Changing the discrete spectrum of the half-line matrix Schr\"odinger operator

TL;DR

This paper reviews how the discrete spectrum of a matrix Schr"odinger operator on the half line can be altered through transformations, without affecting the continuous spectrum, including adding, removing, or changing bound states.

Contribution

It provides a comprehensive analysis of spectral transformations for matrix Schr"odinger operators with explicit examples, expanding understanding of spectral modifications.

Findings

Describes transformations that change the discrete spectrum without affecting the continuous spectrum.

Analyzes the removal, addition, and multiplicity change of bound states.

Provides explicit examples illustrating the spectral transformation processes.

Abstract

We consider the matrix-valued Schr\"odinger operator on the half line with the general selfadjoint boundary condition. When the discrete spectrum is changed without changing the continuous spectrum, we present a review of the transformations of the relevant quantities including the regular solution, the Jost solution, the Jost matrix, the scattering matrix, and the boundary matrices used to describe the selfadjoint boundary condition. The changes in the discrete spectrum are considered when an existing bound state is removed, a new bound state is added, and the multiplicity of a bound state is decreased or increased without removing the bound state. We provide various explicit examples to illustrate the theoretical resultspresented.