Eigenvalue Asymptotics of Narrow Domains

In this paper, we considered the spectrum of the Dirichlet Laplacian $\Delta_\epsilon$ on $\Omega_\epsilon=\{(x,y): -l_1<x<l_2, 0<y<\epsilon h(x)]\}$ where $l_1,l_2>0$ and $h(x)$ is a positive analytic function having $0$ the only point where it achieves its global maximum $M$. In particular we studied in details about the full asymptotics of the eigenvalues.