On the Spectral and Propagation Properties of the Surface Maryland Model

Abstract

We study the discrete Schrödinger operator $H$ in $\ZZ^d$ with the surface potential of the form $V(x)=g δ(x_1) \tan π(α\cdot x_2+ ω)$, where for $x \in \ZZ^d$ we write $x=(x_1,x_2), \quad x_1 \in \ZZ^{d_1}, x_2 \in \mathbb{Z}^{d_2}, α\in \R^{d_2}, ω\in [0,1)$. We first consider the case where the components of the vector $α$ 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. Then we show that generalized eigenfunctions corresponding to this interval have the form of volume (bulk) waves, which are oscillating and non decreasing (or slow decreasing) in all variables. They are the sum of the incident plane wave and of an infinite number of reflected or transmitted plane waves scattered by the "plane" $\ZZ^{d_2}$. These eigenfunctions are orthogonal, complete and verify a natural analogue of the Lippmann-Schwinger equation. We also discuss the case of rational vectors $α$ for $d_1=d_2=1$, i.e. a periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides volume (Bloch) waves there are also surface waves, whose amplitude decays exponentially as $|x_1| \to \infty$. The part of the spectrum corresponding to the surface states consists of a finite number of bands. For large $q$ the bands outside of $[-d,d]$ are exponentially small in $q$, and converge in a natural sense to the pure point spectrum, that was found in [KP] in the case of the Diophantine $α$'s.