Narrow Results

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.

Explicit, analytic and closed expressions for boson realizations of the (m+3)-parameter nonlinearly deformed angular momentum algebra with its highest power m of polynomial function being arbitrary, which combines and generalizes Witten's two deformation schemes, are investigated in terms of the single boson and the single inversion boson, respectively. For each kind, the unitary Holstein–Primakoff-like realization, the non-unitary Dyson–Maléev-like realization and their connections are respectively discussed. Using these realizations, the corresponding representations of as well as their respective acting spaces in the Fock space are obtained.