On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians

TL;DR

This paper rigorously analyzes the mathematical foundations of zero-range quantum interactions modeled by Ter-Martirosyan-Skornyakov conditions, emphasizing the importance of functional space context for self-adjointness.

Contribution

It clarifies the conditions under which Ter-Martirosyan-Skornyakov asymptotics lead to self-adjoint Hamiltonians using Krein-Vishik-Birman extension theory.

Findings

Ter-Martirosyan-Skornyakov asymptotics require suitable functional spaces for self-adjointness

The scheme applies to models with two identical fermions and a third particle

Implications for modeling zero-range quantum interactions

Abstract

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.