2006/12/27 by Ingolf Schäfer, Schäfer, Ingolf
Mathematics · Physics and Astronomy · #22E46 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT #msc:22E46
paper · pdf · doi:10.48550/arxiv.math/0612780
80 pages, 5 figures, phd thesis of the author
arxiv created 2006/12/27 · arxiv updated 2009/12/01
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.