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.

该项目将借助动力系统工具研究光谱问题。研究对象是薛定谔算子，其势是通过沿着紧度量空间上的遍历变换的轨道使用连续函数进行采样来获得的。该框架涵盖了许多有趣的例子，例如近周期势和随机势。此类算子的谱特性与给定遍历变换下 SL(2,R) 值余循环的能量指数族的动态行为密切相关。特别令人感兴趣的是与这些余环相关的李雅普诺夫指数。将研究以下谱问题：任意无理频率的小耦合下准周期势的纯绝对连续谱，适合变换和采样函数类别的康托谱的通用性，李雅普诺夫指数作为能量、扰动准周期势的谱现象以及绝对连续谱的存在对势施加的限制。量子力学是物理学的一个基本分支，其基础是在 20 世纪上半叶建立的。二十世纪。自 20 世纪 50 年代以来，无序环境中的量子力学现象研究一直是一个持续活跃的研究领域。安德森于 1958 年发表了一篇具有里程碑意义的论文。他因对某些随机晶格哈密顿量不存在扩散的研究而获得 1977 年诺贝尔物理学奖。另一个重要事件是 Shechtman 在 1982 年发现了准晶体，他在 1984 年与 Blech、Gratias 和 Cahn 联合撰写的一篇论文中报道了这一发现，这引起了晶体学和固体物理学的范式转变。无序结构电子特性的数学研究是在遍历薛定谔算子的框架内进行的。由于这些算子的势是动态定义的，即通过沿着一个或多个遍历变换的轨道进行采样，因此动力系统工具在此类算子的研究中被证明是有用的，这是很自然的。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，该领域最近取得了重大飞跃。这也导致了跨学科的富有成效的合作，并且这些互动有望取得进一步的成功。