On the Spectrum of the Dirichlet Laplacian in a Narrow Infinite Strip

This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < h(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as → 0, and find the two-term asymptotics in → 0 of each eigenvalue and the one-term asymptotics of the corresponding eigenfunction. The asymptotic formulae obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on R that depends only on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator ∆ .