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arXiv · 1603.07455

On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential

Eric Paturel·Benoît Grébert
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Abstract

We prove that a linear d-dimensional Schr{\"o}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- \Delta u + |x|^2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}^d$ reduces to an autonomous system for most values of the frequency vector $\omega \in \mathbb{R}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

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Eric Paturel, Benoît Grébert. 2016-03-24. On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential. https://arxiv.org/abs/1603.07455

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