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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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
H. Krüger, Probabilistic averages of Jacobi operators. CMP 295, 853–875 (2010)
C. Remling, The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (1) 174(1), 125–171 (2011)
P. Sarnak, Moebius randomness law. Notes
B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices. Comm. Math. Phys. 89(2), 227–234 (1983)
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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© 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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DOI: https://doi.org/10.1007/978-3-642-29849-3_8
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