An upper bound is derived for $\ensuremath{\Delta}$ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit ${k}_{f}a\ensuremath{\rightarrow}\ensuremath{\infty}$ (where
kf is the Fermi momentum,
a is the scattering length, and $\ensuremath{\Delta}$ is a pairing energy: the difference in energy per particle between adding to the system a macroscopic number (but infinitesimal fraction) of particles of one species compared to adding equal numbers of both. The bound is $\ensuremath{\delta}\ensuremath{\le}\frac{5}{3}[2(2\ensuremath{\xi}{)}^{2/5}\ensuremath{-}(2\ensuremath{\xi})]$ where $\ensuremath{\xi}=ϵ/{ϵ}_{\mathrm{FG}}$, $\ensuremath{\delta}=2\ensuremath{\Delta}/{ϵ}_{\mathrm{FG}}$;
ϵ is the energy per particle and
ϵFG is the energy per particle of a noninteracting Fermi gas. If the bound is saturated, then systems with unequal densities of the two species will separate spatially into a superfluid phase with equal numbers of the two species and a normal phase with the excess. If the bound is not saturated, then $\ensuremath{\Delta}$ is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.
The pseudopotential theory is extended to the Bogoliubov-de Gennes equations to determine the excess energy when one atom is added to the trapped superfluid Fermi system with even number of atoms. Particular attention is paid to systems being at the Feshbach resonance point. The results for relatively small particle numbers are in harmony with the Monte Carlo calculations, but are also relevant for systems with larger particle numbers. Concerning the additional one-quasiparticle state we define and determine two new universal numbers to characterize its widths.