Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator

$ipsi_t=(-Delta +{|x|}^{2} +M)psi +partial_2 g(psi,bar psi), quad xin mathbb{R}^{d},, tin mathbb{R},$

where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.