Eigenvalues and resonances for domains with tubes: Neumann boundary conditions. (English) Zbl 0815.35075

Brown, Russell; Hislop, P. D.; Martinez, A.

J. Differ. Equations 115, No. 2, 458-476 (1995).

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”.

Reviewer: Ya.A.Rojtberg (Chernigov)

MSC:

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

Keywords:

unbounded regions; Laplacian; Neumann boundary conditions; spectral resonances; eigenvalues; rates of convergence