Arithmetic Spectral Transitions for the Maryland Model

TL;DR

This paper provides a comprehensive analysis of the spectral properties of the Maryland model, introducing an arithmetic index to precisely describe spectral types for all parameters, including a novel quantization condition for singular continuous spectrum.

Contribution

It offers the first complete spectral description of the Maryland model without parameter restrictions, including a new quantization condition for singular continuous spectrum.

Findings

Complete spectral decomposition for all parameters.

Explicit identification of eigenvalues via quantization condition.

First quantization condition established for singular continuous spectrum.

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 $\sigma_{pp}(h_{\lambda,\alpha,\theta})=\{e:\gamma_{\lambda}(e) \geq \delta (\alpha, \theta) \}$. Since $\sigma_{ac}(h_{\lambda,\alpha,\theta})=\emptyset,\;$ 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.