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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 resolvent estimates, thresholds, scattering theory, Riesz transform
MSC Classifications: 35P25 show english descriptions Scattering theory [See also 47A40] 35P25 - Scattering theory [See also 47A40]
 

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