Reconstruction Ambiguities of Inverse Scattering on the Line

摘要

One dimensional scattering problems governed by the Schroedinger equation or by the impedance equation were studied. In the absence of bound states, there exists a class of potentials that is bijectively related with a class of spectral data, i.e., reflection coefficients as a function of energy. On the other hand, it is known in larger classes several examples of different potentials that are consistent with a given reflection coefficient and no true bound state. It is shown that these ambiguities are related with a Darboux-type transformation which is defined on very wide classes of potentials, leaves invariant the Shroedinger equation whereas the reflection coefficient is flipped, and depends on one arbitrary parameter, so that a one parameter family of equivalent potentials is obtained. If potential classes are defined by their leading asymptotic behavior, the transformation takes a potential from one class to another one. The transformation leaves the transmission coefficient invariant but introduces or suppresses zero energy states or half bound states, so that it is not isospectral.