  In this paper, we consider global solutions for the following nonlinear
Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with
$\lambda\in\R$ and $0\le\alpha<\frac{4}{N-2}$ $(0\le\alpha<\infty$ if $N=1).$
We show that no nontrivial solution can decay faster than the solutions of the
free Schr\"odinger equation, provided that $u(0)$ lies in the weighted Sobolev
space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$
or in $L^2(\R^N),$ according to the different cases.
