Inversion transformation in the Schrödinger equation

The equivalence of Schrödinger equations with different types of potentials is established on the basis of an inversion transformation. It is shown that at zero energy the equation with a spherical potential is equivalent to the equation with an axisymmetric potential of special form. Exact solutions are obtained to problems of the motion of a particle with zero total energy in the field of two “fish-eye” potentials and also in the field of a potential with two Coulomb singularities.