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This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian $H$ close to a completely integrable one and a suitable Cantor set $\Theta$ defined by a Diophantine condition, we find a family ${\Lambda }_{\omega },\omega \in \Theta$, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta$ which is Gevrey smooth with...

This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle $𝒪$ in ${ℝ}^n$, $n\ge 2$, with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency support of...

This paper is devoted to the effective stability estimates (of Nekhoroshev’s type) of the billiard flow for strictly convex bounded domains with analytic boundaries in any dimensions. The main result is that any billiard trajectory with initial data which are $\delta$ - close to the glancing manifold remains close to the glancing manifold in an exponentially large time interval with respect to $1/\delta$. The proof is based on a normal form of the billiard ball map in Gevrey classes. More generally, we prove effective...

Let $S\left(\lambda \right)$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle $𝒪\subset {\mathbf{R}}^n$, $n\ge 3$ with Dirichlet or Neumann boundary conditions on $\partial 𝒪$. The function $s\left(\lambda \right)$, called scattering phase, is determined from the equality $e^{-2\pi is\left(\lambda \right)}=\det S\left(\lambda \right)$. We show that $s\left(\lambda \right)$ has an asymptotic expansion $s\left(\lambda \right)\sim {\sum }_{j=0}^{\infty }{c}_j{\lambda }^{n-j}$ as $\lambda \to +\infty$ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.