Trace formulas for Schrödinger operators on periodic graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a sp...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 508; no. 2; p. 125888
Main Authors Korotyaev, Evgeny, Saburova, Natalia
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.04.2022
Subjects
Discrete Schrödinger operators
Periodic graphs
Trace formulas
Trace formulas
Discrete Schrödinger operators
Periodic graphs
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ISSN0022-247X
1096-0813
DOI10.1016/j.jmaa.2021.125888

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Summary:We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schrödinger operators and the determinant formulas.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2021.125888