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Article Dans Une Revue Annales de l'Institut Henri Poincaré Année : 2017

Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry

Christian Gérard
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Omar Oulghazi
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Michał Wrochna

Résumé

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseu-dodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.
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Dates et versions

hal-01265079 , version 1 (30-01-2016)
hal-01265079 , version 2 (08-01-2017)
hal-01265079 , version 3 (04-03-2017)

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Citer

Christian Gérard, Omar Oulghazi, Michał Wrochna. Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry. Annales de l'Institut Henri Poincaré, 2017, 18 (8), pp.2715-2756. ⟨10.1007/s00023-017-0573-2⟩. ⟨hal-01265079v3⟩
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