We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.1 According to [43], the fact that the Diophantine properties of the frequencies should play a role was first observed in [42].

We give a precise description of spectra of the Maryland model (hλ,α,θu)n=un+1+un−1+λtanπ(θ+nα)un for all values of parameters. We introduce an arithmetically defined index δ(α,θ) and show that for α∉ℚ, σsc(hλ,α,θ)={e:γλ(e)<δ(α,θ)}true¯ and σpp(hλ,α,θ)={e:γλ(e)≥δ(α,θ)}. Since σac(hλ,α,θ)=∅, this gives a complete description of the spectral decomposition for all values of parameters λ, α, and θ, making it the first case of a family where arithmetic spectral transition is described without any parameter exclusion. The set of eigenvalues can be explicitly identified for all parameters, using the quantization condition. We also establish, for the first time for this or any other model, a quantization condition for singular continuous spectrum (an arithmetically defined measure zero set that supports singular continuous measures) for all parameters.© 2017 Wiley Periodicals, Inc.

For almost Mathieu operator (H λ,α,θ u) n = u n+1 + u n−1 + 2λ cos 2π(θ + nα)u n , the dry version of Ten Martini problem predicts that the spectrum Σ λ,α of H λ,α,θ has all gaps open for all λ 0 and α ∈ R\Q. Avila and Jitomirskaya prove that Σ λ,α has all gaps open for Diophantine α and 0 < |λ| < 1. In the present paper, we show that Σ λ,α has all gaps open for all α ∈ R\Q with small λ.

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