The spectra of surface Maryland model for all phases

TL;DR

This paper investigates the spectral properties of surface Maryland models, showing their spectra cover the entire real line for all parameters and analyzing the absolutely continuous spectra and spectral measure dimensions in one dimension.

Contribution

It provides a comprehensive analysis of the spectra of surface Maryland models, establishing their spectra as the entire real line and characterizing spectral measures in one dimension.

Findings

Spectra of the models are the entire real line for all parameters.

Determination of absolutely continuous spectra in one-dimensional case.

Calculation of Hausdorff dimension of spectral measures in one-dimensional case.

Abstract

We study the discrete Schr\"{o}dinger operators $H_{\lambda,\alpha,\theta}$ on $\ell^2(\mathbb{Z}^{d+1})$ with surface potential of the form $V(n,x)=\lambda \delta(x)\tan\pi(\alpha\cdot n+\theta)$, and $H_{\lambda,\alpha,\theta}^{+}$ on $\ell^2(\mathbb{Z}^{d}\times \mathbb{Z}_+)$ with the boundary condition $ \psi_{(n,-1)}=\lambda \tan\pi(\alpha\cdot n+\theta)\psi_{(n,0)} $, where $\alpha\in \mathbb{R}^d$ is rationally independent. We show that the spectra of $H_{\lambda,\alpha,\theta}$ and $H_{\lambda,\alpha,\theta}^{+}$ are $(-\infty,\infty)$ for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if $d=1$.