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

R. Fabbri — 2002

Bollettino dell'Unione Matematica Italiana

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.

On a criterion of Yakubovich type for the absolute stability of nonautonomous control processes.

Fabbri, R.; Impram, S.T.; Johnson, R. — 2003

International Journal of Mathematics and Mathematical Sciences

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