Scattering theory for the Hartree equation

We study the scattering problem for the Hartree equation i partial derivative(t)u = ?1/2 Delta u + f(\u\(2))u, (t, x) is an element of R x R-n with initial data u(0, x) = u(0)(x), x is an element of R-n, where f(\u\(2)) = V * \u\(2), V(x) = lambda\x\(?1), lambda is an element of R, n greater than or equal to 2. We prove that for any u(0) is an element of H-0,H- (gamma) boolean AND H-gamma,H- 0, with 1/2 < gamma < n/2, such that the value epsilon = \\u(0)\\(0, gamma) + \\u(0)\\(gamma,) (0) is sufficiently small, there exist unique u(+/-) is an element of H-sigma,H- (0) boolean AND H-0,H- sigma with 1/2 < sigma < gamma such that for all \t\ greater than or equal to 1 \\u(t) ? exp (-/+ if (\(u) over cap(+/-)\(2)) (x/t) log \t\) U(t)u(+/-)\\( L2) less than or equal to C epsilon\t\(-mu+7 nu)(,) where mu = min(1, gamma/2), 0 < nu < min(1, gamma-sigma/12), <(phi)over cap> denotes the Fourier transform of phi, U(t) is the free Schrodinger evolution group, and H-m,H- s is the weighted Sobolev space defined by H-m,H- (s) = {phi is an element of S'; \\phi\\(m,) (s) = \\(1 + \x\(2))(s/2) (1 ? Delta)(m/2) phi\\(L2) < infinity}.