Use of Partial Knowledge of the Potential in the Phase Problem of Inverse Scattering

We consider the problem of determining a potential V(x) in the one-dimensional Schrödinger equation, given as data the reflectivity r(k) = | R(k)| ², where R(k) denotes the usual quantum mechanical reflection coefficient. It is well known that in the absence of phase information, there can be a considerable degree of nonuniqueness, which is closely connected to the presence of zeros of R(k) in the upper half of the complex plane. Some earlier work of the authors showed that this ambiguity can be resolved by providing a small amount of extra information about the potential. In this article we develop a computational technique, based on an optimization approach to the problem of locating the zeros of R(k). Some numerical examples are given.