  We consider the 1/2-dimensional relativistic Vlasov-Maxwell system that
describes the time-evolution of a plasma. We find a relatively simple criterion
for spectral instability of a wide class of equilibria. This class includes
non-homogeneous equilibria that need not satisfy any additional symmetry
properties (as was the case in previous results), nor should they be monotone
in the particle energy. The criterion is given in terms of the spectral
properties of two Schr\"{o}dinger operators that arise naturally from Maxwell's
equations. The spectral analysis of these operators is quite delicate, and some
general functional analytic tools are developed to treat them. These tools can
be applied to similar systems in higher dimensions, as long as their domain is
finite or periodic.
