Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities

In one of the very few exact quantum mechanical calculations of fugacity coefficients, [L.R. Dodd, A.M. Gibbs. J. Math. Phys. 15 (1974) 41] obtained b2 and b3 for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For b2, we have shown [A. Amaya-Tapia, S.Y. Larsen, M. Lassaut. Mol. Phys. 103 (2005) 1301–1306. ] that Dodd and Gibbs’ result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms, usually contributing only in one dimension. Now, we develop an exact expression for b3-b30 (where b30 is the free particle fugacity coefficient) in terms of sums and differences of three-body eigenphase shifts. Further, we show that if we obtain these eigenphase shifts in a Distorted-Born approximation, then, to first order, we reproduce the leading low temperature behaviour, obtained from an expansion of the twofold integral of Dodd and Gibbs. The contributions of the oscillating terms cancel.The formalism that we propose is not limited to one dimension, but seeks to provide a general method to obtain virial coefficients, fugacity coefficients, in terms of asymptotic quantities. The exact one dimensional results allow us to confirm the validity of our approach in this domain.