Spectral Theory and Universality

该项目涉及光谱理论中的问题，光谱理论是与量子系统的能级和机械系统的振动频率等物理概念相对应的数学理论。该项目的主题之一是普适性，即不同系统的某些普遍（或共同）局部统计行为的出现。这项研究的动机也是可积性，即在某些非线性系统中存在许多守恒量，用于描述它们随时间的行为。该项目的重点是薛定谔算子，中央量子力学和相关系统在一维;数学方法开发有可能照亮其他数学模型和物理应用，如电子导电性在无序材料和信号传输使用孤子。将培训几名研究生和一名博士后，并计划组织会议以及编写一本关于无反射算子的专著。这个项目的一个重点是德布兰日规范系统在正交多项式及相关系统的普适性极限中的应用。由PI共同撰写的一种新方法在规范系统的极限方面重新表述了这个问题，并使用它来获得一个非常普遍的批量普适性标准;将研究这种方法对其他普适性现象的实质性扩展。该项目的另一个重点将是发展无反射薛定谔算子的逆谱理论，用于不具有“直接柯西定理”性质的Dirichlet正则Widom谱;在这种情况下，等谱环面的结构将对具有几乎周期性初始数据的Korteweg-de弗里斯方程产生直接影响。最后，由于最近发展了Stahl-Totik正则性的薛定谔算子，使新的研究方向成为可能，例如研究任意基本谱的薛定谔算子的求和规则。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 questions in spectral theory, which is the mathematical theory that corresponds to physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. One of the topics of this project is universality, which is the appearance of certain universal (or common) local statistical behaviors for different systems. The research is also motivated by integrability, which is the existence of many conserved quantities in certain nonlinear systems used to describe their behavior over time. The project focuses on Schrödinger operators, central to quantum mechanics, and related systems in one dimension; the mathematical methods developed have the potential to illuminate other mathematical models and physical applications such as electron conductivity in disordered materials and signal transmission using solitons. Several graduate students and a postdoc will be trained, and conference organization as well as writing of a monograph on reflectionless operators is planned.One focus of this project is the application of de Branges canonical systems to universality limits for orthogonal polynomials and related systems. A new approach co-authored by the PI reformulates the question in terms of a limit of canonical systems and uses that to obtain a very general criterion for bulk universality; substantial extensions of this approach to other universality phenomena will be studied. Another focus of the project will be the development of inverse spectral theory for reflectionless Schrödinger operators for Dirichlet-regular Widom spectra without the “direct Cauchy theorem” property; the structure of the isospectral torus in this regime will have immediate consequences for the Korteweg-de Vries equation with almost periodic initial data. Finally, of interest will be new research directions made possible by the recent development of Stahl-Totik regularity for Schrödinger operators, such as the study of sum rules for Schrödinger operators with arbitrary essential spectra.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.