Dunkl–Pauli equation for non-reducible reflection groups

Investigate how extending the two-dimensional Dunkl–Pauli equation in a magnetic field from the reducible reflection group \(\mathbb{Z}_2\times\mathbb{Z}_2\) to non-reducible reflection groups affects coordinate separability, the existence and form of the \(\mathfrak{su}(1,1)\) symmetry, and the construction of the Sturmian basis, coherent states, and their time evolution.

Background

The paper formulates the Dunkl–Pauli system using independent coordinate reflections associated with the reducible group W=Z2×Z2W=\mathbb{Z}_2\times\mathbb{Z}_2, which permits separation into angular and radial equations and supports the explicit su(1,1)\mathfrak{su}(1,1) construction developed in the paper. The authors note that non-reducible reflection groups can couple spatial coordinates, potentially preventing reduction to a simple radial equation.

The unresolved issue is to determine whether the separability, dynamical symmetry, Sturmian basis, coherent states, and coherent-state time evolution obtained for the reducible reflection group survive, or how they must be modified, for non-reducible reflection groups.

References

It is important to mention that an extension to non-reducible reflection groups $W$ may affect the separability of the system, since such groups introduce couplings between spatial coordinates. In this case, it may become more difficult to reduce the problem to a simple radial equation as in the present formulation. Furthermore, the structure of the underlying symmetry algebra could also be modified, in the sense that the $\mathfrak{su}(1,1)$ symmetry may not be realized in the same form. These effects could, in turn, impact the construction of the Sturmian basis, coherent states, and their time evolution. However, a detailed assessment of these effects lies beyond the scope of the present work and is left for future investigation.

— Exact Solutions of the Dunkl--Pauli Equation in the Presence of a Magnetic Field from an su(1,1) Algebraic Approach  (2609.31554 - Ramírez et al., 25 Sep 2026) in Concluding Remarks