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Given small initial solutions of the nonlinear quantum harmonic oscillator on$\mathbb{R}$, we are interested in their long time behavior in the energy spacewhich is an adapted Sobolev space. We perturbate the linear part by $V$ takenas multiplicative potentials, in a way that the linear frequencies satisfy anon-resonance condition. More precisely, we prove that for almost allpotentials $V$, the low modes of the solution are almost preserved for verylong times.

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