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

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