Intriguing sub-TeV tails in the pulsed $\gamma$-ray emission from the Crabpulsar have been recently discovered by the MAGIC and VERITAS Collaborations.They were not clearly predicted by any pulsar model. It is at present arguedthat this emission is produced by electrons in the Inverse Compton processoccurring either in the outer gap of the pulsar magnetosphere or in the pulsarwind region at some distance from the light cylinder. We analyse anotherscenario which is consistent with the basic features of this enigmaticemission. It is proposed that this emission is caused by electrons acceleratedvery close to the light cylinder where the $e^\pm$ plasma can not saturateinduced huge electric fields. Electrons reach energies sufficient forproduction of hard $\gamma$-ray spectra in the curvature radiation process. Dueto different curvature radii of the leading and trailing magnetic field lines,the $\gamma$-ray spectra from separate pulses should extend to differentmaximum energies. The scenario can also explain the lower level $\gamma$-rayemission from the interpulse region (between P1 and P2) observed in the Crabpulsar light curve. Moreover, we argue that pulsars with parameters close tothe Vela pulsar should also show pulsed emission with the cut-off at clearlylower energies ($\sim$50 GeV) than that observed in the case of the Crabpulsar. On the other hand, such tail emission is not expected in pulsars withparameters close to the Geminga pulsar. The model also predicts the tail$\gamma$-ray emission extending up to $\sim$50 GeV from some millisecondpulsars with extreme parameters such as PSR J0218+4243 and PSR J1823-3021A.
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