Inverse Schrödinger Cattering on the Line with Partial Knowledge of the Potential

The one-dimensional Schrodinger equation is considered when the potential and its rst moment are absolutely integrable. The potential is uniquely constructed in terms of the scattering data consisting of the re ection coe cient from the right (left) and the knowledge of the potential on the right (left) half line of the real axis. Hence, neither the bound state energies nor the bound state norming constants are needed to determine the potential, and in fact these are uniquely determined by the scattering data. An explicit example is provided. Also considered are two inverse scattering problems for a generalized Schr odinger equation with two potentials when the potential to be recovered is partially known. Mathematics Subject Classi cation (1991): 81U40, 34A55 PACS Numbers: 03.65.Nk, 03.80.+r Short title: Inverse scattering without bound state information