The $L^{p}$-boundedness of wave operators for four-dimensional Schrödinger operators

Abstract

We prove that the low energy parts of the wave operators $W_{\pm}$ for Schrödinger operators $H=-\Delta+V(x)$ on $\mathbb{R}^{4}$ are bounded in $L^{p}(\mathbb{R}^{4})$ for $1<p\leq 2$ and are unbounded for $2<p\leq\infty$ if $H$ has resonances at the threshold. If $H$ has eigenfunctions only at the threshold, it has recently been proved that they are bounded in $L^{p}(\mathbb{R}^{4})$ for $1\leq p<4$ in general and for $1\leq p<\infty$ if all threshold eigenfunctions $\varphi$ satisfy $\int_{\mathbb{R}^{4}}x_{j}V(x)\varphi(x)\,dx=0$ for $1\leq j\leq 4$. We prove in this case that they are unbounded in $L^{p}(\mathbb{R}^{4})$ for $4<p<\infty$ unless the latter condition is satisfied. It is long known that the high energy parts are bounded in $L^{p}(\mathbb{R}^{4})$ for all $1\leq p\leq\infty$ and that the same holds for $W_{\pm}$ if $H$ has no eigenfunctions nor resonances at the threshold.