@MISC{Kotzé92quantumchaos, author = {Antonie Abraham Kotzé}, title = {QUANTUM CHAOS and ANALYTIC STRUCTURE of the SPECTRUM}, year = {1992} }

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Abstract

Quantum chaos is associated with the phenomenon of avoided level crossings on a large scale which leads to a statistical behaviour similar to that of a Gaussian Or-thogonal Ensemble (GOE) of matrices. The same behaviour is seen in a pure quantum one dimensional system consisting of a square well containing δ-function barriers on a Cantor set. For a representative Hamiltonian of the form H0 + λH1, avoided level crossings are associated with branch point singularities when they are continued into the complex λ-plane. These are the exceptional points which are proposed to de-termine the form of the spectrum and thus the spectral fluctuation properties. It is prohibitive to determine the exact positions of the exceptional points but a proce-dure is given to determine their distribution and its implementation is demonstrated considering simple matrix models. By investigating the chaotic quartic oscillator the intricate connection between the distribution of exceptional points and the particular fluctuation properties of level spacings and associated eigenvector statistics is shown. The effect of the coupling matrix elements are also stressed. This means that a GOE,