Blow up versus scattering below the mass-energy threshold for the focusing NLH with potential

In this paper, we study the blow up and scattering result of the solution to the focusing nonlinear Hartree equation with potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-Vu=-(\mathrm{\mid }\cdot {\mathrm{\mid }}^{-3}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^5$$ in the energy space $H^1({\mathbb{R}}^5)$ below the mass-energy threshold. The potential $V$ we considered is an extension of Kato potential in some sense. We extend the results of Meng [26] to nonlinear Hartree equation with potential $V$ under some conditions. By establishing a Virial-Morawetz estimate and a scattering criteria, we obtain the scattering theory based on the method from Dodson-Murphy [11].