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Almost reducibility and oscillatory growth of Sobolev norms

Zhenguo Liang, Zhiyan Zhao, Qi Zhou

TL;DR

The paper analyzes the time-quasiperiodic quadratic perturbation of the 1D quantum harmonic oscillator i∂_t ψ = (J + P(t)) ψ with J = D^2 + X^2 and P(t) a real-analytic quadratic form. Using the Metaplectic representation to translate quantum dynamics into SL(2,R) flows, it proves almost reducibility of the system under small analytic P with Diophantine ω, and derives an o(t^s) upper bound for the H^s-norm in the non-reducible case. To establish sharpness, the authors construct Liouvillean perturbations via a fibred Anosov–Katok scheme, producing explicit oscillatory growth of Sobolev norms that can approach t^s arbitrarily closely while remaining o(t^s) on average; the perturbations can be made with arbitrarily small analytic norm. The combination of KAM-type almost reducibility and AK-type optimality yields a precise picture of long-time Sobolev dynamics for a broad class of time-quasiperiodic quadratic quantum Hamiltonians, with rigorous bounds and explicit constructions. The work advances understanding of energy transfer in quantum systems under quasi-periodic forcing and demonstrates the utility of the Metaplectic framework for quantitative Sobolev analysis in this setting.

Abstract

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-{\rm i}\partial_x)$, 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.

Almost reducibility and oscillatory growth of Sobolev norms

TL;DR

The paper analyzes the time-quasiperiodic quadratic perturbation of the 1D quantum harmonic oscillator i∂_t ψ = (J + P(t)) ψ with J = D^2 + X^2 and P(t) a real-analytic quadratic form. Using the Metaplectic representation to translate quantum dynamics into SL(2,R) flows, it proves almost reducibility of the system under small analytic P with Diophantine ω, and derives an o(t^s) upper bound for the H^s-norm in the non-reducible case. To establish sharpness, the authors construct Liouvillean perturbations via a fibred Anosov–Katok scheme, producing explicit oscillatory growth of Sobolev norms that can approach t^s arbitrarily closely while remaining o(t^s) on average; the perturbations can be made with arbitrarily small analytic norm. The combination of KAM-type almost reducibility and AK-type optimality yields a precise picture of long-time Sobolev dynamics for a broad class of time-quasiperiodic quadratic quantum Hamiltonians, with rigorous bounds and explicit constructions. The work advances understanding of energy transfer in quantum systems under quasi-periodic forcing and demonstrates the utility of the Metaplectic framework for quantitative Sobolev analysis in this setting.

Abstract

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of , we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an upper bound is shown for the 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 norm of the solution is as but arbitrarily ``close" to in an oscillatory way.
Paper Structure (22 sections, 23 theorems, 211 equations)

This paper contains 22 sections, 23 theorems, 211 equations.

Table of Contents

  1. Introduction and main results
  2. Almost reducibility
  3. Growth of Sobolev norm
  4. Ideas of proof
  5. Notations
  6. Description of the remaining of paper
  7. Metaplectic representation and Sobolev norm
  8. Metaplectic representation on ${\rm SL}(2,\mathbb{R})$ -- Introduction
  9. Estimates of ${{\mathcal{H}}}^s-$norms for quantum propagator
  10. Bounds of Sobolev norms -- basic calculations
  11. Upper bound of Meta representation
  12. Lower bounds of Meta representation
  13. Almost reducibility in quantum Hamiltonian systems
  14. General scheme of almost reducibility for Eq. (\ref{['eq_original']})
  15. Concrete construction for non-reducible Eq. (\ref{['eq_original']})
  16. ...and 7 more sections

Key Result

Theorem 1.1

There exists $\varepsilon_*=\varepsilon_*(\gamma, \tau, d, r)>0$ such that if $\|{\mathcal{P}}\|_r <\varepsilon_*$, then Eq. (eq_original) is almost reducible, i.e., for every $j\in{\mathbb N}^*$, there exists an equation where both ${{\mathcal{L}}}_j$ and ${{\mathcal{P}}}_{j+1}(t)$ are self-adjoint quadratic forms of $(X,D)$ as (SA_qua_form), with the coefficients of ${{\mathcal{L}}}_j$ constant

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.3
  • ...and 21 more