On point-like interaction between $n$ fermions and another particle

In this note the point-like interaction of $n$ fermions with a particle of a different nature is considered in a framework of the theory of self-adjoint extensions of symmetric operators. It introduces the family of extensions of the original symmetric operator, most natural from the physical point of view (the so-called Ter-Martirosian–Skornyakov' extensions). Here we prove that for $n\leqslant4$ and large enough values of the mass of the separate particle these extensions are self-adjoint and bounded from below.