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

V. F. Lazutkin

Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 439-466

Résumé

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.