A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators
M. Burak Erdogan, William R. Green
TL;DR
The paper studies endpoint $L^p$-boundedness of wave operators for the higher-order Schrödinger operator $H=(-\Delta)^m+V$ in dimensions with $n>2m$, under decay and spectral assumptions and without smallness on $V$. It unifies even and odd dimensions by using a stationary representation and a low/high energy decomposition, focusing on the low-energy piece $W_{low,k}$ to obtain boundedness for the full range $1\le p\le \infty$. Central to the method is the representation $\mathcal{R}_V^+(\lambda^{2m})V=\mathcal{R}_0^+(\lambda^{2m})vM^+(\lambda)^{-1}v$ with $M^+(\lambda)=U+v\mathcal{R}_0^+(\lambda^{2m})v$, and the introduction of kernels $\Gamma_k(\lambda)$ that control the low-energy contribution via oscillatory integral estimates and cancellations in $\mathcal{R}_0^+(\lambda^{2m})-\mathcal{R}_0^-(\lambda^{2m})$. The results extend prior work to endpoint cases and yield global dispersive bounds, providing a unified, streamlined low-energy argument even in the classical case $m=1$.
Abstract
We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ 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 $m=1$.