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An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

Authors
  • Finster, F.1
  • Kamran, N.2
  • Smoller, J.3
  • Yau, S.-T.4
  • 1 Universität Regensburg, NWF I – Mathematik, Regensburg, 93040, Germany , Regensburg
  • 2 McGill University, Department of Math. and Statistics, Montréal, Québec, H3A 2K6, Canada , Montréal
  • 3 The University of Michigan, Mathematics Department, Ann Arbor, MI, 48109, USA , Ann Arbor
  • 4 Harvard University, Mathematics Department, Cambridge, MA, 02138, USA , Cambridge
Type
Published Article
Journal
Communications in Mathematical Physics
Publisher
Springer-Verlag
Publication Date
Aug 31, 2005
Volume
260
Issue
2
Pages
257–298
Identifiers
DOI: 10.1007/s00220-005-1390-x
Source
Springer Nature
Keywords
License
Yellow

Abstract

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.

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