新拟周期Schr $\ddot{o}$ dinger算子零李雅普诺夫指数与正李雅普诺夫指数的共存性
Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schr$\ddot{o}$dinger operator
摘要：
本文解决了物理学中关于具有势$v(θ)=2λcos2πθ/(1-αcos2πθ),\ (|α|<1)$的Schr $\ddot{o}$ dinger算子的一个问题。借助谱中的李雅普诺夫指数公式，首先证明了对于某些参数，零李雅普诺夫指数和正李雅普诺夫指数共存，并且存在一条曲线将它们分开。正李雅普诺夫指数区域的谱是纯点谱，几乎每个频率和几乎每个相位的本征函数都呈指数衰减。通过研究，我们认识到，如果$0<|λ|<1$，无限势$v(θ)=2λtan^2(πθ)$的某些能量的李雅普诺夫指数为零。
英文摘要：
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 $0<|λ|<1$.
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