Spin–orbit interaction for the double ring-shaped oscillator

•Spin–orbit interaction (SOI) for double ring-shaped potential is studied.•Effect of the SOI on the energy spectrum is discussed.•The degeneracy is broken by SOI.•Around the critical value b0b0, the energy levels are reversed.

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=0m=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≠0m≠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,…|m|=2,3,4,…. We note that around the critical value b0b0, the energy levels are reversed.   We also discuss some special cases for η=2,3,4,5,6,…η=2,3,4,5,6,…, and the b=0,c>0b=0,c>0. It should be pointed out that the parameter b0b0 is relevant for the angular part parameter bb in the single and double ring-shaped potentials and it makes the energy levels changed from positive to negative, but the parameter cc corresponds to the angular part parameter in double ring-shaped potential and the ηη 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.