Exact solution of the Marchenko equation relevant to inverse scattering on the line

This paper presents explicit solutions of the Marchenko equation relevant to the solution of the inverse scattering problem of determining the real potential Q(x) in the I-D Schrodinger equation on the line from the reflection coefficient R(k), which is assumed to be rational. The reflection coefficient is written in the form i C(k - iA)B-1. State space methods are applied to solve the Marchenko equation. both without and with bound states.