\def\mela{\relax} \magnification=\magstep1 %1200 \hsize 14cm \hoffset -0.3cm \baselineskip=16pt \font\sm=cmr8 %%%%%%%%%%%%%%%%%%%%%%%% %file macrodario.tex \catcode`@=11 % %------------------------- comandi riservati --------------------------- % \def\b@lank{ } \newif\if@simboli \newif\if@riferimenti \newif\if@bozze \newif\if@data \def\bozze{\@bozzetrue \immediate\write16{!!! INSERISCE NOME EQUAZIONI !!!}} \newwrite\file@simboli \def\simboli{ \immediate\write16{ !!! Genera il file \jobname.SMB } \@simbolitrue\immediate\openout\file@simboli=\jobname.smb \immediate\write\file@simboli{Simboli di \jobname}} \newwrite\file@ausiliario \def\riferimentifuturi{ \immediate\write16{ !!! Genera il file \jobname.aux } \@riferimentitrue\openin1 \jobname.aux \ifeof1\relax\else\closein1\relax\input\jobname.aux\fi \immediate\openout\file@ausiliario=\jobname.aux} \newcount\eq@num\global\eq@num=0 \newcount\sect@num\global\sect@num=0 \newcount\para@num\global\para@num=0 \newcount\const@num\global\const@num=0 \newcount\lemm@num\global\lemm@num=0 \newif\if@ndoppia \def\numerazionedoppia{\@ndoppiatrue\gdef\la@sezionecorrente{\the\sect@num}} \def\se@indefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\spo@glia#1>{} % si applica a \meaning\xxxxx; butta via tutto quello % che produce \meaning fino al carattere > % (v. manuale TeX, pag. 382, \strip#1>{}). \newif\if@primasezione \@primasezionetrue \def\s@ection#1\par{\immediate \write16{#1}\if@primasezione\global\@primasezionefalse\else\goodbreak \vskip\spaziosoprasez\fi\noindent {\bf#1}\nobreak\vskip\spaziosottosez\nobreak\noindent} %--------------------------- Indice ------------------------------- \newif\if@indice \newif\if@ceindice \newwrite\file@indice \def\indice{ \immediate\write16{Genera il file \jobname.ind} \@indicetrue \immediate\openin2 \jobname.ind \ifeof2\relax\else \closein2\relax \@ceindicetrue\fi \if@ceindice\relax\else \immediate\openout\file@indice=\jobname.ind \immediate\write \file@indice{\string\vskip5pt \string{ \string\bf \string\centerline\string{ Indice \string}\string}\string\par} \fi } \def\quiindice{\if@ceindice\vfill\eject\input\jobname.ind\else\vfill\eject \immediate\write\file@indice{\string{\string\bf\string~ Indice\string}\string\hfill\folio} \null\vfill\eject\null\vfill\eject\relax\fi} % %------------------------------ a disp. dell'utente: sezioni ------------- \def\sezpreset#1{\global\sect@num=#1 \immediate\write16{ !!! sez-preset = #1 } } \def\spaziosoprasez{50pt plus 60pt} \def\spaziosottosez{15pt} \def\sref#1{\se@indefinito{@s@#1}\immediate\write16{ ??? \string\sref{#1} non definita !!!} \expandafter\xdef\csname @s@#1\endcsname{??}\fi\csname @s@#1\endcsname} \def\autosez#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \goodbreak\vskip 48pt plus 60pt \noindent{\bf\the\sect@num.\quad #2} \if@bozze {\tt #1}\fi \par\nobreak\vskip 15pt \nobreak} \def\blankii{\blank\blank} \def\destra#1{{\hfill#1}} \font\titfnt=cmssbx10 scaled \magstep2 \font\capfnt=cmss17 scaled \magstep4 \def\blank{\vskip 12pt} \def\capitolo#1#2\par{ \global\advance\sect@num by 1\if@ndoppia\global\eq@num=0\fi \global\lemm@num=0 \global\para@num=0 \xdef\la@sezionecorrente{\the\sect@num} \def\usa@getta{1}\se@indefinito{@s@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @s@#1\endcsname\la@sezionecorrente\def \usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\sref{#1} !!!}\fi \expandafter\xdef\csname @s@#1\endcsname{\la@sezionecorrente} \immediate\write16{\la@sezionecorrente. #2} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione \la@sezionecorrente : sref. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @s@#1\string\endcsname{\la@sezionecorrente}}\fi \par\vfill\eject \destra{\capfnt {\la@sezionecorrente}\hbox to 10pt{\hfil}} \blankii\noindent{\titfnt\baselineskip=20pt \hfill\uppercase{#2}}\blankii \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf \la@sezionecorrente.#2\string}\string\hfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\par\fi\nobreak} \def\semiautosez#1#2\par{ \gdef\la@sezionecorrente{#1}\if@ndoppia\global\eq@num=0 \fi \global\lemm@num=0 \global\para@num=0 \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ Sezione ** : sref. \expandafter\spo@glia\meaning\la@sezionecorrente} \immediate\write\file@simboli{ }\fi \s@ection#2\par} %------------------paragrafi---------------------------------------- \def\pararef#1{\se@indefinito{@ap@#1} \immediate\write16{??? \string\pararef{#1} non definito !!!} \expandafter\xdef\csname @ap@#1\endcsname {#1} \fi\csname @ap@#1\endcsname} \def\autopara#1#2\par{ \global\advance\para@num by 1 \xdef\il@paragrafo{\la@sezionecorrente.\the\para@num} \vskip10pt \noindent {\bf \il@paragrafo\ #2} \def\usa@getta{1}\se@indefinito{@ap@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @ap@#1\endcsname\il@paragrafo\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 {??? possibili riferimenti errati a \string\pararef{#1} !!!}\fi \expandafter\xdef\csname @ap@#1\endcsname{\il@paragrafo} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\para@num} \if@simboli \immediate\write\file@simboli{ }\immediate\write\file@simboli{ } \immediate\write\file@simboli{ paragrafo \il@paragrafo : pararef. #1} \immediate\write\file@simboli{ } \fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @ap@#1\string\endcsname{\il@paragrafo}}\fi \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\noindent\string\item\string{ \il@paragrafo.\string}#2\string\dotfill\folio\string\par}\fi\fi \if@bozze {\tt #1}\fi\par\nobreak\vskip .3 cm \nobreak} %------------------------------ a disp. dell'utente: equazioni ----------- \def\eqpreset#1{\global\eq@num=#1 \immediate\write16{ !!! eq-preset = #1 } } \def\eqlabel#1{\global\advance\eq@num by 1 \if@ndoppia\xdef\il@numero{\la@sezionecorrente.\the\eq@num} \else\xdef\il@numero{\the\eq@num}\fi \def\usa@getta{1}\se@indefinito{@eq@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @eq@#1\endcsname\il@numero\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\eqref{#1} !!!}\fi \expandafter\xdef\csname @eq@#1\endcsname{\il@numero} \if@ndoppia \def\usa@getta{\expandafter\spo@glia\meaning \il@numero} \else\def\usa@getta{\il@numero}\fi \if@simboli \immediate\write\file@simboli{ Equazione \usa@getta : eqref. #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @eq@#1\string\endcsname{\usa@getta}}\fi} \def\eqsref#1{\se@indefinito{@eq@#1} \immediate\write16{ ??? \string\eqref{#1} non definita !!!} \if@riferimenti\relax \else\eqlabel{#1} ???\fi \fi\csname @eq@#1\endcsname } \def\autoeqno#1{\eqlabel{#1}\eqno(\csname @eq@#1\endcsname)\if@bozze {\tt #1}\else\relax\fi} \def\autoleqno#1{\eqlabel{#1}\leqno(\csname @eq@#1\endcsname)} \def\eqref#1{(\eqsref{#1})} %----------- Lemmi automatici: a disposizione dell'utente ---------------- \def\lemmalabel#1{\global\advance\lemm@num by 1 \xdef\il@lemma{\la@sezionecorrente.\the\lemm@num} \def\usa@getta{1}\se@indefinito{@lm@#1}\def\usa@getta{2}\fi \expandafter\ifx\csname @lm@#1\endcsname\il@lemma\def\usa@getta{2}\fi \ifodd\usa@getta\immediate\write16 { ??? possibili riferimenti errati a \string\lemmaref{#1} !!!}\fi \expandafter\xdef\csname @lm@#1\endcsname{\il@lemma} \def\usa@getta{\expandafter\spo@glia\meaning \la@sezionecorrente.\the\lemm@num} \if@simboli \immediate\write\file@simboli{ Lemma \usa@getta : lemmaref #1}\fi \if@riferimenti \immediate\write\file@ausiliario{\string\expandafter\string\edef \string\csname\b@lank @lm@#1\string\endcsname{\usa@getta}}\fi} \def\autolemma#1{\lemmalabel{#1}\csname @lm@#1\endcsname\if@bozze {\tt #1}\else\relax\fi} \def\lemmaref#1{\se@indefinito{@lm@#1} \immediate\write16{ ??? \string\lemmaref{#1} non definita !!!} \if@riferimenti\else \lemmalabel{#1}???\fi \fi\csname @lm@#1\endcsname} %--------------- bibliografia automatica: riservati ---------------------- \newcount\cit@num\global\cit@num=0 \newwrite\file@bibliografia \newif\if@bibliografia \@bibliografiafalse \newif\if@corsivo \@corsivofalse \def\title#1{{\it #1}} \def\rivista#1{#1} \def\lp@cite{[} \def\rp@cite{]} \def\trap@cite#1{\lp@cite #1\rp@cite} \def\lp@bibl{[} \def\rp@bibl{]} \def\trap@bibl#1{\lp@bibl #1\rp@bibl} \def\refe@renza#1{\if@bibliografia\immediate % scrive su .BIB \write\file@bibliografia{ \string\item{\trap@bibl{\cref{#1}}}\string \bibl@ref{#1}\string\bibl@skip}\fi} \def\ref@ridefinita#1{\if@bibliografia\immediate\write\file@bibliografia{ \string\item{?? \trap@bibl{\cref{#1}}} ??? tentativo di ridefinire la citazione #1 !!! \string\bibl@skip}\fi} \def\bibl@ref#1{\se@indefinito{@ref@#1}\immediate \write16{ ??? biblitem #1 indefinito !!!}\expandafter\xdef \csname @ref@#1\endcsname{ ??}\fi\csname @ref@#1\endcsname} \def\c@label#1{\global\advance\cit@num by 1\xdef % assegna il numero \la@citazione{\the\cit@num}\expandafter \xdef\csname @c@#1\endcsname{\la@citazione}} \def\bibl@skip{\vskip 5truept} %------------------------ bibl. automatica: a disp. dell'utente ------------ \def\stileincite#1#2{\global\def\lp@cite{#1}\global \def\rp@cite{#2}} \def\stileinbibl#1#2{\global\def\lp@bibl{#1}\global \def\rp@bibl{#2}} \def\corsivo{\global\@corsivotrue} \def\citpreset#1{\global\cit@num=#1 \immediate\write16{ !!! cit-preset = #1 } } \def\autobibliografia{\global\@bibliografiatrue\immediate \write16{ !!! Genera il file \jobname.BIB}\immediate \openout\file@bibliografia=\jobname.bib} \def\cref#1{\se@indefinito % se indefinito definisce {@c@#1}\c@label{#1}\refe@renza{#1}\fi\csname @c@#1\endcsname} \def\upcref#1{\null$^{\,\cref{#1}}$} \def\cite#1{\trap@cite{\cref{#1}}} % [5] \def\ccite#1#2{\trap@cite{\cref{#1},\cref{#2}}} % [5,6] \def\ncite#1#2{\trap@cite{\cref{#1}--\cref{#2}}} % [5-8] senza definire \def\upcite#1{$^{\,\trap@cite{\cref{#1}}}$} % ^[5] \def\upccite#1#2{$^{\,\trap@cite{\cref{#1},\cref{#2}}}$} % ^[5,6] \def\upncite#1#2{$^{\,\trap@cite{\cref{#1}-\cref{#2}}}$} % ^[5-8] senza def. \def\clabel#1{\se@indefinito{@c@#1}\c@label % sola definizione {#1}\refe@renza{#1}\else\c@label{#1}\ref@ridefinita{#1}\fi} \def\cclabel#1#2{\clabel{#1}\clabel{#2}} % def. doppia \def\ccclabel#1#2#3{\clabel{#1}\clabel{#2}\clabel{#3}} % def. tripla \def\biblskip#1{\def\bibl@skip{\vskip #1}} % spaziatura nella bibl. \def\insertbibliografia{\if@bibliografia % scrive la bibliografia \immediate\write\file@bibliografia{ } \immediate\closeout\file@bibliografia \if@indice \if@ceindice\relax\else\immediate\write \file@indice{\string\vskip5pt\string{\string\bf\string~ Bibliografia\string}\string\hfill\folio\string\par}\fi\fi \catcode`@=11\input\jobname.bib\catcode`@=12\fi } %--------- per comporre il file con la bibliografia -------------- \def\commento#1{\relax} \def\biblitem#1#2\par{\expandafter\xdef\csname @ref@#1\endcsname{#2}} % ricordare: una lista in chiaro della bibliografia si % ottiene eseguendo $ TEX BIBLIST %---------------- titolo in cima alla pagina, data.----------------- \def\data{\number\day.\number\month.\number\year} \def\datasotto{\@datatrue \footline={\hfil{\rm \data}\hfil}} \def\titoli#1{\if@data\relax\else\footline={\hfil}\fi \xdef\prima@riga{#1}\voffset+20pt \headline={\ifnum\pageno=1 {\hfil}\else\hfil{\sl \prima@riga}\hfil\folio\fi}} \def\duetitoli#1#2{\if@data\relax\else\footline={\hfil}\fi \voffset=+20pt \headline={\ifnum\pageno=1 {\hfil}\else{\ifodd\pageno\hfil{\sl #2}\hfil\folio \else\folio\hfil{\sl #1}\hfil\fi} \fi} } \def\la@sezionecorrente{0} % ------------------COSTANTI --------------------------------- \def\const@label#1{\global\advance\const@num by 1\xdef \la@costante{\the\const@num}\expandafter \xdef\csname @const@#1\endcsname{\la@costante}} \def\cconlabel#1{\se@indefinito{@const@#1} \const@label{#1}\fi} \def\constnum#1{\se@indefinito{@const@#1} \const@label{#1}\fi\csname @const@#1\endcsname} \def\ccon#1{C_{\constnum{#1}}} \catcode`@=12 %------------------ FORMATI TEOREMI E GENERALI -------------------- \def\abstract{ \vskip48pt plus 60pt \noindent {\bf Abstract.}\quad} \def\summary{\vskip48pt plus 60pt \noindent {\bf Summary.}\quad} \def\firma{\noindent \centerline{Dario BAMBUSI}\par\noindent \centerline{Dipartimento di Matematica dell'Universit\`a,}\par\noindent \centerline{Via Saldini 50, 20133 Milano, Italy.}\par} \def\teorema#1#2{\par\vskip4pt \noindent {\bf Teorema \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\esempio#1#2{\par\vskip4pt \noindent {\bf Esempio \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\definizione#1#2{\par\vskip4pt \noindent {\bf Definizione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\dimostrazione{\par\noindent{\bf Dimostrazione.}\ } \def\theorem#1#2{\par\vskip4pt \noindent {\bf Theorem \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\semitheorem#1{\par\vskip4pt \noindent {\bf Theorem.}{\sl \ #1} \par\vskip10pt} \def\lemma#1#2{\par\vskip4pt \noindent {\bf Lemma \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\proof{\par\noindent{\bf Proof.}\ } \def\proposizione#1#2{\par\vskip4pt \noindent {\bf Proposizione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\proposition#1#2{\par\vskip4pt \noindent {\bf Proposition \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\corollario#1#2{\par\vskip4pt \noindent {\bf Corollario \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\osservazione#1#2{\par\vskip4pt \noindent {\bf Osservazione \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\corollary#1#2{\par\vskip4pt \noindent {\bf Corollary \autolemma{#1}.}{\sl \ #2} \par\vskip10pt} \def\remark#1#2{\par\vskip4pt \noindent {\bf Remark \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} \def\definition#1{\par\vskip2pt \noindent {\bf Definition.}{\sl \ #1} \par\vskip2pt} \def\definitionn#1#2{\par\vskip4pt \noindent {\bf Definition \autolemma{#1}.}{\sl \ #2} \par\vskip4pt} %------------------- ROUTINE DI USO GENERALE ----------------------- \def\norma#1{\left\Vert#1\right\Vert} \def\perogni{\forall\hskip1pt} \def\meno{\hskip1pt\backslash} \def\frac#1#2{{#1\over #2}} \def\fraz#1#2{{#1\over #2}} \def\interno{\vbox{\hbox{\vbox to .3 truecm{\vfill\hbox to .2 truecm {\hfill\hfill}\vfill}\vrule}\hrule}\hskip 2pt} \def\quadratino{ \hfill\vbox{\hrule\hbox{\vrule\vbox to 7 pt {\vfill\hbox to 7 pt {\hfill\hfill}\vfill}\vrule}\hrule}\par} \font\strana=cmti10 \def\lie{\hbox{\strana \char'44}} \def\ponesotto#1\su#2{\mathrel{\mathop{\kern0pt #1}\limits_{#2}}} \def\Sup{\mathop{{\rm Sup}}} %\def\Sup#1{\hskip2pt\ponesotto{{\rm Sup}}\su{#1}} \def\tdot#1{\hskip2pt\ddot{\null}\hskip2.5pt \dot{\null}\kern -5pt {#1}} \def\diff#1#2{\frac{\partial #1}{\partial #2}} \def\base#1#2{\frac{\partial}{\partial#1^{#2}}} \def\charslash#1{\setbox2=\hbox{$#1$} \dimen2=\wd2 \setbox1=\hbox{/}\dimen1=\wd1 \ifdim\dimen2>\dimen1 \rlap{\hbox to \dimen2{\hfil /\hfil}} #1 \else \rlap{\hbox to \dimen1{\hfil$#1$\hfil}} / \hfil\fi} \def\Re{{\rm \kern 0.4ex I \kern -0.4 ex R}} \def\Sh{{\rm Sh}\hskip1pt} \def\Ch{{\rm Ch}\hskip1pt} \def\poisson#1#2{\left\{#1 ,#2\right\} } \def\toro{{\bf T}} \def\Na{{\bf N}} \def\ra{{\bf Z}} \def\Ra{{\bf Z}} \def\id{{\bf 1}} \def\Cm{{\bf C}} \def\reale{{\rm Re}\hskip2pt} \def\imma{{\rm Im}\hskip2pt} \def\rin{{\bf Z}} \def\pmb#1{\setbox0=\hbox{#1}\ignorespaces \hbox{\kern-.02em\copy0\kern-\wd0\ignorespaces \kern.05em\copy0\kern-\wd0\ignorespaces \kern-.02em\raise.02em\box0 }} \def\vett#1{\pmb{$#1$}} \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\D{{\cal D}} \def\E{{\cal E}} \def\F{{\cal F}} \def\G{{\cal G}} \def\H{{\cal H}} \def\I{{\cal I}} \def\L{{\cal L}} \def\M{{\cal M}} \def\N{{\cal N}} \def\O{{\cal O}} \def\P{{\cal P}} \def\Q{{\cal Q}} \def\R{{\cal R}} \def\S{{\cal S}} \def\T{{\cal T}} \def\U{{\cal U}} \def\V{{\cal V}} \def\W{{\cal W}} \def\Z{{\cal Z}} \def\J{{\cal J}} \def\sym{\nabla^\Omega} \def\uno{{\kern+.3em {\rm 1} \kern -.22em {\rm l}}} \def\unpo{\vskip3pt} \def\unp{\vskip6pt} \def\a{\`a\ } \def\o{\`o\ } \def\e{\`e\ } %------------------------------ F I N E --------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%% %\input /home/bambusi/macrodario.tex %\input macrodario.tex %\input /home/bambusi/dariobiblio.tex %\input egorov.bib %%%%%%%%%%%%%%%%% CARATTERI \newskip\ttglue \font\dodicirm=cmr12\font\dodicibf=cmbx12\font\dodiciit=cmti12 %%%%%%% \font\ottorm=cmr8\font\ottoi=cmmi8\font\ottosy=cmsy8 \font\ottobf=cmbx8\font\ottott=cmtt8 \font\ottosl=cmsl8\font\ottoit=cmti8 \font\sixrm=cmr6\font\sixbf=cmbx6\font\sixi=cmmi6\font\sixsy=cmsy6 \font\fiverm=cmr5\font\fivesy=cmsy5\font\fivei=cmmi5\font\fivebf=cmbx5 %\def\ottopunti{\def\rm{\fam0\ottorm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \riferimentifuturi \numerazionedoppia %\bozze %\datasotto %\titoli{egorov} \def\Ne{{\rm \kern 0.4ex I \kern -0.4 ex N}} \def\Ce{{\rm \kern 0.4ex I \kern -0.4 ex C}} \def\ha{Ha\-mil\-to\-nian} \def\la{\langle} \def\ra{\rangle} \def\cl{\A} \def\mod#1{\left|#1\right|} \def\op#1{Op^w\left(#1\right)} \def\h{\H} \def\De{\Delta_{\h}} \def\moyal#1#2{\poisson{#1}{#2}_M} \def\indi#1{t-\tau_1,\tau_1,\tau_2,...,\tau_{#1}} \def\st{\sigma e^{-\alpha |t|}} \def\rt{\rho e^{-\alpha |t|}} \def\ds{\displaystyle} \def\Jb{\overline{J}} \def\Hb{\overline{H}} \def\Eb{\overline{E}} \def\om{\omega} \def\Om{\Omega} \def\ep{\epsilon} \def\gk{\tilde{g}_k} \def\Ha{Ha\-mil\-to\-nian} \centerline{\bf LONG TIME SEMICLASSICAL APPROXIMATION OF QUANTUM FLOWS:} \centerline{\bf A PROOF OF THE EHRENFEST TIME} \vskip 24pt \centerline{Dario BAMBUSI\footnote{$^1$} {\sm Dipartimento di Matematica, Universit\`a di Milano, Italy. (bambusi@mat.unimi.it)}, Sandro GRAFFI\footnote{$^2$} {\sm Dipartimento di Matematica, Universit\`a di Bologna, Italy. (graffi@dm.unibo.it)}, Thierry PAUL\footnote{$^3$} {\sm CEREMADE, Universit\'e de Paris-Dauphine, France. (paulth@ceremade.dauphine.fr)}} \abstract{\it Let $ \h$ be a holomorphic \Ha\ of quadratic growth on $ \Re^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the corresponding operators on $L^2(\Re^n)$ generated by Weyl quantization, and $U(t)=\exp{iHt/\hbar}$. It is proved that the $L^2$ norm of the difference between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical approximation of order ${N-1}$ is majorized by $K N^{(6n+1)N}(-\hbar{\rm log}\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $ \ds T_N(\hbar)=-{2{\rm log}\hbar\over {N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{\log|\log\hbar|}$, $0\leq t\leq |\log\hbar|/\log|\log\hbar|$. (Here $K,C$ are constants independent of $N,\hbar$).} \autosez{p}Introduction and statement of the results Denote $\Omega:=\Re^{2n}$ with coordinates $(x,\xi)$. Let $\h(x,\xi)\in C^{\infty}(\Omega;\Re)$, and $b_t(x,\xi):=b\circ \phi^{\h}_t\equiv b(\phi^{\h}_t(x,\xi))$ be the time evolution of any bounded observable $b(x,\xi)\in C^{\infty}(\Omega;\Re)$ under the the flow $\phi^{\h}_t:\Om\leftrightarrow\Om$ generated by the \ha\ $\h$. Denote $H:=Op^W(\h)$ and $B=Op^W(b)$ the self-adjoint operators in $L^2(\Re^n)$ representing the (Weyl) quantization of the symbols $\h,b$ and let $\ds B_t:=e^{iHt/\hbar}Be^{-iHt/\hbar}$ be the Heisenberg observable, i.e. the quantum evolution of the observable $B$ under the unitary group generated by $H$. \par The question of estimating how long the classical and quantum evolutions stay "close" one another or, better, how long the evolution of the quantum observables is determined by the corresponding classical one up to a prescribed error vanishing with $\hbar$ is one of the oldest problems of semiclassical analysis. According to a well known conjecture going back to Chirikov and Zaslavski [Ch,Za], this approximation can be valid on a time interval of maximum duration $T\equiv T(\hbar)$ of order $-{\rm log}\hbar$, called the Ehrenfest time, if the error is required to vanish faster than any power of $\hbar$. The origin of this conjecture, formally verified in some instances[Za] can be understood in the correspondence between symbols $b(x,\xi)$ (classical observables) and operators in Hilbert space $B$ (quantum observables) provided by the Weyl quantization procedure: $$ (Bu)(x)={1\over (2\pi\hbar^n)} \int_{\Re^{2n}}b\left({{x+y}\over 2},\xi\right)e^{i\langle (x-y),\xi\rangle/\hbar}\,u(y)\,dyd\xi, \quad u\in{\cal S}(\Re^n) \autoeqno{Weyl} $$ In this framework the problem can be formulated as follows: $B_t$ solves the Heisenberg equation of motion $$ \dot{B}_t={i\over\hbar}[H,B_t] \autoeqno{Heisenberg} $$ If $B_t$ admits a symbol, denoted $b_t(x,\xi;\hbar)$, by \eqref{Heisenberg} it fulfills the equation $$ \dot{b}_t=\{\h,b_t\}_M \autoeqno{Moyal} $$ with the initial condition $b_0(x,\xi;\hbar)=b(x,\xi)$. Here $\{f,g\}_M(x,\xi)$ is the Moyal bracket of the two observables $f,g\in C^{\infty}(\Re^{2n})$ $$ \{f,g\}_M(x,\xi):=f\#g-g\#f \autoeqno{Moyaldef} $$ where $f\#g$, the symbol of the operator product $FG$, is expressed by the composition of the symbols $f$ and $g$: $$ (f\#g)(x,\xi)={1\over (2\pi\hbar)^{n/2}}\int_{\Re^{4n}}e^{-i\la r,\rho\ra/\hbar\,+i\la w,\tau\ra/\hbar}\,f(x+w,\rho+\xi)g(x+r,\tau+\xi)\,d\rho d\tau\,dr dw \autoeqno{composizione} $$ $\{f,g\}_M$ admits the following formal expansion in powers of $\hbar$ [Fo,Ro,Vo]: $$ \{f,g\}_M(x,\xi)\sim\{f,g\}+{1\over2^j}\sum_{|\alpha+\beta|=j\geq 1}(-1)^{|\beta|}\hbar^j \left(\partial^{\alpha}_{\xi}gD^{\beta}_xg\right)\cdot \left(\partial^{\beta}_{\xi}gD^{\alpha}_xf\right) \autoeqno{Moyalexp} $$ ($\alpha=(\alpha_1,\ldots,\alpha_n)$ is a multi-index, and $|\alpha|:=\alpha_1+\ldots+\alpha_n$; analogous definitions for $\beta$, and $\ds D_x:=-i\hbar\partial_x$). By \eqref{Moyalexp} the differential equation \eqref{Moyal} can be recursively solved in the space of the formal power series in $\hbar$ (for details see [Ro], Chapt.IV.10). The result, known as the semiclassical Egorov theorem, is the formal semiclassical expansion of the symbol $b_t$: $$ b_t(x,\xi;\hbar)\sim (b\circ \phi_t^a)(x,\xi)+\sum_{j=2}^{\infty}b_j(x,\xi;t)\hbar^j \autoeqno{Eg1} $$ Here the term of order zero in $\hbar$, by definition the {\it principal symbol} of $B_t$, is just the evolution of the observable $b$ along the \Ha\ flow generated by $\h$, i.e. the solution of the Liouville equation $\dot{b}_t=\{\h,b_t\}$, and $$ b_j(x,\xi;t)=-i\int_0^t\sum_{|\alpha+\beta|+l=j+1 \atop 0\leq l\leq j-1} (1-(-1)^{|\alpha+\beta|})\Gamma(\alpha,\beta) \left(\partial^{\alpha}_{\xi}\h D^{\beta}_xb_l\right)\circ \phi^{\h}_{t-\tau}(x,\xi)\,d\tau \autoeqno{Eg2} $$ The higher order terms $b_j(x,\xi;t)$ are thus completely determined by the classical evolution but have a polynomial dependence on the derivatives of the flow $\phi^{\h}_t(x,\xi)$ with respect to the inital conditions $(x,\xi)$ up to order $j-1$. If, as it happens in general, there are initial conditions $(x,\xi)$ generating a flow with positive Lyapunov exponents, the difference between the symbol $b_t(x,\xi;\hbar)$ of $B_t$ and any prescribed approximation $\ds (b\circ \phi_t^{\h})(x,\xi)+\sum_{j=2}^{N}b_j(x,\xi;t)\hbar^j$ is expected to increase exponentially in time: hence it can vanish as $\hbar\to 0$ only for a time interval not exceeding $-{\rm log}\hbar$. Put in a different way: the non-local nature of quantum mechanics, embodied in the symbol expansion \eqref{Eg1}, \eqref{Eg2}, can be dominated by its local approximation, the principal symbol $b\circ \phi_t^{\h}(x,\xi)$, only if the the remainder is small. This can be obtained only within the above time span. \par In this paper we work out, in the analytic case, the estimates implying the validity of the above "Ehrenfest time" for a class of flows somewhat restricted but in a sense natural as discussed below. More precisely, for any fixed $\sigma >0$ set $|z|:=\sup|z_k|$ and $\ds \G_{\sigma}:=\left\{z\in \Cm^{2n} \ :\ |\imma z|<\sigma\right\}$. The \Ha\ $\h(x,\xi)\equiv \h(z)$, $z:=(x,\xi)$ is required to fulfill the following properties:\par\noindent \item {(A1)} There exists $\nu>0$ such that $\h$ is real-holomorphic on $\G_{\nu}$. \item{(A2)} Let $Jd\h$ be the symplectic gradient of $\h$. Then there are $A_1>0, A_2>0, \alpha>0$ such that $\left|Jd\h(z+iy)\right|\leq A_1+A_2|z|$ $\forall z, y\in\Re^{2n}$, $\left|y\right|\leq\sigma$. Moreover $|Jd^2\h(z)|\leq \alpha$ on $\G_\sigma$. \item{(A3)} Denote $(\hat\h)(k)$ the Fourier transform of $\h(z)$. Then there are $\rho>0, \sigma>0$ such that $\hat\h(k_1+ik_2)$ is holomorphic on $\G_{\rho}\setminus (0,0)$; moreover $k^3\hat\h(k)$ is holomorphic on $\G_{\rho}$ and $$ |k_1|^3|\hat \h(k_1+ik_2)|\leq C e^{-\sigma|k_1|} \quad {\rm for}\ |k_2|\leq \rho $$ \par\noindent {\bf Remarks}. \item 1 Under the above assumptions $H=Op^W(\h)$ defined by \eqref{Weyl} is essentially self-adjoint in $L^2(\Re^n)$. By a standard abuse of notation we denote $H$ also its self-adjoint closure. \item 2 Within the analyticity and decay assumptions (A1)-(A3), (A2) is the quadratic growth condition ensuring the existence of the Fourier integral operator representing the pro\-pa\-gator $\ds \exp{iHt\over \hbar}$ [Cha] and thus the existence of the symbol of $B_t$ [Ro]. \item 3 In the phase variables $z=(x,\xi)$ Assumption (A3) means that there are $\sigma,\rho>0$ such that $$ \sup_{z+iy\in \G_{\sigma}}\mod{\partial^{|\alpha|+|\beta|=3}_z\h(z+iy)}e^{\rho|z|}<+\infty .\autoeqno{n.13} $$ \vskip 0.1cm\noindent To state the main result of the paper we need some further notation. For $b$ as above set: % % $$ \De b:=\frac1{\hbar^2}\left[\poisson b{\h}-\moyal{b}{\h}\right] \ .\autoeqno{De} $$ and define recursively the two sequences $r_k^t, b^t_k: k\geq 1$ in the following way: $$ r_1^{t-\tau_1}:=\De (b\circ \phi^{t-\tau_1}),\quad r_{k+1}^{\indi{k}}:=\De\left[r_k^{\indi{k-1}}\circ\phi^{\tau_k}\right] \autoeqno{b_j} $$ $$ b_k^t:=\int_0^td\tau_1\int_0^{t-\tau_1}d\tau_2\int_0^{t-\tau_2}d\tau_3... \int_0^{t-\tau_{k-1}}d\tau_kr_k^{\indi{k-1}}\circ \phi^{\tau_k}; b_0:=b\circ\phi^t $$ Moreover, let $b:\G_{\sigma}\to \Cm$ be holomorphic. Set: $$ \mod b_{\sigma,\rho}:=\sup_{x+iy\in \G_{\sigma}}\mod{b(x+iy)}e^{\rho|x|}\ .\autoeqno{n.1} $$ Denote $\cl_{\sigma,\rho}$ the set of all functions $f$ holomorphic on $\G_{\sigma,\rho}$ such that $\mod f_{\sigma,\rho}<+\infty$. Then: \theorem{1.1}{Let there exist $\sigma >0, \rho >0$ and $0<\overline{B}<+\infty$ such that $\mod b_{\sigma,\rho} <\overline{B}$. Then: \item{(1)} The operators $B^j_t:=Op^W(b_j^t)$ are continuous in $L^2$ and the Heisenberg operator $B_t=U(t)BU(-t)$ admits the expansion $$ B_t=\sum_{j=0}^NB_j^t\hbar^{2j}+\hbar^{2(N+1)}S_N^t\ , $$ where $$ S_N^t:=\int_0^td\tau_1\int_0^{t-\tau_1}d\tau_2\int_0^{t-\tau_2}d\tau_3... \int_0^{t-\tau_{k-1}}d\tau_k U(\tau_k)\op{r_k^{\indi{k-1}}}U(-\tau_k) $$ \item{(2)} There are positive constants $E, F$ independent of $j,N$ and $\hbar$ such that for all $j\geq 1$, $N\geq 2$, $t\geq 0$ the following estimates hold: $$ \norma{B^t_j}_{L^2\to L^2}\leq \left[eEe^{7\alpha t}j^{6n+3}\right]^j {\overline{B}F\over j!}e^{(4n+2)\alpha t} \left[\exp\left( \alpha\frac{j(j-1)}2t\right)\right]^{6n+3} \autoeqno{stimaj} $$ $$ \norma{S_N^t}_{L^2\to L^2}\leq \left[eEe^{7\alpha t}N^{6n+3}t\right]^N \frac{\overline{B}F}{N!}e^{(4n+2)\alpha t} \left[\exp\left( \alpha\frac{N(N-1)}2t\right)\right]^{6n+3} \autoeqno{stimaresto} $$} \noindent {\bf Remark.} The holomorphy assumptions are needed to control the remainder to order $\hbar^N$ for all $N$. If we limit ourselves to $N=1$ more general classes of \ha s and of observables can be considered. More precisely, let for instance $\h(x,\xi)$ be a polynomial of order $2p$ such that the subgraph $\Sigma_E:=\{(x,\xi)\in\Re^{2n}|\h(x,\xi)\leq E\}$ is compact for some $E$, and let $b(x,\xi)\in C_0^{\infty}(\Sigma_{{E}})$. Then (proof in the next section) there are $\Gamma>0$ and $\Delta>0$ such that: $$ \norma{B^t-B^t_0}_{L^2\to L^2}=\norma{B^t-Op^W(b\circ\phi^t)}_{L^2\to L^2}\leq \Gamma\hbar^2 t e^{\Delta t}\;. \autoeqno{stima1ord} $$ \vskip 0.4cm\noindent The symbols $b_j^t$ and hence the operators $B_j^t$ are completely determined by the classical flow $\phi^t$ via \eqref{b_j}. The quantum evolution will then stay close to the (semi) classical one as long as the error $S_N^t$ stays small. The estimate \eqref{stimaresto} yields indeed, through a straightforward computation: \corollary{1.12}{Let $\ds T_N(\hbar):=-{2{\rm log}\hbar\over \alpha(N-1)}$, $\ds B^N_t:=\sum_{j=O}^{N-1}B_j^t\hbar^j$. Then, for $0\leq t\leq T_N(\hbar)$: $$ \norma{B_t-B^N_t}\leq \left(\frac{2e^2E}{\alpha }\right)^N \,N^{(6n+1)N}BF \hbar^{2-15/\alpha-(8n+4)/\alpha N} (-\hbar{\rm log}\hbar)^N \autoeqno{stimaN} $$ } \noindent {\bf Remarks.} \item{1} If the Lyapunov numbers are zero for any initial datum $(x,\xi)$, then we can take $\alpha=0$ in formula \eqref{stimaresto}, and by Assertion 2 of Theorem 1.2 one has $$ \norma{B_t-B^{N(\hbar)}_t} =O(\hbar^N) \qquad 0\leq t\leq \tilde{T}_N(\hbar) $$ where $\ds \tilde{T}_N(\hbar):= e^{-1} N^{-6n-1}h^{-1}$. \item{2} Estimates valid for a time interval of duration $-C_N{\rm log}\hbar$ for Hamiltonians admitting polynomial growth of any order (but without control of the constant $C_N$), have been obtained by Combescure and Robert [Co-Ro1] in a weaker sense, i.e. comparing classical and quantum evolutions along coherent states (according to ideas introduced in [He], [BZ] and developed in [Ha], [BIZ], [Co-Ro2]). \item {3} The symbol expansion generated by Assertion (1) of Theorem \lemmaref{1.1}, namely $$ b_t(x,\xi;\hbar)=\sum_{j=0}^N b^t_j\hbar^{2j}+O(\hbar^{2N+1}) $$ differs from \eqref{Eg1} in all terms with $j>0$. This difference makes the present expansion a non formal one, so that its remainder can be estimated. \item{4} Finally, let $T(\hbar)\in C([0,1];\Re_+)$ be an increasing function such that $\ds \lim_{\hbar\to 0}\frac{T(\hbar)}{-{\rm log}\hbar}=0$, and let $\ds N(\hbar):=\left[-\frac{{\rm log}\hbar}{T(\hbar)}\right]$. Then clearly $\norma{B_t-B^{N(\hbar)}_t} =O(\hbar^{\infty}) $, $0\leq t\leq T(\hbar)$. \vskip 0.1cm\noindent To put this result into a more quantitative version, define the function sequence $\{\log^{[k]}(x)\}: k\in\Ne$ by $\log^{[1]}(x):=\log(x)$, $\log^{[k]}(x):=\log(\log^{[k-1]}(x))$. \corollary{hinfinito}{For any integer $k\geq 1$ define $N_k(\hbar):=[\log^{[k]}(|\log(\hbar)|)]$. Then there exist positive constants $C$, $\overline{\hbar}$ such that, for $0<\hbar\leq\overline{\hbar}$ one has $$ \norma{B_t-B^{N_k(\hbar)}_t} \leq C\; \hbar^{\log^{[k]}(|\log(\hbar)|)} $$ for $$ 0\leq t\leq \frac{|\log(\hbar)|}{\log^{[k]}(|\log(\hbar)|)} $$ } \noindent{\bf Acknowledgments.} We thank A.Martinez and D.Robert for reading the paper and several useful remarks. We acknowledge the support of CEREMADE that made possible the collaboration leading to this work. \autosez{p1}Proofs Let $b$ be a Weyl symbol of class $\Sigma_0^1$, and $\h$ an admissible semiclassical symbol (For these notions, see [Ro], Chapter 2; particular examples are all bounded observables $b(x,\xi)\in C^{\infty}(\Re^{2n})$ and the Hamiltonians $\h\in C^{\infty}(\Re^{2n})$ of polynomial growth at infinity). Denote $\phi^t$ the flow generated by $Jd\h$, $J$ the unit $2n\times 2n$ symplectic matrix; let $H=\op{\h}$ be essentially self-adjoint in $L^2(\Re^n)$ and denote also $U(t):=\exp(itH/\hbar)$, $B:=\op b$, $B_t:=U(t)BU(-t)$, and $$ \De b:=\frac1{\hbar^2}\left[\poisson b{\h}-\moyal{b}{\h}\right] \ .\autoeqno{De.1} $$ Our semiclassical expansion is generated by the following simple remark: \lemma{1.2}{The following formula holds $$ B_t:=\op{b\circ\phi^t}+\hbar^2\int_0^td\tau U(\tau)\op{r_1^{t-\tau}}U(-\tau)\ , $$ where $r_1^s:=\De\left(b\circ \phi^s \right)$} \proof Denote $\beta_t:=b\circ \phi^t$. Then: $$ \eqalign{ \frac d{dt}\left[\op{\beta_t}\right]=\op{\frac d{dt}\beta_t}= \op{\poisson{\beta_t}{\h}} \cr =\op{\poisson{\beta_t}{\h}-\poisson{\beta_t}{\h}_M}+\frac i{\hbar}\left[\op{\beta_t}, \op{\h}\right] \cr =\frac i{\hbar}\left[\op{\beta_t}, \op{\h}\right]+\hbar^2\op{r_1^t}\ . }$$ It follows $$ \frac d{dt}\left[\op{\beta_t}-B_t\right]=\frac i{\hbar}\left[\op{\beta_t}-B_t, \op{\h}\right]+\hbar^2\op{r_1^t}\ , $$ and by the variation of parameters formula $$ \op{\beta_t}-B_t=\hbar^2\int_0^t U(t-s)\op{r_1^s}U(-(t-s))ds\ . \autoeqno{diff1} $$ The assertion is now proved performing the change of variable $\tau=t-s$ in the integral.\quadratino \vskip 0.1cm\noindent {\bf Proof of formula \eqref{stima1ord}}. Since $\h$ is a polynomial of degree $2p$ $$ r_1^t=\De (b\circ\phi^t)=\frac1{\hbar^2}\left[\poisson {b\circ\phi^t}{\h}-\moyal{b\circ\phi^t}{\h}\right] =\sum_{{|k|=1\atop k=(k_1,...,k_{2n})}}^{2p}c_k\hbar^{k}\frac{\partial^{|k|}b\circ\phi^t}{\partial z^k} $$ where $c_k(x,\xi)$ is a polynomial of degree $2p-|k|$. Now the smooth functions $\ds \theta_k(x,\xi):=c_k(x,\xi)\frac{\partial^{|k|} b\circ\phi^t}{\partial z^k}$ have compact support in $\Re^{2n}$ and hence define bounded operators in $L^2$ upon Weyl quantization. Denote $\lambda(x,\xi)$ the Lyapunov number of the trajectory $\phi^t$ with any initial datum $(x,\xi)\in\Sigma_{{E}}$. Since $\phi^t(x,\xi)$ is bounded $\forall\,t\in\Re$ we have (see e.g.[Ce], 3.12) $\ds \delta:={\rm sup}_{\Sigma_{{E}}}\lambda(x,\xi)<+\infty$. Hence there are $\gamma_k>0$ such that $\ds {\rm sup}_{\Sigma_{{E}}}\left|\frac{\partial^{|k|}\phi^t}{\partial z^k}\right|\leq \gamma_ke^{\delta t}$. Since $\ds \frac{\partial^{|k|}b\circ\phi^t}{\partial z^k}$ is a polynomial of degree $|k|$ in the variables $\ds \frac{\partial^{|s|}\phi^t}{\partial z^s},\, s=1,\ldots,|k|$ with coefficients depending on $\ds \frac{\partial^{|s|}b}{\partial z^s},\, s=1,\ldots,|k|$, for any fixed $q\in{\bf N}$ depending only on $n$ there are $\Gamma_k(n)>0$ such $\ds {\rm sup}_{\Re^n}\left|\frac{\partial^{|l|}\theta_k}{\partial z^l}\right|\leq \Gamma_ke^{|k|q\delta t}, |l|\leq q$. Hence by the Calderon-Vaillancourt theorem there exists $q>0$ such that $ \ds \|Op^W(\theta_k)\|_{L^2\to L^2}\leq \Gamma_ke^{|k|q\delta t}$. Inserting this estimate in \eqref{diff1} we get \eqref{stima1ord} with $\Delta=2pq, \Gamma={\rm Max}_k \Gamma_k$ .\quadratino \vskip 0.2cm\noindent Recall now the definition of the sequences $r_k^{\indi{k-1}}$ ($k\geq2$) and $\left\{b_k^t\right\}_{k\geq 0}$, $b_0:=b$: $$ \eqalign{ r_{k+1}^{\indi{k}}&:=\De\left[r_k^{\indi{k-1}}\circ\phi^{\tau_i}\right] } $$ $$ b_k^t:=\int_0^td\tau_1\int_0^{t-\tau_1}d\tau_2\int_0^{t-\tau_2}d\tau_3... \int_0^{t-\tau_{k-1}}d\tau_kr_k^{\indi{k-1}}\circ \phi^{\tau_k}\;\; k\geq 1 $$ \lemma{1.3}{Let $B^t_j=Op^W(b^t_j)$. Then: $$ \eqalign{ B_t=\sum_{j=0}^NB_j^t\hbar^{2j}+\hbar^{2(N+1)}S_N\ , } $$ where $$ S_N:=\int_0^td\tau_1\int_0^{t-\tau_1}d\tau_2\int_0^{t-\tau_2}d\tau_3... \int_0^{t-\tau_{k-1}}d\tau_k U(\tau_k)\op{r_k^{\indi{k-1}}}U(-\tau_k) $$ } \proof Just iterate the proof of lemma \lemmaref{1.2}\quadratino \vskip 0.1cm\noindent Let $b:\G_{\sigma}\to \Cm$ be an analytic function; recall the definitions $$ \mod b_{\sigma,\rho}:=\sup_{x+iy\in \G_{\sigma}}\mod{b(x+iy)}e^{\rho|x|}\ .\autoeqno{n.111} $$ and $\cl_{\sigma,\rho}:=\{b\;{\rm holomorphic}\;{\rm in}\; \G_{\sigma}: \mod b_{\sigma,\rho} <+\infty\}$. We will estimate the sequence $r_k$ in the above norm. Clearly we have to estimate the norm of $b\circ \phi^t$ and of $\De b$. We first prove the following \lemma{phi1}{There exists a positive $\sigma$ such that $\phi^t$ extends to a complex analytic function $$ \phi^t:\G_{\sigma e^{-\alpha t}}\to \G_{\sigma} $$ } \proof Denote $f:=Jd\h$, and consider, on $\G_{\sigma}$, the system of equations $$ \dot z=f(z) \ .\autoeqno{1.2} $$ Writing $z=x+iy$ and $f=f_1+if_2$, one has $\dot y=f_2(x+iy)$. Since $f_2=0$ on the real axis, by assumption A2 one has $|f_2(x+iy)|\leq \alpha |y|$. It follows that the inequalities $$ |y|\dot{\null}\leq\alpha|y|\quad\Longrightarrow |y(t)|\leq |y_0|e^{\alpha |t|} \autoeqno{1.1} $$ hold. So one has $\phi^t(\G_{\st})\subset \G_\sigma$. Fix $\tilde t$. Given $\bar z\in \G_{\sigma e^{-\alpha|\tilde t|}}$ we prove that $\phi^{\tilde t}$ is analytic at $\bar z$. By the Cauchy- Kowaleskaya theorem (see e.g.[Pe]) there exists a neighbourhood $ \U$ of $\bar z$ and a time $\bar t$ such that, for any $|\tau|<\bar t$, $\phi^\tau$ is analytic on $\U$. Assume that $\bar t$ is the supremum of such times (so that $\phi^{\bar t}$ is not analytic in $\U$). Assume by contradiction $\bar t<\tilde t$. By \eqref{1.1} the limit $\lim_{\tau\to\bar t}\phi^\tau(z)$ exists on $\U$. Denote $w :=\lim_{\tau\to\bar t}\phi^\tau(\bar z)$. Again by the Cauchy-Kowakeskaya thoerem there exists a neighbourhood $\V$ of $w$ and a $t_1>0$ such that $\phi^{\tau}$ is analytic on $\V$ for $\left|\tau\right|