Remarks on scattering theory and large time asymptotics of solutions to 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 { iu _t = - 1/2Δu + f(|u| ²u, (t, x) ∈ R × R ⁿ, u(0,x) = u ₀(x), x ∈ R ⁿ, n ≥ 2, where the nonlinear interaction term is f(|u| ²) = V * |u| ², V(x) = λ|x| ^{- δ} λ ∈ R,0 < δ <1. We suppose that the initial data u ₀ ∈ H ^0.l and the value ∈= ||u ₀||H ^0.1 is sufficiently small, where l is an integer satisfying l ≥ [n/2] + 3, and [s] denotes the largest integer less than s. Then we prove that there exists a unique final state u ₊ ∈ H ^0,l-2 such that for all t > 1 u(t, x) = 1 û/(it) ^n/2 + (x/t) exp(ix ²/2t-it ^{1- δ}/1- δ f(|û+| ²)(x/t) + 0(1+t ^1-2δ))+0( ^-n/2-δ) uniformly with respect to x ∈ R ⁿ with the following decay estimate ||u(t) ||L ^p ≤ C∈t ^n/p-n/2, for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ <1 there exists a unique final state u + ∈ H ^0,1-2 such that for all t ≥ 1, ||u(t)-exp(-it ^1-δ/1- δ f(|û+| ²(x/t)) U(t)u+||L ² = 0(t ^{1-2 δ}), and uniformly with respect to x ∈ R ⁿ u(t, x) = 1/(it) ^n/2 û+(x/t)exp(ix ²/2t-it ^{1- δ}/1- δ f(|û +