Abstract.
We consider the time-independent quasi-periodic Schrödinger equation
, with a potential function \(V : {\mathbb{T}}^{2} \rightarrow {\mathbb{R}}\) of class C 2 with a unique non-degenerate global minimum, large coupling constants K 2 and energies E in the bottom of the spectrum of the associated Schrödinger operator. We obtain estimates on the Lyapunov exponents and the Lebesgue measure of the spectrum, as well as localization results. Moreover, we show that the projective flow on \({\mathbb{T}}^{2} \times {\mathbb{P}}^{1}\) induced by the Schrödinger equation often is minimal.
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Communicated by Jean Bellissard.
Submitted: May 3, 2006. Accepted: September 20, 2006.
Research partially supported by SVeFUM.
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Bjerklöv, K. Positive Lyapunov Exponent and Minimality for the Continuous 1-d Quasi-Periodic Schrödinger Equation with Two Basic Frequencies. Ann. Henri Poincaré 8, 687–730 (2007). https://doi.org/10.1007/s00023-006-0319-7
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DOI: https://doi.org/10.1007/s00023-006-0319-7