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%%%% KAM Theorem and Quantum Field Theory %%%%
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%%%% by %%%%
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%%%% J. Bricmont, K. Gawedzki, A. Kupiainen %%%%
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%%%% July 20 %%%%
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\begin{document}
\begin{center}
{\Large{\bf{KAM Theorem and Quantum Field Theory}}}
\vs{ 1cm}
{\large{Jean Bricmont}\footnote{Partially supported by
EC grant CHRX-CT93-0411}}
\vs{ 0.2cm}
UCL, FYMA, 2 chemin du Cyclotron,\\
B-1348 Louvain-la-Neuve, Belgium
\vs{0.5cm}
{\large{Krzysztof Gaw\c{e}dzki}}
\vs{ 0.2cm}
CNRS, IHES, 35 route de Chartres,\\
91440 Bures-sur-Yvette, France
\vs{0.5cm}
{\large{Antti Kupiainen}}\footnote{Partially supported by
NSF grant DMS-9501045 and EC grant CHRX-CT93-0411}
\vs{ 0.2cm}
Department of Mathematics,
Helsinki University,\\
P.O. Box 4, 00014 Helsinki, Finland
\end{center}
\date{ }
%\maketitle
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%%
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\vskip 0.3cm
%%\begin{center}
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\vskip 1.3 cm
\begin{abstract}
\vskip 0.3cm
\noindent We give a new proof
of the KAM theorem for analytic Hamiltonians.
The proof is inspired by a quantum field theory
formulation of the problem and is based on a renormalization group
argument treating the small denominators inductively scale by scale.
The crucial cancellations of resonances are shown to follow from
the Ward identities expressing the translation invariance
of the corresponding field theory.
\end{abstract}
\vs{ 1.6cm}
\nsection{Introduction}
\vskip 0.2cm
Consider the Hamiltonian
\qq
H(I,\phi)\,=\,\omega \cdot I + \hf\m I\cdot\mu I+\lambda\, U(\phi,I)
\label{H}
\qqq
with $\phi\in\NR^d/(2\pi\NZ^d)\equiv\NT^d$, $I\in\NR^d$,
$\omega\in\NR^d$ with the components
$\omega_i$ independent over $\NZ^d$ and $\mu$ a real symmetric
$d\times d$ matrix.
It generates the Hamiltonian flow given by the equations of motion
\qq
\dot\phi=\omega +\mu I+\lambda\,\partial_I U\, ,\;\;\dot I=
-\lambda\m\partial_\phi U\m.
\label{eqs}
\qqq
For the parameter $\lambda=0$ and the initial condition
$(\phi_0,0)$, the flow $(\phi_0+\omega t,0)$
is quasiperiodic and spans a $d$-dimensional torus in
$\NT^d\times\NR^d$. KAM-theorem deals with the question
under what conditions such quasiperiodic solutions
persist as the parameter $\lambda$ is turned on.
\vskip 0.2cm
Consider a quasiperiodic solution in the form
$$
(\phi(t),I(t))=(\phi_0+\omega t+\Theta(\phi_0+\omega t),
\, J(\phi_0+\omega t))\m.
$$
Eqs.\,\,(\ref{eqs}) require that $Z=(\Theta,J):\NT^d
\rightarrow \NR^d\times\NR^d$ satisfies the relation
\qq
\CD Z(\phi)=-\lambda\,\partial U(\phi+\Theta(\phi),
\, J(\phi))\m,
\label{Z}
\qqq
where $\partial=(\partial_\phi ,\partial_I)$ and
\qq
\CD=\left(\matrix{0 & \omega\cdot\partial_\phi \cr
-\omega\cdot\partial_\phi & \mu}\right).
\label{G}
\qqq
Note that if $Z$ is a solution of Eq.\,\,(\ref{Z})
then so is $Z_\beta$
for $\beta\in \NR^d$ and
\qq
Z_\beta(\phi)\,=\, Z(\phi-\beta)-(\beta,0)\m.
\label{Ztr}
\qqq
Eq.\,\,(\ref{Z}) is a fixed point problem for the function $Z$
of a difficult type: the straightforward linearization
$\CD+\lambda\da\da U(\phi,0)$ is not invertible
for any interesting $U$ (see e.g. \cite{E}). Also,
one can expect to have a solution only for
sufficiently irrational $\omega\in\NR^d$, e.g.
satisfying a Diophantine condition
\qq
|\omega\cdot q|\, >\, a\m |q|^{-\nu}\quad{\rm for}\quad q\in \NZ^d,
\ q\neq 0
\label{Dio}
\qqq
with some $q$-independent $a,\nu>0$. There
have been traditionally two approaches to the problem:
\vs{2mm}
\no 1. The KAM approach. (\ref{Z}) is solved by a
Newton method that constructs a sequence of
symplectic changes of coordinates defined on
shrinking domains that, in the limit, transform
the problem to the $\lambda=0$ case \cite{A1,A2,Ko,Mo1}.
\vs{2mm}
\no 2. Perturbation theory. For $U$ analytic
(see below) one can attempt to solve (\ref{Z})
by iteration. This leads to a power series in
$\lambda$, the Lindstedt series: $Z=\sum_n Z_n\lambda^n$.
Each $Z_n$ is given as a sum of several
terms (see Sect.\,\,9), some of which are very
large, proportional to $(n!)^a$ with $a>0$,
due to piling up of ``small denominators'' $(\omega\cdot q)$
from the momentum space representation of operator $\CD^{-1}$.
However, the KAM method also yields the analyticity
of $Z$ in $\lambda$ \cite{Mo2}. Thus the Lindstedt series
must converge. To see this directly turned out to be
rather hard and was finally done by
Eliasson \cite{E} who, by regrouping terms,
was able to produce an absolutely convergent
series that gives the quasiperiodic solution.
Subsequently Eliasson's work was simplified and extended
by Gallavotti \cite{G1,G2, GalPar}, by
Chierchia and Falcolini \cite{CF1,CF2} and by
Bonetto, Gentile, Mastropietro
\cite{GGM,GM1,GM2,GM3,BGGM1,BGGM2}.
\vs{4mm}
In the present paper we shall develop a new iterative
scheme to solve Eq.\,\,(\ref{Z}). It is based
on a direct application of the renormalization group (RG) idea
of quantum field theory (QFT) to the problem.
The idea is to split the operator $\CD$ (or rather its
inverse, see Sect.\,\,2) into a small denominator and
large denominator part, where small and large
are defined with respect to a scale of order unity. The next step
is to solve the large denominator problem which results
in a new effective equation of the type (\ref{Z}) for
the small denominator part, with a new right hand side.
The procedure is iterated, with the scale separating
small and large at the $n^{\rm th}$ step equal to
$\eta^{n}$ for some fixed $\eta<1$. As a result we get
a sequence of effective problems that converge to a trivial
one as $n\rightarrow \infty$. A generic step is solved
by a simple application of the Banach Fixed Point Theorem
in a big space of functionals of $Z$
representing the right hand side of Eq.\,\,(\ref{Z})
in the $n^{\rm th}$ iteration step.
\vs{2mm}
Our iteration can be viewed as an iterative resummation
of the Lindstedt series, as will be discussed in Sect.\,\,9.
This iterative approach trivializes the rather formidable
combinatorics of the small denominators.
The functional formulation in terms
of effective problems removes also the mystery
behind the subtle cancellations in the Lindstedt series:
they turn out to be an easy consequence
of a symmetry in the problem as formulated in terms
of the so called Ward identities of QFT.
The QFT analogy of the problem (\ref{Z}) has been
forcefully emphasized by Gallavotti {\it et al.} \cite{GalPar,GGM}. The
proof of Eliasson's theorem by these authors was based
on a separation into scales of the graphical expressions
entering the Lindstedt series and was a direct
inspiration for the present work.
\vs{2mm}
An important part of the standard RG theory is
an approximate scale invariance of the problem
that is exhibited and exploited by the RG
method. The KAM problem also is expected
to have this aspect: as the coupling $\lambda$
is increased the solution with a given
$\omega$ eventually ceases to exist. For
suitable ``scale invariant'' $\omega$ (e.g.
in $d=2$ for $\omega=(1,\gamma)$ with $\gamma$
a ``noble'' irrational) the solution at the critical
$\lambda$ is expected to exhibit a power law decay
of Fourier coefficients and periodic
orbits converging to it have peculiar
``universal'' scaling properties \cite{kad,she}.
We hope that the present approach will
shed some light on these problems in the future.
\vskip 0.4cm
While the main goal of this paper
is to develop a new method, we use
it to reprove the following (classical) result:
\vs{4mm}
\no{\bf Theorem 1}. {\ \it Let $U$ be real analytic in
$\phi$ and analytic in $I$ in a neighborhood
of $I=0$. Assume that $\omega$ satisfies condition}
(\ref{Dio}). {\it Then Eq}.\,\,(\ref{Z}) {\it has a solution
which is analytic in $\lambda$ and real analytic in $\phi$
provided that either}
\vs{2mm}
\no (a) (the non-isochronous case) \ $\mu$ {\it is an invertible
matrix and $\vert\lambda\vert$ is small enough (in a $\mu$-dependent
way)}.
\vs{2mm}
\no (b) (the isochronous case) \ $\mu =0$,
$\,\int_{_{\NT^d}}\hspace{-0.05cm}
\partial_{I} U(\phi,0)\, d\phi =0$, {\it\,the
$d\times d$ matrix with elements
$\int_{\NT^d}\partial_{I_k}\partial_{I_l} U(\phi,0)\, d\phi$,
$\m k,l = 1,\dots, d\m,$ \,is invertible and
$\vert\lambda\vert$ is small enough.}
\vs{2mm}
\noindent{\it The above solutions are unique up to translations
(\ref{Ztr}).}
\vs{4mm}
\no{\bf Remark}. \ Actually, we show that the solution is an
analytic function not only of $\lambda$, but of the potential
$U$, when the latter belongs to a small ball in a Banach
space of analytic functions (see Sect.\,\,3 for
the introduction of such spaces). This allows us to consider
more general Hamiltonians of the form
\qq
H(I,\phi)\,=\,H_0 (I) +\, U(\phi,I).
\nonumber
\qqq
with $H_0$ and $U$ analytic and $U$ small. Indeed, we may
expand $H_0$ around
$I_0$ s.t. $\partial_I H_0 (I_0) = \omega$, with $\omega$
satisfying condition (\ref{Dio}):
$$H_0(I)=H(I_0) +\omega \cdot (I-I_0) +\hf\,(I-I_0)\cdot
\mu(I-I_0)+{\tilde H}_0 (I)$$
and define ${\tilde U}=U +{\tilde H}_0$
so as to include in it all the terms of order higher than
two in the expansion of $H_0$. Replacing $I-I_0$ by $I$,
we may apply Theorem 1 provided that ${\tilde U}$ satisfies
the corresponding hypotheses. Also, more general cases where
$\mu$ is a degenerate matrix can be treated.
\vs{4mm}
The organization of the paper is as follows. In Sect.\,\,2
we explain the RG formalism. In Sect.\,\,3, we introduce
spaces of analytic functions on Banach spaces; such spaces
will be used to solve our RG equations. In
Sect.\,\,4, we state the main inductive estimates which
are proved in Sect.\,\,6 after an interlude on
the Ward identities in Sect.\,\,5. Theorem 1 is proved
then in Sect.\,\,7. Sect.\,\,8 explains the connection
of our formalism to QFT for those familiar with the
latter. We should emphasize that the QFT is solely
a source of intuition, the simple RG formalism of
Sect.\,\,2 is independent of it. Finally, in
Sect.\,\,9, the connection with the Lindstedt series is
explained.
\nsection{Renormalization group scheme}
\vs{2mm}
In this section we explain the iterative RG scheme without spelling
out the technical assumptions that are needed to carry it out.
We refer the reader to Sect.\,\,9 for a graphical representation
of the main quantities introduced here.
\vskip 0.4cm
We shall work with Fourier transforms, denoting by lower case
letters the Fourier transforms of functions of $\phi$,
the latter being denoted
by capital letters:
\qq
F(\phi)=\sum\limits_{q\in{\NZ}^d}\ee^{-i\m q\cdot\phi}
\, f(q)\m,\quad\ {\rm where}\quad\ f(q)=\int_{\NT^d}
\ee^{i\m q\cdot\phi}\, F(\phi)\, d\phi
\nonumber
\qqq
with $d\phi$ standing for the normalized Lebesgue measure
on $\NT^d$.
\vskip 0.4cm
Note first that we may use the translations
(\ref{Ztr}) to limit our search for the solution of Eq.\,\,(\ref{Z})
to the subspace of $\Theta$ with zero average, i.e.
with $\theta(0)=0$ in the Fourier language.
It will be convenient to separate the constant mode of $J$
explicitly by writing $Z=X+(0,\zeta)$ where $X$ has zero average.
Let us define
\qq
W_0(\phi;X,\zeta)=
\lambda\,\partial U((\phi,\zeta)+X(\phi))\m.
\label{W1}
\qqq
Denote by $G_0$ the operator $-\CD^{-1}$
acting on $\NR^{2d}$-valued functions on $\NT^d$
with zero average. In terms of the Fourier transforms,
\qq
(G_0\m x)(q)=\left(\matrix{\mu(\omega\cdot q)^{-2} &
i(\omega\cdot q)^{-1} \cr
-i(\omega\cdot q)^{-1} & 0}\right)x(q) .
\label{G1}
\qqq
for $q\neq 0$ and $(G_0\m x)(0)=0$.
Writing Eq.\,\,(\ref{Z}) separately for the averages (i.e. $q=0$)
and the rest, we may rewrite it as the fixed point
equations
\qq
X &= &G_0\m P\m W_0(X,\zeta)\m,
\label{fp1}\\
(0,\mu\zeta)&=&-\int_{\NT^d} W_0(\phi;X,\zeta)\, d\phi\m,\label{fp12}
\qqq
where $P$ projects out the constants:
$P\m F=F-\int_{_{\NT^d}}F(\phi)\m d\phi$.
Our strategy is to solve
Eq.\,\,(\ref{fp1}) by an inductive RG method for given $\zeta$.
This turns out to be possible quite generally without any
nondegeneracy assumptions on $U$. The latter enter only in
the solution of Eq.\,\,(\ref{fp12}).
Below, we shall treat $W_0$ given by Eq.\,\,(\ref{W1})
as a map on a space of $\NR^{2d}$-valued functions $X$
on $\NT^d$ with arbitrary averages\footnote{That the
solution $X$ of Eq.\,\,(\ref{fp1}) has zero average follows from
the form of the equation.}. The vector $\zeta$ will be treated
as a parameter and we shall often suppress it in the notation
for $W_0$.
\vskip 0.4cm
For the inductive construction of the solution
of Eq.\,\,(\ref{fp1}), we shall decompose
\qq
G_0\ =\ G_1+\Gamma_0\m,
\label{G2}
\qqq
where $\Gamma_0$ will effectively involve only the Fourier
components with $|\omega\cdot q|$ larger than $\CO(1)$
and $G_1$ the ones with $|\omega\cdot q|$
smaller than that (see Sect.\,\,4). In particular, we shall
have $\Gamma_0=\Gamma_0P$. Upon writing $X=Y+\tilde Y$,
\,Eq.\,\,(\ref{fp1}) becomes
\qq
Y+\tilde Y\ =\ (G_1+\Gamma_0)\m P\m W_0(Y+\tilde Y)\m.
\label{dc}
\qqq
Suppose that $\tilde Y=\tilde Y_0$ where $\tilde Y_0$ solves
for fixed $Y$ the ``large denominator'' equation:
\qq
\tilde Y_0\ =\ \Gamma_0 W_0(Y+\tilde Y_0)\m.
\label{lde}
\qqq
Then Eq.\,\,(\ref{dc}) reduces to the relation
\qq
Y\ =\ G_1P\m W_1(Y)
\label{fp2}
\qqq
if we define $W_1(Y)=W_0(Y+\tilde Y_0)$.
We have thus reduced the orginal problem (\ref{fp1})
to the one from which the largest denominators were
eliminated, at the cost of solving the easy large
denominator problem (\ref{lde}) and of replacing
the map $W_0$ by $W_1$.
\vskip 0.4cm
Note that, with these definitions, $\tilde Y_0=\Gamma_0W_1(Y)$
and thus $W_1$ satisfies the fixed point
equation
\qq
W_1(Y)\ =\ W_0(Y+\m\Gamma_0W_1(Y))\m.
\label{W2}
\qqq
Conversely, this equation, which we shall solve for $W_1$
by the Banach Fixed Point Theorem in a suitable space,
implies that $\tilde Y_0=\Gamma_0 W_1(Y)$ satisfies
Eq.\,\,(\ref{lde}) and thus that
\qq
X\,=\, Y+\m\Gamma_0W_1(Y)\,\equiv\, F_1(Y)
\label{F2}
\qqq
is a solution of Eq.\,\,(\ref{fp1}) if and only if
$Y$ solves Eq.\,\,(\ref{fp2}).
\vskip 0.4cm
After $n-1$ inductive steps, the solution of Eq.\,\,(\ref{fp1})
will be given as
\qq
X\,=\, F_{n-1}(Y)\m,
\label{Fn}
\qqq
where $Y$ solves the equation
\qq
Y\,=\, G_{n-1}P\m W_{n-1}(Y)
\label{fpn}
\qqq
and $G_{n-1}$ contains only the denominators
$\vert\omega\cdot q\vert\leq\CO(\eta^n)$
where $\eta$ is a positive number smaller than $1$
fixed once for all. The next inductive
step consists of decomposing $G_{n-1}=G_{n}+\Gamma_{n-1}$
where $\Gamma_{n-1}$ involves $|\omega\cdot q|$
of order $\eta^{n}$ and $G_{n}$ the ones smaller
than that. We define $W_{n}(Y)$ as the solution
of the fixed point equation
\qq
W_{n}(Y)\,=\, W_{n-1}(Y+\m\Gamma_{n-1} W_{n}(Y))
\label{Wn+1}
\qqq
and set
\qq
F_{n}(Y)\,=\, F_{n-1}(Y+\m\Gamma_{n-1} W_{n}(Y))
\label{Fn+1}
\qqq
(which is consistent with relation (\ref{F2}) if we take
$F_0(Y) = Y$).
Then replacing $Y$ in Eqs.\,\,(\ref{Fn}) and (\ref{fpn})
by $\, Y+\m\Gamma_{n-1} W_{n}(Y)\m$, \m we infer that
$X=F_{n}(Y)$ if $Y=G_{n}P\m W_{n}(Y)$ completing
the next inductive step. Note also the cumulative formulas
that follow easily by induction:
\qq
&&W_n(Y)\,=\, W_0(Y\m+\m\Gamma_{0$, $\alpha>0$ and $b<\infty$ such that
the coefficients $U_{m+1}(\phi,\zeta)$, belonging to the space
of $m$-linear maps $\CL(\NC^{2d},\dots,\NC^{2d};\NC^{2d})$,
of the Taylor expansion
$$
\partial U((\phi,\zeta)+Y)=\sum_{m=0}^\infty{_1\over^{m!}}\m
U_{m+1}(\phi,\zeta)
(Y,\dots,Y)
$$
are analytic in $|\zeta|<\rho$
and their Fourier transforms satisfy the bounds
\qq
\sum_q \ee^{\alpha |q|}\, \Vert u_{m+1}(q,\zeta)
\Vert_{_{\CL(\NC^{2d},
\dots,\NC^{2d};\NC^{2d})}}\leq\ b
\, m!\,\rho^{-m}\m.
\label{vmn}
\qqq
For later convenience, we shall use in $\NC^{2d}\cong
\NC^d\times\NC^d$ the norm $\vert\cdot\vert_{_0}$ defined by
\qq
\vert(z_1,z_2)\vert_{_0}\,\equiv\ \vert z_1\vert
+\vert z_2\vert
\label{deff}
\qqq
and the induced norms on the spaces of linear maps.
Inserting the Fourier series for $Y$ we end up
with the expansion
\qq
w_0(q\m ;\m y)&=&\sum\limits_{m=0}^{\infty}
\sum_{\bf q}{_1\over^{m!}}\, u_{m+1}(q-\sum q_i,\m\zeta)
\m(y(q_1),\dots,y(q_m))\nonumber\\
&\equiv& \sum\limits_{m=0}^{\infty}
\sum_{\bf q}w_0^{(m)}(q,q_1,\dots,q_m;\zeta)\m(y(q_1),
\dots,y(q_m))\m,
\label{taylor}
\qqq
where ${\bf q}=(q_1,\dots,q_m)\in \NZ^{md}$.
This formula suggests to consider
$w_0$ as an analytic function of $y$, where $y$ belongs
to a suitable Banach space $h$. We take
$$h\m=\m\{\,y=(y(q))\ \vert\ y(q)\in\NC^{2d}\m,\ \,
{\Vert}y{\Vert}\equiv\sum\limits_{q}|y(q)|_{_0}\m
<\m\infty\,\}\m.$$
Let $B(r_0)$ be the open ball of radius $r_0$ in $h$
centered at the origin and let $H^\infty(B(r_0),h)$ denote
the Banach space of analytic functions \cite{cha}
$w:B(r_0)\rightarrow h$, equipped with the supremum
norm, which we shall denote by $|||w|||$.
The bound (\ref{vmn}) implies that $w_0\in H^\infty(B(r_0),h)$
for $r_0$ small enough, but before stating this,
it is convenient to encode the decay property
of the kernels $w_0^{(m)}$ inherited
from the estimate (\ref{vmn}) as a property
of the functional $w_0$.
\vskip 0.4cm
For that let $\tau_\beta$ denote the translation
by $\beta\in\NR^d$, $(\tau_\beta Y)(\phi)=Y(\phi-\beta)$.
On $h$, $\tau_\beta$ is realized by $(\tau_\beta y)(q)
=e^{i\m \beta\cdot q}\m y(q)$. It induces a map $w\mapsto
w_\beta$ from $H^\infty(B(r_0),h)$ to itself if we set
$$
w_\beta(y)=\tau_\beta(w(\tau_{-\beta}y)).
$$
On the kernels $w^{(m)}$, this is given by
\qq
w_\beta^{(m)}(q\m ;\m q_1,\dots,q_m)\,=\,\ee^{i\m
\beta\cdot(q-\sum q_j)}\, w^{(m)}(q\m ;\m q_1,\dots,q_m)\m.
\label{wbeta}
\qqq
and makes sense also for $\beta\in\NC^d$. We have
$$
|||w_{0\beta}|||\,\leq\, \sum\limits_{m=0}^{\infty}
\sup_{q_1,\dots,q_m}\sum_{ q} \ee^{-{\rm Im}\m\beta
\cdot (q-\sum q_j)}\m
|w_0^{(m)}(q\m ;\m q_1,\dots,q_m;\zeta)|\, r_0^m
$$
Combining this with the bound (\ref{vmn}) we can summarize
the above discussion by
\vskip 0.6cm
\no{\bf Proposition 1}. {\it There exists $r_0>0$,
$\alpha>0$ and $D<\infty$, such that
$w_{0\beta}\in H^\infty(B(r_0),h)$ and it
extends to an analytic function of $\beta$ in the region
$|{\rm Im}\m\beta|<\alpha$ with values in $H^\infty(B(r_0),h)$
satisfying the bound
\qq
|||w_{0\beta}|||\,\leq\, D\m
\vert\lambda\vert\m.
\label{wbeta1}
\qqq
Moreover, $w_{0\beta}$ is analytic in $\zeta$ for $|\zeta|0$, $\vert\lambda\vert<\lambda_{n}(r)$ and $|{\rm Im}\m\beta|
<\alpha$, the equations} (\ref{wnc}) {\it have a unique
solution $w_{n\beta}\in H^\infty(B(r^n),h)$ with
\qq
|||w_{n\m\beta}|||\ \leq\,D\vert\lambda\vert \m,
\label{w2<}
\qqq
where $D$ is as in Proposition 1.
The maps $f_{n\beta}$ defined by Eqs}.\,\,(\ref{fnc})
{\it belong to $H^\infty(B(r^n),h)$. They satisfy the bounds
$|||f_{n\beta}|||\leq{2}\m r^n$. Moreover, $w_{n\beta}$ and
$f_{n\beta}$ are analytic in $\lambda$, $\beta$ and $\zeta$
and they satisfy the recursive relations} (\ref{wn+1})
{\it and} (\ref{fn+1}){\it, respectively.}
\vskip 0.4cm
Postponing the proof to the end of the section,
we shall state the bounds for $w_n$ that will be
inductively established for $\vert\lambda\vert$ small
in an $n$-independent way. Due to the smallness of $\vert
\omega\cdot q\vert$ in the $n^{\rm th}$ scale, $\gamma_n$ will
have very different
effects in the variables $\theta$ and $j$ in $y=(\theta, j)$.
It will be therefore convenient to choose $n$-dependent
norms for $n\geq 1$. Let us first do it for $\NC^{2d}$
by defining
\qq
\vert(z_1,z_2)\vert_{_{\pm n}}\,\equiv\ \vert z_1\vert
+{_1\over^{\eta^{\pm n}}}\vert z_2\vert\m.
\label{deff1}
\qqq
We shall use the notation $|\cdot|_{_{n;m}}$ for
the matrix norms induced by viewing a $2d\times 2d$ matrices
as maps from $\NC^{2d}$ with the norm $|\cdot|_{_n}$
to $\NC^{2d}$ with the norm $|\cdot|_{_m}$.
Next we set
\qq
\Vert y\Vert_{_n}
=\sum\limits_q|y(q)|_{_n}\,
\ee^{\eta^{-n}\vert\omega\cdot q\vert}\m.
\label{norm0}
\qqq
The weight $\ee^{\eta^{-n}\vert\omega\cdot q\vert}$ will
facilitate dealing with non-dangerous large denominators
$\vert\omega\cdot q\vert$.
For $w$, it turns out to be useful to introduce the norms
\qq
\Vert w\Vert_{_{-n}}
=\sum\limits_q|w(q)|_{_{-n}}\,
\ee^{-\eta^{-n}\vert\omega\cdot q\vert}\m.
\label{norm}
\qqq
Let $h_{\pm n}$ denote the corresponding Banach spaces.
Note the natural embeddings for $n\geq 2$
\qq
h_n\ \longrightarrow\ h_{n-1}\ \longrightarrow\ h\,,
\quad\quad\quad\quad\quad h\ \longrightarrow
\ h_{-n+1}\ \longrightarrow\ h_{-n}\ \
\label{embedd}
\qqq
with the norms bounded by $1$:
\qq
\Vert\cdot\Vert\ \leq\ \Vert\cdot\Vert_{_{n-1}}\,
\leq\ \Vert\cdot\Vert_{_{n}}\,,
\quad\quad\quad\Vert\cdot\Vert_{_{-n}}\,\leq
\ \Vert\cdot\Vert_{_{-n+1}}\,\leq\ \Vert\cdot\Vert\,.
\label{embe}
\qqq
For $n\geq2$ (but not for $n=1$), the operator
$\Gamma_{n-1}$ or, more generally, operators
$\Gamma_{n-1}(\kappa)$ may be considered as mapping
$h_{-n}$ into $h_n$. Indeed, it follows
easily with the use of bound (\ref{chib1}) that
\qq
\Vert\Gamma_{n-1}(\kappa)\Vert_{_{-n;n}}
\ \leq\ C\,\eta^{-2n}
\quad\quad{\rm if}\quad\ \ \vert\kappa\vert<\eta^{n-1}B
\label{5.5}
\qqq
with a new ($n$-independent) constant $C$.
\vskip 0.4cm
To simplify notations, we shall denote by $B_n$ the open ball
in $h_n$ of radius $r^n$
and by $\CA_n$ the space $H^\infty(B_n,h_{-n})$
of analytic functions on $B_n$ with the supremum norm
denoted by $|||\cdot|||$.
Finally, for a linear operator $M:h_n\rightarrow h_{m}$ we use
the abbreviated notation $\Vert\cdot\Vert_{_{n;m}}$ for the norm
in $\CL(h_{n},h_{m})$.
Due to the embeddings (\ref{embedd}), we may regard the maps
$w_{n\beta}$, whose existence for sufficiently small
$\vert\lambda\vert$ is claimed in Proposition 2,
as belonging to $\CA_n$.
Note that both sides of relation (\ref{wn+1}) are well
defined for such maps due to the bound (\ref{5.5}) and
that their equality is implied by the results of Proposition 2.
The next proposition states that, viewed as $\CA_n$-valued
functions of $\lambda$, $w_{n\beta}$'s
may be analytically extended to an $n$-independent
disc $\vert\lambda\vert<\lambda_0$ (provided we restrict somewhat
the strip of $\beta$). It also lists the properties of such
extensions.
\vskip 0.6cm
\no{\bf Proposition 3}. (a) \ {\it There exist positive constants
$r$ and $\lambda_0$ with $r<\eta^4$ such that,
for $\vert\lambda\vert<\lambda_0$ and
$\;\vert{\rm Im}\m\beta\vert< {\alpha_n}$, where}
\qq
\alpha_1=\alpha\m,\quad \alpha_{n}=(1-n^{-2})\alpha_{n-1}\m,
\quad n\geq2\m,
\label{an}
\qqq
{\it there exist solutions $w_{n\beta}\equiv w$
of Eqs.}\,\,(\ref{wn+1}) {\it belonging to $\CA_n$, analytic
in $\lambda$, $\beta$ and $\zeta$
and coinciding with the solutions $w_{n\beta}$ of Proposition 2
for $\vert\lambda\vert<\lambda_n(r)$}.
\vs{2mm}
\no (b) \ {\it Writing
\qq
w(y)=w(0)+Dw(0)y+\delta_2w(y),
\label{defs}
\qqq
we have
\qq
&&{\Vert}P w(0){\Vert}_{_{-n}}\ \leq\ \epsilon\, r^{2n}\m,
\label{w0}\\
&&|||\delta_2w|||\quad\,\m \ \ \leq\ \epsilon\, r^{{3\over^2}n}\m,
\label{omega}
\qqq
where $\epsilon\rightarrow 0$ as $\lambda\rightarrow 0$}.
\vs{2mm}
\no(c) \
\qq
\quad{\Vert}Dw(y){\Vert}_{_{n;-n}}\
\leq\ \epsilon\,\eta^{2n}\m.
\label{Dwbound}
\qqq
\vskip 0.5cm
\no{\bf Remarks.} 1. \ If we rescale the maps $w_n$ by
introducing $\m\tilde w_n(y)\,=\, \eta^{-2n}\m r^{-n}\, w_n(r^ny)\m$
then it follows from the above statements that
$\tilde w_{n\beta}\equiv\tilde w$ are analytic maps from
a unit ball in $h_n$ to $h_{-n}$ and $\tilde w(y)
=\tilde w(0)+ D\tilde w(0) y+\delta_2\tilde w(y)$ with
\qq
\Vert P\tilde w(0)\Vert_{_{-n}}\m\leq\m\epsilon\,\eta^{-2n}\m r^n\m,
\quad\ \vert\vert\vert\delta_2\tilde w\vert\vert\vert\m\leq
\m\epsilon\, \eta^{-2n}\m r^{\hf n}\m,\quad\ \Vert D\tilde w(0)
\Vert_{_{n,-n}}\m\leq\m\epsilon\m.
\nonumber
\qqq
Hence with growing $n$, $\m P\tilde w$ becomes an approximately
linear map.
\vskip 0.3cm
\noindent 2. \ Let us explain the idea of the proof of Proposition 3.
Consider the linearization of Eq.\,\,(\ref{wn+1}):
\qq
w_{n}=w_{n-1}+Dw_{n-1}\Gamma_{n-1} w_n\, +\ \dots
\label{linr}
\qqq
In order to solve the above equation one has to invert
the operator $1-Dw_{n-1}\Gamma_{n-1}$:
$$\m w_n=(1-Dw_{n-1}\Gamma_{n-1})^{-1}\m w_{n-1}\,+\,\dots$$
However, operator $\Gamma_{n-1}$ is of order $\eta^{-2n}$
as a map from $h_{-n}$ to $h_{n}$ (recall the bounds
(\ref{5.5})) and we need to show that $Dw_{n-1}$ is effectively
of order $\eta^{2n}$ as a map from $h_n$ to $h_{-n}$, which is,
essentially, what Eq.\,\,(\ref{Dwbound}) says with $n$ shifted
to $n-1$. Altogether, $Dw_{n-1}\Gamma_{n-1}$ remains of order
$\epsilon$ as a map from $h_{-n}$ to $h_{-n}$
(this motivates also our choice of the norms)
and $\Vert(1-Dw_{n-1}\Gamma_{n-1})\Vert_{_{n;-n}}\leq
1+\CO(\epsilon)$. In the proof of the estimate (\ref{Dwbound}),
we shall need the Ward identities discussed in Sect.\,\,5.
This is the only subtle part of our argument.
Indeed, once the bound (\ref{Dwbound}) is shown, the rest
of the proof of Proposition 3 reduces to the standard Banach
Fixed Point Theorem combined with the Diophantine
property of $\omega$.
\vskip 0.2cm
The latter is used in the following way (which is
similar to the way it enters the standard
KAM proof): upon iteration, we consider smaller and smaller
$\vert\omega\cdot q\vert\m$'s, of order $\eta^n$. This means
$|q|$ is of order $\eta^{n/\tau}$, by the Diophantine
condition (\ref{Dio}). On the other hand, the introduction
of the parameter $\beta$ in (\ref{wbeta}) allows to preserve
the exponential decay of the kernels $w_0^{(m)}(q; q_1,
\dots, q_m)$ in the size of $|q-\sum q_j|$. By shrinking
at each step the analyticity region in $\beta$ we show that
the leading contribution to $w_n$'s given by $w_n(0)$
(see Eq.\,\,(\ref{defs})) contracts for $q\neq 0$.
Actually, ${\Vert}Pw_n(0){\Vert}_{_{-n}}$ decays
super-exponentially in $n$, see the estimate (\ref{omega.q}) below,
which explains why we can choose $r$ as small as we want.
\vskip 0.2cm
Finally, the bound (\ref{omega}) is easy to understand.
By definition, $\delta_2w_n$ and its first derivative vanish
for $y=0$, and the norm $|||\delta_2w_n|||$
is defined by taking the supremum over balls of radius $r^n$, hence
one expects $|||\delta_2w_n|||$ to be of order $(1+\CO(\epsilon))^n
\m r^{2n}$ by the Cauchy estimate (\ref{circ2}) (the weaker bound
(\ref{omega}) is sufficient and is a convenient way to control
the non-linear corrections to the iteration). Recall that,
eventually, we construct our solution as a limit of $X_n =F_n(0)$,
for which we need to control $w_n(y)$ only for $y=0$,
see Eq.\,\,(\ref{cum2}). Thus we can let the radius $r^n$
of the ball where our estimates hold tend to zero.
\vs{2mm}
\no 3. \ Combining all the bounds, we get
\qq
|||w_n-(1-P)w_n(0)|||\
\leq\ C\,\epsilon\,\eta^{2n}\m \m.
\label{wbound}
\qqq
The zero mode part $(1-P)w_n(0)$
of $w_n(0)$ will be controlled later, see Eqs.\,\,(\ref{wi1})
and the second of Eqs.\,\,(\ref{smll}) below from which it follows
that it is of the form $(0,\xi_n)$ where $\xi_n=\CO(\lambda)$
converges in $\NR^d$ when $n\to\infty$. Note that, since $w_n$
is multiplied by $\Gamma_{n-1}= \Gamma_{n-1} P$ in the argument
of $w_{n-1}$ in Eq.\,\,(\ref{wn+1}), the constant mode $(1-P)w_n(0)$
may be decoupled from the iteration and we do not need to control
it in order to prove Proposition 3.
\vs{2mm}
\no 4. \ We choose the constants as follows.
$\eta<1$ has been fixed first. $B$ which enters the estimates
(\ref{n_0$ with $n_0$ large
enough, we shall proceed inductively. It will be convenient
to modify slightly the simplified notations of the text
of Proposition 3 and so, below, $w$ will stand
for $w_{(n-1)\beta}$ and $w'$ for $w_{n\beta}$. Finally,
$\Gamma$ will stand for $\Gamma_{n-1}$.
\vskip 0.4cm
\no{\bf Proof of (a)}. \ Consider the recursive equation
(\ref{wn+1}) for $w'$ and use the decomposition (\ref{defs})
to rewrite it as
$$
w'(y)=w(0)+Dw(0)(y+\Gamma w'(y))+\delta_2w(y+\Gamma w'(y))
$$
from which we deduce that
\qq
w'(y)=Hw(0)+HDw(0)y+u(y)\m,
\label{w'}
\qqq
where
\qq
u(y)\equiv H\delta_2w(y+\Gamma w'(y))=H\delta_2w(\Gamma Hw(0)
+{\tilde H}y+\Gamma u(y))
\label{u}
\qqq
with $H=(1-Dw(0)\Gamma)^{-1}$ and ${\tilde H}=1+\Gamma\m
H\m Dw(0)=(1-\Gamma\m Dw(0))^{-1}$\m.
\vskip 0.4cm
In the inductive step, first we assume
that $w$ satisfies the bounds (\ref{w0}), (\ref{omega}) and
(\ref{Dwbound}) with $n$ replaced by $n-1$. The bounds
(\ref{Dwbound}), (\ref{5.5}) and (\ref{embe}) imply then
that the operators $H$ and $\tilde H$ are well defined with
\qq
&{\Vert}H{\Vert}_{_{-n+1;-n+1}}\; ,\;{\Vert} {\tilde H}
{\Vert}_{_{n-1;n-1}}
\ \leq\ 1+C\epsilon\ \leq\ 2,
\label{H2}
\qqq
for $\epsilon$ (i.e.\,\,$\vert\lambda\vert$) small enough.
\vskip 0.4cm
We solve Eq.\,\,(\ref{u}) using the Banach Fixed Point Theorem.
Given its solution $u$, the existence of $w'$ satisfying
Eq.\,\,(\ref{w'}) follows. To solve Eq.\,\,(\ref{u}), we
consider the map $\CG$
defined by
\qq
\CG(u)(y)=H\delta_2w(\tilde y)\quad\ \ {\rm with}\quad\ \
\tilde y=\Gamma Hw(0)+{\tilde H}y+\Gamma u(y)\m.
\label{2}
\qqq
We claim that $\CG$ is a contraction in the ball
\qq
\CB=\{u\in H^\infty(B_\delta,h_{-n+1})\;|\m\;|||u|||\leq
2\m\epsilon\m r^{{_3\over^2}(n-1)}\}\m,
\label{ball}
\qqq
where ${B_\delta}\subset h_{n-1}$ is the open ball of
radius $r^{n-\delta}$ for $0\leq\delta<1$ and $r0$, the sum on the right hand side
is clearly bounded by $C n^2$. However, we may extract
from the sum factors that are super-exponentially small in $n$.
Indeed, for $\vert\omega\cdot q\vert\leq \eta^{\hf n}$,
we may extract from the first exponential under the sum
a factor $\ee^{-\CO(\eta^{-{_n\over^{2\nu}}}n^{-2})}$ due
to the Diophantine condition (\ref{Dio}). On the other hand,
for $\vert\omega\cdot q\vert>\eta^{{\hf n}}$, we may extract
a factor $\ee^{-\CO(\eta^{-\hf n})}$ from the second
exponential. Hence the inductive bound
(\ref{w0}) follows for $n\geq n_0$, and $n_0$ large enough.
\vs{4mm}
Let us now iterate the relation (\ref{omega})
for $\delta_2w'$ equal to $\delta_2u$ (see Eq.\,\,(\ref{w'})).
Recall that ${\Vert}u(y){\Vert}_{_{-n+1}}
\leq 2\m\epsilon\m r^{{_3\over^2}(n-1)}$
for $\Vert y\Vert_{_{n-1}}{3\over4}$ and $r\hf$ and $r\eta^{\hf n}$ or
$\vert\omega\cdot q'\vert>\eta^{\hf n}$. Hence
the bound (\ref{rhobound}) for the off-diagonal operator
$\rho'(\kappa)$.
\vskip 0.4cm
We are left with the proof of the estimate
(\ref{taubound}) for the diagonal operator $\sigma'(\kappa)$.
Let us define
\qq
{s}(z)\,=\, u^{n-1}\m\sigma(0,0;B\eta^{n-1}z)\, u^{n-1}\m,
\label{tild}
\qqq
where $u=(\matrix{_1&_0\cr^0
&^\eta })$ is a block matrix. Similarly, we introduce
the matrix $s'(z)$ related to $\sigma'$.
With the use of symmetry (\ref{sz1}), we write
$s'(z)=\eta^{2(n+1)}(\matrix{_{\wp'_0(z)}
&_{\wp'_1(z)}\cr^{\wp'_1(-z)}&^{\wp'_2(z)} })$.
We shall prove that, for $|z|<1 $,
\qq
&&|\wp'_i(z)-p'_i\, z^{2-i}|\,\m\leq\ A\m |z|^{3-i}
\label{taub0}
\qqq
with $|p'_i|\leq (1-{1\over{n}}){\epsilon\over{32}}$ and
$A\leq {\epsilon\over32}$, assuming inductively similar bounds
for $s(z)$. Note that such inductive assumptions, together
with the identity $|M|_{_{n-1;-n+1}}=|u^{n-1}Mu^{n-1}|_{_{0;0}}$
for the matrix norms imply, in particular, the estimate
\qq
|\sigma(0,0,\kappa)|_{n-1;-n+1} \leq {_1\over^8}\m\epsilon
\,\eta^{2n}
\label{smallk}
\qqq
for $\kappa\in D_{n-1}$.
The bound (\ref{taub0}) will follow from
Lemma 2 expressing the cancellations of resonances.
The leading Taylor coefficients $p_i$ of $\wp_i(z)$
are {\it marginal} in the RG terminology and the higher ones
are {\it irrelevant}. The presence of lower order {\it relevant}
Taylor coefficients would spoil the iterative bounds.
They are, however, forbidden by the Ward identities.
Let us pass to the details.
\vs{4mm}
Let us first prove the estimate (\ref{taubound}) for $\sigma'$
assuming the bound (\ref{smallk}). We shall split
\qq
\sigma'(\kappa)\,
=\,\sigma'_0(\kappa)+\sigma'_1(\kappa)\qquad{\rm with}\qquad
\sigma'_0(\kappa)\ =\ [\m1-\,\sigma(\kappa)\m
\Gamma(\kappa)\m]^{-1}\m\sigma(\kappa)\m,
\label{112}
\qqq
compare with Eq.\,\,(\ref{111}).
Note that $\sigma(\kappa)\m\Gamma(\kappa)$ is an
operator diagonal in Fourier space and hence, so is
$\sigma'_0(\kappa)$. Since, by the inductive hypotheses
(\ref{rhobound}) and (\ref{pibound2}),
\qq
\Vert\sigma(\kappa)-\pi(\kappa;\m\tilde y_0)
\Vert_{_{n-1;-n+1}}\,=\,\Vert\rho(\kappa)
+\delta_1\pi(\kappa;\m\tilde y_0)
\Vert_{_{n-1;-n+1}}\,\leq\,
2\m\epsilon\, r^{{1\over 2}(n-1)}\m,
\label{prnpr1}
\qqq
it follows that
\qq
\Vert\m\sigma'_1(\kappa)\Vert_{_{n;-n}}
\,\leq\,{_1\over^4}\m \m\epsilon\,\eta^{2n}\m,
\label{prnpr}
\qqq
for $r$ small enough.
We pass to the estimation of $\sigma'_0(\kappa)$.
Note that the bound (\ref{5.5}) together with the
inequalities (\ref{embe}) and the inductive
hypothesis (\ref{taubound}) imply that
$\Vert\Gamma(\kappa) \m\sigma(\kappa)
\Vert_{_{-n;-n}}\leq\,\hf
\m C\m\epsilon$ so that
\qq
\Vert\sigma'_0(\kappa)\Vert_{_{n;-n}}\leq\,
2\m\Vert\sigma(\kappa)
\Vert_{_{n;-n}}\m.
\label{tt0}
\qqq
For operators $a$ diagonal in Fourier transform,
$\,{\Vert} a{\Vert}_{_{n;-n}}
=\sup\limits_q\vert a(q,q)\vert_{_{n;-n}}\,\m\ee^{-2\eta^{-n}\vert
\omega\cdot q\vert}\m.\,$
Hence it follows from the bound (\ref{tt0}) that
\qq
\Vert\sigma'_0(\kappa)\Vert_{_{n;-n}}\ \leq\
\sup\limits_q\
2\,\vert\m\sigma(q,q;\m\kappa)\m\vert_{_{n;-n}}
\m\,\ee^{-2\eta^{-n}\vert\omega\cdot q\vert}\m.
\label{esse}
\qqq
For $q$ with $\vert\omega\cdot q\vert<(1-\eta)\eta^{n-1}B$,
we use for $\kappa\in D_n$ the equality
$\sigma(q,q;\m\kappa)=\sigma(0,0;\m\tilde\kappa)$
with $\tilde\kappa=\kappa+\omega\cdot q$
which follows from the second identity (\ref{pi})
(observe that for such $q$'s and for $\kappa\in D_n$,
${\tilde \kappa}\in D_{n-1}$).
By virtue of the inequality (\ref{smallk}),
\qq
|\sigma(0,0;\tilde\kappa)|_{_{n;-n}}
\leq\,|\sigma(0,0;\tilde\kappa)|_{_{n-1;-n+1}}
\leq\,{_1\over^8}\m \m\epsilon\,\eta^{2n}\m.
\hspace{0.3cm}
\qqq
Hence, for $q$ with $\vert\omega\cdot
q\vert<(1-\eta)\eta^{n-1}B$,
we may bound the expression on the right hand side
in the estimate (\ref{esse}) by ${_1\over^4} \epsilon\m
\eta^{2n}$. For $\vert\omega\cdot q\vert\geq(1-\eta)
\eta^{n-1}B$, we instead extract an extra factor
estimating
\qq
2\,\vert\m\sigma(q,q;\m\kappa)\m\vert_{_{n;-n}}
\,\ee^{-2\eta^{-n}\vert\omega\cdot q\vert}\,\leq\,
2\,\Vert\sigma(\kappa)\Vert_{_{n-1;-n+1}}
\,\m\ee^{-2\eta^{-1}(1-\eta)^2 B}\m\cr\cr
\leq\,\ee^{-2\eta^{-1}(1-\eta)^2 B}\, \m\epsilon\m
\eta^{2(n-1)}\,\leq\,{_1\over^4}\m\epsilon\m\eta^{2n}
\label{jb2}
\qqq
for $B$ sufficiently large (this is the only place where
$B$ large is needed). Putting these estimates together
with the inequality (\ref{prnpr}) for $\sigma'_1$,
we infer the bound (\ref{taubound}) for $\sigma'(\kappa)$.
\vskip 0.4cm
We still have to iterate the crucial estimates (\ref{taub0})
which is the only place in the proof of Proposition 3 where
we use the Ward identities.
Writing Eqs.\,\,(\ref{112}) in terms of $s$, see the definition
(\ref{tild}), we obtain
\qq
s'(z)\,=\,[1-(\CL s)(z){\tilde\gamma}(z)]^{-1}(\CL s)(z)\,
+\, s'_1(z)
\nonumber
\qqq
with the ``linearized RG map'' $\CL$,
$$
(\CL s)(z)=u\m s(\eta z)\m u\m,
$$
and ${\tilde\gamma}(z)=u^{-n}\gamma_{n-1}(B\eta^nz)u^{-n}=
\eta^{-2n}(Bz)^{-2}\chi_1(B\eta^2 z)
(\matrix{_\mu &_{iBz}\cr ^{-iBz} & ^0})$, \,see Eqs.\,\,(\ref{chi})
and (\ref{gamma}). The estimate (\ref{prnpr1}) implies that
the remainder $s'_1$ satisfies the bound
$|s'_1(z)|_{_{0,0}}\leq C\m\epsilon\, r^{{1\over 2}n}$.
Combining the definition (\ref{chi0}) (which implies that $\chi_1(z)$
is of order $|z|^6$ for small $z$) and the inductive bound for $s$,
we infer that $|(\CL s)(z){\tilde\gamma}
(z)|_{_{0;0}}\leq C\m\epsilon\m|z|^4$. Thus altogether
$$
|s'(z)-(\CL s)(z)|_{_{0;0}}\m\leq\,\m C\epsilon^2\eta^{2n} |z|^4
+C\m\epsilon\, r^{\hf n}\m.
$$
The map $\CL$ preserves $p_i$ and contracts the constant $A$ to
$\eta\m A$. The Ward identity, Lemma 2, implies that
$\partial^j\wp'_i(0)=0$ for $j<2-i$. Since
$\vert p'_i-p_i\vert\leq C\m\epsilon\,\eta^{-2n}\m r^{\hf n}
\leq{\epsilon\over32 n^2}$ and $\eta\m A+C\m\epsilon^2
+C\m\epsilon\m\eta^{-2n}\m r^{\hf n}\leq A$ \,for $r$
and $\epsilon$ small, we infer that $s'$ satisfies the bound
(\ref{taub0}). This finishes the proof of Lemma 3
and of Proposition 3.
\ \ $\Box$
\vskip 0.2cm
\nsection{ Proof of Theorem 1}
\vskip 0.2cm
We shall first show that $X_n\equiv F_n(0)=\Gamma_{