Counterexamples to the $L^1$ and $L^{\infty}$ boundedness of the one-dimensional wave operators

Mathematical Physics [Submitted on 16 Jun 2026] Title:Counterexamples to the $L^1$ and $L^{\infty}$ boundedness of the one-dimensional wave operators View PDF HTML (experimental)Abstract:It is well established that the wave operators $W_{\pm}(H,-\Delta)$ for the one-dimensional Schrödinger operator $H=-\Delta+V(x)$ are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$ in both generic and exceptional cases. They are also bounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the exceptional case with $\lim\limits_{x\rightarrow-\infty}f_+(0,x)=1$. For the remaining endpoint cases, it has long been expected that they are generally unbounded at the endpoints $p=1,\infty$ due to the presence of the Hilbert transform in the low energy part, yet a rigorous proof has been missing. In this paper, we show that even for a bounded and compactly supported non-zero potential $V$, the wave operators $W_{\pm}(H,-\Delta)$ are unbounded on $L^1(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ in the generic case, as well as in the exceptional case with the condition $\lim\limits_{x\rightarrow-\infty}f_+(0,x)\neq1$. Moreover, in the latter case, they are even unbounded from $L^{\infty}(\mathbb{R})$ to ${\rm BMO}(\mathbb{R})$ (Bounded Mean Oscillation space). Hence together with those known results, our counterexamples complete the picture of the $L^{p}$ boundedness of one-dimensional wave operators. Current browse context: math-ph References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.