We analyze the spectra of one-dimensional Schrodinger operators with potentials which decay rather slowly, and have little smoothness. In an appropriate perturbative range, the generalized eigenfunctions are analyzed, for almost every energy. One consequence is a set of rather sharp conditions for absolutely continuous spectrum to persist under perturbations that are small at infinity, even though dense embedded point spectrum can occur. The WKB approximation describing the asymptotic behavior of the eigenfunctions remains valid for Lebesgue-almost every energy, for a larger class of potentials than those for which it is classically known to be valid (uniformly for every energy). These results are based on maximal function estimates for certain multilinear integral operators, of arbitrary order, which generalize the Fourier transform. A crude, but robust, technique for analyzing a large class of such operators will be outlined. Except for endpoint issues, this technique yields optimal results for several classes of potentials. If time permits, variants will be discussed, including perturbations of a constant electrical field, slowly varying potentials, and perhaps an application to the time-dependent Schrodinger equation. These involve extensions of the WKB approximation, and analysis of energy-dependent potentials. All of this is joint work with Alexander Kiselev.
Copyright. All Rights Reserved.