Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators

We consider the higher order Schrödinger operator $H=(-\mathrm{\Delta }{)}^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb{N}$. We show that for any $\frac{2n}{n-4m+1}<p\le \mathrm{\infty }$ and $0\le \alpha <\frac{n+1}{2}-2m-\frac{n}{p}$, 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 Schrödinger operator $-\mathrm{\Delta }+V$ are unbounded on $L^p({\mathbb{R}}^n)$ for $n>3$ and $\frac{2n}{n-3}<p\le \mathrm{\infty }$ for insufficiently differentiable potentials $V$, and show a failure of $L^{p^{\mathrm{\prime }}}\to L^p$ dispersive estimates that may be of independent interest.