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.
Categories: math.AP