Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials
type: Item
instance of: scholarly article
title: Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials
zbMATH Open document ID: 0745.34046
DOI: 10.1007/BF02099100
author: Eugene Sorets
author: Thomas Spencer
published in: Communications in Mathematical Physics
publication date: 1992-06-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 30558
zbMATH Keywords: quasi-periodic potential
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: finite-difference Schrödinger equation
zbMATH Keywords: spectrum
zbMATH Keywords: Jensen's formula
reviewed by: Yuri S. Kivshar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 30558