RECONSTRUCTION OF A POTENTIAL ON THE LINE THAT IS A-PRIORI KNOWN ON THE HALF LINE

In this paper the authors study the inverse Schrodinger 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 partic ular no information on the norming constants is required. The method i s based on an appropriate trace formula and on the solution of the non linear ordinary differential equation that is obtained when the potent ial is replaced by its trace formula in the Schrodinger equation. The Deift-Trubowitz approach to inverse scattering is followed. The main n ew point is the way in which bound states are treated. In addition to its mathematical interest, the case when the potential is a priori kno wn on the half line is particularly interesting in many applications. One can consider for example a potential that has compact support or t hat it is zero on a half line.