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In \cite{poiret}, we explain how we can construct global solutions for thecubic 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 thebilinear estimate for the harmonic oscillator. This estimate is true in alldimensions $ d \geq 2 $ and we can adapt the proof in these cases. We explainin this paper how we can use the smoothing effect to obtain an analogoustheorem in all dimensions, in particular in dimension 1. The gain of regularityis lower but we can choose any basis of eigenfunctions and random variablesmore general than Gaussian.

Solutions globales pour des \'equations de Schr\"odinger sur-critiques en toutes dimensions