Critical points of discrete periodic operators - Faust, Matthew; Sottile, Frank (Journal of spectral theory, 2024)
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.