Threshold scattering for the focusing NLS with a repulsive potential

TL;DR

This paper proves scattering for the 3D focusing cubic nonlinear Schrödinger equation with a repulsive potential at the critical threshold, extending previous results to short-range and inverse-square potentials.

Contribution

It adapts recent methods to establish sharp threshold scattering for the focusing NLS with various repulsive potentials, including inverse-square potential.

Findings

Established scattering at the sharp threshold for short-range potentials.

Extended results to the inverse-square potential case.

Unified approach applicable to different types of repulsive potentials.

Abstract

We adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as previously considered in the work of Hong, as well as the inverse-square potential, previously considered in the work of the authors.