Study of spin-orbit interaction for the Makarov potential

We study the spin-orbit interaction (SOI) for the Makarov potential to correct the nonrelativistic Schrödinger energy levels and discuss the degeneracy of the energy levels. For a certain principal quantum number n, when we do not consider the spin the degeneracy of the energy levels with magnetic quantum number m = 0 is n, while the degeneracy with the magnetic quantum number m ≠ 0 becomes 2(n − |m|). However, when we consider the spin s = 1/2 the degeneracy of the energy levels with magnetic quantum number m = 0 is 2n, while the degeneracy with the magnetic quantum number m ≠ 0 is 4(n − |m|). After taking SOI into account, the degeneracy in this case is still 2n since the energy levels with the magnetic quantum number m = 0 are not split, but the energy levels with the magnetic quantum number m ≠ 0 will be split into 2(n − |m|) energy levels, each of which has the degeneracy 2. The size and sequence of the energy level splitting are relevant for the potential parameters β and γ except for depending on those quantum numbers n, l, and m. We find that the degenerate energy levels are reversed at the critical values by considering the SOI effect.