  We consider a class of nonlinear Schr\"odinger equations with potential
  \[
  i\partial_t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R}
\times \mathbb{R}^3,
  \]
  where $\frac{4}{3}<\alpha<4$ and $V$ is a Kato-type potential. We establish a
scattering criterion for the equation with non-radial initial data using the
ideas of Dodson-Murphy [Math. Res. Lett. 25(6):1805--1825]. As a consequence,
we prove the energy scattering for the focusing problem with data below the
ground state threshold. Our result extends the recent works of Hong [Commun.
Pure Appl. Anal. 15(5):1571--1601] and Hamano-Ikeda [J. Evol. Equ. 2019]. We
also study long time dynamics of global solutions to the focusing problem with
data at the ground state threshold.
