  In this paper we consider the nonlinear Schr\"o\-din\-ger equation $i u_t
+\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac {2} {N}$ and
$\Im \kappa <0$, then every nontrivial $H^1$-solution blows up in finite or
infinite time. In the case $\alpha >\frac {2} {N}$ and $\kappa \in {\mathbb
C}$, we improve the existing low energy scattering results in dimensions $N\ge
7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} <
\alpha \le \frac {4} {N} $, then small data give rise to global, scattering
solutions in $H^1$.
