On the characteristic polynomial of the almost Mathieu operator
Authors:
Michael P. Lamoureux and James A. Mingo
Journal:
Proc. Amer. Math. Soc. 135 (2007), 32053215
MSC (2000):
Primary 47B39; Secondary 47B15, 46L05
Published electronically:
May 14, 2007
MathSciNet review:
2322751
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be the rotation C*algebra for angle . For with and relatively prime, is the subC*algebra of generated by a pair of unitaries and satisfying . Let be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of is .
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 [A]
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 [AJ]
 A. Avila, S. Jitomirskaya, The Ten Martini Problem, Ann. of Math. to appear, preprint: math.DS/0503363.
 [AK]
 A. Avila, R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles, Ann. of Math. to appear, preprint: math.DS/0306382.
 [AVMS]
 J. Avron, P. H. M. van Mouche, B. Simon, On the Measure of the Spectrum for the Almost Mathieu Operator, Comm. Math. Phy. 132 (1990) 103118. MR 1069202 (92d:39014a)
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 FP. Boca, Rotation C*algebras and Almost Mathieu Operators, Theta, Bucharest, 2001. MR 1895184 (2003e:47063)
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 M.D. Choi, G. A. Elliott, and N. Yui, Gauss Polynomials and the rotation algebras, Invent. Math. 99, (1990), 225  246. MR 1031901 (91b:46067)
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 D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phy. Rev. B, 14 (1976) 22392249.
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 M. Lamoureux, Reflections on the almost Mathieu operator, Integral Equations and Operator Theory 28 (1997), 45  59.MR 1446830 (98d:47068)
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 Y. Last, Zero Measure Spectrum for the Almost Mathieu Operator, Comm. Math. Phy., 164 (1994) 421432.MR 1289331 (95f:47096)
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 J. Puig, Cantor spectrum for the almost mathieu operator, Comm. Math. Phy. 244 (2004), 297234. MR 2031032 (2004k:11129)
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 T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, 1990.MR 1060735 (92a:41016)
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 J. J. Sylvester, On a remarkable modification of Sturm's Theorem, Phil. Mag., 5 (1853), 446  456 (also pp. 609  619 in Mathematical Papers, vol. I, Cambridge University Press, 1904).
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 M. Toda, Theory of Nonlinear Lattices, ed., Springer Series in SolidState Sciences, vol. 20, SpringerVerlag, Berlin, (1989). MR 0971987 (89h:58082)
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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/S0002993907088302
PII:
S 00029939(07)088302
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
