B. Grebert et R. Weder, RECONSTRUCTION OF A POTENTIAL ON THE LINE THAT IS A-PRIORI KNOWN ON THE HALF LINE, SIAM journal on applied mathematics, 55(1), 1995, pp. 242-254
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.