H^1 Scattering for Mass-Subcritical NLS with Short-Range Nonlinearity and Initial Data in Σ

We consider short-range mass-subcritical nonlinear Schrödinger equations, and we show that the corresponding solutions with initial data in Σ scatter in H^1 . Hence we up-grade the classical scattering result proved by Yajima and Tsutsumi from 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 Σ .