Darboux transforms on Band Matrices, Weights and associated Polynomials

TL;DR

This paper investigates how Darboux transformations affect m-periodic weights, associated polynomials, and band matrices within the discrete KP hierarchy, providing a detailed analysis of their interplay and transformations.

Contribution

It offers a precise description of the impact of Darboux transformations on weights, polynomials, and band matrices in the context of the discrete KP hierarchy.

Findings

Darboux transformations induce specific changes in m-periodic weights and associated polynomials.

Vertex operators act on the τ-function to implement Darboux transformations.

The evolution of band matrices follows the discrete KP hierarchy.

Abstract

Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have shown that $m$-periodic sequences of weights lead to "moments", polynomials defined by determinants of matrices involving these moments and $2m+1$-step relations between them, thus leading to $2m+1$-band matrices $L$. Given a Darboux transformations on $L$, which effect does it have on the $m$-periodic sequence of weights and on the associated polynomials ? These questions will receive a precise answer in this paper. The methods are based on introducing time parameters in the weights, making the band matrix $L$ evolve according to the so-called discrete KP hierarchy. Darboux transformations on that $L$ translate into vertex operators acting on the $\tau$-function.