Skew-Orthogonal Polynomials and Universality of Energy-Level Correlations

Random-matrix ensembles serve as models for quantum chaotic systems. We develop the theory of skew-orthogonal polynomials to study matrix ensembles with non-Gaussian weight functions. From the asymptotic properties of these and the orthogonal polynomials, we show that the local energy level correlations in the ensembles become universal properties independent of the global level density. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems.