Finite Rank Perturbations, Scattering Matrices and Inverse Problems

TL;DR

This paper develops a mathematical framework connecting finite rank perturbations of operators to scattering matrices and inverse problems, using boundary triplets and Nevanlinna functions, with explicit solutions for inverse scattering.

Contribution

It introduces a novel extension theoretic approach to express scattering matrices via matrix Nevanlinna functions and solves an inverse scattering problem explicitly.

Findings

Expressed scattering matrix in terms of Nevanlinna functions.

Extended representation to dissipative systems.

Provided explicit solution to inverse scattering problem.

Abstract

In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. The representation results are extended to dissipative scattering systems and an explicit solution of an inverse scattering problem for the Lax-Phillips scattering matrix is presented.