Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schr\"odinger operators

Abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1}<p\leq \infty$ and $0\leq \alpha <\frac{n+1}{2}-2m-\frac{n}p$, there exists a real-valued, compactly supported potential $V\in C^{\alpha}(\mathbb R^n)$ for which the wave operators $W^{\pm}$ are not bounded on $L^p(\mathbb R^n)$. As a consequence of our analysis we show that the wave operators for the usual second order Schr\"odinger operator $-\Delta+V$ are unbounded on $L^p(\mathbb R^n)$ for $n>3$ and $\frac{2n}{n-3}<p\leq \infty$ for insufficiently differentiable potentials $V$, and show a failure of $L^{p'}\to L^p$ dispersive estimates that may be of independent interest.