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 ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ R. We construct the corresponding renormalized quadratic (or energy) form F_α and the socalled Skornyakov–Ter–Martirosyan symmetric extension H_α, 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 F_α is closed and bounded from below. As a consequence, F_α defines a unique self-adjoint and bounded from below extension of H_α and therefore the system is stable. On the other hand, we also show that the form F_α 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.