Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials
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.
scientific article; zbMATH DE number 30558
scientific article