98-244 Jaksic V., Molchanov S.

On the Spectrum of the Surface Maryland Model (431K, postscript) Mar 30, 98

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

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