Numerical Computations in the Inverse-Scattering Problem at Fixed Energy

Constructing potentials from the phase shifts at a given energy yields an infinity of equivalent solutions. The deviations of these solutions from each other can, however, be analyzed according to a priori limitations on the derivatives and other features of "acceptable" potentials. A sketch of this analysis is given together with a numerical comparison of usual potential forms with the equivalent potentials obtained through Newton's method. The observed deviation gives an appraisal of the deviations from each other of all the equivalent potentials with similar bounds on the derivatives. The deviation is small when there are many phase shifts available, all of them definitely smaller than $\frac{\pi }{2}$. For a static potential these conditions can be met for high energies.