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

Luo, Jiawen; Liang, Zhenwen; Zhao, Zhiyan

doi:10.1007/s00220-022-04340-x

Cited by 7 publications

(2 citation statements)

References 37 publications

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“…Basing on a Mourre estimate, Maspero [33] proved similar results for 1-D QHO and half -wave equation on T and the instability is stable in some sense. For a polynomial periodic or quasi-periodic perturbations relative with 1-D QHO we refer to [7], [21], [29] and [31]. For 2-D QHO with perturbation which is decaying in t, Faou-Raphaël [17] constructed a solution whose H 1 −norm presents logarithmic growth with t. For 2-D Schrödinger operator Thomann [38] constructed explicitly a traveling wave whose Sobolev norm presents polynomial growth with t, based on the study in [36] for linear Lowest Landau equations(LLL) with a time-dependent potential.…”

Section: Introduction Of the Main Resultsmentioning

confidence: 99%

Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

Luo

Wang

et al. 2022

J Dyn Diff Equat

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Enlightened by Lemma 1.7 in [25], we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schrödinger equationcan be reduced in H 1 (R) to an autonomous system for most values of the frequency vector ω, where Λ ⊂ R \ {0}, |Λ| < ∞ and x := √ 1 + x 2 . The functions a k (θ) and b k (θ) are analytic on T n σ and µ ≥ 0 will be chosen according to the value of β. Comparing with [25], the novelty is that the phase functions of oscillatory integral are more degenerate when β > 1.

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Section: Introduction Of the Main Resultsmentioning

confidence: 99%

Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

Luo

Wang

et al. 2022

J Dyn Diff Equat

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“…The reducibility results usually imply the boundedness of Sobolev norms. We refer to the papers [6,16,22,26,34,36,37,43] for the growth rate of the solutions of QHO including the upper-lower bound. There are also many literatures, e.g.…”

Section: Discussionmentioning

confidence: 99%

Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials

Liang

Wang

2022

Nonlinearity

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We prove the reducibility of 1D quantum harmonic oscillators in R perturbed by a quasi-periodic in time potential V(x, ωt) under the following conditions, namely there is a C > 0 such that | V ( x , θ ) | ⩽ C , | x ∂ x V ( x , θ ) | ⩽ C , ∀ ( x , θ ) ∈ R × T σ n . The corresponding perturbation matrix ( P i j ( θ ) ) is proved to satisfy ( 1 + | i − j | ) | P i j ( θ ) | ⩽ C and i j | P i + 1 j + 1 ( θ ) − P i j ( θ ) | ⩽ C for any θ ∈ T σ n and i, j ⩾ 1. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element P i j ( θ ) as well as the discrete difference matrix element P i + 1 j + 1 ( θ ) − P i j ( θ ) . For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.

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Reducibility of quantum harmonic oscillator on $$\mathbb {R}^d$$ perturbed by a quasi: periodic potential with logarithmic decay

Liang

Wang

2022

Calc. Var.

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

Cited by 7 publications

References 37 publications

Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

Reducibility of 1-D Quantum Harmonic Oscillator with New Unbounded Oscillatory Perturbations

Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials

Reducibility of quantum harmonic oscillator on $$\mathbb {R}^d$$ perturbed by a quasi: periodic potential with logarithmic decay

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