A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schr\"odinger operators

TL;DR

This paper proves that wave operators for higher order Schrödinger operators are bounded on all $L^p$ spaces for $1 \\leq p \\leq \\infty$ in dimensions greater than twice the order, extending previous results and simplifying the analysis.

Contribution

It extends $L^p$-boundedness of wave operators to all $p$ in the full range for higher order Schrödinger operators without small potential assumptions.

Findings

Wave operators are bounded on $L^p$ for all $1 \\leq p \\leq \\infty$.

The method works uniformly for even and odd dimensions.

Endpoints $p=1$ and $p=\infty$ are included in the analysis.

Abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on $L^p(\mathbb R^n)$ for the full the range $1\leq p\leq \infty$ in both even and odd dimensions without assuming the potential is small. The approach used works without distinguishing even and odd cases, captures the endpoints $p=1,\infty$, and somehow simplifies the low energy argument even in the classical case of $m=1$.