Low-energy atomic scattering: s-wave relation between the interaction potential and the phase shift

TL;DR

The paper investigates the validity of the on-shell (OS) approximation for s-wave scattering across dimensions $D=1,2,3$ by obtaining exact analytical phase shifts for square-well and delta-shell potentials and directly comparing the exact on-shell potential $V_0(k)$ with the OS estimate $V_0(k) \overset{\mathrm{OS}}{=} - \dfrac{2 \hbar^2}{m} k \tan(\delta_0(k))$. It shows that OS accuracy improves with increasing momentum and becomes exact at leading order in the weak-potential limit $\gamma \to 0$, with dimension-dependent deviations at low momentum. The leading-order OS behavior reproduces the linear dependence on the potential strength and provides a practical route to the low-momentum expansion of the Fourier-transformed potential, relevant for constructing effective interactions in quantum-field-theoretic treatments of ultracold gases. These results underscore the OS framework as a controlled approximation for low-energy scattering and EFT matching, and point to future work extending the approach to finite-temperature and many-body settings.

Abstract

We investigate the on-shell approximation in the context of s-wave scattering for ultracold two-body collisions. Our analysis systematically covers spatial dimensions D=1,2,3 , with the aim of identifying the regimes in which the approximation remains valid when applied to commonly used model interaction potentials. Specifically, we focus on the square well and delta shell potentials, both of which admit analytical solutions for the s-wave scattering problem in all dimensions considered. By employing the exact analytical expressions for the s-wave scattering phase shift, we perform a direct comparison between the exact on-shell matrix element of the interaction potential and their corresponding approximations across a range of collision momenta. Particular attention is given to the low-energy regime. Our findings indicate that, although the on-shell approximation generally improves with increasing momentum, its accuracy also improves for weaker potentials. Remarkably, in the limit of weak interactions, we demonstrate that the on-shell approximation becomes exact at leading order. In this regime, the approximation offers a controlled means of deriving the low-momentum expansion of the potential and may serve as a useful tool in constructing effective interactions for quantum field theories.

Figures (4)

• Figure 1: S-wave component of the potential as a function of the scattering energy $k$. $V_0(k)$ is in units of $\hbar^2 R^{D-2}/m$, $\gamma$ is in units of $\hbar^2/(mR^2)$, and $k$ is in units of $1/R$. The regularizer for the $D=2$ case is chosen as $\Lambda = 1/R$. The solid line is the exact s-wave component, Eq. (\ref{['eq:swave-def']}), and the dashed line is the approximation obtained with the on-shell Eq. (\ref{['eq:osa']}).
• Figure 2: Relative error $\varepsilon_{R, 0}$ and $\varepsilon_{R, 2}$ on the approximation of the low-momentum expansion coefficients of the Fourier transform using the on-shell approximation. $\varepsilon_{R, 0}$ is the relative error on the value of $\tilde{g}_0=g_0$, and $\varepsilon_{R, 2}$ is the relative error on $\tilde{g}_2=g_2/2$. $\gamma$ is in units of $\hbar^2/(mR^2)$. The regularizer for the $D=2$ case is chosen as $\Lambda = 1/R$.
• Figure 3: On-shell s-wave component of the potential as a function of the scattering energy $k$. $V_0(k)$ is in units of $\hbar^2R^{D-2}/m$, $\gamma$ is in units of $\hbar^2/(mR)$, and $k$ is in units of $1/R$. The regularizer for the $D=2$ case is chosen as $\Lambda = 1/R$. The solid line is the exact s-wave component $V_0(k)$, and the dashed line is the approximation obtained with the on-shell approximation (\ref{['eq:osa']}).
• Figure 4: Relative error $\varepsilon_{R, 0}$ and $\varepsilon_{R, 2}$ on the approximation of the low-momentum expansion coefficients of the Fourier transform using the on-shell approximation. $\varepsilon_{R, 0}$ is the relative error on the value of $\tilde{g}_0=g_0$, and $\varepsilon_{R, 2}$ is the relative error on $\tilde{g}_2=g_2/2$. Parameters and units are the same as in Fig. \ref{['fig:swe']}.