# A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy

### Ebru Toprak

University of Illinois, Urbana, USA

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## Abstract

We study the two dimensional Schrödinger operator, $H=-\Delta+V$, in the weighted $L^1(\mathbb R^2) \to L^{\infty}(\mathbb R^2)$ setting when there is a resonance of the first kind at zero energy. In particular, we show that if $|V(x)|\lesssim \langle x \rangle ^{-4-}$ and there is only s-wave resonance at zero of $H$, then

with $w(x)=\mathrm {log}^2(2+|x|)$. Here $Ff=-{\frac {1}{4}} \psi\langle \psi,f \rangle$, where $\psi$ is an s-wave resonance function. We also extend this result to wave and matrix Schrödinger equations with potentials under similar conditions.

## Cite this article

Ebru Toprak, A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy. J. Spectr. Theory 7 (2017), no. 4, pp. 1235–1284

DOI 10.4171/JST/189