On the Spectral and Propagation Properties of the Surface Maryland Model

TL;DR

This paper analyzes the spectral and propagation properties of a discrete Schrödinger operator with a surface potential, demonstrating absolute continuity of the spectrum and describing eigenfunctions as volume waves or surface states depending on the potential's rationality.

Contribution

It provides a detailed spectral analysis of the surface Maryland model, including the nature of eigenfunctions and the impact of rational versus irrational parameters on the spectrum.

Findings

Spectrum on [-d,d] is absolutely continuous.

Eigenfunctions are oscillating volume waves.

Surface states exist with exponential decay for rational parameters.

Abstract

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. 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 $\alpha$ for $d_1=d_2=1$, i.e. a periodic surface potential. In this case we show that the spectrum is absolutely continuous and 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 $\alpha$'s.