Fq(t)1984 2918 y FL(1)2037 2885 y FE(\()p FA(v)2114 2893 y FG(j)2146 2885 y FE(\))21 b(=)515 3214 y(=)82 b Fz(W)733 3222 y FG(\033)769 3230 y Fq(t)792 3242 y FL(1)846 3214 y FE(\()p FA(v)923 3222 y FG(j)955 3214 y FH(;)13 b(t)1017 3222 y FF(1)1052 3214 y FE(\))f FH(W)1178 3183 y Fx(y)1167 3227 y FG(\033)1203 3235 y Fq(t)1226 3247 y FL(1)1279 3214 y FE(\()p FA(v)1356 3222 y FG(j)1388 3214 y FE(\))1431 3087 y Fy( )1492 3214 y FH(e)1528 3134 y FG(ig)1582 3113 y FL(2)1625 3094 y Fm(R)1663 3093 y Fq(\033)1697 3104 y(t)1720 3116 y FL(1)1652 3145 y Fq(\033)1686 3156 y(t)1709 3168 y FL(2)1782 3106 y(sin)o(\()q Fk(\000j)p Fl(k)p Fk(j)p Fq(t)2014 3118 y FL(1)2047 3106 y(+)p Fl(k)p Fk(\001)p Fl(x)p FL(\))p 1782 3123 416 3 v 1783 3179 a(2)p Fk(j)p Fl(k)p Fk(j)1882 3167 y FL(3)1915 3187 y FE(\()1944 3179 y FL(1)p Fk(\000)2020 3168 y Fj(b)2017 3179 y Fl(k)q Fk(\001)p Fl(v)2103 3192 y Fq(j)2134 3187 y FE(\))2164 3152 y FL(2)2207 3134 y FG(d)2239 3113 y FL(3)2272 3134 y FG(k)2310 3087 y Fy(!)2383 3064 y(0)2383 3199 y(@)2451 3214 y FH(e)2487 3140 y Fx(\000)p FG(g)2577 3100 y Fm(R)e Fq(\033)2649 3111 y(t)2672 3123 y FL(1)2605 3151 y Fq(\033)2639 3162 y(t)2662 3174 y FL(2)2734 3097 y Fq(a)p FL(\()p Fl(k)p FL(\))2844 3105 y FE(\()2874 3097 y Fq(e)2902 3080 y(i)p Fk(j)p Fl(k)p Fk(j)p Fq(t)3018 3092 y FL(1)3055 3097 y Fk(\000)p Fq(e)3128 3081 y(i)p Fl(k)p Fk(\001)p Fl(x)3240 3105 y FE(\))3270 3097 y Fk(\000)p Fq(h:c:)p 2734 3129 681 3 v 2914 3179 a Fk(j)p Fl(k)p Fk(j)2985 3187 y FE(\()3015 3179 y FL(1)p Fk(\000)3091 3168 y Fj(b)3088 3179 y Fl(k)p Fk(\001)p Fl(v)3173 3192 y Fq(j)3204 3187 y FE(\))3461 3121 y Fq(d)3492 3104 y FL(3)3524 3121 y Fq(k)p 3434 3129 150 3 v 3434 3137 a Fx(p)p 3484 3137 100 3 v 35 x FL(2)p Fk(j)p Fl(k)p Fk(j)3615 3214 y Fz(\000)16 b FH(I)3731 3064 y Fy(1)3731 3199 y(A)3800 3214 y FI(\(3.41\))660 3536 y FE(+)p Fz(W)796 3544 y FG(\033)832 3552 y Fq(t)855 3564 y FL(1)909 3536 y FE(\()p FA(v)986 3544 y FG(j)1018 3536 y FH(;)d(t)1080 3544 y FF(1)1114 3536 y FE(\))g FH(W)1241 3484 y Fk(y)1230 3549 y FG(\033)1266 3557 y Fq(t)1289 3569 y FL(1)1342 3536 y FE(\()p FA(v)1419 3544 y FG(j)1451 3536 y FE(\))1494 3409 y Fy( )1554 3536 y FH(e)1590 3456 y FG(ig)1644 3435 y FL(2)1687 3416 y Fm(R)1726 3415 y Fq(\033)1760 3426 y(t)1783 3438 y FL(1)1715 3467 y Fq(\033)1749 3478 y(t)1772 3490 y FL(2)1844 3428 y(sin\()p Fk(\000j)p Fl(k)p Fk(j)q Fq(t)2077 3440 y FL(1)2109 3428 y(+)p Fl(k)p Fk(\001)p Fl(x)p FL(\))p 1844 3445 416 3 v 1845 3501 a(2)p Fk(j)q Fl(k)p Fk(j)1945 3489 y FL(3)1977 3509 y FE(\()2007 3501 y FL(1)p Fk(\000)2083 3490 y Fj(b)2080 3501 y Fl(k)p Fk(\001)p Fl(v)2165 3514 y Fq(j)2197 3509 y FE(\))2226 3474 y FL(2)2270 3456 y FG(d)2302 3435 y FL(3)2334 3456 y FG(k)2389 3536 y Fz(\000)k FH(I)2506 3409 y Fy(!)3800 3536 y FI(\(3.42\))515 3756 y FO(Since)27 b(the)h(v)n(ector)1479 3848 y FH(W)1552 3856 y FG(\033)1588 3864 y Fq(t)1611 3876 y FL(2)1664 3848 y FE(\()p FA(v)1741 3856 y FG(j)1773 3848 y FE(\))13 b FH(W)1900 3812 y Fx(y)1889 3861 y FG(\033)1925 3869 y Fq(t)1948 3881 y FL(2)2001 3787 y Fy(\000)2036 3848 y Fz(r)p FH(E)2161 3801 y FG(\033)2197 3809 y Fq(t)2220 3821 y FL(2)2157 3869 y Fw(P)2261 3787 y Fy(\001)2309 3848 y FH(')2359 3809 y FF(\()p FG(t)2407 3819 y FL(1)2439 3809 y FF(\))2359 3867 y FG(j;\033)2439 3875 y Fq(t)2462 3887 y FL(1)515 3982 y FO(b)r(elongs)27 b(to)g(the)h(domain)f(of)h(the) g(op)r(erator)1253 4232 y Fz(\000)p FH(g)1364 4130 y Fy(Z)1441 4151 y FG(\033)1477 4159 y Fq(t)1500 4171 y FL(1)1407 4300 y FG(\033)1443 4308 y Fq(t)1466 4320 y FL(2)1564 4154 y FH(a)12 b FE(\()p FA(k)p FE(\))1737 4071 y Fy(\020)1782 4154 y FH(e)1818 4122 y FG(i)p Fx(j)p Fw(k)p Fx(j)q FG(t)1939 4132 y FL(1)1992 4154 y Fz(\000)17 b FH(e)2105 4122 y FG(i)p Fw(k)p Fx(\001)p Fw(x)2222 4071 y Fy(\021)2285 4154 y Fz(\000)g FH(h:c:)p 1564 4215 919 4 v 1769 4310 a Fz(j)q FA(k)p Fz(j)1872 4227 y Fy(\020)1917 4310 y FE(1)h Fz(\000)2052 4290 y Fy(b)2050 4310 y FA(k)e Fz(\001)i FA(v)2199 4318 y FG(j)2231 4227 y Fy(\021)2553 4184 y FH(d)2593 4152 y FF(3)2627 4184 y FH(k)p 2502 4215 218 4 v 2502 4231 a Fy(p)p 2579 4231 141 4 v 63 x FE(2)13 b Fz(j)q FA(k)p Fz(j)515 4485 y FO(w)n(e)27 b(can)g(use)h(the)f(iden)n(tit)n(y)657 4586 y Fy(0)657 4720 y(@)724 4736 y FH(e)760 4662 y Fx(\000)p FG(g)850 4622 y Fm(R)889 4621 y Fq(\033)923 4632 y(t)946 4644 y FL(1)878 4672 y Fq(\033)912 4683 y(t)935 4695 y FL(2)1007 4619 y Fq(a)p FL(\()q Fl(k)p FL(\))1117 4627 y FE(\()1147 4619 y Fq(e)1175 4602 y(i)p Fk(j)p Fl(k)p Fk(j)q Fq(t)1292 4614 y FL(1)1329 4619 y Fk(\000)p Fq(e)1402 4602 y(i)p Fl(k)p Fk(\001)p Fl(x)1513 4627 y FE(\))1543 4619 y Fk(\000)p Fq(h:c:)p 1007 4651 681 3 v 1187 4701 a Fk(j)p Fl(k)p Fk(j)1258 4709 y FE(\()1288 4701 y FL(1)p Fk(\000)1364 4690 y Fj(b)1361 4701 y Fl(k)p Fk(\001)p Fl(v)1446 4714 y Fq(j)1478 4709 y FE(\))1734 4642 y Fq(d)1765 4626 y FL(3)1798 4642 y Fq(k)p 1707 4650 150 3 v 1707 4658 a Fx(p)p 1757 4658 100 3 v 35 x FL(2)p Fk(j)p Fl(k)p Fk(j)1888 4736 y Fz(\000)17 b FH(I)2005 4586 y Fy(1)2005 4720 y(A)2093 4736 y FE(=)1950 5059 y FO(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 3271 523 a FI(\(3.43\))650 740 y FE(=)21 b Fz(\000)p FH(g)843 680 y Fy(R)894 700 y FF(1)879 767 y(0)941 740 y FH(e)977 666 y Fx(\000)p FG(g)r(\025)1104 626 y Fm(R)1143 625 y Fq(\033)1177 636 y(t)1200 648 y FL(1)1132 676 y Fq(\033)1166 687 y(t)1189 699 y FL(2)1261 623 y Fq(a)p FL(\()q Fl(k)p FL(\))1372 631 y FE(\()1401 623 y Fq(e)1429 606 y(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)1546 618 y FL(1)1583 623 y Fk(\000)p Fq(e)1656 606 y(i)p Fl(k)p Fk(\001)p Fl(x)1768 631 y FE(\))1797 623 y Fk(\000)p Fq(h:c:)p 1261 655 681 3 v 1441 705 a Fk(j)p Fl(k)p Fk(j)1512 713 y FE(\()1542 705 y FL(1)p Fk(\000)1618 694 y Fj(b)1615 705 y Fl(k)q Fk(\001)p Fl(v)1701 718 y Fq(j)1732 713 y FE(\))1989 646 y Fq(d)2020 630 y FL(3)2052 646 y Fq(k)p 1962 654 150 3 v 1962 662 a Fx(p)p 2011 662 100 3 v 2011 698 a FL(2)p Fk(j)q Fl(k)p Fk(j)2125 740 y FH(d\025)2222 680 y Fy(R)2274 700 y FG(\033)2310 708 y Fq(t)2333 720 y FL(1)2259 767 y FG(\033)2295 775 y Fq(t)2318 787 y FL(2)2396 692 y FG(a)p FF(\()p Fw(k)p FF(\))2514 698 y FE(\()2544 692 y FG(e)2573 671 y Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)2690 683 y FL(1)2727 692 y Fx(\000)p FG(e)2804 671 y Fq(i)p Fl(k)p Fk(\001)p Fl(x)2916 698 y FE(\))2946 692 y Fx(\000)p FG(h:c:)p 2396 723 700 4 v 2578 776 a Fx(j)p Fw(k)p Fx(j)2652 782 y FE(\()2682 776 y FF(1)p Fx(\000)2762 763 y Fm(b)2760 776 y Fw(k)q Fx(\001)p Fw(v)2852 787 y Fq(j)2883 782 y FE(\))3150 709 y FG(d)3182 688 y FL(3)3215 709 y FG(k)p 3115 723 169 4 v 3115 732 a Fz(p)p 3179 732 105 4 v 50 x FF(2)p Fx(j)q Fw(k)p Fx(j)3311 740 y FH(:)542 940 y FO(Moreo)n(v)n(er)k(w)n(e)i(can)h(estimate)515 1030 y Fy(\015)515 1075 y(\015)515 1120 y(\015)515 1164 y(\015)515 1209 y(\015)515 1254 y(\015)557 1081 y(Z)634 1102 y FG(\033)670 1110 y Fq(t)693 1122 y FL(1)600 1252 y FG(\033)636 1260 y Fq(t)659 1272 y FL(2)757 1105 y FH(a)12 b FE(\()p FA(k)p FE(\))929 1022 y Fy(\020)975 1105 y FH(e)1011 1073 y FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(t)1131 1083 y FL(1)1185 1105 y Fz(\000)17 b FH(e)1298 1073 y FG(i)p Fw(k)p Fx(\001)p Fw(x)1415 1022 y Fy(\021)1478 1105 y Fz(\000)g FH(h:c:)p 757 1166 919 4 v 962 1261 a Fz(j)q FA(k)p Fz(j)1064 1179 y Fy(\020)1110 1261 y FE(1)h Fz(\000)1245 1242 y Fy(b)1243 1261 y FA(k)e Fz(\001)i FA(v)1392 1269 y FG(j)1424 1179 y Fy(\021)1746 1135 y FH(d)1786 1103 y FF(3)1820 1135 y FH(k)p 1695 1166 218 4 v 1695 1182 a Fy(p)p 1772 1182 141 4 v 64 x FE(2)13 b Fz(j)q FA(k)p Fz(j)1922 1183 y FH(W)1995 1191 y FG(\033)2031 1199 y Fq(t)2054 1211 y FL(2)2108 1183 y FE(\()o FA(v)2184 1191 y FG(j)2217 1183 y FE(\))f FH(W)2343 1147 y Fx(y)2332 1196 y FG(\033)2368 1204 y Fq(t)2391 1216 y FL(2)2444 1123 y Fy(\000)2480 1183 y Fz(r)p FH(E)2605 1136 y FG(\033)2641 1144 y Fq(t)2664 1156 y FL(2)2601 1204 y Fw(P)2704 1123 y Fy(\001)2752 1183 y FH(')2802 1145 y FF(\()p FG(t)2850 1155 y FL(1)2882 1145 y FF(\))2802 1203 y FG(j;\033)2882 1211 y Fq(t)2905 1223 y FL(1)2946 1030 y Fy(\015)2946 1075 y(\015)2946 1120 y(\015)2946 1164 y(\015)2946 1209 y(\015)2946 1254 y(\015)3010 1183 y Fz(\024)749 1568 y(\024)948 1459 y Fy(\015)948 1504 y(\015)948 1549 y(\015)948 1594 y(\015)991 1507 y(R)1042 1528 y FG(\033)1078 1536 y Fq(t)1101 1548 y FL(1)1027 1595 y FG(\033)1063 1603 y Fq(t)1086 1615 y FL(2)1164 1519 y FG(a)p FF(\()p Fw(k)p FF(\))1282 1525 y FE(\()1312 1519 y FG(e)1341 1498 y Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)1458 1510 y FL(1)1495 1519 y Fx(\000)p FG(e)1572 1498 y Fq(i)p Fl(k)p Fk(\001)p Fl(x)1684 1525 y FE(\))p 1164 1551 550 4 v 1272 1604 a Fx(j)o Fw(k)p Fx(j)1346 1610 y FE(\()1376 1604 y FF(1)p Fx(\000)1456 1591 y Fm(b)1454 1604 y Fw(k)p Fx(\001)p Fw(v)1545 1615 y Fq(j)1577 1610 y FE(\))1769 1537 y FG(d)1801 1516 y FL(3)1833 1537 y FG(k)p 1734 1551 169 4 v 1734 1560 a Fz(p)p 1798 1560 105 4 v 50 x FF(2)p Fx(j)p Fw(k)p Fx(j)1912 1568 y FH(W)1985 1576 y FG(\033)2021 1584 y Fq(t)2044 1596 y FL(2)2097 1568 y FE(\()p FA(v)2174 1576 y FG(j)2207 1568 y FE(\))g FH(W)2333 1536 y Fx(y)2322 1581 y FG(\033)2358 1589 y Fq(t)2381 1601 y FL(2)2434 1507 y Fy(\000)2469 1568 y Fz(r)p FH(E)2594 1521 y FG(\033)2630 1529 y Fq(t)2653 1541 y FL(2)2590 1589 y Fw(P)2694 1507 y Fy(\001)2742 1568 y FH(')2792 1530 y FF(\()p FG(t)2840 1540 y FL(1)2872 1530 y FF(\))2792 1588 y FG(j;\033)2872 1596 y Fq(t)2895 1608 y FL(1)2936 1459 y Fy(\015)2936 1504 y(\015)2936 1549 y(\015)2936 1594 y(\015)891 1825 y FE(+)964 1717 y Fy(\015)964 1762 y(\015)964 1807 y(\015)964 1852 y(\015)1007 1765 y(R)1058 1786 y FG(\033)1094 1794 y Fq(t)1117 1806 y FL(1)1043 1852 y FG(\033)1079 1860 y Fq(t)1102 1872 y FL(2)1180 1777 y FG(a)1214 1756 y Fk(y)1246 1777 y FF(\()p Fw(k)p FF(\))1330 1783 y FE(\()1360 1777 y FG(e)1389 1756 y Fk(\000)p Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)1551 1768 y FL(1)1588 1777 y Fx(\000)p FG(e)1665 1756 y Fk(\000)p Fq(i)p Fl(k)p Fk(\001)p Fl(x)1822 1783 y FE(\))p 1180 1808 672 4 v 1348 1861 a Fx(j)p Fw(k)p Fx(j)1423 1867 y FE(\()1452 1861 y FF(1)p Fx(\000)1532 1848 y Fm(b)1530 1861 y Fw(k)q Fx(\001)p Fw(v)1622 1872 y Fq(j)1654 1867 y FE(\))1907 1795 y FG(d)1939 1774 y FL(3)1971 1795 y FG(k)p 1871 1809 169 4 v 1871 1818 a Fz(p)p 1935 1818 105 4 v 50 x FF(2)p Fx(j)q Fw(k)p Fx(j)2050 1825 y FH(W)2123 1833 y FG(\033)2159 1841 y Fq(t)2182 1853 y FL(2)2235 1825 y FE(\()p FA(v)2312 1833 y FG(j)2344 1825 y FE(\))h FH(W)2471 1794 y Fx(y)2460 1839 y FG(\033)2496 1847 y Fq(t)2519 1859 y FL(2)2572 1765 y Fy(\000)2607 1825 y Fz(r)p FH(E)2732 1779 y FG(\033)2768 1787 y Fq(t)2791 1799 y FL(2)2728 1847 y Fw(P)2832 1765 y Fy(\001)2880 1825 y FH(')2930 1787 y FF(\()p FG(t)2978 1797 y FL(1)3010 1787 y FF(\))2930 1845 y FG(j;\033)3010 1853 y Fq(t)3033 1865 y FL(1)3074 1717 y Fy(\015)3074 1762 y(\015)3074 1807 y(\015)3074 1852 y(\015)749 2083 y Fz(\024)89 b FE(2)949 1975 y Fy(\015)949 2020 y(\015)949 2065 y(\015)949 2109 y(\015)992 2023 y(R)1043 2044 y FG(\033)1079 2052 y Fq(t)1102 2064 y FL(1)1028 2110 y FG(\033)1064 2118 y Fq(t)1087 2130 y FL(2)1166 2035 y FG(b)p FF(\()p Fw(k)p FF(\))1278 2041 y FE(\()1307 2035 y FG(e)1336 2014 y Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)1453 2026 y FL(1)1486 2014 y Fk(\000)p Fq(i)p Fl(k)p Fk(\001)p Fl(x)1643 2035 y Fx(\000)p FF(1)1721 2041 y FE(\))p 1166 2066 585 4 v 1291 2119 a Fx(j)p Fw(k)p Fx(j)1365 2125 y FE(\()1395 2119 y FF(1)p Fx(\000)1475 2106 y Fm(b)1473 2119 y Fw(k)q Fx(\001)p Fw(v)1565 2130 y Fq(j)1596 2125 y FE(\))1806 2053 y FG(d)1838 2032 y FL(3)1870 2053 y FG(k)p 1771 2067 169 4 v 1771 2076 a Fz(p)p 1835 2076 105 4 v 49 x FF(2)p Fx(j)p Fw(k)p Fx(j)1949 2083 y FH(W)2022 2091 y FG(\033)2058 2099 y Fq(t)2081 2111 y FL(2)2134 2083 y FE(\()p FA(v)2211 2091 y FG(j)2244 2083 y FE(\))12 b FH(W)2370 2052 y Fx(y)2359 2096 y FG(\033)2395 2104 y Fq(t)2418 2116 y FL(2)2471 2023 y Fy(\000)2506 2083 y Fz(r)p FH(E)2631 2037 y FG(\033)2667 2045 y Fq(t)2690 2057 y FL(2)2627 2104 y Fw(P)2731 2023 y Fy(\001)2779 2083 y FH(')2829 2045 y FF(\()p FG(t)2877 2055 y FL(1)2909 2045 y FF(\))2829 2103 y FG(j;\033)2909 2111 y Fq(t)2932 2123 y FL(1)2973 1975 y Fy(\015)2973 2020 y(\015)2973 2065 y(\015)2973 2109 y(\015)3271 2083 y FI(\(3.44\))900 2367 y FE(+)973 2262 y Fy(\034)1030 2307 y(R)1081 2327 y FG(\033)1117 2335 y Fq(t)1140 2347 y FL(1)1067 2394 y FG(\033)1103 2402 y Fq(t)1126 2414 y FL(2)1204 2324 y Fz(j)1225 2318 y FG(e)1254 2298 y Fq(i)p Fk(j)q Fl(k)p Fk(j)p Fq(t)1371 2310 y FL(1)1408 2318 y Fx(\000)p FG(e)1485 2298 y Fq(i)p Fl(k)p Fk(\001)p Fl(x)1597 2324 y Fz(j)1618 2287 y FL(2)p 1204 2350 447 4 v 1212 2409 a FF(2)p Fx(j)q Fw(k)p Fx(j)1317 2392 y FL(3)1349 2415 y FE(\()1379 2409 y FF(1)p Fx(\000)1459 2396 y Fm(b)1457 2409 y Fw(k)q Fx(\001)p Fw(v)1549 2420 y Fq(j)1580 2415 y FE(\))1610 2380 y FL(2)1661 2367 y FH(d)1701 2335 y FF(3)1735 2367 y FH(k)1777 2262 y Fy(\035)1845 2259 y FL(1)p 1845 2267 29 3 v 1845 2300 a(2)1835 2442 y FG(W)1892 2450 y Fq(\033)1926 2461 y(t)1949 2473 y FL(2)1990 2448 y FE(\()2020 2442 y Fw(v)2056 2453 y Fq(j)2087 2448 y FE(\))2117 2442 y FG(W)2182 2417 y Fk(y)2174 2452 y Fq(\033)2208 2463 y(t)2231 2475 y FL(2)2273 2387 y Fm(\020)2307 2442 y Fx(r)p FG(E)2406 2401 y Fq(\033)2440 2412 y(t)2463 2424 y FL(2)2403 2463 y Fl(P)2504 2387 y Fm(\021)2538 2442 y FG(')2578 2412 y FL(\()q Fq(t)2624 2424 y FL(1)2657 2412 y(\))2578 2462 y Fq(j;\033)2655 2473 y(t)2678 2485 y FL(1)2749 2367 y FH(:)501 b FI(\(3.45\))515 2627 y FO(As)21 b(c)n(hec)n(k)n(ed)f(in)i (Lemma)f(A6,)g(the)g(t)n(w)n(o)g(expressions)e(\(3.44\))i(and)g (\(3.45\))f(are)g(logarithmically)515 2718 y(div)n(ergen)n(t)26 b(in)h FH(t)1001 2726 y FF(2)1063 2718 y FO(but)h(v)-5 b(anishing)26 b(with)i(a)f(p)r(o)n(w)n(er)f(la)n(w)g(in)i FH(t)2355 2726 y FF(1)2416 2718 y FO(due)g(to)f(the)h(smo)r(othness)e (of)h(the)515 2810 y FJ(r)l(e)l(gular)g(blo)l(ck)35 b FO(in)26 b(its)f FA(P)p FO(-dep)r(endence)h(and)f(b)r(ecause)g(of)g (the)h(upp)r(er)f(in)n(tegration)f(b)r(ound)h FH(\033)3384 2818 y FG(t)3408 2828 y FL(1)3444 2810 y FO(.)515 2901 y(Concerning)h(the)i(term)f(\(3.42\))g(the)h(corresp)r(onding)e(norm) 796 3010 y Fy(\015)796 3054 y(\015)796 3099 y(\015)796 3144 y(\015)796 3189 y(\015)838 3013 y( )899 3140 y FH(e)935 3060 y FG(ig)989 3039 y FL(2)1032 3020 y Fm(R)1071 3019 y Fq(\033)1105 3030 y(t)1128 3042 y FL(1)1060 3071 y Fq(\033)1094 3082 y(t)1117 3094 y FL(2)1189 3032 y(sin\()p Fk(\000j)p Fl(k)p Fk(j)p Fq(t)1421 3044 y FL(1)1454 3032 y(+)p Fl(k)p Fk(\001)p Fl(x)p FL(\))p 1189 3049 416 3 v 1190 3105 a(2)p Fk(j)p Fl(k)p Fk(j)1289 3093 y FL(3)1322 3113 y FE(\()1352 3105 y FL(1)p Fk(\000)1428 3094 y Fj(b)1425 3105 y Fl(k)p Fk(\001)p Fl(v)1510 3118 y Fq(j)1541 3113 y FE(\))1571 3078 y FL(2)1615 3060 y FG(d)1647 3039 y FL(3)1679 3060 y FG(k)1734 3140 y Fz(\000)17 b FH(I)1851 3013 y Fy(!)1924 3140 y FH(W)1997 3148 y FG(\033)2033 3156 y Fq(t)2056 3168 y FL(2)2109 3140 y FE(\()p FA(v)2186 3148 y FG(j)2218 3140 y FE(\))c FH(W)2345 3104 y Fx(y)2334 3154 y FG(\033)2370 3162 y Fq(t)2393 3174 y FL(2)2446 3080 y Fy(\000)2481 3140 y Fz(r)p FH(E)2606 3094 y FG(\033)2642 3102 y Fq(t)2665 3114 y FL(2)2602 3161 y Fw(P)2706 3080 y Fy(\001)2754 3140 y FH(')2804 3102 y FF(\()p FG(t)2852 3112 y FL(1)2884 3102 y FF(\))2804 3160 y FG(j;\033)2884 3168 y Fq(t)2907 3180 y FL(1)2948 3010 y Fy(\015)2948 3054 y(\015)2948 3099 y(\015)2948 3144 y(\015)2948 3189 y(\015)3271 3140 y FI(\(3.46\))515 3375 y FO(can)27 b(b)r(e)h(treated)f (lik)n(e)g(the)h(expression)e(\(3.45\).)515 3558 y(In)36 b(the)g(end)g(w)n(e)g(obtain)f(that)i(the)f(norm)f(of)h(the)g(term)g FJ(D3.1\))k FO(is)c(surely)f(b)r(ounded)h(b)n(y)g(a)515 3649 y(quan)n(tit)n(y)27 b(of)g(order:)1634 3755 y FH(t)1662 3720 y FF(3)p FG(\017)1662 3769 y FF(1)1739 3755 y Fz(\001)18 b(j)p FE(ln)13 b FH(\033)1920 3763 y FG(t)1944 3773 y FL(2)1980 3755 y Fz(j)k(\001)h FE(\()p FH(\033)2131 3763 y FG(t)2155 3773 y FL(1)2191 3755 y FE(\))2245 3700 y FL(1)p 2231 3708 57 3 v 2231 3741 a(16)2327 3755 y FH(:)923 b FI(\(3.47\))515 3880 y Fp(analysis)34 b(of)g(D3.2\))515 4063 y FO(The)27 b(di\033erence)h(to)f(b)r(e)h(analyzed)f(is)887 4203 y FG(N)5 b FF(\()q FG(t)989 4213 y FL(1)1021 4203 y FF(\))911 4232 y Fy(X)914 4389 y FG(j)s FF(=1)1058 4303 y FH(e)1094 4267 y FG(iH)t(t)1194 4277 y FL(1)1230 4303 y Fz(W)1306 4311 y FG(\033)1342 4319 y Fq(t)1365 4331 y FL(1)1418 4303 y FE(\()p FA(v)1495 4311 y FG(j)1527 4303 y FH(;)13 b(t)1589 4311 y FF(1)1624 4303 y FE(\))f FH(W)1750 4267 y Fx(y)1739 4316 y FG(\033)1775 4324 y Fq(t)1798 4336 y FL(1)1852 4303 y FE(\()o FA(v)1928 4311 y FG(j)1961 4303 y FE(\))g FH(W)2076 4311 y FG(\033)2112 4319 y Fq(t)2135 4331 y FL(2)2188 4303 y FE(\()p FA(v)2265 4311 y FG(j)2297 4303 y FE(\))h FH(W)2424 4267 y Fx(y)2413 4316 y FG(\033)2449 4324 y Fq(t)2472 4336 y FL(2)2525 4242 y Fy(\000)2560 4303 y Fz(r)p FH(E)2685 4256 y FG(\033)2721 4264 y Fq(t)2744 4276 y FL(2)2681 4324 y Fw(P)2785 4242 y Fy(\001)2833 4303 y FH(')2883 4264 y FF(\()p FG(t)2931 4274 y FL(1)2963 4264 y FF(\))2883 4322 y FG(j;\033)2963 4330 y Fq(t)2986 4342 y FL(1)789 4585 y Fz(\000)849 4486 y FG(N)5 b FF(\()q FG(t)951 4496 y FL(1)983 4486 y FF(\))872 4514 y Fy(X)875 4672 y FG(j)s FF(=1)1019 4585 y FH(e)1055 4549 y FG(iH)t(t)1155 4559 y FL(1)1191 4585 y Fz(W)1267 4593 y FG(\033)1303 4601 y Fq(t)1326 4613 y FL(1)1380 4585 y FE(\()p FA(v)1457 4593 y FG(j)1489 4585 y FH(;)13 b(t)1551 4593 y FF(1)1585 4585 y FE(\))g FH(W)1712 4549 y Fx(y)1701 4598 y FG(\033)1737 4606 y Fq(t)1760 4618 y FL(1)1813 4585 y FE(\()p FA(v)1890 4593 y FG(j)1922 4585 y FE(\))g FH(W)2038 4593 y FG(\033)2074 4601 y Fq(t)2097 4613 y FL(2)2150 4585 y FE(\()p FA(v)2227 4593 y FG(j)2259 4585 y FE(\))g FH(W)2386 4549 y Fx(y)2375 4598 y FG(\033)2411 4606 y Fq(t)2434 4618 y FL(2)2487 4525 y Fy(\000)2522 4585 y Fz(r)p FH(E)2647 4539 y FG(\033)2683 4547 y Fq(t)2706 4559 y FL(1)2643 4606 y Fw(P)2747 4525 y Fy(\001)2795 4585 y FH(')2845 4547 y FF(\()p FG(t)2893 4557 y FL(1)2925 4547 y FF(\))2845 4605 y FG(j;\033)2925 4613 y Fq(t)2948 4625 y FL(1)3006 4585 y FH(:)1950 5059 y FO(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 515 523 a FO(F)-7 b(or)26 b(eac)n(h)g(single)g (cell,)h(b)n(y)f(an)h(argumen)n(t)e(similar)h(to)h(that)g(one)f (describ)r(ed)h(in)g(Theorem)f(A2)515 614 y(for)h(the)h(expression)e (\(5.18\),)h(w)n(e)g(ha)n(v)n(e)515 695 y Fy(\015)515 740 y(\015)515 785 y(\015)557 698 y(\020)603 781 y FH(W)687 745 y Fx(y)676 794 y FG(\033)712 802 y Fq(t)735 814 y FL(2)788 721 y Fy(\000)824 781 y Fz(r)p FH(E)949 735 y FG(\033)985 743 y Fq(t)1008 755 y FL(2)945 802 y Fw(P)1048 721 y Fy(\001)1100 781 y Fz(\000)17 b FH(W)1261 745 y Fx(y)1250 794 y FG(\033)1286 802 y Fq(t)1309 814 y FL(2)1362 721 y Fy(\000)1397 781 y Fz(r)p FH(E)1522 735 y FG(\033)1558 743 y Fq(t)1581 755 y FL(1)1518 802 y Fw(P)1622 721 y Fy(\001)1657 698 y(\021)1716 781 y FH(W)1789 789 y FG(\033)1825 797 y Fq(t)1848 809 y FL(1)1901 721 y Fy(\000)1936 781 y Fz(r)p FH(E)2061 735 y FG(\033)2097 743 y Fq(t)2120 755 y FL(1)2057 802 y Fw(P)2161 721 y Fy(\001)2209 781 y FH( )2262 743 y FF(\()p FG(t)2310 753 y FL(1)2342 743 y FF(\))2259 801 y FG(j;\033)2339 809 y Fq(t)2362 821 y FL(1)2403 695 y Fy(\015)2403 740 y(\015)2403 785 y(\015)2467 781 y Fz(\024)82 b FH(C)23 b Fz(\001)17 b FE(\()p FH(\033)2799 789 y FG(t)2823 799 y FL(1)2859 781 y FE(\))2899 726 y FL(1)p 2899 734 29 3 v 2899 767 a(4)2958 781 y Fz(\001)h(j)p FE(ln)13 b FH(\033)3139 789 y FG(t)3163 799 y FL(2)3199 781 y Fz(j)18 b(\001)f FH(t)3304 734 y Fx(\000)3362 715 y FL(3)p Fq(\017)p 3362 723 54 3 v 3374 755 a FL(2)3304 800 y FF(1)3470 781 y FI(\(3.48\))515 945 y FO(so)23 b(that)h(the)h(norm)e(of)h(the)h(v)n(ector)d(corresp)r(onding)g(to)i (the)h(v)-5 b(ariation)22 b FJ(D3.2\))29 b FO(is)23 b(b)r(ounded)i(b)n (y)1588 1111 y FH(C)d Fz(\001)c FH(t)1742 1045 y FL(3)p Fq(\017)p 1741 1053 V 1754 1086 a FL(2)1732 1130 y FF(1)1826 1111 y Fz(\001)g FE(\()o FH(\033)1938 1119 y FG(t)1962 1129 y FL(1)1999 1111 y FE(\))2039 1056 y FL(1)p 2039 1064 29 3 v 2039 1097 a(4)2098 1111 y Fz(\001)f(j)q FE(ln)12 b FH(\033)2278 1119 y FG(t)2302 1129 y FL(2)2339 1111 y Fz(j)h FH(:)877 b FI(\(3.49\))515 1247 y Fp(analysis)34 b(of)g(D3.3\))515 1430 y FO(The)27 b(di\033erence)h(in)n(v)n(olv)n(ed)e (in)i(this)g(step)f(is)515 1619 y FG(N)5 b FF(\()p FG(t)616 1629 y FL(1)648 1619 y FF(\))538 1648 y Fy(X)541 1805 y FG(j)s FF(=1)685 1719 y FH(e)721 1683 y FG(iH)t(t)821 1693 y FL(1)857 1719 y Fz(W)933 1727 y FG(\033)969 1735 y Fq(t)992 1747 y FL(1)1046 1719 y FE(\()o FA(v)1122 1727 y FG(j)1155 1719 y FH(;)13 b(t)1217 1727 y FF(1)1251 1719 y FE(\))g FH(W)1378 1683 y Fx(y)1367 1732 y FG(\033)1403 1740 y Fq(t)1426 1752 y FL(1)1479 1719 y FE(\()p FA(v)1556 1727 y FG(j)1588 1719 y FE(\))g FH(W)1704 1727 y FG(\033)1740 1735 y Fq(t)1763 1747 y FL(2)1816 1719 y FE(\()p FA(v)1893 1727 y FG(j)1925 1719 y FE(\))f FH(W)2051 1683 y Fx(y)2040 1732 y FG(\033)2076 1740 y Fq(t)2099 1752 y FL(2)2153 1658 y Fy(\000)2188 1719 y Fz(r)p FH(E)2313 1672 y FG(\033)2349 1680 y Fq(t)2372 1692 y FL(1)2309 1740 y Fw(P)2412 1658 y Fy(\001)2460 1719 y FH(')2510 1680 y FF(\()p FG(t)2558 1690 y FL(1)2591 1680 y FF(\))2510 1738 y FG(j;\033)2590 1746 y Fq(t)2613 1758 y FL(1)2671 1719 y Fz(\000)1497 2001 y(\000)1557 1902 y FG(N)5 b FF(\()p FG(t)1658 1912 y FL(1)1690 1902 y FF(\))1580 1930 y Fy(X)1582 2088 y FG(j)s FF(=1)1727 2001 y FH(e)1763 1965 y FG(iH)t(t)1863 1975 y FL(1)1899 2001 y Fz(W)1975 2009 y FG(\033)2011 2017 y Fq(t)2034 2029 y FL(1)2087 2001 y FE(\()p FA(v)2164 2009 y FG(j)2196 2001 y FH(;)13 b(t)2258 2009 y FF(1)2293 2001 y FE(\))f FH(e)2371 1961 y FG(i\015)2425 1969 y Fq(\033)2459 1980 y(t)2482 1992 y FL(1)2524 1906 y Fm(\020)2558 1961 y Fw(v)2594 1972 y Fq(j)2626 1961 y FG(;)p Fx(r)p FG(E)2744 1920 y Fq(\033)2778 1931 y(t)2801 1943 y FL(1)2741 1982 y Fl(P)2841 1961 y FG(;t)2884 1971 y FL(1)2916 1906 y Fm(\021)2955 2001 y FH(e)2991 1965 y Fx(\000)p FG(iE)3109 1924 y Fq(\033)3143 1935 y(t)3166 1947 y FL(1)3106 1986 y Fl(P)3206 1965 y FG(t)3230 1975 y FL(1)3266 2001 y FH( )3319 1963 y FF(\()p FG(t)3367 1973 y FL(1)3399 1963 y FF(\))3316 2021 y FG(j;\033)3396 2029 y Fq(t)3419 2041 y FL(1)3478 2001 y FH(:)542 2216 y FO(This)28 b(v)-5 b(ariation)26 b(can)h(b)r(e)h(written)g(as)515 2314 y FG(N)5 b FF(\()p FG(t)616 2324 y FL(1)648 2314 y FF(\))538 2342 y Fy(X)541 2500 y FG(j)s FF(=1)685 2413 y FH(e)721 2377 y FG(iH)t(t)821 2387 y FL(1)857 2413 y Fz(W)933 2421 y FG(\033)969 2429 y Fq(t)992 2441 y FL(1)1046 2413 y FE(\()o FA(v)1122 2421 y FG(j)1155 2413 y FH(;)13 b(t)1217 2421 y FF(1)1251 2413 y FE(\))1294 2330 y Fy(\020)1340 2413 y FH(W)31 b Fz(j)1465 2367 y FG(\033)1501 2375 y Fq(t)1524 2387 y FL(1)1465 2421 y FG(\033)1501 2429 y Fq(t)1524 2441 y FL(2)1587 2413 y FE(\()o FA(v)1663 2421 y FG(j)1696 2413 y FE(\))12 b FH(W)1822 2377 y Fx(y)1876 2413 y Fz(j)1897 2367 y FG(\033)1933 2375 y Fq(t)1956 2387 y FL(1)1897 2421 y FG(\033)1933 2429 y Fq(t)1956 2441 y FL(2)2018 2353 y Fy(\000)2053 2413 y Fz(r)p FH(E)2178 2367 y FG(\033)2214 2375 y Fq(t)2237 2387 y FL(1)2174 2434 y Fw(P)2278 2353 y Fy(\001)2330 2413 y Fz(\000)17 b FH(I)2447 2330 y Fy(\021)2505 2413 y FH(e)2541 2373 y FG(i\015)2595 2381 y Fq(\033)2629 2392 y(t)2652 2404 y FL(1)2693 2318 y Fm(\020)2728 2373 y Fw(v)2764 2384 y Fq(j)2795 2373 y FG(;)p Fx(r)p FG(E)2913 2332 y Fq(\033)2947 2343 y(t)2970 2355 y FL(1)2910 2394 y Fl(P)3011 2373 y FG(;t)3054 2383 y FL(1)3086 2318 y Fm(\021)3124 2413 y FH(e)3160 2377 y Fx(\000)p FG(iE)3278 2336 y Fq(\033)3312 2347 y(t)3335 2359 y FL(1)3275 2398 y Fl(P)3376 2377 y FG(t)3400 2387 y FL(1)3436 2413 y FH( )3489 2375 y FF(\()p FG(t)3537 2385 y FL(1)3569 2375 y FF(\))3486 2433 y FG(j;\033)3566 2441 y Fq(t)3589 2453 y FL(1)3689 2413 y FI(\(3.50\))515 2627 y FO(with)28 b(the)g(de\034nition)p FI(s)762 2826 y FH(W)j Fz(j)887 2779 y FG(\033)923 2787 y Fq(t)946 2799 y FL(1)887 2834 y FG(\033)923 2842 y Fq(t)946 2854 y FL(2)1009 2826 y FE(\()p FA(v)1086 2834 y FG(j)1118 2826 y FE(\))83 b(:=)g FH(W)1479 2790 y Fx(y)1468 2839 y FG(\033)1504 2847 y Fq(t)1527 2859 y FL(1)1580 2826 y FE(\()p FA(v)1657 2834 y FG(j)1689 2826 y FE(\))13 b FH(W)1805 2834 y FG(\033)1841 2842 y Fq(t)1864 2854 y FL(2)1917 2826 y FE(\()p FA(v)1994 2834 y FG(j)2026 2826 y FE(\))21 b(=)g FH(e)2194 2752 y Fx(\000)p FG(g)2284 2712 y Fm(R)2323 2711 y Fq(\033)2357 2722 y(t)2380 2734 y FL(1)2312 2763 y Fq(\033)2346 2774 y(t)2369 2786 y FL(2)2459 2726 y Fq(b)p FL(\()p Fl(k)p FL(\))q Fk(\000)p Fq(b)2634 2709 y Fk(y)2666 2726 y FL(\()q Fl(k)p FL(\))p 2441 2741 321 3 v 2441 2791 a Fk(j)p Fl(k)p Fk(j)2512 2799 y FE(\()2542 2791 y FL(1)p Fk(\000)2618 2780 y Fj(b)2615 2791 y Fl(k)q Fk(\001)p Fl(v)2701 2804 y Fq(j)2732 2799 y FE(\))2809 2733 y Fq(d)2840 2716 y FL(3)2872 2733 y Fq(k)p 2782 2741 150 3 v 2782 2748 a Fx(p)p 2831 2748 100 3 v 2831 2784 a FL(2)p Fk(j)q Fl(k)p Fk(j)3271 2826 y FI(\(3.51\))573 3064 y FH(W)657 3028 y Fx(y)710 3064 y Fz(j)731 3018 y FG(\033)767 3026 y Fq(t)790 3038 y FL(1)731 3072 y FG(\033)767 3080 y Fq(t)790 3092 y FL(2)853 3004 y Fy(\000)888 3064 y Fz(r)p FH(E)1013 3018 y FG(\033)1049 3026 y Fq(t)1072 3038 y FL(1)1009 3085 y Fw(P)1112 3004 y Fy(\001)1231 3064 y FE(:=)83 b FH(W)1479 3028 y Fx(y)1468 3077 y FG(\033)1504 3085 y Fq(t)1527 3097 y FL(2)1580 3004 y Fy(\000)1615 3064 y Fz(r)p FH(E)1740 3018 y FG(\033)1776 3026 y Fq(t)1799 3038 y FL(1)1736 3085 y Fw(P)1840 3004 y Fy(\001)1888 3064 y FH(W)1961 3072 y FG(\033)1997 3080 y Fq(t)2020 3092 y FL(1)2073 3004 y Fy(\000)2108 3064 y Fz(r)p FH(E)2233 3018 y FG(\033)2269 3026 y Fq(t)2292 3038 y FL(1)2229 3085 y Fw(P)2332 3004 y Fy(\001)2389 3064 y FE(=)21 b FH(e)2506 2965 y FG(g)2548 2925 y Fm(R)2587 2924 y Fq(\033)2621 2935 y(t)2644 2947 y FL(1)2576 2976 y Fq(\033)2610 2987 y(t)2633 2999 y FL(2)2791 2939 y Fq(b)p FL(\()q Fl(k)p FL(\))p Fk(\000)p Fq(b)2966 2922 y Fk(y)2998 2939 y FL(\()q Fl(k)p FL(\))p 2705 2954 457 3 v 2705 3017 a Fk(j)p Fl(k)p Fk(j)2777 2958 y Fj(\022)2812 3017 y FL(1)p Fk(\000)2888 3006 y Fj(b)2885 3017 y Fl(k)p Fk(\001r)p Fq(E)3029 2979 y(\033)3063 2990 y(t)3086 3002 y FL(1)3027 3039 y Fl(P)3127 2958 y Fj(\023)3209 2946 y Fq(d)3240 2929 y FL(3)3272 2946 y Fq(k)p 3182 2954 150 3 v 3182 2962 a Fx(p)p 3232 2962 100 3 v 35 x FL(2)p Fk(j)p Fl(k)p Fk(j)3271 3064 y FI(\(3.52\))515 3200 y FO(The)28 b(discussion)f(of)i(this)f(con)n(tribution)g(requires)f(the)h (study)h(of)f(the)g(squared)f(norm)h(of)g(the)515 3291 y(v)n(ector)k(\(3.50\))g(and)h(the)h(con)n(trol)e(of)h(the)g (o\033-diagonal)f(terms,)h(with)g(resp)r(ect)g(to)g(the)h(cell-)515 3382 y(partition)d FE(\000)913 3351 y FF(\()p FG(t)961 3361 y FL(1)993 3351 y FF(\))1021 3382 y FO(,)h(in)f(the)h(corresp)r (onding)e(scalar)g(pro)r(duct.)h(W)-7 b(e)32 b(\034rst)f(c)n(hec)n(k)g (that)h(the)f(sum)515 3474 y(of)e(the)h(o\033-diagonal)d(terms)i(v)-5 b(anishes)28 b(for)h FH(t)1924 3482 y FF(1)1982 3474 y Fz(!)24 b FE(+)p Fz(1)29 b FO(with)g(a)g(rate)g(fast)g(enough.)g (Then)g(w)n(e)515 3565 y(turn)e(to)h(consider)e(the)i(diagonal)e(con)n (tribution.)515 3748 y FJ(o\033-diagonal)32 b(term)6 b FO(s)515 3930 y(Let)37 b(us)h(consider)e(the)i(generic)e FH(l)24 b Fz(\000)f FH(j)42 b FO(term)c(and)f(try)g(to)g(reply)g(the)h (same)f(pro)r(cedure)f(as)515 4022 y(in)29 b(Theorem)f(2.1,)g(b)n(y)g (the)h(insertion)g(of)f(the)h(real)f(parameter)g Fr(\025)h FO(in)g(the)g(dressing)e(op)r(erator)515 4113 y Fz(W)591 4121 y FG(\033)627 4129 y Fq(t)650 4141 y FL(1)703 4113 y FE(\()p FA(v)780 4121 y FG(j)812 4113 y FH(;)14 b(t)875 4121 y FF(1)909 4113 y FE(\))32 b FO(and)f(the)i(subsequen)n(t)e(deriv) -5 b(ativ)n(e)31 b(with)i(resp)r(ect)e(to)h Fr(\025)p FO(.)g(The)g(only)g(obstacles)515 4204 y(in)c(rep)r(eating)e(the)i (usual)g(steps)f(come)g(from)g(the)h(lac)n(k)f(of)g(comm)n(utativit)n (y)g(b)r(et)n(w)n(een)1145 4313 y Fy(R)1196 4334 y FG(\024)1232 4344 y FL(1)1181 4400 y FG(\033)1217 4408 y Fq(t)1240 4420 y FL(1)1304 4325 y FG(a)p FF(\()o Fw(k)p FF(\))q FG(e)1451 4304 y Fq(i)p Fk(j)p Fl(k)p Fk(j)p Fq(s)1576 4325 y FG(\021)1608 4337 y Fq(l;j)1678 4331 y FE(\()1710 4312 y Fm(b)1708 4325 y Fw(k)1745 4331 y FE(\))p 1304 4356 471 4 v 1502 4399 a Fx(j)p Fw(k)p Fx(j)1830 4343 y FG(d)1862 4322 y FL(3)1894 4343 y FG(k)p 1794 4357 169 4 v 1794 4366 a Fz(p)p 1858 4366 105 4 v 50 x FF(2)p Fx(j)q Fw(k)p Fx(j)2056 4373 y FE(and)83 b FH(W)2347 4342 y Fx(y)2400 4373 y Fz(j)2421 4327 y FG(\033)2457 4335 y Fq(t)2480 4347 y FL(1)2421 4382 y FG(\033)2457 4390 y Fq(t)2480 4402 y FL(2)2543 4313 y Fy(\000)2578 4373 y Fz(r)p FH(E)2703 4327 y FG(\033)2739 4335 y Fq(t)2762 4347 y FL(1)2699 4395 y Fw(P)2802 4313 y Fy(\001)1445 4597 y FH(\037)1493 4606 y FG(h)1546 4536 y Fy(\000)1591 4566 y Fw(x)p 1591 4580 37 4 v 1595 4622 a FG(s)1638 4536 y Fy(\001)2056 4597 y FE(and)g FH(W)2347 4565 y Fx(y)2400 4597 y Fz(j)2421 4550 y FG(\033)2457 4558 y Fq(t)2480 4570 y FL(1)2421 4605 y FG(\033)2457 4613 y Fq(t)2480 4625 y FL(2)2543 4536 y Fy(\000)2578 4597 y Fz(r)p FH(E)2703 4550 y FG(\033)2739 4558 y Fq(t)2762 4570 y FL(1)2699 4618 y Fw(P)2802 4536 y Fy(\001)1358 4767 y FG(d\015)1422 4775 y Fq(\033)1456 4786 y(t)1479 4798 y FL(1)1520 4773 y FE(\()1550 4767 y Fw(v)1586 4779 y Fq(l)1610 4767 y FG(;)1639 4748 y Fl(x)p 1639 4756 34 3 v 1642 4788 a Fq(s)1682 4767 y FG(;s)1730 4773 y FE(\))p 1358 4805 403 4 v 1528 4848 a FG(ds)2056 4822 y FE(and)g FH(W)2347 4791 y Fx(y)2400 4822 y Fz(j)2421 4776 y FG(\033)2457 4784 y Fq(t)2480 4796 y FL(1)2421 4831 y FG(\033)2457 4839 y Fq(t)2480 4851 y FL(2)2543 4762 y Fy(\000)2578 4822 y Fz(r)p FH(E)2703 4776 y FG(\033)2739 4784 y Fq(t)2762 4796 y FL(1)2699 4844 y Fw(P)2802 4762 y Fy(\001)1950 5059 y FO(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 515 523 a FO(in)35 b(the)g(study)g(of)g(the)g (limit)h Fr(s)29 b Fu(!)h(1)p FO(.)35 b(Ho)n(w)n(ev)n(er)d(these)j (problems)f(can)h(b)r(e)g(easily)f(circum-)515 614 y(v)n(en)n(ted)29 b(b)n(y)h(means)g(of)g(an)g(\020)1419 587 y FG(\017)p 1416 599 31 3 v 1416 640 a FF(3)1487 614 y FO(argumen)n(t\021,)f(b)n(y) g(exploiting)h(the)h(resolv)n(en)n(t)d(equation)h(\(1.12\))515 706 y(in)g(addition)f(to)h(the)g(usual)f(regularit)n(y)f(prop)r (erties.)g(Therefore)h(an)g(analogous)e(estimate)j(is)515 797 y(obtained:)g(the)h(sum)g(of)g(the)g(absolute)f(v)-5 b(alues)30 b(of)g(the)g(o\033-diagonal)e(terms)h(is)h(b)r(ounded)g(b)n (y)515 888 y Fr(C)20 b Fu(\001)c Fr(t)646 859 y Fx(\000)p FG(\017)646 910 y FF(1)740 888 y Fu(\001)f(j)p Ft(ln)c Fr(\033)905 896 y FG(t)929 906 y FL(2)966 888 y Fu(j)p FJ(.)515 1071 y(diagonal)32 b(terms)515 1254 y FO(Considering)26 b(that)i(the)g(norm)1273 1342 y Fy(\015)1273 1387 y(\015)1273 1432 y(\015)1316 1368 y(\000)1351 1428 y FH(W)j Fz(j)1476 1382 y FG(\033)1512 1390 y Fq(t)1535 1402 y FL(1)1476 1437 y FG(\033)1512 1445 y Fq(t)1535 1457 y FL(2)1598 1428 y FE(\()p FA(v)1675 1436 y FG(j)1707 1428 y FE(\))12 b FH(W)32 b Fz(j)1875 1382 y FG(\033)1911 1390 y Fq(t)1934 1402 y FL(1)1875 1437 y FG(\033)1911 1445 y Fq(t)1934 1457 y FL(2)1996 1368 y Fy(\000)2032 1428 y Fz(r)p FH(E)2157 1382 y FG(\033)2193 1390 y Fq(t)2216 1402 y FL(1)2153 1449 y Fw(P)2256 1368 y Fy(\001)2308 1428 y Fz(\000)17 b FH(I)2425 1368 y Fy(\001)2473 1428 y FH( )2526 1390 y FF(\()o FG(t)2573 1400 y FL(1)2606 1390 y FF(\))2523 1448 y FG(j;\033)2603 1456 y Fq(t)2626 1468 y FL(1)2667 1342 y Fy(\015)2667 1387 y(\015)2667 1432 y(\015)3271 1428 y FI(\(3.53\))542 1607 y FO(can)28 b(b)r(e)g(estimated)f(from)g (ab)r(o)n(v)n(e)g(in)g(terms)h(of)f(a)g(quan)n(tit)n(y)g(of)h(order) 1396 1789 y FE(sup)1339 1877 y Fw(P)p Fx(2)p FF(\000)1466 1847 y FL(\()o Fq(t)1510 1859 y FL(1)1543 1847 y(\))1466 1897 y Fq(j)1582 1789 y Fz(j)q FA(v)1651 1797 y FG(j)1700 1789 y Fz(\000)17 b(r)p FH(E)1902 1753 y FG(\033)1938 1761 y Fq(t)1961 1773 y FL(1)2014 1789 y FE(\()p FA(P)p FE(\))p Fz(j)g(\001)h(j)p FE(ln)13 b FH(\033)2353 1797 y FG(t)2377 1807 y FL(2)2413 1789 y Fz(j)18 b(\001)f FH(t)2518 1742 y Fx(\000)2576 1722 y FL(3)p Fq(\017)p 2576 1730 54 3 v 2588 1763 a FL(2)2518 1808 y FF(1)542 2043 y FO(with)1269 2134 y FE(sup)1211 2222 y Fw(P)p Fx(2)p FF(\000)1338 2192 y FL(\()p Fq(t)1383 2204 y FL(1)1415 2192 y(\))1338 2242 y Fq(j)1455 2071 y Fy(\014)1455 2115 y(\014)1480 2134 y Fz(r)p FH(E)1605 2098 y FG(\033)1641 2106 y Fq(t)1664 2118 y FL(1)1718 2134 y FE(\()p FA(P)p FE(\))g Fz(\000)g(r)p FH(E)2057 2098 y FG(\033)2093 2106 y Fq(t)2116 2118 y FL(1)2169 2074 y Fy(\000)p 2204 2073 93 4 v 60 x FA(P)2264 2142 y FG(j)2297 2074 y Fy(\001)2332 2071 y(\014)2332 2115 y(\014)2379 2134 y Fz(\024)k FH(C)i Fz(\001)17 b FH(t)2604 2087 y Fx(\000)2677 2068 y Fq(\017)p 2662 2076 57 3 v 2662 2108 a FL(16)2604 2153 y FF(1)2749 2134 y FH(;)515 2381 y FO(the)27 b(sum)h(of)f(the)g(diagonal)f(terms)h (amoun)n(ts)f(to)h(a)g(con)n(tribution)g(of)g(order)f FH(t)2948 2334 y Fx(\000)3021 2315 y Fq(\017)p 3005 2323 V 3005 2356 a FL(16)2948 2400 y FF(1)3092 2381 y Fz(\001)17 b(j)p FE(ln)c FH(\033)3272 2389 y FG(t)3296 2399 y FL(2)3332 2381 y Fz(j)41 b FO(at)515 2473 y(most.)515 2564 y(W)-7 b(e)23 b(can)f(conclude)g(that)h(the)f(norm)g(of)h(the)g(v)n(ector)e (corresp)r(onding)f(to)i(the)h(di\033erence)g FJ(D3.3\))515 2655 y FO(is)k(b)r(ounded)h(b)n(y)1692 2764 y FH(C)22 b Fz(\001)c FH(t)1836 2717 y Fx(\000)1909 2698 y Fq(\017)p 1893 2706 V 1893 2739 a FL(16)1836 2783 y FF(1)1981 2764 y Fz(\001)f(j)q FE(ln)c FH(\033)2162 2772 y FG(t)2186 2782 y FL(2)2222 2764 y Fz(j)26 b FH(:)981 b FI(\(3.54\))515 3038 y FD(4)131 b(Scattering)30 b(subspaces)f(and)g(asymptotic)h (observ)-7 b(ables)515 3360 y FO(In)20 b(this)f(Section,)h(at)f (\034rst)h(w)n(e)f(consider)g(the)h(family)f(of)h(v)n(ectors)e FH( )2551 3322 y FG(out)p FF(\()o FG(in)p FF(\))2548 3382 y FG(h;\024)2639 3392 y FL(1)2767 3360 y FE(\()p FH(\034)t(;)13 b FA(a)p FE(\))20 b FO(corresp)r(onding)515 3452 y(to)25 b(the)h(ev)n(olution)f FH(\034)34 b FO(in)25 b(time)h(and)g(to)f(a)g(displacemen)n(t)g FA(a)h FO(in)g(space)e(of)i (the)g(state)f(asso)r(ciated)515 3552 y(to)38 b(the)h(v)n(ector)d FH( )1093 3514 y FG(out)p FF(\()o FG(in)p FF(\))1090 3573 y FG(h;\024)1181 3583 y FL(1)1335 3552 y FO(previously)h (constructed.)h(Then)g(w)n(e)g(construct)g(the)h(co)n(v)-5 b(arian)n(t,)515 3667 y(under)27 b(space-time)f(translations,)g (subspace)g Fz(H)2047 3629 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))2047 3675 y FG(\024)2083 3685 y FL(1)2309 3667 y FO(as)g(the)i(norm)e(closure)g(of)i(the)f(\034nite)515 3779 y(linear)g(com)n(binations)f(of)h(v)n(ectors)f(in)i(the)g(set)1997 3696 y Fy(n)2048 3779 y FH( )2101 3741 y FG(out)p FF(\()o FG(in)p FF(\))2098 3800 y FG(h;\024)2189 3810 y FL(1)2317 3779 y FE(\()p FH(\034)t(;)13 b FA(a)p FE(\))2492 3696 y Fy(o)2543 3779 y FI(:)806 4002 y Fz(H)871 3966 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))871 4015 y FG(\024)907 4025 y FL(1)1127 4002 y FE(:=)p 1230 3906 1930 4 v 1230 3919 a Fy(n)1281 3931 y(_)1379 4002 y FH( )1432 3964 y FG(out)p FF(\()o FG(in)p FF(\))1429 4023 y FG(h;\024)1520 4033 y FL(1)1648 4002 y FE(\()p FH(\034)t(;)g FA(a)p FE(\))38 b(:)99 b FH(h)13 b FE(\()p FA(P)p FE(\))21 b Fz(2)h FH(C)2312 3976 y FF(1)2307 4021 y(0)2359 4002 y FE(\(\006)c Fz(n)f FE(0\))c FH(;)26 b(\034)k Fz(2)21 b FB(R)13 b FH(;)g FA(a)22 b Fz(2)f FB(R)3073 3980 y FF(3)3108 3919 y Fy(o)515 4209 y FO(Later,)28 b(in)i(Theorem)f(4.1,)f (w)n(e)h(de\034ne)h(the)g(v)n(ectors)2167 4126 y Fy(n)2218 4209 y FH( )2271 4171 y FG(out)p FF(\()o FG(in)p FF(\))2268 4230 y FG(h;\026)2474 4126 y Fy(o)2555 4209 y FO(obtained)f(from)g(the) h(strong)515 4330 y(\(time\))19 b(limit)g(of)f(the)h FJ(L.S.Z.)h FO(W)-7 b(eyl)18 b(op)r(erators,)f(with)h(smearing)f (functions)2845 4269 y Fy(\010)2890 4330 y FH(\026)35 b FE(:)j Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))21 b Fz(2)g FH(C)3345 4298 y Fx(1)3340 4343 y FF(0)3423 4269 y Fy(\000)3459 4330 y FB(R)3513 4298 y FF(3)3564 4330 y Fz(n)d 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4778 y FG(out)p FF(\()p FG(in)p FF(\))1950 5059 y FO(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 515 523 a FO(in)27 b(terms)g(of)h(quan)n(tum)f (n)n(um)n(b)r(ers)g(asso)r(ciated)f(with)i(the)f(asymptotic)g(v)-5 b(ariables)26 b(whic)n(h)i(are)515 614 y(w)n(ell)g(de\034ned)i(on)f (them:)g(the)h(asymptotic)e(photon)h(W)-7 b(eyl)29 b(op)r(erators)e (and)i(the)g(asymptotic)515 706 y(electron)37 b(mean)h(v)n(elo)r(cit)n (y)-7 b(.)37 b(In)h(Theorem)f(4.2)h(the)g(asymptotic)f(con)n(v)n (ergence)f(of)i(the)g FH(C)3402 674 y Fx(1)3397 719 y FF(0)515 797 y FO(functions)32 b(of)h(the)f(the)h(v)-5 b(ariable)31 b FH(e)1624 765 y FG(iH)t(t)1737 766 y Fw(x)p 1737 780 37 4 v 1743 822 a FG(t)1784 797 y FH(e)1820 765 y Fx(\000)p FG(iH)t(t)2004 797 y FO(is)h(established)g(on)g(the)g (v)n(ectors)f(of)h Fz(H)3240 765 y FG(out)p FF(\()p FG(in)p FF(\))3444 797 y FO(.)515 906 y(These)i(functions)h(generate)f(the)h (comm)n(utativ)n(e)e(algebra)g Fz(A)2478 868 y FG(out)p FF(\()p FG(in)p FF(\))2478 915 y Fw(v)2514 927 y Fq(el)2682 906 y FO(.)i(In)g(Theorem)f(4.4,)g(w)n(e)515 1011 y(construct)21 b(the)i(canonical)e(W)-7 b(eyl)22 b(algebra)e Fz(A)1921 973 y FG(out)p FF(\()o FG(in)p FF(\))1921 1032 y FG(ph)2125 1011 y FI(,)j FO(generated)e(b)n(y)g(the)i(strong)e(limits)h(of)g(the) 515 1136 y FJ(L.S.Z.)28 b FO(W)-7 b(eyl)27 b(op)r(erators)e(smeared)h (with)i(functions)2213 1053 y Fy(n)2265 1136 y FH(\020)e FE(:)2372 1116 y Fy(e)2368 1136 y FH(\020)17 b FE(\()p FA(k)p FE(\))k Fz(2)h FH(L)2672 1104 y FF(2)2719 1076 y Fy(\000)2755 1136 y FB(R)2809 1104 y FF(3)2843 1136 y FH(;)2877 1076 y Fy(\000)2912 1136 y FE(1)c(+)f Fz(j)p FA(k)p Fz(j)3134 1101 y Fx(\000)p FF(1)3217 1076 y Fy(\001)3265 1136 y FH(d)3305 1104 y FF(3)3339 1136 y FH(k)3381 1076 y Fy(\001)3416 1053 y(o)515 1269 y FO(and)36 b(acting)h(on)f(the)h (space)f Fz(H)1516 1237 y FG(out)p FF(\()p FG(in)p FF(\))1720 1269 y FO(.)h(The)g(algebra)e Fz(A)2321 1230 y FG(out)p FF(\()p FG(in)p FF(\))2321 1290 y FG(ph)2562 1269 y FO(is)i(asso)r (ciated)e(with)i(a)g(free)515 1384 y(massless)29 b(b)r(oson)g(\034eld)i (and)f(comm)n(utes)g(with)h(the)f(algebra)f Fz(A)2519 1345 y FG(out)p FF(\()o FG(in)p FF(\))2519 1393 y Fw(v)2555 1405 y Fq(el)2753 1384 y FO(as)h(a)f(consequence)h(of)515 1475 y(the)e(asymptotic)f(decoupling.)515 1566 y(The)i(sp)r(ectral)g (restriction)g(on)g(the)h(electron)f(\(mean\))g(v)n(elo)r(cit)n(y)g (\(strictly)h(less)f(than)g(1\))h(im-)515 1658 y(plies)k(a)h (restriction)e(of)i Fz(H)1359 1626 y FG(out)p FF(\()o FG(in)p FF(\))1563 1658 y FI(,)g FO(as)f(subspace)g(of)h Fz(H)p FO(,)g(explained)f(with)h(the)g(partial)f(non-)515 1749 y(relativistic)j(c)n(haracter)e(of)j(the)g(mo)r(del.)g(Ho)n(w)n (ev)n(er)d(no)j(issue)f(regarding)f(completeness)h(is)515 1840 y(addressed)32 b(in)i(our)f(discussion,)g(ev)n(en)h(under)f(the)i (restriction)d(on)i(the)g(energy)f(con\034gura-)515 1932 y(tions)27 b(of)h(the)g(system.)515 2297 y Fp(De\034nition)33 b(of)i(the)f(ve)-5 b(ctor)35 b FN( )1588 2267 y FM(out)1585 2320 y(h;\024)1683 2328 y FL(1)1733 2297 y FK(\()p FN(\034)5 b(;)14 b Fg(a)p FK(\))q Fp(.)515 2479 y FO(Applying)27 b(the)h(op)r(erator)e FH(e)1388 2448 y Fx(\000)p FG(i)p Fw(a)p Fx(\001)p Fw(P)1561 2479 y FH(e)1597 2448 y Fx(\000)p FG(iH)t(\034)1787 2479 y FO(to)i(the)g(generic)e(v)n(ector)g FH( )2615 2448 y FG(out)2612 2495 y(h;\024)2703 2505 y FL(1)2768 2479 y FO(,)i(w)n(e)f(obtain:)948 2635 y FH(e)984 2599 y Fx(\000)p FG(i)p Fw(a)p Fx(\001)p Fw(P)1157 2635 y FH(e)1193 2599 y Fx(\000)p FG(iH)t(\034)1355 2635 y FH( )1408 2599 y FG(out)1405 2648 y(h;\024)1496 2658 y FL(1)1554 2635 y FE(=)515 2863 y(=)21 b FH(s)p Fz(\000)42 b FE(lim)692 2908 y FG(t)p Fx(!)p FF(+)p Fx(1)896 2863 y FH(e)932 2827 y Fx(\000)p FG(i)p Fw(a)p Fx(\001)p Fw(P)1105 2863 y FH(e)1141 2827 y Fx(\000)p FG(iH)t(\034)1304 2863 y FH(e)1340 2827 y FG(iH)t(t)1456 2764 y(N)5 b FF(\()p FG(t)p FF(\))1463 2792 y Fy(X)1466 2949 y FG(j)s FF(=1)1594 2863 y Fz(W)1670 2871 y FG(\033)1706 2879 y Fq(t)1750 2863 y FE(\()p FA(v)1827 2871 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3137 y(5)3621 3153 y FH( )3674 3115 y FF(\()p FG(t)p FF(+)p FG(\034)h FF(\))3671 3173 y FG(j;\033)3751 3184 y Fq(t)p FL(+)p Fq(\034)3872 3153 y FE(\()o FH(\034)t(;)14 b FA(a)p FE(\))601 3359 y(=)21 b FH(s)c Fz(\000)59 b FE(lim)811 3404 y FG(t)p Fx(!)p FF(+)p Fx(1)1016 3359 y FH( )1069 3323 y FG(\034)s(;)p Fw(a)1066 3380 y FG(h;k)1154 3390 y FL(1)1204 3359 y FE(\()p FH(t)p FE(\))3310 3484 y FI(\(4.1\))515 3575 y FO(where)1063 3685 y FH( )1116 3647 y FF(\()o FG(t)p FF(+)p FG(\034)6 b FF(\))1113 3705 y FG(j;\033)1193 3716 y Fq(t)p FL(+)p Fq(\034)1313 3685 y FE(\()p FH(\034)t(;)13 b FA(a)p FE(\))21 b(:=)1611 3583 y Fy(Z)1654 3753 y FF(\000)1692 3726 y FL(\()p Fq(t)p FL(+)p Fq(\034)5 b FL(\))1692 3774 y Fq(j)1857 3685 y FH(e)1893 3649 y Fx(\000)p FG(i)p Fw(a)p Fx(\001)p Fw(P)2066 3685 y FH(e)2102 3649 y Fx(\000)p FG(iE)2220 3612 y Fq(\033)2254 3625 y(t)p FL(+)p Fq(\034)2217 3670 y Fl(P)2361 3649 y FG(\034)2400 3685 y FH(h)13 b FE(\()p FA(P)p FE(\))g FH( )2643 3642 y FG(\033)2679 3653 y Fq(t)p FL(+)p Fq(\034)2640 3706 y Fw(P)2786 3685 y FH(d)2826 3649 y FF(3)2860 3685 y FH(P)515 3871 y FI(and)19 b FH( )711 3836 y FG(\034)s(;)p Fw(a)708 3893 y FG(h;\024)799 3903 y FL(1)848 3871 y FE(\()p FH(t)p FE(\))i FO(corresp)r(onds)d(to)j (the)g(appro)n(ximating)e(v)n(ector)g FH( )2475 3880 y FG(h;\024)2566 3890 y FL(1)2616 3871 y FE(\()p FH(t)p FE(\))h FO(with)h(translated)f(w)n(a)n(v)n(e)515 3973 y(function)j(b)r(oth)f(in)h(the)g(electron)e(and)h(the)h(photon)f(v)-5 b(ariables.)21 b(The)h(\034nal)g(equalit)n(y)g(\(4.1\))g(can)515 4064 y(b)r(e)36 b(easily)f(deriv)n(ed)g(exploiting)g(the)i(estimates)e (in)n(v)n(olv)n(ed)f(in)j(the)f(construction)f(of)h FH( )3320 4033 y FG(out)3317 4080 y(h;\024)3408 4090 y FL(1)3444 4064 y FO(.)515 4156 y(Therefore)26 b(the)i(de\034nition)1454 4299 y FH( )1507 4263 y FG(out)1504 4312 y(h;\024)1595 4322 y FL(1)1644 4299 y FE(\()p FH(\034)t(;)14 b FA(a)p FE(\))21 b(:=)g FH(e)1979 4263 y Fx(\000)p FG(i)p Fw(a)p Fx(\001)p Fw(P)2152 4299 y FH(e)2188 4263 y Fx(\000)p FG(iH)t(\034)2351 4299 y FH( )2404 4263 y FG(out)2401 4312 y(h;\024)2492 4322 y FL(1)3310 4299 y FI(\(4.2\))515 4442 y FO(is)27 b(consisten)n(t)g(with)h(the)g(exp)r(ected)g (asymptotic)f(in)n(terpretation.)789 b Fb(\003)515 4625 y FO(The)27 b(de\034nition)h(of)g(the)g(subspace)e(of)i(the)g(minimal)g (asymptotic)f(electron)g(states)g(is)583 4803 y Fz(H)648 4767 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))648 4816 y FG(\024)684 4826 y FL(1)904 4803 y FE(:=)p 1007 4708 2235 4 v 1007 4720 a Fy(n)1058 4732 y(_)1156 4803 y FH( )1209 4777 y FG(out)1206 4825 y(h;\024)1297 4835 y FL(1)1347 4803 y FE(\()o FH(\034)t(;)14 b FA(a)p FE(\))89 b FH(h)13 b FE(\()p FA(P)p FE(\))22 b Fz(2)f FH(C)1942 4777 y FF(1)1937 4822 y(0)1990 4803 y FE(\()o FB(R)2073 4782 y FF(3)2125 4803 y Fz(n)d FE(0\))13 b FH(;)26 b(supph)20 b Fz(\032)h FE(\006)14 b FH(;)25 b(\034)30 b Fz(2)22 b FB(R)12 b FH(;)i FA(a)21 b Fz(2)g FB(R)3155 4782 y FF(3)3190 4720 y Fy(o)3310 4803 y FI(\(4.3\))1950 5059 y FO(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 515 706 a Fp(De\034nition)33 b(of)i(the)f(sc)-5 b(attering)35 b(sp)-5 b(ac)g(es)515 888 y FO(In)20 b(next)g(theorem)f(w)n(e)h(construct)f(the)h(v)n(ector)f FH( )2007 850 y FG(out)p FF(\()n FG(in)p FF(\))2004 910 y FG(h;\026)2230 888 y FO(starting)g(from)g(a)h(v)n(ector)e FH( )3076 850 y FG(out)p FF(\()o FG(in)p FF(\))3073 910 y FG(h;\024)3164 920 y FL(1)3293 888 y FE(\()o FH(\034)t(;)c FA(a)p FE(\))515 1003 y FO(in)30 b Fz(H)679 965 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))679 1012 y FG(\024)715 1022 y FL(1)944 1003 y FO(and)h(a)f(cloud)g(of)g(photons)g(represen)n (ted)f(b)n(y)h(an)g FJ(L.S.Z.)i FO(W)-7 b(eyl)31 b(op)r(erator)d(with) 515 1095 y(smearing)23 b(function)j FH(\026)p FO(.)f(Concerning)f (notations)g(w)n(e)h(omit)g(the)h(dep)r(endence)f(on)g FH(\024)3129 1103 y FF(1)3164 1095 y FH(;)13 b(\034)t(;)g FA(a)25 b FO(due)515 1195 y(to)i FH( )669 1156 y FG(out)p FF(\()o FG(in)p FF(\))666 1216 y FG(h;\024)757 1226 y FL(1)885 1195 y FE(\()p FH(\034)t(;)14 b FA(a)p FE(\))o FO(.)515 1377 y FP(Theorem)30 b(4.1)515 1469 y FO(The)d(strong)g(limit) 1510 1560 y FH(s)17 b Fz(\000)f FH(l)q(im)1757 1568 y FG(t)p Fx(!)p FF(+)p Fx(1)1954 1560 y FH( )2004 1569 y FG(h;\026)2113 1560 y FE(\()p FH(t)p FE(\))21 b(:=)g FH( )2377 1524 y FG(out)2374 1573 y(h;\026)3310 1560 y FI(\(4.4\))515 1684 y FO(exists,)27 b(where:)958 1863 y FH( )1008 1872 y FG(h;\026)1118 1863 y FE(\()p FH(t)p FE(\))21 b(:=)g FH(e)1365 1828 y FG(iH)t(t)1468 1863 y FH(e)1504 1828 y FG(i)1526 1834 y FE(\()1556 1828 y FG(a)p FF(\()p FG(\026)1651 1836 y Fq(t)1679 1828 y FF(\)+)p FG(a)1784 1807 y Fk(y)1815 1828 y FF(\()p FG(\026)1876 1836 y Fq(t)1904 1828 y FF(\))1927 1834 y FE(\))1962 1863 y FH(e)1998 1828 y Fx(\000)p FG(iH)t(t)2149 1863 y FH( )2199 1872 y FG(h;\024)2290 1882 y FL(1)2339 1863 y FE(\()p FH(t)p FE(\))26 b(;)1117 1989 y Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))21 b Fz(2)g FH(C)1432 1953 y Fx(1)1427 2002 y FF(0)1510 1928 y Fy(\000)1545 1989 y FB(R)1599 1953 y FF(3)1651 1989 y Fz(n)d FE(0)1745 1928 y Fy(\001)1814 1989 y FH(;)39 b Fy(e)-47 b FH(\026)1916 1997 y FG(t)1957 1989 y FE(\()p FA(k)p FE(\))20 b(:=)i FH(e)2223 1953 y FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(t)2351 1989 y Fy(e)-47 b FH(\026)13 b FE(\()p FA(k)p FE(\))29 b(;)1015 2170 y FH(a)1056 2135 y Fx(y)1102 2170 y FE(\()o FH(\026)p FE(\))22 b(:=)f(\()p FH(a)13 b FE(\()p FH(\026)p FE(\)\))1551 2135 y Fx(y)1605 2170 y FE(=)1686 2065 y Fy(\022)1742 2069 y(Z)1832 2170 y FH(a)f FE(\()p FA(k)p FE(\))k Fy(e)-47 b FH(\026)14 b FE(\()o FA(k)p FE(\))f FH(d)2223 2135 y FF(3)2257 2170 y FH(k)2299 2065 y Fy(\023)2356 2083 y Fx(y)2414 2170 y FE(;)1033 2343 y FH( )1086 2307 y FG(out)1083 2356 y(h;k)1171 2366 y FL(1)1229 2343 y Fz(2)22 b(H)1367 2307 y FF(1)p FG(out)1367 2356 y(k)1400 2366 y FL(1)1505 2343 y FH(:)542 2503 y FJ(Pr)l(o)l(of)515 2595 y FO(T)-7 b(aking)27 b(in)n(to)i(accoun)n(t)e(Theorem)h(3.1,)g (the)h(function)g FH(\037)2297 2604 y FG(h)2350 2595 y FE(\()p Fz(r)p FH(E)2505 2561 y FG(\033)2541 2569 y Fq(t)2501 2616 y Fw(P)2572 2595 y FE(\))f FO(in)n(tro)r(duced)h(in)f (Theorem)515 2686 y(2.1)35 b(and)h(Huyghens')g(principle,)g(the)h (result)f(follo)n(ws)f(from)h(Co)r(ok's)f(argumen)n(t)g(once)h(the)515 2777 y(constrain)n(t)26 b(\(2.26\))h(is)g(assumed.)1846 b Fb(\003)515 2960 y FO(The)27 b(scattering)g(subspaces)893 3156 y Fz(H)958 3120 y FG(out)p FF(\()o FG(in)p FF(\))1183 3156 y FE(:=)p 1286 3060 1804 4 v 1286 3073 a Fy(n)1337 3085 y(_)1435 3156 y FH( )1488 3117 y FG(out)p FF(\()o FG(in)p FF(\))1485 3177 y FG(h;\026)1730 3156 y FE(:)98 b FH(h)13 b FE(\()p FA(P)p FE(\))22 b Fz(2)f FH(C)2180 3130 y FF(1)2175 3175 y(0)2228 3156 y FE(\(\006)c Fz(n)h FE(0\))13 b FH(;)k Fy(e)-47 b FH(\026)21 b Fz(2)h FH(C)2701 3130 y Fx(1)2696 3175 y FF(0)2779 3156 y FE(\()p FB(R)2863 3134 y FF(3)2914 3156 y Fz(n)c FE(0\))3038 3073 y Fy(o)3310 3156 y FI(\(4.5\))515 3357 y FO(are)24 b(in)n(v)-5 b(arian)n(t)24 b(under)g(space-time)h(translations)e(b)r(ecause)i(the)g(subspaces)f Fz(H)2975 3319 y FF(1)p FG(out)p FF(\()p FG(in)p FF(\))2975 3366 y FG(\024)3011 3376 y FL(1)3235 3357 y FO(are)g(in-)515 3449 y(v)-5 b(arian)n(t)26 b(b)n(y)i(construction.)515 3631 y Fp(Asymptotic)34 b(algebr)-5 b(as)515 3814 y FP(Theorem)30 b(4.2)515 3905 y FO(The)d FH(C)745 3874 y Fx(1)740 3918 y FF(0)837 3905 y FO(functions)h FH(f)36 b FO(of)27 b(the)g(v)-5 b(ariable)27 b FH(e)1852 3874 y FG(iH)t(t)1965 3875 y Fw(x)p 1965 3889 37 4 v 1971 3931 a FG(t)2011 3905 y FH(e)2047 3874 y Fx(\000)p FG(iH)t(t)2199 3905 y FO(,)g(that)g(is)h (the)f(electron)g(mean)g(v)n(elo)r(cit)n(y)515 3997 y(\(at)g(time)h FH(t)p FO(\))f(up)h(to)f(a)g(correction)f(of)h(order)f FH(t)1941 3965 y Fx(\000)p FF(1)2023 3997 y FO(,)h(ha)n(v)n(e)g(strong) f(limits)h(in)h Fz(H)2911 3965 y FG(out)3034 3997 y FO(for)f FH(t)21 b Fz(!)g FE(+)p Fz(1)p FO(,)515 4088 y(namely)1210 4198 y FH(s)16 b Fz(\000)h FH(l)q(im)1457 4206 y FG(t)p Fx(!)p FF(+)p Fx(1)1654 4198 y FH(e)1690 4162 y FG(iH)t(t)1793 4198 y FH(f)1851 4116 y Fy(\020)1907 4150 y FA(x)p 1907 4181 47 4 v 1916 4249 a FH(t)1964 4116 y Fy(\021)2022 4198 y FH(e)2058 4162 y Fx(\000)p FG(iH)t(t)2210 4198 y FH( )2263 4162 y FG(out)2260 4212 y(h;\026)2379 4198 y FE(=:)k FH( )2534 4162 y FG(out)2531 4212 y(h)p Fx(\001)p FG(f)2616 4223 y Fk(r)p Fq(E)2712 4212 y FG(;\026)3310 4198 y FI(\(4.6\))515 4347 y FO(where)27 b FH(f)792 4355 y Fx(r)p FG(E)908 4347 y FE(\()p FA(P)p FE(\))21 b(:=)g FH(l)q(im)1269 4355 y FG(\033)r Fx(!)p FF(0)1403 4347 y FH(f)g FE(\()p Fz(r)p FH(E)1616 4315 y FG(\033)1670 4347 y FE(\()p FA(P)p FE(\)\))p FO(.)515 4530 y FJ(Pr)l(o)l(of)515 4621 y FO(Exploiting)27 b(Theorem)g(3.1)g(and)g(using)g(the)h(fact)g (that)g(the)g(op)r(erators)1362 4801 y FH(f)1420 4718 y Fy(\020)1476 4753 y FA(x)p 1476 4784 V 1485 4852 a FH(t)1533 4718 y Fy(\021)1613 4801 y FH(;)166 b(e)1836 4765 y FG(iH)t(t)1940 4801 y FH(e)1976 4765 y FG(i)1998 4771 y FE(\()2027 4765 y FG(a)p FF(\()p FG(\026)2122 4773 y Fq(t)2150 4765 y FF(\)+)p FG(a)2255 4744 y Fk(y)2286 4765 y FF(\()p FG(\026)2347 4773 y Fq(t)2375 4765 y FF(\))2399 4771 y FE(\))2433 4801 y FH(e)2469 4765 y Fx(\000)p FG(iH)t(t)1950 5059 y FO(40)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 542 523 a FO(are)27 b(uniformly)g(b)r(ounded)h (in)g FN(t)p FO(,)g(w)n(e)f(obtain)897 701 y FH(s)17 b Fz(\000)f FH(l)q(im)1144 709 y FG(t)p Fx(!)p FF(+)p Fx(1)1341 701 y FH(e)1377 665 y FG(iH)t(t)1480 701 y FH(f)1538 618 y Fy(\020)1594 652 y FA(x)p 1594 684 47 4 v 1603 751 a FH(t)1651 618 y Fy(\021)1709 701 y FH(e)1745 665 y Fx(\000)p FG(iH)t(t)1897 701 y FH( )1950 665 y FG(out)1947 714 y(h;\026)2066 701 y FE(=)1015 977 y(=)126 b FH(s)17 b Fz(\000)g FH(l)q(im)1449 985 y FG(t)p Fx(!)p FF(+)p Fx(1)1645 977 y FH(e)1681 945 y FG(iH)t(t)1785 977 y FH(f)1843 917 y Fy(\000)1888 946 y Fw(x)p 1888 960 37 4 v 1894 1002 a FG(t)1934 917 y Fy(\001)1982 977 y FH(e)2018 944 y FG(i)2040 950 y FE(\()2070 944 y FG(a)p FF(\()p FG(\026)2165 952 y Fq(t)2193 944 y FF(\)+)p FG(a)2298 923 y Fk(y)2329 944 y FF(\()p FG(\026)2390 952 y Fq(t)2418 944 y FF(\))2441 950 y FE(\))2476 977 y FH(e)2512 945 y Fx(\000)p FG(iH)t(t)2663 977 y FH( )2716 945 y FG(out)2713 993 y(h;\024)2804 1003 y FL(1)1015 1210 y FE(=)126 b FH(s)17 b Fz(\000)g FH(l)q(im)1449 1218 y FG(t)p Fx(!)p FF(+)p Fx(1)1645 1210 y FH(e)1681 1178 y FG(iH)t(t)1785 1210 y FH(e)1821 1177 y FG(i)1843 1183 y FE(\()1872 1177 y FG(a)p FF(\()p FG(\026)1967 1185 y Fq(t)1995 1177 y FF(\)+)p FG(a)2100 1156 y Fk(y)2131 1177 y FF(\()p FG(\026)2192 1185 y Fq(t)2220 1177 y FF(\))2244 1183 y FE(\))2278 1210 y FH(f)2336 1149 y Fy(\000)2381 1179 y Fw(x)p 2381 1193 V 2387 1235 a FG(t)2428 1149 y Fy(\001)2476 1210 y FH(e)2512 1178 y Fx(\000)p FG(iH)t(t)2663 1210 y FH( )2716 1178 y FG(out)2713 1225 y(h;\024)2804 1235 y FL(1)3310 1210 y FI(\(4.7\))1015 1442 y FE(=)82 b FH(s)17 b Fz(\000)g FH(l)q(im)1405 1450 y FG(t)p Fx(!)p FF(+)p Fx(1)1601 1442 y FH(e)1637 1410 y FG(iH)t(t)1741 1442 y FH(e)1777 1409 y FG(i)1799 1415 y FE(\()1828 1409 y FG(a)p FF(\()p FG(\026)1923 1417 y Fq(t)1951 1409 y FF(\)+)p FG(a)2056 1388 y Fk(y)2087 1409 y FF(\()p FG(\026)2148 1417 y Fq(t)2176 1409 y FF(\))2200 1415 y FE(\))2234 1442 y FH(e)2270 1410 y Fx(\000)p FG(iH)t(t)2421 1442 y FH( )2474 1410 y FG(out)2471 1457 y(h)p Fx(\001)p FG(f)2556 1468 y Fk(r)p Fq(E)2652 1457 y FG(;\024)2707 1467 y FL(1)3310 1442 y FI(\(4.8\))515 1601 y FO(where)26 b(the)g(last)g(step,)h(from)f (\(4.7\))g(to)g(\(4.8\),)h(is)f(pro)n(v)n(ed)f(b)n(y)h(means)g(of)g (the)h(same)f(tec)n(hnique)515 1692 y(used)i(in)h(Theorem)e(A2)h(and)h (the)f(notation)g FH( )1978 1660 y FG(out)1975 1707 y(h)p Fx(\001)p FG(f)2060 1718 y Fk(r)p Fq(E)2156 1707 y FG(;\024)2211 1717 y FL(1)2276 1692 y FO(is)h(justi\034ed)g(starting)e(from)h(the)h (reg-)515 1783 y(ularit)n(y)d(prop)r(erties)h(of)h Fz(r)p FH(E)1393 1752 y FG(\033)1447 1783 y FE(\()p FA(P)p FE(\))p FO(.)515 1875 y(The)c(extension)f(to)h(all)f(of)h Fz(H)1410 1843 y FG(out)1529 1875 y FO(is)g(straigh)n(tforw)n(ard)d(b)r(ecause)i FH(f)2551 1814 y Fy(\000)2596 1844 y Fw(x)p 2596 1858 V 2602 1900 a FG(t)2643 1814 y Fy(\001)2702 1875 y FO(is)h(uniformly)f (b)r(ounded)515 1966 y(in)28 b FN(t)f FO(and)h(the)g(set)1103 1910 y Fy(W)1180 1966 y FH( )1233 1934 y FG(out)1230 1981 y(h;\026)1356 1966 y FO(is)f(dense)g(in)h Fz(H)1827 1934 y FG(out)1923 1966 y FO(,)g(b)n(y)f(construction.)842 b Fb(\003)515 2149 y FO(W)-7 b(e)31 b(call)f Fz(A)877 2110 y FG(out)p FF(\()o FG(in)p FF(\))877 2158 y Fw(v)913 2170 y Fq(el)1111 2149 y FO(the)h(norm)f(closure)g(of)g(the)h(*algebra) d(generated)h(b)n(y)i(the)g FH(C)3042 2117 y Fx(1)3037 2162 y FF(0)3137 2149 y FO(functions)515 2240 y(of)h(the)g(asymptotic)f (electron)g(mean)h(v)n(elo)r(cit)n(y)f(de\034ned)h(in)g Fz(H)2503 2208 y FG(out)p FF(\()o FG(in)p FF(\))2739 2240 y FO(b)n(y)f(the)i(strong)d(limits)515 2331 y(\(4.6\))d(\(for)g (the)h FJ(out)35 b FO(and)28 b(the)g FJ(in)34 b FO(case)27 b(resp)r(ectiv)n(ely\))f(in)i(Theorem)f(4.2.)515 2514 y FP(Corollary)k(4.3)515 2605 y FO(In)c(the)h(space)f Fz(H)1048 2573 y FG(out)1144 2605 y FO(,)h(the)g(unitary)f(op)r (erators)1239 2700 y Fy(n)1290 2783 y Fz(W)1372 2747 y FG(out)1480 2783 y FE(\()p FH(\020)5 b FE(\))22 b(:)1648 2763 y Fy(e)1643 2783 y FH(\020)c FE(\()p FA(k)p FE(\))j Fz(2)g FH(L)1947 2747 y FF(2)1995 2722 y Fy(\000)2030 2783 y FB(R)2084 2747 y FF(3)2119 2783 y FH(;)2153 2722 y Fy(\000)2188 2783 y FE(1)c(+)g Fz(j)q FA(k)p Fz(j)2410 2747 y Fx(\000)p FF(1)2492 2722 y Fy(\001)2540 2783 y FH(d)2580 2747 y FF(3)2615 2783 y FH(k)2657 2722 y Fy(\001)2692 2700 y(o)515 2964 y FO(are)26 b(w)n(ell)i(de\034ned)g(starting)e(from)h (the)h(strong)f(limit:)1197 3142 y Fz(W)1279 3106 y FG(out)1387 3142 y FE(\()p FH(\020)5 b FE(\))22 b(:=)f FH(s)c Fz(\000)59 b FE(lim)1740 3187 y FG(t)p Fx(!)p FF(+)p Fx(1)1944 3142 y FH(e)1980 3106 y FG(iH)t(t)2084 3142 y FH(e)2120 3106 y FG(i)2142 3112 y FE(\()2172 3106 y FG(a)p FF(\()o FG(\020)2257 3114 y Fq(t)2285 3106 y FF(\)+)p FG(a)2390 3085 y Fk(y)2421 3106 y FF(\()p FG(\020)2473 3114 y Fq(t)2501 3106 y FF(\))2525 3112 y FE(\))2559 3142 y FH(e)2595 3106 y Fx(\000)p FG(iH)t(t)2764 3142 y FH(:)525 b FI(\(4.9\))515 3335 y FO(The)27 b(follo)n(wing)g (prop)r(erties)f(hold:)515 3517 y(i\))k(The)h(op)r(erators)1144 3435 y Fy(n)1195 3517 y Fz(W)1277 3486 y FG(out)1385 3517 y FE(\()p FH(\020)5 b FE(\))21 b(:)1553 3498 y Fy(e)1548 3517 y FH(\020)26 b Fz(2)c FH(L)1733 3486 y FF(2)1781 3457 y Fy(\000)1816 3517 y FB(R)1870 3486 y FF(3)1904 3517 y FH(;)1939 3457 y Fy(\000)1974 3517 y FE(1)17 b(+)g Fz(j)p FA(k)p Fz(j)2195 3482 y Fx(\000)p FF(1)2278 3457 y Fy(\001)2326 3517 y FH(d)2366 3486 y FF(3)2400 3517 y FH(k)2442 3457 y Fy(\001)2478 3435 y(o)2559 3517 y FO(satisfy)30 b(the)h(W)-7 b(eyl)30 b(comm)n(u-)515 3632 y(tation)d(rules)1245 3767 y Fz(W)1327 3731 y FG(out)1435 3767 y FE(\()p FH(\020)5 b FE(\))13 b Fz(W)1629 3731 y FG(out)1737 3706 y Fy(\000)1772 3767 y FH(\020)1811 3731 y Fx(0)1833 3706 y Fy(\001)1890 3767 y FE(=)21 b Fz(W)2053 3731 y FG(out)2161 3706 y Fy(\000)2196 3767 y FH(\020)2235 3731 y Fx(0)2275 3767 y FE(+)16 b FH(\020)2390 3706 y Fy(\001)2438 3767 y FH(e)2474 3731 y Fx(\000)2532 3688 y Fq(\032)2562 3696 y FE(\()2592 3688 y Fq(\020)r(;\020)2671 3671 y Fk(0)2693 3696 y FE(\))p 2532 3719 192 3 v 2614 3752 a FL(2)3271 3767 y FI(\(4.10\))515 3922 y FO(where)27 b FH(\032)12 b FE(\()p FH(\020)5 b(;)13 b(\020)949 3890 y Fx(0)971 3922 y FE(\))22 b(=)f(2)p FH(iI)6 b(m)1288 3839 y Fy(\020)1334 3862 y(R)p 1398 3839 40 4 v 1403 3902 a(e)1398 3922 y FH(\020)18 b FE(\()p FA(k)p FE(\))1579 3902 y Fy(e)1569 3922 y FH(\020)1608 3901 y Fx(0)1643 3922 y FE(\()p FA(k)p FE(\))13 b FH(d)1803 3890 y FF(3)1837 3922 y FH(k)1879 3839 y Fy(\021)1925 3922 y FO(;)515 4105 y(ii\))39 b(The)g(mapping)g Fn(R)j Fo(3)g FN(s)g Fo(!)g(W)1661 4075 y FM(out)1776 4105 y FK(\()p FN(s\026)p FK(\))d FO(de\034nes)g(a)f(strongly)g(con)n(tin)n(uous,)g(one)g(para-) 515 4196 y(metric)27 b(group)g(of)g(unitary)g(op)r(erators.)515 4379 y FJ(Pr)l(o)l(of)515 4470 y FO(The)g(existence)h(of)1472 4561 y FH(s)17 b Fz(\000)59 b FE(lim)1602 4606 y FG(t)p Fx(!)p FF(+)p Fx(1)1807 4561 y FH(e)1843 4525 y FG(iH)t(t)1946 4561 y Fz(W)19 b FE(\()p FH(\020)2105 4569 y FG(t)2132 4561 y FE(\))13 b FH(e)2211 4525 y Fx(\000)p FG(iH)t(t)2362 4561 y FH( )2415 4525 y FG(out)2412 4574 y(h;\026)3271 4561 y FI(\(4.11\))515 4719 y FO(with)1583 4810 y Fz(W)19 b FE(\()p FH(\020)1742 4818 y FG(t)1770 4810 y FE(\))i(:=)g FH(e)1959 4774 y FG(i)1981 4780 y FE(\()2011 4774 y FG(a)p FF(\()p FG(\020)2097 4782 y Fq(t)2125 4774 y FF(\)+)p FG(a)2230 4753 y Fk(y)2261 4774 y FF(\()p FG(\020)2313 4782 y Fq(t)2341 4774 y FF(\))2365 4780 y FE(\))1950 5059 y FO(41)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 515 523 a FO(on)35 b(a)g(generic)f FH( )1057 491 y FG(out)1054 539 y(h;\026)1187 523 y FO(implies)i(that)f (the)h(b)r(ounded)f(op)r(erators)f FH(W)2621 491 y FG(out)2728 523 y FE(\()p FH(\020)5 b FE(\))35 b FO(can)g(b)r(e)g(extended)515 614 y(from)27 b(the)h(dense)f(set)1210 558 y Fy(W)1287 614 y FH( )1340 583 y FG(out)1337 630 y(h;\026)1462 614 y FO(to)h(all)f(of)h Fz(H)1839 583 y FG(out)1934 614 y FO(,)g(b)n(y)f(con)n(tin)n(uit)n(y)-7 b(.)515 706 y(In)31 b(order)f(to)h(pro)n(v)n(e)e(the)j(existence)f(of)g(the)g(limit)h (\(4.11\),)f(let)g(us)g(consider)f(for)h FH(t)3111 714 y FF(2)3172 706 y FH(>)c(t)3287 714 y FF(1)3352 706 y FO(the)515 797 y(di\033erence)1143 888 y FH(e)1179 852 y FG(iH)t(t)1279 862 y FL(1)1315 888 y Fz(W)19 b FE(\()p FH(\020)1474 896 y FG(t)1498 906 y FL(1)1534 888 y FE(\))13 b FH(e)1613 852 y Fx(\000)p FG(iH)t(t)1761 862 y FL(1)1796 888 y FH( )1849 852 y FG(out)1846 901 y(h;\026)1961 888 y Fz(\000)k FH(e)2074 852 y FG(iH)t(t)2174 862 y FL(2)2210 888 y Fz(W)i FE(\()p FH(\020)2369 896 y FG(t)2393 906 y FL(2)2429 888 y FE(\))12 b FH(e)2507 852 y Fx(\000)p FG(iH)t(t)2655 862 y FL(2)2691 888 y FH( )2744 852 y FG(out)2741 901 y(h;\026)3271 888 y FI(\(4.12\))515 1003 y FO(and)27 b(a)g(sequence)g(of)h(functions)1542 942 y Fy(\010)1586 1003 y FH(\026)1632 971 y FG(n)1688 1003 y FE(\()p FA(k)p FE(\))21 b Fz(2)g FH(C)1948 971 y Fx(1)1943 1016 y FF(0)2026 942 y Fy(\000)2061 1003 y FB(R)2115 971 y FF(3)2167 1003 y Fz(n)d FE(0)2261 942 y Fy(\001)2330 1003 y FH(n)k Fz(2)f FB(N)2524 942 y Fy(\011)2597 1003 y FO(suc)n(h)27 b(that)1332 1144 y Fz(k)q FH(\020)22 b Fz(\000)16 b FH(\026)1549 1108 y FG(n)1592 1144 y Fz(k)1631 1168 y FG(L)1673 1154 y FL(2)1705 1174 y FE(\()1735 1168 y Fv(R)1771 1154 y FL(3)1803 1168 y FG(;)1822 1174 y FE(\()1852 1168 y FF(1+)p Fx(j)p Fw(k)p Fx(j)2003 1151 y Fk(\000)p FL(1)2081 1174 y FE(\))2111 1168 y FG(d)2143 1154 y FL(3)2175 1168 y FG(k)2209 1174 y FE(\))2264 1144 y Fz(!)2341 1152 y FG(n)p Fx(!)p FF(+)p Fx(1)2573 1144 y FE(0)i FH(:)515 1308 y FO(Then)27 b(w)n(e)h(exploit)f(the)h(follo)n (wing)e(iden)n(tit)n(y)1122 1449 y FH(e)1158 1418 y FG(iH)t(t)1258 1428 y FL(1)1294 1449 y Fz(W)19 b FE(\()p FH(\020)1453 1457 y FG(t)1477 1467 y FL(1)1513 1449 y FE(\))12 b FH(e)1591 1418 y Fx(\000)p FG(iH)t(t)1739 1428 y FL(1)1775 1449 y FH( )1828 1418 y FG(out)1825 1465 y(h;\026)1940 1449 y Fz(\000)17 b FH(e)2053 1418 y FG(iH)t(t)2153 1428 y FL(2)2188 1449 y Fz(W)i FE(\()p FH(\020)2347 1457 y FG(t)2371 1467 y FL(2)2407 1449 y FE(\))13 b FH(e)2486 1418 y Fx(\000)p FG(iH)t(t)2634 1428 y FL(2)2670 1449 y FH( )2723 1418 y FG(out)2720 1465 y(h;\026)2839 1449 y FE(=)878 1682 y(=)92 b FH(e)1066 1650 y FG(iH)t(t)1166 1660 y FL(1)1201 1682 y Fz(W)19 b FE(\()p FH(\020)1360 1690 y FG(t)1384 1700 y FL(1)1420 1682 y FE(\))13 b FH(e)1499 1650 y Fx(\000)p FG(iH)t(t)1647 1660 y FL(1)1683 1682 y FH( )1736 1650 y FG(out)1733 1697 y(h;\026)1848 1682 y Fz(\000)j FH(e)1960 1650 y FG(iH)t(t)2060 1660 y FL(1)2096 1682 y Fz(W)j FE(\()p FH(\020)2255 1690 y FG(t)2279 1700 y FL(1)2315 1682 y FE(\))13 b FH(e)2394 1650 y Fx(\000)p FG(iH)t(t)2542 1660 y FL(1)2577 1682 y FH( )2627 1691 y FG(h;\026)2737 1682 y FE(\()p FH(t)2795 1690 y FF(1)2829 1682 y FE(\))412 b FI(\(4.13\))878 1914 y FE(+)89 b FH(e)1063 1883 y FG(iH)t(t)1163 1893 y FL(1)1199 1914 y Fz(W)19 b FE(\()o FH(\020)1357 1922 y FG(t)1381 1932 y FL(1)1417 1914 y FE(\))13 b FH(e)1496 1883 y Fx(\000)p FG(iH)t(t)1644 1893 y FL(1)1680 1914 y FH( )1730 1923 y FG(h;\026)1839 1914 y FE(\()p FH(t)1897 1922 y FF(1)1931 1914 y FE(\))k Fz(\000)g FH(e)2091 1883 y FG(iH)t(t)2191 1893 y FL(1)2227 1914 y Fz(W)i FE(\()p FH(\026)2398 1883 y FG(n)2398 1927 y(t)2422 1937 y FL(1)2458 1914 y FE(\))13 b FH(e)2537 1883 y Fx(\000)p FG(iH)t(t)2685 1893 y FL(1)2721 1914 y FH( )2771 1923 y FG(h;\026)2880 1914 y FE(\()p FH(t)2938 1922 y FF(1)2972 1914 y FE(\))269 b FI(\(4.14\))878 2147 y FE(+)83 b FH(e)1057 2115 y FG(iH)t(t)1157 2125 y FL(1)1192 2147 y Fz(W)19 b FE(\()p FH(\026)1363 2115 y FG(n)1363 2160 y(t)1387 2170 y FL(1)1424 2147 y FE(\))12 b FH(e)1502 2115 y Fx(\000)p FG(iH)t(t)1650 2125 y FL(1)1686 2147 y FH( )1736 2156 y FG(h;\026)1846 2147 y FE(\()o FH(t)1903 2155 y FF(1)1938 2147 y FE(\))17 b Fz(\000)g FH(e)2098 2115 y FG(iH)t(t)2198 2125 y FL(2)2233 2147 y Fz(W)i FE(\()p FH(\026)2404 2115 y FG(n)2404 2160 y(t)2428 2170 y FL(2)2465 2147 y FE(\))12 b FH(e)2543 2115 y Fx(\000)p FG(iH)t(t)2691 2125 y FL(2)2727 2147 y FH( )2777 2156 y FG(h;\026)2887 2147 y FE(\()o FH(t)2944 2155 y FF(2)2979 2147 y FE(\))262 b FI(\(4.15\))878 2379 y FE(+)89 b FH(e)1063 2347 y FG(iH)t(t)1163 2357 y FL(2)1199 2379 y Fz(W)19 b FE(\()o FH(\026)1369 2347 y FG(n)1369 2392 y(t)1393 2402 y FL(2)1430 2379 y FE(\))13 b FH(e)1509 2347 y Fx(\000)p FG(iH)t(t)1657 2357 y FL(2)1692 2379 y FH( )1742 2388 y FG(h;\026)1852 2379 y FE(\()p FH(t)1910 2387 y FF(2)1944 2379 y FE(\))k Fz(\000)g FH(e)2104 2347 y FG(iH)t(t)2204 2357 y FL(2)2240 2379 y Fz(W)i FE(\()o FH(\020)2398 2387 y FG(t)2422 2397 y FL(2)2458 2379 y FE(\))13 b FH(e)2537 2347 y Fx(\000)p FG(iH)t(t)2685 2357 y FL(2)2721 2379 y FH( )2771 2388 y FG(h;\026)2880 2379 y FE(\()p FH(t)2938 2387 y FF(2)2972 2379 y FE(\))269 b FI(\(4.16\))878 2612 y FE(+)104 b FH(e)1078 2580 y FG(iH)t(t)1178 2590 y FL(2)1214 2612 y Fz(W)19 b FE(\()p FH(\020)1373 2620 y FG(t)1397 2630 y FL(2)1433 2612 y FE(\))13 b FH(e)1512 2580 y Fx(\000)p FG(iH)t(t)1660 2590 y FL(2)1695 2612 y FH( )1745 2621 y FG(h;\026)1855 2612 y FE(\()p FH(t)1913 2620 y FF(2)1947 2612 y FE(\))k Fz(\000)g FH(e)2107 2580 y FG(iH)t(t)2207 2590 y FL(2)2242 2612 y Fz(W)i FE(\()p FH(\020)2401 2620 y FG(t)2425 2630 y FL(2)2461 2612 y FE(\))13 b FH(e)2540 2580 y Fx(\000)p FG(iH)t(t)2688 2590 y FL(2)2724 2612 y FH( )2777 2580 y FG(out)2774 2627 y(h;\026)3271 2612 y FI(\(4.17\))515 2753 y FO(and)27 b(observ)n(e)f(that:)515 2844 y(F)-7 b(or)30 b(\(4.13\))g(and)h(\(4.17\),)f(the)h(corresp)r(onding)e(norms)h (are)g(at)h(most)f(of)h(order)e FH(t)3076 2809 y Fx(\000)p FG(\032)3076 2863 y FF(1)3191 2844 y FO(and)i FH(t)3384 2809 y Fx(\000)p FG(\032)3384 2863 y FF(2)515 2936 y FO(resp)r(ectiv)n(ely)-7 b(,)26 b(for)i(some)e(p)r(ositiv)n(e)i FH(\032)p FO(,)f(b)r(ecause)g(of)g(Theorem)g(4.1;)515 3027 y(F)-7 b(or)27 b(\(4.14\))f(\(and)i(equiv)-5 b(alen)n(tly)27 b(\(4.16\)\),)g(w)n(e)h(can)f(estimate)870 3106 y Fy(\015)870 3151 y(\015)870 3195 y(\015)913 3192 y FH(e)949 3156 y FG(iH)t(t)1049 3166 y FL(1)1085 3192 y Fz(W)19 b FE(\()p FH(\020)1244 3200 y FG(t)1268 3210 y FL(1)1304 3192 y FE(\))12 b FH(e)1382 3156 y Fx(\000)p FG(iH)t(t)1530 3166 y FL(1)1566 3192 y FH( )1616 3201 y FG(h;\026)1726 3192 y FE(\()o FH(t)1783 3200 y FF(1)1818 3192 y FE(\))17 b Fz(\000)f FH(e)1977 3156 y FG(iH)t(t)2077 3166 y FL(1)2113 3192 y Fz(W)j FE(\()p FH(\026)2284 3156 y FG(n)2284 3205 y(t)2308 3215 y FL(1)2345 3192 y FE(\))12 b FH(e)2423 3156 y Fx(\000)p FG(iH)t(t)2571 3166 y FL(1)2607 3192 y FH( )2657 3201 y FG(h;\026)2767 3192 y FE(\()o FH(t)2824 3200 y FF(1)2859 3192 y FE(\))2889 3106 y Fy(\015)2889 3151 y(\015)2889 3195 y(\015)2953 3192 y Fz(\024)515 3475 y(\024)21 b FH(C)h Fz(\001)17 b(k)q FH(\020)22 b Fz(\000)17 b FH(\026)929 3439 y FG(n)971 3475 y Fz(k)1010 3499 y FG(L)1052 3485 y FL(2)1084 3505 y FE(\()1114 3499 y Fv(R)1150 3485 y FL(3)1183 3499 y FG(;)1202 3505 y FE(\()1231 3499 y FF(1+)p Fx(j)p Fw(k)p Fx(j)1383 3483 y Fk(\000)p FL(1)1460 3505 y FE(\))1490 3499 y FG(d)1522 3485 y FL(3)1554 3499 y FG(k)1588 3505 y FE(\))1639 3475 y Fz(\001)1678 3345 y Fy(\015)1678 3389 y(\015)1678 3434 y(\015)1678 3479 y(\015)1678 3524 y(\015)1720 3475 y FH(H)1790 3439 y FG(ph)1868 3396 y FL(1)p 1868 3403 29 3 v 1868 3434 a(2)1927 3370 y Fy(\022)2124 3427 y FE(1)p 1994 3458 298 4 v 1994 3526 a FH(H)2058 3534 y FG(\033)2094 3542 y Fq(t)2117 3554 y FL(1)2174 3526 y FE(+)g FH(a)2302 3370 y Fy(\023)2368 3368 y FL(1)p 2368 3376 29 3 v 2368 3409 a(2)2410 3345 y Fy(\015)2410 3389 y(\015)2410 3434 y(\015)2410 3479 y(\015)2410 3524 y(\015)2470 3475 y Fz(\001)2509 3389 y Fy(\015)2509 3434 y(\015)2509 3479 y(\015)2551 3415 y(\000)2586 3475 y FH(H)2650 3483 y FG(\033)2686 3491 y Fq(t)2709 3503 y FL(1)2767 3475 y FE(+)g FH(a)2885 3415 y Fy(\001)2929 3413 y FL(1)p 2929 3421 V 2929 3454 a(2)2985 3475 y FH(e)3021 3439 y Fx(\000)p FG(iH)t(t)3169 3449 y FL(1)3204 3475 y FH( )3254 3484 y FG(h;\026)3364 3475 y FE(\()p FH(t)3422 3483 y FF(1)3456 3475 y FE(\))3486 3389 y Fy(\015)3486 3434 y(\015)3486 3479 y(\015)515 3684 y FO(where)28 b FN(a)h FO(is)g(a)g(su\036cien)n (tly)g(large)f(p)r(ositiv)n(e)g(n)n(um)n(b)r(er.)h(Both)g(the)g(p)r (ositiv)n(e)g(constan)n(t)f FN(C)36 b FO(and)515 3775 y(the)28 b(t)n(w)n(o)e(norms)1049 3837 y Fy(\015)1049 3882 y(\015)1049 3927 y(\015)1049 3971 y(\015)1049 4016 y(\015)1091 3968 y FH(H)1161 3932 y FG(ph)1239 3888 y FL(1)p 1239 3895 V 1239 3926 a(2)1298 3862 y Fy(\022)1494 3919 y FE(1)p 1365 3951 298 4 v 1365 4018 a FH(H)1429 4026 y FG(\033)1465 4034 y Fq(t)1488 4046 y FL(1)1545 4018 y FE(+)17 b FH(a)1673 3862 y Fy(\023)1739 3860 y FL(1)p 1739 3868 29 3 v 1739 3901 a(2)1781 3837 y Fy(\015)1781 3882 y(\015)1781 3927 y(\015)1781 3971 y(\015)1781 4016 y(\015)1858 3968 y FH(;)1914 3882 y Fy(\015)1914 3927 y(\015)1914 3971 y(\015)1956 3907 y(\000)1991 3968 y FH(H)2055 3976 y FG(\033)2091 3984 y Fq(t)2114 3996 y FL(1)2172 3968 y FE(+)g FH(a)2290 3907 y Fy(\001)2335 3905 y FL(1)p 2335 3913 V 2335 3946 a(2)2390 3968 y FH(e)2426 3932 y Fx(\000)p FG(iH)t(t)2574 3942 y FL(1)2609 3968 y FH( )2659 3977 y FG(h;\026)2769 3968 y FE(\()p FH(t)2827 3976 y FF(1)2861 3968 y FE(\))2891 3882 y Fy(\015)2891 3927 y(\015)2891 3971 y(\015)515 4182 y FO(are)23 b(uniformly)h(b)r (ounded)h(in)g FH(t)1483 4190 y FF(1)1541 4182 y FO(\(in)g FH(t)1695 4190 y FF(2)1754 4182 y FO(for)f(the)g(analogous)e (expression)h(in)i(the)g(case)e(\(4.16\)\);)515 4364 y(F)-7 b(or)35 b(\(4.15\),)h(at)g(\034xed)h FH(n)p FO(,)f(the)h(norm)f (is)g(in\034nitesimal)h(for)e FH(t)2482 4372 y FF(1)2552 4364 y Fz(!)f FE(+)p Fz(1)i FO(b)r(ecause)g(of)g(Theo-)515 4456 y(rem)27 b(4.1.)515 4638 y(Hence)g(the)h(con)n(v)n(ergence)d (\(4.9\))j(trivially)f(follo)n(ws.)f(Moreo)n(v)n(er)f(the)j(so)f (de\034ned)h(op)r(erators)1239 4720 y Fy(n)1290 4803 y Fz(W)1372 4767 y FG(out)1480 4803 y FE(\()p FH(\020)5 b FE(\))22 b(:)1648 4783 y Fy(e)1643 4803 y FH(\020)c FE(\()p FA(k)p FE(\))j Fz(2)g FH(L)1947 4767 y FF(2)1995 4743 y Fy(\000)2030 4803 y FB(R)2084 4767 y FF(3)2119 4803 y FH(;)2153 4743 y Fy(\000)2188 4803 y FE(1)c(+)g Fz(j)q FA(k)p Fz(j)2410 4767 y Fx(\000)p FF(1)2492 4743 y Fy(\001)2540 4803 y FH(d)2580 4767 y FF(3)2615 4803 y FH(k)2657 4743 y Fy(\001)2692 4720 y(o)1950 5059 y FO(42)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 515 523 a FO(are)26 b(unitary)h(in)h Fz(H)1108 491 y FG(out)1204 523 y FI(.)515 706 y FO(Concerning)e(the)i (prop)r(erties)f(i\))h(and)f(ii\):)515 797 y(i\))h(The)f(op)r(erator) 1416 888 y Fz(W)1498 852 y FG(out)1606 888 y FE(\()p FH(\020)5 b FE(\))13 b Fz(W)1800 852 y FG(out)1908 828 y Fy(\000)1943 888 y FH(\020)1982 852 y Fx(0)2005 828 y Fy(\001)2061 888 y FE(:)43 b Fz(H)2190 852 y FG(out)2307 888 y Fz(!)21 b(H)2470 852 y FG(out)515 1013 y FO(is)i(the)g(time)h (limit)g(of)f(the)g(equal)g(time)g(pro)r(duct)g(of)h(the)f(corresp)r (onding)e(appro)n(ximating)g(op-)515 1104 y(erators)28 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y(i\))h(Starting)f(from)g(the)h FH(\034)9 b FO(-ev)n(olv)n(ed)25 b(generators)1321 2201 y FH(e)1357 2165 y FG(iH)t(\034)1471 2201 y Fz(W)1553 2165 y FG(out)p FF(\()p FG(in)p FF(\))1770 2201 y FE(\()p FH(\020)5 b FE(\))13 b FH(e)1918 2165 y Fx(\000)p FG(iH)t(\034)2101 2201 y FE(=)21 b Fz(W)2264 2165 y FG(out)p FF(\()o FG(in)p FF(\))2481 2201 y FE(\()p FH(\020)2545 2209 y Fx(\000)p FG(\034)2631 2201 y FE(\))610 b FI(\(4.18\))515 2372 y FO(where)27 b FH(\020)789 2380 y Fx(\000)p FG(\034)902 2372 y FI(\()962 2352 y Fy(e)958 2372 y FH(\020)992 2380 y Fx(\000)p FG(\034)1091 2372 y FE(\()p FA(k)p FE(\))22 b(:=)g FH(e)1359 2340 y Fx(\000)p FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(\034)1546 2352 y Fy(e)1542 2372 y FH(\020)c FE(\()o FA(k)p FE(\))p FO(\))28 b(is)g(the)h(freely)f (ev)n(olv)n(ed)e(test)j(function)f FH(\020)33 b FO(in)c(the)f(time)515 2477 y FH(\034)9 b FO(,)27 b(the)h(automorphism)e FH(\013)1349 2485 y FG(\034)1416 2477 y FO(of)i Fz(A)1572 2438 y FG(out)p FF(\()o FG(in)p FF(\))1572 2498 y 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4056 y FG(ph)2775 4035 y FI(.)515 4126 y(ii\))i FO(By)f(an)h(appro)n (ximation)e(argumen)n(t,)g(the)i(considered)f(prop)r(ert)n(y)g(follo)n (ws)f(from)i(the)g(de\034-)515 4229 y(nition)i(of)f(the)h(generators)d (of)j Fz(A)1560 4191 y FG(out)p FF(\()o FG(in)p FF(\))1560 4238 y Fw(v)1596 4250 y Fq(el)1793 4229 y FI(and)g Fz(A)2008 4191 y FG(out)p FF(\()o FG(in)p FF(\))2008 4250 y FG(ph)2242 4229 y FO(in)f(Theorem)g(4.2)g(and)g(Corollary)e(4.3)515 4320 y(resp)r(ectiv)n(ely)-7 b(.)2446 b Fb(\003)1950 5059 y FO(43)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 515 523 a FD(5)131 b(APPENDIX)515 697 y FO(In)25 b(the)h(follo)n(wing)e(lemmas)h(and)g(theorems)f(w)n(e)h (assume)f(the)i(prop)r(erties)e(discussed)h(in)g(Sub-)515 788 y(section)36 b(1.3.1.)f(As)h(in)h(the)f(previous)f(Sections,)i(w)n (e)e(use)i(the)f(con)n(v)n(en)n(tion)f(to)h(generically)515 879 y(call)30 b FN(C)38 b FO(the)32 b(constan)n(ts)e(whic)n(h)h(are)f (time)h(indep)r(enden)n(t,)h(uniform)f(in)h(the)f(infrared)g(cut-o\033) 515 971 y(and)h(in)g(the)h(cell)f(partition.)g(The)g(b)r(ounds)h(are)e (in)n(tended)i(from)f(ab)r(o)n(v)n(e,)f(unless)h(otherwise)515 1062 y(indicated.)515 1336 y(W)-7 b(e)35 b(no)n(w)f(pro)n(vide)f(some)h (results)g(ab)r(out)g(the)h(phase)f(factor)g(\020)28 b FH(e)2602 1304 y FG(i\015)2685 1336 y FO(\021)41 b(whic)n(h)35 b(en)n(ters)e(in)i(the)515 1427 y(de\034nition)28 b(of)f(the)h(appro)n (ximating)e(v)n(ector)g FH( )1974 1436 y FG(h;k)2062 1446 y FL(1)2111 1427 y FE(\()p FH(t)p FE(\))p FO(.)515 1610 y FP(Lemma)j(A1)515 1701 y FO(Under)d(the)g(assumptions)g(for)f (the)i(construction)e(\(Subsection)h(1.3.1\))f(and)h(b)r(ecause)g(of)g (the)515 1792 y(de\034nition)i(\(2.6\),)f(the)h(follo)n(wing)e (estimates)i(hold:)570 1877 y Fy(\014)570 1922 y(\014)570 1967 y(\014)596 1963 y FH(\015)636 1971 y FG(\033)672 1979 y Fq(t)695 1991 y FL(2)748 1881 y Fy(\020)794 1963 y FA(v)841 1971 y FG(j)873 1963 y FH(;)13 b Fz(r)p FH(E)1032 1917 y FG(\033)1068 1925 y Fq(t)1091 1937 y FL(2)1028 1984 y 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2804 y(\014)942 2848 y(\014)942 2893 y(\014)968 2890 y FH(\015)1008 2898 y FG(\033)1044 2906 y Fq(t)1088 2890 y FE(\()p FA(v)1165 2898 y FG(j)1197 2890 y FH(;)13 b Fz(r)p FH(E)1356 2854 y FG(\033)1392 2862 y Fq(t)1352 2909 y Fw(P)1423 2890 y FH(;)g(s)p FE(\))k Fz(\000)g FH(\015)1657 2898 y FG(\033)1693 2906 y Fq(t)1737 2807 y Fy(\020)1782 2890 y FA(v)1829 2898 y FG(j)1862 2890 y FH(;)c Fz(r)p FH(E)2021 2854 y FG(\033)2057 2862 y Fq(t)2017 2917 y Fw(P)p FF(+)2119 2893 y Fl(q)p 2119 2906 34 3 v 2122 2939 a Fq(s)2167 2890 y FH(;)g(s)2237 2807 y Fy(\021)2282 2804 y(\014)2282 2848 y(\014)2282 2893 y(\014)2329 2890 y Fz(\024)21 b FH(C)i Fz(\001)17 b FH(s)2562 2854 y Fx(\000\001)2648 2834 y Fq(\013)p 2639 2842 57 3 v 2639 2875 a FL(16)2726 2890 y Fz(\001)h FH(s)2801 2854 y FF(2\(1)p Fx(\000)p FG(\013)p FF(\))3019 2890 y FH(:)270 b FI(\(5.3\))515 3060 y FJ(Pr)l(o)l(of)515 3151 y FO(The)33 b(b)r(ounds)g(can)g(b)r(e)h(obtained)f(b)n(y)g 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y(\034cien)n(tly)i(close) f(to)i FE(1)f FO(and)g FH(\017)22 b(>)f FE(0)28 b FO(su\036cien)n(tly)f (small,)g(the)g(follo)n(wing)g(propagation)e(estimate)515 4156 y(holds)i(true:)515 4253 y Fy(\015)515 4298 y(\015)515 4342 y(\015)557 4278 y(R)628 4339 y(e)-50 b FH(\037)669 4348 y FG(h)722 4339 y FE(\()p FA(q)p FE(\))842 4256 y Fy(\020)887 4339 y FH(e)923 4307 y Fx(\000)p FG(i)p Fw(q)p Fx(\001r)p FG(E)1147 4279 y Fq(\033)1181 4290 y(t)1144 4328 y Fl(P)1234 4339 y Fz(\000)17 b FH(e)1347 4307 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1482 4287 y Fl(x)p 1482 4295 34 3 v 1485 4328 a Fq(s)1529 4256 y Fy(\021)1588 4339 y FH(d)1628 4307 y FF(3)1662 4339 y FH(q)s(e)1735 4307 y Fx(\000)p FG(iE)1853 4279 y Fq(\033)1887 4290 y(t)1850 4328 y Fl(P)1918 4307 y FG(s)1951 4339 y FH(e)1987 4306 y FG(i\015)2041 4314 y Fq(\033)2075 4325 y(t)2107 4312 y FE(\()2137 4306 y Fw(v)2173 4317 y Fq(j)2205 4306 y FG(;)p Fx(r)p FG(E)2323 4277 y Fq(\033)2357 4288 y(t)2320 4327 y Fl(P)2388 4306 y FG(;s)2436 4312 y FE(\))2470 4339 y FH( )2523 4300 y FF(\()p FG(t)p FF(\))2520 4359 y FG(j;\033)2600 4367 y Fq(t)2631 4253 y Fy(\015)2631 4298 y(\015)2631 4342 y(\015)2695 4339 y Fz(\024)1427 4534 y(\024)k FH(C)h Fz(\001)c FH(s)1660 4498 y Fx(\000)p FG(\035)1765 4534 y Fz(\001)f FH(t)1831 4498 y Fx(\000)1889 4478 y FL(3)p Fq(\017)p 1889 4486 54 3 v 1902 4519 a FL(2)1974 4534 y Fz(\001)g(j)q FE(ln)12 b FH(\033)2154 4542 y FG(t)2182 4534 y Fz(j)167 b FH(\035)24 b(>)d FE(0)515 4681 y FO(where)34 b Fy(e)-50 b FH(\037)803 4690 y FG(h)871 4681 y FO(is)27 b(the)h(F)-7 b(ourier)27 b(transformed)f(of)h FH(\037)1991 4690 y FG(h)2032 4681 y FI(,)f FH(s)21 b Fz(\025)g FH(t)g Fz(\035)g FE(1)28 b FO(and)f FA(v)2638 4689 y FG(j)2692 4681 y Fz(\021)21 b(r)p FH(E)2898 4647 y FG(\033)2934 4655 y Fq(t)p 2894 4673 82 3 v 2894 4714 a Fw(P)2941 4725 y Fq(j)3003 4681 y FO(is)27 b(referred)g(to)515 4810 y(the)h(partition)f FE(\000)1052 4778 y FF(\()p FG(t)p FF(\))1128 4810 y FO(.)1950 5059 y(44)p eop end %%Page: 45 45 TeXDict begin 45 44 bop 515 523 a FJ(Pr)l(o)l(of)515 614 y FO(Let)27 b(us)h(start)f(from)g(the)h(follo)n(wing)f(Hilb)r(ert)h (inequalit)n(y:)515 702 y Fy(\015)515 747 y(\015)515 792 y(\015)515 837 y(\015)557 709 y(Z)654 810 y(e)-50 b FH(\037)695 819 y FG(h)748 810 y FE(\()p FA(q)p FE(\))867 728 y Fy(\020)913 810 y FH(e)949 775 y Fx(\000)p FG(i)p Fw(q)p Fx(\001r)p FG(E)1173 746 y Fq(\033)1207 757 y(t)1170 796 y Fl(P)1260 810 y Fz(\000)16 b FH(e)1372 775 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1508 755 y Fl(x)p 1508 763 34 3 v 1511 796 a Fq(s)1555 728 y Fy(\021)1614 810 y FH(d)1654 775 y FF(3)1688 810 y FH(q)s(e)1761 775 y Fx(\000)p FG(iE)1879 746 y Fq(\033)1913 757 y(t)1876 796 y Fl(P)1944 775 y FG(s)1977 810 y FH(e)2013 775 y FG(i\015)2067 783 y Fq(\033)2101 794 y(t)2133 781 y FE(\()2163 775 y Fw(v)2199 786 y Fq(j)2230 775 y FG(;)p Fx(r)p FG(E)2348 746 y Fq(\033)2382 757 y(t)2345 796 y Fl(P)2413 775 y FG(;s)2461 781 y FE(\))2495 810 y FH( )2548 772 y FF(\()p FG(t)p FF(\))2545 830 y FG(j;\033)2625 838 y Fq(t)2657 702 y Fy(\015)2657 747 y(\015)2657 792 y(\015)2657 837 y(\015)2721 810 y Fz(\024)728 1131 y(\024)809 978 y Fy(\015)809 1023 y(\015)809 1067 y(\015)809 1112 y(\015)809 1157 y(\015)809 1202 y(\015)852 1029 y(Z)949 1131 y(e)-50 b FH(\037)990 1140 y FG(h)1043 1131 y FE(\()p FA(q)p FE(\))1162 981 y Fy(0)1162 1115 y(@)1229 1131 y FH(e)1265 1095 y Fx(\000)p FG(i)p Fw(q)p Fx(\001r)p FG(E)1489 1067 y Fq(\033)1523 1078 y(t)1486 1116 y Fl(P)1576 1131 y Fz(\000)17 b FH(e)1689 1061 y FG(i)1711 976 y Fm( )1756 1061 y FG(E)1804 1032 y Fq(\033)1838 1043 y(t)1801 1082 y Fl(P)1869 1061 y Fx(\000)p FG(E)1965 1032 y Fq(\033)1999 1043 y(t)1962 1093 y Fl(P)p FL(+)2056 1072 y Fl(q)p 2056 1084 V 2059 1115 a Fq(s)2104 976 y Fm(!)2149 1061 y FG(s)2182 981 y Fy(1)2182 1115 y(A)2262 1131 y FH(d)2302 1095 y FF(3)2336 1131 y FH(q)s(e)2409 1095 y FG(i\015)2463 1103 y Fq(\033)2497 1114 y(t)2529 1101 y FE(\()2559 1095 y Fw(v)2595 1106 y Fq(j)2627 1095 y FG(;)p Fx(r)p FG(E)2745 1067 y Fq(\033)2779 1078 y(t)2742 1116 y Fl(P)2810 1095 y FG(;s)2858 1101 y FE(\))2892 1131 y FH( )2945 1093 y FF(\()p FG(t)p FF(\))2942 1151 y FG(j;\033)3022 1159 y Fq(t)3053 978 y Fy(\015)3053 1023 y(\015)3053 1067 y(\015)3053 1112 y(\015)3053 1157 y(\015)3053 1202 y(\015)3310 1131 y FI(\(5.4\))863 1442 y FE(+)935 1289 y Fy(\015)935 1334 y(\015)935 1379 y(\015)935 1423 y(\015)935 1468 y(\015)935 1513 y(\015)978 1340 y(Z)1075 1442 y(e)-50 b FH(\037)1116 1451 y FG(h)1169 1442 y FE(\()p FA(q)p FE(\))12 b FH(e)1324 1372 y FG(i)1346 1287 y Fm( )1391 1372 y FG(E)1439 1343 y Fq(\033)1473 1354 y(t)1436 1393 y Fl(P)1504 1372 y Fx(\000)p FG(E)1600 1343 y Fq(\033)1634 1354 y(t)1597 1404 y Fl(P)p FL(+)1692 1383 y Fl(q)p 1692 1395 V 1695 1426 a Fq(s)1739 1287 y Fm(!)1784 1372 y FG(s)1830 1359 y Fy(\020)1876 1442 y FH(e)1912 1406 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)2047 1387 y Fl(x)p 2047 1395 V 2050 1428 a Fq(s)2112 1442 y Fz(\000)k FE(1)2226 1359 y Fy(\021)2285 1442 y FH(d)2325 1406 y FF(3)2360 1442 y FH(q)s(e)2433 1406 y FG(i\015)2487 1414 y Fq(\033)2521 1425 y(t)2553 1412 y FE(\()2583 1406 y Fw(v)2619 1417 y Fq(j)2650 1406 y FG(;)p Fx(r)p FG(E)2768 1378 y Fq(\033)2802 1389 y(t)2765 1427 y Fl(P)2833 1406 y FG(;s)2881 1412 y FE(\))2915 1442 y FH( )2968 1404 y FF(\()p FG(t)p FF(\))2965 1462 y FG(j;\033)3045 1470 y Fq(t)3077 1289 y Fy(\015)3077 1334 y(\015)3077 1379 y(\015)3077 1423 y(\015)3077 1468 y(\015)3077 1513 y(\015)3310 1442 y FI(\(5.5\))515 1658 y FO(In)35 b(order)f(to)h(estimate)g(the)h (in)n(tegrals)e(in)h(\(5.4\))g(and)g(\(5.5\),)h(w)n(e)e(distinguish)i (\020large\021)k(and)515 1749 y(\020small\021)33 b FA(q)p FO(:)515 1858 y(F)-7 b(or)27 b(large)f FA(q)p FO(,)h(that)h(is)1227 1775 y Fy(n)1279 1858 y FA(q)21 b FE(:)38 b Fz(j)q FA(q)p Fz(j)21 b(\025)g FH(s)1634 1826 y FF(\(1)p Fx(\000)p FG(\013)p FF(\))1805 1775 y Fy(o)1856 1858 y FO(,)27 b(w)n(e)h(exploit)f(that)35 b Fy(e)-50 b FH(\037)2531 1867 y FG(h)2584 1858 y FE(\()p FA(q)p FE(\))21 b Fz(2)g FH(S)2848 1798 y Fy(\000)2883 1858 y FB(R)2937 1826 y FF(3)2972 1798 y Fy(\001)3007 1858 y FI(,)28 b FO(therefore)993 2075 y Fz(8)p FH(n)21 b Fz(2)h FB(N)77 b Fz(9)21 b FH(C)1426 2083 y FG(n)1489 2075 y FH(>)g FE(0)78 b FI(s.t)p FH(:)34 b Fz(j)8 b Fy(e)-50 b FH(\037)1892 2084 y FG(h)1945 2075 y FE(\()p FA(q)p FE(\))o Fz(j)22 b FH(<)f(C)2230 2083 y FG(n)2289 2075 y Fz(\001)2384 2026 y FE(1)p 2338 2058 132 4 v 2338 2125 a Fz(j)p FA(q)p Fz(j)2427 2090 y FG(n)2556 2075 y FI(for)91 b Fz(j)p FA(q)p Fz(j)21 b FH(>)g FE(1;)322 b FI(\(5.6\))515 2302 y FO(F)-7 b(or)28 b(small)g FA(q)p FO(,)h(that)g(is)1247 2219 y Fy(n)1298 2302 y FA(q)21 b FE(:)39 b Fz(j)q FA(q)p Fz(j)21 b FH(<)g(s)1654 2270 y FF(\(1)p Fx(\000)p FG(\013)p FF(\))1824 2219 y Fy(o)1875 2302 y FO(,)29 b(the)g(H\366lder)g(prop)r(erties)e(in)i FA(P)g FO(of)g Fz(r)p FH(E)3142 2267 y FG(\033)3178 2275 y Fq(t)3138 2323 y Fw(P)3238 2302 y FO(and)g(of)515 2416 y FH(\036)561 2382 y FG(\033)597 2390 y Fq(t)561 2437 y Fw(P)656 2416 y FO(pro)n(vide)d(the)i(desired)f(result.)515 2615 y FJ(T)-6 b(erm)30 b(\(5.4\))515 2715 y FO(The)d(inequalit)n(y)624 2882 y Fy(\015)624 2927 y(\015)624 2971 y(\015)624 3016 y(\015)624 3061 y(\015)624 3106 y(\015)667 2933 y(Z)763 3035 y(e)-50 b FH(\037)804 3044 y FG(h)857 3035 y FE(\()p FA(q)p FE(\))976 2885 y Fy(0)976 3019 y(@)1044 3035 y FH(e)1080 2999 y Fx(\000)p FG(i)p Fw(q)p Fx(\001r)p FG(E)1304 2971 y Fq(\033)1338 2982 y(t)1301 3020 y Fl(P)1390 3035 y Fz(\000)17 b FH(e)1503 2965 y FG(i)1525 2880 y Fm( )1570 2965 y FG(E)1618 2936 y Fq(\033)1652 2947 y(t)1615 2986 y Fl(P)1683 2965 y Fx(\000)p FG(E)1779 2936 y Fq(\033)1813 2947 y(t)1776 2997 y Fl(P)p FL(+)1871 2976 y Fl(q)p 1871 2988 34 3 v 1874 3019 a Fq(s)1918 2880 y Fm(!)1963 2965 y FG(s)1997 2885 y Fy(1)1997 3019 y(A)2077 3035 y FH(d)2117 2999 y FF(3)2151 3035 y FH(q)s(e)2224 2999 y FG(i\015)2278 3007 y Fq(\033)2312 3018 y(t)2344 3005 y FE(\()2374 2999 y Fw(v)2410 3010 y Fq(j)2441 2999 y FG(;)p Fx(r)p FG(E)2559 2971 y Fq(\033)2593 2982 y(t)2556 3020 y Fl(P)2624 2999 y FG(;s)2672 3005 y FE(\))2706 3035 y FH( )2759 2997 y FF(\()p FG(t)p FF(\))2756 3055 y FG(j;\033)2836 3063 y Fq(t)2868 2882 y Fy(\015)2868 2927 y(\015)2868 2971 y(\015)2868 3016 y(\015)2868 3061 y(\015)2868 3106 y(\015)2932 3035 y Fz(\024)813 3332 y(\024)k FH(C)h Fz(\001)1010 3246 y Fy(\015)1010 3291 y(\015)1010 3336 y(\015)1053 3332 y FH( )1106 3294 y FF(\()o FG(t)p FF(\))1103 3352 y FG(j;\033)1188 3246 y Fy(\015)1188 3291 y(\015)1188 3336 y(\015)1248 3332 y Fz(\001)1286 3231 y Fy(Z)1363 3251 y FF(+)p Fx(1)1329 3401 y FG(s)1358 3385 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))1535 3332 y Fz(j)8 b Fy(e)-50 b FH(\037)1605 3341 y FG(h)1658 3332 y FE(\()p FA(q)p FE(\))o Fz(j)13 b FH(d)1838 3296 y FF(3)1873 3332 y FH(q)20 b FE(+)d FH(C)22 b Fz(\001)2119 3246 y Fy(\015)2119 3291 y(\015)2119 3336 y(\015)2162 3332 y FH( )2215 3294 y FF(\()p FG(t)p FF(\))2212 3352 y FG(j;\033)2298 3246 y Fy(\015)2298 3291 y(\015)2298 3336 y(\015)2357 3332 y Fz(\001)2406 3215 y Fy(R)2457 3236 y FG(s)2486 3215 y FL(\(1)p Fk(\000)p Fq(\013)p FL(\))2442 3303 y FF(0)2663 3276 y Fz(j)q FA(q)p Fz(j)17 b(\001)g(j)7 b Fy(e)-50 b FH(\037)2877 3285 y FG(h)2930 3276 y FE(\()p FA(q)p FE(\))p Fz(j)13 b FH(d)3111 3244 y FF(3)3145 3276 y FH(q)p 2406 3315 777 4 v 2736 3392 a(s)2791 3343 y Fq(\013)p 2781 3351 57 3 v 2781 3384 a FL(16)542 3530 y FO(holds)28 b(b)r(ecause)f(of)g(the)h Fp(Sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(op)g(erties)7 b FO(,)27 b(Subsection)g(1.3.1,)g(whic)n(h)g (imply:)971 3681 y FH(sE)1068 3647 y FG(\033)1104 3655 y Fq(t)1064 3702 y Fw(P)1152 3681 y Fz(\000)17 b FH(sE)1326 3647 y FG(\033)1362 3655 y Fq(t)1322 3711 y Fw(P)p FF(+)1424 3687 y Fl(q)p 1424 3700 34 3 v 1427 3732 a Fq(s)1570 3681 y FE(=)82 b Fz(\000)p FA(q)17 b Fz(\001)g(r)p FH(E)1999 3647 y FG(\033)2035 3655 y Fq(t)1995 3706 y Fw(P)2042 3692 y Fk(0)2221 3681 y FE(with)167 b Fz(j)p FA(P)18 b Fz(\000)f FA(P)2773 3650 y Fx(0)2795 3681 y Fz(j)22 b(\024)2919 3618 y Fy(\014)2919 3663 y(\014)2955 3648 y Fw(q)p 2955 3664 37 4 v 2959 3707 a FG(s)3001 3618 y Fy(\014)3001 3663 y(\014)956 3754 y(\014)956 3799 y(\014)981 3817 y Fz(r)p FH(E)1106 3783 y FG(\033)1142 3791 y Fq(t)1102 3839 y Fw(P)1190 3817 y Fz(\000)17 b(r)p FH(E)1392 3783 y FG(\033)1428 3791 y Fq(t)1388 3842 y Fw(P)1435 3828 y Fk(0)1461 3754 y Fy(\014)1461 3799 y(\014)1570 3817 y Fz(\024)98 b FH(C)23 b Fz(\001)17 b(j)p FA(P)h Fz(\000)f FA(P)2080 3786 y Fx(0)2102 3817 y Fz(j)2148 3759 y FL(1)p 2134 3767 57 3 v 2134 3800 a(16)515 3979 y FO(Therefore)26 b(the)i(term)f(\(5.4\))h(is)f(surely)g(b)r(ounded)h(b)n(y)f(a)g(quan)n (tit)n(y)g(of)h(order)1785 4153 y FH(s)1821 4117 y Fx(\000)1888 4098 y Fq(\013)p 1879 4106 V 1879 4139 a FL(16)1967 4153 y Fz(\001)17 b FH(t)2033 4117 y Fx(\000)2091 4098 y FL(3)p Fq(\017)p 2091 4106 54 3 v 2103 4139 a FL(2)2176 4153 y FH(:)515 4308 y FJ(T)-6 b(erm)30 b(\(5.5\))515 4491 y FO(Let)d(us)h(start)f(from)g(the)h(trivial)f(inequalit)n(y)515 4583 y Fy(\015)515 4627 y(\015)515 4672 y(\015)515 4717 y(\015)515 4762 y(\015)515 4807 y(\015)557 4634 y(Z)654 4736 y(e)-50 b FH(\037)695 4745 y FG(h)748 4736 y FE(\()p FA(q)p FE(\))12 b FH(e)903 4665 y FG(i)925 4581 y Fm( )970 4665 y FG(E)1018 4637 y Fq(\033)1052 4648 y(t)1015 4686 y Fl(P)1083 4665 y FF(\()p Fw(P)p FF(\))p Fx(\000)p FG(E)1274 4637 y Fq(\033)1308 4648 y(t)1271 4698 y Fl(P)p FL(+)1366 4676 y Fl(q)p 1366 4688 34 3 v 1369 4720 a Fq(s)1413 4581 y Fm(!)1458 4665 y FG(s)1504 4653 y Fy(\020)1550 4736 y FH(e)1586 4700 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1721 4680 y Fl(x)p 1721 4688 V 1724 4721 a Fq(s)1785 4736 y Fz(\000)17 b FE(1)1900 4653 y Fy(\021)1959 4736 y FH(d)1999 4700 y FF(3)2034 4736 y FH(q)s(e)2107 4700 y FG(i\015)2161 4708 y Fq(\033)2195 4719 y(t)2227 4706 y FE(\()2256 4700 y Fw(v)2292 4711 y Fq(j)2324 4700 y FG(;)p Fx(r)p FG(E)2442 4672 y Fq(\033)2476 4683 y(t)2439 4721 y Fl(P)2507 4700 y FG(;s)2555 4706 y FE(\))2589 4736 y FH( )2642 4698 y FF(\()p FG(t)p FF(\))2639 4756 y FG(j;\033)2719 4764 y Fq(t)2751 4583 y Fy(\015)2751 4627 y(\015)2751 4672 y(\015)2751 4717 y(\015)2751 4762 y(\015)2751 4807 y(\015)2815 4736 y Fz(\024)1950 5059 y FO(45)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 636 598 a Fz(\024)717 444 y Fy(\015)717 489 y(\015)717 534 y(\015)717 579 y(\015)717 624 y(\015)717 668 y(\015)760 496 y(Z)837 516 y FF(+)p Fx(1)802 666 y FG(s)831 650 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))1016 598 y Fy(e)-50 b FH(\037)1057 607 y FG(h)1110 598 y FE(\()p FA(q)p FE(\))12 b FH(e)1265 527 y FG(i)1287 442 y Fm( )1332 527 y FG(E)1380 499 y Fq(\033)1414 510 y(t)1377 548 y Fl(P)1445 527 y Fx(\000)p FG(E)1541 499 y Fq(\033)1575 510 y(t)1538 560 y Fl(P)p FL(+)1633 538 y Fl(q)p 1633 550 34 3 v 1636 582 a Fq(s)1680 442 y Fm(!)1725 527 y FG(s)1771 515 y Fy(\020)1817 598 y FH(e)1853 562 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1988 542 y Fl(x)p 1988 550 V 1991 583 a Fq(s)2053 598 y Fz(\000)k FE(1)2167 515 y Fy(\021)2226 598 y FH(d)2266 562 y FF(3)2301 598 y FH(q)s(e)2374 562 y FG(i\015)2428 570 y Fq(\033)2462 581 y(t)2494 568 y FE(\()2524 562 y Fw(v)2560 573 y Fq(j)2591 562 y FG(;)p Fx(r)p FG(E)2709 533 y Fq(\033)2743 544 y(t)2706 583 y Fl(P)2774 562 y FG(;s)2822 568 y FE(\))2856 598 y FH( )2909 559 y FF(\()p FG(t)p FF(\))2906 617 y FG(j;\033)2986 625 y Fq(t)3018 444 y Fy(\015)3018 489 y(\015)3018 534 y(\015)3018 579 y(\015)3018 624 y(\015)3018 668 y(\015)3310 598 y FI(\(5.7\))662 912 y FE(+)734 759 y Fy(\015)734 804 y(\015)734 849 y(\015)734 894 y(\015)734 938 y(\015)734 983 y(\015)777 811 y(Z)854 831 y FG(s)883 810 y FL(\(1)p Fk(\000)p Fq(\013)p FL(\))819 981 y FF(0)1067 912 y Fy(e)-50 b FH(\037)1108 921 y FG(h)1161 912 y FE(\()p FA(q)p FE(\))12 b FH(e)1316 842 y FG(i)1338 757 y Fm( )1383 842 y FG(E)1431 814 y Fq(\033)1465 825 y(t)1428 863 y Fl(P)1496 842 y Fx(\000)p FG(E)1592 814 y Fq(\033)1626 825 y(t)1589 874 y Fl(P)p FL(+)1684 853 y Fl(q)p 1684 865 V 1687 897 a Fq(s)1731 757 y Fm(!)1776 842 y FG(s)1822 830 y Fy(\020)1868 912 y FH(e)1904 876 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)2039 857 y Fl(x)p 2039 865 V 2042 898 a Fq(s)2104 912 y Fz(\000)17 b FE(1)2219 830 y Fy(\021)2278 912 y FH(d)2318 876 y FF(3)2352 912 y FH(q)s(e)2425 876 y FG(i\015)2479 884 y Fq(\033)2513 895 y(t)2545 882 y FE(\()2575 876 y Fw(v)2611 887 y Fq(j)2642 876 y FG(;)p Fx(r)p FG(E)2760 848 y Fq(\033)2794 859 y(t)2757 897 y Fl(P)2825 876 y FG(;s)2873 882 y FE(\))2907 912 y FH( )2960 874 y FF(\()p FG(t)p FF(\))2957 932 y FG(j;\033)3037 940 y Fq(t)3069 759 y Fy(\015)3069 804 y(\015)3069 849 y(\015)3069 894 y(\015)3069 938 y(\015)3069 983 y(\015)3141 912 y FH(:)148 b FI(\(5.8\))515 1132 y FO(The)25 b(in)n(tegral)f(in)i(\(5.7\))f(in)n(v)n(olv)n(es)f(large)g FA(q)p FO(,)h(therefore)g(it)g(is)h(easily)e(under)i(con)n(trol)e (thanks)h(to)515 1223 y(\(5.6\).)g(F)-7 b(or)26 b(the)g(second)f(term)h (\(5.8\))g(w)n(e)f(add)h(and)g(subtract)f(the)i(same)e(quan)n(tities)h (to)f(ev)n(en-)515 1315 y(tually)i(obtain)h(three)f(expressions,)f (\(5.10\),)h(\(5.11\))g(and)h(\(5.12\),)f(whic)n(h)g(can)h(b)r(e)g(con) n(trolled)515 1406 y(due)g(to:)515 1497 y(The)f(con)n(v)n(ergence)e (rate)i(of)h(the)g(v)n(ector)e FH(\036)1848 1465 y FG(\033)1848 1512 y Fw(P)1926 1497 y FO(for)h FH(\033)d Fz(!)d FE(0)p FO(;)515 1604 y(The)27 b(regularit)n(y)f(prop)r(erties)h(in)g FA(P)h FO(of)g FH(h)13 b FE(\()p FA(P)p FE(\))p FO(,)28 b FH(e)1998 1571 y FG(i\015)2052 1579 y Fq(\033)2086 1590 y(t)2118 1577 y FE(\()2148 1571 y Fw(v)2184 1582 y Fq(j)2215 1571 y FG(;)p Fx(r)p FG(E)2333 1543 y Fq(\033)2367 1554 y(t)2330 1592 y Fl(P)2398 1571 y FG(;s)2446 1577 y FE(\))2508 1604 y FO(and)f FH(\036)2715 1573 y FG(\033)2715 1619 y Fw(P)2793 1604 y FO(as)g(a)g(v)n(ector)g(in)h Fu(F)3368 1581 y FG(b)3399 1604 y FO(;)515 1707 y(The)d(v)-5 b(anishing)24 b(\(for)h FH(s)c Fz(!)g(1)p FI(\))k FO(v)n(olume)f FH(O)1842 1700 y Fl(q)p 1843 1713 V 1846 1745 a Fq(s)1915 1707 y FO(whic)n(h)h(is)g(the)g(di\033erence)g(b)r(et)n(w)n(een)g(the)h (cell)f FE(\000)3392 1669 y FF(\()p FG(t)p FF(\))3392 1727 y FG(j)515 1811 y FO(and)i(the)h(same)f(cell)h(under)f(a)g (displacemen)n(t)1985 1778 y Fw(q)p 1985 1794 37 4 v 1989 1836 a FG(s)2032 1811 y FO(.)515 1902 y(In)f(the)g(deriv)-5 b(ation)25 b(of)g(\(5.10\),)g(\(5.11\))g(and)g(\(5.12\))g(w)n(e)g(w)n (arn)g(the)h(reader)e(ab)r(out)i(the)g(follo)n(w-)515 1993 y(ing)h(crucial)g(facts:)515 2085 y(i\))20 b(Both)f(the)h(v)n (ectors)e FH( )1250 2051 y FG(\033)1286 2059 y Fq(t)1247 2114 y Fw(P)p Fx(\000)1351 2090 y Fl(q)p 1351 2103 34 3 v 1354 2136 a Fq(s)1412 2085 y FH(;)1472 2065 y Fy(b)1458 2085 y FH( )1511 2051 y FG(\033)1547 2059 y Fq(t)1511 2114 y Fw(P)p Fx(\000)1616 2090 y Fl(q)p 1616 2103 V 1619 2136 a Fq(s)1684 2085 y FE(:=)k FH(e)1823 2053 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1958 2033 y Fl(x)p 1958 2041 V 1961 2074 a Fq(s)2005 2085 y FH( )2058 2051 y FG(\033)2094 2059 y Fq(t)2055 2106 y Fw(P)2145 2085 y FO(b)r(elong)d(to)g(the)h(same)f(\034b)r(er)h(space)f Fz(H)3293 2101 y Fw(P)p Fx(\000)3397 2077 y Fl(q)p 3397 2090 V 3400 2123 a Fq(s)3444 2085 y FO(;)515 2217 y(ii\))28 b(As)f(v)n(ectors)f(in)i(F)-7 b(o)r(c)n(k)27 b(space)p FI(,)1574 2198 y Fy(b)1560 2217 y FH( )1613 2183 y FG(\033)1649 2191 y Fq(t)1613 2247 y Fw(P)p Fx(\000)1718 2223 y Fl(q)p 1718 2236 V 1721 2268 a Fq(s)1793 2217 y FO(and)g FH( )2007 2183 y FG(\033)2043 2191 y Fq(t)2004 2239 y Fw(P)2102 2217 y FO(coincide)1487 2433 y FA(I)1520 2450 y Fw(P)p Fx(\000)1625 2426 y Fl(q)p 1625 2439 V 1628 2471 a Fq(s)1685 2350 y Fy(\020)1731 2433 y FH(e)1767 2397 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)1902 2378 y Fl(x)p 1902 2386 V 1905 2419 a Fq(s)1950 2433 y FH( )2003 2397 y FG(\033)2039 2405 y Fq(t)2000 2452 y Fw(P)2070 2350 y Fy(\021)2137 2433 y Fz(\021)21 b FA(I)2251 2441 y Fw(P)2315 2433 y FE(\()p FH( )2398 2397 y FG(\033)2434 2405 y Fq(t)2395 2452 y Fw(P)2465 2433 y FE(\))815 b FI(\(5.9\))515 2623 y FO(where)27 b(the)h(isomorphism)e FA(I)1416 2631 y Fw(P)1495 2623 y FO(is)h(de\034ned)h(b)n(y)f(\(1.9\))h(in)g(Subsection) f(1.1.)515 2714 y(iii\))h(Inside)f(the)h(in)n(tegral)1164 2870 y Fy(Z)1207 3040 y FF(\000)1245 3012 y FL(\()p Fq(t)p FL(\))1245 3060 y Fq(j)1334 2971 y FH(e)1370 2901 y FG(i)1392 2816 y Fm( )1437 2901 y FG(E)1485 2873 y Fq(\033)1519 2884 y(t)1482 2933 y Fl(P)p Fk(\000)1579 2912 y Fl(q)p 1579 2924 V 1582 2955 a Fq(s)1626 2901 y Fx(\000)p FG(E)1722 2873 y Fq(\033)1756 2884 y(t)1719 2922 y Fl(P)1787 2816 y Fm(!)1833 2901 y FG(s)1866 2971 y FH(h)1910 2979 y Fw(P)1961 2971 y FH(e)1997 2935 y FG(i\015)2051 2943 y Fq(\033)2085 2954 y(t)2117 2941 y FE(\()2147 2935 y Fw(v)2183 2946 y Fq(j)2214 2935 y FG(;)p Fx(r)p FG(E)2332 2907 y Fq(\033)2366 2918 y(t)2329 2956 y Fl(P)2397 2935 y FG(;s)2445 2941 y FE(\))2493 2952 y Fy(b)2479 2971 y FH( )2532 2935 y FG(\033)2568 2943 y Fq(t)2532 2999 y Fw(P)p Fx(\000)2637 2975 y Fl(q)p 2637 2988 V 2640 3020 a Fq(s)2684 2971 y FH(d)2724 2935 y FF(3)2758 2971 y FH(P)515 3209 y FO(the)36 b(in)n(tegration)e(v)-5 b(ariable)35 b FA(P)h FO(is)g(the)g(sp)r(ectral)f(v)-5 b(alue)36 b(of)g(the)g (corresp)r(onding)e(\(v)n(ectorial\))515 3300 y(op)r(erator.)515 3392 y(vi\))28 b(In)f(an)h(expression)e(lik)n(e)1216 3577 y FA(I)1249 3594 y Fw(P)p Fx(\000)1354 3570 y Fl(q)p 1354 3583 V 1358 3616 a(s)1414 3495 y Fy(\020)1460 3577 y FH(W)1544 3541 y Fx(y)1533 3590 y FG(\033)1569 3598 y Fq(t)1612 3495 y Fy(\020)1658 3577 y Fz(r)p FH(E)1783 3541 y FG(\033)1819 3549 y Fq(t)1779 3605 y Fw(P)p Fx(\000)1883 3581 y Fl(q)p 1883 3594 V 1887 3626 a(s)1930 3495 y Fy(\021)1989 3577 y FH(W)2062 3585 y FG(\033)2098 3593 y Fq(t)2142 3495 y Fy(\020)2188 3577 y Fz(r)p FH(E)2313 3541 y FG(\033)2349 3549 y Fq(t)2309 3605 y Fw(P)p Fx(\000)2413 3581 y Fl(q)p 2413 3594 V 2417 3626 a(s)2460 3495 y Fy(\021)2519 3577 y FH( )2572 3541 y FG(\033)2608 3549 y Fq(t)2569 3605 y Fw(P)p Fx(\000)2673 3581 y Fl(q)p 2673 3594 V 2677 3626 a(s)2721 3495 y Fy(\021)515 3762 y Fz(r)p FH(E)640 3728 y FG(\033)676 3736 y Fq(t)636 3792 y Fw(P)p Fx(\000)740 3768 y Fl(q)p 740 3781 V 744 3813 a(s)821 3762 y FO(is)33 b(the)h(v)n(ectorial)e(op)r(erator)g(in)h Fz(H)1910 3779 y Fw(P)p Fx(\000)2015 3755 y Fl(q)p 2015 3768 V 2018 3801 a Fq(s)2096 3762 y FO(obtained)g(b)n(y)g(the)h(m)n(ultiplication)g (of)f(the)515 3879 y(iden)n(tit)n(y)25 b(op)r(erator)f(with)h(the)h (gradien)n(t)e(of)h(the)h(ground)e(state)i(energy)e(ev)-5 b(aluated)25 b(in)g FA(P)13 b Fz(\000)3398 3846 y Fw(q)p 3398 3862 37 4 v 3402 3904 a FG(s)3444 3879 y FO(.)515 3970 y(The)27 b(treatmen)n(t)h(of)f(\(5.8\))g(pro)r(ceeds)g(as)g(follo) n(ws)762 4074 y Fy(\015)762 4119 y(\015)762 4164 y(\015)762 4209 y(\015)762 4253 y(\015)762 4298 y(\015)805 4126 y(Z)881 4146 y FG(s)910 4125 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))847 4296 y FF(0)1095 4227 y Fy(e)-50 b FH(\037)1136 4236 y FG(h)1189 4227 y FE(\()o FA(q)p FE(\))13 b FH(e)1344 4157 y FG(i)1366 4072 y Fm( )1411 4157 y FG(E)1459 4129 y Fq(\033)1493 4140 y(t)1456 4178 y Fl(P)1524 4157 y Fx(\000)p FG(E)1620 4129 y Fq(\033)1654 4140 y(t)1617 4189 y Fl(P)p FL(+)1712 4168 y Fl(q)p 1712 4180 34 3 v 1715 4211 a Fq(s)1759 4072 y Fm(!)1804 4157 y FG(s)1850 4145 y Fy(\020)1896 4227 y FH(e)1932 4191 y Fx(\000)p FG(i)p Fw(q)p Fx(\001)2067 4172 y Fl(x)p 2067 4180 V 2070 4213 a Fq(s)2131 4227 y Fz(\000)k FE(1)2246 4145 y Fy(\021)2305 4227 y FH(d)2345 4191 y FF(3)2380 4227 y FH(q)s(e)2453 4191 y FG(i\015)2507 4199 y Fq(\033)2541 4210 y(t)2573 4197 y FE(\()2603 4191 y Fw(v)2639 4202 y Fq(j)2670 4191 y FG(;)p Fx(r)p FG(E)2788 4163 y Fq(\033)2822 4174 y(t)2785 4212 y Fl(P)2853 4191 y FG(;s)2901 4197 y FE(\))2935 4227 y FH( )2988 4189 y FF(\()p FG(t)p FF(\))2985 4247 y FG(j;\033)3065 4255 y Fq(t)3097 4074 y Fy(\015)3097 4119 y(\015)3097 4164 y(\015)3097 4209 y(\015)3097 4253 y(\015)3097 4298 y(\015)3161 4227 y FE(=)688 4650 y(=)831 4497 y Fy(\015)831 4542 y(\015)831 4587 y(\015)831 4631 y(\015)831 4676 y(\015)831 4721 y(\015)874 4548 y(Z)950 4569 y FG(s)979 4548 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))916 4719 y FF(0)1164 4650 y Fy(e)-50 b FH(\037)1205 4659 y FG(h)1258 4650 y FE(\()p FA(q)p FE(\))1377 4548 y Fy(Z)1420 4719 y FF(\000)1458 4691 y FL(\()p Fq(t)p FL(\))1458 4739 y Fq(j)1547 4650 y FH(e)1583 4580 y FG(i)1605 4495 y Fm( )1650 4580 y FG(E)1698 4551 y Fq(\033)1732 4562 y(t)1695 4612 y Fl(P)p Fk(\000)1792 4591 y Fl(q)p 1792 4603 V 1795 4634 a Fq(s)1839 4580 y Fx(\000)p FG(E)1935 4551 y Fq(\033)1969 4562 y(t)1932 4601 y Fl(P)2000 4495 y Fm(!)2045 4580 y FG(s)2078 4650 y FH(h)2122 4658 y Fw(P)2173 4650 y FH(e)2209 4614 y FG(i\015)2263 4622 y Fq(\033)2297 4633 y(t)2329 4620 y FE(\()2359 4614 y Fw(v)2395 4625 y Fq(j)2427 4614 y FG(;)p Fx(r)p FG(E)2545 4586 y Fq(\033)2579 4597 y(t)2542 4635 y Fl(P)2610 4614 y FG(;s)2658 4620 y FE(\))2705 4630 y Fy(b)2692 4650 y FH( )2745 4614 y FG(\033)2781 4622 y Fq(t)2745 4678 y Fw(P)p Fx(\000)2849 4654 y Fl(q)p 2849 4667 V 2852 4699 a Fq(s)2896 4650 y FH(d)2936 4614 y FF(3)2971 4650 y FH(P)11 b(d)3071 4614 y FF(3)3105 4650 y FH(q)1950 5059 y FO(46)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 841 598 a Fz(\000)913 496 y Fy(Z)990 516 y FG(s)1019 495 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))956 666 y FF(0)1203 598 y Fy(e)-49 b FH(\037)1245 607 y FG(h)1298 598 y FE(\()o FA(q)p FE(\))1417 496 y Fy(Z)1459 666 y FF(\000)1497 638 y FL(\()p Fq(t)p FL(\))1497 686 y Fq(j)1586 598 y FH(e)1622 527 y FG(i)1644 442 y Fm( )1690 527 y FG(E)1738 499 y Fq(\033)1772 510 y(t)1735 548 y Fl(P)1803 527 y Fx(\000)p FG(E)1899 499 y Fq(\033)1933 510 y(t)1896 560 y Fl(P)p FL(+)1990 538 y Fl(q)p 1990 550 34 3 v 1993 582 a Fq(s)2037 442 y Fm(!)2083 527 y FG(s)2116 598 y FH(h)2160 606 y Fw(P)2211 598 y FH(e)2247 562 y FG(i\015)2301 570 y Fq(\033)2335 581 y(t)2367 568 y FE(\()2397 562 y Fw(v)2433 573 y Fq(j)2464 562 y FG(;)p Fx(r)p FG(E)2582 533 y Fq(\033)2616 544 y(t)2579 583 y Fl(P)2647 562 y FG(;s)2695 568 y FE(\))2729 598 y FH( )2782 562 y FG(\033)2818 570 y Fq(t)2779 617 y Fw(P)2850 598 y FH(d)2890 562 y FF(3)2924 598 y FH(P)11 b(d)3024 562 y FF(3)3058 598 y FH(q)3095 444 y Fy(\015)3095 489 y(\015)3095 534 y(\015)3095 579 y(\015)3095 624 y(\015)3095 668 y(\015)688 937 y FE(=)831 784 y Fy(\015)831 829 y(\015)831 874 y(\015)831 919 y(\015)831 963 y(\015)831 1008 y(\015)874 836 y(Z)950 856 y FG(s)979 835 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))916 1006 y FF(0)1164 937 y Fy(e)-50 b FH(\037)1205 946 y FG(h)1258 937 y FE(\()p FA(q)p FE(\))1377 836 y Fy(Z)1420 1006 y FF(\000)1458 978 y FL(\()p Fq(t)p FL(\))1458 1026 y Fq(j)1547 937 y FH(e)1583 867 y FG(i)1605 782 y Fm( )1650 867 y FG(E)1698 839 y Fq(\033)1732 850 y(t)1695 899 y Fl(P)p Fk(\000)1792 878 y Fl(q)p 1792 890 V 1795 921 a Fq(s)1839 867 y Fx(\000)p FG(E)1935 839 y Fq(\033)1969 850 y(t)1932 888 y Fl(P)2000 782 y Fm(!)2045 867 y FG(s)2078 937 y FH(h)2122 945 y Fw(P)2173 937 y FH(e)2209 901 y FG(i\015)2263 909 y Fq(\033)2297 920 y(t)2329 907 y FE(\()2359 901 y Fw(v)2395 912 y Fq(j)2427 901 y FG(;)p Fx(r)p FG(E)2545 873 y Fq(\033)2579 884 y(t)2542 922 y Fl(P)2610 901 y FG(;s)2658 907 y FE(\))2705 918 y Fy(b)2692 937 y FH( )2745 901 y FG(\033)2781 909 y Fq(t)2745 965 y Fw(P)p Fx(\000)2849 941 y Fl(q)p 2849 954 V 2852 986 a Fq(s)2896 937 y FH(d)2936 901 y FF(3)2971 937 y FH(P)11 b(d)3071 901 y FF(3)3105 937 y FH(q)3271 1142 y FI(\(5.10\))831 1352 y Fz(\000)903 1250 y Fy(Z)980 1270 y FG(s)1009 1250 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))946 1420 y FF(0)1193 1352 y Fy(e)-49 b FH(\037)1235 1361 y FG(h)1288 1352 y FE(\()o FA(q)p FE(\))1407 1250 y Fy(Z)1449 1420 y FF(\000)1487 1393 y FL(\()p Fq(t)p FL(\))1487 1440 y Fq(j)1576 1352 y FH(e)1612 1281 y FG(i)1634 1197 y Fm( )1680 1281 y FG(E)1728 1253 y Fq(\033)1762 1264 y(t)1725 1314 y Fl(P)p Fk(\000)1822 1292 y Fl(q)p 1822 1304 V 1825 1336 a Fq(s)1869 1281 y Fx(\000)p FG(E)1965 1253 y Fq(\033)1999 1264 y(t)1962 1302 y Fl(P)2030 1197 y Fm(!)2075 1281 y FG(s)2108 1352 y FH(h)2152 1360 y Fw(P)2203 1352 y FH(e)2239 1316 y FG(i\015)2293 1324 y Fq(\033)2327 1335 y(t)2359 1322 y FE(\()2389 1316 y Fw(v)2425 1327 y Fq(j)2457 1316 y FG(;)p Fx(r)p FG(E)2575 1287 y Fq(\033)2609 1298 y(t)2572 1337 y Fl(P)2640 1316 y FG(;s)2688 1322 y FE(\))2722 1352 y FH( )2775 1316 y FG(\033)2811 1324 y Fq(t)2772 1379 y Fw(P)p Fx(\000)2876 1355 y Fl(q)p 2876 1368 V 2879 1401 a Fq(s)2924 1352 y FH(d)2964 1316 y FF(3)2998 1352 y FH(P)11 b(d)3098 1316 y FF(3)3132 1352 y FH(q)831 1672 y FE(+)903 1571 y Fy(Z)980 1591 y FG(s)1009 1570 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))946 1741 y FF(0)1193 1672 y Fy(e)-49 b FH(\037)1235 1681 y FG(h)1288 1672 y FE(\()o FA(q)p FE(\))1407 1571 y Fy(Z)1449 1741 y FF(\000)1487 1713 y FL(\()p Fq(t)p FL(\))1487 1761 y Fq(j)1576 1672 y FH(e)1612 1602 y FG(i)1634 1517 y Fm( )1680 1602 y FG(E)1728 1574 y Fq(\033)1762 1585 y(t)1725 1634 y Fl(P)p Fk(\000)1822 1613 y Fl(q)p 1822 1625 V 1825 1657 a Fq(s)1869 1602 y Fx(\000)p FG(E)1965 1574 y Fq(\033)1999 1585 y(t)1962 1623 y Fl(P)2030 1517 y Fm(!)2075 1602 y FG(s)2108 1672 y FH(h)2152 1680 y Fw(P)2203 1672 y FH(e)2239 1636 y FG(i\015)2293 1644 y Fq(\033)2327 1655 y(t)2359 1642 y FE(\()2389 1636 y Fw(v)2425 1647 y Fq(j)2457 1636 y FG(;)p Fx(r)p FG(E)2575 1608 y Fq(\033)2609 1619 y(t)2572 1657 y Fl(P)2640 1636 y FG(;s)2688 1642 y FE(\))2722 1672 y FH( )2775 1636 y FG(\033)2811 1644 y Fq(t)2772 1700 y Fw(P)p Fx(\000)2876 1676 y Fl(q)p 2876 1689 V 2879 1721 a Fq(s)2924 1672 y FH(d)2964 1636 y FF(3)2998 1672 y FH(P)11 b(d)3098 1636 y FF(3)3132 1672 y FH(q)3271 1858 y FI(\(5.11\))831 2068 y Fz(\000)903 1966 y Fy(Z)980 1986 y FG(s)1009 1966 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))946 2136 y FF(0)1193 2068 y Fy(e)-49 b FH(\037)1235 2077 y FG(h)1288 2068 y FE(\()o FA(q)p FE(\))1407 1966 y Fy(Z)1449 2136 y FF(\000)1487 2109 y FL(\()p Fq(t)p FL(\))1487 2156 y Fq(j)1576 2068 y FH(e)1612 1997 y FG(i)1634 1913 y Fm( )1680 1997 y FG(E)1728 1969 y Fq(\033)1762 1980 y(t)1725 2030 y Fl(P)p Fk(\000)1822 2008 y Fl(q)p 1822 2020 V 1825 2052 a Fq(s)1869 1997 y Fx(\000)p FG(E)1965 1969 y Fq(\033)1999 1980 y(t)1962 2018 y Fl(P)2030 1913 y Fm(!)2075 1997 y FG(s)2108 2068 y FH(h)2152 2085 y Fw(P)p Fx(\000)2257 2060 y Fl(q)p 2257 2073 V 2260 2106 a Fq(s)2304 2068 y FH(e)2340 1997 y FG(i\015)2394 2005 y Fq(\033)2428 2016 y(t)2460 1913 y Fm( )2506 1997 y Fw(v)2542 2008 y Fq(j)2573 1997 y FG(;)p Fx(r)p FG(E)2691 1969 y Fq(\033)2725 1980 y(t)2688 2030 y Fl(P)p Fk(\000)2785 2008 y Fl(q)p 2785 2020 V 2788 2052 a Fq(s)2832 1997 y FG(;s)2880 1913 y Fm(!)2930 2068 y FH( )2983 2032 y FG(\033)3019 2040 y Fq(t)2980 2095 y Fw(P)p Fx(\000)3084 2071 y Fl(q)p 3084 2084 V 3087 2117 a Fq(s)3132 2068 y FH(d)3172 2032 y FF(3)3206 2068 y FH(P)11 b(d)3306 2032 y FF(3)3340 2068 y FH(q)831 2388 y FE(+)903 2287 y Fy(Z)980 2307 y FG(s)1009 2286 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))946 2457 y FF(0)1193 2388 y Fy(e)-49 b FH(\037)1235 2397 y FG(h)1288 2388 y FE(\()o FA(q)p FE(\))1407 2287 y Fy(Z)1449 2457 y FF(\000)1487 2429 y FL(\()p Fq(t)p FL(\))1487 2477 y Fq(j)1576 2388 y FH(e)1612 2318 y FG(i)1634 2233 y Fm( )1680 2318 y FG(E)1728 2290 y Fq(\033)1762 2301 y(t)1725 2350 y Fl(P)p Fk(\000)1822 2329 y Fl(q)p 1822 2341 V 1825 2373 a Fq(s)1869 2318 y Fx(\000)p FG(E)1965 2290 y Fq(\033)1999 2301 y(t)1962 2339 y Fl(P)2030 2233 y Fm(!)2075 2318 y FG(s)2108 2388 y FH(h)2152 2405 y Fw(P)p Fx(\000)2257 2381 y Fl(q)p 2257 2394 V 2260 2427 a Fq(s)2304 2388 y FH(e)2340 2318 y FG(i\015)2394 2326 y Fq(\033)2428 2337 y(t)2460 2233 y Fm( )2506 2318 y Fw(v)2542 2329 y Fq(j)2573 2318 y FG(;)p Fx(r)p FG(E)2691 2290 y Fq(\033)2725 2301 y(t)2688 2350 y Fl(P)p Fk(\000)2785 2329 y Fl(q)p 2785 2341 V 2788 2373 a Fq(s)2832 2318 y FG(;s)2880 2233 y Fm(!)2930 2388 y FH( )2983 2352 y FG(\033)3019 2360 y Fq(t)2980 2416 y Fw(P)p Fx(\000)3084 2392 y Fl(q)p 3084 2405 V 3087 2437 a Fq(s)3132 2388 y FH(d)3172 2352 y FF(3)3206 2388 y FH(P)11 b(d)3306 2352 y FF(3)3340 2388 y FH(q)3271 2574 y FI(\(5.12\))841 2784 y Fz(\000)913 2682 y Fy(Z)990 2702 y FG(s)1019 2682 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))956 2852 y FF(0)1203 2784 y Fy(e)-49 b FH(\037)1245 2793 y FG(h)1298 2784 y FE(\()o FA(q)p FE(\))1417 2682 y Fy(Z)1459 2852 y FF(\000)1497 2825 y FL(\()p Fq(t)p FL(\))1497 2872 y Fq(j)1586 2784 y FH(e)1622 2713 y FG(i)1644 2629 y Fm( )1690 2713 y FG(E)1738 2685 y Fq(\033)1772 2696 y(t)1735 2734 y Fl(P)1803 2713 y Fx(\000)p FG(E)1899 2685 y Fq(\033)1933 2696 y(t)1896 2746 y Fl(P)p FL(+)1990 2724 y Fl(q)p 1990 2736 V 1993 2768 a Fq(s)2037 2629 y Fm(!)2083 2713 y FG(s)2116 2784 y FH(h)2160 2792 y Fw(P)2211 2784 y FH(e)2247 2748 y FG(i\015)2301 2756 y Fq(\033)2335 2767 y(t)2367 2754 y FE(\()2397 2748 y Fw(v)2433 2759 y Fq(j)2464 2748 y FG(;)p Fx(r)p FG(E)2582 2719 y Fq(\033)2616 2730 y(t)2579 2769 y Fl(P)2647 2748 y FG(;s)2695 2754 y FE(\))2729 2784 y FH( )2782 2748 y FG(\033)2818 2756 y Fq(t)2779 2803 y Fw(P)2850 2784 y FH(d)2890 2748 y FF(3)2924 2784 y FH(P)11 b(d)3024 2748 y FF(3)3058 2784 y FH(q)3095 2630 y Fy(\015)3095 2675 y(\015)3095 2720 y(\015)3095 2765 y(\015)3095 2810 y(\015)3095 2855 y(\015)515 3041 y FO(W)-7 b(e)41 b(no)n(w)g(study)g (the)h(three)f(di\033erences)f(\(5.10\),)h(\(5.11\),)f(\(5.12\))h(whic) n(h)g(are)f(con)n(trolled)515 3132 y(resp)r(ectiv)n(ely)26 b(b)n(y)719 3282 y Fy(Z)796 3302 y FG(s)825 3281 y FL(\(1)p Fk(\000)p Fq(\013)p FL(\))761 3452 y FF(0)1002 3383 y Fz(j)7 b Fy(e)-50 b FH(\037)1071 3392 y FG(h)1124 3383 y FE(\()p FA(q)p FE(\))p Fz(j)1265 3256 y Fy(\()1327 3282 y(Z)1369 3452 y FF(\000)1407 3424 y FL(\()p Fq(t)p FL(\))1407 3472 y Fq(j)1496 3383 y Fz(j)q FH(h)1562 3391 y Fw(P)1613 3383 y Fz(j)1634 3348 y FF(2)1681 3298 y Fy(\015)1681 3342 y(\015)1681 3387 y(\015)1724 3383 y FA(I)1757 3391 y Fw(P)1821 3383 y FE(\()p FH( )1904 3348 y FG(\033)1940 3356 y Fq(t)1901 3403 y Fw(P)1971 3383 y FE(\))17 b Fz(\000)g FA(I)2128 3400 y Fw(P)p Fx(\000)2233 3376 y Fl(q)p 2233 3389 V 2236 3422 a Fq(s)2293 3301 y Fy(\020)2339 3383 y FH( )2392 3348 y FG(\033)2428 3356 y Fq(t)2389 3411 y Fw(P)p Fx(\000)2493 3387 y Fl(q)p 2493 3400 V 2496 3433 a Fq(s)2540 3301 y Fy(\021)2586 3298 y(\015)2586 3342 y(\015)2586 3387 y(\015)2629 3318 y FF(2)2629 3437 y Fx(F)2695 3383 y FH(d)2735 3348 y FF(3)2769 3383 y FH(P)2829 3256 y Fy(\))2901 3254 y FL(1)p 2901 3262 29 3 v 2901 3295 a(2)2956 3383 y FH(d)2996 3348 y FF(3)3030 3383 y FH(q)207 b FI(\(5.13\))580 3614 y Fy(Z)657 3634 y FG(s)686 3614 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))623 3784 y FF(0)863 3716 y Fz(j)8 b Fy(e)-50 b FH(\037)933 3725 y FG(h)986 3716 y FE(\()p FA(q)p FE(\))o Fz(j)1126 3588 y Fy(\()1188 3614 y(Z)1231 3784 y FF(\000)1269 3757 y FL(\()p Fq(t)p FL(\))1269 3804 y Fq(j)1358 3630 y Fy(\014)1358 3675 y(\014)1358 3719 y(\014)1383 3716 y FE(\001)1447 3680 y Fw(P)1447 3735 y(P)p Fx(\000)1552 3711 y Fl(q)p 1552 3724 34 3 v 1555 3756 a Fq(s)1612 3716 y FE([)23 b FH(h)1700 3724 y Fw(P)1751 3716 y FH(e)1787 3680 y FG(i\015)1841 3688 y Fq(\033)1875 3699 y(t)1907 3686 y FE(\()1937 3680 y Fw(v)1973 3691 y Fq(j)2004 3680 y FG(;)p Fx(r)p FG(E)2122 3651 y Fq(\033)2156 3662 y(t)2119 3701 y Fl(P)2188 3680 y FG(;s)2236 3686 y FE(\))2292 3716 y(])2314 3630 y Fy(\014)2314 3675 y(\014)2314 3719 y(\014)2339 3650 y FF(2)2387 3630 y Fy(\015)2387 3675 y(\015)2387 3719 y(\015)2429 3716 y FA(I)2462 3732 y Fw(P)p Fx(\000)2567 3708 y Fl(q)p 2567 3721 V 2570 3754 a Fq(s)2627 3633 y Fy(\020)2673 3716 y FH( )2726 3680 y FG(\033)2762 3688 y Fq(t)2723 3743 y Fw(P)p Fx(\000)2828 3719 y Fl(q)p 2828 3732 V 2831 3765 a Fq(s)2875 3633 y Fy(\021)2921 3630 y(\015)2921 3675 y(\015)2921 3719 y(\015)2964 3650 y FF(2)2964 3769 y Fx(F)3029 3716 y FH(d)3069 3680 y FF(3)3104 3716 y FH(P)3164 3588 y Fy(\))3235 3586 y FL(1)p 3235 3594 29 3 v 3235 3627 a(2)3291 3716 y FH(d)3331 3680 y FF(3)3365 3716 y FH(q)3271 3877 y FI(\(5.14\))654 3981 y Fy(Z)731 4001 y FG(s)760 3981 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))697 4151 y FF(0)937 4083 y Fz(j)7 b Fy(e)-49 b FH(\037)1007 4092 y FG(h)1060 4083 y FE(\()o FA(q)p FE(\))p Fz(j)1200 3929 y Fy(8)1200 3997 y(<)1200 4131 y(:)1268 3981 y(Z)1311 4151 y FG(O)1368 4152 y Fl(q)p 1368 4164 34 3 v 1371 4196 a Fq(s)1432 4083 y Fz(j)p FH(h)1497 4091 y Fw(P)1548 4083 y Fz(j)1569 4047 y FF(2)1617 4083 y Fz(k)p FA(I)1688 4091 y Fw(P)1752 4083 y FE(\()p FH( )1835 4047 y FG(\033)1871 4055 y Fq(t)1832 4102 y Fw(P)1902 4083 y FE(\))p Fz(k)1970 4046 y FF(2)1970 4106 y Fx(F)2036 4083 y FH(d)2076 4047 y FF(3)2111 4083 y FH(P)2171 3929 y Fy(9)2171 3997 y(=)2171 4131 y(;)2249 3930 y FL(1)p 2249 3938 29 3 v 2249 3971 a(2)2304 4083 y FH(d)2344 4047 y FF(3)2378 4083 y FH(q)859 b FI(\(5.15\))515 4297 y FO(where)780 4459 y FE(\001)844 4423 y Fw(P)844 4478 y(P)p Fx(\000)948 4454 y Fl(q)p 948 4467 34 3 v 951 4500 a Fq(s)1008 4459 y FE([)23 b FH(h)1096 4467 y Fw(P)1147 4459 y FH(e)1183 4423 y FG(i\015)1237 4431 y Fq(\033)1271 4442 y(t)1303 4429 y FE(\()1333 4423 y Fw(v)1369 4434 y Fq(j)1401 4423 y FG(;)p Fx(r)p FG(E)1519 4395 y Fq(\033)1553 4406 y(t)1516 4444 y Fl(P)1584 4423 y FG(;s)1632 4429 y FE(\))1688 4459 y(])f(:=)1459 b FI(\(5.16\))966 4690 y FE(=)21 b FH(h)1091 4698 y Fw(P)1142 4690 y FH(e)1178 4655 y FG(i\015)1232 4663 y Fq(\033)1266 4674 y(t)1298 4661 y FE(\()1328 4655 y Fw(v)1364 4666 y Fq(j)1395 4655 y FG(;)p Fx(r)p FG(E)1513 4626 y Fq(\033)1547 4637 y(t)1510 4676 y Fl(P)1579 4655 y FG(;s)1627 4661 y FE(\))1678 4690 y Fz(\000)16 b FH(h)1798 4707 y Fw(P)p Fx(\000)1903 4683 y Fl(q)p 1903 4696 V 1906 4729 a Fq(s)1951 4690 y FH(e)1987 4620 y FG(i\015)2041 4628 y Fq(\033)2075 4639 y(t)2106 4535 y Fm( )2152 4620 y Fw(v)2188 4631 y Fq(j)2219 4620 y FG(;)p Fx(r)p FG(E)2337 4592 y Fq(\033)2371 4603 y(t)2334 4653 y Fl(P)p Fk(\000)2431 4631 y Fl(q)p 2431 4643 V 2434 4675 a Fq(s)2478 4620 y FG(;s)2526 4535 y Fm(!)2593 4690 y FH(:)1950 5059 y FO(47)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 515 523 a FO(and)28 b FH(O)745 516 y Fl(q)p 745 529 34 3 v 748 561 a Fq(s)821 523 y FO(is)g(the)g(di\033erence)g(b)r(et)n(w)n(een)g(the)h(cell)f FE(\000)2081 485 y FF(\()p FG(t)p FF(\))2081 543 y FG(j)2185 523 y FO(and)g(the)h(same)e(cell)i(under)f(a)f(displace-)515 625 y(men)n(t)735 592 y Fw(q)p 735 608 37 4 v 739 650 a FG(s)781 625 y FO(.)515 807 y FJ(Di\033er)l(enc)l(e)i(\(5.10\))515 990 y FO(Using)e(the)h(fact)g(that)g FA(P)21 b Fz(2)h FE(\000)1437 952 y FF(\()p FG(t)p FF(\))1437 1010 y FG(j)1534 990 y Fz(\032)f FE(\006)p FO(,)28 b(w)n(e)f(estimate:)842 1099 y Fy(\015)842 1143 y(\015)842 1188 y(\015)885 1185 y FA(I)918 1193 y Fw(P)982 1185 y FE(\()p FH( )1065 1149 y FG(\033)1101 1157 y Fq(t)1062 1204 y Fw(P)1132 1185 y FE(\))17 b Fz(\000)g FA(I)1289 1201 y Fw(P)p Fx(\000)1394 1177 y Fl(q)p 1394 1190 34 3 v 1397 1223 a Fq(s)1454 1102 y Fy(\020)1499 1185 y FH( )1552 1149 y FG(\033)1588 1157 y Fq(t)1549 1212 y Fw(P)p Fx(\000)1654 1188 y Fl(q)p 1654 1201 V 1657 1234 a Fq(s)1701 1102 y Fy(\021)1747 1099 y(\015)1747 1143 y(\015)1747 1188 y(\015)1790 1238 y Fx(F)1864 1185 y FE(=)702 1473 y(=)845 1387 y Fy(\015)845 1432 y(\015)845 1477 y(\015)888 1473 y FA(I)921 1481 y Fw(P)984 1390 y Fy(\020)1030 1473 y FH(W)1114 1437 y Fx(y)1103 1486 y FG(\033)1139 1494 y Fq(t)1183 1473 y FE(\()p Fz(r)p FH(E)1338 1437 y FG(\033)1374 1445 y Fq(t)1334 1492 y Fw(P)1405 1473 y FE(\))c FH(W)1521 1481 y FG(\033)1557 1489 y Fq(t)1600 1473 y FE(\()p Fz(r)p FH(E)1755 1437 y FG(\033)1791 1445 y Fq(t)1751 1492 y Fw(P)1822 1473 y FE(\))g FH( )1918 1437 y FG(\033)1954 1445 y Fq(t)1915 1492 y Fw(P)1985 1390 y Fy(\021)2044 1473 y Fz(\000)1354 1706 y(\000)p FA(I)1447 1722 y Fw(P)p Fx(\000)1552 1698 y Fl(q)p 1552 1711 V 1555 1744 a Fq(s)1612 1623 y Fy(\020)1658 1706 y FH(W)1742 1670 y Fx(y)1731 1719 y FG(\033)1767 1727 y Fq(t)1810 1623 y Fy(\020)1856 1706 y Fz(r)p FH(E)1981 1670 y FG(\033)2017 1678 y Fq(t)1977 1733 y Fw(P)p Fx(\000)2081 1709 y Fl(q)p 2081 1722 V 2084 1755 a Fq(s)2128 1623 y Fy(\021)2187 1706 y FH(W)2260 1714 y FG(\033)2296 1722 y Fq(t)2340 1623 y Fy(\020)2386 1706 y Fz(r)p FH(E)2511 1670 y FG(\033)2547 1678 y Fq(t)2507 1733 y Fw(P)p Fx(\000)2611 1709 y Fl(q)p 2611 1722 V 2614 1755 a Fq(s)2658 1623 y Fy(\021)2717 1706 y FH( )2770 1670 y FG(\033)2806 1678 y Fq(t)2767 1733 y Fw(P)p Fx(\000)2871 1709 y Fl(q)p 2871 1722 V 2874 1755 a Fq(s)2919 1623 y Fy(\021)2965 1620 y(\015)2965 1664 y(\015)2965 1709 y(\015)3007 1759 y Fx(F)702 1938 y Fz(\024)845 1852 y Fy(\015)845 1897 y(\015)845 1942 y(\015)888 1938 y FA(I)921 1946 y Fw(P)984 1938 y FE(\()p FH(W)1087 1946 y FG(\033)1123 1954 y Fq(t)1167 1938 y FE(\()p Fz(r)p FH(E)1322 1902 y FG(\033)1358 1910 y Fq(t)1318 1957 y Fw(P)1389 1938 y FE(\))g FH( )1485 1902 y FG(\033)1521 1910 y Fq(t)1482 1957 y Fw(P)1552 1938 y FE(\))k Fz(\000)g FA(I)1709 1955 y Fw(P)p Fx(\000)1814 1931 y Fl(q)p 1814 1944 V 1817 1976 a Fq(s)1874 1855 y Fy(\020)1919 1938 y FH(W)1992 1946 y FG(\033)2028 1954 y Fq(t)2072 1855 y Fy(\020)2118 1938 y Fz(r)p FH(E)2243 1902 y FG(\033)2279 1910 y Fq(t)2239 1966 y Fw(P)p Fx(\000)2343 1942 y Fl(q)p 2343 1955 V 2346 1987 a Fq(s)2390 1855 y Fy(\021)2449 1938 y FH( )2502 1902 y FG(\033)2538 1910 y Fq(t)2499 1966 y Fw(P)p Fx(\000)2604 1942 y Fl(q)p 2604 1955 V 2607 1987 a Fq(s)2651 1855 y Fy(\021)2697 1852 y(\015)2697 1897 y(\015)2697 1942 y(\015)2739 1991 y Fx(F)3271 1938 y FI(\(5.17\))845 2170 y FE(+)917 2085 y Fy(\015)917 2129 y(\015)917 2174 y(\015)960 2170 y FA(I)993 2187 y Fw(P)p Fx(\000)1098 2163 y Fl(q)p 1098 2176 V 1102 2209 a(s)1158 2088 y Fy(\020\020)1250 2170 y FH(W)1334 2135 y Fx(y)1323 2184 y FG(\033)1359 2192 y Fq(t)1402 2170 y FE(\()p Fz(r)p FH(E)1557 2135 y FG(\033)1593 2143 y Fq(t)1553 2190 y Fw(P)1625 2170 y FE(\))f Fz(\000)h FH(W)1832 2135 y Fx(y)1821 2184 y FG(\033)1857 2192 y Fq(t)1901 2088 y Fy(\020)1947 2170 y Fz(r)p FH(E)2072 2135 y FG(\033)2108 2143 y Fq(t)2068 2198 y Fw(P)p Fx(\000)2172 2174 y Fl(q)p 2172 2187 V 2175 2220 a Fq(s)2219 2088 y Fy(\021\021)2324 2170 y FH(W)2397 2178 y FG(\033)2433 2186 y Fq(t)2476 2088 y Fy(\020)2522 2170 y Fz(r)p FH(E)2647 2135 y FG(\033)2683 2143 y Fq(t)2643 2198 y Fw(P)p Fx(\000)2747 2174 y Fl(q)p 2747 2187 V 2750 2220 a Fq(s)2795 2088 y Fy(\021)2853 2170 y FH( )2906 2135 y FG(\033)2942 2143 y Fq(t)2903 2198 y Fw(P)p Fx(\000)3008 2174 y Fl(q)p 3008 2187 V 3011 2220 a Fq(s)3055 2088 y Fy(\021)3101 2085 y(\015)3101 2129 y(\015)3101 2174 y(\015)3144 2224 y Fx(F)3271 2170 y FI(\(5.18\))515 2360 y FO(No)n(w)27 b(notice)g(that:)515 2480 y(The)i(norm)f FI(\(5.17\))i FO(is)e(b)r(ounded)h(b)n(y)g(a)f (quan)n(tit)n(y)g(of)h(order)2388 2397 y Fy(\020)2444 2444 y Fx(j)p Fw(q)p Fx(j)p 2444 2463 75 4 v 2467 2505 a FG(s)2528 2397 y Fy(\021)2598 2395 y FL(1)p 2584 2403 57 3 v 2584 2436 a(16)2684 2480 y FO(as)f(consequence)g(of)g(the)515 2599 y(H\366lder)e(regularit)n(y)f(in)i FA(P)g FO(of)g(the)g(v)n(ector) e FH(\036)1875 2565 y FG(\033)1911 2573 y Fq(t)1875 2620 y Fw(P)1969 2599 y FO(\()p Fp(Sp)-5 b(e)g(ctr)g(al)34 b(pr)-5 b(op)g(erties)7 b FO(,)27 b(Subsection)g(1.3.1\);)515 2690 y(The)g(norm)g FI(\(5.18\))i FO(can)e(b)r(e)h(estimated)g (starting)f(from)g(the)h(norm)f(of)g(the)h(v)n(ector)515 2954 y FH(I)549 2971 y Fw(P)p Fx(\000)653 2947 y Fl(q)p 653 2960 34 3 v 656 2993 a Fq(s)713 2804 y Fy(0)713 2939 y(@)780 2954 y FH(g)832 2853 y Fy(Z)909 2873 y FG(\024)875 3023 y(\033)911 3031 y Fq(t)974 2854 y Fy(b)972 2874 y FA(k)17 b Fz(\001)1074 2791 y Fy(\020)1120 2874 y Fz(r)p FH(E)1245 2840 y FG(\033)1281 2848 y Fq(t)1241 2895 y Fw(P)1329 2874 y Fz(\000)g(r)p FH(E)1531 2840 y FG(\033)1567 2848 y Fq(t)1527 2903 y Fw(P)p Fx(\000)1631 2879 y Fl(q)p 1631 2892 V 1634 2925 a Fq(s)1678 2791 y Fy(\021)1737 2813 y(\000)1772 2874 y FH(b)c FE(\()p FA(k)p FE(\))j Fz(\000)h FH(b)2051 2842 y Fx(y)2097 2874 y FE(\()p FA(k)p FE(\))2203 2813 y Fy(\001)p 972 2937 1267 4 v 989 3033 a Fz(j)q FA(k)p Fz(j)1091 2950 y Fy(\020)1137 3033 y FE(1)h Fz(\000)1272 3013 y Fy(b)1270 3033 y FA(k)e Fz(\001)i(r)p FH(E)1497 2998 y FG(\033)1533 3006 y Fq(t)1493 3054 y Fw(P)1564 2950 y Fy(\021)13 b(\020)1668 3033 y FE(1)18 b Fz(\000)1803 3013 y Fy(b)1801 3033 y FA(k)e Fz(\001)i(r)p FH(E)2028 2998 y FG(\033)2064 3006 y Fq(t)2024 3062 y Fw(P)p Fx(\000)2128 3038 y Fl(q)p 2128 3051 34 3 v 2131 3083 a Fq(s)2175 2950 y Fy(\021)2309 2906 y FH(d)2349 2874 y FF(3)2383 2906 y FH(k)p 2258 2937 218 4 v 2258 2953 a Fy(p)p 2335 2953 141 4 v 64 x FE(2)13 b Fz(j)q FA(k)p Fz(j)2485 2954 y FH(W)2558 2962 y FG(\033)2594 2970 y Fq(t)2638 2872 y Fy(\020)2684 2954 y Fz(r)p FH(E)2809 2918 y FG(\033)2845 2926 y Fq(t)2805 2982 y Fw(P)p Fx(\000)2909 2958 y Fl(q)p 2909 2971 34 3 v 2912 3003 a Fq(s)2956 2872 y Fy(\021)3015 2954 y FH( )3068 2918 y FG(\033)3104 2926 y Fq(t)3065 2982 y Fw(P)p Fx(\000)3170 2958 y Fl(q)p 3170 2971 V 3173 3003 a Fq(s)3217 2804 y Fy(1)3217 2939 y(A)3412 2954 y FI(\(5.19\))515 3220 y FO(therefore)26 b(in)i(terms)f(of)h(the)g(follo)n(wing)e(quan)n(tities:)974 3346 y Fy(0)974 3477 y(B)974 3525 y(@)1041 3417 y(Z)1118 3437 y FG(\024)1084 3587 y(\033)1120 3595 y Fq(t)1171 3368 y Fy(0)1171 3503 y(@)1552 3438 y FH(g)1594 3418 y Fy(b)1592 3438 y FA(k)16 b Fz(\001)1694 3355 y Fy(\020)1739 3438 y Fz(r)p FH(E)1864 3404 y FG(\033)1900 3412 y Fq(t)1860 3459 y Fw(P)1949 3438 y Fz(\000)g(r)p FH(E)2150 3404 y FG(\033)2186 3412 y Fq(t)2146 3467 y Fw(P)p Fx(\000)2251 3443 y Fl(q)p 2251 3456 V 2254 3489 a Fq(s)2298 3355 y Fy(\021)p 1248 3501 1400 4 v 1248 3535 a Fz(p)p 1312 3535 39 4 v 62 x FE(2)e Fz(j)p FA(k)p Fz(j)1463 3542 y FL(3)p 1463 3550 29 3 v 1463 3583 a(2)1518 3514 y Fy(\020)1564 3597 y FE(1)j Fz(\000)1698 3577 y Fy(b)1696 3597 y FA(k)g Fz(\001)g(r)p FH(E)1923 3563 y FG(\033)1959 3571 y Fq(t)1919 3618 y Fw(P)1990 3514 y Fy(\021)c(\020)2095 3597 y FE(1)k Fz(\000)2229 3577 y Fy(b)2227 3597 y FA(k)g Fz(\001)g(r)p FH(E)2454 3563 y FG(\033)2490 3571 y Fq(t)2450 3626 y Fw(P)p Fx(\000)2554 3602 y Fl(q)p 2554 3615 34 3 v 2557 3648 a Fq(s)2602 3514 y Fy(\021)2657 3368 y(1)2657 3503 y(A)2725 3372 y FF(2)2772 3518 y FH(d)2812 3483 y FF(3)2846 3518 y FH(k)2888 3346 y Fy(1)2888 3477 y(C)2888 3525 y(A)2966 3341 y FL(1)p 2966 3349 29 3 v 2966 3382 a(2)3271 3518 y FI(\(5.20\))542 3793 y FO(whic)n(h)28 b(is)f(b)r(ounded)h(b)n(y) 1065 3981 y FH(C)22 b Fz(\001)1181 3895 y Fy(\014)1181 3940 y(\014)1181 3985 y(\014)1206 3981 y Fz(r)p FH(E)1331 3945 y FG(\033)1367 3953 y Fq(t)1327 4009 y Fw(P)p Fx(\000)1431 3985 y Fl(q)p 1431 3998 34 3 v 1434 4030 a Fq(s)1496 3981 y Fz(\000)16 b(r)p FH(E)1697 3945 y FG(\033)1733 3953 y Fq(t)1693 4000 y Fw(P)1765 3895 y Fy(\014)1765 3940 y(\014)1765 3985 y(\014)1807 3981 y Fz(\001)i(j)p FE(ln)13 b FH(\033)1988 3989 y FG(t)2016 3981 y Fz(j)2047 3926 y FL(1)p 2047 3934 29 3 v 2047 3967 a(2)2111 3981 y Fz(\024)21 b FH(C)h Fz(\001)2307 3918 y Fy(\014)2307 3963 y(\014)2333 3981 y FH(s)2369 3945 y Fx(\000)p FG(\013)2461 3918 y Fy(\014)2461 3963 y(\014)2510 3914 y FL(1)p 2496 3922 57 3 v 2496 3955 a(16)2584 3981 y Fz(\001)17 b(j)q FE(ln)c FH(\033)2765 3989 y FG(t)2792 3981 y Fz(j)2824 3926 y FL(1)p 2824 3934 29 3 v 2824 3967 a(2)2896 3981 y FE(;)632 4180 y Fy(\022)689 4184 y(Z)766 4204 y FG(\024)731 4354 y(\033)767 4362 y Fq(t)819 4200 y Fy(\015)819 4244 y(\015)819 4289 y(\015)861 4285 y FH(I)895 4302 y Fw(P)p Fx(\000)999 4278 y Fl(q)p 999 4291 34 3 v 1002 4324 a Fq(s)1060 4203 y Fy(\020)1105 4285 y FH(b)g FE(\()p FA(k)p FE(\))f FH(W)1343 4293 y FG(\033)1379 4301 y Fq(t)1423 4203 y Fy(\020)1469 4285 y Fz(r)p FH(E)1594 4250 y FG(\033)1630 4258 y Fq(t)1590 4313 y Fw(P)p Fx(\000)1694 4289 y Fl(q)p 1694 4302 V 1697 4335 a Fq(s)1741 4203 y Fy(\021)1800 4285 y FH( )1853 4250 y FG(\033)1889 4258 y Fq(t)1850 4313 y Fw(P)p Fx(\000)1955 4289 y Fl(q)p 1955 4302 V 1958 4335 a Fq(s)2002 4203 y Fy(\021)2048 4200 y(\015)2048 4244 y(\015)2048 4289 y(\015)2090 4220 y FF(2)2090 4339 y Fx(F)2156 4285 y FH(d)2196 4250 y FF(3)2231 4285 y FH(k)2273 4180 y Fy(\023)2339 4178 y FL(1)p 2339 4186 29 3 v 2339 4219 a(2)2403 4285 y FE(=)808 b FI(\(5.21\))551 4617 y Ft(=)626 4444 y Fs(0)626 4561 y(B)626 4601 y(B)626 4643 y(@)687 4527 y(Z)758 4547 y FG(\024)727 4679 y(\033)763 4687 y Fq(t)810 4441 y Fs(\015)810 4481 y(\015)810 4521 y(\015)810 4561 y(\015)810 4601 y(\015)810 4640 y(\015)810 4680 y(\015)810 4720 y(\015)849 4617 y Fr(I)880 4637 y Fw(P)p Fx(\000)984 4613 y Fl(q)p 984 4626 34 3 v 987 4658 a Fq(s)1043 4444 y Fs(0)1043 4561 y(B)1043 4601 y(B)1043 4643 y(@)1105 4617 y Fr(W)1172 4625 y FG(\033)1208 4633 y Fq(t)1251 4524 y Fs(\022)1303 4617 y Fu(r)p Fr(E)1418 4586 y FG(\033)1454 4594 y Fq(t)1414 4649 y Fw(P)p Fx(\000)1518 4625 y Fl(q)p 1518 4638 V 1521 4670 a Fq(s)1565 4524 y Fs(\023)1629 4444 y(0)1629 4561 y(B)1629 4601 y(B)1629 4643 y(@)1691 4617 y Fr(b)12 b Ft(\()p Fh(k)p Ft(\))k(+)2215 4565 y Fr(g)r(\037)2295 4542 y FG(\024)2295 4582 y(\033)2331 4590 y Fq(t)2374 4565 y Ft(\()q Fh(k)p Ft(\))p 1927 4602 834 3 v 1927 4652 a Fu(p)p 1985 4652 36 3 v 1985 4707 a Ft(2)c Fu(j)p Fh(k)p Fu(j)2125 4659 y FL(3)p 2125 4667 29 3 v 2125 4699 a(2)2179 4614 y Fs(\022)2231 4707 y Ft(1)k Fu(\000)2354 4690 y Fs(b)2352 4707 y Fh(k)g Fu(\001)f(r)p Fr(E)2561 4676 y FG(\033)2597 4684 y Fq(t)2557 4739 y Fw(P)p Fx(\000)2661 4715 y Fl(q)p 2661 4728 34 3 v 2664 4760 a Fq(s)2708 4614 y Fs(\023)2770 4444 y(1)2770 4561 y(C)2770 4601 y(C)2770 4643 y(A)2844 4617 y Fr( )2892 4586 y FG(\033)2928 4594 y Fq(t)2890 4649 y Fw(P)p Fx(\000)2995 4625 y Fl(q)p 2995 4638 V 2998 4670 a Fq(s)3042 4444 y Fs(1)3042 4561 y(C)3042 4601 y(C)3042 4643 y(A)3104 4441 y(\015)3104 4481 y(\015)3104 4521 y(\015)3104 4561 y(\015)3104 4601 y(\015)3104 4640 y(\015)3104 4680 y(\015)3104 4720 y(\015)3143 4462 y FF(2)3143 4775 y Fx(F)3208 4617 y Fr(d)3244 4590 y FF(3)3278 4617 y Fr(k)3317 4444 y Fs(1)3317 4561 y(C)3317 4601 y(C)3317 4643 y(A)3389 4431 y FL(1)p 3389 4439 29 3 v 3389 4472 a(2)1950 5059 y FO(48)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 542 523 a FO(for)27 b(whic)n(h,)h(using)f(the)h (resolv)n(en)n(t)e(equation)h(\(1.12\))g(in)g(Subsection)h(1.2)768 729 y FH(b)13 b FE(\()o FA(k)p FE(\))g FH( )983 737 y Fw(P)p FG(;\033)1085 745 y Fq(t)1137 729 y FE(=)1317 680 y FH(g)p 1228 712 218 4 v 1228 727 a Fy(p)p 1305 727 141 4 v 64 x FE(2)g Fz(j)q FA(k)p Fz(j)1468 624 y Fy(\022)1859 680 y FE(1)p 1535 712 687 4 v 1535 779 a FH(E)1596 745 y FG(\033)1632 753 y Fq(t)1592 801 y Fw(P)1680 779 y Fz(\000)k(j)p FA(k)p Fz(j)g(\000)g FH(H)2004 788 y Fw(P)p Fx(\000)p Fw(k)p FG(;\033)2190 796 y Fq(t)2231 624 y Fy(\023)2300 729 y FH( )2353 693 y FG(\033)2389 701 y Fq(t)2350 748 y Fw(P)2574 729 y FH(\033)2618 737 y FG(t)2667 729 y Fz(\024)k(j)p FA(k)p Fz(j)h(\024)f FH(\024)13 b(;)253 b FI(\(5.22\))515 977 y FO(it)26 b(is)f(easy)g(to)g (pro)n(vide)f(a)h(b)r(ound)h(b)n(y)g(a)f(quan)n(tit)n(y)g Fz(O)2158 894 y Fy(\020)2203 977 y Fz(j)q FE(ln)13 b FH(\033)2346 985 y FG(t)2373 977 y Fz(j)2405 921 y FL(1)p 2405 929 29 3 v 2405 962 a(2)2464 977 y Fz(\001)18 b FH(t)2531 945 y Fx(\000)2588 925 y FL(3)p Fq(\017)p 2588 933 54 3 v 2601 966 a FL(2)2656 894 y Fy(\021)2728 977 y FO(\(uniform)25 b(in)h FH(s)p FO(\))g(whic)n(h)515 1091 y(is)h(enough)g(for)g(our)g(purp)r(oses.)515 1182 y(The)g(di\033erence)h(\(5.10\),)f(through)f(the)i(term)g(\(5.13\),)f (can)g(b)r(e)h(estimated)f(in)h(terms)g(of)1217 1373 y FH(C)23 b Fz(\001)1333 1312 y Fy(\000)1368 1373 y FH(s)1404 1337 y Fx(\000)p FG(\013)1496 1312 y Fy(\001)1555 1306 y FL(1)p 1541 1314 57 3 v 1541 1347 a(16)1629 1373 y Fz(\001)17 b FH(t)1695 1337 y Fx(\000)1753 1317 y FL(3)p Fq(\017)p 1753 1325 54 3 v 1766 1358 a FL(2)1838 1373 y FE(+)g FH(C)22 b Fz(\001)2030 1312 y Fy(\000)2065 1373 y FH(s)2101 1337 y Fx(\000)p FG(\013)2193 1312 y Fy(\001)2253 1306 y FL(1)p 2239 1314 57 3 v 2239 1347 a(16)2326 1373 y Fz(\001)c FH(t)2393 1337 y Fx(\000)2450 1317 y FL(3)p Fq(\017)p 2450 1325 54 3 v 2463 1358 a FL(2)2535 1373 y Fz(\001)g(j)p FE(ln)13 b FH(\033)2716 1381 y FG(t)2744 1373 y Fz(j)515 1532 y FO(so)27 b(that)g(it)h(is)g(surely)f(b)r(ounded) h(b)n(y)f(a)g(quan)n(tit)n(y)g(of)h(order)1648 1707 y FH(s)1684 1672 y Fx(\000\001)1770 1652 y Fq(\013)p 1761 1660 57 3 v 1761 1693 a FL(16)1848 1707 y Fz(\001)18 b FH(t)1915 1672 y Fx(\000)1973 1652 y FL(3)p Fq(\017)p 1973 1660 54 3 v 1985 1693 a FL(2)2057 1707 y Fz(\001)g(j)p FE(ln)13 b FH(\033)2238 1715 y FG(t)2266 1707 y Fz(j)26 b FH(:)937 b FI(\(5.23\))515 1958 y FJ(Di\033er)l(enc)l(es)29 b(\(5.11\))j(and)e(\(5.12\))515 2141 y FO(They)d(are)g(easily)f(under)i (con)n(trol)e(b)r(ecause:)515 2232 y FH(h)k Fz(2)h FH(C)731 2201 y FF(1)726 2246 y(0)778 2172 y Fy(\000)813 2232 y FB(R)867 2201 y FF(3)919 2232 y Fz(n)17 b FE(0)1012 2172 y Fy(\001)1081 2232 y FO(and)34 b(the)f(estimate)h(\(5.3\))f(in)g (Lemma)g(A2)g(holds;)g(this)h(implies)g(that)f(for)515 2324 y(\(5.14\))27 b(and)g(then)h(for)f(\(5.11\))g(w)n(e)g(can)g (surely)g(pro)n(vide)f(a)h(b)r(ound)h(with)g(the)g(quan)n(tit)n(y)1572 2503 y FH(C)22 b Fz(\001)17 b FH(s)1723 2467 y Fx(\000\001)1809 2448 y Fq(\013)p 1800 2456 57 3 v 1800 2489 a FL(16)1888 2503 y Fz(\001)g FH(s)1962 2467 y FF(2\(1)p Fx(\000)p FG(\013)p FF(\))2180 2503 y Fz(\001)g FH(t)2246 2467 y Fx(\000)2304 2448 y FL(3)p Fq(\017)p 2304 2456 54 3 v 2317 2489 a FL(2)2389 2503 y FE(;)861 b FI(\(5.24\))515 2663 y FO(Starting)24 b(from)h(a)f(di\033erence)h(b)r(et)n(w)n(een)g(v) n(olumes,)f(the)i(expression)d(\(5.15\))h(can)h(b)r(e)g(b)r(ounded)515 2754 y(b)n(y)i(a)g(quan)n(tit)n(y)g(of)h(order)1485 2981 y FE(sup)1387 3055 y Fx(j)p Fw(q)p Fx(j\024)p FG(s)1538 3039 y FL(\()q(1)p Fk(\000)p Fq(\013)p FL(\))1711 2876 y Fy(\022)1778 2933 y Fz(j)p FA(q)p Fz(j)p 1778 2964 90 4 v 1805 3032 a FH(s)1877 2876 y Fy(\023)1944 2874 y FL(1)p 1944 2882 29 3 v 1944 2915 a(2)2003 2981 y Fz(\001)17 b FH(t)2069 2945 y Fx(\000)p FG(\017)2169 2981 y Fz(\024)k FH(s)2286 2945 y Fx(\000)2343 2926 y Fq(\013)p 2343 2934 38 3 v 2348 2967 a FL(2)2413 2981 y Fz(\001)c FH(t)2479 2945 y Fx(\000)p FG(\017)2574 2981 y FH(:)676 b FI(\(5.25\))515 3249 y FJ(Conclusion)p FO(.)515 3448 y(F)-7 b(or)35 b FH(\013)p FO(,)i FE(0)d FH(<)g(\013)14 b FE(\()o FH(<)21 b FE(1\))q FI(,)36 b FO(su\036cien)n(tly)g(close)f(to)h FE(1)h FO(and)f FH(\017)e(>)g FE(0)j FO(su\036cien)n(tly)f(small,)f (there)h(ex-)515 3548 y(ists)27 b FH(\035)d(>)d FE(0)28 b FO(suc)n(h)f(that)h(the)g(sum)g(of)f(the)h(terms)f(\(5.4\))h(and)f (\(5.5\))g(is)h(b)r(ounded)g(b)n(y)1598 3740 y FN(C)d Fo(\001)18 b FN(s)1762 3706 y Fa(\000)p FM(\035)1876 3740 y Fo(\001)g(j)p FK(ln)c FN(\033)2070 3752 y FM(t)2100 3740 y Fo(j)k(\001)h FN(t)2213 3706 y Fa(\000)2275 3684 y FL(3)p Fq(\017)p 2275 3693 54 3 v 2287 3726 a FL(2)2361 3740 y FN(:)3403 3899 y Fb(\003)515 4057 y FP(Corollary)29 b(A3)515 4145 y FO(Under)c(the)h(same)f(assumptions)g(as)f(in)i (Theorem)f(A2,)g(for)g FH(s)c Fz(\035)g FE(1)26 b FO(and)f(suc)n(h)h (that)f FH(s)3178 4113 y Fx(\000)p FG(\013)3292 4145 y Fz(\025)c FH(\033)3417 4153 y FG(t)3444 4145 y FO(,)515 4224 y(the)28 b(norm)f(of)g(the)h(v)n(ector)564 4285 y Fs(")605 4308 y(Z)676 4328 y FG(s)p Fx(\001)p FG(s)753 4308 y Fk(\000)p Fq(\013)645 4460 y FG(\033)681 4468 y Fq(t)708 4460 y Fx(\001)p FG(s)856 4285 y Fs( )922 4345 y Ft(cos)1028 4292 y Fs(\000)1061 4345 y Fh(q)15 b Fu(\001)1164 4318 y Fw(x)p 1164 4330 37 3 v 1168 4371 a FG(s)1227 4345 y Fu(\000)g(j)p Fh(q)p Fu(j)1379 4292 y Fs(\001)p 922 4383 490 3 v 994 4448 a Ft(\(1)h Fu(\000)1145 4447 y Fs(b)1143 4448 y Fh(q)f Fu(\001)h Fh(v)1280 4458 y FG(j)1312 4448 y Ft(\))1437 4398 y Fu(\000)1518 4348 y Ft(cos)1624 4295 y Fs(\000)1656 4348 y Fh(q)g Fu(\001)f(r)p Fr(E)1865 4317 y FG(\033)1901 4325 y Fq(t)1861 4371 y Fw(P)1948 4348 y Fu(\000)g(j)p Fh(q)p Fu(j)2100 4295 y Fs(\001)p 1518 4383 615 3 v 1652 4448 a Ft(\()q(1)h Fu(\000)1803 4447 y Fs(b)1801 4448 y Fh(q)g Fu(\001)f Fh(v)1938 4458 y FG(j)1971 4448 y Ft(\))2143 4285 y Fs(!)2220 4352 y Fr(d)p Ft(\012)p Fr(d)e Fu(j)o Fh(q)p Fu(j)p 2220 4383 218 3 v 2313 4448 a Fr(s)2448 4285 y Fs(#)2501 4398 y Fr(e)2534 4370 y Fx(\000)p FG(iE)2652 4342 y Fq(\033)2686 4353 y(t)2649 4391 y Fl(P)2716 4370 y FG(s)2750 4398 y Fr(e)2783 4370 y FG(i\015)2837 4378 y Fq(\033)2871 4389 y(t)2903 4374 y Ft(\()2930 4370 y Fw(v)2966 4381 y Fq(j)2998 4370 y FG(;)p Fx(r)p FG(E)3116 4342 y Fq(\033)3150 4353 y(t)3113 4391 y Fl(P)3181 4370 y FG(;s)3229 4374 y Ft(\))3260 4398 y Fr( )3308 4363 y FF(\()q FG(t)p FF(\))3306 4420 y FG(j;\033)3386 4428 y Fq(t)515 4600 y FO(\(with)28 b FA(v)783 4608 y FG(j)837 4600 y Fz(\021)21 b(r)p FH(E)1043 4565 y FG(\033)1079 4573 y Fq(t)p 1039 4591 47 3 v 1039 4633 a Fw(P)1085 4644 y Fq(j)1120 4600 y FO(\))28 b(is)f(surely)g(b)r (ounded)h(b)n(y)f(a)h(quan)n(tit)n(y)f(of)g(order)1597 4810 y FH(s)1633 4774 y Fx(\000)p FF(1)1733 4810 y Fz(\001)17 b FH(s)1807 4774 y Fx(\000)p FG(\035)1912 4810 y Fz(\001)h(j)p FE(ln)13 b FH(\033)2093 4818 y FG(t)2121 4810 y Fz(j)k(\001)g FH(t)2225 4774 y Fx(\000)2283 4755 y FL(3)p Fq(\017)p 2283 4763 54 3 v 2296 4796 a FL(2)3271 4810 y FI(\(5.26\))1950 5059 y FO(49)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 515 523 a FO(for)27 b(some)g FH(\035)c(>)e FE(0)p FO(.)515 614 y FJ(Pr)l(o)l(of)515 706 y FO(The)27 b(pro)r(of)f(pro)r(ceeds)h(along)f(the)h(same)g(lines)g (as)f(for)h(the)g(terms)g(\(5.4\))g(and)g(\(5.8\))g(in)g(Theo-)515 797 y(rem)g(A2.)2595 b Fb(\003)515 1071 y FO(In)39 b(the)g(next)f (lemma)h(w)n(e)f(pro)n(vide)g(some)g(upp)r(er)h(estimates)f(for)g(the)h (absolute)f(v)-5 b(alue)39 b(of)515 1162 y(the)28 b(function)1084 1346 y FH(')1134 1362 y FG(t)1158 1348 y Fk(\000)p Fq(\013)1245 1362 y FG(;)p Fw(v)1300 1373 y Fq(j)1349 1346 y FE(\()o FA(x)p FH(;)13 b(s)p FE(\))21 b(:=)h FH(g)1689 1310 y FF(2)1735 1244 y Fy(Z)1812 1264 y FG(\024)1848 1274 y FL(1)1778 1414 y FG(t)1802 1400 y Fk(\000)p Fq(\013)1906 1244 y Fy(Z)46 b(Z)2095 1297 y FE(cos)13 b(\()p FA(k)k Fz(\001)g FA(x)g Fz(\000)g(j)p FA(k)p Fz(j)c FH(s)p FE(\))p 2095 1329 556 4 v 2170 1341 a Fy(\020)2216 1424 y FE(1)k Fz(\000)2350 1404 y Fy(b)2348 1424 y FA(k)g Fz(\001)g FA(v)2497 1432 y FG(j)2530 1341 y Fy(\021)2661 1346 y FH(d)p FE(\012)p FH(d)c Fz(j)p FA(k)p Fz(j)515 1609 y FO(where)36 b FE(1)h FH(>)f(\013)g(>)g FE(0)p FO(,)h FH(t)f Fz(\035)g FE(1)i FO(and)f FH(s)f(>)f(t)p FO(,)i FA(v)1940 1617 y FG(j)2009 1609 y Fz(\021)e(r)p FH(E)2229 1574 y FG(\033)2265 1582 y Fq(t)p 2225 1600 82 3 v 2225 1642 a Fw(P)2272 1653 y Fq(j)2307 1609 y FO(.)i(The)g(pro)r(of)g(only)f (requires)g(some)515 1726 y(in)n(tegrations)26 b(b)n(y)h(parts;)g (therefore)f(it)i(is)g(left)g(to)g(the)f(reader.)515 1908 y FP(Lemma)i(A4)515 2000 y FO(The)e(follo)n(wing)g(t)n(w)n(o)g(b)r (ounds)g(hold)h(for)f FH(t)21 b Fz(\035)g FE(1)28 b FO(and)f FH(s)21 b(>)h(t)p FO(:)515 2182 y(Uniformly)27 b(in)h FA(x)21 b Fz(2)h FB(R)1201 2151 y FF(3)1555 2221 y Fy(\014)1555 2266 y(\014)1555 2311 y(\014)1580 2307 y FH(')1630 2323 y FG(t)1654 2309 y Fk(\000)p Fq(\013)1741 2323 y FG(;)p Fw(v)1796 2334 y Fq(j)1845 2307 y FE(\()p FA(x)p FH(;)12 b(s)p FE(\))2021 2221 y Fy(\014)2021 2266 y(\014)2021 2311 y(\014)2068 2307 y Fz(\024)21 b FH(C)h Fz(\001)2275 2259 y FE(ln)12 b FH(t)p 2275 2290 105 4 v 2309 2358 a(s)2406 2307 y FE(;)844 b FI(\(5.27\))542 2463 y FO(In)28 b(the)g(region)1038 2403 y Fy(\010)1083 2463 y FA(x)21 b Fz(2)h FB(R)1278 2431 y FF(3)1334 2463 y FE(:)64 b(\(1)17 b Fz(\000)g FH(\032)1621 2431 y Fx(0)1643 2463 y FE(\))c FH(s)21 b(<)g Fz(j)p FA(x)p Fz(j)g FH(<)g FE(\(1)d Fz(\000)f FH(\032)p FE(\))12 b FH(s)76 b(;)90 b FE(0)22 b FH(<)f(\032)g(<)g(\032) 2806 2431 y Fx(0)2849 2463 y FH(<)g FE(1)2968 2403 y Fy(\011)1538 2584 y(\014)1538 2628 y(\014)1538 2673 y(\014)1564 2670 y FH(')1614 2686 y FG(t)1638 2672 y Fk(\000)p Fq(\013)1725 2686 y FG(;)p Fw(v)1780 2697 y Fq(j)1828 2670 y FE(\()p FA(x)p FH(;)13 b(s)p FE(\))2005 2584 y Fy(\014)2005 2628 y(\014)2005 2673 y(\014)2051 2670 y Fz(\024)22 b FH(C)2188 2682 y FG(\032;\032)2271 2668 y Fk(0)2314 2670 y Fz(\001)2362 2621 y FH(t)2390 2590 y FG(\013)p 2362 2652 72 4 v 2363 2720 a FH(s)2399 2699 y FF(2)3271 2670 y FI(\(5.28\))515 2859 y FO(where)27 b(the)h(p)r(ositiv)n(e)f(constan)n(t)g FH(C)1595 2871 y FG(\032;\032)1678 2857 y Fk(0)1731 2859 y FO(dep)r(ends)h(on)f FH(\032)p FI(,)f FH(\032)2294 2827 y Fx(0)2316 2859 y FO(.)515 3041 y(W)-7 b(e)35 b(no)n(w)g(discuss) f(some)h(prop)r(erties)f(for)g(the)i(annihilation)e(op)r(erator)g(asso) r(ciated)f(to)i(the)515 3133 y(asymptotic)27 b(b)r(oson)g(\034eld)h (when)f(the)h(Hamiltonian)f(is)h FH(H)2344 3141 y FG(\033)2380 3149 y Fq(t)2411 3133 y FO(,)g FH(\033)2506 3141 y FG(t)2555 3133 y FH(>)21 b FE(0)p FO(.)515 3315 y FP(Theorem)30 b(A5)515 3406 y FO(The)d(limit)678 3650 y FH(s)16 b Fz(\000)63 b FE(lim)807 3695 y FG(s)p Fx(!)p FF(+)p Fx(1)1018 3650 y FH(e)1054 3614 y FG(iH)1126 3622 y Fq(\033)1160 3633 y(t)1191 3614 y FG(s)1237 3548 y Fy(Z)1314 3568 y FG(\024)1350 3578 y FL(1)1279 3718 y FG(\033)1315 3726 y Fq(t)1409 3571 y FH(a)13 b FE(\()o FA(k)p FE(\))g FH(e)1618 3540 y FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(s)1747 3571 y FH(\021)1785 3580 y FG(l;j)1870 3489 y Fy(\020)1918 3552 y(b)1916 3571 y FA(k)1962 3489 y Fy(\021)p 1409 3632 600 4 v 1664 3700 a Fz(j)p FA(k)p Fz(j)2078 3601 y FH(d)2118 3570 y FF(3)2153 3601 y FH(k)p 2028 3632 218 4 v 2028 3648 a Fy(p)p 2105 3648 141 4 v 64 x FE(2)g Fz(j)p FA(k)p Fz(j)2255 3650 y FH(e)2291 3614 y Fx(\000)p FG(iH)2411 3622 y Fq(\033)2445 3633 y(t)2477 3614 y FG(s)2510 3650 y FH( )2563 3611 y FF(\()p FG(t)p FF(\))2560 3669 y FG(j;\033)2640 3677 y Fq(t)2693 3650 y FE(=:)21 b FH(a)2836 3614 y FG(out)2836 3663 y(\033)2872 3671 y Fq(t)2944 3650 y FE(\()5 b(\024)-43 b FH(\021)3012 3659 y FG(l;j)3083 3650 y FE(\))13 b FH( )3179 3611 y FF(\()p FG(t)p FF(\))3176 3669 y FG(j;\033)3256 3677 y Fq(t)515 3871 y FO(is)27 b(w)n(ell)h(de\034ned)p FI(,)g FO(with)1509 4058 y FE(\024)-43 b FH(\021)1542 4067 y FG(l;j)1626 4058 y FE(\()p FA(k)p FE(\))21 b(:=)1917 3980 y FH(\021)1955 3989 y FG(l;j)2040 3897 y Fy(\020)2088 3960 y(b)2086 3980 y FA(k)2132 3897 y Fy(\021)p 1867 4041 363 4 v 1867 4120 a Fz(j)p FA(k)p Fz(j)c(\001)2011 4057 y Fy(p)p 2088 4057 141 4 v 63 x FE(2)c Fz(j)q FA(k)p Fz(j)2239 4058 y FH(\037)2287 4022 y FG(\024)2323 4032 y FL(1)2287 4071 y FG(\033)2323 4079 y Fq(t)2372 4058 y FE(\()p FA(k)p FE(\))792 b FI(\(5.29\))515 4242 y FO(where)25 b FH(\037)801 4210 y FG(\024)837 4220 y FL(1)801 4255 y FG(\033)837 4263 y Fq(t)887 4242 y FE(\()p FA(k)p FE(\))h FO(is)g(the)g(c)n(haracteristic)f(function)h(of)h(the)f(set)g Fz(f)q FA(k)34 b FE(:)43 b FH(\033)2667 4250 y FG(t)2716 4242 y FH(<)21 b Fz(j)p FA(k)p Fz(j)h(\024)f FH(\024)3033 4250 y FF(1)3067 4242 y Fz(g)q FO(.)26 b(The)h(v)n(ec-)515 4347 y(tor)g FH(a)690 4309 y FG(out)p FF(\()n FG(in)p FF(\))690 4355 y FG(\033)726 4363 y Fq(t)906 4347 y FE(\()5 b(\024)-43 b FH(\021)974 4356 y FG(l;j)1045 4347 y FE(\))13 b FH( )1141 4309 y FF(\()p FG(t)p FF(\))1138 4367 y FG(j;\033)1218 4375 y Fq(t)1277 4347 y FO(b)r(elongs)27 b(to)h FH(D)15 b FE(\()p FH(H)1849 4355 y FG(\033)1885 4363 y Fq(t)1916 4347 y FE(\))p FO(.)515 4461 y(Under)31 b(the)g(assumption)g(that)g FA(P)20 b FE(+)f FA(k)26 b Fz(2)h FE(\006)32 b FO(for)e FA(P)i FO(and)e FA(k)h FO(b)r(elonging)g(resp)r(ectiv)n(ely)f(to)h FE(\000)3392 4422 y FF(\()p FG(t)p FF(\))3392 4480 y FG(j)515 4552 y FO(and)c Fz(f)q FA(k)33 b FE(:)43 b(0)22 b FH(<)f Fz(j)p FA(k)p Fz(j)h(\024)f FH(\024)1236 4560 y FF(1)1271 4552 y Fz(g)p FO(,)28 b(the)g(follo)n(wing)e(iden)n(tit)n (y)i(holds:)1601 4735 y FH(a)1642 4699 y FG(out)p FF(\()o FG(in)p FF(\))1642 4748 y FG(\033)1678 4756 y Fq(t)1858 4735 y FE(\()6 b(\024)-44 b FH(\021)1926 4744 y FG(l;j)1998 4735 y FE(\))12 b FH( )2093 4696 y FF(\()p FG(t)p FF(\))2090 4754 y FG(j;\033)2170 4762 y Fq(t)2223 4735 y FE(=)21 b(0)d FH(:)1950 5059 y FO(50)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 515 614 a FJ(Pr)l(o)l(of)515 706 y FO(The)40 b(existence)f(of)h(the)h(limit)g(is)e(a)h(simple)g (application)g(of)f(the)i(propagation)d(estimate)515 797 y(in)f(Theorem)f(A2)h(and)g(of)f(Co)r(ok's)g(argumen)n(t)g(b)n(y)h (using)f(the)i(function)f FH(\037)2945 806 y FG(h)2998 797 y FE(\()p Fz(r)p FH(E)3153 763 y FG(\033)3189 771 y Fq(t)3149 818 y Fw(P)3220 797 y FE(\))g FO(as)f(in)515 888 y(Theorem)26 b(2.1)h(and)h(exploiting)f(the)h(estimates)f(\(5.27\)) g(and)g(\(5.28\))g(in)h(Lemma)f(A4.)515 993 y(The)g(v)n(ector)g FH(a)976 955 y FG(out)p FF(\()n FG(in)p FF(\))976 1001 y FG(\033)1012 1009 y Fq(t)1192 993 y FE(\()5 b(\024)-43 b FH(\021)1260 1002 y FG(l;j)1331 993 y FE(\))13 b FH( )1427 955 y FF(\()p FG(t)p FF(\))1424 1013 y FG(j;\033)1504 1021 y Fq(t)1536 993 y FO(b)r(elongs)26 b(to)i FH(D)15 b FE(\()p FH(H)2107 1001 y FG(\033)2143 1009 y Fq(t)2174 993 y FE(\))p FJ(.)28 b FO(Indeed,)g(for)f(eac)n(h)g FH(s)p FO(,)g(the)h(v)n(ector)1259 1177 y FH(H)1323 1185 y FG(\033)1359 1193 y Fq(t)1390 1177 y FH(e)1426 1141 y FG(iH)1498 1149 y Fq(\033)1532 1160 y(t)1563 1141 y FG(s)1596 1177 y FH(e)1632 1141 y Fx(\000)p FG(iH)1756 1120 y Fq(ph)1825 1141 y FG(s)1858 1177 y FH(a)12 b FE(\()6 b(\024)-44 b FH(\021)1979 1186 y FG(l;j)2051 1177 y FE(\))13 b FH(e)2130 1141 y FG(iH)2206 1120 y Fq(ph)2274 1141 y FG(s)2308 1177 y FH(e)2344 1141 y Fx(\000)p FG(iH)2464 1149 y Fq(\033)2498 1160 y(t)2529 1141 y FG(s)2562 1177 y FH( )2615 1139 y FF(\()p FG(t)p FF(\))2612 1197 y FG(j;\033)2692 1205 y Fq(t)3271 1177 y FI(\(5.30\))542 1329 y FO(is)28 b(w)n(ell)f(de\034ned)h(b)r(ecause)1581 1421 y FH(a)13 b FE(\()5 b(\024)-43 b FH(\021)1703 1430 y FG(l;j)1774 1421 y FE(\))13 b FH( )1870 1382 y FF(\()p FG(t)p FF(\))1867 1440 y FG(j;\033)1947 1448 y Fq(t)2000 1421 y Fz(\032)21 b FH(D)15 b FE(\()p FH(H)2253 1429 y FG(\033)2289 1437 y Fq(t)2320 1421 y FE(\))30 b FH(:)870 b FI(\(5.31\))515 1549 y FO(The)27 b(inclusion)h(\(5.31\))e(is)i(pro)n(v)n(ed)e(b)n(y)h (an)g(appro)n(ximation)f(argumen)n(t,)g(exploiting)i(the)f(fact)515 1660 y(that)35 b Fz(j)p FA(k)p Fz(j)801 1605 y FL(1)p 801 1613 29 3 v 801 1646 a(2)865 1660 y Fz(\001)27 b FE(\024)-43 b FH(\021)946 1669 y FG(l;j)1030 1660 y FE(\()p FA(k)p FE(\))32 b Fz(2)g FH(L)1304 1628 y FF(2)1352 1600 y Fy(\000)1387 1660 y FB(R)1441 1628 y FF(3)1476 1660 y FH(;)13 b(d)1550 1628 y FF(3)1584 1660 y FH(k)1626 1600 y Fy(\001)1696 1660 y FO(and)35 b FH( )1918 1622 y FF(\()p FG(t)p FF(\))1915 1680 y FG(j;\033)1995 1688 y Fq(t)2059 1660 y Fz(\032)d FH(D)2229 1600 y Fy(\000)2264 1660 y FH(H)2334 1628 y FF(2)2328 1673 y FG(\033)2364 1681 y Fq(t)2395 1600 y Fy(\001)2430 1660 y FO(.)j(No)n(w,)f(since)h FH(H)2982 1668 y FG(\033)3018 1676 y Fq(t)3083 1660 y FO(is)g(a)f(closed)515 1751 y(op)r(erator,)22 b(it)j(is)f(enough)g(to)g (pro)n(v)n(e)f(the)i(con)n(v)n(ergence)d(for)h FH(s)e Fz(!)h FE(+)p Fz(1)p FI(.)i FO(W)-7 b(e)25 b(rewrite)e(the)i(v)n(ector) 515 1843 y(\(5.30\))i(as)903 2018 y FH(H)967 2026 y FG(\033)1003 2034 y Fq(t)1033 2018 y FH(e)1069 1982 y FG(iH)1141 1990 y Fq(\033)1175 2001 y(t)1207 1982 y FG(s)1240 2018 y FH(e)1276 1982 y Fx(\000)p FG(iH)1400 1961 y Fq(ph)1469 1982 y FG(s)1502 2018 y FH(a)12 b FE(\()6 b(\024)-44 b FH(\021)1623 2027 y FG(l;j)1695 2018 y FE(\))13 b FH(e)1774 1982 y FG(iH)1850 1961 y Fq(ph)1918 1982 y FG(s)1952 2018 y FH(e)1988 1982 y Fx(\000)p FG(iH)2108 1990 y Fq(\033)2142 2001 y(t)2173 1982 y FG(s)2206 2018 y FH( )2259 1980 y FF(\()p FG(t)p FF(\))2256 2038 y FG(j;\033)2336 2046 y Fq(t)2451 2018 y FE(=)760 b FI(\(5.32\))973 2198 y FE(=)130 b FH(e)1199 2166 y FG(iH)1271 2174 y Fq(\033)1305 2185 y(t)1336 2166 y FG(s)1369 2198 y FH(e)1405 2166 y Fx(\000)p FG(iH)1529 2145 y Fq(ph)1598 2166 y FG(s)1631 2198 y FH(a)12 b FE(\()6 b(\024)-44 b FH(\021)1752 2207 y FG(l;j)1824 2198 y FE(\))13 b FH(e)1903 2166 y FG(iH)1979 2145 y Fq(ph)2047 2166 y FG(s)2081 2198 y FH(e)2117 2166 y Fx(\000)p FG(iH)2237 2174 y Fq(\033)2271 2185 y(t)2302 2166 y FG(s)2335 2198 y FH(E)2396 2166 y FG(\033)2432 2174 y Fq(t)2476 2198 y FE(\()p FA(P)p FE(\))g FH( )2662 2160 y FF(\()p FG(t)p FF(\))2659 2218 y FG(j;\033)2739 2226 y Fq(t)1116 2405 y FE(+)p FH(e)1212 2373 y FG(iH)1284 2381 y Fq(\033)1318 2392 y(t)1349 2373 y FG(s)1395 2322 y Fy(h)1431 2405 y FH(H)1495 2413 y FG(\033)1531 2421 y Fq(t)1579 2405 y Fz(\000)k FH(H)1726 2373 y FG(ph)1798 2405 y FH(;)c(e)1868 2373 y Fx(\000)p FG(iH)1992 2352 y Fq(ph)2061 2373 y FG(s)2094 2405 y FH(a)g FE(\()5 b(\024)-44 b FH(\021)2215 2414 y FG(l;j)2287 2405 y FE(\))13 b FH(e)2366 2373 y FG(iH)2442 2352 y Fq(ph)2510 2373 y FG(s)2544 2322 y Fy(i)2593 2405 y FH(e)2629 2373 y Fx(\000)p FG(iH)2749 2381 y Fq(\033)2783 2392 y(t)2814 2373 y FG(s)2847 2405 y FH( )2900 2367 y FF(\()p FG(t)p FF(\))2897 2425 y FG(j;\033)2977 2433 y Fq(t)1130 2631 y FE(+)p FH(e)1226 2599 y FG(iH)1298 2607 y Fq(\033)1332 2618 y(t)1363 2599 y FG(s)1409 2548 y Fy(h)1446 2631 y FH(H)1516 2599 y FG(ph)1605 2631 y FH(;)30 b(e)1692 2599 y Fx(\000)p FG(iH)1816 2578 y Fq(ph)1884 2599 y FG(s)1918 2631 y FH(a)12 b FE(\()6 b(\024)-44 b FH(\021)2039 2640 y FG(l;j)2111 2631 y FE(\))12 b FH(e)2189 2599 y FG(iH)2265 2578 y Fq(ph)2334 2599 y FG(s)2367 2548 y Fy(i)2416 2631 y FH(e)2452 2599 y Fx(\000)p FG(iH)2572 2607 y Fq(\033)2606 2618 y(t)2638 2599 y FG(s)2671 2631 y FH( )2724 2592 y FF(\()p FG(t)p FF(\))2721 2651 y FG(j;\033)2801 2659 y Fq(t)2845 2631 y FH(:)515 2773 y FO(Because)24 b(of)i(the)g(\034rst)f(part)g(of)h(the)g(theorem)f(and)h(the)g(follo)n (wing)e(equalit)n(y)h(whic)n(h)h(holds)f(on)515 2864 y FH(D)15 b FE(\()p FH(H)687 2872 y FG(\033)723 2880 y Fq(t)754 2864 y FE(\))938 2973 y Fy(h)975 3056 y FH(H)1045 3020 y FG(ph)1134 3056 y FH(;)30 b(e)1221 3020 y Fx(\000)p FG(iH)1345 2999 y Fq(ph)1413 3020 y FG(s)1447 3056 y FH(a)12 b FE(\()6 b(\024)-44 b FH(\021)1568 3065 y FG(l;j)1640 3056 y FE(\))12 b FH(e)1718 3020 y FG(iH)1794 2999 y Fq(ph)1863 3020 y FG(s)1896 2973 y Fy(i)1954 3056 y FE(=)21 b Fz(\000)2108 2954 y Fy(Z)2184 2974 y FG(\024)2220 2984 y FL(1)2150 3124 y FG(\033)2186 3132 y Fq(t)2270 3056 y FH(a)12 b FE(\()p FA(k)p FE(\))h FH(e)2479 3020 y FG(i)p Fx(j)o Fw(k)p Fx(j)q FG(s)2618 2977 y FH(\021)2656 2986 y FG(l;j)2740 2894 y Fy(\020)2788 2958 y(b)2786 2977 y FA(k)2833 2894 y Fy(\021)p 2618 3038 261 4 v 2640 3054 a(p)p 2717 3054 141 4 v 64 x FE(2)g Fz(j)p FA(k)p Fz(j)2889 3056 y FH(d)2929 3020 y FF(3)2963 3056 y FH(k)19 b(;)515 3240 y FO(eac)n(h)28 b(term)g(on)g(the)h(righ)n(t)f(hand)h(side)f(of)h (the)g(expression)e(\(5.32\))g(has)h(a)h(w)n(ell)f(de\034ned)h(limit) 515 3331 y(for)e FH(s)21 b Fz(!)g FE(+)p Fz(1)p FO(.)515 3514 y(W)-7 b(e)36 b(denote)g(as)1052 3431 y Fy(\020)1098 3514 y FH( )1151 3475 y FF(\()o FG(t)p FF(\))1148 3533 y FG(j;\033)1228 3541 y Fq(t)1259 3431 y Fy(\021)1305 3566 y Fw(P)p FF(+)p Fw(k)1439 3514 y FO(the)g(pro)5 b(jection)35 b(of)g FH( )2148 3475 y FF(\()p FG(t)p FF(\))2145 3533 y FG(j;\033)2225 3541 y Fq(t)2293 3514 y FO(on)g(the)h(\034b)r(er) g(space)f Fz(H)3062 3523 y Fw(P)p FF(+)p Fw(k)3196 3514 y FO(.)h(Start-)515 3639 y(ing)g(from)g(the)h(sp)r(ectral)f(decomp)r (osition)g(of)h Fz(H)k FO(with)c(resp)r(ect)g(to)f(the)h FA(P)g FO(op)r(erators)e(and)515 3730 y(b)r(ecause)27 b(of)g(the)h(equation)f(\(5.32\),)g(w)n(e)g(deduce)h(that)1342 3882 y Fy(Z)1419 3903 y FG(\024)1455 3913 y FL(1)1384 4053 y FG(\033)1420 4061 y Fq(t)1504 3984 y FH(a)1545 3948 y FG(out)1545 3997 y(\033)1581 4005 y Fq(t)1653 3984 y FE(\()p FA(k)p FE(\))1811 3906 y FH(\021)1849 3915 y FG(l;j)1934 3823 y Fy(\020)1982 3886 y(b)1980 3906 y FA(k)2026 3823 y Fy(\021)p 1782 3967 320 4 v 1782 4047 a Fz(j)p FA(k)p Fz(j)1884 3983 y Fy(p)p 1961 3983 141 4 v 64 x FE(2)13 b Fz(j)p FA(k)p Fz(j)2124 3901 y Fy(\020)2170 3984 y FH( )2223 3946 y FF(\()p FG(t)p FF(\))2220 4004 y FG(j;\033)2300 4012 y Fq(t)2331 3901 y Fy(\021)2377 4037 y Fw(P)p FF(+)p Fw(k)2524 3984 y FH(d)2564 3948 y FF(3)2598 3984 y FH(k)633 b FI(\(5.33\))515 4195 y FO(is)39 b(a)f(v)n(ector)g(in)h Fz(H)1124 4203 y Fw(P)1213 4195 y FO(and)g(it)g(b)r(elongs)g(to)f(the)i(domain)e(of)h FH(H)2533 4203 y Fw(P)p FG(;\033)2635 4211 y Fq(t)2665 4195 y FO(.)g(Then)g(the)h(pro)r(cedure)515 4287 y(consists)26 b(in)i(studying)g(the)g(mean)f(v)-5 b(alue)27 b(of)h(the)g(p)r(ositiv)n (e)f(op)r(erator)1636 4439 y FH(H)1700 4447 y Fw(P)p FG(;\033)1802 4455 y Fq(t)1850 4439 y Fz(\000)17 b FH(E)1988 4403 y FG(\033)2024 4411 y Fq(t)2068 4439 y FE(\()p FA(P)p FE(\))g(+)g(\001)542 4591 y FO(on)28 b(the)g(giv)n(en)e(v)n(ector)h(in) g Fz(H)1428 4599 y Fw(P)1479 4591 y FO(:)1100 4743 y Fz(h)p FH(H)1194 4751 y Fw(P)p FG(;\033)1296 4759 y Fq(t)1343 4743 y Fz(\000)17 b FH(E)1481 4707 y FG(\033)1517 4715 y Fq(t)1561 4743 y FE(\()p FA(P)p FE(\))g(+)g(\001)p Fz(i)1869 4768 y Fm(R)1907 4778 y Fq(\024)1941 4790 y FL(1)1897 4818 y Fq(\033)1931 4829 y(t)1989 4808 y FG(a)2023 4794 y Fq(out)2023 4825 y(\033)2057 4836 y(t)2114 4808 y FF(\()p Fw(k)p FF(\))2222 4765 y Fq(\021)2253 4780 y(l;j)2323 4773 y FE(\()2356 4754 y Fj(b)2353 4765 y Fl(k)2386 4773 y FE(\))p 2209 4797 221 3 v 2209 4840 a Fk(j)p Fl(k)p Fk(j)2280 4804 y Fx(p)p 2330 4804 100 3 v 36 x FL(2)p Fk(j)p Fl(k)p Fk(j)2439 4753 y Fm(\020)2474 4808 y FG( )2517 4781 y FL(\()p Fq(t)p FL(\))2515 4828 y Fq(j;\033)2592 4839 y(t)2623 4753 y Fm(\021)2658 4845 y Fl(P)p FL(+)p Fl(k)2780 4808 y FG(d)2812 4794 y FL(3)2844 4808 y FG(k)3271 4743 y FI(\(5.34\))1950 5059 y FO(51)p eop end %%Page: 52 52 TeXDict begin 52 51 bop 515 523 a FO(where)27 b FE(\001)g FO(is)h(a)f(prop)r(erly)g(small)g(p)r(ositiv)n(e)g(n)n(um)n(b)r(er.)515 614 y(The)g(condition)h(\(1.23\))e(implies)i(the)g(inequalit)n(y)1451 776 y FH(E)1512 740 y FG(\033)1548 748 y Fq(t)1592 776 y FE(\()p FA(P)17 b FE(+)g FA(k)p FE(\))g Fz(\000)f(j)q FA(k)p Fz(j)h(\000)g FH(E)2191 740 y FG(\033)2227 748 y Fq(t)2270 776 y FE(\()p FA(P)p FE(\))22 b FH(<)f FE(0)740 b FI(\(5.35\))515 938 y FO(for)29 b FA(k)21 b Fz(2)13 b(f)q FA(k)34 b FE(:)43 b(0)21 b FH(<)g Fz(j)q FA(k)p Fz(j)g(\024)g FH(\024)1336 946 y FF(1)1371 938 y Fz(g)p FO(,)31 b(so)e(that)i(w)n(e)e(can)h(conclude)g(that)g(the)h(original)e (v)n(ector)f(\(5.33\))515 1029 y(is)f(zero.)2633 b Fb(\003)515 1303 y FO(Next)28 b(Lemma)g(pro)n(vides)f(the)i(estimates)f(of)g(the)h (expressions)e(\(3.44\))g(and)h(\(3.45\))g(in)n(v)n(olv)n(ed)515 1395 y(in)g(the)g(con)n(trol)e(of)h(the)h(di\033erence)g FJ(D3.1)11 b FO(\))29 b(in)f(Subsection)f(3.1.4.)515 1577 y FP(Lemma)i(A6)515 1669 y FO(Under)40 b(the)g(assumptions)f(for)g (the)i(construction)e(\(Subsection)h(1.3.1\))f(and)g(for)h FH(\013)p FO(,)g FE(0)h FH(<)515 1760 y(\013)13 b FE(\()p FH(<)21 b FE(1\))p FI(,)28 b FO(su\036cien)n(tly)f(close)g(to)h FE(1)p FO(,)g(the)g(estimates)f(b)r(elo)n(w)g(are)g(v)-5 b(alid:)515 1869 y Fy(\015)515 1913 y(\015)515 1958 y(\015)515 2003 y(\015)515 2048 y(\015)515 2093 y(\015)557 2022 y FH(g)609 1920 y Fy(Z)686 1941 y FG(\033)722 1949 y Fq(t)745 1961 y FL(1)652 2090 y FG(\033)688 2098 y Fq(t)711 2110 y FL(2)809 1944 y FH(b)13 b FE(\()o FA(k)p FE(\))974 1861 y Fy(\020)1020 1944 y FH(e)1056 1912 y FG(i)p Fx(j)o Fw(k)p Fx(j)q FG(t)1176 1922 y FL(1)1208 1912 y Fx(\000)p FG(i)p Fw(k)p Fx(\001)p Fw(x)1391 1944 y Fz(\000)k FE(1)1506 1861 y Fy(\021)p 809 2005 744 4 v 927 2100 a Fz(j)p FA(k)p Fz(j)1029 2017 y Fy(\020)1075 2100 y FE(1)g Fz(\000)1209 2080 y Fy(b)1207 2100 y FA(k)g Fz(\001)g FA(v)1356 2108 y FG(j)1388 2017 y Fy(\021)1622 1974 y FH(d)1662 1942 y FF(3)1697 1974 y FH(k)p 1572 2005 218 4 v 1572 2021 a Fy(p)p 1649 2021 141 4 v 63 x FE(2)c Fz(j)p FA(k)p Fz(j)1799 2022 y FH(W)1872 2030 y FG(\033)1908 2038 y Fq(t)1931 2050 y FL(2)1984 2022 y FE(\()p FA(v)2061 2030 y FG(j)2093 2022 y FE(\))g FH(W)2220 1986 y Fx(y)2209 2035 y FG(\033)2245 2043 y Fq(t)2268 2055 y FL(2)2321 1961 y Fy(\000)2356 2022 y Fz(r)p FH(E)2481 1975 y FG(\033)2517 1983 y Fq(t)2540 1995 y FL(2)2477 2043 y Fw(P)2581 1961 y Fy(\001)2629 2022 y FH(')2679 1984 y FF(\()p FG(t)2727 1994 y FL(1)2759 1984 y FF(\))2679 2042 y FG(j;\033)2759 2050 y Fq(t)2782 2062 y FL(1)2823 1869 y Fy(\015)2823 1913 y(\015)2823 1958 y(\015)2823 2003 y(\015)2823 2048 y(\015)2823 2093 y(\015)2887 2022 y Fz(\024)21 b FH(C)5 b Fz(\001)p FH(t)3077 1986 y FF(1)p Fx(\000)p FG(\017)3077 2038 y FF(1)3186 2022 y Fz(\001j)q FE(ln)12 b FH(\033)3349 2030 y FG(t)3373 2040 y FL(2)3410 2022 y Fz(j\001)p FE(\()p FH(\033)3526 2030 y FG(t)3550 2040 y FL(1)3586 2022 y FE(\))h(;)515 2302 y Fy(*)577 2327 y(Z)653 2348 y FG(\033)689 2356 y Fq(t)712 2368 y FL(1)619 2498 y FG(\033)655 2506 y Fq(t)678 2518 y FL(2)827 2265 y Fy(\014)827 2310 y(\014)827 2355 y(\014)853 2351 y FH(e)889 2319 y FG(i)p Fx(j)o Fw(k)p Fx(j)q FG(t)1009 2329 y FL(1)1062 2351 y Fz(\000)k FH(e)1175 2319 y FG(i)p Fw(k)p Fx(\001)p Fw(x)1293 2265 y Fy(\014)1293 2310 y(\014)1293 2355 y(\014)1318 2285 y FF(2)p 776 2412 628 4 v 776 2519 a FE(2)c Fz(j)p FA(k)p Fz(j)916 2483 y FF(3)964 2436 y Fy(\020)1010 2519 y FE(1)k Fz(\000)1144 2499 y Fy(b)1142 2519 y FA(k)g Fz(\001)g FA(v)1291 2527 y FG(j)1323 2436 y Fy(\021)1369 2454 y FF(2)1414 2429 y FH(d)1454 2393 y FF(3)1488 2429 y FH(k)1530 2302 y Fy(+)1602 2254 y FL(1)p 1602 2262 29 3 v 1602 2295 a(2)1592 2572 y FG(W)1649 2580 y Fq(\033)1683 2591 y(t)1706 2603 y FL(2)1747 2578 y FE(\()1777 2572 y Fw(v)1813 2583 y Fq(j)1845 2578 y FE(\))1875 2572 y FG(W)1940 2546 y Fk(y)1932 2582 y Fq(\033)1966 2593 y(t)1989 2605 y FL(2)2030 2517 y Fm(\020)2064 2572 y Fx(r)p FG(E)2163 2531 y Fq(\033)2197 2542 y(t)2220 2554 y FL(2)2160 2593 y Fl(P)2261 2517 y Fm(\021)2296 2572 y FG(')2336 2542 y FL(\()p Fq(t)2381 2554 y FL(1)2414 2542 y(\))2336 2592 y Fq(j;\033)2413 2603 y(t)2436 2615 y FL(1)2502 2429 y Fz(\024)k FH(C)5 b Fz(\001)p FH(t)2702 2362 y FL(1)p 2703 2370 V 2703 2403 a(2)2741 2382 y Fx(\000)2799 2362 y FL(5)p Fq(\017)p 2799 2370 54 3 v 2811 2403 a FL(4)2692 2448 y FF(1)2867 2429 y Fz(\001)p FE(\()p Fz(j)p FE(ln)13 b FH(\033)3060 2437 y FG(t)3084 2447 y FL(2)3120 2429 y Fz(j)p FE(\))3181 2374 y FL(1)p 3181 2382 29 3 v 3181 2415 a(2)3224 2429 y Fz(\001)p FE(\()p FH(\033)3319 2437 y FG(t)3343 2447 y FL(1)3379 2429 y FE(\))3433 2374 y FL(1)p 3419 2382 57 3 v 3419 2415 a(16)542 2725 y FO(where)27 b FA(v)829 2733 y FG(j)883 2725 y Fz(\021)21 b(r)p FH(E)1089 2679 y FG(\033)1125 2687 y Fq(t)1148 2699 y FL(1)p 1085 2717 82 3 v 1085 2758 a Fw(P)1132 2769 y Fq(j)1189 2725 y FO(.)515 2842 y FJ(Pr)l(o)l(of)515 2934 y FO(The)29 b(pro)r(of)g(is)g(only)g(outlined)g(b)r(ecause)g(the)h(estimates)f(in)n (v)n(olv)n(e)e(similar)i(pro)r(cedures)f(as)g(in)515 3025 y(Theorem)c(A2)h(on)f(the)i(basis)e(of)h(the)g(kno)n(wn)f(sp)r (ectral)h(prop)r(erties.)e(The)i(k)n(ey)g(ingredien)n(ts)f(to)515 3116 y(b)r(e)k(exploited)f(are:)515 3208 y(The)g(pull-through)g(form)n (ula)g(for)g(the)h(action)f(of)g FH(b)13 b FE(\()p FA(k)p FE(\))o FO(;)515 3299 y(The)27 b FA(k)p FO(-regularit)n(y)e(of)1155 3397 y Fy(\020)1201 3480 y FH(e)1237 3444 y FG(i)p Fx(j)p Fw(k)p Fx(j)p FG(t)1357 3454 y FL(1)1394 3480 y FH(e)1430 3444 y Fx(\000)p FG(i)p Fw(k)p 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FO(I)40 b(w)n(ould)g(lik)n(e)g(to)g(thank:)g(G.Dell'An)n(tonio)h (for)f(man)n(y)g(helpful)h(discussions)e(during)h(m)n(y)515 4362 y(Ph.d.)f(in)g(S.I.S.S.A.-T)-7 b(rieste;)38 b(J.F)-7 b(r\366hlic)n(h)38 b(for)f(his)i(in)n(terest)f(in)g(m)n(y)g(w)n(ork,)f (his)i(criticism)515 4462 y(and)28 b(our)g(presen)n(t,)g(fruitful)h (collab)r(oration;)d(G.Morc)n(hio)i(for)f(all)i(he)f(taugh)n(t)g(me)h (ab)r(out)f(the)515 4561 y(infrared)e(problem)h(in)h(Q.E.D.)1950 5059 y(52)p eop end %%Page: 53 53 TeXDict begin 53 52 bop 515 523 a FD(References)515 705 y FO([Bl.])182 b(Ph.Blanc)n(hard.)27 b FJ(Comm.)j(Math.Phys.)h FO(15:)26 b(156)h(\(1969\).)515 871 y([B.N.])120 b(F.Blo)r(c)n(h)27 b(and)h(A.Nordsiec)n(k.)e FJ(Helv.)k(Phys.)i(A)l(cta.)27 b FO(45:)g(303)f(\(1972\))515 1037 y([Bu1.])117 b(D.Buc)n(hholz.)27 b(Comm.)h(Math.)f(Ph)n(ys.)h(52:)f(147)f(\(1977\))515 1203 y([Bu.2])117 b(D.Buc)n(hholz.)27 b(Comm.)h(Math.)f(Ph)n(ys.)h(85:) f(49)f(\(1982\))515 1369 y([B.P)-7 b(.S.])64 b(D.Buc)n(hholz,)27 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FO(.)g(14:)d(1165-1280)c(\(2002\))515 2863 y([G.J.])133 b(J.Glimm)28 b(and)f(A.Ja\033e.)h FJ(A)n(nn.)h(Math.) g FO(91:)d(362)h(\(1970\))515 3029 y([Ha])183 b(Haag.)62 b(Lo)r(cal)g(Quan)n(tum)g(Ph)n(ysics,)h(Fields)g(P)n(articles,)f (Algebras.)g(Berlin:)848 3129 y(Springer)26 b(V)-7 b(erlag)27 b(\(1992\).)515 3295 y([He])188 b(K.Hepp.)28 b FJ(Comm.)i(Math.Phys.)h FO(1:)c(95)g(\(1965-1966\).)515 3461 y([K.F.])122 b(P)-7 b(.P)g(.Kulish)28 b(and)g(L.D.F)-7 b(adeev.)27 b FJ(The)l(or.)32 b(Math.Phys.)e FO(4:153)c(\(1970\).)515 3627 y([L.S.Z.])69 b(Lehmann,)27 b(Symanzik)h(and)f(Zimmermann.)g FJ(Nuovo)j(Cimento)j FO(1,425)26 b(\(1955\))515 3793 y([Ne.])165 b(E.)28 b(Nelson.)f FJ(J.)j(Math.)h(Phys.)e FO(5:1190-1197)23 b(\(1964\))515 3959 y([Pi.])185 b(A.Pizzo.)28 b FJ(A)n(nn.)h(Henri.)h(Poinc)l(ar)l (\351.)f FO(4)p FJ(,)f FO(439-486)c(\(2003\))515 4125 y([R.S.])134 b(M.Reed)25 b(and)g(B.Simon.)f FJ(Metho)l(ds)29 b(of)f(mo)l(dern)f(mathematic)l(al)i(physics:)g(V)-6 b(olume)848 4225 y(3.)28 b FO(A)n(cademic)g(Press.)515 4391 y([Sc.])181 b(B.Sc)n(hro)r(er.)26 b FJ(F)-6 b(ortschr.)31 b(Physik)38 b FO(11:1-31)25 b(\(1963\).)1950 5059 y(53)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0408270802292--