Absolutely continuous spectrum of Stark operators

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 vertical bar q(x)vertical bar <= C(1+vertical bar x vertical bar)(-1/4-epsilon); in the smoothness direction, a sufficient condition in Holder classes is q epsilon C1/2+epsilon(R). On the other hand, we show that there exist potentials which both satisfy vertical bar q(x)vertical bar <= C(1+vertical bar x vertical bar)(-1/4) and belong to C-1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.