The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities : Odd dimensions

Michael Goldberg, William R. Green

Published 2015-08-25Version 1

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n>3$ odd, and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^n)$ for all $1\leq p\leq \infty$ if zero is not an eigenvalue, and on $\frac{n} {n-2}<p<\frac{n}{2}$ if zero is an eigenvalue. We show, that if $\int_{\mathbb R^n} V(x) \phi(x) \, dx=0$ for all eigenfunctions $\phi$, then the wave operators are bounded for $\frac{n}{n-1}<p<n$. If, in addition $\int_{\mathbb R^n} xV(x) \phi(x) \, dx=0$, then the wave operators are bounded for $1<p<\infty$.