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Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials. (English) Zbl 0745.34046

The authors present a new, simple way to estimate the Lyapunov exponent of solutions of the finite-difference Schrödinger equation, \[ ((H- E)\psi)(n){\buildrel{\text{def}}\over =}-[\psi(n+1)+\psi(n-1)]+[\lambda f(\alpha n+\theta)]\psi(n), \] where \(f\) is a non-constant real-analytic function of period 1 and \(\alpha\) is irrational. For \(\lambda\) large they prove that the Lyapunov exponent is positive for every energy \(E\) in the spectrum of \(H\) and a.e. \(\theta\). The main idea of the approach is to deform the phase \(\theta\) to the complex plane, and then to recover the information for real \(\theta\) using an elementary extension of Jensen’s formula.

MSC:

34D08 Characteristic and Lyapunov exponents of ordinary differential equations
39A11 Stability of difference equations (MSC2000)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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