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

It is well established that the wave operators $W_{\pm }(H,-\mathrm{\Delta })$ for the one-dimensional Schrödinger operator $H=-\mathrm{\Delta }+V(x)$ are bounded on $L^p(\mathbb{R})$ for all $1<p<\mathrm{\infty }$ in both generic and exceptional cases. They are also bounded on $L^1(\mathbb{R})$ and $L^{\mathrm{\infty }}(\mathbb{R})$ in the exceptional case with $\underset{x\to -\mathrm{\infty }}{\lim }{f}_{+}(0,x)=1$. For the remaining endpoint cases, it has long been expected that they are generally unbounded at the endpoints $p=1,\mathrm{\infty }$ 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 $W_{\pm }(H,-\mathrm{\Delta })$ are unbounded on $L^1(\mathbb{R})$ and $L^{\mathrm{\infty }}(\mathbb{R})$ in the generic case, as well as in the exceptional case with the condition $\underset{x\to -\mathrm{\infty }}{\lim }{f}_{+}(0,x)\mathrm{\ne }1$. Moreover, in the latter case, they are even unbounded from $L^{\mathrm{\infty }}(\mathbb{R})$ to $\mathrm{B}\mathrm{M}\mathrm{O}(\mathbb{R})$ (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.