On spectral bands of discrete periodic operators

TL;DR

This paper investigates the spectral properties of discrete periodic operators on integer lattices, identifying conditions for spectral gaps and analyzing the structure of spectral band edges across various lattice configurations.

Contribution

It characterizes lattices that allow spectral gaps with small potentials and analyzes the dimensionality of spectral band edge level sets.

Findings

Certain lattices permit spectral gaps with arbitrarily small potentials.

The dimension of spectral band edge level sets is at most d-2 for many lattices.

The class of lattices affecting spectral gap existence is explicitly described.

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 $d-2$.