Arithmetic Spectral Transitions for the Maryland Model

Abstract

We give a precise description of spectra of the Maryland model $(h_{\lambda ,\alpha ,\theta }u{)}_n=u_{n+1}+u_{n-1}+\lambda \tan \pi (\theta +n\alpha )u_n$ for all values of parameters. We introduce an arithmetically defined index $\delta (\alpha ,\theta )$ and show that for $\alpha \notin \mathbb{Q},$ ${\sigma }_{sc}(h_{\lambda ,\alpha ,\theta })=\overline{\{e:{\gamma }_{\lambda }(e)<\delta (\alpha ,\theta )\}}$ and ${\sigma }_{pp}(h_{\lambda ,\alpha ,\theta })=\{e:{\gamma }_{\lambda }(e)\ge \delta (\alpha ,\theta )\}$. Since ${\sigma }_{ac}(h_{\lambda ,\alpha ,\theta })=\varnothing ,$ this gives complete description of the spectral decomposition for {\it all} values of parameters $\lambda ,\alpha ,\theta$, 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.

arXiv commentCPAM to appear

References in corpus (3)

• Complex one-frequency cocycles
• An exact renormalization formula for the Maryland model
• Critical integer quantum Hall topology and the integrable Maryland model as a topological quantum critical point

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