Critical points of discrete periodic operators
Matthew Faust
Texas A&M University, USAFrank Sottile
Texas A&M University, USA
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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 - and -periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some -periodic graphs.
Cite this article
Matthew Faust, Frank Sottile, Critical points of discrete periodic operators. J. Spectr. Theory 14 (2024), no. 1, pp. 1–35
DOI 10.4171/JST/503