Critical points of discrete periodic operators

Matthew Faust, Frank Sottile

Published 2022-06-27Version 1

We study the spectrum 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 achieved. We show that this criterion holds for periodic graphs of dimensions 2 and 3 with sufficiently many edges and use our results to establish the spectral edges conjecture for some periodic graphs of dimension 2. Our larger goal is to develop new methods to address questions in spectral theory.