On the possible mechanism to form the radio emission spectrum of the   Crab pulsar
  In the present paper a self-consistent theory, explaining shape of the observed phase-averaged radio spectrum in the frequency range from 100MHz to 10GHz is presented. The radio waves are assumed to be generated near the light cylinder through the cyclotron resonance. The cyclotron instability provides excitement of the electron-positron plasma eigen-waves, which come in radio domain when the resonant particles are the most energetic primary beam electrons. It is widely accepted that the distribution function of relativistic particles is one-dimensional at the pulsar surface. The generated waves react back on the resonant particles causing their diffusion in the perpendicular direction to the magnetic field and violating the one-dimensionality, which switches on the synchrotron radiation process. The synchrotron emission of the beam electrons provides generation of high-energy $\gamma$-rays simultaneously with the radio emission, that explains the observed pulse phase-coincidence in these energy domains. The theory provides a power-law radio spectrum with the spectral index equal to $3.7$, coming in agreement with the observations. 
George Machabeli and Nino Chkheidze
astro-ph.HE astro-ph.SR