The spectra of surface Maryland model for all phases

We study the discrete Schrödinger operators $H_{λ,α,θ}$ on $\ell^2(\mathbb{Z}^{d+1})$ with surface potential of the form $V(n,x)=λδ(x)\tanπ(α\cdot n+θ)$, and $H_{λ,α,θ}^{+}$ on $\ell^2(\mathbb{Z}^{d}\times \mathbb{Z}_+)$ with the boundary condition $ψ_{(n,-1)}=λ\tanπ(α\cdot n+θ)ψ_{(n,0)}$, where $α\in \mathbb{R}^d$ is rationally independent. We show that the spectra of $H_{λ,α,θ}$ and $H_{λ,α,θ}^{+}$ are $(-\infty,\infty)$ for all parameters. We can also determine the absolutely continuous spectra and Hausdorff dimension of the spectral measures if $d=1$.

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