Scattering Matrices for Close Singular Selfadjoint Perturbations of Unbounded Selfadjoint Operators

TL;DR

This paper derives explicit formulas for the scattering matrix of unbounded selfadjoint operators with singular perturbations, using Krein's formula, and applies it to Laplace operators with zero-range potentials.

Contribution

It provides a new explicit expression for the scattering matrix in the context of singular perturbations of unbounded selfadjoint operators, extending previous results.

Findings

Explicit scattering matrix formula derived for singular perturbations

Application to Laplace operator with zero-range potentials

Resolvent difference is trace class under assumptions

Abstract

In this paper, we consider an unbounded selfadjoint operator $A$ and its selfadjoint perturbations in the same Hilbert space $\mathcal{H}$. As S.Albeverio and P. Kurosov (2000), we call a selfadjoint operator $A_{1}$ the singular perturbation of $A$ if $A_{1}$ and {A} have different domains $\mathcal{D}(A),\mathcal{D}(A_{1})$ but $A=A_{1}$ on $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$. Assuming that $A$ has absolutely continuous spectrum and the difference of resolvents $R_{z}(A_{1}) -R_{z}(A)$ of $A_{1}$ and $A$ for non-real $z$ is a trace class operator we find the explicit expression for the scattering matrix for the pair $A, A_{1}$ through the constituent elements of the Krein formula for the resolvents of this pair. As an illustration, we find the scattering matrix for the standardly defined Laplace operator in $L_{2}\left(\mathbf{R}_{3}\right)$ and its singular perturbation in the form of an infinite sum of zero-range potentials.