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

Sorets, Eugene; Spencer, Thomas

Commun. Math. Phys. 142, No. 3, 543-566 (1991).

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.

Reviewer: Y.Kivshar (Düsseldorf)

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.)

Keywords:

quasi-periodic potential; Schrödinger operator; Lyapunov exponent; finite-difference Schrödinger equation; spectrum; Jensen’s formula

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