Linear stability of Vlasov-Poisson electron plasma in crossed fields. Perturbations propagating parallel to the magnetic field

It is shown that electrostatic Vlasov-Poisson perturbations that propagate parallel to the magnetic field in a planar magnetron are stable for both an isotropic and also for a particular anisotropic (T_y = 3T_x) temperature distribution. The inhomogeneity of the electron density is fully incorporated in the analysis. The proof makes use of only the dispersion relation of Trivelpiece-Gould type, without actually solving the eigenvalue equation. These results suggest, not unexpectedly, that these modes should be stable for all such anisotropic velocity distributions.