Polynomial decay of the gap length for C quasi-periodic Schrödinger operators and spectral application

Cai, Ao; Wang, Xueyin

doi:10.1016/j.jfa.2021.109035

Cited by 11 publications

(10 citation statements)

References 33 publications

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“…Proof. The proof follows a similar line as Lemma 6.1 in [24] (see also [11]), we will give the proof in Appendix B for completeness.…”

Section: Proof Of Theorem 12mentioning

confidence: 83%

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Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Li¹

2021

Preprint

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We establish a quantitative version of strong almost reducibility result for sl(2, R) quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays subexponentially with respect to its labelling, the long range duality operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases and the spectrum is an interval for discrete Schrödinger operator acting on Z d with small separable potentials. All these results are based on a refined KAM scheme, and thus are perturbative.

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“…Proof. The proof follows a similar line as Lemma 6.1 in [24] (see also [11]), we will give the proof in Appendix B for completeness.…”

Section: Proof Of Theorem 12mentioning

confidence: 83%

“…If the regularity of v is weaker than analytic, one can not expect an exponential decay rate [13]. In finite differentiable regime, Cai-Wang [11] showed that the gap lengths decay polynomially. In this paper, we improve this result to sub-exponential decay for small Gevrey potentials.…”

Section: Introductionmentioning

confidence: 99%

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Li¹

2021

Preprint

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show abstract

“…Proof. The proof follows a similar line as lemma 6.1 in [43] (see also [19]), we will give the proof in appendix B for completeness.…”

Section: Proof Of Theorem 13mentioning

confidence: 84%

“…If the regularity of v is weaker than analytic, one can not expect an exponential decay rate [22]. If v is finitely differentiable, Cai-Wang [19] showed that the gap lengths decay polynomially. In this paper, we improve this result to sub-exponential decay for small Gevrey potentials.…”

Section: Estimates On Spectral Gapsmentioning

confidence: 99%

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

2022

Nonlinearity

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We establish a quantitative version of strong almost reducibility result for S L ( 2 , R ) quasi-periodic cocycle close to a constant in the Gevrey class. We prove that, if the frequency is Diophantine, the long range operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases; for the one dimensional quasi-periodic Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling; and the spectrum is an interval for discrete Schrödinger operators acting on Z d with small separable potentials.

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“…Back then, I just finished my PhD with my dissertation "Reducibility of finitely differentiable quasi-periodic cocycles and its applications" based on Kolmogorov-Arnold-Moser (KAM) theory under the supervision of You. Obviously, I was a purely quasiperiodic person [1,2,8,9] while Duarte and Klein had already many collaborations on random cocycles as well as quasi-periodic ones, see the two excellent books [10,11] and the references therein.…”

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confidence: 99%

Randomness versus quasi-periodicity

Cai¹

2023

Preprint

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This paper serves as an extended road map for our long-term project "Mixed Random-quasiperiodic Cocycles" [3, 4, 5, 6, 7] with Pedro Duarte and Silvius Klein. Despite exhibiting totally different natures, the random world and the quasi-periodic one may still have potential relations that we are keen to reveal. This was inspired by Jiangong You's intriguing question on the stability of the Lyapunov exponent of quasi-periodic Schrödinger operators under random noise in 2018. Contents

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Polynomial decay of the gap length for C quasi-periodic Schrödinger operators and spectral application

Cited by 11 publications

References 33 publications

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Randomness versus quasi-periodicity

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