Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions

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.