par Baye, Daniel Jean 
;Sparenberg, Jean-Marc ;Pupasov, Andrey ;Samsonov, Boris
Référence Journal of Physics A: Mathematical and Theoretical, 47, 24, 243001
Publication Publié, 2014-06-02
;Sparenberg, Jean-Marc ;Pupasov, Andrey ;Samsonov, Boris Référence Journal of Physics A: Mathematical and Theoretical, 47, 24, 243001
Publication Publié, 2014-06-02
Article révisé par les pairs
| Résumé : | The present status of the three-dimensional inverse-scattering method withsupersymmetric transformations is reviewed for the coupled-channel case.We first revisit in a pedagogical way the single-channel case, where thesupersymmetric approach is shown to provide a complete, efficient andelegant solution to the inverse-scattering problem for the radial Schrödingerequation with short-range interactions. A special emphasis is put on thedifferences between conservative and non-conservative transformations, i.e.transformations that do or do not conserve the behaviour of solutions of theradial Schrödinger equation at the origin. In particular, we show that for thezero initial potential, a non-conservative transformation is always equivalentto a pair of conservative transformations. These single-channel results areillustrated on the inversion of the neutron–proton triplet eigenphase shifts forthe S- and D-waves. We then summarize and extend our previous works on thecoupled-channel case, i.e. on systems of coupled radial Schrödinger equations,and stress remaining difficulties and open questions of this problem by puttingit in perspective with the single-channel case. We mostly concentrate on two-channel examples to illustrate general principles while keeping mathematics assimple as possible. In particular, we discuss the important difference betweenthe equal-threshold and different-threshold problems. For equal thresholds,conservative transformations can provide non-diagonal Jost and scatteringmatrices. Iterations of such transformations in the two-channel case are studiedand shown to lead to practical algorithms for inversion. A convenient particulartechnique where the mixing parameter can be fitted without modifying theeigenphases is developed with iterations of pairs of conjugate transformations.This technique is applied to the neutron–proton triplet S–D scattering matrix, forwhich exactly-solvable matrix potential models are constructed. For differentthresholds, conservative transformations do not seem to be able to provide anon-trivial coupling between channels. In contrast, a single non-conservativetransformation can generate coupled-channel potentials starting from the zeropotential and is a promising first step towards a full solution to the coupled-channel inverse problem with threshold differences. |
