High-Energy analysis and Levinson's theorem for the selfadjoint matrix Schroedinger operator on the half line

TL;DR

This paper analyzes the high-energy behavior of the matrix Schrödinger operator with selfadjoint boundary conditions on the half line, deriving asymptotics and Levinson's theorem relating bound states to scattering data.

Contribution

It establishes high-energy asymptotics for the Jost and scattering matrices and derives Levinson's theorem for matrix Schrödinger operators with integrable potentials.

Findings

High-energy asymptotics for Jost and scattering matrices

Levinson's theorem relating bound states to scattering phase shifts

Results applicable to selfadjoint boundary conditions on the half line

Abstract

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the related Jost matrix, the inverse of the Jost matrix, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, Levinson's theorem is derived, relating the number of bound states to the change in the argument of the determinant of the scattering matrix.