Modified Wave operators for the Hartree equation with repulsive Coulomb potential

TL;DR

This paper extends the theory of modified wave operators to the Hartree equation with a repulsive Coulomb potential, demonstrating the existence of solutions that scatter to prescribed asymptotic profiles for small localized data.

Contribution

It introduces a novel extension of the work on wave operators to the Hartree equation with Coulomb potential, establishing existence and uniqueness of scattering solutions.

Findings

Existence of modified wave operators for the Hartree equation with Coulomb potential.

Construction of global solutions scattering to prescribed profiles.

Applicability to small localized scattering data.

Abstract

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.