Ground State Energy of Dilute Bose Gases in 1D

We study the ground state energy of a gas of 1D bosons with density \(\rho \), interacting through a general, repulsive 2-body potential with scattering length a, in the dilute limit \(\rho |a|\ll 1\). The first terms in the expansion of the thermodynamic energy density are \((\pi ^2\rho ^3/3)(1+2\rho a)\), where the leading order is the 1D free Fermi gas. This result covers the Tonks–Girardeau limit of the Lieb–Liniger model as a special case, but given the possibility that \(a>0\), it also applies to potentials that differ significantly from a delta function. We include extensions to spinless fermions and 1D anyonic symmetries, and discuss an application to confined 3D gases.