Asymptotic Completeness and S-Matrix for Singular Perturbations

TL;DR

This paper establishes a criterion for asymptotic completeness and provides a representation of the scattering matrix for certain pairs of semi-bounded self-adjoint operators, with applications to Laplacians on rough hypersurfaces.

Contribution

It introduces a new criterion for asymptotic completeness without requiring trace-class conditions on resolvent differences.

Findings

Derived a criterion for asymptotic completeness.

Represented the scattering matrix explicitly.

Applied results to Laplacians with boundary conditions on rough hypersurfaces.

Abstract

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.