Darboux-Crum-Krein transformations and generalized Bargmann potentials

In this talk we deal with the inverse scattering problem for non-relativistic quantum mechanical systems. For local spherically symmetric potentials V(r) the corresponding Schroedinger equation at a given energy k2 reduces to an ordinary second-order differential equation in the radius r for each angular momentum quantum number l=λ-1/2. The S-matrices for such elastic scattering processes therefore depend on the two continuous variables λ and k. Since the potential V(r) depends only on one continuous variable r, it is plausible that the corresponding inverse scattering problem is overdetermined. In order to overcome this difficulty one variable, either λ or k, is usually kept constant. Thus, one arrives at the standard inverse scattering problems at fixed angular momentum and fixed energy, respectively. (orig.)