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On the spectrum of the surface Maryland model. (English) Zbl 0916.47027

Summary: We study spectral properties of the discrete Laplacian \(H\) on the half space \(\mathbb{Z}^{d+1}_+= \mathbb{Z}^d\times \mathbb{Z}_+\) with a boundary condition \(\psi(n,-1)= \lambda\tan(\pi\alpha\cdot n+\theta)\psi(n,0)\), where \(\alpha\in[0, 1]^d\). We denote by \(H_0\) the Dirichlet Laplacian on \(\mathbb{Z}^{d+1}_+\). Whenever \(\alpha\) is independent over rationals, \(\sigma(H)= \mathbb{R}\). Khoruzenko and Pastur have shown for a set of \(\alpha\)’s of Lebesgue measure 1 that the spectrum of \(H\) on \(\mathbb{R}\setminus \sigma(H_0)\) is pure point and that corresponding eigenfunctions decay exponentially. In this letter, we show that if \(\alpha\) is independent over rationals, then the spectrum of \(H\) on the set \(\sigma(H_0)\) is purely absolutely continuous.

MSC:

47B80 Random linear operators
60H25 Random operators and equations (aspects of stochastic analysis)
47N55 Applications of operator theory in statistical physics (MSC2000)
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