Dynamics of Schroedinger Cocycles and Applications to Spectral Theory

这个项目将在动力系统工具的帮助下研究光谱问题。研究的对象是薛定谔算子，其势是在紧度量空间上沿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有趣的例子，比如几乎周期势和随机势。这些算子的谱性质与给定遍历变换上的能量指标SL（2，R）值环族的动力学行为密切相关。我们特别感兴趣的是与这些环相关的李雅普诺夫指数。本文将研究以下谱问题：任意不合理频率下小耦合准周期势的纯绝对连续谱，适当变换和采样函数的康托尔谱的一般性，李雅普诺夫指数作为能量函数的不规则性，摄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，其基础是在20世纪上半叶建立起来的。自20世纪50年代以来，无序环境中量子力学现象的研究一直是一个活跃的研究领域。安德森在1958年发表了一篇具有里程碑意义的论文。1977年，他因研究某些随机晶格哈密顿量的无扩散而获得诺贝尔物理学奖。另一个重要事件是谢赫特曼在1982年发现准晶体，这一发现在1984年他与布莱希、格拉提亚斯和卡恩联合撰写的一篇论文中得到了报道，并引起了晶体学和固态物理学的范式转变。无序结构的电子性质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采样来定义，因此很自然地，动力系统工具应该证明在研究这些算子方面是有用的。在一些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卓有成效的合作，并且这些互动有进一步成功的希望。

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.