Critical points of discrete periodic operators

TL;DR

This paper investigates the spectra of operators on periodic graphs using algebraic geometry, providing bounds on critical points and criteria for their attainment, with applications to the spectral edges conjecture.

Contribution

It introduces bounds on complex critical points of the Bloch variety and an effective criterion for when these bounds are achieved, advancing spectral analysis of periodic graphs.

Findings

Bound on the number of complex critical points of the Bloch variety.

Effective criterion for attaining the critical point bound.

Verification of the spectral edges conjecture for certain Z^2-periodic graphs.

Abstract

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z^2- and Z^3-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z^2-periodic graphs.