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The present status of the coupled-channel inverse-scattering method withsupersymmetric transformations is reviewed. We first revisit in a pedagogicalway the single-channel case, where the supersymmetric approach is shown toprovide a complete solution to the inverse-scattering problem. A specialemphasis is put on the differences between conservative and non-conservativetransformations. In particular, we show that for the zero initial potential, anon-conservative transformation is always equivalent to a pair of conservativetransformations. These single-channel results are illustrated on the inversionof the neutron-proton triplet eigenphase shifts for the S and D waves. We thensummarize and extend our previous works on the coupled-channel case and stressremaining difficulties and open questions. We mostly concentrate on two-channelexamples to illustrate general principles while keeping mathematics as simpleas possible. In particular, we discuss the difference between theequal-threshold and different-threshold problems. For equal thresholds,conservative transformations can provide non-diagonal Jost and scatteringmatrices. Iterations of such transformations are shown to lead to practicalalgorithms for inversion. A convenient technique where the mixing parameter isfitted independently of the eigenphases is developed with iterations of pairsof conjugate transformations and applied to the neutron-proton triplet S-Dscattering matrix, for which exactly-solvable matrix potential models areconstructed. For different thresholds, conservative transformations do not seemto be able to provide a non-trivial coupling between channels. In contrast, asingle non-conservative transformation can generate coupled-channel potentialsstarting from the zero potential and is a promising first step towards a fullsolution to the coupled-channel inverse problem with threshold differences.

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