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

Part of:Elliptic equations and systemsSpecial classes of linear operators

Published online by Cambridge University Press:  17 April 2019

R. HAN,
M. LEMM and
W. SCHLAG
R. HAN
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA email rui.han@math.gatech.edu
M. LEMM
Affiliation:
Harvard University, Department of Mathematics, 1 Oxford Street, Cambridge, MA 02138, USA email mlemm@math.harvard.edu
W. SCHLAG
Affiliation:
Yale University, Department of Mathematics, 10 Hillhouse Ave, New Haven, CT 06511, USA email willsg69@gmail.com
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Abstract

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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.

Keywords

Hamiltonian dynamicsrandom dynamicsLyapunov exponentSkew-shift

MSC classification

Primary:47B36: Jacobi (tridiagonal) operators (matrices) and generalizations35J10: Schrödinger operator

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020, pp. 2788 - 2853
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Copyright
© Cambridge University Press, 2019

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