  We prove that a system of $N$ fermions interacting with an additional
particle via point interactions is stable if the ratio of the mass of the
additional particle to the one of the fermions is larger than some critical
$m^*$. The value of $m^*$ is independent of $N$ and turns out to be less than
$1$. This fact has important implications for the stability of the unitary
Fermi gas. We also characterize the domain of the Hamiltonian of this model,
and establish the validity of the Tan relations for all wave functions in the
domain.
