Quasimodes with exponentially small errors associated with elliptic periodic rays

Article type: Research Article

Authors: Cardoso, Fernando^; | Popov, Georgi

Affiliations: Universidade Federal de Pernambuco, Departamento de Matemática, 50.540‐740 Recife‐Pe, Brazil E‐mail: fernando@dmat.ufpe.br | Département de Mathématiques, Université de Nantes, UMR 6629 du CNRS, 2, rue de la Houssinière, BP 92208, 44072 Nantes cedex 03, France E‐mail: popov@math.univ‐nantes.fr

Note: [] Partially supported by CNPq (Brazil).

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.

Journal: Asymptotic Analysis, vol. 30, no. 3-4, pp. 217-247, 2002