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Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base

  • R. HAN (a1), M. LEMM (a2) and W. SCHLAG (a3)


We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below $1$ , which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.



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[BelSim] Béllissard, J. and Simon, B.. Cantor spectrum for the almost Mathieu equation. J. Funct. Anal. 48(3) (1982), 408419.
[Bou1] Bourgain, J.. Green’s Function Estimates for Lattice Schrödinger Operators and Applications (Annals of Mathematics Studies, 158) . Princeton University Press, Princeton, NJ, 2005.
[Bou2] Bourgain, J.. On the spectrum of lattice Schrödinger operators with deterministic potential. J. Anal. Math. 87 (2002), 3775.
[Bou3] Bourgain, J.. Positive Lyapounov exponents for most energies. Geometric Aspects of Functional Analysis (Lecture Notes in Mathematics, 1745) . Springer, Berlin, 2000, pp. 3766.
[BouGol] Bourgain, J. and Goldstein, M.. On nonperturbative localization with quasi-periodic potential. Ann. of Math. (2) 152(3) (2000), 835879.
[BouGolSch] Bourgain, J., Goldstein, M. and Schlag, W.. Anderson localization for Schrödinger operators on ℤ with potentials given by the skew-shift. Comm. Math. Phys. 220(3) (2001), 583621.
[Dam] Damanik, D.. Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications. Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday (Proceedings of Symposia in Pure Mathematics, 76, Part 2) . American Mathematical Society, Providence, RI, 2007, pp. 539563.
[DuaKle] Duarte, P. and Klein, S.. Lyapunov exponents of linear cocycles. Continuity via Large Deviations (Atlantis Studies in Dynamical Systems, 3) . Atlantis Press, Paris, 2016.
[DuaKle2] Duarte, P. and Klein, S.. Continuity of the Lyapunov Exponents of Linear Cocycles. Associação Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2017.
[Fur] Fürstenberg, H.. Noncommuting random products. Trans. Amer. Math. Soc. 108 (1963), 377428.
[GolSch] Goldstein, M. and Schlag, W.. Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154(1) (2001), 155203.
[Hea] Heath-Brown, D. R.. Pair correlation for fractional parts of 𝛼n 2 . Math. Proc. Cambridge Philos. Soc. 148(3) (2010), 385407.
[Her] Herman, M.-R.. Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’d et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3) (1983), 453502.
[Kat] Katznelson, Y.. An Introduction to Harmonic Analysis (Cambridge Mathematical Library) , 3rd edn. Cambridge University Press, Cambridge, 2004.
[Kru1] Krüger, H.. Multiscale analysis for ergodic Schrödinger operators and positivity of Lyapunov exponents. J. Anal. Math. 115 (2011), 343387.
[Kru2] Krüger, H.. On positive Lyapunov exponent for the skew-shift potential. Preprint.
[MarStr] Marklof, J. and Strömbergsson, A.. Equidistribution of Kronecker sequences along closed horocycles. Geom. Funct. Anal. 13(6) (2003), 12391280.
[Mon] Montgomery, H. L.. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis (CBMS Regional Conference Series in Mathematics, 84) . American Mathematical Society, Providence, RI, 1994.
[RudSarZah] Rudnick, Z., Sarnak, P. and Zaharescu, A.. The distribution of spacings between the fractional parts of n 2𝛼. Invent. Math. 145(1) (2001), 3757.
[Sch] Schlag, W.. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. J. Mod. Dyn. 7(4) (2013), 619637.
[SorSpe] Sorets, E. and Spencer, T.. Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Comm. Math. Phys. 142(3) (1991), 543566.
[Via] Viana, M.. Lectures on Lyapunov Exponents (Cambridge Studies in Advanced Mathematics, 145) . Cambridge University Press, Cambridge, 2014.


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Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base

  • R. HAN (a1), M. LEMM (a2) and W. SCHLAG (a3)


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