Normal Forms for Semilinear Quantum Harmonic Oscillators

TL;DR

This paper proves the existence of Birkhoff normal forms for semilinear quantum harmonic oscillators, leading to insights into their long-term dynamics and stability, especially in low dimensions.

Contribution

It establishes the Birkhoff normal form for these oscillators at any order and analyzes the resulting integrability and bounded dynamics under generic conditions.

Findings

Normal form exists near the origin at any order.

Integrability of the normal form in one dimension.

Almost global existence and Sobolev norm control for small data.

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.