Skip to main content
SpringerLink
Log in

Analytic quasi-periodic Schrödinger operators and rational frequency approximants

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Consider a quasi-periodic Schrödinger operator H α,θ with analytic potential and irrational frequency α. Given any rational approximating α, let S + and S − denote the union, respectively, the intersection of the spectra taken over θ. We show that up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from S − of the periodic operators associated with the continued fraction expansion of α. This proves a conjecture of Y. Last in the analytic case. Similarly, from the asymptotics of S +, one recovers the spectrum of H α,θ .

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. Aubry S., Andre G.: Analyticity breaking and Anderson localization in incommensurate lattices. Annals of the Israeli Physical Society 3, 133–164 (1980)

    MathSciNet  Google Scholar 

  2. A. Avila. Global theory of one-frequency operators I: Stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity. Preprint (2009).

  3. Avila A., Damanik D.: Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling. Inventiones Mathematicae 172, 439–453 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Avila, B. Fayad and R. Krikorian. A KAM scheme for SL(2,\({\mathbb{R}}\)) cocycles with Liouvillian frequencies. Geometric and Functional Analysis, to appear.

  5. Avila A., Krikorian R.: Reducibility and non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Annals of Mathematics 164, 911–940 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Avila and R. Krikorian. Reducibility and non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles, http://w3.impa.br/~avila/regular.pdf. An extended preprint of [AK06].

  7. A. Avila and S. Jitomirskaya. The ten Martini problem. Annals of Mathematics, 170 (2009), 303–342.

    Google Scholar 

  8. J. Avron and B. Simon. Almost periodic Schrödinger operators. II. The integrated density of states. Duke Mathematical Journal, 50 (1983), 369–391.

    Google Scholar 

  9. J. Avron, P.H.M.v. Mouche, and B. Simon. On the measure of the spectrum for the almost mathieu operator. Communications in Mathematical Physics, 132 (1990), 103–118.

    Google Scholar 

  10. Bellissard J., Simon B.: Cantor spectrum for the almost Mathieu equation. Journal of Functional Analysis 48, 408–419 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bourgain J., Jitomirskaya S.: Continuity of the Lyapunov exponent for quasi-periodic operators with analytic potential. Journal of Statistical Physics 108(5–6), 1203–1218 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chambers W.: Linear network model for magnetic breakdown in two dimensions. Physical Review A 140, 135–143 (1965)

    Google Scholar 

  13. V. Chulaevsky and F. Delyon. Purely absolutely continuous spectrum for almost Mathieu operators. Journal of Statistical Physics, 55 (1989), 1279–1284.

    Google Scholar 

  14. H. Cycon, R. Froese, W. Kirsch, and B. Simon. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics. Springer-Verlag, Berlin (1987).

  15. M.I. Ganzburg. Polynomial inequalities on measurable sets and their application. Constructive Approximation, 17 (2001), 275–306.

    Google Scholar 

  16. A.Y. Gordon, S. Jitomirskaya, Y. Last and B. Simon. Duality and singular continuous spectrum in the almost Mathieu equation. Acta Mathematica, 178 (1997), 169–183.

    Google Scholar 

  17. Hofstadter D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B 14, 2239–2249 (1976)

    Article  Google Scholar 

  18. S. Ya. Jitomirskaya. Metal-insulator transition for the almost Mathieu operator. Annals of Mathematics, 150 (1999), 1159–1175.

  19. S. Jitomirskaya and I.V. Krasovsky. Continuity of the measure of the spectrum for discrete quasiperiodic operators. Mathematical Research Letters, 9 (2002), 413–421.

    Google Scholar 

  20. S. Jitomirskaya and R. Mavi. Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials. Preprint (2012).

  21. Y. Last. Zero measure spectrum for the almost Mathieu operator. Communications in Mathematical Physics, 164 (1994), 421–432.

    Google Scholar 

  22. Y. Last. A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Communications in Mathematical Physics, 151 (1993), 183–192.

    Google Scholar 

  23. Y. Last. On the measure of gaps and spectra for discrete 1d Schrödinger operators. Communications in Mathematical Physics, 149 (1992), 347–360.

    Google Scholar 

  24. Last Y., Simon B.: essential spectrum of Schrödinger, Jacobi, and CMV operators. Journal d’Analyse Mathématique 98, 183–220 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Inventiones Mathematicae, (2)135 (1999), 329–367.

    Google Scholar 

  26. V.A. Mandelshtam and S. Ya. Zhitomirskaya. 1D-Quaisperiodic operators. latent symmetries. Communications in Mathematical Physics, 139 (1991), 589–604.

    Google Scholar 

  27. C.A. Marx. Singular components of spectral measures for ergodic Jacobi matrices. Journal of Mathematical Physics, 52 (2011), 073508.

    Google Scholar 

  28. L. Pastur. Spectral properties of disordered systems in one-body approximation. Communications in Mathematical Physics, 75 (1980), 179.

    Google Scholar 

  29. F. Peherstorfer and K. Schiefermayr. Description of the extremal polynomials on several intervals and their computation. I. Acta Mathematica Hungarica, (1–2)83 (1999), 27–58.

  30. G. Pólya. Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhangenden Gebieten. In: it Sitzungsberichte der Kóniglich Preussischen Akademie der Wissenschaften zu Berlin, (1928), pp. 228–232.

  31. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. IV. Academic Press Inc., London (1978).

  32. Remling C.: The absolutely continuous spectrum of Jacobi matrices. Annals of Mathematics 174, 125–171 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Shamis. Some connections between almost periodic and periodic discrete Schrödinger operators with analytic potentials. Journal of Spectral Theory, (3)1 (2011), 349–362.

    Google Scholar 

  34. M. Shamis, S. Sodin, On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices. Journal of Approximation Theory, (4)163 (2011), 491–504.

    Google Scholar 

  35. B. Simon. Fifteen Problems in Mathematical Physics. Oberwolfach Anniversary Volume (1984), pp. 423–454

  36. B. Simon. Schrödinger operators in the twenty-first century. Mathematical Physics (2000), 283–288. Imperial College Press, London (2000).

  37. G. Teschl. Jacobi Operators and Completely Integrable Nonlinear Lattices. In: Mathematical Surveys and Monographs, Vol. 72, American Mathematical Society, Providence (2000).

  38. Toda M.: Theory of Nonlinear Lattices. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  39. D.J. Thouless. Bandwidth for a quasiperiodic tight binding model. Physics Review B, 28 (1983), 42724276.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92717, USA

    S. Jitomirskaya & C. A. Marx

Authors
  1. S. Jitomirskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. Marx
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Jitomirskaya.

Additional information

The work was supported by NSF Grant DMS-1101578.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jitomirskaya, S., Marx, C.A. Analytic quasi-periodic Schrödinger operators and rational frequency approximants. Geom. Funct. Anal. 22, 1407–1443 (2012). https://doi.org/10.1007/s00039-012-0179-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-012-0179-2

Keywords

Search

Navigation