Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

Liu, Wencaı; Shi, Yunfeng

doi:10.4171/jst/275

Cited by 12 publications

(15 citation statements)

References 33 publications

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“…(i) For any | | ≤ 1 C(α)||R|| 2 2δ , there exist some B 1, , P 1, ∈ C ω (R/Z, SL(2, R)) and , there exist some B 2, , P 2, ∈ C ω (R/Z, SL(2, R)) and P 2, ∈ SL(2, R) such that (6.49) B −1 2, (x + α)(P 1, + 2 P 1, (x))B 2, (x) = P 2, + 3 P 2, (x) and Proof. The proof can be found in [22]. Trace(P 1, ) = 2 − a m R 2 11 .…”

Section: 2mentioning

confidence: 96%

“…In this section, we will prove that the lengths of the spectral gaps decay exponentially. The proofs are similar to that of [22]. For reader's convenience, we include the details below.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…In a recent work by Parnovski and Shterenberg [24], they got the asymptotic expansion for the length of some spectral gap.All results mentioned above are attached to the Schrödinger type operators and little is known about the Jacobi type operators (such as the EHM). In [13], Han proved the spectrum of the non-self dual EHM with weak Diophantine frequency contains no collapsed spectral gap.For a more detailed exposition of the history of the spectral gaps studying, we refer the reader to [20][21][22].The methods of the present paper follow that of [3,21], but more subtle estimates and technical differences. More precisely, using ideas of [3,21], we first establish (at the boundary of some spectral gap) quantitative reducibility results for the extended Harper's cocycles.…”

mentioning

confidence: 99%

“…For a more detailed exposition of the history of the spectral gaps studying, we refer the reader to [20][21][22].…”

mentioning

confidence: 99%

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Exponential Decay of the Lengths of the Spectral Gaps for the Extended Harper’s Model with a Liouvillean Frequency

Shi

Yuan

2018

J Dyn Diff Equat

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In this paper, we study the non-self dual extended Harper's model with Liouvillean frequency. By establishing quantitative reducibility results together with the averaging method, we prove that the lengths of the spectral gaps decay exponentially.arXiv:1708.01762v4 [math.SP] 30 Dec 2017 Remark 1.2. If λ 2 max{λ 1 +λ 3 ,1} > e Cβ(α) , then L λ > Cβ(α). Remark 1.3. Based on this theorem, Jian-Shi [16] proved the 1 2 -Hölder continuity of the integrated density of states for the EHM. They also obtained the Carleson homogeneity of the spectrum.The investigations of the spectral gaps for the AMO (i.e., λ 1 = λ 3 = 0) are closely related to the Cantor set structure of the spectrum Σ λ 2 ,α . In fact, the famous Ten Martini problem says that Σ λ 2 ,α is a Cantor set for all λ 2 = 0, α ∈ R \ Q. Much effort [6,7,14,25] was expended to solve the Ten Martini problem and finally it was proved by Avila and Jitomirskaya [2]. A stronger assertion which is called the dry Ten Martini problem suggests that Σ λ 2 ,α contains no collapsed spectral gap for all λ 2 = 0, α ∈ R \ Q. To the best of our knowledge, the dry Ten Martini problem still remains open and only partial results were obtained [2,3,5,7,23,25]. In fact, Avila-You-Zhou [5] proved the dry Ten Martini problem for the non-critical AMO.The first result concerning upper bounds of the lengths of the spectral gaps for the lattice quasi-periodic Schrödinger operators was proved by Amor [12] in which she showed that the lengths of the spectral gaps decay sub-exponentially. She used the KAM techniques developed by Eliasson [11]. Thus the frequency must satisfy the Diophantine condition. Recently, Leguil-You-Zhao-Zhou [21] proved that the lengths of the spectral gaps for the general Schrödinger operators with weak Diophantine frequency decay exponentially. Moreover, they obtained the lower bounds of the lengths of the spectral gaps for the AMO with Diophantine frequency. Based on some results of [23], Liu and Shi [22] generalized a result of [21] to the Liouvillean frequency case.For the continuous quasi-periodic Schrödinger operators, Damanik-Goldstein [8] and Damanik-Goldstein-Lukic [9] obtained the upper bounds of the lengths of the spectral gaps. In a recent work by Parnovski and Shterenberg [24], they got the asymptotic expansion for the length of some spectral gap.All results mentioned above are attached to the Schrödinger type operators and little is known about the Jacobi type operators (such as the EHM). In [13], Han proved the spectrum of the non-self dual EHM with weak Diophantine frequency contains no collapsed spectral gap.For a more detailed exposition of the history of the spectral gaps studying, we refer the reader to [20][21][22].The methods of the present paper follow that of [3,21], but more subtle estimates and technical differences. More precisely, using ideas of [3,21], we first establish (at the boundary of some spectral gap) quantitative reducibility results for the extended Harper's cocycles. Then using the averaging method, we can show that the fibered r...

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Section: 2mentioning

confidence: 96%

“…In this section, we will prove that the lengths of the spectral gaps decay exponentially. The proofs are similar to that of [22]. For reader's convenience, we include the details below.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

“…For a more detailed exposition of the history of the spectral gaps studying, we refer the reader to [20][21][22].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Exponential Decay of the Lengths of the Spectral Gaps for the Extended Harper’s Model with a Liouvillean Frequency

Shi

Yuan

2018

J Dyn Diff Equat

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“…if we choose ε not very small, then the remainder in the expansion will be larger than the asymptotic terms). Somewhat similar problems were considered in [3] and, in the discrete setting, in [4] (see also [5] and references there). However, there is a significant difference between these papers and our results.…”

Section: Introductionmentioning

confidence: 95%

Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case

Parnovski

Shterenberg

2019

Commun. Math. Phys.

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We consider the family of operators H (ε) := − d 2 dx 2 + εV in R with almostperiodic potential V . We study the behaviour of the integrated density of states (IDS) N (H (ε) ; λ) when ε → 0 and λ is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each λ the IDS has a complete asymptotic expansion in powers of ; these powers are either integer, or in some special cases half-integer. These results are new even for periodic V . We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set S of energies (which we call the super-resonance set) such that for any √ λ ∈ S there is a complete power asymptotic expansion of IDS, and when √ λ ∈ S, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set S is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of H (ε) has a complete asymptotic expansion in natural powers of ε when ε → 0.

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Sharp Hölder Continuity of the Integrated Density of States for Extended Harper’s Model with a Liouville Frequency

Jian

Shi

2019

Acta Math Sci

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In this paper, we study the non-self-dual extended Harper's model with a Liouville frequency. Based on the work of [20], we show that the integrated density of states (IDS for short) of the model is 1 2 -Hölder continuous. An an application, we also obtain the Carleson homogeneity of the spectrum.We call λ = (λ 1 , λ 2 , λ 3 ) ∈ R 3 + the coupling, α ∈ R \ Q the frequency and x ∈ R the phase. The EHM was originally proposed by Thouless [22] and if λ 1 = λ 3 = 0 it reduces to the famous almost Mathieu operator (AMO for short). Physically, the EHM describes the influence of a transversal magnetic field of flux α on a single tight-binding electron in a 2-dimensional crystal layer (see [4,22]).

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Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

Abstract: We prove that the size of the spectral gaps of weakly coupled quasi-periodic Schrödinger operators with Liouville frequencies decays exponentially. As an application, we obtain the homogeneity of the spectrum.

Cited by 12 publications

References 33 publications

Exponential Decay of the Lengths of the Spectral Gaps for the Extended Harper’s Model with a Liouvillean Frequency

Exponential Decay of the Lengths of the Spectral Gaps for the Extended Harper’s Model with a Liouvillean Frequency

Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case

Sharp Hölder Continuity of the Integrated Density of States for Extended Harper’s Model with a Liouville Frequency

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