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arXiv · 2206.13649

Critical points of discrete periodic operators

Matthew Faust·Frank Sottile
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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.

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Matthew Faust, Frank Sottile. 2022-06-27. Critical points of discrete periodic operators. https://arxiv.org/abs/2206.13649

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