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On the spectrum of the Dirichlet Laplacian in a narrow strip

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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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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, Az, USA

    Leonid Friedlander

  2. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Michael Solomyak

Authors
  1. Leonid Friedlander
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  2. Michael Solomyak
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Correspondence to Leonid Friedlander.

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Friedlander, L., Solomyak, M. On the spectrum of the Dirichlet Laplacian in a narrow strip. Isr. J. Math. 170, 337–354 (2009). https://doi.org/10.1007/s11856-009-0032-y

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  • DOI: https://doi.org/10.1007/s11856-009-0032-y

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