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

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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.