  We study and develop the stationary scattering theory for a class of one-body
Stark Hamiltonians with short-range potentials, including the Coulomb
potential, continuing our study in [AIIS1,AIIS2]. The classical scattering
orbits are parabolas parametrized by asymptotic orthogonal momenta, and the
kernel of the (quantum) scattering matrix at a fixed energy is defined in these
momenta. We show that the scattering matrix is a classical type
pseudodifferential operator and compute the leading order singularities at the
diagonal of its kernel. Our approach can be viewed as an adaption of the method
of Isozaki-Kitada [IK] used for studying the scattering matrix for one-body
Schr\"odinger operators without an external potential. It is more flexible and
more informative than the more standard method used previously by
Kvitsinsky-Kostrykin [KK1] for computing the leading order singularities of the
kernel of the scattering matrix in the case of a constant external field (the
Stark case). Our approach relies on Sommerfeld's uniqueness result in Besov
spaces, microlocal analysis as well as on classical phase space constructions.
