L^p-L^p' Estimates for the Nonlinear Schroedinger Equation on the Line and Inverse Scattering for the Nonlinear Schroedinger Equation with a Potential

摘要

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