One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: II. $N$ Particles

TL;DR

This paper classifies the symmetries of N identical particles in one-dimensional traps with two-body interactions, revealing how these symmetries influence spectral degeneracies and their manipulation through trap and interaction tuning.

Contribution

It extends the symmetry classification to N particles, providing algebraic solutions for spectra and degeneracies in various trap geometries and interaction limits.

Findings

Symmetries explain spectral degeneracies in N-particle systems.

Algebraic solutions for spectra are derived in specific interaction limits.

Symmetry considerations determine universality of energy shifts.

Abstract

This is the second 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 analyzes the symmetries of $N$ particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.