Threshold behavior of the Jost function for the atomic polarization potential

We evaluate the l-wave Jost function for the polarization potential and calculate its effective range expansion for l=0 up to the order k⁴. We investigate not only the phase but also the modulus of the Jost function, generalizing previous works for the expansion up to the order k² of the s-wave phase shift. We discuss the validity domain of this generalized expansion for the interaction of an electron with a Helium atom in any of its three long-lived He states, i.e., 1¹S, 2¹S, and 2³S. Even though this expansion is shown to be appropriate for most of the applications in electron-atom collisions, we propose here a more precise and simpler approach that provides much better results in a broader energy range when comparing with elaborated ab initio calculations.