We investigate the interplay between fractional kinetic dispersion and dipolar angular anisotropy in two-dimensional Aharonov--Bohm quantum rings within an effective semi-analytical and numerical framework. The orbital dynamics is formulated through a gauge-covariant fractional kinetic operator built from the spectral power of the magnetic kinetic operator on a bounded annular domain, while the impurity-induced anisotropy is modeled by a D/ (r^2+a^2) potential regularized near the origin. For moderate departures from the parabolic limit, an effective Mathieu--Floquet angular reduction is used to analyze low-lying states and is benchmarked against full numerical calculations in conventional limiting cases. The formulation avoids the additive combination of a fractional kinetic term with a separate parabolic magnetic Hamiltonian and does not introduce energy-dependent mass renormalization, thereby preventing gauge inconsistency and double counting of non-parabolicity. The kinetic prefactor is chosen so that the standard Schr"odinger--Pauli form is recovered exactly at =2. The spectral fractional kinetic operator transforms covariantly under smooth, single-valued gauge transformations, and the spectrum is gauge invariant in the sense that different gauge choices for the same physical flux produce identical eigenvalues. Within this effective description, we find that decreasing the fractional order lowers the dipolar threshold for angular localization and suppresses the amplitude of Aharonov--Bohm oscillations while preserving the flux periodicity. The same fractional--dipolar interplay enhances angular localization and modifies optical transition strengths in the low-lying spectrum. These results suggest that the fractional order can serve as an additional phenomenological tuning parameter for quantum-ring spectroscopy, provided the model is interpreted within its effective single-particle range of validity.
Sek et al. (Wed,) studied this question.
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