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Low Frequency Estimates for Long Range Perturbations in Divergence Form

Published:2011-04-25
Printed: Oct 2011
• Jean-Marc Bouclet,
Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France, F-31062
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Abstract

We prove a uniform control as $z \rightarrow 0$ for the resolvent $(P-z)^{-1}$ of long range perturbations $P$ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension $d \geq 3$ when $P$ is defined on $\mathbb{R}^d$ and in dimension $d \geq 2$ when $P$ is defined outside a compact obstacle with Dirichlet boundary conditions.
 Keywords: resolvent estimates, thresholds, scattering theory, Riesz transform