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

本课题研究紧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，薛定谔算子的特征值统计，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化问题。（iii）矩阵模型中空间分布和双标度极限的通用性。首席研究员与Freeman Dyson、Jean Bourgain（普林斯顿高等研究院）、Joel Lebowitz（罗格斯大学）、Yakov Sinai（普林斯顿大学）等人在这一领域合作了几年。他的重要贡献包括可积量子系统的Weyl谱误差函数的几乎周期性的证明，特征值分布的极限定理，特征值统计的Berry饱和猜想的证明，以及其他几个结果。在这个项目中，首席研究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尺度极限的普适性。这些问题对于量子混沌理论、玻色子弦和量子引力、介观系统和多元统计分析等领域的许多应用都是非常重要的，本课题针对这些应用进行了研究。这些工具包括可积系统理论的方法，半经典渐近，矩阵黎曼-希尔伯特问题，和其他现代数学的方法。拟议的研究在很大程度上基于计算机实验。

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.