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

We study the discrete Schroedinger operator H in Zd with the surface potential of the form V(x)=gδ(x1)tan π(α·x2+ω), where for x(set-membership sign)Zd we write x=(x1,x2), x1(set-membership sign)Zd1, x2(set-membership sign)Zd2, α(set-membership sign)Rd2, ω(set-membership sign)[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 nondecreasing (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 subspace Zd2. These eigenfunctions are orthogonal, complete and verify a natural analog of the Lippmann-Schwinger equation. We discuss also the case where d1=d2=1 and α=p/q is a rational number, i.e., a q-periodic surface potential. In this case we show that the spectrum is absolutely continuous and besides the volume (Bloch) waves there are also the surface waves, whose amplitude decays exponentially as |x1|→∞. The part of the spectrum corresponding to the surface waves 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 [B. Khoruzhenko and L. Pastur, Phys. Rep. 288, 109-125 (1997)] in the case of the Diophantine α's