Zero measure spectrum for multi-frequency Schrödinger operators
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Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.
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Chaika, Jon, Damanik, David, Fillman, Jake, et al.. "Zero measure spectrum for multi-frequency Schrödinger operators." Journal of Spectral Theory, 12, no. 2 (2022) EMS Press: 573-590. https://doi.org/10.4171/jst/411.