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Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes

About this Title

Thierry Daudé, Laboratoire AGM, Département de Mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France and François Nicoleau, Laboratoire Jean Leray, UMR 6629, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1170
ISBNs: 978-1-4704-2376-6 (print); 978-1-4704-3701-5 (online)
Published electronically: December 28, 2016
Keywords: Inverse scattering, black holes, Dirac equation
MSC: Primary 81U40, 35P25; Secondary 58J50

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Table of Contents


  • 1. Introduction
  • 2. Kerr-Newman-de-Sitter black holes
  • 3. The massless charged Dirac equation
  • 4. The direct scattering problem
  • 5. Uniqueness results in the inverse scattering problem at fixed energy
  • 6. The angular equation and partial inverse result
  • 7. The radial equation: complexification of the angular momentum
  • 8. Large $z$ asymptotics of the scattering data
  • 9. The inverse scattering problem
  • A. Growth estimate of the eigenvalues $\mu _{kl}(\lambda )$
  • B. Limiting Absorption Principles and scattering theory for $H_0$ and $H$


In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitly by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.

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