A note on endpoint $L^p$-continuity of wave operators for classical and 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 $n>2m$, $m\in \mathbb{N}$. 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 $1\le p\le \infty$ 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 $m=1$.

16 pages, revised. This paper extends the authors' work in arXiv:2107.09620 to capture endpoints in even dimensions. Updated according to referee comments, to appear in the J. of Differential Equations

References in corpus (3)

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

Cited by in corpus (2)

• $L^p$-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