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

Abstract In this paper, we study the dynamics of the focusing nonlinear Hartree equation with a Kato potential under some assumptions on the potential . We prove the blow up versus global existence dichotomy for solutions beyond the threshold, based on the method from Duyckaerts–Roudenko in [6]. Our result extends the work of [13], which studied the blow up and scattering theory for solutions of below that threshold.