Small-Energy Analysis for the Selfadjoint Matrix Schroedinger Operator on the Half Line

TL;DR

This paper analyzes the low-energy behavior of the selfadjoint matrix Schrödinger operator on the half line, providing explicit formulas and continuity results for the scattering matrix at zero energy, with implications for inverse scattering.

Contribution

It offers new explicit formulas and continuity results for the scattering matrix at zero energy for matrix Schrödinger operators with general boundary conditions.

Findings

Scattering matrix is continuous at zero energy.

Explicit formula for the scattering matrix at zero energy.

Established small-energy asymptotics for related quantities.

Abstract

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the corresponding Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.