Bloch varieties and quantum ergodicity for periodic graph operators

TL;DR

This paper links the irreducibility of Bloch varieties to the absence of non-trivial periods in spectral band functions for periodic graph operators, providing new criteria and answering a prior open question.

Contribution

It introduces criteria based on Bloch varieties to analyze spectral band functions, demonstrating irreducibility implies no non-trivial periods for a broad class of periodic graph operators.

Findings

Irreducibility of Bloch varieties implies no non-trivial periods in spectral band functions.

Established criteria for spectral overlaps in periodic graph operators.

Answered an open question regarding periodic Schrödinger operators on  lattices.

Abstract

For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph operators, the irreducibility of Bloch varieties implies no non-trivial periods for spectral band functions. This particularly shows that spectral band functions of discrete periodic Schr\"odinger operators on $\mathbb{Z}^d$ have no non-trivial periods, answering a question asked by Mckenzie and Sabri [Quantum ergodicity for periodic graphs. arXiv preprint arXiv:2208.12685 (2022)].