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
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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.