Carleman Estimates and Absence of Embedded Eigenvalues

Let L = −Δ− W be a Schrödinger operator with a potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W\in L^{\frac{n+1}{2}}(\mathbb{R}^n)$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geq 2$$\end{document}. We prove that there is no positive eigenvalue. The main tool is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}-L^{p^\prime}$$\end{document} Carleman type estimate, which implies that eigenfunctions to positive eigenvalues must be compactly supported. The Carleman estimate builds on delicate dispersive estimates established in [7]. We also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials.