Content-Type: multipart/mixed; boundary="-------------0405211228482" This is a multi-part message in MIME format. ---------------0405211228482 Content-Type: text/plain; name="04-159.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-159.keywords" KAM, elliptic tori, resonances, divergent series ---------------0405211228482 Content-Type: application/postscript; name="ellittici.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ellittici.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: ellittici.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR10 CMTI10 CMBX10 CMBX12 CMSY7 CMCSC10 CMSY10 CMR7 %%+ CMMI10 CMMIB10 MSBM10 CMMI7 CMR5 CMMI5 CMEX10 MSBM5 CMBX7 EUFM10 %%+ EUFM7 CMR8 CMMI8 EUFM5 CMMI6 CMSY6 CMSY5 CMR6 CMSY8 CMBX6 CMTI8 %%+ CMBX8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o ellittici.ps 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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 (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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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM5 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 118 /v put dup 119 /w put readonly def /FontBBox{0 -251 1451 753}readonly def /UniqueID 5031998 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 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 118 /v put dup 119 /w put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueID 5031992 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 114 /r put dup 118 /v put dup 119 /w put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM5 %!PS-AdobeFont-1.1: MSBM5 2.1 %%CreationDate: 1992 Oct 17 08:36:07 % 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 (MSBM5) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM5 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 90 /Z put readonly def /FontBBox{0 -574 3027 1074}readonly def /UniqueID 5032013 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 D10F04E3F63AB8B2990864357741C8BCF55C8EDA33D52F7F9C684835918F6011 2EC854A55A33F19A50452987C9020931C4C2408014F23F35192290EEBC032DEE 239B4EE853F8348C7059A8AD96FAF2C80BE34ABF15B14C0A2B5F08BD893E06DE 317BA94CD4196567069E2E1BE9379BC1568553382FF29FF8E7250D4B4E871E81 FD8341BCDC7F5A7DAF565D673F0817C68A5544DF0F588A27E30A29695DAA019A 07069DBA1B2EC5B38CFDE9D7C76E71D03995E511EE119EC04CA8BCA43B26193F D3DBE0C4D9AA8800D00CEF7AC58DDBDA94963D6695D245943511D71E2C58230E 5913A781F5509A087D258F6C82E0C9FA31C81E2DAFA1643C86BDD322A9D492A3 0C3C62A6867E24FA96442A4E9FB8D5300B8332678362DDC162D8172F8CB8BCD7 AB110BF40CAB1AB0C73FB4A7CC4D452AA1EF368E6A107B9F98DA506F5213D9D7 71A77F2171A806106EF70872156C42885E2D11BE329D2D616EEAB9E15D3C8418 CF8535E2F35024E45F8FAC2351666C56C0E6A4B786FB68E21220BEDEEFAC32CC B805D1E33BF67CB7A1EAA19580CD5B7AB4EE2D2621235264E14D93073FD21728 89F41561A2B4CAC651B21161E82E29DA7693BF3ED07FC10446E70F5E02D72151 6844A20BE89A630B5BD53EB59F6B3FDF302406DCDC48C4DFAF610D59591598B8 4C94D4D7F60F0329462D2F7F93D55634100468C1D94328C4AF36EA5E5726B81B 7983D241BF7F78928EC9CCE2B27BA69C32BACCAF546140820EAD0771F2CFF97D 1B3C20DF91E5AC5AC537C1EA758960AF0E6C18C21CA0ACD3223CE922228BE6A9 B205E334973A8F36EA704622EB99C26FA34C89883657E288E6D751F729CDE267 01EC34DEFF7771CF7919C6A5944C6FBA4ADF093486489503A1C44C73AF48A9E9 A92A14AF1A95C205AD68435214648F8DD34F5163223F0B0B98573570EE0FB546 1D018EB21CDBEBB29FCB9F407D752607FEE5A72665EA5FFE99A0FF2B073BA1FE 7476461394FEB8906F1234A0910FA5914F171A4CF111321BFEB9B5EA08917296 810257262E2C5314153428065D7DA7542B6F7FC4D419F1F361BE4436D038EDDD 696E1D5DBB5294C41B6214E30F0FDB2CC54D843A3CC02E8ED938AB70B0AA0E93 C8510F7FA17B738562432BC3B1F0E0EFCAA67D08C0A183F999E02657FDA82CD8 D5220482FFC47E1FC0A6D8B8D8D9CC67360FC685E0AB208697E29084F3450502 129D56F816F521A3A6B202C44877C3EAA4987C08AF254E34FC326237D510DFF2 3869FFBA56AFE864802F4A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueID 5087392 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D004B836D34E88C25F6CE738846C8E2E59A2BCF 4ACF80A26D78872E9343A0537BC3BD7715F32ACD958D5AAED865BFE129278935 063A31C2634DE2F9077E0AAAAEB224466B779096D8E3FF0A12AD5157F6603DED 1A82F3511359143311179080C510740B401C930C96270FD1AB3ECBCFEF5DE53F E846BAAE95828D5790922640EF8AB9D7CEBE7669FEA02B72F86872D3D8754A18 A1629C40A7C00C956F140BC63362478279C36EE353638CD3E249897207A94504 4400668C8E702058EBF7284C9BDF830A3FC79C7EE900CC4C3664F9767A237275 CEE3671644A75F1E696DA906B4C66870DBE87F5B4A176920C078ADBE24F55C09 3D18CDE21B5FBC1C6A8AB18E05EDBEF9D1C1C18B3E6377BA2A688579D4F708F9 A5CF4F56C5E39E2726106E9713E638775E606464CD674E5DC25CE9A696A65806 C8E9D206B421E246F18013ACC6C7B2985BA93B1B7D7745CCB25B09957F50128C B523A55ACA6A7A2A0193A536E590291ED9D577B527CAD0372E05BFCA1829FED1 662D06144A5FFA628C587A4FA05B179F1A7E3B23B47765FDC054271A0DBF9C2B B4F4771F80D1F7AAD9024868C30DAD5CF728DB2A71D86D53B0E674996E8C01F7 EF97B225A28872F8AD4752A466E078C2B020EB832F237CB9B5631EB2D2EDDB00 709D3864CA3A6C3EF18085EAEABC011E9F35C9BE4B5D0B608361F329B5784DAC 5557A602E9E3C204909D84DB988F0BAB914E87CD685C7DA55C5E0B9F0176184F FC39B570873BBF346A0D1DE3942DA05434949A65CE64D8BAB0A091C40F7FF47A 4FC57CB4420221C7B3EB8B891044B5FB0227009F0F6028D3F28545E6396CA41F 228D8839A35712ED56A98AF56EB1FC55B662BB1DBEC166EEC9D5B0B61391E394 831AAC9D49C2DA84E03F2313E1F8FF53F287268589291FF1B79D9D1AAD045464 9B510168771CF93A5D438AF5DB83C512B57566396F1615B6D1AD034F5215070C F5193B20838C5DED83D1EB2473CAC0D1F8339428A5B98E5C88281461E0419C46 FDBAB512FE612C437371822150AE79EC2EB2D1F4501D9F9DE15D6AFB0A020401 91FB4474DF088AD245265B73F5D01108DA4F89694557807EBBBA6894D2A75F29 40B052859CDF3E69052CFA2E1A60DB51145C7C91FEB64F82DEFBDE21431B128C 2C47AABF3DE42AF31A0A977FCE870411B852FEC674AC91836E6C7BC947E9F2F7 B892D6DADD5DC5AF9B0D89FC3393320FD920137E8771E2A9DF6698A124EAE663 AF4DCAFE6C8BBD41B20981AC9D5370312B74D7D920AC44FD3CBB034EBCFC5FB0 0F23871CFB51D10A2DB8B775F3145C9BA49285E53CC93213B5E698DD7F10411E 8E596530248F97E754FDF8304EBE3D58F1AF15AEB50B7B369DA5CE886F342E45 207D846E35C4D55EA2AA2319CB7D09221FBAAE80F576FD9E69C54059BEC406D8 0244E1CFD7156D5B6D114B6388B68575551589B02616808AD262898ACC8B3A93 2FEC34EC6FADD13EB1AFBE6772DA17EE6CABA32AF3572817B998983636139B5D C0110CCF3087093A702C2EA6CB9D7BE0FB016D9F1A2EFAD44DCCA933420FE9B8 EFA0C90008513F3206C9ACE118D036D0DD35E7B5DB673666866C65AE3ED03328 1F24A5FDA73D63E1631A6D3160387F666BA54CEF1F42231A6BEF36C7BA9DFA09 3166E4C002FBC9C94AAB45C795F8535DF417ACDBA497D43E649F510327671D0A BCA6EDDA0B440B4B7B5FB5921D6527DF0AD11E05526CD5952C250F7757094B16 75BC2EA8BC166FB0DA505534474FC52DAFFE66F893A9F8CE82B78E816ED09B7E 9EBF2344CF8C1B234BEF12B08FB73458BD0AC43FB72B4E187AE8F9CBBC22A143 29B1B8D47BFD04CEB88A115AC4B1E03903863E2AFDE294CF0C364C5D6CAE12BD 4D1A7C1C8676B5BD10837F4933EE4964B2355C19D8943F14072D2115C7F89269 C89B5A78118DCC52E229700B5BAB069AFB355D7697B1060FB021BFE69B2483F5 15492E4C5833D6248D8F9FBFB60B21A0041228D5F32B2F45E96B0B21FA227291 B09B479FDC96D54056608F6F95FE714E5FB1F923665FF80B542B7712C6027FE6 A29B25D6E57FC50267F0861BFA442CC93C0889E0AA2144D9A402A9C2E743975D B27E647E2DB3DDD787ED7E05E275FF5DBF648BFBADB4BD166ABFE369057641A2 FAA937D6852D2113A56ABBE374D225555474893E89511A8A0E600698639BBBE7 665B325625E4ACFF9EA516555D24D2B990710B0760678D5246685575DDBC7D47 746D4C5FE2633267ED07C008A106CBEE574FABFD56DF569E5C52B00F81D52461 AF5FC0A417371272B2847FB099D6E67188CCA4BE585CB6E3EEC391782B286327 F7406F9715AC109C863A10DE205CD37C1EB455E526ECCF76D929CC09F3799395 B2A1F3CA2534817DE38A3DD2F4A924FB2CE2C5F44D0EE76589D73A98C763912D E84F7CCE155AD232149860FB076EEE11EA32C9F30B1B0966167BD629535628A6 119462D043F891E63B91C2E6E1278D69C1113A4982AC812A4B46939D09CA9A6D 7C0AF3B93CC78CFB491A75E87004EA06F051FFD58F1DF49B029A476C94899760 C27707A884474FF127151D7DE7DD76B45C22A6794100CE580AE4E8E25E263CDB F59B2ADFE49E8300C49D59DC4709E50DC87174BFEF274DED64D21F6CE301C4FB 582CE3D97A235720C925ACC1C70B5F559D00385D3AAB27BB82CBB137A53FB671 EA1FD8E8CA42F2435B53284F90FD7F93B32DD3668AA6DB873279AF7A4C8534A2 4CBBE8A8F6A16A1922626040B77526AF7D5BFF6FCDAB7379F30A2A6087A1438E 010F1964B21F0F079AD2547C005D1740DE05C3CB65E5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 98 /hatwide put dup 101 /tildewide put dup 102 /tildewider put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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. 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%%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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%%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 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American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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%%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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y(the)k(rotation)f(v)n(ector)g(as)g (\()p Fw(!)s Fx(;)14 b Fy(0)p FG(\))47 b FA(\021)f FG(\()p Fx(!)1442 4653 y Fz(1)1479 4641 y Fx(;)14 b(:)g(:)g(:)g(;)g(!)1716 4653 y Fu(r)1752 4641 y Fx(;)g FG(0)p Fx(;)g(:)g(:)g(:)f(;)h FG(0\))42 b(and)g(parameterize)e(the)i(in)n(v)-5 b(arian)n(t)41 b(torus)g(for)118 4747 y Fx(")e FG(=)g(0)e(with)h(the)g(action)f(v)-5 b(alue)37 b Fy(I)j FG(=)f Fy(0)e FG(then,)k(after)c(translating)f(the)i (origin)e(in)i Fv(R)2910 4706 y Fu(d)2986 4747 y FG(b)n(y)f(\()p Fw(!)s Fx(;)14 b Fy(0)p FG(\))38 b(and)f(set-)118 4854 y(ting)28 b Fy(I)23 b FG(=)g(\()p Fy(A)p Fx(;)14 b Fy(B)p FG(\))23 b FA(2)h Fv(R)843 4812 y Fu(r)899 4854 y FA(\002)18 b Fv(R)1046 4812 y Fu(s)1082 4854 y Fx(;)c Fw(')23 b FG(=)g(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))23 b FA(2)g Fv(T)1675 4812 y Fu(r)1730 4854 y FA(\002)18 b Fv(T)1873 4812 y Fu(s)1909 4854 y FG(,)27 b(the)h(Hamiltonian)g (\(1.1\))f(b)r(ecomes)1121 5076 y FA(H)d FG(=)f Fw(!)e FA(\001)d Fy(A)h FG(+)1609 5020 y(1)p 1609 5057 V 1609 5133 a(2)1660 5076 y Fy(A)g FA(\001)g Fy(A)f FG(+)1976 5020 y(1)p 1976 5057 V 1976 5133 a(2)2028 5076 y Fy(B)g FA(\001)g Fy(B)h FG(+)f Fx("f)9 b FG(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))p Fx(;)832 b FG(\(1)p Fx(:)p FG(2\))118 5291 y(where)22 b(\()p Fw(\013)p Fx(;)14 b Fy(A)p FG(\))24 b FA(2)f Fv(T)751 5250 y Fu(r)796 5291 y FA(\002)8 b Fv(R)933 5250 y Fu(r)993 5291 y FG(and)22 b(\()p Fw(\014)s Fx(;)14 b Fy(B)p FG(\))23 b FA(2)g Fv(T)1537 5250 y Fu(s)1581 5291 y FA(\002)8 b Fv(R)1718 5250 y Fu(s)1776 5291 y FG(are)21 b(conjugate)h(v)-5 b(ariables,)22 b(with)h Fx(r)10 b FG(+)e Fx(s)24 b FG(=)e Fx(d)p FG(,)i(and)e FA(\001)h FG(denotes)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1151 b FG(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 118 356 a FG(2:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(the)c(inner)f(pro)r(duct)h(b)r (oth)g(in)g Fv(R)1143 514 y Fu(r)1207 555 y FG(and)g(in)f Fv(R)1530 514 y Fu(s)1566 555 y FG(.)37 b(Here)27 b(w)n(e)g(imp)r(ose)h (that)f Fw(!)k FG(is)c(a)h(v)n(ector)e(in)i Fv(R)3057 514 y Fu(r)3122 555 y FG(satisfying)1211 752 y FA(j)p Fw(!)21 b FA(\001)d Fw(\027)6 b FA(j)23 b(\025)g Fx(C)1603 764 y Fz(0)1640 752 y FA(j)p Fw(\027)6 b FA(j)1740 718 y FC(\000)p Fu(\034)1823 726 y Ft(0)2025 752 y FA(8)14 b Fw(\027)29 b FA(2)23 b Fv(Z)2300 711 y Fu(r)2355 752 y FA(n)18 b(f)p Fy(0)p FA(g)p Fx(;)921 b FG(\(1)p Fx(:)p FG(3\))118 949 y(with)27 b Fx(C)365 961 y Fz(0)425 949 y Fx(>)c FG(0)j(and)g Fx(\034)777 961 y Fz(0)837 949 y FA(\025)d Fx(r)18 b FA(\000)d FG(1,)26 b(whic)n(h)h(is)f(called)f (the)i FF(Diophantine)j(c)l(ondition)p FG(;)e(w)n(e)d(shall)h(de\014ne) g(b)n(y)g Fx(D)3433 961 y Fu(\034)3464 969 y Ft(0)3500 949 y FG(\()p Fx(C)3591 961 y Fz(0)3629 949 y FG(\))118 1055 y(the)i(set)g(of)f(rotation)g(v)n(ectors)f(in)i Fv(R)1247 1014 y Fu(r)1312 1055 y FG(satisfying)f(\(1.3\).)36 b(W)-7 b(e)28 b(also)f(write)1408 1261 y Fx(f)9 b FG(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))23 b(=)1808 1182 y Fr(X)1791 1357 y Fq(\027)5 b FC(2)p Fp(Z)1912 1340 y Fs(r)1959 1261 y Fx(e)1998 1227 y Fu(i)p Fq(\027)g FC(\001)p Fq(\013)2138 1261 y Fx(f)2179 1273 y Fq(\027)2226 1261 y FG(\()p Fw(\014)s FG(\))p Fx(:)1120 b FG(\(1)p Fx(:)p FG(4\))118 1526 y(W)-7 b(e)32 b(shall)f(supp)r(ose)h(that)g Fx(f)40 b FG(is)31 b(analytic)g(in)h(a)f(strip)h(around)e(the)i(real)f (axis)g(of)h(the)g(v)-5 b(ariables)30 b Fw(\013)p Fx(;)14 b Fw(\014)s FG(,)32 b(so)f(that)118 1632 y(there)24 b(exist)h(constan)n (ts)f Fx(F)939 1644 y Fz(0)976 1632 y Fx(;)14 b(F)1066 1644 y Fz(1)1104 1632 y Fx(;)g(\024)1189 1644 y Fz(0)1250 1632 y FG(suc)n(h)25 b(that)f FA(j)p Fx(@)1683 1592 y Fu(q)1678 1663 y Fq(\014)1729 1632 y Fx(f)1770 1644 y Fq(\027)1817 1632 y FG(\()p Fw(\014)s FG(\))p FA(j)f(\024)f Fx(q)s FG(!)p Fx(F)2188 1644 y Fz(0)2226 1632 y Fx(F)2291 1592 y Fu(q)2279 1654 y Fz(1)2328 1632 y Fx(e)2367 1602 y FC(\000)p Fu(\024)2458 1610 y Ft(0)2490 1602 y FC(j)p Fq(\027)5 b FC(j)2601 1632 y FG(for)24 b(all)g Fw(\027)29 b FA(2)23 b Fv(Z)3052 1591 y Fu(r)3113 1632 y FG(and)i(all)f Fw(\014)i FA(2)d Fv(T)3603 1591 y Fu(s)3638 1632 y FG(.)189 1738 y(There)i(are)g(quite)h(a)g(few)g(results)f(on)g(the)i(ab)r(o)n(v) n(e)d(problem,)i(essen)n(tially)e(solv)n(ed,)i(under)f(the)h (assumptions)g(of)118 1845 y(Theorem)20 b(1)g(b)r(elo)n(w,)h(in)g(Ref.) 35 b([JLZ],)20 b(and)h(on)f(closely)f(related)h(problems.)34 b(W)-7 b(e)21 b(summarize)e(our)h(understanding)118 1951 y(of)28 b(the)g(existing)f(results)g(in)h(App)r(endix)g(A1.)189 2057 y(The)g(equations)e(of)i(motion)f(for)g(the)h(system)g(\(1.2\),)f (written)h(in)g(terms)f(of)h(the)g(angle)e(v)-5 b(ariables)27 b(alone,)g(are)1145 2253 y(\177)1134 2254 y Fw(\013)c FG(=)g FA(\000)p Fx("@)1456 2266 y Fq(\013)1510 2254 y Fx(f)9 b FG(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))p Fx(;)1993 2232 y FG(\177)1985 2254 y Fw(\014)25 b FG(=)e FA(\000)p Fx("@)2301 2271 y Fq(\014)2351 2254 y Fx(f)9 b FG(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))p Fx(;)845 b FG(\(1)p Fx(:)p FG(5\))118 2450 y(so)23 b(that,)h(once)f(a)g (solution)g(of)g(\(1.5\))g(is)g(found,)i(the)f(action)e(v)-5 b(ariables)23 b(are)f(immediately)h(obtained)g(b)n(y)g(a)g(simple)118 2556 y(di\013eren)n(tiation:)37 b Fy(A)23 b FG(=)889 2555 y(_)869 2556 y Fw(\013)18 b FA(\000)g Fw(!)s FG(,)28 b Fy(B)23 b FG(=)1342 2535 y(_)1325 2556 y Fw(\014)s FG(.)189 2663 y(W)-7 b(e)31 b(lo)r(ok)f(for)g(solutions)g(of)g (\(1.5\),)h(for)f Fx(")e FA(6)p FG(=)f(0,)k(conjugated)f(to)h(the)g (free)f(solution)g(\()p Fw(\013)2955 2675 y Fz(0)3013 2663 y FG(+)20 b Fx(!)s(t;)14 b Fw(\014)3275 2675 y Fz(0)3312 2663 y Fx(;)g FG(0)p Fx(;)g FG(0\),)30 b(i.e.)118 2769 y(w)n(e)d(lo)r(ok)g(for)g(solutions)g(of)h(the)g(form)946 2966 y Fw(\013)p FG(\()p Fx(t)p FG(\))c(=)e Fw( )g FG(+)c Fy(a)p FG(\()p Fw( )s Fx(;)c Fw(\014)1618 2978 y Fz(0)1655 2966 y FG(;)g Fx(")p FG(\))p Fx(;)180 b Fw(\014)s FG(\()p Fx(t)p FG(\))23 b(=)g Fw(\014)2284 2978 y Fz(0)2339 2966 y FG(+)c Fy(b)p FG(\()p Fw( )s Fx(;)14 b Fw(\014)2666 2978 y Fz(0)2703 2966 y FG(;)g Fx(")p FG(\))p Fx(;)657 b FG(\(1)p Fx(:)p FG(6\))118 3162 y(for)28 b(some)g(functions)h Fy(a)g FG(and)f Fy(b)p FG(,)h(real)f(analytic)f(and)i(2)p Fx(\031)s FG(-p)r(erio)r(dic)f(in)g Fw( )g FA(2)d Fv(T)2565 3121 y Fu(r)2601 3162 y FG(,)k(suc)n(h)f(that)h(the)g(motion)f(in)h (the)118 3268 y(v)-5 b(ariable)27 b Fw( )j FG(is)e(go)n(v)n(erned)d(b)n (y)j(the)g(equation)1574 3246 y(_)1553 3268 y Fw( )e FG(=)c Fw(!)s FG(.)37 b(W)-7 b(e)28 b(shall)f(pro)n(v)n(e)f(the)i (follo)n(wing)f(result.)118 3446 y Fy(Theorem)35 b(1.)45 b FF(Consider)34 b(the)e(Hamiltonian)i(\(1.2\),)h(with)e Fw(!)d FA(2)f Fx(D)2289 3458 y Fu(\034)2320 3466 y Ft(0)2356 3446 y FG(\()p Fx(C)2447 3458 y Fz(0)2484 3446 y FG(\))k FF(and)g Fx(f)41 b FF(analytic)34 b(and)f(p)l(erio)l(dic)h(in)118 3552 y(b)l(oth)c(variables.)41 b(Supp)l(ose)30 b Fw(\014)1044 3564 y Fz(0)1111 3552 y FF(to)f(b)l(e)h(such)g(that)1632 3748 y Fx(@)1676 3766 y Fq(\014)1727 3748 y Fx(f)1768 3760 y Fo(0)1809 3748 y FG(\()p Fw(\014)1896 3760 y Fz(0)1933 3748 y FG(\))24 b(=)e Fy(0)p Fx(;)1344 b FG(\(1)p Fx(:)p FG(7\))118 3945 y FF(and)23 b(assume)e(that)h(the)h(eigenvalues)g Fx(a)1318 3957 y Fz(1)1355 3945 y Fx(;)14 b(:)g(:)g(:)f(;)h(a)1583 3957 y Fu(s)1641 3945 y FF(of)22 b(the)h(matrix)e Fx(@)2168 3915 y Fz(2)2163 3974 y Fq(\014)2214 3945 y Fx(f)2255 3957 y Fo(0)2296 3945 y FG(\()p Fw(\014)2383 3957 y Fz(0)2420 3945 y FG(\))i FF(ar)l(e)f(p)l(airwise)i(distinct)e(and)h(p)l(ositive,) 118 4051 y(i.e.)40 b(for)30 b(some)h(c)l(onstant)e Fx(a)23 b(>)f FG(0)29 b FF(one)h(has)h Fx(a)1522 4063 y Fu(i)1549 4051 y Fx(;)d(a)1644 4063 y Fu(j)1698 4051 y FA(\000)18 b Fx(a)1825 4063 y Fu(i)1889 4051 y Fx(>)36 b(a)h(>)g FG(0)29 b FF(for)h(al)t(l)h Fx(j)d(>)23 b(i)g FG(=)f(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(s)p FF(.)118 4157 y(Then)31 b(ther)l(e)e(exist)h(a)g(c)l(onstant)p 1138 4112 39 4 v 29 w Fx(")23 b(>)f FG(0)30 b FF(and)g(a)g(set)f FA(E)i(\032)22 b FG(\(0)p Fx(;)p 1994 4112 V 14 w(")p FG(\))30 b FF(such)f(that)h(the) g(fol)t(lowing)i(holds.)118 4264 y(\(i\))d(F)-6 b(or)29 b(al)t(l)h Fx(")23 b FA(2)g(E)37 b FF(ther)l(e)29 b(ar)l(e)g(solutions) g(of)g(\(1.5\))i(of)e(the)g(form)h(\(1.6\),)h(wher)l(e)e(the)g(two)g (functions)g Fy(a)p FG(\()p Fw( )s Fx(;)14 b Fw(\014)3516 4276 y Fz(0)3554 4264 y FG(;)g Fx(")p FG(\))118 4370 y FF(and)30 b Fy(b)p FG(\()p Fw( )s Fx(;)14 b Fw(\014)522 4382 y Fz(0)560 4370 y FG(;)g Fx(")p FG(\))29 b FF(ar)l(e)h(r)l(e)l(al) g(analytic)h(and)g FG(2)p Fx(\031)s FF(-p)l(erio)l(dic)g(in)f(the)g (variables)h Fw( )c FA(2)c Fv(T)2716 4329 y Fu(r)2753 4370 y FF(.)118 4476 y(\(ii\))31 b(The)f(r)l(elative)h(L)l(eb)l(esgue)f (me)l(asur)l(e)f(of)i FA(E)26 b(\\)18 b FG(\(0)p Fx(;)c(")p FG(\))30 b FF(with)g(r)l(esp)l(e)l(ct)g(to)f FG(\(0)p Fx(;)14 b(")p FG(\))30 b FF(tends)f(to)h FG(1)f FF(for)i Fx(")23 b FA(!)g FG(0)p FF(.)118 4583 y(\(iii\))31 b(The)g(functions)f Fy(a)p Fx(;)14 b Fy(b)30 b FF(c)l(an)f(b)l(e)h(extende)l(d)g(to)g (Lipschitz)h(functions)e(of)i Fx(";)14 b Fw( )32 b FF(in)e FG([0)p Fx(;)p 2865 4537 V 14 w(")o FG(])19 b FA(\002)f Fv(T)3088 4541 y Fu(r)3124 4583 y FF(.)118 4760 y(R)l(emarks.)77 b FG(\(1\))41 b(F)-7 b(rom)40 b(the)i(literature)e(one)g(migh)n(t)h (exp)r(ect)g(that)g(the)g(non-resonance)e(condition)i(on)f(the)118 4866 y(eigen)n(v)-5 b(alues)27 b(of)g Fx(@)686 4884 y Fq(\014)736 4866 y Fx(f)777 4878 y Fo(0)819 4866 y FG(\()p Fw(\014)906 4878 y Fz(0)943 4866 y FG(\))h(could)f(b)r(e)h(a)n(v)n (oided;)f(see)g(App)r(endix)h(A1.)118 4972 y(\(2\))22 b(The)g(case)f(of)h(negativ)n(e)e Fx(")i FG(w)n(as)f(dealt)h(with)g(in) g(Ref.)35 b([GG],)23 b(with)f(tec)n(hniques)g(close)f(to)g(the)i(ones)e (in)n(tro)r(duced)118 5079 y(here,)27 b(and)h(it)g(corresp)r(onds)d(to) j(the)g(case)f(of)g(h)n(yp)r(erb)r(olic)g(tori.)118 5185 y(\(3\))34 b(The)g(case)e(of)i(mixed)g(stationarit)n(y)-7 b(,)34 b(i.e.)55 b(det)14 b Fx(@)1781 5155 y Fz(2)1776 5214 y Fq(\014)1826 5185 y Fx(f)1867 5197 y Fo(0)1909 5185 y FG(\()p Fw(\014)1996 5197 y Fz(0)2033 5185 y FG(\))33 b FA(6)p FG(=)g(0)g(and)h(eigen)n(v)-5 b(alues)32 b(of)i Fx(@)3024 5155 y Fz(2)3019 5214 y Fq(\014)3069 5185 y Fx(f)3110 5197 y Fo(0)3151 5185 y FG(\()p Fw(\014)3238 5197 y Fz(0)3276 5185 y FG(\))g(of)f(mixed)118 5291 y(signs)24 b(\(with)i(non-degeneracy)c(of)j(the)g(p)r(ositiv)n(e)f(ones\),)h(can)g (b)r(e)g(treated)f(in)h(exactly)f(the)h(same)f(w)n(a)n(y)g(discussed) 118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1151 b FG(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 118 356 a FG(3:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(in)c(this)g(pap)r(er)f(and)g (the)h(ab)r(o)n(v)n(e)f(result)g(extends)h(to)f(this)h(case;)f(cf.)37 b(Theorem)27 b(2)g(in)h(Section)f(7.)118 662 y(\(4\))32 b(F)-7 b(or)32 b Fx(")e FA(62)h(E)40 b FG(the)32 b(smo)r(oth)g (extension)g(in)g(\(iii\))h(do)r(es)f(not)g(represen)n(t)f(parametric)g (equations)h(of)g(in)n(v)-5 b(arian)n(t)118 768 y(tori:)54 b(it)37 b(just)h(sa)n(ys)d(that)i(their)f(v)-5 b(alues)37 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y(prop)r(er)g(frequencies,)i(i.e.)55 b(the)34 b(comp)r(onen)n(ts)f(of)h Fw(!)s FG(,)h(and)e(the)h(normal)f(frequencies,)i(i.e.)55 b(the)34 b(square)f(ro)r(ots)118 2404 y(of)k(the)g(eigen)n(v)-5 b(alues)35 b(of)i Fx("@)1004 2374 y Fz(2)999 2433 y Fq(\014)1049 2404 y Fx(f)1090 2416 y Fo(0)1131 2404 y FG(\()p Fw(\014)1218 2416 y Fz(0)1256 2404 y FG(\),)i(are)d(still)h(irrelev)-5 b(an)n(t.)63 b(The)37 b(analysis)e(of)i(suc)n(h)f(singularities)g (leads)g(to)118 2510 y(what)30 b(w)n(e)g(call)g FF(non-r)l(esonant)g FG(or)g FF(high)j(fr)l(e)l(quency)g(r)l(esummations)p FG(,)e(whic)n(h)f(can)g(b)r(e)h(treated)f(b)n(y)g(the)h(metho)r(d)118 2617 y(of)36 b(Ref.)64 b([GG],)37 b(i.e.)63 b(of)37 b(the)g(h)n(yp)r (erb)r(olic)e(case,)j(in)f(whic)n(h)f(no)g(resonances)f(at)h(all)g(w)n (ere)f(p)r(ossible)i(b)r(et)n(w)n(een)118 2723 y(prop)r(er)32 b(frequencies)h(and)g(normal)f(frequencies)h(\(simply)h(b)r(ecause,)g (for)f(the)g(Hamiltonian)g(\(1.2\))g(the)h(latter)118 2829 y(did)26 b(not)g(exist\).)36 b(F)-7 b(urther)26 b(probing)f(of)g(the)h(singularities)f(leads)g(to)g(what)h(w)n(e)f (call)h(the)g FF(r)l(esonant)f FG(\(or)g FF(infr)l(ar)l(e)l(d)p FG(\))118 2936 y FF(r)l(esummations)p FG(:)42 b(the)31 b(analysis)e(is)i(more)e(elab)r(orated)h(and)g(it)h(requires)e(new)h (ideas,)h(obtained)f(b)n(y)g(com)n(bining)118 3042 y(the)e(ideas)f(in)h (Ref.)37 b([GG])28 b(with)h(the)e(ones)g(in)n(tro)r(duced)h(in)g(Ref.) 37 b([Ge].)189 3150 y(In)e(Section)h(5)e(w)n(e)h(discuss)g(the)h (non-resonan)n(t)d(resummations)h(while)h(the)h(new)f(infrared)g (resummations)118 3256 y(are)26 b(studied)h(in)h(Section)f(6)f(where)h (a)f(\\fully)h(renormalized)f(series")f(is)i(obtained,)g(i.e.)36 b(a)27 b(resummation)f(of)h(the)118 3362 y(series)e(de\014ning)h(the)g (formal)f(expansion)g(of)h(the)h(quasi-p)r(erio)r(dic)d(solution)i (\(1.6\))g(of)f(the)i(equations)e(of)h(motion)118 3469 y(\(1.5\),)i FF(which)j(we)f(pr)l(ove)h(to)e(b)l(e)h(absolutely)h(c)l (onver)l(gent)p FG(.)189 3576 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y Fk(=)h(2)p Fj(;)12 b(p)2282 2716 y Fi(v)2321 2726 y Ft(1)2391 2708 y Fk(=)33 b(2)p Fj(;)12 b(p)2582 2716 y Fi(v)2622 2726 y Ft(2)2692 2708 y Fk(=)33 b(3)p Fj(;)12 b(p)2883 2716 y Fi(v)2922 2726 y Ft(3)2992 2708 y Fk(=)472 2783 y(2)p Fj(;)g(p)575 2791 y Fi(v)615 2801 y Ft(4)674 2783 y Fk(=)22 b(2.)36 b(The)25 b(length)h(of)f(the)i(lines)d(should)i(b)r(e)f(the)h (same)g(but)g(it)f(is)f(dra)n(wn)i(of)f(arbitrary)472 2857 y(size.)37 b(The)26 b(endno)r(des)h Fm(v)1139 2867 y Fh(i)1165 2857 y Fk(,)e Fj(i)e Fk(=)g(5)p Fj(;)11 b(:)h(:)f(:)g(;)h Fk(11)26 b(can)g(b)r(e)g(either)g(no)r(des)g(or)f(lea)n(v)n(es)h(of)f (the)i(tree.)37 b(The)472 2932 y(separated)25 b(line)f(illustrates)f (the)i(w)n(a)n(y)f(to)g(think)g(of)f(the)i(lab)r(el)f Fj(\021)e Fk(=)d(\()p Fj(\015)2358 2909 y Fg(0)2382 2932 y Fj(;)11 b(\015)t Fk(\).)189 3145 y FG(A)29 b(tree)f Fx(\022)j FG(\(see)d(Fig.)39 b(1\))28 b(is)h(de\014ned)f(as)g(a)g (partially)f(ordered)h(set)g(of)g(p)r(oin)n(ts,)h(connected)f(b)n(y)g (orien)n(ted)g 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Fz(2)3528 3482 y FG(and)118 3588 y(en)n(ters)i Fn(v)408 3600 y Fz(1)445 3588 y FG(.)37 b(More)26 b(generally)g(w)n(e)h(write)g Fn(v)1453 3600 y Fz(2)1513 3588 y FA(\036)c Fn(v)1648 3600 y Fz(1)1713 3588 y FG(if)k Fn(v)1835 3600 y Fz(2)1900 3588 y FG(is)g(on)g(the)h(path)f(of)h(lines)f(connecting)g Fn(v)3180 3600 y Fz(1)3244 3588 y FG(to)h(the)f(ro)r(ot.)118 3695 y(The)h(p)r(oin)n(ts)f(di\013eren)n(t)h(from)f(the)h(ro)r(ot)f (will)h(b)r(e)g(called)f(the)h FF(no)l(des)g FG(of)g(the)g(tree.)189 3844 y(Eac)n(h)e(line)h(from)g Fn(v)g FG(to)g Fn(v)969 3814 y FC(0)1019 3844 y FG(carries)e(a)i(pair)f Fx(\021)k FG(of)d(lab)r(els)g Fx(\021)f FG(=)d(\()p Fx(\015)5 b(;)14 b(\015)2239 3814 y FC(0)2262 3844 y FG(\))27 b(ranging)f(in)h FA(f)p FG(1)p Fx(;)14 b(:)g(:)g(:)e(;)i(d)p FA(g)27 b FG(\(mark)n(ed)f(in)i(Fig.)118 3950 y(1)g(only)f(on)h(some)f(of)h(the)g (lines)g(for)f(clarit)n(y)g(of)h(the)g(dra)n(wing\).)37 b(The)27 b(lab)r(els)h Fx(\015)33 b FG(and)27 b Fx(\015)2817 3920 y FC(0)2868 3950 y FG(should)h(b)r(e)g(regarded)e(as)118 4056 y(asso)r(ciated)k(with)h Fn(v)h FG(and)f Fn(v)999 4026 y FC(0)1023 4056 y FG(,)h(resp)r(ectiv)n(ely;)g(hence)f(with)h (eac)n(h)e(no)r(de)h Fn(v)g FG(with)h Fx(p)2693 4068 y Fm(v)2770 4056 y FG(en)n(tering)f(lines)g Fx(`)3323 4068 y Fz(1)3360 4056 y Fx(;)14 b(:)g(:)g(:)f(;)h(`)3579 4068 y Fu(p)3613 4076 y Fi(v)118 4163 y FG(one)24 b(can)f(asso)r(ciate) f Fx(p)802 4175 y Fm(v)860 4163 y FG(+)11 b(1)22 b(lab)r(els)i Fx(\015)1273 4175 y Fz(0)1310 4163 y Fx(;)14 b(\015)1390 4175 y Fz(1)1427 4163 y Fx(;)g(:)g(:)g(:)g(;)g(\015)1655 4175 y Fu(p)1689 4183 y Fi(v)1738 4163 y FG(,)24 b(with)h Fx(\015)2014 4175 y Fz(0)2074 4163 y FG(=)e Fx(\015)2205 4175 y Fu(`)2233 4183 y Fi(v)2305 4163 y FG(and)g Fx(\015)2505 4175 y Fu(j)2563 4163 y FG(=)g Fx(\015)2699 4133 y FC(0)2694 4186 y Fu(`)2722 4194 y Fs(i)2752 4163 y FG(.)36 b(Also)23 b(the)i(ro)r(ot)e(line)h(\(from)118 4269 y Fn(v)165 4281 y Fz(0)233 4269 y FG(to)31 b(the)h(ro)r(ot\))f(carries)e(t)n(w)n(o)h (suc)n(h)h(lab)r(els)g(and)g(the)g(one)g(asso)r(ciated)f(with)i(the)f (\014nal)g(extreme)g(of)g(the)g(ro)r(ot)118 4375 y(line)d(will)g(b)r(e) g(called)f(the)h FF(r)l(o)l(ot)i(lab)l(el)p FG(.)189 4525 y(Fixed)d(an)n(y)f Fx(`)611 4537 y Fm(v)680 4525 y FA(2)e Fx(\022)r FG(,)j(w)n(e)g(shall)f(sa)n(y)g(that)h(the)g(subset) g(of)f Fx(\022)j FG(con)n(taining)d Fx(`)2482 4537 y Fm(v)2555 4525 y FG(as)h(w)n(ell)f(as)g(all)h(no)r(des)f Fn(w)d FA(\026)g Fn(v)k FG(and)118 4631 y(all)g(lines)h(connecting)f (them)h(is)g(a)f FF(subtr)l(e)l(e)g FG(of)g Fx(\022)j FG(with)e(ro)r(ot)f Fn(v)2055 4601 y FC(0)2078 4631 y FG(;)h(of)g(course)e(a)h(subtree)h(is)f(a)g(tree.)189 4780 y(Giv)n(en)i(a)g(tree,)h(with)g(eac)n(h)f(no)r(de)h Fn(v)f FG(w)n(e)g(asso)r(ciate)f(a)h FF(harmonic)j FG(or)c FF(mo)l(de)p FG(,)j(as)e(called)g(in)h(Ref.)43 b([GG],)30 b(i.e.)43 b(a)118 4886 y(lab)r(el)26 b Fw(\027)365 4898 y Fm(v)435 4886 y FA(2)d Fv(Z)573 4845 y Fu(r)610 4886 y FG(.)36 b(W)-7 b(e)26 b(shall)f(denote)h(b)n(y)f Fx(V)19 b FG(\()p Fx(\022)r FG(\))27 b(the)f(set)g(of)f(no)r(des)h(and)g(b)n(y) f(\003\()p Fx(\022)r FG(\))h(the)g(set)g(of)g(lines.)36 b(The)26 b(n)n(um)n(b)r(er)118 4993 y Fx(k)g FG(=)d FA(j)p Fx(V)c FG(\()p Fx(\022)r FG(\))p FA(j)27 b FG(of)g(no)r(des)f(in)h(the) g(tree)f Fx(\022)r FG(,)h(equal)g(to)f(the)h(n)n(um)n(b)r(er)f FA(j)p FG(\003\()p Fx(\022)r FG(\))p FA(j)i FG(of)e(lines,)h(will)g(b)r (e)g(called)f(the)h FF(or)l(der)h FG(of)e Fx(\022)r FG(.)189 5142 y(W)-7 b(e)28 b(call)f(a)g(no)r(de)h(with)g(one)f(en)n(tering)g (line)h(and)f Fy(0)h FG(harmonic)e(lab)r(el)i(a)f FF(trivial)k(no)l(de) p FG(.)189 5291 y(With)38 b(an)n(y)e(line)h Fx(`)i FG(=)f Fx(`)957 5303 y Fm(v)1041 5291 y FG(w)n(e)e(asso)r(ciate)g(\(b)r (esides)h(the)h(ab)r(o)n(v)n(e)e(men)n(tioned)h(pair)f Fx(\021)2889 5303 y Fu(`)2960 5291 y FG(=)i(\()p Fx(\015)3138 5303 y Fu(`)3170 5291 y Fx(;)14 b(\015)3255 5261 y FC(0)3250 5315 y Fu(`)3282 5291 y FG(\))38 b(of)f(lab)r(els)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1151 b FG(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 118 356 a FG(5:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(assuming)27 b(v)-5 b(alues)27 b(in)h FA(f)p FG(1)p Fx(;)14 b(:)g(:)g(:)e(;)i(d)p FA(g)p FG(\))28 b(a)f FF(momentum)i(lab)l(el)g Fw(\027)1970 567 y Fu(`)2025 555 y FA(2)23 b Fv(Z)2163 514 y Fu(r)2227 555 y FG(de\014ned)28 b(as)1479 767 y Fw(\027)1527 779 y Fu(`)1582 767 y FA(\021)23 b Fw(\027)1718 779 y Fu(`)1746 796 y Fm(v)1819 767 y FG(=)1968 688 y Fr(X)1917 863 y Fm(w)p Ff(2)p Fs(V)13 b Ft(\()p Fs(\022)q Ft(\))1953 920 y Fm(w)p Ff(\026)p Fm(v)2163 767 y Fw(\027)2211 779 y Fm(w)2277 767 y Fx(:)1191 b FG(\(2)p Fx(:)p FG(2\))118 1099 y FF(We)30 b(shal)t(l)g(assume)g(that)f(no)h(tr)l(e)l(e)e(c)l (ontains)i(trivial)h(no)l(des)e(with)h(the)g(entering)f(line)h(with)g Fy(0)f FF(momentum)p FG(:)36 b(this)118 1205 y(is)e(an)g(imp)r(ortan)n (t)g(restriction,)h(as)f(w)n(e)g(shall)g(see.)56 b(W)-7 b(e)35 b(call)f FF(de)l(gr)l(e)l(e)g Fx(P)12 b FG(\()p Fx(\022)r FG(\))35 b(of)g(a)e(tree)h(the)h(order)e(of)h(the)h(tree)118 1311 y(min)n(us)28 b(the)g(n)n(um)n(b)r(er)f(of)h Fy(0)f FG(momen)n(tum)h(lines,)g(so)e(that)i FA(j)p Fx(V)19 b FG(\()p Fx(\022)r FG(\))p FA(j)h(\000)e Fx(P)12 b FG(\()p Fx(\022)r FG(\))28 b(is)f(their)h(n)n(um)n(b)r(er.)189 1417 y(W)-7 b(e)25 b(call)e(\002)542 1429 y Fq(\027)5 b Fu(;k)q(;\015)728 1417 y FG(the)24 b(set)g(of)g(trees)g Fx(\022)i FG(of)f(order)d Fx(k)s FG(,)j(i.e.)36 b(with)25 b FA(j)p Fx(V)19 b FG(\()p Fx(\022)r FG(\))p FA(j)24 b FG(=)e Fx(k)27 b FG(no)r(des,)e(and)f(\002)2948 1387 y Fu(o)2948 1441 y Fq(\027)5 b Fu(;k)q(;\015)3133 1417 y FG(the)25 b(set)f(of)g(trees)118 1524 y(of)k(degree)e Fx(k)s FG(,)i(i.e.)37 b(with)28 b Fx(P)12 b FG(\()p Fx(\022)r FG(\))24 b(=)e Fx(k)s FG(.)37 b(One)27 b(has)g(\002)1675 1536 y Fq(\027)5 b Fu(;k)q(;\015)1860 1524 y FA(6)p FG(=)22 b(\002)2012 1494 y Fu(o)2012 1547 y Fq(\027)5 b Fu(;k)q(;\015)2173 1524 y FG(.)189 1630 y(Eac)n(h)21 b(tree)h Fx(\022)j FG(\\decorated")c(b)n(y)h(lab)r(els)g(in)h(the)g(describ)r(ed)f(w)n(a)n (y)f(will)i(ha)n(v)n(e)e(a)h FF(value)h FG(whic)n(h)g(is)f(de\014ned)h (in)g(terms)118 1736 y(of)28 b(a)f(pro)r(duct)g(of)h(sev)n(eral)e (factors.)118 1843 y FA(\017)h FG(With)i(eac)n(h)e(no)r(de)g Fn(v)h FG(w)n(e)f(asso)r(ciate)f(a)h FF(no)l(de)k(factor)1512 2044 y Fx(F)1565 2056 y Fm(v)1636 2044 y FG(=)1723 1965 y Fr(Y)1761 2142 y Fu(j)1843 2044 y Fx(@)1887 2056 y Fu(\015)1922 2064 y Fs(j)1957 2044 y Fx(f)1998 2056 y Fq(\027)2036 2073 y Fm(v)2087 2044 y FG(\()p Fw(\014)2174 2056 y Fz(0)2212 2044 y FG(\))p Fx(;)1224 b FG(\(2)p Fx(:)p FG(3\))118 2323 y(where)25 b(the)h(lab)r(els)f Fx(\015)771 2335 y Fu(j)831 2323 y FG(are)f(the)i Fx(p)1150 2335 y Fm(v)1210 2323 y FG(+)14 b(1)25 b(lab)r(els)g(asso)r(ciated)f (with)i(the)f(extreme)g Fn(v)g FG(of)h(the)f Fx(p)2968 2335 y Fm(v)3040 2323 y FG(lines)h(en)n(tering)e(the)118 2429 y(no)r(de)f Fn(v)h FG(and)f(of)g(the)h(line)f(exiting)g(it,)i(and) e(the)h(deriv)-5 b(ativ)n(es)22 b Fx(@)2045 2441 y Fu(\015)2087 2429 y FG(,)j(with)e Fx(\015)28 b FG(=)23 b(1)p Fx(;)14 b FG(2)p Fx(;)g(:)g(:)g(:)e(;)i(r)r FG(,)25 b(ha)n(v)n(e)d(to)h(b)r(e)h (in)n(terpreted)118 2536 y(as)j(factors)g(\()p Fx(i)p Fw(\027)600 2548 y Fm(v)646 2536 y FG(\))678 2548 y Fu(\015)721 2536 y FG(.)37 b(Hence)28 b Fx(F)1081 2548 y Fm(v)1156 2536 y FG(is)f(a)g(tensor)g(of)h(rank)e Fx(p)1886 2548 y Fm(v)1951 2536 y FG(+)18 b(1.)118 2642 y FA(\017)31 b FG(With)h(eac)n(h)e(line)h Fx(`)g FG(carrying)e(lab)r(els)i Fx(\021)1432 2654 y Fu(`)1492 2642 y FG(=)e(\()p Fx(\015)1661 2654 y Fu(`)1693 2642 y Fx(;)14 b(\015)1773 2654 y Fu(`)1801 2638 y Ff(0)1827 2642 y FG(\))32 b(and)e(momen)n(tum)i Fw(\027)2543 2654 y Fu(`)2606 2642 y FG(w)n(e)e(asso)r(ciate)g(a)g (matrix,)i(called)118 2748 y FF(pr)l(op)l(agator)p FG(,)901 2882 y Fx(G)966 2894 y Fu(`)1021 2882 y FA(\021)23 b Fx(\016)1146 2896 y Fu(\015)1181 2905 y Fs(`)1209 2896 y Fu(;\015)1268 2877 y Ff(0)1264 2919 y Fs(`)1438 2826 y FG(1)p 1307 2863 305 4 v 1307 2939 a(\()p Fw(!)e FA(\001)e Fw(\027)1510 2951 y Fu(`)1542 2939 y FG(\))1574 2915 y Fz(2)1621 2882 y Fx(;)746 b FG(if)35 b Fw(\027)2521 2894 y Fu(`)2576 2882 y FA(6)p FG(=)23 b Fy(0)p Fx(;)901 3066 y(G)966 3078 y Fu(`)1021 3066 y FA(\021)g(\000)p Fx(")1213 3032 y FC(\000)p Fz(1)1315 3066 y FG(\()p Fx(@)1396 3032 y Fz(2)1391 3091 y Fq(\014)1442 3066 y Fx(f)1483 3078 y Fo(0)1524 3066 y FG(\()p Fw(\014)1611 3078 y Fz(0)1648 3066 y FG(\)\))1712 3031 y FC(\000)p Fz(1)1712 3094 y Fu(\015)1747 3103 y Fs(`)1777 3094 y Fu(;\015)1836 3074 y Ff(0)1832 3116 y Fs(`)1878 3066 y Fx(\037)p FG(\()p Fx(\015)2005 3078 y Fu(`)2037 3066 y Fx(;)14 b(\015)2122 3032 y FC(0)2117 3087 y Fu(`)2172 3066 y Fx(>)23 b(r)r FG(\))p Fx(;)181 b FG(if)34 b Fw(\027)2665 3078 y Fu(`)2720 3066 y FG(=)23 b Fy(0)p Fx(;)3491 2965 y FG(\(2)p Fx(:)p FG(4\))118 3243 y(where)k Fx(\037)p FG(\()p Fx(\015)485 3255 y Fu(`)517 3243 y Fx(;)14 b(\015)602 3213 y FC(0)597 3266 y Fu(`)652 3243 y Fx(>)23 b(r)r FG(\))29 b(is)e(1)g(if)h(b)r(oth)g Fx(\015)1307 3255 y Fu(`)1367 3243 y FG(and)f Fx(\015)1576 3213 y FC(0)1571 3266 y Fu(`)1631 3243 y FG(are)g(strictly)g(greater)f (than)h Fx(r)r FG(,)i(and)e(0)h(otherwise.)189 3349 y(Giv)n(en)e(the)h (de\014nitions)f(\(2.3\))g(and)g(\(2.4\))g(de\014ne)h(a)f FF(value)j(function)d FG(V)-7 b(al,)27 b(whic)n(h)f(with)h(eac)n(h)e (tree)h Fx(\022)j FG(of)d(order)118 3456 y Fx(k)31 b FG(asso)r(ciates)26 b(a)h FF(tr)l(e)l(e)i(value)1224 3591 y FG(V)-7 b(al\()p Fx(\022)r FG(\))23 b(=)1570 3534 y Fx(")1609 3504 y Fu(k)p 1570 3571 80 4 v 1575 3647 a Fx(k)s FG(!)1660 3498 y Fr(\020)1783 3512 y(Y)1723 3694 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(\022)r Fz(\))1963 3591 y Fx(F)2016 3603 y Fu(v)2056 3498 y Fr(\021)o(\020)2218 3512 y(Y)2169 3694 y Fu(`)p FC(2)p Fz(\003\()p Fu(\022)r Fz(\))2386 3591 y Fx(G)2451 3603 y Fu(`)2483 3498 y Fr(\021)2533 3591 y Fx(;)935 b FG(\(2)p Fx(:)p FG(5\))118 3839 y(where,)25 b(b)n(y)g(the)g(de\014nitions,)g(all)g(lab)r(els)f Fx(\015)1440 3851 y Fu(i)1493 3839 y FG(asso)r(ciated)f(with)j(the)f(no)r(des)f(app) r(ear)g(t)n(wice)h(b)r(ecause)f(they)h(app)r(ear)118 3945 y(also)j(in)h(the)h(propagators:)36 b(w)n(e)29 b(mak)n(e)f(in)h (\(2.5\))g(the)g FF(summation)i(c)l(onvention)e FG(that)g(rep)r(eated)g Fx(\015)k FG(lab)r(els)c(asso-)118 4051 y(ciated)k(with)g(no)r(des)g (and)f(lines)h(are)f(summed)h(o)n(v)n(er,)f(with)i(the)f(exception)f (of)h(the)g(lab)r(el)g Fx(\015)38 b FG(asso)r(ciated)31 b(with)118 4157 y(the)26 b(ro)r(ot)e(\(b)r(ecause)i(w)n(e)f(do)g(not)g (consider)f(it)i(a)f(no)r(de)g(and)h(the)f(corresp)r(onding)f(lab)r(el) h Fx(\015)30 b FG(app)r(ears)24 b(only)h(once)g(in)118 4264 y(\(2.5\)\).)37 b(Therefore)26 b(\(2.5\))i(is)f(a)g(n)n(um)n(b)r (er)h(lab)r(eled)f(b)n(y)h Fx(\015)f FG(=)c(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(d)p FG(,)28 b(i.e.)37 b(V)-7 b(al\()p Fx(\022)r FG(\))28 b(is)g(a)f(v)n(ector.)118 4441 y FF(R)l(emarks.)36 b FG(\(1\))22 b(The)h(trees)f(can)g(b)r(e)h(dra)n(wn)f(in)h(v)-5 b(arious)21 b(w)n(a)n(ys:)33 b(w)n(e)22 b(can)g(limit)h(the)g (arbitrariness)d(b)n(y)j(demanding)118 4547 y(that)g(the)f(length)h(of) f(the)g(segmen)n(ts)g(represen)n(ting)f(the)h(lines)h(is)f(1)g(\(unlik) n(e)g(the)h(dra)n(wings)d(in)j(the)f(ab)r(o)n(v)n(e)f(\014gures\))118 4654 y(and)38 b(that)f(the)h(angles)f(b)r(et)n(w)n(een)h(the)g(lines)f (are)g(irrelev)-5 b(an)n(t.)66 b FF(The)40 b(c)l(ombinatorics)g(b)l (eing)f(very)h(imp)l(ortant)p FG(,)118 4760 y(b)r(ecause)26 b(it)g(matters)g(in)g(the)g(c)n(hec)n(k)f(of)h(cancellations)f(essen)n (tial)g(for)h(the)g(analysis,)f(w)n(e)h(adopt)g(the)g(con)n(v)n(en)n (tion)118 4866 y(that)g(trees)e(are)g(dra)n(wn)g(on)h(a)g(plane,)g(ha)n (v)n(e)f(lines)h(of)g(unit)h(length)f FF(and)j(c)l(arry)g(an)g (identi\014er)g(lab)l(el)p FG(,)f(that)e(w)n(e)g(call)118 4972 y FF(numb)l(er)31 b(lab)l(el)h FG(\(not)e(sho)n(wn)f(in)i(the)f (ab)r(o)n(v)n(e)f(\014gures\))h(whic)n(h)g(distinguishes)g(the)g(lines) g(from)g(eac)n(h)f(other)h(ev)n(en)118 5079 y(if)f(w)n(e)f(ignore)g (the)h(other)f(lab)r(els)g(attac)n(hed)g(to)g(them.)41 b(F)-7 b(urthermore)27 b(t)n(w)n(o)h(trees)g(that)h(can)f(b)r(e)h(sup)r (erp)r(osed)f(b)n(y)118 5185 y(piv)n(oting)c(the)g(lines)h(merging)e (in)n(to)h(the)h(same)f(no)r(de)g Fn(v)p FG(,)h(around)e Fn(v)i FG(itself,)g(are)f(considered)f FF(identic)l(al)p FG(.)37 b(This)25 b(is)f(a)118 5291 y(con)n(v)n(en)n(tion)j(whic)n(h)h (is)f(useful)i(for)e(c)n(hec)n(king)g(cancellations:)37 b(ho)n(w)n(ev)n(er)26 b(it)i(is)g(b)n(y)g(no)f(means)h(the)g(only)g(p)r (ossible)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)i(19:31)1151 b FG(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 118 356 a FG(6:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(one.)45 b(Others)29 b(are)g(p)r(ossible)h(and)g(often)h(v)n(ery)e(con)n(v)n(enien)n(t)g(in) i(other)e(resp)r(ects)h([G3],)h([GM1],)g(but)g(in)f(a)g(giv)n(en)118 662 y(w)n(ork)c(a)h(c)n(hoice)g(has)g(to)h(b)r(e)g(made)f(once)g(and)h (for)f(all.)118 768 y(\(2\))33 b(A)f(line)h Fx(`)f FG(carrying)f Fy(0)h FG(momen)n(tum)g(is)h(somewhat)e(sp)r(ecial.)51 b(W)-7 b(e)33 b(could)f(visualize)g(the)h(part)f(of)g(the)h(tree)118 874 y(preceding)e(suc)n(h)g(lines)h(b)n(y)f(encircling)g(it)h(in)n(to)g (a)f(dotted)h(circle:)44 b(suc)n(h)31 b(a)h(represen)n(tation)d(has)j (b)r(een)g(used)f(in)118 981 y(earlier)e(pap)r(ers,)h(e.g.)44 b(in)30 b(Ref.)45 b([GG],)31 b(calling)f(the)g(subtree)g Fx(\022)2100 993 y Fu(`)2162 981 y FG(with)h Fx(`)e FG(as)h(ro)r(ot)f (line)i(a)e FF(le)l(af)p FG(.)46 b(Here,)31 b(ho)n(w)n(ev)n(er,)118 1087 y(w)n(e)c(shall)h(a)n(v)n(oid)e(using)h(a)g(sp)r(ecial)g(w)n(ord)g (for)g(the)h Fy(0)f FG(momen)n(tum)h(lines)g(and)f(the)h(subtrees)f (preceding)g(them.)118 1193 y(\(3\))h(W)-7 b(e)28 b(can)f(think)h(of)g (the)g(propagators)c(as)j(matrices)g(of)g(the)h(form)1467 1433 y Fx(G)1532 1445 y Fu(`)1588 1433 y FG(=)1675 1341 y Fr(\020)1739 1383 y Fx(G)1804 1395 y Fu(`;\013\013)2025 1383 y Fx(G)2090 1395 y Fu(`;\013\014)1740 1483 y Fx(G)1805 1495 y Fu(`;\014)s(\013)2026 1483 y Fx(G)2091 1495 y Fu(`;\014)s(\014)2240 1341 y Fr(\021)2289 1433 y Fx(;)1179 b FG(\(2)p Fx(:)p FG(6\))118 1668 y(where)27 b Fx(G)423 1680 y Fu(`;\013\013)561 1668 y FG(,)h Fx(G)677 1680 y Fu(`;\013\014)813 1668 y FG(,)g Fx(G)929 1680 y Fu(`;\014)s(\013)1092 1668 y FG(and)f Fx(G)1318 1680 y Fu(`;\014)s(\014)1479 1668 y FG(are)g Fx(r)21 b FA(\002)d Fx(r)r FG(,)29 b Fx(r)21 b FA(\002)d Fx(s)p FG(,)28 b Fx(s)18 b FA(\002)g Fx(r)30 b FG(and)e Fx(s)18 b FA(\002)g Fx(s)28 b FG(matrices.)118 1775 y(\(4\))h(The)f(v)-5 b(alue)28 b(of)g(a)g(tree)g Fx(\022)j FG(de\014ned)d(ab)r(o)n(v)n(e)f(has)h(no)g(p)r(ole)g(at)g Fx(")c FG(=)g(0)j(if)i(V)-7 b(al\()p Fx(\022)r FG(\))25 b FA(6)p FG(=)f(0)k(b)r(ecause)g(ev)n(ery)f(line)h(with)118 1881 y Fy(0)k FG(momen)n(tum)g(is)g(preceded)f(b)n(y)h(at)g(least)f(t)n (w)n(o)g(no)r(des,)i(so)e(that)i(the)f(total)f(p)r(o)n(w)n(er)g(of)h Fx(")g FG(to)f(whic)n(h)h(the)h(v)-5 b(alue)118 1987 y(is)29 b(prop)r(ortional)d(is)j(alw)n(a)n(ys)e(non-negativ)n(e)g(and,) h(in)h(fact,)g(it)g(is)g(necessarily)e(p)r(ositiv)n(e:)38 b(w)n(e)29 b(need)f(to)h(tak)n(e)f(in)n(to)118 2093 y(accoun)n(t)e (that)h Fx(@)647 2111 y Fq(\014)697 2093 y Fx(f)738 2105 y Fo(0)780 2093 y FG(\()p Fw(\014)867 2105 y Fz(0)904 2093 y FG(\))c FA(\021)g FG(0)j(and)h(that)g(our)f(trees)g(con)n(tain)g (no)g(trivial)g(no)r(des)h(with)g(one)f(en)n(tering)g(line)h(with)118 2200 y Fy(0)h FG(momen)n(tum.)37 b(Note)27 b(that)h(V)-7 b(al\()p Fx(\022)r FG(\))29 b(is)e(a)g(monomial)g(in)h Fx(")f FG(of)h(degree)e Fx(P)12 b FG(\()p Fx(\022)r FG(\).)118 2306 y(\(5\))27 b(In)h(the)f(case)g(of)g(maximal)f(tori)h(and)g(if)h(V) -7 b(al)o(\()p Fx(\022)r FG(\))24 b FA(6)p FG(=)f(0)k(there)g(are)f FF(no)j(lines)h(with)g Fy(0)f FF(momentum)d FG(for)h(systems)118 2412 y(describ)r(ed)d(b)n(y)g(the)g(Hamiltonians)g(\(1.1\):)35 b(indeed)24 b Fx(s)f FG(=)g(0,)h(see)g(also)f([Ga].)35 b(In)25 b(this)f(case)f(the)i(n)n(um)n(b)r(er)f(of)g(no)r(des,)118 2519 y(i.e.)37 b(the)26 b(tree)g(order,)f(coincides)g(with)i(the)g(p)r (o)n(w)n(er)d(of)i Fx(")g FG(asso)r(ciated)f(with)i(the)f(monomial)g (in)g Fx(")g FG(de\014ned)g(b)n(y)g(the)118 2625 y(tree)j(v)-5 b(alue,)30 b(i.e.)43 b(with)30 b(the)g(tree)f(degree.)41 b(In)30 b(general,)e(ho)n(w)n(ev)n(er,)g(the)i(order)e FA(j)p Fx(V)19 b FG(\()p Fx(\022)r FG(\))p FA(j)31 b FG(of)e(a)g(tree)g(can)h(b)r(e)f(larger)118 2731 y(than)f(its)g(degree) e Fx(P)12 b FG(\()p Fx(\022)r FG(\):)38 b FA(j)p Fx(V)19 b FG(\()p Fx(\022)r FG(\))p FA(j)24 b(\025)e Fx(P)12 b FG(\()p Fx(\022)r FG(\))24 b FA(\025)1538 2699 y Fz(1)p 1538 2713 34 4 v 1538 2760 a(2)1581 2731 y FA(j)p Fx(V)19 b FG(\()p Fx(\022)r FG(\))p FA(j)p FG(.)189 2908 y(The)34 b(ab)r(o)n(v)n(e)f(de\014nitions)h(uniquely)g(attribute)g(a)g(v)-5 b(alue)34 b(to)g(eac)n(h)f(tree.)56 b(The)34 b(follo)n(wing)f(result)h (states)f(the)118 3015 y(existence)28 b(of)h(formal)f(solutions)g(to)g (\(1.5\))h(whic)n(h)f(are)g(conjugated)g(to)g(the)h(unp)r(erturb)r(ed)h (motion)e(\(1.4\),)h(pro-)118 3121 y(vided)j(the)h(v)-5 b(alue)32 b Fw(\014)768 3133 y Fz(0)837 3121 y FG(is)h(suitably)f (\014xed.)50 b(The)33 b(pro)r(of)e(is)h(an)g(algebraic)f(c)n(hec)n(k)g (whic)n(h)h(do)r(es)g(not)g(distinguish)118 3227 y(the)c(p)r(ossible)f (signs)g(of)h Fx(")f FG(and)h(can)f(b)r(e)h(tak)n(en)f(from)g(Ref.)37 b([GG])28 b(where)g(it)g(is)f(done)g(in)h(the)g(case)f Fx(")c(<)f FG(0.)118 3404 y FB(Lemma)27 b(1.)35 b FF(The)26 b(F)-6 b(ourier)27 b(tr)l(ansform)f(of)g(the)g(p)l(ower)h(series)f (solution)g Fy(h)d FG(=)g(\()p Fy(a)p Fx(;)14 b Fy(b)p FG(\))27 b FF(of)f(\(1.5\))h(of)g(the)f(form)g(\(2.1\))118 3511 y(is)k(obtaine)l(d)h(by)f(writing)h(\(the)e(de\014nition)i(of)f FG(\002)1633 3481 y Fu(o)1633 3534 y(k)q(;)p Fq(\027)5 b Fu(;\015)1824 3511 y FF(fol)t(lows)32 b(\(2.2\)\))1463 3740 y Fx(")1502 3705 y Fu(k)1543 3740 y Fx(h)1591 3697 y Fz(\()p Fu(k)q Fz(\))1591 3754 y Fq(\027)5 b Fu(;\015)1719 3740 y FG(=)1882 3661 y Fr(X)1807 3840 y Fu(\022)r FC(2)p Fz(\002)1937 3819 y Fs(o)1937 3862 y(k)q(;\027)s(;\015)2090 3740 y FG(V)-7 b(al\()p Fx(\022)r FG(\))1176 b(\(2)p Fx(:)p FG(7\))118 4051 y FF(for)31 b(al)t(l)g Fw(\027)d FA(2)c Fv(Z)584 4010 y Fu(r)621 4051 y FF(,)30 b(al)t(l)h Fx(k)26 b FA(2)d Fv(N)30 b FF(and)g Fx(\015)e FG(=)22 b(1)p Fx(;)14 b(:)g(:)g(:)g(;)g(d)p FF(.)189 4228 y FG(The)27 b(expression)f(\(2.7\))h(is)h(w)n(ell)f(de\014ned)h(at)f(\014xed)g Fx(k)j FG(and)e(the)f(sum)h(o)n(v)n(er)d Fx(k)31 b FG(giv)n(es)26 b(what)h(w)n(e)g(call)g(the)h FF(formal)118 4335 y(p)l(ower)c(series)g (solution)d FG(for)g(the)g(equations)e(for)i(the)g(parametric)e (represen)n(tation)g(\(2.1\),)j(\(1.6\))e(of)h(the)g(in)n(v)-5 b(arian)n(t)118 4441 y(tori.)1144 4689 y FD(3.)100 b(The)37 b(simplest)i(resummation)118 4866 y FG(The)26 b(p)r(o)n(w)n(er)g (series)f(in)h Fx(")g FG(in)h(\(2.1\))f(and)g(its)g(F)-7 b(ourier)26 b(transform)f(de\014ned)i(b)n(y)f(the)g(sum)h(o)n(v)n(er)d Fx(k)29 b FG(of)e(\(2.7\))f(ma)n(y)f(b)r(e)118 4972 y(not)k(con)n(v)n (ergen)n(t)e FF(as)32 b(a)f(p)l(ower)i(series)d FG(\(as)e(far)h(as)g(w) n(e)g(kno)n(w\).)41 b(The)29 b(problem)g(is)g(di\016cult)h(b)r(ecause)f (if)g(in)h(\(2.7\))118 5079 y(w)n(e)d(replace)g(V)-7 b(al\()p Fx(\022)r FG(\))28 b(with)g FA(j)p FG(V)-7 b(al\()p Fx(\022)r FG(\))p FA(j)29 b FG(the)f(series)f(certainly)f(div)n(erges.) 189 5185 y(Our)36 b(aim,)j(as)e(stated)f(in)i(the)f(in)n(tro)r (duction,)i(is)e(to)g(sho)n(w)f(that)h(nev)n(ertheless)f(a)g(meaning)g (to)h(the)h(series)118 5291 y(can)29 b(b)r(e)g(giv)n(en.)40 b(W)-7 b(e)29 b(shall)g(sho)n(w)f(that)h(the)g(tree)g(v)-5 b(alues)28 b(can)h(b)r(e)g(further)g(decomp)r(osed)f(in)n(to)h(sums)f (of)h(sev)n(eral)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)h(19:31)1151 b FG(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 118 356 a FG(7:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(other)j(quan)n(tities)h(and)g (that)g(the)g(v)-5 b(arious)35 b(con)n(tributions)g(to)h(the)g(series)f (can)h(b)r(e)g(rearranged)d(b)n(y)j(suitably)118 662 y(collecting)28 b(them)h(in)n(to)f(families:)39 b(the)29 b(sums)f(of)g(the)h(con)n(tributions)f(from)g(eac)n(h)f(family)i(lea)n (v)n(e)e(us)h(with)h(a)f(new)118 768 y(series)d(\(no)i(longer)e(a)h(p)r (o)n(w)n(er)f(series)g(in)h Fx(")p FG(\))h(whic)n(h)f(is)g(in)h(fact)f (con)n(v)n(ergen)n(t)e(and)i(its)h(sum)f(solv)n(es)f(the)i(problem)f (of)118 874 y(constructing)j(the)h(parametric)f(represen)n(tations)f Fy(h)f FG(=)f(\()p Fy(a)p Fx(;)14 b Fy(b)p FG(\),)32 b(\(2.1\),)e(of)g(the)g(in)n(v)-5 b(arian)n(t)29 b(tori)g(at)h(least)f (for)h(all)118 981 y Fx(")23 b FA(2)g(E)7 b FG(,)28 b(with)g FA(E)35 b FG(a)27 b(set)h(with)g(0)f(as)g(a)g(densit)n(y)h(p)r(oin)n(t) g(\(i.e.)37 b(a)27 b(Leb)r(esgue)g(p)r(oin)n(t\).)189 1087 y(F)-7 b(or)21 b(this)g(purp)r(ose)g(w)n(e)g(need)h(to)f(de\014ne) h(and)f(consider)f(more)h(in)n(v)n(olv)n(ed)f(trees)h(and)g(more)f(in)n (v)n(olv)n(ed)g(de\014nitions)118 1193 y(of)34 b(their)g(v)-5 b(alues.)57 b(W)-7 b(e)35 b(b)r(egin)f(b)n(y)g(remarking)e(that)j (trees)e(ma)n(y)h(con)n(tain)f FF(trivial)38 b(no)l(des)p FG(,)e(i.e.)57 b(no)r(des)34 b(with)h Fy(0)118 1300 y FG(harmonic)27 b(separating)f(t)n(w)n(o)h(lines)g(with)h(equal)f(momen) n(tum)h Fw(\027)h FA(6)p FG(=)22 b Fy(0)p FG(.)189 1406 y(One)g(can)f(supp)r(ose)h(that)g FF(no)j(tr)l(e)l(e)f(c)l(ontains)h (trivial)g(no)l(des)e FG(pro)n(vided)e(w)n(e)h(use)f(for)h(all)g (lines,)h(with)f(momen)n(tum)118 1512 y Fw(\027)29 b FA(6)p FG(=)22 b Fy(0)28 b FG(and)f(lab)r(els)h Fx(\015)5 b(;)14 b(\015)886 1482 y FC(0)936 1512 y FG(asso)r(ciated)26 b(with)i(the)g(extremes,)f(the)h FF(new)g FG(propagators)p 625 1710 43 4 v 625 1755 a Fx(g)r FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))868 1704 y Fu(def)889 1755 y FG(=)33 b(\()p Fx(x)1066 1721 y Fz(2)1123 1755 y FA(\000)18 b Fx(M)1287 1767 y Fz(0)1324 1755 y FG(\))1356 1721 y FC(\000)p Fz(1)1445 1755 y Fx(;)180 b(x)1709 1704 y Fu(def)1730 1755 y FG(=)33 b Fw(!)21 b FA(\001)e Fw(\027)6 b Fx(;)179 b(M)2288 1767 y Fz(0)2339 1704 y Fu(def)2359 1755 y FG(=)34 b Fx(")2511 1638 y Fr(\022)2585 1699 y FG(0)229 b(0)2585 1799 y(0)83 b Fx(@)2759 1769 y Fz(2)2754 1828 y Fq(\014)2804 1799 y Fx(f)2845 1811 y Fo(0)2886 1799 y FG(\()p Fw(\014)2973 1811 y Fz(0)3011 1799 y FG(\))3057 1638 y Fr(\023)3132 1755 y Fx(:)336 b FG(\(3)p Fx(:)p FG(1\))118 1999 y(This)32 b(is)g(a)g FF(r)l(esummation)h(of)i(many)f(diver)l(gent)g(series)f FG(obtained)f(b)n(y)g(adding)f(the)i(v)-5 b(alues)31 b(of)h(trees)g(obtained)118 2105 y(from)i(a)g(tree)h(without)f(trivial) g(no)r(des)h(b)n(y)f(\\insertion")f(of)h(an)h(arbitrary)d(n)n(um)n(b)r (er)i(of)h(trivial)f(no)r(des)g(on)g(the)118 2211 y(branc)n(hes)c(with) h(momen)n(tum)g Fw(\027)j FA(6)p FG(=)28 b Fy(0)p FG(:)43 b(this)31 b(requires)e(summing)i(series,)g(one)f(p)r(er)h(branc)n(h)f (of)g(a)h(tree)f FF(without)118 2318 y(trivial)f(no)l(des)p FG(,)d(whic)n(h)f(are)f(geometric)f(series)h(with)i(ratio)e(giv)n(en)g (b)n(y)g(the)i Fx(d)13 b FA(\002)g Fx(d)25 b FG(matrix)f Fx(z)j FG(=)3108 2284 y Fu(M)3171 2292 y Ft(0)p 3057 2299 198 4 v 3057 2346 a Fz(\()p Fq(!)r FC(\001)p Fq(\027)5 b Fz(\))3222 2330 y Ft(2)3264 2318 y FG(;)26 b FA(j)p Fx(z)t FA(j)f FG(can)f(b)r(e)118 2437 y(larger)c(than)h(1)g(b)r(ecause) g(the)h Fx(s)f FG(non-zero)f(eigen)n(v)-5 b(alues)20 b Fx("a)1933 2449 y Fu(j)1967 2437 y FG(,)j Fx(j)28 b FG(=)23 b(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(s)p FG(,)23 b(of)e Fx(M)2643 2449 y Fz(0)2701 2437 y FG(are)f(unrelated)h(to)h Fx(x)h FG(=)g Fw(!)9 b FA(\001)d Fw(\027)g FG(.)3625 2406 y Fz(1)189 2543 y FG(Therefore)33 b(replacing)934 2481 y Fr(P)1021 2501 y FC(1)1021 2568 y Fu(p)p Fz(=0)1158 2543 y Fx(z)1201 2513 y Fu(p)1273 2543 y FG(b)n(y)g(\(1)23 b FA(\000)f Fx(z)t FG(\))1653 2513 y FC(\000)p Fz(1)1776 2543 y FF(is)36 b(not)g(rigor)l(ous)g(and)h(ne)l(e)l(ds)e(to)h(b)l(e)g (eventual)t(ly)g(justi\014e)l(d)p FG(.)118 2649 y(Certainly)24 b(w)n(e)g(m)n(ust)h(at)g(least)f(supp)r(ose)g(that)h Fx(x)1627 2619 y Fz(2)1677 2649 y FA(\000)12 b Fx(M)1835 2661 y Fz(0)1897 2649 y FG(can)24 b(b)r(e)h(in)n(v)n(erted:)35 b(otherwise)23 b(the)i(v)-5 b(alues)24 b(of)h(the)g(trees)118 2756 y(represen)n(ting)i(the)i(new)g(series)f(migh)n(t)h(ev)n(en)f(b)r (e)h(meaningless!)39 b(\(i.e.)i(if)29 b(some)g(lines)f(will)h(ha)n(v)n (e)f(momen)n(tum)h Fw(\027)118 2862 y FG(suc)n(h)f(that)g(det\()p Fx(x)680 2832 y Fz(2)737 2862 y FA(\000)18 b Fx(M)901 2874 y Fz(0)938 2862 y FG(\))24 b(=)f(0\).)38 b(This)28 b(happ)r(ens)g(for)f(a)g(dense)h(set)g(of)g Fx(")p FG('s)g(and)f(w)n(e) h(ha)n(v)n(e)f(to)g(exclude)h(suc)n(h)g Fx(")p FG('s)118 2978 y(b)n(y)f(imp)r(osing)h(conditions)f(on)g(the)h(eigen)n(v)-5 b(alues)27 b Fx(\025)1720 2934 y Fz([0])1720 3001 y Fu(r)r Fz(+)p Fu(j)1862 2978 y FA(\021)22 b Fx("a)2032 2990 y Fu(j)2067 2978 y Fx(;)28 b(j)g FG(=)22 b(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(s)p FG(,)28 b(i.e.)37 b(on)27 b Fx(")p FG(.)189 3084 y(F)-7 b(or)23 b(uniformit)n(y)g(of)g(notations)g(it)g 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Fx(;)457 b FG(\(3)p Fx(:)p FG(2\))118 3609 y(where)21 b Fx(C)411 3621 y Fz(0)470 3609 y FG(is)g(the)g(Diophan)n(tine)h(constan)n(t)e(in)h(\(1.3\))g (\(\014xed)h(throughout)e(the)i(analysis\);)g(th)n(us)f Fx(I)3125 3621 y Fu(C)3203 3609 y FG(is)g(an)g(in)n(terv)-5 b(al)118 3715 y(of)28 b(size)f Fx(O)r FG(\()p Fx(C)532 3685 y Fz(2)571 3715 y FG(\))h(\(i.e.)817 3683 y Fz(3)p 817 3697 34 4 v 817 3744 a(4)860 3715 y Fx(C)925 3685 y Fz(2)962 3715 y FG(\).)38 b(In)28 b(other)f(w)n(ords)g(w)n(e)g (\014nd)i(it)f(con)n(v)n(enien)n(t)f(to)g(measure)g Fx(")h FG(in)g(units)g(of)g Fx(C)3355 3685 y Fz(2)3349 3736 y(0)3392 3715 y Fx(a)3436 3685 y FC(\000)p Fz(1)3436 3736 y Fu(s)3553 3715 y FG(via)118 3822 y(an)f(in)n(teger)g Fx(n)558 3834 y Fz(0)595 3822 y FG(.)37 b FF(We)30 b(a)g(priori)i (assume,)e(for)g(simplicity,)i(the)e(r)l(estrictions)g Fx(a)2582 3834 y Fu(s)2618 3822 y Fx(")22 b FA(\024)h Fx(C)2832 3791 y Fz(2)2826 3842 y(0)2899 3822 y FF(and)31 b Fx(")22 b FA(\024)h FG(1.)189 3928 y(T)-7 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b(the)g(measure)f(of)h(the)g (complemen)n(t)f(of)h(the)118 1465 y(set)26 b FA(E)p 290 1443 V 13 x Fu(n)331 1486 y Ft(0)363 1478 y FC(\000)p Fz(1)478 1465 y FG(where)f(\(3.3\))h(is)f(v)n(eri\014ed)g(is)h(a)f FF(smal)t(l)k(fr)l(action)d FG(of)g(order)e Fx(C)2328 1434 y Fz(1)p Fu(=)p Fz(2)2459 1465 y FG(of)h(the)i(measure)d(of)i(the) g(in)n(terv)-5 b(al)25 b Fx(I)3582 1477 y Fu(C)3638 1465 y FG(,)118 1571 y(whose)i(size)g(is)h(prop)r(ortional)e(to)h Fx(C)1248 1541 y Fz(2)1286 1571 y FG(,)g(in)h(whic)n(h)g(w)n(e)f(let)h Fx(")f FG(v)-5 b(ary)e(,)27 b(at)h(least)f(if)p 2552 1525 50 4 v 28 w Fx(n)2602 1583 y Fz(0)2667 1571 y FG(is)g(large.)504 1819 y FD(4.)50 b(Resummations:)i(seman)m(tic)38 b(and)g(heuristic)g (considerations)118 1996 y FG(Replacing)g(the)h(propagators)c Fx(x)1182 1966 y FC(\000)p Fz(2)1310 1996 y FG(of)j(the)h(lines)f(b)n (y)g(\()p Fx(x)1974 1966 y Fz(2)2038 1996 y FA(\000)25 b Fx(M)2209 2008 y Fz(0)2246 1996 y FG(\))2278 1966 y FC(\000)p Fz(1)2405 1996 y FG(w)n(e)38 b(obtain)g(a)g(represen)n (tation)f(of)h(the)118 2102 y(parametric)29 b(equations)g Fy(h)i FG(in)n(v)n(olving)d(simpler)i(trees)g(\(i.e.)45 b(trees)30 b(with)h(no)f(trivial)f(no)r(des\).)45 b(The)30 b(new)h(repre-)118 2209 y(sen)n(tation)c(is)h(a)g(series)f(in)h(whic)n (h)g(eac)n(h)f(term)h(is)g(w)n(ell)g(de\014ned)h(if)f Fx(")g FG(is)g(in)g(the)h(large)d(set)i FA(E)p 2959 2187 42 4 v 13 x Fu(n)3000 2230 y Ft(0)3033 2222 y FC(\000)p Fz(1)3145 2209 y FA(\032)c Fx(I)3270 2221 y Fu(C)3354 2209 y FG(in)29 b(whic)n(h)118 2315 y(\(3.3\))e(holds.)37 b(This)27 b(is)g(quite)h(di\013eren)n(t)g(from)f(the)g(original)f (Lindstedt)i(series)f(in)g(\(2.7\))g(whose)g(terms)g(are)g(w)n(ell)118 2421 y(de\014ned)h(for)f(all)g Fx(")p FG(.)189 2528 y(W)-7 b(e)23 b(should)g(also)e(stress)h(that)h(the)g(resummed)g(series)e(is)i (in)g(a)f(sense)g(more)g(natural:)34 b(the)23 b Fy(0)g FG(momen)n(tum)g(lines)118 2634 y(no)n(w)c(app)r(ear)g(as)h(less)f (anomalous)f(b)r(ecause)i(their)g(propagator)d(is)j(m)n(uc)n(h)f(more)g (closely)g(related)h(to)f(\()p Fx(x)3290 2604 y Fz(2)3332 2634 y FA(\000)s Fx(M)3481 2646 y Fz(0)3517 2634 y FG(\))3549 2604 y FC(\000)p Fz(1)3638 2634 y FG(.)118 2740 y(One)26 b(can)g(sa)n(y)f(that)h(it)h(is)f(just)h(the)f(latter)g(ev)-5 b(aluated)26 b(at)g Fx(x)d FG(=)g(0)j(with)g(the)h(meaningless)e(en)n (tries)g(\(i.e.)37 b(the)27 b(\014rst)118 2846 y Fx(r)34 b FG(diagonal)c(en)n(tries\))h(replaced)g(b)n(y)g(0.)48 b(Another)31 b(w)n(a)n(y)g(of)g(sa)n(ying)f(the)i(latter)f(prop)r(ert)n (y)f(is)i(that)f(lines)h Fx(`)f FG(with)118 2953 y Fy(0)f FG(momen)n(tum)h(and)f(lab)r(els)g Fx(\015)1078 2965 y Fu(`)1110 2953 y Fx(;)14 b(\015)1195 2923 y FC(0)1190 2976 y Fu(`)1250 2953 y FA(\024)27 b Fx(r)33 b FG(are)d(forbidden.)45 b(One)30 b(should)g(not)g(b)r(e)h(surprised)e(b)n(y)i(this)f(fact:)43 b(it)31 b(is)118 3059 y(the)d(generalization)e(of)h(the)h(corresp)r (onding)d(prop)r(ert)n(y)i(in)g(the)h(case)f(of)g(maximal)g(tori)g(\()p Fx(r)f FG(=)d Fx(d)p FG(\))28 b(in)g(whic)n(h)f(this)118 3165 y(means)33 b(that)h(lines)f(with)h Fy(0)g FG(momen)n(tum)f(are)g (forbidden.)54 b(The)34 b(latter)f(prop)r(ert)n(y)f(go)r(es)h(bac)n(k)g (to)g(P)n(oincar)n(\023)-39 b(e's)118 3272 y(theory)24 b(of)g(the)g(Lindstedt)h(series)e(and)h(is)g(the)h(k)n(ey)f(to)g(the)g (pro)r(of)g(of)g(the)h(KAM)f(theorem)g(and)g(of)g(cancellations)118 3378 y(whic)n(h)i(mak)n(e)f(the)h(formal)f(Lindstedt)h(series)f(for)g (maximal)g(tori)g(absolutely)g(con)n(v)n(ergen)n(t;)f(see)h(Refs.)37 b([E1])25 b(and)118 3484 y([Ga].)42 b(Ho)n(w)n(ev)n(er)27 b(the)i(new)h(series)e(is)h(still)g(only)g(a)g(formal)f(represen)n (tation)g(b)r(ecause)g(it)i(is)f(b)n(y)g(no)g(means)g(clear)118 3591 y(that)f(it)g(is)f(absolutely)g(con)n(v)n(erges.)189 3697 y(The)k(next)g(natural)f(idea)g(is)h(to)f(try)h(to)g(establish)f (con)n(v)n(ergence)e(b)n(y)j(further)f(mo)r(difying)h(the)g (propagators,)118 3803 y(c)n(hanging)g(at)i(the)g(same)f(time)i(the)f (trees)f(structure,)h(un)n(til)g(one)g(ac)n(hiev)n(es)e(a)h(formal)g (represen)n(tation)f(whose)118 3909 y(con)n(v)n(ergence)e(will)j(b)r(e) h(\\easy")d(to)h(c)n(hec)n(k.)49 b(Once)31 b(ac)n(hiev)n(ed)g(a)g (formal)g(represen)n(tation)f(whic)n(h)i(is)g(con)n(v)n(ergen)n(t)118 4016 y(w)n(e)27 b(shall)h(ha)n(v)n(e)e(to)h(c)n(hec)n(k)g(that)h(it)g (really)f(solv)n(es)f(the)i(equations)e(for)i Fy(h)p FG(.)189 4122 y(The)i(mo)r(di\014cation)f(of)h(the)g(trees)f(structure) g(will)h(b)r(e)g(p)r(erformed)f(b)n(y)g(steps:)41 b(at)30 b(eac)n(h)e(step,)j(lab)r(eled)e(b)n(y)h(an)118 4228 y(in)n(teger)24 b Fx(n)e FG(=)h(0)p Fx(;)14 b FG(1)p Fx(;)g(:)g(:)g(:)o FG(,)25 b(the)g(propagators)c(of)j(the)h(lines)f (with)h(non-zero)e(momen)n(tum)i(will)f(ha)n(v)n(e)f(b)r(een)i(mo)r (di\014ed)118 4335 y(acquiring)30 b(lab)r(els)g([0])p Fx(;)14 b FG([1])p Fx(;)g(:)g(:)g(:)f FG([)p Fx(n)21 b FA(\000)f FG(1],)31 b(or)f(the)h(lab)r(el)g([)p FA(\025)d Fx(n)p FG(],)k(indicating)f(that)g(they)g(are)f(giv)n(en)g(no)g(longer) g(b)n(y)118 4441 y(\()p Fx(x)197 4411 y Fz(2)255 4441 y FA(\000)19 b Fx(M)420 4453 y Fz(0)457 4441 y FG(\))489 4411 y FC(\000)p Fz(1)608 4441 y FG(but)30 b(b)n(y)f(a)g(matrix)g(prop) r(ortional)f(to)h(\()p Fx(x)1884 4411 y Fz(2)1942 4441 y FA(\000)19 b(M)2126 4411 y Fz([)p FC(\024)p Fu(p)p Fz(])2253 4441 y FG(\))2285 4411 y FC(\000)p Fz(1)2375 4441 y FG(,)30 b(if)g(their)f(lab)r(el)h(is)f([)p Fx(p)p FG(],)h(with)g Fx(p)c(<)g(n)p FG(,)j(or)118 4547 y(\(with)g(a)f (di\013eren)n(t)h(prop)r(ortionalit)n(y)d(factor\))i(to)g(\()p Fx(x)1766 4517 y Fz(2)1823 4547 y FA(\000)19 b(M)2007 4517 y Fz([)p FC(\024)p Fu(n)p Fz(])2141 4547 y FG(\))2173 4517 y FC(\000)p Fz(1)2263 4547 y FG(,)29 b(if)f(their)h(lab)r(el)f(is) h([)p FA(\025)24 b Fx(n)p FG(];)29 b(here)e FA(M)3394 4517 y Fz([)p FC(\024)p Fu(p)p Fz(])3550 4547 y FG(are)118 4654 y(suitable)34 b(matrices.)56 b FF(Her)l(e)35 b(and)h(in)g(the)g (fol)t(lowing)i(the)e(symb)l(ols)g FG([)p FA(\024)e Fx(n)p FG(])h FF(and)h FG([)p FA(\025)e Fx(n)p FG(])h FF(ar)l(e)h(c)l (onsistently)g(use)l(d.)118 4760 y(Henc)l(e)28 b FG([)p FA(\025)23 b Fx(n)p FG(])29 b FF(do)l(es)g(not)f(denote)h(the)g(set)f (of)i(sc)l(ales)f FG([)p Fx(p)p FG(])f FF(with)h Fx(p)23 b FA(\025)g Fx(n)p FF(,)29 b(and)g(in)g(fact)g(it)f(is)h(just)f(a)h (di\013er)l(ent)g(sc)l(ale;)118 4866 y(likewise)k FG([)p FA(\024)27 b Fx(n)p FG(])k FF(do)l(es)h(not)g(\\include")f FG([)p Fx(p)p FG(])h FF(even)g(if)g Fx(p)26 b FA(\024)g Fx(n)p FF(.)44 b(In)32 b(other)g(wor)l(ds)g(one)g(has)g(to)g(r)l(e)l (gar)l(d)g(the)g(symb)l(ols)118 4972 y FG([)p FA(\024)23 b Fx(n)p FG(])p FF(,)30 b FG([)p Fx(n)p FG(])f FF(and)h FG([)p FA(\025)23 b Fx(n)p FG(])29 b FF(as)h(unr)l(elate)l(d)f(symb)l (ols.)39 b(This)31 b(might)e(app)l(e)l(ar)i(unusual)e(but)f(it)i(turns) e(out)h(to)g(b)l(e)h(a)f(go)l(o)l(d)118 5079 y(notation)h(for)h(our)e (purp)l(oses)p FG(.)189 5185 y(The)41 b(prop)r(ortionalit)n(y)e(factor) h(dep)r(ends)i(on)e Fx(x)i FG(and)f(con)n(tains)f(cut-o\013)h (functions)g(whic)n(h)g(v)-5 b(anish)41 b(unless)118 5291 y Fx(x)165 5261 y Fz(2)227 5291 y FA(\000)24 b(M)416 5261 y Fz([)p FC(\024)p Fu(p)p Fz(])580 5291 y FG(has)35 b(smallest)h(eigen)n(v)-5 b(alue)35 b(of)i(order)d Fx(O)r FG(\(2)1937 5261 y FC(\000)p Fz(2)p Fu(p)2061 5291 y Fx(C)2126 5261 y Fz(2)2120 5312 y(0)2164 5291 y FG(\);)41 b(the)36 b(cut-o\013s)g(are)f(so)h(devised)g(that)g(if)h(the)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1151 b FG(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 118 356 a FG(9:)27 b FF(De)l(gener)l(ate)j(el)t (liptic)i(r)l(esonanc)l(es)118 555 y FG(propagator)21 b(do)r(es)h(not)i(v)-5 b(anish)23 b(its)g(denominator)f(has)h(a)g (minim)n(um)g(size)g(prop)r(ortional)f(to)h(2)3049 525 y FC(\000)p Fz(2)p Fu(p)3195 555 y FG(and)g(the)h(ratio)118 662 y(b)r(et)n(w)n(een)29 b(its)f(minim)n(um)h(and)g(maxim)n(um)f(v)-5 b(alues)28 b(will)h(b)r(e)g(b)r(ounded)g(ab)r(o)n(v)n(e)e(and)h(b)r (elo)n(w)g(b)n(y)g(a)g Fx(p)p FF(-indep)l(endent)118 768 y(c)l(onstant)p FG(.)34 b(No)21 b(mo)r(di\014cation)g(will)g(b)r(e) g(made)g(of)g(the)g(propagators)d(of)j(the)h Fy(0)e FG(momen)n(tum)i (lines:)33 b(for)21 b(uniformit)n(y)118 874 y(of)28 b(notation)f(w)n(e) g(shall)g(attac)n(h)g(a)g(lab)r(el)h([)p FA(1)p FG(])g(to)f(suc)n(h)h (lines.)189 1014 y(Considering)41 b(trees)h(with)h(no)f(trivial)g(no)r (des)g(in)h(whic)n(h)f(eac)n(h)g(line)h(carries)d(also)i(an)g(extra)f FF(sc)l(ale)i FG(lab)r(el)118 1120 y([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g FG([1])p Fx(;)g(:)g(:)g(:)f FG([)p Fx(n)24 b FA(\000)g FG(1])p Fx(;)14 b FG([)p FA(\025)36 b Fx(n)p FG(])h(a)f(new)g(formal)f(represen)n(tation)g(of)h Fy(h)h FG(will)f(b)r(e)h(obtained)f(b)n(y)g(assigning,)h(to)118 1226 y(the)c(trees,)g(v)-5 b(alues)33 b(de\014ned)g(b)n(y)f(the)h(same) f(form)n(ula)g(in)h(\(2.5\),)g(with)h(the)f(propagators)c Fx(G)3028 1238 y Fu(`)3093 1226 y FG(replaced)j(b)n(y)g(the)118 1342 y(new)g(propagators,)e(that)j(w)n(e)e(denote)h Fx(g)1405 1299 y Fz([)p Fu(p)p Fz(])1402 1367 y Fu(`)1513 1342 y FG(if)g(the)h(line)f Fx(`)f FG(carries)g(the)h(lab)r(el)g([)p Fx(p)p FG(],)h(with)g Fx(p)d FG(=)g FA(1)p Fx(;)14 b FG(0)p Fx(;)g(:)g(:)g(:)f(;)h(n)21 b FA(\000)g FG(1,)118 1462 y(and)29 b Fx(g)324 1419 y Fz([)p FC(\025)p Fu(n)p Fz(])321 1487 y Fu(`)486 1462 y FG(if)h(the)e(line)h(carries)e(the)i (lab)r(el)g([)p FA(\025)24 b Fx(n)p FG(].)40 b(When)29 b(the)g(line)g Fx(`)f FG(is)g(on)h(scale)f([)p Fx(p)p FG(],)g(with)h Fx(p)c FG(=)f(0)p Fx(;)14 b(:)g(:)g(:)f(;)h(n)19 b FA(\000)g FG(1,)118 1568 y(or)35 b([)p FA(\025)i Fx(n)p FG(])g(or)e([)p FA(1)p FG(],)k(then)d(the)h(corresp)r(onding)d (propagator)g(will)i(b)r(e)h(prop)r(ortional)d(to)i(\()p Fx(x)3051 1538 y Fz(2)3113 1568 y FA(\000)24 b(M)3302 1538 y Fz([)p FC(\024)p Fu(p)p Fz(])3430 1568 y FG(\))3462 1538 y FC(\000)p Fz(1)3587 1568 y FG(or)118 1675 y(\()p Fx(x)197 1644 y Fz(2)254 1675 y FA(\000)18 b(M)437 1644 y Fz([)p FC(\024)p Fu(n)p Fz(])571 1675 y FG(\))603 1644 y FC(\000)p Fz(1)720 1675 y FG(or)27 b(\()p Fx("@)942 1644 y Fz(2)937 1704 y Fq(\014)987 1675 y Fx(f)1028 1687 y Fo(0)1069 1675 y FG(\()p Fw(\014)1156 1687 y Fz(0)1194 1675 y FG(\)\))1258 1644 y FC(\000)p Fz(1)1347 1675 y FG(.)189 1832 y(The)e(construction)f(will)h(b)r(e)g(p)r(erformed)g(in)g (suc)n(h)f(a)h(w)n(a)n(y)e(that)i(the)h(matrices)e(\()p Fx(x)2726 1802 y Fz(2)2777 1832 y FA(\000)13 b(M)2955 1802 y Fz([)p FC(\024)p Fu(p)p Fz(])3082 1832 y FG(\))25 b(will)g(b)r(e)g(de\014ned)118 1938 y(b)n(y)f(series)f(whic)n(h)h FF(wil)t(l)j(b)l(e)g(pr)l(ove)l(d)g(to)g(b)l(e)f(c)l(onver)l(gent)p FG(;)f(furthermore)e(if)i(w)n(e)e(only)h(considered)f(the)h(con)n (tributions)118 2044 y(to)h(the)h(formal)e(represen)n(tation)g(of)h Fy(h)h FG(coming)e(from)h(trees)g FF(in)j(which)h(no)e(pr)l(op)l (agator)j(c)l(arries)e(the)g(\\last)g(lab)l(el")118 2151 y FG([)p FA(\025)23 b Fx(n)p FG(])29 b FF(then)h(the)g(c)l(orr)l(esp)l (onding)h(series)f(would)h(b)l(e)f(c)l(onver)l(gent)p FG(.)189 2290 y(W)-7 b(e)26 b(express)f(the)h(latter)g(prop)r(ert)n(y)f (b)n(y)g(sa)n(ying)g(that)h FF(the)i(p)l(erforme)l(d)i(r)l(esummations) e(r)l(e)l(gularize)h(the)f(formal)118 2396 y(r)l(epr)l(esentation)37 b(of)g Fy(h)f FF(down)h(to)f(sc)l(ale)h FG([)p Fx(n)22 b FA(\000)h FG(1])p FF(,)38 b(or)e(that)h(the)f(pr)l(op)l(agators)i (singularities)f(ar)l(e)g(pr)l(ob)l(e)l(d)g(down)118 2502 y(to)e(sc)l(ale)h FG([)p Fx(n)22 b FA(\000)g FG(1].)53 b(The)34 b(problem)e(of)i(course)e(remains)g(to)h(understand)g(the)h (con)n(tributions)e(from)h(the)h(trees)118 2609 y(con)n(taining)e (lines)i(with)f(lab)r(el)h([)p FA(\025)e Fx(n)p FG(].)54 b(The)33 b(construction)f(will)i(b)r(e)f(suc)n(h)g(that)h(their)f (propagators)d(are)j(also)118 2715 y(prop)r(erly)c(de\014ned)h(b)r (ecause)f(the)h(matrices)f FA(M)1630 2685 y Fz([)p FC(\024)p Fu(n)p Fz(])1795 2715 y FG(will)h(alw)n(a)n(ys)d(b)r(e)k(w)n(ell)e (de\014ned)h(b)n(y)g(con)n(v)n(ergen)n(t)d(series)i(\(as)118 2821 y(w)n(e)g(shall)f(see\).)41 b(Ho)n(w)n(ev)n(er)27 b(for)h(the)i(lines)f(whose)f(lab)r(el)h(is)g([)p FA(\025)c Fx(n)p FG(])j(no)h(useful)g(p)r(ositiv)n(e)g(lo)n(w)n(er)e(b)r(ound,)j (not)f(ev)n(en)118 2928 y Fx(n)p FG(-dep)r(enden)n(t,)36 b(can)e(b)r(e)g(giv)n(en)f(on)h(the)g(smallest)g(eigen)n(v)-5 b(alue)33 b(of)h(the)g(denominators)f(in)h(the)g(corresp)r(onding)118 3034 y(propagators.)189 3173 y(W)-7 b(e)40 b(shall)g(sa)n(y)f(that)h (the)g(lines)g(with)h(scale)e([)p FA(\025)k Fx(n)p FG(])d(prob)r(e)f (the)i(singularit)n(y)d(all)i(the)g(w)n(a)n(y)f(do)n(wn)g(to)h(the)118 3279 y(smallest)26 b(frequencies)f(or)g(all)h(the)g(w)n(a)n(y)f(do)n (wn)g(in)h(the)h FF(infr)l(ar)l(e)l(d)g FG(scales.)35 b(Th)n(us)26 b(in)g(spite)g(of)g(the)g(con)n(v)n(ergence)e(of)118 3386 y(the)30 b(con)n(tributions)f(to)g Fy(h)h FG(coming)f(from)g (trees)g(with)h(lab)r(els)g([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g FG([1])p Fx(;)g(:)g(:)g(:)f(;)h FG([)p Fx(n)19 b FA(\000)g FG(1])30 b(the)g(represen)n(tation)e(of)118 3492 y Fy(h)g FG(remains)f(formal.)189 3631 y(Therefore)18 b(w)n(e)h(shall)g(pro)r(ceed)g(b)n(y)h(increasing)e(the)h(v)-5 b(alue)20 b(of)f Fx(n)h FG(trying)f(to)g(tak)n(e)g(the)h(limit)g Fx(n)j FA(!)g(1)p FG(.)34 b(This)20 b(is)f(the)118 3737 y(pro)r(cedure)24 b(follo)n(w)n(ed)h(in)g(the)h(case)e(of)h(the)h (theory)e(of)h(h)n(yp)r(erb)r(olic)g(tori)g(in)g(Ref.)37 b([GG].)26 b(In)f(that)h(case,)f(ho)n(w)n(ev)n(er,)118 3844 y(the)30 b(propagators)c(denominators)i(\()p Fx(x)1320 3814 y Fz(2)1378 3844 y FA(\000)19 b(M)1562 3814 y Fz([)p FC(\024)p Fu(n)p Fz(])1696 3844 y FG(\))30 b(had)f(eigen)n(v)-5 b(alues)28 b FF(always)33 b(b)l(ounde)l(d)e(b)l(elow)i(pr)l(op)l (ortional)t(ly)118 3950 y(to)e Fx(x)266 3920 y Fz(2)304 3950 y FG(.)41 b(Indeed)30 b(the)f(last)g Fx(s)g FG(eigen)n(v)-5 b(alues)28 b(of)h FA(M)1638 3920 y Fz([)p FC(\024)p Fu(n)p Fz(])1802 3950 y FG(w)n(ere)f(negativ)n(e)g(whereas)g(the)h(\014rst)g Fx(r)j FG(remained)c(close)h(to)118 4056 y(zero)h(within)i Fx(O)r FG(\()p Fx("x)742 4026 y Fz(2)780 4056 y FG(\))f(\(a)g (non-trivial)f(prop)r(ert)n(y)-7 b(,)31 b(ho)n(w)n(ev)n(er,)f(due)h(to) g(remark)-5 b(able)30 b(cancellations)f(w)n(ell)i(kno)n(wn)118 4163 y(in)d(the)g(KAM)g(theory)-7 b(,)27 b([Ga]\).)189 4302 y(Here)h(the)g(matrices)f Fx(x)909 4272 y Fz(2)966 4302 y FA(\000)18 b(M)1149 4272 y Fz([)p FC(\024)p Fu(n)p Fz(])1311 4302 y FG(will)28 b(b)r(e)h(sho)n(wn)e(to)h(ha)n(v)n(e)f(the) h(\014rst)g Fx(r)j FG(eigen)n(v)-5 b(alues)26 b(di\013ering)i(b)n(y)g (a)f(factor)118 4408 y(\(1)e(+)f Fx(O)r FG(\()p Fx(")442 4378 y Fz(2)480 4408 y FG(\)\))37 b(and)g(the)h(last)e Fx(s)h FG(di\013ering)g(b)n(y)g Fx(O)r FG(\()p Fx(")1743 4378 y Fz(2)1781 4408 y FG(\))g(with)g(resp)r(ect)g(to)g(those)g(of)g Fx(x)2831 4378 y Fz(2)2893 4408 y FA(\000)24 b Fx(M)3063 4420 y Fz(0)3137 4408 y FG(\(whic)n(h)37 b(has)g(b)n(y)118 4514 y(construction)25 b Fx(r)j FG(eigen)n(v)-5 b(alues)24 b Fx(x)1131 4484 y Fz(2)1195 4514 y FG(and)h Fx(s)g FG(eigen)n(v)-5 b(alues)25 b Fx(x)1893 4484 y Fz(2)1945 4514 y FA(\000)13 b Fx("a)2106 4526 y Fu(j)2167 4514 y Fx(j)28 b FG(=)22 b(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(s)p FG(\).)37 b(Th)n(us)25 b(the)h(denominators)e(can)118 4621 y(b)r(ecome)32 b(small)f(b)r (ecause)g FF(either)i Fx(x)1238 4591 y Fz(2)1307 4621 y FG(gets)e(close)g(to)h(0)f FF(or)h FG(b)r(ecause)f(it)i(gets)e(close) g(to)g Fx("a)2938 4633 y Fz(1)2975 4621 y Fx(;)14 b(:)g(:)g(:)g("a)3206 4633 y Fu(s)3241 4621 y FG(.)49 b(Therefore)118 4727 y(the)32 b(regularization)e(will)i(ha)n(v)n(e)f(to)g(b)r(e)h(split)h (in)f(t)n(w)n(o)f(parts.)49 b(The)32 b(\014rst)f(part)g(will)i(concern) d(regularizing)g(the)118 4833 y(scales)24 b([)p Fx(p)p FG(])i(with)g Fx(p)f FG(suc)n(h)g(that)g(the)h(eigen)n(v)-5 b(alues)24 b(of)i Fx(x)1785 4803 y Fz(2)1836 4833 y FA(\000)14 b(M)2015 4803 y Fz([)p FC(\024)p Fu(p)p Fz(])2168 4833 y FG(remains)24 b(b)r(ounded)i(b)r(elo)n(w)f(prop)r(ortionally)f(to)118 4940 y Fx(x)165 4909 y Fz(2)203 4940 y FG(;)i(w)n(e)g(shall)f(call)h (this)g(part)f(of)h(the)g(analysis)f(the)h FF(high)k(fr)l(e)l(quencies) e(r)l(esummation)p FG(.)37 b(The)26 b(other)f(part,)h(whic)n(h)118 5046 y(w)n(e)g(shall)h(call)f(the)h FF(infr)l(ar)l(e)l(d)j(r)l (esummation)p FG(,)c(will)h(concern)f(the)h(regularization)d(of)j(the)g (scales)f([)p Fx(p)p FG(],)g(in)h(whic)n(h)g Fx(x)118 5152 y FG(can)e(b)r(e)h(so)f(close)g(to)h(some)f Fx("a)1065 5164 y Fu(j)1125 5152 y FG(that)h(the)g(denominators)f(cannot)g(b)r(e)h (b)r(ounded)g(b)r(elo)n(w)f(prop)r(ortionally)f(to)i Fx(x)3601 5122 y Fz(2)3638 5152 y FG(.)189 5291 y(W)-7 b(e)31 b(asso)r(ciate)f(with)h(eac)n(h)f(momen)n(tum)h Fw(\027)37 b FG(the)31 b(frequency)f Fx(x)f FG(=)f Fw(!)c FA(\001)c Fw(\027)37 b FG(and)31 b(w)n(e)f(measure)g(the)h(strength)g (of)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)f(19:31)1151 b FG(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 118 356 a FG(10:)27 b FF(De)l(gener)l(ate)i(el)t (liptic)j(r)l(esonanc)l(es)118 555 y FG(this)c(resonance)e(b)n(y)h(the) h(in)n(teger)f Fx(p)g FG(if)h Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b FA(')f Fx(C)1772 525 y Fz(2)1766 576 y(0)1809 555 y FG(2)1851 525 y FC(\000)p Fz(2)p Fu(p)1974 555 y FG(,)28 b(with)1044 782 y Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b(=)e(min)1467 834 y Fu(j)1565 687 y Fr(\014)1565 736 y(\014)1565 786 y(\014)1593 782 y Fx(x)1640 748 y Fz(2)1696 782 y FA(\000)c Fx(\025)p 1779 795 49 4 v 1828 739 a Fz([0])1828 805 y Fu(j)1903 782 y FG(\()p Fx(")p FG(\))2006 687 y Fr(\014)2006 736 y(\014)2006 786 y(\014)2047 731 y Fu(def)2067 782 y FG(=)2166 687 y Fr(\014)2166 736 y(\014)2166 786 y(\014)2194 782 y Fx(x)2241 748 y Fz(2)2297 782 y FA(\000)g Fx(\025)p 2380 795 V -43 x Fz([0])2428 810 y Fu(j)2455 818 y Fs(")2488 810 y Fz(\()p Fu(x)p Fz(\))2582 782 y FG(\()p Fx(")p FG(\))2685 687 y Fr(\014)2685 736 y(\014)2685 786 y(\014)2713 782 y Fx(:)755 b FG(\(4)p Fx(:)p FG(1\))118 1018 y(Therefore)25 b(the)i(condition)f(that)h(the)g(resonance)d(strength)i(of)h(the)g (frequency)f Fx(x)g FG(b)r(e)h(b)r(ounded)g(b)r(elo)n(w)f(prop)r(or-) 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b(with)p 463 1649 V 463 1695 a Fx(n)512 1707 y Fz(0)573 1695 y FG(=)22 b Fx(n)710 1707 y Fz(0)766 1695 y FG(+)p 849 1649 V 18 w Fx(n;)p 1102 1649 V 180 w(n)1165 1644 y Fu(def)1185 1695 y FG(=)53 b FA(\000)18 b FG(1)g(+)1539 1638 y(1)p 1539 1675 42 4 v 1539 1752 a(2)1604 1695 y(log)1711 1715 y Fz(2)1773 1638 y FG(1)p 1772 1675 43 4 v 1772 1752 a Fx(\032)1825 1695 y(;)180 b(\032)23 b FG(=)2192 1638 y(1)p 2192 1675 42 4 v 2192 1752 a(4)2243 1695 y Fx(a)2287 1660 y FC(\000)p Fz(1)2287 1715 y Fu(s)2390 1695 y FG(min)p FA(f)p Fx(a)2614 1707 y Fz(1)2651 1695 y Fx(;)14 b FG(min)2742 1747 y Fu(j)2826 1695 y FA(f)p Fx(a)2912 1707 y Fu(j)s Fz(+1)3049 1695 y FA(\000)k Fx(a)3176 1707 y Fu(j)3211 1695 y FA(gg)p Fx(:)173 b FG(\(4)p Fx(:)p FG(2\))118 1931 y(In)32 b(fact)g(the)g (requiremen)n(t)f(could)g(b)r(e)h(ful\014lled)h(with)p 1844 1886 50 4 v 32 w Fx(n)f FG(one)f(unit)h(larger:)43 b(the)33 b(in)n(terest)e(of)g(using)h(the)g(ab)r(o)n(v)n(e)118 2038 y(v)-5 b(alue)28 b(of)p 427 1992 V 27 w Fx(n)g 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23 w Fx(n)2522 2570 y Fz(0)2560 2558 y Fx(;)908 b FG(\(4)p Fx(:)p FG(3\))118 2772 y(so)27 b(that)g(if)h(the)g(lab)r(el)g Fx(p)f FG(of)g(the)h(line)g (with)f(frequency)g Fx(x)h FG(is)f Fx(p)c FA(\024)p 2157 2727 V 23 w Fx(n)2207 2784 y Fz(0)2271 2772 y FG(then)28 b(one)f(has,)g(if)2869 2740 y Fz(1)p 2869 2754 34 4 v 2869 2801 a(2)2912 2772 y Fx(a)2956 2784 y Fu(s)2992 2772 y FG(2)3034 2742 y FC(\000)p Fz(2\()p 3145 2707 42 4 v Fu(n)o Fz(+1\))3317 2772 y FA(\000)18 b Fx(\015)5 b(")22 b FA(\025)h FG(0,)589 3017 y FA(j)p Fx(x)659 2983 y Fz(2)715 3017 y FA(\000)18 b Fx(\025)846 2974 y Fz([)p Fu(p)p Fz(])846 3040 y Fu(j)923 3017 y FG(\()p Fx(x;)c(")p FG(\))p FA(j)23 b(\025)1254 2961 y FG(1)p 1254 2998 V 1254 3074 a(2)1305 3017 y Fx(D)r FG(\()p Fx(x;)14 b(")p FG(\))19 b(+)1675 2961 y(1)p 1675 2998 V 1675 3074 a(2)1727 3017 y Fx(D)r FG(\()p Fx(x;)14 b(")p FG(\))19 b FA(\000)f Fx(\015)5 b(")2174 2983 y Fz(2)2234 3017 y FA(\025)2331 2961 y FG(1)p 2331 2998 V 2331 3074 a(2)2383 3017 y Fx(D)r 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b(\014rst)g(graph)f(is)g Fj( )1643 3892 y Fe(0)1678 3883 y Fk(,)h(the)h(second)f(is)f Fj(\037)2162 3892 y Fe(0)2214 3883 y Fk(and)h(the)h(third)e(is)p 2696 3845 44 4 v 17 w Fj(\037)2740 3901 y Fe(0)2794 3883 y Fd(\021)j Fj( )2915 3892 y Fe(1)2949 3883 y Fj(\037)2993 3892 y Fe(0)3028 3883 y Fk(.)189 4078 y FG(Instead)25 b(of)h(the)f(sharp)g(m)n (ultiscale)g(decomp)r(osition)g(considered)f(in)i(Ref.)36 b([GG])26 b(here)f(it)h(will)g(b)r(e)g(con)n(v)n(enien)n(t)118 4184 y(to)k(w)n(ork)f(with)i(a)e(smo)r(oth)h(one)g(as)g(in)g(Ref.)45 b([Ge].)g(Let)31 b Fx( )s FG(\()p Fx(D)r FG(\))g(b)r(e)f(a)g Fx(C)2380 4154 y FC(1)2481 4184 y FG(non-decreasing)e(compact)i(supp)r (ort)118 4290 y(function)e(de\014ned)g(for)f Fx(D)e FA(\025)e FG(0,)k(see)h(Fig.)36 b(2,)27 b(suc)n(h)h(that)804 4443 y Fx( )s FG(\()p Fx(D)r FG(\))c(=)e(1)p Fx(;)97 b FG(for)82 b Fx(D)25 b FA(\025)e Fx(C)1698 4409 y Fz(2)1692 4464 y(0)1735 4443 y Fx(;)180 b( )s FG(\()p Fx(D)r FG(\))24 b(=)f(0)p Fx(;)96 b FG(for)82 b Fx(D)26 b 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Fx(;)181 b FG(for)27 b(all)g Fx(n)c FA(\025)g FG(0)p Fx(;)719 b FG(\(5)p Fx(:)p FG(2\))118 5008 y(for)29 b(all)h(c)n(hoices)f(of)h (the)g(function)g(\001\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b FA(\025)e FG(0:)41 b(in)31 b(particular)d(for)h(\001\()p Fx(x;)14 b(")p FG(\))28 b(=)f Fx(D)r FG(\()p Fx(x)p FG(\))k(with)f Fx(D)r FG(\()p Fx(x)p FG(\))h(de\014ned)f(in)118 5114 y(\(5.3\))d(b)r(elo)n(w.)37 b(W)-7 b(e)28 b(set)g(the)g(follo)n(wing)e (notations.)118 5291 y FB(Definition)32 b(1.)k FF(L)l(et)p 841 5246 50 4 v 30 w Fx(n)890 5303 y Fz(0)928 5291 y Fx(;)p 965 5246 V 14 w(n)29 b FF(b)l(e)h(as)g(in)g(\(4.2\))h(and)f Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))30 b FF(as)h(in)e (\(4.1\).)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)h(19:31)1130 b FG(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 118 356 a FG(12:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y(\(i\))g(Divide)i(the)e(interval)h Fx(I)991 567 y Fu(C)1074 555 y FA(\021)27 b FG([)p Fx(")1228 567 y Fz(min)1342 555 y Fx(;)14 b FG(4)p Fx(")1460 567 y Fz(min)1573 555 y FG(])p FF(,)33 b(wher)l(e)g Fx(")f FF(varies,)i(se)l(e)e(\(3.2\),)i(into)e(a)h(\014nite)e(numb)l(er)h(of)g (smal)t(l)118 662 y(intervals)37 b Fx(I)43 b FF(of)37 b(size)g(smal)t(ler)g(than)1320 629 y Fz(1)p 1320 643 34 4 v 1320 690 a(2)1363 662 y Fx(")1402 674 y Fz(min)1516 662 y Fx(\032)p FF(,)h(se)l(e)e(\(4.2\),)j(i.e.)60 b(smal)t(ler)37 b(than)f(a)h(fr)l(action)g(of)g(the)g(minimum)118 768 y(sep)l(ar)l(ation)31 b(b)l(etwe)l(en)e(the)h(eigenvalues)h FG(0)p Fx(;)14 b(a)1515 780 y Fz(1)1552 768 y Fx(;)g(:)g(:)g(:)f(;)h(a) 1780 780 y Fu(s)1816 768 y FF(.)38 b(De\014ne)654 923 y Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(I)7 b FG(\))24 b(=)f(min)1042 976 y Fu(")p FC(2)p Fu(I)1180 923 y Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b(=)e(min)1563 976 y Fu(")p FC(2)p Fu(I)1701 923 y FG(min)1755 975 y Fu(j)1854 827 y Fr(\014)1854 877 y(\014)1854 927 y(\014)1881 923 y Fx(x)1928 889 y Fz(2)1985 923 y FA(\000)c Fx(\025)p 2068 936 49 4 v -43 x Fz([0])2116 946 y Fu(j)2191 923 y FG(\()p Fx(")p FG(\))2294 827 y Fr(\014)2294 877 y(\014)2294 927 y(\014)2345 923 y FG(=)k(min)2447 976 y Fu(")p FC(2)p Fu(I)2585 827 y Fr(\014)2585 877 y(\014)2585 927 y(\014)2612 923 y Fx(x)2659 889 y Fz(2)2715 923 y FA(\000)d Fx(\025)p 2799 936 V -43 x Fz([0])2847 951 y Fu(j)s Fz(\()p Fu(x)p Fz(\))2971 923 y FG(\()p Fx(")p FG(\))3074 827 y Fr(\014)3074 877 y(\014)3074 927 y(\014)3102 923 y Fx(:)366 b FG(\(5)p Fx(:)p FG(3\))118 1100 y FF(wher)l(e)30 b Fx(j)5 b FG(\()p Fx(x)p FG(\))30 b FF(is)f(the)h(smal)t(lest)f(value)h(of)g Fx(j)k FF(for)c(which)h(the)e(last)g(e)l(quality)h(holds:)40 b(exc)l(eptional)t(ly)30 b(ther)l(e)g(might)f(b)l(e)118 1207 y FG(2)g FF(such)h(lab)l(els.)40 b(The)30 b Fx(j)5 b FG(\()p Fx(x)p FG(\))31 b FF(is)f Fx(")p FF(-indep)l(endent,)h(by)f (c)l(onstruction,)f(for)i Fx(")23 b FA(2)g Fx(I)7 b FF(.)118 1384 y(R)l(emarks.)57 b FG(\(1\))34 b(Note)g(that,)i(as)d(a)h (consequence)f(of)h(the)g(de\014nition)h(of)f(the)g(in)n(terv)-5 b(als)33 b Fx(I)41 b FG(and)34 b(of)g Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(I)7 b FG(\))35 b(as)118 1490 y(giv)n(en)27 b(b)n(y)g(\(5.3\),)h(one)f(has,)g(for)g(all)g Fx(")c FA(2)g Fx(I)7 b FG(,)1223 1714 y(min)1277 1767 y Fu(j)1375 1619 y Fr(\014)1375 1669 y(\014)1375 1719 y(\014)1403 1714 y Fx(x)1450 1680 y Fz(2)1506 1714 y FA(\000)18 b Fx(\025)p 1589 1727 V 1638 1671 a Fz([0])1638 1738 y Fu(j)1712 1714 y FG(\()p Fx(")p FG(\))1815 1619 y Fr(\014)1815 1669 y(\014)1815 1719 y(\014)1866 1714 y FA(\025)1964 1658 y FG(1)p 1964 1695 42 4 v 1964 1771 a(2)2016 1619 y Fr(\014)2016 1669 y(\014)2016 1719 y(\014)2043 1714 y Fx(x)2090 1680 y Fz(2)2146 1714 y FA(\000)g Fx(\025)p 2229 1727 49 4 v 2278 1671 a Fz([0])2278 1743 y Fu(j)s Fz(\()p Fu(x)p Fz(\))2402 1714 y FG(\()p Fx(")p FG(\))2505 1619 y Fr(\014)2505 1669 y(\014)2505 1719 y(\014)2533 1714 y Fx(;)935 b FG(\(5)p Fx(:)p FG(4\))118 1956 y(\(2\))26 b(If)g Fx(")g FG(is)g(in)g(one)f(of)h(the)g(in)n(terv)-5 b(als)25 b Fx(I)33 b FG(and)25 b Fx(x)i FG(v)n(eri\014es)d Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(I)7 b FG(\))24 b FA(\024)f Fx(C)2304 1926 y Fz(2)2298 1977 y(0)2342 1956 y FG(2)2384 1926 y FC(\000)p Fz(2)p 2469 1891 42 4 v Fu(n)2509 1934 y Ft(0)2572 1956 y FG(then)j(there)f(is)h(only)f(one)h(v)-5 b(alue)26 b(of)118 2063 y Fx(j)33 b FG(for)27 b(whic)n(h)g(last)h (equalit)n(y)f(in)g(\(5.3\))h(holds.)118 2169 y(\(3\))g FF(We)i(shal)t(l)h(\014x,)f(fr)l(om)g(now)g(on,)g Fx(")g FF(in)g(one)g(of)h(the)f(intervals)g Fx(I)g FA(\022)23 b Fx(I)2345 2181 y Fu(C)2401 2169 y FG(.)38 b(Remark)27 b(that)h Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(I)7 b FG(\))28 b(is)g(piecewise)118 2275 y(linear)e(in)h Fx(x)491 2245 y Fz(2)555 2275 y FG(with)g(slop)r(e)f(equal)g(to)h(1)f(in)h(absolute)f (v)-5 b(alue)26 b(for)g Fx(x)h FG(in)g(the)g(regions)e(where)h(it)h (will)g(b)r(e)g(considered)118 2382 y(\(see)h(b)r(elo)n(w\))f(and)h(w)n (e)f(simplify)h(the)g(notation)f(b)n(y)g(setting)1589 2545 y Fx(D)r FG(\()p Fx(x)p FG(\))1785 2494 y Fu(def)1806 2545 y FG(=)34 b Fx(D)r FG(\()p Fx(x)p FG(;)14 b Fx(I)7 b FG(\))p Fx(:)1301 b FG(\(5)p Fx(:)p FG(5\))118 2793 y(\(4\))28 b(The)g(n)n(um)n(b)r(er)f(of)h(in)n(terv)-5 b(als)27 b Fx(I)k FA(\032)23 b Fx(I)1344 2805 y Fu(C)1428 2793 y FG(can)k(and)h(will)g(b)r(e)g(tak)n(en)f(indep)r(enden)n(t)i(of) f Fx(")2843 2805 y Fz(min)2957 2793 y FG(,)g(i.e.)37 b(of)28 b(the)g(in)n(terv)-5 b(al)118 2899 y Fx(I)154 2911 y Fu(C)238 2899 y FG(where)27 b Fx(")h FG(v)-5 b(aries,)26 b(and)i(equal)f(to)g(a)g(\014xed)h(in)n(teger)f FA(\024)22 b FG(6)p Fx(\032)2000 2869 y FC(\000)p Fz(1)2089 2899 y FG(.)118 3005 y(\(5\))28 b(F)-7 b(rom)27 b(no)n(w)g(on)g(w)n(e)h (only)f(consider)f(trees)h(with)h(no)g(trivial)f(no)r(des.)189 3182 y(A)d(simple)g(w)n(a)n(y)e(to)i(represen)n(t)e(the)i(v)-5 b(alue)24 b(of)f(a)g(tree)h(as)f(sum)g(of)h(man)n(y)f(terms)g(is)h(to)f (mak)n(e)g(use)h(of)f(the)h(iden)n(tit)n(y)118 3307 y(in)f(\(5.2\).)35 b(The)22 b(propagator)e Fx(g)s FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))23 b FA(\021)g Fx(g)1408 3276 y Fz([)p FC(\025)p Fz(0])1534 3307 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))1735 3256 y Fu(def)1756 3307 y FG(=)34 b(\()p Fx(x)1934 3276 y Fz(2)1980 3307 y FA(\000)8 b Fx(M)2134 3319 y Fz(0)2170 3307 y FG(\))2202 3276 y FC(\000)p Fz(1)2314 3307 y FG(of)23 b(eac)n(h)f(line)g(with)h(non-zero)e(momen)n(tum)118 3413 y(\(hence)28 b(with)g Fx(x)c FA(6)p FG(=)e(0\))28 b(is)f(written)h(as)322 3585 y Fx(g)365 3551 y Fz([)p FC(\025)p Fz(0])491 3585 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b(=)f Fx( )844 3597 y Fz(0)881 3585 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))14 b Fx(g)1184 3551 y Fz([)p FC(\025)p Fz(0])1312 3585 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))19 b(+)f Fx(\037)1653 3597 y Fz(0)1690 3585 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))c Fx(g)1993 3551 y Fz([)p FC(\025)p Fz(0])2121 3585 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))2322 3534 y Fu(def)2342 3585 y FG(=)34 b Fx(g)2484 3551 y Fz([0])2558 3585 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))20 b(+)e Fx(g)2891 3480 y Fr(\010)2939 3547 y FC(\025)p Fz(1)3024 3480 y Fr(\011)3076 3585 y FG(\()p Fx(x)p FG(;)c Fx(")p FG(\))p Fx(;)205 b FG(\(5)p Fx(:)p FG(6\))118 3727 y(and)28 b(w)n(e)f(note)g(that)h Fx(g)809 3697 y Fz([0])884 3727 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b(v)-5 b(anishes)27 b(if)h Fx(D)r FG(\()p Fx(x)p FG(\))h(is)e(smaller)g (than)h(\()p Fx(C)2367 3739 y Fz(0)2405 3727 y Fx(=)p FG(2\))2521 3697 y Fz(2)2557 3727 y FG(,)g(see)f(Fig.)37 b(2.)189 3833 y(If)28 b(w)n(e)e(replace)g(eac)n(h)h Fx(g)904 3803 y Fz([)p FC(\025)p Fz(0])1030 3833 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b(with)g(the)f(sum)g(in)h(\(5.6\))f(then)g (the)h(v)-5 b(alue)27 b(of)g(eac)n(h)f(tree)h(of)g(order)f Fx(k)k FG(is)d(split)118 3939 y(as)d(a)f(sum)h(of)g(up)h(to)f(2)802 3909 y Fu(k)866 3939 y FG(terms)1070 3909 y Fz(2)1131 3939 y FG(whic)n(h)g(can)g(b)r(e)g(iden)n(ti\014ed)h(b)n(y)f(a\016xing) f(on)h(eac)n(h)f(line)h(with)h(momen)n(tum)f Fw(\027)29 b FA(6)p FG(=)23 b Fy(0)118 4046 y FG(a)k(lab)r(el)h([0])f(or)605 3978 y Fr(\010)676 4046 y FA(\025)c FG(1)806 3978 y Fr(\011)854 4046 y FG(.)37 b(F)-7 b(urther)27 b(splittings)h(of)g(the)g(tree)f(v)-5 b(alues)27 b(can)g(b)r(e)h(ac)n(hiev)n(ed)f(as)g(follo)n(ws.)118 4241 y FB(Definition)46 b(2.)75 b FF(F)-6 b(or)42 b Fx(p)i FG(=)h(1)p Fx(;)14 b(:)g(:)g(:)f(;)p 1339 4195 50 4 v 14 w(n)1388 4253 y Fz(0)1426 4241 y FF(,)45 b(let)c FA(M)1724 4210 y Fz([)p Fu(p)p Fz(])1800 4241 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))42 b FF(b)l(e)g(matric)l(es)g(with)g (eigenvalues)h Fx(\025)3176 4197 y Fz([)p Fu(p)p Fz(])3176 4264 y Fu(j)3253 4241 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FF(,)45 b Fx(p)g FG(=)118 4357 y(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(n)p FF(;)30 b(we)g(set)g FA(M)802 4327 y Fz([0])876 4357 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b FA(\021)e Fx(M)1255 4369 y Fz(0)1322 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FF(,)g(and)g(furthermor)l(e)2231 883 y Fr(P)2319 970 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2578 945 y Fw(\027)2626 957 y Fm(v)2696 945 y FG(=)22 b Fy(0)p FF(.)118 1052 y(\(ii\))31 b(The)f(de)l(gr)l(e)l (e)h(of)f(a)g(self-ener)l(gy)h(cluster)e(is)h(the)g(numb)l(er)f(of)i (no)l(des.)118 1229 y(R)l(emark.)38 b FG(The)27 b(essen)n(tial)g(prop)r (ert)n(y)g(of)g(a)h(self-energy)e(cluster)h(is)h(that)g(it)g(has)f (necessarily)f(just)i(one)f(en)n(tering)118 1335 y(line)j(and)g(one)g (exiting)g(line,)h(and)e(b)r(oth)i(ha)n(v)n(e)e FF(e)l(qual)j(momentum) d FG(\(b)r(ecause)2599 1273 y Fr(P)2687 1360 y Fm(v)o FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2946 1335 y Fw(\027)2994 1347 y Fm(v)3067 1335 y FG(=)27 b Fy(0)p FG(\).)44 b(Note)30 b(that)118 1441 y(b)r(oth)d(scales)f(of)h(the)g(external)f(lines)h(of)g (a)g(self-energy)e(cluster)i Fx(T)38 b FG(are)25 b(strictly)i(larger)e (than)i(the)g(scale)f(of)h Fx(T)38 b FG(as)118 1548 y(a)28 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FF(the)f(set)h(of)g (self-ener)l(gy)g(clusters)g(of)g(or)l(der)g Fx(k)j FF(and)d(sc)l(ale)g FG([)p Fx(n)p FG(])f FF(which)j(do)e(not)f(c)l(ontain)118 2256 y(any)34 b(other)g(self-ener)l(gy)g(cluster)f(nor)h(any)g(trivial) h(no)l(de;)h(we)e(c)l(al)t(l)g(them)g FG(renormalized)c(self-energy)g (clusters)118 2363 y FF(on)g(sc)l(ale)g Fx(n)p FF(.)118 2469 y(\(iii\))h(Given)f(a)g(self-ener)l(gy)g(cluster)f Fx(T)34 b FA(2)23 b(S)1501 2439 y FC(R)1495 2492 y Fu(k)q(;n)1627 2469 y FF(we)29 b(shal)t(l)i(de\014ne)f(the)f(self-ener)l(gy)h(value)g (of)h Fx(T)40 b FF(as)30 b(the)f(matrix)3624 2439 y Fz(3)981 2691 y FA(V)1032 2703 y Fu(T)1084 2691 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))23 b(=)1620 2635 y Fx(")1659 2605 y Fu(k)p 1522 2672 277 4 v 1522 2748 a FG(\()p Fx(k)e FA(\000)d FG(1\)!)1808 2599 y Fr(\020)1927 2612 y(Y)1871 2794 y Fu(`)p FC(2)p Fz(\003\()p Fu(T)9 b Fz(\))2103 2691 y Fx(g)2146 2648 y Fz([)p Fu(n)2206 2657 y Fs(`)2235 2648 y Fz(])2143 2716 y Fu(`)2258 2599 y Fr(\021)o(\020)2438 2612 y(Y)2371 2794 y Fm(v)o FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2625 2691 y Fx(F)2678 2703 y Fm(v)2726 2599 y Fr(\021)2775 2691 y Fx(;)693 b FG(\(5)p Fx(:)p FG(8\))118 2964 y FF(wher)l(e)34 b Fx(g)399 2921 y Fz([)p Fu(n)459 2930 y Fs(`)488 2921 y Fz(])396 2990 y Fu(`)540 2964 y FG(=)29 b Fx(g)677 2934 y Fz([)p Fu(n)737 2943 y Fs(`)766 2934 y Fz(])789 2964 y FG(\()p Fw(!)24 b FA(\001)d Fw(\027)997 2976 y Fu(`)1029 2964 y FG(;)14 b Fx(")p FG(\))p FF(.)49 b(Note)33 b(that,)i(ne)l(c)l(essarily,)h Fx(n)2109 2976 y Fu(`)2170 2964 y FA(\024)29 b Fx(n)p FF(.)50 b(The)34 b Fx(k)2605 2976 y Fu(T)2679 2964 y FA(\000)20 b FG(1)33 b FF(lines)h(of)g(the)g (self-ener)l(gy)118 3071 y(cluster)29 b Fx(T)41 b FF(wil)t(l)31 b(b)l(e)f(imagine)l(d)h(as)f(distinct)g(and)h(to)e(c)l(arry)i(a)f(numb) l(er)f(lab)l(el)i(r)l(anging)f(in)g FA(f)p FG(1)p Fx(;)14 b(:)g(:)g(:)e(;)i(k)3233 3083 y Fu(T)3304 3071 y FA(\000)k FG(1)p FA(g)p FF(.)189 3248 y FG(The)28 b(recursiv)n(e)d(de\014nition)j (of)g(the)g(matrices)f FA(M)1746 3218 y Fz([)p Fu(n)p Fz(])1828 3248 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)29 b Fx(n)23 b FA(\025)f FG(1,)27 b(will)h(b)r(e)g(\(if)h(the)f(series)e (con)n(v)n(erges\))339 3479 y FA(M)439 3445 y Fz([)p Fu(n)p Fz(])522 3479 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b(=)820 3387 y Fr(\020)884 3375 y Fu(n)p FC(\000)p Fz(1)894 3400 y Fr(Y)888 3576 y Fu(p)p Fz(=0)1024 3479 y Fx(\037)1076 3491 y Fu(p)1114 3479 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))1360 3387 y Fr(\021)1452 3375 y FC(1)1425 3400 y Fr(X)1425 3579 y Fu(k)q Fz(=2)1649 3400 y Fr(X)1559 3587 y Fu(T)9 b FC(2S)1697 3567 y Ff(R)1693 3609 y Fs(k)q(;n)p Ff(\000)p Ft(1)1872 3479 y FA(V)1923 3491 y Fu(T)1975 3479 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))2176 3428 y Fu(def)2197 3479 y FG(=)2296 3387 y Fr(\020)2359 3375 y Fu(n)p FC(\000)p Fz(1)2369 3400 y Fr(Y)2363 3576 y Fu(p)p Fz(=0)2499 3479 y Fx(\037)2551 3491 y Fu(p)2589 3479 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))2835 3387 y Fr(\021)2886 3479 y Fx(M)2976 3445 y Fz([)p Fu(n)p Fz(])3059 3479 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))p Fx(;)222 b FG(\(5)p Fx(:)p FG(9\))118 3748 y(where)34 b(the)i(self-energy)d(v)-5 b(alues)35 b(are)f(ev)-5 b(aluated)34 b(b)n(y)h(means)f(of)h(the)g(propagators)d (on)j(scales)f([)p Fx(p)p FG(],)i(with)g Fx(p)f FG(=)118 3855 y(0)p Fx(;)14 b(:)g(:)g(:)f(;)h(n)p FG(,)29 b(whic)n(h)f(mak)n(es) f(sense)h(b)r(ecause)f(w)n(e)h(ha)n(v)n(e)f(already)g(de\014ned)i(the)f (propagators)e(on)i(scale)f([0])h(and)g(the)118 3961 y(matrices)f FA(M)551 3931 y Fz([0])625 3961 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b FA(\021)f Fx(M)1005 3973 y Fz(0)1069 3961 y FG(\(cf.)38 b(De\014nition)28 b(2\).)189 4138 y(With)g(the)g(ab)r(o)n(v)n(e)f(new)g(de\014nitions)h (w)n(e)f(ha)n(v)n(e)g(the)h(formal)f(iden)n(tities)1424 4336 y Fx(h)1472 4348 y Fq(\027)5 b Fu(;\015)1600 4336 y FG(=)1715 4232 y FC(1)1688 4257 y Fr(X)1688 4436 y Fu(k)q Fz(=1)1898 4257 y Fr(X)1823 4444 y Fu(\022)r FC(2)p Fz(\002)1953 4424 y Ff(R)1953 4466 y Fs(k)q(;\027)s(;\015)2106 4336 y FG(V)-7 b(al\()p Fx(\022)r FG(\))p Fx(;)1095 b FG(\(5)p Fx(:)p FG(10\))118 4613 y(where)27 b(w)n(e)g(ha)n(v)n(e)g (rede\014ned)g(the)h FF(value)g FG(of)g(a)f(tree)g Fx(\022)f FA(2)d FG(\002)1920 4583 y FC(R)1920 4636 y Fu(k)q(;)p Fq(\027)5 b Fu(;\015)2109 4613 y FG(as)1024 4835 y(V)-7 b(al)o(\()p Fx(\022)r FG(\))24 b(=)1370 4779 y Fx(")1409 4749 y Fu(k)p 1370 4816 80 4 v 1375 4892 a Fx(k)s FG(!)1459 4743 y Fr(\020)1571 4756 y(Y)1523 4938 y Fu(`)p FC(2)p Fz(\003\()p Fu(\022)r Fz(\))1740 4835 y Fx(g)1783 4801 y Fz([)p Fu(\021)1836 4810 y Fs(`)1865 4801 y Fz(])1887 4835 y FG(\()p Fw(!)e FA(\001)c Fw(\027)2090 4847 y Fu(`)2122 4835 y FG(;)c Fx(")p FG(\))2230 4743 y Fr(\021\020)2403 4756 y(Y)2343 4938 y Fm(v)p FC(2)p Fu(V)g Fz(\()p Fu(\022)r Fz(\))2583 4835 y Fx(F)2636 4847 y Fm(v)2683 4743 y Fr(\021)2733 4835 y Fx(;)693 b FG(\(5)p Fx(:)p FG(11\))p 118 5018 1200 4 v 109 5080 a Fz(3)189 5110 y Fk(This)25 b(is)g(a)g(matrix)g(b)r (ecause)i(the)g(self-energy)e(cluster)g(inherits)g(the)i(lab)r(els)e Fj(\015)t(;)12 b(\015)2376 5087 y Fg(0)2424 5110 y Fk(attac)n(hed)27 b(to)g(the)f(endno)r(de)h(of)e(the)h(en)n(tering)118 5185 y(line)d(and)i(to)f(the)g(initial)f(no)r(de)i(of)e(the)i(exiting)f (line.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 118 356 a FG(15:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(with)c([)p Fx(\021)371 567 y Fu(`)403 555 y FG(])c(=)e([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g(:)g(:)g(:)f(;)h FG([)p 998 510 50 4 v Fx(n)1048 567 y Fz(0)1104 555 y FA(\000)k FG(1])p Fx(;)c FG([)p FA(\025)p 1399 510 V 22 w Fx(n)1449 567 y Fz(0)1486 555 y FG(].)37 b(Note)28 b(that)g(\(5.10\))f(is)g(not)h(a)f(p)r(o)n(w)n (er)f(series)h(in)h Fx(")p FG(.)189 683 y(The)g(statemen)n(t)g(in)g (\(5.10\))f(requires)g(some)g(though)n(t,)h(but)h(it)f(turns)g(out)g (to)g(b)r(e)g(a)g(tautology)-7 b(,)27 b(see)g(also)g(Ref.)118 789 y([GG],)38 b(and)e(Ch.)65 b(VI)r(I)r(I)38 b(in)f(Ref.)65 b([GBG],)37 b FF(if)i(one)g(ne)l(gle)l(cts)e(c)l(onver)l(genc)l(e)i(pr) l(oblems)f FG(whic)n(h,)h(ho)n(w)n(ev)n(er,)e(will)118 895 y(o)r(ccup)n(y)27 b(us)h(in)g(the)g(rest)f(of)g(this)h(pap)r(er.)37 b(A)27 b(sk)n(etc)n(h)g(of)h(the)g(argumen)n(t)e(is)i(as)f(follo)n(ws.) 189 1023 y(Imagine)35 b(that)g(w)n(e)g(ha)n(v)n(e)f(only)h(scales)f([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g(:)g(:)g(:)g(;)g FG([)p Fx(n)23 b FA(\000)g FG(1])p Fx(;)14 b FG([)p FA(\025)35 b Fx(n)p FG(],)j(i.e.)60 b(w)n(e)35 b(ha)n(v)n(e)f(p)r(erformed)h(the)g (scale)118 1129 y(decomp)r(osition)25 b(of)g(the)g(propagators)d(up)k (to)f(scale)f([)p Fx(n)13 b FA(\000)g FG(1])25 b(and)g(w)n(e)g(ha)n(v)n (e)e(not)j(decomp)r(osed)e(the)i(propagators)118 1235 y(on)h(scale)g([)p FA(\025)c Fx(n)p FG(])k(and)h(that)g(w)n(e)f(ha)n(v) n(e)f(c)n(hec)n(k)n(ed)h(the)h(statemen)n(t)f(\(5.9\))h(and)f(\(5.10\)) g(\(trivially)g(true)h(for)f Fx(n)c FG(=)f(0\).)189 1362 y(Giv)n(en)35 b(a)h(tree)f Fx(\022)k FA(2)e FG(\002)924 1332 y FC(R)924 1386 y Fu(k)q(;)p Fq(\027)5 b Fu(;\015)1121 1362 y FG(with)36 b(lines)g(carrying)e(lab)r(els)h([)p Fx(p)p FG(])h(with)g Fx(p)g FG(=)g(0)p Fx(;)14 b(:)g(:)g(:)f(;)h(n)24 b FA(\000)f FG(1)36 b(or)f([)p FA(\025)h Fx(n)p FG(])f(or)g([)p FA(1)p FG(],)118 1514 y(w)n(e)c(can)f(split)i(the)f(propagators)d Fx(g)1238 1484 y Fz([)p FC(\025)p Fu(n)p Fz(])1373 1514 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))31 b(as)g Fx(g)1740 1484 y Fz([)p Fu(n)p Fz(])1822 1514 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))21 b(+)g Fx(g)2159 1409 y Fr(\010)2207 1476 y FC(\025)p Fu(n)p Fz(+1)2384 1409 y Fr(\011)2436 1514 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))32 b(as)e(in)i(\(5.6\))e (with)i Fx(g)3298 1484 y Fz([)p Fu(n)p Fz(])3380 1514 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))30 b(=)118 1652 y Fx( )172 1664 y Fu(n)217 1652 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))p Fx(g)506 1622 y Fz([)p FC(\025)p Fu(n)p Fz(])642 1652 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))27 b(and)f Fx(g)1059 1547 y Fr(\010)1107 1614 y FC(\025)p Fu(n)p Fz(+1)1284 1547 y Fr(\011)1337 1652 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))23 b(=)g Fx(\037)1687 1664 y Fu(n)1732 1652 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))p Fx(g)2021 1622 y Fz([)p FC(\025)p Fu(n)p Fz(])2157 1652 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\).)37 b(In)26 b(this)h(w)n(a)n(y) e(w)n(e)g(get)h(new)g(trees)g FF(which)118 1758 y(in)32 b(gener)l(al)g(c)l(ontain)f(self-ener)l(gy)h(clusters)f(of)h(sc)l(ale)g FG([)p Fx(n)p FG(].)43 b(W)-7 b(e)29 b(can)g(in)h(fact)f(construct)g (in\014nitely)h(man)n(y)f(trees)118 1864 y(with)g(self-energy)f (clusters)g(of)h(scale)f([)p Fx(n)p FG(])h(simply)g(b)n(y)f FF(inserting)h FG(an)g(arbitrary)e(n)n(um)n(b)r(er)h(of)h(them)h(on)e (an)n(y)g(line)118 1971 y Fx(`)f FG(with)h(scale)f FA(f\025)22 b Fx(n)d FG(+)f(1)p FA(g)p FG(.)189 2098 y(The)27 b(v)-5 b(alues)28 b(of)f(the)h(trees)f(obtained)g(b)n(y)g Fx(k)f FA(\025)c FG(0)27 b(suc)n(h)g(self-energy)f FF(insertions)i FG(on)f(a)g(giv)n(en)g(line)h(of)f(a)g(tree)g(in)118 2204 y(\002)183 2174 y FC(R)183 2228 y Fu(k)q(;)p Fq(\027)5 b Fu(;\015)374 2204 y FG(can)29 b(b)r(e)h(arranged)d(in)n(to)j(a)f (geometric)f(progression:)38 b(in)30 b(fact)g(they)g(di\013er)f(only)g (b)n(y)h(a)f(factor)g(equal)g(to)118 2356 y(the)k(v)-5 b(alue)33 b(of)g(the)g(in)n(teger)f(p)r(o)n(w)n(er)f Fx(k)36 b FG(in)d Fx(g)1485 2251 y Fr(\010)1533 2318 y FC(\025)p Fu(n)p Fz(+1)1710 2251 y Fr(\011)1762 2356 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))1949 2289 y Fr(\000)1988 2356 y Fx(M)2078 2326 y Fz([)p Fu(n)p Fz(+1])2244 2356 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))p Fx(g)2474 2251 y Fr(\010)2523 2318 y FC(\025)p Fu(n)p Fz(+1)2700 2251 y Fr(\011)2753 2356 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))2940 2289 y Fr(\001)2978 2306 y Fu(k)q Fz(+1)3136 2356 y FG(if)33 b Fx(M)3307 2326 y Fz([)p Fu(n)p Fz(+1])3474 2356 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))118 2462 y(is)36 b(de\014ned)h(as)e(in)h (\(5.9\),)i(where)e(the)g FA(V)1403 2474 y Fu(T)1456 2462 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))36 b(are)g(ev)-5 b(aluated)35 b(b)n(y)h(using)g(as)f(propagators)e Fx(g)3175 2432 y Fz([)p Fu(p)p Fz(])3251 2462 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)39 b(with)118 2569 y(0)23 b FA(\024)g Fx(p)g FA(\024)g Fx(n)28 b FG(or)f Fx(p)c FG(=)g FA(1)p FG(,)29 b(for)e(the)h(lines)g(carrying)e(a)h(scale)g(lab)r(el)h([)p Fx(p)p FG(].)38 b FF(Summation)29 b(over)i Fx(k)i FF(wil)t(l)e(simply)g (change)118 2706 y Fx(g)161 2601 y Fr(\010)209 2668 y FC(\025)p Fu(n)p Fz(+1)386 2601 y Fr(\011)439 2706 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))34 b FF(into)g Fx(g)879 2676 y Fz([)p FC(\025)p Fu(n)p Fz(+1])1097 2706 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))34 b FF(and)h(at)e(the)h(same)g (time)g(one)g(shal)t(l)h(have)g(to)f(c)l(onsider)g(only)h(tr)l(e)l(es)e (with)118 2812 y(no)h(self-ener)l(gy)h(cluster)f(of)h(sc)l(ale)f FG([)p Fx(n)p FG(])g FF(nor)g(of)h(sc)l(ale)g FG([)p Fx(p)p FG(])f FF(with)g Fx(p)d(<)f(n)k FF(and)h(with)f(lines)h(c)l (arrying)g(sc)l(ale)f(lab)l(els)118 2919 y FG([)p FA(1)p FG(])p Fx(;)14 b(:)g(:)g(:)g(;)g FG([)p Fx(n)p FG(])33 b FF(or)g FG([)p FA(\025)28 b Fx(n)21 b FG(+)f(1])p FF(.)47 b FG(In)32 b(this)f(w)n(a)n(y)f(w)n(e)h(pro)n(v)n(e)e(\(5.10\))h(for)h (all)f Fx(n)f FA(\024)p 2542 2873 V 28 w Fx(n)2592 2931 y Fz(0)2630 2919 y FG(:)43 b(w)n(e)31 b(could)g(con)n(tin)n(ue,)g(but)h (for)118 3025 y(the)c(reasons)e(outlined)i(in)g(Section)f(4,)g(w)n(e)h (decide)f(to)h(stop)f(the)h(resummations)f(at)g(this)h(scale.)189 3152 y(In)34 b(other)g(w)n(ords)e(the)j(ab)r(o)n(v)n(e)e(is)g(a)h (generalization)e(of)i(the)h(simple)f(resummation)f(considered)g(in)i (Section)118 3259 y(3.)42 b(The)29 b(result)g(is)g(still)h FF(as)h(formal)i(as)e(the)h(Lindste)l(dt)f(series)h(we)g(starte)l(d)f (with)f FG(ev)n(en)f(assuming)f(con)n(v)n(ergence)118 3365 y(of)34 b(the)g(series)f(in)h(\(5.9\).)54 b(In)34 b(fact)g(the)g(consequen)n(t)f(expression)f(for)i Fy(h)f FG(cannot)h(ev)n(en)f(b)r(e,)j(if)e(tak)n(en)f(literally)-7 b(,)118 3471 y(correct)24 b(b)r(ecause)i(as)f(in)h(Section)f(3)h FF(the)i(denominators)h(in)f(the)g(new)g(expr)l(essions)h(c)l(ould)f (even)g(vanish)h(b)l(e)l(c)l(ause)118 3578 y(no)h(lower)h(cut-o\013)e (op)l(er)l(ates)h(on)g(the)g(lines)g(with)g(sc)l(ale)h FG([)p FA(\025)p 1986 3532 V 23 w Fx(n)2035 3590 y Fz(0)2073 3578 y FG(])e FF(as)h(the)g(thir)l(d)h(of)f(\(5.7\))h(shows)p FG(.)189 3776 y(T)-7 b(o)34 b(pro)r(ceed)g(w)n(e)g(m)n(ust)h(\014rst)f (c)n(hec)n(k)g(that)h(the)g(series)e(\(5.9\))h(de\014ning)h Fx(M)2565 3746 y Fz([)p Fu(n)p Fz(])2647 3776 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))36 b(are)d(really)h(con)n(v)n(ergen)n(t.) 118 3882 y(In)k(spite)f(of)g(the)h(last)f(commen)n(t)g(this)h(will)f(b) r(e)h(true)f(b)r(ecause)g(in)h(the)g(ev)-5 b(aluation)36 b(of)i Fx(M)3057 3852 y Fz([)p Fu(n)p Fz(])3139 3882 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))38 b FF(the)h(only)118 3988 y(pr)l(op)l(agators)c(ne)l(e)l(de)l(d)e(have)h(sc)l(ales)g FG([)p Fx(p)p FG(])f FF(with)g Fx(p)c FA(\024)g Fx(n)20 b FA(\000)h FG(1)30 b(so)h(that,)h(see)f(the)g(factors)f Fx( )2886 4000 y Fu(n)2932 3988 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))p Fx(;)14 b(\037)3267 4000 y Fu(n)3313 3988 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\))33 b(in)118 4095 y(\(5.7\),)h(their)f(denominators)f(not)h(only)f(do)h(not)g(v)-5 b(anish)33 b(but)g(ha)n(v)n(e)f(con)n(trolled)g(sizes)g(that)h(can)g(b) r(e)g(b)r(ounded)118 4201 y(b)r(elo)n(w)27 b(prop)r(ortionally)f(to)i Fx(x)1047 4171 y Fz(2)1112 4201 y FG(b)n(y)f(\(4.4\),)h(i.e.)37 b(simply)27 b(b)n(y)h(a)f(constan)n(t)g(times)g Fx(C)2663 4171 y Fz(2)2657 4221 y(0)2701 4201 y FA(j)p Fw(\027)6 b FA(j)2801 4171 y FC(\000)p Fz(2)p Fu(\034)2917 4179 y Ft(0)2953 4201 y FG(,)28 b(see)f(\(1.3\).)189 4328 y(In)c(Ref.)36 b([GG])23 b(it)h(has)e(b)r(een)h(sho)n(wn)g FF(by)i(a)h(pur)l(ely)g(algebr)l(aic)i(symmetry)d(ar)l(gument)d FG(that,)j(as)d(long)g(as)g(one)h(can)118 4434 y(pro)n(v)n(e)g(con)n(v) n(ergence)f(of)i(the)h(series)e(in)i(\(5.9\),)g(the)g(matrices)e Fx(M)2117 4404 y Fz([)p Fu(n)p Fz(])2200 4434 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))25 b(are)e(Hermitian)i(and)f(\()p Fx(M)3219 4404 y Fz([)p Fu(n)p Fz(])3302 4434 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\)\))3521 4404 y Fu(T)3597 4434 y FG(=)118 4541 y Fx(M)208 4511 y Fz([)p Fu(n)p Fz(])290 4541 y FG(\()p FA(\000)p Fx(x)p FG(;)g Fx(")p FG(\).)37 b(F)-7 b(urthermore)25 b(w)n(e)h(should)g(exp)r(ect)g(that)h (the)f(eigen)n(v)-5 b(alues)25 b(of)h(the)h(matrix)f FA(M)3078 4511 y Fz([)p FC(\024)p Fu(n)p Fz(])3212 4541 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))27 b(should)118 4647 y(b)r(e)h(appro)n(ximately)e(lo)r(cated)h(either)h(near)e(0)i(or)e (near)h Fx("a)1925 4659 y Fz(1)1962 4647 y Fx(;)14 b(:)g(:)g(:)f(;)h ("a)2229 4659 y Fu(s)2292 4647 y FG(at)28 b(least)f(within)h Fx(O)r FG(\()p Fx(")2982 4617 y Fz(2)3020 4647 y FG(\).)189 4774 y(The)h(exp)r(ectation)g(relies)g(on)g(Ref.)43 b([GG])29 b(\(see)h(Eq.)41 b(\(3.25\)\))29 b(where)g(the)g(follo)n(wing)g(\\)p FF(c)l(anc)l(el)t(lations)j(r)l(esult)8 b FG(")118 4881 y(is)24 b(deriv)n(ed)e(for)h Fx(n)657 4893 y Fz(0)718 4881 y FG(large)f(enough)h(\(hence)h(for)f Fx(")h FG(small)f(b)r (ecause)g(2)2202 4850 y FC(\000)p Fz(2)p Fu(n)2328 4858 y Ft(0)2360 4850 y FC(\000)p Fz(2)2472 4881 y Fx(<)g("a)2643 4893 y Fu(s)2701 4881 y FA(\024)f FG(2)2830 4850 y FC(\000)p Fz(2)p Fu(n)2956 4858 y Ft(0)2993 4881 y Fx(C)3058 4850 y Fz(2)3052 4901 y(0)3095 4881 y FG(\):)35 b(w)n(e)23 b(repro)r(duce)118 4987 y(the)28 b(pro)r(of)f(in)h(App)r(endix)g(A3)g (b)r(elo)n(w,)f(adapting)g(it)h(to)g(the)g(presen)n(t)f(notations.)118 5185 y FB(Lemma)37 b(2.)52 b FF(If)35 b Fx(n)700 5197 y Fz(0)772 5185 y FF(is)g(lar)l(ge)g(enough)g(and)h Fx(n)31 b FA(\024)p 1703 5139 V 32 w Fx(n)1753 5197 y Fz(0)1822 5185 y FG(=)g Fx(n)1968 5197 y Fz(0)2027 5185 y FG(+)p 2114 5139 V 22 w Fx(n)j FF(\(se)l(e)h(\(4.2\)\))h(then)e(the)h(fol)t (lowing)i(pr)l(op)l(erties)118 5291 y(hold.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 118 356 a FG(16:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y(\(i\))e(The)h(matric)l(es)f FA(M)845 525 y Fz([)p FC(\024)p Fu(n)p Fz(])979 555 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FF(,)31 b Fx(x)24 b FG(=)e Fw(!)f FA(\001)e Fw(\027)6 b FF(,)30 b(ar)l(e)g(Hermitian)g (and)g(c)l(an)g(b)l(e)g(written)f(as)1079 820 y FA(M)1179 785 y Fz([)p FC(\024)p Fu(n)p Fz(])1314 820 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))23 b(=)1612 678 y Fr( )1692 762 y FA(M)1792 719 y Fz([)p FC(\024)p Fu(n)p Fz(])1792 772 y Fu(\013\013)1926 762 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))86 b FA(M)2299 719 y Fz([)p FC(\024)p Fu(n)p Fz(])2299 787 y Fu(\013\014)2433 762 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))1692 893 y FA(M)1792 850 y Fz([)p FC(\024)p Fu(n)p Fz(])1792 918 y Fu(\014)s(\013)1926 893 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))86 b FA(M)2299 850 y Fz([)p 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FG(0)k(are)f Fx(n;)14 b(n)998 3546 y Fz(0)1035 3534 y FG(-indep)r(enden)n(t)28 b(constan)n(ts,)f(and) g Fx(\034)33 b FG(=)23 b Fx(\034)2278 3546 y Fz(0)2315 3534 y FG(.)118 3712 y FF(R)l(emarks.)36 b FG(\(1\))23 b(The)h(\014rst)f(three)g(b)r(ounds)g(on)g(the)h(eigen)n(v)-5 b(alues)22 b(in)i(\(5.14\),)f(follo)n(w)g(from)g(the)g(\014rst)g(line)h (of)f(\(5.13\))118 3818 y(b)n(y)29 b(using)g(the)g(self-adjoin)n(tness) f(of)h(the)h(matrices)e FA(M)1850 3788 y Fz([)p FC(\024)p Fu(n)p Fz(])1985 3818 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\);)30 b(see)f(App)r(endix)h(A4.)41 b(The)29 b(other)g(b)r(ounds)g (in)118 3924 y(\(5.13\))e(imply)h(the)g(last)f(b)r(ound)h(in)g (\(5.14\);)f(see)g(App)r(endix)i(A4.)118 4030 y(\(2\))41 b(The)f(natural)g(domain)g(of)g(de\014nition)h(in)f Fx(x)h FG(of)g FA(M)1954 4000 y Fz([)p Fu(n)p Fz(])2036 4030 y FG(\()p Fx(x;)14 b(")p FG(\),)45 b Fx(n)f(>)g FG(0,)f(will)d(turn)h (out)f(to)h(b)r(e)f Fx(D)r FG(\()p Fx(x)p FG(\))46 b FA(\024)118 4137 y FG(2)160 4107 y FC(\000)p Fz(2\()p Fu(n)p FC(\000)p Fz(1\))427 4137 y Fx(C)492 4107 y Fz(2)486 4157 y(0)529 4137 y FG(,)26 b(but)g(w)n(e)f(imagine)f(that)i(it)f(is)h (de\014ned)f(for)g(all)g Fx(x)h FG(b)n(y)e(con)n(tin)n(uing)h(it)h(as)e (a)h(constan)n(t)f(from)h(its)h(limit)118 4243 y(v)-5 b(alue.)54 b(In)34 b(fact)f(this)h(is)f(not)g(imp)r(ortan)n(t)g(b)r (ecause,)i(as)d(w)n(e)h(shall)g(see,)i(only)e(the)g(v)-5 b(alues)33 b(of)g FA(M)3196 4213 y Fz([)p Fu(n)p Fz(])3279 4243 y FG(\()p Fx(x;)14 b(")p FG(\))34 b(with)118 4349 y Fx(D)r FG(\()p Fx(x)p FG(\))e FA(\024)f FG(2)470 4319 y FC(\000)p Fz(2\()p Fu(n)p FC(\000)p Fz(1\))736 4349 y Fx(C)801 4319 y Fz(2)795 4370 y(0)871 4349 y FG(en)n(ter)h(in)n(to)g (the)h(analysis.)50 b(Smo)r(othness)32 b(means)g(di\013eren)n(tiabilit) n(y)g(in)h Fx(")e FA(2)g Fx(I)3365 4361 y Fu(C)3454 4349 y FG(and)h(a)118 4456 y(righ)n(t)i(and)g(left)h(di\013eren)n(tiabilit)n (y)f(in)g Fx(x)p FG(.)58 b(The)34 b(lac)n(k)f(of)h(di\013eren)n (tiabilit)n(y)h(in)f Fx(x)p 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b(instance)h(if)e Fj(r)k Fk(=)d Fj(s)g Fk(=)g(2)h(and)f Fj(f)7 b Fk(\()p Fq(\013)p Fj(;)12 b Fq(\014)r Fk(\))31 b(=)f Fj(f)1555 5088 y Fc(0)1594 5079 y Fk(\()p Fq(\014)r Fk(\))21 b(+)e Fj(f)1823 5088 y Fe(1)1858 5079 y Fk(\()p Fq(\014)r Fk(\))13 b(cos)f Fj(\013)2122 5088 y Fe(1)2177 5079 y Fk(+)20 b Fj(f)2286 5088 y Fe(2)2320 5079 y Fk(\()p Fq(\014)s Fk(\))12 b(cos)g Fj(\013)2584 5088 y Fe(2)2619 5079 y Fk(,)31 b Fb(to)h(lowest)g(or)l(der)h(in)e Fj(x;)11 b(")p Fk(,)31 b(one)g(has)118 5185 y Fj(M)193 5150 y Fe([)p Fg(\024)p Fh(n)p Fe(])186 5197 y Fh(\013\013)318 5185 y Fk(\()p Fj(x)p Fk(;)12 b Fj(")p Fk(\))20 b(=)f(3)p Fj(")639 5162 y Fe(2)674 5185 y Fj(x)714 5162 y Fe(2)748 5185 y Fk(\(2)p Fj(!)856 5162 y Fe(4)854 5202 y Fh(u)895 5185 y Fk(\))922 5162 y Fg(\000)p Fe(1)1005 5185 y Fk([)p Fj(f)1066 5162 y Fe(2)1059 5202 y Fh(u)1101 5185 y Fk(\()p Fq(\014)r Fk(\))q(+)q Fd(j)p Fj(@)1315 5193 y Fs(\014)1354 5185 y Fj(f)1388 5193 y Fh(u)1428 5185 y Fk(\()p Fq(\014)s Fk(\))p Fd(j)1549 5162 y Fe(2)1584 5185 y Fk(])p Fj(\016)1635 5193 y Fh(u;v)1727 5185 y Fk(,)e Fj(M)1839 5150 y Fe([)p Fg(\024)p Fh(n)p Fe(])1832 5210 y Fh(\013\014)1984 5185 y Fk(=)j Fj(i")2116 5162 y Fe(2)2150 5185 y Fj(x)p Fk(\(2)p Fj(!)2298 5162 y Fe(3)2296 5202 y Fh(v)2333 5185 y Fk(\))2360 5162 y Fg(\000)p Fe(1)2443 5185 y Fj(@)2480 5197 y Fh(\014)2515 5205 y Fs(v)2555 5185 y Fk([\()p Fj(f)2643 5162 y Fe(2)2636 5202 y Fh(u)2678 5185 y Fk(\()p Fq(\014)r Fk(\))q(+)q Fd(j)p Fj(@)2892 5193 y Fs(\014)2931 5185 y Fj(')2977 5193 y Fh(u)3018 5185 y Fk(\()p Fq(\014)r Fk(\))p Fd(j)3138 5162 y Fe(2)3173 5185 y Fk(\)],)d(and)g Fj(M)3462 5150 y Fe([)p Fg(\024)p Fh(n)p Fe(])3455 5210 y Fh(\014)s(\014)3607 5185 y Fk(=)118 5291 y Fj("@)192 5268 y Fe(2)188 5310 y Fs(\014)227 5291 y Fj(f)261 5300 y Fe(0)295 5291 y Fk(\()p Fq(\014)r Fe(\))p Fk(,)24 b Fj(u;)11 b(v)23 b Fk(=)c(1)p Fj(;)12 b Fk(2.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 118 356 a FG(17:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(b)r(elo)n(w)23 b(prop)r(ortionally)f(to)h Fx(D)r FG(\()p Fx(x)p FG(\).)37 b(But)24 b(this)g(w)n(ould)f(mak)n(e)g(the)h(discussion)e(needlessly)h (notationally)g(in)n(v)n(olv)n(ed)118 662 y(and)28 b(w)n(e)f(a)n(v)n (oid)f(it.)118 768 y(\(3\))21 b(One)f(should)h(also)f(remark)f(that,)j (although)f(w)n(e)f(excluded)h(some)f(v)-5 b(alues)20 b(of)h Fx(")g FG(\(i.e.)35 b(w)n(e)20 b(required)g Fx(")j FA(2)g(E)p 3476 746 42 4 v 13 x Fu(n)3517 789 y Ft(0)3549 781 y FC(\000)p Fz(1)3638 768 y FG(,)118 874 y(see)33 b(\(3.3\)\),)h(here)f(all)f Fx(")g FA(2)g Fx(I)1018 886 y Fu(C)1108 874 y FG(are)g(allo)n(w)n(ed.)51 b(The)33 b(restriction)f(on)h Fx(")g FG(pla)n(ys)f(no)g(role)g(in)i(the)f(high)g (frequency)118 981 y(resummations:)50 b(so)34 b(far)g(its)h(only)f (purp)r(ose)g(is)h(to)f(a)n(v)n(oid)g(divisions)g(b)n(y)g(0)g(and)h(to) f(assign)g(a)g(\014nite)h(v)-5 b(alue)35 b(to)118 1087 y(con)n(tributions)40 b(to)g Fy(h)h FG(from)f(trees)g(with)h(with)h (propagators)37 b(on)j(scale)g([)p FA(\025)p 2614 1041 50 4 v 45 w Fx(n)2663 1099 y Fz(0)2701 1087 y FG(])g(\(whic)n(h)h (could)g(b)r(e)g(in\014nite)118 1193 y(b)r(ecause)27 b(of)h(the)g(lac)n(k)e(of)i(an)f(infrared)g(cut-o\013)h(in)g(their)f (expressions;)f(see)h(the)h(third)g(of)g(\(5.7\)\).)118 1300 y(\(4\))38 b(The)f(b)r(ounds)h(on)f(the)h(en)n(tries)f(of)h FA(M)1500 1269 y Fz([)p Fu(n)p Fz(])1582 1300 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))39 b(in)e(the)h(second)f(and)h(third)g (lines)f(of)h(\(5.13\))e(arise)h(from)118 1406 y(cancellations)26 b(that)h(are)f(c)n(hec)n(k)n(ed)f(in)i(Ref.)37 b([GG])28 b(via)e(a)g(sequence)g(of)h(algebraic)e(iden)n(tities)i(on)g(the)g (Lindstedt)118 1512 y(series)g(co)r(e\016cien)n(ts)g(and)g FF(the)j(r)l(e)l(al)g(di\016culty)h(lies)f(in)g(the)g(pr)l(o)l(of)h(of) f(c)l(onver)l(genc)l(e)p FG(.)38 b(The)27 b(algebraic)f(mec)n(hanism) 118 1618 y(for)h(the)h(cancellations)f(is)g(brie\015y)g(recalled)g(in)h (App)r(endix)g(A3,)g(for)f(completeness.)118 1725 y(\(5\))j(Lo)r(osely) e(sp)r(eaking)h(\(as)g(men)n(tioned)h(in)g(Section)f(4\))h(the)g (reason)e(wh)n(y)h(the)h(ab)r(o)n(v)n(e)e(result)h(holds)h(with)p 3547 1679 V 30 w Fx(n)3596 1737 y Fz(0)3634 1725 y FG(-)118 1831 y(indep)r(enden)n(t)e(constan)n(ts,)e(and)h(wh)n(y)g(its)g(pro)r (of)g(can)f(b)r(e)i(tak)n(en)e(from)h(Ref.)37 b([GG],)28 b(is)f(that)g(if)h(the)f(scales)f(of)h(the)118 1937 y(propagators)c (are)j(constrained)f(to)h(b)r(e)h([)p Fx(p)p FG(])f(with)h Fx(p)c(<)p 1820 1892 V 22 w(n)1870 1949 y Fz(0)1934 1937 y FG(the)k(propagators)c(denominators)i(can)h(b)r(e)g(estimated)118 2044 y(b)n(y)36 b(2)284 2014 y FC(\000)p Fz(2\()p 395 1979 42 4 v Fu(n)p Fz(+1\))p FC(\000)p Fz(2)635 2044 y Fx(x)682 2014 y Fz(2)756 2044 y FG(b)n(y)g(\(4.4\))g(and)h(b)n(y)f (the)h(Remark)e(\(1\))i(after)f(De\014nition)h(1.)63 b(This)36 b(means)g(that)h(one)f(can)118 2150 y(pro)r(ceed)27 b(as)g(in)g(the)h(h)n(yp)r(erb)r(olic)f(tori)g(cases)g(in)g(whic)n(h)h (b)r(oundedness,)f(from)g(b)r(elo)n(w,)g(prop)r(ortionally)f(to)h Fx(x)3529 2120 y Fz(2)3595 2150 y FG(of)118 2256 y(the)33 b(propagators)c(denominators)i(w)n(as)g(the)i(main)f(feature)g (exploited)g(and)g FF(no)i(r)l(estriction)f FG(on)f Fx(")g FG(had)g(to)g(b)r(e)118 2363 y(required,)27 b(other)g(than)h(suitable)f (smallness.)189 2547 y(The)38 b(lemma)g(can)f(b)r(e)i(pro)n(v)n(ed)d(b) n(y)h(imitating)h(the)h(con)n(v)n(ergence)c(pro)r(of)j(of)f(the)i(KAM)f (theorem,)i(see)d(for)118 2653 y(instance)g(Ref.)66 b([GG];)37 b(ho)n(w)n(ev)n(er)e(in)i(the)h(follo)n(wing)e(App)r(endix)i(A3)f(the)g (part)g(of)g(the)g(pro)r(of)g(whic)n(h)g(is)f(not)118 2759 y(reducible)27 b(to)h(a)f(purely)g(algebraic)f(c)n(hec)n(k)h(is)g (rep)r(eated,)h(for)f(completeness.)189 2872 y(W)-7 b(e)34 b(ha)n(v)n(e)f(therefore)g(constructed)g(a)h(new)g(represen)n(tation)e (of)i(the)g(formal)f(series)g(for)h(the)g(function)h Fy(h)f FG(of)118 2979 y(the)c(parametric)f(equations)g(for)g(the)h(in)n (v)-5 b(arian)n(t)29 b(torus:)41 b(in)30 b(it)g(only)g(trees)f(with)h (lines)g(carrying)e(a)h(scale)g(lab)r(el)118 3085 y([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g(:)g(:)g(:)f(;)h FG([)p 579 3039 50 4 v Fx(n)629 3097 y Fz(0)688 3085 y FA(\000)21 b FG(1])32 b(or)g([)p FA(\025)p 1096 3039 V 30 w Fx(n)1146 3097 y Fz(0)1183 3085 y FG(])h(and)f FF(no)i(self-ener)l(gy)h(clusters)d FG(are)f(presen)n(t.)51 b(The)32 b(ab)r(o)n(v)n(e)f(lemma)i(will)f(b)r(e)118 3191 y(the)c(starting)f(blo)r(c)n(k)g(of)g(the)h(construction)f(that)h (follo)n(ws.)749 3439 y FD(6.)50 b(Renormalization:)i(the)37 b(infrared)h(resummation)118 3623 y FG(Con)n(v)n(ergence)30 b(problems)h(still)i(arise)e(from)h(the)h(propagators)c Fx(g)2185 3593 y Fz([)p FC(\025)p 2256 3558 42 4 v Fu(n)2296 3601 y Ft(0)2329 3593 y Fz(])2351 3623 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)35 b(whic)n(h)d(b)r(ecome)g(uncon)n(trollably) 118 3730 y(large)24 b(for)h Fx(x)e FG(=)g Fw(!)17 b FA(\001)d Fw(\027)31 b FG(close)24 b(to)h(the)h(eigen)n(v)-5 b(alues)24 b(of)i Fx(M)1832 3742 y Fz(0)1894 3730 y FG(b)r(ecause)f(the)g(b)r (ound)h(\(4.4\))f(whic)n(h)h(allo)n(w)n(ed)e(con)n(trol)g(of)118 3836 y(the)31 b(divisors)e(in)i(terms)g(of)f(the)h(classical)f(small)g (divisors)f(\(i.e.)46 b(in)31 b(terms)g(of)f FA(j)p Fx(x)p FA(j)p FG(\))i(do)r(es)e(not)h(hold)f(an)n(y)g(more.)118 3942 y(Hence)e(w)n(e)f(m)n(ust)h(c)n(hange)e(strategy)-7 b(.)118 4126 y FB(Definition)32 b(6.)37 b FF(Given)31 b Fx(d)19 b FA(\002)f Fx(d)30 b FF(Hermitian)g(matric)l(es)h FA(M)1994 4096 y Fz([)p FC(\024)p Fu(n)p Fz(])2128 4126 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FF(,)31 b Fx(n)23 b FG(=)p 2532 4080 50 4 v 23 w Fx(n)2582 4138 y Fz(0)2619 4126 y Fx(;)p 2656 4080 V 14 w(n)2706 4138 y Fz(0)2762 4126 y FG(+)18 b(1)p Fx(;)c(:)g(:)g(:)o FF(,)31 b(with)f(eigenvalues) 118 4242 y Fx(\025)166 4199 y Fz([)p Fu(n)p Fz(])166 4265 y Fu(j)249 4242 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FF(,)31 b(we)f(intr)l(o)l(duc)l(e)g(the)g(fol)t(lowing)i (notations.)118 4371 y(\(i\))e(The)h(se)l(quenc)l(e)e(of)i FG(self-energies)d Fx(\025)p 1308 4384 49 4 v -43 x Fz([)p Fu(n)p Fz(])1356 4395 y Fu(j)1439 4371 y FG(\()p Fx(")p FG(\))i FF(is)g(de\014ne)l(d)g(for)h Fx(n)23 b FA(\025)p 2234 4326 50 4 v 22 w Fx(n)2284 4383 y Fz(0)2351 4371 y FF(by)933 4646 y Fx(\025)p 933 4659 49 4 v 982 4603 a Fz([)p Fu(n)p Fz(])982 4669 y Fu(j)1064 4646 y FG(\()p Fx(")p FG(\))1181 4595 y Fu(def)1202 4646 y FG(=)33 b Fx(\025)1348 4603 y Fz([)p Fu(n)p Fz(])1348 4669 y Fu(j)1432 4554 y Fr(\020)1481 4541 y(q)p 1564 4541 320 4 v 105 x Fx(\025)p 1564 4659 49 4 v 1613 4603 a Fz([)p Fu(n)p FC(\000)p Fz(1])1613 4669 y Fu(j)1780 4646 y FG(\()p Fx(")p FG(\))q Fx(;)14 b(")1960 4554 y Fr(\021)2009 4646 y Fx(;)183 b(\025)p 2215 4659 V 2264 4603 a Fz([)p 2283 4568 42 4 v Fu(n)2324 4611 y Ft(0)2356 4603 y FC(\000)p Fz(1])2264 4669 y Fu(j)2464 4646 y FG(\()p Fx(")p FG(\))2581 4595 y Fu(def)2601 4646 y FG(=)34 b Fx(\025)2748 4603 y Fz([0])2748 4669 y Fu(j)2823 4646 y Fx(;)645 b FG(\(6)p Fx(:)p FG(1\))118 4915 y FF(pr)l(ovide)l(d)32 b Fx(\025)p 446 4928 49 4 v -43 x Fz([)p Fu(n)p Fz(])494 4939 y Fu(j)577 4915 y FG(\()p Fx(")p FG(\))23 b FA(\025)g FG(0)p FF(,)30 b Fx(n)23 b FG(=)p 1048 4870 50 4 v 22 w Fx(n)1098 4927 y Fz(0)1135 4915 y Fx(;)p 1172 4870 V 14 w(n)1222 4927 y Fz(0)1278 4915 y FG(+)18 b(1)p Fx(;)c(:)g(:)g(:)o FF(.)118 5022 y(\(ii\))31 b(The)f FG(propagator)25 b(divisors)j FF(ar)l(e)j(de\014ne)l(d)f(for)g Fx(n)23 b FA(\025)p 1881 4976 V 23 w Fx(n)1931 5034 y Fz(0)1998 5022 y FF(by)1383 5271 y FG(\001)1452 5236 y Fz([)p Fu(n)p Fz(])1535 5271 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))1736 5220 y Fu(def)1757 5271 y FG(=)1855 5175 y Fr(\014)1855 5225 y(\014)1855 5275 y(\014)1883 5271 y Fx(x)1930 5236 y Fz(2)1986 5271 y FA(\000)k Fx(\025)p 2069 5284 49 4 v 2118 5227 a Fz([)p Fu(n)p Fz(])2118 5299 y Fu(j)s Fz(\()p Fu(x)p Fz(\))2242 5271 y FG(\()p Fx(")p FG(\))2346 5175 y Fr(\014)2346 5225 y(\014)2346 5275 y(\014)2373 5271 y Fx(;)1095 b FG(\(6)p Fx(:)p FG(2\))118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 118 356 a FG(18:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 568 y(wher)l(e)f Fx(j)5 b FG(\()p Fx(x)p FG(\))30 b FF(is)g(the)g(lab)l(el)h(wher)l(e)f(the)g (minimum)g(of)1792 472 y Fr(\014)1792 522 y(\014)1792 572 y(\014)1820 568 y Fx(x)1867 538 y Fz(2)1923 568 y FA(\000)18 b Fx(\025)p 2006 581 49 4 v 2055 525 a Fz([)p Fu(n)p Fz(])2055 591 y Fu(j)2137 568 y FG(\()p Fx(")p FG(\))2241 472 y Fr(\014)2241 522 y(\014)2241 572 y(\014)2298 568 y FF(is)30 b(r)l(e)l(ache)l(d.)118 767 y(R)l(emarks.)37 b FG(\(1\))25 b(The)h(self-energies)e(are)h(de\014ned)h(recursiv)n(ely) d(starting)i(from)g(those)g(of)h(the)g(matrix)f Fx(M)3382 779 y Fz(0)3444 767 y FG(whose)118 873 y(\014rst)33 b Fx(r)j FG(eigen)n(v)-5 b(alues)32 b(are)h(0.)53 b(Hence,)35 b FF(as)g(long)h(as)f(one)g(c)l(an)g(extend)g(the)g(last)g(of)f FG(\(5.14\))e(and)i(as)e(long)h(as)f FF(the)118 989 y(self-ener)l(gies) i Fx(\025)p 589 1002 V 638 946 a Fz([)p Fu(n)p Fz(])638 1012 y Fu(j)721 989 y FG(\()p Fx(")p FG(\))f FF(r)l(emain)h(close)g(to) f(the)h(original)h(value)f Fx(\025)2170 946 y Fz([0])2170 1012 y Fu(j)2245 989 y FG(,)e(as)f(w)n(e)g(shall)g(c)n(hec)n(k)g(for)g Fx(")g FG(small)g(enough,)118 1118 y(one)c(has)g Fx(\025)p 418 1131 V 467 1075 a Fz([)p Fu(n)p Fz(])467 1141 y Fu(j)550 1118 y FG(\()p Fx(")p FG(\))c(=)g(0)k(for)g Fx(j)h FG(=)22 b(1)p Fx(;)14 b(:)g(:)g(:)g(;)g(r)30 b FG(and)d Fx(\025)p 1564 1131 V 1613 1075 a Fz([)p Fu(n)p Fz(])1613 1141 y Fu(j)1696 1118 y FG(\()p Fx(")p FG(\))c Fx(>)g FG(0)k(for)g Fx(j)h(>)22 b(r)r FG(.)118 1235 y(\(2\))37 b(Under)g(the)g(same)f (conditions)g(and)g(if)i(\001)1622 1205 y Fu(n)p Fz(])1686 1235 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))39 b FA(')e FG(2)2056 1205 y FC(\000)p Fz(2)p Fu(n)2186 1235 y Fx(C)2251 1205 y Fz(2)2245 1255 y(0)2325 1235 y FG(the)g(lab)r(el)g Fx(j)5 b FG(\()p Fx(x)p FG(\))38 b(dep)r(ends)f(only)f(on)h Fx(M)3602 1247 y Fz(0)3638 1235 y FG(,)118 1341 y(hence)30 b(it)g(is)f Fx(n)p FG(-indep)r(enden)n(t,)h(and)g(furthermore)e(it)i (is)f(constan)n(t)g(at)g Fx(x)h FG(\014xed,)h(as)d Fx(")i FG(v)-5 b(aries)28 b(in)i(the)g(in)n(terv)-5 b(als)28 b Fx(I)118 1447 y FG(in)n(tro)r(duced)e(in)g(De\014nition)h(1)f(\(b)r (ecause)g(for)f(large)g Fx(n)h FG(the)h(frequency)e Fx(x)i FG(is)f(constrained)f(to)h(b)r(e)h(close)e(to)h(one)g(of)118 1563 y(the)i Fx(\025)p 261 1576 V 310 1520 a Fz([)p Fu(n)p Fz(])310 1586 y Fu(j)392 1563 y FG(\()p Fx(")p FG(\)\).)118 1680 y(\(3\))j(The)g(name)f(of)h FF(pr)l(op)l(agator)j(divisor)f FG(assigned)d(to)g(\001)1944 1649 y Fz([)p Fu(n)p Fz(])2027 1680 y FG(\()p Fx(x;)14 b(")p FG(\))32 b(in)f(\(6.2\))f(re\015ects)g (its)h(later)f(use)h(as)f(a)g(lo)n(w)n(er)118 1786 y(b)r(ound)e(on)f (the)h(denominator)f(of)g(a)h(propagator,)d(see)i(Remark)f(\(7\))i(to)g (the)g(inductiv)n(e)f(assumption)h(b)r(elo)n(w.)189 1966 y(By)23 b(rep)r(eating)h(the)g(analysis)e(of)i(Section)g(4)f(w)n(e)h (can)f(represen)n(t)g(the)h(function)g Fy(h)g FG(via)f(sums)h(of)g(v)-5 b(alues)23 b(of)h(trees)118 2073 y(in)34 b(whic)n(h)f(lines)h(can)f (carry)f(scale)h(lab)r(els)g([)p FA(1)p FG(])p Fx(;)14 b FG([0])p Fx(;)g(:)g(:)g(:)f(;)h FG([)p 1944 2027 50 4 v Fx(n)1994 2085 y Fz(0)2054 2073 y FA(\000)22 b FG(1])p Fx(;)14 b FG([)p 2266 2027 V Fx(n)2315 2085 y Fz(0)2353 2073 y FG(])p Fx(;)g FG([)p 2436 2027 V Fx(n)2485 2085 y Fz(0)2545 2073 y FG(+)22 b(1])p Fx(;)14 b(:)g(:)g(:)33 b FG(and)g(whic)n(h)h(con)n(tain)f(no)118 2179 y(self-energy)f (clusters)h(and)g(no)g(trivial)g(no)r(des)g(\(i.e.)55 b(are)32 b(renormalized)g(trees,)j(see)e(De\014nition)h(5)f(in)h (Section)118 2285 y(5\).)41 b(The)30 b(new)f(propagators)d(will)j(b)r (e)h(de\014ned)f(b)n(y)g(the)g(same)g(pro)r(cedure)f(used)h(to)g (eliminate)g(the)h(self-energy)118 2392 y(clusters)c(of)h(scales)e([)p Fx(n)p FG(])i(with)g Fx(n)22 b FA(\024)p 1213 2346 V 23 w Fx(n)1263 2404 y Fz(0)1317 2392 y FA(\000)15 b FG(1.)37 b(Ho)n(w)n(ev)n(er)24 b(w)n(e)i(shall)g(no)n(w)g(determine)h(the)g (scale)f(of)g(a)g(line)h(in)g(terms)118 2498 y(of)f(the)h(recursiv)n (ely)d(de\014ned)i(\001)1118 2468 y Fz([)p Fu(n)p Fz(])1201 2498 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))27 b(rather)e(than)h(in)g (terms)g(of)g Fx(D)r FG(\()p Fx(x)p FG(\):)37 b(the)27 b(latter)e(b)r(ecomes)h(to)r(o)g(rough)f(to)118 2604 y(resolv)n(e)h(the)i(separation)e(b)r(et)n(w)n(een)h(the)h(eigen)n(v)-5 b(alues)27 b(and)g(their)h(v)-5 b(ariations.)189 2732 y(Let)31 b Fx(X)p 410 2710 42 4 v 13 x Fu(n)450 2753 y Ft(0)483 2745 y FC(\000)p Fz(1)572 2732 y FG(\()p Fx(x)p FG(\))697 2681 y Fu(def)718 2732 y FG(=)830 2669 y Fr(Q)p 909 2655 V 909 2690 a Fu(n)950 2698 y Ft(0)982 2690 y FC(\000)p Fz(1)909 2757 y Fu(m)p Fz(=0)1085 2732 y Fx(\037)1137 2744 y Fu(m)1200 2732 y FG(\()p Fx(D)r FG(\()p Fx(x)p FG(\)\),)i Fx(Y)1550 2744 y Fu(n)1595 2732 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))1796 2681 y Fu(def)1817 2732 y FG(=)1930 2669 y Fr(Q)2008 2690 y Fu(n)2008 2757 y(m)p Fz(=)p 2118 2722 V Fu(n)2159 2765 y Ft(0)2209 2732 y Fx(\037)2261 2744 y Fu(m)2324 2732 y FG(\(\001)2425 2702 y Fz([)p Fu(m)p Fz(])2526 2732 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))32 b(for)e Fx(n)d FA(\025)p 3077 2686 50 4 v 28 w Fx(n)3126 2744 y Fz(0)3194 2732 y FG(and)j Fx(Y)p 3406 2710 42 4 v 13 x Fu(n)3448 2753 y Ft(0)3480 2745 y FC(\000)p Fz(1)3597 2732 y FA(\021)118 2838 y FG(1:)37 b(the)28 b(de\014nition)g(of)f(the)h(new)g(propagators)c(will) k(b)r(e)371 3072 y Fx(g)414 3038 y Fz([)p 433 3003 V Fu(n)474 3046 y Ft(0)506 3038 y Fz(])543 3021 y Fu(def)563 3072 y FG(=)105 b Fx(X)p 802 3050 V 13 x Fu(n)843 3093 y Ft(0)875 3085 y FC(\000)p Fz(1)964 3072 y FG(\()p Fx(x)p FG(\))14 b Fx( )p 1143 3050 V 13 x Fu(n)1185 3093 y Ft(0)1222 3072 y FG(\(\001)1323 3038 y Fz([)p 1342 3003 V Fu(n)1383 3046 y Ft(0)1415 3038 y Fz(])1438 3072 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))g(\()p Fx(x)1750 3038 y Fz(2)1807 3072 y FA(\000)k(M)1990 3038 y Fz([)p FC(\024)p 2061 3003 V Fu(n)2102 3046 y Ft(0)2134 3038 y Fz(])2157 3072 y FG(\()p Fx(x)p FG(;)c Fx(")p FG(\)\))2376 3038 y FC(\000)p Fz(1)2466 3072 y Fx(;)287 3234 y(g)330 3199 y Fz([)p 349 3164 V Fu(n)390 3207 y Ft(0)422 3199 y Fz(+1])543 3183 y Fu(def)563 3234 y FG(=)105 b Fx(X)p 802 3212 V 13 x Fu(n)843 3255 y Ft(0)875 3247 y FC(\000)p Fz(1)964 3234 y FG(\()p Fx(x)p FG(\))14 b Fx(\037)p 1141 3212 V 13 x Fu(n)1183 3255 y Ft(0)1219 3234 y FG(\(\001)1320 3199 y Fz([)p 1339 3164 V Fu(n)1381 3207 y Ft(0)1413 3199 y Fz(])1436 3234 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))g Fx( )p 1723 3212 V 13 x Fu(n)1765 3255 y Ft(0)1797 3247 y Fz(+1)1886 3234 y FG(\(\001)1987 3199 y Fz([)p 2006 3164 V Fu(n)2047 3207 y Ft(0)2079 3199 y Fz(+1])2186 3234 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))g(\()p Fx(x)2498 3199 y Fz(2)2556 3234 y FA(\000)k(M)2739 3199 y Fz([)p FC(\024)p 2810 3164 V Fu(n)2850 3207 y Ft(0)2882 3199 y Fz(+1])2989 3234 y FG(\()p Fx(x)p FG(;)c Fx(")p FG(\)\))3208 3199 y FC(\000)p Fz(1)3298 3234 y Fx(;)676 3365 y(:)g(:)g(:)404 3527 y(g)447 3493 y Fz([)p Fu(n)p Fz(])543 3476 y Fu(def)563 3527 y FG(=)105 b Fx(X)p 802 3505 V 13 x Fu(n)843 3548 y Ft(0)875 3540 y FC(\000)p Fz(1)964 3527 y FG(\()p Fx(x)p FG(\))14 b Fx(Y)1137 3539 y Fu(n)p FC(\000)p Fz(1)1268 3527 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\))g Fx( )1523 3539 y Fu(n)1569 3527 y FG(\(\001)1670 3493 y Fz([)p Fu(n)p Fz(])1753 3527 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))g(\()p Fx(x)2065 3493 y Fz(2)2122 3527 y FA(\000)k(M)2305 3493 y Fz([)p FC(\024)p Fu(n)p Fz(])2440 3527 y FG(\()p Fx(x)p FG(;)c Fx(")p FG(\)\))2659 3493 y FC(\000)p Fz(1)2749 3527 y Fx(;)3491 3285 y FG(\(6)p Fx(:)p FG(3\))118 3748 y(and)30 b(so)f(on,)i(using)e(inde\014nitely)i(the)f(iden)n(tit)n(y)g (1)d FA(\021)f Fx( )1849 3760 y Fu(n)1894 3748 y FG(\(\001)1995 3718 y Fz([)p Fu(n)p Fz(])2079 3748 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\)\))21 b(+)e Fx(\037)2455 3760 y Fu(n)2500 3748 y 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y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b(=)e(min)825 4565 y Fr(n)894 4657 y FG(min)948 4709 y Fu(j)1046 4561 y Fr(\014)1046 4611 y(\014)1046 4661 y(\014)1074 4657 y Fx(x)d FA(\006)1223 4551 y Fr(q)p 1306 4551 253 4 v 106 x Fx(\025)p 1306 4670 49 4 v -43 x Fz([)p Fu(m)p Fz(])1354 4680 y Fu(j)1455 4657 y FG(\()p Fx(")p FG(\))1558 4561 y Fr(\014)1558 4611 y(\014)1558 4661 y(\014)1600 4657 y Fx(;)41 b FG(min)1680 4710 y Fu(j)s FC(\025)p Fu(i)1816 4561 y Fr(\014)1816 4611 y(\014)1816 4661 y(\014)1844 4657 y Fx(x)19 b FA(\006)1993 4551 y Fr(q)p 2076 4551 253 4 v 106 x Fx(\025)p 2076 4670 49 4 v -43 x Fz([)p Fu(m)p Fz(])2124 4680 y Fu(j)2225 4657 y FG(\()p Fx(")p FG(\))g FA(\006)2430 4546 y Fr(q)p 2513 4546 253 4 v 111 x Fx(\025)p 2513 4670 49 4 v -43 x Fz([)p Fu(m)p Fz(])2561 4680 y Fu(i)2662 4657 y FG(\()p Fx(")p FG(\))2765 4561 y Fr(\014)2765 4611 y(\014)2765 4661 y(\014)2807 4565 y(o)2885 4657 y FA(\025)k FG(2)3015 4622 y FC(\000)3076 4600 y Ft(1)p 3076 4609 29 4 v 3076 4642 a(2)3114 4622 y Fu(m)3223 4601 y Fx(C)3282 4613 y Fz(0)p 3187 4638 167 4 v 3187 4714 a FA(j)p Fw(\027)6 b FA(j)3287 4690 y Fu(\034)3318 4698 y Ft(1)3364 4657 y Fx(;)409 4851 y FA(jE)483 4817 y Fu(o)476 4871 y(m)539 4851 y FA(j)23 b(\024)f Fx(K)6 b FG(2)791 4817 y FC(\000)852 4794 y Ft(1)p 852 4803 29 4 v 852 4837 a(2)890 4817 y Fu(m)954 4851 y Fx(C)1019 4817 y Fz(2)1056 4851 y Fx(;)3491 4730 y FG(\(6)p Fx(:)p FG(7\))118 5008 y FF(for)31 b(al)t(l)g Fx(m)22 b FA(\024)h Fx(n)18 b FA(\000)g FG(1)30 b FF(and)g(al)t(l)h Fx(x)p FF(.)118 5185 y(R)l(emarks.)37 b FG(Assuming)28 b(v)-5 b(alidit)n(y)27 b(of)h(the)g(h)n(yp)r(othesis)f(for)g Fx(m)c(<)g(n)k FG(w)n(e)g(note)h(a)f(few)h(of)g(its)f(implications.)118 5291 y(\(1\))d(So)g(far)g(w)n(e)f(ha)n(v)n(e)g(only)h(c)n(hec)n(k)n(ed) f(the)h(h)n(yp)r(othesis)g(for)f(scales)g([)p Fx(m)p FG(])i(with)f Fx(m)f FA(\024)p 2679 5246 50 4 v 23 w Fx(n)2729 5303 y Fz(0)2766 5291 y FG(,)i(as)e(expressed)g(b)n(y)h (Lemma)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 118 356 a FG(20:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(2)27 b(in)g(Section)g(5,)f (i.e.)37 b(for)27 b(the)g(high)g(frequency)f(propagators.)34 b(If)27 b(\(i\))h(is)f(pro)n(v)n(ed)e(also)h(for)g Fx(m)d FG(=)g Fx(n)k FG(then)g(w)n(e)g(can)118 662 y(imp)r(ose)32 b(\(6.7\))g(immediately)h(b)n(y)f(excluding)g(a)g(set)g FA(E)1838 632 y Fu(o)1831 682 y(m)1927 662 y FG(of)g Fx(")p FG('s)g(of)g(measure)f(estimated)i(b)n(y)f(2)3126 632 y FC(\000)3187 609 y Ft(1)p 3187 618 29 4 v 3187 652 a(2)3226 632 y Fu(m)3289 662 y Fx(C)3354 632 y Fz(2)3391 662 y Fx(K)38 b FG(with)118 768 y Fx(K)e FG(a)29 b(constan)n(t)h(that)g (can)g(b)r(e)g(b)r(ounded)g(in)h(terms)f(of)g Fx(A)1920 738 y FC(0)1943 768 y Fx(;)14 b(A)30 b FG(b)n(y)g(in)n(tro)r(ducing)g (the)g(constan)n(ts)f Fx(\032)3194 780 y Fu(m)3287 768 y FG(and)h Fx(\032)3494 738 y FC(0)3494 789 y Fu(m)3587 768 y FG(as)118 884 y(in)i(\(A2.1\),)h(with)f Fx(\025)p 701 897 49 4 v -44 x Fz([0])749 907 y Fu(j)824 884 y FG(\()p Fx(")p FG(\))g(replaced)f(b)n(y)g Fx(\025)p 1410 897 V 1459 840 a Fz([)p Fu(m)p Fz(])1459 907 y Fu(j)1559 884 y FG(\()p Fx(")p FG(\),)j(and)d(pro)r(ceeding)g(as)g(done)g(in)h (App)r(endix)h(A2)e(for)g(the)h(case)118 1013 y Fx(n)e FA(\024)p 292 967 50 4 v 29 w Fx(n)342 1025 y Fz(0)379 1013 y FG(.)49 b(Note)31 b(that)h(since)f(the)h(self-energies)e Fx(\025)p 1657 1026 49 4 v 1706 970 a Fz([)p Fu(m)p Fz(])1706 1036 y Fu(j)1806 1013 y FG(\()p Fx(")p FG(\))i(are)f FA(\021)e Fx(\025)p 2178 1026 V -43 x Fz([0])2226 1036 y Fu(j)2301 1013 y FG(\()p Fx(")p FG(\))j(for)f(all)g Fx(m)f FG(=)f(0)p Fx(;)14 b(:)g(:)g(:)f(;)p 3109 967 50 4 v 14 w(n)3159 1025 y Fz(0)3217 1013 y FA(\000)21 b FG(1)31 b(one)g(will)118 1119 y(ha)n(v)n(e,)e(for)g(suc)n(h)g Fx(m)p FG('s,)h FA(E)886 1089 y Fu(o)879 1140 y(m)968 1119 y FA(\021)c Fx(I)1095 1131 y Fu(C)1151 1119 y Fx(=)p FA(E)p 1237 1097 42 4 v 13 x Fu(n)1277 1141 y Ft(0)1310 1132 y FC(\000)p Fz(1)1399 1119 y FG(,)k(see)f(\(3.3\).)42 b(It)30 b(is)f(v)n(ery)g(imp)r(ortan)n(t)g(to)g(k)n(eep)g(in)h(mind,)g (in)g(the)g(ab)r(o)n(v)n(e)118 1226 y(argumen)n(t,)g(that)h(the)g (self-energies)e(either)i(are)e(0)i(\(for)f Fx(j)j FA(\024)28 b Fx(r)r FG(\))j(or)f(are)g(close)f(within)j Fx(O)r FG(\()p Fx(")3031 1196 y Fz(2)3069 1226 y FG(\))e(to)h(the)g(p)r(ositiv)n(e)118 1332 y(eigen)n(v)-5 b(alues)30 b(of)i Fx(M)731 1344 y Fz(0)767 1332 y FG(,)h(and)e(they)g(are)g FF(di\013er)l(entiable)i FG(in)e Fx(")g FG(and)h(to)f(the)h(righ)n(t)e(and)h(left)h(of)g(eac)n (h)e Fx(x)i FG(b)n(y)f(\(i\);)j(see)118 1438 y(\(5.14\).)118 1545 y(\(2\))25 b(If)h(a)f(line)g(with)h(a)f(scale)f([)p Fx(n)p FG(])h(has)g(v)-5 b(anishing)25 b(propagator)d(\(i.e.)36 b Fx(g)2272 1514 y Fz([)p Fu(n)p Fz(])2355 1545 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))23 b(=)g(0)i(b)r(ecause)g(of)g(the)g Fx(\037;)14 b( )28 b FG(cut-o\013)118 1651 y(functions)k(in)f(the)h (de\014nition)g(\(6.4\)\))f(but)h Fx(n)f FG(di\013ers)g(at)h(most)f(b)n (y)g(one)g(unit)h(from)f(the)g(in)n(teger)g Fx(n)3264 1621 y FC(0)3318 1651 y FG(suc)n(h)g(that)118 1757 y Fx(g)161 1727 y Fz([)p Fu(n)221 1702 y Ff(0)243 1727 y Fz(])266 1757 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))30 b FA(6)p FG(=)e(0.)47 b(Th)n(us)31 b(if)h(w)n(e)f(consider)f(\001)1504 1727 y Fz([)p Fu(n)p Fz(])1587 1757 y FG(\()p Fx(x;)14 b(")p FG(\))32 b(w)n(e)f(can)f(b)r(ound)i(it)g(b)n(y)e(c)n(hanging)g (the)i(inequalities)f(\(6.6\))118 1863 y(in)n(to)24 b Fx(C)348 1833 y Fz(2)342 1884 y(0)400 1863 y FG(2)442 1833 y FC(\000)p Fz(2\()p Fu(n)p Fz(+2\))730 1863 y Fx(<)f FG(\001)887 1833 y Fz([)p Fu(n)p Fz(])970 1863 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b FA(\024)e Fx(C)1333 1833 y Fz(2)1327 1884 y(0)1385 1863 y FG(2)1427 1833 y FC(\000)p Fz(2\()p Fu(n)p FC(\000)p Fz(2\))1693 1863 y FG(.)36 b(The)25 b(remark)e(will)i(b)r(e)g(useful)g(later)f(when)g(w) n(e)h(shall)f(exploit)118 1970 y(it)k(in)g(the)g(discussion)f(of)g(the) h(cancellations)f(whic)n(h)g(w)n(e)g(shall)h(study)f(to)h(c)n(hec)n(k)f (the)h(inductiv)n(e)f(h)n(yp)r(othesis.)118 2076 y(\(3\))f(By)f (\(5.13\))f(and)i(\(5.14\),)f(and)g(\(I\))h(in)g(App)r(endix)g(A4,)g(w) n(e)f(deduce)g(that)h Fx(\025)2561 2033 y Fz([)p Fu(m)p Fz(])2561 2099 y Fu(j)2662 2076 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))q(,)26 b(hence)f Fx(\025)p 3127 2089 49 4 v -43 x Fz([)p Fu(m)p Fz(])3175 2099 y Fu(j)3276 2076 y FG(\()p Fx(")p FG(\),)h(do)f(not)118 2205 y(c)n(hange)i(b)n(y)g(more) g(than)g Fx(C)20 b(B)e(")1108 2175 y Fz(2)1159 2143 y Fr(P)1247 2230 y Fu(n)p FC(\025)p 1340 2195 42 4 v Fu(n)1381 2238 y Ft(0)1431 2205 y Fx(e)1470 2175 y FC(\000)p Fu(\024)1561 2183 y Ft(1)1593 2175 y Fz(2)1626 2150 y Fs(n=)p Ft(\(2)p Fs(\034)1772 2162 y Ft(1)1805 2150 y(\))1835 2205 y FG(,)28 b(with)g(resp)r(ect)g(to)f Fx(\025)2509 2162 y Fz([0])2509 2229 y Fu(j)2584 2205 y FG(\()p Fx(")p FG(\))q(,)g(if)i Fx(")22 b(<)p 2964 2160 39 4 v 23 w(")3003 2217 y Fz(1)3068 2205 y FG(\(and)27 b Fx(m)c FA(\025)p 3445 2160 50 4 v 23 w Fx(n)3495 2217 y Fz(0)3532 2205 y FG(\).)118 2336 y(\(4\))28 b(Hence)f(if)h Fx(")f FG(is)h(small)f(enough)g(the)g (self-energies,)f(i.e.)37 b Fx(\025)p 1995 2349 49 4 v 2044 2293 a Fz([)p Fu(m)p Fz(])2044 2359 y Fu(j)2144 2336 y FG(\()p Fx(")p FG(\))q(,)27 b(ha)n(v)n(e)g(distance)g(b)r (ounded)h(ab)r(o)n(v)n(e)e(b)n(y)h(2)p Fx(a)3588 2348 y Fu(s)3623 2336 y Fx(")118 2442 y FG(and)h(b)r(elo)n(w)g(b)n(y)642 2410 y Fz(1)p 642 2424 34 4 v 642 2471 a(2)685 2442 y Fx(")14 b FG(min)890 2375 y Fr(\010)939 2442 y Fx(a)983 2454 y Fz(1)1020 2442 y Fx(;)g FG(min)1195 2454 y Fu(j)1230 2442 y FA(f)p Fx(a)1316 2454 y Fu(j)s Fz(+1)1453 2442 y FA(\000)19 b Fx(a)1581 2454 y Fz(1)1618 2442 y FA(g)1660 2375 y Fr(\011)1732 2442 y FG(=)24 b(2)p Fx(\032)14 b("a)2003 2454 y Fu(s)2066 2442 y FG(with)29 b Fx(\032)f FG(de\014ned)h(in)f (\(4.2\),)h(if)g Fx(")f FG(is)g(small)g(enough,)118 2549 y(sa)n(y)f Fx(")22 b(<)p 411 2503 39 4 v 23 w(")450 2561 y Fz(2)487 2549 y FG(.)118 2655 y(\(5\))f(Therefore)e(b)n(y)i(Remark)f (\(4\))g(w)n(e)h(see)f(that)h(the)g(distance)g(of)f FA(j)p Fx(x)p FA(j)2204 2625 y Fz(2)2263 2655 y FG(from)g(the)h(closest)f(v)-5 b(alue)21 b Fx(\025)p 3053 2668 49 4 v -43 x Fz([)p Fu(m)p Fz(])3101 2678 y Fu(j)3202 2655 y FG(\()p Fx(")p FG(\))g(is)f(smaller) 118 2785 y(than)28 b(one)f(fourth,)h(up)g(to)f(corrections)f Fx(O)r FG(\()p Fx(")1517 2754 y Fz(2)1555 2785 y FG(\),)i(the)g (distance)f(b)r(et)n(w)n(een)g(the)h(distinct)h(v)-5 b(alues)27 b(of)g Fx(\025)p 3209 2798 V 3258 2741 a Fz([)p Fu(m)p Fz(])3258 2808 y Fu(j)3358 2785 y FG(\()p Fx(")p FG(\))q(,)g(if)i Fx(m)118 2891 y FG(is)23 b(large)f(enough)h(compared)f (to)h Fx(n)1199 2903 y Fz(0)1236 2891 y FG(,)h(i.e.)36 b(if)23 b(2)p Fx(C)1603 2861 y Fz(2)1597 2911 y(0)1640 2891 y FG(2)1682 2861 y FC(\000)p Fz(2)p Fu(m)1853 2891 y Fx(<)g(\032"a)2067 2903 y Fu(s)2125 2891 y FG(\(or)f Fx(m)10 b FA(\000)g Fx(n)2462 2903 y Fz(0)2521 2891 y FA(\025)p 2609 2845 50 4 v 23 w Fx(n)23 b 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b(of)k(scale)e([)p FA(\025)d Fx(n)p FG(]\))28 b(ha)n(v)n(e)e(momen)n(ta)h Fw(\027)34 b FG(w)n(e)27 b(b)r(egin)h(b)n(y)f(sho)n(wing)f(that)1483 4134 y Fr(X)1422 4316 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))1677 4213 y FA(j)p Fw(\027)1748 4225 y Fm(v)1795 4213 y FA(j)23 b Fx(>)g FG(2)1971 4179 y Fz(\()p Fu(n)p FC(\000)p Fz(6\))p Fu(=)p Fz(\(2)p Fu(\034)2273 4187 y Ft(1)2304 4179 y Fz(\))2334 4213 y Fx(:)1092 b FG(\(6)p Fx(:)p FG(10\))118 4501 y(Indeed)31 b(the)f(cluster)g(con)n (tains)g(at)g(least)g(one)g(line)g Fx(`)d FG(=)h Fx(`)1941 4513 y Fm(v)2017 4501 y FG(with)j(propagator)d(whic)n(h)i(w)n(e)g(can)g (supp)r(ose)g(to)g(b)r(e)118 4607 y(not)h(v)-5 b(anishing)30 b(and)g(whic)n(h)h(has)f(scale)f([)p Fx(n)21 b FA(\000)f FG(1].)45 b(W)-7 b(e)31 b(can)f(write)h Fw(\027)2273 4619 y Fu(`)2333 4607 y FG(=)c Fw(\027)2479 4577 y Fz(0)2473 4630 y Fu(`)2536 4607 y FG(+)20 b Fx(\033)2668 4619 y Fu(`)2701 4607 y Fw(\027)6 b FG(,)31 b(where)f Fx(\033)3099 4619 y Fu(`)3159 4607 y 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y Fu(j)s Fz(\()p Fu(x)1716 1522 y Fs(`)1746 1513 y Fz(\))1790 1485 y FG(\()p Fx(")p FG(\))1893 1389 y Fr(\014)1893 1439 y(\014)1893 1489 y(\014)1939 1485 y FG(+)2022 1389 y Fr(\014)2022 1439 y(\014)2022 1489 y(\014)2050 1485 y FA(j)p Fx(x)p FA(j)g(\000)2245 1383 y Fr(q)p 2328 1383 320 4 v 102 x Fx(\025)p 2328 1498 49 4 v 2377 1442 a Fz([)p Fu(n)p FC(\000)p Fz(2])2377 1513 y Fu(j)s Fz(\()p Fu(x)p Fz(\))2544 1485 y FG(\()p Fx(")p FG(\))2648 1389 y Fr(\014)2648 1439 y(\014)2648 1489 y(\014)3449 1485 y FG(\(6)p Fx(:)p FG(12\))1148 1680 y FA(\025)1236 1584 y Fr(\014)1236 1634 y(\014)1236 1684 y(\014)1264 1680 y Fx(x)1311 1692 y Fu(`)1362 1680 y FA(\000)f Fx(x)h FG(+)f Fx(\021)1635 1692 y Fu(`)1667 1578 y Fr(q)p 1750 1578 217 4 v 102 x Fx(\025)p 1750 1693 49 4 v -44 x Fz([)p Fu(n)p FC(\000)p Fz(2])1798 1708 y Fu(j)s Fz(\()p Fu(x)1892 1717 y Fs(`)1922 1708 y Fz(\))1966 1680 y FG(\()p Fx(")p FG(\))h(+)f Fx(\021)2215 1578 y Fr(q)p 2298 1578 320 4 v 102 x Fx(\025)p 2298 1693 49 4 v -44 x Fz([)p 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b(giv)n(en)g(in)h(App)r(endix)g(A3)f(\(cf.)52 b(in)33 b(particular)e(Section)i(A3.4\),)g(and)f(giv)n(es)g FA(N)2706 3472 y Fu(m)2769 3460 y FG(\()p Fx(T)12 b FG(\))30 b FA(\024)h Fx(E)3081 3472 y Fu(m)3158 3397 y Fr(P)3246 3485 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))3505 3460 y FA(j)p Fw(\027)3576 3472 y Fu(v)3615 3460 y FA(j)p FG(,)118 3581 y(with)25 b Fx(E)365 3593 y Fu(m)451 3581 y FG(=)e(2)581 3551 y Fz(\(6)p FC(\000)p Fu(m)p Fz(\))p Fu(=)p Fz(\(2)p Fu(\034)901 3559 y Ft(1)932 3551 y Fz(\))986 3581 y FG(and)h Fx(\034)1180 3593 y Fz(1)1242 3581 y FG(in)g(\(3.4\),)g(whic)n(h)g(sho)n(ws)f(con)n(v)n(ergence)f(of)i(the)g (series)f(in)h(\(6.13\))g(if)g Fx(")g FG(is)g(small)118 3687 y(enough,)j(sa)n(y)g Fx(")22 b(<)p 720 3642 39 4 v 23 w(")p FG(.)189 3794 y(W)-7 b(e)31 b(can)e(and)h(shall)g(assume)g (that)p 1322 3748 V 30 w Fx(")g FG(do)r(es)g(not)g(exceed)g(min)p FA(f)p 2179 3748 V Fx(")2217 3806 y Fz(1)2255 3794 y Fx(;)p 2292 3748 V 14 w(")2330 3806 y Fz(2)2367 3794 y Fx(;)p 2404 3748 V 14 w(")2443 3806 y Fz(3)2480 3794 y FA(g)p FG(,)h(with)p 2767 3748 V 30 w Fx(")2806 3806 y Fz(1)2843 3794 y FG(,)p 2897 3748 V 31 w Fx(")2936 3806 y Fz(2)3004 3794 y FG(and)p 3168 3748 V 30 w Fx(")3206 3806 y Fz(3)3274 3794 y FG(in)n(tro)r(duced)118 3900 y(earlier)d(\(see)i(Remarks)e(3,)i(4,)g(7)f(after)g(the)h(inductiv)n(e) g(h)n(yp)r(othesis\).)43 b FF(The)32 b(r)l(est)g(of)g(the)g(ar)l (gument)f(r)l(ep)l(e)l(ats)g(the)118 4006 y(analysis)i(in)f(App)l (endix)g(A3)g(with)g(minor)g(notational)g(changes)p FG(:)42 b(w)n(e)30 b(only)f(hin)n(t)h(at)f(the)h(details)g(in)g(App)r(endix)118 4113 y(A3.4.)189 4219 y(Under)20 b(the)h(considered)e(h)n(yp)r(otheses) g(the)i(matrices)e FA(M)1944 4189 y Fz([)p Fu(n)p Fz(])2026 4219 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))21 b(are)e(w)n(ell)h (de\014ned,)i(b)n(y)e(the)h(ab)r(o)n(v)n(e)d(discussion)118 4325 y(on)29 b(con)n(v)n(ergence)f(of)h(the)h(de\014ning)g(series)e(on) i(the)g(set)f FA(\\)1927 4290 y Fu(n)p FC(\000)p Fz(1)1927 4351 y Fu(m)p Fz(=)p 2037 4317 42 4 v Fu(n)2079 4359 y Ft(0)2111 4351 y FC(\000)p Fz(1)2200 4325 y FA(E)2244 4337 y Fu(m)2307 4325 y FG(.)43 b(The)30 b(symmetry)f(in)h(item)g (\(i\))g(is)g(due)g(to)118 4431 y(algebraic)f(iden)n(tities)i(v)-5 b(alid)30 b(for)g(the)h(Lindstedt)g(series.)45 b(They)30 b(are)g(detailed)g(in)h(Ref.)46 b([GG],)31 b(App)r(endix)h(A5,)118 4538 y(for)e Fx(")e(<)g FG(0:)43 b(b)r(eing)31 b(of)g(algebraic)e (nature)h(the)h(argumen)n(t)f(do)r(es)g(not)h(dep)r(end)h(on)e(the)h (sign)g(of)f Fx(")h FG(and)g(it)g(holds)118 4644 y(unc)n(hanged)c(in)h (the)g(presen)n(t)f(case.)189 4750 y(The)g(second)g(and)g(third)g (lines)h(of)f(inequalities)g(in)g(\(5.13\))g(em)n(b)r(o)r(dy)g(the)h (cancellations.)35 b(W)-7 b(e)28 b(need)f(to)g(c)n(hec)n(k)118 4857 y(the)32 b(cancellations)e(to)h(mak)n(e)g(sure)g(for)g(instance)g (that)g(the)h(structure)f(of)g(the)h(matrix)f FA(M)3031 4827 y Fz([)p Fu(n)p Fz(])3113 4857 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))33 b(preserv)n(es)118 4963 y(the)d(eigen)n(v)-5 b(alues)28 b(and)h(the)h(Whitney)g(smo)r(othness:)39 b(a)29 b(danger)f(b)r(eing)h(that)h(the)f(\014rst)g Fx(r)k FG(eigen)n(v)-5 b(alues)28 b(b)r(ecome)118 5069 y(\\detac)n(hed")j (from)g(0,)i(i.e.)50 b(no)32 b(longer)f(can)g(b)r(e)h(b)r(ounded)h(b)n (y)f Fx("x)2205 5039 y Fz(2)2242 5069 y FG(,)h(see)f(\(5.14\).)49 b(F)-7 b(or)31 b(instance)h(a)g(b)r(ound)g(lik)n(e)118 5185 y Fx(O)r FG(\()p Fx(")254 5155 y Fz(2)292 5185 y FG(\))e(w)n(ould)g(not)g(b)r(e)h(enough)e(as)h(it)g(w)n(ould)g(imply)h (that)f(the)h(self-energies)d Fx(\025)p 2614 5198 49 4 v 2663 5142 a Fz([)p Fu(n)p Fz(])2663 5208 y Fu(j)2745 5185 y FG(\()p Fx(")p FG(\))j(ma)n(y)e(b)r(ecome)h(di\013eren)n(t)118 5291 y(from)d(zero)g(for)g Fx(j)h FA(\024)23 b Fx(r)r FG(.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 118 356 a FG(23:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)189 555 y FG(Since)23 b(the)g(function)g FA(M)959 525 y Fz([)p Fu(n)p Fz(])1042 555 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))23 b(is)g(de\014ned)g(on)g(the)g (complemen)n(t)f(of)h(a)f(dense)h(op)r(en)g(set)g(di\013eren)n (tiabilit)n(y)f(in)118 662 y(the)j(sense)g(of)g(Whitney)g(can)g(b)r(e)g (pro)n(v)n(ed)e(\(as)i(usual\))g(b)n(y)f(computing)h(a)f(formal)h (deriv)-5 b(ativ)n(e)24 b(and)g(then)i(sho)n(wing)118 768 y(that)i(it)g(is)f(con)n(tin)n(uous)g(and)h(that)f(it)h(can)g(also) e(b)r(e)i(used)g(as)f(a)g(b)r(ound)h(in)g(in)n(terp)r(olations.)2973 738 y Fz(6)189 874 y FG(The)d(computation)f(of)h(the)g(formal)g(deriv) -5 b(ativ)n(es)23 b(pro)r(ceeds)h(as)g(the)i(computation)e(of)h(the)g (actual)g(deriv)-5 b(ativ)n(es)118 981 y(done)35 b(in)g(the)h(pro)r(of) e(of)h(Lemma)g(2)g(\(in)g(App)r(endix)h(A3\).)59 b(One)35 b(pro)n(v)n(es)e(formal)i(righ)n(t)f(and)h(left)h(con)n(tin)n(uous)118 1087 y(di\013eren)n(tiabilit)n(y)28 b(of)h(the)g(matrices)e FA(M)1373 1057 y Fz([)p Fu(n)p Fz(])1456 1087 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))29 b(on)f(the)h(closed)f(set)g FA(\\)2363 1051 y Fu(n)p FC(\000)p Fz(1)2363 1113 y Fu(m)p Fz(=)p 2473 1078 42 4 v Fu(n)2514 1121 y Ft(0)2547 1113 y FC(\000)p Fz(1)2636 1087 y FA(E)2680 1099 y Fu(m)2771 1087 y FG(simply)h(b)n(y)f(di\013eren)n(tiating)118 1204 y(term)35 b(b)n(y)g(term)g(the)g(v)-5 b(alue)35 b(of)g(eac)n(h)f (cluster)g(con)n(tributing)h(to)g FA(M)2286 1173 y Fz([)p Fu(n)p Fz(])2368 1204 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\).)60 b(This)35 b(in)n(v)n(olv)n(es)e(di\013eren)n(tiating)118 1310 y(matrices)24 b(lik)n(e)f(\()p Fx(x)675 1280 y Fz(2)725 1310 y FA(\000)12 b(M)902 1280 y Fz([)p FC(\024)p Fu(p)p Fz(])1029 1310 y FG(\()p Fx(x)p FG(;)i Fx(")p FG(\)\))1248 1280 y FC(\000)p Fz(1)1338 1310 y FG(,)25 b(i.e.)36 b(the)25 b(matrices)e FA(M)2097 1280 y Fz([)p Fu(p)p Fz(])2173 1310 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))25 b(with)g Fx(p)e(<)f(n)p FG(,)j(whic)n(h)f(are)g(di\013eren)n(tiable)118 1416 y(b)n(y)38 b(the)h(inductiv)n(e)g(assumption,)i(or)d(it)h(in)n(v)n (olv)n(es)e(di\013eren)n(tiating)h(the)h(cut-o\013)f(functions)h Fx( )3169 1428 y Fu(p)3208 1416 y Fx(;)14 b(\037)3297 1428 y Fu(p)3374 1416 y FG(and)38 b(the)118 1535 y(lo)r(cations)28 b Fx(\025)p 468 1548 49 4 v 517 1492 a Fz([)p Fu(p)p Fz(])517 1558 y Fu(j)593 1535 y FG(\()p Fx(")p FG(\))h(with)g Fx(j)h(>)25 b(r)32 b FG(\(b)r(ecause)d Fx(\025)p 1479 1548 V -43 x Fz([)p Fu(p)p Fz(])1527 1558 y Fu(j)1603 1535 y FG(\()p Fx(")p FG(\))d FA(\021)f FG(0)j(for)g Fx(j)j FA(\024)25 b Fx(r)r FG(\))30 b(whic)n(h)e(app)r(ear)g(in)i(the)f (form)f(\001)3299 1505 y Fz([)p Fu(p)p Fz(])3376 1535 y FG(\()p Fx(x;)14 b(")p FG(\))29 b(in)118 1641 y(the)c(argumen)n(ts)f (of)h(the)g(cut-o\013)g(functions.)37 b(All)25 b(suc)n(h)g(quan)n (tities)f(are)g(di\013eren)n(tiable)h(in)g Fx(")g FG(and)g(righ)n(t)f (and)h(left)118 1748 y(di\013eren)n(tiable)k(in)g Fx(x)g FG(b)n(y)f(the)i(inductiv)n(e)e(assumption;)h(furthermore)f(all)h (terms)f(arising)f(from)i(di\013eren)n(tiation)118 1854 y(either)37 b(of)f FA(M)566 1824 y Fz([)p Fu(p)p Fz(])642 1854 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))38 b(or)d Fx(\025)p 977 1867 V 1026 1811 a Fz([)p Fu(p)p Fz(])1026 1877 y Fu(j)1102 1854 y FG(\()p Fx(")p FG(\),)k(with)f Fx(p)g(<)g(n)p FG(,)h(app)r(ear)d(m)n(ultiplied)h(b)n(y)g(some)f(p)r(o) n(w)n(er)f(of)i Fx(")p FG(,)i(so)d(that)h(the)118 1960 y(inductiv)n(e)28 b(assumption)f(is)g(found)h(to)g(hold)f(also)g(for)g Fx(p)c FG(=)f Fx(n)28 b FG(\(for)f(a)g(similar)g(discussion)g(see)g (Ref.)38 b([Ge]\).)189 2066 y(Note)27 b(that)g(\001)637 2036 y Fz([)p Fu(n)p Fz(])720 2066 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b(dep)r(end)g(on)f Fx(j)5 b FG(\()p Fx(x)p FG(\))28 b(but)f(as)g Fx(")g FG(v)-5 b(aries)26 b(within)h(the)h(in)n(terv)-5 b(al)26 b Fx(I)7 b FG(,)28 b(see)e(\(ii\))i(in)f(de\014nition)h(1,)118 2173 y Fx(j)5 b FG(\()p Fx(x)p FG(\))28 b(is)e(not)g(only)g Fx(")p FG(-indep)r(enden)n(t)h(but)g(it)g(is)f(also)g(constan)n(t)f(in)i Fx(x)g FG(for)f Fx(x)h FG(v)-5 b(arying)25 b(in)i(small)f(in)n(terv)-5 b(als)26 b(near)f(the)118 2279 y(eigen)n(v)-5 b(alues)24 b(of)g Fx(M)717 2291 y Fz(0)779 2279 y FG(and,)h(therefore,)f(in)h(in)n (terv)-5 b(als)24 b(widely)g(spaced)g(b)r(ecause)h Fx(n)d FA(\025)p 2741 2233 50 4 v 23 w Fx(n)2791 2291 y Fz(0)2828 2279 y FG(:)36 b(this)25 b(is)f(due)h(to)f(the)h(cut-)118 2385 y(o\013)g(functions)g(whic)n(h)g(force)g Fx(x)g FG(to)g(b)r(e)g(close)f(to)h(a)g(single)f(eigen)n(v)-5 b(alue)24 b(if)i(the)f(propagator)d(of)j(the)g(corresp)r(onding)118 2492 y(line)g(is)g(di\013eren)n(t)g(from)g(0.)35 b(Hence)25 b(for)g Fx(n)e FA(\025)p 1501 2446 V 22 w Fx(n)1551 2504 y Fz(0)1613 2492 y FG(w)n(e)i(do)f(not)h(ha)n(v)n(e)f(to)h(di\013eren)n (tiate)g(the)g(function)g Fx(j)5 b FG(\()p Fx(x)p FG(\))26 b(\(neither)118 2598 y(with)31 b(resp)r(ect)g(to)f Fx(x)h FG(nor)f(with)h(resp)r(ect)g(to)f Fx(")h FG(from)f(whic)n(h)g(it)h(do)r (es)g(not)f(dep)r(end\);)j(for)e Fx(n)c(<)p 3100 2552 V 28 w(n)3150 2610 y Fz(0)3218 2598 y FG(the)k(function)118 2704 y Fx(j)5 b FG(\()p Fx(x)p FG(\))29 b(is)e(constan)n(t)g(to)h(the)g (righ)n(t)e(and)i(to)f(the)h(left)h(of)e(ev)n(ery)f(p)r(oin)n(t.)189 2811 y(The)33 b Fx(n)p FG(-indep)r(endence)h(of)f(the)h(constan)n(ts)e Fx(A)1647 2780 y FC(0)1671 2811 y Fx(;)14 b(A;)g(B)37 b FG(app)r(earing)32 b(in)i(the)g(inductiv)n(e)f(h)n(yp)r(othesis)g(is) g(pro)n(v)n(ed)118 2917 y(w)n(ord)23 b(b)n(y)h(w)n(ord)f(as)h(the)g (corresp)r(onding)e(argumen)n(t)h(in)i(App)r(endix)g(A3;)g(the)g (constan)n(t)e Fx(\024)2912 2929 y Fz(1)2973 2917 y FG(has)h(b)r(een)g (estimated)118 3023 y(ab)r(o)n(v)n(e)i(\(see)i Fx(G)585 3035 y Fz(2)650 3023 y FG(in)g(\(6.13\)\))f(and)g(is)h Fx(n)p FG(-indep)r(enden)n(t.)189 3129 y(The)k(in)n(terp)r(olation)g(b) r(ound,)i(see)e(fo)r(otnote)1619 3099 y Fz(6)1656 3129 y FG(,)i(necessary)c(for)i(de\014ning)h(the)f(Withney)h(deriv)-5 b(ativ)n(es,)33 b(holds)118 3236 y(b)r(ecause)c(in)g(comparing)f(t)n(w) n(o)g(con)n(tributions)h(to)g FA(M)1802 3206 y Fz([)p Fu(n)p Fz(])1884 3236 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))30 b(with)g(di\013eren)n(t)f Fx(x)h FG(or)e(di\013eren)n(t)i Fx(")f FG(the)g(di\016cult)n(y)118 3342 y(migh)n(t)i(only)g(come)f (from)h(the)g(comparison)e(of)i(\()p Fx(x)1725 3312 y Fz(2)1725 3366 y Fu(`)1784 3342 y FA(\000)20 b(M)1969 3312 y Fz([)p FC(\024)p Fu(p)p Fz(])2097 3342 y FG(\()p Fx(x)2176 3354 y Fu(`)2208 3342 y Fx(;)14 b(")p FG(\)\))2348 3312 y FC(\000)p Fz(1)2469 3342 y FG(ev)-5 b(aluated)30 b(at)h(t)n(w)n(o)f(di\013eren)n(t)i(p)r(oin)n(ts)118 3448 y(and)c(for)f(one)g(line)h Fx(`)f FG(at)g(a)h(time:)37 b(this)28 b(can)f(b)r(e)h(done)f(algebraically)f(b)n(y)h(using)g(the)h (resolv)n(en)n(t)e(iden)n(tit)n(y)516 3553 y Fr(\020)566 3645 y Fx(x)613 3611 y Fz(2)613 3665 y Fu(`)669 3645 y FA(\000)18 b(M)852 3611 y Fz([)p FC(\024)p Fu(p)p Fz(])980 3645 y FG(\()p Fx(x)1059 3657 y Fu(`)1091 3645 y Fx(;)c(")p FG(\))1199 3553 y Fr(\021)1249 3570 y FC(\000)p Fz(1)1357 3645 y FA(\000)1440 3553 y Fr(\020)1489 3645 y Fx(x)1536 3611 y FC(0)1560 3599 y 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4878 y Fz(0)730 4866 y FG(+)r Fy(b)p FG(\)\))20 b(where)f Fx(g)j FG(is)d(the)h(pseudo-di\013eren)n(tial)f(op)r(erator)f (\()p Fw(!)5 b FA(\001)r Fw(\027)h FG(\))2658 4836 y FC(\000)p Fz(2)2747 4866 y FG(.)35 b(The)19 b(pro)r(of)g(is)h(of)f (algebraic)118 4972 y(nature)26 b(and)h(ultimately)g(follo)n(ws)f(from) g(the)h(fact)g(that)g(the)g(series)e(w)n(e)i(are)e(considering)h(is)g (a)h(resummation)f(of)118 5079 y(Lindstedt's)21 b(series)e(whic)n(h)h (is)h(a)e(formal)h(solution)g(of)g(the)h(problem.)34 b(This)20 b(explains)g(wh)n(y)g(the)g(v)-5 b(arious)19 b(algebraic)118 5185 y(iden)n(tities)31 b(necessary)e(for)h(the)i(c)n (hec)n(k)e(actually)g(hold)g(and)h(the)g(pro)r(of)f(pro)r(ceeds)g (exactly)g(as)g(in)h(Section)g(8)g(of)118 5291 y(Ref.)41 b([Ge]:)e(w)n(e)28 b(repro)r(duce)g(the)h(argumen)n(t)e(and)i(the)g(c)n (hain)f(of)g(iden)n(tities)h(in)g(App)r(endix)h(A5.)40 b(Therefore)27 b(the)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)j(19:31)1130 b FG(24)p eop end %%Page: 25 25 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b(this)h(sense)f(\()p FF(unlikely)p FG(\))i(con)n(v)n(ergence)c(of)i(the)118 4535 y(Lindstedt)23 b(series)f(has)g(not)h(b)r(een)g(ruled)f(out)h(\()p FF(yet)p FG(\).)36 b(Nor)22 b(there)g(is)h(an)n(y)f(uniqueness)g (result)h(on)f(the)h(v)-5 b(alue)22 b(of)h(the)118 4642 y(renormalized)30 b(series.)47 b(The)31 b(latter)g(dep)r(ends)h(on)f (quite)h(a)e(few)i(arbitrary)d(c)n(hoices)i(\(ev)n(en)g(in)g(the)h(h)n (yp)r(erb)r(olic)118 4748 y(cases\):)j(for)25 b(instance)h(the)g (cut-o\013)g(shap)r(es)f(in)h(Fig.)36 b(2)26 b(are)e(quite)i(arbitrary) e(and)i(in)g(principle)f(the)i(allo)n(w)n(ed)d Fx(")p FG('s)118 4854 y(will)k(c)n(hange)e(with)i(the)g(c)n(hoice.)189 4972 y(F)-7 b(urthermore,)29 b(although)g(w)n(e)g(ha)n(v)n(e)g(not)g (really)g(c)n(hec)n(k)n(ed)f(all)h(necessary)f(details,)i(it)g(seems)f (to)h(us)f(that)h(our)118 5079 y(metho)r(d)22 b(also)e(sho)n(ws)h (that,)i(giv)n(en)d(a)h(v)-5 b(alue)22 b Fx(")1526 5091 y Fz(0)1584 5079 y FG(for)f(whic)n(h)h(the)f(renormalized)f(series)h (con)n(v)n(erges,)f(one)h(can)g(\014nd)h(a)118 5185 y(complex)e(domain) g(of)g Fx(")g FG(whic)n(h)g(is)g(op)r(en,)i(reac)n(hes)c(the)j(real)e (axis)h(with)g(a)g(v)n(ertical)f(cusp)h(at)g Fx(")2973 5197 y Fz(0)3031 5185 y FG(and)g(extends)g(to)g(an)118 5291 y(op)r(en)k(region)e(including)i(a)g(segmen)n(t)f(\()p FA(\000)p Fx(\021)s(;)14 b FG(0\))24 b(on)f(the)h(negativ)n(e)f(real)g (axis.)35 b(In)23 b(this)i(domain)e(the)h(renormalized)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 118 356 a FG(26:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(series)19 b(should)g(con)n(v)n (erge)e(taking)j(on)f(the)h(real)f(axis)g(real)f(v)-5 b(alues)20 b(parameterizing)d(an)j FF(hyp)l(erb)l(olic)i FG(torus)d(with)h(the)118 662 y(same)j(rotation)g(v)n(ector.)34 b(Ho)n(w)n(ev)n(er)22 b(since)i(there)f(are)g(no)g(uniqueness)h(pro)r (ofs)f(w)n(e)g(cannot)g(guaran)n(tee)f(that)i(eac)n(h)118 768 y(suc)n(h)29 b(extension)f(do)r(es)h(not)g(corresp)r(ond)e(to)i(a)f FF(di\013er)l(ent)h FG(torus)g(\(close)f(within)i(an)n(y)e(p)r(o)n(w)n (er)g(of)g Fx(")h FG(to)g(an)n(y)f(other)118 874 y(torus)i(of)h(the)h (same)e(t)n(yp)r(e\).)48 b(This)31 b(w)n(ould)f(signal)g(a)h(\\gian)n (t)f(bifurcation")g(that)h(one)g(w)n(ould)f(lik)n(e)h(to)g(exclude:)118 981 y(in)j(Ref.)56 b([GG])35 b(an)e(attempt)i(w)n(as)d(made)i(to)g(sho) n(w)f(uniqueness)g(b)n(y)h(estimating)f(the)h(size)g(of)g(the)g (Lindstedt)118 1087 y(series)27 b(co)r(e\016cien)n(ts)g(aiming)g(at)h (applying)f(the)h(theory)f(of)g(Borel)g(transforms.)35 b(Ho)n(w)n(ev)n(er)26 b(w)n(e)h(could)h(not)f(pro)n(v)n(e)118 1193 y(go)r(o)r(d)g(enough)f(b)r(ounds.)37 b(W)-7 b(e)28 b(obtained)f Fx(k)s FG(!)1480 1163 y Fu(\013)1555 1193 y FG(gro)n(wth)f(with)i(a)e(to)r(o)h(large)f Fx(\013)i FG(\(giv)n(en)f(our)f(estimated)i(size)f(of)g(the)118 1300 y(domain)g(of)h(analyticit)n(y)f(in)h Fx(")p FG(\))f(to)h(apply)f (uniqueness)g(results)g(from)h(the)g(theory)e(of)i(Borel)e(summations)i (.)727 1548 y FD(App)s(endix)39 b(A1.)99 b(A)37 b(brief)h(review)g(of)f (earlier)h(results)118 1725 y FG(The)44 b(system)g(whic)n(h)f(is)h (usually)f(studied)i(in)f(literature)f(when)h(the)g(problem)f(of)h(p)r (ersistence)g(of)f(lo)n(w)n(er-)118 1831 y(dimensional)27 b(elliptic)h(tori)f(is)h(studied,)g(is)f(of)h(the)g(form)951 2027 y FA(H)c FG(=)e Fw(!)s FG(\()p Fx(\030)t FG(\))d FA(\001)g Fy(A)f FG(+)1578 1923 y Fu(s)1534 1948 y Fr(X)1533 2127 y Fu(k)q Fz(=1)1668 2027 y FG(\012)1728 2039 y Fu(k)1769 2027 y FG(\()p Fx(\030)t FG(\))1887 1960 y Fr(\000)1925 2027 y Fx(q)1965 1993 y Fz(2)1962 2048 y Fu(k)2022 2027 y FG(+)g Fx(p)2147 1993 y Fz(2)2147 2048 y Fu(k)2187 1960 y Fr(\001)2244 2027 y FG(+)g Fx(P)12 b FG(\()p Fw(\013)p Fx(;)i Fy(A)p Fx(;)g Fy(q)p Fx(;)g Fy(p)p FG(\))p Fx(;)600 b FG(\()p Fx(A)p FG(1)p Fx(:)p FG(1\))118 2244 y(where)32 b(\()p Fw(\013)p Fx(;)14 b Fy(A)p Fx(;)g Fy(p)p Fx(;)g Fy(q)p FG(\))33 b FA(2)f Fv(T)956 2203 y Fu(r)1015 2244 y FA(\002)21 b Fv(R)1166 2203 y Fu(r)1225 2244 y FA(\002)g Fv(R)1376 2203 y Fu(s)1433 2244 y FA(\002)h Fv(R)1584 2203 y Fu(s)1620 2244 y FG(.)52 b(The)33 b(function)h Fx(P)44 b FG(is)33 b(analytic)f(in)h(its)g(argumen)n(ts,)g(and)g Fx(\030)k FG(is)118 2350 y(a)d(parameter)e(in)j Fv(R)765 2309 y Fu(r)802 2350 y FG(;)i(the)e(function)f Fx(P)46 b FG(is)34 b(a)g FF(p)l(erturb)l(ation)p FG(:)50 b(this)34 b(means)g(that)g(a)f(rescaling)g(of)h(the)g(actions)118 2457 y(could)27 b(allo)n(w)e(us)i(to)g(in)n(tro)r(duce)g(a)f(small)g (parameter)g Fx(")g FG(in)i(fron)n(t)e(of)h(the)g(function)h Fx(P)12 b FG(.)36 b(The)27 b(frequencies)f(of)h(the)118 2563 y(harmonic)i(oscillators)f(are)h(called)h FF(normal)j(fr)l(e)l (quencies)p FG(;)f(the)e(case)g(\012)2375 2575 y Fu(k)2415 2563 y FG(\()p Fx(\030)t FG(\))e(=)f(\012)2699 2575 y Fu(k)2767 2563 y FG(=)j(constan)n(t)f(\(that)i(is)f(with)118 2669 y(the)h(normal)f(frequencies)g(indep)r(enden)n(t)i(of)f Fx(\030)t FG(\))g(is)g(a)f(particular)g(case,)h(and)f(it)i(is)e (usually)h(referred)e(to)i(as)f(the)118 2776 y(\\constan)n(t)c (frequency)h(case".)35 b(Existence)26 b(of)h(in)n(v)-5 b(arian)n(t)26 b(tori)h(for)f(the)i(system)f(\(A1.1\))g(w)n(as)f (originally)f(pro)n(v)n(ed)118 2882 y(b)n(y)j(Mel'nik)n(o)n(v)f([Me1],) i([Me2],)f(new)g(pro)r(ofs)g(w)n(ere)f(pro)r(duced)h(b)n(y)g(Eliasson)f ([E1],)h(Kuksin)f([Ku],)i(and)f(P\177)-42 b(osc)n(hel)118 2988 y([P1].)36 b(The)26 b(case)f Fx(s)e FG(=)f(1)k(is)g(easier,)f(and) h(it)g(w)n(as)f(earlier)g(solv)n(ed)g(b)n(y)g(Moser)g([Mo].)36 b(Later)25 b(pro)r(ofs)h(w)n(ere)f(giv)n(en)g(b)n(y)118 3095 y(R)r(\177)-44 b(ussmann,)28 b(see)f(for)g(instance)g(Ref.)37 b([R].)28 b(See)g(also)f(the)h(v)n(ery)e(recen)n(t)h(Ref.)37 b([L)-9 b(W].)189 3201 y(F)i(or)39 b Fx(P)54 b FG(=)42 b(0)d(the)h(dimension)f(of)g(the)h(tori)f(is)g Fx(r)45 b(<)e(d)c FG(and)g(the)h(v)-5 b(ariables)38 b(\()p Fy(q)p Fx(;)14 b Fy(p)p FG(\))40 b(mo)n(v)n(e)e(around)h(stable)118 3307 y(equilibrium)28 b(p)r(oin)n(ts,)f(hence)h(suc)n(h)f(tori)g(are)g (called)g FF(el)t(liptic)32 b(lower-dimensional)g(tori)p FG(.)189 3413 y(The)j(conditions)f(under)g(whic)n(h)g(the)h(quoted)f (results)g(are)g(pro)n(v)n(ed)f(are,)i(b)r(esides)f(the)h(usual)f (Diophan)n(tine)118 3520 y(condition)26 b(\(1.3\))f(on)h Fw(!)s FG(,)g(t)n(w)n(o)f(non-resonance)f(conditions)h(in)n(v)n(olving) f(one)i(and)g(t)n(w)n(o)f(normal)g(frequencies)g(\(the)118 3626 y(so)i(called)h(\014rst)g(and)g(second)f(Mel'nik)n(o)n(v)g (conditions,)h(originally)e(in)n(tro)r(duced)i(in)g(Ref.)38 b([Me1]\);)29 b(in)f(particular)118 3732 y(one)23 b(has)g(to)g(imp)r (ose)g(that)h(the)f(normal)g(frequencies)f(are)h(non-degenerate)e (\(i.e.)36 b(they)23 b(ha)n(v)n(e)g(to)g(b)r(e)g(all)g(di\013eren)n(t) 118 3839 y(from)k(eac)n(h)g(other\).)189 3945 y(Recen)n(tly)40 b(pro)r(ofs)g(of)h(existence)f(of)h(elliptic)g(lo)n(w)n(er-dimensional) d(tori)i(w)n(ere)g(giv)n(en)g(b)n(y)g(requesting)g(only)118 4051 y(the)g(\014rst)g(Mel'nik)n(o)n(v)e(conditions:)61 b(this)40 b(allo)n(ws)e(treating)h(degenerate)f(frequencies.)73 b(The)39 b(\014rst)h(result)f(in)118 4157 y(this)e(direction)e(is)h (due)h(to)f(Bourgain)e([B3],)k(where)e(the)h(ideas)e(in)n(tro)r(duced)h (in)h(Refs.)63 b([CrW])36 b(and)g([B1])g(to)118 4264 y(pro)n(v)n(e)31 b(existence)i(of)g(p)r(erio)r(dic)g(and)g(quasi-p)r (erio)r(dic)f(solutions)g(in)i(nearly)e(in)n(tegrable)g(Hamiltonian)g (partial)118 4370 y(di\013eren)n(tial)i(equations)g(w)n(ere)f(adapted)h (to)g(construct)g(lo)n(w)n(er-dimensional)e(tori)i(in)g(the)h (\014nite-dimensional)118 4476 y(Hamiltonian)22 b(systems)h(\(A1.1\))f (corresp)r(onding)f(to)h(the)h(case)f(of)g(constan)n(t)g(normal)g (frequencies.)34 b(New)23 b(pro)r(ofs,)118 4583 y(extending)28 b(the)h(results)f(also)f(to)h(the)h(case)e(of)h(non-constan)n(t)f (normal)h(frequencies,)g(are)f(due)h(to)h(Xu)f(and)g(Y)-7 b(ou)118 4689 y([Y],)28 b([XY].)189 4795 y(An)d(extension)e(of)h(the)h (results)f(of)g(existence)g(of)g(p)r(erio)r(dic)g(and)g(quasi-p)r(erio) r(dic)f(solutions)g(describing)g(lo)n(w)n(er-)118 4902 y(dimensional)32 b(in)n(v)-5 b(arian)n(t)31 b(tori)h(for)g (in\014nite-dimensional)g(PDE)g(systems)g(has)g(b)r(een)g(pro)n(vided)g (in)g(a)g(series)g(of)118 5008 y(pap)r(ers,)27 b(whic)n(h)h(include)g (Refs.)37 b([Ku],)27 b([W)-7 b(a],)28 b([CrW],)g([KP],)f([P2],)f([B1],) i([B2],)f([B4],)g([GM2])h(and)f([GMP].)189 5185 y(On)j(the)h(other)f (hand)g(the)h(problem)f(\(1.2\))g(has)f(not)i(b)r(een)g(widely)f (studied)h(in)f(literature.)45 b(It)30 b(corresp)r(onds)118 5291 y(to)35 b(a)f(degenerate)f(case)h(b)r(ecause)g(in)h(absence)f(of)h (p)r(erturbations)f(the)h(lo)n(w)n(er-dimensional)d(tori)i(are)g (neither)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)c(19:31)1130 b FG(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 118 356 a FG(27:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(elliptic)f(nor)e(h)n(yp)r(erb) r(olic:)41 b(it)30 b(is)g(the)g(p)r(erturbation)f(itself)i(whic)n(h)f (determines)f(if)i(the)f(tori,)g(when)g(con)n(tin)n(uing)118 662 y(to)e(exist,)f(b)r(ecome)h(elliptic)g(or)e(h)n(yp)r(erb)r(olic)i (\(or)f(of)g(mixed)h(t)n(yp)r(e)g(or)e(parab)r(olic\).)118 768 y(\(i\))39 b(The)g(case)f(of)h(h)n(yp)r(erb)r(olic)f(tori)g(is)g (easier,)i(and)f(it)g(w)n(as)f(the)h(\014rst)f(to)h(b)r(e)g(studied,)i (b)n(y)e(T)-7 b(reshc)n(h)n(\177)-39 b(ev)36 b([T].)118 874 y(Recen)n(tly)23 b(the)h(problem)f(w)n(as)f(reconsidered)g(in)h (Ref.)36 b([GG],)24 b(where)f(the)g(analyticit)n(y)g(domain)g(of)g(the) h(in)n(v)-5 b(arian)n(t)118 981 y(tori)33 b(w)n(as)g(studied)h(in)g (more)e(detail.)55 b(In)34 b(the)g(case)f(of)g(elliptic)h(tori)f(the)h (problem)f(w)n(as)g(considered)f(in)i(Refs.)118 1087 y([ChW])24 b(and)f([W)n(C],)h(where)f(T)-7 b(reshc)n(h)n(\177)-39 b(ev's)21 b(approac)n(h)g(to)j(the)g(study)f(of)h(the)f(case)g(of)g(h)n (yp)r(erb)r(olic)g(tori,)h(in)n(v)n(olving)118 1193 y(a)31 b(preliminary)g(c)n(hange)f(of)i(co)r(ordinates,)f(is)g(used)h(to)f (cast)g(the)h(Hamiltonian)g(in)f(a)g(form)h(whic)n(h)f(is)h(suitable) 118 1300 y(for)i(applying)g(P\177)-42 b(osc)n(hel's)33 b(results)h(on)g(elliptic)i(tori:)50 b(in)35 b(particular)e(this)i(imp) r(oses)f(the)h(same)f(conditions)g(as)118 1406 y(in)d(Ref.)45 b([P1])29 b(on)h(the)h(normal)e(frequencies)h(whic)n(h)g(app)r(ear)f (after)h(the)h(canonical)e(c)n(hange)g(of)h(co)r(ordinates)f(is)118 1512 y(p)r(erformed.)118 1618 y(\(ii\))34 b(The)g(existence)f(problem)g (has)g(b)r(een)h(also)f(considered)f(in)i(Ref.)55 b([JLZ],)33 b FF(wher)l(e)j(el)t(liptic)h(and)f(hyp)l(erb)l(olic)118 1725 y(tori)h(wer)l(e)g(studie)l(d)g(simultane)l(ously)p FG(,)g(again)d(b)n(y)g(imp)r(osing)h(some)g(non-degeneracy)d (conditions)j(on)g(normal)118 1831 y(frequencies.)49 b(Ref.)h([JLZ])31 b(do)r(es)h(not)g(in)n(v)n(estigate)e(resummations)h (of)h(Lindstedt's)g(series;)h(it)f(is)g(based)f(on)h(a)118 1937 y(rapid)h(con)n(v)n(ergence)d(metho)r(d,)35 b(close)d(in)h(spirit) g(to)g(the)h(original)d(pro)r(ofs)h(of)h(the)h(KAM)f(theorem:)47 b(a)32 b(concise)118 2044 y(existence)27 b(pro)r(of)g(of)h(lo)n(w)n (er-dimensional)d(tori)i(is)h(ac)n(hiev)n(ed)e(in)i(b)r(oth)g(the)g (elliptic)g(and)f(h)n(yp)r(erb)r(olic)h(cases.)189 2221 y(W)-7 b(e)27 b(stress)e(that)i(in)g(all)f(quoted)g(pap)r(ers,)g (except)h(Ref.)36 b([JLZ])26 b(and)h([T],)f(the)h(problem)f(is)g (considered)g(with)h Fx(")118 2327 y FG(\(i.e.)44 b(the)31 b(size)e(of)h(the)g(p)r(erturbation\))g(\014xed)g(and)g(the)g(study)g (deals)f(with)i(estimates)e(of)h(the)g(measure)f(of)h(the)118 2433 y(rotation)25 b(v)n(ectors)f Fw(!)k FG(for)e(whic)n(h)f(there)h (exist)g(in)n(v)-5 b(arian)n(t)24 b(tori.)36 b(W)-7 b(e)26 b(supp)r(ose,)g(instead,)g(that)g Fw(!)j FG(is)c(\014xed,)h(hence)118 2540 y(w)n(e)g(study)g(the)h(dep)r(endence)g(on)f Fx(")g FG(of)g(the)g(lo)n(w)n(er-dimensional)e(in)n(v)-5 b(arian)n(t)25 b(tori)h(and,)g(in)h(particular,)e(the)h(set)h(of)118 2646 y(v)-5 b(alues)27 b(of)h Fx(")f FG(for)g(whic)n(h)h(the)g(tori)f (surviv)n(e.)189 2823 y(Our)j(tec)n(hniques)g(extend)h(those)f(in)h (Refs.)46 b([GG])31 b(and)g([Ge],)g(and)g(are)e(based)h(on)h(the)g (metho)r(d)g(in)n(tro)r(duced)118 2929 y(in)h(Refs.)48 b([E2])30 b(and)h([Ga].)48 b(With)32 b(resp)r(ect)f(to)g(Ref.)49 b([Ge],)32 b(where)f(existence)g(of)g(quasi-p)r(erio)r(dic)f(solutions) h(is)118 3036 y(pro)n(v)n(ed)g(for)h(the)h FF(gener)l(alize)l(d)i(R)n (ic)l(c)l(ati)f(e)l(quation)f FG(considered)f(in)g(Ref.)52 b([Ba],)33 b(the)g(main)f(di\016cult)n(y)h(is)f(due)h(to)118 3142 y(the)c(presence)f(of)g(sev)n(eral)f(normal)g(frequencies.)39 b(It)29 b(is)f(not)h(surprising)e(that)i(this)f(generates)f(extra)h (tec)n(hnical)118 3248 y(di\016culties:)49 b(as)32 b(already)g(noted,)j (it)f(is)f(w)n(ell)g(kno)n(wn)g(that)g(the)h(case)e Fx(s)h FG(=)f(1)h(is)g(easier;)i(see)d(Refs.)55 b([Mo])33 b(and)118 3355 y([C].)d(An)h(adv)-5 b(an)n(tage)28 b(of)i(the)h(presen)n(t)e (metho)r(d)i(is)f(that)g(it)h(is)f(fully)g(constructiv)n(e)f(and)h(giv) n(es)f(a)h(v)n(ery)e(detailed)118 3461 y(kno)n(wledge)e(of)i(the)g (solution.)1027 3709 y FD(App)s(endix)38 b(A2.)50 b(Excluded)38 b(v)-6 b(alues)38 b(of)g Fx(")118 3886 y FG(De\014ne)610 4005 y Fx(\032)p 653 3983 42 4 v 13 x Fu(n)693 4026 y Ft(0)726 4018 y FC(\000)p Fz(1)829 3954 y Fu(def)849 4005 y FG(=)948 3889 y Fr(r)p 1031 3889 100 4 v 1061 3948 a Fx(")p 1041 3986 80 4 v 1041 4062 a(a)1085 4074 y Fu(s)1144 4005 y FG(min)1296 3912 y Fr(n)1365 4005 y FG(min)1380 4057 y Fu(i>r)1517 4005 y FA(j)p Fx(@)1584 4017 y Fu(")1620 3894 y Fr(q)p 1703 3894 227 4 v 111 x Fx(\025)p 1703 4018 49 4 v 1752 3962 a Fz([0])1752 4028 y Fu(i)1826 4005 y FG(\()p Fx(")p FG(\))q FA(j)p Fx(;)26 b FG(min)2025 4047 y Fs(i)p Ff(6)p Ft(=)p Fs(j)2000 4095 y(i;j)r(>r)2166 4005 y FA(j)p Fx(@)2233 4017 y Fu(")2268 3899 y Fr(q)p 2351 3899 227 4 v 106 x Fx(\025)p 2351 4018 49 4 v 2400 3962 a Fz([0])2400 4028 y Fu(j)2475 4005 y FG(\()p Fx(")p FG(\))18 b FA(\000)g Fx(@)2723 4017 y Fu(")2759 3894 y Fr(q)p 2842 3894 227 4 v 111 x Fx(\025)p 2842 4018 49 4 v -43 x Fz([0])2890 4028 y Fu(i)2965 4005 y FG(\()p Fx(")p FG(\))q FA(j)3092 3912 y Fr(o)3147 4005 y Fx(;)610 4256 y(\032)653 4222 y FC(0)p 653 4242 42 4 v 653 4277 a Fu(n)693 4285 y Ft(0)726 4277 y FC(\000)p Fz(1)829 4205 y Fu(def)849 4256 y FG(=)1030 4200 y(1)p 958 4237 188 4 v 958 4260 a FA(p)p 1027 4260 119 4 v 53 x Fx("a)1110 4325 y Fu(s)1169 4256 y FG(max)1231 4309 y Fu(j)1337 4164 y Fr(n)1392 4151 y(q)p 1475 4151 227 4 v 105 x Fx(\025)p 1475 4269 49 4 v 1524 4213 a Fz([0])1524 4279 y Fu(j)1599 4256 y FG(\()p Fx(")p FG(\))1702 4164 y Fr(o)1757 4256 y Fx(;)3428 4132 y FG(\()p Fx(A)p FG(2)p Fx(:)p FG(1\))118 4465 y(and)24 b(note)g(that)g Fx(\032)p 676 4443 42 4 v 13 x Fu(n)717 4486 y Ft(0)749 4478 y FC(\000)p Fz(1)862 4465 y FG(is)g(b)r(ounded)g(from)g(b)r(elo)n (w)f(prop)r(ortionally)f(to)i Fx(\032)p FG(,)h(as)e(de\014ned)h(in)g (\(4.2\),)h(and)f Fx(\032)3324 4435 y FC(0)p 3324 4454 V 3324 4489 a Fu(n)3365 4497 y Ft(0)3397 4489 y FC(\000)p Fz(1)3509 4465 y FG(=)f(1.)118 4571 y(Then)29 b(\(3.3\))f(excludes,)g (for)g(eac)n(h)g Fw(\027)6 b FG(,)28 b(an)g(in)n(terv)-5 b(al)28 b(in)h Fx(")f FG(whose)g(measure)f(is)i(b)r(ounded)f(\(using) 3137 4517 y FA(p)p 3206 4517 119 4 v 54 x Fx(a)3250 4583 y Fu(s)3285 4571 y Fx(")d FA(\024)f Fx(C)6 b FG(;)29 b(see)118 4677 y(\(3.2\)\))f(b)n(y)1433 4784 y(2)1475 4749 y FC(\000)p Fz(\()p 1553 4714 42 4 v Fu(n)1593 4757 y Ft(0)1625 4749 y FC(\000)p Fz(1\))p Fu(=)p Fz(2)1808 4784 y Fx(C)20 b(C)1946 4796 y Fz(0)1997 4784 y Fx(K)2068 4796 y Fz(0)2105 4784 y FA(j)p Fw(\027)6 b FA(j)2205 4749 y FC(\000)p Fu(\034)2288 4757 y Ft(1)2324 4784 y Fx(;)1081 b FG(\()p Fx(A)p FG(2)p Fx(:)p FG(2\))118 4945 y(where)27 b(the)h(constan)n(t)f Fx(K)907 4957 y Fz(0)971 4945 y FG(can)h(b)r(e)g(estimated)f(b)n(y)h Fx(K)1802 4957 y Fz(0)1861 4945 y FG(=)23 b Fx(s)14 b(a)2046 4914 y FC(\000)p Fz(1)2046 4965 y Fu(s)2135 4945 y Fx(\032)2178 4909 y FC(\000)p Fz(1)p 2178 4936 V 2178 4971 a Fu(n)2219 4979 y Ft(0)2251 4971 y FC(\000)p Fz(1)2340 4945 y FG(.)189 5051 y(The)28 b(Diophan)n(tine)f(condition)h(on)f Fw(!)j FG(implies)e(that)g(if)g(\(3.3\))f(is)h(in)n(v)-5 b(alid)27 b(then)i FA(j)p Fw(\027)6 b FA(j)27 b FG(cannot)g(b)r(e)h(to)r(o)f (small)943 5271 y(2)985 5178 y Fr(q)p 1068 5178 324 4 v 93 x Fx("a)1151 5283 y Fu(s)1186 5271 y Fx(\032)1229 5242 y FC(0)p 1229 5262 42 4 v 1229 5297 a Fu(n)1270 5305 y Ft(0)1302 5297 y FC(\000)p Fz(1)1410 5271 y FG(+)18 b(2)1535 5236 y FC(\000)p Fz(\()p 1613 5201 V Fu(n)1653 5244 y Ft(0)1685 5236 y FC(\000)p Fz(1\))p Fu(=)p Fz(2)1868 5271 y Fx(C)1927 5283 y Fz(0)1964 5271 y FA(j)p Fw(\027)6 b FA(j)2064 5236 y FC(\000)p Fu(\034)2147 5244 y Ft(1)2206 5271 y FA(\025)23 b(j)p Fx(x)p FA(j)g(\025)g Fx(C)2557 5283 y Fz(0)2595 5271 y FA(j)p Fw(\027)6 b FA(j)2695 5236 y FC(\000)p Fu(\034)2778 5244 y Ft(0)2813 5271 y Fx(:)592 b FG(\()p Fx(A)p FG(2)p Fx(:)p FG(3\))118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 118 356 a FG(28:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 567 y FG(Therefore)504 479 y Fr(q)p 587 479 324 4 v 88 x Fx("a)670 579 y Fu(s)705 567 y Fx(\032)748 539 y FC(0)p 748 559 42 4 v 748 594 a Fu(n)789 602 y Ft(0)821 594 y FC(\000)p Fz(1)949 567 y FA(\025)1062 535 y Fz(1)p 1062 549 34 4 v 1062 596 a(4)1105 567 y Fx(C)1164 579 y Fz(0)1202 567 y FA(j)p Fw(\027)6 b FA(j)1302 537 y FC(\000)p Fu(\034)1385 545 y Ft(0)1458 567 y FG(if)p 1543 522 50 4 v 37 w Fx(n)1593 579 y Fz(0)1669 567 y FA(\025)38 b FG(3,)h(hence)e(in)g(this)g(case)g (w)n(e)f(only)h(ha)n(v)n(e)e(to)i(consider)f(the)118 728 y(v)-5 b(alues)41 b(of)f Fw(\027)47 b FG(with)41 b FA(j)p Fw(\027)6 b FA(j)45 b(\025)g FG(\()p Fx(C)1129 740 y Fz(0)1167 728 y Fx(=)p FG(\(4)1283 640 y Fr(q)p 1365 640 324 4 v 1365 728 a Fx("a)1448 740 y Fu(s)1483 728 y Fx(\032)1526 700 y FC(0)p 1526 720 42 4 v 1526 755 a Fu(n)1567 763 y Ft(0)1600 755 y FC(\000)p Fz(1)1689 728 y FG(\)\))1753 698 y Fz(1)p Fu(=\034)1851 706 y Ft(0)1888 728 y FG(.)76 b(Since)41 b Fx(C)t(=)p FG(2)j Fx(<)2518 675 y FA(p)p 2587 675 119 4 v 53 x Fx("a)2670 740 y Fu(s)2751 728 y FA(\024)g Fx(C)52 b FG(=)44 b(2)3122 698 y FC(\000)p Fu(n)3215 706 y Ft(0)3251 728 y Fx(C)3310 740 y Fz(0)3348 728 y FG(,)g(w)n(e)d(get)118 889 y(the)33 b(b)r(ound)f(\(3.5\))g(with)g Fx(\034)958 901 y Fz(1)1026 889 y FG(=)e Fx(\034)h FG(+)21 b Fx(r)k FG(+)c(1)31 b(and)h Fx(K)k FG(=)30 b Fx(K)1934 901 y Fz(0)1985 889 y Fx(C)2044 901 y Fz(0)2095 889 y FG(\(4)p Fx(C)2234 801 y Fr(q)p 2317 801 206 4 v 88 x Fx(\032)2360 861 y FC(0)p 2360 881 42 4 v 2360 916 a Fu(n)2401 924 y Ft(0)2434 916 y FC(\000)p Fz(1)2537 889 y Fx(C)2602 854 y FC(\000)p Fz(1)2596 912 y(0)2691 889 y FG(\))2723 859 y Fz(\()p Fu(\034)2780 867 y Ft(1)2812 859 y FC(\000)p Fu(r)r FC(\000)p Fz(1\))p Fu(=\034)3073 867 y Ft(0)3123 827 y Fr(P)3210 914 y Fq(\027)5 b FC(6)p Fz(=)p Fo(0)3447 857 y Fz(1)p 3370 871 187 4 v 3370 918 a FC(j)p Fq(\027)g FC(j)3453 902 y Fs(r)q Ft(+1)3597 889 y FG(=)118 1026 y(4)p Fx(K)231 1038 y Fz(0)267 962 y Fr(p)p 350 962 229 4 v 64 x Fx(\032)393 1002 y FC(0)p 417 1011 42 4 v 417 1047 a Fu(n)458 1055 y Ft(0)490 1047 y FC(\000)p Fz(1)593 964 y Fr(P)681 1051 y Fq(\027)g FC(6)p Fz(=)p Fo(0)917 994 y Fz(1)p 840 1008 187 4 v 840 1055 a FC(j)p Fq(\027)g FC(j)923 1038 y Fs(r)q Ft(+1)1037 1026 y FG(.)77 b(Note)42 b(that)f(a)g(condition)g(lik)n(e)f Fx(\034)2207 1038 y Fz(1)2291 1026 y Fx(>)45 b(\034)37 b FG(+)27 b Fx(r)44 b FG(is)d(su\016cien)n(t)g(to)g(obtain)g(b)r(oth) 118 1133 y(summabilit)n(y)33 b(o)n(v)n(er)d Fw(\027)38 b FG(and)33 b(a)f(measure)f(\(of)i(the)g(excluded)f(set\))h(relativ)n (ely)e(small)h(with)h(resp)r(ect)f(to)h(that)g(of)118 1239 y Fx(I)154 1251 y Fu(C)210 1239 y FG(.)k(If)p 351 1193 50 4 v 26 w Fx(n)401 1251 y Fz(0)461 1239 y Fx(<)23 b FG(3,)j(hence)g Fx(n)919 1251 y Fz(0)979 1239 y Fx(<)d FG(3,)j(the)g(same)g(conclusion)f(trivially)g(holds)h(p)r(ossibly)f (increasing)g(the)h(v)-5 b(alue)26 b(of)g Fx(K)118 1345 y FG(b)n(y)h(a)h(factor)e(4.)420 1593 y FD(App)s(endix)38 b(A3.)50 b(Resummations:)h(con)m(v)m(ergence)38 b(and)h(smo)s(othness) 118 1770 y FG(T)-7 b(o)37 b(pro)n(v)n(e)e(Lemma)i(2,)i(w)n(e)d(\014rst) h(sho)n(w)f(that)h(the)h(series)e(de\014ning)h Fx(M)2407 1740 y Fz([)p Fu(n)p Fz(])2489 1770 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))38 b(for)e(0)j FA(\024)f Fx(n)g FA(\024)p 3226 1725 V 39 w Fx(n)3276 1782 y Fz(0)3350 1770 y FG(con)n(v)n(erge)118 1877 y(and)33 b(then)h(w)n(e)e(c)n(hec)n (k)h(smo)r(othness)f(and)h(the)g(b)r(ounds.)54 b(This)33 b(is)g(done)g(for)f(completeness)h(as)f(the)i(argumen)n(t)118 1983 y(is)h(almost)g(a)g(w)n(ord)f(b)n(y)h(w)n(ord)f(rep)r(etition)h (of)g(the)h(analysis)e(in)h(Ref.)60 b([GG],)36 b(with)g(a)e(few)i (sligh)n(t)f(c)n(hanges)f(of)118 2089 y(notations)c(necessary)f(to)h (adapt)g(it)h(to)g(our)e(presen)n(t)h(notations)g(and)g(scop)r(e.)46 b(T)-7 b(o)30 b(study)h(con)n(v)n(ergence)d(of)i(the)118 2196 y(series)36 b(de\014ning)g Fx(M)762 2165 y Fz([)p Fu(n)p Fz(])845 2196 y FG(\()p Fx(x;)14 b(")p FG(\),)39 b Fx(n)f FA(\024)p 1285 2150 V 38 w Fx(n)1335 2208 y Fz(0)1372 2196 y FG(,)h(w)n(e)d(remark)f(that)i(w)n(e)g(ha)n(v)n(e)e (to)i(consider)e(only)h(trees)g(in)h(whic)n(h)g(all)118 2302 y(propagators)29 b(ha)n(v)n(e)i(scales)f([)p Fx(p)p FG(])i(with)h Fx(p)d FA(\024)p 1491 2256 V 29 w Fx(n)1541 2314 y Fz(0)1578 2302 y FG(.)50 b(Therefore)31 b(the)h(propagators)d (whic)n(h)j(do)f(not)h(v)-5 b(anish)32 b(will)g(b)r(e)118 2408 y(suc)n(h)h(that)h(their)g(denominators)e(satisfy)h Fx(D)r FG(\()p Fx(x)p FG(\))i Fx(>)d FG(2)1846 2378 y FC(\000)p Fz(2\()p 1957 2343 42 4 v Fu(n)p Fz(+1\))2112 2408 y FA(j)p Fx(x)p FA(j)2205 2378 y Fz(2)2243 2408 y FG(,)j(see)e(\(4.4\),)i(so)e(that)h(they)g(are)f(e\013ectiv)n(ely)118 2515 y(estimated)i(from)f(b)r(elo)n(w)g(b)n(y)h FA(j)p Fx(x)p FA(j)1165 2484 y Fz(2)1237 2515 y FG(times)g(a)f(constan)n(t.)57 b(Note)35 b(that)g(the)g(case)e Fx(n)i FG(=)f(0)g(is)h(ob)n(vious)e (\(and)i(it)g(is)118 2621 y(treated)27 b(in)h(Section)g(3\).)118 2798 y Fy(A3.1.)35 b FF(Conver)l(genc)l(e.)j FG(W)-7 b(e)28 b(supp)r(ose)e(that)i(the)f(eigen)n(v)-5 b(alues)26 b(of)h FA(M)2287 2768 y Fz([)p FC(\024)p Fu(p)p Fz(])2414 2798 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)28 b Fx(n)23 b FG(=)f(0)p Fx(;)14 b(:)g(:)g(:)g(;)g(n)i FA(\000)h FG(1,)27 b(di\013er)g(from)118 2914 y(the)k(corresp)r(onding)e(ones)h (of)h FA(M)1187 2884 y Fz([)p FC(\024)p Fz(0])1313 2914 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))29 b FA(\021)f Fx(M)1703 2926 y Fz(0)1771 2914 y FG(so)i(that)h FA(j)p Fx(\025)2130 2871 y Fz([)p Fu(p)p Fz(])2130 2937 y Fu(j)2206 2914 y FG(\()p Fx(x;)14 b(")p FG(\))21 b FA(\000)f Fx(\025)2547 2871 y Fz([0])2547 2937 y Fu(j)2623 2914 y FA(j)28 b Fx(<)g(\015)5 b(")2854 2884 y Fz(2)2921 2914 y FG(for)31 b(some)f Fx(\015)j(>)27 b FG(0,)32 b(and)118 3020 y(that)25 b Fx(")e FG(is)h(small)g(enough)g(so)f(that)i Fx(\015)5 b(")1296 2990 y Fz(2)1355 3020 y Fx(<)1453 2987 y Fz(1)p 1453 3001 34 4 v 1453 3049 a(2)1496 3020 y Fx("a)1579 3032 y Fu(s)1614 3020 y FG(2)1656 2990 y FC(\000)p Fz(2)p 1741 2955 42 4 v Fu(n)o FC(\000)p Fz(2)1895 3020 y FG(and,)24 b(therefore)g(\(see)g(\(4.4\)\),)h(if)f(a)g(line)g(with)h(frequency)118 3144 y Fx(x)j FG(has)f(scale)g([)p Fx(p)p FG(],)h Fx(p)23 b(<)f(n)p FG(,)28 b(then)g FA(j)p Fx(x)1191 3114 y Fz(2)1247 3144 y FA(\000)18 b Fx(\025)1378 3101 y Fz([)p Fu(p)p Fz(])1378 3167 y Fu(j)1454 3144 y FG(\()p Fx(x;)c(")p FG(\))p FA(j)24 b Fx(>)f FG(2)1818 3114 y FC(\000)p Fz(2\()p 1929 3079 V Fu(n)o Fz(+2\))2083 3144 y Fx(x)2130 3114 y Fz(2)2168 3144 y FG(.)189 3260 y(W)-7 b(e)26 b(shall)g(use)g(that)g (if)h(a)e(the)i(propagator)c(of)j(a)g(line)g(is)g(on)g(a)f(scale)h([)p Fx(n)p FG(])g(then)g(one)g(has)g Fx(D)r FG(\()p Fx(x)p FG(\))e FA(\024)e FG(2)3269 3230 y FC(\000)p Fz(2\()p Fu(n)p FC(\000)p Fz(2\))3536 3260 y Fx(C)3601 3230 y Fz(2)3595 3281 y(0)3638 3260 y FG(,)118 3366 y(ev)n(en)i(though)g FF(we)j(c)l(ould)g(al)t(low)f FG(also)d(a)g(b)r(ound)i Fx(D)r FG(\()p Fx(x)p FG(\))f FA(\024)f FG(2)1941 3336 y FC(\000)p Fz(2\()p Fu(n)p FC(\000)p Fz(1\))2207 3366 y Fx(C)2272 3336 y Fz(2)2266 3387 y(0)2310 3366 y FG(.)36 b(The)24 b(reason)e(for)i(this)h(is)f(again)f(for)g(later)118 3473 y(use)28 b(in)f(b)r(ounds)h(necessary)e(to)h(establish)h(the)g (needed)f(cancellations)g(as)g(commen)n(ted)g(in)h(Section)g(A3.2.)189 3579 y(Consider)h(a)g(renormalized)f(self-energy)h(cluster)g Fx(T)38 b FA(2)27 b(S)2014 3549 y FC(R)2008 3603 y Fu(k)q(;n)p FC(\000)p Fz(1)2195 3579 y FG(,)j(and)g(de\014ne)g(\003)2712 3591 y Fu(m)2775 3579 y FG(\()p Fx(T)12 b FG(\))26 b(=)g FA(f)p Fx(`)g FA(2)h FG(\003\()p Fx(T)12 b FG(\))40 b(:)g Fx(n)3538 3591 y Fu(`)3597 3579 y FG(=)118 3685 y Fx(m)p FA(g)p FG(,)27 b(for)g Fx(m)c FA(\024)g Fx(n)18 b FA(\000)g FG(1,)27 b(and)h FA(P)7 b FG(\()p Fx(T)12 b FG(\))27 b(the)h(set)f(of)h(lines)f(\(path\))i(connecting)e(the)h(external)f (lines)g(of)h Fx(T)12 b FG(.)189 3792 y(If)20 b Fw(\027)26 b FG(is)20 b(the)h(momen)n(tum)f(\015o)n(wing)f(in)h(the)h(line)f(en)n (tering)f Fx(T)31 b FG(then)21 b(the)f(momen)n(tum)h(\015o)n(wing)e(in) h(a)g(line)g Fx(`)j FA(2)g FG(\003\()p Fx(T)12 b FG(\))118 3898 y(of)28 b(scale)f([)p Fx(p)p FG(],)h Fx(p)23 b FA(\024)g Fx(n)18 b FA(\000)g FG(1,)28 b(will)g(b)r(e)g Fw(\027)1272 3868 y Fz(0)1266 3921 y Fu(`)1328 3898 y FG(+)18 b Fx(\033)1458 3910 y Fu(`)1490 3898 y Fw(\027)6 b FG(,)28 b Fx(\033)1642 3910 y Fu(`)1697 3898 y FG(=)23 b(0)p Fx(;)14 b FG(1,)27 b(where)h Fw(\027)2251 3868 y Fz(0)2245 3921 y Fu(`)2315 3898 y FG(is)g(the)g(momen)n(tum)g(that)g(w)n(ould)g(\015o)n(w)f(on)118 4004 y Fx(`)g FG(if)h Fw(\027)h FG(=)23 b Fy(0)p FG(.)36 b(The)28 b(corresp)r(onding)e(frequency)h(will)h(b)r(e)g Fx(x)1926 3974 y FC(0)1926 4028 y Fu(`)1981 4004 y FG(=)23 b Fx(x)2116 3974 y Fz(0)2116 4028 y Fu(`)2172 4004 y FG(+)18 b Fx(\033)2302 4016 y Fu(`)2334 4004 y Fx(x)p FG(,)29 b(with)f(ob)n(vious)e(notations.)189 4111 y(First)h(of)h(all)f (w)n(e)h(shall)f(pro)n(v)n(e)f(the)i(b)r(ound)1479 4229 y Fr(X)1418 4411 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))1673 4308 y FA(j)p Fw(\027)1744 4320 y Fm(v)1791 4308 y FA(j)23 b(\025)g FG(2)1967 4273 y Fz(\()p Fu(n)p FC(\000)p 2086 4238 V Fu(n)o FC(\000)p Fz(5\))p Fu(=\034)2302 4281 y Ft(0)2338 4308 y Fx(:)1067 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(1\))118 4602 y(for)32 b Fx(T)42 b FA(2)31 b(S)483 4572 y FC(R)477 4626 y Fu(k)q(;n)p FC(\000)p Fz(1)664 4602 y FG(.)51 b(If)32 b(there)h(is)f(a)g(line)g Fx(`)e FA(2)i FG(\003)1576 4614 y Fu(n)p FC(\000)p Fz(1)1706 4602 y FG(\()p Fx(T)12 b FG(\))31 b(whic)n(h)i(do)r(es)f(not)g(b)r (elong)g(to)g FA(P)7 b FG(\()p Fx(T)12 b FG(\))31 b(then)i Fx(x)3287 4614 y Fu(`)3350 4602 y FG(=)e Fx(x)3493 4572 y Fz(0)3493 4626 y Fu(`)3531 4602 y FG(,)i(so)118 4709 y(that)d(\(A3.1\))g(follo)n(ws)e(from)i(the)g(Diophan)n(tine)f (condition)h(on)f Fw(!)s FG(.)43 b(If)30 b(all)f(lines)g(in)h(\003)2817 4721 y Fu(n)p FC(\000)p Fz(1)2947 4709 y FG(\()p Fx(T)12 b FG(\))30 b(b)r(elong)f(to)g FA(P)7 b FG(\()p Fx(T)12 b FG(\))118 4815 y(consider)29 b(the)h(one)f(among)g(them,)i(sa)n(y)d Fx(`)p FG(,)i(whic)n(h)g(is)f(closest)g(to)h Fx(`)2214 4785 y Fz(2)2214 4838 y Fu(T)2266 4815 y FG(,)g(i.e.)43 b(the)30 b(en)n(tering)f(line)h(of)g Fx(T)12 b FG(.)42 b(Then)30 b(call)118 4921 y Fx(T)167 4933 y Fz(1)233 4921 y FG(the)g(connected)f(set)g(of)g(no)r(des)g(and)g(lines)h(b)r(et) n(w)n(een)1877 4891 y Fz(7)1943 4921 y Fx(`)f FG(and)g Fx(`)2205 4891 y Fz(2)2205 4944 y Fu(T)2257 4921 y FG(.)42 b(If)30 b Fx(T)2456 4933 y Fz(1)2522 4921 y FG(is)f(a)g(single)f(no)r (de)i Fn(v)f FG(then)h Fw(\027)3428 4933 y Fm(v)3500 4921 y FA(6)p FG(=)c Fy(0)p FG(,)p 118 5018 1200 4 v 110 5080 a Fz(7)189 5110 y Fk(The)d(lines)e(b)r(et)n(w)n(een)j(t)n(w)n (o)f(lines)f Fj(`)1084 5119 y Fe(1)1140 5110 y Fk(and)h Fj(`)1305 5119 y Fe(2)1362 5110 y Fk(with)f Fj(`)1550 5119 y Fe(2)1604 5110 y Fj(<)d(`)1707 5119 y Fe(1)1764 5110 y Fk(are)j(all)g(the)h(lines)e(whic)n(h)i(precede)g Fj(`)2739 5119 y Fe(1)2796 5110 y Fk(but)g(whic)n(h)f(do)h(not)g (precede)g Fj(`)3627 5119 y Fe(2)118 5185 y Fk(nor)h(coincide)g(with)g (it.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 118 356 a FG(29:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(otherwise)26 b Fn(v)h FG(w)n(ould)g(b)r(e)g(a)g(trivial)f(no)r(de;)h(if)h Fx(T)1578 567 y Fz(1)1641 555 y FG(is)f(not)g(a)g(single)f(no)r(de)h (then)h(b)n(y)e(construction)h(all)f(the)i(lines)f(of)118 662 y Fx(T)167 674 y Fz(1)228 662 y FG(ha)n(v)n(e)c(scales)h(strictly)g (smaller)f(than)i Fx(n)p FG(,)g(hence)f Fx(x)1772 674 y Fu(`)1827 662 y FA(6)p FG(=)f Fx(x)i FG(otherwise)e Fx(T)2402 674 y Fz(1)2463 662 y FG(w)n(ould)h(b)r(e)h(a)f(self-energy)f (cluster.)35 b(In)118 768 y(b)r(oth)23 b(cases)e(one)i(has)f FA(j)p Fx(x)873 780 y Fu(`)913 768 y FA(\000)8 b Fx(x)p FA(j)24 b FG(=)f FA(j)p Fx(x)1238 738 y Fz(0)1238 792 y Fu(`)1275 768 y FA(j)g Fx(>)g(C)1468 780 y Fz(0)1506 768 y FA(j)1543 706 y Fr(P)1630 793 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)1836 801 y Ft(1)1869 793 y Fz(\))1913 768 y Fw(\027)1961 780 y Fm(v)2007 768 y FA(j)2030 738 y FC(\000)p Fu(\034)2113 746 y Ft(0)2149 768 y FG(.)36 b(On)22 b(the)h(other)f(hand)g(b)r(oth)h Fx(D)r FG(\()p Fx(x)p FG(\))h(and)e Fx(D)r FG(\()p Fx(x)3596 780 y Fu(`)3629 768 y FG(\))118 890 y(m)n(ust)31 b(b)r(e)f FA(\024)e FG(\()p Fx(C)626 902 y Fz(0)664 890 y FG(2)706 859 y FC(\000)p Fz(\()p Fu(n)p FC(\000)p Fz(2\)+1)1023 890 y FG(\))1055 859 y Fz(2)1123 890 y FG(hence,)j(b)n(y)g(\(4.4\))f FA(j)p Fx(x)p FA(j)p Fx(;)14 b FA(j)p Fx(x)1900 902 y Fu(`)1933 890 y FA(j)27 b(\024)h Fx(C)2135 902 y Fz(0)2172 890 y FG(2)2214 859 y FC(\000)p Fu(n)p Fz(+)p 2358 824 42 4 v Fu(n)p Fz(+3)2487 890 y FG(,)j(so)f(that)h FA(j)p Fx(x)21 b FA(\000)f Fx(x)3052 902 y Fu(`)3084 890 y FA(j)28 b(\024)f Fx(C)3286 902 y Fz(0)3324 890 y FG(2)3366 859 y FC(\000)p Fu(n)p Fz(+)p 3510 824 V Fu(n)o Fz(+4)3638 890 y FG(,)118 996 y(and)h(\(A3.1\))f(follo)n(ws)g(also)f(in)i(suc)n(h) f(a)h(case.)189 1103 y(The)k(next)g(task)f(will)i(b)r(e)f(to)g(sho)n(w) f(that)h(the)g(n)n(um)n(b)r(er)g FA(N)2038 1115 y Fu(m)2101 1103 y FG(\()p Fx(T)12 b FG(\))32 b(of)g(lines)f(on)h(scale)f([)p Fx(m)p FG(],)i(with)g Fx(m)d FA(\024)g Fx(n)21 b FA(\000)g FG(1,)118 1209 y(con)n(tained)39 b(in)i(a)e(cluster)h Fx(T)51 b FG(is)40 b(b)r(ounded)g(b)n(y)g FA(N)1721 1221 y Fu(m)1784 1209 y FG(\()p Fx(T)12 b FG(\))43 b FA(\024)g FG(max)p FA(f)p Fx(E)2318 1221 y Fu(m)2394 1147 y Fr(P)2482 1234 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2741 1209 y FA(j)p Fw(\027)2812 1221 y Fm(v)2859 1209 y FA(j)27 b(\000)f FG(1)p Fx(;)14 b FG(0)p FA(g)p FG(,)41 b(with)g Fx(E)3490 1221 y Fu(m)3597 1209 y FG(=)118 1330 y Fx(E)19 b FG(2)240 1300 y FC(\000)p Fu(m=\034)416 1308 y Ft(0)486 1330 y FG(for)34 b(a)h(suitably)f(c)n(hosen)g(constan)n(t)g Fx(E)5 b FG(;)38 b(as)c(it)h(will)g(emerge)e(from)i(the)g(pro)r(of)f (one)g(can)g(tak)n(e)g Fx(E)40 b FG(=)118 1437 y(2)14 b(2)216 1407 y Fz(\()p 242 1372 V Fu(n)o Fz(+4\))p Fu(=\034)457 1415 y Ft(0)493 1437 y FG(.)189 1543 y(Before)30 b(considering)f (clusters)h(w)n(e)g(adapt)h(to)f(our)g(con)n(text)g(the)h(classical)e (b)r(ound)i(\(Siegel-Bryuno-P\177)-42 b(oshel;)118 1650 y(see)24 b(Ref.)36 b([Ga])25 b(and)f(references)f(quoted)i(therein\),)g (stating)f(that,)h(if)g FA(N)2366 1662 y Fu(m)2430 1650 y FG(\()p Fx(\022)r FG(\))g(denotes)f(the)h(n)n(um)n(b)r(er)f(of)g (lines)h(on)118 1756 y(scales)17 b([)p Fx(m)p FG(],)k(then)e(b)n(y)f (induction)h(on)f(the)g(n)n(um)n(b)r(er)g(of)h(no)r(des)f(of)g Fx(\022)j FG(one)d(sho)n(ws:)31 b FA(N)2613 1768 y Fu(m)2676 1756 y FG(\()p Fx(\022)r FG(\))24 b FA(\024)f FG(max)o FA(f)p Fx(E)3150 1768 y Fu(m)3227 1694 y Fr(P)3315 1781 y Fm(v)o FC(2)p Fu(V)15 b Fz(\()p Fu(\022)r Fz(\))3559 1756 y FA(j)p Fw(\027)3630 1768 y Fm(v)3677 1756 y FA(j\000)118 1862 y FG(1)p Fx(;)f FG(0)p FA(g)p FG(.)34 b(Indeed)25 b(if)f Fx(\022)j FG(con)n(tains)c(only)h(one)g(no)r(de)g Fn(v)1637 1874 y Fz(0)1698 1862 y FG(and)g(the)h(frequency)f Fx(x)f FG(=)g Fw(!)14 b FA(\001)e Fw(\027)2684 1874 y Fm(v)2726 1883 y Ft(0)2787 1862 y FG(of)24 b(the)h(ro)r(ot)e(line)i (has)e(scale)118 1969 y([)p Fx(m)p FG(])28 b(one)f(has)349 2176 y(2)391 2142 y FC(\000)p Fu(m)p Fz(+1)590 2176 y Fx(C)649 2188 y Fz(0)709 2176 y FA(\025)797 2101 y Fr(p)p 880 2101 184 4 v 75 x Fx(D)r FG(\()p Fx(x)p FG(\))d FA(\025)f FG(2)1216 2142 y FC(\000)p Fz(\()p 1294 2107 42 4 v Fu(n)o Fz(+1\))1448 2176 y FA(j)p Fx(x)p FA(j)h(\025)f FG(2)1695 2142 y FC(\000)p Fz(\()p 1773 2107 V Fu(n)o Fz(+1\))1927 2176 y Fx(C)1986 2188 y Fz(0)2024 2176 y FA(j)p Fw(\027)2095 2188 y Fm(v)2137 2197 y Ft(0)2174 2176 y FA(j)2197 2142 y FC(\000)p Fu(\034)2280 2150 y Ft(0)2353 2176 y FA(\))g(j)p Fw(\027)2530 2188 y Fm(v)2573 2197 y Ft(0)2609 2176 y FA(j)g Fx(>)g FG(2)2785 2142 y Fz(\()p Fu(m)p FC(\000)p 2922 2107 V Fu(n)o FC(\000)p Fz(2\))p Fu(=\034)3138 2150 y Ft(0)3174 2176 y Fx(;)231 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(2\))118 2383 y(hence)28 b Fx(E)410 2395 y Fu(m)473 2383 y FA(j)p Fw(\027)544 2395 y Fm(v)587 2404 y Ft(0)623 2383 y FA(j)19 b(\000)f FG(1)k FA(\025)h FG(2)k(and)h(the)g(b)r(ound)g (holds)f(in)h(this)g(simple)f(case.)189 2490 y(If)38 b Fx(\022)h FG(has)e Fx(k)j FG(no)r(des)d(and)g(the)g(ro)r(ot)f(line)i FF(do)l(es)h(not)e FG(ha)n(v)n(e)f(scale)g([)p Fx(m)p FG(])h(the)h(inductiv)n(e)f(assumption,)i(if)f(it)g(is)118 2596 y(assumed)27 b(for)g(the)h(cases)f(of)g Fx(k)1070 2566 y FC(0)1116 2596 y Fx(<)c(k)31 b FG(no)r(des,)c(giv)n(es)f(the)i (b)r(ound)g(for)f Fx(k)s FG(-no)r(des)g(trees.)189 2703 y(If)f(the)g(ro)r(ot)e(line)i(has)f(scale)f([)p Fx(m)p FG(])i(then)g(on)f(eac)n(h)f(path)i(of)f(tree)g(lines)h(leading)f(to)g (the)h(ro)r(ot)e(w)n(e)h(select)g(the)h(line)118 2810 y(among)k(the)h(ones)e(on)i(scales)e([)p Fx(m)1171 2779 y FC(0)1194 2810 y FG(])i(with)g Fx(m)1513 2779 y FC(0)1564 2810 y FA(\025)d Fx(m)i FF(closest)j(to)g(the)f(r)l(o)l(ot)f FG(\(if)g(an)n(y)f(is)h(found)f(on)h(the)g(path\))g(and)118 2916 y(w)n(e)e(call)h(the)g(selected)f(lines)h Fx(`)1080 2928 y Fz(1)1117 2916 y Fx(;)14 b(:)g(:)g(:)f(;)h(`)1336 2928 y Fu(q)1372 2916 y FG(.)44 b(If)30 b Fx(q)f FA(6)p FG(=)d(1)k(either)f(the)h(b)r(ound)g(follo)n(ws)f(just)h(as)f(in)h(the) g(case)f(of)g Fx(k)h FG(=)c(1)118 3022 y(\(when)i Fx(q)e FG(=)d(0\))k(or)g(from)g(the)h(inductiv)n(e)g(h)n(yp)r(othesis)f (\(when)h Fx(q)e FA(\025)d FG(2\).)189 3129 y(The)36 b(case)g Fx(q)k FG(=)e(1)e(and)g([)p Fx(n)1054 3141 y Fu(`)1082 3149 y Ft(1)1118 3129 y FG(])i(=)f([)p FA(1)p FG(])f(\(i.e.)64 b Fw(\027)1696 3141 y Fu(`)1724 3149 y Ft(1)1798 3129 y FG(=)37 b Fy(0)p FG(\))f(can)g(b)r(e)h(treated)f(as) g(the)g(case)g Fx(q)k FG(=)e(0.)62 b(If)37 b Fx(q)k FG(=)c(1)118 3235 y(and)31 b Fw(\027)331 3247 y Fu(`)359 3255 y Ft(1)424 3235 y FA(6)p FG(=)e Fy(0)p FG(,)j(b)n(y)f(construction)f(all)h(lines)g (b)r(et)n(w)n(een)g(the)h(ro)r(ot)e(line)h Fx(`)g FG(and)g Fx(`)2609 3247 y Fz(1)2646 3235 y FG(,)h(see)f(fo)r(otnote)3170 3205 y Fz(7)3207 3235 y FG(,)i(ha)n(v)n(e)d(scales)118 3342 y([)p Fx(m)214 3311 y FC(0)237 3342 y FG(],)36 b(with)e Fx(m)587 3311 y FC(0)644 3342 y Fx(<)e(m)p FG(,)k(so)d(that)h(suc)n(h)f (lines,)i(together)e(with)i(the)f(no)r(des)f(they)h(connect,)h(form)f (a)f(cluster)g Fx(T)12 b FG(.)118 3448 y(The)32 b(frequencies)f Fx(x)769 3460 y Fu(`)833 3448 y FG(and)g Fx(x)1045 3460 y Fu(`)1073 3468 y Ft(1)1142 3448 y FG(m)n(ust)g(b)r(e)h(di\013eren)n (t)g(b)r(ecause)f(the)h(tree)f Fx(\022)j FG(con)n(tains)d(no)g (self-energy)f(clusters.)118 3554 y(On)j(the)g(other)f(hand)845 3483 y Fr(p)p 928 3483 216 4 v 71 x Fx(D)r FG(\()p Fx(x)1078 3566 y Fu(`)1111 3554 y FG(\))p Fx(;)1180 3483 y Fr(p)p 1263 3483 248 4 v 71 x Fx(D)r FG(\()p Fx(x)1413 3566 y Fu(`)1441 3574 y Ft(1)1478 3554 y FG(\))g FA(\024)f FG(2)1680 3524 y FC(\000)p Fu(m)p Fz(+1)1879 3554 y Fx(C)1938 3566 y Fz(0)1976 3554 y FG(,)j(hence)f FA(j)p Fx(x)2339 3566 y Fu(`)2371 3554 y FA(j)p Fx(;)14 b FA(j)p Fx(x)2501 3566 y Fu(`)2529 3574 y Ft(1)2566 3554 y FA(j)32 b(\024)f FG(2)2759 3524 y FC(\000)p Fu(m)p Fz(+)p 2921 3489 42 4 v Fu(n)o Fz(+3)3050 3554 y Fx(C)3109 3566 y Fz(0)3179 3554 y FG(b)n(y)i(\(4.4\),)h(and)118 3660 y Fx(C)177 3672 y Fz(0)215 3660 y FA(j)p Fw(\027)286 3672 y Fu(`)331 3660 y FA(\000)13 b Fw(\027)457 3672 y Fu(`)485 3680 y Ft(1)521 3660 y FA(j)544 3630 y FC(\000)p Fu(\034)627 3638 y Ft(0)686 3660 y FA(\024)22 b(j)p Fx(x)843 3672 y Fu(`)889 3660 y FA(\000)13 b Fx(x)1014 3672 y Fu(`)1042 3680 y Ft(1)1078 3660 y FA(j)24 b(\024)e FG(2)1254 3630 y FC(\000)p Fu(m)p Fz(+)p 1416 3595 V Fu(n)o Fz(+4)1545 3660 y Fx(C)1604 3672 y Fz(0)1641 3660 y FG(,)k(so)e(that)h(w)n(e)g (get)2222 3598 y Fr(P)2309 3685 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2568 3660 y FA(j)p Fw(\027)2639 3672 y Fm(v)2686 3660 y FA(j)23 b(\025)g FG(\(2)2894 3630 y FC(\000)p Fu(m)p Fz(+)p 3056 3595 V Fu(n)o Fz(+4)3185 3660 y FG(\))3217 3630 y FC(\000)p Fz(1)p Fu(=\034)3367 3638 y Ft(0)3403 3660 y FG(,)j(whic)n(h)118 3767 y(giv)n(es)k FA(N)393 3779 y Fu(m)456 3767 y FG(\()p Fx(\022)r FG(\))g FA(\024)e FG(1)20 b(+)h Fx(E)893 3779 y Fu(m)970 3704 y Fr(P)1058 3792 y Fm(v)o FC(2)p Fu(V)14 b Fz(\()p Fu(\022)r Fz(\))1302 3767 y FA(j)p Fw(\027)1373 3779 y Fm(v)1420 3767 y FA(j)20 b(\000)h Fx(E)1610 3779 y Fu(m)1687 3704 y Fr(P)1774 3792 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))2033 3767 y FA(j)p Fw(\027)2104 3779 y Fm(v)2151 3767 y FA(j)21 b(\000)f FG(1)29 b FA(\024)f Fx(E)2505 3779 y Fu(m)2582 3704 y Fr(P)2670 3792 y Fm(v)o FC(2)p Fu(V)15 b Fz(\()p Fu(\022)r Fz(\))2914 3767 y FA(j)p Fw(\027)2985 3779 y Fm(v)3032 3767 y FA(j)21 b(\000)f FG(1,)31 b(so)g(that)g(the)118 3873 y(b)r(ounds)d(is)f(completely)h (pro)n(v)n(ed.)118 4051 y FF(R)l(emark.)70 b FG(The)38 b(ab)r(o)n(v)n(e)f(discussion)h(exploits)g(the)g(prop)r(ert)n(y)g(that) g(the)h(tree)f 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Fx(G)1079 4666 y Fz(0)1143 4654 y FA([)25 b Fx(`)1258 4666 y Fz(in)1343 4654 y FA([)h Fx(`)1459 4666 y Fz(out)1559 4654 y FG(.)69 b(Let)38 b([)p Fx(p)1875 4666 y Fz(in)1934 4654 y FG(])p Fx(;)14 b FG([)p Fx(p)2059 4666 y Fz(out)2159 4654 y FG(])39 b(b)r(e)f(the)h(scales)e(of)h(the)h(lines)f Fx(`)3234 4666 y Fz(out)3372 4654 y FG(and)g Fx(`)3579 4666 y Fz(in)3638 4654 y FG(,)118 4760 y(resp)r(ectiv)n(ely)-7 b(,)38 b(and)e(supp)r(ose) f(that)i Fx(p)1319 4772 y Fz(in)1378 4760 y Fx(;)14 b(p)1457 4772 y Fz(out)1594 4760 y FA(\025)37 b Fx(m)p FG(,)h(while)e(all)g (lines)g(in)h Fx(G)2548 4772 y Fz(0)2621 4760 y FG(\(if)g(an)n(y\))f (ha)n(v)n(e)f(scales)g([)p Fx(p)p FG(])h(with)118 4866 y Fx(p)26 b FA(\024)g Fx(n)19 b FA(\000)h FG(1.)42 b(Note)30 b(that)f(in)h(general)e Fx(G)1374 4878 y Fz(0)1441 4866 y FG(is)i(not)f(ev)n(en)g(a)g(cluster)g(unless)h Fx(p)2498 4878 y Fz(in)2557 4866 y Fx(;)14 b(p)2636 4878 y Fz(out)2762 4866 y FA(\025)26 b Fx(n)p FG(.)42 b(Then)30 b(w)n(e)f(can)g(pro)n(v)n (e)118 4972 y(that)34 b FA(N)372 4984 y Fu(m)435 4972 y FG(\()p Fx(G)532 4984 y Fz(0)570 4972 y FG(\))f FA(\024)g FG(max)o FA(f)p Fx(E)990 4984 y Fu(m)1067 4910 y Fr(P)1155 4997 y Fm(v)o FC(2)p Fu(V)15 b Fz(\()p Fu(G)1374 5005 y Ft(0)1406 4997 y Fz(\))1450 4972 y FA(j)p Fw(\027)1521 4984 y Fm(v)1567 4972 y FA(j)23 b(\000)f FG(1)p Fx(;)14 b FG(0)p FA(g)p FG(,)34 b(where)f Fx(V)18 b FG(\()p Fx(G)2329 4984 y Fz(0)2367 4972 y FG(\))34 b(is)g(the)g(set)f(of)h(no)r(des)f (preceding)g Fx(`)3562 4984 y Fz(out)118 5079 y FG(and)24 b(follo)n(wing)g Fx(`)659 5091 y Fz(in)718 5079 y FG(,)h(and)g Fx(E)986 5091 y Fu(m)1074 5079 y FG(is)f(de\014ned)h(ab)r(o)n(v)n(e.)34 b(If)25 b Fx(G)1848 5091 y Fz(0)1910 5079 y FG(has)f(zero)g(lines)g (then)h(the)g(harmonic)e Fw(\027)3149 5091 y Fz(0)3211 5079 y FG(of)h(the)h(\(only\))118 5185 y(no)r(de)33 b(in)h Fx(V)19 b FG(\()p Fx(G)591 5197 y Fz(0)629 5185 y FG(\))33 b(is)g(large,)h FA(j)p Fw(\027)1088 5197 y Fz(0)1125 5185 y FA(j)e(\025)g FG(2)1319 5155 y Fz(\()p Fu(m)p FC(\000)p 1456 5120 42 4 v Fu(n)p FC(\000)p Fz(2\))p Fu(=\034)1673 5163 y Ft(0)1742 5185 y FG(\(b)n(y)h(the)h(Diophan)n (tine)f(prop)r(ert)n(y\))g(and)g(the)g(statemen)n(t)h(is)118 5291 y(true.)j(Hence)28 b(w)n(e)f(pro)r(ceed)g(inductiv)n(ely)g(on)h (the)g(n)n(um)n(b)r(er)f(of)h(lines)f(in)h Fx(G)2435 5303 y Fz(0)2473 5291 y FG(.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)i(19:31)1130 b FG(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 118 356 a FG(30:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)189 555 y FG(If)27 b(no)f(line)g(of)h Fx(G)699 567 y Fz(0)763 555 y FG(on)f(the)g(path)h FA(P)7 b FG(\()p Fx(G)p FG(\))27 b(connecting)f(the)g(external)g(lines)g(of)g Fx(G)h FG(has)f(scale)g([)p Fx(m)p FG(])g(then)h(the)g(lines)118 662 y(in)g Fx(G)279 674 y Fz(0)344 662 y FG(on)g(scale)f([)p Fx(m)p FG(])h(\(if)h(an)n(y\))f(b)r(elong)g(to)g(trees)f(with)i(ro)r (ot)e(on)h FA(P)7 b FG(\()p Fx(G)p FG(\),)28 b(and)f(the)g(statemen)n (t)g(follo)n(ws)f(from)h(the)118 768 y(b)r(ound)h(on)f(trees.)189 875 y(Supp)r(ose)g(that)f Fx(`)d FA(2)g(P)7 b FG(\()p Fx(G)p FG(\))27 b(is)g(a)f(line)h(on)f(scale)f([)p Fx(m)p FG(],)i(then)g(call)f Fx(G)2240 887 y Fz(1)2304 875 y FG(and)h Fx(G)2530 887 y Fz(2)2594 875 y FG(the)g(disjoin)n(t)f (subsets)h(of)f Fx(G)h FG(suc)n(h)118 982 y(that)k Fx(G)366 994 y Fz(1)423 982 y FA([)21 b Fx(G)564 994 y Fz(2)621 982 y FA([)g Fx(`)27 b FG(=)f Fx(G)p FG(.)45 b(Then)31 b Fx(G)1268 994 y Fz(1)1325 982 y FA([)21 b Fx(`)29 b FG(and)h Fx(G)1694 994 y Fz(2)1752 982 y FA([)21 b Fx(`)29 b FG(ha)n(v)n(e)g(the)i(same)e(structure)h(of)g Fx(G)g FG(itself)h(but)g(eac)n(h)e(has)118 1088 y(less)e(lines:)37 b(and)28 b(again)e(the)i(inductiv)n(e)g(assumption)f(yields)g(the)h (result.)189 1195 y(Therefore,)34 b(as)e(a)h(particular)f(case,)i(b)n (y)f(c)n(ho)r(osing)f Fx(G)1902 1207 y Fz(0)1972 1195 y FG(=)g Fx(T)12 b FG(,)34 b(with)g Fx(T)44 b FA(2)32 b(S)2618 1165 y FC(R)2612 1219 y Fu(k)q(;n)p FC(\000)p Fz(1)2799 1195 y FG(,)j(the)f(b)r(ound)g(for)e FA(N)3468 1207 y Fu(m)3532 1195 y FG(\()p Fx(G)p FG(\))118 1302 y(implies)c(the)g(b)r(ound)g(on)f FA(N)982 1314 y Fu(m)1045 1302 y FG(\()p Fx(T)12 b FG(\))28 b(w)n(e)f(are)g(lo)r(oking)f(for.)189 1409 y(The)31 b(ab)r(o)n(v)n(e)e(analysis)h(is)g(tak)n(en)g(from)h (Ref.)46 b([Ge])31 b(and)g(di\013ers)g(from)f(Ref.)47 b([GG])31 b(b)r(ecause)f(here)g(the)i(scales)118 1515 y(dep)r(end)26 b(on)e Fx(")h FG(and)f(it)i(is)e(not)h(clear)f(ho)n(w)g (to)h(de\014ne)g(a)f(\\strong)f(Diophan)n(tine)i(condition",)g(whic)n (h)f(w)n(ould)h(allo)n(w)118 1622 y(a)i(one-to-one)f(corresp)r(ondence) g(b)r(et)n(w)n(een)i(line)f(scales)g(and)g(line)h(momen)n(ta.)189 1729 y(The)g(b)r(ound)g(on)f(the)h(con)n(tribution)f(of)g(a)h(single)f (self-energy)f(cluster)h Fx(T)34 b FA(2)24 b(S)2639 1699 y FC(R)2633 1752 y Fu(k)q(;n)p FC(\000)p Fz(1)2847 1729 y FG(is)k(then)300 1957 y Fx(")339 1926 y Fu(k)p 300 1994 80 4 v 305 2070 a Fx(k)s FG(!)389 2013 y Fx(C)454 1977 y FC(\000)p Fz(2)p Fu(k)448 2035 y Fz(0)580 2013 y Fx(F)645 1978 y Fz(2)p Fu(k)719 2013 y Fx(e)758 1973 y FC(\000)820 1950 y Ft(1)p 820 1959 29 4 v 820 1993 a(2)858 1973 y Fu(\024)897 1981 y Ft(0)940 1917 y Fr(P)1028 2004 y Fm(v)1086 1973 y FC(j)p Fq(\027)1144 1989 y Fm(v)1191 1973 y FC(j)1215 1921 y Fr(\020)1304 1908 y Fu(m)1363 1916 y Ft(0)1297 1934 y Fr(Y)1278 2110 y Fu(m)p Fz(=0)1435 2013 y FG(2)1477 1978 y Fz(\()p Fu(m)p Fz(+3\)2)p FC(N)1759 1986 y Fs(m)1814 1978 y Fz(\()p Fu(T)9 b Fz(\))1918 1921 y Fr(\021)1968 2013 y FA(\001)1437 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(3\))308 2288 y FA(\001)350 2196 y Fr(\020)399 2288 y Fx(e)438 2238 y FC(\000)500 2216 y Ft(1)p 500 2225 V 500 2258 a(2)538 2238 y Fu(\024)577 2246 y Ft(0)620 2182 y Fr(P)708 2269 y Fi(v)p Ff(2)p Fs(V)12 b Ft(\()p Fs(T)c Ft(\))936 2238 y FC(j)p Fq(\027)994 2254 y Fm(v)1040 2238 y FC(j)1188 2185 y(1)1168 2209 y Fr(Y)1078 2385 y Fu(m)p Fz(=)p Fu(m)1247 2393 y Ft(0)1279 2385 y Fz(+1)1377 2288 y FG(2)1419 2238 y Fz(2\()p Fu(m)p Fz(+3\)2)1680 2213 y Ff(\000)p Fs(m=\034)1835 2225 y Ft(0)1871 2238 y Fu(E)1934 2182 y Fr(P)2022 2269 y Fi(v)p Ff(2)p Fs(V)k Ft(\()p Fs(T)c Ft(\))2250 2238 y FC(j)p Fq(\027)2308 2254 y Fm(v)2354 2238 y FC(j)2378 2196 y Fr(\021)2479 2288 y FA(\024)2604 2232 y Fx(")2643 2202 y Fu(\024)2686 2232 y Fx(G)2751 2202 y Fu(k)p 2604 2269 188 4 v 2663 2345 a Fx(k)s FG(!)2802 2288 y Fx(e)2841 2238 y FC(\000)2902 2216 y Ft(1)p 2902 2225 29 4 v 2902 2258 a(2)2940 2238 y Fu(\024)2979 2246 y Ft(0)3023 2182 y Fr(P)3111 2269 y Fi(v)p Ff(2)p Fs(V)k Ft(\()p Fs(T)7 b Ft(\))3338 2238 y FC(j)p Fq(\027)3396 2254 y Fm(v)3443 2238 y FC(j)3467 2288 y Fx(;)118 2571 y FG(with)22 b Fx(F)33 b FG(an)21 b(upp)r(er)h(b)r(ound)f(on)g(the)h(constan)n(ts)e Fx(F)1637 2583 y Fz(0)1675 2571 y Fx(;)14 b(F)1765 2583 y Fz(1)1824 2571 y FG(b)r(ounding)21 b(the)h(F)-7 b(ourier)20 b(transform)g(of)h (the)h(p)r(erturbation)118 2677 y(\(see)29 b(\(1.4\)\),)h(while)f Fx(m)833 2689 y Fz(0)899 2677 y FG(is)g(de\014ned)g(so)g(that)g(log)14 b(2)1733 2615 y Fr(P)1820 2702 y Fu(m>m)1990 2710 y Ft(0)2040 2677 y FG(2\()p Fx(m)20 b FG(+)f(3\)2)2407 2647 y FC(\000)p Fu(m=\034)2583 2655 y Ft(0)2618 2677 y Fx(E)30 b FA(\024)2809 2644 y Fz(1)p 2809 2658 34 4 v 2809 2705 a(2)2852 2677 y Fx(\024)2900 2689 y Fz(0)2967 2677 y FG(and)f Fx(G)g FG(is)g(a)g(suitable)118 2783 y(constan)n(t.)189 2890 y(The)37 b(n)n(um)n(b)r(er)g(of)g(trees)g(can)g(b)r(e)g(b)r(ounded)h(b) n(y)f(4)1794 2860 y Fu(k)1834 2890 y Fx(k)s FG(!,)j(and)d(the)h(sum)f (o)n(v)n(er)e(the)j(scale)e(lab)r(els)h(in)n(v)n(olv)n(es)f(at)118 2997 y(most)27 b(2)g(p)r(ossible)g(v)-5 b(alues)27 b(p)r(er)g(line)h(b) r(ecause)f(of)g(the)h(upp)r(er)f(and)h(lo)n(w)n(er)d(cut-o\013s)j (presen)n(t)e(in)i(the)g(propagators)118 3103 y(de\014nition.)37 b(The)25 b(sum)g(o)n(v)n(er)e(the)j(harmonics)e(can)g(b)r(e)i (estimated)f(b)n(y)g(making)f(use)h(of)g FF(p)l(art)g FG(of)h(the)f(exp)r(onen)n(tial)118 3209 y(factor)c(in)i(\(A3.3\))f (\(sa)n(y)877 3177 y Fz(1)p 877 3191 V 877 3238 a(4)920 3209 y Fx(\024)968 3221 y Fz(0)1005 3209 y FG(\))h(while)f(the)h(other) 1630 3177 y Fz(1)p 1630 3191 V 1630 3238 a(4)1673 3209 y Fx(\024)1721 3221 y Fz(0)1781 3209 y FG(will)f(b)r(e)g(used)h(as)e(a) h(factor)f(b)r(ounded)i(b)n(y)e Fx(e)3098 3179 y FC(\000)3160 3157 y Ft(1)p 3160 3166 29 4 v 3160 3199 a(4)3198 3179 y Fu(\024)3237 3187 y Ft(0)3269 3179 y Fz(2)3302 3154 y Ft(\()p Fs(n)p Ff(\000)p 3406 3126 37 4 v Fs(n)p Ff(\000)p Ft(5\))p Fs(=\034)3597 3166 y Ft(0)3638 3209 y FG(,)118 3316 y(b)n(y)27 b(\(A3.1\).)189 3423 y(Hence)f(w)n(e)g(get)f(con)n(v)n (ergence)f(at)h(exp)r(onen)n(tial)h(rate)f(2)1903 3393 y FC(\000)p Fz(1)2018 3423 y FG(for)g Fx(")e(<)f(")2331 3435 y Fz(1)2394 3423 y FG(\(and)k Fx(")2625 3435 y Fz(1)2688 3423 y FG(is)g(an)g(explicitly)g(computable)118 3529 y(constan)n(t\))h(and)h(the)g(matrix)f Fx(M)1150 3499 y Fz([)p Fu(n)p Fz(])1232 3529 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b(is)g(de\014ned)g(b)n(y)f(a)g(con)n(v)n(ergen)n(t)f (series)g(and)i(it)g(is)f(b)r(ounded)h(b)n(y)1240 3755 y FA(k)p Fx(M)1372 3721 y Fz([)p Fu(n)p Fz(])1454 3755 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FA(k)23 b Fx(<)p 1794 3688 68 4 v 23 w(B)t(")1900 3721 y Fz(2)1937 3755 y Fx(e)1976 3721 y FC(\000)2038 3698 y Ft(1)p 2038 3707 29 4 v 2038 3741 a(4)2076 3721 y Fu(\024)2115 3729 y Ft(0)2147 3721 y Fz(2)2180 3696 y Ft(\()p Fs(n)p Ff(\000)p 2284 3668 37 4 v Fs(n)p Ff(\000)p Ft(4\))p Fs(=\034)2475 3708 y Ft(0)2516 3755 y Fx(;)889 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(4\))118 3964 y(for)27 b(a)g(suitable)p 623 3897 68 4 v 27 w Fx(B)k FG(whic)n(h)d(can)f(b)r(e)g(read)g(from)g (\(A3.3\),)g(i.e.)37 b(w)n(e)27 b(get)g(the)g(\014rst)g(of)h(the)f (\014rst)g(line)h(in)f(\(5.13\))g(with)118 4070 y(the)d(constan)n(t)f Fx(B)28 b FG(replaced)23 b(b)n(y)p 1114 4004 V 23 w Fx(B)t FG(,)i Fx(\034)32 b FG(=)23 b Fx(\034)1421 4082 y Fz(0)1458 4070 y FG(,)i(and)e Fx(\024)1711 4082 y Fz(1)1772 4070 y FG(=)1869 4038 y Fz(1)p 1869 4052 34 4 v 1869 4099 a(4)1912 4070 y Fx(\024)1960 4082 y Fz(0)1997 4070 y Fx(e)2036 4040 y FC(\000)p Fz(\()p 2114 4005 42 4 v Fu(n)p Fz(+4\))p Fu(=\034)2330 4048 y Ft(0)2366 4070 y FG(.)36 b(The)23 b Fx(")2630 4040 y Fz(2)2691 4070 y FG(factor)g(is)g(due)h(to) g(the)g(parallel)118 4176 y(remark)31 b(that,)j(in)f(an)n(y)f (self-energy)f(cluster)h(whose)g(v)-5 b(alue)32 b(con)n(tributes)g(to)g FA(M)2681 4146 y Fz([)p Fu(n)p Fz(])2764 4176 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)34 b Fx(k)i FG(is)c(certainly)g FA(\025)f FG(2)118 4283 y(\(see)d(Remark)e(to)i(De\014nition)g(4)f(in)h (Section)g(5\).)189 4390 y(Therefore)e(if)i Fx(")g FG(is)f(small)h (enough)f(\(that)h(is)f(smaller)g(than)h(a)f(constan)n(t)g(indep)r (enden)n(t)h(of)g Fx(n)22 b FA(\024)p 3200 4344 50 4 v 23 w Fx(n)3250 4402 y Fz(0)3287 4390 y FG(\))874 4659 y FA(kM)1016 4625 y Fz([)p FC(\024)p Fu(n)p Fz(])1150 4659 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))19 b FA(\000)f Fx(M)1520 4671 y Fz(0)1557 4659 y FA(k)k(\024)p 1709 4592 68 4 v 23 w Fx(B)t(")1815 4625 y Fz(2)1895 4555 y FC(1)1868 4580 y Fr(X)1866 4756 y Fu(n)p Fz(=1)2005 4659 y Fx(e)2044 4625 y FC(\000)2105 4602 y Ft(1)p 2105 4611 29 4 v 2105 4645 a(4)2143 4625 y Fu(\024)2182 4633 y Ft(0)2215 4625 y Fz(2)2248 4600 y Ft(\()p Fs(n)p Ff(\000)p 2352 4572 37 4 v Fs(n)o Ff(\000)p Ft(4\))p Fs(=\034)2542 4612 y Ft(0)2597 4608 y Fu(def)2618 4659 y FG(=)33 b Fx(B)2783 4625 y FC(0)2807 4659 y Fx(")2846 4625 y Fz(2)2883 4659 y Fx(;)522 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(5\))118 4949 y(so)24 b(that)g(the)h(eigen)n(v)-5 b(alues)22 b(of)j FA(M)1150 4919 y Fz([)p FC(\024)p Fu(n)p Fz(])1284 4949 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))25 b(will)f(b)r(e)h(shifted)f (with)h(resp)r(ect)f(to)g(the)h(corresp)r(onding)d(eigen)n(v)-5 b(alues)118 5078 y(of)28 b Fx(M)294 5090 y Fz(0)358 5078 y FG(b)n(y)f Fx(\015)5 b(")560 5048 y Fz(2)625 5078 y FG(at)27 b(most,)h(with)g Fx(\015)1203 5027 y Fu(def)1223 5078 y FG(=)34 b Fx(B)1389 5048 y FC(0)1412 5078 y Fx(C)6 b FG(,)28 b(see)g(\(I\))g(in)g(App)r(endix)g(A4.)189 5185 y(Hence)d(if)h(w)n(e)f(de\014ne)g Fx(\015)30 b FG(as)25 b Fx(B)1104 5155 y FC(0)1127 5185 y Fx(C)32 b FG(and)25 b Fx(")g FG(is)g(c)n(hosen)f(small)h(enough,)g(sa)n(y)f Fx(")f(<)g(")2637 5197 y Fz(2)2674 5185 y FG(,)i(so)g(that)g Fx(\015)5 b(")3086 5155 y Fz(2)3146 5185 y Fx(<)3244 5152 y Fz(1)p 3244 5166 34 4 v 3244 5214 a(2)3287 5185 y Fx("a)3370 5197 y Fu(s)3405 5185 y FG(2)3447 5155 y FC(\000)p Fz(2)p 3532 5120 42 4 v Fu(n)o FC(\000)p Fz(2)118 5291 y FG(\(as)32 b(it)h(m)n(ust)g(b)r(e)g(in)g(order)e(that)i(the)g (ab)r(o)n(v)n(e)e(argumen)n(t)g(b)r(e)i(consisten)n(t,)g(see)g(the)f(b) r(eginning)h(of)f(the)h(curren)n(t)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)d(19:31)1130 b FG(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 118 356 a FG(31:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(Section\))c(w)n(e)f(obtain)g (the)h(v)-5 b(alidit)n(y)28 b(of)f(the)h(assumed)f(inductiv)n(e)h(h)n (yp)r(othesis)f(for)g(all)h Fx(n)22 b FA(\024)p 3004 510 50 4 v 23 w Fx(n)3054 567 y Fz(0)3119 555 y FG(and)27 b(of)h(the)g(\014rst)118 662 y(inequalit)n(y)f(in)h(the)g(\014rst)f (line)h(of)g(\(5.13\))f(where)g Fx(B)32 b FG(can)27 b(b)r(e)h(c)n (hosen)e(equal)h(to)p 2592 595 68 4 v 28 w Fx(B)32 b FG(ab)r(o)n(v)n(e.)189 839 y(The)e(symmetries)g(in)g(items)g(\(i\))h (and)f(\(ii\))g(are)f(an)h(algebraic)e(consequence)h(of)h(the)h(form)e (of)h(the)h(Lindstedt)118 945 y(series:)36 b(hence)28 b(they)f(are)g(a)g(necessary)f(consequence)h(of)g(the)h(pro)n(v)n(ed)e (con)n(v)n(ergence,)f(see)j(Ref.)37 b([GG].)118 1122 y Fy(A3.2.)47 b FF(Smo)l(othness.)i FG(The)31 b(function)h Fx(M)1472 1092 y Fz([)p Fu(n)p Fz(])1554 1122 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))32 b(whic)n(h)g(w)n(e)f(ha)n(v)n(e)f (just)i(sho)n(wn)e(to)h(b)r(e)h(w)n(ell)f(de\014ned)h(for)f(all)118 1229 y Fx(")e FG(small)g(enough)f(will)h(b)r(e)h(smo)r(oth)e(in)i Fx(";)14 b(x)p FG(.)41 b(W)-7 b(e)29 b(assume)g(inductiv)n(ely)g(that)g (this)g(is)g(the)h(case)e(for)g Fx(M)3375 1199 y Fz([)p Fu(p)p Fz(])3451 1229 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\),)118 1335 y(0)26 b FA(\024)h Fx(p)f(<)h(n)20 b FA(\000)f FG(1,)30 b(and)g(that)g(the)g(b)r(ounds)g(in)h(the)f(\014rst) f(line)i(of)e(\(5.13\))g(hold)h(for)g(suc)n(h)f Fx(p)p FG('s)h(\(the)g(case)f Fx(p)e FG(=)f(0)k(is)118 1441 y(ob)n(vious)c(as)h FA(M)620 1411 y Fz([0])695 1441 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))24 b FA(\021)e Fx(M)1074 1453 y Fz(0)1111 1441 y FG(\).)189 1548 y(Eac)n(h)34 b(deriv)-5 b(ativ)n(e)35 b(with)h(resp)r(ect)f(to)h Fx(x)g FG(or,)g(resp)r(ectiv)n(ely)-7 b(,)37 b(to)e Fx(")h FG(will)g(replace)e (the)i(v)-5 b(alue)35 b(of)h(a)f(self-energy)118 1654 y(cluster)27 b(with)h Fx(k)j FG(no)r(des)c(b)n(y)h(a)f(sum)h(of)f Fx(k)k FG(terms)c(whic)n(h)g(can)h(b)r(e)g(b)r(ounded)g(b)n(y)f(a)g(b)r (ound)h(lik)n(e)f(\(A3.3\).)189 1760 y(In)e(fact,)g(giv)n(en)f(a)g (self-energy)f(cluster)h Fx(T)12 b FG(,)25 b(the)g(righ)n(t)e(deriv)-5 b(ativ)n(e)24 b Fx(@)2300 1730 y Fz(+)2295 1781 y Fu(x)2380 1760 y FG(ma)n(y)f(fall)i(on)f(a)g(denominator)g(of)g(one)h(of)118 1866 y(the)g Fx(k)14 b FA(\000)d FG(1)23 b(cluster)h(lines.)36 b(If)24 b(its)h(frequency)e(is)h Fx(x)11 b FG(+)g Fx(x)1767 1878 y Fz(0)1830 1866 y FG(with)24 b(scale)g(lab)r(el)g([)p Fx(m)p FG(],)h(deriv)-5 b(ation)23 b(yields,)i(up)f(to)g(a)g(sign,)118 1973 y(a)h(pro)r(duct)g(of)g(t)n(w)n(o)g(matrices)f(\(\()p Fx(x)1182 1985 y Fz(0)1234 1973 y FG(+)14 b Fx(x)p FG(\))1392 1943 y Fz(2)1443 1973 y FA(\000)g(M)1622 1943 y Fz([)p FC(\024)p Fu(m)p Fz(])1774 1973 y FG(\()p Fx(x)1853 1985 y Fz(0)1904 1973 y FG(+)g Fx(x)p FG(;)g Fx(")p FG(\)\))2170 1943 y FC(\000)p Fz(1)2285 1973 y FG(times)25 b(2)14 b(\()p Fx(x)2639 1985 y Fz(0)2690 1973 y FG(+)g Fx(x)p FG(\))g FA(\000)g Fx(@)2990 1943 y FC(\006)2985 1993 y Fu(x)3045 1973 y FA(M)3145 1943 y Fz([)p FC(\024)p Fu(m)p Fz(])3297 1973 y FG(\()p Fx(x)3376 1985 y Fz(0)3428 1973 y FG(+)g Fx(x)p FG(;)g Fx(")p FG(\))118 2079 y(with)27 b(an)e(appropriate)f(order)h(of)h(m)n(ultiplication.)36 b(The)26 b(term)g(2)14 b(\()p Fx(x)h FG(+)g Fx(x)2381 2091 y Fz(0)2419 2079 y FG(\))f(\(\()p Fx(x)2576 2091 y Fz(0)2629 2079 y FG(+)h Fx(x)p FG(\))2788 2049 y Fz(2)2841 2079 y FA(\000)g(M)3021 2049 y Fz([)p FC(\024)p Fu(m)p Fz(])3173 2079 y FG(\()p Fx(x)3252 2091 y Fz(0)3305 2079 y FG(+)g Fx(x)p FG(;)f Fx(")p FG(\)\))3572 2049 y FC(\000)p Fz(2)118 2185 y FG(can)34 b(b)r(e)h(b)r(ounded)f(prop)r(ortionally)f (to)h(\()p Fx(C)1499 2150 y FC(\000)p Fz(2)1493 2208 y(0)1589 2185 y FG(2)1631 2155 y Fz(2\()p Fu(m)p FC(\000)p Fz(1\))1863 2185 y FG(\))1895 2155 y Fz(3)p Fu(=)p Fz(2)2034 2185 y FA(\024)g FG(\()p Fx(C)2230 2150 y FC(\000)p Fz(2)2224 2208 y(0)2320 2185 y FG(2)2362 2155 y Fz(2\()p Fu(m)p FC(\000)p Fz(1\))2594 2185 y FG(\))2626 2155 y Fz(2)2664 2185 y FG(,)i(while)e(the)h(remaining)f(term)118 2303 y(can)27 b(b)r(e)g(studied)g(b)n(y)g(making)f(use)g(of)h(the)h (inductiv)n(e)f(assumption)f FA(k)p Fx(@)2341 2315 y Fu(x)2382 2303 y FA(M)2482 2272 y Fz([)p FC(\024)p Fu(m)p Fz(])2635 2303 y FG(\()p Fx(x)2714 2315 y Fz(0)2768 2303 y FG(+)17 b Fx(x)p FG(;)d Fx(")p FG(\))p FA(k)23 b(\024)g Fx(B)t(")3264 2272 y Fz(2)3301 2303 y Fx(a)3345 2259 y FC(\000)p Fz(1)p Fu(=)p Fz(2)3345 2312 y Fu(s)3528 2303 y FG(and)118 2409 y(it)28 b(leads)f(to)h(the)g(same)f(b)r(ound)h (found)g(for)f(the)h(\014rst)f(term,)h(i.e.)37 b(\()p Fx(C)2254 2373 y FC(\000)p Fz(2)2248 2431 y(0)2343 2409 y FG(2)2385 2379 y Fz(2\()p Fu(m)p FC(\000)p Fz(1\))2618 2409 y FG(\))2650 2379 y Fz(2)2687 2409 y FG(,)28 b FF(multiplie)l(d)h FG(b)n(y)e Fx(B)t(")3339 2379 y Fz(2)3376 2409 y FG(.)3399 2379 y Fz(8)189 2515 y FG(If)34 b(the)g(deriv)-5 b(ativ)n(e)32 b(falls)h(on)g(either)h(a)f Fx( )1484 2527 y Fu(p)1556 2515 y FG(or)f(a)h Fx(\037)1790 2527 y Fu(p)1862 2515 y FG(function,)i(w)n(e)f(can)f(use)g(that)g(suc)n(h)h(deriv)-5 b(ativ)n(e)32 b(can)h(b)r(e)118 2621 y(b)r(ounded)25 b(prop)r(ortionally)e(to)i Fx(C)1160 2586 y FC(\000)p Fz(1)1154 2644 y(0)1250 2621 y FG(2)1292 2591 y Fu(p)1354 2621 y FG(and)1513 2559 y Fr(P)1601 2580 y Fu(m)p FC(\000)p Fz(1)1601 2646 y Fu(p)p Fz(=0)1763 2621 y FG(2)1805 2591 y Fu(p)1866 2621 y FG(=)d(2)1995 2591 y Fu(m)2058 2621 y FG(,)j(to)g(obtain)g(again)e(the)j(same)e(b)r(ound)h(as)f(the)i (\014rst)118 2728 y(case.)189 2834 y(Hence)g(the)h(\014nal)f(b)r(ound)h (has)f(the)h(form)f Fx(B)1561 2846 y Fz(1)1614 2834 y FG(+)15 b Fx(")1733 2804 y Fz(2)1770 2834 y Fx(B)t(b)26 b FG(with)h Fx(B)2150 2846 y Fz(1)2187 2834 y Fx(;)14 b(b)26 b FG(suitable)h(constan)n(ts,)e(pro)n(vided)h Fx(")g FG(is)g(small)118 2940 y(enough,)32 b(sa)n(y)e Fx(")e(<)h(")779 2952 y Fz(3)816 2940 y FG(.)48 b(The)31 b(v)-5 b(alue)31 b(of)g(the)g(constan)n(ts)g Fx(B)1958 2952 y Fz(1)1995 2940 y Fx(;)14 b(b)31 b FG(do)f(not)i(dep)r(end)f(on)g (the)h(inductiv)n(ely)f(assumed)118 3047 y(v)-5 b(alue)35 b(for)g Fx(B)t FG(:)52 b(in)35 b(particular)f Fx(B)1175 3059 y Fz(1)1247 3047 y FG(can)h(b)r(e)g(obtained)g(\(see)g(Remark)f (\(2\))h(b)r(elo)n(w)g(for)g(a)f(smarter)g(b)r(ound\))i(b)n(y)118 3153 y(replacing)30 b(2)519 3123 y Fz(\()p Fu(m)p Fz(+3\))749 3153 y FG(in)i(the)g(t)n(w)n(o)e(factors)g(in)i(the)g FF(l.h.s.)h FG(of)f(\(A3.3\))f(b)n(y)g(2)2408 3123 y Fz(2\()p Fu(m)p Fz(+3\))2671 3153 y FG(and)g(b)n(y)g(inserting)g(a)g (factor)g Fx(k)118 3259 y FG(times)e(a)f(constan)n(t)f(\(to)h(k)n(eep)g (trac)n(k)f(of)i(all)f(the)g(constan)n(t)g(factors)f(arising)g(from)h (di\013eren)n(tiation\).)39 b(Therefore)118 3366 y(if)29 b Fx(B)f FG(=)c(2)p Fx(B)480 3378 y Fz(1)545 3366 y FG(the)29 b(estimate)g(on)f Fx(@)1188 3335 y Fz(+)1183 3386 y Fu(x)1242 3366 y FA(M)1342 3335 y Fz([)p FC(\024)p Fu(n)p Fz(])1477 3366 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))29 b(follo)n(ws)e(if)i Fx(")f FG(is)g(small)g(enough,)g(sa)n(y)g Fx(")c(<)g(")3058 3378 y Fz(4)3094 3366 y FG(.)40 b(The)28 b(same)g(can)118 3472 y(b)r(e)g(said)f(ab)r(out)h(the)g(left)g(deriv)-5 b(ativ)n(e)27 b Fx(@)1358 3442 y FC(\000)1353 3492 y Fu(x)1414 3472 y FG(.)189 3578 y(The)32 b(righ)n(t)g(and)g(left)h (di\013eren)n(tiabilit)n(y)f(of)g FA(M)1671 3548 y Fz([)p Fu(n)p Fz(])1753 3578 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))33 b(with)g(resp)r(ect)f(to)g Fx(x)h FG(is)f(due)g(to)g(the)h (dep)r(endence)g(of)118 3684 y FA(M)218 3654 y Fz([)p Fu(n)p Fz(])301 3684 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))33 b(on)f(the)i(function)f Fx(D)r FG(\()p Fx(x)p FG(\):)48 b(the)33 b(latter)g(has)f(a)g(discon)n(tin)n(uous)g(deriv)-5 b(ativ)n(e)32 b(at)g(a)h(\014nite)g(n)n(um)n(b)r(er)g(of)118 3800 y(p)r(oin)n(ts)f(\(roughly)g(at)g(midp)r(oin)n(ts)h(b)r(et)n(w)n (een)f(the)h(eigen)n(v)-5 b(alues)31 b Fx(\025)2166 3757 y Fz([0])2166 3823 y Fu(j)2273 3800 y FG(of)i Fx(M)2454 3812 y Fz(0)2490 3800 y FG(\).)2545 3770 y Fz(9)2634 3800 y FG(Note)f(that)h(the)g(denominators)118 3917 y(in)i(the)g (self-energy)f(v)-5 b(alues)34 b(de\014ning)h Fx(M)1456 3887 y Fz([)p Fu(n)p Fz(])1538 3917 y FG(\()p Fx(x)p FG(;)14 b Fx(")p 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b(without)h(an)n(y)f(further)h(restriction)f(on)g Fx(")p FG(:)34 b(and)23 b(a)f(similar)g(argumen)n(t)g(holds)h(for)f (the)h Fx(")p FG(-deriv)-5 b(ativ)n(es.)118 1707 y Fy(A3.3.)48 b FF(Canc)l(el)t(lations.)j FG(Only)32 b(the)g(b)r(ound)g(in)g(the)g (fourth)f(line)h(of)g(\(5.13\))f(follo)n(ws)f(from)i(those)f(in)h(the)g (\014rst)118 1813 y(line.)48 b(The)31 b(b)r(ounds)g(in)h(the)f(second)g (and)g(third)g(lines)g(express)f(remark)-5 b(able)30 b(prop)r(erties)g(of)h(Lindstedt)h(series)118 1920 y(and)g(are)f(essen) n(tially)g(algebraic)f(prop)r(erties:)45 b(they)32 b(are)f(the)i (\\same")d(cancellations)h(whic)n(h)h(o)r(ccur)f(in)h(KAM)118 2026 y(theory)-7 b(,)31 b(see)g(Refs.)47 b([Ga],)31 b([GM1],)h(and)f (are)f(based)g(on)h(the)g(remark)e(that)i(if)h Fx(T)42 b FG(is)31 b(a)f(self-energy)g(cluster)g(the)118 2132 y(en)n(tering)j(and)h(exiting)g(lines)g(ha)n(v)n(e)f(the)i(same)f (momen)n(tum)g Fw(\027)6 b FG(:)50 b(hence)34 b(the)g(sum)h(of)f(the)g (harmonics)f(of)h(the)118 2239 y(no)r(des)27 b(of)h Fx(T)39 b FG(v)-5 b(anishes)860 2176 y Fr(P)947 2264 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))1206 2239 y Fw(\027)1254 2251 y Fm(v)1324 2239 y FG(=)23 b Fy(0)p FG(.)189 2347 y(W)-7 b(e)28 b(start)f(b)n(y)g(dealing)g(with)h(the)g(trivial)f (cases.)36 b(Consider)27 b(\014rst)g(self-energy)f(clusters)h Fx(T)39 b FG(suc)n(h)27 b(that)1431 2479 y Fr(X)1370 2661 y Fm(v)p FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))1625 2558 y FA(j)p Fw(\027)1696 2570 y Fm(v)1743 2558 y FA(j)23 b(\025)g FG(\()p Fx(C)1968 2570 y Fz(0)2006 2558 y Fx(=)p FG(2)2090 2524 y Fz(6)2126 2558 y FA(j)p Fx(x)p FA(j)p FG(\))2251 2524 y Fz(1)p Fu(=\034)2349 2532 y Ft(0)2386 2558 y Fx(:)1019 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(6\))118 2859 y(F)-7 b(or)30 b(suc)n(h)h(a)f(self-energy)g(cluster)g Fx(T)42 b FG(one)31 b(can)f(use)h(part)f(\(sa)n(y)g(1)p Fx(=)p FG(8\))g(of)h(the)g(exp)r(onen)n(tial)f(deca)n(y)g(of)h(the)g (no)r(de)118 2995 y(factors)i(to)g(obtain)g(a)h(b)r(ound)g Fx(e)1143 2944 y 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FC(0)1245 4681 y Fu(i;j)1323 4661 y FG(\()p Fx(T)12 b FG(\))18 b(+)g Fx(xM)1686 4618 y FC(0)p Fz(\(1\))1677 4684 y Fu(i;j;)p Fm(v)1814 4661 y FG(\()p Fx(x;)c(T)e FG(\))p Fx(;)676 b(i)23 b FA(\024)f Fx(r)n(;)28 b(r)e(<)d(j;)3428 4578 y FG(\()p Fx(A)p FG(3)p Fx(:)p FG(7\))118 4866 y(so)g(that)h (after)g(p)r(erforming)f(the)h(sum)g(o)n(v)n(er)e(the)j(self-energy)d (clusters)h(of)h FA(F)2486 4878 y Fu(T)2538 4866 y FG(,)h(i.e.)36 b(after)23 b(p)r(erforming)g(the)i(sums)118 4910 y Fr(P)206 4997 y Fm(v)o Fu(;)p Fm(w)q FC(2)p Fu(V)14 b Fz(\()p Fu(T)9 b Fz(\))559 4972 y FG(or,)25 b(resp)r(ectiv)n(ely)-7 b(,)1149 4910 y Fr(P)1237 4997 y Fm(v)o FC(2)p Fu(V)15 b Fz(\()p Fu(T)9 b Fz(\))1482 4972 y FG(,)26 b(the)g(\014rst)f(t)n(w)n (o)g(terms)g(in)h(the)g(\014rst)f(line)h(and)f(the)h(\014rst)g(term)f (in)h(the)118 5079 y(second)h(line)h(do)f(not)h(con)n(tribute)f(b)r (ecause)1512 5016 y Fr(P)1600 5104 y Fm(v)1661 5079 y Fw(\027)1709 5091 y Fm(v)1778 5079 y FG(=)c Fy(0)p FG(.)37 b(Ho)n(w)n(ev)n(er)26 b(one)h(has)g(to)g(sho)n(w)g(that)h(the)g (matrices)f Fx(M)118 5185 y FG(and)e Fx(M)367 5155 y FC(0)414 5185 y FG(in)g(the)g FF(r.h.s.)38 b FG(of)24 b(\(A3.7\))h(satisfy)f(appropriate)f(b)r(ounds)i(once)f(the)h(factors)e Fx(x)i FG(determining)g(the)g(order)118 5291 y(of)j(zero)g(at)g Fx(x)d FG(=)f(0)k(are)f(extracted.)38 b(F)-7 b(rom)28 b(the)h(con)n(v)n(ergence)d(one)i(exp)r(ects)g(that)h(the)g(b)r(ounds)f (should)g(still)h(b)r(e)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)h(19:31)1130 b FG(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 118 356 a FG(33:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(prop)r(ortional)27 b(to)h Fx(")738 525 y Fz(2)803 555 y FG(while)g(the)h(deriv)-5 b(ativ)n(es)27 b Fx(@)1627 525 y FC(\006)1622 576 y Fu(x)1711 555 y FG(or)g Fx(@)1857 567 y Fu(")1921 555 y FG(should)h(satisfy)f(b)r (ounds)i(prop)r(ortional)d(to)i Fx(")3353 525 y Fz(2)3418 555 y FG(or)g(to)g Fx(")118 662 y FG(resp)r(ectiv)n(ely)-7 b(.)189 776 y(The)40 b(\(A3.7\))g(are)f(pro)n(v)n(ed)g(b)n(y)h(means)f (of)h(in)n(terp)r(olations,)j(see)c([GM1],)44 b(b)r(et)n(w)n(een)c(the) g(con)n(tributions)g(of)118 882 y(the)32 b(self-energy)d(clusters)i(in) g(the)h(family)f FA(F)1548 894 y Fu(T)1600 882 y FG(.)48 b(When)32 b(w)n(e)f(collect)f(together)h(the)g(v)-5 b(alues)31 b(of)g(the)h(self-energy)118 988 y(clusters)27 b(in)h FA(F)576 1000 y Fu(T)655 988 y FG(then)h(the)f(argumen)n(ts)e(of)h (some)g(of)h(the)g(propagators)c(can)j(fall)h(outside)f(the)h(supp)r (orts)g(of)f(the)118 1095 y(resp)r(ectiv)n(e)c(cut-o\013)g(function)h (\(b)r(ecause)f(the)g(lines)g(are)g(shifted)h(but)f(their)h(scale)e (lab)r(els)h(are)f(k)n(ept)h(\014xed)h(so)e(that)118 1201 y(scales)k(of)h(the)g(propagators)d(of)j(the)h(self-energy)d (clusters)h Fx(T)2050 1171 y FC(0)2096 1201 y FA(2)d(F)2234 1213 y Fu(T)2313 1201 y FG(are)j(the)i(ones)e(inherited)h(b)n(y)g Fx(T)38 b FG(while)27 b(the)118 1307 y(momen)n(tum)h(\015o)n(wing)f(in) 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y(larger)27 b(than)i(what)f(app)r(eared)g(there)h(as)f (necessary)f(so)h(that)h(the)g(estimate)g(\(4.4\))f(is)h(apparen)n(tly) e(w)n(orse)g(than)118 2591 y(it)h(should.)189 2705 y(In)35 b(some)g(cases,)h(ho)n(w)n(ev)n(er,)f(a)g(serious)f(problem)g(seems)h (to)g(arise)f(when)h(actually)g(attempting)g(to)g(deriv)n(e)118 2811 y(b)r(ounds:)44 b(namely)30 b(the)i(b)r(ounds)f(on)f(the)i (matrices)e(whic)n(h)h(app)r(ear)f(as)g(co)r(e\016cien)n(ts)h(in)g (\(A3.7\))g(can)f(really)g(b)r(e)118 2917 y(c)n(hec)n(k)n(ed)23 b(as)g(just)i(outlined)f(b)n(y)f(the)i(ab)r(o)n(v)n(e)d(hin)n(ts,)j (and)f(without)g(a\013ecting)g(the)g(v)-5 b(alues)24 b(of)g Fx(")f FG(for)h(whic)n(h)g(one)f(has)118 3023 y(con)n(v)n(ergence,)28 b(only)h(if)h Fx(x)f FG(v)n(eri\014es)g(the)g (condition)h(of)f(b)r(eing)h(so)e(small)h(that)h(the)g(v)-5 b(ariations)28 b(of)h(the)h(momen)n(ta)118 3130 y(\015o)n(wing)c(in)g (the)h(inner)g(lines)f(of)h Fx(T)12 b FG(,)25 b(when)i(the)g(en)n 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4208 y FA(j)f(\025)f(j)p Fx(x)2830 4220 y Fu(`)2862 4208 y FA(j)h(\025)3023 4176 y Fz(1)p 3023 4190 34 4 v 3023 4237 a(2)3066 4208 y FA(j)p Fx(x)3136 4178 y Fz(0)3136 4232 y Fu(`)3174 4208 y FA(j)p FG(,)i(so)e(that)h(the) 118 4315 y(scales)h(can)h(c)n(hange)f(b)n(y)g(at)h(most)g(one)g(unit)g (b)n(y)g(shifting)g(the)h(external)e(lines)h(of)g Fx(T)12 b FG(.)58 b(Then)35 b(the)h(quan)n(tities)118 4421 y Fx(D)r FG(\()p Fx(x)268 4433 y Fu(`)301 4421 y FG(\))30 b(do)f(not)h(c)n(hange)f(m)n(uc)n(h)g(for)h(all)f(lines)h Fx(`)c FA(2)h FG(\003\()p Fx(T)12 b FG(\),)30 b(and)g(w)n(e)f(shall)g (ha)n(v)n(e)g(the)h(cancellation)f(through)g(the)118 4527 y(men)n(tioned)k(mec)n(hanism.)54 b(Therefore)32 b(the)i(con)n(tribution)f(of)g FA(M)2214 4497 y Fz([)p Fu(p)p Fz(])2290 4527 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))34 b(to)f FA(M)2718 4497 y Fz([)p FC(\024)p Fu(n)p Fz(])2852 4527 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))34 b(can)f(b)r(e)h(b)r(ounded)118 4646 y(in)e(b)r(oth)h(cases)d(prop)r 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y(Finally)i(w)n(e)g(note)g(that)h (in)g(the)f(estimates)g(of)h(the)f Fx(M)9 b FG('s)28 b(in)h(\(A3.7\))f(w)n(e)g(ha)n(v)n(e)f(to)h(sum)h(o)n(v)n(er)d(the)j (scale)f(lab)r(els)118 5079 y(and)h(this)h(giv)n(es)e(a)h(factor)g(p)r (er)g(line)h(larger)d(than)j(the)f(one)g(arising)f(in)i(the)g(b)r(ound) g(\(A3.3\))f(\(whic)n(h)h(w)n(as)e(2\);)i(in)118 5185 y(fact)g(w)n(e)f(ha)n(v)n(e)g(to)h(consider)e(also)h(trees)g(with)i(v) -5 b(anishing)29 b(v)-5 b(alue:)41 b(but)31 b(the)f(scales)e(of)i(the)g (divisors)f(asso)r(ciated)118 5291 y(with)j(their)g(lines)g(can)g(c)n (hange)e(at)i(most)g(b)n(y)f(one)h(unit)g(with)g(resp)r(ect)g(to)g(the) g(scale,)g(hence)g(w)n(e)g(can)f(ha)n(v)n(e)g(at)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)f(19:31)1130 b FG(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 118 356 a FG(34:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(most)c(4)f(scale)f(lab)r(els)i (p)r(er)f(line.)118 733 y FF(R)l(emark.)36 b FG(W)-7 b(e)24 b(stress)f(once)g(more)g(that)h(the)g(ab)r(o)n(v)n(e)f(analysis) f(holds)i(if)g Fx(")f FG(is)h(small)f(enough,)h(sa)n(y)f Fx(")g(<)p 3300 687 39 4 v 22 w(")3339 745 y Fz(1)3400 733 y FG(with)p 3585 687 V 24 w Fx(")3624 745 y Fz(1)118 839 y FG(determined)28 b(b)n(y)g(collecting)f(all)g(the)i(\(three\))f (restrictions)e(imp)r(osed)i(b)n(y)g(requiring)e Fx(")i FG(to)f(b)r(e)i(\\small)e(enough",)118 945 y(deriv)n(ed)22 b(ab)r(o)n(v)n(e)g(and)p 792 900 V 23 w Fx(")831 957 y Fz(1)891 945 y FG(is)h FF(indep)l(endent)h FG(of)f Fx(n)1564 957 y Fz(0)1624 945 y FG(\(otherwise)f(it)i(w)n(ould)e(b)r(e) i(unin)n(teresting\).)35 b(The)23 b(reason)e(is)i(that)118 1052 y(as)31 b(long)f(as)g(w)n(e)h(do)g(not)g(deal)g(with)h Fx(x)p FG('s)f(whic)n(h)g(are)f(to)r(o)h(close)f(to)h(the)h(eigen)n(v) -5 b(alues)30 b(of)h Fx(M)3012 1064 y Fz(0)3049 1052 y FG(,)h(so)e(that)i(the)f(k)n(ey)118 1158 y(inequalit)n(y)f(\(4.4\))g (holds,)g(w)n(e)g(do)g(not)g(really)f(see)h(the)g(di\013erence)g(b)r 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(discussed)f(exactly)g(as)g(for)g(the)h(scales)e([)p Fx(n)p FG(])i(with)118 1866 y Fx(n)25 b FA(\024)p 282 1821 50 4 v 24 w Fx(n)332 1878 y Fz(0)369 1866 y FG(,)k(with)h(the)f (only)f(di\013erence)g(that)h(no)n(w)f(one)h(has)f(to)g(use)h(also)f (the)h(second)f(part)g(of)g(the)h(Diophan)n(tine)118 1973 y(conditions)j(\(6.7\),)h(as)e(already)g(done)h(in)g(the)g (argumen)n(t)f(leading)h(to)g(\(6.12\);)h(in)g(particular)d(the)j(role) e(of)h(the)118 2079 y(exp)r(onen)n(t)f Fx(\034)515 2091 y Fz(0)583 2079 y FG(is)f(no)n(w)g(pla)n(y)n(ed)g(b)n(y)g(2)p Fx(\034)1305 2091 y Fz(1)1373 2079 y FG(\(b)r(ecause)g(of)h(the)f (Diophan)n(tine)h(conditions)f(in)h(\(6.7\))f(whic)n(h)h(replaces)118 2185 y(\(1.3\))g(in)h(the)g(discussion\),)g(while)g(in)g(the)g (analogues)d(of)j(\(A3.1\))f(and)h(the)g(follo)n(wing)e(b)r(ounds)i(no) p 3310 2140 V 31 w Fx(n)f FG(app)r(ear,)118 2292 y(as)g(the)h (propagator)c(divisors)i(are)h(b)r(ounded)h(directly)f(in)g(terms)h(of) f(the)h(corresp)r(onding)d(scales,)j(and)f(not)g(in)118 2398 y(terms)c(of)h(the)g(frequencies.)189 2504 y(Also)37 b(the)g(argumen)n(t)f(giv)n(en)g(ab)r(o)n(v)n(e)g(ab)r(out)g(the)i (cancellations)d(extends)i(easily)g(to)f(the)i(scales)e([)p Fx(n)p FG(],)j(with)118 2611 y Fx(n)26 b FA(\025)p 285 2565 V 26 w Fx(n)335 2623 y Fz(0)372 2611 y FG(.)43 b(The)29 b(only)g(di\013erence)h(is)f(that)h(in)g(\(A3.6\))f(the)h(exp)r(onen)n (t)g(1)p Fx(=\034)2421 2623 y Fz(0)2487 2611 y FG(has)f(to)g(b)r(e)h (replaced)f(with)h(1)p Fx(=)p FG(\(2)p Fx(\034)3570 2623 y Fz(1)3606 2611 y FG(\),)118 2748 y(in)g(suc)n(h)f(a)g(w)n(a)n(y)f (that)i(for)f(an)n(y)f(line)i Fx(`)c FA(2)g FG(\003\()p Fx(T)12 b FG(\))29 b(one)g(has)g FA(jj)p Fx(x)2026 2717 y Fz(0)2026 2771 y Fu(`)2064 2748 y FA(j)20 b(\000)2191 2646 y Fr(q)p 2274 2646 395 4 v 102 x FA(j)p Fx(\025)2345 2704 y Fz([)p Fu(n)2405 2713 y Fs(`)2434 2704 y FC(\000)p Fz(1])2345 2771 y Fu(j)2542 2748 y FG(\()p Fx(")p FG(\))p FA(jj)27 b(\025)e FG(4)p FA(j)p Fx(x)p FA(j)p FG(,)30 b(hence)g(the)g(c)n(hain)f(of)118 2854 y(inequalities)434 3098 y(2)490 2978 y Fr(\014)490 3028 y(\014)490 3078 y(\014)490 3127 y(\014)517 3098 y FA(j)p Fx(x)587 3064 y Fz(0)587 3119 y Fu(`)624 3098 y FA(j)19 b(\000)749 2993 y Fr(q)p 832 2993 V 105 x FA(j)p Fx(\025)903 3055 y Fz([)p Fu(n)963 3064 y Fs(`)992 3055 y FC(\000)p Fz(1])903 3121 y Fu(j)1100 3098 y FG(\()p Fx(")p FG(\))p FA(j)1227 2978 y Fr(\014)1227 3028 y(\014)1227 3078 y(\014)1227 3127 y(\014)1277 3098 y FA(\025)1365 2978 y Fr(\014)1365 3028 y(\014)1365 3078 y(\014)1365 3127 y(\014)1393 3098 y FA(j)p Fx(x)1463 3110 y Fu(`)1495 3098 y FA(j)g(\000)1620 2993 y Fr(q)p 1703 2993 V 105 x FA(j)p Fx(\025)1774 3055 y Fz([)p Fu(n)1834 3064 y Fs(`)1863 3055 y FC(\000)p Fz(1])1774 3121 y Fu(j)1971 3098 y FG(\()p Fx(")p FG(\))p FA(j)2097 2978 y Fr(\014)2097 3028 y(\014)2097 3078 y(\014)2097 3127 y(\014)2148 3098 y FA(\025)2245 3042 y FG(1)p 2245 3079 42 4 v 2245 3155 a(2)2311 2978 y Fr(\014)2311 3028 y(\014)2311 3078 y(\014)2311 3127 y(\014)2338 3098 y FA(j)p Fx(x)2408 3064 y Fz(0)2408 3119 y Fu(`)2446 3098 y FA(j)g(\000)2571 2993 y Fr(q)p 2654 2993 395 4 v 105 x FA(j)p Fx(\025)2725 3055 y Fz([)p Fu(n)2785 3064 y Fs(`)2814 3055 y FC(\000)p Fz(1])2725 3121 y Fu(j)2922 3098 y FG(\()p Fx(")p FG(\))p FA(j)3048 2978 y Fr(\014)3048 3028 y(\014)3048 3078 y(\014)3048 3127 y(\014)3090 3098 y Fx(;)315 b FG(\()p Fx(A)p FG(3)p Fx(:)p FG(8\))118 3343 y(follo)n(ws,)27 b(and)h(again)f(b)n(y)h(shifting)g(the)g (external)f(lines)h(of)g Fx(T)39 b FG(the)29 b(scales)e(of)h(the)g(in)n (ternal)f(lines)h(can)g(c)n(hange)f(at)118 3449 y(most)h(b)n(y)f(one)g (unit,)h(when)g(\(A3.6\))g(is)f(not)h(satis\014ed)1084 3697 y FD(App)s(endix)39 b(A4.)49 b(Matrix)38 b(prop)s(erties)118 3874 y Fy(\(I\))21 b FF(L)l(et)i Fx(M)467 3886 y Fz(0)527 3874 y FF(b)l(e)g(a)h Fx(d)t FA(\002)t Fx(d)f FF(Hermitian)h(matrix)f (with)h(eigenvalues)g Fx(\025)2174 3886 y Fz(1)2235 3874 y Fx(<)f(:)14 b(:)g(:)22 b(<)h(\025)2578 3886 y Fu(p)2640 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y(remarks)c(that)i(the)g(matrix)f Fx(E)1089 1549 y Fu(j)1124 1537 y Fx(M)9 b(E)1275 1549 y Fu(j)1338 1537 y FG(has)27 b Fx(n)1536 1549 y Fu(j)1598 1537 y FG(eigen)n(v)-5 b(alues)27 b(and)g(that)h(it)g(has)f(the)h(form)f Fx(\025)2987 1549 y Fu(j)3041 1537 y FG(+)18 b Fx(")3173 1516 y Fr(f)3163 1537 y Fx(M)8 b FG(\()p Fx(")p FG(\).)189 1648 y(So)37 b(the)g(problem)f(is)h(reduced)g(to)f(the)i(case)e(in)h(whic)n(h)g Fx(M)2092 1660 y Fz(0)2166 1648 y FG(is)f(the)i(iden)n(tit)n(y)f(p)r (erturb)r(ed)g(b)n(y)g(an)f(analytic)118 1754 y(matrix.)41 b(Either)692 1733 y Fr(f)682 1754 y Fx(M)8 b FG(\()p Fx(")p FG(\))30 b(is)f(prop)r(ortional)f(to)h(the)g(iden)n(tit)n(y)h (and)f(there)g(is)g(nothing)g(more)f(to)h(do,)g(or)g(it)g(is)g(not:)118 1861 y(hence)f(there)f(will)g(b)r(e)h(an)f(order)f(in)i Fx(")f FG(at)h(whic)n(h)f(the)h(degeneracy)e(is)h(remo)n(v)n(ed)f(and)h (rep)r(eating)g(the)h(argumen)n(t)118 1967 y(w)n(e)i(reduce)f(the)i (problem)e(to)h(a)g(similar)f(one)h(for)f(matrices)h(of)g(dimension)g (lo)n(w)n(er)e(than)i Fx(n)3007 1979 y Fu(j)3042 1967 y FG(:)42 b(and)30 b(so)f(on)h(un)n(til)118 2073 y(w)n(e)d(\014nd)h(a)g (matrix)f(\(p)r(ossibly)g(one)g(dimensional\))h(prop)r(ortional)d(to)j (the)g(iden)n(tit)n(y)g(to)f(all)g(orders.)p 3620 2065 42 42 v 189 2255 a(In)h(our)f(analysis)f(w)n(e)h(need)h(the)g(follo)n (wing)e(corollary)-7 b(.)118 2437 y Fy(\(I)s(I\))23 b FF(L)l(et)i Fx(M)510 2449 y Fz(0)572 2437 y FF(b)l(e)g(Hermitian)h (with)g Fx(r)i FF(de)l(gener)l(ate)e(eigenvalues)g(e)l(qual)g(to)f FG(0)f FF(and)i Fx(s)d FG(=)g Fx(d)9 b FA(\000)g Fx(r)27 b FF(simple)f(eigenvalues)118 2543 y Fx("a)201 2555 y Fu(j)236 2543 y FF(,)k Fx(j)e FG(=)22 b(1)p Fx(;)14 b(:)g(:)g(:)g(;)g (s)p FF(.)118 2649 y(\(i\))29 b(The)h(matrix)f Fx(M)756 2661 y Fz(0)809 2649 y FG(+)15 b Fx(")928 2619 y Fz(2)965 2649 y Fx(M)1046 2661 y Fz(1)1112 2649 y FF(with)29 b Fx(M)1372 2661 y Fz(1)1438 2649 y FF(Hermitian)g(and)g(di\013er)l (entiable)i(in)d Fx(")h FF(with)g(b)l(ounde)l(d)g(derivative)i(wil)t(l) 118 2756 y(have)36 b Fx(s)f FF(non-de)l(gener)l(ate)g(eigenvalues)h Fx("a)1488 2768 y Fu(j)1544 2756 y FG(+)22 b Fx(O)r FG(\()p Fx(")1767 2725 y Fz(2)1805 2756 y FG(\))p FF(,)37 b Fx(j)g FG(=)32 b(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(s)p FF(,)36 b(and)f Fx(r)j FF(eigenvalues)e Fx(\025)3122 2768 y Fz(1)3160 2756 y FG(\()p Fx(")p FG(\))p Fx(;)14 b(:)g(:)g(:)g(;)g(\025)3496 2768 y Fu(r)3533 2756 y FG(\()p Fx(")p FG(\))p FF(,)118 2862 y(al)t(l)35 b(analytic)g(in)f Fx(")p FF(,)i(with)e(the)g(pr)l(op)l (erty)h(that)f(for)h(al)t(l)g Fx(k)e FG(=)e(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(r)36 b FF(one)f(has)f FA(j)p Fx(\025)2710 2874 y Fu(k)2752 2862 y FG(\()p Fx(")p FG(\))p FA(j)d Fx(<)f(C)20 b(")3122 2832 y Fz(2)3159 2862 y FF(,)36 b(if)e Fx(")g FF(is)g(smal)t(l)118 2968 y(enough)c(and)h Fx(C)36 b FF(is)30 b(a)g(suitable)g(c)l(onstant.)118 3074 y(\(ii\))h(If)f Fx(M)435 3086 y Fz(1)501 3074 y FF(dep)l(ends)h(on)f(a)g(p)l(ar)l(ameter)g Fx(x)h FF(and)f(is)g (di\013er)l(entiable)h(also)g(in)f Fx(x)g FF(with)g(b)l(ounde)l(d)g (derivative)i(then)1010 3289 y FA(j)p Fx(@)1077 3301 y Fu(x)1119 3289 y Fx(\025)1167 3301 y Fu(j)1202 3289 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FA(j)24 b(\024)f Fx(C)6 b(")1628 3254 y Fz(2)1665 3289 y Fx(;)99 b FA(j)p Fx(@)1854 3301 y Fu(")1889 3289 y Fx(\025)1937 3301 y Fu(j)1973 3289 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p FA(j)24 b(\024)e Fx(C)q(;)185 b(j)28 b(>)22 b(r)n(;)1010 3429 y FA(j)p Fx(\025)1081 3441 y Fu(j)1117 3429 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))19 b FA(\000)f Fx(\025)1454 3441 y Fu(j)1489 3429 y FG(\()p Fx(x)1568 3394 y FC(0)1592 3429 y FG(;)c Fx(")p FG(\))p FA(j)23 b(\024)g Fx(C)6 b(")1938 3394 y Fz(2)1975 3429 y FA(j)p Fx(x)19 b FA(\000)f Fx(x)2194 3394 y FC(0)2218 3429 y FA(j)2241 3394 y Fz(1)p Fu(=r)2345 3429 y Fx(;)183 b(j)28 b FA(\024)23 b Fx(r)n(;)3428 3358 y FG(\()p Fx(A)p FG(4)p Fx(:)p FG(2\))118 3637 y FF(if)31 b Fx(")e FF(is)h(smal)t(l)h(enough)f(and)g 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y Fz(0)2695 4213 y FG(+)19 b Fx(")2818 4183 y Fz(2)2855 4213 y Fx(M)2936 4225 y Fz(1)2972 4213 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))29 b(with)f(those)g(of)118 4319 y Fx(M)199 4331 y Fz(0)256 4319 y FG(+)20 b Fx(")380 4289 y Fz(2)417 4319 y Fx(M)498 4331 y Fz(1)535 4319 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))20 b(+)g Fx(")866 4289 y Fz(2)903 4319 y Fx(O)r FG(\()p FA(j)p Fx(x)i FA(\000)d Fx(x)1223 4289 y FC(0)1247 4319 y FA(j)p FG(\).)45 b(By)30 b(the)h(ab)r(o)n(v)n(e)d(expression)h(for)h(the)g(pro)5 b(jection)30 b(on)f(the)i(plane)f(of)g(the)118 4426 y(\014rst)e Fx(r)i FG(eigen)n(v)-5 b(alues)27 b(this)h(is)g(reduced)g(to)f(the)h (problem)g(of)g(comparing)e(t)n(w)n(o)h Fx(r)22 b FA(\002)c Fx(r)30 b FG(matrices)d(of)h(order)f Fx(")3463 4396 y Fz(2)3528 4426 y FG(and)118 4532 y(di\013ering)g(b)n(y)g Fx(O)r FG(\()p FA(j)p Fx(x)19 b FA(\000)e Fx(x)873 4502 y FC(0)897 4532 y FA(j)p FG(\).)37 b(The)28 b(p)r(o)n(w)n(er)e(1)p Fx(=r)j 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Fx(x)2641 5067 y Fz(2)2679 5097 y Fx(N)94 b(")2879 5067 y Fz(2)2916 5097 y Fx(xP)2523 5196 y(")2562 5166 y Fz(2)2599 5196 y Fx(xP)2711 5166 y FC(\003)2863 5196 y Fx(")2902 5166 y Fz(2)2939 5196 y Fx(Q)3042 5029 y Fr(\023)3103 5147 y FF(,)28 b(with)h Fx(N)t(;)14 b(Q)27 b FF(two)118 5291 y Fx(r)17 b FA(\002)d Fx(r)30 b FF(and)e Fx(r)17 b FA(\002)d Fx(s)28 b FF(matric)l(es)g(and)g Fx(P)40 b FF(a)28 b Fx(r)17 b FA(\002)c Fx(s)28 b FF(matrix)f(then)h (the)g(\014rst)f Fx(r)j FF(eigenvalues)f(of)g Fx(M)2961 5303 y Fz(0)3012 5291 y FG(+)13 b Fx(M)3171 5303 y Fz(1)3236 5291 y FF(ar)l(e)28 b(b)l(ounde)l(d)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)i(19:31)1130 b FG(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 118 356 a FG(36:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y(by)e FA(j)p Fx(\025)297 567 y Fu(j)333 555 y FG(\()p Fx(x;)14 b(")p FG(\))p FA(j)24 b Fx(<)e(C)6 b(")758 525 y Fz(2)795 555 y Fx(x)842 525 y Fz(2)880 555 y FF(,)30 b(for)h Fx(j)d FG(=)23 b(1)p 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Fu(h)1427 2237 y FG(,)28 b Fx(h)22 b FG(=)h(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(r)30 b FG(consists)d(of)g(v)n(ectors)f(of)h(the)h(form)f Fy(e)3088 2249 y Fu(h)3149 2237 y FG(+)3232 2120 y Fr(\022)3307 2187 y Fx(")3346 2157 y Fz(2)3383 2187 y Fx(x)3430 2157 y Fz(2)3467 2187 y Fy(u)3520 2199 y Fu(h)3325 2287 y Fx(")3364 2257 y Fz(2)3401 2287 y Fx(x)p Fy(u)3501 2257 y FC(0)3501 2310 y Fu(h)3577 2120 y Fr(\023)3638 2237 y FG(,)118 2386 y(so)h(that)h(one)f(c)n(hec)n(ks)f(that)i(the)g(matrix) f(\()p Fy(v)1490 2398 y Fu(h)1533 2386 y Fx(;)14 b FG(\()p Fx(M)1683 2398 y Fz(0)1739 2386 y FG(+)19 b Fx(M)1904 2398 y Fz(1)1941 2386 y FG(\))p Fy(v)2023 2398 y Fu(h)2062 2382 y Ff(0)2089 2386 y FG(\))29 b(is)f(a)g Fx(r)22 b FA(\002)d Fx(r)31 b FG(matrix)d(whic)n(h)h(is)f(prop)r(ortional)f(to) 118 2493 y Fx(")157 2462 y Fz(2)194 2493 y Fx(x)241 2462 y Fz(2)307 2493 y FG(\(i.e.)40 b(it)29 b(has)f(the)h(form)f Fx(")1098 2462 y Fz(2)1135 2493 y Fx(x)1182 2462 y Fz(2)1220 2493 y Fx(M)1301 2505 y Fz(2)1337 2493 y FG(\()p Fx(x;)14 b(")p FG(\),)30 b(with)f Fx(M)1848 2505 y Fz(2)1913 2493 y FG(b)r(ounded)g(for)f Fx(")g FG(small)g(and)h(for)f FA(j)p Fx(x)p FA(j)d Fx(<)f FG(1\))k(and)g(whic)n(h,)118 2599 y(b)n(y)f(construction,)g(has)g(the)h(same)f(eigen)n(v)-5 b(alues)27 b(as)g(the)h(\014rst)f Fx(r)j FG(eigen)n(v)-5 b(alues)27 b(of)g(the)h(matrix)f Fx(M)3162 2611 y Fz(0)3218 2599 y FG(+)18 b Fx(M)3382 2611 y Fz(1)3418 2599 y FG(.)189 2829 y(F)-7 b(or)27 b(the)h(ab)r(o)n(v)n(e)e(prop)r(erties)h(see)g (also)g([RS])h(and)f([Ka].)240 3077 y FD(App)s(endix)39 b(A5.)99 b(Algebraic)38 b(iden)m(tities)g(for)f(the)h(renormalized)g (expansion)118 3308 y FG(W)-7 b(e)37 b(sho)n(w)f(that)h(the)h(function) f Fy(h)g FG(de\014ned)g(through)f(the)i(renormalized)d(expansion)h (solv)n(es)f(the)i(equations)118 3414 y(of)d(motion)f(\(1.5\))g(for)g (all)g Fx(")g FA(2)g(E)7 b FG(.)55 b(This)34 b(is)f(essen)n(tially)f(a) i(rep)r(etition)f(of)h(Ref.)55 b([Ge].)g(W)-7 b(e)34 b(shall)f(c)n(hec)n(k)f(that)118 3520 y Fy(h)23 b FG(=)g Fx("g)s(@)408 3532 y Fq(')461 3520 y Fx(f)9 b FG(\()p Fw( )20 b FG(+)c Fy(a)p Fx(;)e Fw(\014)845 3532 y Fz(0)899 3520 y FG(+)i Fy(b)p FG(\),)27 b(where)f Fw(')d FG(=)g(\()p Fw(\013)p Fx(;)14 b Fw(\014)s FG(\))27 b(and)g Fx(g)i FG(is)e(the)g(pseudo-di\013eren)n(tial)e(op)r(erator)g(with)i(k)n (ernel)118 3627 y Fx(g)s FG(\()p Fw(!)22 b FA(\001)d Fw(\027)6 b FG(\))25 b(=)g(\()p Fw(!)d FA(\001)e Fw(\027)6 b FG(\))761 3597 y FC(\000)p Fz(2)850 3627 y FG(.)41 b(W)-7 b(e)29 b(can)f(write)h Fy(h)c FG(=)1593 3565 y Fr(P)1681 3652 y Fq(\027)5 b FC(2)p Fp(Z)1801 3635 y Fs(r)1853 3627 y Fx(e)1892 3597 y Fu(i)p Fq(\027)g FC(\001)p Fq( )2035 3627 y Fy(h)2088 3639 y Fq(\027)2135 3627 y FG(,)29 b Fy(h)2240 3639 y Fq(\027)2312 3627 y FG(=)2402 3565 y Fr(P)2490 3585 y FC(1)2490 3652 y Fu(n)p Fz(=0)2633 3627 y Fy(h)2686 3639 y Fu(n;)p Fq(\027)2823 3627 y FG(\(only)f(t)n(w)n (o)g(terms)h(in)g(this)118 3733 y(series)e(are)h(di\013eren)n(t)g(from) g(0)g(for)f(eac)n(h)h Fw(\027)6 b FG(\),)28 b(with)h Fy(h)1771 3745 y Fu(n;)p Fq(\027)1903 3733 y FG(=)1991 3671 y Fr(P)2079 3691 y FC(1)2079 3758 y Fu(k)q Fz(=1)2218 3671 y Fr(P)2305 3758 y Fu(\022)r FC(2)p Fz(\002)2435 3738 y Ff(R)2435 3780 y Fs(k)q(;\027)2522 3758 y Fz(\()p Fu(n)p Fz(\))2633 3733 y FG(V)-7 b(al\()p Fx(\022)r FG(\),)30 b(where)d(\002)3216 3703 y FC(R)3216 3757 y Fu(k)q(;)p Fq(\027)3319 3733 y FG(\()p Fx(n)p FG(\))i(is)f(the)118 3866 y(set)i(of)g(trees)f(in)h(\002)712 3836 y FC(R)712 3890 y Fu(k)q(;)p Fq(\027)845 3866 y FG(suc)n(h)g(that)g(the)g(ro)r(ot) f(line)h(has)g(scale)f Fx(n)p FG(.)43 b(With)31 b(resp)r(ect)f(to)g (the)g(previous)f(sections)g(w)n(e)118 3973 y(ha)n(v)n(e)23 b(dropp)r(ed)g(the)h(comp)r(onen)n(t)f(lab)r(el)h Fx(\015)k FA(2)23 b(f)p FG(1)p Fx(;)14 b(:)g(:)g(:)f(;)h(d)p FA(g)23 b FG(in)h(the)g(de\014nition)g(of)f(the)h(set)g(of)f(trees,)h(for)f (notational)118 4079 y(con)n(v)n(enience.)189 4239 y(Note)28 b(that,)g(for)f(all)g Fx(x)c FA(6)p FG(=)g(0)k(and)h(for)f(all)g Fx(p)c FA(\025)g 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Fz(1)1344 5313 y Fu(p)p Fz(=0)1488 5288 y Fx(\037)1540 5300 y Fu(p)1579 5288 y FG(\(\001)1680 5258 y Fz([)p Fu(p)p Fz(])1756 5288 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))31 b(for)e Fx(n)d FA(\025)g FG(1,)j(\011)2461 5300 y Fz(0)2498 5288 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))27 b(=)f Fx( )2857 5300 y Fz(0)2894 5288 y FG(\(\001)2995 5258 y Fz([0])3071 5288 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\)\):)42 b(b)n(y)29 b(using)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)h(19:31)1130 b FG(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 118 356 a FG(37:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y FG(\(A5.1\))c(one)f(can)g(write,) h(in)f(F)-7 b(ourier)27 b(space)g(and)g(ev)-5 b(aluating)27 b(the)h(functions)g(of)g Fw(')f FG(at)h Fw(')23 b FG(=)g(\()p Fw( )e FG(+)d Fw(\013)p Fx(;)c Fw(\014)3399 567 y Fz(0)3455 555 y FG(+)k Fw(\014)s FG(\),)370 816 y Fx(g)s FG(\()p Fw(!)j FA(\001)d Fw(\027)6 b FG(\))667 749 y Fr(\002)702 816 y Fx("@)785 828 y Fq(')838 816 y Fx(f)j 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Fx(f)9 b FG(\()p Fw(')p FG(\))3073 1024 y Fr(\003)3108 1125 y Fq(\027)725 1367 y FG(=)23 b Fx(g)s FG(\()p Fw(!)e FA(\001)d Fw(\027)6 b FG(\))1139 1263 y FC(1)1112 1288 y Fr(X)1110 1464 y Fu(n)p Fz(=0)1249 1275 y Fr(\020)1298 1367 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(\))1539 1333 y Fz(2)1595 1367 y FA(\000)18 b(M)1778 1333 y Fz([)p FC(\024)p Fu(n)p Fz(])1912 1367 y FG(\()p Fw(!)k FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2229 1275 y Fr(\021)2292 1367 y Fx(g)2335 1333 y Fz([)p Fu(n)p Fz(])2417 1367 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2748 1300 y Fr(\002)2782 1367 y Fx("@)2865 1379 y Fq(')2919 1367 y Fx(f)9 b FG(\()p Fw(')p FG(\))3095 1300 y Fr(\003)3130 1400 y Fq(\027)725 1643 y FG(=)23 b Fx(g)s FG(\()p Fw(!)e FA(\001)d Fw(\027)6 b FG(\))1139 1539 y FC(1)1112 1564 y Fr(X)1110 1740 y Fu(n)p Fz(=0)1249 1550 y Fr(\020)1298 1643 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(\))1539 1608 y Fz(2)1595 1643 y FA(\000)18 b(M)1778 1608 y Fz([)p FC(\024)p 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y Fu(n)p Fz(=1)1259 3365 y Fr(\020)1309 3457 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))1550 3423 y Fz(2)1605 3457 y FA(\000)18 b(M)1788 3423 y Fz([)p FC(\024)p Fu(n)p Fz(])1923 3457 y FG(\()p Fw(!)j FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2240 3365 y Fr(\021)2303 3457 y Fx(g)2346 3423 y Fz([)p Fu(n)p Fz(])2428 3457 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))683 b(\()p Fx(A)p FG(5)p Fx(:)p FG(3\))932 3641 y FC(1)905 3666 y Fr(X)901 3841 y Fu(p)p Fz(=)p Fu(n)1042 3641 y(n)p FC(\000)p Fz(1)1045 3666 y Fr(X)1046 3842 y Fu(q)r Fz(=0)1181 3745 y Fx(M)1271 3711 y Fz([)p Fu(q)r Fz(])1345 3745 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))1703 3641 y FC(1)1676 3666 y Fr(X)1676 3845 y Fu(k)q Fz(=1)1902 3666 y Fr(X)1810 3853 y Fu(\022)r FC(2)p Fz(\002)1940 3833 y Ff(R)1940 3875 y Fs(k)q(;\027)2027 3853 y Fz(\()p Fu(p)p Fz(\))2127 3745 y FG(V)-7 b(al\()p Fx(\022)r FG(\))740 4062 y(+)18 b Fx(g)s FG(\()p Fw(!)j FA(\001)e Fw(\027)6 b FG(\))1150 3958 y FC(1)1123 3983 y Fr(X)1121 4159 y Fu(n)p Fz(=2)1259 3970 y Fr(\020)1309 4062 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))1550 4028 y Fz(2)1605 4062 y FA(\000)18 b(M)1788 4028 y Fz([)p FC(\024)p Fu(n)p Fz(])1923 4062 y FG(\()p Fw(!)j FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2240 3970 y Fr(\021)2303 4062 y Fx(g)2346 4028 y Fz([)p Fu(n)p Fz(])2428 4062 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))901 4250 y Fu(n)p FC(\000)p Fz(1)905 4275 y Fr(X)905 4451 y Fu(p)p Fz(=1)1042 4246 y Fu(p)p FC(\000)p Fz(1)1041 4275 y Fr(X)1043 4451 y Fu(q)r Fz(=0)1175 4354 y Fx(M)1265 4319 y Fz([)p Fu(q)r Fz(])1339 4354 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))1697 4250 y FC(1)1670 4275 y Fr(X)1670 4454 y Fu(k)q Fz(=1)1895 4275 y Fr(X)1804 4462 y Fu(\022)r FC(2)p Fz(\002)1934 4442 y Ff(R)1934 4484 y Fs(k)q(;\027)2021 4462 y Fz(\()p Fu(p)p Fz(\))2121 4354 y FG(V)-7 b(al\()p Fx(\022)r FG(\))p Fx(;)118 4660 y FG(whic)n(h,)28 b(b)n(y)f(the)h(de\014nitions)g(of)f Fy(h)p FG(,)h(can)f(b)r(e)h(written)g(as)184 4916 y Fx(g)s FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))482 4849 y Fr(\002)516 4916 y Fx("@)599 4928 y Fq(')652 4916 y Fx(f)j FG(\()p Fw(')p FG(\))828 4849 y Fr(\003)863 4949 y Fq(\027)933 4916 y FG(=)23 b Fx(g)s FG(\()p Fw(!)e FA(\001)d Fw(\027)6 b FG(\))1304 4824 y Fr(h)1387 4812 y FC(1)1360 4837 y Fr(X)1357 5013 y Fu(n)p Fz(=0)1496 4824 y Fr(\020)1546 4916 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))1787 4882 y Fz(2)1842 4916 y FA(\000)18 b(M)2025 4882 y Fz([)p FC(\024)p Fu(n)p Fz(])2160 4916 y FG(\()p Fw(!)j FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2477 4824 y Fr(\021)2540 4916 y Fy(h)2593 4928 y Fu(n;)p Fq(\027)3428 4916 y FG(\()p Fx(A)p FG(5)p Fx(:)p FG(4\))369 5208 y(+)481 5104 y FC(1)454 5129 y Fr(X)452 5305 y Fu(n)p Fz(=1)591 5208 y FG(\011)656 5220 y Fu(n)700 5208 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))1061 5104 y FC(1)1034 5129 y Fr(X)1031 5303 y Fu(p)p Fz(=)p Fu(n)1171 5104 y(n)p FC(\000)p Fz(1)1174 5129 y Fr(X)1175 5305 y Fu(q)r Fz(=0)1310 5208 y Fx(M)1400 5174 y Fz([)p Fu(q)r Fz(])1474 5208 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))p Fy(h)1844 5220 y Fu(p;)p Fq(\027)1963 5208 y FG(+)2076 5104 y FC(1)2049 5129 y Fr(X)2046 5305 y Fu(n)p Fz(=2)2185 5208 y FG(\011)2250 5220 y Fu(n)2295 5208 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2626 5104 y Fu(n)p FC(\000)p Fz(1)2628 5129 y Fr(X)2629 5305 y Fu(p)p Fz(=1)2765 5100 y Fu(p)p FC(\000)p Fz(1)2765 5129 y Fr(X)2767 5305 y Fu(q)r Fz(=0)2899 5208 y Fx(M)2989 5174 y Fz([)p Fu(q)r Fz(])3063 5208 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))p Fy(h)3433 5220 y Fu(p;)p Fq(\027)3533 5116 y Fr(i)3572 5208 y Fx(:)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 118 356 a FG(38:)27 b Fb(De)l(gener)l(ate)e(el) t(liptic)g(r)l(esonanc)l(es)118 555 y FG(The)j(terms)f(in)h(the)g (second)f(line)h(of)f(\(A5.4\))h(can)f(b)r(e)h(written)g(as)330 726 y FC(1)303 751 y Fr(X)304 927 y Fu(p)p Fz(=1)437 738 y Fr(\020)501 722 y Fu(p)p FC(\000)p Fz(1)501 751 y Fr(X)502 927 y Fu(q)r Fz(=0)721 722 y Fu(p)679 751 y Fr(X)634 927 y Fu(n)p Fz(=)p Fu(q)r Fz(+1)857 830 y Fx(M)947 795 y Fz([)p Fu(q)r Fz(])1021 830 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\011)1403 842 y Fu(n)1447 830 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))k(+)1865 722 y Fu(p)p FC(\000)p Fz(1)1865 751 y Fr(X)1866 927 y Fu(q)r Fz(=0)2070 726 y FC(1)2044 751 y Fr(X)1998 927 y Fu(n)p Fz(=)p Fu(p)p Fz(+1)2223 830 y Fx(M)2313 795 y Fz([)p Fu(q)r Fz(])2386 830 y FG(\()p Fw(!)k FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\011)2768 842 y Fu(n)2813 830 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))3054 738 y Fr(\021)3103 830 y Fy(h)3156 842 y Fu(p;)p Fq(\027)479 1133 y FG(=)593 1029 y FC(1)566 1054 y Fr(X)567 1230 y Fu(p)p Fz(=1)700 1025 y Fu(p)p FC(\000)p Fz(1)700 1054 y Fr(X)702 1230 y Fu(q)r Fz(=0)834 1133 y Fx(M)924 1099 y Fz([)p Fu(q)r Fz(])998 1133 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))1399 1029 y FC(1)1372 1054 y Fr(X)1329 1230 y Fu(n)p Fz(=)p Fu(q)r Fz(+1)1550 1133 y FG(\011)1615 1145 y Fu(n)1660 1133 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))g Fy(h)2044 1145 y Fu(p;)p Fq(\027)3428 984 y FG(\()p Fx(A)p FG(5)p Fx(:)p FG(5\))118 1407 y(and,)28 b(b)n(y)f(c)n(hanging)f Fx(p)d FA(!)g Fx(n)28 b FG(and)f Fx(n)c FA(!)g Fx(s)p FG(,)28 b(w)n(e)f(obtain)524 1569 y FC(1)497 1594 y Fr(X)494 1770 y Fu(n)p Fz(=1)633 1581 y Fr(\020)697 1569 y Fu(n)p FC(\000)p Fz(1)700 1594 y Fr(X)702 1770 y Fu(q)r Fz(=0)837 1673 y Fx(M)927 1639 y Fz([)p Fu(q)r Fz(])1001 1673 y FG(\()p Fw(!)21 b FA(\001)d Fw(\027)6 b FG(;)14 b Fx(")p FG(\))p Fx(\037)1369 1685 y Fz(0)1406 1673 y FG(\(\001)1507 1639 y Fz([0])1582 1673 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))g Fx(:)g(:)g(:)g(\037)2108 1685 y Fu(q)2144 1673 y FG(\(\001)2245 1639 y Fz([)p Fu(q)r Fz(])2320 1673 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))g FA(\001)997 1960 y(\001)1105 1856 y FC(1)1078 1881 y Fr(X)1039 2057 y Fu(s)p Fz(=)p Fu(q)r Fz(+1)1251 1960 y Fx(\037)1303 1972 y Fu(q)r Fz(+1)1424 1960 y FG(\(\001)1525 1925 y Fz([)p Fu(q)r Fz(+1])1684 1960 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))g Fx(:)g(:)g(:)f( )2211 1972 y Fu(s)2247 1960 y FG(\(\001)2348 1925 y Fz([)p Fu(s)p Fz(])2421 1960 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))2770 1868 y Fr(\021)2820 1960 y Fy(h)2873 1972 y Fu(n;)p Fq(\027)670 2259 y FG(=)787 2155 y FC(1)760 2180 y Fr(X)757 2356 y Fu(n)p Fz(=1)896 2155 y Fu(n)p FC(\000)p Fz(1)899 2180 y Fr(X)901 2356 y Fu(q)r Fz(=0)1036 2259 y Fx(M)1126 2225 y Fz([)p Fu(q)r Fz(])1200 2259 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\))p Fx(\037)1569 2271 y Fz(0)1606 2259 y FG(\(\001)1707 2225 y Fz([0])1782 2259 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))g Fx(:)g(:)g(:)f(\037)2307 2271 y Fu(q)2344 2259 y FG(\(\001)2445 2225 y Fz([)p Fu(q)r Fz(])2520 2259 y FG(\()p Fw(!)21 b FA(\001)d Fw(\027)6 b FG(;)14 b Fx(")p FG(\)\))g Fy(h)2935 2271 y Fu(n;)p Fq(\027)3043 2259 y Fx(;)3428 1970 y FG(\()p Fx(A)p FG(5)p Fx(:)p FG(6\))118 2538 y(where)23 b(the)h(iden)n(tit)n(y)f(\(A5.1\))g (has)g(b)r(een)h(used)f(in)h(the)f(last)g(line)h(\(with)g(the)f (correct)f(in)n(terpretation)h(of)g(the)h(term)118 2644 y(with)k Fx(s)23 b FG(=)g Fx(j)g FG(+)18 b(1)27 b(explained)h(after)f (\(A5.1\)\).)37 b(By)27 b(the)h(de\014nition)g(of)g(the)g(matrices)f FA(M)2872 2614 y Fz([)p FC(\024)p Fu(n)p Fz(])3006 2644 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))28 b(one)g(has)755 2811 y Fu(n)p FC(\000)p Fz(1)758 2836 y Fr(X)759 3011 y Fu(q)r Fz(=0)895 2914 y Fx(M)985 2880 y Fz([)p Fu(q)r Fz(])1058 2914 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))p Fx(\037)1427 2926 y Fz(0)1464 2914 y FG(\(\001)1565 2880 y Fz([0])1640 2914 y FG(\()p Fx(x)p FG(;)g Fx(")p FG(\)\))g Fx(:)g(:)g(:)h(\037)2037 2926 y Fu(q)2073 2914 y FG(\(\001)2174 2880 y Fz([)p Fu(q)r Fz(])2249 2914 y FG(\()p Fx(x)p FG(;)f Fx(")p FG(\)\))24 b(=)f FA(M)2680 2880 y Fz([)p FC(\024)p Fu(n)p Fz(])2814 2914 y FG(\()p Fx(x)p FG(;)14 b Fx(")p FG(\))p Fx(;)404 b FG(\()p Fx(A)p FG(5)p Fx(:)p FG(7\))118 3193 y(so)27 b(that,)h(b)n(y)f(inserting)g(\(A5.6\))h(in)g(\(A5.3\),)f (after)h(ha)n(ving)e(used)i(\(A5.7\),)f(w)n(e)h(obtain)286 3451 y Fx(g)s FG(\()p Fw(!)21 b FA(\001)d Fw(\027)6 b FG(\))583 3384 y Fr(\002)618 3451 y Fx("@)701 3463 y Fq(')754 3451 y Fx(f)j FG(\()p Fw(')p FG(\))930 3384 y Fr(\003)965 3484 y Fq(\027)1035 3451 y FG(=)23 b Fx(g)s FG(\()p Fw(!)e FA(\001)d Fw(\027)6 b FG(\))1449 3347 y FC(1)1423 3372 y Fr(X)1420 3548 y Fu(n)p Fz(=0)1559 3359 y Fr(h)14 b(\020)1661 3451 y FG(\()p Fw(!)22 b FA(\001)c Fw(\027)6 b FG(\))1902 3417 y Fz(2)1958 3451 y FA(\000)18 b(M)2141 3417 y Fz([)p FC(\024)p Fu(n)p Fz(])2275 3451 y FG(\()p Fw(!)k FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))2592 3359 y Fr(\021)2660 3451 y FG(+)k FA(M)2843 3417 y Fz([)p FC(\024)p Fu(n)p Fz(])2977 3451 y FG(\()p Fw(!)k FA(\001)c Fw(\027)6 b FG(;)14 b Fx(")p FG(\))3294 3359 y Fr(i)3333 3451 y Fy(h)3386 3463 y Fu(n;)p Fq(\027)1035 3727 y FG(=)23 b Fx(g)s FG(\()p Fw(!)e FA(\001)d Fw(\027)6 b FG(\))1449 3623 y FC(1)1423 3648 y Fr(X)1420 3824 y Fu(n)p Fz(=0)1545 3727 y FG(\()p Fw(!)21 b FA(\001)e Fw(\027)6 b FG(\))1786 3693 y Fz(2)1823 3727 y Fy(h)1876 3739 y Fu(n;)p Fq(\027)2007 3727 y FG(=)2124 3623 y FC(1)2097 3648 y Fr(X)2095 3824 y Fu(n)p Fz(=0)2234 3727 y Fy(h)2287 3739 y Fu(n;)p Fq(\027)2417 3727 y FG(=)23 b Fy(h)2558 3739 y Fq(\027)2605 3727 y Fx(;)800 b FG(\()p Fx(A)p FG(5)p Fx(:)p FG(8\))118 3990 y(so)27 b(that)h(the)g(assertion)e(is)h (pro)n(v)n(ed.)118 4167 y FF(R)l(emark.)52 b FG(Note)33 b(that)g(at)f(eac)n(h)g(step)g(only)g(absolutely)g(con)n(v)n(erging)e (series)h(ha)n(v)n(e)h(b)r(een)h(dealt)f(with,)i(so)e(that)118 4273 y(the)c(ab)r(o)n(v)n(e)e(analysis)h(is)g(rigorous)e(and)j(not)f (only)g(formal.)118 4521 y Fy(Ac)m(kno)m(wledgmen)m(ts.)35 b FG(W)-7 b(e)25 b(are)e(indebted)i(to)f(V.)h(Mastropietro)d(for)i(man) n(y)f(discussions)h(and,)g(in)h(particular,)118 4628 y(to)j(A.)g(Giuliani)f(for)g(critical)g(reading)g(and)g(sev)n(eral)f (suggestions.)1628 5005 y FD(References)118 5150 y Fk([B1])111 b(J.)32 b(Bourgain,)h Fb(Construction)g(of)h(quasi-p)l(erio)l(dic)g (solutions)g(for)f(Hamiltonian)i(p)l(erturb)l(ations)f(of)f(line)l(ar)h (e)l(quations)g(and)354 5225 y(applic)l(ations)28 b(to)e(nonline)l(ar)h (PDE)p Fk(,)c(In)n(ternatational)j(Mathematics)f(Researc)n(h)f(Notices) h Fa(11)e Fk(\(1994\))i(475{497.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 118 356 a FG(39:)27 b Fb(De)l(gener)l(ate)e(el) t(liptic)g(r)l(esonanc)l(es)118 555 y Fk([B2])111 b(J.)29 b(Bourgain,)i Fb(Construction)g(of)g(p)l(erio)l(dic)h(solutions)g(of)f (nonline)l(ar)h(wave)f(e)l(quations)h(in)e(higher)h(dimension)p Fk(,)g(Geometric)354 630 y(and)25 b(F)-6 b(unctional)24 b(Analysis)f Fa(5)h Fk(\(1995\),)h(629{639.)118 788 y([B3])111 b(J.)24 b(Bourgain,)f Fb(On)i(Melnikov's)h(p)l(ersistency)f(pr)l(oblem) p Fk(,)g(Mathematical)f(Researc)n(h)h(Letters)f Fa(4)g Fk(\(1997\),)h(445{458.)118 946 y([B4])111 b(J.)22 b(Bourgain,)g Fb(Quasi-p)l(erio)l(dic)k(solutions)f(of)f(Hamiltonian)i(p)l(erturb)l (ations)f(of)g(2D)f(line)l(ar)h(Schr\177)-36 b(odinger)25 b(e)l(quations)p Fk(,)e(Annals)354 1020 y(of)h(Mathematics)g Fa(148)f Fk(\(1998\),)i(no.)32 b(2,)23 b(363{439.)118 1178 y([Ba])111 b(J.)26 b(C.)e(A.)h(Barata,)h Fb(On)h(formal)i(quasi-p) l(erio)l(dic)f(solutions)h(of)e(the)h(Schr\177)-36 b(odinger)28 b(e)l(quation)f(for)h(a)g(two-level)f(system)g(with)g(a)354 1253 y(Hamiltonian)g(dep)l(ending)g(quasi-p)l(erio)l(dic)l(al)t(ly)g (on)f(time)p Fk(,)d(Reviews)h(in)f(Mathematical)i(Ph)n(ysics)f Fa(12)e Fk(\(2000\),)k(no.)31 b(1,)23 b(25{64.)118 1411 y([BaG])56 b(M.V.)20 b(Bartuccelli,)i(G.)f(Gen)n(tile,)h Fb(Lindste)l(dt)i(series)f(for)h(p)l(erturb)l(ations)h(of)e(iso)l(chr)l (onous)k(systems.)k(A)24 b(r)l(eview)f(of)h(the)f(gener)l(al)354 1485 y(the)l(ory)p Fk(,)h(Reviews)g(in)f(Mathematical)i(Ph)n(ysics)e Fa(14)g Fk(\(2002\),)j(no.)31 b(2,)23 b(121{171.)118 1643 y([BGGM])42 b(F.)31 b(Bonetto,)k(G.)c(Galla)n(v)n(otti,)j(G.)d (Gen)n(tile,)j(Mastropietro,)f Fb(Lindste)l(dt)f(series,)j(ultr)l (aviolet)e(diver)l(genc)l(es)g(and)g(Moser's)354 1718 y(the)l(or)l(em)p Fk(,)25 b(Annali)e(della)h(Scuola)g(Normale)f(Sup)r (eriore)h(di)f(Pisa)h(Classe)f(di)g(Scienze)i Fa(26)e Fk(\(1998\),)i(no.)32 b(3,)23 b(545{593.)118 1876 y([C])145 b(C.-Q.)26 b(Cheng,)i Fb(L)l(ower-dimensional)j(invariant)e(tori)f(in)h (the)g(r)l(e)l(gions)g(of)g(instability)f(for)h(ne)l(arly)g(inte)l(gr)l (able)g(Hamiltonian)354 1950 y(systems)p Fk(,)23 b(Comm)n(unications)i (in)e(Mathematical)i(Ph)n(ysics)f Fa(203)e Fk(\(1999\),)j(no.)32 b(2,)23 b(385{419.)118 2108 y([ChW])34 b(C.-Q.)21 b(Cheng,)h(S.)g(W)-6 b(ang,)23 b Fb(The)h(surviving)g(of)g(lower-dimensional)i(tori)e(fr)l (om)h(a)f(r)l(esonant)h(torus)g(of)f(Hamiltonian)h(systems)p Fk(,)354 2183 y(Journal)f(of)g(Di\013eren)n(tial)f(Equations)i Fa(155)d Fk(\(1999\),)j(no.)32 b(2,)23 b(311{326.)118 2340 y([CrW])45 b(W.)25 b(Craig)g(and)g(C.E.)f(W)-6 b(a)n(yne,)26 b Fb(Newton)-7 b('s)27 b(metho)l(d)h(and)g(p)l(erio)l(dic)g(solutions)g (of)f(nonline)l(ar)h(wave)f(e)l(quations)p Fk(,)e(Comm)n(unica-)354 2415 y(tions)f(in)g(Pure)f(and)h(Applied)g(Mathematics)h Fa(46)e Fk(\(1993\),)i(no.)31 b(11,)24 b(1409{1501.)118 2573 y([E1])113 b(L.H.)31 b(Eliasson,)i Fb(Perturb)l(ations)i(of)e (stable)g(invariant)h(tori)f(for)h(Hamiltonian)g(systems)p Fk(,)g(Annali)d(della)h(Scuola)h(Normale)354 2648 y(Sup)r(eriore)24 b(di)f(Pisa)h(Classe)f(di)g(Scienze)i Fa(15)e Fk(\(1988\),)i(no.)32 b(1,)23 b(115{147.)118 2805 y([E2])113 b(L.H.)24 b(Eliasson,)g Fb(A)n(bsolutely)i(c)l(onver)l(gent)g(series)g(exp)l(ansions)i(for)f (quasi-p)l(erio)l(dic)h(motions)p Fk(,)d(Mathematical)g(Ph)n(ysics)g (Elec-)354 2880 y(tronic)f(Journal)g Fa(2)f Fk(\(1996\),)i(pap)r(er)f (4,)g(1{33)h(\(Preprin)n(t)e(1988\).)118 3038 y([F)-6 b(e])125 b(C.)23 b(F)-6 b(e\013erman,)24 b Fb(Pointwise)i(c)l(onver)l (genc)l(e)g(of)g(F)-5 b(ourier)26 b(series)p Fk(,)d(Annals)h(of)f (Mathematics)i Fa(98)p Fk(,)d(551{571,)j(1973.)118 3192 y([G3])106 b(G.)27 b(Galla)n(v)n(otti,)h Fb(Twistless)g(KAM)g(tori,)h (quasi)g(\015at)f(homo)l(clinic)i(interse)l(ctions,)f(and)g(other)f(c)l (anc)l(el)t(lations)i(in)e(the)g(p)l(ertur-)354 3267 y(b)l(ation)d(series)f(of)h(c)l(ertain)f(c)l(ompletely)h(inte)l(gr)l (able)g(hamiltonian)h(systems.)32 b(A)24 b(r)l(eview)p Fk(,)e(Reviews)h(in)f(Mathematical)h(Ph)n(ysics)354 3341 y Fa(6)p Fk(,)g(343{411,)i(1994.)118 3495 y([Ga])106 b(G.)24 b(Galla)n(v)n(otti,)g Fb(Twistless)i(KAM)f(tori)p Fk(,)e(Comm)n(unications)i(in)e(Mathematical)i(Ph)n(ysics)e Fa(164)g Fk(\(1994\),)i(no.)32 b(1,)23 b(145{156.)118 3653 y([Ga01])36 b(G.)19 b(Galla)n(v)n(otti:)30 b Fb(R)l (enormalization)23 b(gr)l(oup)f(in)f(Statistic)l(al)h(Me)l(chanics)f (and)h(Me)l(chanics:)31 b(gauge)21 b(symmetries)g(and)h(vanishing)354 3728 y(b)l(eta)k(functions)p Fk(,)d(Ph)n(ysics)h(Rep)r(orts)g Fa(352)p Fk(,)e(251{272,)j(2001.)118 3882 y([Ga02])36 b(G.)f(Galla)n(v)n(otti:)55 b Fb(Exact)37 b(R)l(enormalization)h(Gr)l (oup)p Fk(,)h(P)n(aris)34 b(IHP)-6 b(,)35 b(12)h(o)r(ctob)r(er)g(2002,) j(Seminaire)c(P)n(oincar)n(\023)-33 b(e,)38 b(Editors)d(B.)354 3957 y(Duplan)n(tier,)24 b(V.)f(Riv)l(asseau,)h(Institut)h(H.)e(P)n (oincar)n(\023)-33 b(e-CNRS-CEA.)118 4111 y([GG])86 b(G.)19 b(Galla)n(v)n(otti,)h(G.)e(Gen)n(tile,)i Fb(Hyp)l(erb)l(olic)h (low-dimensional)i(tori)e(and)g(summations)i(of)e(diver)l(gent)f (series)p Fk(,)f(Comm)n(unications)354 4185 y(in)24 b(Mathematical)g (Ph)n(ysics)g Fa(227)f Fk(\(2002\),)i(no.)31 b(3,)24 b(421{460.)118 4343 y([GBG])36 b(G.)29 b(Galla)n(v)n(otti,)i(F.)e (Bonetto,)j(G.)d(Gen)n(tile,)h Fb(Asp)l(e)l(cts)i(of)e(the)h(er)l(go)l (dic,)h(qualitative)f(and)g(statistic)l(al)g(pr)l(op)l(erties)h(of)f (motion)354 4418 y Fk(Springer{V)-6 b(erlag,)24 b(Berlin,)e(2004.)118 4572 y([Ge])110 b(G.)17 b(Gen)n(tile,)h Fb(Quasi-p)l(erio)l(dic)i (solutions)g(for)f(two-level)g(systems)p Fk(,)e(Comm)n(unications)g(in) f(Mathematical)i(Ph)n(ysics)f Fa(242)e Fk(\(2003\),)354 4647 y(no.)32 b(1{2,)24 b(221{250.)118 4801 y([GM1])41 b(G.)18 b(Gen)n(tile,)i(V.)e(Mastropietro,)h Fb(Metho)l(ds)i(for)g(the) f(analysis)i(of)f(the)f(Lindste)l(dt)h(series)f(for)h(KAM)e(tori)i(and) g(r)l(enormalizability)354 4875 y(in)31 b(classic)l(al)h(me)l(chanics.) 49 b(A)30 b(r)l(eview)h(with)g(some)g(applic)l(ations)p Fk(,)h(Reviews)e(in)f(Mathematical)h(Ph)n(ysics)f Fa(8)g Fk(\(1996\),)j(no.)48 b(3,)354 4950 y(393{444.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 118 356 a FG(40:)27 b FF(De)l(gener)l(ate)i(el) t(liptic)j(r)l(esonanc)l(es)118 555 y Fk([GM2])41 b(G.)34 b(Gen)n(tile,)j(V.)c(Mastropietro,)j Fb(Construction)f(of)g(p)l(erio)l (dic)h(solutions)h(of)e(nonline)l(ar)h(wave)f(e)l(quations)h(with)f (Dirichlet)354 630 y(b)l(oundary)g(c)l(onditions)g(by)e(the)h(Lindste)l (dt)g(series)f(metho)l(d)p Fk(,)j(to)d(app)r(ear)g(on)f(Journal)h(de)g (Math)n(\023)-33 b(ematiques)34 b(Pures)e(et)h(Ap-)354 705 y(pliqu)n(\023)-33 b(ees.)118 859 y([GMP])28 b(G.)e(Gen)n(tile,)i (V.)d(Mastropietro,)i(M.)e(Pro)r(cesi,)i Fb(Perio)l(dic)i(solutions)g (for)f(c)l(ompletely)h(r)l(esonant)g(wave)f(e)l(quations)p Fk(,)f(Preprin)n(t,)354 934 y(2004.)118 1075 y([JLZ])73 b(A.)20 b(Jorba,)g(R.)g(de)g(la)g(Lla)n(v)n(e,)g(M.)f(Zou,)i Fb(Lindste)l(dt)h(series)g(for)h(lower-dimensional)h(tori)p Fk(,)c(in)f Fb(Hamiltonian)24 b(systems)e(with)g(thr)l(e)l(e)354 1150 y(or)j(mor)l(e)f(de)l(gr)l(e)l(es)h(of)f(fr)l(e)l(e)l(dom)g Fk(\(S'Agar\023)-35 b(o,)22 b(1995\),)h(151{167,)h(NA)-6 b(TO)21 b(Adv.)31 b(Sci.)g(Inst.)g(Ser.)f(C)22 b(Math.)31 b(Ph)n(ys.)g(Sci.,)21 b(533,)i(Ed.)354 1224 y(C.)g(Sim\023)-35 b(o,)23 b(Klu)n(w)n(er)h(Acad.)31 b(Publ.,)23 b(Dordrec)n(h)n(t,)h (1999.)118 1378 y([Ka])106 b(T.)21 b(Kato,)h Fb(Perturb)l(ation)i(the)l (ory)f(for)h(line)l(ar)g(op)l(er)l(ators)p Fk(,)f(Grundlehren)f(der)f (Mathematisc)n(hen)i(Wissensc)n(haften,)f(Band)g(132,)354 1453 y(Springer-V)-6 b(erlag,)23 b(Berlin-New)f(Y)-6 b(ork,)23 b(1976.)118 1607 y([Ku])102 b(S.B.)24 b(Kuksin,)h Fb(Ne)l(arly)h(inte)l(gr)l(able)h(in\014nite-dimensional)g(Hamiltonian) h(systems)p Fk(,)c(Lecture)i(Notes)g(in)e(Mathematics)i(1556,)354 1682 y(Springer-V)-6 b(erlag,)23 b(Berlin,)f(1993.)118 1836 y([KP])93 b(S.B.)24 b(Kuksin,)f(J.)h(P\177)-35 b(osc)n(hel,)24 b Fb(Invariant)j(Cantor)g(manifolds)g(of)g(quasi-p)l(erio)l(dic)g (oscil)t(lations)g(for)f(a)g(nonline)l(ar)i(Schr\177)-36 b(odinger)354 1911 y(e)l(quation)p Fk(,)24 b(Annals)g(of)f(Mathematics) i Fa(143)d Fk(\(1996\),)k(no.)31 b(1,)23 b(149{179.)118 2068 y([L)-8 b(W])88 b(R.)31 b(de)h(la)g(Lla)n(v)n(e,)i(C.E.)c(W)-6 b(a)n(yne,)35 b Fb(Whisker)l(e)l(d)f(and)g(low)g(dimensional)h(tori)e (in)g(ne)l(arli)h(inte)l(gr)l(able)f(Hamiltonian)h(systems)p Fk(,)354 2143 y(Mathematical)25 b(Ph)n(ysics)f(Electronic)g(Journal,)f (2004.)118 2297 y([Me1])65 b(V.K.)28 b(Mel'nik)n(o)n(v,)i Fb(On)g(c)l(ertain)h(c)l(ases)g(of)g(c)l(onservation)g(of)g(c)l (onditional)t(ly)h(p)l(erio)l(dic)f(motions)h(under)f(a)g(smal)t(l)g (change)g(of)354 2372 y(the)c(Hamiltonian)g(function)p Fk(,)d(Doklady)i(Ak)l(ademii)e(Nauk)h(SSSR)g Fa(165)e Fk(\(1965\),)k(1245{1248;)g(english)d(translation)h(in)f(So)n(viet)354 2447 y(Mathematics)h(Doklady)f Fa(6)g Fk(\(1965\),)h(1592{1596.)118 2604 y([Me2])65 b(V.K.)20 b(Mel'nik)n(o)n(v,)i Fb(A)h(c)l(ertain)g (family)h(of)g(c)l(onditional)t(ly)g(p)l(erio)l(dic)h(solutions)g(of)e (a)h(Hamiltonian)h(systems)p Fk(,)c(Doklady)h(Ak)l(ade-)354 2679 y(mii)h(Nauk)h(SSSR)g Fa(181)f Fk(\(1968\),)i(546{549;)g(english)e (translation)i(in)e(So)n(viet)i(Mathematics)g(Doklady)f Fa(9)f Fk(\(1968\),)i(882-886.)118 2837 y([Mo])96 b(J.)26 b(Moser,)g Fb(Conver)l(gent)i(series)g(exp)l(ansions)i(for)e(quasi-p)l (erio)l(dic)h(motions)p Fk(,)f(Mathematisc)n(he)g(Annalen)f Fa(169)e Fk(\(1967\),)j(136{)354 2911 y(176.)118 3053 y([P1])113 b(J.)26 b(P\177)-35 b(osc)n(hel,)26 b Fb(On)h(el)t(liptic)g (lower-dimensional)i(tori)e(in)h(Hamiltonian)g(systems)p Fk(,)e(Mathematisc)n(he)h(Zeitsc)n(hrift)e Fa(202)g Fk(\(1989\),)354 3127 y(no.)32 b(4,)23 b(559{608.)118 3281 y([P2])113 b(J.)33 b(P\177)-35 b(osc)n(hel,)36 b Fb(Quasi-p)l(erio)l(dic)g (solutions)g(for)f(a)g(nonline)l(ar)h(wave)f(e)l(quation)p Fk(,)h(Commen)n(tarii)d(Mathematici)h(Helv)n(etici)g Fa(71)354 3356 y Fk(\(1996\),)25 b(no.)32 b(2,)23 b(269{296.)118 3514 y([RS])105 b(M.)21 b(Reed,)h(B.)f(Simon,)g Fb(Metho)l(ds)k(of)e (mo)l(dern)i(mathematic)l(al)g(physics.)33 b(IV.)23 b(A)n(nalysis)h(of) g(op)l(er)l(ators)p Fk(,)f(Academic)g(Press,)d(New)354 3589 y(Y)-6 b(ork-London,)24 b(1978.)118 3743 y([R])144 b(H.)29 b(R)r(\177)-37 b(ussmann,)31 b Fb(Invariant)h(tori)f(in)g (non-de)l(gener)l(ate)g(ne)l(arly)h(inte)l(gr)l(able)f(Hamiltonian)h (systems)p Fk(,)f(Regular)e(and)h(Chaotic)354 3817 y(Dynamics)24 b Fa(6)f Fk(\(2001\),)i(119{204.)118 3975 y([T])145 b(D.V.)37 b(T)-6 b(reshc)n(h)n(\177)-33 b(ev,)41 b Fb(A)e(me)l(chanism)h(for)f (the)f(destruction)h(of)g(r)l(esonanc)l(e)g(tori)g(in)f(Hamiltonian)i (systems)p Fk(,)g(Rossi)-8 b(\025)-27 b(\020sk)l(a)n(y)n(a)354 4050 y(Ak)l(ademiy)n(a)26 b(Nauk.)35 b(Matematic)n(heski)-8 b(\025)-27 b(\020Sb)r(ornik)27 b Fa(180)c Fk(\(1989\),)k(no.)35 b(10,)26 b(1325{1346;)h(english)e(translation)h(in)e(Mathematics)354 4124 y(of)g(the)g(USSR-Sb)r(ornik)g Fa(68)f Fk(\(1991\),)i(no.)31 b(1,)24 b(181{203.)118 4282 y([W)n(C])75 b(S.)26 b(W)-6 b(ang,)27 b(C.-Q.)d(Cheng,)j Fb(L)l(ower-dimensional)i(tori)f(for)g (generic)f(Hamiltonian)h(systems)p Fk(,)e(Chinese)g(Science)i(Bulletin) d Fa(44)354 4357 y Fk(\(1999\),)g(no.)32 b(13,)24 b(1187{1191.)118 4515 y([W)-6 b(a])95 b(C.E.)24 b(W)-6 b(a)n(yne,)26 b Fb(Perio)l(dic)i(and)g(quasi-p)l(erio)l(dic)g(solutions)f(of)g(nonline) l(ar)h(wave)f(e)l(quations)h(via)e(KAM)g(the)l(ory)p Fk(,)f(Comm)n(unica-)354 4589 y(tions)f(in)g(Mathematical)g(Ph)n(ysics) g Fa(127)f Fk(\(1990\),)i(no.)31 b(3,)24 b(479{528.)118 4747 y([XY])90 b(J.)17 b(Xu,)i(J.)e(Y)-6 b(ou:)28 b Fb(Persistenc)l(e) 20 b(of)g(lower)g(dimensional)i(tori)e(under)g(the)g(\014rst)f (Melnikov's)h(non-r)l(esonanc)l(e)h(c)l(ondition)p Fk(,)e(Journal)354 4822 y(de)24 b(Math)n(\023)-33 b(ematiques)26 b(Pures)d(et)h(Appliqu)n (\023)-33 b(ees)24 b Fa(80)f Fk(\(2001\),)i(no.)32 b(10,)24 b(1045{1067.)118 4980 y([Y])143 b(J.)20 b(Y)-6 b(ou:)29 b Fb(Perturb)l(ations)23 b(of)g(lower-dimensional)h(tori)e(for)g (Hamiltonian)h(systems)p Fk(,)d(Journal)g(of)g(Di\013eren)n(tial)g (Equations)g Fa(152)354 5054 y Fk(\(1999\),)25 b(no.)32 b(1,)23 b(1{29.)118 5491 y Ft(21)p Fs(=mag)q(g)q(io=)p Ft(2004;)30 b(19:31)1130 b FG(40)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0405211228482--