Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schr\"odinger operators

TL;DR

This paper constructs specific counterexamples demonstrating the unboundedness of wave operators on certain L^p spaces for higher order Schrödinger operators, revealing limitations in their boundedness properties.

Contribution

It provides explicit counterexamples for wave operator unboundedness in higher order Schrödinger operators and extends the analysis to classical second order cases with less smooth potentials.

Findings

Wave operators are unbounded on L^p for certain p and potentials.

Counterexamples exist for higher order Schrödinger operators with specific regularity.

Failure of dispersive estimates for less smooth potentials.

Abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1}<p\leq \infty$ and $0\leq \alpha <\frac{n+1}{2}-2m-\frac{n}p$, there exists a real-valued, compactly supported potential $V\in C^{\alpha}(\mathbb R^n)$ for which the wave operators $W^{\pm}$ are not bounded on $L^p(\mathbb R^n)$. As a consequence of our analysis we show that the wave operators for the usual second order Schr\"odinger operator $-\Delta+V$ are unbounded on $L^p(\mathbb R^n)$ for $n>3$ and $\frac{2n}{n-3}<p\leq \infty$ for insufficiently differentiable potentials $V$, and show a failure of $L^{p'}\to L^p$ dispersive estimates that may be of independent interest.