Nonlinear scattering with nonlocal interaction

We consider the scattering problem for the Hartree type equation in ℝⁿ with n≧2: {Mathematical expression} where {Mathematical expression} and * denotes the convolution in ℝⁿ. We prove the existence of wave operators in H^{0, k} = {ψ∈L²(ℝⁿ);|x|^kψ∈L²(ℝⁿ)} for any positive integer k under the assumption 1<γ₁, γ₂<2. This is an optimal result in the sense that the existence of wave operators breaks down if min (γ₁, γ₂≢1. The case where 1<γ₁, γ₂ = 2 is also treated according to the sign of λ₂.