Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics

The exact solutions of the Schrödinger equation with the double ring-shaped Coulomb potential are presented, including the bound states, continuous states on the “κ/2π scale”, and the calculation formula of the phase shifts. The polar angular wave functions are expressed by constructing the so-called super-universal associated Legendre polynomials. Some special cases are discussed in detail.