Exact dynamical decay rate for the almost Mathieu operator. (English) Zbl 1461.37029
The authors consider the almost Mathieu operator
\[
(H_{\lambda,\alpha,\theta}u)(n)= u(n+1)+ u(n- 1)+2\lambda\cos 2\pi(\alpha n+\theta)u(n)
\]
by its action on \(u \in\ell^2(\mathbb{Z})\). Here \(\lambda\neq0\) is the coupling, \(\alpha\in\mathbb{R}\backslash\mathbb{Q}\) is the frequency, and \(\theta\in\mathbb{R}\) is the phase. They prove that for supercritical almost Mathieu operators with Diophantine frequencies the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent.
Reviewer: Erdogan Sen (Tekirdağ)
MSC:
37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |