Lp-Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator

Given a one dimensional perturbed Schrödinger operator H =  − d2/dx2 + V(x), we consider the associated wave operators W ± , defined as the strong L2 limits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}$$\end{document}. We prove that W ±  are bounded operators on Lp for all 1 < p < ∞, provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+|x|)^{2}V(x)\in L^{1}$$\end{document}, or else \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+|x|)V(x)\in L^{1}$$\end{document} and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.