96-470 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) Oct 2, 96

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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.

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