In this paper we consider the nonlinear Schr\"o\-din\-ger equation iut+Δu+κ∣u∣αu=0. We prove that if α<N2 and
ℑκ<0, then every nontrivial H1-solution blows up in finite or
infinite time. In the case α>N2 and κ∈C, we improve the existing low energy scattering results in dimensions N≥7. More precisely, we prove that if N+N2+16N8<α≤N4, then small data give rise to global, scattering
solutions in H1