Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

TL;DR

This paper constructs a self-adjoint, bounded Hamiltonian for three bosons in three dimensions with zero-range interactions, avoiding the Thomas effect through a regularization involving an effective three-body force.

Contribution

It develops a regularized Hamiltonian for three-boson systems with zero-range forces, ensuring mathematical well-posedness and physical relevance, extending previous models.

Findings

The Hamiltonian is self-adjoint and bounded from below.

The domain and action of the Hamiltonian are explicitly characterized.

The Hamiltonian is the norm resolvent limit of rescaled non-local interaction Hamiltonians.

Abstract

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan--Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.