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.