Abstract
We consider the spectrum of the almost Mathieu operator \({H_\alpha}\) with frequency \({\alpha}\) and in the case of the critical coupling. Let an irrational \({\alpha}\) be such that \({|\alpha - p_n/q_n| < c q_n^{-\varkappa}}\), where \({p_n/q_n}\), \({n=1,2,\ldots}\), are the convergents to \({\alpha}\), and \({c}\), \({\varkappa}\) are positive absolute constants, \({\varkappa < 56}\). Assuming certain conditions on the parity of the coefficients of the continued fraction of \({\alpha}\), we show that the central gaps of \({H_{p_n/q_n}}\), \({n=1,2,\ldots}\), are inherited as spectral gaps of \({H_\alpha}\) of length at least \({c'q_n^{-\varkappa/2}}\), \({c' > 0}\).
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Krasovsky, I. Central Spectral Gaps of the Almost Mathieu Operator. Commun. Math. Phys. 351, 419–439 (2017). https://doi.org/10.1007/s00220-016-2774-9
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DOI: https://doi.org/10.1007/s00220-016-2774-9