Spectral statistics of orthogonal and symplectic ensembles

This article describes a direct approach for computing scalar and matrix kernels, respectively for the unitary ensembles on the one hand and the orthogonal and symplectic ensembles on the other hand, leading to correlation functions and gap probabilities. In the classical orthogonal polynomials (Hermite, Laguerre, and Jacobi), the matrix kernels for the orthogonal and symplectic ensemble are expressed in terms of the scalar kernel for the unitary case, using the relation between the classical orthogonal polynomials going with the unitary ensembles and the skew-orthogonal polynomials going with the orthogonal and symplectic ensembles. The article states the fundamental theorem relating the orthonormal and skew-orthonormal polynomials that enter into the Christoffel-Darboux kernels

On the commutative factorization of n × n matrix Wiener–Hopf kernels with distinct eigenvalues

In this article, we present a method for factorizing n × n matrix Wiener–Hopf kernels where n >2 and the factors commute. We are motivated by a method posed by Jones (Jones 1984 a Proc. R. Soc. A 393 , 185–192) to tackle a narrower class of matrix kernels; however, no matrix of Jones' form has yet been found to arise in physical Wiener–Hopf models. In contrast, the technique proposed herein should find broad application. To illustrate the approach, we consider a 3×3 matrix kernel arising in a problem from elastostatics. While this kernel is not of Jones' form, we shall show how it can be factorized commutatively. We discuss the essential difference between our method and that of Jones and explain why our method is a generalization. The majority of Wiener–Hopf kernels that occur in canonical diffraction problems are, however, strictly non-commutative. For 2×2 matrices, Abrahams has shown that one can overcome this difficulty using Padé approximants to rearrange a non-commutative kernel into a partial-commutative form; an approximate factorization can then be derived. By considering the dynamic analogue of Antipov's model, we show for the first time that Abrahams' Padé approximant method can also be employed within a 3×3 commutative matrix form.