The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations
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| Format: | Preprint |
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2025
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| _version_ | 1866916885356871680 |
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| author | Cheng, Han Huang, Shanlin Soffer, Avy Wu, Zhao |
| author_facet | Cheng, Han Huang, Shanlin Soffer, Avy Wu, Zhao |
| contents | 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 $1 \leq p \leq \infty$.
\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} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations Cheng, Han Huang, Shanlin Soffer, Avy Wu, Zhao Analysis of PDEs Classical Analysis and ODEs 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 $1 \leq p \leq \infty$. \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} |
| title | The $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.05533 |