Positive Lyapunov exponents for a class of ergodic Schrödinger operators

Abstract

We present a simple method to estimate the Lyapunov exponent γ(E) for the system

$$ - (\psi _{j + 1} + \psi _{j - 1} ) + v_j (\omega )\psi _j = E\psi _j$$

, where {v _j (ω)}_ω∈Ω is an ergodic family of potentials defined forj∈ℤ. We assume that there is a constant ζ>2 and large positive integersl, L such that for almost every ω and everyE there is an infinite sequence of disjoint intervalsJ _n ⊂ℤ with the following properties:

• 1) The length of each interval is large than 2l.

• 2) The distance between any two adjacent intervals is less thanL.

• 3) |v _j (ω)−E|≧ζ for$j \in \cup _n J_n$.

Under these conditions we prove that meas {E: γ(E)=O}≦Be ^-βl/6 where \gb andB are positive constants and \ldmeas\rd refers to Lebesgue measure.