Spin Splitting in Dunkl Molecular Potentials Without an External Field: A Pauli–Dunkl Framework with Reflection-Induced Spin–Orbit Coupling

Abstract

We propose a spin-1/2 Dunkl molecular framework in which spin splitting can arise without an external magnetic field, subject to two explicitly stated hypotheses concerning the spectral theory of the spin–Dunkl angular operator. All prior Dunkl–Pauli work requires an external field to lift spin degeneracy [1, 2]; the scalar Dunkl–Schrödinger equation with Kratzer–Mie and pseudoharmonic potentials has been solved but only for spinless particles [3]. We close this gap by introducing the intrinsic coupling \(\lambda r^{-2}\mathcal {S}_D\), where \(\mathcal {S}_D\) couples Clifford spin generators to Dunkl angular momenta through the Coxeter reflection geometry. This interaction is a phenomenologically motivated, separability-preserving field-free spin–Dunkl coupling, introduced by analogy with the standard relativistic spin–orbit term rather than derived from a Dunkl–Dirac equation (that derivation remains an open problem, Sect. 11). It can be interpreted as a reflection-induced spin–orbit term whose pattern is dictated by the Coxeter root system rather than any applied field. Using this coupling we derive closed-form radial bound-state energy spectra for spin-1/2 particles in both the Kratzer–Mie and pseudoharmonic molecular potentials within a unified Dunkl framework. These radial spectra are exact for any angular eigenvalue of the spin–Dunkl operator \(\mathcal {A}_D = -\Delta _{D,S} + 2M\lambda \mathcal {S}_D\) that satisfies a stated spectral hypothesis: we prove \(\mathcal A_D\) is formal self-adjoint, but essential self-adjointness—required for a unique self-adjoint extension with real discrete spectrum—is not proved and is adopted throughout as an explicit hypothesis. All quantitative spectral statements are conditional on this hypothesis. For the \(A_2\) Coxeter group in \(d=2\) we further propose a phenomenological ansatz for the angular eigenvalues. This ansatz is supported only at leading order by first-order perturbation theory and has not been derived from the exact spectrum of \(\mathcal A_D\); the full \(A_2\) angular spectrum remains an open problem. Subject to this ansatz, we illustrate numerically that the resulting spin splitting grows with \(\lambda \) and, for the Kratzer–Mie potential, is asymptotically suppressed as \(O(k^{-3})\) with the Coxeter multiplicity k; for the pseudoharmonic potential the splitting instead saturates to a nonzero constant as \(k\rightarrow \infty \), reflecting the different functional dependence of the two spectra on the angular eigenvalue. We emphasize that all numerical results in this paper are illustrative consequences of an unproved ansatz, not first-principles predictions, and would need revision once the true \(A_2\) angular spectrum is known.