On the Lyapounov Exponents of Schrödinger Operators Associated with the Standard Map

Bourgain, J.

doi:10.1007/978-1-4614-6406-8_3

J. Bourgain⁵

Part of the book series: Fields Institute Communications ((FIC,volume 68))

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Abstract

It is shown that Schrödinger operators defined from the standard map have positive (mean) Lyapounov exponents for almost all energies.

Mathematical classification subject (2010): 37D25, 37D50

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References

P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincaré Anal. Non linéaire 11(4), 359–409 (1994)
MathSciNet MATH Google Scholar
S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, in Stochastic analysis (Katata/Kyoto, 1982). North-Holland Math. Library, vol. 32 (North-Holland, Amsterdam, 1984), pp. 225–247
Google Scholar
S. Kotani, One-dimensional random Schrödinger operators and Herglotz functions, in Probabilistic Methods in Mathematical Physic (Katata/Kyoto 1985) (Academic Press, Boston, 1987), pp. 219–250
Google Scholar
S. Kotani, Jacobi matrices with random potentials taking finitely many values. Ref. Math. Phys. 1, 129–133 (1989)
Article MathSciNet MATH Google Scholar
S. Kotani, Generalized Floquet theory for stationary Schrödinger operators in one dimension. Chaos Sol. Fract. 8, 1817–1854 (1997)
Article MathSciNet MATH Google Scholar
Y. Last, B. Simon, Eigenfunctions, transfer matrices and absolutely continuous spectrum of one-dimensional Schrödinger operators. Inventiones Math. 135(2), 329–367 (1999)
Article MathSciNet MATH Google Scholar
C. Remling, The absolutely continuous spectrum of Jacobi matrices. Annals of Math 174(2), 125–171 (2011)
Article MathSciNet MATH Google Scholar

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Acknowledgements

The author is grateful to A. Avila, S. Sodin and T. Spencer for some discussions on this topic, and the referee for comments on the presentation.

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Institute for Advanced Study, Princeton, NJ, 08540, USA
J. Bourgain

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J. Bourgain
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Correspondence to J. Bourgain .

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Editors and Affiliations

TU Wien, Wiedner Hauptstrasse 8–10, Wien, 1040, Austria
Monika Ludwig
Fac. Exact Sciences, School of Mathematical Sciences, University of Tel Aviv, Schreiber Building 334, Tel Aviv, 69978, Israel
Vitali D. Milman
University of Ottawa, Ottawa, K1N 6N5, Canada
Vladimir Pestov
, Department of Math and Stat Scis, University of Alberta, Central Academic Bldg 632, Edmonton, T6G 2G1, Alberta, Canada
Nicole Tomczak-Jaegermann

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Bourgain, J. (2013). On the Lyapounov Exponents of Schrödinger Operators Associated with the Standard Map. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds) Asymptotic Geometric Analysis. Fields Institute Communications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6406-8_3

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DOI: https://doi.org/10.1007/978-1-4614-6406-8_3
Published: 02 February 2013
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6405-1
Online ISBN: 978-1-4614-6406-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)

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