Spectral Theory of Periodic and Quasiperiodic Quantum Systems

周期和准周期量子系统的谱论

• 批准号: 1758326
• 负责人: Ilya Kachkovskiy
• 金额: $ 7.18万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-16 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1758326&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Schrodinger操作员的光谱理论，开发和研究定期和准碘介质（例如晶体或准晶体）中量子颗粒的数学模型。周期性培养基对应于晶体，例如金属或半导体，它们可以在某些能量下自由地进行电子。建议研究在“禁区”边缘附近的数学严格模型的电子传输模型，并开发出有效质量近似的新方法。准碘操作员是无序系统的示例，取决于政权，看起来像纯晶体或具有随机杂质的晶体，同时是完全确定性的。研究中的一个模型表明，在任意小型疾病中的随机行为，并且有可能成为随机环境的合适替代品，而无需使用较大的参数空间。多维和多粒子模型将特别强调，并可能应用于量子自旋系统和量子信息理论。该项目为本科生和研究生提供了研究机会。该研究项目的活动属于由正在研究的运营商类别及其光谱类型的几个小组。在Anderson定位的准操作员（“随机行为”）中，该项目研究具有扰动性较大的疾病和低规律性模型的分析势的多粒子模型，后者的结果预计将是非驱动性的。这里的方法包括操作者理论，谐波分析，实际代数几何以及用于次谐波或分段单音函数的大偏差定理。在绝对连续的光谱区域（“结晶行为”）中，该项目研究了相应的Schrodinger操作员的Schrodinger Cocycles低规律性降低与强烈的弹道传输之间的关系，这又与量子自旋系统的运输特性有关。在定期运算符的领域，旨在研究在2D和3D病例中，在光谱带边缘的Bloch品种可能的奇异性。最后，从更抽象的一面来看，该项目旨在开发拓扑非平凡的情况下几乎通勤矩阵的定量分类，这证明了与cantor spectra既与cantor spectra contor Spectra contor oderiodic operators and cantor spectra conterrics conterrics contrices contrices contrices contrices contrices and quasiperiodic operators又与某些量子旋转系统进行了连接。