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

TL;DR

This paper proves the $L^p$ boundedness of wave operators for Laplace operators with finite rank perturbations across all dimensions, resolving endpoint cases and revealing a dichotomy based on the integral of perturbation functions.

Contribution

It extends previous results by establishing $L^p$ boundedness at endpoints and in lower dimensions, and uncovers a new dichotomy related to the integral properties of the perturbation functions.

Findings

Wave operators are bounded on $L^p$ for all $1 \\le p \\le \\infty$ in dimensions $d \\ge 3$.

In dimensions $d=1,2$, $L^p$ boundedness of wave operators is established for the first time.

A dichotomy at $p=1$ depends on whether the perturbation functions have zero integral, affecting boundedness and weak type estimates.

Abstract

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_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 $\int_{\mathbb{R}^d} \varphi_i(x) \, \d x = 0$ holds for every $1\le i\le N$, then the wave operators are bounded on $L^p(\mathbb{R}^d)$ for all $1 \leq p \leq \infty$. \item If there exists at least one $i$ ($1\le i\le N$) such that $\int_{\mathbb{R}^d}\varphi_i(x)\d x\ne0$, 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}