98-434 Weder R.

L^p-L^p' Estimates for the Schroedinger Equation on the Line and Inverse Scattering for the Nonlinear Schroedinger Equation with a Potential (75K, LATeX 2e) Jun 11, 98

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Abstract. 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.

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