Spectral Theory and Dynamics of Ergodic Schrodinger Operators

本研究项目的目标是开发遍历薛定谔算符的谱分析动力学技术，这是在模拟量子粒子在某些无序介质中的运动时出现的。动力系统理论的很大一部分涉及某些数学或物理系统（如双曲系统或哈密顿系统）中典型轨迹的长期行为。遍历薛定谔算子谱分析的一个关键任务是研究相关特征值方程解的渐近行为。由于“解的渐近行为”可以解释为“某些系统的长期行为”，因此可以在两个不同领域之间建立桥梁。这个项目的目标是开发动力技术，通过建造这样的桥梁来驱动，这可能对两个领域都有利。不同的无序介质导致不同类型的遍历碱体系。著名的、被广泛研究的安德森模型对应的是全移位产生的i.i.d随机变量。本课题研究的两种基本系统是拟周期系统，即典型的几乎周期系统和双曲系统，即典型的强混合系统。各种程度的随机性可以通过一个动态对象Lyapunov指数来检测，这是本项目的主要研究对象。该项目的一个重点是研究李亚普诺夫指数的正性和大偏差估计。这些性质对基本动力学的随机性非常敏感，通常难以获得，因此是动力学系统的中心主题之一。从谱理论的角度来看，它们是安德森局域化现象的有力指示，并立即暗示了李亚普诺夫指数和态的积分密度的一定规律性。深入研究准周期基动力学和双曲基动力学的这两个性质，可以为如何获得标准映射的正Lyapunov指数提供启示。这是动力系统中最著名的未解问题之一，其难点恰恰在于椭圆型和双曲型行为的复杂共存。动力系统和谱理论之间的另一个基本关系是康托谱现象。关于这一现象最著名的物理例子是霍夫施塔特蝴蝶。在动力系统中，康托谱现象可以看作是一致双曲系统的一种普遍性。这个项目的另一个重点是研究康托谱。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this research project is to develop dynamical techniques for the spectral analysis of the ergodic Schroedinger operators, which arise in modeling the motion of quantum particles in certain disordered media. A large part of the theory of dynamical systems deals with long-term behaviors of typical trajectories in certain mathematical or physical systems, such as hyperbolic systems or Hamiltonian systems. A key task of spectral analysis of the ergodic Schroedinger operators is to study the asymptotic behaviors of the solutions of the associated eigenvalue equations. Bridges between two different areas can then be built since ``asymptotic behaviors of solutions'' may be interpreted as ``long-term behaviors of certain systems''. The goal of this project is to develop dynamical techniques that are driven by building such bridges and that may benefit both areas.Different disordered media lead to different type of ergodic base systems. The famous and intensively studied Anderson model corresponds to i.i.d. random variables which can be generated by full shift. Two types of base systems with which this project is concerned are quasi-periodic systems, typical almost periodic systems, and hyperbolic systems, classic type of strongly mixing systems. Various levels of randomness may be detected by a dynamical object, the Lyapunov exponent, which is the main object of study of this project. One focus of this project is the study of positivity and large deviation estimates of the Lyapunov exponent. These properties are super sensitive to the randomness of the base dynamics, are generally difficult to obtain, and are thus among central topics in dynamics systems. From the side of spectral theory, they are strong indications of the Anderson Localization phenomenon and imply immediately certain regularity of both the Lyapunov exponent and the integrated density of states. Deep investigation of the two properties for both quasi-periodic and hyperbolic base dynamics may shed light on how to obtain positive Lyapunov exponent of the standard map. This is one of the most notorious unsolved problems in dynamical systems where the difficulty lies exactly in the complicated coexistence of both elliptic and hyperbolic behaviors. Another fundamental relation between dynamical systems and spectral theory is the Cantor Spectrum phenomenon. The most famous physical example regarding this phenomenon is the Hofstadter's butterfly. In dynamical systems, Cantor spectrum phenomenon may be viewed as some kind of ubiquity of uniformly hyperbolic systems. Another focus of this project is then to investigate Cantor spectrum.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.