  We study the stability problem for a non-relativistic quantum system in
dimension three composed by $ N \geq 2 $ identical fermions, with unit mass,
interacting with a different particle, with mass $ m $, via a zero-range
interaction of strength $ \alpha \in \R $. We construct the corresponding
renormalised quadratic (or energy) form $ \form $ and the so-called
Skornyakov-Ter-Martirosyan symmetric extension $ H_{\alpha} $, which is the
natural candidate as Hamiltonian of the system. We find a value of the mass $
m^*(N) $ such that for $ m > m^*(N)$ the form $ \form $ is closed and bounded
from below. As a consequence, $ \form $ defines a unique self-adjoint and
bounded from below extension of $ H_{\alpha}$ and therefore the system is
stable. On the other hand, we also show that the form $ \form $ is unbounded
from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this
suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas
effect occurs.
