Some remarks on effective range formulae in potential scattering

In this paper, we consider the radial Schrodinger equation with a real-valued potential having spherical symmetry, and we present alternative proofs of very recent results on the necessary, as well as sufficient, conditions on the decrease of the potential at infinity for the validity of effective range formulae in 3D in low energy potential scattering (Khuri et al 2008 arXiv:0812.4054v1. See theorem 1 below. This paper also contains a careful study of the 2D case, with some amazing new results). Our proofs are based on compact formulae for the phase shifts. The sufficiency conditions have been well known for a long time. But the necessity of the same conditions for potentials keeping a constant sign at large distances are new. All these conditions are established here for dimension 3 and for all angular momenta > = 0.