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

Fabbri, R.

Fabbri, R.

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 149-161 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

In this paper we study the Lyapunov exponent $β (E)$ for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let $Γ \subset R^{k}$ be the set of frequency vectors whose components are rationally independent. Let $Γ \subset R^{k}$ , and consider the complement in $Γ C^{r} (T^{k})$ of the set $D$ where exponential dichotomy holds. We show that $β = 0$ 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 $β (E)$ per l'operatore di Schrödinger in una dimensione con potenziale quasi periodico. Indicato con $Γ \subset R^{k}$ l'insieme delle frequenze le cui componenti sono razionalmente indipendenti e considerato $0 \leq r < 1$ , si fa vedere come $β (E)$ risulti zero sul complementare in $Γ C^{r} (T^{k})$ dell'insieme $D$ 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.

Publié le : 2002-02-01

MR 1881929 | Zbl 1177.34108

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     author = {R. Fabbri},
     title = {On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schr\"odinger operator},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {149-161},
     zbl = {1177.34108},
     mrnumber = {1881929},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_1_149_0}
}

Fabbri, R. On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 149-161. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_1_149_0/

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