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 alpha is an element of R. We construct the corresponding renormalized quadratic (or energy) form F-alpha 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 F-alpha is closed and bounded from below. As a consequence, F-alpha 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 F-alpha 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.