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 [Formula: see text] and the so-called 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 [Formula: see text] is closed and bounded from below. As a consequence, [Formula: see text] 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 [Formula: see text] 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.

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