Exactly complete solutions of the Coulomb potential plus a new ring-shaped potential

A new exactly solvable ring-shaped potential is proposed. The energy eigenvalues and eigenfunctions of the bound states for the Schrödinger equation with this potential are presented analytically. The exact solutions of the continuous states of this quantum system are also obtained. The calculation formula of phase shifts is derived. Analytical properties of the scattering amplitude are discussed. We find that the energy of the particle and the radial wave functions of continuous states reduce to the energy eigenvalues and the radial wave functions of the bound states at the poles of the scattering amplitude.