  The majority of X-ray-detected rotation-powered millisecond pulsars (MSPs)
appear to exhibit predominantly thermal emission, believed to originate from
the heated magnetic polar caps of the pulsar. In the nearest MSP, J0437--4715 a
faint PL is also observed at >3 keV, usually associated with magnetospheric
emission processes. However, the hard emission in this and other similar MSPs
may instead be due to weak Comptonization of the thermal polar cap emission by
energetic electrons/positrons of small optical depth most likely in the pulsar
magnetosphere. This spectral model implies that all soft X-rays are of purely
thermal origin, which has important implications in the study of neutron stars.
