On the spectrum of the Dirichlet Laplacian in a narrow strip |
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Authors: | Leonid Friedlander Michael Solomyak |
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Affiliation: | (1) Department of Mathematics, University of Arizona, Tucson, Az, USA;(2) Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel |
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Abstract: | We consider the Dirichlet Laplacian Δ∈ in a family of bounded domains {−a < x < b, 0 < y < εh(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 ε → 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 ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ∈. |
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