  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.
