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We study the discrete Schr\"odinger operator $H$ in $\ZZ^d$ with the surfacepotential 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 thecase 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 discreteLaplacian) is absolutely continuous. Then we show that generalizedeigenfunctions corresponding to this interval have the form of volume (bulk)waves, which are oscillating and non decreasing (or slow decreasing) in allvariables. They are the sum of the incident plane wave and of an infinitenumber of reflected or transmitted plane waves scattered by the "plane"$\ZZ^{d_2}$. These eigenfunctions are orthogonal, complete and verify a naturalanalogue of the Lippmann-Schwinger equation. We also discuss the case ofrational vectors $\alpha$ for $d_1=d_2=1$, i.e. a periodic surface potential.In this case we show that the spectrum is absolutely continuous and besidesvolume (Bloch) waves there are also surface waves, whose amplitude decaysexponentially as $|x_1| \to \infty$. The part of the spectrum corresponding tothe surface states consists of a finite number of bands. For large $q$ thebands outside of $[-d,d]$ are exponentially small in $q$, and converge in anatural sense to the pure point spectrum, that was found in [KP] in the case ofthe Diophantine $\alpha$'s.Comment: 44page

On the spectral and wave propagation properties of the surface Maryland model