Spectral Theory of Periodic and Quasiperiodic Quantum Systems

本研究项目的主要目标是基于薛定谔算子的谱理论，建立和研究周期和准周期介质（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，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”边缘附近电子输运的数学严谨模型，并开发有效质量近似的新方法。准周期算符是无序系统的例子，它可以看起来像纯晶体，或者具有随机杂质的晶体，同时是完全确定的。其中一个正在研究的模型在任意小的混乱下表现出随机行为，并且可以潜在地取代随机环境，而无需使用大的参数空间。将特别强调多维和多粒子模型，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。本研究项目的活动按所研究的算符的类别及其光谱的类型分为若干组。在准周期算子的安德森局域化（“类随机行为”）领域，该项目研究了具有微扰大无序和低正则模型分析势的多粒子模型，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。在绝对连续谱（“晶体行为”）领域，该项目研究了薛定谔环的低规则约化性与对应薛定谔算子的强弹道输运之间的关系，这反过来又与量子自旋系统的输运性质有关。在周期算符的领域，我们打算研究在二维和三维情况下光谱带边缘的Bloch变异的可能奇点。最后，在更抽象的方面，该项目旨在发展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尔谱和一些量子自旋系统的联系。

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.