  The probability amplitude for $N$ particles in a quantum gas with negligible
range of interparticle interaction potentials to come to a small region of size
$r$ scales like $r^\gamma$. It is shown that $\gamma$ is quantitatively related
to the ground state energy of these $N$ fermions in the unitarity limit,
confined by an isotropic harmonic potential. For large $N$, the short range
density distribution of these $N$ particles is predominantly the same as the
Thomas-Fermi profile of the gas in the unitarity limit confined by such a
harmonic potential. These results may shed light on strongly interacting
ultracold atomic Fermi gases, in a trap or on an optical lattice.
