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Moebius Schrödinger

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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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References

  1. J. Bourgain, Positive Lyapounov Exponents for Most Energies. Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1745 (Springer, Berlin, 2000), pp. 37–66

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  2. H. Krüger, Probabilistic averages of Jacobi operators. CMP 295, 853–875 (2010)

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  3. C. Remling, The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (1) 174(1), 125–171 (2011)

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  4. P. Sarnak, Moebius randomness law. Notes

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  5. B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices. Comm. Math. Phys. 89(2), 227–234 (1983)

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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.

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Correspondence to Jean Bourgain .

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© 2012 Springer-Verlag Berlin Heidelberg

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Bourgain, J. (2012). Moebius Schrödinger. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_8

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