2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217785In this paper we consider the Schrödinger 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 Schrödinger operator with confining potential and the decay of the associate spectral projectors.29 pages, 0 figureAnalysis of PDEsMathematical Physics35A07; 35B35; 35B05; 37L50; 35Q55text