Content-Type: multipart/mixed; boundary="-------------0204110826153" This is a multi-part message in MIME format. ---------------0204110826153 Content-Type: text/plain; name="02-181.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-181.keywords" Quasimodes, Birkhoff normal form, resonances ---------------0204110826153 Content-Type: application/x-tex; name="f-g-revised.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="f-g-revised.tex" \documentclass[11pt]{article} \usepackage[latin1]{inputenc} \usepackage{latexsym} \textheight22cm \textwidth15cm \voffset=-1cm \hoffset=-1,3cm \def\sm{\Sigma_\lambda} \newcommand{\Ge}{\varepsilon} \newcommand{\Gr}{\varrho} \newcommand{\Gp}{\varphi} \newcommand{\nbd}{neighborhood } \newcommand{\Ga}{\alpha} \newcommand{\Id}{{\rm Id}} \newcommand{\rgl}{\rangle} \newcommand{\lgl}{\langle } \newcommand{\n}{\noindent} \newcommand{\dy}{\displaystyle} \newcommand{\midskip}{\vspace{8pt}} \newcommand{\noi}{\noindent} \newcommand{\pn}{\par\noindent} \newcommand{\CaixaPreta}{\vrule Depth0pt height5pt width5pt} \newcommand{\startproof}{\noindent {\em Proof.} \hspace{2mm}} \newcommand{\finishproof}{\hfill $\Box$ \vspace{5mm}} \newcommand{\compl}{\mbox{I${\!\!\!}$C}} \newcommand{\real}{\mbox{\it I${\!}$R}} \newcommand{\relN}{\mbox{\it I${\!}$N}} \newcommand{\intZ}{\mbox{Z${\!\!}$Z}} \newtheorem{theo}{Theorem}[section] \newtheorem{Theorem}{Theorem} \newtheorem{lemma}{Lemma}[section] \newtheorem{corol}{Corollary}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{rem}{Remark}[section] \newtheorem{cor}{Corollary} \renewcommand{\theequation}{\thesection .\arabic{equation}} \renewcommand{\arraystretch}{1.3} %\baselineskip20pt \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 nondegenerate 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. \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 nondegenerate. \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 $00$ 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 Hamitonian 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 $j0$ 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 $00$ 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 $0 n-1$ 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 $Q(x,I) = - K(I) +R(\varphi,I)$ 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}\, \alpha ! C^{|\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, $I_\nu = O(a)$, $00$, 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 $00$. 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 $00$ 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 $q>0$. 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 $\mu-1 >\sigma(\tau'+1)$, and $C,\, C_0\ge 1$. 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.} \hspace{2mm} A natural idea would be to change the variables putting $\omega = \omega(I)$ and then to apply the Faa de Bruno formula for the composition. Unfortunately, we lose control on the Gevrey constant in (\ref{eq:6.5}) in this way. We consider \[ \displaystyle z_k(I) = 1 - e^{-i\langle \omega(I),k\rangle} +\ i\kappa |k|^{-\tau}\, \psi\left(|\langle \omega(I),k\rangle|\, |k|^{\tau}\right)\, ,\quad 0\neq k \in {\bf Z}^n\, . \] Here $\psi\in G^{1+\varepsilon}(D)$, satisfies $0\le \psi \le 1$, $\psi(x) = 0$ for $x\ge \kappa/2$, and $\psi(x) = 1$ for $x\le \kappa/4$, and we fix $0<\varepsilon < \mu-1 -\sigma(\tau'+1) $. Choosing the constant $C_0>1$ sufficiently big we can suppose that $|D_x^p\psi(x)|\le C_0^{p+1}p!^{1+\varepsilon }$, $x\in{\bf R}$, $p\in {\bf Z}_+$. It follows from (\ref{eq:3.5}) 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 $\kappa_0= c_1\kappa$, $c_1>0$, $|k| = \sum_{j=1}^{n-1}|k_j|$. 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}). We are going to find suitable Gevrey estimates for $u_k(I)$. For this 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+\varepsilon} \, \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 $I\in D,\ k\in{\bf Z}^{n-1}\backslash\{0\},\ \alpha\in{\bf N}^{n-1}_+$. \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 $C_1 = 2^{n+1} C_0$ and using (\ref{eq:6.4}) we get (\ref{eq:6.7}) for $ z_{k1}$. Setting $\psi_k(x)=\kappa|k|^{-\tau}\psi(x|k|^{\tau})$, we get $|D_x^p\psi_k(x)|\le C_0^{p+1}p!^{1+\varepsilon}|k|^{\tau p-\tau}$, $x\in{\bf R}$, $p\in {\bf Z}_+$, and using the Faa de Bruno formula for $z_{k2}(I) = \kappa |k|^{-\tau}\, \psi\left(|\langle \omega(I),k\rangle|\, |k|^{\tau}\right) = \psi_k(|\langle \omega(I),k\rangle|)$, we get \[ \left|D_I^\alpha z_{k2}(I) \right| \le \displaystyle{\sum_{p=1}^{|\alpha|} \sum_{\scriptstyle{ {\alpha^1+\cdots+\alpha^p=\alpha} \atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\ C_0^{p+1}p!\, ^{\varepsilon}\, |k|^{\tau p +p -\tau}\, C_0^{|\alpha|} \alpha!\, \left((|\alpha^1|-1)!\cdots (|\alpha^p|-1)!\right)^{\tau'+1} , \] which implies (\ref{eq:6.7}) for $z_{k2}$ as above. \finishproof \begin{lemma} There exists a positive constant $C_1$ depending only on $C_0$, $\kappa_0$ and $n$ such that \begin{equation} |D_I^\alpha (z_k(I)^{-1})|\ \leq \ C_1^{|\alpha|+1}\, \alpha! \, ^{1+\varepsilon}\, \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 \[ \left|D_I^\alpha\left(z_k(I)^{-1}\right)\right| \ \le \ \left(\frac{\widetilde C_0}{\kappa_0}\right) ^{|\alpha|} \, \alpha !\,^{1+\varepsilon}\,|k|^{\tau} \displaystyle{ \sum_{p=1}^{|\alpha|}\sum_{\scriptstyle{ {\alpha^1+\cdots+\alpha^p=\alpha} \atop{|\alpha^1|\ge 1,\ldots, |\alpha^p| \ge 1}}}}\, \prod_{s=1}^{p}\, |k|^{(\tau +1)j_s}(|\alpha^s|-j_s)!^{\, \tau'+1} \, , \] where $1\le j_s\le |\alpha^s|$. Choosing $C_1 = 2^{n+1}\kappa_0^{-1}\widetilde C_0$ we prove (\ref{eq:6.9*}). \finishproof For any $m>0$ we set $\langle k\rangle_m = 1 + |k_1|^m +\cdots + |k_n|^m,\ k\in {\bf Z}^{n-1}$. Next, for any $j\in {\bf N}$ we denote $m(j)\ =\ [(\tau' +1)j + \tau] + n$, where $[p]$, $p\le [p] 0,\ k\in {\bf Z}^{n-1},$ 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+\varepsilon} \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 \[ k^\beta \langle k\rangle_m D^\alpha_I f_k( I)\ =\ (2\pi)^{1-n}\int_{{\bf T}^{n-1}}^{} e^{-i\langle k,\varphi \rangle}\, D_\varphi^\beta \langle D_\varphi \rangle_m D^\alpha_I f(\varphi,I)\, d\varphi \, . \] 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 $I \in D,\ k\in {\bf Z}^{n-1}$, 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 \[ W(k)\left|D^\gamma_I \left(z_k(I)^{-1} \right)\right|\, \left| k^\beta D^{\alpha-\gamma}_I f_k( I) \right| \] \[ \ \le \ (C_2C_3)^{|\gamma|+1}\, \gamma ! \, ^{1 + \varepsilon}\, \Gamma((\tau' +1)|(|\gamma|-j_\gamma )+1)\, \left|\langle k\rangle_{m(j_\gamma)}k^\beta D^{\alpha-\gamma}_I f_k( I) \right|\, , \] where $0\leq j_\gamma\leq |\gamma|$. On the other hand, $\gamma ! ^{\, \varepsilon}\ \leq\ \widetilde C_3^{|\gamma|+1}\, \Gamma (\varepsilon |\gamma| + 1)$, and using (\ref{eq:gamma}), we estimate $\gamma ! \, ^{\varepsilon}\, \Gamma((\tau' +1)|(|\gamma|-j_\gamma )+1) \le 2 \widetilde C_3^{|\gamma| + 1} \Gamma((\tau' +1)|(|\gamma|-j_\gamma )+\varepsilon |\gamma|+1)$. Taking into account (\ref{eq:6.11*}) and using again (\ref{eq:gamma}) we obtain \[ A_k \ \leq \ n d_0\, \alpha !\, \sum_{0\leq\gamma\leq\alpha}^{} C_4^{|\gamma| +1} \, C^t\, \Gamma (s)\, \, , \] where $C_4=C_2C_3 \widetilde C_3$ and \[ t= |\alpha-\gamma| +|\beta|+m(j_\gamma)\ , \] \[ s= (\mu-1) |\alpha-\gamma| + \sigma|\beta|+q + \sigma m(j_\gamma) + (\tau'+1)(|\gamma|-j_\gamma) + \varepsilon |\gamma|. \] Set $\delta =\mu -1 - \sigma(\tau'+1) -\varepsilon >0$. We have \[ s\ \le\ (\mu-1) |\alpha -\gamma| + \sigma |\beta| + \sigma(\nu-1) + q + \sigma (\tau'+1)j_\gamma + (\tau'+1)(|\gamma|-j_\gamma )+ \varepsilon |\gamma| \] \[ = \ (\mu-1) |\alpha| +\sigma |\beta| + \sigma(\nu-1) + q- \delta|\gamma| - (\sigma-1)(\tau'+1)(|\gamma|-j_\gamma)\, . \] Moreover, $t \le |\alpha| +|\beta|+m(|\gamma|)-|\gamma|$. Since $\delta >0$, $\sigma >1$ and $|\gamma|\ge j_\gamma$, 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 $d_1 = \displaystyle n\, \max_{1\le x\le 2}\Gamma (x)^{-1}$. 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} \end{document} ---------------0204110826153--