Algebraic Aspects of Periodic Graph Operators

Stephen P. Shipman () and Frank Sottile ()
Additional contact information
Stephen P. Shipman: Louisiana State University, Department of Mathematics
Frank Sottile: Texas A&M University, Department of Mathematics

Chapter 111 in Operator Theory, 2026, pp 3513-3553 from Springer

Abstract: Abstract A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, the Bloch and Fermi varieties. These encode the relation between the eigenvalues of the translation group and the eigenvalues of the operator. As they are algebraic varieties, algebraic methods may be used to study the spectrum of the operator. This paper establishes a framework in which commutative algebra directly comes to bear on the spectral theory of periodic operators, helping to distinguish their algebraic and analytic aspects. It also discusses reducibility of the Fermi variety and nondegeneracy of spectral band edges.

Keywords: Periodic operator; Graph operator; Bloch variety; Fermi variety; Spectral theory; Toric variety (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_80

Ordering information: This item can be ordered from
http://www.springer.com/9783032163561

DOI: 10.1007/978-3-032-16356-1_80

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().