Reconstruction of a Potential on the Line That is a Priori Known on the Half Line

Abstract

In this paper the authors study the inverse Schrödinger scattering on the real line. A method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies. In particular no information on the forming constants is required. The method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the Schrödinger equation. The Deift–Trubowitz approach to inverse scattering is followed. The main new point is the way in which bound states are treated. In addition to its mathematical interest, the case when the potential is a priori known on the half line is particularly interesting in many applications. One can consider for example a potential that has compact support or that it is zero on a half line.