Discretisation of inverse scattering on the half-line and conformal mappings

We consider Schrödinger operators with compactly supported potentials on the half line. We prove the following results about discretization of inverse scattering: i) if we know the Jost function at some sequence of real numbers then we can recover the potential. ii) If we know the S-matrix at some sequence of real numbers then we can recover the potential.
With each compactly supported potential we associate a conformal mapping from the upper half-plane onto some domain. We describe this conformal mapping in terms of inverse problems i) and ii). For example, if we know the value of conformal mapping at some sequence of real numbers then we can recover the conformal mapping and the potential.