  Accretion-powered X-ray pulsars are among the most luminous X-ray sources in
the Galaxy. However, despite decades of theoretical and observational work
since their discovery, no satisfactory model for the formation of the observed
X-ray spectra has emerged. In this paper, we report on a self-consistent
calculation of the spectrum emerging from a pulsar accretion column that
includes an explicit treatment of the bulk and thermal Comptonization occurring
in the radiation-dominated shocks that form in the accretion flows. Using a
rigorous eigenfunction expansion method, we obtain a closed-form expression for
the Green's function describing the upscattering of monochromatic radiation
injected into the column. The Green's function is convolved with
bremsstrahlung, cyclotron, and blackbody source terms to calculate the emergent
photon spectrum. We show that energization of photons in the shock naturally
produces an X-ray spectrum with a relatively flat continuum and a high-energy
exponential cutoff. Finally, we demonstrate that our model yields good
agreement with the spectra of the bright pulsar Her X-1 and the low luminosity
pulsar X Per.
