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Wave operators for the surface Maryland model. (English) Zbl 0983.82007
Summary: We study scattering properties of the discrete Laplacian \(H\) on the half-space \(\mathbb{Z}_+^{d+1} =\mathbb{Z}^d \times \mathbb{Z}_+\) with the boundary condition \(\pi(n,-1)= \lambda\text{ tan} (\pi\alpha \cdot n+ \theta)\psi (n,0)\), where \(\alpha\in [0,1]^d\). We denote by \(H_0\) the Dirichlet Laplacian on \(\mathbb{Z}^{d+1}_+\). B. A. Khoruzenko and L. Pastur [Phys. Rep. 288, 109-126 (1997)] have shown that if \(\alpha\) has typical Diophantine properties then the spectrum of \(H\) on \(\mathbb{R}\setminus \sigma (H_0)\) is pure point and that corresponding eigenfunctions decay exponentially. We demonstrated in an earlier paper [Lett. Math. Phys. 45, 185-193 (1998; Zbl 0916.47027), see also Helv. Phys. Acta 71, 629-657 (1998; Zbl 0939.60074) and Commun. Math. Phys. 208, 153-172 (1999; Zbl 0952.60059)] that for every \(\alpha\) independent over the rationals the spectrum of \(H\) on \(\sigma(H_0\)) is purely absolutely continuous.
In this paper, we continue the analysis of \(H\) on \(\sigma (H_0)\) and prove that whenever \(\alpha\) is independent over the rationals, the wave operators \(\Omega^\pm (H,H_0)\) exist and are complete on \(\sigma(H_0)\). Moreover, we show that under the same conditions \(H\) has no surface states on \(\sigma (H_0)\).

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
60H25 Random operators and equations (aspects of stochastic analysis)
39A70 Difference operators
47B39 Linear difference operators
47N50 Applications of operator theory in the physical sciences
47N55 Applications of operator theory in statistical physics (MSC2000)
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References:
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