Remarks on $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities

Abstract

We consider the continuity property in Lebesgue spaces $L^p({\mathbb{R}}^m)$ of the wave operators $W_{\pm }$ of scattering theory for Schrödinger operators $H=-\Delta +V$ on ${\mathbb{R}}^m$, $|V(x)|\le C⟨x{⟩}^{-\delta }$ for some $\delta >2$ when $H$ is of exceptional type, i.e. $\mathcal{N}=\{u\in ⟨x{⟩}^s{L}^2({\mathbb{R}}^m):(1+(-\Delta {)}^{-1}V)u=0\}\ne \{0\}$ for some $1/2<s<\delta -1/2$. It has recently been proved by Goldberg and Green for $m\ge 5$ that $W_{\pm }$ are in general bounded in $L^p({\mathbb{R}}^m)$ for $1\le p<m/2$, for $1\le p<m$ if all $\varphi \in \mathcal{N}$ satisfy ${\int }_{{\mathbb{R}}^m}V\varphi dx=0$ and, for $1\le p<\infty$ if ${\int }_{{\mathbb{R}}^m}{x}_{i}V\varphi dx=0,i=1,\dots ,m$ in addition. We make the results for $p>m/2$ more precise and prove in particular that these conditions are also necessary for the stated properties of $W_{\pm }$. We also prove that, for $m=3,W_{\pm }$ are bounded in $L^p({\mathbb{R}}^3)$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\varphi \in \mathcal{N}$ satisfy ${\int }_{{\mathbb{R}}^3}V\varphi dx=0$ and ${\int }_{{\mathbb{R}}^3}{x}_{i}V\varphi dx=0$, $i=1,2,3$, simultaneously.