  A theory of resonant Raman scattering spectroscopy of one dimensional
electronic systems is developed on the assumptions that (i) the excitations of
the one dimensional electronic system are described by the Luttinger Liquid
model, (ii) Raman processes involve virtual excitations from a filled valence
band to an empty state of the one dimensional electronic system and (iii)
excitonic interactions between the valence and conduction bands may be
neglected. Closed form analytic expressions are obtained for the Raman
scattering cross sections, and are evaluated analytically and numerically for
scattering in the polarized channel, revealing a "double-peak" structure with
the lower peak involving multispinon excitations with total spin S=0 and the
higher peak being the conventional plasmon. A key feature of our results is a
nontrivial power law dependence, involving the Luttinger Liquid exponents, of
the dependence of the Raman cross sections on the difference of the laser
frequency from resonance. We find that near resonance the calculated ratio of
intensity in the lower energy feature to the intensity in the higher energy
feature saturates at a value of the order of unity (times a factor of the ratio
of the velocities of the two modes). We explicate the differences between the
'Luttinger liquid' and 'Fermi liquid' calculations of RRS spectra and argue
that excitonic effects, neglected in all treatments so far, are essential for
explaining the intensity ratios observed in quantum wires. We also discuss
other Luttinger liquid features which may be observed in future RRS
experiments.
