$H^1$ scattering for mass-subcritical NLS with short-range nonlinearity and initial data in $\Sigma$

TL;DR

This paper proves that solutions to short-range mass-subcritical nonlinear Schrödinger equations with initial data in  in the energy space scatter, extending previous $L^2$ results to $H^1$ and providing new insights into moment scattering.

Contribution

It extends classical scattering results from $L^2$ to $H^1$ for mass-subcritical NLS with short-range nonlinearity and initial data in .

Findings

Solutions in  scatter in $H^1$

Partial results on scattering of first order moments

Short proof of classical scattering result using lens transform

Abstract

We consider short-range mass-subcritical nonlinear Schr\"odinger equations and we show that the corresponding solutions with initial data in $\Sigma$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and Tsutsumifrom $L^2$ to $H^1$.We also provide some partial results concerning the scattering of the first order moments, as well as a short proof via lens transform of a classical result due to Tsutsumi and Cazenave-Weissler on the scattering in $\Sigma$.