Spectral Theory and Dynamics of Ergodic Schrodinger Operators

遍历薛定谔算子的谱理论和动力学

• 批准号: 1764154
• 负责人: Zhenghe Zhang
• 金额: $ 20.99万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764154&HistoricalAwards=false
• 关键词: Spectral; Theory; Dynamics; Ergodic; Schrodinger

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.

该研究项目的目标是开发用于遍历薛定谔算子光谱分析的动力学技术，该技术在模拟某些无序介质中量子粒子的运动时出现。动力系统理论的很大一部分涉及某些数学或物理系统（例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。然后可以在两个不同领域之间建立桥梁，因为“解的渐近行为”可以被解释为“某些系统的长期行为”。该项目的目标是开发由建造此类桥梁驱动的动力学技术，并且可能使这两个领域受益。不同的无序介质导致不同类型的遍历基础系统。著名且经过深入研究的安德森模型对应于 i.i.d.可以通过全班生成的随机变量。该项目涉及的两种基本系统是准周期系统（典型的几乎周期系统）和双曲系统（经典类型的强混合系统）。动态对象李亚普诺夫指数可以检测到各种级别的随机性，这是该项目的主要研究对象。该项目的重点之一是研究 Lyapunov 指数的正性和大偏差估计。这些属性对基础动力学的随机性非常敏感，通常很难获得，因此是动力学系统的中心主题之一。从谱理论的角度来看，它们强烈表明了安德森局域化现象，并且立即暗示了李亚普诺夫指数和积分态密度的某些规律性。对准周期和双曲基动力学的两个性质的深入研究可能有助于了解如何获得标准映射的正李亚普诺夫指数。这是动力系统中最臭名昭著的未解决问题之一，其困难恰恰在于椭圆和双曲行为的复杂共存。动力系统和谱理论之间的另一个基本关系是康托谱现象。关于这种现象最著名的物理例子是霍夫施塔特蝴蝶。在动力系统中，康托谱现象可以被视为某种普遍存在的一致双曲系统。 该项目的另一个重点是研究康托谱。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators

均匀双曲性及其与一维离散薛定谔算子谱分析的关系

• DOI: 10.4171/jst/333
• 发表时间: 2020
• 期刊: Journal of Spectral Theory
• 影响因子: 1
• 作者: Zhang, Zhenghe

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

• DOI: 10.1090/tran/7832
• 发表时间: 2017-06
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Valmir Bucaj;D. Damanik;J. Fillman;Vitaly Gerbuz;Tom VandenBoom;Fengpeng Wang;Zhenghe Zhang