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Eigenvalues and Resonances for Domains with Tubes: Neumann Boundary Conditions

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
115
Issue
2
Identifiers
DOI: 10.1006/jdeq.1995.1023

Abstract

Abstract We consider unbounded regions which consist of a bounded domain C 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 C , 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.

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