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.
Authors
Nock, AndreCollections
- Theses [4222]