Critical points of discrete periodic operators

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...

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Bibliographic Details
Published inJournal of spectral theory Vol. 14; no. 1; pp. 1 - 35
Main Authors Faust, Matthew, Sottile, Frank
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.01.2024
Subjects
Graph theory
Mathematical research
Schrodinger equation
United States
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ISSN1664-039X
1664-0403
DOI10.4171/jst/503

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Summary: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.
ISSN:1664-039X
1664-0403
DOI:10.4171/jst/503