Arithmetic Spectral Transitions for the Maryland Model

Abstract

We give a precise description of spectra of the Maryland model $ (h_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+ λ\tan π(θ+nα)u_n$ for all values of parameters. We introduce an arithmetically defined index $δ(α, θ)$ and show that for $α\notin\mathbb{Q},\,$ $σ_{sc}(h_{λ,α,θ})=\overline{\{e:γ_λ(e) <δ(α, θ) \}}$ and $σ_{pp}(h_{λ,α,θ})=\{e:γ_λ(e) \geq δ(α, θ) \}$. Since $σ_{ac}(h_{λ,α,θ})=\emptyset,\;$ this gives complete description of the spectral decomposition for {\it all} values of parameters $λ,α,θ$, 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 {\it 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.