Scattering theory with both regular and singular perturbations

Abstract

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_{\mathsf{B}},A)$, where both $A$ and $A_{\mathsf{B}}$ are self-adjoint operator and $A_{\mathsf{B}}$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $({\Delta }_{\mathsf{B}},\Delta )$, where $\Delta$ is the free self-adjoint Laplacian in $L^2({\mathbb{R}}^3)$ and ${\Delta }_{\mathsf{B}}$ is a self-adjoint operator in a class of Laplacians with both a regular perturbation, given by a short-range potential, and a singular one describing boundary conditions (like Dirichlet, Neumann and semi-transparent $\delta$ and ${\delta }^{\prime }$ ones) at the boundary of an open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_{\mathsf{B}}$ and a Kreĭn-like formula for the resolvent difference $(-A_{\mathsf{B}}+z{)}^{-1}-(-A+z{)}^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato–Rellich potential suffices) and the singular perturbations.