  We consider the following family of Cauchy problems: {equation*} i\partial_t
u= \Delta u - u|u|^\alpha, (t,x) \in \R \times \R^d {equation*}
$$u(0)=\varphi\in H^1(\R^d)$$ where $0<\alpha<\frac 4{d-2}$ for $d\geq 3$ and
$0<\alpha<\infty$ for $d=1,2$. We prove that the $L^r$-norms of the solutions
decay as $t\to \pm \infty$, provided that $2<r<\frac{2d}{d-2}$ when $d\geq 3$
and $2<r<\infty$ when $d=1,2$. In particular we extend previous results
obtained by Ginibre and Velo for $d\geq 3$ and by Nakanishi for $d=1,2$, where
the same decay results are proved under the extra assumption $\alpha >\frac
4d$.
