Mathematics

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 Kreĭn,
Višiik, and Birman. We identify the explicit `Kreĭn-Višik-Birman extension param-
eter' as an operator on the `space of charges' for this model (the `Kreĭn 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 re-
produce for the first time the previous partial constructions obtained by means of
an alternative quadratic form approach.