Lᵖ-continuity of wave operators for higher order Schr\"odinger operators with threshold eigenvalues in high dimensions

We consider the higher order Schr\"odinger operator H=(-Δ)ᵐ+V(x) in n dimensions with real-valued potential V when $n>4m$, m∈ N. We adapt our recent results for $m>1$ to show that when H has a threshold eigenvalue the wave operators are bounded on Lᵖ( Rⁿ) for the natural range 1≤ p<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.