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On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator. (English) Zbl 1177.34108

Summary: 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 \(0 \leq r < 1\), and consider the complement in \(\Gamma \times C^{r} (\mathbb{T}^{k})\) of the set \(\mathcal A\) 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.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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