Approximating spectral invariants of Harper operators on graphs II

Authors:
Varghese Mathai, Thomas Schick and Stuart Yates

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1917-1923

MSC (2000):
Primary 58J50, 39A12

Published electronically:
September 20, 2002

MathSciNet review:
1955281

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Abstract: 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. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

**[Ad]**Toshiaki Adachi,*A note on the Følner condition for amenability*, Nagoya Math. J.**131**(1993), 67–74. MR**1238633****[Bel]**Jean Bellissard,*Gap labelling theorems for Schrödinger operators*, From number theory to physics (Les Houches, 1989) Springer, Berlin, 1992, pp. 538–630. MR**1221111****[DLMSY]**J. Dodziuk, P. Linnell, V. Mathai, T. Schick and S. Yates, Approximating -invariants, and the Atiyah conjecture,*Commun. in Pure and Applied Math.*(to appear).**[Eck]**Beno Eckmann,*Approximating 𝑙₂-Betti numbers of an amenable covering by ordinary Betti numbers*, Comment. Math. Helv.**74**(1999), no. 1, 150–155. MR**1677086**, 10.1007/s000140050081**[El]**G. Elek, On the analytic zero divisor conjecture of Linnell,`math.GR/0111180`.**[MY]**V. Mathai and S. Yates, Approximating spectral invariants of Harper operators on graphs,*J. Functional Analysis***188**(2002), no. 1, 111-136.**[Sh]**M. A. Shubin,*Discrete magnetic Laplacian*, Comm. Math. Phys.**164**(1994), no. 2, 259–275. MR**1289325****[Sun]**Toshikazu Sunada,*A discrete analogue of periodic magnetic Schrödinger operators*, Geometry of the spectrum (Seattle, WA, 1993) Contemp. Math., vol. 173, Amer. Math. Soc., Providence, RI, 1994, pp. 283–299. MR**1298211**, 10.1090/conm/173/01831

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Additional Information

**Varghese Mathai**

Affiliation:
Department of Mathematics, University of Adelaide, Adelaide 5005, Australia

Email:
vmathai@maths.adelaide.edu.au

**Thomas Schick**

Affiliation:
FB Mathematik, Universität Göttingen, Bunsenstrasse 3, 37073 Göttingen, Germany

Email:
schick@uni-math.gwdg.de

**Stuart Yates**

Affiliation:
Department of Mathematics, University of Adelaide, Adelaide 5005, Australia

Email:
syates@maths.adelaide.edu.au

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06739-4

Keywords:
Harper operator,
discrete magnetic Laplacian,
DML,
approximation theorems,
amenable groups,
von Neumann algebras,
graphs,
integrated density of states

Received by editor(s):
January 12, 2002

Published electronically:
September 20, 2002

Additional Notes:
The first and third authors acknowledge support from the Australian Research Council.

Communicated by:
Jozef Dodziuk

Article copyright:
© Copyright 2002
American Mathematical Society