Schrodinger Operators with Spectral Transitions

该项目涉及准周期和其他确定性和随机结构的反常光谱和扩散特性。这是对作为含杂质系统模型的无序系统基本性质的研究。 准周期算子为整数量子霍尔效应、实验准晶体和量子混沌理论提供了中心或重要的模型。 严格理论的发展预计将有助于理解所有上述现象，特别是可能导致寻找具有所需物理性质的新材料。 无序系统还用于模拟许多其他微观和宏观效应：从量子局域化到地震。拟议的主题包括研究量子力学的高度无序和弱无序系统的特性，这些系统表现出某些异常行为。 该项目的一个组成部分涉及教育研究生和其他年轻研究人员。该项目由几个部分组成。 一是证明多维准周期算子的扩展态。另一个是研究紧束缚准周期模型中相互作用的影响。另一个是开发一种新的、决定性的多维安德森定位方法。 其他重要目标是研究与某些量子混沌模型相关的问题。该项目涉及持续开发非微扰方法，用于证明薛定谔算子和量子自旋系统中的局域型效应、无序环境中的渗流和接触过程，以及绝对连续光谱的研究。该奖项反映了 NSF 的法定使命，并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。

The project concerns the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is research on the fundamental properties of disordered systems that serve as models of systems with impurities. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals and the quantum chaos theory. The development of the rigorous theory is expected to contribute to the understanding of all the above phenomena, and in particular, 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. An integral part of the project concerns educating graduate students and other young researchers. The project consists of several parts. One is to prove the extended states for multidimensional quasiperiodic operators. Another is to study the effects of interaction in tight-binding quasi-periodic models. One more is to develop a new, determinantal, approach to multidimensional Anderson localization. Other important objectives are the study of issues related to certain models of quantum chaos. The project involves the continuing development of non-perturbative methods for the proofs of localization type effects both in Schrodinger operators and in quantum spin systems, percolation and contact processes in disordered environments, as well as for the study of absolutely continuous 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.