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= (-) 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<n2m 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.