Normal Forms for Semilinear Quantum Harmonic Oscillators

Abstract

We consider the semilinear harmonic oscillator $$iψ_t=(-Δ+\va{x}^{2} +M)ψ+\partial_2 g(ψ,\bar ψ), \quad x\in \R^d, t\in \R$$ 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 $d\ge 2$. 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.

References in corpus (1)

• Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations

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