Communications in Mathematical Physics

, Volume 162, Issue 3, pp 529–537 | Cite as

Positive Lyapunov exponents for a class of ergodic Schrödinger operators

  • Steve Surace


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. 1)

    The length of each interval is large than 2l.

  2. 2)

    The distance between any two adjacent intervals is less thanL.

  3. 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.


Neural Network Statistical Physic Complex System Positive Constant Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Steve Surace
    • 1
  1. 1.Department of MathematicsDrew UniversityMadisonUSA

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