Approximating spectral invariants of Harper operators on graphs

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free...

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Bibliographic Details
Published in:Journal of Functional Analysis
Main Authors: Varghese, M., Yates, S.
Format: Article in Journal/Newspaper
Language:English
Published: Academic Press Inc 2002
Subjects:
Harper operator
approximation theorems
amenable groups
von Neumann algebras
graphs
Fuglede–Kadison determinant
algebraic number theory
Harper
DML
Online Access:http://hdl.handle.net/2440/3596
https://doi.org/10.1006/jfan.2001.3841
Description
Summary:We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory. Varghese Mathai and Stuart Yates http://www.elsevier.com/wps/find/journaldescription.cws_home/622879/description#description

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