Non-radial scattering theory for nonlinear Schr\"odinger equations with potential

We consider a class of nonlinear Schr"odinger equations with potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-Vu=\pm \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^3,$$ 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.