The phenomenology of anomalous X-ray pulsars is usually interpreted within the paradigm of very highly magnetized neutron stars, also known as magnetars. According to this paradigm, the persistent emission of anomalous X-ray pulsars (AXPs) is powered by the decay of the magnetic field. However, an alternative scenario in which the persistent emission is explained through accretion is also discussed in literature. In particular AXP 4U 0142+61has been suggested to be either an accreting neutron star or a white dwarf. Here we rule out this scenario based on the the observed X-ray variability properties of the source. We directly compare the observed power spectra of 4U 0142+61 and of two other magnetars, 1RXS J170849.0$-$400910 and 1E 1841-045 with that of the X-ray pulsar 1A 0535+262, and of the intermediate polar GK Persei. In addition, we include a bright young radio pulsar PSR B1509-58 into comparison. We show that, unlike accreting sources, no aperiodic variability within the expected frequency range is observed in the power density spectrum of the magnetars and the radio pulsar. Considering that strong variability is an established features of all accreting systems from young stellar objects to super-massive black holes, and absence of the variability reports from other magnetars, we conclude that our results indicates also that magnetars in general are not powered by accretion.
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