The stability of nonlinear Schr\"odinger equations with a potential in high Sobolev norms revisited

We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}