Communications in Mathematical Physics

, Volume 319, Issue 3, pp 791–811

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two


DOI: 10.1007/s00220-012-1640-7

Cite this article as:
Erdoğan, M.B. & Green, W.R. Commun. Math. Phys. (2013) 319: 791. doi:10.1007/s00220-012-1640-7


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 L2 spaces with polynomially growing weights.

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of Mathematics and Computer ScienceEastern Illinois UniversityCharlestonUSA
  3. 3.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA