On spectral bands of discrete periodic operators

Abstract

We consider discrete periodic operator on ${\mathbb{Z}}^d$ with respect to lattices $\Gamma \subset {\mathbb{Z}}^d$ of full rank. We describe the class of lattices $\Gamma$ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed $d-2$.

14 pages