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)
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.