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

DOI:
https://doi.org/10.1090/S0002-9939-07-08830-2

Published electronically:
May 14, 2007

MathSciNet review:
2322751

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the rotation C*-algebra for angle . For with and relatively prime, is the sub-C*-algebra of generated by a pair of unitaries and satisfying . Let

**[A]**William Arveson,*Improper filtrations for 𝐶*-algebras: spectra of unilateral tridiagonal operators*, Acta Sci. Math. (Szeged)**57**(1993), no. 1-4, 11–24. MR**1243265****[A]**William Arveson,*𝐶*-algebras and numerical linear algebra*, J. Funct. Anal.**122**(1994), no. 2, 333–360. MR**1276162**, https://doi.org/10.1006/jfan.1994.1072**[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 non-uniform hyperbolicity for quasi-periodic schrodinger co-cycles,*Ann. of Math.*to appear, preprint: math.DS/0306382.**[AVMS]**J. Avron, P. H. M. van Mouche, and B. Simon,*On the measure of the spectrum for the almost Mathieu operator*, Comm. Math. Phys.**132**(1990), no. 1, 103–118. MR**1069202**

J. E. Avron, P. van Mouche, and B. Simon,*Erratum: “On the measure of the spectrum for the almost Mathieu operator”*, Comm. Math. Phys.**139**(1991), no. 1, 215. MR**1116417****[BS]**J. Béllissard and B. Simon,*Cantor spectrum for the almost Mathieu equation*, J. Funct. Anal.**48**(1982), no. 3, 408–419. MR**678179**, https://doi.org/10.1016/0022-1236(82)90094-5**[B]**Florin-Petre Boca,*Rotation 𝐶*-algebras and almost Mathieu operators*, Theta Series in Advanced Mathematics, vol. 1, The Theta Foundation, Bucharest, 2001. MR**1895184****[CEY]**Man Duen Choi, George A. Elliott, and Noriko Yui,*Gauss polynomials and the rotation algebra*, Invent. Math.**99**(1990), no. 2, 225–246. MR**1031901**, https://doi.org/10.1007/BF01234419**[H]**D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,*Phy. Rev. B*,**14**(1976) 2239-2249.**[L]**Michael P. Lamoureux,*Reflections on the almost Mathieu operator*, Integral Equations Operator Theory**28**(1997), no. 1, 45–59. MR**1446830**, https://doi.org/10.1007/BF01198795**[LT]**Y. Last,*Zero measure spectrum for the almost Mathieu operator*, Comm. Math. Phys.**164**(1994), no. 2, 421–432. MR**1289331****[P]**Joaquim Puig,*Cantor spectrum for the almost Mathieu operator*, Comm. Math. Phys.**244**(2004), no. 2, 297–309. MR**2031032**, https://doi.org/10.1007/s00220-003-0977-3**[R]**Theodore J. Rivlin,*Chebyshev polynomials*, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR**1060735****[S]**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).**[T]**Morikazu Toda,*Theory of nonlinear lattices*, 2nd ed., Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989. MR**971987**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B39,
47B15,
46L05

Retrieve articles in all journals with MSC (2000): 47B39, 47B15, 46L05

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:
https://doi.org/10.1090/S0002-9939-07-08830-2

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