  This article is concerned with the small data problem for the cubic nonlinear
Schr\"odinger equation (NLS) in one space dimension, and short range
modifications of it. We provide a new, simpler approach in order to prove that
global solutions exist for data which is small in $H^{0,1}$. In the same
setting we also discuss the related problems of obtaining a modified scattering
expansion for the solution, as well as asymptotic completeness.
