Dynamics of Schroedinger Cocycles and Applications to Spectral Theory

薛定谔余循环动力学及其在谱理论中的应用

• 批准号: 0800100
• 负责人: David Damanik
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800100&HistoricalAwards=false
• 关键词: Dynamics; Schroedinger; Cocycles; Applications; Spectral

This project will investigate spectral problems with the help of dynamical systems tools. The object of study are Schroedinger operators whose potentials are obtained by sampling with a continuous function along the orbits of an ergodic transformation on a compact metric space. This framework covers many examples of interest, such as almost-periodic potentials and random potentials. The spectral properties of such operators are closely linked to the dynamical behavior of an energy-indexed family of SL(2,R)-valued cocycles over the given ergodic transformation. Of interest are in particular the Lyapunov exponents associated with these cocycles. The following spectral problems will be investigated: purely absolutely continuous spectrum for quasi-periodic potentials at small coupling for arbitrary irrational frequency, the genericity of Cantor spectrum for suitable classes of transformations and sampling functions, the irregularity of the Lyapunov exponent as a function of the energy, spectral phenomena for perturbed quasi-periodic potentials, and restrictions put on the potentials by the existence of absolutely continuous spectrum.Quantum mechanics is a fundamental branch of physics whose foundations were established during the first half of the twentieth century. The study of quantum mechanical phenomena in disordered environments has been an area of ongoing active study since the 1950's. A landmark paper was published by Anderson in 1958. He was awarded the Nobel Prize in Physics in 1977 for his work on the absence of diffusion for certain random lattice Hamiltonians. Another event of importance was the discovery of quasicrystals by Shechtman in 1982, which was reported in a 1984 paper he wrote jointly with Blech, Gratias and Cahn, and which caused a paradigm shift in crystallography and solid state physics. The mathematical study of electronic properties of disordered structures is carried out within the framework of ergodic Schroedinger operators. Since the potentials of these operators are defined dynamically, namely by sampling along the orbits of one or more ergodic transformations, it is quite natural that dynamical systems tools should prove to be useful in the study of such operators. The field has recently taken major leaps after a number of very talented young researchers from dynamical systems entered it. This has also lead to fruitful collaborations across the disciplines and there is promise for further success of these interactions.

这个项目将借助动力系统工具来研究光谱问题。本文的研究对象是薛定谔算子，它的位势是在紧致度量空间上沿着遍历变换的轨道用连续函数采样得到的。这个框架涵盖了许多感兴趣的例子，例如概周期势和随机势。这类算子的谱性质与给定遍历变换上的SL(2，R)值余循环族的动力学行为密切相关。令人感兴趣的是与这些余循环相关的李亚普诺夫指数。我们将研究以下频谱问题：对于任意无理频率，小耦合的准周期势的纯绝对连续谱，适当的变换和采样函数的康托尔谱的一般性，作为能量函数的Lyapunov指数的不规则性，扰动准周期势的谱现象，以及绝对连续谱的存在对势的限制。量子力学是物理学的一个基本分支，其基础建立于20世纪上半叶。无序环境中的量子力学现象的研究自20世纪50年代S以来一直是一个活跃的领域。安德森于1958年发表了一篇里程碑式的论文。1977年，他因在某些随机晶格哈密顿量的无扩散方面的工作而获得诺贝尔物理学奖。另一个重要的事件是1982年谢克特曼发现准晶，他在1984年与Blech、Gatias和Cahn共同撰写的一篇论文中报道了这一发现，这导致了结晶学和固体物理学的范式转变。在遍历薛定谔算符的框架下，对无序结构的电子性质进行了数学研究。由于这些算符的势是动态定义的，即通过沿一个或多个遍历变换的轨道采样来定义的，所以动态系统工具在研究这些算符时应该被证明是很自然的。最近，在许多来自动力系统的非常有才华的年轻研究人员进入该领域后，该领域取得了重大飞跃。这也导致了跨学科的卓有成效的合作，这些互动有望进一步取得成功。