The generalized Marchenko method in the inverse scattering problem for a first-order linear system

TL;DR

This paper extends the Marchenko inverse scattering method to a broader class of first-order linear systems, providing a unified framework for potential reconstruction and explicit solutions, including reflectionless cases.

Contribution

The paper generalizes the Marchenko method to new linear systems, deriving integral equations applicable to systems lacking existing inverse scattering solutions.

Findings

Derived Marchenko integral equations for generalized systems

Explicit formulas for reflectionless potentials and solutions

Described bound-state data in terms of matrix triplets

Abstract

The Marchenko method is developed in the inverse scattering problem for a linear system of first-order differential equations containing potentials proportional to the spectral parameter. The corresponding Marchenko system of integral equations is derived in such a way that the method can be applied to some other linear systems for which a Marchenko method is not yet available. It is shown how the potentials and the scattering solutions to the linear system are constructed from the solution to the Marchenko system. The bound-state information for the linear system with any number of bound states and any multiplicities is described in terms of a pair of constant matrix triplets. When the potentials in the linear system are reflectionless, some explicit solution formulas are presented in closed form for the potentials and for the scattering solutions to the linear system. The theory is illustrated with some explicit examples.