Spectral Theory of Periodic and Quasiperiodic Quantum 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，例如金属或半导体，它们可以在某些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格模型，并开发新的有效质量近似方法。准周期算符是无序系统的例子，根据状态，它可以看起来像纯晶体，或者是具有随机杂质的晶体，同时是完全确定的。研究中的一个模型在任意小的无序状态下表现出类似随机的行为，并且可能是随机环境的合适替代品，而不必采用大的参数空间。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。该研究项目的活动分为几个组，由所研究的算子类和它们的谱类型区分。在准周期算子（“类随机行为”）的安德森局部化领域，该项目研究了在大扰动无序和低正则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，真实的代数几何，以及次调和或分段单调函数的大偏差定理。在绝对连续谱（“结晶行为”）领域，该项目研究薛定谔上循环的低正则性约化与相应薛定谔算子的强弹道输运之间的关系，而后者又与量子自旋系统的输运性质有关。在该地区的周期运营商，它的目的是研究可能的奇异性的布洛赫品种在边缘的频谱带，无论是在2D和3D的情况下。最后，在更抽象的一面，该项目的目的是发展一个定量分类几乎交换矩阵在拓扑非平凡的情况下，这表明连接与康托谱准周期运营商和一些量子自旋系统。

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.