The spectra of the 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 $d=1$.