Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

Abstract

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $\Sigma$ of) compact hypersurfaces $\Gamma =\partial \Omega ,\Omega \subset {\mathbb{R}}^n$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space ${\mathbb{R}}^n$. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, $\delta$ and $\delta \prime$-type, either assigned on $\Gamma$ or on $\Sigma \subset \Gamma$.