The three-body scattering hypervolume of identical fermions in one dimension

TL;DR

This paper investigates the three-body scattering hypervolume for identical fermions in one dimension, deriving asymptotic wave function expansions, calculating hypervolume values for specific potentials, and exploring implications for energy, pressure, and recombination rates.

Contribution

It introduces a detailed analysis of the three-body scattering hypervolume in 1D fermions, including derivations, numerical calculations, and physical implications, which is novel in this context.

Findings

Derived asymptotic expansions of the three-body wave function.

Computed the hypervolume for specific potentials like square-well and Gaussian.

Quantified effects on energy, pressure, and recombination rates in 1D Fermi gases.

Abstract

We study the zero-energy collision of three identical spin-polarized fermions with short-range interactions in one dimension. We derive the asymptotic expansions of the three-body wave function when the three fermions are far apart or one pair and the third fermion are far apart, and the three-body scattering hypervolume $D_F$ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, $D_F$ acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schr\"{o}dinger equation to compute $D_F$ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero $D_F$ and the three-body recombination rate in one dimension.