Landau poles, branch cuts and supra-luminous waves in a one-dimensional plasma

A simple initial value problem is set up for a one-dimensional, mobile, electron plasma coexisting with a positive charged, immobile background. It is solved relativistically, a la Landau, and it is demonstrated that for a stable plasma the perturbation electric field consists of three contributions: (1) the usual damped Landau poles from waves with phase velocities less than the speed of light; (2) a damped branch cut contribution with branch points at +or-c; (3) undamped supra-luminous poles from waves with phase velocities greater than the speed of light. The modified Nyquist diagrams are sketched and, in the limit of long times, it is shown that the branch cut terms can, for particular plasma distribution functions, exceed the Landau pole terms.