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 
Similar Articles 
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 .
 [A]
William
Arveson, Improper filtrations for 𝐶*algebras: spectra of
unilateral tridiagonal operators, Acta Sci. Math. (Szeged)
57 (1993), no. 14, 11–24. MR 1243265
(94i:46071)
 [A]
William
Arveson, 𝐶*algebras and numerical linear algebra, J.
Funct. Anal. 122 (1994), no. 2, 333–360. MR 1276162
(95i:46083), http://dx.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 nonuniform hyperbolicity for quasiperiodic schrodinger cocycles, 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
(92d:39014a)
 [BS]
J.
Béllissard and B.
Simon, Cantor spectrum for the almost Mathieu equation, J.
Funct. Anal. 48 (1982), no. 3, 408–419. MR 678179
(84h:81019), http://dx.doi.org/10.1016/00221236(82)900945
 [B]
FlorinPetre
Boca, Rotation 𝐶*algebras and almost Mathieu
operators, Theta Series in Advanced Mathematics, vol. 1, The
Theta Foundation, Bucharest, 2001. MR 1895184
(2003e:47063)
 [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
(91b:46067), http://dx.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) 22392249.
 [L]
Michael
P. Lamoureux, Reflections on the almost Mathieu operator,
Integral Equations Operator Theory 28 (1997), no. 1,
45–59. MR
1446830 (98d:47068), http://dx.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 (95f:47096)
 [P]
Joaquim
Puig, Cantor spectrum for the almost Mathieu operator, Comm.
Math. Phys. 244 (2004), no. 2, 297–309. MR 2031032
(2004k:11129), http://dx.doi.org/10.1007/s0022000309773
 [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
(92a:41016)
 [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 SolidState Sciences, vol. 20, SpringerVerlag, Berlin, 1989. MR 971987
(89h:58082)
 [A]
 W. Arveson, Improper Filtrations for C*algebras: spectra of unilateral tridiagonal operators, Acta Sci. Math (Szeged), 57 (1993), 1124.MR 1243265 (94i:46071)
 [A]
 W. Arveson, C*algebras and numerical linear algebra, J. Functional Analysis, 122 (1994), 333360. MR 1276162 (95i:46083)
 [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)
 [BS]
 J. Bellissard and B. Simon, Cantor spectrum for the Almost Mathieu Operator, J. Functional Analysis 48, (1982) 408419. MR 0678179 (84h:81019)
 [B]
 FP. Boca, Rotation C*algebras and Almost Mathieu Operators, Theta, Bucharest, 2001. MR 1895184 (2003e:47063)
 [CEY]
 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)
 [H]
 D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phy. Rev. B, 14 (1976) 22392249.
 [L]
 M. Lamoureux, Reflections on the almost Mathieu operator, Integral Equations and Operator Theory 28 (1997), 45  59.MR 1446830 (98d:47068)
 [LT]
 Y. Last, Zero Measure Spectrum for the Almost Mathieu Operator, Comm. Math. Phy., 164 (1994) 421432.MR 1289331 (95f:47096)
 [P]
 J. Puig, Cantor spectrum for the almost mathieu operator, Comm. Math. Phy. 244 (2004), 297234. MR 2031032 (2004k:11129)
 [R]
 T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, 1990.MR 1060735 (92a:41016)
 [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]
 M. Toda, Theory of Nonlinear Lattices, ed., Springer Series in SolidState Sciences, vol. 20, SpringerVerlag, Berlin, (1989). MR 0971987 (89h:58082)
Similar Articles
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:
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
