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Comptes Rendus Mathématique
Volume 339, n° 5
pages 345-350 (septembre 2004)
Doi : 10.1016/j.crma.2004.06.016
Received : 25 June 2004 ;  accepted : 29 June 2004
Paradoxe de Klein pour lʼéquation de Klein-Gordon chargée : superradiance et opérateur de diffusion
Klein paradox for the charged Klein-Gordon equation: superradiance and scattering.
 

Alain Bachelot
Université Bordeaux-1, institut de mathématiques, UMR CNRS 5466, 33405 Talence cedex, France 

Résumé

Nous développons la théorie de la diffusion pour lʼéquation de Klein-Gordon chargée sur   en présence dʼun potentiel électrostatique   admettant des limites distinctes   quand  . Dans ce cas, lʼénergie conservée nʼest pas définie positive (paradoxe de Klein). Nous faisons lʼanalyse spectrale de lʼéquation harmonique, et établissons lʼexistence dʼun opérateur de diffusion dont la norme du symbole est strictement supérieur à 1 pour les fréquences dans  . Ces résultats sʼappliquent à la métrique de DeSitter-Reissner-Nordstrm, pour justifier la notion de superradiance des trous noirs chargés. Pour citer cet article : A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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Abstract

We develop the scattering theory for the charged Klein-Gordon equation on  , when the electrostatic potential   has different asymptotics   as  . In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in  . These results can be applied to the DeSitter-Reissner-Nordstrm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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