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 ∈ . We construct the corresponding renormalized quadratic (or energy) form $mathcal{F}-$ and the so-called SkornyakovTerMartirosyan 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 $mathcal{F}- $ is closed and bounded from below. As a consequence, $mathcal{F}-$ 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 $mathcal{F}-$ 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. © 2012 World Scientific Publishing Company.