The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations

Abstract

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-Δ+\sum\limits_{i=1}^N\langle\cdot\,, φ_i\rangle φ_i \qquad \mbox{on}\,\,\, \R^d. \end{equation*} For dimensions $d\ge 3$, we prove that the wave operators $W_{\pm }(H,H_0)$ are bounded on $L^p$ for the full range $1\le p\le \infty$. This extends the work of Nier and the third author \cite{NS} by resolving the previously unexplored question of boundedness at the endpoint cases $p=1$ and $p=\infty$. In lower dimensions $d=1,2$, we establish the $L^p$-boundedness of the wave operators for the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case $p=1$: \begin{itemize} \item If $\int_{\mathbb{R}^d} φ_i(x) \, \d x = 0$ holds for every $1\le i\le N$, then the wave operators are bounded on $L^p({\mathbb{R}}^d)$ for all $1\le p\le \infty$. \item If there exists at least one $i$ ($1\le i\le N$) such that $\int_{\mathbb{R}^d}φ_i(x)\d x\ne0$, then the wave operators remain bounded for $1<p<\infty$ and satisfy weak type $(1,1)$ estimates, but fail to be bounded on $L^1({\mathbb{R}}^d)$. \end{itemize}

33 pages