一般Hamilton算子经典极限和谱统计的表示论构造
Representation Theoretical Construction of the Classical Limit and Spectral Statistics of Generic Hamiltonian Operators
摘要：
本文从半单复李群的泛包络代数中的一个算子出发，讨论了该算子的谱沿一系列表示的最近邻统计量。在第一章的简短介绍之后，这个问题的动机是量子力学系统的经典极限的一般构造，这是在第二章中采用的。在第三章中，我们证明了对于简单算子，即，李代数算子沿着一列不可约表示的最近邻统计收敛到Dirac测度。在适当地完备了泛包络代数之后，第四章证明了一般算子的指数收敛于Poisson统计量。证明利用的组合不等式的卡茨-Sarnak型环面，这是证明了第5章。在附录中收集了来自群论和最近邻分布理论的必要事实。
英文摘要：
Starting with an operator in the universal enveloping algebra of a semi-simple, complex Lie group the nearest neighbor statistics of the spectra of this operator along a sequence of representations are discussed. After a short introduction in chapter 1 this problem is motivated by a general construction of the classical limit for quantum mechanical systems, which is adopted to this setting, in chapter 2. In chapter 3 it is shown that for simple operators, i.e., operators of the Lie algebra the nearest neighbor statistics along a sequence of irreducible representations converge to the Dirac measure. After a suitable completion of the universal enveloping algebra the convergence to Poisson statistics is proved in chapter 4 for the exponentials of generic operators. The proof makes use of a combinatorial inequality of the Katz-Sarnak type for tori, which is proved in chapter 5. In the appendix the necessary facts from group theory and the theory of nearest neighbor distributions are gathered.
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