We present the exact solutions of the angular Teukolsky equation with m ≠ 0 given by a confluent Heun function. This equation is first transformed to a confluent Heun differential equation through some variable transformations. The Wronskian determinant, which is constructed by two linearly dependent solutions, is used to calculate the eigenvalues precisely. The normalized eigenfunctions can be obtained by substituting the calculated eigenvalues into the unnormalized eigenfunctions. The relations among the linearly dependent eigenfunctions are also discussed. When c2=cR2+i0.5emcI2, the eigenvalues are approximately expressed as Alm≈l()l+1+()cR2+i0.12emcI20.12em[]1−m2/l()l+1/2 for small |c|2 but large l. The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c2. It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c2, which is real or imaginary.

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented.The visualizations of the space probability distribution (SPD) are illustrated for the two-(contour) and three-dimensional (isosurface) cases. The quantum numbers (n, l, m) are mainly relevant for those quasi quantum numbers ( ' n , ' l , ' m ) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n, l, m)=(6, 5, 0), the SPD moves towards two poles of axis z when the P increases. Finally, we discuss the different cases for the potential parameter b which is taken as negative and positive values for 0 c > . Compared with the particular case =0 b , the SPDs are shrunk for 0.5 b = − while spread out for 0.5 b = .

A few important integrals involving the product of two universal associated Legendre polynomials , and x2a(1 − x2)−p−1, xb(1 ± x)−p−1 and xc(1 −x2)−p−1 (1 ± x) are evaluated using the operator form of Taylor’s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e. l′ ≠ k′ and m′ ≠ n′. Their selection rules are also given. We also verify the correctness of those integral formulas numerically.

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.