Global dynamics below excited solitons for the non-radial NLS with potential

We consider the global dynamics of solutions to the $3d$ cubic nonlinear Schr\"odinger equation in the presence of an external potential, in the setting in which the equation admits both ground state solitons and excited solitons at small mass. We prove that small mass solutions with energy below that of the excited solitons either scatter to the ground states or grow their $H^1$-norm in time. In particular, we give an extension of the result of Nakanishi [19] from the radial to the non-radial setting.