  In this paper I prove a L^p-L^p' estimate for the solutions of the
one-dimensional Schroedinger equation with a potential in L^1_gamma where in
the generic case gamma > 3/2 and in the exceptional case (i.e. when there is a
half-bound state of zero energy) gamma > 5/2. I use this estimate to construct
the scattering operator for the nonlinear Schroedinger equation with a
potential. I prove moreover, that the low-energy limit of the scattering
operator uniquely determines the potential and the nonlinearity using a method
that allows as well for the reconstruction of the potential and of the
nonlinearity.
