Absolutely Continuous Spectrum for One-dimensional Schr Odinger Operators with Slowly Decaying Potentials: Some Optimal Results

and some self-adjoint boundary condition at the origin. We assume that U is some bounded function for which HU has absolutely continuous spectrum. The presence of the absolutely continuous spectrum has direct consequences for the physical properties of the quantum particle described by the operator HU (see, e.g. [23, 2]). If we perturb this operator by some decaying potential V (x), the Weyl criterion implies that the essential spectra of the operators HU and HU+V coincide. We seek conditions on the rate of decay of V (x) which ensure that the absolutely continuous spectrum of the unperturbed operator HU is also preserved. This problem has a long history as one of the most natural questions in quantum mechanics, and we briefly recall the main results. It has long been known that if the perturbation V (x) is absolutely integrable, then the absolutely continuous spectrum of the original operator is preserved. Until recently, little more was known concerning the preservation of the absolutely continuous spectrum of Schrödinger operators under decaying perturbations in the general situation. Substantially more information is available in the case when U(x) = 0. There has been much work on proving the absolute continuity of the spectrum for Schrödinger operators with potentials of slower decay, but satisfying some additional special assumptions. For example, by a result going back to Weidmann [31], if a potential V may be represented as a sum of a function of bounded variation and an absolutely integrable function, then the spectrum of the operator HV on R = (0,∞) is purely absolutely continuous. Many authors developed a scattering theory for longrange potentials whose derivatives satisfy certain bounds; see for example [1, 5, 12].