The $L^p$-continuity of wave operators for higher order Schr\"odinger operators

TL;DR

This paper studies the boundedness of wave operators for higher order Schrödinger operators with potential, establishing $L^p$-continuity results depending on dimension, order, and potential smallness, extending known results for second order cases.

Contribution

It extends $L^p$-boundedness of wave operators to higher order Schrödinger operators for odd and even dimensions under new potential conditions.

Findings

Wave operators are bounded on $L^p$ for all $1 extless p extless \infty$ in odd dimensions.

In even dimensions, small potential assumptions ensure boundedness on the full $L^p$ range.

Removing smallness assumptions in even dimensions still yields boundedness for $1 extless p extless \infty$.

Abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$, $m>1$. When $n$ is odd, we prove that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under $n$ and $m$ dependent conditions on the potential analogous to the case when $m=1$. Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave operators are bounded on the same range for even $n$. We further show that if the smallness assumption is removed in even dimensions the wave operators remain bounded in the range $1<p<\infty$.