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.

该项目将借助动态系统工具研究光谱问题。研究的对象是施罗辛格运营商，其电势是通过沿着紧凑型公制空间上沿着千古转化的轨道进行的连续函数获得的。该框架涵盖了许多感兴趣的例子，例如几乎有周期的潜力和随机潜力。此类运算符的光谱特性与给定的ergodic变换上的SL（2，r）值的cocycles的能量指数家族的动力学行为紧密相关。感兴趣的是与这些共生相关的Lyapunov指数。将研究以下光谱问题：对于任意非理性频率的小耦合时，纯绝对连续的光谱，用于任意非理性频率的小耦合，适当的转换和取样函数的cantor频谱的一般性，以及lyapunov的不规则性的不规则性，该功能是构成势，光谱的效果，构成势，疾病的效果，构成了疾病的效果，以及呈危险的现象。绝对连续的光谱。Quantum Mechanics是物理学的基本分支，其基础是在20世纪上半叶建立的。自1950年代以来，对无序环境中量子机械现象的研究一直是正在进行的活跃研究的领域。安德森（Anderson）于1958年发表了具有里程碑意义的论文。1977年，他因缺乏某些随机晶格汉密尔顿人的扩散而获得了诺贝尔物理奖。另一个重要的事件是Shechtman在1982年发现了准晶体，这是他与Blech，Gratias和Cahn共同撰写的一篇论文中的报道，这引起了晶体学和固态物理学的范式转变。无序结构的电子特性的数学研究是在厄贡施罗辛格运营商的框架内进行的。由于这些操作员的电势是动态定义的，即通过沿一个或多个千古转换的轨道进行采样，因此，动态系统工具应被证明在研究此类操作员的研究中有用是很自然的。在许多来自动力学系统的非常有才华的年轻研究人员进入该领域之后，该领域取得了巨大的飞跃。这也导致了整个学科的富有成果的合作，并且有望进一步取得这些互动的成功。