Geodesic
On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 149-161.

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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.
In questo lavoro viene studiato l'esponente di Lyapunov $\beta(E)$ per l'operatore di Schrödinger in una dimensione con potenziale quasi periodico. Indicato con $\Gamma\subset \mathbb{R}^{k}$ l'insieme delle frequenze le cui componenti sono razionalmente indipendenti e considerato $0\leq r 1$, si fa vedere come $\beta(E)$ risulti zero sul complementare in $\Gamma \times C^{r} (\mathbb{T}^{k} )$ dell'insieme $\mathcal{D}$ in cui si ha dicotomia esponenziale (D.E.). Le tecniche ed i metodi usati sono basati sulle proprieta' del numero di rotazione e della D.E. per l'operatore considerato. 
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Fabbri, R. On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 149-161. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_1_a6/

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