Scattering solutions to nonlinear Schr\"odinger equation with a long range potential

TL;DR

This paper proves that solutions to a nonlinear Schrödinger equation with long-range potentials scatter if initial data is below a certain ground state and has positive virial, extending results to long-range interactions.

Contribution

It establishes scattering results for nonlinear Schrödinger equations with long-range potentials, including cases with long-range inverse-power potentials.

Findings

Solutions with initial data below the ground state scatter.

The results apply to both short-range and long-range potentials.

Positive virial functional is key to scattering behavior.

Abstract

In this paper, we consider a nonlinear Schr\"odinger equation with a repulsive inverse-power potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schr\"odinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials.