Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $\mathrm{\Sigma }$ of) compact hypersurfaces $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$. 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 ${\delta }^{\mathrm{\prime }}$-type, either assigned on $\mathrm{\Gamma }$ or on $\mathrm{\Sigma }\subset \mathrm{\Gamma }$.