  We prove several new results on the absolutely continuous spectra of
perturbed one-dimensional Stark operators. First, we find new classes of
perturbations, characterized mainly by smoothness conditions, which preserve
purely absolutely continuous spectrum. Then we establish stability of the
absolutely continuous spectrum in more general situations, where imbedded
singular spectrum may occur. We present two kinds of optimal conditions for the
stability of absolutely continuous spectrum: decay and smoothness. In the decay
direction, we show that a sufficient (in the power scale) condition is $|q(x)|
\leq C(1+|x|)^{-{1/4}-\epsilon};$ in the smoothness direction, a sufficient
condition in H\"older classes is $q \in C^{{1/2}+\epsilon}(\reals)$. On the
other hand, we show that there exist potentials which both satisfy $|q(x)| \leq
C(1+|x|)^{-1/4}$ and belong to $C^{1/2}(\reals)$ for which the spectrum becomes
purely singular on the whole real axis, so that the above results are optimal
within the scales considered.
