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 $\tilde{V}(k)$ of the interaction potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\tilde{g}}_0$ and ${\tilde{g}}_2$ of $\tilde{V}(k)={\tilde{g}}_0+{\tilde{g}}_2{k}^2+...$ 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 2D uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.

8 pages, 2 figures, 1 table, accepted for publication in Phys. Rev. A

References in corpus (12)

• Extension of Bogoliubov theory to quasi-condensates
• Ground state energy of the two-dimensional weakly interacting Bose gas: First correction beyond Bogoliubov theory
• Energy-dependent effective interactions for dilute many-body systems
• A simple model for interactions and corrections to the Gross-Pitaevskii Equation
• Low-energy potential scattering in two and three dimensions
• Non-Universal Equation of State of the Two-Dimensional Bose Gas
• Ground state energy of the interacting Bose gas in two dimensions: an explicit construction
• Condensation and superfluidity of dilute Bose gases with finite-range interaction
• Finite-Range Corrections to the Thermodynamics of the One-Dimensional Bose Gas
• Zero-temperature equation of state of a two-dimensional bosonic quantum fluid with finite-range interaction
• Effective-range signatures in quasi-1D matter waves: sound velocity and solitons
• Low-energy scattering parameters: A theoretical derivation of the effective range and scattering length for arbitrary angular momentum