# On the $L^p$ boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions

## TL;DR

This paper investigates the boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions, establishing $L^p$ boundedness under certain zero-energy conditions.

## Contribution

It demonstrates that wave operators remain bounded on $L^p$ spaces for $1<p<
finite$ when zero is an eigenvalue or resonance, extending known results to cases with threshold obstructions in 2D.

## Key findings

- Wave operators are bounded on $L^p$ for $1<p<
finite$ with zero eigenvalue or resonance.
- Contrast with higher dimensions where zero obstructions limit $p$ range.
- Results apply under sufficient decay of the potential.

## Abstract

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^2)$ for all $1< p< \infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p(\mathbb R^2)$ for $1 < p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents $p$.