Nonlinear scattering with nonlocal interaction

Abstract

We consider the scattering problem for the Hartree type equation in ℝⁿ withn≧2:

$$i\frac{{\partial u}}{{\partial t}} + \frac{1}{2}\Delta u = (V * |u|^2 )u,$$

where$V(x) = \sum\limits_{j = 1}^2 {\lambda _j |x|^{ - \gamma j} ,(\lambda _1 ,\lambda _2 ) \ne (0,0),\lambda _j \in \mathbb{R}} ,\gamma _j > 0$ and * denotes the convolution in ℝⁿ. We prove the existence of wave operators inH ^0,k = {ψ∈L ²(ℝⁿ);|x|^kψ∈ L ²(ℝⁿ)} for any positive integerk 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.