Method of obtaining a closed-form exact 3D scattering solution of the Schrodinger equation

The aim of the present paper is to obtain a closed-form exact 3D scattering solution of the stationary Schrodinger equation. A method to construct a pair of isometric orthogonal similarity variables (u, v) is developed. Using these variables a family of potentials is found for which the corresponding similarity 3D scattering wave function is constructed. The method is illustrated by an example that preserves all the features of the general case. The studied example provides the only known 3D potential with a well followed by a barrier for which the exact scattering wave function is constructed. This is very important because the most potentials used in nuclear, atomic and molecular physics are of this form. The scattering amplitude and all its poles are determined. The poles correspond to the natural modes of a system of a particle interacting with the potential. It is shown that the exact 3D scattering solution for the chosen example coincides with its uniform asymptotic approximation devoted to remove the singularities of the 3D WKB approximation of the scattering wave function. The dependence of the scattering amplitude and of its poles on the parameters of the potential is stressed.