The spectra of surface Maryland model for all phases

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 $d=1$.