Quasimodes with exponentially small errors associated with elliptic periodic rays

Abstract

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 γ in X. We find a Cantor family Λ of invariant tori of the corresponding Poincaré map which is Gevrey smooth with respect to the transversal variables (the frequencies). Quantizing the Gevrey family Λ, 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.