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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Approximating spectral invariants of Harper operators on graphs II
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by Varghese Mathai, Thomas Schick and Stuart Yates PDF
Proc. Amer. Math. Soc. 131 (2003), 1917-1923 Request permission

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.
References
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Additional Information
  • Varghese Mathai
  • Affiliation: Department of Mathematics, University of Adelaide, Adelaide 5005, Australia
  • MR Author ID: 231404
  • Email: vmathai@maths.adelaide.edu.au
  • Thomas Schick
  • Affiliation: FB Mathematik, Universität Göttingen, Bunsenstrasse 3, 37073 Göttingen, Germany
  • MR Author ID: 635784
  • Email: schick@uni-math.gwdg.de
  • Stuart Yates
  • Affiliation: Department of Mathematics, University of Adelaide, Adelaide 5005, Australia
  • Email: syates@maths.adelaide.edu.au
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 1917-1923
  • MSC (2000): Primary 58J50, 39A12
  • DOI: https://doi.org/10.1090/S0002-9939-02-06739-4
  • MathSciNet review: 1955281