Low-Energy Atomic Scattering: S-Wave Relation Between the Interaction Potential and the Phase Shift

Year: 2025

Authors: Lorenzi F., Salasnich L.

Autors Affiliation: Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via Marzolo 8, I-35131 Padua, Italy; Ist Nazl Fis Nucleare INFN, Sez Padova, Via Marzolo 8, I-35131 Padua, Italy; CNR, Ist Nazl Ott INO, Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy.

Abstract: The on-shell approximation is investigated in the context of s-wave scattering for ultracold two-body collisions. This analysis systematically covers spatial dimensions D=1,2,3$D=1,2,3$, with the aim of identifying the regimes in which the approximation remains valid when applied to commonly used model interaction potentials. Specifically, the square well and delta shell potentials are focused, 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, a direct comparison is performed 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. These 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, it is demonstrated 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.

Journal/Review: ANNALEN DER PHYSIK

More Information: L.S. acknowledges the BIRD Project Ultra-cold atoms in curved geometries of the University of Padova. L.S. was partially supported by the European Union NextGenerationEU within the National Center for HPC, Big Data, and Quantum Computing (Project No. CN00000013, CN1 Spoke 10: Quantum Computing) and by the European Quantum Flagship Project PASQuanS 2. L.S. acknowledges Iniziativa Specifica Quantum of Istituto Nazionale di Fisica Nucleare, the Project Frontiere Quantistiche within the 2023 funding programme ’Dipartimenti di Eccellenza’ of the Italian Ministry for Universities and Research, and the PRIN 11 2022 Project Quantum Atomic Mixtures: Droplets, Topological Structures, and Vortices.
KeyWords: scattering theory; two-body physics; ultracold atoms
DOI: 10.1002/andp.202500227