Renyi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields

One-parameter functionals of the Renyi R_ρ,γ(α) and Tsallis T_ρ,γ(α) types are calculated in the position (subscript ρ) and momentum (γ) spaces for the 2D nanoring placed into the combination of the uniform magnetic field B and the AB flux Φ_AB. Ring potential is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. Position (momentum) Renyi entropy depends on B as a negative (positive) logarithm of ω_eff=(ω₀²+ω_c²/4)^1/2 , where ω₀ determines the quadratic steepness of the confining potential and ω_c is a cyclotron frequency. This makes the sum R_ρnm(α)+R_γnm(α/(2α-1)) a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and satisfies corresponding uncertainty relation. In the limit α->1 both entropies in either space tend to their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient α where the momentum entropies exist reveals that it depends on the ring geometry, Φ_AB and m. There is the only orbital for which both uncertainty relations turn into the identity at α=1/2 and which is not necessarily the lowest-energy level. At any coefficient α, the R_ρ-Φ_AB curve mimics the energy variation with Φ_AB.