  We prove that the ground state for the Dirac equation on Minkowski space in
static, smooth external potentials satisfies the Hadamard condition. We show
that it follows from a condition on the support of the Fourier transform of the
corresponding positive frequency solution. Using a Krein space formalism, we
establish an analogous result in the Klein-Gordon case for a wide class of
smooth potentials. Finally, we investigate overcritical potentials, i.e. which
admit no ground states. It turns out, that numerous Hadamard states can be
constructed by mimicking the construction of ground states, but this leads to a
naturally distinguished one only under more restrictive assumptions on the
potentials.
