Scattering theory and large time asymptotics of solutions to the Hartree type equations with a long range potential

We study the scattering problem and asymptotics for large time of solutions to the Hartree type equations [formula] where the nonlinear interaction term [formula] We suppose that in the case n ≥ 2 the initial data [formula] and the value [formula] is sufficiently small and in one-dimensional case (n = 1) we assume that [formula] and the value [formula] is sufficiently small. Then we prove that there exists a unique final state [formula] such that the asymptotics [formula] is true as t → ∞ uniformly with respect [formula] to with the following decay estimate [formula] for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for [formula] there exists a unique final state [formula] such that [formula] for all t ≥ 1, and the asymptotic formula [formula] is valid as t → ∞ uniformly with respect to [formula] where ϕ denotes the Fourier transform of the function ϕ [formula] Analogous results are obtained for the following NLS equation [formula] with cubic nonlinearity and growing with time coefficient, where [formula].