Asymptotics of the eigenvalues of the Laplacian and quasimodes. A series of quasimodes corresponding to a system of caustics close to the boundary of the domain
Abstract:
For a bounded convex domain in the plane, asymptotic formulas with error tending to zero are constructed for a certain series of eigenvalues of the Laplacian with zero boundary conditions. The boundary of the domain is assumed to be sufficiently smooth. It is proved that
$$
\varliminf_{\lambda\to+\infty}N^*(\lambda)/N(\lambda)>0,
$$
where $N(\lambda)$ is the number of eigenvalues (with multiplicities taken into account) less than $\lambda$ and $N^*(\lambda)$ is the number of those eigenvalues for which an asymptotic expansion has been found.
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