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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 repulsivezero-range even-wave interactions are mapped exactly to a 1DLieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending onisotropic Heisenberg spin-spin interactions, such that the complete SFG and LLHenergy spectra are identical. The ground state in the ferromagnetic phase isgiven exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in theantiferromagnetic phase by a variational method combining Bethe ansatzsolutions of the LL and 1D Heisenberg models. There are excitation branchescorresponding to LL type I and II phonons and spin waves, the latter behavingquadratically for small wave number in the ferromagnetic phase and linearly inthe antiferromagnetic phase.Comment: 4 pages, 2 figures, revtex4. To be submitted to Phys. Rev. Let

Ground and Excited States of Spinor Fermi Gases in Tight Waveguides and the Lieb-Liniger-Heisenberg Model