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On the spectrum of the Dirichlet Laplacian in a narrow strip. (English) Zbl 1173.35090
Summary: We consider the Dirichlet Laplacian \(\Delta_\varepsilon\) in a family of bounded domains \(\{-a<x<b\), \(0<y<\varepsilon h(x)\}\). The main assumption is that \(x = 0\) is the only point of global maximum of the positive, continuous function \(h(x)\). We find the two-term asymptotics in \(\varepsilon \rightarrow 0\) of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on \(\mathbb R\) that depends on the behavior of \(h(x)\) as \(x \rightarrow 0\). The proof is based on a detailed study of the resolvent of the operator \(\Delta_\varepsilon\).

35P15 Estimates of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
Full Text: DOI
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