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On the spectrum of the Dirichlet Laplacian in a narrow infinite strip. (English) Zbl 1170.35487
Suslina, T. (ed.) et al., Spectral theory of differential operators. M. Sh. Birman 80th anniversary collection. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4738-1/hbk). Translations. Series 2. American Mathematical Society 225; Advances in the Mathematical Sciences 62, 103-116 (2008).
Summary: This is a continuation of the paper [L. Friedlander and M. Solomyak, Isr. J. Math. 170, 337–354 (2009; Zbl 1173.35090)]. We consider the Dirichlet Laplacian in a family of unbounded domains \(\{x\in\mathbb R\), \(0<y<\varepsilon h(x)\}\). The main assumption is that \(x=0\) is the only point of global maximum of the positive, continuous function \(h(x)\). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as \(\varepsilon\to 0\), and find the two-term asymptotics in \(\varepsilon\to 0\) 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 \(\mathbb R\) that depends only on the behavior of \(h(x)\) as \(x\to0\).
The proof is based on a detailed study of the resolvent of the operator \(\Delta_\varepsilon\).
For the entire collection see [Zbl 1152.47002].

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
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