Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Regularity of the spectrum for the almost Mathieu operator

Author: Norbert Riedel
Journal: Proc. Amer. Math. Soc. 129 (2001), 1681-1687
MSC (2000): Primary 47A10, 47B39, 46L05, 31A05, 11J04
Published electronically: October 31, 2000
MathSciNet review: 1814097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the logarithmic potential associated with the integrated density of states is constant on the spectrum of the almost Mathieu operator in case the irrational frequency is sufficiently well approximable by rationals in terms of a diophantine condition.

References [Enhancements On Off] (What's this?)

  • [A] Bernard Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Mathematics, vol. 735, Springer, Berlin, 1979 (French). MR 549769
  • [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, 10.1007/BF01234419
  • [N] Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
  • [P] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
  • [R1] Norbert Riedel, On the topological stable rank of irrational rotation algebras, J. Operator Theory 13 (1985), no. 1, 143–150. MR 768308
  • [R2] Norbert Riedel, Spectra and resolvents of certain polynomials in the irrational rotation algebra, Indiana Univ. Math. J. 39 (1990), no. 4, 937–945. MR 1087179, 10.1512/iumj.1990.39.39044
  • [R3] N. Riedel; The spectrum of a class of almost periodic operators; preprint.
  • [SY] M. Sodin, P. Yuditskii; Almost periodic Jacobi matrices with homogeneous spectrum, and Hardy classes of character automorphic functions; J. Geom. Anal. 7 (1997), 387-435. CMP 99:09
  • [T] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894
  • [W] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR 0218588

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A10, 47B39, 46L05, 31A05, 11J04

Retrieve articles in all journals with MSC (2000): 47A10, 47B39, 46L05, 31A05, 11J04

Additional Information

Norbert Riedel
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118

Received by editor(s): September 9, 1999
Published electronically: October 31, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society