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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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
Online AccessGet full text
ISSN1664-039X
1664-0403
1664-0403
DOI10.4171/jst/503

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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 {\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.
AbstractList 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.
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. Keywords: Bloch variety, Schrodinger operator, Kushnirenko Theorem, Toric variety, Newton polytope.
Audience Academic
Author Faust, Matthew
Sottile, Frank
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SubjectTerms Graph theory
Mathematical research
Schrodinger equation
Title Critical points of discrete periodic operators
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