Eigenvalues and Resonances for Domains with Tubes: Neumann Boundary Conditions

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20 Scopus citations


We consider unbounded regions which consist of a bounded domain 풞 joined to an unbounded region ℰ by a tube T(ε) whose cross-section is of small diameter ε. On such a region, we consider the Laplacian with Neumann boundary conditions. We show that as ε → 0+, the spectral resonances converge to eigenvalues of 풞, resonances of ℰ, or eigenvalues for a two point boundary value problem on an interval of the same length as the tube. The main goal of our work is to give estimates for the rates of convergence.

Original languageEnglish
Pages (from-to)458-476
Number of pages19
JournalJournal of Differential Equations
Issue number2
StatePublished - Jan 20 1995

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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