From Lippmann-Schwinger formulations to a general formula for absolute asymptotic scattering phase functions and shifts: a unified framework for potentials of any range

Abstract

Appropriately handling the Lippmann-Schwinger equation can bring substantial benefits in quantum scattering. Here, the advantages of a parametric equation for the system's normalised wavefunction, ψ, supplied with regular $f^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and irregular $g^{\mathrm{r}\mathrm{e}\mathrm{f}}$ wavefunctions of a reference Hamiltonian for arbitrary $V^{\mathrm{r}\mathrm{e}\mathrm{f}}=U^{\mathrm{e}\mathrm{f}\mathrm{f}}-V$ subsystems are discussed to extend previous results [H. Piel and M. Chrysos, Mol. Phys., 116, 2364 (2018)] to any interaction-range for the effective $U^{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ‘excess’ V potentials. While the idea of using reference wavefunctions to solve scattering problems is not new [J. P. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998); B. P. Ruzic, C. H. Greene, and J. L. Bohn, Phys. Rev. A 87, 032706 (2013)], convincing evidence that a wide variety of problems can benefit from the careful choice of $f^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and $g^{\mathrm{r}\mathrm{e}\mathrm{f}}$ and the demonstration of how their occurrence in a compact formula permits the absolute definition of asymptotic scattering phases are among our highlights. Various examples employing Coulomb-van-der-Waals interactions in helium, along with an insightful rephrasing of ideas originally introduced in quantum defect theory for ultracold alkali-alkali collisions are given, adding to the recognition of these works and further promoting the need for properly selected reference wavefunctions in scattering calculations.

GRAPHICAL ABSTRACT

Keywords:

• Elastic scattering
• atomic collisions
• helium
• van der Waals interactions
• Coulomb interactions

Acknowledgements

We wish to acknowledge thesis fellowship support from the French Ministry of Education to one of us (HP).

Disclosure statement

No potential conflict of interest was reported by the authors.