NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS

Benoît Grébert (1) , Rafik Imekraz (1) , Eric Paturel (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R$$ 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 $d\geq 2$.\\ 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.

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Benoît Grébert, Rafik Imekraz, Eric Paturel. NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS. Communications in Mathematical Physics, 2009, 291, pp.763-798. ⟨hal-00309695v3⟩