Eigenvalue Asymptotics of Narrow Domains

Abstract

We considered the spectrum of the Dirichlet Laplacian $\Delta_\epsilon$ on the planar domain $$\Omega_\epsilon=\{(x,y): -l_1<x<l_2, 0<y<\epsilon h(x)]\}$$ 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. First we decompose $\Delta_\epsilon$ corresponding to the decomposition of the vertical $L^2$ space into the fundamental mode and remaining higher modes. Then we analyze model operator corresponding to the fundamental mode. In the end we investigate the difference between the model operator and the original $\Delta_\epsilon$.