An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
Francisco J. Herranz, Danilo Latini
TL;DR
This work unifies Dunkl-type superintegrable systems with coalgebra symmetry by introducing a novel differential-difference realization of $\mathfrak{sl}(2,\mathbb{R})$ tied to the $\mathbb{Z}_2^N$ reflection group. It proves that a broad class of $N$-dimensional Dunkl Hamiltonians $\hat{H}=F(\hat{\boldsymbol{\pi}}^2+\sum_i(\beta_i+\gamma_i\hat{R}_i)/\hat{x}_i^2,\hat{\boldsymbol{x}}^2,\hat{\boldsymbol{x}}\cdot\hat{\boldsymbol{\pi}}-\imath\hbar(\tfrac{N}{2}+\sum_i\mu_i\hat{R}_i))$ possess $2N-3$ universal quantum integrals arising from left and right partial Casimirs, making them QMS in any dimension. Specializing this framework yields a spectrum of MS Dunkl oscillators and KC-type systems on Euclidean, curved, and nonuniform spaces, including new models on $\mathbb{S}^N$, $\mathbb{H}^N$, Darboux III, and Taub-NUT geometries, each accompanied by explicit quadratic quantum integrals such as Dunkl-Demkov-Fradkin tensors or Dunkl-LRL vectors. The results bridge Dunkl operators with coalgebra methods, enabling curvature and non-central deformations while preserving maximal or quasi-maximal superintegrability, and offer a versatile route to construct and solve further Dunkl MS systems. The work thus broadens the landscape of exactly solvable quantum systems in multiple geometries and opens avenues for deeper algebraic understanding of their spectra.
Abstract
This work aims to bridge the gap between Dunkl superintegrable systems and the coalgebra symmetry approach to superintegrability, and subsequently to recover known models and construct new ones. In particular, an infinite family of $N$-dimensional quasi-maximally superintegrable quantum systems with reflections, sharing the same set of $2N-3$ quantum integrals, is introduced. The result is achieved by introducing a novel differential-difference realization of $\mathfrak{sl}(2, \mathbb{R})$ and then applying the coalgebra formalism. Several well-known maximally superintegrable models with reflections appear as particular cases of this general family, among them, the celebrated Dunkl oscillator and the Dunkl-Kepler-Coulomb system. Furthermore, restricting to the case of "hidden" quantum quadratic symmetries, maximally superintegrable curved oscillator and Kepler-Coulomb Hamiltonians of Dunkl type, sharing the same underlying $\mathfrak{sl}(2, \mathbb{R})$ coalgebra symmetry, are presented. Namely, the Dunkl oscillator and the Dunkl-Kepler-Coulomb system on the $N$-sphere and hyperbolic space together with two models which can be interpreted as a one-parameter superintegrable deformation of the Dunkl oscillator and the Dunkl-Kepler-Coulomb system on non-constant curvature spaces. In addition, maximally superintegrable generalizations of these models, involving non-central potentials, are also derived on flat and curved spaces. For all specific systems, at least an additional quantum integral is explicitly provided, which is related to the Dunkl version of a (curved) Demkov-Fradkin tensor or a Laplace-Runge-Lenz vector.