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 similar to(k) of the interaction potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters g similar to 0 and g similar to 2 of V similar to(k) = g similar to 0 + g similar to 2k2 + center dot center dot center dot 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.