Eigenvalues and Resonances for Domains with Tubes: Neumann Boundary Conditions

We consider unbounded regions which consist of a bounded domain joined to an unbounded region 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 , resonances of , 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.