The Inverse Scattering Problem for the Matrix Schr\"odinger Equation

TL;DR

This paper generalizes the inverse scattering problem for the matrix Schrödinger equation on the half line, establishing conditions for the unique reconstruction of potentials and boundary conditions from scattering data.

Contribution

It extends classical inverse scattering results to general selfadjoint boundary conditions, providing necessary and sufficient conditions for data correspondence.

Findings

Characterized scattering data sets for the matrix Schrödinger equation.

Proved existence and uniqueness of potential and boundary matrices from scattering data.

Provided explicit examples illustrating the theory.

Abstract

The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix potential is integrable, is selfadjoint, and has a finite first moment. The corresponding scattering data set is constructed, and such scattering data sets are characterized by providing a set of necessary and sufficient conditions assuring the existence and uniqueness of the correspondence between the scattering data set and the input data set containing the potential and boundary matrices. The work presented here provides a generalization of the classical result by Agranovich and Marchenko from the Dirichlet boundary condition to the general selfadjoint boundary condition. The theory presented is illustrated with various explicit examples.