Spectral and Transport Theory of Schrodinger Operators

Pi：加州大学IrvineDMS-0300974*********************************************************************This分校的Svetlana Jitomirskaya项目围绕着与遍历薛定谔算符的局域型效应相关的问题，以及在准周期和安德森模型类型设置下的反常量子输运研究。它包括继续发展非微扰方法来证明局部化和其他性质，以及研究证明非微扰结果的局限性的现象。另一个主要目标是发展研究反常量子输运的方法，以同时应用于确定性和随机势。我们的研究涉及准周期和其他确定性和随机性结构的反常光谱和扩散特性。因此，这是对无序系统的基本性质的研究，无序系统是含有杂质的系统的模型。确定性的遍历势能，特别是准周期势能经常被用来模拟准晶。研究微观模型的光谱和量子输运性质对于理解和解释大量准晶实验数据具有重要意义。这样的理解可能会导致找到具有所需物理性能的新材料。无序系统也被用来模拟其他许多微观和宏观效应：从量子局域化到地震。建议的主题包括研究表现出某些反常行为的高度无序和弱无序量子力学系统的性质。

PI: Svetlana Jitomirskaya, University of California, IrvineDMS-0300974*********************************************************************This project is centered around issues related to localization type effects for ergodic Schroedinger operators and the study of anomalous quantum transport in the quasiperiodic and Anderson model-type settings. It involves the continuing development of non-perturbative methods for theproofs of localization and other properties as well as studying phenomena that demonstrate the limitations of non-perturbative results. Another major goal is the development of methods to study anomalous quantum transport to be applied to both deterministic and random potentials.Our research concerns the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is therefore research on the fundamental properties of disordered systems that serve as models of systems with impurities. Deterministic ergodic,particularly quasiperiodic, potentials are often used to model quasicrystals. Study of the spectral and quantum transport properties of the microscopic models is important to understand and interpret much of the experimental data on quasicrystals. Such understanding may lead to finding new materials with desired physical properties. Disordered systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. The proposed topics include studying properties of both highly and weakly disordered systems of Quantum Mechanics that demonstrate certain anomalous behavior.