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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
Article

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. 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.

Keywords

Neural Network Statistical Physic Complex System Positive Constant Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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