Discrete Ambiguities and Equivalent Potentials

Ambiguities arising in the construction of potentials from the scattering amplitude at a given energy are studied. The functional space of all possible potentials is explored in general by means of the JWKB approximation and in particular by the generalized Born approximation. Typical kinds of ambiguities are exhibited. Certain of them ("window" ambiguities) arise from the fact that wave functions are unable to test analytical details over distances much smaller than a wavelength. Others are owing to the parts of the potential that are located between turning points and therefore not "visible" in the classical limit. These parts can be deformed slightly without perturbing the phase shifts (continuous ambiguity) and besides can also be modified drastically so as to include more nodes of the wave function. This strong modification yields classes of equivalent potentials which are layered near the origin. As a first approximation, it is very remarkable that the layering does not depend on any detail of the shape of the potential.