The spectra of surface Maryland model for all phases

Abstract

We study the discrete Schrödinger 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 $d=1$.