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.