Scattering matrices and Dirichlet-to-Neumann maps

TL;DR

This paper derives a general formula for the scattering matrix using operator-valued Titchmarsh-Weyl functions and applies it to various Schrödinger operator problems, expressing the scattering matrix explicitly via Dirichlet-to-Neumann maps.

Contribution

It introduces a unified representation formula for scattering matrices in terms of an abstract operator-valued Titchmarsh-Weyl m-function and applies it to multiple Schrödinger operator scenarios.

Findings

Explicit scattering matrix formulas for Schrödinger operators on unbounded domains

Representation of scattering matrices using Dirichlet-to-Neumann maps

Unified approach applicable to different self-adjoint realizations

Abstract

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl $m$-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schr\"{o}dinger operators on unbounded domains, Schr\"{o}dinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schr\"{o}dinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.