02-181 Fernando Cardoso, Georgi Popov

Quasimodes with Exponentially Small Errors Associated with Elliptic Periodic Rays (106K, LaTeX 2e) Apr 11, 02

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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 $\gamma$ in $X$. We find a Cantor family $\Lambda$ of invariant tori of the corresponding Poincar 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.

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