On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator

Fabbri, R.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 149-161

Résumé

In this paper we study the Lyapunov exponent $\beta(E)$ for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let $\Gamma\subset \mathbb{R}^{k}$ be the set of frequency vectors whose components are rationally independent. Let $\Gamma\subset \mathbb{R}^{k}$, and consider the complement in $\Gamma \times C^{r} (\mathbb{T}^{k} )$ of the set $\mathcal{D}$ where exponential dichotomy holds. We show that $\beta=0$ is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.