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.
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