  An upper bound is derived for $\Delta$ for a cold dilute fluid of equal
amounts of two species of fermion in the unitary regime $k_f a \to \infty$
(where $k_f$ is the Fermi momentum and $a$ the scattering length, and $\Delta$
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 $\delta \leq
{5/3} (2 (2 \xi)^{2/5} - (2 \xi))$ where $\xi=\epsilon/\epsilon_{\rm FG}$,
$\delta= 2 \Delta/\epsilon_{\rm FG}$; $\epsilon$ is the energy per particle and
$\epsilon_{\rm FG}$ 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 $\Delta$ is the usual superfluid gap. If the superfluid gap
exceeds the maximum allowed by the inequality phase separation occurs.
