Distribution of Eigenvalues of the Schrodinger Operator on Compact Manifolds. Matrix Model and Asymptotics of Orthogonal Polynomials

Bleher 9623214本项目致力于研究紧致流形上薛定谔算子的半经典迹公式，量子能级的极限概率分布，以及随机矩阵理论中的相关问题。研究的主题包括：(I)Weyl谱公式和周期测地线中振荡部分的渐近性，薛定谔算符本征值的统计，以及量子混沌理论的数学问题。(Ii)不稳定不变流形附近的一致半经典量子化问题。(3)矩阵模型中间距分布和双标度极限的普适性。首席研究员与弗里曼·戴森和让·布尔加恩(普林斯顿高级研究所)、乔尔·勒博维茨(罗格斯大学)、雅科夫·西奈(普林斯顿大学)等人合作，在这一领域工作了几年。他的重要贡献包括证明了可积量子系统Weyl谱误差函数的几乎周期性，证明了本征值分布的极限定理，证明了本征值统计的Berry饱和猜想，以及其他几个结果。在这个项目中，主要的研究人员将研究量子系统中能级和本征态的统计，以及矩阵模型中本征值分布和双标度极限的普适性。这些问题在量子混沌理论、玻色弦和量子引力、介观系统和多元统计分析中的大量应用中都是非常重要的，该项目是针对这些应用的。这些工具包括可积系统理论的方法、半经典渐近法、矩阵Riemann-Hilbert问题和其他现代数学方法。这项拟议的研究在很大程度上是基于计算机实验。

Abstract Bleher 9623214 The project is devoted to the semiclassical trace formulae for the Schrodinger operator on compact manifolds, limiting probability distributions of quantum energy levels, and related problems in the theory of random matrices. Topics of investigation include: (i) Asymptotics of the oscillatory part in the Weyl spectral formula and periodic geodesics, statistics of eigenvalues of the Schrodinger operator, and mathematical problems of the theory of quantum chaos. (ii) The problem of the uniform semiclassical quantization near unstable invariant manifolds. (iii) Universality of the spacing distribution and double scaling limit in the matrix model. The principal investigator has been working in this area for several years in collaboration with Freeman Dyson and Jean Bourgain (Institute for Advanced Study, Princeton), Joel Lebowitz (Rutgers University), Yakov Sinai (Princeton University), and others. Among his important contributions are the proof of almost periodicity of the Weyl spectral error function for integrable quantum systems, limit theorems for the distribution of eigenvalues, the proof of the Berry saturation conjecture for the eigenvalue statistics, and several other results. In the project the principal investigator is going to study the statistics of energy levels and eigenstates in quantum systems, and the universality of the eigenvalue distribution and double scaling limit in the matrix model. These problems are very important for numerous applications in the theory of quantum chaos, bosonic strings and quantum gravity, mesoscopic systems, and multivariate statistical analysis, and the project is directed to the applications. The tools include the methods of the theory of integrable systems, semiclassical asymptotics, the matrix Riemann-Hilbert problem, and other methods of modern mathematics. The proposed study is heavily based on computer experiments.