Communications in Mathematical Physics

, Volume 164, Issue 2, pp 421–432 | Cite as

Zero measure spectrum for the almost Mathieu operator



We study the almost Mathieu operator: (H α, λ, θ u)(n)=u(n+1)+u(n-1)+λ cos (2παn+θ)u(n), onl2(Z), and show that for all λ,θ, and (Lebesgue) a.e. α, the Lebesgue measure of its spectrum is precisely |4–2|λ‖. In particular, for |λ|=2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational α's (and |λ|=2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Lebesgue Measure 


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  1. 1.
    Aubry, S., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc.3, 133–164 (1980)Google Scholar
  2. 2.
    Avron, J., Simon, B.: Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J.50, 369–391 (1983)Google Scholar
  3. 3.
    Avron, J., van Mouche, P., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys.132, 103–118 (1990)Google Scholar
  4. 4.
    Bellissard, J., Lima, R., Testard, D.: A metal-insulator transition for the almost Mathieu model. Commun. Math. Phys.88, 207–234 (1983)Google Scholar
  5. 5.
    Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal.48, 408–419 (1982)Google Scholar
  6. 6.
    Chambers, W.: Linear network model for magnetic breakdown in two dimensions. Phys. Rev. A140, 135–143 (1965)Google Scholar
  7. 7.
    Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math.99, 225–246 (1990)Google Scholar
  8. 8.
    Chulaevsky, V., Delyon, F.: Purely absolutely continuous spectrum for almost Mathieu operators. J. Stat. Phys.55, 1279–1284 (1989)Google Scholar
  9. 9.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  10. 10.
    Delyon, F.: Absence of localization for the almost Mathieu equation. J. Phys. A20, L21-L23 (1987)Google Scholar
  11. 11.
    Harper, P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. Lond. A68, 874–892 (1955)Google Scholar
  12. 12.
    Helffer, B., Sjostrand, J.: Semi-classical analysis for Harper's equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France (N.S.)39, 1–139 (1989)Google Scholar
  13. 13.
    Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B14, 2239–2249 (1976)Google Scholar
  14. 14.
    Falconer, K.J.: The geometry of fractal sets. Cambridge: Cambridge University Press 1985Google Scholar
  15. 15.
    Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990)Google Scholar
  16. 16.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, Fifth ed. Oxford: Oxford University Press, 1979Google Scholar
  17. 17.
    Last, Y.: A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys.151, 183–192 (1993)Google Scholar
  18. 18.
    Last, Y., Wilkinson, M.: A sum rule for the dispersion relations of the rational Harper's equation. J. Phys. A25, 6123–6133 (1992)Google Scholar
  19. 19.
    Simon, B.: Almost periodic Schrödinger operators: a review. Adv. Appl. Math.3, 463–490 (1982)Google Scholar
  20. 20.
    Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys.46, 861–909 (1987)Google Scholar
  21. 21.
    Thouless, D.J.: Bandwidth for a quasiperiodic tight binding model. Phys. Rev. B28, 4272–4276 (1983)Google Scholar
  22. 22.
    Thouless, D.J.: Scaling for the discrete Mathieu equation. Commun. Math. Phys.127, 187–193 (1990)Google Scholar
  23. 23.
    Thouless, D.J., Tan, Y.: Total bandwidth for the Harper equation. III. Corrections to scaling. J. Phys. A24, 4055–4066 (1991)Google Scholar
  24. 24.
    Thouless, D.J., Tan, Y.: Scaling, localization and bandwidths for equations with competing periods. Physica A177, 567–577 (1991)Google Scholar
  25. 25.
    Toda, M.: Theory of nonlinear lattices, 2nd Ed., Chap. 4. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  26. 26.
    Watson, G.I.: WKB analysis of energy band structure of modulated systems. J. Phys. A24, 4999–5010 (1991)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Y. Last
    • 1
  1. 1.Department of PhysicsTechnion-Israel Institute of TechnologyHaifaIsrael

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