Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

Abstract

We present a new, simple way to estimate the rate of exponential growth (Lyapunov exponent) of solutions of the finite-difference Schrödinger equation:

$$((H - E)\psi )(n)\mathop = \limits^{def} - [\psi (n + 1) + \psi (n - 1)] + [\lambda f(\alpha n + \theta )]\psi (n).$$

Heref is a non-constant real-analytic function of period 1 and α is irrational. For λ large we prove that the Lyapunov exponent is positive for every energyE in the spectrum ofH and a.e. θ. In particular, the absolutely continuous spectrum ofH is empty. In the continuum we study the quasi-periodic operator onL ²(R)

$$H = - \frac{{d^2 }}{{dx^2 }} - K^2 [\cos x + \cos (\alpha x + \theta )]$$

for largeK and show that for wide intervals of low energies the Lyapunov exponent is positive. The main idea, which originated from M. Herman's subharmonic argument [11], is to deform the phase θ to the complex plane. This enables us to avoid small denominator problems by moving them off the axis, making estimates much easier to perform. We recover the information for real θ using an elementary extension of Jensen's formula (subharmonicity).