A weighted dispersive estimate for Schrödinger operators in dimension two

Abstract

Let $H=-\Delta +V$, where $V$ is a real valued potential on ${\mathbb{R}}^2$ 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.

23 pages

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• Dispersive Estimates in R^3 with Threshold Resonances
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• The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates
• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions
• Scattering operator and wave operators for 2D Schrödinger operators with threshold obstructions