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.
We present a method for solving few-body problems for trapped particles and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e., the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying
C6v symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of
C6v. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.
This paper introduces a model for a few repulsively interacting particles trapped in a one-dimensional harmonic well and provides exact solutions for the three-particle case. This model is a hybrid of two other well-known systems, the Calogero model and the contact-interaction model, and coincides with them in limiting cases. However, those models have purely two-body interactions whereas this model has intrinsically few-body interactions. Comparing these three models provides clarifying distinctions among the properties of symmetry, separability, and integrability. The model's analytic solutions provide a useful basis to improve approximation schemes, especially near the unitary limit of hard-core contact interactions.