The spectra of surface Maryland model for all phases

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