%% ------------------- MACROS
%file inizio.tex
\def\mela{\relax}
\magnification1200
\hsize 14cm
%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{
\centerline{{\bf Summary.}}\par}
\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\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\proposition#1#2{\par\vskip4pt
\noindent {\bf Proposition \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}
%------------------- 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{\nobreak\par\nobreak
\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\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\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 ---------------------------------
\numerazionedoppia
\autobibliografia
\def\cmp{Commun. Math. Phys.}
\def\cmf{Commun. Math. Phys.}
\commento{cmf\cmf}
\def\gio{A. Giorgilli}
\def\gal{L. Galgani}
\def\bgg{G. Benettin, \gal, \gio}
\def\zamp{J. Appl. Math. Phys. (ZAMP)}
\def\catene{bam93a}
\def\giro{bam91b}
\def\breather{bam95?}
\biblitem{Nel85}E.~Nelson: Quantum Fluctuations, Princeton U.P.,
(Princeton, 1985).
\biblitem{boh48}D. Bohm, M. Weinstein: \title{The Self Oscillations of a
Charged Particle.} \rivista{Phys. Rev.} {\bf 74}, 1789-1798 (1948).
\biblitem{col62}S. Coleman, R. E. Norton: \title{Runaway Modes in Model
Field Theories} \rivista{Phys. Rev.} {\bf 125}, 1422-1428 (1962).
\biblitem{Jack}J.D. Jackson: Classical Electrodynamics, John Wiley \&
Sons (New York 1975).
\biblitem{Schweber}S.~Schweber: An Introduction to Relativistic Quantum
Field Theory. Row Peterson (New York 1961)
\biblitem{pau38}W. Pauli, M. Fierz:\title{
Zur Theorie der Emission
langwelliger Lichtquanten.} \rivista{Nuovo Cimento},
{\bf 15}, 167-188 (1938).
\biblitem{Lor09}H.A. Lorentz: The theory of Electrons, Dover, (New York
1952); first edition 1909.
\biblitem{nor59}R.E.~Norton, W.K.R.~Watson:\title{ Commutation Rules
and
Spurious Eigenstates.} Phys. Rev. {\bf 116} 1567--1603 (1959).
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reaction} \rivista{Fortschritte der Physik} {\bf 9} 343 (1961)
\biblitem{FeyII}R.P.~Feynman, R.B.~Leighton, M.~Sands: The Feynman
lectures in physics, Mainly Electromagnetis and Matter, Vol. II.
Addison--Wesley (Redwood City, 1963)
\biblitem{Barut}A.O.~Barut: Electrodynamics and Classical theory of
Fields and Particles. Dover (New York, 1964)
\biblitem{dir38}P.A.M. Dirac: \title{Classical Theory of Radiating
Electrons. }\rivista{Proc. Roy. Soc.} {\bf A167}, 148-168 (1938).
\biblitem{Pazy}A. Pazy: Semigroups of Linear Operators and
Applications to Partial Differential Equations. Springer Verlag.
(New York 1983).\commento{coll 282.44}
\biblitem{GelShi}I.M.~Gel'fand, G.E.~Shilov: Generalized Functions,
Vol.~3. Academic Press (New York and London, 1967)
\biblitem{abr09}M. Abraham: Theorie der Elektrizit\"at, vol. II,
Teubner (Leipzig--Berlin 1908).
\biblitem{bam91a}D. Bambusi, L. Galgani: \title{Some Rigorous Results
on
the Pauli-Fierz Model of Classical Electrodynamics.} \rivista{ Ann.
Inst. H. Poincar\'e, } Physique th\'eorique {\bf 58} 155-171 (1993).
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(1900)
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Damping. }\rivista{Phys. Rev.} {\bf 76} 466-477 (1947)
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on Euclidean Spaces. Princeton University Press (Princeton, New Jersey
1971)
\riferimentifuturi
%------------- definizioni --------------------------
\def\trivial#1{\relax}
\def\rot{{\rm rot}\hskip1pt}
\def\ald{Abraham--Lorentz--Dirac}
\def\bold#1{{\bf #1}}
\def\intre{\int_{\Re^3}}
\def\sob#1{H_*^{\{#1\}}}
\def\aomega{\bold A^{l}_{\omega}}
\def\arun{\bold A^l_r}
\def\Ao{\bold{A}^{o,l}_{\omega}}
\def\Ae{\bold{A}^{e,l}_{\omega}}
\def\ao#1#2{\bold{A}^{o,#1}_{#2}}
\def\ae#1#2{\bold{A}^{e,#1}_{#2}}
\def\rhoadiomega{\left|\hat\rho_a\left(\frac{\omega}{c} \right)\right|^2}
\def\rhodiomega{\left|\hat\rho\left(\frac{\omega}{c} \right)\right|^2}
\def\rhidiomega{\left|\hat\D\left(\frac{a\omega}{c} \right)\right|^2}
\def\rhodik{\left|\hat\rho(k)\right|^2}
\def\conf{\Q}
\def\scal#1#2{\left\langle #1;#2\right\rangle_{\conf}}
\def\vk{{\bf k}}
\def\ve{\bold e_j(\vk)}
\def\sldue#1#2{\left\langle #1;#2\right\rangle_{L^2}}
\def\mdi#1{m_{tot}(#1){#1}^2-\alpha}
\def\massa{(\mdi{\omega})^2+\gamma\omega^6}
\def\ottoterzi#1{\frac83\pi\frac{e}{c^4}{#1}^2\alpha\hat
\rho\left(\frac{#1}{c} \right) }
\def\intzero{\int_0^\infty}
\def\interre{\int_{-\infty}^{\infty}}
\def\piu#1#2{\frac{#1}{#1+#2}}
\def\menom#1#2{\frac{#1}{#1-#2}}
\def\effe{f(\nu)\left|\hat\rho(k)\right|^2}
\def\ba{{\bf A}}
\def\bw{\bold w}
\def\bv#1{\bold v_{#1}}
\def\gr#1{\bold #1}
\def\lea{\A}
%----------- fine definizioni --------------------------
%----------- TEXT
\duetitoli{D.~Bambusi, D.~Noja}{On the Electrodynamics of point particles}
{\bf \centerline{ON CLASSICAL ELECTRODYNAMICS OF POINT PARTICLES}
\centerline{AND MASS RENORMALIZATION}
\centerline{Some preliminary results}
}
\vskip48pt
\centerline{Dario BAMBUSI, Diego NOJA}
\centerline{Dipartimento di Matematica dell'Universit\`a,}
\centerline{ Via Saldini 50, 20133 MILANO, Italy\footnote{*}{e-mail:
``bambusi@mat.unimi.it'' and ``noja@mat.unimi.it'' }.}\par
\abstract {\it We consider the problem of finding rigorous results for
the dynamics of a classical charged point particle interacting with the
electromagnetic field, as described by the standard Maxwell--Lorentz
equations. Some results are given for the corresponding linearized
system, i.e. the so called dipole approximation, in the presence of a
mechanical linear restoring force. We regularize the system by taking a
form factor for the particle (Pauli--Fierz model) and
study the limit of the particle's motion as the regularization is
removed. We prove that (i) if the regularization is removed but mass is not
renormalized the motion is trivial (i.e. the particle does
not move at all); (ii) if the regularization is removed and mass is
renormalized, the particle's motion corresponding to smooth initial data
for the field has a well defined nontrivial limit; (iii) in the case of
vanishing initial field the limit motion satisfies exactly the \ald\
equation; (iv) for generic initial data the limit motion is runaway.}
\autosez{intro}Introduction
Concerning the interaction of a charged point particle with the
electromagnetic field in classical electrodynamics, the following
quotation from E.~Nelson
might be appropriate: ``With suitable ultraviolet
and infrared cutoffs, this is a dynamical system of finitely many
degrees of freedom and we have global existence and uniqueness.... Is it
an exaggeration to say that nothing whatever is known about the behavior
of this system as the cutoffs are removed, and there is not one single
theorem that has been proven?"\cite{Nel85} (p. 65). In the present paper
we give some preliminary results on the limit motion of the particle
when the cutoff is removed. The main limitation is due to the
fact that, instead of the complete Maxwell--Lorentz system (namely
Maxwell equations for the field with a current due to the particle's
motion, and the relativistic Newton equation for the particle with
Lorentz force due to the field), we consider the corresponding
linearized, and thus non--relativistic, version, i.e. the so called
dipole approximation. This is indeed a model very much studied both in
its classical and in its quantum version (see e.g. [\cref{boh48},
\cref{col62}]), which presumably allows to give preliminary information
also on the complete non linear system. Moreover, we recall that a large
part of physical effects ranging from Thomson scattering to Lamb shift
(see e.g. [\cref{Jack}, \cref{Schweber}]) are well described within such
an approximation.
As is well known, in the Coulomb gauge the only unknowns are the vector
potential $\bold A$ for the field (with ${\rm div} \bold A=0$) and the
particle's position $\bold q$; the formal equations of motion are
$$
\eqalign{
\frac1{c^2}\ddot \bold A-\bigtriangleup \bold A=\frac{4\pi}{c^2}\bold
j_{tr}\ ,
\cr
m_0 \ddot \bold q=-\frac ec \dot\bold A(0)-\alpha\bold q\ ,
}\autoeqno{d1.1}
$$
where $\bold j_{tr}$ is the transversal part of the current $\bold j(\bold x)
:=
e\dot\bold q
\delta(\bold x)$, with $\delta(\bold x)$ the usual delta function, while
$e,$ and $m_0$ are respectively the
particle's charge and bare mass; here we also added a linear restoring
force $-\alpha\bold q$ (with $\alpha>0$).
In order to regularize the system, following a long tradition (see for
example Pauli and Fierz\upcite{pau38}), instead of the cutoffs indicated
by Nelson we take a form factor corresponding to a rigid extended
particle. Moreover, we concentrate our attention just on the particle's
motion, which,
due to the non trivial coupling with the field, clearly depends also on the
initial data for the field variables.
The first result we prove
is that the particle's motions corresponding to solutions
of the Cauchy problem with regular data for the field (namely, in the
Schwartz space) have a limit (in the $C^0$ topology) when the
regularization is
removed, i.e. that the particle performs regular motions when the ``radius''
of the form factor tends to zero.
In this connection let us recall that, as is well known, a
crucial point for the discussion of
the point-like limit is mass renormalization, the need of which is
expected from heuristic considerations [\cref{Lor09}, \cref{nor59},
\cref{erb61},
\cref{FeyII}, \cref{Barut}]. We first show
that an infinite mass
renormalization, leading to a negative bare mass,
is actually needed. Indeed we prove that if mass is not
renormalized (i.e. the bare mass is taken positive and independent of
the form factor), the limit particle's motion is trivial, in the sense
that the
particle remains forever at rest even if initially displaced from the
equilibrium position of the spring. So,
in order to obtain non trivial limiting
particle's motions one has to take a negative bare mass (for any radius
smaller than a certain critical radius coinciding with the so called
``classical radius''), which moreover has to tend to infinity as
the radius tends to zero. Therefore we study the limit of the particle's
motion when the radius tends to zero, and simultaneously mass is
renormalized, and prove the convergence result mentioned
above. We also prove continuous dependence of the particle's motion on
intial data. So it turns out that {\it classical electrodynamics of
point particles, at
least in the dipole approximation, is well defined}, even if the
equations of motions might not have a well defined point--like limit
(due to mass rinormalization).
A second kind of results concerns a description of the limiting
particle's motions. We deal only with the simplest case, namely that of
a particle initially at rest, displaced from the position of mechanical
equilibrium, with vanishing initial data for the field. In such a case
we prove that the
limiting particle's motion satisfies exactly the well known
\ald\ (or ALD) equation;
in this sense we can say that we have here a rigorous
proof of such an equation, which was up to now obtained only by
heuristic arguments. The exact form of the solution is a damped
oscillation superimposed to an exponentially increasing motion; thus we
have here a {\it runaway motion}. We recall that runaway motions are
well known to appear in the context of the \ald\
equation\upcite{dir38}.
However, since the standard deductions of such an equation from the
Maxwell--Lorentz system involve many unclear approximations, there
could remain a doubt concerning the actual existence of runaways in
classical electrodynamics. The present result shows that, at least in
the linear approximation, in classical electrodynamics runaways appear
for generic initial data
(for a discussion of the analogous phenomenon in quantum mechanics see
e.g. \cite{col62}). So the conclusion is that non trivial point
particle's motions in the Maxwell--Lorentz system can be obtained only
by paying the price of having generically runaway solutions,
which are obviously
unacceptable form a physical point of view. We postpone a discussion of
this problem to future work. A further comment is that
we expect that the results described above for the linearized system can
be essentially extended to the complete nonlinear system.
Work is in progress in this direction.
\unp
We come now to a discussion of the technical aspects of the paper. For
the regularized system existence and uniqueness results are deduced by
means of standard semigroup theory\upcite{Pazy}. But such a theory fails
in the point--like limit, essentially due to the difficulty of giving a
meaning to the formal expression defining the Lorentz force in the that
limit; indeed this is related to the old problem of defining the product
of two singular distributions.
We are able to overcome the problem by providing directly a
representation formula for the solution of the Cauchy problem for the
regularized system, so that we can study explicitly the limit of the
particle's motion. The representation formula is based on the explicit
calculation of the spectral resolution of the linear operator describing
the dipole approximation. The formulation of the spectral
resolution theorem used here is essentially that of
the book \cite{GelShi} by Gel'fand and
Shilov.
\unp
The paper is divided into two parts. The
first part consists of two sections which are devoted to the statement
of the results. Precisely, in section \sref{mai}
we present the model, discuss mass renormalization, and state the
well posedness results for the dynamics of a point particle. In
sect.~\sref{run} we give an explicit formula for the motion of a point
particle, corresponding to vanishing initial field;
we also compare
these solutions with the solutions of the ALD equation.
Part~II consists of
the proofs of the above results and is divided into three sections:
in sect.~\sref{sol} we give the representation formula for the solution
of the Cauchy problem for an extended particle;
in sect.~\sref{tec} we prove the result on the
dynamics of a rigid extended particle, and in sect.~\sref{tec2} those
concerning the point--like limit.
\unp
\noindent
{\it Acknowledgements} We thank Andrea Carati, Luigi Galgani, Andrea
Posilicano, and Jacopo Sassarini for many useful discussions and for
several comments on preliminary versions of the paper.
\vfill\eject
\noindent {\bf PART I: Statement of the results }
\autosez{mai}Well posedness of the dynamics of a point particle
We consider a relativistic charged particle
interacting with the electromagnetic field, and subjected to an external
linear restoring force.
We recall that, working in the Coulomb gauge,
the complete nonlinear
Maxwell--Lorentz system takes the form \cite{Jack}
$$
\eqalign{
\frac1{c^2}\ddot \bold A-\bigtriangleup \bold A&= \frac{4\pi e}c
\Pi(\dot {\bold q}\delta_q)\ ,
\cr
\frac{d}{dt}\left( \frac{m_0\dot \bold q}{\sqrt{1-(\dot q/c)^2}
} \right) &=-\frac ec \dot \bold A(\bold q)+e\frac{\dot\bold q}c
\wedge \rot \bold A(\bold q)-\alpha\bold q
,}\autoeqno{a}
$$
where the vector potential $\gr A=\gr A(\gr x)$ is subjected to the
constraint ${\rm div} \bold A=0$, and $\alpha>0$ is a constant
characterizing the external linear force, $\delta_q$ is the distribution
translated with respect to the $\delta$ function centered at the origin,
formally $\delta_{q}(\bold x):=\delta(\bold x-\bold q)$. Finally $\Pi$
is the projector on the subspace of vector fields with vanishing
divergence, i.e. $\Pi({\bf j})$ is the so called transversal part of the
current $\bold j$, often denoted by $\bold j_t$. We take now the so
called dipole approximation, namely linearize the system about the
equilibrium point $\bold q=\dot{\bold q}=\bold A=\dot{\bold A}=0$,
obtaining system \eqref{d1.1}. We then regularize the system by
substituting the $\delta$ function by a smooth normalized (in $L^1$)
charge distribution $\rho$, obtaining the system
$$
\eqalign{
&\frac1{c^2}\ddot {\bold A}-\bigtriangleup \bold A=
\frac{4\pi e}c \Pi(\dot {\bold q}\rho)\ ,
\cr
&m_0\ddot {\bold q} =-\intre \frac{e}c \rho(x)\dot {\bold A}
(x)d^3x-\alpha {\bold q}\ ,
}\autoeqno{ml}
$$
{\it which is the one that will be studied here}. For the sake of
brevity, {\it the system \eqref{ml} will be called simply the
Maxwell--Lorentz system,} omitting the qualification ``in the dipole
approximation'' and also the qualification ``regularized''.
It will be studied
in the configuration
space $\Q_0$ defined by
$$
\Q_0:=\S_*(\Re^3,\Re^3)\oplus \Re^3\ni (\bold A,\bold q)\
,\autoeqno{sp.1}
$$
where $\S_*$ denotes the subset of the vector fields belonging to the
Schwartz space $\S$ ($C^\infty$ functions decaying at infinity faster
than any power) having vanishing divergence.
Concerning $\rho$
we will assume that (i) it is $C^{\infty}$,
(ii) it
decays at least exponentially fast at infinity, (iii)
it is spherically symmetric,
and (iv)
its Fourier transform $\hat\rho$, defined by
$$
\rho(\bold x)=\frac1{(2\pi)^{3/2}}\intre \hat \rho(\vk)e^{i\vk \bold
x}d^3k\ ,
$$
is everywhere non--vanishing ({\it i.e.}
$\hat\rho(\vk)\not=0$, $\perogni \vk\in\Re^3$); finally, in order to
simplify the discussion of the point--like limit we
will assume (v) that $\rho$ has the form
$$
\rho_{a}(x):=\frac1{a^3}\D\left(\frac xa\right)\ , \autoeqno{pa.1}
$$
where $\D$ is a positive, normalized (in $L^1$) function.
Notice that the pointlike limit is obtained
by letting the ``radius" $a$ of the particle tend to zero (so that
$\rho_a$ tends to a delta concentrated at the origin), and that for $a>0$
the Cauchy problem for system \eqref{ml} is well posed in the phase
space $\Q_0\times \Q_0$.
We discuss now the limit of the particle's motion
as the ``radius'' $a$ of the charge distribution tends to zero.
\unp
To begin with, we consider the special class of initial data with the
particle initially at rest in some position $q_0\not=0$, and vanishing
initial field.
First we show the need of mass renormalization. We have the following
\proposition{p.1}{Having fixed $m_0>0$, for each $a>0$ denote by $\bold
q_a(t)$ the solution of \eqref{ml} corresponding to initial data $\bold
A_0=\dot {\bold A}_0=\dot{\bold q}_0=0$, and $\bold q_0\not=0$; assume
that $\bold q_a(.)$ converges weakly to a distribution $\bold q(.)$ as
$a\to 0$. Then there exists a constant vector $\bar {\bold q}$ such that
$\bold q(t)\equiv \bar {\bold q}$.}
This means that for positive bare mass, if the dynamics of the
particle admits a point--like limit, such a limit dynamics is trivial.
This could be expected since it is well known that the electromagnetic
mass of a point particle is infinite; the above proposition proves
exactly that a point particle is unaffected by the presence of a
force ($-\alpha \bold q_0$) no matter how large it is, so that it
behaves as if its mass were infinite.
>From the mathematical point of view this is seen as follows.
As already anticipated in the introduction, the proof
of all results of this section is based on the use of a
representation formula for the solution of the Cauchy problem of
\eqref{ml}. Now, in
such a formula the bare mass $m_0$ appears only summed
to the quantity
$$
m_{em}:=\frac{32}3 \pi^2\frac{e^2}{c^2}\int_0^\infty|\hat\rho(k)|^2 dk\
, \autoeqno{mem}
$$
which is usually interpreted as the electromagnetic mass corresponding
to the given charge distribution. So it can be expected that the
particle behaves as if its experimental mass were the sum of $m_0$ and
$m_{em}$.
So, we renormalize mass, i.e. we consider $m_0$ as a
function of $a$, precisely we put
$$
m_0:=m-m_{em}=m-\frac1a\left[\frac{32}3\pi^2\frac{e^2}{c^2}
\intzero\left|\hat\D(k)\right|^2dk
\right]\ ,\autoeqno{s.5}
$$
where $m$ is a fixed parameter to be identified with the physical mass
of the particle.
Notice that \eqref{s.5} requires to consider negative bare masses
$m_0$, which, as $a$ tends to zero, diverge to minus infinity.
Concerning the behaviour of the system in the limit $a\to0$ we have
\proposition{p.2}{Consider the Cauchy problem for system \eqref{ml} with
form factor $\rho$ given by \eqref{pa.1}, $m_0$ given by \eqref{s.5},
and initial data $\bold A_0=\dot \bold A_0=\dot \bold q_0=0$, with
$\bold q_0\not=0$. For each $a>0$ let $\bold q^{(a)}(t)$ be the
corresponding particle's motion. Then, for any $T>0$, as $a\to0$
the function $\bold q^{(a)}(.)$ converges in $C^1([-T,T],\Re^3)$ to a
non constant function.}
So, the particular solution corresponding to the above initial data has
a point--like limit which is nontrivial, {\it provided mass is
renormalized}.
We consider now the case where the initial particle's velocity too is
different from zero; this is a nontrivial generalization. Indeed in such
a case ($\bold{\dot q_0}\not=0$), if one takes a vanishing initial
field, it turns out that the trajectory of the particle has no
point--like limit; precisely one has
\proposition{p.23}{Consider the Cauchy problem for system \eqref{ml} with
form factor $\rho$ given by \eqref{pa.1}, $m_0$ given by \eqref{s.5},
and initial data $\bold A_0=\dot \bold A_0=\bold q_0=0$, with
$\dot \bold q_0\not=0$. For each $a>0$ let $\bold q_{(a)}(t)$ be the
corresponding particle's motion. Then, for any $T>0$, as $a\to0$,
one has
$$
|\bold q_{(a)}(t)|\to\infty\ ,\quad \perogni t\in [-T,T]\meno \N\ ,
$$
where $\N$ is a finite (possibly empty) set.}
It is not difficult to prove that the same happens also if one takes as
initial data for the field any regular function (i.e. without
singularities). On the other hand the result of proposition
\lemmaref{p.23} is not astonishing, because it is known (at least in the
case of a uniform motion) that a particle moving with some velocity
carries a field [\cref{abr09}, \cref{bam91a}] which in the case of a
point--like particle has a singularity at the particle's position. So,
it seems natural to study the particular class of initial data such that
a particle with non vanishing velocity is accompanied by ``its own
field''. In order to give a precise statement we recall
that\upcite{bam91a} in the non--linear Maxwell--Lorentz system a free
particle can move uniformly with velocity $\bold w$, only if accompanied
by the field $\bold X$, which vanishes at infinity and solves the
equation
$$
\bigtriangleup \bold X -\frac1{c^2}
\sum_{i,l=1}^3w_iw_l\frac{\partial^2}{\partial x_i\partial x_l}
\bold X=-4\pi\frac ec \Pi(\rho \bold w)\ .
$$
In
the dipole approximation, i.e. after a linearization in the velocity
and in the field, such an equation reduces to
$$
\bigtriangleup \bold X =-4\pi\frac ec \Pi(\rho \bold w)\ .\autoeqno{e.3}
$$
We denote by $\bold X_w$ the unique solution of equation \eqref{e.3}
vanishing at infinity, and
study the point--like limit of the solutions of the Cauchy problem
corresponding to initial data of the form $\dot q_0\not=0, {\bf
A}_0=\bold X_{\dot q_0}$. Initial data of the form $(\bold q_0,\dot
\bold q_0, \bold X_{\dot q_0}, 0)$ will be called of ``congruent type''.
\proposition{p.3}{Consider the Cauchy problem for system \eqref{ml} with
form factor $\rho$ given by \eqref{pa.1}, $m_0$ given by \eqref{s.5},
and initial data $(\bold q_0,\dot \bold q_0, \bold A_0, \dot \bold
A_0)=(\bold q_0,\dot \bold q_0, \bold X_{\dot q_0}, 0) $. For
each $a>0$
let $\bold q^{(a)}(t)$ be the corresponding particle's motion.
Then, for any $T>0$, as $a\to0$ the function $\bold q^{(a)}(.)$
converges in $C^1([-T,T],\Re^3)$ to a non constant function.
}
We come now to the case of general initial data for the field. The
following theorem holds.
\theorem{p.44}{Consider the Cauchy problem for system \eqref{ml} with
form factor $\rho$ given by \eqref{pa.1}, $m_0$ given by \eqref{s.5},
and initial data $(\bold q_0,\dot \bold q_0, \bold A_0, \dot \bold
A_0)=(\bold q_0,\dot \bold q_0, \bold X_{\dot q_0}+\bold A'_0, \dot
\bold A_0)$ with $ (\bold A'_0, \dot \bold
A_0)\in\S_*(\Re^3,\Re^3)\times\S_*(\Re^3,\Re^3)$. For each $a>0$ let
$\bold q^{(a)}(t)$ be the corresponding particle's motion. Then,
for any $T>0$, as $a\to0$ the function $\bold q^{(a)}(.)$ converges in
$C^0([-T,T],\Re^3)$. Moreover, the limiting particle's motion depends
continuously on
$$
(\bold q_0,\dot\bold q_0,\bold A'_0, \dot \bold
A_0)\in\Re^3\times\Re^3\times\S_*(\Re^3,\Re^3)\times\S_*(\Re^3,\Re^3)\ .
$$
}
So, even if (due to mass renormalization) the Maxwell-- Lorentz system
has no point--like limit, the above theorem shows that the dynamics of
a point particle is well defined, at least for regular initial data for
the field. Moreover, the Cauchy problem is well posed in the sense of
Hadamard. By the way, our technique allows to generalize the existence
result to the case of initial fields $(\bold A'_0,\dot\bold A_0)$ which
are only $C^0$, and decay at infinity faster than $x^{-3/2}$; further
generalizations are also possible, but we did not try
to characterize the most general allowed initial fields.
\autosez{run}On the solutions corresponding to congruent initial
data, and comparison with the Abraham--Lorentz--Dirac equation.
In the case of congruent initial data it is possible to calculate
explicitly the point--limit of the solution of the Maxwell-Lorentz
system. In order to come to the corresponding formula we introduce some
notations. We define
$$
\omega_0^2:=\frac\alpha{m}\ ,\quad \tau_0:=\frac23\frac{e^2}{m c^3}\ ,
$$
then we consider the equation
$$
\tau_0\nu^3-\nu^2-\omega_0^2=0\ ,\autoeqno{e.7}
$$
and denote by $\nu_r$, $\nu_+=\nu_3+i\nu_2$,
$\nu_-=\nu_3-i\nu_2$ its three solutions ($\nu_2,\nu_3>0$).
\theorem{qo}{The point--like limit of the particle's motion
corresponding to the solution of the Cauchy
problem for the Maxwell--Lorentz system with initial data
$$
(\bold q_0,\dot \bold q_0, \bold {A}_0,\dot\bold A_0)=
(\bold q_0,\dot \bold
q_0,\bold X_{\dot q_0},0)
$$
is given by
$$
\bold q(t)=
\left\{
\eqalign{
e^{-\nu_3t}[\lea_1^+ \cos(\nu_2t)+
\lea^+_2\sin(\nu_2t)
]+\lea^+_3 e^{\nu_rt}\ ,\quad {\rm if }\ t>0
\cr
e^{\nu_3t}[\lea_1^- \cos(\nu_2t)+
\lea^-_2\sin(\nu_2t)
]+\lea^-_3 e^{-\nu_rt}\ ,\quad {\rm if }\ t<0
}\right.\ ,
\autoeqno{p.30}
$$
where $\lea^{\pm}_1$, $\lea^{\pm}_2$, $\lea^{\pm}_3$
are real vector constants
depending on the initial data, and on $e$, $m$, $\omega_0$.
Moreover, one has the following asymptotics
$$
\left\{\eqalign{\nu_r&=\displaystyle{\frac{\omega_0}\epsilon}+O(\epsilon)
\cr
\nu_2&=\omega_0+O(\epsilon^2)
\cr
\nu_3&=\omega_0\epsilon/2+O(\epsilon^2) }\right.\ ,\quad
\left\{\eqalign{
\lea_1^{\pm}&=\bold q_0+O(\epsilon^2)
\cr
\lea_2^{\pm}&=\frac{\dot \bold q_0}{\omega_0}+O(\epsilon^2)
\cr
\lea_3^{\pm}&=O(\epsilon)
}\right.
\autoeqno{p.33}
$$
as
$\epsilon\to0$, in terms of the dimensionless parameter
$$
\epsilon:=\omega_0\tau_0\ . \autoeqno{ep}
$$
}
We point out that, from the proof of this theorem, it is not difficult
to deduce also a representation formula for the limit particle's motion,
in the case of a nonvanishing initial field.
Qualitatively we have thus that the Maxwell--Lorentz system in the
dipole approximation predicts that the motion of a point--like charged
harmonic oscillator is the superposition of a damped oscillation and a
runaway motion, at least when the initial ``free'' field is vanishing.
Moreover, it is possible to show that the particle's motion
corresponding to generic initial deta is runaway. It turns out that it
is possible to select initial field in such a way that the corresponding
particle's motion is non runaway. A detailed discussion of this point is
beyond the aim of this paper, and is deferred to forthcoming work.
We
concentrate now on the comparison of \eqref{p.30} with the solutions of the
Abraham--Lorentz--Dirac (ALD) equation
$$
m\tau_0\tdot {\bold q}=m\ddot {\bold q}+\alpha\bold q\
.\autoeqno{ald.1}
$$
Such a comparison is particularly interesting since all available
deductions of the ALD equation appear to be just
heuristic$^{[\cref{Lor09},\cref{boh48},\cref{Mor}]}$.
We have
the following
\theorem{ald}{The point limit of the particle's motion \eqref{p.30} in
the ML system is also a solution of the following problem
$$
\matrix{
& -m\tau_0\tdot {\bold q}=m\ddot {\bold q}+\alpha\bold q\ , &t<0\ ,
\cr
& m\tau_0\tdot {\bold q}=m\ddot {\bold q}+\alpha\bold q\ , &t>0\ .
}
$$
}
So, for positive times the particle's motion satisfies exactly the ALD
equation, and a different one for negative times. In
this connection we point out that a careful analysis of the usual
deductions of \ald\ leads to the same conclusion. By the way the fact
that different equations arise for negative and positive times is not a
particular feature of the present model. A classical case where this
happens is that of Boltzmann equation, and a simple and enlightening
mechanical model where this phenomenon appears was given by Lamb at the
beginning of the century (see \cite{lam00}).
Now, an important problem remains open concerning the practical use of
ALD equation. In order to determine a definite solution of the \ald\
equation, one has to assign initial data for position, velocity and
acceleration of the particle. It is natural to choose the same initial
particle's velocity and position as for the complete Maxwell--Lorentz
system, while the problem of the determination of the initial
acceleration is non trivial since the acceleration due to the Lorentz
force is not defined in the pointlike limit. Usually the initial
acceleration for the ALD equation is determined by imposing the so
called nonrunaway condition, namely by choosing it in such a way that
the corresponding solution has bounded acceleration as $t\to \infty$. We
do not discuss here the problem of justifying such a procedure, since it
would invove the discussion of a non runaway condition for the ML
system.
\vskip\spaziosoprasez
\noindent{\bf PART II: Proof of the results}
\autosez{sol}Solution of the Cauchy problem in the case of a rigid
extended particle.
We state here our result
concerning the solutions of the Cauchy problem in the case of a rigid
extended particle. We will begin the study in the configuration space
\eqref{sp.1}.
We recall that, since $\rho$ is spherically symmetric, $\hat \rho$ is
spherically symmetric too, and moreover one has,
$\hat\rho(\vk)=\hat\rho(k)=\hat\rho^{*}(k)$, where $k=|\vk|$, and the
star denotes complex conjugation.
\unpo
To begin with
we remark that, due to the spherical symmetry of the form factor,
there exists a subspace of $\S_*$ in which the dynamics of the field is
decoupled from that of the particle. Indeed, the only interacting fields
are the divergence free part of spherically symmetric fields. Precisely,
consider in $\S(\Re^3,\Re^3)$ the subset $\F_0$ of the fields depending
only on the distance $x= |\bold x|$ of the point from the origin:
$$
\F_0:=\left\{\ba \in \S(\Re^3,\Re^3)\ :\ \ba(\bold x)=\ba(x)\right\}
\ ,\autoeqno{f0}
$$
and define the set
$$
\F_*:=\Pi(\F_0)\ ,\autoeqno{pint}
$$
namely the set of the vector fields which are the divergence
free part of some spherically symmetric (in the above sense) vector
field. Then the following simple result, to be proven in
sect.~\sref{tec}, holds
\proposition{int}{The configuration space $\Q_0$ (cf. eq. \eqref{sp.1})
splits into two subspaces
$$
\Q_0=\Q\oplus\Q_1\ ,
$$
with
$$
\Q:=\F_*\oplus\Re^3 \ ,\autoeqno{3}
$$
which are invariant for the dynamics of the Maxwell--Lorentz system
\eqref{ml}. Moreover, $\Q$ and $\Q_1$ are mutually orthogonal with
respect to the $L^2\oplus\Re^3$ metric, and
the projection of \eqref{ml} on $\Q_1$ is
the free wave equation.}
We thus confine our study of the Maxwell--Lorentz system to the
configuration space $\Q$ where the nontrivial dynamics takes place.
In order to obtain a formula for the solution of the Cauchy problem
we need to
introduce a few more notations.
A fundamental role, is played by the
electromagnetic mass $m_{em}$ corresponding to
a given charge distribution (which was defined by \eqref{mem}).
In addition,
we introduce a function $m_1(\omega)$, having
the dimension of a mass, by
$$
m_1(\omega):=\frac{16}3\pi^2\frac{e^2}{c^2}\omega\int_{-\infty}^\infty
\frac{|\hat\rho(k)|^2}{\omega_k-\omega}dk\ ,
$$
where $\omega_k=ck$, and the integral has to be interpreted as a principal
value. Notice that $m_1(\omega)$ is proportional to the Hilbert
transform of the real function $|\hat\rho(k)|^2$ (which is extended to
$\Re_-$ by symmetry) and that, in the case of a point--like particle, $m_1$
vanishes identically. We will denote by
$m_{tot}=m_{tot}(\omega)$ the function
$$
m_{tot}(\omega):=m_0+m_{em}+m_1(\omega)\ .\autoeqno{m}
$$
which will play the role of total (renormalized) mass.
Finally we introduce the following relevant function
$$
C(\omega):=\frac2\pi \Omega^2\tau 8\pi^3 \rhodiomega
\frac{\omega^2}{(\omega^2-\Omega^2)^2+\left(8\pi^3\rhodiomega
\right)^2 \omega^6\tau^2}\ ,
\autoeqno{s.4}
$$
where
$$
\Omega^2=\Omega^2(\omega):=\frac{\alpha}{m_{tot}(\omega)}\ ,\quad
\tau:=\frac23\frac{e^2}{m_{tot}(\omega) c^3}\ .
$$
Using these notations we can state the theorem concerning the solution
of the Cauchy problem for the Maxwell--Lorentz system. In order to
simplify its reading we point out that the solution of
the equations of motion will be given as a superposition of simple
harmonic oscillations corresponding to the normal modes of the system
(for a previous application of this method to field theory see
\cite{bla47}); in the case of negative bare mass (which is needed in
order to discuss the point--like limit) the solution contains a part
with real exponentials which correspond to normal modes with
imaginary frequencies (runaway modes). The
distributions $\bold A_\omega^l$ and the functions $\bold A_r^l$
appearing in the forthcoming theorem are just the field components of
the improper and of the proper eigenvectors of the linear operator
describing the Maxwell--Lorentz system. Correspondingly $C(\omega)$
plays the role of a normalization constant for the improper
eigenfunction corresponding to the frequency $\omega$, while the
constant $C_r$ is the normalization constant for the runaway
eigenvector.
\theorem{t.1}{Consider the Cauchy problem for system \eqref{ml} with
initial data $((\bold A_0,\bold q_0), $
$(\dot {\bold A}_0,\dot\bold
q_0))\in\Q\times\Q$,
and let $\bold v_l$, $l=1,2,3$ be a fixed orthonormal
basis of $\Re^3$. Then there exists a family of distributions
$\left\{\aomega\right\}^{l=1,2,3}_{\omega\in[0,\infty)}$,
$\aomega\in\S'$ (the dual of $\S$)
such that
the solution of the Cauchy problem is given by
$$
\eqalign{
\bold q(t)&:=\sum_{l=1,2,3}\bold
v_l\int_0^\infty C(\omega) \left[-\xi^l_\omega
\cos(\omega t)+\omega \eta^l_\omega \sin(\omega t)
\right]d\omega+\theta(-m_0)
\bold q_r(t)\ ,
\cr
\bold A(t)&:=\sum_{l=1,2,3}\int_0^\infty \alpha C(\omega) \left[\eta^l_\omega
\cos(\omega t)+\frac{\xi^l_\omega}{\omega} \sin(\omega t)
\right]\aomega d\omega+\theta(-m_0)
\bold A_r(t)
\ ,}\autoeqno{s.1}
$$
where $\theta$ is the usual step function, while $\bold q_r(t), \bold
A_r(t)$, which define the ``runaway part''
of the solution (which is present only in the
case of negative bare mass), are given by
$$
\eqalign{
\bold q_{r}(t)&:=\sum_{l=1,2,3}\bold v_l C_r \left[-\xi^l_r
\Ch(\sqrt{-\lambda_r} t)-\sqrt{-\lambda_r} \eta^l_r
\Sh(\sqrt{-\lambda_r} t) \right]\ ,
\cr
\bold A_r(t)&:=\sum_{l=1,2,3} C_r\alpha \left[\eta^l_r
\Ch(\sqrt{-\lambda_r} t)+\frac{\xi^l_r}{\sqrt{-\lambda_r}}
\Sh(\sqrt{-\lambda_r} t) \right]\bold A^{l}_r\ ,
}\autoeqno{sa.1}
$$
$\bold A_r^l$ are functions of the space point ${\bf x}$, while
$C_r$ and $\lambda_r<0$ are real constants depending on the
form factor, on $m_0$ and on $e$.
Moreover, the functions $\xi_\omega^l,\eta_\omega^l$,
can be expressed in terms
of the initial data by
$$
\eqalign{
\xi_\omega^l&:=\frac 1{4\pi c^2}\left\langle\aomega, \dot{\bold A}_0
\right\rangle -\bold q_0\cdot \bold v_l\ ,
\cr
\eta_\omega^l&:=\frac 1{4\pi c^2}
\left\langle\aomega, {\bold A}_0
\right\rangle
+\frac{m_0}\alpha \bold v_l\cdot \dot \bold q_0 +\frac e{c\alpha} \bold
v_l\cdot
\intre \rho(x) \bold A_0(x)d^3x\ ,
}\autoeqno{s.2}
$$
where $\langle.,. \rangle$ denotes the pairing of a distribution with a
smooth function, and
$\xi_r^l,\eta_r^l$ are given by the same expressions with $\bold A^l_r$
in place
of $\bold A^l_\omega$. Explicitly $\bold A^l_\omega$ and $\bold A^l_r$
are given by eqs.~\eqref{b.4} and \eqref{s.8} below.
}
We point out that, in the case of negative bare mass ($m_0<0$), the
above solution is the sum of a runaway part \eqref{sa.1}
and a ``regular'' part; in particular,
the runaway part has a characteristic exponential dependence on time.
This is compatible
with energy conservation because the Hamiltonian is indefinite when
$m_0<0$.
We remark that the field $\bold X_{\dot q}$ adapted to the velocity
$\bold{\dot q}$ does not belong to our phase space, so in principle the
formulas of the theorem do not apply to the case where $\bold X_{\dot
q}$ is taken as initial an datum. However, it is easy to see that the
above formulas define a unitary evolution operator on a larger space.
Precisely, consider the family of Hilbert spaces $\sob s$ defined as the
completion of $\F_*$ (cf.~\eqref{pint}) in the norm
$\norma{|\bigtriangleup|^{s/2}f}_{L^2}$. By standard semigroup theory
\cite{Pazy} it is immediate to show that the Cauchy problem for
system~\eqref{ml} is well posed in each of the phase spaces
$$
\Q^{\{s\}}:=\left(\sob {s-1}\oplus \Re^3\right)\oplus \left(\sob
{s}\oplus \Re^3\right)\ni ((\dot\bold A_0,\dot \bold q_0),(\bold A_0,
\bold q_0))\ ,\quad (s\geq1)
$$
Moreover, remark that eqs.~\eqref{s.1}, \eqref{sa.1}, \eqref{s.2}
describe the action of a linear continuous operator (the group action)
on a dense subset of $\Q^{\left\{s\right\}}$,
and therefore these formulas can be extended to the whole of
$\Q^{\{s\}}$. The field $\bold X_{\dot q}$ belongs to $\sob 1$.
\autosez{tec}Proof of the results on the dynamics of an extended
particle.
First we will prove proposition
\lemmaref{int}, and give a representation lemma for the fields of the
``interacting subspace''.
Then we will come to the heart of the section, namely the
proof of theorem \lemmaref{t.1}, which
is based on the application of the
spectral resolution theorem for self--adjoint operators in Hilbert
spaces. The first step consists in determining a suitable Hilbert space
such that the Maxwell--Lorentz system turns out to be equivalent to the
equation
$$
\ddot\zeta+B\zeta=0\ ,\autoeqno{t.2}
$$
where $\zeta$ is a point of the Hilbert space, and $B$ a self--adjoint
operator in it. Then we will calculate the spectrum and the
eigenfunctions, of the operator $B$. Subsequently we will show that the
singular continuous part of the spectrum of $B$ is empty, and we
will calculate the ``normalization constants'' for proper and improper
eigenfunctions obtaining
the spectral resolution of the identity.
Finally we will write explicitly the solution of the Cauchy
problem, obtaining theorem \lemmaref{t.1}.
We recall that the Maxwell--Lorentz system \eqref{ml} is
Hamiltonian with Hamiltonian function given by
$$
\eqalign{
H&=\intre \left(2\pi c^2 \bold E^2(x)
-\frac1{8\pi}\bold A(x)\cdot \bigtriangleup \bold A(x)
\right)d^3x
\cr
&+\frac1{2m_0}\left( \bold p-\frac ec\intre\rho(x) \bold A(x)d^3x
\right)^2+\frac12\alpha \bold q^2\ ,
}\autoeqno{1}
$$
where $\bold E=\bold{\dot A}/(4\pi c^2)$
is the momentum conjugated to $\bold A$, and $\bold p$
is the momentum
conjugated to $\bold q$.
Using the above Hamiltonian formulation of the problem it is easy to
obtain the
\unp
\noindent {\bf Proof of Proposition \lemmaref{int}}.
The key point is that, denoting by
$$
I(\bold p,\bold A):=\left( \bold p-\frac ec\intre\rho(\bold x) \bold
A(\bold x)d^3x
\right)^2
$$
the term of the Hamiltonian describing the interaction between field
and particle, and by $\Pi_*$ the orthogonal projector (in the
$L^2$ metric) of $\S_*$ onto
$\F_*$ it is easy to see that
$$
I(\bold p,\bold A)=I(\bold p,\Pi_*(\bold A))\ .\autoeqno{t.45}
$$
\trivial{
Indeed for fixed $\bold p$ and $\bold A$, denote
$$
\bw=\bold p-\frac ec\int_{\Re^3}\rho(x) \bold A(\bold x)d^3x\ ,
$$
so that we have
$$
I(\bold p,\bold A)
=\bw\cdot \bold p-\frac ec\sldue{\rho \bw}{\bold A}\ .
$$
\trivial{a maggior chiarezza de passaggio sopra
$$
=\bw\cdot\bw=\bw\cdot \left(\bold p-
\frac ec\int_{\Re^3}\rho(x) \bold A(\bold x)d^3x
\right)
=\bw\cdot\bold p-\frac ec\intre (\bw\rho(x))\cdot \bold A(\bold x)
d^3x
$$
%fine trivial
}
But,
since $\bold A\in \S_*$ and $\Pi$ is an orthogonal projector in the $L^2$
norm the above quantity is equal to
$$
\bw\cdot \bold p-\frac ec\sldue{\Pi(\rho \bw)}{\bold A}\ ,\autoeqno{c.10}
$$
but $\Pi(\rho\bw)\in\F_*$, so that \eqref{c.10} is equivalent to
$$
\bw\cdot \bold p-\frac ec\sldue{\Pi(\rho \bw)}{\Pi_*(\bold
A)}=\bold w(\bold p,\bold A)
\cdot \bold w(\bold p,\Pi_*(\bold A))\ ;
$$
then,
repeating the argument for the first factor we obtain \eqref{t.45}.
%fine trivial
}
The
proof of the proposition follows from the remark that also the term in
the first line of \eqref{1} does not mix the subspace $\F_*$
and its orthogonal complement. Thus the dynamics in the orthogonal
complement of $\F_*$ is decoupled from the dynamics of the rest of the
system.
\quadratino
We give now some simple results which are
useful in order to deal in practice
with the space $\F_*$. First we fix the notation concerning the
Fourier transform of the fields: we put
$$
\bold A(\bold x)=\sum_{j=1,2}\frac1{(2\pi)^{3/2}}\intre \hat
A_j(\vk)\ve e^{i\bold k\bold x}d^3k\ ,
$$
where, as usual $\ve$ are polarization vectors, which have the
property of constituting together with the vector $\bold k$ an orthonormal
basis of $\Re^3$.
\lemma{rep}{Let $\bold A\in\F_*$ be a field, and
fix an orthonormal basis $\bold v_j$ of $\Re^3$; correspondingly there
exists a unique triple $(A^1_c,A^2_c,A^3_c)$
of symmetric scalar functions such that
$$
\bold A=\sum_{i=1,2,3}\Pi(\bold v_iA^i_c)\ .\autoeqno{c.3}
$$
}
\proof Since $\F_*=\Pi(\F_0)$, it is enough to show
that $\Pi$ is injective, $i.e.$ that the only solution of the equation
$$
\sum_{i}\Pi(\bold v_iA^i_c)=0\ ,\autoeqno{c.4}
$$
is zero. Passing to the Fourier transform, and performing the angular
integration the result is easily obtained.
\trivial{
$$
\eqalign{
\sldue{\Pi(\bv i A^i_c)}{\Pi(\bv l A^l_c)}=\sum_{j}\intre [\ve\cdot \bv i]
[\ve\cdot\bv l]\hat A^i_c(k)\hat A^l_c(k)d^3k
\cr
=\frac83\pi \delta_{i,l}\int_0^\infty k^2\hat A^i_c(k)\hat A^l_c(k)dk
\ ,
}
$$
i.e. the three terms of \eqref{c.4} are orthogonal to each
other. So \eqref{c.4} implies
$$
\Pi(\bv iA^i_c)=0\ ,\quad \perogni i=1,2,3\ ,\autoeqno{c.5}
$$
and the Fourier transform of this formula gives
$$
\hat A^i_c(k)[\ve\cdot\bv i ]=0\ ,\quad \perogni i=1,2,3\ ,
$$
which implies the thesis.
fine trivial}
\quadratino
Notice also that the Fourier components $\hat A_j$ of $\bold A$ are
given by
$$
\hat A_j(\bold k)=\sum_{i=1,2,3}[\bold v_i\cdot\ve ]\hat A_c^i(k)\ ,
\autoeqno{c.2}
$$
where $A_c^i$ are defined by \eqref{c.3}. In the following we will
always use the above notation: {\it i.e.} $A_c^i$ {\it will always
denote the unique functions such that \eqref{c.3} holds}; the
symmetrical functions corresponding to a given field will always be
denoted by eliminating the boldface from the symbol of the field, by
adding a lower index $c$ and by adding the upper index of the
corresponding direction. Moreover, it will be often useful to extend the
functions $A_c^i$ to the whole real axis, which will be done by
symmetry. We also fix the vectors $\bv l$.
Finally we remark that one has
$$
\sldue{\bold A}{\bold A'}=\frac83\pi\sum_l\int_0^\infty k^2\hat A^l_c(k)
\hat A^{\prime l}_c(k)dk\ .\autoeqno{c.11}
$$
We turn now to the determination of a self adjoint operator $B$ and of a
Hilbert space such that \eqref{ml} turns out to be equivalent to
\eqref{t.2}. To this end we exploit a Lagrangian structure of system
\eqref{ml}, although not the trivial one. In fact Hamiltonian \eqref{1}
comes (via Legendre transform) from a Lagrangian which is not suitable
for our purposes; indeed, its kinetic energy is indefinite when $m_0<0$,
and so it cannot be used properly as a metric for the configuration
space (one might use the theory of spaces with indefinite metric, but
this is quite cumbersome). So, we first perform the canonical coordinate
transformation $\bold q=-\bold P\ ,\quad \bold p=\bold Q$, and then a
Legendre transform. We thus obtain the Lagrangian
$$
\L(\zeta,\dot\zeta)=
\frac12\scal{\dot\zeta}{\dot\zeta}-\frac12\scal{\zeta}{B\zeta}\
,
$$
where $\zeta=(\bold A,\bold Q)$, the scalar product $\scal..$ is defined
by
$$
\scal{(\bold A,\bold Q)}{(\bold A',\bold Q')}:=
\frac1{4\pi c^2}
\intre\bold A(x)\cdot \bold A'(x) d^3x+\frac1\alpha
\bold Q\cdot \bold Q'\ ,\autoeqno{t1.3}
$$
and the operator $B$ is explicitly given by
$$
B\zeta=\left\{
\eqalign{
&4\pi c^2\left\{-\frac{e}{m_0c}\Pi\left[\rho \left(\bold Q-\frac
ec\intre\rho(x)\bold
A(x)d^3x \right) \right] -\frac1{4\pi}\bigtriangleup \bold A\right\}
\cr
&\frac\alpha{m_0}\left(\bold Q-\frac
ec\intre\rho(x)\bold A(x)d^3x \right)\ .
}
\right.\ \autoeqno{t.3}
$$
We now complete the configuration space $\Q$ in the topology induced by
the metric \eqref{t1.3} obtaining a Hilbert space that will be denoted
by $\bar\Q$, and we complexify it. It is now easy to see that the
operator $B$ is self adjoint and that the equations of motions are of
the form \eqref{t.2}. Moreover, such an operator satisfies the
hypotheses of the spectral resolution theorem in the form by Gel'fand,
namely in the form ensuring the completeness of the generalized
eigenfunction expansion. Indeed, we have that $\Q$ is a nuclear space
dense in $\bar \conf$, which is left invariant by $B$.
Concerning the spectrum of $B$ we have the following
\lemma{s.1}{The continuous spectrum of the operator $B$ coincides with
$[0,\infty)$; and, for
$\lambda=\omega^2$ in the continuous spectrum, the
corresponding generalized eigenfunctions are given by
$\zeta_\omega^l:=(\bold A^l_{\omega},\bold v_l)$.
If $m_0>0$, this is the whole spectrum. If $m_0<0$,
the spectrum
of $B$ contains also a proper eigenvalue $\lambda_r$, given by
the unique negative solution of the equation
$$
m_0+\frac{32}3\pi^2\int_0^\infty
{e^2}|\hat\rho(k)|^2\frac{k^2}{c^2k^2-\lambda_r} dk =
\frac\alpha{\lambda_r}\ ;\autoeqno{s.7}
$$
the corresponding eigenvectors are given by
$\zeta_r^l:=(\bold A^{l}_r,\bold v_l)$. Explicitely $\aomega$ and $\bold
A^{l}_r$ are given by
$\aomega=\Ao+\Ae$, where $\Ae$, $\Ao$ are
distributions belonging to the dual of $\F_*$ defined by
$$
\eqalign{
\left\langle \Ae; \bold A\right\rangle &:=\frac{16}3\pi^2\frac{ec}\alpha
\omega\interre\frac{k^2\hat A^l_c(k)\hat\rho(k)}{\omega_k-\omega}dk
\cr
\left\langle \Ao; \bold A\right\rangle
&:=-\frac{c}{e\alpha\hat\rho\left(\omega/c\right)}(m_{tot}(\omega)
\omega^2-\alpha)\hat A^l_c\left(\frac\omega c\right)
\ ,}\autoeqno{b.4}
$$
(the integral is just a symbol for the Hilbert transform), and
$\bold A^{l}_r$ is given,
in terms of Fourier
components, by
$$
\hat A^{l}_{r,j}(\bold k)=\frac{4\pi ec\lambda_r}{\alpha}
\frac{\hat \rho(k)[\bold v_l\cdot \ve]}
{c^2k^2-\lambda_r}\ .\autoeqno{s.8}
$$
The spectrum contains no other points.}
\proof Consider the eigenvalue equation for $B$
$$
\left(\eqalign{&\lambda \bold A\cr &\lambda \bold Q}
\right)=B \left(\eqalign{&\bold A\cr &\bold Q}
\right)\ .\autoeqno{b.1}
$$
By using the second of these equations ($\lambda Q=...$), passing to
the
Fourier transform, and to the symmetric fields $\hat A^i_c$ as in
\eqref{c.2},
one obtains that the first equation $(\lambda A=...)$ can be given the
form
$$
(\lambda-c^2k^2)\hat A^i_c(k)=-\frac{4\pi e c}{\alpha}\lambda\hat\rho(k)
[\bold
Q\cdot \bold v_i]\ ,
$$
whose general solution in the distribution space is
$$
\hat A^i_c(k)=4\pi \frac{ec\lambda}{\alpha}\frac{\hat \rho(k)[\bold
Q\cdot\bold v_i]}
{\omega_k^2-\lambda} +K^i(k)\delta(ck-\sqrt{\lambda})\ ,\autoeqno{b.3}
$$
where $K^i$ are functions.
We look now for negative eigenvalues corresponding to which the above
$\delta$ function is always zero. Substituting \eqref{b.3} in the second
equation \eqref{b.1}, and performing the angular integration, one
obtains equation \eqref{s.7}, which has a solution only if $m_0$ is
negative. So, if this is the case one has that $\lambda_r$ is a proper
eigenvalue, and $(\bold A^{l}_{r},\bold v_l)\in\Q$ are the corresponding
eigenfunctions. We turn now to the case $\lambda>0$. In
this case the r.h.s. of \eqref{b.3}
has a singularity; so, in order to use it to
define a distribution, we have to choose a prescription for the
calculation of its integral with a regular function. In fact
different prescriptions would lead to different values of the
functions $K^i$. We choose here the prescription
``calculate the integrals as Cauchy principal
values''.
Substituting in the second of equations
\eqref{b.1}, performing the angular integration, and where possible
performing the radial integration, one gets the equation
$$
\eqalign{
m_0\frac{\omega^2}\alpha \bold Q=\bold Q-\bold Q\frac{32}3
\pi^2e^2\frac{\omega^2}\alpha
\int_0^\infty \frac{k^2\rhodiomega}{\omega_k^2-\omega^2}dk
-\frac83 \pi\frac{\omega^2}{c^4}e \hat\rho^*\left(\frac\omega c\right)
\sum_{l}
K^l\left(\frac\omega c\right) \bold v_l\ ,}
$$
where we have put $\omega:=\sqrt\lambda$.
For any choice of $\bold Q$ this equation (for $K^l$) has a solution.
Three independent solutions are obtained by choosing $\bold
Q=\bv l$, $l=1,2,3$, from which
$$
K^l\left(\frac\omega
c\right)=-\frac{m_{tot}(\omega)\omega^2-\alpha}{\frac83\pi
\frac{\omega^2\alpha}{c^4}\hat\rho^*\left(\frac\omega c\right)e}\ ,\quad
l=1,2,3\ ,
$$
where $m_{tot}(\omega)$ is defined by \eqref{m}. The thesis immediately
follows.
\quadratino
\lemma{sin}{The operator $B$ describing the Maxwell--Lorentz system has
no singular continuous spectrum.}
\proof We will apply theorem XIII.20 of ref.~\cite{ReeSimIV}, according
to which, if there exists a $p>1$, a positive $\delta$ and a dense
subset $D$ of $\bar\conf$ such that $\perogni \zeta\in D$ one has
$$
\sup_{0<\epsilon<\delta}\int_a^b|\imma\scal{\zeta}{R(x+i\epsilon)\zeta}|^p
dx<\infty\ ,\autoeqno{r.9}
$$
where $R(\lambda):=(B-\lambda)^{-1}$ is the resolvent of $B$, then $B$
has a purely absolutely continuous spectrum on $(a,b)$.
So we calculate the resolvent of $B$ by solving the equation
$\zeta=(B-\lambda)\zeta_1$. Denoting $\zeta=(\bold A,\bold Q)$,
one obtains, for $\lambda$ with non vanishing
imaginary part,
$$
\eqalign{
\scal{R(\lambda)\left(\matrix{\bold A\cr\bold Q}\right) }{\left(
\matrix{\bold A\cr\bold Q}\right)
}=
\frac2{3c^2}&\sum_l\intzero\frac{k^2\hat
A_{c}^l(k)^2}{\omega_k^2-\lambda}dk
\cr
-\frac{64\lambda \pi^2e^2}
{9c^2(m_{tot}\lambda-\alpha)}
&\sum_{l}\left[\intzero\frac{k^2\hat
A_c^l(k)\hat\rho(k)}{\omega^2_k-\lambda}dk \right]^2
\cr
-\frac{16}3\sum_l\frac{\pi e}c[\bold Q\cdot\bv
l]&\frac1{m_{tot}\lambda-\alpha }\intzero\frac{k^2\hat
A_c^l(k)\hat\rho(k)}{\omega^2_k-\lambda}dk-\frac1\alpha\frac{m_{tot}
\bold Q\cdot\bold Q}{m_{tot}\lambda-\alpha}\ ,
}\autoeqno{r.5}
$$
where
$$
m_{tot}:=m_0+\frac{32}3\pi^2 e^2\intzero
\frac{k^2\hat\rho(k)}{\omega_k^2-\lambda}dk\ .
$$
Let us analyze \eqref{r.5}. We begin by the first term at the
r.h.s.; it is an expression of the form
$$
\intzero\frac{k^2g(k)}{\omega_k^2-\lambda}dk\ ,\autoeqno{r.6}
$$
where $g$ is a function of class $\S$, provided $A\in\S$, i.e.
$\zeta\in\Q$ (which is dense in $\bar\conf$).
Extending $g$ to
$(-\infty,0)$ by symmetry, we have that \eqref{r.6} is proportional to
$$
\interre\frac{kg(k) }{\omega_k-\sqrt\lambda}dk\ .\autoeqno{r.7}
$$
The above function of $\lambda$ is exactly the composition of the
function
$$
\tilde g(z):=\interre\frac{k g(k) }{\omega_k-z}dk\ ,\autoeqno{r.10}
$$
with the function square root. Moreover, \eqref{r.10} is just the
complex extension of the Hilbert transform of $kg(k)$, and it is well
known that, if $\norma{kg(k)}_{L^p}<\infty$, then one has (see e.g.
\cite{SteWei})
$$
\sup_{0\leq\epsilon<\delta}\interre |\tilde g(x+i\epsilon)|^p dx\leq
C\norma{kg(k)}_{L^p}^p\ ,\quad \perogni p>1\ ,
$$
for some positive $C$ (recall that the Hilbert transform is an operator
of type $(p,p)$ for any $p>1$). From this,
it follows immediately that the (supremum for $0<\imma\lambda<\delta$ of the)
integral of the $p-$th power ($p>1$) of this term over any finite
interval is finite.
We come to the other terms of \eqref{r.5}. We have to show that, as
functions of $\lambda$, also all these terms satisfy the above property.
First we consider the quantity $m_{tot}\lambda-\alpha$ appearing at the
denominator of all these terms, and prove that it is an analytic
function of $\lambda$. We recall that the Fourier transform
of an $L^1$ function which decays exponentially at infinity is
analytical;
viceversa, if a real analytic function (over the whole real space)
is $L^1$ then its Fourier transform
decays exponentially at infinity. From this it is easy to conclude that
$\hat\rho^2$ is an analytic function, and therefore also its Hilbert
transform is. It follows that $m_1(\omega)$ is analytic, and so
that also $m_{tot}\lambda-\alpha$ is analytic as a function of
$\lambda$, provided $|\imma \lambda|$ is small enough.
>From this we have that the zeroes of the above denominator cannot have
accumulation points. Denote such zeros
by $\lambda_1,...,\lambda_n,...$, fix a
positive $\epsilon$, denote by $B_i$ the closed ball of radius
$\epsilon$ and center $\lambda_i$. In the complementary of the union of
these balls the above denominator is bounded. So, in this set we can
repeat the argument used for the first term of \eqref{r.5} and deduce
that if the operator $B$ has a singular continuous spectrum, then such
a singular continuous spectrum is contained in the union of the above
balls. However, this has to hold for any $\epsilon,$ so the singular
continuous spectrum has to be concentrated in the points $\lambda_i$.
Moreover, these points are isolated, so the singular continuous spectrum
is empty. By the way, we remark that the appearance of the quantity
$m_{tot}\lambda-\alpha $ at the denominator of the resolvent could
induce to think that the points $\lambda_i$ belong to the point spectrum
of $B$. This is not true. In fact a careful analysis of the
calculation of the resolvent
shows that in correspondence to these points the resolvent
exists, but has a formal expression different from that used in
\eqref{r.5}; moreover, its domain is dense, but does not contain $\Q$.
\quadratino
Next we calculate the functions $C^l(\omega)$
such that, for regular enough $f$ one has
$$
\sum_{l}\langle \zeta^n_\nu,\int_{0}^\infty f^l(\omega)\zeta^l_\omega
d\omega
\rangle_{\Q}=\frac{f^n(\nu)}{C^n(\nu)}\ .\autoeqno{t.4}
$$
These quantities exist due to the spectral resolution theorem.
It is
very useful to remark that the l.h.s. of this equation can be
interpreted as the definition of the family of distributions (labelled by
$\nu$) $\scal{\zeta^n_\nu}{\zeta^l_\omega}$:
$$
\int_0^\infty\scal{\zeta^n_\nu}{\zeta^l_\omega} f(\omega) d\omega:=
\scal{\zeta_\nu^n}{\int_0^\infty \zeta_\omega^l f(\omega)d\omega} \ ,
$$
and, with this definition eq.~\eqref{t.4} can be rewritten in the form
$$
\scal{\zeta^n_\nu}{\zeta^l_\omega}=\frac{\delta(\nu-\omega)}{C^n(\omega)}
\delta_{l,n}\ .\autoeqno{d.2}
$$
In particular, this relation states that the distribution on
the l.h.s. is singular.
\lemma{norm}{The functions $C^n(\omega)$ of equation \eqref{t.4}
coincide with $\alpha C(\omega)$ where $C(\omega)$ is the function
defined by equation \eqref{s.4}.
}
\proof We take $f^l(\omega)=f(\omega)\delta_{l,n}$, with $n$ fixed, so
that, taking into
account the structure of the pairing $\scal..$ and of the eigenfunctions
$A^l_\omega$, the l.h.s. of \eqref{t.4} can be given the form
$$
\eqalign{
&\frac{\delta_{m,n}}{\alpha}\int_{0}^\infty f(\omega)d\omega+
\frac1{4\pi c^2}\sldue{\ae n\nu}{\int_0^\infty f(\omega)\ae m\omega
d\omega}
\cr
&+\frac1{4\pi c^2}\sldue{\ae n\nu}{\int_0^\infty f(\omega)\ao m\omega
d\omega}
+\frac1{4\pi c^2}\sldue{\ao n\nu}{\int_0^\infty f(\omega)\ae m\omega
d\omega}
\cr
&
+\frac1{4\pi c^2}\sldue{\ao n\nu}{\int_0^\infty f(\omega)\ao m\omega
d\omega}\ .
}\autoeqno{d.1}
$$
In the same way as in \eqref{d.2} one can interpret all
integrals in \eqref{d.1} as defining some distributions. We know by the
general theorem stating eq.~\eqref{d.2} (but this can also be verified
explicitly) that the regular parts of these distributions cancel
each other. So we will calculate just their singular part, and in the whole
following calculations we shall retain only terms contributing to
this singular part.
We begin by calculating
the singular part of $\sldue{\Ao}{\ao{n}{\nu}}$.
\trivial{By a simple calculation
we have
$$
\eqalign{
\frac1{4\pi c^2}\int_0^\infty \sldue{\Ao}{\ao{n}{\nu}}f(\nu)d\nu=
\frac{\left[\mdi{\omega} \right]^2}{\rhodiomega
e^2\frac{32}3\pi^2\frac{\omega^2}{c^3}\alpha^2}f(\omega)\
,
}
$$
\trivial{
$$
\eqalign{
\frac
1{4\pi c^2}\int_0^\infty d\nu\intre
\frac{\mdi{\omega}}{\ottoterzi{\omega}}[\bold
v_l\cdot \ve] \delta(\omega_k-
\omega)
\cr
\times\frac{\mdi{\nu}}{\ottoterzi{\nu}}\delta(\omega_k-\nu)f(\nu)d^3k\ ,
}
$$
performing the angular integration, and the $\nu$ integration, we obtain
that the above expression coincides with
$$
\eqalign{
\frac{\mdi{\omega}}{\frac{32}3\pi^2\frac{e^2}{c^6}\omega^2\alpha^2\hat
\rho\left(\frac{\omega}c\right) } \frac1c\intzero dk
k^2\delta(\omega_k-\omega) \frac{\mdi{\omega_k}}{\omega_k^2\hat\rho(k) }
f(\omega_k)
\cr
}=
$$
}
%fine trivial
which implies
fine trivial}
It is easy to see that
$$
\frac1{4\pi c^2} \sldue{\Ao}{\ao{n}{\nu}}
=\frac{\left[\mdi{\omega} \right]^2}{\rhodiomega
e^2
\frac{32}3\pi^2\frac{\omega^2}{c^3}\alpha^2}\delta
(\omega-\nu)\delta_{l,n}
\ .
$$
With a similar calculation it is easily proved that the singular part of
$\sldue{\Ae}{\ao n\nu}$ vanishes. We calculate now the singular part of
$\sldue{\Ae}{\ae n\nu}$. Extending the functions $f$ and $\hat \rho$ to
$(-\infty,0)$ by symmetry, we have
$$
\eqalign{
&\frac1{4\pi c^2}\intzero \sldue{\Ae}{\bold A^{e,l}_\nu}f(\nu) d\nu=
\cr
&\frac83\pi^2\frac{e^2}{\alpha^2}\frac{\omega^2}{c^3} \interre \nu^2
f(\nu)\interre \omega^2_kd\omega_k\frac{\left|
\hat\rho(k)\right|^2}{(\omega_k^2-\omega^2)
(\omega^2_k-\nu^2)}\ .}
\autoeqno{t.11}
$$
We then calculate the integral. Splitting the denominator, exploiting the
symmetries of the arguments, by simple manipulations one obtains that
the integral at the r.h.s. of \eqref{t.11} coincides with
$$
\eqalign{
\frac12 \interre d\nu \interre d\omega_k \menom{\omega_k}\omega \effe
\cr
+\frac12 \interre d\nu \interre d\omega_k \menom{\omega_k}{\omega}
\frac{\omega_k}{\nu-\omega_k}\effe
\cr
+\frac12 \interre d\nu \interre d\omega_k \piu{\omega_k}{\omega}\effe
\cr
+\frac12 \interre d\nu \interre d\omega_k
\piu{\omega_k}{\omega}\frac{\omega_k}{\nu-\omega_k}\effe\ .
}\autoeqno{t.12}
$$
The first and the third of these integrals have the form of a regular
function of $\omega$ (the first one is proportional to the Hilbert transform
of $\omega_k|\hat\rho(k)|^2$)
multiplied by $f(\nu)$ and integrated over $\nu$,
so they do not contribute to the singular part of our distribution,
and we just forget about them. The calculation of the remaining two
integrals can be obtained by
introducing the Hilbert transform $f$,
exploiting the symmetries of the various terms, isolating the part of
the integrals contributing to the singular part of the distributions,
and exploiting the fact that the square of the Hilbert transform is the
identity. One thus obtains
$$
\frac1{4\pi c^2}\sldue{\Ae}{\bold A^{e,l}_{\nu}}= \frac83
\pi^4\frac{e^2}{\alpha^2}\frac{\omega^4}{c^3}\rhodiomega
\delta(\omega-\nu)\ .
$$
>From this, using \eqref{d.1} and the definitions of $\Omega$ and $\tau$,
the thesis easily follows.
\quadratino
\noindent{\bf Proof of theorem~\lemmaref{t.1}} First notice that,
by the
results of lemmas \lemmaref{s.1}, \lemmaref{sin}, \lemmaref{norm},
using also the spectral resolution theorem, any
$\zeta\in\Q$ can be written as
$$
\zeta=\sum_l\left[\int_0^\infty\scal{\zeta}{\zeta_\omega^l}\zeta_\omega^l
\alpha C(\omega)d\omega+ \scal{\zeta}{\zeta_r^l}\zeta_r^l\alpha C_r
\right]\ ,
$$
where $\zeta_r^l$ are the runaway eigenvectors, and
$$
C_r:=(\norma{\zeta_r^l}^2\alpha)^{-1} \autoeqno{t.145}
$$
(here and in the definition of $C(\omega)$ the division by $\alpha$ is
introduced for future convenience).
Then, using Stone theorem and the relations
$$
\bold Q=\bold p=m_0\bold{\dot q}+\frac ec \intre \rho(\bold x)\bold
A(\bold x)d^3x\ ;\quad -\alpha\bold q=\alpha\bold P=\bold {\dot Q}\ .
$$
in order to go back to the original coordinates one obtains theorem
\lemmaref{t.1}.
\quadratino
\autosez{tec2}Proof of the Results Concerning the Point--like Limit.
We will denote by $C_a(\omega)$ the function $C(\omega)$
(cf.~\eqref{s.4}) corresponding to the charge distribution \eqref{pa.1},
and we notice that
$$
C_a(\omega)=\frac2\pi \frac{ 8\pi^3\rhidiomega\omega^2}{\frac1\gamma
\left(\frac{m_{tot}}{\alpha}\omega^2-1\right)^2+\gamma
\left(8\pi^3\rhidiomega\right)^2\omega^6}\ ,\autoeqno{pu.3}
$$
where $\gamma=2e^2/(3\alpha c^3)$.
The nonrunaway part of
the particle's motion corresponding to $\dot\bold q_0=\bold
A_0=\dot\bold A_0=0$ is given by
$$
\bold q_0 \int_0^\infty C_a(\omega)\cos(\omega t)d\omega .\autoeqno{p.2}
$$
We begin by the
\noindent {\bf Proof of proposition \lemmaref{p.1}.} Notice that the
Fourier transform of the motion $\bold q_{a}(t)$ is given by the
function $C_a(\omega)$, and recall that weak convergence of $\bold
q_{a}$ is equivalent to weak convergence of $C_a(\omega)$. From
\eqref{pu.3} one has that, as $a\to0$ with fixed $m_0$,
$C_a(\omega)\to0$ pointwise, uniformly on $[\epsilon,\infty)$, and
therefore it tends to a (possibly vanishing) singular distribution
concentrated at the origin $\omega=0$. So the limit, which is assumed to
exist, is a finite linear combination of the delta and of its
derivatives. Taking the Fourier transform one has
$$
\bold q(t)=\bold q_0 \sum_{k=0}^{n}C_k t^k\ .
$$
By conservation of energy one has the uniform estimate
$$
\frac12m_0\dot{q_{a}}^2(t)+\frac12 \alpha q_a^2(t)\leq\frac12\alpha
q_0^2\ ,
$$
from which $C_k=0$ $\perogni k\geq1$.
\quadratino
We come now to the proof of proposition \lemmaref{p.2}.
First we prove the convergence of the runaway part of the solution.
This follows from the following
\lemma{pu.2}{When $a\to0$ the negative eigenvalue $\lambda_r$ of the
Maxwell--Lorentz system converges to
$$
\lambda_r=-\nu_r^2\ ,
$$
where $\nu_r$ is the only real solution of equation \eqref{e.7}. The
constant $C_r$ then converges to
$$
C_r^0:=\left( \frac32+\frac{\nu_r^2}{2\omega_0^2}\right)^{-1}
$$
}
\proof The equation defining $\lambda_r$ can be given
the form
$$
m+\frac{32}3\pi^2\frac{e^2}{c^3}\lambda_r \intzero\frac{|\hat\D(ak)|^2}
{\omega_k^2-\lambda_r}dk=\frac{\alpha}{\lambda_r}\ ,\autoeqno{pu.8}
$$
which is of the form
$$
m+f(a,\lambda_r)=\frac{\alpha}{\lambda_r}\ ,
$$
where $f$ is a regular function of $a\in[0,\infty)$, and $\lambda_r$.
Therefore, by the implicit function theorem $\lambda_r$ depends
regularly on $a$, so the limiting value of $\lambda_r$ is just obtained
by solving the limiting equation. So,
one obtains eq.~\eqref{e.7}. The proof of the
convergence of $C_r$ is very similar and is omitted.
\quadratino
The convergence (as $a\to 0$) of the non--runaway part is an immediate
consequence of the following
\lemma{pu.1}{Consider the function $C_a(\omega)$ cf.~\eqref{pu.3}, with
$m_0$ given by \eqref{s.5}. There exist positive constants $\bar a$,
$M$, $K_1$, $K_2$, $\bar\omega$ such that
the following estimate holds
$$
C_a(\omega)\leq g(\omega)\ ,\perogni 0\leq a\leq\bar a\ ,
$$
where $g(\omega)$ is defined by
$$
g(\omega):=\left\{\matrix{M & \omega \in[0,\bar\omega]
\cr
\max\left\{
\displaystyle{\frac1{(K_1\omega^2-\alpha)
\omega},\frac1{K_2\omega^4}}\right\}
&\omega>\bar \omega
}\right.\ ,
$$
Moreover one has $g\in L^1(\Re^+)$}
\proof To obtain the estimate of $C_a(\omega)$ in the domain
$\omega\leq\bar\omega$ it is enough to show that its denominator is
uniformly bounded away from zero in this domain. To this end notice that
$\rhidiomega$ does not vanish in this domain, so that the second term in
the denominator of $C_a(\omega)$ vanishes only at $\omega=0$, while for
$\omega$ in a fixed neighbourhood of the origin the first term of this
denominator is uniformly bounded away from zero (notice that
$m_1(\omega)\leq K\omega$ for suitable $K$ and $\omega$ in a
neighbourhood of the origin).
Consider now the set $\omega>\bar\omega$. Using the standard inequality
$\gamma a^2+b^2/\gamma\geq 2a b$, one obtains
$$
C_a(\omega)\leq \frac2\pi \frac{1}{(m_{tot}\omega^2-\alpha)\omega}\ ,
$$
and,
provided $m_{tot}$ is uniformly bounded away from zero,
$$
C_a(\omega)
\leq
\frac2\pi\frac1{(K_1\omega^2-\alpha)\omega}\ .\autoeqno{pu.6}
$$
Moreover, if $\bar\omega$ is large enough this is integrable. However,
$m_{tot}(\omega)$ generally has zeroes which tend to infinity as
$a\to0$, so that the above inequality is not true in
$(\bar\omega,\infty)$ and more work is needed. We will use \eqref{pu.6}
in a domain where $m_{tot}$ is uniformly bounded away from zero, and we
will show that in the complementary domain the quantity $\rhidiomega$
does not vanish, so that here one can use the inequality
$$
C_a(\omega)\leq\frac2\pi \frac{1}{\gamma8\pi^3\rhidiomega\omega^4}\ .
$$
To make this precise we first make the change of variables
$(\omega,a)\mapsto(b,a)$, with $b=\omega a/c$ and
define the set
$$
S:=\left\{(a,b)\in[0,\bar a]\times\Re^+\ :\ m_{tot}(b,a)=0
\right\}\ ;
$$
then we consider a closed neighbourhood depending on a small parameter
$\epsilon$ of this set, defined by
$$
S_\epsilon:=\bigcup_{(a,b)\in S}B_{\epsilon}(a,b)\ ,
$$
where
$$
B_\epsilon(a,b):=\left\{\tilde a,\tilde b\ :\ |a-\tilde
a|\leq\epsilon\ \ |\tilde b-b|\leq\epsilon\right\}\ .
$$
Since (as is easily seen) $m_1(b,a)$
is a regular function of its variables, in $(S_\epsilon)^c$ one has
$$
m_{tot}(b,a)\geq k_1>0\ ,
$$
for some $k_1$, so that going back to the original variables one obtains
$$
|m_{tot}(\omega)\omega^2-\alpha|\geq \big| |k_1\omega^2|-\alpha\big|\ ,
$$
which, provided $\bar\omega>\sqrt{\alpha/k_1}$, is uniformly away from
zero, so \eqref{pu.6} holds on this set.
Consider now $S_\epsilon$. We show that there exists a positive $K_3$
such that, in this set, one has
$$
\hat\rho_a\left(\frac\omega c\right)=\hat \D(b)\geq K_3\ .
$$
To this end we have to discuss in some detail the structure of this set.
The equation which defines $S$ is
$$
m_{1}(b,a)=-m\ .\autoeqno{pu.5}
$$
Now, the quantity $m_1(b,a)$
is proportional (through inessential
constants) to $b[H(|\hat\D|^2)](b)/a$, where
$H(|\hat\D|^2)$ denotes the Hilbert transform of the function
$|\hat\D^2|$.
So, equation \eqref{pu.5} has the form
$$
K_4 b H(|\hat\D|^2)(b) =-\frac{ma}c\ ,\autoeqno{pu.4}
$$
where $K_4$ is a positive constant.
Using the formula
$$
[H(xf)](y)=y[H(f)](y)+\frac1\pi \interre f(x)dx\ ,
$$
which holds for any regular $f$,
and the regularity of $|\hat\D|^2$, it is easy to see that
the Hilbert transform of this function tends to zero exactly as $1/b$ when
$b\to\infty$; it follows that the
l.h.s. of \eqref{pu.4} tends to a finite strictly negative
limit (obviously independent
of $a$) as $b\to\infty$. Then, there exists $\bar b$ and $\bar a$ such that
for any $a<\bar a$ all solutions of \eqref{pu.4} are in the region
$b\leq\bar b$. It follows that $\hat\rho_a\left(\omega/c\right)=\hat\D(b)$,
which is a function of $b$ vanishing only at infinity,
is uniformly and strictly positive in $\S$, and therefore also in
$S_\epsilon$, provided $\epsilon$ is small enough.
\quadratino
>From this lemma, exploiting Lebesgue dominated convergence theorem, one
obtains that the particle's motion converges in $C^1[-T,T]$ (as $a\to0$) to
$$
\bold q(t):=\bold
q_0 \frac2\pi \omega_0^2\tau_0\int_0^\infty \frac{\omega^2\cos(\omega
t)}{ (\omega^2-\omega_0^2)^2+\omega^6\tau_0^2}d\omega +\bold q_0
C^0_r\Ch(\nu_r t)
\
.\autoeqno{p.5}
$$
Concerning the case of non--vanishing initial velocity, first we remark
that proposition \lemmaref{p.23} is a simple consenquence of the fact
that the coefficients $\eta^l_\omega$ diverge (due to mass
renormalization) as $a\to0$. For the proof of proposition \lemmaref{p.3}
one has to calculate the coefficients $\eta_\omega^l$ and $\eta_r^l$
corresponding to the initial data $\dot \bold q_0$, $\bold A_0=\bold
X_{\dot q_0}$. This gives
$$
\eta_\omega^l=\frac{\dot \bold q_0\cdot \bold v_l}{\omega^2}\ ,\quad
\eta_r^l=\frac{\dot \bold q_0\cdot \bold v_l}{\lambda_r}\
.\autoeqno{e.41}
$$
It is then a simple variant of the proof of eq.~\eqref{p.5} to show
that corresponding to the initial data considered in proposition
\lemmaref{p.3}, the limit of the particle's motion is
$$
\eqalign{
\bold q\left(t\right)&:= \frac2\pi \omega_0^2\tau_0\int_0^\infty
\omega^2 \frac{\bold q_0 \cos(\omega
t)+(\dot \bold q_0/\omega)\sin(\omega t)}
{ (\omega^2-\omega_0^2)^2+\omega^6\tau_0^2}d\omega
\cr
&+C^0_r[\bold q_0\Ch(\nu_r t)+\frac{\bold {\dot q}_0}{\nu_r}\Sh(\nu_r t)]
\
.}\autoeqno{p.6}
$$
\noindent{\bf Proof theorem \lemmaref{qo}}. One has to
calculate the integral~\eqref{p.5}. Denoting by
$P(\omega)$ the denominator of the argument of the integral one has
$$
P(\omega)=(\omega^2-\omega^2_0+i\tau_0\omega^3)
(\omega^2-\omega^2_0-i\tau_0\omega^3)\ ,
$$
so that the roots with positive imaginary part of $P(\omega)$ are given
by $\omega_1=i\nu_r$, $\omega_2=\nu_2+i\nu_3$, $\omega_3=-\nu_2+i\nu_3$.
For the explicit calculation of the integrals one has to compute the
residues of their arguments in the poles $\omega_1$, $\omega_2$,
$\omega_3$. A long but straightforward calculation allows
to calculate all the residues and to put the various expressions in real
form obtaining that the integral in \eqref{p.5} is given by
$$
e^{-\nu_3|t|}[\lea_1 \cos(\nu_2|t|)+
\lea_2\sin(\nu_2|t|)
]+\lea_4 e^{-\nu_r|t|}\ .\autoeqno{pu.22}
$$
The explicit form of the constants $\lea_1$ and $\lea_2$ is not
important here (except for the asymptotics \eqref{p.33}),
while $\lea_4$ is given by
$$
\lea_4=
\frac{\omega_0^2}{\tau_0}\frac{1}{\nu^4+4\nu_2^2\nu_3^2}[\dot \bold q_0-
\bold q_0 \nu_r]\ ,
$$
where $\nu^2:=\nu_2^2-\nu_3^2+\nu_r^2$.
Then, one has to show that the term proportional to $\lea_4$ in
\eqref{pu.22} exactly cancels the decreasing exponential of the
hyperbolic sine and cosine present in \eqref{p.6}. This is a quite
complicated calculation which makes use of some known identities among
the roots of an equation of third order. Precisely
the identities needed are
$$
\eqalign{
&\nu_2^2+\nu_3^2-2\nu_r\nu_3=0\ ,
\cr
&(\nu_2^2+\nu_3^2)\nu_r=\frac{\omega_0^2}{\tau_0}\ .
}
$$
The explicit calculation is omitted.
\quadratino
Due to the linearity of the problem the proof of theorem \lemmaref{p.44}
requires only the study of the limit of motions with initial data of the
form $(\bold q_0,\dot \bold q_0, \bold A_0, \dot \bold A_0)=(0,0,\bold
A'_0, \dot \bold A_0)$ with $ (\bold A'_0, \dot \bold
A_0)\in\S_*(\Re^3,\Re^3)\times\S_*(\Re^3,\Re^3)$. This is a simple
variant of the proof of equation \eqref{p.5} and of theorem
\lemmaref{qo}, and therefore is omitted.
\vskip\spaziosoprasez
\centerline{{\bf References}}
\vskip 10pt
\insertbibliografia
\vfill\eject
\bye