Irregular solutions of the s-wave radial Schrödinger equation for van der Waals potentials

Abstract

Simple ideas based on quantum chromodynamics suggest that the strong nuclear force might possess a long-range van der Waals component. Such a component, if it exists, would be most apparent in low-energy interactions between pairs of hadrons, at least one of which is neutral. In this article, as the initial part of a program to set limits on the magnitude of the strength of such a van der Waals interaction, we construct low-energy series expansions of the two irregular solutions f0(±k,r) to the s-wave Schrödinger equation that describes the interaction of a pair of particles interacting through a potential with a van der Waals tail of the form V(r)∼−λℏcrN−10 /rN, for arbitrary values of the potential parameters, λ, r0, and N. Accurate approximation schemes based on the series expansions are developed, and the results are used to obtain an expression for the low-energy elastic cross section. The effects of the van der Waals component of this interaction are clearly evident in the energy dependence of the cross section.