  Observations of accreting neutron stars (NS) with strong magnetic fields can
be used not only for studying the accretion flow interaction with NS
magnetospheres, but also for understanding the physical processes inside NSs
and for estimating their fundamental parameters. Of particular interest are (i)
the interaction of a rotating neutron star (magnetosphere) with the in-falling
matter at different accretion rates, and (ii) the theory of deep crustal
heating and the influence of a strong magnetic field on this process. Here, we
present results of the first systematic investigation of 16 X-ray pulsars with
Be optical companions during their quiescent states, based on data from the
Chandra, XMM-Newton and Swift observatories. The whole sample of sources can be
roughly divided into two distinct groups: i) relatively bright objects with a
luminosity around ~10^34 erg/s and (hard) power-law spectra, and ii) fainter
ones showing thermal spectra. X-ray pulsations were detected from five objects
in group i) with quite a large pulse fraction of 50-70 per cent. The obtained
results are discussed within the framework of the models describing the
interaction of the in-falling matter with the neutron star magnetic field and
those describing heating and cooling in accreting NSs.
