  We reconstruct the whole family of self-adjoint Hamiltonians of
Ter-Martirosyan-- Skornyakov type for a system of two identical fermions
coupled with a third particle of different nature through an interaction of
zero range. We proceed through an operator-theoretic approach based on the
self-adjoint extension theory of Krein, Visik, and Birman. We identify the
explicit Krein-Visik-Birman extension parameter as an operator on the "space of
charges" for this model (the "Krein space") and we come to formulate a sharp
conjecture on the dimensionality of its kernel. Based on our conjecture, for
which we also discuss an amount of evidence, we explain the emergence of a
multiplicity of extensions in a suitable regime of masses and we reproduce for
the first time the previous partial constructions obtained by means of an
alternative quadratic form approach.
