  In \cite{poiret}, we explain how we can construct global solutions for the
cubic Schr\"odinger equation in three dimensional with initial data in $
L^2(\mathds{R}^3) $. The main ingredient of this proof is the existence of the
bilinear estimate for the harmonic oscillator. This estimate is true in all
dimensions $ d \geq 2 $ and we can adapt the proof in these cases. We explain
in this paper how we can use the smoothing effect to obtain an analogous
theorem in all dimensions, in particular in dimension 1. The gain of regularity
is lower but we can choose any basis of eigenfunctions and random variables
more general than Gaussian.
