Inverse Scattering on the Line for the Nonlinear Klein Gordon Equation with a Potential1

In this paper we obtain a L p L p estimate for the Klein Gordon equation with a potential on the line. This result follows from the previously known estimates in the case without potential and from the continuity on the Sobolev spaces of the Schr¨odinger wave operators that we proved in a previous paper. Using this estimate we construct the scattering operator for the nonlinear Klein Gordon equation with a potential. Moreover, we prove that the small amplitude scattering operator uniquely determines both the potential and the nonlinearity. Our proof also gives a method for the reconstruction of the potential and of the nonlinearity.