Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

TL;DR

This paper introduces discrete spectral transformations for skew orthogonal polynomials, deriving associated integrable systems in multiple dimensions and extending to matrix form, with implications for random matrix theory.

Contribution

It presents novel discrete spectral transformations for skew orthogonal polynomials and derives new integrable systems, including a matrix extension in 2+1 dimensions.

Findings

Discrete spectral transformations are established for skew orthogonal polynomials.

Discrete integrable systems are derived in 1+1 and 2+1 dimensions.

A factorization theorem for the Christoffel kernel in random matrix theory is provided.

Abstract

Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2x2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.