  We present mid-infrared spectroscopy of a sample of 16 optically faint
infrared luminous galaxies obtained with the Infrared Spectrograph (IRS) on the
Spitzer Space Telescope. These sources were jointly selected from Spitzer and
Chandra imaging surveys in the NDWFS Bootes field and were selected from their
bright X-ray fluxes to host luminous AGN. None of the spectra show significant
emission from polycyclic aromatic hydrocarbons (PAHs; 6.2um equivalent widths
<0.2um), consistent with their infrared emission being dominated by AGN. Nine
of the X-ray sources show 9.7um silicate absorption features. Their redshifts
are in the range 0.9<z<2.6, implying infrared luminosities of
log(L{IR})=12.5-13.6 solar luminosities. The average silicate absorption
strength is not as strong as that of previously targeted optically faint
infrared luminous galaxies with similar mid-infrared luminosities implying that
the X-ray selection favors sources behind a smaller column of Si-rich dust than
non-X-ray selection. Seven of the X-ray sources have featureless power-law
mid-IR spectra. We argue that the featureless spectra likely result from the
sources having weak or absent silicate and PAH features rather than the sources
lying at higher redshifts where these features are shifted out of the IRS
spectral window. We investigate whether there are any correlations between
X-ray and infrared properties and find that sources with silicate absorption
features tend to have fainter X-ray fluxes and harder X-ray spectra, indicating
a weak relation between the amount of silicate absorption and column density of
X-ray-absorbing gas.
