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|<< 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.