The quantum characteristics of a class of complicated double ring-shaped non-central potential

Firstly we outline how to obtain the normalized polar angular wave functions of the Schrödinger equation with a class of complicated double ring-shaped non-central potential by introducing the super-universal associated Legendre polynomials, and present the exact energy equation and normalized radial wave functions for the given central potentials, such as the harmonic oscillator and Coulomb potential. We then discuss in detail the quantum characteristics of the polar and radial wave functions and energy equation, and their reducing problems. These features include bound state relations between the harmonic oscillator and Coulomb potential quantum systems, and degeneracy of their energy levels. We also observe that the bound state relations, including the energy levels and wave functions for the harmonic oscillator and Coulomb potential systems, can be obtained from each other by mapping the system parameters. Special cases related to reduction questions are discussed in detail.