$L^p$-continuity of wave operators for higher order Schr\"odinger operators with threshold eigenvalues in high dimensions

TL;DR

This paper proves that wave operators for higher order Schrödinger operators with threshold eigenvalues are bounded on certain L^p spaces in high dimensions, extending classical results and simplifying previous proofs.

Contribution

It extends L^p-continuity results of wave operators to higher order Schrödinger operators with threshold eigenvalues in high dimensions, unifying even and odd cases.

Findings

Wave operators are bounded on L^p for 1 ≤ p < n/(2m) in high dimensions.

Results apply to both even and odd dimensions without distinction.

The approach simplifies proofs and matches classical m=1 case.

Abstract

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.