  The ground and excited states of a one-dimensional (1D) spin-1/2 Fermi gas
(SFG) with both attractive zero-range odd-wave interactions and repulsive
zero-range even-wave interactions are mapped exactly to a 1D
Lieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending on
isotropic Heisenberg spin-spin interactions, such that the complete SFG and LLH
energy spectra are identical. The ground state in the ferromagnetic phase is
given exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in the
antiferromagnetic phase by a variational method combining Bethe ansatz
solutions of the LL and 1D Heisenberg models. There are excitation branches
corresponding to LL type I and II phonons and spin waves, the latter behaving
quadratically for small wave number in the ferromagnetic phase and linearly in
the antiferromagnetic phase.
