Random data Cauchy problem for supercritical Schr\"odinger equations

In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$. Here we use the randomisation method of the inital conditions, introduced by N. Burq-N. Tzvetkov and we are able to show that the equation admits strong solutions for data in $\H^{s}$ for some $s<s_{c}$. In the appendix we prove the equivalence between the smoothing effect for a Schr\"odinger operator with confining potential and the decay of the associate spectral projectors.