On the spectrum of the Dirichlet Laplacian in a narrow strip

Leonid Friedlander, Michael Solomyak

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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.

Original languageEnglish (US)
Pages (from-to)337-354
Number of pages18
JournalIsrael Journal of Mathematics
Volume170
Issue number1
DOIs
StatePublished - Mar 1 2009

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ASJC Scopus subject areas

  • Mathematics(all)

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