Complex Jacobi matrices generated by Darboux transformations

TL;DR

This paper investigates complex Jacobi matrices derived from Darboux transformations, analyzing their properties, invariances, and implications for orthogonal polynomials and recurrence relations.

Contribution

It introduces new invariance properties of complex Jacobi matrices under Christoffel and Geronimus transformations, and explores their effects on orthogonal polynomials and recurrence relations.

Findings

Nevai class invariance under transformations

Spectrum may include an extra point after transformation

Geronimus transformations lead to R_{II}-recurrence relations

Abstract

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to $R_{II}$-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.