Scattering theory for the Hartree equation

We study the scattering problem for the Hartree equation i∂_tu = - 1/2Δu +f(\u\²)u, (t,x) ∈ R x Rⁿ, with initial data u(0,x) = U₀(X), x ∈ Rⁿ, where f(|u|²) = V * |u|², V(x) = λ|x|^-1, λ ∈ R, n ≥ 2. We prove that for any U₀ ∈ H^0,γ∩ H^γ,0, with 1/2 < γ < n/2, such that the value ∈ = ∥u₀∥0,γ + ∥u₀∥γ,0 is sufficiently small, there exist unique u± 6 H'0 n /f0''7 with | < CT < 7 such that for all |t| > 1 ||u(t) - exp (Ti/(|u± 2) (y ) log |i|) l/(t)U± L² < C'eltrμ+7, where JJL = min(l, γ), O < v < min(l, -1γf-), <f denotes the Fourier transform of if, U(t) is the free Schrödinger evolution group, and Hm<s is the weighted Sobolev space defined by Hm<s = {<fe S'; IMIm1, = Ii(I + N2)s/2(l - Ar/VllL2 < oo}.