Scattering and blowup beyond the mass-energy threshold for the cubic NLS with a potential

We consider the focusing cubic nonlinear Schrödinger equation with a potential, $i \partial _{t}u+(\Delta -V)u+|u|^{2}u=0$, with finite energy and finite variance initial data in dimension $3$. We investigate the scattering versus blow up dichotomy above the mass-energy threshold for finite variance solutions. Key ingredients in the proof include variational analysis, a virial argument and the concentration-compactness/rigidity method.