On-shell approximation for the s -wave scattering theory

We investigate the scattering theory of two particles in a generic D-dimensional space. For the s-wave problem, by adopting an on-shell approximation for the T-matrix equation, we derive analytical formulas which connect the Fourier transform V (k) of the interaction potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters g0 and g2 of V(k)=g0+g2k2+» in terms of the s-wave scattering length as and the s-wave effective range rs for D=3, D=2, and D=1. Our results, which are strongly dependent on the spatial dimension D, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.