论文标题
新的Quasi-periodic Schr $ \ ddot {o} $ lyapunov指数和正lyapunov指数的共存
Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schr$\ddot{o}$dinger operator
论文作者
论文摘要
在本文中，我们解决了有关具有$ v（θ）=2λCOS2πθ/（1-αcos2πθ），\（|α| <1）$的Schr $ \ ddot {O} $ Dinger操作员的问题。借助频谱中Lyapunov指数的公式，首先证明了某些参数的零Lyapunov指数和正lyapunov指数的共存，并且存在将它们分开的曲线。正lyapunov指数区域的光谱纯粹是纯点光谱，几乎每个频率和几乎每个阶段都具有指数衰减的特征函数。从研究中，我们意识到，如果$ 0 <|λ| <1 $，则无限电位$ v（θ）=2λtan^2（πθ）$对于某些能量的lyapunov指数为零。
In this paper we solve a problem about the Schr$\ddot{o}$dinger operator with potential $v(θ)=2λcos2πθ/(1-αcos2πθ),\ (|α|<1)$ in physics. With the help of the formula of Lyapunov exponent in the spectrum, the coexistence of zero Lyapunov exponent and positive Lyapunov exponent for some parameters is first proved, and there exists a curve that separates them. The spectrum in the region of positive Lyapunov exponent is purely pure point spectrum with exponentially decaying eigenfunctions for almost every frequency and almost every phase. From the research, we realize that the infinite potential $v(θ)=2λtan^2(πθ)$ has zero Lyapunov exponent for some energies if $0<|λ|<1$.
