Low-Temperature Behavior for the Quantum Virial Coefficients

The low-temperature behavior of all the quantum cluster integrals $b_l$ is calculated by using a perturbation scheme in which a hyperspherical representation for the many-body problem is used. Definitions of the usual equations for the $T$, $R$, and $S$ scattering matrices as well as the many-body eigenphase shift are given. The technique for calculating the energy eigenvalues is outlined and a relationship between the level shift and the eigenphase shift is given. It is found that for an everywhere-finite short-ranged potential without bound states, $b_l$ behaves as a polynomial in $\lambda =\frac{h}{{\left(2\mathrm{}\pi mkT\right)}^{\frac{1}{2}}}$ in the low-temperature limit.