Non-radial scattering theory for nonlinear Schrodinger equations with potential

In this paper, we study a class of nonlinear Schrodinger equations (NLS) with potential i partial derivative(t)u + Delta u - Vu = +/-vertical bar u vertical bar(alpha)u, (t, x) is an element of R x R-3, where 4/3 < alpha < 4 and V is a Kato-type potential including the genuine Yukawa potential as a special case. By using variational analysis and interaction Morawetz estimates, we establish a scattering criterion for the equation with non-radial initial data. 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, 2016) and Hamano and Ikeda (J Evolut Equ 20:1131-1172, 2020). As a by product of the scattering criterion and the concentration-compactness lemma `a la P. L. Lions, we study long time dynamics of global solutions to the focusing problem with data at the ground state threshold. Our result is robust and can be applicable to show the energy scattering for the focusing NLS with Coulomb potential.