Extension of the one-dimensional scattering theory, and ambiguities

Publisher Website
Google Scholar
Abstract
Extensions of the one-dimensional scattering theory are made in two directions. In the first, one uses the Schrodinger equation. Starting from the Fadeev class of potentials (V mod integral - infinity infinity (1+ mod x mod ) mod V(x) mod dx< infinity ) or from narrower classes, the authors introduce a special x-2 behaviour at infinity. In the second, they use the impedance equation, which reduces to the Schrodinger equation if the impedance factor alpha is positive everywhere and twice differentiable, but not if the impedance factor is only piecewise differentiable-this is the working assumption used in the present paper. Behaviour of the impedance at infinity which correspond to the x-2 behaviour of the potential are also admitted. They proceed through repeated applications of a transformation which is a limiting case of a Darboux transformation and acts on the parameters in a simple algebraic way, whereas the effect on the reflection coefficient R+(k) is an overall flip Rx to -R+. Because of these extensions of the scattering theory, ghosts, singularities, etc, may appear in the analysis. Ambiguities also appear in the inverse problem; these are related to zero-energy bound states. They are discussed in detail and the theory is shown to unify all the previously published material on potential ambiguities. The full inverse problem, however, is not treated by the authors.