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Zero measure spectrum for the almost Mathieu operator

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Abstract

We study the almost Mathieu operator: (H α, λ, θ u)(n)=u(n+1)+u(n-1)+λ cos (2παn+θ)u(n), onl 2 (Z), and show that for all λ,θ, and (Lebesgue) a.e. α, the Lebesgue measure of its spectrum is precisely |4–2|λ‖. In particular, for |λ|=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational α's (and |λ|=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.

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Authors and Affiliations

  1. Department of Physics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Y. Last

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  1. Y. Last
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Communicated by T. Spencer

Work partially supported by the GIF

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Last, Y. Zero measure spectrum for the almost Mathieu operator. Commun.Math. Phys. 164, 421–432 (1994). https://doi.org/10.1007/BF02101708

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  • DOI: https://doi.org/10.1007/BF02101708

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