A study of stability of the "plasma - beam" system for the Lorentz plasma

The stability of the system comprising the cold immobile Lorentz plasma of density $N_e$ and the directed nonrelativistic (velocity $\vec{V}$) electronic beam of a small density $N'_e << N_e$ is investigated. The instability increment of the system is found via analysis of the electric permittivity $\epsilon(\omega)$ in the limiting case of $\omega$ being much greater than the effective frequency of collisions. A conclusion is made that the system in question is stable with respect to oscillations with relatively small values of $\vec{k} \vec{V}$.