  For quantum systems of zero-range interaction we discuss the mathematical
scheme within which modelling the two-body interaction by means of the
physically relevant ultra-violet asymptotics known as the
"Ter-Martirosyan-Skornyakov condition" gives rise to a self-adjoint realisation
of the corresponding Hamiltonian. This is done within the self-adjoint
extension scheme of Krein, Visik, and Birman. We show that the
Ter-Martirosyan-Skornyakov asymptotics is a condition of self-adjointness only
when is imposed in suitable functional spaces, and not just as a pointwise
asymptotics, and we discuss the consequences of this fact on a model of two
identical fermions and a third particle of different nature.
