%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% 93 K, Plain Tex, 17 pages, 2 figures (automatically generated) for a %% postscript printer driven by dvips: %% see instructions (in the first few lines below) %% for other solutions. The figures are generated %% with the names fig1.ps and fig2.ps. BODY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %TO PRINT THE POSTCRIPT FIGURES THE DRIVER NUMBER MIGHT HAVE TO BE %ADJUSTED. IF the 4 choices 0,1,2,3 do not work set in the following line %the \driver variable to =5. Setting it =0 works with dvilaser setting it %=1 works with dvips, =2 with psprint, =3 with dvitps, (hopefully). %Using =5 prints incomplete figures (but still understandable from the %text). The value MUST be set =5 if the printer is not a postscript one. \newcount\driver \driver=1 %%%this is the value to set!!! %%% the values =0,1 have been tested. The figures are automatically %%% generated. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FORMATO \newcount\mgnf\newcount\tipi\newcount\tipoformule \mgnf=0 %ingrandimento \tipi=2 %uso caratteri: 2=cmcompleti, 1=cmparziali, 0=amparziali \tipoformule=1 %=0 da numeroparagrafo.numeroformula; se no numero %assoluto %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCIPIT\ \ifnum\mgnf=0 \magnification=\magstep0\hoffset=-0.5cm \voffset=-0.5truecm\hsize=15truecm\vsize=24.truecm \parindent=12.pt\fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.truecm \voffset=-0.5truecm\hsize=16.5truecm\vsize=24.truecm \baselineskip=14pt plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\overfullrule=10pt % %%%%%GRECO%%%%%%%%% % \let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi \let\ps=\psi \let\o=\omega \let\y=\upsilon \let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\U=\Upsilon %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% Numerazione pagine %%%%%%%%%%%%%%%%%%%%% NUMERAZIONE PAGINE {\count255=\time\divide\count255 by 60 \xdef\oramin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\oramin{\oramin:\ifnum\count255<10 0\fi\the\count255}} \def\ora{\oramin } \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\ \ora} \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglioa\hss} % %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3} \else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} 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dilatare la figura in accordo al valore di % \magnification, correggendo i parametri P1 e P2 nell'istruzione % \special{#4.ps P1 P2 scale} % \def\insertplot#1#2#3#4{ \par \xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \vbox{ \line{} \hbox{ \hskip\xshift \vbox to #2{\vfil \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate} \special{ps: plotfile #4.ps} \special{ps::[end]grestore} \fi \ifnum\driver=1 #3 \special{psfile=#4.ps} \fi \ifnum\driver=2 #3 \ifnum\mgnf=0 \special{#4.ps 1. 1. scale}\fi \ifnum\mgnf=1 \special{#4.ps 1.2 1.2 scale}\fi\fi \ifnum\driver=3 \ifnum\mgnf=0 \psfig{figure=#4.ps,height=#2,width=#1,scale=1.} \kern-\baselineskip #3\fi \ifnum\mgnf=1 \psfig{figure=#4.ps,height=#2,width=#1,scale=1.2} \kern-\baselineskip #3\fi \ifnum\driver=5 #3 \fi \fi} \hfil}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newskip\ttglue %%cm semplificato \def\TIPI{ \font\ottorm=cmr8 \font\ottoi=cmmi8 \font\ottosy=cmsy8 \font\ottobf=cmbx8 \font\ottott=cmtt8 %\font\ottosl=cmsl8 \font\ottoit=cmti8 %%%%% cambiamento di formato%%%%%% \def \ottopunti{\def\rm{\fam0\ottorm}% passaggio a tipi da 8-punti \textfont0=\ottorm \textfont1=\ottoi \textfont2=\ottosy \textfont3=\ottoit \textfont4=\ottott \textfont\itfam=\ottoit \def\it{\fam\itfam\ottoit}% \textfont\ttfam=\ottott \def\tt{\fam\ttfam\ottott}% \textfont\bffam=\ottobf \normalbaselineskip=9pt\normalbaselines\rm} \let\nota=\ottopunti} %%%%%%%% %%am \def\TIPIO{ \font\setterm=amr7 %\font\settei=ammi7 \font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7 %%%%% cambiamenti di formato %%% \def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti \textfont0=\setterm %\textfont1=\settei \textfont2=\settesy %\textfont3=\setteit %\textfont\itfam=\setteit \def\it{\fam\itfam\setteit} \textfont\bffam=\settebf \def\bf{\fam\bffam\settebf} \normalbaselineskip=9pt\normalbaselines\rm }\let\nota=\settepunti} %%%%%%% %%cm completo \def\TIPITOT{ \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 scaled\magstep1 \font\twelveex=cmex10 scaled\magstep1 \font\twelveit=cmti12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 \font\twelvesl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\sixrm=cmr6 \font\sixbf=cmbx6 \font\sixi=cmmi6 \font\sixsy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\twelvetruecmr=cmr10 scaled\magstep1 \font\twelvetruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\sixtruecmr=cmr6 \font\sixtruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% cambio grandezza %%%%%% \def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\sixtruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\sixtruecmsy }\let\nota=\eightpoint} \newfam\msbfam %per uso in \TIPITOT \newfam\truecmr %per uso in \TIPITOT \newfam\truecmsy %per uso in \TIPITOT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%Scelta dei caratteri %\newcount\tipi \tipi=0 %e' definito all'inizio \newskip\ttglue \ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi \def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI VARIE % \def\V#1{\vec#1} \def\T#1{#1\kern-4pt\lower9pt\hbox{$\widetilde{}$}\kern4pt{}} \let\dpr=\partial\let\io=\infty\let\ig=\int \def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle} \let\0=\noindent \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hglue3.pt${\scriptstyle #1}$\hglue3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \let\implica=\Rightarrow\def\tto{{\Rightarrow}} \def\pagina{\vfill\eject}\def\acapo{\hfill\break} \def\qed{\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} \let\ciao=\bye %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%LATINORUM \def\etc{\hbox{\sl etc}}\def\eg{\hbox{\sl e.g.\ }} \def\ap{\hbox{\sl a priori\ }}\def\aps{\hbox{\sl a posteriori\ }} \def\ie{\hbox{\sl i.e.\ }} \def\fiat{{}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%DEFINIZIONI LOCALI \def\AA{{\V A}}\def\aa{{\V\a}}\def\nn{{\V\n}}\def\oo{{\V\o}} \def\BB{{\V B}}\def\bb{{\V\b}}\def\gg{{\V g}}\def\mm{{\V\m}} \def\mm{{\V m}}\def\nn{{\V\n}}\def\lis#1{{\overline #1}} \def\NN{{\cal N}}\def\FF{{\cal F}}\def\VV{{\cal V}}\def\EE{{\cal E}} \def\CC{{\cal C}}\def\RR{{\cal R}}\def\LL{{\cal L}}\def\UU{{\cal U}} \def\TT{{\cal T}} \def\={{ \; \equiv \; }}\def\su{{\uparrow}}\def\giu{{\downarrow}} \def\Dpr{{\V \dpr}\,} \def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}} \def\sign{{\rm sign\,}} \def\atan{{\,\rm arctg\,}} \def\pps{{\V\ps{\,}}} \let\dt=\displaystyle \def\2{{1\over2}} \def\txt{\textstyle}\def\OO{{\cal O}} \def\tst{\textstyle} \def\st{\scriptscriptstyle} \let\\=\noindent \def\*{\vskip0.3truecm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% FIGURA FIG2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\%=12\catcode`\}=12\catcode`\{=12 \catcode`\<=1\catcode`\>=2 \openout13=fig1.ps \write13 \write13 \write13<0 90 punto > \write13<70 90 punto > \write13<120 60 punto > \write13<160 130 punto > \write13<200 110 punto > \write13<240 170 punto > \write13<240 130 punto > \write13<240 90 punto > \write13<240 0 punto > \write13<240 30 punto > \write13<210 70 punto > \write13<240 70 punto > \write13<240 50 punto > \write13<0 90 moveto 70 90 lineto> \write13<70 90 moveto 120 60 lineto> \write13<70 90 moveto 160 130 lineto> \write13<160 130 moveto 200 110 lineto> \write13<160 130 moveto 240 170 lineto> \write13<200 110 moveto 240 130 lineto> \write13<200 110 moveto 240 90 lineto> \write13<120 60 moveto 240 0 lineto> \write13<120 60 moveto 240 30 lineto> \write13<120 60 moveto 210 70 lineto> \write13<210 70 moveto 240 70 lineto> \write13<210 70 moveto 240 50 lineto> \write13 \write13 \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% FIGURA FIG2.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\%=12\catcode`\}=12\catcode`\{=12 \catcode`\<=1\catcode`\>=2 \openout13=fig2.ps \write13<%%BoundingBox: 0 0 240 170> \write13<% fig.pst> \write13 \write13<2 0 360 newpath arc fill stroke grestore} def> \write13 \write13<4 0 360 newpath arc fill stroke grestore} def> \write13<0 90 punto > \write13<70 90 punto > \write13<120 60 punto > \write13<160 130 punto > \write13<200 110 punto > \write13<200 150 tondo > \write13<240 130 punto > \write13<240 84 tondo > \write13<200 10 punto > \write13<180 60 tondo > \write13<200 90 punto > \write13<0 90 moveto 70 90 lineto> \write13<70 90 moveto 120 60 lineto> \write13<70 90 moveto 160 130 lineto> \write13<160 130 moveto 200 110 lineto> \write13<160 130 moveto 197 148 lineto> \write13<200 110 moveto 240 130 lineto> \write13<200 110 moveto 236 86 lineto> \write13<120 60 moveto 200 10 lineto> \write13<120 60 moveto 176 60 lineto> \write13<120 60 moveto 200 90 lineto> \write13 \write13 \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\%=12\catcode`\}=12\catcode`\{=12 \catcode`\<=1\catcode`\>=2 \openout13=gvnn2.ps \write13 \write13<2 0 360 newpath arc fill stroke grestore} def> \write13 \write13<3 0 360 newpath arc fill stroke grestore} def> \write13 \write13 \write13 \write13<4 2 roll moveto lineto stroke grestore} def> \write13 \write13<-20 80 20 80 linea 20 80 puntone> \write13<140 80 180 80 linea 180 80 punto> \write13<180 80 220 100 linea 220 100 puntone> \write13<180 80 220 90 tlinea 220 90 puntone> \write13<180 80 220 60 linea 220 60 puntone> \write13<180 80 220 120 linea 220 120 puntone> \write13 \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\input cfiat %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} \vglue0.truecm \BOZZA \font\titolo=cmbx12 \null \vskip2.truecm \0{\titolo Tree expansion and multiscale analysis for KAM tori} %\footnote{${}^*$}{\nota Archived in %{\tt mp\_arc@math.utexas.edu} \#94-??; to get a TeX version, send an empty %E-mail message.} \vskip1.truecm \0{\bf G.Gentile}\footnote{${}^1$}{\nota E-mail: {\tt gentileg\%39943.hepnet@lbl.gov}; address: Dipartimento di Fisica, Universit\`a di Roma ``La Sa\-pi\-en\-za", P. Moro 2, 00185 Roma, Italia.}, {\bf V.Mastropietro}\footnote{${}^2$}{\nota Address: Dipartimento di Matematica, Universit\`a di Roma II ``Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italia.} \vskip0.5truecm \0{\bf Abstract:} {\it We prove that the perturbative expansion for the KAM invariant tori of the Thirring model (with interaction depending also on the action variables) is convergent by using techniques usual in quantum field theory like the multiscale decomposition and the tree expansion. The proof follows the ideas of Eliasson, [3], and extends the results found in the case of an action-independent interaction potential, [4].} \vskip1.truecm \0{\sl Keywords}: {\it KAM theorem, dynamical systems, re\-nor\-mal\-iza\-tion group, quan\-tum field the\-ory, mul\-ti\-scale de\-composi\-tion, form factors} \vskip2.5truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\{\bf 1. Introduction} \*\numsec=1\numfor=1\pgn=1 \\The KAM theorem proves in a indirect way that the formal perturbation series for the invariant tori of quasi integrable systems are convergent if some conditions are fulfilled by the hamiltonian, [11], [1], [12]. Of course one should be able to prove the convergence studing directly the perturbative series and this has been performed in recent times by Eliasson, [3]. However the work by Eliasson has not enjoyed a wide circulation, maybe because of the excessive generality with which the considered problem is attacked, (see also the regarding comments in [5]). Then in [4] the convergence of the perturbative expansion of the tori is proven by using the Eliasson's ideas in a special model, the {\sl Thirring model}, [13], {\sl with the further simplification that the interaction does not depend on the actions}: in this way the discussion becomes much less involved, and so is suitable for a more easy understanding. The proof is implemented with methods usual in the renormalization group approach to quantum field theory, like the multiscale decomposition of the propagator and the tree expansion. Some minor technicalities of the proof are resolved in [9], where the strong diophantine hypothesis used in [4] is completely relaxed, (see [4], [9], for a detailed review). In this paper the proof is extended to cover the case in which the interaction potential of the Thirring model does not depend only on the angle variables, as in [4], but also on the action variables. As a function of the angles, the potential is supposed to be still an even trigonometric polynomial of degree $N<\io$, while, as a function of the actions, we assume that it is analytical, since this does yields no further technical work with respect to the case of a polynomial-like interaction. The idea is always to confine ourselves to a not too tangled case, in such a way to emphasize the interesting features of the method, and to point out the analogies with quantum field theory. We shall define a perturbative expansion for the KAM tori in terms of Feynman's graphs. However the ``dimensional bounds" are not sufficient to prove the series convergence: then we perform a resummation so obtaining a multiple series, in terms of the coupling constant and of suitable functions called ``form factors". If such functions are uniformly bounded, the series is convergent; endly the form factors boundedness is proven by using some cancellation mechanisms between diagrams operating to all perturbative orders. The analogy with quantum field theory is striking; for other results in this direction, see also [7], [8]. With respect to [4] and [9], the use of a smooth decomposition of the propagator and the introduction of a ``renormalized expansion" via a localization operation makes simpler the proof. The hamiltonian function of the Thirring model is % $$\fra12 J^{-1}\AA\cdot\AA\,+\,\e f(\aa,\AA) \; , \Eq(1.1)$$ % where $J$ is the non singular matrix of the inertia moments, $\AA=(A_1,\ldots,A_l)\in {\bf R}^l$ are their angular momenta and $\aa=(\a_1,\ldots,\a_l)\in {\bf T}^l$ are the angles describing their positions. We shall consider a ``rotation vector" $\oo_0=(\o_1,\ldots,\o_l)\in {\bf R}^l$ verifying the {\sl diophantine property} with diophantine constants $C_0,\t>0$; this means that % $$ C_0|\oo_0\cdot\nn|\ge |\nn|^{-\t} \; , \kern2.5cm\V0\ne\nn\in {\bf Z}^l \; , \Eq(1.2) $$ % and it is easy to see that the {\sl diophantine vectors} have full measure in ${\bf R}^l$ if $\t$ is fixed $\t>l-1$. We suppose $f$ to be an even trigonometric polynomial of degree $N$ in the angle variables and an analytic function in the angular momenta variables, \ie % $$ f(\aa,\AA)=\sum_{|\nn|\le N} f_\nn(\AA)\,\cos\nn\cdot\aa \; , \qquad f_\nn(\AA)=f_{-\nn}(\AA) \; , \Eq(1.3) $$ % with $f_\nn(\AA)$ analytic in $\AA$ in a domain $W(\AA_0,\r)= \{ \AA\in{\bf R}^l \; : \; |\AA-\AA_0|/|\AA_0| \le \r\}$, for any $\nn$, being $\AA_0=J\oo_0$. Finally, if $J_j$, $j=1,\ldots,l$ are the eigenvalues of the matrix $J$, we define % $$ J_m=\min_{j=1,\ldots,l} J_j \; , \qquad J_M=\max_{j=1,\ldots,l} J_j \; , \qquad F=\max_{|\nn|\le N, \AA\in W(\AA_0,\r)} f_\nn(\AA) \; , $$ % The fundamental result of this work is the following one. \* \\{\bf Theorem 1.1.} {\it The hamiltonian model \equ(1.1) admits an $\e$--analytic family of motions starting at $\aa=\V0$ and having the form % $$\AA=\AA_0+\V H(\AA_0,\oo_0t;\e)+\V \m(\AA_0,\e) \; , \qquad\qquad \aa=\oo_0t+\V h(\AA_0,\oo_0 t;\e) \; , \Eq(1.4) $$ % with $\V H(\AA,\pps;\e)$, $\V h(\AA,\pps;\e)$ analytic in $\pps$ with} $\hbox{Re}\pps\in {\bf T}^l$, {\it and} $|\hbox{Im}\pps_j|<\x$, {\it and in $\AA\in W(\AA_0,\r)$, where $\AA_0=J\oo_0$, and with vanishing average in ${\bf T}^l$, and with $\V H(\AA_0,\pps;\e)$, $\V h(\AA_0,\pps;\e)$ and $\V\m(\AA_0,\e)$ analytic for $|\e|<\e_0$ with a suitable $\e_0$ close to $0$: % $$ \e_0 = E_0 \, [J_MJ_m^{-2}C_0^2Fe^{\x N}\r^{-2} ]^{-1} \; , \Eq(1.5) $$ % where $E_0$ is a dimensionless quantity depending only on $N$ and $l$. This means that the set $(\AA,\aa)$ described as $\pps$ varies in ${\bf T}^l$ is, for $\e$ small enough, an invariant torus for \equ(1.1), which is run quasi periodically with angular velocity vector $\oo_0$. It is a family of invariant tori coinciding, for $\e=0$, with the torus $\AA=\AA_0,\,\aa=\pps\in {\bf T}^l$.} \* \\{\it Remark 1.} One recognizes a version of the KAM theorem. The proof that follows extends the one reported in [4] to the more general case in wich the interaction depends also on the angular momenta. \* \\{\it Remark 2.} Note that, in distinction to [4], the result is {\sl not uniform} in the twist rate $T$, defined as $T=J_M^{-1}$: this is known from the KAM theorem, and follows from the fact that the interaction depends also on the action variables. In other words the {\sl twistless} property in [4] is a consequence of the special choice of the interaction potential, which is of the form \equ(1.2), with $f_\nn(\AA)$ replaced with $f_\nn$ independent on $\AA$. \* Calling $\V H^{(k)}$, $\V h^{(k)}$, $\V\m^{(k)}$ the $k$-th order coefficients of the Taylor expansion of $\V H$, $\V h$, $\V\m$ in powers of $\e$, and writing the equation of motion as $\dot\a_j=(J^{-1}\AA)_j+\e\dpr_{A_j} f(\aa,\AA)$, and $\dot A_j=-\e\dpr_{\a_j} f(\aa,\AA)$, $j=1,\ldots,l$, we get immediately recursion relations for $\V H^{(k)},\V h^{(k)}$; for $k=1$: % $$ \eqalign{ & \V\o_0\cdot\dpr_\aa\,H^{(1)}_j = - \dpr_{\a_j} f \; , \cr % & \V\o_0\cdot\dpr_\aa\,h^{(1)}_j = \left(J^{-1}[\V H^{(1)}+ \V\m^{(1)}]\right)_j + \dpr_{A_j} f \; , \cr} \Eq(1.6) $$ % where $\oo_0\cdot\dpr_\aa=\sum_{i=1}^l\oo_{0i}\dpr_{\a_i}$, and, for $k>1$: % $$ \eqalign{ \V\o_0\cdot\dpr_\aa\,H^{(k)}_j = & - \sum_{m>0} \sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s + q_s)= m} \fra1{ \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr & \cdot \; \dpr_{\a_j}\,\dpr^{p_1}_{\a_1}\ldots\dpr^{p_l}_{\a_l} \dpr^{q_1}_{A_1}\ldots\dpr^{q_l}_{A_l}\,f(\oo_0t,\AA_0) \; \cdot \cr & \cdot \; {\sum}^* \prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} h^{(k_{sj})}_s \prod_{i=1}^{q_s} \left( H^{(k_{si}')}_s + \m^{(k_{si}')}_s \right) \Big] \; , \cr % \V\o_0\cdot\dpr_\aa\,h^{(k)}_j = \; & \left( J^{-1} [\V H^{(k)} + \V\m^{(k)}]\right)_j + \sum_{m>0} \sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m} \fra1{\prod_{s=1}^l p_s!\,q_s!} \quad \cdot \cr & \cdot \; \dpr_{A_j}\,\dpr^{p_1}_{\a_1}\ldots\dpr^{p_l}_{\a_l} \dpr^{q_1}_{A_1}\ldots\dpr^{q_l}_{A_l}f(\oo_0t,\AA_0) \; \cdot \cr & \cdot \; {\sum}^* \prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} h^{(k_{sj})}_s \prod_{i=1}^{q_s} \left( H^{(k_{si}')}_s + \m^{(k_{si}')}_s \right) \Big] \; , \cr} \Eq(1.7) $$ % where the $\sum^*$ denotes summation over the integers $k_{sj}\ge1$, $k_{si}'\ge1$, with: $\sum_{s=1}^l(\sum_{j=1}^{p_s}k_{sj}$ $+$ $\sum_{i=1}^{q_s}k_{si}') =$ $k-1$. In fact from the equations of motion for the angular momenta, we obtain immediately the first recursive relation in \equ(1.7). Then suppose that $\V h^{(k)}(\pps)$ and $\V H^{(k)}(\pps)$ are trigonometric polynomials of degree $\le k N$, respectively odd and even in $t$, for $1\le k< k_0$: we see immediately that the r.h.s. of the first equation in \equ(1.7) is odd in $t$. Then the first equation in \equ(1.7) can be solved for $k=k_0$. It yields an even function $\V H^{(k_0)}(\pps) + \V\m^{(k_0)}$ which is defined up to the constant $\V\m^{(k_0)}$, (which we call ``counterterm").\footnote{${}^3$}{\nota Note that $\V H^{(k)}$ has to have zero average over $\pps$ by construction.} Such a constant, however, must be taken so that the equation for the angle variables, \ie the second of \equ(1.7), has zero average, in order to be soluble. Hence the equation for $\V h^{(k)}$ can be solved and its solution is a trigonometric polynomial in $\pps$, defined up to a constant: such a constant has to be chosen to be vanishing so that $\V h^{(k_0)}$ is odd in $t$ and the procedure can be iterated. Therefore the equations for $\V \m^{(k)}$ will have to be obtained recursively by imposing that, for all $k$'s, the average over $\pps$ of the r.h.s of the second equation in \equ(1.7) is identically vanishing and requiring $\V h^{(k)}_{\V0}\,\=\,\V0$, $\forall k$: then the trigonometric polynomial $\V h^{(k)}(\pps)$ will be completely determined (if possible at all) from the second of \equ(1.7). If, given a function $F(\pps)$, we define by $F_\nn$ its $\nn$-th Fourier series component, % $$ F_\nn = \int_{{\bf T}^l} {d\pps\over(2\p)^2}\,F(\pps)\, e^{-i\nn\cdot\pps} \; , \qquad\qquad F(\pps) = \sum_\nn F_\nn \, e^{i\nn\cdot\pps} \; , \Eq(1.8) $$ % one easily finds, for $k=1$, from \equ(1.6) % $$ \eqalign{ h^{(1)}_{j\nn} & = \left( -iJ^{-1}\nn \right)_j \left[ i\oo_0\cdot\nn \right]^{-2} f_\nn(\AA_0) + \left[ i\oo_0\cdot\nn \right]^{-1} \dpr_{A_j} f_\nn(\AA_0)\;, \qquad \nn\neq\V0 \; ,\cr % H^{(1)}_{j\nn} & = \left( -i\nn_j \right) \left[ i\oo_0\cdot\nn \right]^{-1} f_\nn(\AA_0)\;, \qquad \nn\neq\V0 \; , \cr % \m^{(1)}_j & = - (J\dpr_{\AA})_j f_{\V0}(\AA) \; . \cr} \Eq(1.9)$$ % The \equ(1.7) provides an algorithm to evaluate a formal power series solution to our problem. It has been remarked, [3], [4], [14], that \equ(1.7) yields a {\sl diagrammatic expansion} of $\V h^{(k)}$, $\V H^{(k)}$ and $\V \m^{(k)}$, (we simply ``i\-te\-ra\-te" it until only $\V h^{(1)}$ and $\V H^{(1)}$, given by \equ(1.9), and $\V\m^{(k')}$, $k'0} \sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s + q_s)= m} {\sum}^* {1 \over \prod_{s=1}^l p_s!\,q_s! } \; \cdot \cr & \cdot \; (i\nn_0)_j \prod_{j=1}^l (i\nn_0)_j^{p_j} \prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0) \; \cdot \cr & \cdot \; \prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}} \prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'} + \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr % (i\oo\cdot\nn)^2 \, \bar h^{(k)}_{j\nn} = \; & (i\oo\cdot\nn)\, (\h^{-1}_m {\bar H^{(k)}}_{\nn})_j \cr & + (i\oo\cdot\nn) \Big({F\,C_0^2 \over J_m} \Big) \sum_{m>0} \sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m} {\sum}^* {1\over \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr & \cdot \; \dpr_{B_j}\,\prod_{j=1}^l (i\nn_0)_j^{p_j} \prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0) \; \cdot \cr & \cdot \; \prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}} \prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'} + \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr} \Eq(2.1) $$ % and, for $\nn=\V0$, % $$ \eqalign{ \bar \m^{(k)}_{j} & = \Big({F\,J_M\,C_0^2 \over J_m^2} \Big) \sum_{m>0} \sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m} {\sum}^* {1\over \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr & \cdot \; (\h_M \dpr_{\BB})_j\,\prod_{j=1}^l (i\nn_0)_j^{p_j} \prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0) \; \cdot \cr & \cdot \; \prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}} \prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'} + \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr} \Eq(2.2) $$ % where the $\sum^*$ denotes summation over the integers $k_{sj}\ge1$, $k_{si}'\ge1$, with: $\sum_{s=1}^l(\sum_{j=1}^{p_s} k_{sj}$ $+$ $\sum_{i=1}^{q_s}k_{si}') =$ $k-1$, and over the integers $\nn_0$, $\nn_{sj}$, $\nn_{si}'$, $s=1,\ldots,l$, $j=1,\ldots,p_s$ and $i=1,\ldots,q_s$, with: $\nn_0+\sum_{s=1}^l(\sum_{j=1}^{p_s} \nn_{sj}+$ $\sum_{i=1}^{q_s}\nn_{si}') =\nn$. \* For the time being we ignore the presence of the ``constant part" of the angular momenta, \ie $\V \m^{(k)}$, $k\ge1$, \ie we reason as if it was $\V\m^{(k)}\,=\,\V0$, $\forall k\ge1$: we shall see below how the discussion has to be modified when also such terms are taken into account. A tree diagram $\th$ will consist of a family of lines ({\sl branches} or {\sl lines}) arranged to connect a partially ordered set of points ({\sl vertices} or {\sl nodes}), with the higher vertices to the right. The branches are naturally ordered as well; all of them have two vertices at their extremes, (possibly one of them is a top vertex), except the lowest or {\sl first branch} which has only one vertex, the {\sl first vertex} $v_0$ of the tree. The other extreme $r$ of the first branch will be called the {\sl root} of the tree and will not be regarded as a vertex; we shall call the first branch also {\sl root branch}. If $v_1$ and $v_2$ are two vertices of the tree we say that $v_1v_0$ can be considered the first vertex of the tree consisting of the vertices following $v$: such a tree will be called a subtree of $\th$. To each branch $\l_v$ and to each vertex $v$ we attach a finite set of labels: $\z_{\l_v}$, $R_{\l_v}$, $j_{\l_v}$, and, respectively, $\d_v$, $m_v$, $k_v$, $\nn_v$, and a {\sl vertex function} $\EE_v$, which are defined as follows. \acapo (1) The label $\z_{\l_v}$ can assume the symbolic values $\z_{\l_v}=h,H$; \acapo (2) $R_{\l_v}=1$ if $\z_{\l_v}=H$ and $R_{\l_v}=2$ if $\z_{\l_v}=h$, and it is called the {\sl degree} of the line $\l_v$; \acapo (3) $j_{\l_v}=1,\ldots,l$; \acapo (4) $\d_v=0,1$, if $\z_{\l_v}=h$, and it is identically $0$ if $\z_{\l_v}=H$; \acapo (5) $m_v$ is the number of branches emerging from $v$, and can be written as % $$ m_v=\sum_{i=1}^3 \sum_{j=1}^l q_{v,j}^i \; , $$ % if $q_{v,j}^i$, $i=1,2,3$, are the branches leading to vertices $w$ with $\p(w)=v$ and carrying a $\z_{\l_w}$ label equal, respectively, to $h$, $H$, $\m$; \acapo (6) the {\sl order label} $k_v$ is given by the number of vertices of the subtree with first vertex $v$; \acapo (7) the {\sl mode label} $\nn_v$ is such that $\nn_v\in{\bf Z}^l$, $|\nn_v|\le N$; \acapo (8) the vertex function is defined as % $$ \eqalign{ \EE_v = & {F\,C_0^2 \over J_m} \Big\{ \Big[ \Big( (-i(\h_m)^{-1}\nn_v)_{j_{\l_v}} (1-\d_v) + (i\oo\cdot\nn(v)) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h} \cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} \Big] \; \cdot \cr & \cdot \; \prod_{w: \p(w)=v \atop \z_{\l_w}=h} (i\nn_v)_{j_{\l_w}} \prod_{w: \p(w)=v \atop \z_{\l_w}=H} \dpr_{B_{j_{\l_w}}} \Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.3) $$ % where $\d_{\z_{\l_v},x}$ denotes the Kronecker's delta (which is 1 only if $\z_{\l_v}=x$), and the last $m_v$ factors are missing if $v$ is a top vertex, (in this case $m_v=0$). Each of the $m_v+1$ factors appearing in \equ(2.3) will be called a {\sl vertex factor}. \* The labels, ``decorating'' the tree, will be called also decorations (\ie the labels attached to the tree and the decorations are synonimous below). Finally to the branch $\l_v$ leading from the vertex $\p(v)$ to the vertex $v$ we associate a ``propagator" $g(\oo\cdot\nn_{\l_v})$, which is given by: % $$ g(\oo\cdot\nn_{\l_v}) = {1 \over \left[i\oo\cdot\nn_{\l_v} \right]^{R_{\l_v}} } \; , \Eq(2.4) $$ % where $\nn_{\l_v}=\sum_{w\ge v}\nn_w$ is defined as the {\sl momentum} entering $v$, and $R_{\l_v}$ is called the {\sl degree of the line}. A possible tree is drawn in Fig.2.1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURA 2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \midinsert \insertplot{240pt}{170pt}{%fig1.tex \ins{-35pt}{90pt}{\sl root} \ins{10pt}{110pt}{$j_{\l_{v_0}}\,R_{\l_{v_0}}$} \ins{10pt}{80pt}{$\nn(v_0)\,\b_{\l_{v_0}}$} \ins{60pt}{85pt}{$v_0$} \ins{50pt}{125pt}{$\matrix{\d_{v_0}\,m_{v_0}\cr k_{v_0}\,\nn_{v_0}\cr}$} \ins{95pt}{132pt}{$j_{\l_{v_1}}\,R_{\l_{v_1}}$} \ins{120pt}{110pt}{$\nn(v_1)\,\b_{\l_{v_1}}$} \ins{152pt}{120pt}{$v_1$} \ins{135pt}{160pt}{$\matrix{\d_{v_1}\,m_{v_1}\cr k_{v_1}\,\nn_{v_1}\cr}$} \ins{110pt}{50pt}{$v_2$} \ins{190pt}{100pt}{$v_3$} \ins{230pt}{160pt}{$v_5$} \ins{230pt}{120pt}{$v_6$} \ins{230pt}{85pt}{$v_7$} \ins{230pt}{-10pt}{$v_{11}$} \ins{230pt}{20pt}{$v_{10}$} \ins{200pt}{65pt}{$v_4$} \ins{230pt}{65pt}{$v_8$} \ins{230pt}{45pt}{$v_9$} }{fig1} \kern1.3cm \didascalia{Fig.2.1: A tree $\th$ with $m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2,m_{v_4}=2$ and $m=12$, $\prod_v m_v!=2^4\cdot6$, and some decorations. The line numbers, distinguishing the lines, are not shown.} \endinsert % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We imagine that all the tree lines have the same length (even though they are drawn with arbitrary length in Fig.2.1). A group acts on the set of trees, generated by the permutations of the subtrees having the same root. Two diagrams that can be superposed by the action of a transformation of the group will be regarded as identical, (the superpositon has to be such that all the decorations of the diagram match). To order $k$, not considering the decorations, the number of trees is bounded by $2^{2k}$. We shall imagine that each branch carries also an arrow pointing to the root (``gravity" direction, opposite to the order). \* Then, if $X^{(k)}_{j\nn}(h)=\bar h^{(k)}_{j\nn}$ and $X^{(k)}_{j\nn}(H)= \bar H^{(k)}_{j\nn}$, it follows that \equ(2.1) can be rewritten as % $$ X^{(k)}_{j\nn}(\z) = {\sum_\th}^* \prod_{v_0\le v\in\th} \prod_{j=1}^l {1\over q_{v,j}^1!q_{v,j}^2!} \, g(\oo\cdot\nn_{\l_v}) \; \EE_v \; , \Eq(2.5) $$ % where the sum runs over all the tree $\th$'s of order $k$, with $\nn_{\l_{v_0}}=\nn$, $j_{\l_{v_0}}=j$ and $\z_{\l{v_0}}=\z$, and the $*$ recalls that the diagram $\th$ can and will be supposed such that $\nn_{\l_v}\ne\V0$ for all $v\in\th$, (by the parity properties remarked in \S 1 and because the counterterms are assumed to be zero, so that $q_{v,j}^3\=0$). \* If also the contributions $\bar\m^{(k)}_j$'s are considered, the above discussion has to be modified as follows. Some of the top vertices $v$ are drawn as fat points, and represent the quantities $\bar\m^{(k_v)}_{j_v}$: the corresponding $\z_{\l_v}$ label is set $\z_{\l_v}=\m$, and the label $k_v$ is such that $k_vv_0$, which is not a top vertex, $\z_{\l_v}\neq\m$, and \equ(2.3) has to be replaced with % $$ \eqalign{ \EE_v = & {F\,C_0^2 \over J_m} \Big\{ \Big[ \Big( (-i(\h_m)^{-1}\nn_v)_{j_{\l_v}} (1-\d_v) + (i\oo\cdot\nn_{\l_v}) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h} \cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} \Big] \; \cdot \cr & \cdot \; \prod_{w: \p(w)=v \atop \z_{\l_w}=h} (i\nn_v)_{j_{\l_w}} \prod_{w: \p(w)=v \atop \z_{\l_w}=H} \dpr_{B_{j_{\l_w}}} \prod_{w: \p(w)=v \atop \z_{\l_w}=\m} \dpr_{B_{j_{\l_w}}} \Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.6) $$ % while, if $v>v_0$ is a top vertex, % $$ \eqalign{ \EE_v = & {F\,C_0^2 \over J_m} \Big\{ \Big[ \Big( (-i(\h_m)_j^{-1}\nn_v)_{j_{\l_v}} (1-\d_v) + (i\oo\cdot\nn(v)) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h} \cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} + \bar \m_{j_{\l_v}}^{(k_v)} \d_{\z_{\l_v},\m} \Big] \Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.7) $$ % Then a formula analogous to \equ(2.5) still holds, with the constraint that the label $k$ is given by the number of the free vertices plus the sum of the orders $k_v$ of all the leaves $v\in\th$: % $$ X^{(k)}_{j\nn}(\z) = {\sum_\th}^* \prod_{v_0\le v\in\th} \prod_{i=1}^3 \prod_{j=1}^l {1\over q_{v,j}^i!} \, g(\oo\cdot\nn_{\l_v}) \, \EE_v \; , \Eq(2.8) $$ % where the sum runs over all the trees of order $k$, (with $\nn_{\l_{v_0}}=\V0$ and $j_{\l_{v_0}}=j$), having $k_0(\th)$ free vertices and $\NN_\FF(\th)$ fruits $v_i$ of order $k_i$, $i=1,\ldots,\NN_\FF(\th)$, with the constraint that their orders add to $k-k_0(\th)$. If $\z_{\l_{v_0}}=\m$, then \equ(2.8) has to be replaced by the following equation: % $$ \bar\m^{(k)}_j = - \Big( {J_M\,F\,C_0^2\over J_m^2} \Big) {\sum}^* (\h_M\dpr_{\BB})_j \, \f_{\nn}(\BB_0) \prod_{v_0 < v\in\th} \prod_{i=1}^3 \prod_{j=1}^l {1\over q_{v,j}^i!} \, g(\oo\cdot\nn_{\l_v}) \; \EE_v \; , \Eq(2.9) $$ % Note that \equ(2.8) and \equ(2.9) can depend on $\bar\m_{j'}^{(k')}$, only if $k'From \equ(2.8) and \equ(2.9) we deduce that each tree can be considerd as representing a contribution to $X^{(k)}_{j\nn}(\z)$, $\z=h,H$, or $\m^{(k)}_j$: such a contribution will be called the {\sl value of the tree}. Then, if $v_0$ is the first vertex of the tree, and $\z_{\l_{v_0}}=h$, the value of the label $\d_{v_0}$ tells us which term we are considering among the two of the first equation in \equ(2.1); the argument can be repeated for each following vertex. The interpretation of the other labels is obvious. \vskip1.truecm \\{\bf 3. Dimensional bounds} \*\numsec=3\numfor=1\pgn=1 \\The following Lemma shows that the estimates for the KAM tori can be reduced to the study of the contributions of the fruitless trees. The proof can be found in Appendix A1. \* \\{\bf Lemma 3.1.} {\it Suppose that we can prove that the contribution to $\bar\mu_j^{(k)}$ and the one to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$, arising from a single tree stripped of the fruits, ({\rm i.e.} the contribution we obtain by deleting the fruits from the tree), is bounded by $D_1^{k_0}$ for some constant $D_1$, if $k_0$ is the number of the free vertices; then a bound $B_0^k$ for the complete values $|\bar\mu_j^{(k)}|$ and $|X_{j\nn}^{(k)}(\z)|$, $\z=h,H$, follows immediately, for a suitable constant $B_0=2^3\r^{-1}B_1D_1$, being $\r$ defined after \equ(1.3) and $B_1=2^5l(2N+1)^l$.} \* \\{\it Remark 1}. The value of $B_1$ is found below, and is due to the trees counting, (see in particular the discussion after (3.5)): $B_1$ counts the possible decorated fruitless trees. \* \\{\it Remark 2}. Note that, given a fruitless tree, contributing to $\bar\mu_j^{(k)}$, the bound we obtain for it contains an extra factor $J_M/J_m$ with respect to the same bound we would obtain if it had contributed to $X_j^{(k)}(\z)$, $\z=h,H$, as a comparison between \equ(2.2) and the first equation in \equ(2.1) shows. Therefore we can confine ourselves to study the contributions to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$, arising only from fruitless trees, and, if a bound $\tilde B_0$ is found for them, then it will be $B_0=J_MJ_m^{-1}\tilde B_0$. \* In order to bound the fruitless tree values, we introduce a multiscale decomposition of the propagator. Let $\chi(x)$ be a $C^\io$ not increasing function such that $\chi(x)=0$, if $|x|\ge 2$ and $\chi(x)=1$ if $|x|\le 1$, and let $\chi_n(x)=\chi(2^{-n}x)-\chi(2^{-(n-1)}x)$, $n\le 0$, and $\chi_1(x)=1-\chi(x)$: such functions realize a $C^\io$ partition of unity, for $x\in[0,\io)$, in the following way. Let us write % $$ 1=\chi_1(x)+\sum_{n=-\io}^0\chi_n(x)\=\sum_{n=-\io}^1\chi_n(x) \; . \Eq(3.1) $$ % Then we can decompose the propagator in the following way: % $$ g(\oo\cdot\nn_{\l_v})={1\over [i\oo\cdot\nn_{\l_v}]^{R_{\l_v}}}\= \sum_{n=-\io}^1{\chi_n(x)\over [i\oo\cdot\nn_{\l_v}]^{R_{\l_v}}}\= \sum_{n=-\io}^1 g^{(n)}(\oo\cdot\nn_{\l_v}) \; \Eq(3.2) $$ % where $g^{(n)}(\oo\cdot\nn_{\l_v})$ is the ``propagator at scale $n$". If $n<0$, $g^{(n)}(\oo\cdot\nn_{\l_v})$ is a $C^{\io}$ compact support function different from $0$ for $2^{n-1}<|\oo\cdot\nn_{\l_v}|\le 2^{n+1}$, while $g^{(1)}(\oo\cdot\nn_{\l_v})$ has support for $1<|\oo\cdot\nn_{\l_v}|$. In the domain where it is different from zero, the propagator verifies the bound % $$ \Big| {\partial^p\over\partial x^p} g^{(n)}(x) \Big|_{x=\oo\cdot\nn_{\l_v}} \le a_{R_{\l_v}}(p) 2^{-n(R_{\l_v}+p)} \; , \qquad p\in {\bf N} \; , $$ % where $a_{R_{\l_v}}(p)$ is a suitable constant, such that $a_{R_{\l_v}}(0)=2^{R_{\l_v}}$, which depends on the form of the function $\chi(x)$.\footnote{${}^4$}{\nota The consatnt $a_{R_{\l_v}}(p)$ has a bad dependence on $p$, but we shall see that in our bounds $p$ does not increase ever beyond 2: see in particular (4.4) below and, especially, the right after remark.} Proceeding as in quantum field theory, see [5], given a tree $\th$ we can attach a {\sl scale label} $n_{\l_v}$ to each branch $\l_v$ in $\th$, which is equal to the scale of the propagator associated to the branch via \equ(2.4) and \equ(2.2). Looking at such labels we identify the connected cluster $T$ of vertices which are linked by a continuous path of branches with the same scale labels $n_T$ or a higher one and which are maximal: we shall say that {\sl the cluster $T$ has scale $n_T$}. Therefore an inclusion relation is established between the clusters, in such a way that the innermost clusters are the clusters with the highest scale, and so on. Each cluster can have an arbitrary number of branches entering it, ({\sl incoming lines}), but only one line exiting, ({\sl outgoing lines}); we use that the branches carry an arrow pointing to the root: this gives a meaning to the words ``incoming" and ``outgoing". The multiscale decomposition \equ(3.2) of the propagator allows us to rewrite \equ(2.5) as % $$ X_{j\nn}^{(k)}(\z)={\sum_{\th}}^*\prod_{v_0\le v\in\th} \prod_{j=1}^l \prod_{i=1}^3 {1\over q_{v,j}^i!} g^{(n_{\l_v})}(\oo\cdot\nn_{\l_v})\,\EE_v \; , \Eq(3.3) $$ % where the sum is over the labeled trees, and therefore, with respect to \equ(2.8), we have the extra labels $n_\l$ associated to the lines $\l$'s: a (not optimal bound) of the number of terms appearing in the sum over the $\nn_v$ and $n_v$ labels is given by $[2(2N+1)^l]^k$, as $|\n_i|\le N$, $\forall i=1,\ldots,l$, and because of the support compact property of the propagators. \* \\{\bf Definition 3.1 (Resonance).} {\it Among the clusters we consider the ones with the property that there is only one incoming line, carrying the same momentum of the outgoing line, and we define them {\rm resonances}. If $V$ is one such cluster we denote by $\l_V$ the incoming line and $K(V)$ the number of vertices contained in $V$ ({\rm resonance order}). The line scale $n=n_{\l_V}$ is lower than the lowest scale $n'=n_V$ of the lines inside $V$: we call $n_{\l_V}$ the {\rm resonance-scale}, and $\l_V$ a {\rm resonant line}.} \* \\{\it Remark.} Note that a resonance $V$, as a cluster, has an its own scale $n_V$, which is higher than its resonance-scale $n_{\l_V}$, $n_V\ge n_{\l_V}+1$. \* Recall that $R_\l$ is the {\sl degree} of the line $\l$: it is the exponent of the propagator corresponding to the line, (see \equ(2.2)). Given a resonance $V$, let us define the {\sl resonance degree} $D_V=1,2$ as the degree $R_{\l_V}=1,2$ of the resonant line, \ie $D_V=R_{\l_V}$. Given a tree $\th$, let us define $N_n$ the number of lines with scale $n\le 0$, and $N^j_n$, $j=1,2$, the number of lines $\l$ with scale $n\le 0$ and having $R_\l=j$, (\ie with degree $j$). Then it is easy to check that the scaling properties of the propagators and the definitions \equ(2.2) and \equ(2.5) immediately imply that the contribution to $\V X^{(k)}(\z)$ arising from a given tree $\th$ can be bounded by: % $$ \CC^k\prod_{n\le 0} 2^{-(2n N_n^2+n N_n^1)} \prod_{v}2^{n_{\l_v}\d_v} \; , \Eq(3.4) $$ % for a suitable constant $\CC$; if we look at \equ(2.6), we can write % $$ \CC=2^2J_m^{-1} C_0^2 N^2 F\r^{-1} \; , \Eq(3.5) $$ % where $C_0$ is the diophantine constant introduced in \equ(1.3), $\r$ is introduced after \equ(1.3), $N$ is a bound on the mode values $\nn_v$ of the vertices $v\in\th$, and the factor $2^2$ is due to the definition of the compact support of the propagators. The last product in \equ(3.4) arises because for each line $\l_v$ of degree $2$ such that $\d_v=1$ there is an extra factor $(i\oo\cdot\nn(v))$, (see \equ(2.6)). Therefore we only have to multiply \equ(3.4) by the number of diagram shapes for $\th$ having vertices with given bifurcation numbers $m_v$, $v\in\th$, ($\le 2^{2k}$, see [10]), by the number of ways of attaching the labels, ($\le [2\cdot 2 \cdot 2\cdot l\cdot (2N+1)^l]^k$), so that the number of trees of order $k$ can be bounded by $B_1^k$, if $B_1$ is defined as in Lemma 3.1. The following bound holds for the number of lines with scale $n\le 0$: % $$ N_n^1+N_n^2 \= N_n \le {2k\over E2^{-n\t^{-1} } } + \sum_{T,n_T=n} \sum_{j=1}^2 m_T^j \; , \Eq(3.6) $$ % where $m^j_T$ is the number of resonances $V$'s inside the cluster $T$, having resonance-scale $n_{\l_V}=n_T$ and degree $D_V=j$, and $E$ can be chosen $E=2^{-3\t^{-1}}N^{-1}$. This is a slightly modified version of the Brjiuno's lemma, [2]: a proof is in Appendix A2, and it is taken from [4], [9], with some minor changes. Therefore if there was no resonance, \ie if it was $m^j_T=0$ for any $T$, then we would obtain a (not optimal) bound $G_0^k$ for a suitable constant $G_0>0$; the labeled trees counting and \equ(3.5) give % $$ G_0=b_12^7l(2N+1)^l F\r^{-1}J_m^{-1} C_0^2 N^2 \; , \Eq(3.7) $$ % where $b_1=\exp[\sum_{n=1}^\io 4(\ln 2)nE^{-1}2^{-n\t^{-1}}]$. However {\sl there are resonances}, and we have to deal with them. \* \\{\bf Definition 3.2 (Resonance factor).} {\it Let us consider a resonance $V$; let $\l_V$ be the incoming line, as in Definition 3.1. We denote by $w_0$ the vertex which the outgoing line of $V$ leads to, (recall that the ordering of the tree is opposite to the gravity direction), and by $w_2$ the vertex which the incoming line (resonant line) leads to: such lines are characterized by the labels, respectively, $\z_{\l_{w_0}}$ and $\z_{\l_{w_2}}$. Such a couple of values $(\z_{\l_{w_0}}, \z_{\l_{w_2}})$ can assume the values $(H,H)$, $(H,h)$, $(h,H)$ and $(h,h)$, and we can introduce a label $s_V$ in order to distinguish the four above possible cases. Let us define the {\rm resonance factor} $\VV_{s_V,j_{\l_{w_0}}j_{\l_{w_2}}}(\oo\cdot\nn_{\l_V}))$ as the quantity % $$ \VV_{s_V,j_{\l_{w_0}}j_{\l_{w_2}}}(\oo\cdot\nn(v_{\l_V}))=\EE_{w_0} \prod_{w_00$: to such an aim the next section is devoted. \vskip1.truecm \\{\bf 4. Boundedness of the form factors and convergence of the perturbative series} \*\numsec=4\numfor=1\pgn=1 \\In this section we prove that $|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|\le C^k$, for some $C>\e_1^{-1}$ and any value of $\oo\cdot\nn$. This will be done by writing the form factor as a sum over diagrams which can be thought as resonances with their incoming line, (see Definition 3.3), so that, in order to obtain the contribution arising from a single diagram, we have to compute the resonance factor times the propagator associated to its incoming line. The resonance factor is expressed in terms of the original trees, ({\sl not of the resumed trees}): the corresponding resonance $V$ can be interpretated as a tree having an endpoint $w_2$, (see Definition 3.2), from which a mode $\nn_{w_2}+\nn_{\l_V}$ is emitted, instead of a mode $\nn_{w_2}$ simply. \* The first step in order to prove the boundedness of the form factor is to note that $\sum_V \VV^{n_{\l_V}}_{s_V,jj'}(\oo\cdot\nn_{\l_V})= O((\oo\cdot\nn_{\l_V})^{R_{\l_V}})$, if $R_{\l_V}$ is the degree of the propagator in \equ(3.2). This is in fact the case, as the following lemma shows. \* \\{\bf Lemma 4.1.} {\it The form factor introduced in Definition 3.3 through \equ(3.9) can be written in the following way: % $$ \s^{n(k)}_{s,jj'}(\oo\cdot\nn) = \sum_{V : \nn_{\l_V}=\nn \atop k(V)=k,n_{\l_V}=n} \int_0^1 dt \; t^{R_{\l_V}-1} \left[ {\dpr^{R_{\l_V}} \over \dpr (\oo\cdot\nn)^{R_{\l_V}} } \, \VV^{n_{\l_V}}_{s,jj'}(t\oo\cdot\nn) \right] \, \ch_n(\oo\cdot\nn) \; . \Eq(4.1) $$ % This means that the first $R_{\l_V}$ terms of the Taylor expansion of the resonance factor $\VV^{n_{\l_V}}_{s,jj'}(\oo\cdot\nn)$ in powers of $\oo\cdot\nn$ add to zero when summed to give the form factor.} \* We have taken into account explicitly the expression giving the propagator on scale $n$, $g^{(n)}(\oo\cdot\nn)$ $=$ $\ch_n(\oo\cdot\nn)[i\oo\cdot\nn]^{-R_{\l_V}}$, (see \equ(3.2)). The proof of Lemma 4.1 is given in Appendix A3. \* In order to prove that $|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|\le C^k$, for some constant $C$, we modify the rules how to construct the trees by splitting each resonance factor $\VV$ as $\VV=\LL\VV+(1-\LL)\VV$, where % $$ \eqalign{ \LL \VV^n_{1,jj'}(\oo\cdot\nn) & = \VV^n_{1,jj'}(0) \; , \cr % \LL \VV^n_{2,jj'}(\oo\cdot\nn) & = \VV^n_{2,jj'}(0) + [\oo\cdot\nn]\,\dot\VV^n_{2,jj'}(0) \; , \cr % \LL \VV^n_{3,jj'}(\oo\cdot\nn) & = \VV^n_{3,jj'}(0) \; , \cr % \LL \VV^n_{4,jj'}(\oo\cdot\nn) & = \VV^n_{4,jj'}(0) + [\oo\cdot\nn]\,\dot\VV^n_{2,jj'}(0) \; , \cr} \Eq(4.2) $$ % where $\dot\VV^n_{s,jj'}(0)$ denotes the first derivative of $\VV^n_{s,jj'}$ with respect to $\oo\cdot\nn$, computed in $\oo\cdot\nn=0$. Note that the resonant factors depend on $\oo\cdot\nn$ only through the propagators, (see \equ(3.2)). Then, for each line $\l$ inside the resonance, the momentum flowing in it is given by $\nn_\l\=\nn_\l^0+\e_\l\nn$, where $\nn_\l^0$ is the sum of the mode labels corresponding to the vertices following $\l$ but inside the resonance, and $\e_\l=0,1$, so that, even if we set $\oo\cdot\nn=0$, (\ie $\oo\cdot\nn_\l= \oo\cdot\nn_\l^0$ for each $\l$ inside the resonance), no too small divisor appears because of the presence of the compact support functions $\ch_{n'}(\oo\cdot\nn_\l)$, $n'>n$. Given a tree, on any cluster the $\LL$ or $1-\LL\=\RR$ operators apply; however for the cancellations seen in Lemma 4.1 the sum over the trees of order $k$ containing one or more resonances on which the $\LL$ operator applies is vanishing, so that we can rule out all such contributions and consider simply the trees with resonances on which the operator $\RR$ applies. It is convenient to write the effect of $\RR$ on a resonance $V$ as % $$ \eqalign{ \RR\VV^n_{s,jj'}(\oo\cdot\nn) & = (\oo\cdot\nn) \ig_0^1 dt\; \dot\VV^n_{s,jj'}(t\oo\cdot\nn) \qquad \qquad (\hbox{first order zero}: s=1,3) \; , \cr % \RR\VV ^n_{s,jj'}(\oo\cdot\nn) & = (\oo\cdot\nn)^2 \ig_0^1 dt\;t\; \ddot\VV^n_{s,jj'}(t\oo\cdot\nn) \qquad (\hbox{second order zero}: s=2,4) \; , \cr} \Eq(4.3) $$ % where $\ddot\VV^n_{s,jj'}$ denotes the second derivative with respect to the variable $\oo\cdot\nn$. As there are resonances enclosed in other resonances the above formula can suggest that there are propagators derived up to $\approx k$ times, if $k$ is the order of the graph. This would be of course a source of problems, as $a_{R_{\l_V}}(p) > p!$, where $a_{R_{\l_V}}(p)$ is defined after \equ(4.2). However it is not so: in fact the propagators are derived at most two times. This can be seen as follows. Let $n$ be the resonance-scale of the maximal resonance $V$, and let us define $V_0$ as the collection of lines and vertices in $V$ not contained in any resonance internal to $V$. Then we can write $\RR\VV(\oo\cdot\nn_{\l_V})$, (we do not write explicitly the labels of the resonance factor), as % $$ \RR \Big( \prod_{\l\in V_0} g_\l^{(n_\l)} \prod_{V'\subset V} [\RR\VV (\oo\cdot\nn_{\l_{V'}})] \prod_{v\in V_0} \EE_v \Big) \; , \Eq(4.4) $$ % being the second product over the resonances $V' \subset V$ which are maximal; in \equ(4.4), for any resonance $V' \subseteq V$, $\RR\VV(\oo\cdot\nn_{\l_{V'}})$ can be written either as in \equ(4.3) or as a difference $\RR\VV(\oo\cdot\nn_{\l_{V'}})$ $=$ $\VV(\oo\cdot\nn_{\l_{V'}})-\LL\VV(\oo\cdot\nn_{\l_{V'}})$, in according to which expression turns out to be more convenient to deal with. Then the first step is to write the action of $\RR$ on the maximal cluster as in \equ(4.3), leaving the other terms $\RR\VV (\oo\cdot\nn_{\l_{V'}})$ written as differences: so \equ(4.4) can be written by the Leibniz's rule as a sum of terms, and the derivatives of $\RR$ apply either on some propagator $g_\l^{(n)}$ or on some $\RR\VV (\oo\cdot\nn_{\l_{V'}})$. In the end there can be either no derivative, or one derivative, or two derivatives applied on each $\RR\VV (\oo\cdot\nn_{\l_{V'}})$. If only one derivative acts on $\RR \VV(\oo\cdot\nn)$, $\nn=\nn_{\l_{V'}}$, and, \eg, $s=2,4$, then we write, when such a term is not vanishing, % $$ \dpr \RR \VV(\oo\cdot\nn)=\dpr \VV(\oo\cdot\nn)- \dot\VV(0)=(\oo\cdot\nn)\ig_0^1 dt\, \ddot \VV(t\oo\cdot\nn) \; , $$ % because in such a case the derivative with respect to $\oo\cdot\nn$ is equal to the dervative with respect to $\oo\cdot\nn_{\l_{V}}$, while if two derivatives act on $\RR\VV (\oo\cdot\nn_{\l_{V'}})$, then we write % $$ \dpr^2 \RR \VV(\oo\cdot\nn)= \ddot \VV(\oo\cdot\nn) \; . $$ % The case $s=1,3$ is easier, and can be discussed in the same way. Then no more than two derivatives can act on each resonance $V'$ in any case, and the procedure can be iterated, since the resonances $V'$ can be dealt with as the resonance $V$. The effect of the $\RR$ operator is to obtain a gain factor either $2^{n-n'}$ or $2^{n-n'}2^{n'-n{'}{'}}$, where $n'$ and $n{'}{'}$ are the scales of two lines $\l'$ and $\l{'}{'}$ contained in some clusters $T'$ and $T{'}{'}$ inside $V$; the line $\l{'}{'}$ can coincide with $\l'$, or also be absent, if $s_V=1,3$. So we can rewrite, \eg, the first factor as $2^{n-n'}=2^{n-n_1}\ldots 2^{n_q-n'}$, where $n_i$ is the scale of the cluster $T_i\supset T_{i+1}$, with $T_0=V$ and $T_{q+1}=T'$. Analogous considerations hold for $n{'}{'}$, so that we can conclude that: (1) no more than two derivatives can ever act on any propagators; (2) a gain $2^{D_{V'}(n_{\l_{V'}}-n_{V'})}$ is obtained for any resonance $V'\subseteq V$; (3) the total number of terms generated by the derivation operations is bounded by $k(V)^2$, if $k(V)$ is the order of the resonance $V$, (see Definition 3.1). Therefore, for the value of the diagram formed by the resonance plus its incoming line, we find the bound % $$ \eqalign{ 2^{-D_Vn_V} \Big[ & \tilde \CC^k {\prod_{v}}^* 2^{n_{\l_v}\d_v} \prod_{n\ge n_V}2^{-(2nN_{n}^2+nN_{n}^1)}\Big] \cdot \; \cr & \cdot \; \Big[ \prod_{n\ge n_V} \prod_{T \atop n_T=n}\prod_{j=1}^2 \prod_{i=1}^{m_T^j}\,2^{D_{V_i}(n-n_{V_i}) } \Big] \; , \cr} \Eq(4.5) $$ % where $n_V$ is the scale of the resonance, $D_V$ is the degree of the resonance, the product with $*$ is over all the lines not exiting from any resonance,\footnote{${}^5$}{\nota If a line $\l_{w_0}$ comes out from a resonance, and $\d_{w_0}=1$, then the factor $(i\oo\cdot\nn(w_0))$ appearing in the first vertex factor corresponding to $w_0$, (see \equ(3.6)), is used in order to implement the cancellation of the form factor, (as proof of Lemma 4.1 shows, see Appendix A3), and then the bound improvement \equ(4.4); therefore we have no more the factor $2^{n_{\l_{w_0}}}$ in \equ(3.4) corresponding to such a line.} and the second square bracket is the part coming from the resummations, and follows from the above discussion about the gain factors. The constant $\tilde\CC$ differs from $\CC$ in \equ(3.5) as it takes into account the bound on the derivatives of the propagators: we can set $\tilde\CC=\CC\,e^2\,a_2(2)$, as the sum over all the outer resonances $V$'s of the factors $[2k(V)]^2$ can be bounded by $e^{2k}$, and $a_R(p)\le a_2(2)$, for any $R=1,2$, and $0\le p \le 2$. In Appendix A2, we show that, if $N_n^j(V)$ is the number of lines on scale $n$ and of degree $j$ contained inside a resonance $V$, then the following bound holds: %From \equ(3.6) and the fact that, for any tree of order $k$, %the number of clusters on scale $n$ verifies the bound %$\sum_{T,n_T=n}1\le (2k)^{-1}[E\,2^{-n\t^{-1}}]^{-1}$, %(see Appendix A2), we obtain % $$ \sum_{j=1}^2 j N_n^j (V) \le \fra{8k(V)}{E\,2^{-n\t^{-1}}} + \sum_{T \subseteq V \atop n_T=n} \Big[ -2 + \sum_{j=1}^2 j \, m_T^j \Big] \; . \Eq(4.6) $$ % Substituting \equ(4.6) into \equ(4.5), we see that the $j m_T^j$ is taken away by the first factor in $\,2^{D_{V_i}n}$ $2^{-D_{V_i}n_{V_i}}$, being $n=n_{\l_{V_i}}$, while the remaining $\,2^{-D_{V_i}n_{V_i}}$ are compensated by the $-2$ before the $\sum_j m_T^j$ in \equ(4.6), taken from the factors with $T=V_i$, (note that there are always enough $-2$'s); in particular we can get rid of the factor $2^{-D_Vn_V}$ in \equ(4.5). Hence \equ(4.5) is bounded by % $$ a_2(2)^k \CC^k e^{2k} \prod_n\,2^{-8 n k E^{-1}\,2^{n/\t}}\le D_1^k \; , \Eq(4.7) $$ % if $k=k(V)$, with a suitable constant $D_1$; the previous bounds give % $$ D_1 =b_1^2 \,e^2 \, a_2(2)\,\CC= 2^2\,a_2(2)\,e^2\,F\,J_m^{-1}\,C_0^2\,N^2\,\r^{-1}\, \exp[\sum_{n=1}^\io 8\,n(\ln 2)\,E^{-1}\,2^{-\t^{-1}n}] \; . $$ % As in the preceding section, the number of trees contributing to $\s^{n(k)}_{s,jj'}(\oo\cdot\nn)$ is bounded by $B_1^k$, so that, if we reacall the Remark 2 after Lemma 3.1, we can bound $|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|$ by $B_0^k$, for a suitable constant $B_0$ given by % $$ B_0 \= 2^3\r^{-1}J_MJ_m^{-1}B_1D_1= 2^{10}\,e^2\,a_2(2)\,b_1^2\,l\,(2N+1)^l\,F\,\r^{-2}\, J_M\,J_m^{-2}\,C_0^2\,N^2 \; . \Eq(4.8) $$ % Obviously an analogous bound holds also for the contributions to $\bar\m^{(k)}_j$ and to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$, so that the proof of Theorem 1.1 is complete, and the value \equ(1.5) for the KAM tori convergence radius is obtained. \qed \vskip1.truecm \\{\bf Acknowledgements.} We thank G.Gallavotti for having proposed us this work, and for many suggestions and useful discussions. \vskip1.truecm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% APPENDICI A1,A2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip1.truecm \\{\bf Appendix A1. Proof of Lemma 3.1} \* \\Let us consider first the contribution to $\bar\m_j^{(k)}$. A value $\bar\m_{j}^{(k)}$ can be represented as a sum of tree values, as shown in \equ(2.9). Then each tree whose value contributes to $\bar\m^{(k_i)}_{j_i}$ can be enclosed inside a bubble. Since $\bar\m_{j}^{(k)}$ depends on other fruits, we can iterate the procedure until no fruit is left: at each step we obtain some new bubbles which are contained in a bubble drawn in the previous step. We consider a single tree appearing in the multiple sums obtained through the above procedure: we note that the values arising from the different bubbles factorize, since the momentum flowing in the first branch of the maximal tree inside a bubble is identically vanishing, and therefore the momenta running in any tree branches do not depend on the bubbles encircling the following vertices. Then for each innermost bubble $b_0$, which does not contain any bubbles, a bound $D_1^{\tilde k_{b_0}}$, if $\tilde k_{b_0}$ is the number of (free) vertices of the tree inside the bubble, is obtained, by assumption. Hence we pass to the next to the innermost bubbles: for each such bubble $b_1$ a bound $D_1^{\tilde k_{b_1}}$ $2^{\tilde k_{b_1}}$ $\prod_{b_0\subset b_1} 2\r^{-1}$ is found, where $\tilde k_{b_1}$ now is the number of vertices contained in the considered bubble, (\ie is the number of free vertices of the tree inside the considered bubble but outside the inner bubbles). In fact, for each vertex inside $b_1$ we can bound % $$ \left\{ { \prod_{j=1}^l \dpr_{B_j}^{q_{v,j}^2+q_{v,j}^3} \f(\BB_0) \over q_{v,j}^2!q_{v,j}^3! } \right\} \le \left[ \prod_{j=1}^l 2^{q_{v,j}^2+q_{v,j}^3} \r^{-q_{v,j}^3} \right] \r^{-q_{v,j}^2} \; , $$ % where $\r^{-q_{v,j}^2}$ can be interpretated as a bound for $[q_{v,j}^2!]^{-1} \prod_{j=1}^l \dpr_{B_j}^{q_{v,j}^2} \f(\BB_0)$, because of the assumption of analiticity in the angular momenta variables of the interaction potential. And so on, until in the end a bound $\prod_b [ (2D_1)^{\tilde k_b} 2\r^{-1} ]$ follows for the considered tree value, being the product over the bubbles $b$ and $\tilde k_b$ being the number of vertices inside the bubble $b$ but outside the inner bubbles. This means that the value of a single tree contributing to $\bar\m_j^{(k)}$ is bounded by $[4\r^{-1}D_1]^k$, as $\sum_b \tilde k_b\le k$. Then we have only to perform the sum over the trees, but from the above discussion we conclude that such a sum is arranged in the following way: we sum over all the trees with $k$ vertices, and over all the possible ways to put bubbles around the vertices of the tree $\th$. The first sum is bounded by $B_1^k$, (as it is proven in \S 3, see the discussion after \equ(3.5)), while the second one is trivially bounded by $2^k$, as there can be at worst one bubble per vertex. Then, if we take $B_0=2^3\r^{-1}B_1 D_1$, the stated result follows. If the trees contribute to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$, the proof remains the same: simply we do not draw any bubble around the entire tree. \qed \vskip1.truecm \\{\bf Appendix A2. Resonant Siegel-Brjiuno bound.} \* \\Let us call $N^*_n\,\=\,N_n^*(\th)$ the number of non resonant lines of a tree $\th$ carrying a scale label $\le n$, \ie $N_n^* +\sum_T(m_T^1+m_T^2)=\sum_{n'\le n} N_n$. We shall prove first that $N^*_n\le 2k (E 2^{-n/\t})^{-1}-1$ if $N_n>0$. If $\th$ has the root line with scale $>n$ then calling $\th_1,\th_2,\ldots,\th_m$ the subdiagrams of $\th$ emerging from the first vertex of $\th$ and with $k_j>E2^{-n/\t}$ lines, it is $N^*_n(\th)=N^*_n(\th_1)+\ldots+N^*_n(\th_m)$ and the statement is inductively implied from its validity for $k'k-2^{-1} E\,2^{-n/\t}$. Finally, and this is the real problem as the analysis of a few examples shows, we claim that in the latter case the root line of $\th_1$ is either a resonant line or it has scale $>n$. Accepting the last statement we have: $N^*_n(\th)=1+N^*_n(\th_1)= 1+N^*_n(\th'_1)+\ldots+N^*_n(\th'_{m'})$, with $\th'_j$ being the $m'$ subdiagrams emerging from the first node of $\th'_1$ with orders $k'_j>E\,2^{-n/\t}$: this is so because the root line of $\th_1$ will not contribute its unit to $N^*_n(\th_1)$. Going once more through the analysis the only non trivial case is if $m'=1$ and in that case $N^*_n(\th'_1)=N^*_n(\th{'}{'}_1)+\ldots+N^*_n(\th{'}{'}_{m{'}{'}})$, \etc., until we reach a trivial case or a diagram of order $\le k-2^{-1} E\,2^{-n/\t}$. It remains to check that if $k_1>k-2^{-1}E\,2^{-n/\t}$ then the root line of $\th_1$ has scale $>n$, unless it is entering a resonance. Suppose that the root line of $\th_1$ has scale $\le n$ and is not entering a resonance. Note that $|\oo\cdot\nn(v_0)|\le\,2^{n+1},|\oo\cdot\nn(v_1)|\le \,2^{n+1}$, if $v_0,v_1$ are the first vertices of $\th$ and $\th_1$ respectively. Hence $\d\=|(\oo\cdot(\nn(v_0)-\nn(v_1))|\le2\,2^{n+1}$ and the diophantine assumption implies that $|\nn(v_0)-\nn(v_1)|> (2\,2^{n+1})^{-1/\t}$, or $\nn(v_0)=\nn(v_1)$. The latter case being discarded as we are not considering the resonances, it follows that $k-k_1<2^{-1}E\,2^{-n/\t}$ is inconsistent: it would in fact imply that $\nn(v_0)-\nn(v_1)$ is a sum of $k-k_1$ vertex modes and therefore $|\nn(v_0)-\nn(v_1)|< 2^{-1}NE\,2^{-n/\t}$ hence $\d>2^3\,2^n$ which is contradictory with the above opposite inequality. \qed \* Analogously, we can prove that, if $N_n>0$, then the number $p_n(\th)$ of clusters of scale $n$ verifies the bound $p_n(\th) \le 2k \,(E2^{-n/\t})^{-1}-1$. In fact this is true for $k \le E2^{n/\t}$, (see footnote 6). Otherwise, if the first tree vertex $v_0$ is not in a cluster of scale $n$, it is $p_n(\th)=p(\th_1)+\ldots+p_n(\th_m)$, with the above notation, and the statement follows by induction. If $v_0$ is in a cluster on scale $n$ we call $\tilde\th_1, \ldots, \tilde\th_m$ the subdiagrams emerging from the cluster containing $v_0$ and with orders $k_j>E2^{-n/\t}$, $j=1,\ldots,m$. It will be $p_n(\th)=1+p(\tilde\th_1)+\ldots+p_n(\tilde\th_m)$. Again we can assume $m=1$, the other cases being trivial. But in such a case there will be only one branch entering the cluster $T$ on scale $n$ containing $v_0$ and it will have a momentum of scale $n'\le n-1$. Therefore the cluster $T$ must contain at least $E2^{-n/\t}$ vertices, (otherwise, if $\l$ is a line on scale $n$ contained in $T$, and $\nn_\l^0$ is the sum of the mode labels corresponding to the vertices following $v_0$ but inside $T$, we would have $|\oo\cdot\nn_\l|\le 2^{n+1}$ and, simultaneously, $|\oo\cdot\nn_\l|\ge 2^{n+3}-2^{n-1}>2^{n+2}$, which would lead to a contradiction). This means that $k_1\le k -E2^{-n/\t}$. \qed \* Let us consider now a resonance $V$, and let us call $N_n(V)$ and $N_n^*(V)$ the number of lines on scale $n$ in $V$, and, respectively, the number of non resonant lines inside $V$ carrying a scale label $\le n$. Again a bound $N_n^*(V)\le2k(V)(E2^{-n/\t})^{-1}-1$ holds, if $k(V)$ is the order of the resonance $V$; analogously $p_n(V)\le2k(V)(E2^{-n/\t})^{-1}-1$, if $p_n(V)$ is the number of clusters on scale $n$ contained in $V$. The proofs of such statements can be easily adapted from the previous ones, by noting that $N_n(V)\neq0$ requires $k(V)>E2^{-n/\t}$. We give them explicitly, only for completeness. Given a subdiagram $T$, let us denote by $k(T)$ the number of vertices contained in $T$, and by $N_n^*(T)$ the number of non resonant lines inside $T$ with a scale label $\le n$. We prove by induction that $N_n^*(T)\le2k(T)(E2^{-n/\t})^{-1}-1$, each time only one line on scale $n'n'$ for every line $\l$ inside $T$, (note that the resonance $V$ satisfies such a requirement, but it is not necessary that $T$ is a cluster to make the statement to hold). Let us consider a subdiagram $T$, verifying the above described properties, \eg a resonance $V$ on scale $n_V$, with $n'\le n_V-1$. By assumption there is only one line entering the subdiagram $T$, and essentially by definition there is only one line exiting; by analogy to Definition 3.2, let us call $w_0$ and $w_2$ the two vertices which the two lines, respectively, lead to. Let us call $T_1$, $\ldots$, $T_m$ the subdiagrams emerging from the vertex $w_0$ and with $k(T_j)>E2^{-n/\t}$: by construction, one of such diagram, say the first one, contains the vertex $\p(w_2)$, while the other ones are trees. Let us consider the first branch of the subdiagram $T_1$. If the considered branch scale label is $>n$, then the assertion follows by induction, by using also the previous results on $N_n^*(\th)$ for trees, and the fact that $k(T)>E2^{-n/\t}$ if $N_n(T)\neq0$. Otherwise, it is $N_n^*(T)=1+\sum_{i=1}^mN_n^*(T_i)$; the case $m\ge2$ can be again inductively studied and the statement easily follows. If $m=1$, then $N_n^*(T)=1+N_n^*(T_1)$, and, if $v_1 \in T_1$ is the vertex to which the outgoing line of $T_1$ leads to, we consider the $m'$ subdiagrams emerging from $v_1$: again one of them, say the first one, contains the vertex $\p(w_2)$ of $V$, while the other ones are trees. If $m'\ge1$, again an inductive proof can be performed. If $m'=1$, we have once more a trivial case unless it is $k(T_1)> k(T)-2^{-1}E2^{-n/\t}$. But in this case we can reason as along the proof of the bound on $N_n^*(\th)$, and check that there are only two possibilities: either the line leading to $v_1$ is a resonant line on scale $n$, or it has scale $>n$. The proof of such a statement can be carried out exactly in the same way as to $N_n(\th)$, so that we do not repeat it here. This means that we can write: $N_n^*(T)=1+N_n^*(T_1)=1+N_n^*(T_1')+\ldots+N_n^*(T_{m{'}{'}}')$, and the above analysis can be iterated as many times as it is needed to reach either a trivial case or a subdiagram $\tilde T$ such that $k(\tilde T)\le k(T)-2^{-1}E2^{-n/\t}$. This proves the statement about the number of lines on scale $n$ contained in a resonance, (note the bound we have obtained can be replaced by zero if the scale is $nw_0$, $w\in V$), and let us study its dependence on the mode labels. We see from \equ(3.3) that, if $R_{\l_w}=2$ we can associate to such a line a {\sl line factor} which is given by the product of a factor linear in the mode labels arising from the vertex $w$, times a factor $(i\nn_{\p(w)})_{j_{\l_w}}$ arising from the vertex $\p(w)$, times a propagator $g^{(n_{\l_w})}(\oo\cdot\nn(w))$; if $R_{\l_w}=1$ we associate to it a {\sl line factor} which is given by the product of a factor linear in the mode labels arising from the vertex $w$, times a factor independent on the the mode labels arising from the vertex $\p(w)$, times a propagator $g^{(n_{\l_w})}(\oo\cdot\nn(w))$. Then, for each line inside $V$, the line factor is a homogeneous function of even degree in the mode labels. To the first vertex $w_0$ we associate a factor $(-i\nn_{w_0})_{j_{\l_{w_0}}}(1-\d_{w_0})$ $+(i\oo\cdot\nn(w_0)) \dpr_{B_{j_{\l_{w_0}}}}\,\d_{w_0}$. Since the function $\f_{\nn}(\BB_0)$ is supposed to be even in $\nn$, no other factor has to be considered in order to obtain the behaviour of the resonance, when $\oo\cdot\nn_{\l_V}=0$ and the signs of the mode labels of all the vertices in $V$ are simultaneously changed. When such an operation is performed we see that, if $\z_{\l_{w_0}}=H$, (recall that $\d_{w_0}\=0$ if $\z_{\l_{w_0}}=H$), or $\z_{\l_{w_0}}=h$, with $\d_{w_0}=0$, the overall sign of the resonance factor changes, while, if $\z_{\l_{w_0}}=h$, with $\d_{w_0}=1$, the overall sign of the resonance factor does not change. Now, let us consider separately the possible kinds of resonance, see \equ(3.8) above. \\(1) If $\z_{\l{w_0}}=\z_{\l_{w_2}}=H$, then we deduce from the above discussion that the sign of $\VV_1^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$ changes when all the signs of the mode labels of the vertices in $V$ are changed; then, fixed a set of compatible values of $\nn_w$, $w\in V$, if we sum together the two contributions $\{\nn_w\}_{w\in V}$ and $\{ -\nn_w\}_{w\in V}$, we obtain zero. \\(2) If $\z_{\l_{w_0}}=H$ and $\z_{\l_{w_2}}=h$, we consider all the trees we obtain by detaching from the resonance the subtree with first vertex $w_2$, then reattaching it to all the remaining vertices $w\in V$, and we add also the contributions obtained by the previous ones by an overall sign reversal of the mode labels $\nn_w$: if $\oo\cdot\nn_{\l_V}=0$, no propagator changes, and the only effect of our operation is that one of the vertex factors changes by taking successively the values $(\nn_w)_{j_{\l_{w_2}}}$, $w\in V$. Then we build in this way a quantity proportional to $\sum_{w\in V} (\nn_w)_{j_{\l_{w_2}}}=$ $[\nn(w_2)-\nn(w_0)]_{j_{\l_{w_2}}}\,\=\,0$. If we sum also on a overall change of sign of the $\nn_w$'s, and we take into account the parity in the mode labels of the resonance factor, we obtain a second order zero. \\(3) If $\z_{\lambda_{W_0}}=h$ and $\z_{\lambda_{W_2}}=H$ we note that, when $\d_{w_0}=0$, then the difference from the contribution with $\z_{\l_{w_0}}=H$ reduces to the label $R_{\l_{w_0}}$ which now is $2$ instead of $1$: then the results of the the first item still hold. If $\d_{w_0}=1$, then $\VV_3^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$ vanishes to first order, as it contains a factor $i\oo\cdot\nn_{\l_V}$, see \equ(3.4). \\(4) If $\z_{\lambda_{W_0}}=\z_{\lambda_{W_2}}=h$ we note that, when $\d_{w_0}=0$, then the difference from the contribution with $\z_{\l_{w_0}}=H$ reduces to the label $R_{\l_{w_0}}$ which now is $2$ instead of $1$: then the results of the the second item still hold. If $\d_{w_0}=1$, then $\VV_2^{n_{\l_V}}(0)\,\=\,0$, as it contains a factor $\oo\cdot\nn_{\l_V}$, (as to the first order in the previous item for $\d_{w_0}=1$), and the first derivative with respect to $\oo\cdot\nn_{\l_V}$ in $\oo\cdot\nn_{\l_V}=0$ is still vanishing for parity reasons analogous to those of the case $\VV_2^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$ in fact the only difference is that a factor $(-i\nn_{w_0})_{j_{\l_{w_0}}}$ is missing, (replaced by a derivative $\dpr_{B_{j_{\l_{w_0}}}}$), and a factor $(i\nn_{\p(w_2)})_{j_{\l_{w_2}}}$ replaces $\dpr_{B_{j_{\l_{w}}}}$. 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