Scattering Theory for the matrix Schr\"odinger operator on the half line with general boundary conditions

TL;DR

This paper develops a comprehensive scattering theory for matrix Schr"odinger operators on the half line with general boundary conditions, including spectral analysis, wave operators, and Levinson's theorem, for integrable potentials.

Contribution

It introduces a general boundary condition framework for matrix Schr"odinger operators and establishes key spectral and scattering properties, including the spectral shift function and Levinson's theorem.

Findings

Proved the limiting absorption principle.

Constructed generalized Fourier maps and wave operators.

Derived high-energy asymptotics of the spectral shift function.

Abstract

We study the stationary scattering theory for the matrix Schr\"odinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space the subspace of absolute continuity of the matrix Schr\"odinger operator and final space $L^2((0, \infty))$. We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We also construct the spectral shift function and we give its high-energy asymptotics. Furthermore, assuming that the potential also has a finite first moment, we prove a Levinson's theorem for the spectral shift function.