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

Abstract

We consider the continuity property in Lebesgue spaces $L^{p}(\R^{m})$ of the wave operators $W_\pm$ of scattering theory for Schrödinger operators $H=-\lap + V$ on $\R$^m, $|V(x)|\le C\ax$^-delta for some $\delta>2$ when $H$ is of exceptional type, i.e. $\Ng={u \in \ax^{s} L^{2}(\R^{m}) \colon (1+ (-\lap)^{-1}V)u=0 }\not={0}$ 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 $L^{p}(\R^{m})$ for $1\le p<m/2$, for $1\le p<m$ if all $\f\in \Ng$ satisfy $\int_{\R^{m}} V\f dx=0$ and, for $1\le p<\infty$ if $\int_{\R^{m}} x_{i} V\f dx=0, i=1, \dots, m$ 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 $L^{p}(\R^{3})$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy $\int_{\R^{3}}V\f dx=0$ and $\int_{\R^{3}} x_{i} V\f dx=0, i=1, 2, 3$, simultaneously.