Abstract
Let β∈(0,1) be an irrational, and [a 1,a 2,...] be the continued fraction expansion of β. Let H β be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum σ(H β) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim inf k→∞(a 1 a 2⋅⋅⋅a k ))1/k<∞.
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Liu, QH., Wen, ZY. Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials. Potential Analysis 20, 33–59 (2004). https://doi.org/10.1023/A:1025537823884
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DOI: https://doi.org/10.1023/A:1025537823884