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

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.