Efficient approach to determining the eigenenergies and eigenstates of three and four identical particles under external harmonic confinement

At ultralow temperatures, low-energy properties are captured by a zero-range interaction that is characterized by one or two atomic physics parameters. This work takes advantage of this simplification and considers identical particles (bosons or fermions) with two-body zero-range interactions under one-dimensional external harmonic confinement. Building on earlier work (S.E. Gharashi and D. Blume, Phys. Rev. Lett. 111, 045302 (2013)), we solve the Lippmann-Schwinger equation for three- and four-particle systems. We show that the use of recursive relationships (P.Q. Son et al., arXiv.2411.15541) significantly reduces the computational cost of evaluating relevant matrix elements, without compromising accuracy or stability. Possible extensions of the algorithm to higher-dimensional systems will be discussed. We will also report on our progress of leveraging the tools developed to describe one-dimensional few-particle systems with generalized exchange statistics.