Inverse scattering for the matrix Schroedinger operator and Schroedinger operator on graphs with general self-adjoint boundary conditions

TL;DR

This paper develops a method for inverse scattering of matrix Schrödinger operators with general self-adjoint boundary conditions, using a parameterization approach and Darboux transformations, with applications to quantum graphs.

Contribution

It introduces a novel scheme for factorizing matrix Schrödinger operators and solving inverse problems with general boundary conditions, extending previous methods.

Findings

Constructed Darboux transformations that modify potentials and boundary conditions.

Provided a solution to the inverse scattering problem using a Marchenko equation.

Discussed applications to quantum graphs with trivial compact parts.

Abstract

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schroedinger operator and hence construct a Darboux transformation an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discusss the specialisation to the case of graph with trivial compact part, ie. diagonal matrix potential.