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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the characteristic polynomial of the almost Mathieu operator


Authors: Michael P. Lamoureux and James A. Mingo
Journal: Proc. Amer. Math. Soc. 135 (2007), 3205-3215
MSC (2000): Primary 47B39; Secondary 47B15, 46L05
Published electronically: May 14, 2007
MathSciNet review: 2322751
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Abstract: Let $ A_\theta$ be the rotation C*-algebra for angle $ \theta$. For $ \theta = p/q$ with $ p$ and $ q$ relatively prime, $ A_\theta$ is the sub-C*-algebra of $ M_q(C(\mathbb{ T}^2))$ generated by a pair of unitaries $ u$ and $ v$ satisfying $ uv = e^{2 \pi i \theta} v u$. Let

$\displaystyle h_{\theta, \lambda} = u + u^{-1} + \lambda/2(v + v^{-1})$

be the almost Mathieu operator. By proving an identity of rational functions we show that for $ q$ even, the constant term in the characteristic polynomial of $ h_{\theta, \lambda}$ is $ (-1)^{q/2}(1 + (\lambda/2)^q) - (z_1^q + z_1^{-q} + (\lambda/2)^q(z_2^q + z_2^{-q}))$.


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Additional Information

Michael P. Lamoureux
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2T 1A1
Email: mikel@math.ucalgary.ca

James A. Mingo
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email: mingo@mast.queensu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08830-2
PII: S 0002-9939(07)08830-2
Received by editor(s): April 3, 2006
Received by editor(s) in revised form: June 19, 2006
Published electronically: May 14, 2007
Additional Notes: Research supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society