Five is More: Comments on Symmetry, Integrability, and Solvability for a Few Particles in a One-Dimensional Trap

TL;DR

This paper reviews how symmetry, integrability, and solvability of a few-particle quantum system in one-dimensional traps depend on system parameters, highlighting the loss of algebraic solvability for five or more particles with spin.

Contribution

It provides a detailed analysis of the minimal symmetries and the conditions under which algebraic solvability is lost in multi-particle, spinful systems.

Findings

Symmetry and solvability depend on trap shape and interactions.

Algebraic solvability is lost for five or more particles with spin.

Minimal symmetries are characterized for composite systems.

Abstract

This contributed conference proceeding reviews some results about a system of a few identical particles with spin trapped in one-dimensional potentials and experiencing two-body interactions. The focus is on how symmetry, integrability, and solvability depend on the trap shape, two-body interaction, the number of particles, and the number of spin components. A series of comments are presented that characterize the minimal symmetries possible for a composite system constructed from interacting single particles, with special focus on the contact interaction. For five and more particles with internal components like spin, a kind of universality called algebraically solvability is lost.