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quasimodes, resonances, KAM theorem
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\begin{document}



\title{
Quasimodes with Exponentially Small Errors
Associated with Elliptic Periodic Rays 
%(Running title: Quasimodes)
}
\author{
\begin{tabular}{c}
Fernando Cardoso\thanks{Partially supported by CNPq
(Brazil)}\\ 
Universidade Federal\\
 de Pernambuco\\
Departamento de Matem\'atica\\
50.540-740  Recife-Pe \\
 Brazil\\
e-mail: fernando@dmat.ufpe.br
\end{tabular}\qquad
\begin{tabular}{c}
Georgi Popov\\
Universit\'e de Nantes\\
 D\'epartement de Math\'ematiques\\   
UMR 6629 du CNRS\\
2, rue de la Houssini\`ere, BP 92208, \\
44072 Nantes  
Cedex 03, France\\
e-mail: popov@math.univ-nantes.fr
\end{tabular}}

\date{}
\maketitle

\begin{abstract}
\noindent


The aim of this paper is to construct compactly supported 
Gevrey quasimodes with exponentially
small discrepancy for the Laplace 
operator with Dirichlet boundary conditions in a domain $X$ with
analytic boundary. 
The quasimodes are associated with a non-degenerate elliptic closed
broken geodesic $\gamma$ in $X$. We find a Cantor family $\Lambda$ of
invariant tori of the corresponding Poincaré map which is Gevrey
smooth with respect to the transversal variables (the frequencies). 
Quantizing the  Gevrey family $\Lambda$, we construct quasimodes with
exponentially small discrepancy. As a consequence,  we  
obtain a large amount of resonances
exponentially close to the real axis for suitable compact obstacles. 

This is a new  version of mparc 02-181, where the beginning of the proof of
Proposition 6.1 is revised. 
\end{abstract}




\setcounter{section}{0}

%**************************************************************************
%******************18.11.2000********************
%**************************************************************************





\section{Introduction}
\setcounter{equation}{0}

The motivation for this paper is the modified Lax-Phillips conjecture
about the  existence of resonances near the real axis for trapping obstacles. 
It is known by a result of Stefanov
\cite{S} that if there exists a  compactly supported 
$C^\infty$-quasimode for
the corresponding exterior problem, then the counting function $N(r)$ 
of resonances of
modulus less or equal $r$ lying sufficiently 
``close'' to the real axis  can be estimated from below by  the counting
function of the quasimode. A sharp
lower bound  $N(r) \ge C(1 +r^n)$ has been obtained in this way in \cite{S}
for the so called Helmholtz resonator 
 in ${\bf R}^n$, $n\ge 2$,  with a smooth compact 
boundary, where $N(r)$ counts with multiplicities all the
resonances $\lambda$ of modulus  $\le r$  such that  
$0< {\rm Im}\, \lambda \le C_N |\lambda|^{-N}$ for  each $N\in {\bf
N}$ and some $C_N>0$.  
The aim of this paper is to construct 
Gevrey  quasimodes, that is quasimodes with exponentially small 
discrepancy,  for the Helmholtz resonator with analytic boundary. 
As an consequence we localize resonances exponentially
close to the real axis and we 
obtain a sharp lower bound  for their 
counting function. 

Consider the Laplace
operator $\Delta_D$  with Dirichlet boundary conditions in a   domain  
$X \subset {\bf R}^n$, $n\ge 2$, with a smooth compact boundary $\Gamma$.   
We are going to construct quasimodes for $\Delta_D$ 
with exponentially small discrepancy. They will be   associated to
appropriate  periodic broken geodesics $\gamma$ in $X$. We impose the
following two conditions:
\begin{itemize}
\item[(H$_1$)] $\gamma$ is a  broken closed  elliptic geodesic 
and the corresponding Poincaré map is  $N$-elementary, $N\ge 4$,
and non-degenerate.
\item[(H$_2$)] $\Gamma$ is analytic in a neighborhood of the vertices
of $\gamma$.
\end{itemize}
The precise meaning of  (H$_1$) will be explained in Sect. 3. 
We note that  (H$_1$) is stable under small $C^\infty$ perturbations of the
boundary.  



We define a Gevrey quasimode for the Laplace operator $\Delta_D$ in
$X$ with Dirichlet boundary conditions as follows: 
Given  $\varrho > 1$, we say that  
${\cal Q} = \{(u_\nu, \lambda_\nu),
\nu \in {\cal M}\}$, ${\cal M} \subset {\bf Z}^{n}$ being  an
unbounded index set, is a 
{\em Gevrey $G^\varrho$-quasimode }
of  $\Delta_D$    if  
$u_\nu \in C_0^\infty(\overline X)$, 
$\lambda_\nu > 0$, 
$\lim_{\nu\rightarrow \infty}\lambda_\nu~\rightarrow~+~\infty$ 
and
\begin{equation} 
\left\{ 
\begin{array}{rcll} 
||(\Delta+\lambda^2_\nu)u_\nu||_{L^2(X)}& = &
O(e^{-c\lambda_\nu^{1/\varrho}})\quad\mbox{in}\quad X
\nonumber \\ [0.3cm]
u_\nu|_{\Gamma}&=&0
\nonumber \\ [0.3cm]
\langle u_{\nu}, u_{\nu'}\rangle_{L^2( X )}&=&\delta_{\nu,\nu'},\ 
\nu,\nu' \in {\cal M} 
                                         \label{eq:1.1} 
\end{array}
\right. 
\end{equation}
for some constant $c>0$, where
$\delta_{\nu,\nu'}=0$ if $\nu\neq\nu'$ and $\delta_{\nu,\nu}=1$. 
We define the support of ${\cal Q}$ by 
supp${\cal Q} = 
\cap_{p= 1}^\infty\overline{\cup_{|\nu| \ge p}\,  {\rm supp}\,
u_\nu}$. 
Fix  $\varrho > 2n+1$ if $n \ge 3$ and $\varrho > 11/2$ if $n=2$. 
Our main result is: 
\begin{Theorem}  
Let (H$_1$) and (H$_2$)  be 
fulfilled. Then for  each compact neighborhood
$K$ of $\gamma$ in $\overline X$  
there exists a 
Gevrey $G^\varrho$-quasimode 
${\cal Q}=\{(u_\nu,\lambda_\nu),\nu\in {\cal M}\}$ 
for $\Delta_D$ with support in $K$. Moreover, 
\begin{equation}
\# \{\nu \in {\cal M}:\, \lambda_\nu \leq \lambda\}\ = \ V
\lambda^n \, +\, o(\lambda^n)\, ,\quad \lambda \rightarrow +\infty,
                                         \label{eq:1.2} 
\end{equation}
where $V$ is a positive constant. 
\end{Theorem}
The quasimodes we are going to construct are associated with a Cantor
family $\Lambda$ of Kolmogorov-Arnold-Moser (KAM) invariant tori of the
Poincaré map $P$ of $\gamma$. 
We are going to find $\lambda_\nu$ by means of the   
quantization relations  (\ref{eq:5.2}) and  (\ref{eq:5.8a})  
involving   ``Birkhoff invariants'' on $\Lambda$ 
of the ``quantum Poincaré map'' associated with $\gamma$. 
The constant $V$ in (\ref{eq:1.2}) is given by the
volume of the ``flow-out'' of $\Lambda$ by the
billiard flow in $S^\ast X:=\{(x,\xi)\in T^\ast X:\, |\xi|=1\}$ (see
(\ref{eq:5.6})).   
Quasimodes satisfying (\ref{eq:1.2}) and having  discrepancy of the form
$O_N(\lambda_\nu^{-N})$ for each $N>0$ 
have been constructed  by Lazutkin (see \cite{La} and the
references there) and Colin de
Verdière \cite{CV}, see also \cite{C-P} and \cite{P1}. 


As a consequence of Theorem 1 (see \cite{La} and  \cite{P2}), 
we can localize eigenvalues of $\Delta_D$ in each
interval 
$[\lambda_\nu^2 - \exp(-{c\over 2}|\lambda_\nu|^{1/\varrho}),
\lambda_\nu^2 + \exp(-{c\over 2}|\lambda_\nu|^{1/\varrho})$, $|\nu|\gg 1$, 
provided that $X$ is bounded.  
Suppose now that $X={\bf R}^n \setminus {\cal O}$, where ${\cal O}$ is
a bounded open set with analytic  boundary. 
Combining Theorem 1 with a result of Stefanov \cite{S}, 
we obtain a sharp lower bound of the counting function of 
resonances  (counted with multiplicities)  
of $\Delta_D$ in the upper half plane which are
exponentially close to the real axis. 
Recall that the multiplicity of each resonance 
$0\neq\lambda \in {\rm Res}(\Delta_D)$ is given by 
$m(\lambda) =\displaystyle {\rm rank}\, \int_{|z-\lambda| = \varepsilon} 
z R_\chi(z) dz \, ,\ 0<\varepsilon \ll 1,$
where $R_\chi(z)$ is the meromorphic continuation of $\chi (\Delta_D -
z^2)^{-1}\chi$, from Im$\,\lambda<0$ to the whole complex plane ${\bf C}$ if
$n$ is odd, and to the logarithmic Riemann surface if $n$ is
even, $\chi \in C_0^\infty({\bf R}^n)$, $\chi =1$ in a neighborhood of
${\cal O}$,  
and $m(\lambda)$ does not depend on $\chi$. 
By a result of Burq \cite{B}, there exist positive constants $c_0$ and
$C_1$ such that all the resonances $\lambda$ of  $\Delta_D$ satisfy 
${\rm Im}\, \lambda\, \ge  \,
C_1e^{-c_0|\lambda|}$. Consider  the sequence
of resonances
${\bf N}\ni j \to z_j\in {\rm Res}(\Delta_D)$  of $\Delta_D$ 
counted with multiplicities 
(each $z\in {\rm Res}(\Delta_D)$ appears exactly 
 $m(z)$ times successively in the sequences $\{z_j\}_{j\in {\bf
N}}$). 
Fix $0<c_1<c/2$, where 
$c>0$ is introduced in (\ref{eq:1.1}).  Using Corollary 1 and Remark 5 in \cite{S}, we localize
resonances near $\lambda_\nu$ and obtain a sharp lower bound of 
their counting function. 
\begin{cor}
Let  $X= {\bf R}^n\backslash {\cal O}$,  where ${\cal O}$ is a
compact obstacle with  analytic  boundary. 
Suppose that    (H$_1$) 
holds. Then there 
exists an one-to-one (injective) maping ${\cal M} \ni \nu \to j_\nu
\in {\bf N}$ 
 such that  
\[
C_1'\,  e^{-c_0\lambda_{\nu}}  < {\rm Im}\, z_{j_\nu} \le  
C_2\,  \lambda_{\nu}^{n+2} e^{-c\lambda_{\nu}^{1/\varrho}}\ ,\quad  
|{\rm Re}\, z_{j_\nu} - \lambda_{\nu}| \le 
C_2\,  e^{-c_1\lambda_{\nu}^{1/\varrho}}\ , 
\]
for each $\nu\in {\cal M}$,  where $C_1'>0$ and   $C_2\gg 1$. 
Moreover, the counting function of these resonances satisfies
$\#\{\nu\in{\cal M}:   0<{\rm Re}\, z_{j_\nu} \le r\} =  
V\, r^n + o(r^n)$ as $r\rightarrow +\infty$, 
where $V$ is the constant in Theorem 1. 
\end{cor}
We  point out that the statement of Corollary 1 is stable under small
analytic perturbations of the boundary. Moreover, Theorem 1 and
Corollary 1  hold also if we replace the Euclidean metric with a
smooth Riemannian metric which is analytic in a neighborhood of the 
closed geodesic $\gamma$. We stress also  that 
we give in Sect. 5.1 a   procedure of finding  $\lambda_\nu$ from 
the quantization relations  (\ref{eq:5.2}) and  (\ref{eq:5.8a}).

Resonances exponentially close to the real axis have been found for
the reduced wave equation in ${\bf R}^n$ in the case of 
spherically symmetric media by Ralston \cite{Ra} and
recently by Vodev \cite{V2}. Such resonances were obtained by
Vodev \cite{V1} for the Neumann problem in linear
elasticity for obstacles with analytic boundaries. The reason for
the existence of resonances exponentially close to the real axis in
that  case is that  Rayleigh waves are supported in the elliptic
region of the boundary. Semi-classical 
resonances for Schr\"odinger type operators 
exponentially close to the real axis and associated with Gevrey
families of KAM tori 
have been found in \cite{P2}. Here we extend  
some of the techniques developed in \cite{P2} to the case of discrete
Hamiltonian systems such as the Poincaré and the quantum Poincaré maps 
associated to $\gamma$. We would like to note that 
in general  KAM results are easier to state for
discrete 
Hamiltonian systems than for Hamiltonians but  
more difficult to prove.  


The article is organized as follows: 
In Sect. 2 we reduce the problem to a spectral problem at the boundary
$\Gamma$ following a standard procedure in  geometric optics.  
Consider the periodic broken geodesic 
$\gamma:[0,T]\rightarrow X $ 
with vertices $\gamma_j=\gamma(t_j)\in \Gamma$, $0 = t_1 < t_2 <
\cdots < t_m < t_{m+1} =T$. 
Recall that $\gamma(t)$ moves on a straight line with unit velocity in
each interval $(t_j,t_{j+1}),\, 1\le j \le m$, and it reflects at the
boundary $\Gamma$ of $X$ 
by the usual law of  geometric optics. 
Denote by $\widetilde\gamma(t) = (\gamma(t),\xi(t))$, $t\neq t_j$,  
the corresponding broken bicharacteristic
in  $S^\ast X = \{(x,\xi)\in T^\ast X:\, |\xi| = 1\}$,
and by  $\rho_j=(\gamma_j,\eta_j)$, $\rho_{m+1} = \rho_1$, 
its ``vertices'' in 
$B^\ast\Gamma = \{(y,\eta)\in T^\ast\Gamma:\, |\eta|\le 1\}$, 
where 
$\eta_j = \xi(t_j -0)|_{T_{\gamma_j}\Gamma} =  
\xi(t_j + 0)|_{T_{\gamma_j}\Gamma}\, .$ 
We fix $\sigma>1$ such that $\sigma -1\ll 1$, and 
consider an outgoing  parametrix of the reduced wave equation 
$(\Delta + \lambda^2)U_1(\lambda) = O(e^{-c|\lambda|^{1/\sigma}})$ in
$X$  with initial data in a small neighborhood $\Gamma_1$ of $\gamma_1$
such that the restriction $U_1(\lambda)|_{\Gamma_1}$ is a
pseudodifferential operator $\Psi_1(\lambda)$ with a large parameter $\lambda$
($\lambda$-PDO) with a Gevrey symbol of class $S^\sigma$ (see
Appendix), the $G^\sigma$-frequency set of which,  
${\rm WF}^\sigma\, (\Psi_1(\lambda))$,  is contained in a small
neighborhood of $(\gamma_1,\eta_1)\in B^\ast \Gamma$. Moreover, we suppose
that $\Psi_1(\lambda)$ coincides with the identity mapping
microlocally near $(\gamma_1,\eta_1)$. The operator $U_1(\lambda)$ is a
Fourier integral operator with a large parameter $\lambda$
($\lambda$-FIO) which can be represented by oscillatory integrals with
analytic phase functions and amplitudes in $S^\sigma$. When reaching
the boundary again in a small neighborhood $\Gamma_2$ of $\gamma_2$, we
take   an outgoing  parametrix $U_2(\lambda)$ in $X$ such that 
$(U_1(\lambda) +U_2(\lambda)) |_{\Gamma_2}= 0$. Repeating this
procedure $m$
times we come back to $\Gamma_1$. Then $U(\lambda):= U_1(\lambda) +
\cdots + U_m (\lambda)$ satisfies  
$(\Delta + \lambda^2)U(\lambda) = O(e^{-c|\lambda|^{1/\sigma}})$ in
$X$ and by construction  $U(\lambda)_{\Gamma} = \Psi_1(\lambda) -
\bar M(\lambda) + O(e^{-c|\lambda|^{1/\varrho}})$, 
where $\bar M(\lambda)$ is a $\lambda$-FIO. The
corresponding canonical relation is just the graph of the Poincaré
map associated to $\widetilde\gamma$ 
and it can be represented as an oscillatory
integral with an analytic phase function and amplitude in $S^\sigma$. 
The operator $\bar M(\lambda)$ is often referred   to as a  quantum
Poincaré map associated to $\gamma$. 
To obtain $(u_\nu,\lambda_\nu)$ in (\ref{eq:1.1}) it will be
sufficient  to solve the spectral problem $\bar
M(\lambda_\nu)u_\nu = u_\nu$ modulo
$O(e^{-c|\lambda|^{1/\varrho}})u_\nu$, where the $G^\varrho$-frequency set of 
$\{(u_\nu,\lambda_\nu)\}$ is contained in a small neighborhood of
$\rho_1$.  
To do this we shall obtain a ``normal'' form of $\bar
M(\lambda)$, in other words, we shall separate microlocally 
the variables near a large invariant set of the Poincaré map. 

In Sect. 3 we obtain a large Cantor family 
of {\em analytic} KAM tori of the
Poincaré map $P$ which is {\em Gevrey smooth} with respect to the
``transversal variables'' (the frequencies) $\omega\in \Omega_\kappa$ in
the sense of Whitney. The Cantor set  $\Omega_\kappa\subset {\bf R}^{n-1}$ is 
given by the  Diophantine condition (\ref{eq:3.5}), where the exponent
$\tau >n-1$ and the small constant $\kappa>0$ are fixed,  and it has a positive
Lebesgue measure. Moreover, for each $\tau'>\max (\tau, 5/2)$, 
we obtain a symplectic normal form of $P$
near the family of KAM tori in the Gevrey class $G^{\tau^\prime +
2}$. 
In other words, we find polar symplectic coordinates 
$(\varphi,I)\in {\bf T}^{n-1}\times D$,  ${\bf T}:= {\bf
R}/2\pi {\bf Z}$,  $D\subset {\bf R}_+^{n-1}$, an  open set, and 
a Gevrey function $K\in G^{\tau^\prime + 2}(D)$  
such that  
\[
P(\varphi,I)\ =\ (\varphi + \nabla K(I), I) + R_0(\varphi,I),
\]
where $R_0 \in G^{1,\tau^\prime + 2}({\bf T}^{n-1}\times D)$ 
(analytic with respect to $\varphi$ and $G^{\tau^\prime + 2}$ with
respect to $I$) and $R_0$
is {\em flat} at ${\bf T}^{n-1}\times E_\kappa$, where $ E_\kappa$ is
a Cantor set of positive measure in $D$ such that the map   
$\omega: E_\kappa \rightarrow \Omega_\kappa$, 
$\omega(I) =  \nabla K(I)$,  is
bijective. 
Then each ${\bf T}^{n-1}\times \{I\}$, $I\in E_\kappa$, 
is  an invariant torus of
$P$ with a vector of rotation $\omega/2\pi =  \nabla K(I)/2\pi$. 
We denote that
torus by $\Lambda_\omega$ and by $\Lambda$ the family of all such tori. 
The set $\Lambda \subset B^\ast \Gamma$ has a positive Lebesgue
measure. 
Since $R_0$ is Gevrey smooth and flat at ${\bf T}^{n-1}\times
E_\kappa$,  it is also {\em exponentially small} near 
${\bf T}^{n-1}\times E_\kappa$ (see (\ref{eq:3.6})). 
As in \cite{P1}, Corollary I.2, this leads to ``effective stability''
of the quasiperiodic motion 
of the billiard flow near the flow-out ${\cal T}\subset S^\ast X$  
 of the family
$\Lambda$ of  invariant tori. In  other words, we have 
stability in  finite but exponentially large  time intervals. 
To obtain the normal form, we 
use a similar result for analytic Hamiltonians \cite{P1} and a
result of Kuksin and P\"oschel \cite{KP}. We give the following
geometric interpretation of $K(I)$, $I\in E_\kappa$. Denote by ${\cal
T}_\omega$ the flow-out of $\Lambda_\omega$, $\omega = \nabla K(I)\in
\Omega_\kappa$, by the broken bicharacteristic flow. Then 
${\cal T}_\omega$ is homeomorphic to  ${\bf T}^n$ and
$I=(I_1,\ldots,I_{n-1})$  is just the action along suitable cycles in
$\Lambda_\omega$. It turns out that $K(I)$, $I\in E_\kappa$, is
uniquely determined modulo a constant, which can be fixed so that 
$-K(I)$ is the action along a suitable cycle of   ${\cal T}_\omega$
(see Sect. 3).  

Starting from the symplectic normal form of $P$, we obtain in Sect. 4
a ``Birkhoff normal form'' 
of the quantum Poincaré map $\overline M(\lambda)$
near $\Lambda$. 
We have $\overline M(\lambda) = e^{i\pi(m +
\vartheta/4)}M(\lambda)$, where $\vartheta$ is  the Maslov's index
corresponding to $\gamma$ and the principal symbol of $M(\lambda)$
equals one. Conjugating $M(\lambda)$ with suitable $\lambda$-FIOs 
with Gevrey symbols we obtain a $\lambda$-FIO, $M_0(\lambda)$, 
the distribution kernel $M_0(\lambda,x,y)$ of which is  of the form
\begin{equation}
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{n-1}} 
\, \exp({i\lambda(\langle x-y,I\rangle - K(I) + R(x,I)}))\, 
(p^0(I,\lambda) + r(x,I,\lambda))\, dI  \
                                       \label{eq:1.3}
\end{equation}
modulo $O( e^{-c|\lambda|^{1/\varrho}})$. 
Here $R\in G^{1,\mu}({\bf T}^{n-1}\times D)$, $\mu >\tau' +2$, 
$p^0$ and $r$ are Gevrey symbols in 
$S_{\ell}({\bf T}^{n-1},D)$, $\ell =(\sigma,\nu,\varrho)$ (see
Appendix), 
compactly supported with respect to $I$ in $D$,  
and the ``perturbations'' $R$ and $r$ are {\em flat} at 
${\bf T}^{n-1}\times E_\kappa$. In other words, we have a separation
of the variables in (\ref{eq:1.3}) at ${\bf T}^{n-1}\times E_\kappa$. 
The operator $M_0(\lambda)$ acts in
the space $L^2({\bf T}^{n-1}, {\bf L})$, where ${\bf L}$ is a 
flat Hermitian line bundle  over
${\bf T}^{n-1}$. 
An orthonormal basis 
$e_p,\, p\in {\bf Z}^{n-1}$, of $L^2({\bf T}^{n-1},{\bf L})$ 
is given by 
$e_p(x)= \exp( i \langle p + \vartheta_0/4,x\rangle)$, 
  where $\vartheta_0 = (2,\ldots,2)$. 
We call (\ref{eq:1.3}) 
``quantum Birkhoff normal form'' of $M(\lambda)$ and 
$p(I,\lambda), I\in E_\kappa$, quantum Birkhoff invariants. 
The construction of the normal form 
(\ref{eq:1.3}) is the main technical part in the paper.  
We look for  formal Gevrey symbols 
\[
a(\varphi, I,\lambda) \ =\ \sum_{j=0}^\infty\, a_j(\varphi,
I)\lambda^{-j}\ ,\quad p^0(I,\lambda) \ =\ \sum_{j=0}^\infty\,
p^0_j(I)\lambda^{-j}
\]
in $S_{\ell}({\bf T}^{n-1},D)$. We have $a_0=p_0^0 = 1$, while the  
functions $a_j$, $j\ge 1$,   are solutions
of the ``homological equation''
\begin{equation}
\frac{1}{i}\left[a_j(\varphi - \nabla K(I),I) - a_j(\varphi,I)\right]\
= \ f_j(\varphi,I)
                                                 \label{eq:1.4}
\end{equation}
in ${\bf T}^{n-1}\times E_\kappa$. The functions $p_j^0(I)$, $j\ge 1$,
are chosen so that the average of $f_j(\varphi,I)$ on 
${\bf T}^{n-1}$ is $0$. Here $f_j$
does not depend on $a_j$, $p_j^0$, but it does depend on $a_k$ and
$p_k^0$ for $1\le k\le j-1$. The main  difficulty is to find
Gevrey estimates for $a_j$,  $p_j^0$, and  $f_j$ which are 
uniform with respect to $j$ (see (\ref{eq:4.9})).  
Note that the 
Gevrey index $\varrho$  in  (\ref{eq:1.1}) and 
 (\ref{eq:1.3}) is related to  the loss of derivatives in the 
the ``homological equation'' (see Proposition 6.1), which can not be
avoided because of small divisors.  In this section we use some of the
techniques developed in \cite{P2}. The main difference comes from the
fact that here we look for a normal form of a $\lambda$-FIO, while in 
\cite{P2} we have found a normal form of a $\lambda$-PDO which is much
easier. A natural idea would be to write $M_0(\lambda)$ as an exponent
of a $\lambda$-PDO modulo an operator the full symbol of which
vanishes to infinite order at ${\bf T}^{n-1}\times E_\kappa$ 
and then to use the results in \cite{P2}. This
could be done but it involves ones more the Diophantine conditions
and we lose Gevrey regularity. 

In Sect. 5 we obtain the quasimode ${\cal Q}$. We are looking for 
eigenvalues $Z_\nu(\lambda)$ 
of  $M_0(\lambda)$ with eigenfunctions $e_p$, where 
 $|\lambda -\lambda_\nu^0|  \le C$, $C= {\rm const.} \gg
1$, and  $ \lambda_\nu^0$,  $\nu = (p,q)\in {\bf N}^{n-1}\times {\bf
N}$, 
is given by the quantization condition (\ref{eq:5.2}). This condition
makes the contribution of  $R$ and $r$ in $M_0(\lambda)e_p$ 
exponentially small  and for
such $\lambda$ we obtain  $M_0(\lambda)e_p =
Z_\nu(\lambda)e_p$ modulo $O( e^{-c|\lambda|^{1/\varrho}})$. To find
$\lambda_\nu$ we solve the equation 
$e^{i\pi(m + \vartheta/4)}Z_\nu(\lambda) = 1$  
modulo $O( e^{-c|\lambda|^{1/\varrho}})$ writing it in the form 
(\ref{eq:5.8a}) and then using the implicit function theorem. 
We prove also the orthogonality relations.   

In Sect. 6 we solve the homological equation in Gevrey classes. 
In the Appendix  we 
collect some facts about Gevrey symbols.  



\section{Reduction at the boundary.}
\setcounter{equation}{0}
To prove Theorem 1 we use an outgoing parametrix for the Helmholtz
equation near $\Gamma$. Fix $\sigma>1$. 
Let $\Gamma_j\subset\Gamma$ be  
small compact neighborhoods of the vertices $\gamma_j$ of $\gamma$, 
$j=1,\dots,m+1$, where
$\rho_j=(\gamma_j,\eta_j)$ and $\rho_{m+1}=\rho_1$. Denote by $\Psi_j$ 
a  $\lambda$-pseudodifferential operator 
 with a Gevrey symbol in $S^\sigma$ (see Appendix), 
supported in a small compact neighborhood  of $\rho_j,\
j=1,\dots,m+1$, and  with a large parameter $\lambda\in 
{\cal D} := \{z \in {\bf C}: |{\rm Im}\, z| \le D_0, {\rm Re}\, z
\ge 1\}$, 
$D_0>0$ being  fixed. 
Consider the parametrix $H_j:C_0^\infty(\Gamma)\rightarrow 
C^\infty( X )$, of the problem 
\[
\left\{ 
\begin{array}{rcl}
(\Delta+\lambda^2)H_j(\lambda)u&=&O_s(e^{-C|\lambda|^{1/\sigma}})u,
\quad C>0,\\ [0.3cm]
H_j(\lambda)u|_{\Gamma_j}&=&\Psi_ju\, . 
\end{array}
\right.
\]
This  is a Fourier integral operator with a large parameter
$\lambda\in {\cal D}$, the distribution kernel of 
which can be written microlocally as an oscillatory integral 
with an  analytic non-degenerate phase function and Gevrey amplitude in
$S^\sigma$.   
Hereafter 
\[
O_s(e^{-C|\lambda|^{1/\sigma}})\, :\ L_2(\Gamma)
\longrightarrow  H^s(X)
\] 
stands for a family of bounded operators depending
on $\lambda$ with norms $\le C_1e^{-C|\lambda|^{1/\sigma}}$,
$C_1>0$ and $H^s(X)$, $s \ge 0$ is the Sobolev space.  
Denote by $\imath^\ast$ the map of restriction on $\Gamma,\ \imath^\ast
u=u|_{\Gamma}$. 
The canonical relation of the $\lambda$-FIO $\imath^\ast H_j(\lambda)$
is just the graph of the billiard ball map $B$ defined in a neighborhood
of $(\gamma_j,\eta_j)$.  
We may  suppose that the Gevrey $G^\sigma$-frequency 
set $WF^\sigma$  of
$\imath^\ast H_j(\lambda)$ (see Appendix) is contained in 
$T^\ast \Gamma_j\times T^\ast \Gamma_{j+1}$ for any $j=1,\dots,m$ and that
$(\Psi_{j+1}\imath^\ast H_j(\lambda)-\imath^\ast H_j(\lambda))u|_{\Gamma_{j+1}}=
O(e^{-C|\lambda|^{1/\sigma}})u.$ 
Consider the operator
\[
G(\lambda)u=H_1(\lambda)u-H_2(\lambda)\imath^\ast H_1(\lambda)u+\cdots+
(-1)^{m-1}H_m(\lambda)\imath^\ast
H_{m-1}(\lambda)\cdots \imath^\ast H_1(\lambda)u
\]
for $u\in C_0^\infty(\Gamma_1)$. Obviously
\[
\left\{
\begin{array}{rcll}
(\Delta+\lambda^2)G(\lambda)u\ &=&\ 
O_s(e^{-C|\lambda|^{1/\sigma}})u\, , \\
\imath^\ast G(\lambda) u \ &=&\ \Psi_1(\lambda) u -
\bar{M}(\lambda)u +O_s(e^{-C|\lambda|^{1/\sigma}})u \, ,
\end{array}
\right.
\]
for any $u\in C_0^\infty (\Gamma)$,  where $\bar{M}(\lambda)=
(-1)^m\Psi_{m+1}\imath^\ast H_m(\lambda)\imath^\ast H_{m-1}(\lambda)\cdots
\imath^\ast H_1(\lambda)$, 
and the symbol of $\Psi_{m+1}$ is 1 in a neighborhood
of the support of the symbol of $\Psi_1$ and equals 0 outside
$\Gamma_{m+1}$.
We are going to find $\lambda_\nu>0$ and $v_\nu\in L^2(\Gamma)$ such that 
$G(\lambda_\nu)v_\nu|_{\Gamma}=
O_s(e^{-c\lambda_\nu^{1/\varrho}})v_\nu$, $c>0$. 
To do so we have to solve the equation
\begin{equation}
(\bar{M}(\lambda)-\Psi_1)u=O_s(e^{-c\lambda^{1/\varrho} })u\, .
                                                      \label{eq:2.1}
\end{equation}
 Observe that $\bar{M}(\lambda)$ is a $\lambda$-FIO 
of order 0 with a
large parameter $\lambda\in {\cal D}$. The canonical relation of 
$\bar{M}(\lambda)$ is given by
the graph of the Poincaré map 
$P= B^m:W_0 \to W$, where $W_0$ and $W$ are neighborhoods of $\rho_1$ 
in $B^\ast \Gamma$.
Moreover, the principal symbol of $\bar{M}(\lambda)$ can be
identified with the function
\[
\sigma(\bar{M})(y,\eta,\lambda)\ =\  \exp(i[\lambda L(y,\eta) + 
(m +\vartheta/4)\pi])\, ,\ 
(y,\eta)\in W_0\, ,
\]
where  $\vartheta$ is the Maslov 
index  corresponding to the broken closed
bicharacteristic $\widetilde\gamma$.   Here 
$L$ is the ``return time function'', i.e. 
$\Phi^{L(y,\eta)}(y,\eta) = P(y,\eta)$, $(y,\eta)\in W_0$, 
$t\rightarrow \Phi^t(y,\eta)$ being the 
broken bicharacteristic issuing from $(y,\eta)$. 
We set $\bar M(\lambda) =  e^{i(m+\vartheta/4) \pi} M(\lambda)$. 
Then  the
principal symbol of $M(\lambda)$ equals one, 
modulo the Liouville factor $\exp(i\lambda L(y,\eta))$. 



\section{Birkhoff  normal form of $P$ in Gevrey classes}
\setcounter{equation}{0}

We are going to explain (H$_1$) in more details. Consider a periodic 
broken geodesics $\gamma:[0,T]\rightarrow X $ 
with vertices $\gamma_j=\gamma(t_j)\in \Gamma$, $0 = t_1 < t_2 <
\cdots < t_m < t_{m+1} =T$, $1 \le j \le m$. 
Let  $\widetilde\gamma(t) = (\gamma(t),\xi(t))$, $t\neq t_j$,  
be the corresponding broken bicharacteristic
of $S^\ast X$ with  ``vertices'' $\rho_j=(\gamma_j,\eta_j)$,
  $\rho_{m+1} = \rho_1$, in 
$B^\ast\Gamma$.  The billiard ball
map $B$ is {\em analytic} and  symplectic 
in a neighborhood of $\rho_j$ in $B^\ast\Gamma$ because of (H$_2$). 
Recall that for $(y,\eta)\in B^\ast \Gamma$ close to $\rho_j$, 
$B(y,\eta)$
is obtained as follows: Consider the bicharacteristic $(x(t),\xi(t))$,
$t\ge 0$, 
in  $S^\ast X$ issuing from $(y,\eta)$, i.e.  $x(0) = y$ and 
$\xi |_{T_x\Gamma} = \eta$, $\langle \xi, n(x)\rangle >0$, 
$n(x)$ being the inward unit normal to $\Gamma$ and denote by 
$t_1(y,\eta)>0$ the first time of intersection with the boundary. The 
function $t_1$ is 
obtained by the implicit function theorem and it is analytic near
$\rho_j$.   We set 
$B(y,\eta)= (x(t_1(y,\eta)), \xi(t_1(y,\eta))|_{T_x\Gamma})$. 
Obviously, $\{\rho_1,\rho_2,\dots,\rho_m\}$ is a periodic
orbit of $B$, 
$B\rho_j=\rho_{j+1}$, for $j<m$, 
$B\rho_m=\rho_1$,  and $\rho_1$ is a fixed point of the (local)
Poincar\'e map $P = B^m: W_0 \rightarrow W$, where $W_0$ and $W$ are 
neighborhoods of $\rho_1$ in  $B^\ast\Gamma$. 
We assume that $\gamma$ is elliptic, which means that the eigenvalues
of the linear Poincar\'e map $DP_{\rho_1}$ are of the form $e^{\pm
i \alpha_j^0}$, $j=1,\ldots,n-1$, $0 < \alpha_j^0 \le \pi$. 
We suppose also that $P$ is $N$-elementary for some $N\ge 4$, which
means that 
\begin{equation}
 \sum_{j=1}^{n-1} \alpha_j^0\,  k_j \notin  2\pi {\bf Z}, \ \forall \, 
k=(k_1,\ldots,k_{n-1})\in {\bf Z}^{n-1},\  0<|k_1|+\cdots
+|k_{n-1}| \le N.  
\label{eq:3.1}
\end{equation}
Then $P$ admits an analytic  Birkhoff normal form near
$\rho_1=(0,0)$. In other words, there exist analytic  symplectic
local coordinates $(q,p):W_0\rightarrow  T^\ast {\bf R}^{n-1}$, 
$(q(\rho_1),p(\rho_1)) = (0,0)$, such that in the corresponding polar 
symplectic coordinates  
$(\theta,r)\in {\bf T}^{n-1}\times V$, $V$ being the intersection of a
ball of radius $r_0 \ll 1$ with a proper 
open cone in ${\bf R}_+^{n-1}$ having vertex at $0$, the map $P$ becomes 
\begin{equation}
P_0(\theta,r) = (\theta + \nabla S(r) + O(|r|^{(N-1)/2}), r +
O(|r|^{(N+1)/2})),\quad  r\rightarrow 0,
                                         \label{eq:3.2}
\end{equation}
where $S$ is a polynomial of degree $\le N/2$ and  
$\nabla S(0)=(\alpha_1^0,\dots,\alpha_{n-1}^0)$. 
  Denote by $B_1$, the matrix
$(\partial_i\partial_jS(0))^{n-1}_{i,j=1}$ and make the non-degeneracy
assumption 
\begin{equation}
det\ B_1\ne0.\label{eq:3.3}
\end{equation}

We consider Gevrey functions of the following type: 
Given  a domain 
$D \subset {\bf R}^{n-1}$ 
and reals  $\sigma, \mu \ge 1$, 
we say that $f\in G^{\sigma,\mu}({\bf T}^{n-1}\times D)$ if  for every 
compact subset $Y$ of $ D$ there exists $C = C(Y)>0$ such that 
\begin{equation} 
\sup\limits_{(\varphi,I)\in {\bf T}^n\times Y}\, 
|\partial_\varphi^{\beta}\partial_I^\alpha f(\varphi,I)|\ 
\leq\ C^{|\alpha |+|\beta|+1}\, 
\beta !\, ^\sigma\, \alpha!\, ^{\mu}\ , \  
\forall \, \alpha,\, \beta \in {\bf N}^{n-1}\, , 
                                      \label{eq:3.4} 
\end{equation} 
where ${\bf N}$ stands for the set of all non-negative integers. 
Hereafter, given a multi-index $\alpha \in {\bf N}^d$, $d\ge 1$, we
write $|\alpha| = \sum_{j=1}^d\, |\alpha_j|$. 
The solutions $u_\nu,\lambda_\nu$ of
(\ref{eq:2.1}) will be  associated with a Gevrey smooth family 
of invariant tori of the Poincar\'e
map $P$ in a neighborhood of $\rho_1$. Fix  $C_1 >1$ and 
for each $0<a\leq a_0\ll 1$  set 
$$ 
D\ =\ D_a\ =\ \{I\in\Gamma:\, C_1^{-1}a \leq I_j\leq C_1a,\ j=1,\ldots,n-1\}, 
$$ 
and denote by $\Omega = \Omega_a\ = \nabla S ( D_a)$. Choosing $a_0>0$
sufficiently small, the map $\nabla S : D_a \rightarrow \Omega_a$
becomes a diffeomorphism. Fix  $\tau> n-1$,  
and $0< \varepsilon \ll 1$ and set 
$\kappa =\kappa_a = \varepsilon a$.  Consider the set of all 
$\omega\in{\bf R}^{n-1}$   defined by the  Diophantine condition:   
\begin{equation}
|\langle \omega, k \rangle - 2\pi k_n| \ \ge \  
\frac{\kappa} { (\sum_{j=1}^{n}|k_j|)^{\tau}},\quad \forall\, 
(k,k_n)\in{\bf Z}^n \backslash\{0\}.
                                        \label{eq:3.5}
\end{equation}
Let $\Xi_\kappa$ be the set of all  $\omega\in \Omega$ satisfying 
(\ref{eq:3.5}) and  having 
distance $\geq\kappa$ to the boundary of $\Omega$. 
We define $\Omega_\kappa$ to be the set of  points  of  positive Lebesgue 
density in $\Xi_\kappa$. In other words, $\omega\in\Omega_\kappa$ if for  
any neighborhood $U$ of $\omega$ in $\Omega$ the 
Lebesgue measure of $U\cap\Xi_\kappa$ 
is positive. Obviously, $\Omega_\kappa$ and $\Xi_\kappa$ have the same 
Lebesgue measure,  which is positive if  $\varepsilon >0$ 
is sufficiently small. Fix $\tau' >\max(\tau,5/2)$. 
Recall that a symplectic diffeomorphism 
$\chi: {\bf T}^{n-1}\times D \rightarrow {\bf T}^{n-1}\times D$ is
called exact symplectic if $\chi$ admits  a generating function 
$\Phi\in C^\infty({\bf T}^{n-1}\times D)$, which 
is determined modulo a constant by
\[
\chi (\varphi + \nabla_I \Phi(\varphi,I), I) = 
(\varphi, I + \nabla_\varphi  \Phi(\varphi,I))\ ,\ 
(\varphi,I)\in {\bf T}^{n-1}\times D\, . 
\]
\begin{theo} Suppose that (\ref{eq:3.3}) holds.  
Then for  each $0<a\le a_0$, $a_0$ sufficiently small, there exists 
an exact  symplectic diffeomorphism $\chi_0$ of Gevrey class 
$G^{1,\tau' +2}$  mapping 
${\bf T}^{n-1}\times D$ into itself, a $G^{\tau' +2}$-diffeomorphism 
$\omega:D\rightarrow \Omega$, and $K\in G^{\tau' +2}(D)$ 
 such that the exact symplectic map 
\[
P^0 = \chi_0^{-1}\circ P_0 \circ \chi_0
\in G^{1,\tau' +2}({\bf T}^{n-1}\times D;{\bf T}^{n-1}\times D )
\] 
has a generating function  
$Q (\varphi, I)\ =\ -  K(I)\ +\ R(\varphi, I)$ in $G^{1,\tau' +2}$,  
where  
\[
D_I^\alpha (\omega(I)-\nabla K(I))\ =\ 0 \quad  {\rm and}\quad  
D_I^\alpha R(\varphi, I)\ =\ 0\ ,  \quad  \forall\, \alpha \in {\bf
N}^{n-1}\, ,
\]
on $E_\kappa:=\omega^{-1}(\Omega_\kappa)$ and  
${\bf T}^{n-1}\times E_\kappa$ respectively. 
The symplectic map $\chi_0$ is exact symplectic with 
generating function $\Phi\in G^{1,\tau' +2} 
({\bf T}^{n-1}\times D)$ and for  each $|\alpha|\leq (N-1)/2$ and $0<q<1$,
we have 
$$ 
\left| D^\beta_\varphi D^\alpha_I \Phi(\varphi,I)
\right|\ 
+\ \left|D^\alpha_I (\omega(I) - \nabla S(I))\right|\ 
$$
$$
\leq\ 
C^{|\alpha|+|\beta|+1}\, a^{q(N-1)/2-|\alpha|}\ \beta ! \, ,\ 
\forall\beta\in {\bf N}^{n-1}\, ,
$$  
for any $(\varphi,I)\in {\bf T}^{n-1}\times D$, where  $C=C_q$ does not depend
on $a$. 
\end{theo} 
To prove the theorem we use Theorem 2, \cite{KP}. We write the real
analytic map $P$ as a 
symplectic map in a neighborhood of the origin in ${\bf R}^{2n-2}$ of the
form $z \rightarrow \Lambda z + \hat F (z)$, where $\Lambda= DP(0)$ 
and  $\hat F (z) = O(z^2)$. Moreover,  
$\Lambda = \exp(JA)$, where $A$ is a real symmetric matrix and 
\[
J\ =\ \pmatrix{\mbox{0\quad   Id}\cr \mbox{-\rm Id\quad  0} \cr}\, .
\]
Applying  Theorem 2,
\cite{KP}, we find a $2\pi$-periodic Hamiltonian $\widetilde H(t,z) = 
\langle Az,z\rangle + \hat H(t,z)$ which is jointly real analytic in
some neighborhood of the origin in ${\bf R}^{2n-2}$ in $z$, and in
$t$, and such
that $P$ is just the flow $\exp(tX_{\widetilde{H}})_{|t=2\pi}$ at time
$2\pi$. Because of (\ref{eq:3.1}) we can find a Birkhoff 
normal form up to  order $N$
of the $2\pi$-periodic Hamiltonian $\tilde H$ \cite{A}, 
which in fact coincides
with $S$. This gives a normal form of the non-degenerated Hamiltonian 
$H = \tau + \widetilde H (t,z) +  (\tau + \widetilde H (t,z))^2$, 
$(t,\tau,z)\in {\bf T}\times {\bf R}\times {\bf R}^{2n-2}$ 
in a neighborhood of $\{(t,0,0): t\in {\bf T}\}$, and we deduce Theorem
3.1 from Theorem I.1, \cite{P1}, using the same argument as in the 
proof of Corollary I.3,  \cite{P1}. 


Set $\chi = \chi_1\circ\chi_0$, where $\chi_0$ is given by Theorem 3.1
and $\chi_1$ is the symplectic map which puts $P$ in the  Birkhoff normal
form (\ref{eq:3.2}). Then $\chi$ is an exact symplectic
diffeomorphism. 
Let $\Omega\ni \omega\rightarrow I(\omega)\in D$ be the inverse map to 
the diffeomorphism $I\rightarrow \omega(I)$.  
Then $E_\kappa= I(\Omega_\kappa)$ and   each  
$$ 
\Lambda_\omega\  =\ \chi({\bf T}^{n-1}\times\{I(\omega)\})\, ,\ 
\omega\in\Omega_\kappa \, , 
$$ 
is an invariant torus of $P$ with  frequency $\omega$.   
Moreover, we can find  
two positive constants $C_1$ and  $c$ depending only on the 
constant $C$ in 
(\ref{eq:3.4}) of  $R$,  such that for every 
$\alpha,\beta \in {\bf N}^{n-1}$ the following estimate holds 
\begin{equation} 
|\partial_\varphi^\beta\partial_{I}^{\alpha} R(\varphi, I)|\ \leq\ 
C_1^{|\alpha|+|\beta|+1}\, \beta !\,  \alpha !\, ^{\tau' +2}\,  
\exp\left(- c\, |E - I|^{\, -\frac{1}{\tau' +1}}\right)\, ,\    
                                         \label{eq:3.6} 
\end{equation}  
for each $(\varphi,I)\in {\bf T}^{n-1}\times Y$, $I\notin E$,                  where 
$|E - I| = \inf_{I'\in E}\, |I'-I|$ is the distance to the 
compact set $E$ (see \cite{P1}, (I,3)). Using (\ref{eq:3.6}) we can obtain
as in \cite{P1}, Corollary I.2,  
effective stability of the quasiperiodic motion of 
the discrete dynamical system $P^k$, $k\in {\bf Z}$,  
near the union 
$\Lambda$ of $\Lambda_\omega$, $\omega\in \Omega_\kappa$.  

We are going to provide a geometric interpretation of $K(I)$, 
$I = (I_1,\ldots, I_{n-1}) \in
E_\kappa$. Obviously $I_j = (2\pi )^{-1}\int _{\gamma_j} \xi d x$,
where $\gamma_j$, $j=1,\ldots, n-1$, form a basis of cycles in
$\Lambda_\omega$, $\omega = \nabla K(I)$. Consider the ``flow-out'' 
${\cal T}_\omega$ of $\Lambda_\omega$ by the bicharacteristic flow. 
Let $\rho=\chi(\varphi,I)\in \Lambda_\omega$. 
We denote by $\gamma_{n1}$ the broken bicharacteristic arc in ${\cal
T}_\omega$ issuing from $\rho$ and having  endpoint at $P(\rho)$, and by 
$\gamma_{n2}$ an arc in $\Lambda_\omega$ connecting $P(\rho)$ and
$\rho$ such that $\gamma_{n2}^0:= \chi^{-1}(\gamma_{n2})$ is given by 
$s\rightarrow (\varphi + (1-s)\nabla K(I),I)$, $s\in [0,1]$. 
Let $\gamma_n$ be the union of the two arcs. 
We are going to show that $K(I)$, $I\in E_\kappa$,  is 
uniquely determined modulo a constant and  we have
\begin{equation} 
K(I)\ =\ -\, \int_{\gamma_n}\, \xi d x\, , \ I \in E_\kappa\,  
                                         \label{eq:3.7} 
\end{equation} 
choosing 
the constant appropriately. 
Note that the right hand side of (\ref{eq:3.7}) depends only on the
homotopy class of the loop  $\gamma_n$ in the ``Lagrangian'' manifold
${\cal T}_\omega$. 
To prove (\ref{eq:3.7}) we use the Poincaré identity
\[
P^\ast  (\xi dx) = \xi dx + dL,
\]
where $\xi dx$ is the fundamental one form at $T^\ast\Gamma$ and
$L(\rho)$ stands for the action along the broken bicharacteristic
$\gamma_{n1}$. Since $\chi$ is exact symplectic  
we have $\chi^\ast(\xi dx) = I d\varphi +
d\Psi$ with a suitable smooth function $\Psi$. 
Combining the two equalities
we obtain 
\[
(P^0)^\ast  (I d\varphi) - I d\varphi = d((L\circ \chi) + \Psi -
\Psi\circ P^0).
\]                        
This implies 
\begin{equation}
I\nabla K(I) - K(I) = L(\chi(\varphi,I)) + \Psi(\varphi,I) -
\Psi(P^0(\varphi,I)) + C\ ,\ (\varphi,I) \in {\bf T}^{n-1}\times
E_\kappa\, ,
                                          \label{eq:3.8} 
\end{equation} 
and we choose $K$ so that $C=0$.  
On the other hand, 
\[
 \int_{\gamma_{n2}}\, \xi d x \ =\ \int_{\gamma_{n2}^0}\, (I d\varphi 
+  d\Psi)\ =\ - I\nabla K(I)+ \Psi(\varphi,I) -
\Psi(P^0(\varphi,I))\, ,
\]
and we obtain (\ref{eq:3.7}).  
 
 \section{Quantum Birkhoff  normal form of $M(\lambda)$ in Gevrey classes}
\setcounter{equation}{0}
 

Consider the map $\chi = \chi_1\circ\chi_0$. 
 Recall that $\chi_1^{-1}$ is a composition of an
analytic symplectic map $W_0\ni (y,\eta) \rightarrow (q,p)\in T^\ast
{\bf R}^{n-1}$ and the map $(q,p) \rightarrow (\theta,r)$, which puts
$(q,p)$ in polar symplectic coordinates, $q_j = \sqrt{2r_j} \cos
\theta_j$,  $p_j = \sqrt{2r_j} \sin \theta_j$, $j=1,\ldots, n-1$. 
Using the restriction
of $\chi$ to ${\bf T}^{n-1}\times \{I(\omega)\}$, $\omega\in
\Omega_\kappa$, we identify the first cohomology groups 
$H^1(\Lambda_\omega,{\bf Z}) =  H^1(T^{n-1},{\bf Z})={\bf Z}^{n-1}$,
and we denote by $\vartheta_0\in {\bf Z}^{n-1}$ 
the Maslov class of the invariant
torus $\Lambda(\omega)$. It is easy to see that $\vartheta_0
=(2,\ldots,2)$, 
which comes in fact from the map $(\theta,r) \rightarrow (q,p)$. 
We set up the parameters of  the Gevrey classes.  
In (\ref{eq:3.5}) we fix $\tau > n-1$ 
when $n\ge 3$ and $\tau > 3/2$ when $n=2$. 
Denote $\nu =\tau+n+1$ and fix  $\tau'$ such that 
$\tau+n-1\ >  \tau' \ >\ \max (\tau,5/2)$. Then fix $\mu$ such that 
$\nu \ >\ \mu\ >\ \tau' + 2$, choose $\sigma >1$ 
sufficiently close to
$1$ such that 
\begin{equation}
\nu \ >\ \mu\ >\ \sigma(\tau' + 1) + 1 \, ,
                                              \label{eq:4.1}
\end{equation}
and set 
\[
\varrho\ =\  \sigma (\nu + 1) \, .
\]
Then $\varrho$ could be any number
bigger than $\nu +1 $ and sufficiently close to $\nu +1$, hence, 
$\varrho$ 
could be any number $> 2n +1$ when $n\ge 3$ and  $> 11/2$ when $n=2$. 


As in \cite{CV} we consider 
the flat Hermitian line bundle ${\bf L}$ over
${\bf T}^{n-1}$ which is associated to the class $\vartheta_0$. 
The sections $f$
in ${\bf L}$ can be identified canonically with functions
$\widetilde{f}:{\bf R}^{n-1}
\rightarrow {\bf C}$ so that
\begin{equation}
\widetilde{f}(x +2\pi p)\ =\ e^{i\frac{\pi}{2}\langle 
\vartheta_0,p\rangle}
\widetilde{f}(x)
                                              \label{eq:4.2}
\end{equation}
for each $x\in {\bf R}^{n-1}$ and $p\in {\bf Z}^{n-1}$. 
An orthonormal basis of $L^2({\bf T}^{n-1},{\bf L})$ 
is given by
$e_m,\ m\in {\bf Z}^{n-1}$,  where
$$
\widetilde e_m (x)\ =\ 
\exp\left( i \langle m + \vartheta_0/4,x\rangle
\right).
$$ 
As in \cite{P2}, Sect. II, using Theorem 3.1, we  conjugate 
$M(\lambda)$ with a $\lambda$-FIO
\[
T: C^\infty({\bf T}^{n-1},{\bf L}) \rightarrow 
C^\infty(\Gamma_1)\, .
\]
We look for $T=T_1\circ T_0$, where
$T_1 $ is associated to $\chi_1$ and $T_0$ to $\chi_0$. As in
\cite{P2}, we write microlocally the distribution kernel of 
\[
T_1: C^\infty({\bf T}^{n-1},{\bf L})
\rightarrow  C^\infty(\Gamma_1)
\] 
as an oscillatory integral 
with  analytic phase function and full symbol in the class
$S^\sigma$. The operator 
\[
T_0: C^\infty({\bf T}^{n-1},{\bf L})
\rightarrow C^\infty({\bf T}^{n-1},{\bf L})
\] 
is given by
\[
\widetilde{T_0(\lambda)u}(x) = 
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{2n-2}} 
\, e^{i\lambda (\langle x-y,I\rangle + \Phi(x,I))} \, b(x,I,\lambda)\,
\widetilde u (y) \, dI dy \, ,
\]
$u\in C^\infty({\bf T}^{n-1},{\bf L})$, where 
$\Phi\in
G^{1,\tau'+2}$ is given by Theorem 3.1, $b$ is a Gevrey symbol 
in the class $S_{\widetilde \ell} ({\bf T}^{n-1}\times D)$, 
$\widetilde \ell = (\sigma,\mu,\sigma +\mu -1)$, and $b$ is uniformly
compactly supported with respect to $I$ in $D$. As in \cite{P2}, Sect.
II, we
obtain an  operator 
$M_1(\lambda) = T(\lambda)^{-1}  M(\lambda) T(\lambda)$ of  the form
\begin{equation}
\widetilde{M_1(\lambda)u}(x) = 
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{2n-2}} 
\, e^{i\lambda (\langle x-y,I\rangle + Q(x,I))} \, p(x,I,\lambda)\,
\widetilde u (y) \, dI dy ,\ 
                                  \label{eq:4.3}
\end{equation}
$u\in C^\infty({\bf T}^{n-1},{\bf L})$, 
where $p$ is a Gevrey symbol 
in the class $S_{\widetilde \ell} ({\bf T}^{n-1}\times D)$, 
\[
p(\varphi,I;\lambda)\sim\sum^\infty_{j=0}p_j(\varphi,I)\lambda^{-j}\ ,
\]
and $p_0(\varphi,I)= 1$. The phase function 
$Q(x,I) = - K(I) +R(x,I)$ is given by Theorem 3.1, 
and it is uniquely defined modulo a constant. Comparing the Liouville
factors in the principal symbols of $M_1(\lambda)$ and $M(\lambda)$
and using (\ref{eq:3.8}), we obtain as in \cite{C-P}, Sect. 3.3, that 
$K$ is given by (\ref{eq:3.7}). 
For each  $a\in C^\infty ({\bf T}^{n-1}\times D)$
we set 
\[
{\cal L}  a(\varphi,I) =  a(\varphi - \omega(I),I) -  
a(\varphi,I)\, ,
\] 
with $\omega(I)$ introduced  by Theorem
3.1, $\omega(I) = \nabla K(I)$ on $E_\kappa$. 
Denote 
\[
K_0(I,\xi) = \int_0^1 \nabla_I K(I +\tau \xi) d \tau
\]
and  set 
$D_\xi = (1/i) \partial_\xi$.  


\begin{lemma}
 Let $M_0(\lambda)$ be a $\lambda$-FIO of the
form  (\ref{eq:4.3}) with  amplitude
$p^0(I,\lambda)$, independent of the angle variable, and let $A(\lambda)$
be a $\lambda$-PDO with full symbol $a(\varphi,I,\lambda)$ acting on 
$C^\infty({\bf T}^{n-1},{\bf L})$ such that
\begin{equation}
a\ \sim\ \sum^\infty_{j=0}a_j(\varphi,I)\lambda^{-j},\quad
p^0\ \sim\ \sum^\infty_{j=0}p^0_j(I)\lambda^{-j} \, ,
                                  \label{eq:4.4}
\end{equation}
are uniformly compactly supported with respect to $I$ in $D$ and 
$a_0=p_0^0 = 1$
in a neighborhood of $E_\kappa$. 
Suppose that  $a$ and $p^0$ belong to 
$ S_{\ell} ({\bf T}^{n-1}\times D)$, $\ell =(\sigma,\mu,\varrho)$. 
Then
\begin{equation}
M_1(\lambda)A(\lambda) - A(\lambda)M_0(\lambda) = R^0(\lambda) + R^1(\lambda),
                                  \label{eq:4.5}
\end{equation}
where $R^0$ and $R^1$ are $\lambda$-FIOs of the form 
(\ref{eq:4.3}) with 
amplitudes $r^0,r^1\in S_{\ell}({\bf T}^{n-1} \times D)$. 
 Moreover, $r^1$ vanishes to infinite order for 
$I\in E_\kappa$ and 
\[
r^0\ \sim\ \sum^\infty_{j=0}r_j(\varphi,I)\lambda^{-j}, 
\]
where
\[
r_j(\varphi,I)\ =\ \frac{1}{i}({\cal L}\,
a_{j})(\varphi,I)\, +\, p_j(\varphi,I)\, -\, p^0_j(I)\,
+\, F_j(\varphi,I)\, ,
\]
\[
F_1(\varphi,I)=0,\ 
F_j(\varphi,I)\ =\ F_{j1}(\varphi,I)\, -\, F_{j2}(\varphi,I)\, ,\ j\geq 2\, , 
\]
and 
\[
F_{j1}(\varphi,I)\ =\  \sum_{{\scriptstyle{
{r+s+|\gamma|=j }\atop{1\le s < j}}}} 
\frac{1}{\gamma!}\, \left[ D^\gamma_\xi \left(p_r(\varphi,I + \xi)\, 
\partial^\gamma_\varphi\,
a_s(\varphi - K_0(I,\xi),I)\right)\right]_{|\xi =0}\, , \ 
\]
\[
F_{j2}(\varphi,I)\ =\ \sum^{j-1}_{s=1}a_s(\varphi,I)\, p^0_{j-s}(I)\, . 
\]
\end{lemma}

\noindent
{\em Proof}. We write the  operator $M_1(\lambda)A(\lambda)$ in 
the form (\ref{eq:4.3}) with amplitude given by 
\[
g(x,I,\lambda) = 
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{2n-2}} 
\, e^{i\lambda (\langle x-z,\xi-I\rangle + Q(x,\xi) -Q(x,I))} \, 
p(x,\xi,\lambda)
a(z,I,\lambda)\, d\xi dz \ ,
\] 
which belongs to $C^\infty ({\bf T}^{n-1}\times D)$ for each $\lambda$ fixed. 
Changing the variables we write $g(x,I,\lambda)$ in the form 
\[
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{2n-2}} 
\, e^{-i\lambda \langle v,\eta\rangle } \, p(x,I + \eta,\lambda)
a(v +x - K_0(I,\eta) + H_0(x,I,\eta) ,I,\lambda)\, d\eta dv \, ,
\] 
where $ H_0(x,I,\eta) = \int_0^1 \nabla_I R(x,I +\tau \eta) d \tau$ 
and all its derivatives 
vanish for $\eta=0$, $I\in E_\kappa$. Using the Taylor formula 
for the amplitude at $v=0$ and integrating by parts we get
$g(x,I,\lambda) \sim \sum_{j=1}^\infty g_j(x,I)\lambda^{-j}\, 
\in S_{\ell} ({\bf T}^{n-1}\times D)$, where
\[
g_{j}(x,I)\ =\ 
\sum_{r+s+|\gamma|=j}\,
\frac{1}{\gamma!}\, \left[ D^\gamma_\eta \left(p_r(x,I + \eta)\, 
\partial^\gamma_x\,
a_s(x - K_0(I,\eta) +H_0(x,I,\eta) ,I)\right)\right]_{|\eta =0}\, . 
\]
In the same way we write 
 $A(\lambda)M_0(\lambda)$ in 
the form (\ref{eq:4.3}) with amplitude $h(x,I,\lambda)$ 
given by the oscillatory integral 
\[
\left(\frac{\lambda}{2\pi}\right)^{n-1}\,   
p^0(I,\lambda)\, \int_{{\bf R}^{2n-2}} 
\, e^{i\lambda (\langle x-z,\xi-I\rangle + Q(z,I) -Q(x,I))} \, 
a(x,\xi,\lambda)
 d\xi dz \, .
\] 
Changing the variables we obtain $h = h^0 + h^1$, where 
$h^0(x,I,\lambda) = a(x,I,\lambda) p^0(I,\lambda)$ and
$h^1(x,I,\lambda)$ is given by 
\[
\left(\frac{\lambda}{2\pi}\right)^{n-1}\,  p^0(I,\lambda)\,  
\int_{{\bf R}^{2n-2}} 
\, e^{-i\lambda \langle v,\eta\rangle } 
[a(x, \eta  + I + H_1(x,v,I), \lambda) - a(x, \eta + I, \lambda)]\, 
d\eta dv \ ,
\] 
where $ H_1(x,v,I) = \int_0^1 \nabla_x R(x+\tau v,I) d \tau$
and all its derivatives 
vanish for $I\in E_\kappa$. Then 
$h^1 \in S_{\ell} ({\bf T}^{n-1}\times D)$ and 
$\partial^\beta_Ih^1 =0$ 
for each $I\in E_\kappa$ and any $\beta$. This proves the
lemma. \finishproof

 
We are going to find $a, p^0 \in S_{\ell} ({\bf T}^{n-1}\times D)$, 
$p^0$ independent of $\varphi$ such that $r^0 \equiv 0$ on 
${\bf T}^{n-1}\times E_\kappa$. 
\begin{prop}
There exist Gevrey symbols $a, p^0 \in S_{\ell} ({\bf T}^{n-1}\times
D)$ of the form (\ref{eq:4.4}) such that $r^0\equiv 0$ on   
${\bf T}^{n-1}\times E_\kappa\times {\cal D}$. 
\end{prop}
{\em Proof}. We take  $a_0 = p_0^0 =1$ in 
${\bf T}^{n-1}\times D$. We are going  to find $a_j$ and  $p_j^0$ for
$j\ge 1$.  








We solve the equations $r_j=0,\ j\geq 1$, as in \cite{P2}, Sect. 5.
  First we set 
\begin{equation}
p^0_j(I)\ =\ (2\pi)^{1-n}\int_{{\bf T}^{n-1}}(p_j(\varphi,I)\, +\, F_j
(\varphi,I))\, d\varphi\, ,
                                         \label{eq:4.6}
\end{equation}
then, using Proposition 6.1, we find $a_{j}$ such that
\begin{equation}
\left\{
\begin{array}{rcll} 
\frac{1}{i}{\cal L} a_{j}(\varphi,I)\ &=&\ f_j(\varphi,I)\, , \
(\varphi,I)\in {\bf T}^{n-1}\times E_\kappa\, ,
\\
\int_{{\bf T}^{n-1}}a_{j}(\varphi,I)d\varphi\ &=&\ 0\, ,\ I\in D\, ,
                                         \label{eq:4.7}
\end{array}
\right.
\end{equation}
where $f_j(\varphi,I)=p^0_j(I)-p_j(\varphi,I)-F_j(\varphi,I)$.
For $j=1$ we obtain
$$
p^0_1(I)\ =\ (2\pi)^{1-n}\int_{{\bf T}^{n-1}}p_1(\varphi,I)\, d\varphi,
$$
and
\[
\frac{1}{i}{\cal L}a_1(\varphi,I)\ =\ 
p^0_1(I)\, -\, p_1(\varphi,I)\, ,\quad
\int_{{\bf T}^{n-1}}a_1(\varphi,I)\, d\varphi\ =\ 0.
\]
Denote by ${\bf N}_+ = {\bf N}\backslash \{0\}$ the set of all positive
integers.  
Since $p \in S_{\widetilde\ell}({\bf T}^{n-1}\times D)$, the functions 
$p_j$, $j\in {\bf N}_+$, satisfy the 
estimates
\[
|D^\alpha_I D^\beta_\varphi
p_j(\varphi,I)| 
\leq
\widetilde C_0^{|\alpha|+|\beta|+j} \alpha!^\mu \beta!^\sigma 
(j !)^{\sigma+\mu-1} 
\]
\begin{equation}
\leq 
C_0^{|\alpha| +|\beta|+j} \alpha ! \beta !
\Gamma((\mu-1)|\alpha|+(\sigma-1)|\beta| + (\sigma+\mu-1) (j-1) +1) ,
                                       \label{eq:4.8}
\end{equation}
for $(\varphi,I)\in {\bf T}^{n-1}\times D$ 
and  each $\alpha,\,  
\beta\in{\bf N}^{n-1}$, where
$\Gamma(s)$ 
is the Euler $\Gamma$ function, 
and $\widetilde C_0$ and $C_0$ are suitable
positive constants. 
In particular, using Proposition 6.1  
we find $a_1$ satisfying 
\[
|D^\alpha_I D^\beta_\varphi a_1(\varphi,I)|\ \leq\
2RC_0\, \alpha !\, C^{ |\alpha| + |\beta|}\, 
\Gamma((\mu-1) |\alpha|+\sigma|\beta|+\varrho)
\, ,
\]
choosing $C> C_0$. 
Fix $j\geq 2$ and suppose that there exist  
$p^0_k(I)$ and $a_k(\varphi,I)$, $1\leq k\leq j-1$,
satisfying (\ref{eq:4.6}) and (\ref{eq:4.7}), and such that
\begin{equation}
\begin{array}{rcll}  
|D^\alpha_I   p^0_k(I)|\ &\leq& \ d^{k-1/2} C^{|\alpha|}\, \alpha !\, 
\Gamma((\mu-1) |\alpha| + (k-1)\varrho +2\sigma)\, , \\
|D^\alpha_I  D^\beta_\varphi a_k(\varphi,I)|\ &\leq& \ d^k
C^{|\alpha| + |\beta|}\, \alpha ! \, 
\Gamma( (\mu -1) |\alpha| +\sigma|\beta|+ k \varrho)\, ,
                                       \label{eq:4.9}
\end{array} 
\end{equation}
for any $(\varphi,I)\in {\bf T}^{n-1}\times D$ and 
$\alpha,\beta\in{\bf N}^{n-1}$, 
where $d\geq2RC_0$. ,
We shall prove that $p^0_j$ and $a_{j}$
satisfy the same estimates for $k=j$ with $d\gg 1$ independent of $j$. 

First we estimate the derivatives of 
\[
b_{k,\gamma}(\varphi,I) = \frac{1}{\gamma!}\, \left[ \partial^\gamma_\xi 
a_k(\varphi -  K_0(I,\xi),I)\right]_{|\xi =0}\, ,\ I\in D\, . 
\]
For that we fix  $C_1>0$ such that for each $\alpha, \gamma 
\in {\bf N}^{n-1}$, $|\alpha|+|\gamma|\neq
0$, we have
\begin{equation}
|D^\alpha_ID^\gamma_\xi K_0(I,\xi)|\ \le \ 
C_1^{|\alpha +\gamma|}\, \alpha! \gamma!\,  
((|\alpha +\gamma| -1) !)\, ^{\mu-1}\, ,\ 
(I,\xi)\in D\times D\, .
                                     \label{eq:4.10}
\end{equation}
\begin{lemma}
Let $C>4C_1$. Then there is a constant $C_2>0$ independent of $j$ and
$d$, such that for each $\alpha,\beta,\gamma \in {\bf N}^{n-1}$ and
$1 \le k \le j-1$ we have 
\begin{equation}
|D^\alpha_I D^\beta_\varphi b_{k,\gamma} (\varphi,I)|\ \le \
C_2\, d^k \, 
C^{|\alpha| + |\beta| +|\gamma| }\,  \alpha  !\, 
\Gamma( (\mu-1) (|\alpha|+|\gamma|) +\sigma|\beta|+ 
k\varrho +\sigma)\, .  
                                     \label{eq:4.11}
\end{equation}
Moreover, if $\gamma \neq 0$ we have 
\begin{equation}
|D^\alpha_I D^\beta_\varphi b_{k,\gamma} (\varphi,I)|\ \le \
C_2\, d^k \,  
C^{|\alpha| + |\beta| + |\gamma|}\,  \alpha  !\, 
\Gamma( (\mu-1) (|\alpha|+|\gamma|-1) +\sigma|\beta| + 
k\varrho +\sigma)\, .  
                                     \label{eq:4.12}
\end{equation}
\end{lemma}

\noindent {\em Proof.} \hspace{2mm}
We write 
\begin{equation}
D^\alpha_I D^\beta_\varphi b_{k,\gamma} (\varphi,I)\ =\ \frac{1}{\gamma !}\, 
\sum_{\alpha^\prime \le \alpha}\,  
\pmatrix{\alpha\cr\alpha^\prime\cr}\,
\left(D^{\alpha^\prime}_\eta  \partial^\gamma_\xi
D^{\alpha-\alpha^\prime}_I D^\beta_\varphi 
 a_k(\varphi - K_0(\eta,\xi),I)\right)_{|\eta=I,\, \xi=0}\,. 
                                     \label{eq:4.13}
\end{equation}
We are going to  estimate each term  $A_{\alpha^\prime}$ of
(\ref{eq:4.13}). First, assuming  that $\gamma \neq 0$, we shall prove
(\ref{eq:4.12}). 
 Set $z=(\eta,\xi)$ and $\delta = (\alpha^\prime,\gamma)$.
Then  $\delta \neq 0$ and  
using Faa de Bruno formula  (see \cite{L-Y},
Corollary 5.5) we get  
\[
\begin{array}{rcll}
|A_{\alpha^\prime}| \ &\le&  \ \displaystyle \frac{1}{\gamma !}\, 
 \sum_{\cal B(\alpha')}\ 
\pmatrix{\alpha\cr\alpha^\prime\cr}\, 
\frac{ \alpha' !\gamma ! }{\ell !\, \delta^1! \cdots \delta^p!}\, 
\left| \left(D^{\ell +\beta}_\varphi D^{\alpha -\alpha'}_I a_k\right)
(\varphi -  \nabla K(I), I)\right|\\  [0.7cm]
&\times& \ \left|\left(D^{\delta^1}_z K_0\right)(I,0)\right|\ldots  
\left|\left(D^{\delta^p}_z K_0\right)(I,0)\right| \, , 
\end{array}
\]
where the index set ${\cal B}(\alpha')$ 
consists of   all $\ell \in {\bf N}^{n-1}$, 
$\delta^1,\ldots,\delta^p\in {\bf N}^{2n-2}$, $p\le |\delta|$, 
such that
\[ 
 1 \le |\ell| =  p  \le   |\delta|= |\alpha'|+|\gamma|\, ,\
\delta^1+\cdots+\delta^p  =   \delta\, ,  \ 
|\delta^1|\ge 1,\, \ldots,\, |\delta^p| \ge \,  1\, .
\]
Using 
(\ref{eq:4.9}) and (\ref{eq:4.10}), we obtain  
\begin{equation}
\begin{array}{rcll}
|A_{\alpha^\prime}| \ &\le& \ \alpha ! \, d^k \, 
\displaystyle \sum_{\cal B(\alpha')}\ 
C^{|\alpha| -|\alpha^\prime|+p +|\beta| }\, C_1^{|\alpha'|+|\gamma|}\\ [0.5cm]
&\times& \ \Gamma\left((\mu-1)|\alpha -\alpha^\prime|+\sigma(p+|\beta|) +
k\rho\right)
\left((|\delta^1|-1) ! \cdots (|\delta^p| -1) !\right)^{\mu-1} \, .
\end{array}
                                     \label{eq:4.14}
\end{equation}
Since $|\delta^j| \ge  1$, $j=1,\ldots,p$,  and $\mu >2$,  
we get by Stirling's formula
\[
\left((|\delta^1|-1) ! \cdots (|\delta^p| -1) !\right)^{\mu-1} \ \le \
(|\delta|-p) ! \, ^{\mu-1} \ \le \ A\, 
 \Gamma((\mu-1)(|\delta|-p) +
1)\, ,
\]
where $A>0$ depends only on  $\mu$. 
Then using the inequality 
\begin{equation}
\Gamma(x)\Gamma(y) \le 2\Gamma(x+y-1), \ x,y\ge 1\, ,
                                       \label{eq:gamma}
\end{equation}
 we estimate   the second  line in  (\ref{eq:4.14}) by
\[
2A\, 
\Gamma\left((\mu-1)|\alpha -\alpha^\prime|+\sigma(p +|\beta|) +
(\mu-1)((|\delta|-p) + k\rho\right)
\]
\[
\le \ 4 A\, 
\frac{\Gamma\left((\mu-1)|\alpha|+\sigma|\beta| +(\mu-1)(|\gamma|-1)
+ k\rho + \sigma\right)} 
{\Gamma((\mu-\sigma -1)(p-1) +1)}, 
\]
where $\mu-\sigma -1 > \sigma \tau' >0$. 
Observe that for any $m\in {\bf N}$, the number of multi-indices 
$\alpha =(\alpha_1,\ldots, \alpha_p)$ with $|\alpha|=m$ is given by 
$\pmatrix{m + p -1\cr p-1\cr}$ (see \cite{Ro}, Sect. 1.2). 
Then  the
number of partitions $(\delta^1,\cdots, \delta^p)$ of 
$\delta=(\delta_1,\ldots, \delta_{2n-2})$ in the
index set ${\cal B(\alpha')}$  is dominated by
\[
\pmatrix{\delta_1 + p -1\cr p-1\cr}\cdots \pmatrix{\delta_{2n-2} + p -1\cr
p-1\cr}\ \le \ 2^{|\alpha'| +|\gamma| } 2^{2n(p-1)}\, .
\]
  Summing up with respect to $p\ge 1$ 
in (\ref{eq:4.14}) and  using that $C>4C_1$  we obtain
\[
|A_{\alpha^\prime}|\ \le \  C_3
d^k\,  C^{|\alpha|+|\beta| +|\gamma|}\, 
2^{-|\alpha'|}\,  \alpha !\, 
\Gamma\left((\mu-1)|\alpha|+\sigma|\beta| +(\mu-1)(|\gamma|-1) +  k\rho
+ \sigma\right)\,  ,
\]
where $C_3>0$ depends only on $C,\ C_1$, $\mu$, $\sigma$ and $n$. 
Summing up with respect to $\alpha^\prime$ we prove (\ref{eq:4.12}).  

It remains to prove (\ref{eq:4.11}). 
If $\gamma=\alpha'=0$,  we just use 
(\ref{eq:4.9}). 
If $\gamma=0$ but $\alpha^\prime \neq 0$ we use the
argument above. This completes the proof of Lemma 4.2. 
\finishproof

We are able now  to estimate the derivatives of $F_j$.
\begin{lemma}
Let $d>C>4C_0>1$, where   $C_0$ is 
the constant in  (\ref{eq:4.8}). Then there is $R_1>0$ 
independent of $d$ and $j$ such that 
for any  $\alpha,\beta\in
{\bf N}^{n-1}$  we  have 
$$
|D^\alpha_I D^\beta_\varphi F_{j1}(\varphi,I)|\ \leq\
R_1d^{j-1}C^{ \alpha +|\beta|} \, \alpha !\, 
\Gamma((\mu-1) \alpha+\sigma|\beta|+ (j-1)\varrho + 2\sigma)
$$
for $(\varphi,I)\in {\bf T}^{n-1}\times D$.   
\end{lemma}

\noindent {\em Proof.} \hspace{2mm} Consider
\[
B_{r,s,\gamma}(\varphi,I) = 
\frac{1}{\gamma!}\, \left[ D^\gamma_\xi (p_r(\varphi,I + \xi)\, 
\partial^\gamma_\varphi\,
a_s(\varphi - K_0(I,\xi),I))\right]_{|\xi =0}\, .
\]
where
\begin{equation}
2\leq r+s+|\gamma|=j\ ,\quad 1\leq s\leq j-1\, . 
                                        \label{eq:4.15}
\end{equation}
First we suppose that $r=0$. Since $s\le j-1$ we have 
$\gamma\neq 0$. Then using (\ref{eq:4.12})
we get
\[
|D^\alpha _ID^\beta_\varphi
B_{0,s,\gamma}(\varphi,I)|
\]
\begin{equation}
\begin{array}{rcll}
&\leq&  \displaystyle
C_2\, d^s\, 
C^{|\alpha| + |\beta| + |\gamma|}\, \alpha !\, 
\Gamma( (\mu-1) (|\alpha|+|\gamma|-1) +\sigma(|\beta|+|\gamma|)
+ s\varrho + \sigma)\\
&\le& C_2\, d^{j-1}\,  
C^{|\alpha| + |\beta|}\,  \alpha !\,  
\Gamma( (\mu-1) |\alpha|+\sigma|\beta|+(\mu +\sigma -1)(|\gamma|-1) 
+ s\varrho + 2 \sigma)\, .
\end{array}
                                        \label{eq:4.16a}
\end{equation}
 Suppose now that $r\ge 1$.
We have
\[
|D^\alpha _I D^\beta_\varphi 
B_{r,s,\gamma}(\varphi,I)|\\ 
\le \   
\displaystyle\sum_{\alpha^\prime \leq\alpha}^{} 
\sum_{\beta^\prime \leq\beta}^{}\,\sum_{\gamma^\prime\leq\gamma}^{}\,
\frac{1}{\gamma!}\pmatrix{\alpha\cr\alpha^\prime\cr}
\pmatrix{\beta\cr\beta^\prime\cr}
\pmatrix{\gamma\cr\gamma^\prime\cr}\, (\gamma - \gamma') !\, 
\]
\begin{equation}
\times\  |D^{\alpha^\prime+\gamma^\prime}_I
D^{\beta^\prime}_\varphi
p_r(\varphi,I)|\, |D^{\alpha - \alpha^\prime}_I
\partial^{\gamma+\beta-\beta^\prime}_\varphi
b_{s,\gamma-\gamma^\prime}\, 
(\varphi,I)| \, .  
                                        \label{eq:4.16}
\end{equation} 
Taking into account (\ref{eq:4.8}), (\ref{eq:4.9}) and (\ref{eq:4.11}), we
estimate each term $\Sigma$ of   (\ref{eq:4.16})   by
\[
\begin{array}{rcll}
|\Sigma|\ &\le& \ 
  \displaystyle \frac{d^s }{\gamma!}\, \displaystyle
\pmatrix{\alpha\cr \alpha^\prime\cr}\pmatrix{\beta\cr\beta^\prime\cr}
\pmatrix{\gamma\cr\gamma^\prime\cr}\, 
(\alpha' +\gamma^\prime) !\, \beta' !\, (\alpha-\alpha^\prime)! 
\, (\gamma-\gamma^\prime)!\\   [0.5cm] 
&\times& \Gamma((\mu -1)|\gamma^\prime+\alpha^\prime| + 
(\sigma-1)|\beta^\prime| + 
(\sigma+\mu-1)(r-1) +1) \\   [0.5cm]
&\times&\ 
\Gamma((\mu-1)|\alpha- \alpha^\prime|+\sigma|\gamma +\beta-\beta^\prime| + 
(\mu-1)|\gamma -\gamma^\prime| + s\varrho + \sigma)\,\\   [0.5cm]
&\times&\ 
C_2\,  C_0^{r}\,
C^{|\alpha| +|\beta| + |\gamma|}\,
\left(\frac{C_0}{C}\right)^{|\alpha^\prime|+|\gamma^\prime|+|\beta^\prime|}
\, .
\end{array}
\]
We have 
\[ 
\frac{1}{\gamma !}\, 
\pmatrix{\alpha\cr \alpha^\prime\cr}\pmatrix{\beta\cr\beta^\prime\cr}
\pmatrix{\gamma\cr\gamma^\prime\cr}\, 
(\gamma^\prime + \alpha^\prime)!\, \beta'!\,
(\alpha-\alpha^\prime)!\, 
(\gamma-\gamma^\prime)!
\]
\[
\le \ \alpha !\, 2^{|\alpha^\prime|+|\gamma^\prime|}\, 
|\beta| \cdots (|\beta -\beta^\prime| +1)\, . 
\]
Multiplying it by 
\[
\Gamma(|\beta-\beta^\prime| 
+ (\mu-1)|\alpha-\alpha^\prime| +\sigma|\gamma |+
(\sigma-1)|\beta-\beta^\prime| + 
(\mu-1)|\gamma -\gamma^\prime| + s \varrho +\sigma) ,
\] 
and using that 
$x\Gamma(x) = \Gamma (x+1)$, $x>0$,  we estimate the product by
\[
\alpha!\, 2^{|\alpha^\prime|+|\gamma^\prime|}\, 
\Gamma ((\mu-1)|\alpha-\alpha^\prime|+\sigma|\gamma|+\sigma|\beta| -
(\sigma-1)|\beta^\prime| + 
(\mu-1)|\gamma -\gamma^\prime| +  s \varrho +\sigma) \, .  
\]
Then multiplying it  by 
\[
\Gamma((\mu-1)(|\alpha^\prime | +|\gamma^\prime|)+(\sigma-1)|\beta^\prime| + 
(\sigma + \mu  -1)(r-1) +1)\] 
and taking into account (\ref{eq:gamma}), 
we estimate the product by
\[
\alpha!\, 
2^{|\alpha^\prime| +|\gamma^\prime|}\, 
\Gamma ((\mu-1)|\alpha|+ \sigma|\beta| + 
(\sigma + \mu  -1)(|\gamma| +r - 1) +  s \varrho +\sigma) \, .  
\]
Since  $d>C>4C_0>1$, we estimate $|\Sigma|$ by
\[
d^{j-1} \, C_2^\prime  \,
C^{ |\alpha|+|\beta|}\,
\alpha!\, 
\Gamma ((\mu-1)|\alpha|+ \sigma|\beta| + 
(\sigma + \mu  -1)(|\gamma| +r-1) + s \varrho) \, 
2^{-|\alpha^\prime +\beta^\prime+\gamma^\prime|}\, . 
\]

Summing up with respect to
$m=(\alpha^\prime,\beta^\prime,\gamma^\prime)$ we estimate
\[
|D^{\alpha}_I \partial^\beta_\varphi B_{r,s,\gamma}(\varphi,I)|
\]
\begin{equation} 
\le \   d^{j-1} \, \alpha!\, 
C_3\,    C^{ |\alpha|+|\beta|}\, 
\Gamma ((\mu-1)|\alpha|+ \sigma|\beta| + 
(\sigma + \mu  -1)(|\gamma| +r-1) + s \varrho +\sigma) \, , 
                                        \label{eq:4.16b}
\end{equation}
where $C_3 = C_2' \, \sum_{m\in {\bf N}^{3n-3}}\ 2^{-m}$.
On the other hand, 
\[
1 < s\varrho < (\sigma +\mu-1)(|\gamma|+r-1)+s\varrho = 
(j-1)\varrho-\delta(|\gamma|+r-1),  
\]
where $\delta = \varrho -\mu -\sigma +1 >  \mu(\sigma-1)+1 > 0$, since 
$\varrho = \sigma(\nu+1)$ and $\nu > \mu $. Using  (\ref{eq:gamma}),  we get  
 for $|\gamma|+r \ge 1$, 
\[
\Gamma((\mu-1) |\alpha|+\sigma|\beta|+
(\sigma + \mu -1)(|\gamma|+r-1) +s\varrho +2\sigma) 
\]
\[
\leq 
\displaystyle\frac{\Gamma((\mu-1)|\alpha|+\sigma |\beta|+(j-1)\varrho+2\sigma)}
{\Gamma(\delta(|\gamma|+r-1) + 1)}\, .
\]  
Applying it to (\ref{eq:4.16a}) and  
(\ref{eq:4.16b}) we obtain
$$
|D^{\alpha}_I \partial^\beta_\varphi B_{r,s,\gamma}(\varphi,I)|
\leq
d^{j-1}\, C_4\, C^{ |\alpha| +|\beta|}\,  \alpha!\, \frac{
\Gamma((\mu-1) |\alpha| +\sigma|\beta|+ (j-1)\nu +2\sigma)}
{\Gamma(\delta(|\gamma|+r-1) + 1) }\, .
$$
where   $C_4$ does not depend on  $d$. 
Summing up with respect to $\gamma$ and $r$ we get
\[
|D^{\alpha}_I  D^\beta_\varphi F_{j1}(\varphi,I)| \leq
R_1\, d^{j-1}\, C^{|\alpha|+|\beta|}\, \alpha!\, 
\Gamma((\mu-1)|\alpha|+\sigma|\beta|+(j-1)\varrho +2\sigma)\, ,
\]
where
$R_1 = C_4\, \sum_{|\gamma|+r\ge 1}
 \Gamma(\delta(|\gamma|+r-1) + 1)^{-1} \, .$
We have proved the lemma. 
\finishproof



We are ready to estimate $p^0_j(I)$. In view of (\ref{eq:4.7}) we have
$$
p^0_j(I)\ =\ (2\pi)^{1-n}\int_{{\bf T}^{n-1}}
(p_j(\varphi,I)+F_{j1}(\varphi,I))
d\varphi\, .
$$
Then  (\ref{eq:4.8})  and Lemma 4.3 imply
$$
\begin{array}{rcl}
|D^\alpha_I p^0_j(I)|&\leq& R_1\, d^{j-1}\, C^{|\alpha|}\, \alpha
!\,  \Gamma((\mu-1)|\alpha|+ (j-1)\varrho +2\sigma)\\ [0.3cm]
&+&
C_0^{|\alpha|+j+1}\, \alpha !\, 
\Gamma((\mu-1)|\alpha| + (\sigma + \mu -1)(j-1) +1) \\ [0.3cm]
&\leq&
d^{j-1/2}C^{|\alpha|}\, \alpha !\, \Gamma((\mu-1) |\alpha| 
+ (j-1)\varrho + 2\sigma) \, ,
\end{array}
$$
since $\varrho = \sigma\nu + \sigma > 
\sigma + \mu - 1$. 
Here we choose $d$ sufficiently large as a function of $R_1$ and $C_0$. 
This proves the first part of (\ref{eq:4.9}) for $k=j$.

It remains to estimate
$F_{j2}(\varphi,I)$ and $a_{j}(\varphi,I)$. 
As in \cite{P2}, Lemma V.2, we obtain:

\begin{lemma}
There is $R_2>0$ independent of $d$ and $j$ such that 
for any  $\alpha,\beta\in {\bf
Z}_+^{n-1}$  we have
$$
|D^\alpha_I D^\beta_\varphi F_{j2}(\varphi,I)|\ \leq\
R_2d^{j-{1\over 2}} C^{|\alpha|+|\beta|}\,  \alpha!\, 
\Gamma((\mu-1)|\alpha|+\sigma|\beta|+ (j-1)\varrho + 2\sigma)\, 
$$
for $(\varphi,I)\in {\bf T}^{n-1}\times D$. 
\end{lemma}


\noindent {\em Proof.} \hspace{2mm} 
Using  (\ref{eq:4.9}) for $k\le j-1$,  we obtain
$$
\begin{array}{c}
|D^\alpha_I  D^\beta_\varphi(a_s(\varphi,I)p^0_{j-s}(I))|
\leq\
d^{j-\frac{1}{2}}C^{|\alpha|+|\beta|}
\displaystyle\sum_{\gamma\leq\alpha}
\pmatrix{\alpha\cr\gamma\cr}\, \gamma !\, \Gamma((\mu-1)|\gamma| 
+\sigma|\beta|+s\varrho)\\ [0.5cm]
\times
\, (\alpha -\gamma) !\, \Gamma((\mu-1)|\alpha-\gamma|+(j-s-1)\varrho +2\sigma).
\end{array}
$$
Recall that $1\le s\le j-1$ and $\mu -1> \tau' + 1 > 7/2$.
Using Lemma 4.5 below we complete the proof of the lemma as in
\cite{P2}, Lemma V.2. \finishproof

As in \cite{P2}, Lemma A.3 we prove
\begin{lemma}
Let $\mu>9/2$. 
Then there exists a positive constant $M$ such that
for any $x,y\in {\bf N}$ and $p\geq1,\ q\geq1$, we have
$$
\Gamma((\mu-1) x+p)\, \Gamma((\mu-1) y+q)\ \leq\
M\, \Gamma((\mu-1)(x+y)+p+q)\, B(p,q)^{1/3}\, \pmatrix{x+y\cr x\cr}^{-5/6}\,.
$$
\end{lemma}



Finally, combining Lemma 4.3 and Lemma 4.4  we estimate the 
right hand side of the first equation in (\ref{eq:4.7}) as follows:
$$
|D^\alpha_I D^\beta_\varphi f_{j}(\varphi,I)|\ \leq\
M_3\, d^{j-\frac{1}{2}}\, C^{|\alpha|+|\beta|}\,\, \alpha !\, 
\Gamma((\mu-1)|\alpha|+\sigma|\beta|+(j-1)\varrho+2\sigma)\, ,\ 
\forall \alpha,\beta
\in {\bf N}^{n-1}\, ,
$$
where $M_3$ does not depend on  $j$ and  $d$. 
Now applying Proposition 6.1 we find $a_{j}$ satisfying 
 (\ref{eq:4.7}) and   the second inequality in 
(\ref{eq:4.9}) on ${\bf T}^{n-1}\times D$ for $k=j$, 
choosing $d\gg 1$. 
 This proves Proposition 4.1. \finishproof

Using Lemma 4.1 we
obtain symbols $a$ and $p^0$ in 
$ S_{\ell} ({\bf T}^{n-1}\times D)$, $\ell =(\sigma,\mu,\varrho)$,
such that  
\begin{equation}
M_1(\lambda)A(\lambda) - A(\lambda)M_0(\lambda) = R_1(\lambda) +  R'(\lambda)
\, ,
                                  \label{eq:4.17}
\end{equation}
where $R_1(\lambda)$ and $R'(\lambda)$ are  
$\lambda$-FIOs, $\lambda\in {\cal D}$, of the form 
(\ref{eq:4.3}). The  
amplitude $r \in S_{\ell} ({\bf T}^{n-1}\times D)$ of $R_1(\lambda)$ 
vanishes to infinite order for $I\in E_\kappa$, and the   amplitude
$r'$ of $R'(\lambda)$ is in the residue class 
$S_{\ell}^{-\infty} ({\bf T}^{n-1}\times D)$ (see Appendix) and we have
\begin{equation}
R'(\lambda)= O_{s,s'}\left(e^{-c|\lambda|^{1/\rho}}\right)\, :\
H^s({\bf T}^{n-1}; {\bf L})\ \longrightarrow \ 
H^{s'}({\bf T}^{n-1}; {\bf L})\ ,\ s,s'
\in {\bf R}\, ,
                                  \label{eq:4.18}
\end{equation}
where $H^s$ is the corresponding Sobolev space. Recall that the
distribution kernel of $\widetilde{M_0(\lambda)}$ is 
\[
\widetilde{M_0}(x,y,I,\lambda) = 
\left(\frac{\lambda}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{2n-2}} 
\, e^{i\lambda (\langle x-y,I\rangle + Q(x,I))} \, p^0(I,\lambda)\,
dI\, . 
\]
One can consider 
$e^{i\pi (m+\vartheta/4)}\widetilde{M_0(\lambda)}$ as a ``quantum Birkhoff 
normal form'' of the ``quantum Poincaré map'' 
$\overline{M_0}(\lambda)$ and the function  $p^0(I,\lambda), I\in
E_\kappa$, 
as its Birkhoff invariants. 


















 
 \section{Construction  of quasimodes. Orthogonality relations.}
\setcounter{equation}{0}
\subsection{Construction of quasimodes} 


Consider the $\lambda$-FIO $R_1(\lambda)$ having the  form (\ref{eq:4.3})
with phase function $Q$ and 
amplitude $r\in S_{\ell}$,  which is uniformly compactly supported  with
respect to $I$ in $D$. 
Given $C_1>0$ we extend $Q$ and $r$ for complex $I$ as in (\ref{eq:A4}). 
Changing the contour of integration and using (\ref{eq:A7}) we write
the distribution kernel of $R_1(\lambda)$ as follows
\[
\left(\frac{1}{2\pi}\right)^{n-1}\, \int_{{\bf R}^{n-1}} 
\, e^{i(\langle x-y,I\rangle + \lambda Q(x,I/\lambda))} \, 
r(x,I/\lambda,\lambda)\, dI  \, + \, 
O\left( e^{-c|\lambda|^{1/(\mu -1)}}\right)\, 
,\ \lambda \in {\cal D}\, .
\]
This implies 
\begin{equation}
\widetilde{R_1(\lambda)e_p}(x)\ =\  
e^{i\lambda Q(x,(p + \vartheta_0/4)/\lambda)} 
r(x, (p + \vartheta_0/4)/\lambda,\lambda) \widetilde e_p(x)\,  + \, 
O\left( e^{-c|\lambda|^{1/(\mu -1)}}\right)\widetilde e_p(x)\, ,
                                  \label{eq:5.1}
\end{equation}
for $\lambda \in {\cal D}$, where for each $s,s'\in {\bf R}$ 
the reminder term  is $O( e^{-c|\lambda|^{1/(\mu -1)}}): 
H^s \rightarrow H^{s'}$. 


The index set ${\cal M}$ and the frequencies $I_\nu$ of the
quasimode ${\cal Q}$ we are going to construct are determined as
follows: The pair
$\nu=(p,q)\in{\bf N}^{n-1}\times{\bf N}$ belongs to $\cal M$ if there
exists  $I_\nu\in E_\kappa$ and $\lambda^0_\nu>0$ such that the following
quantization conditions hold:
\begin{equation}
\lambda^0_\nu I_\nu\ =\ p+\vartheta_0/4+O(1)\ ,\quad 
- \lambda^0_\nu K(I_\nu) + \pi(m + \vartheta/4)\ =\  2\pi q +O(1)\,  ,
                                                     \label{eq:5.2}
\end{equation} 
as $|p|+|q|\to\infty$. We set 
\[
\lambda^0_\nu\ =\ -   \frac{2\pi q -\pi(m + \vartheta/4)}{K(I_\nu)}\, . 
\]
Recall that $K|_{E_\kappa} < 0$ is given by 
(\ref{eq:3.7}) and
$m$ is the number of  vertices of the broken geodesic $\gamma$.  
We set $\lambda^0 = \lambda^0_\nu$ in (\ref{eq:A4}) 
and suppose that $\lambda$ satisfies 
$|\lambda - \lambda^0_\nu| < C_2$, where $C_2\gg 1$ is fixed. Then 
(\ref{eq:5.2}) holds if $\lambda^0_\nu$ is replaced by $\lambda$, and 
using   (\ref{eq:5.1}) and Remark A.1, we obtain
$R_1(\lambda)e_p\ =\ 
O\left( e^{-c|\lambda|^{1/(\mu -1)}}\right) \, e_p$, 
where $c>0$. 
In the same way, using (\ref{eq:A4}) and Remark A.1 for $p^0$ and 
$Q(x, I/\lambda) + K(I/\lambda)$, we get
\[
\widetilde{M_0(\lambda)e_p}(x)\ =\ Z_\nu(\lambda)\,  e_p\, + \, 
O\left( e^{-c|\lambda|^{1/(\mu -1)}}\right)\widetilde e_p(x)\, ,
\]
where
$Z_\nu(\lambda)\ =\ 
e^{ -i\lambda K((p + \vartheta_0/4)/\lambda)} 
p^0((p + \vartheta_0/4)/\lambda,\lambda, \lambda^0_\nu)$. 

More precisely,   $p^0(I,\lambda)= 1 + p_1^0(I)\lambda^{-1} + \cdots$ 
is given by  (\ref{eq:A3}) with $\lambda^0=\lambda^0_\nu$,  and 
$K(z, \lambda^0_\nu)$ and $p^0(z,\lambda, \lambda^0_\nu)$, $z=I+iY$,  
are defined by (\ref{eq:A4}).  
We write 
\[
Z_\nu(\lambda)\ =\ 
\exp\, (-i\lambda K((p + \vartheta_0/4)/\lambda, \lambda^0_\nu)
+  {\rm Log}\, p^0((p + \vartheta_0/4)/\lambda,\lambda, \lambda^0_\nu))\, . 
\]
We are going to solve the equation 
$e^{i\pi (m+\vartheta/4)}Z_\nu(\lambda) = 1$
modulo $O\left( e^{-c|\lambda^0_\nu|^{1/(\mu -1)}}\right)$. 
For that 
we are looking for a small perturbation
$\lambda=\lambda_\nu$ of $\lambda^0_\nu$ such that
\begin{equation}
\begin{array}{rcll}
-\lambda K((p + \vartheta_0/4)/\lambda, \lambda^0_\nu) + \pi
(m+\vartheta/4)
&+& \, {\rm Log}\, p^0((p + \vartheta_0/4)/\lambda,\lambda,
\lambda^0_\nu) \\ 
&=& 
 2\pi q + O\left( e^{-c|\lambda_\nu|^{1/(\mu -1)}}\right).
\end{array}
                                                     \label{eq:5.3}
\end{equation}
Suppose for a moment that $\lambda_\nu$ exist. Then  
for each  $\nu =(p,q)\in {\cal M}$ we set
\begin{equation}
v_\nu^0\ :=\ T(\lambda_\nu)A(\lambda_\nu)e_p \quad {\rm and} \quad 
u_\nu^0\ :=\  G(\lambda_\nu)v_\nu^0\ =\ 
G(\lambda_\nu)T(\lambda_\nu)A(\lambda_\nu)e_p\, .
                                                     \label{eq:5.3a}
\end{equation} 
In view of  (\ref{eq:5.3}) 
\begin{equation}
(\bar M(\lambda_\nu) -\,  {\rm Id}\, )v_\nu^0 = 
O\left( e^{-c|\lambda_\nu|^{1/\varrho}}\right)\, v_\nu^0\, ,
                                                     \label{eq:5.3b}
\end{equation} 
and we get
\[
\left|
\begin{array}{rcll}  
\left(\, \Delta \, +\, \lambda_\nu^2 \, \right)\, u_\nu^0(x)\ &=&  
O_s\left( e^{-c|\lambda_\nu|^{1/\varrho}}\right)\,u_\nu^0\, ,\
{\rm in}\  X\, ,\\
u_\nu^0 |_\Gamma \ &=&\ 
O_s\left( e^{-c|\lambda_\nu|^{1/\varrho}}\right)\, u_\nu^0
\end{array} 
\right.
\]
Then modifying $u_\nu^0$ by an exponentially small function,  if
necessary, 
we    obtain
\begin{equation} 
\left|
\begin{array}{rcll}  
\left\|\left(\, \Delta \, +\, \lambda_\nu^2 \, \right)\, 
u_\nu^0(x)\right\|_{H^s}\ &=& \
 O_s\left( e^{-c|\lambda_\nu|^{1/\varrho}}\right)\, \|u_\nu^0\|_{L^2(X)}\, ,\\
u_\nu^0 |_\Gamma \ &=&\ 0\, . 
\end{array} 
\right.
                                       \label{eq:5.4}
\end{equation}
We are going to solve (\ref{eq:5.3}). Introduce a small parameter 
$\varepsilon_\nu = \left( \lambda^0_\nu\right)^{-1}$ and put
\[
\left\{
\begin{array}{rcll} 
\lambda_\nu\ &=& \ \lambda_\nu^0(1  + \varepsilon_\nu v_\nu)\, ,\\
p + \vartheta_0/4\ &=& \ \lambda_\nu (I_\nu + \varepsilon_\nu w_\nu)\, ,
\end{array} 
\right.
\]
where $|v_\nu|,\, \|w_\nu\| = O(1)$. Substituting it in  (\ref{eq:5.3})
and using  (\ref{eq:5.2}) we obtain 
\begin{equation} 
\left\{
\begin{array}{rcll}
w_\nu + v_\nu I_\nu  + \varepsilon v_\nu w_\nu \ &=&\
r_\nu^1 \\
K( I_\nu)v_\nu +  \langle \nabla K(I_\nu),w_\nu\rangle + \varepsilon 
F_\nu(v_\nu,w_\nu)  \ &=&\ r_\nu^2

\end{array} 
\right.
                                       \label{eq:5.5}
\end{equation}
for $\varepsilon =\varepsilon_\nu$,   where $r_\nu^j \in {\bf C}$, 
$r_\nu^j = O(1)$ and ${\rm Im}\, r_\nu^j = O(\varepsilon_\nu)$,
$j=1,2$. Here $F_\nu:{\bf C}^{n-1}\times {\bf C}^{n-1} \rightarrow
{\bf C}$ are uniformly compactly supported with respect to $\nu$, 
\[
\overline\partial_v F_\nu(v,w)\ ,\ 
\overline\partial_w F_\nu(v,w)\ =\  O\left( e^{-c|\lambda_\nu|^{1/(\mu
-1)}}\right)\ {\rm for}\ |{\rm Im}\, v|\, ,\  |{\rm Im}\, w| =
O(1)\, , 
\]
uniformly with respect to
$\nu$. Moreover, for $|\alpha| + |\beta| \le 2$ we have
\begin{equation}
|\partial_v^\alpha \partial_w^\beta F_\nu(v,w)|\, \le\,  
\widetilde C\, ,  
                                       \label{eq:5.5a}
\end{equation}
where $\widetilde C$ does not depend on $\nu$. 
On the other hand,  
$I_\nu = O(a)$, $0<a\le a_0\ll 1$, and $-K(I) = L(\varrho_1)
+ O(a)$ in view of (\ref{eq:3.7}), where $L(\varrho_1)$ is the length
of $\gamma$. Fix $0<a\ll 1$. 
Applying the implicit function
theorem we get $v_\nu(\varepsilon)$ and $w_\nu(\varepsilon)$, 
$0<\varepsilon\le \varepsilon^0$, satisfying (\ref{eq:5.5}), where 
$\varepsilon^0$ is independent of  $\nu$. 
Then  setting $\varepsilon =
\varepsilon_\nu$, we obtain $v_\nu$ and $w_\nu$ for $|\nu| \gg 1$.  


\subsection{Asymptotics of the counting function} 
In view of (\ref{eq:5.1}), we have
$\nu = \lambda_\nu^0 (I_\nu, -K(I_\nu)/2\pi) + O(1)$, $\nu\in {\cal
M}$, $|\nu|\rightarrow \infty$. 
Set $D(r) = \{\tau (I, -K(I)/2\pi):\ I\in E_\kappa, 1\le \tau\le r\}$
and denote by $D(r) +C$, $C>0$, the set of all $x\in {\bf R}^n$ within
a distance $\le C$ to  $D(r)$. Then we have 
\[
N^\ast (r)\ 
:=\ \# \{\nu \in {\cal M}:\, \lambda_\nu \leq r\}\ =  {\rm Vol}\, 
(D(r) +C) + o(r^n)\, ,\ r\rightarrow +\infty\, 
\]
for some $C>0$. 
As in \cite{C-P}, Sect. 4.4,  we obtain $ {\rm Vol}\, 
((D(r) +C)\backslash D(r))  = o(r^n)$, and calculating the volume of
$D(r)$ we find 
\[
N^\ast (r)\ =  
\frac{r^n}{2\pi n}\ 
\int_{E_\kappa}\, (\langle I,\nabla K(I)\rangle - K(I))\,
dI\ +\, o(r^n)\, .
\]
On the other hand,  using (\ref{eq:3.8}) and taking into account that 
$P^0$ and $\chi$ are  symplectic,
hence, volume preserving, we evaluate the integral above by
\[
\displaystyle
\frac{1}{(2\pi)^{n-1}}\, \int_{{\bf T}^{n-1}\times E_\kappa} \, 
L(\chi(\varphi,I))\, d\varphi dI\ 
=\ 
\frac{1}{(2\pi)^{n-1}}\, \int_\Lambda \, L(\rho)\, d\rho\, , 
\]
$L$ being the ``return time function'' for the Poincaré map $P$, i.e. 
$\Phi^{L(\rho)}(\rho)= P(\rho)$, where $t\rightarrow \Phi^{t}(\rho)$
is the broken bicharacteristic starting from $\rho$. This implies
\begin{equation}
N^\ast (r)\ =\ 
\left(\frac{r}{2\pi}\right)^{n}\, 
\frac{\rm Vol\, {\cal T}} {n}\, + \,  o(r^n)\, ,\
r \rightarrow +\infty\, ,
                     \label{eq:5.6}
\end{equation}
where  ${\cal T}$ is the flow-out by the broken bicharacteristic flow 
of the union $\Lambda$  of the 
invariant   tori $\Lambda_\omega$, $\omega\in \Omega_\kappa$, and 
$\rm Vol\, {\cal T}$ stands for the Liouville measure of  ${\cal T}$
on $S^\ast X$.                   

\subsection{Orthogonality relations} 
First we  show that $\lambda_\nu$ can be chosen
real-valued. 
Taking into account  (\ref{eq:5.3a}), we obtain
\begin{equation}
\|u_\nu^0\|_{L^2(X)}\ \le C\, ,\ \forall \nu\in  {\cal M}\, ,
                                       \label{eq:5.7a}
\end{equation}
where $C>0$ is a constant. 
Then using (\ref{eq:5.4}) we estimate the normal derivative
$\partial_\eta u_\nu^0|_\Gamma$, $\eta$ being the unit inner normal to
$\Gamma$, as follows
\[
\|(\partial_\eta u_\nu^0)|_\Gamma\|_{L^2(\Gamma)}\ \le \ 
O(1) \|u_\nu^0\|_{H^2(X)}\ \le \ O(|\lambda_\nu|^2)\, .
\]
On the other hand, 
$(\partial_\eta u_\nu^0)|_\Gamma = 
\imath^\ast\partial_\eta G(\lambda_\nu)v_\nu^0$, 
$v_\nu^0=T(\lambda_\nu)A(\lambda_\nu)e_p$.  
It is easy to see that 
\[
\imath^\ast\partial_\eta G(\lambda)\ =\ 
B(\lambda)(\bar M(\lambda)\, + \, {\rm Id}\, )\, +\, O\left(
e^{-c|\lambda_\nu|^{1/\varrho}}\right)\, ,
\] 
where $B(\lambda)$ is a
$\lambda$-PDO on $\Gamma$ which is elliptic in a neighborhood $W_1$ of
$\rho_1$. Moreover, 
the $G^\varrho$-frequency set of $\{v_\nu^0\}_{\nu\in {\cal M}}$ is
contained in $W_1$ choosing $0<a\le a_0\ll 1$. Then  using
(\ref{eq:5.3b}) and (\ref{eq:5.7a}) we get 
\begin{equation}
\|u_\nu^0\|_{L^2(X)}\ \ge \ \tilde C(1 + |\lambda_\nu|)^{-2}
\|T(\lambda_\nu)A(\lambda_\nu)e_p\|_{L^2(\Gamma)} - 
O\left(e^{-c|\lambda_\nu|^{1/\varrho}}\right)
\ \ge \ C (1 + |\lambda_\nu|)^{-2}
                                       \label{eq:5.7}
\end{equation}
for $\nu =(p,q)\in {\cal M}$, 
where $C$ and $\tilde C$ are positive constants. By Green's
formula we get
\[
C|\lambda_\nu^2 - \overline{\lambda_\nu}\, ^2|\ \le\ 
|\langle \lambda_\nu^2 u_\nu^0, u_\nu^0 \rangle\,  -\,  
\langle  u_\nu^0, \lambda_\nu^2 u_\nu^0 \rangle|\ \le\ 
O\left(e^{-c|\lambda_\nu|^{1/\varrho}}\right)\, ,
\]
which allows us to take $\lambda_\nu$ in ${\bf R}$. From now we
suppose that $\lambda_\nu>0$. In view of (\ref{eq:5.3}), $\lambda_\nu$
is a small perturbation of $\lambda^0_\nu$, and it satisfies
\[
-\lambda_\nu K((p + \vartheta_0/4)/\lambda_\nu) + \pi
(m+\vartheta/4)
\]
\begin{equation}
+ \, {\rm Log}\, p^0((p + \vartheta_0/4)/\lambda_\nu,
\lambda_\nu) 
\
= \  \ 2\pi q + 
O\left( e^{-c|\lambda_\nu|^{1/(\mu -1)}}\right)\, .
                                                     \label{eq:5.8a}
\end{equation} 


To obtain the orthogonality relations, we consider three cases. First 
we suppose  that 
$\lambda_\nu - \lambda_{\nu'} \ge \lambda_\nu /2$. As in 
\cite{C-P}, Sect. 4.2, we write $\langle u_\nu^0,u_{\nu'}^0\rangle$ as
an oscillatory integral with a large parameter $\lambda_\nu$ and an
additional parameter $\lambda_{\nu'}/\lambda_{\nu}\in (0,1/2]$. The 
amplitude of that integral  belongs to $S_{\mu,\varrho}$ and the phase function
is Gevrey $G^\mu$ smooth and it has no critical points. Then
integrating by parts we get 
\[
|\langle u_\nu^0,u_{\nu'}^0\rangle _{L^2(X)}|\ \le \ C
e^{-c|\lambda_\nu|^{1/\varrho}}\, ,
\]
where $c$ and $C$ are positive constants. Suppose now that
\[
\exp\left(-\frac{c}{3}|\lambda_\nu|^{1/\varrho}\right)\ \le\ 
\lambda_\nu\, -\, \lambda_{\nu'}\ \le\ {\lambda_\nu \over 2} \, .
\]
Applying Green's
formula we get
\[
|\langle u_\nu^0,u_{\nu'}^0\rangle _{L^2(X)}|\ \le \ 
\exp\left(\frac{c}{3}|\lambda_\nu|^{1/\varrho}\right)\, 
|(\lambda_\nu^2 - \lambda_{\nu'}\, ^2)\, 
\langle u_\nu^0,u_{\nu'}^0\rangle | \ \le\ 
C \exp\left(-{c\over 3}|\lambda_\nu|^{1/\varrho}\right)\, .
\]
Consider the case 
\[
|\lambda_\nu\, -\, \lambda_{\nu'}|\ \le \ \min\, 
\left[ \exp\left(-{c \over 3}|\lambda_\nu|^{1/\varrho}\right), 
 \exp\left(-{c \over 3}|\lambda_{\nu^\prime}|^{1/\varrho}\right)\right]\, .
\]
Given $\nu \in {\cal M}$ we denote by ${\cal I}_\nu$ the connected
component of the union 
\[
\displaystyle\cup_{\nu' \in {\cal M}}\, 
\left[\lambda_{\nu'} \, -\, 
\exp\left(-{c \over 3}\lambda_{\nu^\prime}^{1/\varrho}\right)\, ,\, 
\lambda_{\nu'} \, +\, 
\exp\left(-{c \over 3}\lambda_{\nu^\prime}^{1/\varrho}\right)\right]\, 
\]
which contains $\lambda_\nu$ and set 
${\cal M}_\nu = \{\nu'\in \cal M\, :\ \lambda_{\nu'} \in {\cal
I}_\nu\}$. 
Using (\ref{eq:5.6}) we estimate the length of ${\cal I}_\nu$ by 
\[
|{\cal I}_\nu| \le C \lambda_\nu^n 
\exp\left(-{c \over 3}\lambda_{\nu}^{1/\varrho}\right)\, .
\]
We consider now the linear space 
${\cal F}_\nu \ =\ {\rm Span}\, \{u^0_{\nu^\prime}:\, 
\nu^\prime \in {\cal M}_\nu\}$ and set   $J_\nu = \#{\cal M}_\nu$ 
(the cardinality of ${\cal M}_\nu$). 
Let 
\begin{equation}
v_\delta\ =\ \sum_j \delta_j
u^0_{\nu_j}\, \in \, {\cal F}_\nu \ 
,\ \delta = (\delta_1,\ldots, \delta_{J_\nu}) \neq 0\ .   
                                     \label{eq:5.11}
\end{equation}
Fix $0<c_1<c/3$ and 
note that 
\[
u^0_{\nu^\prime}\  =\ 
G(\lambda_\nu)T(\lambda_\nu)A(\lambda_\nu)e_{p^\prime}\, +\, 
O\left(\exp (- c_1\lambda_{\nu}^{1/\varrho})\right)
e_{p^\prime}\, ,\ 
\]
uniformly with respect to $\nu\in {\cal
M}_\nu$ and $\nu^\prime  \in {\cal
M}_\nu$.  It follows from  (\ref{eq:5.3}) that $q'=q$ for each 
$\nu^\prime = (p',q') \in {\cal
M}_\nu$. 
Then  the dimension of ${\cal F}_\nu$ is  $J_\nu = \#{\cal M}_\nu$ 
 and we obtain $J_\nu =
O(\lambda_\nu^n)$ in view of (\ref{eq:5.6}). 
Moreover,  
\[
\|v_\delta\|_{L^2(X)} \le C_1\|\delta\|\ ,  \quad  
\|\delta\| = (\sum |\delta_j|^2)^{1/2}\, ,
\] 
where $C_1$ is independent of $\nu$. 
By (\ref{eq:5.4}) we get 
\begin{equation}
\left|
\begin{array}{rcll}  
\left\|\left(\, \Delta \, +\, \lambda_\nu^2 \, \right)\, 
v_\delta(x)\right\|_{H^s}\ &\le& \
 C_s\,  \|\delta\|\, e^{-c_1|\lambda_\nu|^{1/\varrho}}\, , \\
v_\delta |_\Gamma \ &=&\ 0\, .   
\end{array} 
\right.
                                      \label{eq:5.12}
\end{equation}
On the other hand, as in the proof of (\ref{eq:5.7}) we get
\begin{equation}
\begin{array}{rcll}  
\|v_\delta\|_{L^2(X)}\ &\ge& \ C_2(1 + \lambda_\nu)^{-2}\, 
\|(\partial_\eta v_\delta)|_\Gamma\|_{L^2(\Gamma_1)}\\
&\ge&\ C_3(1 + \lambda_\nu)^{-2}\, 
\|\sum_j \delta_j e_{p_j}\|_{L^2({\bf T}^{n-1}, {\bf L})}\\
&-& \ 
O\left(e^{-c_1|\lambda_\nu|^{1/\varrho}}\right)\,  \|v_\delta
\|_{L^2(X)}\ 
\ge \ C_4(1 + \lambda_\nu)^{-2}\,  \|\delta\|\, ,
\end{array} 
                                     \label{eq:5.13}
\end{equation}
where the constants are independent of $\nu$. We fix in each  
${\cal F}_\nu$ an
orthonormal basis $v_{\delta(\nu,k)}$, $k=1,\ldots, J_\nu$, 
 given by (\ref{eq:5.11}) with some 
$\delta=\delta(\nu,k)$.  Using (\ref{eq:5.13}) we get
\begin{equation}
C\ \le \  \|\delta(\nu,k)\| \ \le \  C^{-1}(1 + \lambda_\nu)^2
                                      \label{eq:5.14}
\end{equation}
where $C>0$ does not depend on $\nu$ and $k$. Let 
$v_{\delta({\nu',k'})}$ belongs to the space ${\cal F}_{\nu'}$
corresponding to ${\nu'}\notin {\cal M}_\nu$. 
Then using the estimates  in the first two cases as well as  
(\ref{eq:5.14}) and the relation $J_\nu = O(\lambda_\nu^n)$, we
estimate the scalar product $\langle v_{\delta({\nu,k})},
v_{\delta({\nu',k'})}\rangle$ by 
\[
\min\,  (\lambda_\nu^{2n+4}\exp(-{c\over 3}\lambda_\nu^{1/\varrho}) ,\, 
\lambda_{\nu'}^{2n+4} \exp(-{c\over 3}\lambda_{\nu^\prime}^{1/\varrho}))\, .
\] 
Finally, using Gramm-Schmidt
procedure
we obtain $u_\nu$ in (\ref{eq:1.1}) 
as linear combinations of $v_{\delta({\nu,k})}$ taking $c=c_1/2$. 

By construction the $G^\varrho$ micro-support 
${\rm WF}^\varrho ({\cal Q})$ of ${\cal Q}$ (see Appendix) is
contained in the flow-out by the bicharacteristic flow in $S^\star X$
of the union of the invariant tori $\Lambda$. 





\section{On the homological equation in Gevrey classes}
\setcounter{equation}{0}
Suppose that
$f\in G^{\sigma,\mu} ({\bf T}^{n-1}\times D)$ satisfies
\begin{equation}
\left|D^\alpha_I  D_\varphi^\beta f(\varphi, I)\right|\
\leq\ d_0\, C^{ |\alpha|+|\beta|}\, \, \alpha !\, 
\Gamma ((\mu-1) |\alpha| + \sigma |\beta| + q)\ ,\ 
(\varphi, I)\in {\bf T}^{n-1}\times D, 
                                      \label{eq:6.1}
\end{equation}
for any $\alpha, \beta \in {\bf N}^{n-1}$, 
and some $C\ge 1$ and $q>0$.
Let
\begin{equation}
\int_{{\bf T}^{n-1}}^{} f(\varphi,I)d\varphi\ =\ 0\, , \quad 
(\varphi, I)\in {\bf T}^{n-1}\times E_\kappa .
                                      \label{eq:6.2}
\end{equation}
We are going to find $u\in  G^{\sigma,\mu} ({\bf T}^{n-1}\times D)$, 
which solves the  ``homological'' equation
\begin{equation}
\left\{
\begin{array}{lcr}
 \ u(\varphi - \omega(I),I) -
u(\varphi,I)\ =\ f(\varphi,I)\, ,\  
(\varphi, I)\in {\bf T}^{n-1}\times E_\kappa,   \\
\int_{{\bf T}^{n-1}}^{} u(\varphi,I)d\varphi\ = \ 0\, ,\ I\in D, 
\end{array} 
\right.                                    
                                  \label{eq:6.3}
\end{equation}
and  provide the corresponding Gevrey estimates for the derivatives of 
$u$. Recall that  
$\omega: D \to \Omega$ is a diffeomorphism of  Gevrey class 
$G^{\tau'+2}(D,\Omega)$ given by Theorem 3.1, and  
there exists $C_0>0$ such that for each  
$\alpha\in {\bf N}^{n-1}_+$ we have: 
\begin{equation}
\left|D_I^\alpha \omega (I)\right|\ \leq\ C_0^{|\alpha|}\,  \alpha!\, 
((|\alpha|-1) !)^{\tau'+1}\ , \quad I\in D\, .
                                               \label{eq:6.4}
\end{equation}

\begin{prop}
Let $f\in G^{\sigma,\mu} ({\bf T}^{n-1}\times D)$ satisfy 
(\ref{eq:6.1}) and (\ref{eq:6.2}),
where $\sigma>1$ and $\mu-1 >\sigma(\tau'+1)$. 
Then there exists 
$u\in  G^{\sigma,\mu}  ({\bf T}^{n-1}\times D)$ satisfying (\ref{eq:6.3}) 
and  the estimate
\begin{equation}
\left|D^\alpha_I  D_\varphi^\beta u(\varphi, I)\right|
\ \leq\ R\, d_0\, C^{ \alpha+|\beta|+\nu}\,\, \alpha !\, 
\Gamma\left((\mu-1) | \alpha| + \sigma|\beta| + \sigma(\nu-1) +q\right) \, ,
                                  \label{eq:6.5}
\end{equation}
for any $(\varphi, I)\in  {\bf T}^{n-1}\times D$ and  
$\alpha,\beta \in {\bf N}^{n-1}$, 
where  the constant $R>0$ does not depend on $d_0$.    
\end{prop}
\noindent {\em Proof.} 
Define  the residue $[z]_{2\pi}$ of $z\in {\bf R}$ by  
$z = [z]_{2\pi} + 2\pi k_n\, ,\ -\pi< [z]_{2\pi} \le \pi\, ,\
k_n\in {\bf Z}$, 
and  for  $0\neq k \in {\bf Z}^n$ consider 
\[
z_k(I) = 1  - e^{-i\langle \omega(I),k\rangle}
+\  \kappa |k|^{-\tau}\,  
\psi\left( |k|^{\tau} [\langle \omega(I),k\rangle]_{2\pi}\right)\,. 
\]
Here
$\psi\in G^{1+\epsilon}(D)$, 
satisfies  $0\le \psi \le 1$, $\psi(x) = 0$ for $|x|\ge \kappa/2$, 
 and 
$\psi(x) = 1$ for $|x|\le \kappa/4$, $0<\kappa  \ll 1$ being the small
constant in (\ref{eq:3.5}), and we  
fix
$0<\epsilon < \mu-1 -\sigma(\tau'+1) $. 
Choosing  $C_0\gg 1$  
sufficiently big we 
suppose that  $|D_x^p\psi(x)|\le   C_0^{p+1}p!^{1+\epsilon }$, 
$x\in{\bf R}$, $p\in {\bf Z}_+$. 
It is easy to see that
\begin{equation}
|z_k(I)|\ \geq\ \frac{\kappa_0}{|k|^{\tau}}\ ,\quad 
I\in D,\
k\in{\bf Z}^{n-1}\backslash\{0\}\, ,
                                              \label{eq:6.6}
\end{equation}
where $\kappa_0= \kappa/2\pi$. Indeed, writing   
\[
z_k(I) = 1  - \cos\left([\langle \omega(I),k\rangle]_{2\pi}\right) 
+\  \kappa |k|^{-\tau}\,  
\psi\left( |k|^{\tau} [\langle \omega(I),k\rangle]_{2\pi}\right)
+ i \sin\left([\langle \omega(I),k\rangle]_{2\pi}\right)\, ,
\] 
and estimating  
below ${\rm Re}\, z_k(I)$, respectively $|{\rm Im}\, z_k(I)|$,  
when $\left|[\langle \omega(I),k\rangle]_{2\pi}\right|$
belongs to  the union 
$[0,\kappa |k|^{-\tau}/4]\cup [\pi/2, \pi]$, respectively to the
interval 
$[\kappa |k|^{-\tau}/4, \pi/2]$, we obtain (\ref{eq:6.6}). 

  

Consider the Fourier expansions of $f$ and $u$
\[
f(\varphi, I)\ =\ \sum_{k\in{\bf Z}^{n-1}} 
e^{i\langle k,\varphi \rangle}\, f_k(I)\ ,\ 
u(\varphi, I)\ =\ \sum_{k\in{\bf Z}^{n-1}} 
e^{i\langle k,\varphi \rangle}\, u_k(I),
\]
where
$$
f_k(I)\ =\ (2\pi)^{1-n}\int_{{\bf T}^{n-1}}^{}
e^{-i\langle k,\varphi \rangle}\,  g(\varphi,I)\, d\varphi,
$$
and $u_k(I)$ is defined in the same way. 
We set  $u_0=0$, and
$$
u_k(I)\ =\ - z_k(I)^{-1}\, f_k(I),\quad  I\in D,\
0\neq k\in {\bf Z}^{n-1}\, .
$$
Then $u$ satisfies (\ref{eq:6.3}) since  
$z_k(I) = 1  - \exp(-i\langle \omega(I),k\rangle)$ on $E_\kappa$. 
We are going to find suitable  Gevrey estimates for $u_k(I)$. 
To this end we estimate the derivatives of $z_k(I)^{-1}$. 
\begin{lemma} There is  $\widetilde C_0>0$ depending only on $C_0$,
$\kappa$,  and
$n$ such that 
\begin{equation}
|D_I^\alpha z_k(I)|\leq \widetilde C_0^{|\alpha|}\, \alpha!\,
^{1+\epsilon}  \, 
\max_{1\leq p\leq|\alpha|}\left(|k|^{\tau p +p -\tau} 
((|\alpha|-p)!)^{\tau'+1}\right),
                                             \label{eq:6.7}
\end{equation}
for any $I\in D,\ 0\neq  k\in{\bf Z}^{n-1},\ 
0 \neq \alpha\in{\bf N}^{n-1}$.
\end{lemma}
\startproof 
Applying Faa de Bruno formula for    
 $z_{k1}(I) = 1  - e^{-i\langle \omega(I),k\rangle}$, 
$\alpha \neq 0$, $k  \neq 0$,  we get  
\[
\left|D_I^\alpha z_{k1}(I) \right|
 \ \le  \ 
\displaystyle{ \sum_{p=1}^{|\alpha|}\sum_{\scriptstyle{
{\alpha^1+\cdots+\alpha^p=\alpha}
\atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\ 
\frac{\alpha!}{\alpha^1! \cdots \alpha^p!}\, 
 \left|\langle D^{\alpha^1}_I  \omega(I),k\rangle  \right|\ldots  
\left|\langle D^{ \alpha^p}_I  \omega(I),k\rangle \right| \, .
\]
As in the proof of Lemma 4.2, given  $1\le p \le |\alpha|$, 
we estimate the number of partitions 
$\alpha^1,\, \ldots,\, \alpha^p$ of $\alpha$ in the index set 
of the second sum  
by $2^{|\alpha|}2^{n(p-1)}$. Then choosing $C_1 = 2^{n+2} C_0$ and
using (\ref{eq:6.4}) we get (\ref{eq:6.7}) for $ z_{k1}$. 
Set 
$\psi_k(x)=\kappa|k|^{-\tau}\psi(x|k|^{\tau})$
and $z_{k2}(I) = \kappa |k|^{-\tau}\,  
\psi\left(|k|^{\tau} [\langle \omega(I),k\rangle]_{2\pi}
\right) = \psi_k([\langle \omega(I),k\rangle]_{2\pi})$. 
For any $k\neq 0$ fixed and $I$ in the support of $z_{k2}(I)$ we have 
$[\langle \omega(I),k\rangle]_{2\pi} = \langle \omega(I),k\rangle
- 2\pi k_n$, where $k_n\in {\bf Z}$ is locally  constant. 
Moreover, 
$|D_x^p\psi_k(x)|\le  C_0^{p+1}p!^{1+\epsilon}|k|^{\tau p-\tau}$, 
$x\in{\bf R}$,
$p\in {\bf Z}_+$, and using the 
Faa de Bruno formula for  $z_{k2}(I)$, we get 
\[
\left|D_I^\alpha z_{k2}(I) \right| \le 
 \displaystyle{\sum_{p=1}^{|\alpha|}  
\sum_{\scriptstyle{
{\alpha^1+\cdots+\alpha^p=\alpha}
\atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\ 
 C_0^{p+1}p!\, ^{\epsilon}\, |k|^{\tau p +p -\tau}\, 
 C_0^{|\alpha|} \alpha!\, 
\left((|\alpha^1|-1)!\cdots (|\alpha^p|-1)!\right)^{\tau'+1} , 
\]
which implies  (\ref{eq:6.7}) for $z_{k2}$ as above.  
\finishproof 
\begin{lemma} There exists a positive constant
$C_1$ depending only on $C_0$,   $\kappa$ and $n$ such that
\begin{equation}
|D_I^\alpha (z_k(I)^{-1})|\ \leq \  C_1^{|\alpha|+1}\, 
\alpha! \, ^{1+\epsilon}\,  
\max_{0\leq j\leq|\alpha|}\left(|k|^{\tau j + j + \tau}
(|\alpha|-j)!^{\, \tau'+1}\right),
                                                    \label{eq:6.9*}
\end{equation}
for any $I\in D,\ 
k\in{\bf Z}^{n-1}\backslash\{0\},\ \alpha\in{\bf N}^{n-1}$.
\end{lemma}
{\em Proof}.  
Applying Faa de Bruno formula for $\alpha \neq 0$  we get  
\[
\left|D_I^\alpha\left(z_k(I)^{-1}\right)\right|
 \ \le  \ 
\displaystyle{ \sum_{p=1}^{|\alpha|}\sum_{\scriptstyle{
{\alpha^1+\cdots+\alpha^p=\alpha}
\atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\ 
\frac{\alpha!}{\alpha^1! \cdots \alpha^p!}\, 
| z_k(I)|^{-p-1}
 \left|D^{\alpha^1}_I z_k(I)  \right|\ldots  
\left|D^{\alpha^p}_I z_k(I) \right| \, .
\]
Using (\ref{eq:6.6}) and Lemma 6.1 we estimate 
\[
\left|D_I^\alpha\left(z_k(I)^{-1}\right)\right|
 \ \le  \ \left(\frac{\widetilde C_0}{\kappa_0}\right) 
^{|\alpha|+1} \,  \alpha !\,^{1+\epsilon}\,|k|^{\tau}
\displaystyle{ \sum_{p=1}^{|\alpha|}\sum_{\scriptstyle{
{\alpha^1+\cdots+\alpha^p=\alpha}
\atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\,  
\prod_{s=1}^{p}\, |k|^{(\tau +1)j_s}(|\alpha^s|-j_s)!^{\, \tau'+1}
 \, ,
\]
where $1\le j_s\le |\alpha^s|$. Choosing 
$C_1 = 2^{n+2}\kappa_0^{-1}\widetilde C_0$ 
we prove  (\ref{eq:6.9*}) as above. \finishproof 

For any $m \in {\bf N}$ we set $\langle k\rangle_m = 1 + |k_1|^m +\cdots +
|k_n|^m,\ 0\neq k\in {\bf Z}^{n-1}$.
Next, for any $j\in {\bf N}$ we denote $m(j)\ =\ [(\tau' +1)j + \tau]  + n$, 
where $[p]$, $p-1 < [p] \le p$, stands for the integer part
of $p\in {\bf R}$. Set $W(k) = (1 + |k|)^{n-1 +\varepsilon_0} $ with
$0\ <\ \varepsilon_0\ <\ \min\{\tau' -\tau, [\tau]-\tau+1\}.$
Then
$$
\tau j +j+\tau + n - 1 +\varepsilon_0\ <\ m(j) \ 
\leq\ (\tau' +1)j +\nu-1\, ,\ 
\forall j\in {\bf N} \, ,
$$
with $\nu =\tau + n + 1$.
Taking into account the inequality
$|k|^m\ \leq\ n^m \langle k\rangle_m\, ,\ m  \in {\bf N},\ 
0\neq  k\in {\bf Z}^{n-1},$
and using Lemma 6.2, we get
\begin{equation}
W(k)\left|D_I^\alpha 
\left(z_k(I)^{-1}\right)\right|\ \leq\ C_2^{|\alpha|+1}\, 
\alpha ! \, ^{1+\epsilon}
\max_{0\leq j\leq |\alpha|}\left( (|\alpha|-j )!\, ^{\tau'  +1}
\langle k\rangle_{m(j)}\, \right)\, ,
                                      \label{eq:6.10*}
\end{equation}
for any $I\in D,\ \alpha\in {\bf N}^{n-1}$, 
and $0\neq k\in {\bf Z}^{n-1}$,
with a constant $C_2>0$ depending only on $n,\ \kappa$, and $C_1$.
Integrating by parts, we obtain 
\[
k^\beta k_j^m D^\alpha_I
f_k( I)\ =\ 
(2\pi)^{1-n}\int_{{\bf T}^{n-1}}^{}
e^{-i\langle k,\varphi \rangle}\, 
D_\varphi^\beta D_{\varphi_j}^m D^\alpha_I
f(\varphi,I)\,
d\varphi \, .
\]
Then  using (\ref{eq:6.1})     we get 
\begin{equation}
\left|k^\beta \langle k\rangle_m D^\alpha_I  f_k( I)\right|\
\leq\ nd_0\, C^{|\alpha|+|\beta|+m}\, \alpha !\, 
\Gamma( (\mu-1)  |\alpha| + \sigma|\beta| + \sigma m + q)\,  ,
                                      \label{eq:6.11*}
\end{equation}
for any $I \in D,\ 
0\neq k\in {\bf Z}^{n-1}$, and any $m\in {\bf N}$, 
$\alpha, \beta \in {\bf N}^{n-1}$. Consider
\[
A_k\ :=\ W(k)\left|k^\beta D^\alpha_I
u_k( I) \right|\ 
=\ \sum_{0\leq\gamma\leq\alpha}^{} \ \frac{\alpha !}{\gamma ! (\alpha
-\gamma)!} W(k)\left|D^\gamma_I \left(z_k(I)^{-1}
\right)\right|\,  \left| 
k^\beta D^{\alpha-\gamma}_I f_k( I) \right|\, .
\]
By Stirling's formula we have
$p !  ^{\, \tau' + 1}\ \leq\ C_3^{p}\, \Gamma ((\tau'+1) p + 1)$, 
$p\geq 0$,  
with some constant $C_3>0$. Then using 
(\ref{eq:6.10*}) we obtain 
\[
 W(k)\left|D^\gamma_I \left(z_k(I)^{-1}
\right)\right|\,  \left| 
k^\beta D^{\alpha-\gamma}_I f_k( I) \right|
\]
\[
\ \le \ (C_2C_3)^{|\gamma|+1}\, \gamma ! \, ^{1 + \epsilon}\,  
\Gamma((\tau'  +1)(|\gamma|-j_\gamma )+1)\,
\left|k^\beta \langle k\rangle_{m(j_\gamma)} D^{\alpha-\gamma}_I
f_k( I)  \right|\,  , 
\]
where   $0\leq j_\gamma\leq |\gamma|$.  On the other hand, 
$\gamma !  ^{\, \epsilon}\ \leq\ \widetilde C_3^{|\gamma|+1}\, 
\Gamma (\epsilon |\gamma| 
+ 1)$,  and using  (\ref{eq:gamma}), we estimate 
$\gamma ! \, ^{\epsilon}\,  
\Gamma((\tau'  +1)(|\gamma|-j_\gamma )+1) \le 2 \widetilde C_3^{|\gamma| +
1} \Gamma((\tau'  +1)(|\gamma|-j_\gamma )+\epsilon |\gamma|+1)$. 
Taking into account  (\ref{eq:6.11*}) and using  again 
(\ref{eq:gamma}) we obtain 
\[
A_k \ \leq \ 
4 n d_0\,  \alpha !\, \sum_{0\leq\gamma\leq\alpha}^{}
C_4^{|\gamma| +1} \,  C^t\,
\Gamma (s)\,  \, ,
\]
where   $C_4=C_2C_3 \widetilde C_3$ and   
\[
t= |\alpha-\gamma| +|\beta|+m(j_\gamma)\ ,
\]
\[ 
s=  (\mu-1) |\alpha-\gamma| + \sigma|\beta|+q + 
\sigma m(j_\gamma) + (\tau'+1)(|\gamma|-j_\gamma) + \epsilon |\gamma|. 
\]
Set  $\delta =\mu -1 - \sigma(\tau'+1) -\epsilon >0$. 
We have 
\[
s\ \le\  (\mu-1) |\alpha -\gamma| +
\sigma |\beta| + \sigma(\nu-1) + q + \sigma (\tau'+1)j_\gamma + 
(\tau'+1)(|\gamma|-j_\gamma )+ \epsilon |\gamma|
\]
\[
= \ (\mu-1) |\alpha| 
+\sigma |\beta| + \sigma(\nu-1) + q-  \delta|\gamma| - 
(\sigma-1)(\tau'+1)(|\gamma|-j_\gamma)\, . 
\]  
Moreover, $t \le  |\alpha| +|\beta|+m(|\gamma|)-|\gamma|$. 
Since $\delta >0$, $\sigma >1$, $|\gamma|\ge j_\gamma$ and $C\ge 1$, 
we obtain 
\[
A_k\ \leq\   d_0\, d_1\,  C^{ |\alpha| +|\beta|}\, \alpha !\, 
\sum_{0\leq \gamma\leq\alpha}^{}\ C_4^{|\gamma|+1}\, 
C^{m(|\gamma|)-|\gamma|}\,   \frac{
\Gamma ((\mu-1)|\alpha| +\sigma|\beta|+\sigma(\nu-1) +q )}
{\Gamma (\delta|\gamma| +1 ) }\, ,
\]
where 
$d_1 = \displaystyle 16 n\, \max_{1\le x\le 2}\Gamma (x)^{-1}$. This  implies 
\[
A_k\ \leq\  R_1\,  d_0\, C^{ |\alpha| +|\beta|}\, \alpha !\,
\Gamma ((\mu-1)|\alpha| +\sigma|\beta| +\sigma(\nu-1) +q )\, ,  
\]
where $R_1  =  d_1 
\sum_{\gamma \in {\bf N}^{n-1}}^{} \
\frac{ C_4^{|\gamma|+1}\, C^{m(|\gamma|)-|\gamma|}}
{\Gamma (\delta|\gamma| +1 ) }$ and $m(|\gamma|) -|\gamma| \le \tau' |\gamma| + \nu -1$. 
Finally, summing up $W(k)^{-1}A_k$,  we obtain
$$
\left|D_I^\alpha D_\varphi^\beta v(\varphi, I)\right|\  
\leq \  R\, d_0\, C^{|\alpha| +|\beta|}\, \alpha !\, 
\Gamma ((\mu-1) |\alpha| +\sigma|\beta| +\sigma(\nu-1) +q)\, ,
$$
where
$R\ =\ R_1\, \sum_{k\in{\bf Z}^{n-1}}W(k)^{-1}$.
The constant $R>0$ is independent of $d_0$.   
The proof of the proposition is complete. 
\hfill $\Box$ \vspace{5mm} 




\appendix
\section*{Appendix}
\setcounter{equation}{0}
\newtheorem{rema}{Remark A.}


\renewcommand{\theequation}{A.\arabic{equation}}


We collect here some facts about Gevrey symbols.  

Let $M$ be either ${\bf T}^{m}$,  $m\ge 1$, or 
a bounded  domain in ${\bf R}^{m}$. 
Denote by $D$  a bounded domain in ${\bf R}^{m}$ 
and ${\cal D}:= \{z\in {\bf C}: |z| \ge 1,\, |{\rm Im}\, z| \le \widetilde
C \}$, where  $\widetilde C>0$ is fixed.  
Fix $\sigma,\mu>1$, $\varrho\geq \sigma + \mu -1$,  and
set $\ell =(\sigma,\mu,\varrho)$. 
We use  a class of formal Gevrey symbols $FS_{\ell}(M\times D)$
defined  as follows. 
Consider a sequence of smooth functions $g_j\in C_0^\infty(M\times D)$,
$j\in {\bf Z}_+$ such that ${\rm supp}\, g_j$ is contained in a fixed
compact subset of $M\times D$. We say that
\begin{equation}
\sum_{j=0}^{\infty}\, g_j(\varphi,I)\, \lambda^{-j}\ ,\quad \lambda\in
{\cal D}\, ,
                              \label{eq:A1}
\end{equation}
is a formal Gevrey symbol in $FS_{\ell}(M\times D)$ if there exists
a positive constant $C$ such that $g_j$ satisfies the estimates
\begin{equation}
\sup_{M\times D}|\partial^\beta_\varphi\partial^\alpha_Ig_j(\varphi,I)|
\ \leq\ C^{j+|\alpha|+|\beta|+1}\, 
\beta!\, ^\sigma\, \alpha!\, ^\mu\, j!\, ^\varrho
                               \label{eq:A2}
\end{equation}
for any $\alpha, \beta$ and $j$.

The function $g(\varphi,I,\lambda)$, $(\varphi,I)\in M\times {\bf R}^{m}$,
  is called a realization of the formal symbol
(\ref{eq:A1}) in $M\times D$ if for each 
$\lambda \in {\cal D}$ 
it is smooth with respect 
to $(\varphi,I)$ and has  compact support in  $M\times D$,  
and if there exists a positive constant $C_1$ such that
\[
\sup_{\bf Q}\big|\partial^\beta_\varphi\partial^\alpha_I
(g(\varphi,I,\lambda)-\sum^N_{j=0}
g_j(\varphi,I)\lambda^{-j})\big|\ \leq\ |\lambda|^{-N-1}\, 
C_1^{N+|\alpha|+|\beta|+2}\, 
\beta!\, ^\sigma\,
\alpha!\, ^\mu\, (N+1)!\, ^\varrho\,
\]
for any multi-indices $\alpha,\beta$ and
$N\in {\bf N}$, where ${\bf Q}=M\times D\times {\cal D}$.
For example, one can take
\[
g(\varphi,I,\lambda)\ =\ 
\sum^{}_{j\leq \, \varepsilon |\lambda|^{1/\varrho}}\, 
g_j(\varphi,I)\, \lambda^{-j}\, ,
\]
where $0<\varepsilon \le \varepsilon_0 \ll 1$ 
depends only on the constant $C_1$ 
and the
dimension of $M$ and $D$. 
We denote by $S_{\ell}(M\times D)$ the
corresponding class of symbols.
In order to work  with   symbols  holomorphic  with respect to
$\lambda$ we fix $0<\varepsilon < 2\varepsilon_0$,  
choose $\lambda^0\ge 1$ such that 
$|\lambda -\lambda^0| <  \lambda^0/2$ and take 
\begin{equation}
g^0(\varphi,I,\lambda)\ =\ \displaystyle \sum_{j < \varepsilon
(\lambda^0)^{1/\varrho}}\, g_j(\varphi,I)\lambda^{-j}\, . 
                                    \label{eq:A3}
\end{equation}
Note that 
\[
\left|D_I^\alpha(g(\varphi,I,\lambda) -g^0(\varphi, I,\lambda))\right|\ \le \
C^ {|\alpha| +1}\, \alpha !^{\, \mu}\,
e^{-c|\lambda|^{1/\varrho}}\, ,\ |\lambda -\lambda^0| < 
 \lambda^0/2\, ,
\]
where $c,C>0$ are independent of $\lambda^0$.   
We say that $g\in S^{-\infty}_{\ell}(M\times D)$
if 
$$
\sup_{\bf Q}|\partial^\beta_\varphi\partial^\alpha_I\,
g(\varphi,I,\lambda)|\
\leq\ |\lambda|^{-N}\, C^{N+|\alpha+\beta|+1}\, 
\beta!\,^\sigma\, \alpha!\,^\mu\, N !\, ^\varrho
$$
for $\lambda \in {\cal D},\ \forall N\in{\bf N}$, 
and for any multi-indices
$\alpha,\beta\in{\bf N}^m$, or equivalently
$$
\sup_{\bf Q}|\partial^\beta_\varphi\partial^\alpha_I\,
g(\varphi,I,\lambda)|
\ \leq\
C_1^{|\alpha+\beta|+1}\, \beta!\, ^\sigma\, \alpha!\, ^\mu\, 
\exp(-c|\lambda|^{1/\varrho})
$$
for some $C_1,\ c>0$, 
and any $\lambda\in {\cal D},\, \alpha,\, \beta\, 
\in{\bf N}^m$.
Moreover, given $f, g\in S_{\ell}(X\times D)$, 
we say that $f$ is
equivalent to $g$ ($f\sim g$) if 
$f-g\in S^{-\infty}_{\ell}(X\times D)$. 
It is not hard to prove that any two realizations of
$\sum^\infty_{j=0}g_j\lambda^{-j}$ in  $S_{\ell}(X\times D)$ 
are equivalent.
When $\sigma=\mu$ and $\varrho=2\sigma -1$, we set 
$S^{\sigma}=S_{\ell}$ and $S^{\sigma,-\infty}=
S^{-\infty}_{\ell}$.







To each symbol $g\in S_{\ell}(M\times D)$, ${\rm dim}\, M = {\rm
dim}\, D = m$,   we associate an
$\lambda$-PDO  by
$$
P(\lambda)u(x)\ =\ \left(\frac{\lambda}{2\pi}\right)^{m}\, 
\int_{{\bf R}^{2m}}\, 
e^{i\lambda\langle x-y,\xi\rangle }\,
g(x,\xi,\lambda)\, u(y)d\xi dy,\  u\in C_0^\infty(M). 
$$
It is well defined modulo $\exp(-c|\lambda|^{1/\varrho})$. Indeed, 
for any  $g\in S^{-\infty}_{\ell}$ we have 
$$
||P(\lambda)u||_{L^2}\ \le\ C  \exp(-c|\lambda|^{1/\varrho}) ||u||_{L^2}\, 
,\  u\in C_0^\infty(M),
$$
with some positive constants $c$ and $C$. 

Let $u(x,\lambda)$ be a family of smooth functions in $M$ 
for $\lambda \in {\cal D}$. 
The $G^\varrho$ micro-support
${\rm WF}^\varrho(u)\, \subset\, T^\ast(M)$ of $u$
is defined as follows: 
$(x_0,\xi_0)\notin {\rm WF}^\varrho(u)$ if there exists 
$c>0$ and compact neighborhoods
$U$ of $x_0$ and $V$ of $\xi_0$ such that for any
 $G^\varrho$ function
$v$ with compact support in $U$ 
$$
\int\, e^{ - i\lambda \langle x, \xi\rangle} \, v(x)u(x,h)dx\
=\ O\left(e^{ - c |\lambda|^{ 1/\varrho}}\right)\ 
,
$$
uniformly with respect to $\xi\in V$. Obviously, 
$(x_0,\xi_0,x_0,-\xi_0)$ does
not belong to the  $G^\varrho$ microsupport
of the distribution kernel of the   $\lambda$-PDO 
$P(\lambda)$ above if its amplitude $p\in S_\ell$
belongs to $S^{-\infty}_\ell
$ in a neighborhood of $(x_0,\xi_0)$. 


We are going  to extend symbols in 
$S_{\ell} ({\bf T}^{n-1}\times D)$ in the complex space with
respect to $I$. Given $g\in S_{\ell}$ with a compact support with
respect to $I$ in $D$ and $\delta >0$, we set 
\begin{equation}
g(\varphi, I+iY,\lambda,\lambda_0)\ =\ \displaystyle \sum_{|\gamma| < (\delta
\lambda^0)^{1/(\mu -1)}}\, \frac{i^{|\gamma|}Y^{\gamma}}{\gamma !}\, 
\partial^\gamma_I \, g(\varphi, I, \lambda)\, ,\ Y\in {\bf R}^{n-1}\,
,\   |Y|\le \widetilde C|\lambda|^{-1}\, ,
                                             \label{eq:A4}
\end{equation}
where $\lambda\in {\cal D}$, $\lambda^0 \ge 1$ and  
$\widetilde C > 0$. 
We write also 
$g(\varphi,z,\lambda)$ instead of $ g(\varphi,z,\lambda,\lambda_0)$. 
Since $g\in S_{\ell}$, we have
\[
|D^\alpha_I D^\beta_\varphi g(\varphi,I,\lambda,\lambda_0)|\ \le \ 
C^{|\alpha| +
|\beta| + 1}\, \alpha ! ^{\, \mu} \beta ! ^{\, \sigma}\, ,
\]
for $(\varphi,I)\in {\bf T}^{n-1}\times D$ and $\lambda \in {\cal
D}$. Choose  $\lambda^0\ge 1$ and $\delta >0$ so that 
\begin{equation}
|\lambda-\lambda^0| < \lambda^0/2 \quad , \quad 0<\delta < \frac{1}{e C
\widetilde C 2^{\mu +1}}\, .  
                                             \label{eq:A5}
\end{equation}  
Then we have
\begin{equation}  
|D^\alpha_I D^\beta_\varphi g(\varphi,z,\lambda, \lambda_0)|\ 
\le \ d_0\, 
(2^{\mu}C)^{|\alpha| +
|\beta| + 1}\, \alpha ! ^{\, \mu} \beta ! ^{\, \sigma}\ ,\ z = I +
iY\, ,
                            \label{eq:A6} 
\end{equation}
\begin{equation}  
|D^\alpha_I D^\beta_\varphi \overline \partial_z 
g(\varphi,z,\lambda,\lambda_0)|\ \le \ 
d_0\, e\,  (2^{\mu}C)^{|\alpha| +
|\beta| + 1}\, \alpha ! ^{\, \mu} \beta ! ^{\, \sigma}\, 
e^{-c |\lambda|^{1/(\mu -1)}}\,  ,
                                             \label{eq:A7}
\end{equation}
 for $(\varphi,I)\in {\bf T}^{n-1}\times D$, 
$|Y|\le \widetilde C|\lambda|^{-1}$,  
and $\lambda \in {\cal D}$, $|\lambda|\gg L$, 
where $c= (2\delta/3)^{1/(\mu -1)}$, 
the constant $d_0>0$ depends only on
$n$, and $L>0$ is a  constant. 
To prove (\ref{eq:A6}), we estimate the 
$D^\alpha_I D^\beta_\varphi$-derivative of each term in (\ref{eq:A4}) by
\begin{equation}
\frac {(\alpha + \gamma)!^{\, \mu}}{\gamma !} \beta !^{\, \sigma}
C^{|\alpha| + |\beta| +|\gamma| + 1}\,
\left(\frac{\widetilde C}{|\lambda|}\right)^{|\gamma|}\ 
\le \ e^{-|\gamma|}\,
C^{|\alpha| + |\beta|  + 1}\, 2^{\mu |\alpha|}\,   
\alpha ! ^{\, \mu} \beta ! ^{\, \sigma}\, . 
                                             \label{eq:A8}
\end{equation}
Indeed, since $ \gamma !\, \le |\gamma|^{ |\gamma|}$,  we obtain for  
$|\lambda|\ge L \gg 1$ 
\[
\frac {(\alpha + \gamma)!^{\, \mu}}{\gamma !}\, 
C^{|\gamma|}\, \left(\frac{\widetilde C}{|\lambda|}\right)^{|\gamma|}\
\ \le \ d_1 \alpha !\,^\mu\,  2^{\mu|\alpha|}\, 
\left( \frac{2^\mu C \widetilde C
|\gamma|^{\mu-1}}{|\lambda|}\right)^{|\gamma|}\, . 
\]
Then  using (\ref{eq:A5}) we obtain
\[
 \frac{2^\mu C \widetilde C |\gamma|^{\mu-1}}{|\lambda|} \ \le \ 
\frac{\delta 2^\mu C \widetilde C \lambda^0}{|\lambda|}\ \le \ \frac{1}{e}
\, ,
\]
which proves  (\ref{eq:A8}). 

Notice  that the
$\overline\partial_z$-derivatives of all terms in   (\ref{eq:A4})
with $|\gamma| < (\delta
\lambda^0)^{1/(\mu -1)} -1$ cancel, and then using  (\ref{eq:A8}) for
the remaining terms we  obtain (\ref{eq:A7}). 

\vspace{0.5cm}
\noindent
{\em Remark A.1. } 
{\em Suppose that $g(\varphi,I,\lambda)$ vanishes up to
infinite order at each $I\in E_\kappa$. Then $g$ satisfies
(\ref{eq:3.6}) uniformly with respect to $\lambda\in {\cal D}$. 
Choose $\delta$ as in  (\ref{eq:A5})
with $C$ replaced by $C_2=\max(C,C_1)$, where $C_1$ is the corresponding 
constant in (\ref{eq:3.6}) which depends only on $C$. Then for each 
$z =I + iY$ with $|E_\kappa - z| \le C_0 |\lambda|^{-1}$ and $C_0>0$
fixed,  we have}
\[
|D^\alpha_I D^\beta_\varphi   
g(\varphi,I + iY,\lambda,\lambda_0)|\ \le \ 
d_2\, (2^{\mu}C_2)^{|\alpha| +
|\beta| + 1}\, \alpha ! ^{\, \mu} \beta ! ^{\, \sigma}\, 
e^{-c |\lambda|^{1/(\mu -1)}}\,  . 
\]
To prove the remark, we apply (\ref{eq:3.6}) to each term of
(\ref{eq:A4}) and then use  (\ref{eq:A8}). 








\begin{thebibliography}{999}

\bibitem{A} V. Arnold, {\em  Mathematical methods of classical
mechanics}, 
Graduate Texts in Mathematics, {\bf 60}, Springer, 
New York - Heidelberg - Berlin, 1978. 

\bibitem{B} N. Burq,  Décroissance de l'énergie locale de
l'équation des ondes pour le problème extérieur et absence de
résonance au voisinage du réel, {\em Acta Math.} {\bf 180} (1998),
1-29. 

\bibitem{C-P} F. Cardoso, G. Popov,  Rayleigh quasimodes in
linear elasticity, {\em Comm. in Part. Diff. Equations} {\bf17} (1992),
1327--1367. 

\bibitem{CV} Y. Colin de Verdi\`ere,  Quasimodes sur les
vari\'et\'es Riemannienes, {\em Inventiones Math.} {\bf43} (1977), 15--52.



\bibitem{KP} S. Kuksin, J. P\"oschel, 
On the inclusion of analytic symplectic maps in analytic
Hamiltonian  flows and its applications, 
Kuksin, S. (ed.) et al., Seminar on dynamical systems,  
 Prog. Nonlinear Differ. Equ. Appl. {\bf 12} 
(1994), 96-116 , Birkh\"auser,

\bibitem{La} V. Lazutkin, {\em   KAM theory and semiclassical approximations
to eigenfunctions}, Springer, Berlin, 1993. 

\bibitem{L-Y} N. Lerner,   D. Yafaev,  On trace theorems for
pseudo-differential operators, 
Semin. Equ. Deriv. Partielles, Ec. Polytech., 
Cent. Math., Palaiseau Semin. 1995-1996, Exp. No.2, 18 p. (1996)

\bibitem{P} G. Popov,  Quasimodes for the Laplace operator and
glancing hypersurfaces, Proceedings of the Conference on Microlocal
Analysis and Nonlinear Waves, Ed. M. Beals, R. Melrose, J. Rauch,
Springer, 1991.

\bibitem{P1} G. Popov,  Invariant tori, effective stability and
quasimodes with exponentially small error terms I, Birkhoff normal
forms, {\em Ann. Henri Poincar\'e} {\bf 1} (2000), 223-248.

\bibitem{P2} G. Popov,  Invariant tori, effective stability and
quasimodes with exponentially small error terms II, Quantum Birkhoff normal
forms, {\em Ann. Henri Poincar\'e} {\bf 1} (2000), 249-279.

\bibitem{Ra} J. Ralston,  Trapped rays in spherically symmetric
media and poles of the scattering matrix, {\em Comm. Pure Appl. Math.} 
{\bf 24} (1971), 571-582. 

\bibitem{Ro} L. Rodino, {\em  Linear partial differential operators in
Gevrey spaces}, World Scientific, Singapore, 1993.

\bibitem{S} P. Stefanov,  Quasimodes and resonances: Sharp lower
bounds, {\em Duke Math. J.} {\bf 99}, (1999), 75-92. 

\bibitem{V1} G. Vodev,  Existence of Rayleigh resonances
exponentially close to the real axis, {\em Ann. Inst. H. Poincaré
(Physique Théorique)} {\bf 67} (1997), 41-57. 

\bibitem{V2} G. Vodev,  Resonances in the Euclidean scattering,
 {\em Cubo Mathem\'atica Educational} {\bf 3} (2001), 317-360.




                               
\end{thebibliography}

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