Spectral Theory of Periodic and Quasiperiodic Quantum Systems

周期和准周期量子系统的谱论

• 批准号: 1600422
• 负责人: Ilya Kachkovskiy
• 金额: $ 10.11万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2017-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600422&HistoricalAwards=false
• 关键词: Spectral; Theory; Periodic; Quasiperiodic; Quantum

The main goals of this research project are to develop and study mathematical models of quantum particles in periodic and quasiperiodic media, such as crystals or quasicrystals, based on spectral theory of Schrodinger operators. Periodic media correspond to crystalline structures, such as metals or semiconductors, which can conduct electrons freely at certain energies. It is proposed to study mathematically rigorous models of electron transport near the edges of the "forbidden zones" and develop new approaches to the effective mass approximation. Quasiperiodic operators are examples of disordered systems which, depending on the regime, can look like pure crystals, or crystals with random impurities, while being completely deterministic. One of the models under study demonstrates random-like behavior at arbitrarily small disorder and can potentially be a suitable replacement for a random environment without having to employ a large parameter space. Special emphasis will be given to multi-dimensional and multi-particle models, with possible applications to quantum spin systems and quantum information theory. The project provides research opportunities for undergraduate and graduate students.The activities of this research project fall into several groups distinguished by the classes of the operators under study and the types of their spectra. In the area of Anderson localization for quasiperiodic operators ("random-like behavior"), the project studies multi-particle models with analytic potentials at perturbatively large disorder and low regularity models, with the latter results expected to be non-perturbative. The methods here include operator theory, harmonic analysis, real algebraic geometry, and large deviation theorems for subharmonic or piecewise-monotonic functions. In the area of absolutely continuous spectrum ("crystalline behavior"), the project investigates the relation between low regularity reducibility of Schrodinger cocycles and strong ballistic transport for the corresponding Schrodinger operators, which, in turn, is related to transport properties of quantum spin systems. In the area of periodic operators, it is intended to study possible singularities of the Bloch varieties at the edges of spectral bands, both in 2D and 3D cases. Finally, on the more abstract side, the project aims to develop a quantitative classification of almost commuting matrices in topologically non-trivial cases, which demonstrates connections both with Cantor spectra for quasiperiodic operators and with some quantum spin systems.

该研究项目的主要目标是基于薛定谔算子的谱理论，开发和研究周期性和准周期性介质（例如晶体或准晶体）中量子粒子的数学模型。周期性介质对应于晶体结构，例如金属或半导体，它们可以在一定能量下自由传导电子。建议研究“禁区”边缘附近电子传输的严格数学模型，并开发有效质量近似的新方法。准周期算子是无序系统的示例，根据状态的不同，它们可能看起来像纯晶体或具有随机杂质的晶体，同时又是完全确定性的。正在研究的模型之一展示了在任意小的无序情况下的类似随机行为，并且有可能成为随机环境的合适替代品，而无需使用大的参数空间。将特别强调多维和多粒子模型，并可能应用于量子自旋系统和量子信息论。该项目为本科生和研究生提供研究机会。该研究项目的活动分为几组，根据所研究的算子类别及其光谱类型进行区分。在准周期算子的安德森定位领域（“类随机行为”），该项目研究了在微扰大无序和低规律性模型下具有解析势的多粒子模型，后者的结果预计是非微扰的。这里的方法包括算子理论、调和分析、实代数几何以及次调和或分段单调函数的大偏差定理。在绝对连续光谱（“晶体行为”）领域，该项目研究了薛定谔余循环的低正则性还原性与相应薛定谔算子的强弹道输运之间的关系，而这又与量子自旋系统的输运性质有关。在周期算子领域，旨在研究 2D 和 3D 情况下布洛赫簇在谱带边缘可能存在的奇点。最后，在更抽象的方面，该项目旨在开发拓扑非平凡情况下几乎交换矩阵的定量分类，这证明了与准周期算子的康托谱和一些量子自旋系统的联系。