Phase recovery with nondecaying potentials

The one-dimensional Schrodinger equation is considered when the potential is asymptotic to a positive constant on the right-half line and may possess hound states. The recovery of such a potential supported in the right-half line is studied in terms of the scattering data consisting of the magnitude of the reflection coefficient from the left, a known potential placed to the left of the unknown potential, and the magnitude of the reflection coefficient for the combined potential. It is shown that there are at most two distinct potentials corresponding to the scattering data, that the uniqueness holds in the absence of bound states, and that in the case of nonuniqueness the reflection coefficients of the two potentials are related to each other in a specific way. A quadrature is presented to obtain the analytic continuation of the reflection coefficient. The theory is illustrated with some explicit examples.