Almost reducibility and oscillatory growth of Sobolev norms

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of (x, -i partial differential 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(ts)-upper bound is shown for the Hs-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 Hs-norm of the solution is o(ts) as t -> infinity but arbitrarily close to ts in an oscillatory way.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.