, Volume 319, Issue 3, pp 791-811
Date: 21 Dec 2012

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

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Abstract

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

Communicated by B. Simon