**
Below is the ascii version of the abstract for 96-470.
The html version should be ready soon.**A.Kiselev
Preservation of the absolutely continuous spectrum of Schr\"odinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators
(81K, LaTeX)
ABSTRACT. We prove a new criteria of stability of the absolutely continuous spectrum
of one-dimensional Schr\"odinger operators under slowly decaying
perturbations. As applications, we show that the absolutely continuous
spectrum of the free and periodic Schr\"odinger operators is preserved
under perturbations by all potentials $V(x)$ satisfying $|V(x)| \leq
C(1+x)^{-\frac{2}{3}-\epsilon}.$ The main new technique includes an
a.e. convergence theorem for a class of integral operators.