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Communications in Mathematical Physics

, Volume 352, Issue 2, pp 519–583 | Cite as

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

  • Christian Gérard
  • Omar Oulghazi
  • Michał WrochnaEmail author
Article

Abstract

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 spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Christian Gérard
    • 1
  • Omar Oulghazi
    • 1
  • Michał Wrochna
    • 2
    Email author
  1. 1.Département de MathématiquesUniversité Paris-Sud XIOrsay CedexFrance
  2. 2.Université Grenoble Alpes, Institut Fourier, UMR 5582 CNRS, CS 40700Grenoble Cedex 09France

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