Scattering for the generalized Hartree equation with a potential

Abstract

We consider the focusing generalized Hartree equation in $H^1({\mathbb{R}}^3)$ with a potential, \begin{equation*} iu_t + Δu - V(x)u + (I_γ\ast |u|^p )|u|^{p-2} u=0, \end{equation*} where $I_{\gamma }=\frac{1}{|x{|}^{3-\gamma }}$, $p\ge 2$ and $\gamma <3$. In this paper, we prove scattering for the generalized Hartree equation with a potential in the intercritical case assuming radial initial data. The novelty of our approach lies in the use of a general mass-potential condition, incorporating the potential V, which extends the standard mass-energy framework. To this end, we employ a simplified method inspired by Dodson and Murphy \cite{Dod-Mur}, based on Tao's scattering criteria and Morawetz estimates. This approach provides a more straightforward proof of scattering compared to the traditional concentration-compactness/rigidity method of Kenig and Merle \cite{KENIG}.

20 pages