# Scattering for the non-radial inhomogeneous Hartree equation with a potential

## TL;DR

This paper proves scattering for the non-radial inhomogeneous Hartree equation with a potential in the intercritical regime, extending previous results by handling inhomogeneous weights and the potential term.

## Contribution

It introduces new techniques to handle the inhomogeneous weight and potential in the scattering analysis of the Hartree equation, generalizing prior work to more complex settings.

## Key findings

- Established scattering for non-radial initial data in the intercritical case.
- Proved global well-posedness for small initial data with potential.
- Adapted Tao's scattering criterion and Morawetz estimates to the weighted, potential-including setting.

## Abstract

In this work, we consider the focusing generalized inhomogeneous Hartree equation with potential \[ i u_t + \Delta u - V(x)u + \left(I_{\gamma} * |x|^{-b}|u|^{p}\right)|x|^{-b}|u|^{p-2}u = 0, \] where $0<\gamma<3$ and $0<b<\frac{1+\gamma}{2}$. We prove scattering in the intercritical case for nonradial initial data, under a mass-potential condition that generalizes the usual mass-energy threshold. The main new points compared to previous works are the inhomogeneous weight $|x|^{-b}$ and the presence of a potential $V$, which lead us to study the perturbed operator $-\Delta + V$. Our proof follows the general strategy of Murphy, but we need to adapt several steps to deal with the weight and the potential. We use Tao's scattering criterion together with localized Morawetz estimates in this setting. As a preliminary step, we establish global well-posedness for small data, which, in the presence of $V$, requires careful analysis using appropriate admissible Strichartz pairs.