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Bound states in weakly deformed strips and layers. (English) Zbl 1043.35046

This paper deals with the spectra of Dirichlet Laplacian on straight strips in \(\mathbb{R}^2\) or layers in \(\mathbb{R}^3\) with a weak local deformation. Of course, the problem is trivial as long as the strip or layer is straight because then one can employ separation of variables. However, already a local perturbation such as bending, deformation, or a change of boundary conditions can produce a non-empty discrete spectrum. The goal of the present paper is to show the effect of a local deformation of the strip or layer, which is more subtle than the bending or boundary condition modification. The main difference is that the effective interaction induced by deformation can be of different signs, both attractive and repulsive.

MSC:

35C20 Asymptotic expansions of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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