Dual Christoffel transformations

TL;DR

This paper extends Crum's theorem and Krein-Adler modifications to discrete quantum mechanics, linking eigenfunction transformations with Christoffel transformations of dual orthogonal polynomials, enabling finite degree deletions.

Contribution

It formulates dual Christoffel transformations within discrete quantum mechanics, connecting polynomial modifications with eigenfunction transformations.

Findings

Formulation of Crum's theorem for discrete quantum mechanics

Development of Krein-Adler modifications in this context

Establishment of Christoffel transformations for dual orthogonal polynomials

Abstract

Crum's theorem and its modification a la Krein-Adler are formulated for the discrete quantum mechanics with real shifts, whose eigenfunctions consist of orthogonal polynomials of a discrete variable. The modification produces the associated polynomials with a finite number of degrees deleted. This in turn provides the well known Christoffel transformation for the dual orthogonal polynomials with the corresponding positions deleted.