INVERSE BORN APPROXIMATION FOR THE GENERALIZED NONLINEAR SCHRÖDINGER OPERATOR IN TWO DIMENSIONS

This work deals with the inverse scattering problems for the two-dimensional Schrödinger equation [Formula: see text] with a power-like nonlinearity, where the real-valued unknown functions αl on belong to [Formula: see text] with certain special behaviour at infinity. We prove Saito's formula which implies the uniqueness result and a representation formula for a sum of the functions αl in the sense of tempered distributions. What is more, we prove that the leading order singularities of this sum can be obtained exactly by the inverse Born approximation method from general scattering data at arbitrarily large energies. Especially, we show for the functions in Lp, for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces. In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by this scattering data.