Physical significance of Dunkl two-photon eigenvalue equations

Determine the physical significance, in the context of quantum optics, of the differential-difference equations obtained from the Dunkl–Fock–Bargmann realization of the Dunkl two-photon algebra, namely equations (\ref{eq:defeq}) and (\ref{eq:defeqR}), including the version with an explicit reflection-operator term.

Background

The paper constructs a Dunkl two-photon algebra by adjoining a reflection generator to the standard two-photon algebra and realizes it on a Dunkl–Fock–Bargmann space. Applying this realization to the two-photon algebra eigenvalue equation produces differential-difference equations involving the Dunkl derivative and reflected argument f(-z).

Equation (\ref{eq:defeq}) is the direct Dunkl counterpart of the standard two-photon eigenstate equation, while equation (\ref{eq:defeqR}) additionally includes the reflection generator with coefficient \beta_6. Although these equations are derived algebraically and reduce to the standard differential equation in the appropriate limit, their interpretation and relevance for quantum-optical systems are not resolved in the paper.