On the simultaneous generation of radio and soft X-ray emission by AXP 4U 0142+61

In the present paper we study the possibility of a simultaneous generation of radio waves and soft $X$-rays by means of the quasi-linear diffusion (QLD) in the anomalous pulsar AXP 4U 0142+61. Considering the magnetosphere composed of the so-called beam component and the plasma component respectively, we argue that the frozen-in condition will inevitably lead to the generation of the unstable cyclotron waves. These waves, via the QLD, will in turn influence the particle distribution function, leading to certain values of the pitch angles, thus to an efficient synchrotron mechanism, producing soft $X$-ray photons. We show that for physically reasonable parameters of magnetospheric plasmas, the QLD can provide generation of radio waves in the following interval $40$ MHz-$111$ MHz connected to soft $X$-rays for the domain $0.3$keV-$1.4$keV.