The dynamics of the focusing NLH with a potential beyond the mass-energy threshold

We study the dynamics of the focusing nonlinear Hartree equation with a Kato potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-Vu=-(\mathrm{\mid }\cdot {\mathrm{\mid }}^{-\gamma }\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u,x\in {\mathbb{R}}^d$$ under some assumptions on the potential $V$. We prove the blow up versus global existence dichotomy for solutions beyond the threshold, based on the method from Duyckaerts-Roudenko [6]. Furthermore, our result compensates for the one of in [13] below that threshold.