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On the density of certain spectral points for a class of \(C^2\) quasiperiodic Schrödinger cocycles. (English) Zbl 1496.37029

Summary: For \(C^2\) cos-type potentials, large coupling constants, and fixed \(Diophantine\) frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point \(E , \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta\), where \(\beta\in [\frac{1}{2},1]\). Our approach is a further improvement on the papers [J. Xu et al., The Hölder continuity of Lyapunov exponents for a class of Cos-type quasiperiodic Schrödinger cocycles, Preprint, arXiv:2006.03381] and [Y. Wang and Z. Zhang, Int. Math. Res. Not. 2017, No. 8, 2300–2336 (2017; Zbl 1405.37086)].

MSC:

37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37A30 Ergodic theorems, spectral theory, Markov operators
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems

Citations:

Zbl 1405.37086
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References:

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