Characteristic Polynomials of Random Matrices and Quantum Chaotic Scattering
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Scattering is a fundamental phenomenon in physics, e.g. large parts of the knowledge about quantum systems stem from scattering experiments. A scattering process can be completely characterized by its K-matrix, also known as the \Wigner reaction matrix" in nuclear scattering or \impedance matrix" in the electromagnetic wave scattering. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory (RMT), where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. These two approaches are believed to lead to the same results due to a universality conjecture by P. Brouwer, which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution in the limit of large matrix dimension of H. For unitarily invariant ensembles, this conjecture will be proved in the thesis by explicit calculation, utilising results about ensemble averages of characteristic polynomials. This thesis furthermore analyses various characteristics of the K-matrix such as the distribution of a diagonal element at the spectral edge or the distribution of an off-diagonal element in the bulk of the spectrum. For the latter it is necessary to know correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices for the Gaussian Orthogonal Ensemble (GOE), which is an interesting and important topic in itself, as they frequently arise in various other applications of RMT to physics of quantum chaotic systems, and beyond. A larger part of the thesis is dedicated to provide an explicit evaluation of the large-N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method.
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