Lp−Lp Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential

In this paper I prove a Lp−Lp estimate for the solutions to the one-dimensional Schrödinger equation with a potential in L1γ where in the generic caseγ>3/2 and in the exceptional case (i.e., when there is a half-bound state of zero energy) γ>5/2. I use this estimate to construct the scattering operator for the nonlinear Schrödinger equation with a potential. I prove moreover, that the low-energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity.