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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
##### Keywords:
Dirichlet Laplacian; spectrum; infinite strip; resolvent
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