Advertisement

Generic Continuous Spectrum for Ergodic Schrödinger Operators

  • Michael Boshernitzan
  • David DamanikEmail author
Article

Abstract

We consider families of discrete Schrödinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sampling function, the set of elements in the associated family of Schrödinger operators that have no eigenvalues is large in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.

Keywords

Continuous Spectrum Spectral Type Arithmetic Progression Point Spectrum Sampling Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Avila, A., Bochi, J., Damanik, D.: Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. http://arxiv.org/abs/0709.2667, 2007, to appear in Duke Math. J.
  2. 2.
    Avila A., Damanik D.: Generic singular spectrum for ergodic Schrödinger operators. Duke Math. J. 130, 393–400 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avron J., Simon B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6, 81–85 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Berthé V., Holton C., Zamboni L.: Initial powers of Sturmian sequences. Acta Arith. 122, 315–347 (2006)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bjerklöv K.: Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent. Geom. Funct. Anal. 16, 1183–1200 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Boshernitzan M.: A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52, 723–752 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Boshernitzan M.: A condition for unique ergodicity of minimal symbolic flows. Ergod. Th. & Dynam. Sys. 12, 425–428 (1992)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Boshernitzan M.: Quantitative recurrence results. Invent. Math. 113, 617–631 (1993)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Boshernitzan, M., Damanik, D.: The repetition property for sequences on tori generated by polynomials or skew-shifts. http://arxiv.org/abs/0708.3234 , 2007, to appear in Israel J. Math.
  10. 10.
    Boshernitzan, M., Damanik, D.: Pinned repetitions in symbolic flows. In preparationGoogle Scholar
  11. 11.
    Bourgain J.: On the spectrum of lattice Schrödinger operators with deterministic potential. J. Anal. Math. 87, 37–75 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bourgain J.: On the spectrum of lattice Schrödinger operators with deterministic potential II. J. Anal. Math. 88, 221–254 (2002)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Bourgain J.: Estimates on Green’s functions, localization and the quantum kicked rotor model. Ann. of Math. 156, 249–294 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, 158. Princeton: Princeton, NJ: University Press, 2005Google Scholar
  15. 15.
    Bourgain J., Goldstein M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. 152, 835–879 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bourgain J., Goldstein M., Schlag W.: Anderson localization for Schrödinger operators on \({\mathbb Z}\) with potentials given by the skew-shift. Commun. Math. Phys. 220, 583–621 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Bourgain J., Jitomirskaya S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148, 453–463 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Carmona R., Lacroix J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  19. 19.
    Chan, J., Goldstein, M., Schlag, W.: On non-perturbative Anderson localization for C α potentials generated by shifts and skew-shifts. http://arxiv.org/list/math/0607302, 2006
  20. 20.
    Cornfeld, I., Fomin, S., Sinaĭ, Ya.: Ergodic Theory, Grundlehren der Mathematischen Wissenschaften 245, New York: Springer-Verlag, 1982Google Scholar
  21. 21.
    Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, Berlin: Springer-Verlag, 1987Google Scholar
  22. 22.
    Damanik, D.: Strictly ergodic subshifts and associated operators, In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposia in Pure Mathematics 74, Providence, RI: Amer. Math. Soc. pp. 505–538Google Scholar
  23. 23.
    Delyon F., Petritis D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential. Commun. Math. Phys. 103, 441–444 (1986)zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Furstenberg H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, 573–601 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gordon A.: On the point spectrum of the one-dimensional Schrödinger operator. Usp. Math. Nauk. 31, 257–258 (1976)zbMATHGoogle Scholar
  26. 26.
    Jitomirskaya S., Simon B.: Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165, 201–205 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Keane M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Khintchine A.: Continued Fractions. Dover, Mineola, NY (1997)Google Scholar
  29. 29.
    Last Y., Simon B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one- dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)zbMATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Masur H.: Interval exchange transformations and measured foliations. Ann. of Math. 115, 168–200 (1982)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Mignosi F.: Infinite words with linear subword complexity. Theoret. Comput. Sci. 65, 221–242 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ornstein D., Weiss B.: Entropy and data compression schemes. IEEE Trans. Inform. Theory 39, 78–83 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Veech W.: Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. 115, 201–242 (1982)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Veech W.: The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106, 1331–1359 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Veech W.: Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation. Ergod. Th. & Dynam. Sys. 7, 149–153 (1987)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsRice UniversityHoustonUSA

Personalised recommendations