  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.
