$L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

We consider the higher order Schrödinger operator $H=(-\mathrm{\Delta }{)}^m+V(x)$ 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 $1\le p<\frac{n}{2m}$ 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.