Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials

Abstract

We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in $L^p(\reals)$, $p<2$. If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.