  We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in
$n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We
adapt our recent results for $m>1$ to show that when $H$ has a threshold
eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural
range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The approach used
works without distinguishing even and odd cases, and matches the range of
boundedness in the classical case when $m=1$. The proof applies in the
classical $m=1$ case as well and simplifies the argument.
