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.

该研究项目的目的是开发动力学技术，用于对ergodic schroedinger操作员的光谱分析，这在对某些无序介质中量子颗粒的运动进行建模时产生。动态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。对千古施罗辛格运营商的光谱分析的关键任务是研究相关特征值方程解决方案的渐近行为。然后可以建立两个不同领域之间的桥梁，因为``解决方案的渐近行为''可以解释为``某些系统的长期行为''。该项目的目的是开发由建造此类桥梁驱动的动态技术，并可能使两个领域受益。不同的媒体导致不同类型的Ergodic基本系统。著名和深入研究的安德森模型对应于I.I.D.随机变量可以通过完全偏移生成。涉及该项目的两种类型的基本系统是准周期系统，典型的几乎是周期系统以及双曲系统，经典类型的强烈混合系统。可以通过动力学对象，Lyapunov指数检测到各种级别的随机性，这是该项目研究的主要对象。该项目的一个重点是研究Lyapunov指数的阳性和较大偏差估计。这些属性对基础动力学的随机性非常敏感，通常很难获得，因此是动态系统中的中心主题之一。从光谱理论的一面来看，它们是安德森本地化现象的有力迹象，并立即暗示了莱普诺夫指数和状态综合密度的某些规律性。对准周期性和双曲基碱基动力学的两种特性的深入研究可能会阐明如何获得标准图的阳性Lyapunov指数。这是动态系统中最臭名昭著的未解决的问题之一，在动态系统中，困难完全在于椭圆形和双曲线行为的复杂共存。动力学系统与光谱理论之间的另一个基本关系是Cantor Spectrum现象。关于这种现象的最著名的物理例子是霍夫史塔特的蝴蝶。在动态系统中，康托频谱现象可能被视为某种统一双曲线系统的普遍性。 然后，该项目的另一个重点是调查Cantor Spectrum。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

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