Single- and coupled-channel radial inverse scattering with supersymmetric transformations

Abstract

The present status of the three-dimensional inverse-scattering method with supersymmetric transformations is reviewed for the coupled-channel case. We first revisit in a pedagogical way the single-channel case, where the supersymmetric approach is shown to provide a complete, efficient and elegant solution to the inverse-scattering problem for the radial Schrödinger equation with short-range interactions. A special emphasis is put on the differences between conservative and non-conservative transformations, i.e. transformations that do or do not conserve the behaviour of solutions of the radial Schrödinger equation at the origin. In particular, we show that for the zero initial potential, a non-conservative transformation is always equivalent to a pair of conservative transformations. These single-channel results are illustrated on the inversion of the neutron–proton triplet eigenphase shifts for the S- and D-waves. We then summarize and extend our previous works on the coupled-channel case, i.e. on systems of coupled radial Schrödinger equations, and stress remaining difficulties and open questions of this problem by putting it in perspective with the single-channel case. We mostly concentrate on two-channel examples to illustrate general principles while keeping mathematics as simple as possible. In particular, we discuss the important difference between the equal-threshold and different-threshold problems. For equal thresholds, conservative transformations can provide non-diagonal Jost and scattering matrices. Iterations of such transformations in the two-channel case are studied and shown to lead to practical algorithms for inversion. A convenient particular technique where the mixing parameter can be fitted without modifying the eigenphases is developed with iterations of pairs of conjugate transformations. This technique is applied to the neutron–proton triplet S–D scattering matrix, for which exactly-solvable matrix potential models are constructed. For different thresholds, conservative transformations do not seem to be able to provide a non-trivial coupling between channels. In contrast, a single non-conservative transformation can generate coupled-channel potentials starting from the zero potential and is a promising first step towards a full solution to the coupled-channel inverse problem with threshold differences.