Mathieu function solutions to the radial Schrödinger equation for the − f 2/ r 4 interaction

Source abstract

We describe the Mathieu function solutions to the radial Schrödinger equation for the −f2/r4 potential with reference to adiabatic elastic scattering of electrons from neutral atoms. By an appropriate choice of boundary conditions, the total scattering phase shift δl decomposes into two parts: δl = γl + ρl. The ``polarizability phase shift'' ρl depends solely on parameters of the interaction and is easily calculated. The ``polarizability extracted phase shift'' γl contains information of the core interaction. We suggest that such a parametrization of this scattering problem is convenient for data analysis or for potential scattering calculations. We find that the Mathieu function solutions are characterized by a ``polarizability range'' rt=√f/k For r > rt, the functions resemble spherical waves with phase shift ρl with respect to the free particle wavefunctions; while for r < rt they deviate strongly from spherical wave forms. The relationship of rt with respect to the atomic radius grossly determines the behavior of the scattering phase shifts.