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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 $$\Delta_\varepsilon$$.

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