Representation theoretical construction of the classical limit and spectral statistics of generic Hamiltonian operators [Elektronische Ressource] / von Ingolf Schäfer

RepresentationTheoreticalConstructionoftheClassicalLimitandSpectralStatisticsofGenericHamiltonianOperatorsvon Ingolf SchäferNovember 21, 2006Dissertationzur Erlangung des Grades eines Doktors derNaturwissenschaften– Dr. rer. nat. –Datum des Promotionskolloquiums: 09.11.2006Gutachter: Prof. Dr. E. Oeljeklaus (Universität Bremen)Prof. Dr. A. T. Huckleberry (Ruhr-Universität Bochum)Contents1 Introduction 72 ConstructionoftheClassicalLimit 112.1 The Classical Limit in the Simple Case . . . . . . . . . . . . . . . . . 112.2 Thel Limit in the General Case . . . . . . . . . . . . . . . . 122.3 Realizing the Classical Limit as an Analytical Limit . . . . . . . . . . 143 SpectralStatisticsofSimpleOperators 213.1 A Convergence Theorem for Simple Operators . . . . . . . . . . . . . 213.2 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Rescaling and Spectral Statistics . . . . . . . . . . . . . . . . 253.2.2ling and exp . . . . . . . . . . . . . . . . . . . . . . . . . 274 SpectralStatisticsofGenericOperators 294.1 Topology and Completion ofU(g) . . . . . . . . . . . . . . . . . . . . 29n4.2 A Notion of Hermitian Operators forO(C ) . . . . . . . . . . . . . . 314.3 Examples of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Rational Independence of the Spectra in Representations . . . . . . . 354.5 Ergodic Properties ofH . . . . . . . . . . . . . . . . . . . . . . . . 36gen4.6 The Sets B . . . . . . .