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.
From: Xiaohua Yao [view email]
[v1]
Tue, 16 Jun 2026 13:20:38 UTC (21 KB)
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