Infinite Barriers and Symmetries for a Few Trapped Particles in One Dimension

This article analyzes symmetries for models describing a few particles in a one-dimensional, few-well trap. In addition to parity and particle permutation symmetry, the symmetry of separability, well permutation symmetry, and ordering permutation symmetry are necessary to explain how spectral degeneracies depend on the trap shape, the strength of interaction, and the amount of tunneling. Generally, introducing impassable barriers into configuration space increases the amount of symmetry and makes the energy spectrum more degenerate. This effect is especially pronounced in models with thin, symmetrically-placed barriers and contact interactions. The simplest case of two particles in two wells is analyzed in depth, and algebraic solutions and energy level mappings are provided for the infinite square well and harmonic traps as examples.