Almost reducibility for finitely differentiable SL(2,\mathbb{R}) -valued quasi-periodic cocycles

Chavaudret, Claire

doi:10.1088/0951-7715/25/2/481

Cited by 7 publications

(11 citation statements)

References 9 publications

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“…which is enough for our purposes. Another improvement is the lower bound on the degree of differentiability k (we obtain an integer k which only depends on τ , which fits the intuition from classical KAM theory), motivated by the estimate (3.19) above, which is more explicit than the one in [18].…”

Section: 5mentioning

confidence: 66%

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

et al. 2018

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We show that if the base frequency is Diophantine, then the Lyapunov exponent of a C k quasi-periodic SL(2, R) cocycle is 1/2-Hölder continuous in the almost reducible regime, if k is large enough. As a consequence, we show that if the frequency is Diophantine, k is large enough, and the potential is C k small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is 1/2-Hölder continuous.

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Section: 5mentioning

confidence: 66%

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

et al. 2018

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“…If A + F(θ ) ∈ C k (T d , SL(2, R)) and A ∈ SL(2, R), then there exists f (θ ) ∈C k (T d , sl(2, R)) such that A + F(θ ) = Ae f (θ) .Remark 3.4. Proposition 3.1 is indeed similar to[11, Theorem 1.2], where the dependence of ε on A is implicit; one can also check that in[11] ε does not depend on A if A is a rotation matrix. Global-local reduction.…”

mentioning

confidence: 53%

“…Note that all the above are under the assumption that the cocycles are analytic. For C k cocycles, one can see [2,11] and [9].…”

Section: Remark 13mentioning

confidence: 99%

“…It is worth noting that C k -local theory has been recently built by Chavaudret [ Continuity of the spectrum of quasi-periodic Schrödinger operators 571 PROPOSITION 3.1. [9,11] Let α ∈ DC(κ, τ ), A ∈ SL(2, R) and f ∈ C k (T d , sl(2, R)) with k Dτ for some large D ∈ N. There exists ε depending on κ, τ, d, k, A such that if f (θ ) k ε, and ρ(α, Ae f (θ) ) is Diophantine with respect to α, then (α, Ae f (θ) ) is C k,k 1 reducible.…”

Section: Aubry Dualitymentioning

confidence: 99%

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Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials

Zhao¹

2018

Ergod. Th. Dynam. Sys.

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In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on $\ell ^{2}(\mathbb{Z})$ with the potentials $v\in C^{k}(\mathbb{T})$. For sufficiently large $k$, we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in Jitomirskaya and Mavi [Continuity of the measure of the spectrum for quasiperiodic schrödinger operator with rough potentials. Comm. Math. Phys.325 (2014), 585–601]. Our results are based on a generalization of the rigidity theorem in Avila and Krikorian [Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2)164 (2006), 911–940]; more precisely, we prove that in the $C^{k}$ case, for almost every frequency $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and for almost every $E$, the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.

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“…The quantity ρ L (n, m; {ξ ·,k } k>|m(·)| , χ) is studied in the same way as before. The change of variables used above transforms it (for, say, m = 0 and n > 0) into an expression like (10), which can again be estimated with the help of Lemma 2.4. Using the assumptions of the present theorem, we obtain in this way the estimate…”

Section: Notice That This Change Of Variables Can Be Represented As Amentioning

confidence: 99%

An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

Damanik

Gorodetski

2016

Advances in Mathematics

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We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented.On the one hand, we show that the Schrödinger operators on ℓ 2 (Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z).On the other hand, we show that Schrödinger operators on ℓ 2 (Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasiperiodic potentials with · ∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.

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Almost reducibility for finitely differentiable SL(2,\mathbb{R}) -valued quasi-periodic cocycles

Cited by 7 publications

References 9 publications

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Continuity of the spectrum of quasi-periodic Schrödinger operators with finitely differentiable potentials

An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

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