Semi-Analytical Hyperspherical Potential Curves of Four Particles with Zero-Range Interactions

In few-body systems, hyperspherical adiabatic potentials encapsulate critical information about ultracold behavior and enable the description of a wide range of quantum phenomena [1]. In this work, we systematically investigate the four-body problem with s-wave zero-range interactions using the adiabatic hyperspherical formalism [2]. The hyperangular adiabatic Schrödinger equation is solved by constructing the relevant Green’s function and employing it within the corresponding Lippmann-Schwinger equation. The Bethe-Peierls two-body boundary condition is applied at pairwise coalescence points, ensuring the appropriate short-range behavior of the wave function. Currently, we are working towards deriving a transcendental equation that can be numerically solved for the hyperspherical potential curves in the zero-range limit. These four-body potentials can serve as the starting point for analyzing universal threshold scaling laws, bound and resonant structures, and scattering processes in ultracold atomic and molecular ensembles.

[1] C. H. Greene, P. Giannakeas, and J. Pérez-Ríos, Rev. Mod. Phys. 89, 035006 (2017).

[2] S. T. Rittenhouse, N. P. Mehta, and C. H. Greene, Phys. Rev. A 82, 022706 (2010).