On the quantum mechanical three-body problem with zero-range interactions

TL;DR

This paper reviews the quantum three-body problem with zero-range interactions in three dimensions, focusing on the construction of the Hamiltonian as a self-adjoint operator for bosons and fermions, and proving self-adjointness under certain restrictions.

Contribution

It provides a comprehensive review of the current methods for constructing the Hamiltonian and introduces a quadratic form approach to establish self-adjointness for three identical bosons with specific restrictions.

Findings

Constructed the Hamiltonian as a self-adjoint operator for the three-body problem.

Proved self-adjointness and boundedness from below for three identical bosons with restrictions.

Reviewed existing methods for zero-range interactions in quantum three-body systems.

Abstract

In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave" subspace.