  We systematically investigate the mode dispersion and spectral weight of the
elementary excitation spectra in one-dimensional continuum and lattice electron
systems by using the RPA, the Luttinger liquid model, and the Hubbard model.
Both charge and spin excitations are studied in details and compared among the
theoretical models. For the lattice Hubbard model we use both Bethe-ansatz
equations and Lanczos-Gagliano method to calculate dispersion and spectral
weight separately. We discuss the theoretically calculated elementary
excitation spectra in terms of the experimental inelastic light (Raman)
scattering spectroscopy of 1D semiconductor quantum wire systems. Our results
show that in the polarized (i.e. non-spin-flip) Raman scattering spectroscopy,
only the 1D charge density excitations should show up with observable spectral
weight with the single particle excitations (in RPA) or singlet spin
excitations (in the Luttinger model and the Hubbard model) having negligible
spectral weight. The depolarized (spin-flip) Raman scattering spectra manifest
the spin density or the triplet spin excitations. We also provide a qualitative
comparison between the continuum and the lattice 1D elementary excitation
spectra.
