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

TL;DR

This paper classifies all self-adjoint Hamiltonians for a three-particle fermionic system with zero-range interactions, revealing multiple solutions depending on mass ratios, using an operator extension framework.

Contribution

It provides a complete characterization of self-adjoint extensions for the (2+1)-fermionic model via an operator-theoretic approach, connecting to previous partial results.

Findings

Identification of the extension parameter as an operator on the charge space

Conjecture on the kernel's dimensionality affecting extension multiplicity

Reproduction of earlier partial constructions through a new framework

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.