Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan-- Skornyakov type

Abstract

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.