Spectral and Transport Theory of Schrodinger Operators

项目负责人：Svetlana Jitomirskaya，加州大学欧文分校edms -0300974*********************************************************************本项目主要围绕遍历薛定谔算子的局域化效应以及准周期和安德森模型类型设置下异常量子输运的研究。它包括继续发展非微扰方法来证明局域性和其他性质，以及研究证明非微扰结果局限性的现象。另一个主要目标是发展研究异常量子输运的方法，以应用于确定性和随机势。我们的研究涉及到准周期和其他确定性和随机结构的异常光谱和扩散性质。因此，这是对作为含杂质系统模型的无序系统的基本性质的研究。确定性遍历势，特别是准周期势常用于模拟准晶体。研究微观模型的光谱和量子输运性质对于理解和解释准晶体的许多实验数据是非常重要的。这种理解可能会导致发现具有理想物理特性的新材料。无序系统也用于模拟许多其他微观和宏观效应：从量子局域化到地震。提出的主题包括研究量子力学中表现出某些异常行为的高度和弱无序系统的性质。

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.