Lp-Lp estimates for the Schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential

In this paper I prove a L-p-L-p estimate for the solutions to the one-dimensional Schrodinger equation with a potential in L-y(1) where in the generic case y > 3/2 and in the exceptional case (i.e., when there is a half-bound stare of zero energy) > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schrodinger 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. (C) 2000 Academic Press.