  We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in
$n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$,
$m>1$. When $n$ is odd, we prove that the wave operators extend to bounded
operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under $n$ and $m$
dependent conditions on the potential analogous to the case when $m=1$.
Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave
operators are bounded on the same range for even $n$. We further show that if
the smallness assumption is removed in even dimensions the wave operators
remain bounded in the range $1<p<\infty$.
