On-shell approximation for the s -wave scattering theory

Abstract

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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the $s$-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the $s$-wave scattering length ${a}_{s}$ and the $s$-wave effective range ${r}_{s}$ 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.