Skip to main content
SpringerLink
Log in

Cantor Spectrum for the Almost Mathieu Operator

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \({{ {{\left({{H_{{b,\phi}} x}}\right)}}_n= x_{{n+1}} +x_{{n-1}} + b \cos{{\left({{2 \pi n \omega + \phi}}\right)}}x_n }}\) on l 2(ℤ) and its associated eigenvalue equation to deduce that for b≠0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ‘‘Ten Martini Problem’’ for these values of b and ω. Moreover, we prove that for |b|≠0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnol’d, V.I.: Geometrical methods in the theory of ordinary differential equations. Vol. 250 of Grundlehren der Mathematischen Wissenschaften. New York: Springer-Verlag, 1983

  2. Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Preprint, 2003

  3. Avron, J., Simon B.: Almost periodic Schrödinger operators II. The integrated density of states. Duke Math. J. 50, 369–391 (1983)

    MATH  Google Scholar 

  4. Azbel, M.Ya.: Energy spectrum of a conduction electron in a magnetic field. Soviet Phys. JETP. 19, 634–645 (1964)

    Google Scholar 

  5. Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3), 408–419 (1982)

    MATH  Google Scholar 

  6. Broer, H.W., Puig, J., Simó, C.: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. Commun. Math. Phys. 241 (2-3), 467–503 (2003)

    Google Scholar 

  7. Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. The Probability and its Applications. Basel-Boston: Birkhäuser, 1990

  8. Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99(2), 225–246 (1990)

    MATH  Google Scholar 

  9. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955

  10. Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, Vol. 629, Berlin: Springer-Verlag, 1978

  11. DeConcini, C., Johnson R.A.: The algebraic-geometric AKNS potentials. Ergodic Theory Dynam. Syst. 7(1), 1–24 (1987)

    Google Scholar 

  12. Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. 89(3), 415–426 (1983)

    MATH  Google Scholar 

  13. Dinaburg, E.I., Sinai, Y.G.: The one-dimensional Schrödinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975)

    MATH  Google Scholar 

  14. Eliasson, L.H.: One-dimensional quasi-periodic Schrödinger operators – dynamical systems and spectral theory. In: European Congress of Mathematics, Vol. I (Budapest, 1996), Basel: Birkhäuser, 1998, pp. 178–190

  15. Eliasson, L.H.: Reducibility and point spectrum for linear quasi-periodic skew-products. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, (electronic), 1998, pp. 779–787

  16. Eliasson, L.H.: On the discrete one-dimensional quasi-periodic Schrödinger equation and other smooth quasi-periodic skew products. In: Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995), Volume 533 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Dordrecht: Kluwer Acad. Publ., 1999, pp. 55–61

  17. Eliasson, L.H.: Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Herman, M.R.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), (1983)

  19. Ince, E.L.: Ordinary Differential Equations. New York: Dover Publications, 1944

  20. Jitomirskaya, S.Y.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. (2) 150(3), 1159–1175 (1999)

    Google Scholar 

  21. Johnson, R.: The recurrent Hill’s equation. J. Diff. Eq. 46, 165–193 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger equation. J. Diff. Eq. 91, 88–110 (1991)

    MathSciNet  MATH  Google Scholar 

  23. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438, (1982)

    MathSciNet  MATH  Google Scholar 

  24. Johnson, R.A.: A review of recent work on almost periodic differential and difference operators. Acta Appl. Math. 1(3), 241–261 (1983)

    MATH  Google Scholar 

  25. Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on T × S L(2, R). Preprint

  26. Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164(2), 421–432 (1994)

    MATH  Google Scholar 

  27. Last, Y.: Almost everything about the almost Mathieu operator. I. In: XIth International Congress of Mathematical Physics (Paris, 1994), Cambridge MA: Internat. Press, 1995, pp. 366–372

  28. Moser, J.: An example of schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)

    MathSciNet  MATH  Google Scholar 

  29. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helv. 59, 39–85 (1984)

    MathSciNet  MATH  Google Scholar 

  30. Puig, J.: Reducibility of linear differential equations with quasi-periodic coefficients: A survey. Barcelona: Preprint University of Barcelona, 2002, Available at http://www.maia. ub.es/∼puig/preprints/qpred.ps

  31. Puig, J, Simó, C.: Analytic families of reducible linear quasi-periodic equations. In progress 2003

  32. Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math. 3(4), 463–490 (1982)

    MATH  Google Scholar 

  33. Simon, B.: Schrödinger operators in the twenty-first century. In: Mathematical physics 2000, London: Imp. Coll. Press, 2000, pp. 283–288

  34. Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Statist. Phys. 46(5-6), 861–909 (1987)

    Google Scholar 

  35. van Mouche, P.: The coexistence problem for the discrete Mathieu operator. Commun. Math. Phys. 122(1), 23–33 (1989)

    MATH  Google Scholar 

  36. Yakubovich, V.A., Starzhinskii V.M.: Linear differential equations with periodic coefficients. 1, 2. New York-Toronto, Ont: Halsted Press [John Wiley & Sons], 1975

Download references

Author information

Authors and Affiliations

  1. Dept. de Matemàtica Aplicada i Anàlisi, Univ. de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Joaquim Puig

Authors
  1. Joaquim Puig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joaquim Puig.

Additional information

Communicated by B. Simon

Rights and permissions

Reprints and permissions

About this article

Cite this article

Puig, J. Cantor Spectrum for the Almost Mathieu Operator. Commun. Math. Phys. 244, 297–309 (2004). https://doi.org/10.1007/s00220-003-0977-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0977-3

Keywords

Search

Navigation