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\0{\titolo Tree expansion and multiscale analysis for KAM tori}
%\footnote{${}^*$}{\nota Archived in
%{\tt mp\_arc@math.utexas.edu} \#94-??; to get a TeX version, send an empty
%E-mail message.}
\vskip1.truecm
\0{\bf G.Gentile}\footnote{${}^1$}{\nota
E-mail: {\tt gentileg\%39943.hepnet@lbl.gov}; address:
Dipartimento di Fisica, Universit\`a di Roma ``La Sa\-pi\-en\-za",
P. Moro 2, 00185 Roma, Italia.},
{\bf V.Mastropietro}\footnote{${}^2$}{\nota
Address: Dipartimento di Matematica, Universit\`a di Roma II
``Tor Vergata", Via della Ricerca Scientifica,
00133 Roma, Italia.}
\vskip0.5truecm
\0{\bf Abstract:} {\it We prove that the perturbative expansion for
the KAM invariant tori of the Thirring model (with interaction
depending also on the action variables) is convergent by using
techniques usual in quantum field theory like the multiscale
decomposition and the tree expansion. The proof follows the
ideas of Eliasson, [3], and extends the results found
in the case of an action-independent interaction potential, [4].}
\vskip1.truecm
\0{\sl Keywords}: {\it KAM theorem, dynamical systems,
re\-nor\-mal\-iza\-tion group, quan\-tum field the\-ory,
mul\-ti\-scale de\-composi\-tion, form factors}
\vskip2.5truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\{\bf 1. Introduction}
\*\numsec=1\numfor=1\pgn=1
\\The KAM theorem proves in a indirect way that the formal
perturbation series for the invariant tori of quasi integrable systems
are convergent if some conditions are fulfilled by the
hamiltonian, [11], [1], [12]. Of course one should be able to prove the
convergence studing directly the perturbative series and this has been
performed in recent times by Eliasson, [3]. However the work
by Eliasson has not enjoyed a wide circulation, maybe because of the
excessive generality with which the considered problem is attacked,
(see also the regarding comments in [5]). Then in [4] the
convergence of the perturbative expansion of the tori is proven
by using the Eliasson's ideas in a special model,
the {\sl Thirring model}, [13], {\sl with
the further simplification that the interaction does not depend on
the actions}: in this way the discussion becomes much less
involved, and so is suitable for a more easy understanding. The proof
is implemented with methods usual in the renormalization
group approach to quantum field theory, like the multiscale decomposition
of the propagator and the tree expansion. Some minor technicalities of
the proof are resolved in [9], where the strong diophantine hypothesis
used in [4] is completely relaxed, (see [4], [9],
for a detailed review). In this paper the proof
is extended to cover the case in which the interaction potential
of the Thirring model does not depend only on the angle variables, as
in [4], but also on the action variables. As a function of the angles,
the potential is supposed to be still an even trigonometric polynomial
of degree $N<\io$, while, as a function of the actions, we assume that
it is analytical, since this does yields no further technical
work with respect to the case of a polynomial-like interaction.
The idea is always to confine ourselves to a not too tangled
case, in such a way to emphasize the interesting features of the
method, and to point out the analogies with quantum
field theory. We shall define a perturbative expansion
for the KAM tori in terms of Feynman's graphs. However
the ``dimensional bounds" are not sufficient to prove
the series convergence: then we perform a resummation so
obtaining a multiple series, in terms of the coupling constant
and of suitable functions called ``form factors". If such functions
are uniformly bounded, the series is convergent; endly
the form factors boundedness is proven by using some
cancellation mechanisms between diagrams operating to all
perturbative orders. The analogy with quantum field theory
is striking; for other results in this direction, see also [7], [8].
With respect to [4] and [9], the use of a
smooth decomposition of the propagator and the
introduction of a ``renormalized expansion" via a localization
operation makes simpler the proof.
The hamiltonian function of the Thirring model is
%
$$\fra12 J^{-1}\AA\cdot\AA\,+\,\e f(\aa,\AA) \; , \Eq(1.1)$$
%
where $J$ is the non singular matrix of the inertia moments,
$\AA=(A_1,\ldots,A_l)\in {\bf R}^l$ are their angular momenta and
$\aa=(\a_1,\ldots,\a_l)\in {\bf T}^l$ are the angles describing their
positions. We shall consider a ``rotation vector"
$\oo_0=(\o_1,\ldots,\o_l)\in {\bf R}^l$ verifying the {\sl diophantine
property} with diophantine constants $C_0,\t>0$; this means that
%
$$ C_0|\oo_0\cdot\nn|\ge |\nn|^{-\t} \; ,
\kern2.5cm\V0\ne\nn\in {\bf Z}^l \; , \Eq(1.2) $$
%
and it is easy to see that the {\sl diophantine vectors} have
full measure in ${\bf R}^l$ if $\t$ is fixed
$\t>l-1$.
We suppose $f$ to be an even trigonometric polynomial of degree
$N$ in the angle variables and an analytic function in the angular
momenta variables, \ie
%
$$ f(\aa,\AA)=\sum_{|\nn|\le N} f_\nn(\AA)\,\cos\nn\cdot\aa \; , \qquad
f_\nn(\AA)=f_{-\nn}(\AA) \; , \Eq(1.3) $$
%
with $f_\nn(\AA)$ analytic in $\AA$ in a domain $W(\AA_0,\r)=
\{ \AA\in{\bf R}^l \; : \; |\AA-\AA_0|/|\AA_0| \le \r\}$, for any
$\nn$, being $\AA_0=J\oo_0$. Finally, if $J_j$, $j=1,\ldots,l$ are
the eigenvalues of the matrix $J$, we define
%
$$ J_m=\min_{j=1,\ldots,l} J_j \; , \qquad
J_M=\max_{j=1,\ldots,l} J_j \; , \qquad
F=\max_{|\nn|\le N, \AA\in W(\AA_0,\r)} f_\nn(\AA) \; , $$
%
The fundamental result of this work is the following one.
\*
\\{\bf Theorem 1.1.} {\it The hamiltonian model \equ(1.1) admits
an $\e$--analytic family of motions starting at
$\aa=\V0$ and having the form
%
$$\AA=\AA_0+\V H(\AA_0,\oo_0t;\e)+\V \m(\AA_0,\e) \; , \qquad\qquad
\aa=\oo_0t+\V h(\AA_0,\oo_0 t;\e) \; , \Eq(1.4) $$
%
with $\V H(\AA,\pps;\e)$, $\V h(\AA,\pps;\e)$ analytic
in $\pps$ with} $\hbox{Re}\pps\in {\bf T}^l$, {\it and}
$|\hbox{Im}\pps_j|<\x$, {\it and in $\AA\in W(\AA_0,\r)$, where
$\AA_0=J\oo_0$, and with vanishing average
in ${\bf T}^l$, and with $\V H(\AA_0,\pps;\e)$, $\V h(\AA_0,\pps;\e)$ and
$\V\m(\AA_0,\e)$ analytic for $|\e|<\e_0$
with a suitable $\e_0$ close to $0$:
%
$$ \e_0 = E_0 \, [J_MJ_m^{-2}C_0^2Fe^{\x N}\r^{-2} ]^{-1} \; , \Eq(1.5) $$
%
where $E_0$ is a dimensionless quantity depending only on $N$ and $l$.
This means that the set $(\AA,\aa)$
described as $\pps$ varies in ${\bf T}^l$
is, for $\e$ small enough, an invariant torus for \equ(1.1), which is run
quasi periodically with angular velocity vector $\oo_0$. It is a family of
invariant tori coinciding, for $\e=0$, with the torus
$\AA=\AA_0,\,\aa=\pps\in {\bf T}^l$.}
\*
\\{\it Remark 1.} One recognizes a version of the KAM
theorem. The proof that follows extends the one reported in [4]
to the more general case in wich the interaction depends also
on the angular momenta.
\*
\\{\it Remark 2.} Note that, in distinction to [4], the result is
{\sl not uniform} in the twist rate $T$, defined as $T=J_M^{-1}$:
this is known from the KAM theorem, and follows from the fact
that the interaction depends also on the action variables. In other
words the {\sl twistless} property in [4] is a consequence of the
special choice of the interaction potential, which is of the
form \equ(1.2), with $f_\nn(\AA)$ replaced with
$f_\nn$ independent on $\AA$.
\*
Calling $\V H^{(k)}$, $\V h^{(k)}$, $\V\m^{(k)}$ the $k$-th order
coefficients of the Taylor expansion of $\V H$, $\V h$, $\V\m$ in powers of
$\e$, and writing the equation of motion as
$\dot\a_j=(J^{-1}\AA)_j+\e\dpr_{A_j} f(\aa,\AA)$, and
$\dot A_j=-\e\dpr_{\a_j} f(\aa,\AA)$, $j=1,\ldots,l$, we get immediately
recursion relations for $\V H^{(k)},\V h^{(k)}$; for $k=1$:
%
$$ \eqalign{
& \V\o_0\cdot\dpr_\aa\,H^{(1)}_j = - \dpr_{\a_j} f \; , \cr
%
& \V\o_0\cdot\dpr_\aa\,h^{(1)}_j = \left(J^{-1}[\V H^{(1)}+
\V\m^{(1)}]\right)_j + \dpr_{A_j} f \; , \cr} \Eq(1.6) $$
%
where $\oo_0\cdot\dpr_\aa=\sum_{i=1}^l\oo_{0i}\dpr_{\a_i}$,
and, for $k>1$:
%
$$ \eqalign{
\V\o_0\cdot\dpr_\aa\,H^{(k)}_j = & -
\sum_{m>0}
\sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop
\sum_{s=1}^l (p_s + q_s)= m}
\fra1{ \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr
& \cdot \; \dpr_{\a_j}\,\dpr^{p_1}_{\a_1}\ldots\dpr^{p_l}_{\a_l}
\dpr^{q_1}_{A_1}\ldots\dpr^{q_l}_{A_l}\,f(\oo_0t,\AA_0) \; \cdot \cr
& \cdot \; {\sum}^*
\prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} h^{(k_{sj})}_s
\prod_{i=1}^{q_s} \left( H^{(k_{si}')}_s + \m^{(k_{si}')}_s \right)
\Big] \; , \cr
%
\V\o_0\cdot\dpr_\aa\,h^{(k)}_j = \; & \left( J^{-1} [\V H^{(k)}
+ \V\m^{(k)}]\right)_j + \sum_{m>0}
\sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m}
\fra1{\prod_{s=1}^l p_s!\,q_s!} \quad \cdot \cr
& \cdot \;
\dpr_{A_j}\,\dpr^{p_1}_{\a_1}\ldots\dpr^{p_l}_{\a_l}
\dpr^{q_1}_{A_1}\ldots\dpr^{q_l}_{A_l}f(\oo_0t,\AA_0) \; \cdot \cr
& \cdot \; {\sum}^*
\prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} h^{(k_{sj})}_s
\prod_{i=1}^{q_s} \left( H^{(k_{si}')}_s + \m^{(k_{si}')}_s \right)
\Big] \; , \cr} \Eq(1.7) $$
%
where the $\sum^*$ denotes summation over the integers $k_{sj}\ge1$,
$k_{si}'\ge1$, with: $\sum_{s=1}^l(\sum_{j=1}^{p_s}k_{sj}$ $+$
$\sum_{i=1}^{q_s}k_{si}') =$ $k-1$.
In fact from the equations of motion for the angular momenta, we obtain
immediately the first recursive relation in \equ(1.7). Then
suppose that $\V h^{(k)}(\pps)$ and $\V H^{(k)}(\pps)$ are
trigonometric polynomials of degree
$\le k N$, respectively odd and even in $t$, for $1\le k< k_0$:
we see immediately that the r.h.s. of the first equation in
\equ(1.7) is odd in $t$. Then the first equation in \equ(1.7)
can be solved for $k=k_0$. It yields an even function
$\V H^{(k_0)}(\pps) + \V\m^{(k_0)}$ which is defined up to the constant
$\V\m^{(k_0)}$, (which we call ``counterterm").\footnote{${}^3$}{\nota
Note that
$\V H^{(k)}$ has to have zero average over $\pps$ by construction.}
Such a constant, however, must be taken so that the equation for
the angle variables, \ie the second of \equ(1.7),
has zero average, in order to be soluble. Hence the equation
for $\V h^{(k)}$ can be solved
and its solution is a trigonometric polynomial in $\pps$,
defined up to a constant: such a constant has to be chosen to be
vanishing so that $\V h^{(k_0)}$ is odd in $t$ and the procedure can
be iterated. Therefore the equations
for $\V \m^{(k)}$ will have to be
obtained recursively by imposing that, for all $k$'s,
the average over $\pps$
of the r.h.s of the second equation in \equ(1.7) is
identically vanishing and requiring $\V h^{(k)}_{\V0}\,\=\,\V0$,
$\forall k$:
then the trigonometric polynomial
$\V h^{(k)}(\pps)$ will be completely
determined (if possible at all) from the second of \equ(1.7).
If, given a function $F(\pps)$, we define by $F_\nn$ its $\nn$-th
Fourier series component,
%
$$ F_\nn = \int_{{\bf T}^l} {d\pps\over(2\p)^2}\,F(\pps)\,
e^{-i\nn\cdot\pps} \; , \qquad\qquad F(\pps) =
\sum_\nn F_\nn \, e^{i\nn\cdot\pps} \; , \Eq(1.8) $$
%
one easily finds, for $k=1$, from \equ(1.6)
%
$$ \eqalign{
h^{(1)}_{j\nn} & = \left( -iJ^{-1}\nn \right)_j \left[ i\oo_0\cdot\nn
\right]^{-2} f_\nn(\AA_0)
+ \left[ i\oo_0\cdot\nn
\right]^{-1} \dpr_{A_j} f_\nn(\AA_0)\;, \qquad \nn\neq\V0 \; ,\cr
%
H^{(1)}_{j\nn} & = \left( -i\nn_j \right) \left[ i\oo_0\cdot\nn
\right]^{-1} f_\nn(\AA_0)\;,
\qquad \nn\neq\V0 \; , \cr
%
\m^{(1)}_j & = - (J\dpr_{\AA})_j f_{\V0}(\AA) \; . \cr}
\Eq(1.9)$$
%
The \equ(1.7) provides an algorithm to evaluate a formal power
series solution to our problem. It has been remarked, [3], [4], [14],
that \equ(1.7) yields a {\sl diagrammatic expansion} of $\V h^{(k)}$,
$\V H^{(k)}$ and $\V \m^{(k)}$, (we simply ``i\-te\-ra\-te"
it until only $\V h^{(1)}$ and $\V H^{(1)}$,
given by \equ(1.9), and $\V\m^{(k')}$, $k'0}
\sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop
\sum_{s=1}^l (p_s + q_s)= m} {\sum}^*
{1 \over \prod_{s=1}^l p_s!\,q_s! } \; \cdot \cr
& \cdot \; (i\nn_0)_j \prod_{j=1}^l (i\nn_0)_j^{p_j}
\prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0)
\; \cdot \cr & \cdot \;
\prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}}
\prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'}
+ \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr
%
(i\oo\cdot\nn)^2 \, \bar h^{(k)}_{j\nn} = \; &
(i\oo\cdot\nn)\, (\h^{-1}_m {\bar H^{(k)}}_{\nn})_j \cr & +
(i\oo\cdot\nn) \Big({F\,C_0^2 \over J_m} \Big) \sum_{m>0}
\sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m}
{\sum}^* {1\over \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr
& \cdot \; \dpr_{B_j}\,\prod_{j=1}^l (i\nn_0)_j^{p_j}
\prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0)
\; \cdot \cr & \cdot \;
\prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}}
\prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'}
+ \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr} \Eq(2.1) $$
%
and, for $\nn=\V0$,
%
$$ \eqalign{
\bar \m^{(k)}_{j} & =
\Big({F\,J_M\,C_0^2 \over J_m^2} \Big) \sum_{m>0}
\sum_{p_1,\ldots,p_l,q_1,\ldots,q_l \atop \sum_{s=1}^l (p_s+ q_s)= m}
{\sum}^* {1\over \prod_{s=1}^l p_s!\,q_s! } \quad \cdot \cr
& \cdot \; (\h_M \dpr_{\BB})_j\,\prod_{j=1}^l (i\nn_0)_j^{p_j}
\prod_{i=1}^l \dpr_{B_i}^{q_i} \f_{\nn_0}(\BB_0)
\; \cdot \cr & \cdot \;
\prod_{s=1}^l \Big[ \prod_{j=1}^{p_s} \bar h^{(k_{sj})}_{s,\nn_{sj}}
\prod_{i=1}^{q_s} \left( \bar H^{(k_{si}')}_{s,\nn_{si}'}
+ \bar \m^{(k_{si}')}_{s,\nn_{si}'} \right) \Big] \; , \cr} \Eq(2.2) $$
%
where the $\sum^*$ denotes summation over the integers $k_{sj}\ge1$,
$k_{si}'\ge1$, with: $\sum_{s=1}^l(\sum_{j=1}^{p_s} k_{sj}$ $+$
$\sum_{i=1}^{q_s}k_{si}') =$ $k-1$, and over the integers
$\nn_0$, $\nn_{sj}$, $\nn_{si}'$, $s=1,\ldots,l$, $j=1,\ldots,p_s$
and $i=1,\ldots,q_s$, with: $\nn_0+\sum_{s=1}^l(\sum_{j=1}^{p_s}
\nn_{sj}+$ $\sum_{i=1}^{q_s}\nn_{si}') =\nn$.
\*
For the time being we ignore the presence of the ``constant part"
of the angular momenta, \ie $\V \m^{(k)}$, $k\ge1$,
\ie we reason as if it was $\V\m^{(k)}\,=\,\V0$, $\forall k\ge1$:
we shall see below how the discussion
has to be modified when also such terms are taken into account.
A tree diagram $\th$ will consist of a family of lines ({\sl branches} or
{\sl lines})
arranged to connect a partially ordered set of points ({\sl vertices}
or {\sl nodes}), with the higher vertices to the right. The branches are
naturally ordered as well; all of them have two vertices at their
extremes, (possibly one of them is a top vertex), except the lowest
or {\sl first branch} which has only one vertex, the {\sl first vertex}
$v_0$ of the tree. The other extreme $r$ of the first branch will be called
the {\sl root} of the tree and will not be regarded as a vertex; we
shall call the first branch also {\sl root branch}.
If $v_1$ and $v_2$ are two vertices of the tree we say that $v_1v_0$ can
be considered the first vertex of the tree consisting of the vertices
following $v$: such a tree will be called a subtree of $\th$.
To each branch $\l_v$ and to each vertex $v$
we attach a finite set of labels: $\z_{\l_v}$, $R_{\l_v}$, $j_{\l_v}$,
and, respectively,
$\d_v$, $m_v$, $k_v$, $\nn_v$, and a {\sl vertex function} $\EE_v$,
which are defined as follows.
\acapo
(1) The label $\z_{\l_v}$ can assume the symbolic values
$\z_{\l_v}=h,H$;
\acapo
(2) $R_{\l_v}=1$ if $\z_{\l_v}=H$ and
$R_{\l_v}=2$ if $\z_{\l_v}=h$, and it is called the {\sl degree} of
the line $\l_v$;
\acapo
(3) $j_{\l_v}=1,\ldots,l$;
\acapo
(4) $\d_v=0,1$, if $\z_{\l_v}=h$,
and it is identically $0$ if $\z_{\l_v}=H$;
\acapo
(5) $m_v$ is the number of branches
emerging from $v$, and can be written as
%
$$ m_v=\sum_{i=1}^3 \sum_{j=1}^l q_{v,j}^i \; , $$
%
if $q_{v,j}^i$, $i=1,2,3$, are the branches leading
to vertices $w$ with $\p(w)=v$ and
carrying a $\z_{\l_w}$ label equal, respectively,
to $h$, $H$, $\m$;
\acapo
(6) the {\sl order label}
$k_v$ is given by the number
of vertices of the subtree with first vertex $v$;
\acapo
(7) the {\sl mode label} $\nn_v$ is such that $\nn_v\in{\bf Z}^l$,
$|\nn_v|\le N$;
\acapo
(8) the vertex function is defined as
%
$$ \eqalign{
\EE_v = & {F\,C_0^2 \over J_m}
\Big\{ \Big[ \Big( (-i(\h_m)^{-1}\nn_v)_{j_{\l_v}} (1-\d_v)
+ (i\oo\cdot\nn(v)) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h}
\cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} \Big] \; \cdot \cr
& \cdot \;
\prod_{w: \p(w)=v \atop \z_{\l_w}=h} (i\nn_v)_{j_{\l_w}}
\prod_{w: \p(w)=v \atop \z_{\l_w}=H} \dpr_{B_{j_{\l_w}}}
\Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.3) $$
%
where $\d_{\z_{\l_v},x}$ denotes the Kronecker's delta (which
is 1 only if $\z_{\l_v}=x$), and the last $m_v$ factors are
missing if $v$ is a top vertex, (in this case $m_v=0$).
Each of the $m_v+1$ factors appearing in \equ(2.3) will be called a
{\sl vertex factor}.
\*
The labels, ``decorating'' the tree, will be called also
decorations (\ie the labels attached to the tree
and the decorations are synonimous below).
Finally to the branch $\l_v$ leading from the vertex $\p(v)$
to the vertex $v$ we associate a ``propagator" $g(\oo\cdot\nn_{\l_v})$,
which is given by:
%
$$ g(\oo\cdot\nn_{\l_v}) = {1 \over
\left[i\oo\cdot\nn_{\l_v} \right]^{R_{\l_v}} }
\; , \Eq(2.4) $$
%
where $\nn_{\l_v}=\sum_{w\ge v}\nn_w$ is defined
as the {\sl momentum} entering $v$, and $R_{\l_v}$ is called the {\sl
degree of the line}.
A possible tree is drawn in Fig.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURA 2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\midinsert
\insertplot{240pt}{170pt}{%fig1.tex
\ins{-35pt}{90pt}{\sl root}
\ins{10pt}{110pt}{$j_{\l_{v_0}}\,R_{\l_{v_0}}$}
\ins{10pt}{80pt}{$\nn(v_0)\,\b_{\l_{v_0}}$}
\ins{60pt}{85pt}{$v_0$}
\ins{50pt}{125pt}{$\matrix{\d_{v_0}\,m_{v_0}\cr k_{v_0}\,\nn_{v_0}\cr}$}
\ins{95pt}{132pt}{$j_{\l_{v_1}}\,R_{\l_{v_1}}$}
\ins{120pt}{110pt}{$\nn(v_1)\,\b_{\l_{v_1}}$}
\ins{152pt}{120pt}{$v_1$}
\ins{135pt}{160pt}{$\matrix{\d_{v_1}\,m_{v_1}\cr k_{v_1}\,\nn_{v_1}\cr}$}
\ins{110pt}{50pt}{$v_2$}
\ins{190pt}{100pt}{$v_3$}
\ins{230pt}{160pt}{$v_5$}
\ins{230pt}{120pt}{$v_6$}
\ins{230pt}{85pt}{$v_7$}
\ins{230pt}{-10pt}{$v_{11}$}
\ins{230pt}{20pt}{$v_{10}$}
\ins{200pt}{65pt}{$v_4$}
\ins{230pt}{65pt}{$v_8$}
\ins{230pt}{45pt}{$v_9$}
}{fig1}
\kern1.3cm
\didascalia{Fig.2.1: A tree $\th$ with
$m_{v_0}=2,m_{v_1}=2,m_{v_2}=3,m_{v_3}=2,m_{v_4}=2$ and $m=12$,
$\prod_v m_v!=2^4\cdot6$, and some decorations. The line numbers,
distinguishing the lines, are not shown.}
\endinsert
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We imagine that all the tree lines have the same length (even though
they are drawn with arbitrary length in Fig.2.1). A group acts on the
set of trees, generated by the permutations of the subtrees having
the same root. Two diagrams that can be superposed by the
action of a transformation of the group will be regarded as identical,
(the superpositon has to be such that all the decorations
of the diagram match). To order $k$, not considering the decorations,
the number of trees is bounded by $2^{2k}$.
We shall imagine that each branch carries also an
arrow pointing to the root (``gravity" direction, opposite to the order).
\*
Then, if $X^{(k)}_{j\nn}(h)=\bar h^{(k)}_{j\nn}$ and $X^{(k)}_{j\nn}(H)=
\bar H^{(k)}_{j\nn}$, it follows that \equ(2.1) can be rewritten as
%
$$ X^{(k)}_{j\nn}(\z) = {\sum_\th}^* \prod_{v_0\le v\in\th}
\prod_{j=1}^l {1\over q_{v,j}^1!q_{v,j}^2!} \, g(\oo\cdot\nn_{\l_v})
\; \EE_v \; , \Eq(2.5) $$
%
where the sum runs over all the tree $\th$'s of order $k$, with
$\nn_{\l_{v_0}}=\nn$, $j_{\l_{v_0}}=j$ and $\z_{\l{v_0}}=\z$,
and the $*$ recalls that the diagram $\th$ can and will be
supposed such that $\nn_{\l_v}\ne\V0$ for all $v\in\th$, (by the parity
properties remarked in \S 1 and because the counterterms
are assumed to be zero, so that $q_{v,j}^3\=0$).
\*
If also the contributions $\bar\m^{(k)}_j$'s are considered, the above
discussion has to be modified as follows. Some of the top vertices $v$ are
drawn as fat points, and represent the quantities $\bar\m^{(k_v)}_{j_v}$:
the corresponding $\z_{\l_v}$ label is set $\z_{\l_v}=\m$, and the label
$k_v$ is such that $k_vv_0$, which is not a top vertex, $\z_{\l_v}\neq\m$, and
\equ(2.3) has to be replaced with
%
$$ \eqalign{
\EE_v = & {F\,C_0^2 \over J_m}
\Big\{ \Big[ \Big( (-i(\h_m)^{-1}\nn_v)_{j_{\l_v}} (1-\d_v)
+ (i\oo\cdot\nn_{\l_v}) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h}
\cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} \Big] \; \cdot \cr
& \cdot \;
\prod_{w: \p(w)=v \atop \z_{\l_w}=h} (i\nn_v)_{j_{\l_w}}
\prod_{w: \p(w)=v \atop \z_{\l_w}=H} \dpr_{B_{j_{\l_w}}}
\prod_{w: \p(w)=v \atop \z_{\l_w}=\m} \dpr_{B_{j_{\l_w}}}
\Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.6) $$
%
while, if $v>v_0$ is a top vertex,
%
$$ \eqalign{
\EE_v = & {F\,C_0^2 \over J_m}
\Big\{ \Big[ \Big( (-i(\h_m)_j^{-1}\nn_v)_{j_{\l_v}} (1-\d_v)
+ (i\oo\cdot\nn(v)) \dpr_{B_{j_{\l_v}}} \d_v \Big) \d_{\z_{\l_v},h}
\cr & + (-i\nn_v)_{j_{\l_v}} \d_{\z_{\l_v},H} +
\bar \m_{j_{\l_v}}^{(k_v)} \d_{\z_{\l_v},\m} \Big]
\Big\} \; \f_{\nn_v}(\BB) \Big|_{\BB=\BB_0} \; , \cr} \Eq(2.7) $$
%
Then a formula analogous to \equ(2.5) still holds,
with the constraint that the label $k$ is
given by the number of the free vertices plus the sum of the orders
$k_v$ of all the leaves $v\in\th$:
%
$$ X^{(k)}_{j\nn}(\z) = {\sum_\th}^* \prod_{v_0\le v\in\th}
\prod_{i=1}^3 \prod_{j=1}^l {1\over q_{v,j}^i!} \,
g(\oo\cdot\nn_{\l_v}) \, \EE_v \; , \Eq(2.8) $$
%
where the sum runs over all the trees of order $k$, (with
$\nn_{\l_{v_0}}=\V0$ and $j_{\l_{v_0}}=j$), having $k_0(\th)$ free
vertices and $\NN_\FF(\th)$ fruits $v_i$ of order
$k_i$, $i=1,\ldots,\NN_\FF(\th)$, with the
constraint that their orders add to $k-k_0(\th)$.
If $\z_{\l_{v_0}}=\m$, then \equ(2.8) has to be replaced by the
following equation:
%
$$ \bar\m^{(k)}_j = - \Big( {J_M\,F\,C_0^2\over J_m^2} \Big)
{\sum}^* (\h_M\dpr_{\BB})_j \, \f_{\nn}(\BB_0)
\prod_{v_0 < v\in\th}
\prod_{i=1}^3 \prod_{j=1}^l {1\over q_{v,j}^i!}
\, g(\oo\cdot\nn_{\l_v}) \; \EE_v \; , \Eq(2.9) $$
%
Note that \equ(2.8) and \equ(2.9) can depend on $\bar\m_{j'}^{(k')}$,
only if $k'From \equ(2.8) and \equ(2.9) we deduce that each tree
can be considerd as representing a contribution to $X^{(k)}_{j\nn}(\z)$,
$\z=h,H$, or $\m^{(k)}_j$: such a contribution
will be called the {\sl value of the tree}. Then, if $v_0$ is the first
vertex of the tree, and $\z_{\l_{v_0}}=h$, the value of the label
$\d_{v_0}$ tells us which term we are considering among the two
of the first equation in \equ(2.1); the argument can be repeated
for each following vertex. The interpretation of the other labels
is obvious.
\vskip1.truecm
\\{\bf 3. Dimensional bounds}
\*\numsec=3\numfor=1\pgn=1
\\The following Lemma shows that the estimates for the KAM tori
can be reduced to the study of the contributions of the
fruitless trees. The proof can be found in Appendix A1.
\*
\\{\bf Lemma 3.1.} {\it Suppose that we can prove that the
contribution to $\bar\mu_j^{(k)}$ and the one to $X_{j\nn}^{(k)}(\z)$,
$\z=h,H$, arising from
a single tree stripped of the fruits, ({\rm i.e.} the
contribution we obtain by deleting the fruits from the tree), is
bounded by $D_1^{k_0}$ for some constant $D_1$,
if $k_0$ is the number of the free vertices;
then a bound $B_0^k$ for the
complete values $|\bar\mu_j^{(k)}|$ and
$|X_{j\nn}^{(k)}(\z)|$, $\z=h,H$, follows immediately, for a
suitable constant $B_0=2^3\r^{-1}B_1D_1$, being $\r$ defined
after \equ(1.3) and $B_1=2^5l(2N+1)^l$.}
\*
\\{\it Remark 1}. The value of $B_1$ is found below, and
is due to the trees counting, (see in particular the discussion
after (3.5)): $B_1$ counts the possible decorated fruitless trees.
\*
\\{\it Remark 2}. Note that, given a fruitless tree, contributing to
$\bar\mu_j^{(k)}$, the bound we obtain for it contains an extra factor
$J_M/J_m$ with respect to the same bound
we would obtain if it had contributed to
$X_j^{(k)}(\z)$, $\z=h,H$, as a comparison between \equ(2.2) and the first
equation in \equ(2.1) shows. Therefore we can confine ourselves to
study the contributions to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$,
arising only from fruitless trees, and, if a bound $\tilde B_0$ is
found for them, then it will be $B_0=J_MJ_m^{-1}\tilde B_0$.
\*
In order to bound the fruitless tree values, we introduce a multiscale
decomposition of the propagator. Let $\chi(x)$ be a $C^\io$ not increasing
function such that $\chi(x)=0$, if $|x|\ge 2$ and $\chi(x)=1$ if $|x|\le 1$,
and let $\chi_n(x)=\chi(2^{-n}x)-\chi(2^{-(n-1)}x)$, $n\le 0$, and
$\chi_1(x)=1-\chi(x)$: such functions realize a $C^\io$ partition
of unity, for $x\in[0,\io)$, in the following way. Let us write
%
$$ 1=\chi_1(x)+\sum_{n=-\io}^0\chi_n(x)\=\sum_{n=-\io}^1\chi_n(x)
\; . \Eq(3.1) $$
%
Then we can decompose the propagator in the following way:
%
$$ g(\oo\cdot\nn_{\l_v})={1\over [i\oo\cdot\nn_{\l_v}]^{R_{\l_v}}}\=
\sum_{n=-\io}^1{\chi_n(x)\over [i\oo\cdot\nn_{\l_v}]^{R_{\l_v}}}\=
\sum_{n=-\io}^1 g^{(n)}(\oo\cdot\nn_{\l_v}) \; \Eq(3.2) $$
%
where $g^{(n)}(\oo\cdot\nn_{\l_v})$ is the ``propagator at scale $n$".
If $n<0$, $g^{(n)}(\oo\cdot\nn_{\l_v})$ is a $C^{\io}$
compact support function different from $0$ for
$2^{n-1}<|\oo\cdot\nn_{\l_v}|\le 2^{n+1}$,
while $g^{(1)}(\oo\cdot\nn_{\l_v})$ has
support for $1<|\oo\cdot\nn_{\l_v}|$. In the domain where
it is different from zero, the propagator verifies the bound
%
$$ \Big| {\partial^p\over\partial x^p} g^{(n)}(x)
\Big|_{x=\oo\cdot\nn_{\l_v}}
\le a_{R_{\l_v}}(p) 2^{-n(R_{\l_v}+p)} \; , \qquad p\in {\bf N} \; , $$
%
where $a_{R_{\l_v}}(p)$ is a suitable constant,
such that $a_{R_{\l_v}}(0)=2^{R_{\l_v}}$,
which depends on the form of the function $\chi(x)$.\footnote{${}^4$}{\nota
The consatnt $a_{R_{\l_v}}(p)$ has a bad dependence on $p$, but we shall
see that in our bounds $p$ does not increase ever beyond 2: see in
particular (4.4) below and, especially, the right after remark.}
Proceeding as in quantum field theory, see [5], given a tree $\th$
we can attach a {\sl scale label} $n_{\l_v}$ to each branch
$\l_v$ in $\th$, which is equal to the scale of the propagator
associated to the branch via \equ(2.4) and \equ(2.2).
Looking at such labels we identify the connected
cluster $T$ of vertices which are
linked by a continuous path of branches with the same scale labels
$n_T$ or a higher one and which are maximal: we shall say that
{\sl the cluster $T$ has scale $n_T$}. Therefore an inclusion relation is
established between the clusters, in such a way that the innermost clusters
are the clusters with the highest scale, and so on.
Each cluster can have an arbitrary number of branches entering it,
({\sl incoming lines}), but only one line exiting, ({\sl outgoing lines});
we use that the branches carry an arrow pointing to the root: this gives a
meaning to the words ``incoming" and ``outgoing".
The multiscale decomposition \equ(3.2) of the propagator
allows us to rewrite \equ(2.5) as
%
$$ X_{j\nn}^{(k)}(\z)={\sum_{\th}}^*\prod_{v_0\le v\in\th}
\prod_{j=1}^l \prod_{i=1}^3 {1\over q_{v,j}^i!}
g^{(n_{\l_v})}(\oo\cdot\nn_{\l_v})\,\EE_v \; , \Eq(3.3) $$
%
where the sum is over the labeled trees, and therefore, with respect
to \equ(2.8), we have the extra labels $n_\l$
associated to the lines $\l$'s: a (not optimal bound) of the number
of terms appearing in the sum over
the $\nn_v$ and $n_v$ labels is given by $[2(2N+1)^l]^k$,
as $|\n_i|\le N$, $\forall i=1,\ldots,l$, and because of
the support compact property of the propagators.
\*
\\{\bf Definition 3.1 (Resonance).} {\it Among the clusters we
consider the ones with the property that there is only one
incoming line, carrying the same momentum of the outgoing line,
and we define them {\rm resonances}. If $V$ is one such cluster
we denote by $\l_V$ the incoming line and $K(V)$ the number
of vertices contained in $V$ ({\rm resonance order}). The line scale
$n=n_{\l_V}$ is lower than the lowest scale $n'=n_V$
of the lines inside $V$: we call $n_{\l_V}$ the
{\rm resonance-scale}, and $\l_V$ a {\rm resonant line}.}
\*
\\{\it Remark.} Note that a resonance $V$, as
a cluster, has an its own scale $n_V$, which is higher
than its resonance-scale $n_{\l_V}$, $n_V\ge n_{\l_V}+1$.
\*
Recall that $R_\l$ is the {\sl degree} of the line $\l$: it is the
exponent of the propagator corresponding to the line, (see \equ(2.2)).
Given a resonance $V$, let us define the {\sl resonance degree}
$D_V=1,2$ as the degree $R_{\l_V}=1,2$ of the resonant line,
\ie $D_V=R_{\l_V}$.
Given a tree $\th$, let us define $N_n$ the number of lines with scale
$n\le 0$, and $N^j_n$, $j=1,2$, the number of lines $\l$ with scale $n\le 0$
and having $R_\l=j$, (\ie with degree $j$). Then it is easy to check that the
scaling properties of the propagators and the definitions \equ(2.2)
and \equ(2.5) immediately imply that the contribution to $\V X^{(k)}(\z)$
arising from a given tree $\th$ can be bounded by:
%
$$ \CC^k\prod_{n\le 0}
2^{-(2n N_n^2+n N_n^1)} \prod_{v}2^{n_{\l_v}\d_v} \; ,
\Eq(3.4) $$
%
for a suitable constant $\CC$; if we look at \equ(2.6), we can write
%
$$ \CC=2^2J_m^{-1} C_0^2 N^2 F\r^{-1} \; , \Eq(3.5) $$
%
where $C_0$ is the diophantine constant introduced in \equ(1.3),
$\r$ is introduced after \equ(1.3), $N$ is a
bound on the mode values $\nn_v$ of the vertices $v\in\th$, and
the factor $2^2$ is due to the definition of the compact support
of the propagators. The last product in \equ(3.4)
arises because for each line $\l_v$ of degree
$2$ such that $\d_v=1$ there is an extra factor $(i\oo\cdot\nn(v))$,
(see \equ(2.6)).
Therefore we only have to multiply \equ(3.4) by the number of
diagram shapes for $\th$ having vertices with given
bifurcation numbers $m_v$, $v\in\th$, ($\le 2^{2k}$, see [10]),
by the number of ways of attaching the labels,
($\le [2\cdot 2 \cdot 2\cdot l\cdot (2N+1)^l]^k$),
so that the number of trees of order $k$ can be bounded by
$B_1^k$, if $B_1$ is defined as in Lemma 3.1.
The following bound holds
for the number of lines with scale $n\le 0$:
%
$$ N_n^1+N_n^2 \= N_n \le {2k\over E2^{-n\t^{-1} } } + \sum_{T,n_T=n}
\sum_{j=1}^2 m_T^j \; , \Eq(3.6) $$
%
where $m^j_T$ is the number of resonances $V$'s inside the cluster $T$,
having resonance-scale $n_{\l_V}=n_T$ and degree $D_V=j$, and $E$
can be chosen
$E=2^{-3\t^{-1}}N^{-1}$. This is a slightly modified version of the
Brjiuno's lemma, [2]: a proof is in Appendix A2, and it is
taken from [4], [9], with some minor changes.
Therefore if there was no resonance, \ie if it was $m^j_T=0$ for any $T$,
then we would obtain a (not optimal) bound $G_0^k$ for a suitable constant
$G_0>0$; the labeled trees counting and \equ(3.5) give
%
$$ G_0=b_12^7l(2N+1)^l F\r^{-1}J_m^{-1} C_0^2 N^2 \; , \Eq(3.7) $$
%
where $b_1=\exp[\sum_{n=1}^\io 4(\ln 2)nE^{-1}2^{-n\t^{-1}}]$.
However {\sl there are resonances}, and we have to deal with them.
\*
\\{\bf Definition 3.2 (Resonance factor).}
{\it Let us consider a resonance $V$;
let $\l_V$ be the incoming line, as in Definition 3.1.
We denote by $w_0$ the vertex
which the outgoing line of $V$ leads to, (recall that the ordering
of the tree is opposite to the gravity direction), and by $w_2$ the vertex
which the incoming line (resonant line) leads to: such lines are
characterized by the labels, respectively, $\z_{\l_{w_0}}$ and
$\z_{\l_{w_2}}$. Such a couple of values
$(\z_{\l_{w_0}}, \z_{\l_{w_2}})$ can assume the values $(H,H)$,
$(H,h)$, $(h,H)$ and $(h,h)$, and we can introduce a label $s_V$ in order to
distinguish the four above possible cases. Let us define the {\rm
resonance factor}
$\VV_{s_V,j_{\l_{w_0}}j_{\l_{w_2}}}(\oo\cdot\nn_{\l_V}))$
as the quantity
%
$$ \VV_{s_V,j_{\l_{w_0}}j_{\l_{w_2}}}(\oo\cdot\nn(v_{\l_V}))=\EE_{w_0}
\prod_{w_00$: to such an aim the next section
is devoted.
\vskip1.truecm
\\{\bf 4. Boundedness of the form factors and convergence
of the perturbative series}
\*\numsec=4\numfor=1\pgn=1
\\In this section we prove that $|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|\le
C^k$, for some $C>\e_1^{-1}$ and any value of $\oo\cdot\nn$.
This will be done by writing the form factor as a sum over diagrams
which can be thought as resonances with their incoming line, (see
Definition 3.3), so that,
in order to obtain the contribution arising from a single diagram,
we have to compute the resonance factor times
the propagator associated to its incoming line.
The resonance factor is expressed in terms of the original trees,
({\sl not of the resumed trees}): the corresponding resonance $V$ can
be interpretated as a tree having an endpoint $w_2$, (see
Definition 3.2), from which a mode $\nn_{w_2}+\nn_{\l_V}$ is
emitted, instead of a mode $\nn_{w_2}$ simply.
\*
The first step in order to prove the boundedness of the form factor
is to note that $\sum_V \VV^{n_{\l_V}}_{s_V,jj'}(\oo\cdot\nn_{\l_V})=
O((\oo\cdot\nn_{\l_V})^{R_{\l_V}})$, if $R_{\l_V}$ is the
degree of the propagator in \equ(3.2). This is in fact the case,
as the following lemma shows.
\*
\\{\bf Lemma 4.1.} {\it The form factor introduced in
Definition 3.3 through \equ(3.9) can be written in the
following way:
%
$$ \s^{n(k)}_{s,jj'}(\oo\cdot\nn) =
\sum_{V : \nn_{\l_V}=\nn \atop k(V)=k,n_{\l_V}=n}
\int_0^1 dt \; t^{R_{\l_V}-1} \left[
{\dpr^{R_{\l_V}} \over \dpr (\oo\cdot\nn)^{R_{\l_V}} }
\, \VV^{n_{\l_V}}_{s,jj'}(t\oo\cdot\nn) \right]
\, \ch_n(\oo\cdot\nn) \; . \Eq(4.1) $$
%
This means that the first $R_{\l_V}$ terms of the Taylor expansion
of the resonance factor $\VV^{n_{\l_V}}_{s,jj'}(\oo\cdot\nn)$
in powers of $\oo\cdot\nn$ add to zero when summed to give the
form factor.}
\*
We have taken into account explicitly the expression giving the
propagator on scale $n$, $g^{(n)}(\oo\cdot\nn)$ $=$
$\ch_n(\oo\cdot\nn)[i\oo\cdot\nn]^{-R_{\l_V}}$, (see
\equ(3.2)). The proof of Lemma 4.1 is given in Appendix A3.
\*
In order to prove that $|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|\le
C^k$, for some constant $C$, we modify the rules how to
construct the trees by splitting each resonance factor
$\VV$ as $\VV=\LL\VV+(1-\LL)\VV$, where
%
$$ \eqalign{
\LL \VV^n_{1,jj'}(\oo\cdot\nn) & = \VV^n_{1,jj'}(0) \; , \cr
%
\LL \VV^n_{2,jj'}(\oo\cdot\nn) & = \VV^n_{2,jj'}(0)
+ [\oo\cdot\nn]\,\dot\VV^n_{2,jj'}(0) \; , \cr
%
\LL \VV^n_{3,jj'}(\oo\cdot\nn) & = \VV^n_{3,jj'}(0) \; , \cr
%
\LL \VV^n_{4,jj'}(\oo\cdot\nn) & = \VV^n_{4,jj'}(0)
+ [\oo\cdot\nn]\,\dot\VV^n_{2,jj'}(0) \; , \cr} \Eq(4.2) $$
%
where $\dot\VV^n_{s,jj'}(0)$ denotes the first derivative of
$\VV^n_{s,jj'}$ with respect to $\oo\cdot\nn$, computed in
$\oo\cdot\nn=0$. Note that the resonant factors depend on
$\oo\cdot\nn$ only through the propagators, (see \equ(3.2)).
Then, for each line $\l$ inside the resonance, the momentum
flowing in it is given by $\nn_\l\=\nn_\l^0+\e_\l\nn$,
where $\nn_\l^0$ is the sum of the mode labels corresponding to
the vertices following $\l$ but inside the resonance, and $\e_\l=0,1$,
so that, even if we set $\oo\cdot\nn=0$, (\ie $\oo\cdot\nn_\l=
\oo\cdot\nn_\l^0$ for each $\l$ inside the resonance), no too
small divisor appears because of the presence of the compact support
functions $\ch_{n'}(\oo\cdot\nn_\l)$, $n'>n$.
Given a tree, on any cluster the $\LL$ or $1-\LL\=\RR$
operators apply; however for the cancellations seen
in Lemma 4.1 the sum over the trees of order $k$ containing one or
more resonances on which the $\LL$ operator applies
is vanishing, so that we can rule out all such contributions and
consider simply the trees with resonances on which the operator
$\RR$ applies.
It is convenient to write the effect of $\RR$ on a resonance $V$ as
%
$$ \eqalign{
\RR\VV^n_{s,jj'}(\oo\cdot\nn) & = (\oo\cdot\nn) \ig_0^1 dt\;
\dot\VV^n_{s,jj'}(t\oo\cdot\nn) \qquad
\qquad (\hbox{first order zero}: s=1,3) \; , \cr
%
\RR\VV ^n_{s,jj'}(\oo\cdot\nn) & = (\oo\cdot\nn)^2 \ig_0^1 dt\;t\;
\ddot\VV^n_{s,jj'}(t\oo\cdot\nn)
\qquad (\hbox{second order zero}: s=2,4) \; , \cr}
\Eq(4.3) $$
%
where $\ddot\VV^n_{s,jj'}$ denotes the second derivative with respect to
the variable $\oo\cdot\nn$.
As there are resonances enclosed in other resonances the above formula
can suggest that there are propagators derived up to $\approx k$
times, if $k$ is the order of the graph. This would be of course
a source of problems, as $a_{R_{\l_V}}(p) > p!$, where $a_{R_{\l_V}}(p)$
is defined after \equ(4.2). However it is not so: in fact the propagators
are derived at most two times. This can be seen as follows.
Let $n$ be the resonance-scale of the
maximal resonance $V$, and let us define $V_0$
as the collection of lines and vertices in $V$ not contained
in any resonance internal to $V$.
Then we can write $\RR\VV(\oo\cdot\nn_{\l_V})$, (we do not
write explicitly the labels of the resonance factor), as
%
$$ \RR \Big( \prod_{\l\in V_0} g_\l^{(n_\l)} \prod_{V'\subset V}
[\RR\VV (\oo\cdot\nn_{\l_{V'}})] \prod_{v\in V_0} \EE_v \Big)
\; , \Eq(4.4) $$
%
being the second product over the resonances $V' \subset V$ which
are maximal; in \equ(4.4), for any resonance $V' \subseteq V$,
$\RR\VV(\oo\cdot\nn_{\l_{V'}})$ can be written either
as in \equ(4.3) or as a difference
$\RR\VV(\oo\cdot\nn_{\l_{V'}})$ $=$
$\VV(\oo\cdot\nn_{\l_{V'}})-\LL\VV(\oo\cdot\nn_{\l_{V'}})$,
in according to which expression turns out to be more convenient to
deal with.
Then the first step is to write the action of $\RR$ on the maximal
cluster as in \equ(4.3), leaving the other terms
$\RR\VV (\oo\cdot\nn_{\l_{V'}})$ written
as differences: so \equ(4.4) can be written by the Leibniz's rule
as a sum of terms, and the derivatives of $\RR$ apply either on
some propagator $g_\l^{(n)}$ or on some
$\RR\VV (\oo\cdot\nn_{\l_{V'}})$. In the end there
can be either no derivative, or one derivative, or two derivatives
applied on each $\RR\VV (\oo\cdot\nn_{\l_{V'}})$.
If only one derivative acts on
$\RR \VV(\oo\cdot\nn)$, $\nn=\nn_{\l_{V'}}$, and, \eg,
$s=2,4$, then we write, when such a term is not vanishing,
%
$$ \dpr \RR \VV(\oo\cdot\nn)=\dpr \VV(\oo\cdot\nn)-
\dot\VV(0)=(\oo\cdot\nn)\ig_0^1 dt\, \ddot \VV(t\oo\cdot\nn) \; , $$
%
because in such a case the derivative with respect to
$\oo\cdot\nn$ is equal to the dervative with respect to
$\oo\cdot\nn_{\l_{V}}$, while if two derivatives act on
$\RR\VV (\oo\cdot\nn_{\l_{V'}})$, then we write
%
$$ \dpr^2 \RR \VV(\oo\cdot\nn)= \ddot \VV(\oo\cdot\nn) \; . $$
%
The case $s=1,3$ is easier, and can be discussed in the same way.
Then no more than two derivatives can act on each resonance $V'$
in any case, and the procedure can be iterated, since the resonances
$V'$ can be dealt with as the resonance $V$.
The effect of the $\RR$ operator is to obtain a gain factor either
$2^{n-n'}$ or $2^{n-n'}2^{n'-n{'}{'}}$, where $n'$ and $n{'}{'}$ are
the scales of two lines $\l'$ and $\l{'}{'}$ contained in some
clusters $T'$ and $T{'}{'}$ inside $V$; the line $\l{'}{'}$ can
coincide with $\l'$, or also be absent, if $s_V=1,3$. So we
can rewrite, \eg, the first factor as $2^{n-n'}=2^{n-n_1}\ldots
2^{n_q-n'}$, where $n_i$ is the scale of the cluster $T_i\supset T_{i+1}$,
with $T_0=V$ and $T_{q+1}=T'$. Analogous considerations hold for $n{'}{'}$,
so that we can conclude that: (1) no more than two derivatives can
ever act on any propagators; (2) a gain
$2^{D_{V'}(n_{\l_{V'}}-n_{V'})}$ is obtained for any resonance
$V'\subseteq V$; (3) the
total number of terms generated by the derivation operations is
bounded by $k(V)^2$, if $k(V)$ is the order of the resonance $V$,
(see Definition 3.1).
Therefore, for the value of the diagram formed by the resonance
plus its incoming line, we find the bound
%
$$ \eqalign{
2^{-D_Vn_V} \Big[ & \tilde \CC^k
{\prod_{v}}^* 2^{n_{\l_v}\d_v}
\prod_{n\ge n_V}2^{-(2nN_{n}^2+nN_{n}^1)}\Big] \cdot \; \cr &
\cdot \; \Big[ \prod_{n\ge n_V} \prod_{T \atop n_T=n}\prod_{j=1}^2
\prod_{i=1}^{m_T^j}\,2^{D_{V_i}(n-n_{V_i}) }
\Big] \; , \cr} \Eq(4.5) $$
%
where $n_V$ is the scale of the resonance, $D_V$ is the degree of
the resonance, the product with $*$ is over all the lines
not exiting from any resonance,\footnote{${}^5$}{\nota
If a line $\l_{w_0}$ comes out from a resonance, and $\d_{w_0}=1$,
then the factor $(i\oo\cdot\nn(w_0))$ appearing in the first vertex
factor corresponding to $w_0$, (see \equ(3.6)), is used in order to
implement the cancellation of the form factor, (as proof of
Lemma 4.1 shows, see Appendix A3), and then the bound improvement
\equ(4.4); therefore we have no more the factor
$2^{n_{\l_{w_0}}}$ in \equ(3.4) corresponding to such a line.}
and the second square bracket is the part coming from the resummations,
and follows from the above discussion about the gain factors. The constant
$\tilde\CC$ differs from $\CC$ in \equ(3.5) as it takes into account
the bound on the derivatives of the propagators: we can set
$\tilde\CC=\CC\,e^2\,a_2(2)$, as the sum over all the outer resonances $V$'s
of the factors $[2k(V)]^2$ can be bounded by $e^{2k}$, and
$a_R(p)\le a_2(2)$, for any $R=1,2$, and $0\le p \le 2$.
In Appendix A2, we show that, if $N_n^j(V)$ is the number of lines
on scale $n$ and of degree $j$
contained inside a resonance $V$, then the following
bound holds:
%From \equ(3.6) and the fact that, for any tree of order $k$,
%the number of clusters on scale $n$ verifies the bound
%$\sum_{T,n_T=n}1\le (2k)^{-1}[E\,2^{-n\t^{-1}}]^{-1}$,
%(see Appendix A2), we obtain
%
$$ \sum_{j=1}^2 j N_n^j (V) \le \fra{8k(V)}{E\,2^{-n\t^{-1}}} + \sum_{T
\subseteq V \atop n_T=n}
\Big[ -2 + \sum_{j=1}^2 j \, m_T^j \Big] \; . \Eq(4.6) $$
%
Substituting \equ(4.6) into \equ(4.5), we see that the $j m_T^j$
is taken away by the first factor in $\,2^{D_{V_i}n}$
$2^{-D_{V_i}n_{V_i}}$, being $n=n_{\l_{V_i}}$, while the
remaining $\,2^{-D_{V_i}n_{V_i}}$ are compensated by the $-2$ before
the $\sum_j m_T^j$ in \equ(4.6), taken from the factors with $T=V_i$,
(note that there are always enough $-2$'s); in particular
we can get rid of the factor $2^{-D_Vn_V}$ in \equ(4.5).
Hence \equ(4.5) is bounded by
%
$$ a_2(2)^k \CC^k e^{2k} \prod_n\,2^{-8 n k E^{-1}\,2^{n/\t}}\le
D_1^k \; , \Eq(4.7) $$
%
if $k=k(V)$, with a suitable constant $D_1$; the previous bounds give
%
$$ D_1 =b_1^2 \,e^2 \, a_2(2)\,\CC=
2^2\,a_2(2)\,e^2\,F\,J_m^{-1}\,C_0^2\,N^2\,\r^{-1}\,
\exp[\sum_{n=1}^\io 8\,n(\ln 2)\,E^{-1}\,2^{-\t^{-1}n}] \; . $$
%
As in the preceding section, the number of trees contributing to
$\s^{n(k)}_{s,jj'}(\oo\cdot\nn)$ is bounded by $B_1^k$, so that,
if we reacall the Remark 2 after Lemma 3.1, we can bound
$|\s^{n(k)}_{s,jj'}(\oo\cdot\nn)|$ by $B_0^k$, for a suitable
constant $B_0$ given by
%
$$ B_0 \= 2^3\r^{-1}J_MJ_m^{-1}B_1D_1=
2^{10}\,e^2\,a_2(2)\,b_1^2\,l\,(2N+1)^l\,F\,\r^{-2}\,
J_M\,J_m^{-2}\,C_0^2\,N^2 \; . \Eq(4.8) $$
%
Obviously an analogous bound holds also for the contributions
to $\bar\m^{(k)}_j$ and to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$,
so that the proof of Theorem 1.1 is complete, and the value
\equ(1.5) for the KAM tori convergence radius is obtained. \qed
\vskip1.truecm
\\{\bf Acknowledgements.} We thank G.Gallavotti for having proposed
us this work, and for many suggestions and useful discussions.
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% APPENDICI A1,A2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1.truecm
\\{\bf Appendix A1. Proof of Lemma 3.1}
\*
\\Let us consider first the contribution to $\bar\m_j^{(k)}$.
A value $\bar\m_{j}^{(k)}$
can be represented as a sum of tree values, as shown in \equ(2.9).
Then each tree whose value contributes
to $\bar\m^{(k_i)}_{j_i}$ can be enclosed inside a bubble. Since
$\bar\m_{j}^{(k)}$ depends on other fruits, we can iterate
the procedure until no fruit is left: at each step we obtain some new
bubbles which are contained in a bubble drawn in the previous step.
We consider a single tree appearing in the multiple sums
obtained through the above procedure: we note that the values arising
from the different bubbles factorize, since the momentum flowing in
the first branch of the maximal tree inside a bubble is identically
vanishing, and therefore the momenta running in any tree branches
do not depend on the bubbles encircling the following vertices.
Then for each innermost bubble $b_0$, which does not
contain any bubbles, a bound $D_1^{\tilde k_{b_0}}$,
if $\tilde k_{b_0}$ is the number of (free) vertices of the tree
inside the bubble, is obtained, by assumption.
Hence we pass to the next to the innermost bubbles: for each
such bubble $b_1$ a bound $D_1^{\tilde k_{b_1}}$
$2^{\tilde k_{b_1}}$ $\prod_{b_0\subset b_1} 2\r^{-1}$
is found, where $\tilde k_{b_1}$ now is the number of vertices
contained in the considered bubble, (\ie is the number of free vertices
of the tree inside the considered bubble but outside the inner bubbles).
In fact, for each vertex inside $b_1$ we can bound
%
$$ \left\{ { \prod_{j=1}^l \dpr_{B_j}^{q_{v,j}^2+q_{v,j}^3}
\f(\BB_0) \over q_{v,j}^2!q_{v,j}^3! } \right\} \le
\left[ \prod_{j=1}^l 2^{q_{v,j}^2+q_{v,j}^3} \r^{-q_{v,j}^3} \right]
\r^{-q_{v,j}^2} \; , $$
%
where $\r^{-q_{v,j}^2}$ can be interpretated as a bound for
$[q_{v,j}^2!]^{-1} \prod_{j=1}^l \dpr_{B_j}^{q_{v,j}^2}
\f(\BB_0)$, because of the assumption of analiticity in the
angular momenta variables of the interaction potential.
And so on, until in the end a bound $\prod_b [ (2D_1)^{\tilde k_b}
2\r^{-1} ]$ follows for the considered tree value, being the
product over the bubbles $b$ and $\tilde k_b$ being the number of
vertices inside the bubble $b$ but outside the inner bubbles.
This means that the value of a single tree contributing to
$\bar\m_j^{(k)}$ is bounded by $[4\r^{-1}D_1]^k$, as
$\sum_b \tilde k_b\le k$.
Then we have only to perform the sum over the trees, but from the above
discussion we conclude that such a sum is arranged in the following way: we
sum over all the trees with $k$ vertices, and over all the possible ways to
put bubbles around the vertices of the tree $\th$. The first sum is
bounded by $B_1^k$, (as it is proven in \S 3, see the discussion
after \equ(3.5)),
while the second one is trivially bounded by
$2^k$, as there can be at worst one bubble per vertex. Then, if we take
$B_0=2^3\r^{-1}B_1 D_1$, the stated result follows.
If the trees contribute to $X_{j\nn}^{(k)}(\z)$, $\z=h,H$, the
proof remains the same: simply we do not draw any bubble around
the entire tree.
\qed
\vskip1.truecm
\\{\bf Appendix A2. Resonant Siegel-Brjiuno bound.}
\*
\\Let us call $N^*_n\,\=\,N_n^*(\th)$ the number of non resonant lines
of a tree $\th$ carrying a scale label $\le n$, \ie $N_n^*
+\sum_T(m_T^1+m_T^2)=\sum_{n'\le n}
N_n$. We shall prove first that $N^*_n\le 2k (E 2^{-n/\t})^{-1}-1$ if
$N_n>0$.
If $\th$ has the root line with scale $>n$ then calling
$\th_1,\th_2,\ldots,\th_m$ the subdiagrams of $\th$ emerging from the
first vertex of $\th$ and with $k_j>E2^{-n/\t}$ lines, it is
$N^*_n(\th)=N^*_n(\th_1)+\ldots+N^*_n(\th_m)$ and the statement is inductively
implied from its validity for $k'k-2^{-1} E\,2^{-n/\t}$. Finally, and this is the real
problem as the analysis of a few examples shows, we claim that in the
latter case the root line of $\th_1$ is either a resonant line or it has
scale $>n$.
Accepting the last statement we have: $N^*_n(\th)=1+N^*_n(\th_1)=
1+N^*_n(\th'_1)+\ldots+N^*_n(\th'_{m'})$, with $\th'_j$ being the $m'$
subdiagrams emerging from the first node of $\th'_1$ with orders
$k'_j>E\,2^{-n/\t}$: this is so because the root line of $\th_1$ will
not contribute its unit to $N^*_n(\th_1)$. Going once more through the
analysis the only non trivial case is if $m'=1$ and in that case
$N^*_n(\th'_1)=N^*_n(\th{'}{'}_1)+\ldots+N^*_n(\th{'}{'}_{m{'}{'}})$,
\etc., until we reach a
trivial case or a diagram of order $\le k-2^{-1} E\,2^{-n/\t}$.
It remains to check that if $k_1>k-2^{-1}E\,2^{-n/\t}$ then the root line of
$\th_1$ has scale $>n$, unless it is entering a resonance.
Suppose that the root line of $\th_1$ has scale $\le n$ and is not
entering a resonance. Note
that $|\oo\cdot\nn(v_0)|\le\,2^{n+1},|\oo\cdot\nn(v_1)|\le
\,2^{n+1}$, if $v_0,v_1$ are the first vertices of $\th$ and $\th_1$
respectively. Hence $\d\=|(\oo\cdot(\nn(v_0)-\nn(v_1))|\le2\,2^{n+1}$ and
the diophantine assumption implies that $|\nn(v_0)-\nn(v_1)|>
(2\,2^{n+1})^{-1/\t}$, or $\nn(v_0)=\nn(v_1)$. The latter case being
discarded as we are not considering the resonances, it follows
that $k-k_1<2^{-1}E\,2^{-n/\t}$ is inconsistent:
it would in fact imply that $\nn(v_0)-\nn(v_1)$ is a sum of $k-k_1$
vertex modes and therefore $|\nn(v_0)-\nn(v_1)|< 2^{-1}NE\,2^{-n/\t}$
hence $\d>2^3\,2^n$ which is contradictory with the above opposite
inequality. \qed
\*
Analogously, we can prove that, if $N_n>0$,
then the number $p_n(\th)$ of clusters of scale $n$ verifies the bound
$p_n(\th) \le 2k \,(E2^{-n/\t})^{-1}-1$. In fact this is true
for $k \le E2^{n/\t}$, (see footnote 6). Otherwise, if the
first tree vertex $v_0$ is not in a cluster of scale $n$, it is
$p_n(\th)=p(\th_1)+\ldots+p_n(\th_m)$, with the above notation,
and the statement follows by induction. If $v_0$ is in a cluster
on scale $n$ we call $\tilde\th_1, \ldots, \tilde\th_m$ the
subdiagrams emerging from the cluster containing $v_0$ and
with orders $k_j>E2^{-n/\t}$, $j=1,\ldots,m$. It will be
$p_n(\th)=1+p(\tilde\th_1)+\ldots+p_n(\tilde\th_m)$. Again
we can assume $m=1$, the other cases being trivial. But in
such a case there will be only one branch entering the cluster $T$
on scale $n$ containing $v_0$ and it will have a momentum of
scale $n'\le n-1$. Therefore the cluster $T$ must contain at least
$E2^{-n/\t}$ vertices, (otherwise, if $\l$ is a line on scale $n$
contained in $T$, and $\nn_\l^0$ is the sum of the mode labels
corresponding to the vertices following $v_0$ but inside $T$, we would have
$|\oo\cdot\nn_\l|\le 2^{n+1}$ and, simultaneously,
$|\oo\cdot\nn_\l|\ge 2^{n+3}-2^{n-1}>2^{n+2}$, which would lead to a
contradiction). This means that $k_1\le k -E2^{-n/\t}$. \qed
\*
Let us consider now a resonance $V$, and let us call $N_n(V)$ and
$N_n^*(V)$ the number of lines on scale $n$ in $V$, and,
respectively, the number of non resonant lines inside $V$ carrying a
scale label $\le n$. Again a bound $N_n^*(V)\le2k(V)(E2^{-n/\t})^{-1}-1$
holds, if $k(V)$ is the order of the resonance $V$; analogously
$p_n(V)\le2k(V)(E2^{-n/\t})^{-1}-1$, if $p_n(V)$ is the number of
clusters on scale $n$ contained in $V$. The proofs of such
statements can be easily adapted from the previous ones,
by noting that $N_n(V)\neq0$ requires $k(V)>E2^{-n/\t}$.
We give them explicitly, only for completeness.
Given a subdiagram $T$, let us denote by $k(T)$ the number
of vertices contained in $T$, and by $N_n^*(T)$ the number of
non resonant lines inside $T$ with a scale label $\le n$.
We prove by induction that $N_n^*(T)\le2k(T)(E2^{-n/\t})^{-1}-1$,
each time only one line on scale $n'n'$ for every line $\l$ inside $T$,
(note that the resonance $V$ satisfies such
a requirement, but it is not necessary that $T$ is a cluster
to make the statement to hold).
Let us consider a subdiagram $T$, verifying the above described
properties, \eg a resonance $V$ on scale $n_V$, with $n'\le n_V-1$.
By assumption there is only one line entering the subdiagram $T$,
and essentially by definition there is only one line exiting; by
analogy to Definition 3.2, let us call $w_0$ and $w_2$ the
two vertices which the two lines, respectively, lead to.
Let us call $T_1$, $\ldots$, $T_m$ the subdiagrams emerging from the
vertex $w_0$ and with $k(T_j)>E2^{-n/\t}$: by construction,
one of such diagram, say the first one, contains the vertex $\p(w_2)$,
while the other ones are trees.
Let us consider the first branch
of the subdiagram $T_1$. If the considered branch
scale label is $>n$, then the assertion follows by induction,
by using also the previous results on $N_n^*(\th)$ for trees, and the
fact that $k(T)>E2^{-n/\t}$ if $N_n(T)\neq0$.
Otherwise, it is $N_n^*(T)=1+\sum_{i=1}^mN_n^*(T_i)$;
the case $m\ge2$ can be again
inductively studied and the statement easily follows.
If $m=1$, then
$N_n^*(T)=1+N_n^*(T_1)$, and, if $v_1 \in T_1$ is the vertex to
which the outgoing line of $T_1$ leads to, we consider the
$m'$ subdiagrams emerging from $v_1$: again one of them, say the first
one, contains the vertex $\p(w_2)$ of $V$, while the other ones
are trees. If $m'\ge1$, again an inductive proof can be performed.
If $m'=1$, we have once more a trivial
case unless it is $k(T_1)> k(T)-2^{-1}E2^{-n/\t}$. But in this
case we can reason as along the proof of the bound on $N_n^*(\th)$,
and check that there are only two possibilities: either the
line leading to $v_1$ is a resonant line on scale $n$, or it
has scale $>n$. The proof of such a statement can be carried
out exactly in the same way as to $N_n(\th)$, so that we do not
repeat it here. This means that we can write:
$N_n^*(T)=1+N_n^*(T_1)=1+N_n^*(T_1')+\ldots+N_n^*(T_{m{'}{'}}')$,
and the above analysis can be iterated as many times as it
is needed to reach either a trivial case or a subdiagram $\tilde T$
such that $k(\tilde T)\le k(T)-2^{-1}E2^{-n/\t}$.
This proves the statement about the number of lines on scale $n$
contained in a resonance, (note the bound we have obtained can
be replaced by zero if the scale is $nw_0$, $w\in V$),
and let us study its dependence on the mode labels.
We see from \equ(3.3) that, if $R_{\l_w}=2$ we
can associate to such a line a {\sl line factor} which is
given by the product of a factor
linear in the mode labels arising from the vertex $w$,
times a factor $(i\nn_{\p(w)})_{j_{\l_w}}$
arising from the vertex $\p(w)$,
times a propagator $g^{(n_{\l_w})}(\oo\cdot\nn(w))$;
if $R_{\l_w}=1$ we associate to it
a {\sl line factor} which is given by the product of a factor
linear in the mode labels arising from the vertex $w$, times a factor
independent on the the mode labels arising from the vertex $\p(w)$,
times a propagator $g^{(n_{\l_w})}(\oo\cdot\nn(w))$.
Then, for each line inside $V$, the line
factor is a homogeneous function of even degree in the mode labels.
To the first vertex $w_0$ we associate a factor
$(-i\nn_{w_0})_{j_{\l_{w_0}}}(1-\d_{w_0})$ $+(i\oo\cdot\nn(w_0))
\dpr_{B_{j_{\l_{w_0}}}}\,\d_{w_0}$.
Since the function $\f_{\nn}(\BB_0)$
is supposed to be even in $\nn$, no other factor has to be considered
in order to obtain the behaviour of the resonance,
when $\oo\cdot\nn_{\l_V}=0$ and the signs of the mode labels of all
the vertices in $V$ are simultaneously changed.
When such an operation is performed we see that,
if $\z_{\l_{w_0}}=H$, (recall that $\d_{w_0}\=0$ if $\z_{\l_{w_0}}=H$),
or $\z_{\l_{w_0}}=h$, with $\d_{w_0}=0$,
the overall sign of the resonance factor
changes, while, if $\z_{\l_{w_0}}=h$, with $\d_{w_0}=1$,
the overall sign of the resonance factor does not change.
Now, let us consider separately the possible kinds of
resonance, see \equ(3.8) above.
\\(1) If $\z_{\l{w_0}}=\z_{\l_{w_2}}=H$, then we deduce from
the above discussion that the sign of
$\VV_1^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$ changes when all
the signs of the mode labels of the vertices in $V$ are changed;
then, fixed a set of compatible values of $\nn_w$, $w\in V$,
if we sum together the two contributions $\{\nn_w\}_{w\in V}$ and
$\{ -\nn_w\}_{w\in V}$, we obtain zero.
\\(2) If $\z_{\l_{w_0}}=H$ and $\z_{\l_{w_2}}=h$,
we consider all the trees we obtain by detaching from the resonance
the subtree with first vertex $w_2$, then reattaching it to
all the remaining vertices $w\in V$, and we add also the contributions
obtained by the previous ones by an overall sign reversal of the
mode labels $\nn_w$: if $\oo\cdot\nn_{\l_V}=0$,
no propagator changes, and
the only effect of our operation is that one of the vertex factors
changes by taking successively the values $(\nn_w)_{j_{\l_{w_2}}}$,
$w\in V$.
Then we build in this way a quantity proportional to $\sum_{w\in V}
(\nn_w)_{j_{\l_{w_2}}}=$ $[\nn(w_2)-\nn(w_0)]_{j_{\l_{w_2}}}\,\=\,0$.
If we sum also on a overall change of sign of the $\nn_w$'s,
and we take into account the parity in the mode labels of the resonance
factor, we obtain a second order zero.
\\(3) If $\z_{\lambda_{W_0}}=h$ and $\z_{\lambda_{W_2}}=H$ we note
that, when $\d_{w_0}=0$, then the difference from the
contribution with $\z_{\l_{w_0}}=H$ reduces to
the label $R_{\l_{w_0}}$ which now is $2$ instead of $1$: then
the results of the the first item still hold.
If $\d_{w_0}=1$, then $\VV_3^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$
vanishes to first order, as it
contains a factor $i\oo\cdot\nn_{\l_V}$, see \equ(3.4).
\\(4) If $\z_{\lambda_{W_0}}=\z_{\lambda_{W_2}}=h$
we note that, when
$\d_{w_0}=0$, then the difference from the
contribution with $\z_{\l_{w_0}}=H$ reduces to
the label $R_{\l_{w_0}}$ which now is $2$ instead of $1$: then
the results of the the second item still hold.
If $\d_{w_0}=1$, then
$\VV_2^{n_{\l_V}}(0)\,\=\,0$,
as it contains a factor $\oo\cdot\nn_{\l_V}$,
(as to the first order in the previous item for $\d_{w_0}=1$),
and the first derivative with respect
to $\oo\cdot\nn_{\l_V}$ in $\oo\cdot\nn_{\l_V}=0$
is still vanishing for parity reasons
analogous to those of the case $\VV_2^{n_{\l_V}}(\oo\cdot\nn_{\l_V})$
in fact the only difference
is that a factor $(-i\nn_{w_0})_{j_{\l_{w_0}}}$ is missing, (replaced
by a derivative $\dpr_{B_{j_{\l_{w_0}}}}$), and a factor
$(i\nn_{\p(w_2)})_{j_{\l_{w_2}}}$ replaces $\dpr_{B_{j_{\l_{w}}}}$.
Then from the above discussion and from the definition of
factor form given in Definition 3.3, (see in particular \equ(3.9)),
the results stated in Lemma 4.1 are proven. \qed
\vskip1.truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% BIBLIOGRAFIA %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\ciao
ENDBODY