Preservation of the absolutely continuous spectrum of Schr\"odinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators

Abstract

We prove 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.