Scattering for NLS with a potential on the line

TL;DR

This paper proves scattering in the energy space for a 1D nonlinear Schrödinger equation with a specific class of potentials, using concentration-compactness and rigidity methods.

Contribution

It extends scattering results to NLS with non-negative, repulsive potentials satisfying certain regularity conditions, in the mass-supercritical regime.

Findings

Established $H^{1}$ scattering for NLS with potential on the line.

Applied concentration-compactness/rigidity approach to this setting.

Demonstrated scattering under conditions on the potential $V$.

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.