Almost reducibility and oscillatory growth of Sobolev norms

Abstract

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-\mathrm{i}{\partial }_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the $\CH^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal{H}}^s-$norm of the solution is $o(t^s)$ as $t\to \infty$ but arbitrarily ``close" to $t^s$ in an oscillatory way.

42 pages

References in corpus (8)

• Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Growth of Sobolev norms for coupled Lowest Landau Level equations
• Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation
• Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials
• Growth of Sobolev norms in quasi integrable quantum systems
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb{R}$

Cited by in corpus (1)

• Growth of Sobolev norms for completely resonant quantum harmonic oscillators on ${\mathbb{R}}^2$