The interaction-sensitive states of a trapped two-component ideal Fermi gas and application to the virial expansion of the unitary Fermi gas

We consider a two-component ideal Fermi gas in an isotropic harmonic potential. Some eigenstates have a wavefunction that vanishes when two distinguishable fermions are at the same location, and would be unaffected by s-wave contact interactions between the two components. We determine the other, interaction-sensitive eigenstates, using a Faddeev ansatz. This problem is nontrivial, due to degeneracies and to the existence of unphysical Faddeev solutions. As an application we present a new conjecture for the fourth-order cluster or virial coefficient of the unitary Fermi gas, in good agreement with the numerical results of Blume and coworkers.