Scattering theory with both regular and singular perturbations

TL;DR

This paper develops a comprehensive scattering theory framework for operators with combined regular and singular perturbations, including boundary conditions, providing new formulas and criteria applicable to Laplacians with complex boundary interactions.

Contribution

It introduces an asymptotic completeness criterion and a representation formula for the scattering matrix involving both regular and singular perturbations, applicable to Laplacians with boundary conditions.

Findings

Established an asymptotic completeness criterion.

Derived a Krein-like resolvent formula for combined perturbations.

Applied results to Laplacians with boundary conditions like Dirichlet, Neumann, and delta interactions.

Abstract

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(\Delta_B,\Delta)$, where $\Delta$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $\Delta_B$ 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 $\delta'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations.