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: $$i{\mathrm{\partial }}_{t}u+\frac{1}{2}\mathrm{\Delta }u-\frac{1}{\mathrm{\mid }x\mathrm{\mid }}u=((-\mathrm{\Delta }{)}^{-1}\mathrm{\mid }u\mathrm{\mid }{)}^{2}u$$ 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.