On the relation between Darboux transformations and polynomial mappings

TL;DR

This paper explores the connection between Darboux transformations and polynomial mappings through the lens of Stieltjes functions, matrix transformations, and Jacobi matrices, revealing new insights into their interrelations.

Contribution

It demonstrates that the unwrapping transformation of Stieltjes functions corresponds to a Darboux transformation of Jacobi matrices, linking polynomial mappings to spectral transformations.

Findings

Unwrapping transformation is a Darboux transformation of Jacobi matrices

Chihara's solutions to the Carlitz problem are shifted Darboux transformations

Matrix interpretation clarifies the relation between polynomial mappings and spectral theory

Abstract

Let d\mu(t) be a probability measure on [0,+\infty) such that its moments are finite. Then the Cauchy-Stieltjes transform S of d\mu(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(\lambda) to \lambda S(\lambda^2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation.