**
Below is the ascii version of the abstract for 97-361.
The html version should be ready soon.**Christ, M., Kiselev, A., Remling, C.
The absolutely continuous spectrum of one-dimensional Schr\"odinger
operators with decaying potentials.
(14K, LATeX)
ABSTRACT. This is an announcement of the proof of some optimal results
on the presevation of the absolutely continuous spectrum
under perturbations by decaying potentials. We show that
if |V(x)| \leq C(1+x)^{-\alpha} with \alpha > 1/2, the
whole positive semi-axis is an essential support of the
absolutely continuous spectrum. This result is optimal
on the power scale. We also derive a new
general criterion for the stability of the a.c. spectrum.
Another result is that if limsup_{x \goto \infty}x|V(x)| < C,
the spectrum is purely a.c. on ((2C/\pi)^{2},\infty).
This is also optimal.