Perturbative correction to the ground state properties of one-dimensional strongly interacting bosons in a harmonic trap

We calculate the first-order perturbation correction to the ground state energy and chemical potential of a harmonically trapped boson gas with contact interactions about the infinite repulsion Tonks-Girardeau limit. With $c$ denoting the interaction strength, we find that for a large number of particles $N$ the $1\mathrm{/}c$ correction to the ground state energy increases as $N^{5\mathrm{/}2}$, in contrast to the unperturbed Tonks-Girardeau value that is proportional to $N^2$. We describe a thermodynamic scaling limit for the trapping frequency that yields an extensive ground state energy and reproduces the zero temperature thermodynamics obtained by a local density approximation.