Normal Forms for Semilinear Quantum Harmonic Oscillators

Details Normal Forms for Semilinear Quantum Harmonic Oscillators Normal Forms for Semilinear Quantum Harmonic Oscillators

2008-08-07

arxiv.org/abs/0808.0995v3

Communications in Mathematical Physics 291 (2009) 763-798

Mathematics

Analysis of PDEs

Scientific paper

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.

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