Mathieu function solutions to the radial Schrödinger equation for the −f2/r4 interaction

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 δ_l decomposes into two parts: δ_l = γ_l + ρ_l. The ``polarizability phase shift'' ρ<sub>l</sub> depends solely on parameters of the interaction and is easily calculated. The ``polarizability extracted phase shift'' γ_l 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'' $r_t\mathrm{=\surd f/k}$ For r > r_t, the functions resemble spherical waves with phase shift ρ_l with respect to the free particle wavefunctions; while for r < r_t they deviate strongly from spherical wave forms. The relationship of r_t with respect to the atomic radius grossly determines the behavior of the scattering phase shifts.