Free boson realization of the Dunkl intertwining operator in one dimension

TL;DR

This paper demonstrates a realization of the Dunkl intertwining operator in one dimension using free boson (oscillator) operators, linking it to Hermite and generalized Hermite polynomials.

Contribution

It provides a novel bosonic realization of the Dunkl intertwining operator, connecting algebraic operators with oscillator representations in one dimension.

Findings

Dunkl intertwining operator maps Hermite to generalized Hermite polynomials.

Realization in terms of oscillator operators.

Establishes a link between Dunkl operators and quantum harmonic oscillators.

Abstract

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials.