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Eigenvalues and resonances for domains with tubes: Neumann boundary conditions. (English) Zbl 0815.35075
Authors’ summary: “We consider unbounded regions which consist of a bounded domain $${\mathcal C}$$ joined to an unbounded region $${\mathcal E}$$ by a tube $$T(\varepsilon)$$ whose cross-section is of small diameter $$\varepsilon$$. On such a region, we consider the Laplacian with Neumann boundary conditions. We show that as $$\varepsilon \to 0^ +$$, the spectral resonances converge to eigenvalues of $${\mathcal C}$$, resonances of $${\mathcal E}$$, 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”.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
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