On the Riemann–Hilbert problem for the one-dimensional Schrödinger equation

Abstract

A matrix Riemann–Hilbert problem associated with the one-dimensional Schrödinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann–Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann–Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.