One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles

TL;DR

This paper classifies the symmetries of few-body quantum systems in one-dimensional traps with two-body interactions, revealing how these symmetries influence energy degeneracies and how they can be manipulated through trap and interaction tuning.

Contribution

It provides a comprehensive symmetry classification for one, two, and three particles in 1D traps, enabling algebraic solutions for spectra and degeneracies, and explores universality in energy shifts.

Findings

Symmetry groups explain degeneracies in few-body spectra.

Trap shape and interactions can be tuned to manipulate degeneracies.

Algebraic solutions are possible in non-interacting and unitary limits.

Abstract

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the $N$ particle case.