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arXiv · 2206.12929

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

M. Burak Erdogan·Michael Goldberg·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 $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1} 3$ 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.

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M. Burak Erdogan, Michael Goldberg, William R. Green. 2022-06-26. Counterexamples to $L^p$ boundedness of wave operators for classical and higher order Schr\"odinger operators. https://doi.org/10.1016/j.jfa.2023.110008

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