A nonrelativistic quantum field theory with point interactions in three dimensions

TL;DR

This paper develops a rigorous Hamiltonian model for nonrelativistic particles in three dimensions interacting through point interactions involving bosonic particle creation and annihilation, with a focus on self-adjointness and boundary conditions.

Contribution

It introduces a mathematically rigorous construction of the Hamiltonian for a quantum field theory with point interactions in three dimensions, including domain and boundary condition definitions.

Findings

Defined the Hamiltonian operator with point interactions.

Established self-adjointness via boundary conditions.

Provided a framework for analyzing particle interactions at points.

Abstract

We construct a Hamiltonian for a quantum-mechanical model of nonrelativistic particles in three dimensions interacting via the creation and annihilation of a second type of nonrelativistic particles, which are bosons. The interaction between the two types of particles is a point interaction concentrated on the points in configuration space where the positions of two different particles coincide. We define the operator, and its domain of self-adjointness, in terms of co-dimension-three boundary conditions on the set of collision configurations relating sectors with different numbers of particles.