The $L^p$-continuity of wave operators for higher order Schrödinger operators

Abstract

We consider the higher order Schrödinger 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 $1\le p\le \infty$ 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 $1<p<\infty$.

35 pages. We fixed an error in the statement of Lemma 4.2 that appears in the published version

References in corpus (3)

• A counterexample to dispersive estimates for Schrödinger operators in higher dimensions
• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators
• $L^p$-boundedness of wave operators for bi-Schrödinger operators on the line

Cited by in corpus (4)

• A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators
• Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schrödinger operators
• $L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions
• Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials