Mathematics

We provide a limiting absorption principle for the self-adjoint realisations of Laplace operators corresponding to boundary conditions on (relatively open parts Σ of) compact hyper-surfaces Г=∂Ω, Ω⊂ℝ^n. For any of such self-adjoint operators we also provide a generalised eigenfunction expansion and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Kreĭn-type formula which provides the resolvent difference between the free Laplacian on the whole space ℝ^n and the one corresponding to boundary conditions on the hyper-surface. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, δ, and δ'-type, either assigned on Г or on Σ⊂Г.