Positive Lyapunov Exponent and Minimality for the Continuous 1-d Quasi-Periodic Schrödinger Equation with Two Basic Frequencies
Positive Lyapunov Exponent and Minimality for the Continuous 1-d Quasi-Periodic Schrödinger Equation with Two Basic Frequencies
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DOI：
10.1007/s00023-006-0319-7
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发表时间：
2007-06
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通讯作者：
Kristian Bjerklöv
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文献类型：
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作者：
Kristian Bjerklöv
本文考虑了与时间无关的拟周期薛定谔方程$$-u^{\prime\prime}（x）+ K^{2}V（x，\theta + \omega x）u（x）= Eu（x），\，x \in {\mathbb{R}}$$，势函数为C2，具有唯一的非退化全局极小值，大的耦合常数K2和能量E在相应的薛定谔算子谱的底部.我们得到的估计的李雅普诺夫指数和勒贝格措施的频谱，以及本地化的结果。此外，我们还证明了由薛定谔方程引起的投影流往往是最小的。
We consider the time-independent quasi-periodic Schrödinger equation $$-u^{\prime\prime}(x) + K^{2}V (x, \theta + \omega x)u(x) = Eu(x), \,\,\,\, x \in {\mathbb{R}}$$ , with a potential functionof classC2with a unique non-degenerate global minimum, large coupling constantsK2and energiesEin 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 oninduced by the Schrödinger equation often is minimal.