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arXiv · 2207.14264

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

M. Burak Erdogan·William R. Green
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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 $m=1$.

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M. Burak Erdogan, William R. Green. 2022-07-28. A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schr\"odinger operators. https://doi.org/10.1016/j.jde.2023.01.028

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