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We first consider the case where the components of the vector $\\alpha$ are rationally independent, i.e. the case of the quasi periodic potential. We prove that the spectrum of $H$ on the interval $[-d,d]$ (coinciding with the spectrum of the discrete Laplacian) is absolutely continuous. 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Bentosela, L. Pastur, Ph. Briet","submitted_at":"2002-06-13T13:01:01Z","abstract_excerpt":"We study the discrete Schr\\\"odinger operator $H$ in $\\ZZ^d$ with the surface potential of the form $V(x)=g \\delta(x_1) \\tan \\pi(\\alpha \\cdot x_2+ \\omega)$, where for $x \\in \\ZZ^d$ we write $x=(x_1,x_2), \\quad x_1 \\in \\ZZ^{d_1}, x_2 \\in \\mathbb{Z}^{d_2}, \\alpha \\in \\R^{d_2}, \\omega \\in [0,1)$. We first consider the case where the components of the vector $\\alpha$ are rationally independent, i.e. the case of the quasi periodic potential. We prove that the spectrum of $H$ on the interval $[-d,d]$ (coinciding with the spectrum of the discrete Laplacian) is absolutely continuous. 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