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

Abstract Given small initial solutions of the nonlinear quantum harmonic oscillator on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the low modes of the solution are almost preserved for very long times.