Modified Wave operators for the Hartree equation with repulsive Coulomb potential

We study the final state problem for the Hartree equation with repulsive Coulomb potential: $$i\partial_t u+\frac{1}{2}\Delta u-\frac{1}{|x|}u=((-\Delta)^{-1}|u|)^2u$$ We show the work in \cite{KaMi} can be extended to the Hartree nonlinearity: Given a prescribed asymptotic profile, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data.