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

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.