Singular problems in quantum and elastic waveguides via Dirichlet-to-Neumann analysis

The present work is concerned with the existence and the behaviour of discrete eigenvalues arising from a perturbation of an infinite waveguide in quantum mechanics or elasticity theory. Physically, these eigenvalues correspond to bound states in quantum mechanics or harmonic oscillations near imperfections such as cracks and holes in elasticity theory. They play an important role in applications in science and industry such as non-destructive testing theory of sensitive structures (e. g. wings of air planes, nuclear components etc.). More specifically we consider the asymptotic behaviour of the discrete eigenvalues for small perturbations which arise from a change of the boundary conditions on a small subset of the boundary. To determine the asymptotic behaviour of the discrete eigenvalues we shall use a pseudo-differential approach which is based on an analysis of the corresponding Dirichlet-to-Neumann operator. This method is of benefit mainly because it uses the symbol expansion of the Dirichlet-to-Neumann operator in local charts and the behaviour of the unperturbed resolvent near the spectral minimum. In this way we are able to reformulate the singular perturbation of the boundary conditions into an additive perturbation of the Dirichlet-to-Neumann operator. The asymptotic formula for the eigenvalues follows by a subsequent Birman-Schwinger analysis of the Dirichet-to-Neumann operator.