Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases
N=3−−5, are provided.
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional $\ensuremath{\kappa}$-component fermions with repulsive and attractive $\ensuremath{\delta}$-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the $\ensuremath{\kappa}$-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You's result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as $\ensuremath{\kappa}\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as for spinless Bosons.
We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional
κ-component fermions with repulsive and with attractive delta-function interaction in strong and weak coupling regimes. We thus obtain a highly accurate result for the ground state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair correlation functions shed light on the quantum statistic effects of the
κ-component interacting fermions. For the balanced spin case with repulsive interaction, the ground state energy obtained confirms Yang and You's result [Chin. Phys. Lett. {\bf 28}, 020503 (2011)] that the energy per particle as
κ→∞ is the same as for spinless Bosons.