n Distinguishable Particles on the Real Line interacting via Two Body Delta Potentials

TL;DR

This paper analyzes a quantum system of distinguishable particles on the real line with delta interactions, establishing the algebraic structure of its Hamiltonian and the existence of a stable dynamical subalgebra.

Contribution

It proves the Hamiltonian's affiliation to a resolvent algebra and constructs a stable C*-dynamical system for the particle interactions.

Findings

Hamiltonian affiliated to resolvent algebra

Existence of a stable C*-dynamical system

Identification of a time-evolution invariant subalgebra

Abstract

This paper studies a system of $n \in \mathbb{N}: \, n \geq 2$ non-relativistic, spinless quantum particles moving on the real line and interacting via a two-body delta potential. The Hamiltonian of such a system is proved to be affiliated to the resolvent algebra of the case, $\mathcal{R}\left( \mathbb{R}^{2n},\sigma \right)$; it is further shown the existence of a $\text{C}^{\ast}-$dynamical system and of a subalgebra $\pi_S\left( \mathfrak{S}_0 \right)^{-1} \subset \mathcal{R}\left( \mathbb{R}^{2n},\sigma \right)$, stable under time evolution, where $\pi_S$ is the Schr{\"o}dinger representation of the algebra.