A weighted dispersive estimate for Schr\"{o}dinger operators in dimension two

TL;DR

This paper establishes a decay estimate for the Schrödinger evolution in two dimensions with a potential, showing a specific logarithmic decay rate under certain spectral conditions.

Contribution

It proves a weighted dispersive estimate for Schrödinger operators in 2D, extending Murata's results to weighted $L^1$-$L^ty$ spaces with logarithmic weights.

Findings

Decay rate of $1/|t|\, ext{log}^2(|t|)$ for Schr6dinger evolution

Valid under zero being a regular point of the spectrum

Applicable to potentials with decay $|V(x)| \, ext{lesssim} \,\langle x\rangle^{-3-}$

Abstract

Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying $|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H=-\Delta+V$, then $$ \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)}, |t| >2, $$ with $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights.