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

32 pages, augmented version: with respect to published version, paragraph added at the end of section 5.3 and figure 3 added (comparison of our conjecture for the fourth cluster coefficient of the unitary Fermi gas to the Monte Carlo diagrammatic results of K. van Houcke et al, Nature Physics 8, 366 (2012))International audienceWe 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