Spin-orbit interaction for the double ring-shaped oscillator

The spin-orbit interactions (SOI) for the single and double ring-shaped oscillator potentials are studied as an energy correction to the Schrödinger equation. We find that the degeneracy for the energy levels with angular quantum number m=0 keeps invariant in the case of the SOI. The degeneracy is still 2 for single ring-shaped potential and 4 for double ring-shaped potential. However, for the energy levels with angular quantum number m≠0 the degeneracy is reduced from original 4 for the single ring-shaped potential and 8 for the double ring-shaped potential to 2. That is, their energy levels in the case of the SOI are split to 2 (single) and 4 (double) sublevels. There exists an accidental degeneracy for the cases |m|=2, 3, 4, . . .. We note that around the critical value b₀, the energy levels are reversed. We also discuss some special cases for η=2, 3, 4, 5, 6, . . ., and the b=0, c>0. It should be pointed out that the parameter b₀ is relevant for the angular part parameter b in the single and double ring-shaped potentials and it makes the energy levels changed from positive to negative, but the parameter c corresponds to the angular part parameter in double ring-shaped potential and the η is related to it. This model can be useful for investigations of axial symmetric subjects like the ring-shaped molecules or related problems and may also be easily extended to a many-electron theory.