Fernando Cardoso, Georgi Popov
Quasimodes with Exponentially Small Errors
Associated with Elliptic Periodic Rays
(106K, LaTeX 2e)
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 non-degenerate 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.
This is a new version of mparc 02-181, where the beginning of
the proof of Proposition 6.1 is revised.