Scattering for NLS with a potential on the line

Abstract

We show the $H^{1}$ scattering for a one dimensional nonlinear Schr\"odinger equation with a non-negative, repulsive potential $V$ such that $V,xV\in W^{1,1}$, and a mass-supercritical non-linearity. We follow the approach of concentration-compacity/rigidity first introduced by Kenig and Merle.