  We study the Schwinger-Dyson equation for the fermion self-energy in massless
and massive $QED_2$, in the ladder approximation. When the fermion is massless
(and the photon massless or massive), we check the reliability of this
approximation by comparing its solutions with the exact ones. They agree only
when the photon is massless. For a massive fermion and massless photon, we show
that there is no consistent solution at all, the infrared divergences
introduced by the approximation forbidding even the trivial solution. When both
fermion and photon are massive, we find a non-perturbative (extra) fermion mass
generation (which survives in the limit when the bare mass of the fermion tends
to zero). We argue that, in this case, the ladder approximation will provide
reliable results if the bare masses of the fields are large compared with the
(dimensionful) coupling constant.
