Abstract
We consider an m-dimensional analytic cocycle \({\mathbb{T} \times \mathbb{R}^m \ni (x, \vec{\psi}) \mapsto (x + \omega, A (x) \cdot \vec{\psi}) \in \mathbb{T} \times \mathbb{R}^m}\), where \({\omega \notin \mathbb{Q}}\) and \({A \in C^\omega (\mathbb{T}, \mathrm{Mat}_m (\mathbb{R}))}\). Assuming that the d × d upper left corner block of A is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets–Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrödinger operators.
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References
Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton (2005)
Bourgain, J., Jitomirskaya, S.: Anderson localization for the band model, Geometric aspects of functional analysis. In: Lecture Notes in Mathematics, vol. 1745, pp. 67–79. Springer, Berlin (2000)
Bruns, W., Vetter, U.: Determinantal rings. In: Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)
Chan J.: Method of variations of potential of quasi-periodic Schrödinger equations. Geom. Funct. Anal. 17(5), 1416–1478 (2008)
Duren, Peter L.: Theory of H p spaces. In: Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)
Federer, H.; Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)
Gol’dsheÄd I.Ya., Sorets E.: Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip. Commun. Math. Phys. 145(3), 507–513 (1992)
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), 453–502 (1983)
Hunt B.R., Sauer T., Yorke A.J.: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Am. Math. Soc. (N.S.) 27(2), 217–238 (1992)
Kotani S., Simon B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Commun. Math. Phys. 119(3), 403–429 (1988)
Levin, B.Ya.: Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, (1996), in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, translated from the Russian manuscript by Tkachenko.
Schlag W.: Regularity and convergence rates for the Lyapunov exponents of linear co-cycles. J. Mod. Dyn. 7(4), 619–637 (2013)
Sorets E., Spencer T.: Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Commun. Math. Phys. 142(3), 543–566 (1991)
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Duarte, P., Klein, S. Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles. Commun. Math. Phys. 332, 189–219 (2014). https://doi.org/10.1007/s00220-014-2082-1
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DOI: https://doi.org/10.1007/s00220-014-2082-1