  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.
