On Lax-Phillips scattering matrix of the abstract wave equation

TL;DR

This paper investigates how the singularities of scattering matrices for the abstract wave equation depend on the choice of asymptotic subspaces, with applications to radial wave equations involving nonlocal potentials.

Contribution

It introduces a detailed analysis of the dependence of scattering matrix singularities on asymptotic subspace choices and applies the results to radial wave equations with nonlocal potentials, utilizing the concept of associated inner functions.

Findings

Singularities of scattering matrices depend on asymptotic subspace choices.

Application of the theory to radial wave equations with nonlocal potentials.

Use of associated inner functions in the analysis.

Abstract

The dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces is studied. The obtained results are applied to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function introduced in the Douglas-Shapiro-Shields work \cite{DSS} plays an essential role.