The $L^p$-boundedness of wave operators for four dimensional Schrödinger operators with threshold resonances

Abstract

We prove that the low energy parts of the wave operators $W_{\pm }$ for Schrödinger operators $H = -\lap + V(x)$ on ${\mathbb{R}}^4$ are bounded in $L^p({\mathbb{R}}^4)$ for $1<p\le 2$ and are unbounded for $2<p\le \infty$ 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 $1\le p<4$ in general and for $1\le p<\infty$ if all threshold eigenfunctions $\ph$ satisfy $\int_{\R^4}x_j V(x) \ph(x)dx=0$ for $1\le j\le 4$. 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 $1\le p\le \infty$ and that the same holds for $W_{\pm }$ if $H$ has no eigenfunctions nor resonances at the threshold.