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

摘要Bleher 9623214 该项目致力于半经典迹公式的 紧流形上的薛定谔算子，极限概率 量子能级的分布和相关问题 随机矩阵理论调查的主题包括： (i)Weyl谱中振荡部分的渐近性 公式和周期测地线，统计特征值的薛定谔算子，和数学问题的 量子混沌理论(ii)一致半经典问题 不稳定不变流形附近的量子化(iii)普遍性 矩阵中的间距分布和双标度极限 模型首席研究员一直在这方面工作 与弗里曼·戴森和吉恩· Bourgain（普林斯顿高等研究院），Joel Lebowitz （罗格斯大学），雅科夫西奈（普林斯顿大学），和其他人。他的重要贡献之一是证明了 可积量子系统的Weyl谱误差函数的概周期性，本征值分布的极限定理， 特征值统计和其他一些结果。 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，以及本征值分布和双标度的普适性 矩阵模型中的极限。这些问题对于量子混沌理论中的许多应用都是非常重要的， 弦和量子引力，介观系统，和多元 统计分析，该项目是针对应用程序。这些工具包括理论的方法， 可积系统，半经典渐近性，矩阵 Riemann-Hilbert问题，以及其他现代数学方法。 这项研究主要基于计算机实验。

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.