Critical points of discrete periodic operators

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 ${\mathbb{Z}}^2$- and ${\mathbb{Z}}^3$-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some ${\mathbb{Z}}^2$-periodic graphs.