Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation

Abstract

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is a polynomial of degree $2$ in $(x,-\mathrm{i}{\partial }_x)$, with coefficients quasi-periodically depending on time. By establishing the reducibility results, we describe the growth of Sobolev norms. In particular, the $t^{2s}-$polynomial growth of ${\mathcal{H}}^s-$norm is observed in this model if the original time quasi-periodic equation is reduced to a constant Stark Hamiltonian.

arXiv comment19 pages, 1 figure. arXiv admin note: text overlap with arXiv:2003.13034

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• Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials
• Almost reducibility and oscillatory growth of Sobolev norms
• Growth of Sobolev norms for completely resonant quantum harmonic oscillators on ${\mathbb{R}}^2$
• One dimensional energy cascades in a fractional quasilinear NLS
• Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations