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.