Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillator

Given small initial solutions of the nonlinear quantum harmonic oscillator on R , we are interested in their long time behaviour in the energy space which is an adapted Sobolev space. We perturbate the linear part by V taken as multiplicative potentials, in a way that the linear frequencies satisfy a non-resonance condition. More precisely, we prove that for almost all potentials V , the low modes of the solution are almost preserved for very long times.