Factorization for the full-line matrix Schr\"odinger equation and a unitary transformation to the half-line scattering

TL;DR

This paper develops a factorization approach for the matrix Schrödinger equation on the full line, establishes a unitary transformation to the half-line problem, and relates their scattering matrices, providing new insights into their spectral properties.

Contribution

It introduces a factorization formula for the scattering matrix of the full-line matrix Schrödinger equation and constructs a unitary transformation linking full-line and half-line operators.

Findings

Explicit examples show left and right transmission coefficients can differ.

A unitary transformation relates full-line and half-line Schrödinger operators.

Levinson's theorem is extended from the half-line to the full-line context.

Abstract

The scattering matrix for the full-line matrix Schr\"odinger equation is analyzed when the corresponding matrix-valued potential is selfadjoint, integrable, and has a finite first moment. The matrix-valued potential is decomposed into a finite number of fragments, and a factorization formula is presented expressing the matrix-valued scattering coefficients in terms of the matrix-valued scattering coefficients for the fragments. Using the factorization formula, some explicit examples are provided illustrating that in general the left and right matrix-valued transmission coefficients are unequal. A unitary transformation is established between the full-line matrix Schr\"odinger operator and the half-line matrix Schr\"odinger operator with a particular selfadjoint boundary condition and by relating the full-line and half-line potentials appropriately. Using that unitary transformation, the relations are established between the full-line and the half-line quantities such as the Jost solutions, the physical solutions, and the scattering matrices. Exploiting the connection between the corresponding full-line and half-line scattering matrices, Levinson's theorem on the full line is proved and is related to Levinson's theorem on the half line.