Inverse scattering with singular potentials: A supersymmetric approach

Abstract

By using potentials with a singularity at the origin, the inverse scattering problem at fixed orbital momentum $l$ can be decomposed in two parts. First, the unique singular potential without a bound state corresponding to a given phase shift is constructed; then, bound states may be added without modifying the phase shift. The first step, called the singular inverse problem, is discussed. When the phase shift is smaller at high energies than at zero energy, the obtained effective potential has a repulsive core of the form $\ensuremath{\nu}(\ensuremath{\nu}{+1)r}^{\ensuremath{-}2}$ where $\ensuremath{\nu}$ is larger than $l$. If the $S$ matrix can be approximated by the product of the $S$ matrix of a reference potential by a rational function of the wave number $k$, the singular potential is a generalized Bargmann potential. It can be constructed with supersymmetric transformations of the reference potential. Each transformation adds a pole to the $S$ matrix. The repulsive core parameter $\ensuremath{\nu}$ of the final potential is equal to $l$ plus the difference of the number of added poles in the upper and lower half $k$ planes. This generalized Bargmann potential as well as its solutions can be expressed in terms of the reference potential and of Wronskians of its solutions. As an application, we invert the phase shifts of neutron-proton and proton-proton ${}^{1}{S}_{0}$ elastic scatterings and obtain in both cases a $\ensuremath{\nu}=1$ singular nuclear potential with two wells. In the neutron-proton case, this potential is compared with a regular potential obtained from a Gel'fand-Levitan-Marchenko inversion method.