Arithmetic Spectral Transitions for the Maryland Model

• Published in: Communications on pure and applied mathematics Vol. 70; no. 6; pp. 1025 - 1051
• Main Authors: Jitomirskaya, Svetlana, Liu, Wencai
• Format: Journal Article
• Language: English
• Published: New York John Wiley and Sons, Limited 01.06.2017
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21688

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)}¯ 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.