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Moebius Schrödinger

  • Jean BourgainEmail author
Chapter
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Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

Consider the one-dimensional lattice Schrödinger operator with potential given by the Moebius function. It is shown that the Lyapounov exponent is strictly positive for almost all energies, answering a question posed by P. Sarnak.

Keywords

Moebius Function Lyapounov Exponents Pointwise Closure Fubini Argument Banach Limit 
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Notes

Acknowledgements

The author is grateful to P. Sarnak for bringing the problem to his attention and several discussions. He also thanks H. Krüger for comments on how Theorem 1 in this note may be derived directly from Lemma 2 and the results in [2]. The author was partially supported by NSF Grants DMS-0808042 and DMS 0835373.

References

  1. 1.
    J. Bourgain, Positive Lyapounov Exponents for Most Energies. Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1745 (Springer, Berlin, 2000), pp. 37–66Google Scholar
  2. 2.
    H. Krüger, Probabilistic averages of Jacobi operators. CMP 295, 853–875 (2010)Google Scholar
  3. 3.
    C. Remling, The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (1) 174(1), 125–171 (2011)Google Scholar
  4. 4.
    P. Sarnak, Moebius randomness law. NotesGoogle Scholar
  5. 5.
    B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices. Comm. Math. Phys. 89(2), 227–234 (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA

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