Arithmetic Spectral Transitions for the Maryland Model

Abstract: We give a precise description of spectra of the Maryland model $$(h_{\lambda,\alpha,\theta}u)<em>n=u</em>{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$$ for all values of parameters. We introduce an arithmetically defined index $\delta (\alpha, \theta)$ and show that for $\alpha\notin\mathbb{Q},\,$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{{e:\gamma_{\lambda}(e) &lt;\delta (\alpha, \theta) }}$ 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.