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hal-00309695, version 3

## NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS

Benoit Grebert () 1, Rafik Imekraz 1, Eric Paturel 1

Communications in Mathematical Physics 291 (2009) 763-798

Abstract: 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.

• 1:  Laboratoire de Mathématiques Jean Leray (LMJL)
• CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes

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• Submitted on: Monday, 14 December 2009 11:48:07
• Updated on: Monday, 14 December 2009 13:40:51