**Memoirs of the American Mathematical Society**

2016;
113 pp;
Softcover

MSC: Primary 81; 35;
Secondary 58

**Print ISBN: 978-1-4704-2376-6
Product Code: MEMO/247/1170**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

**Electronic ISBN: 978-1-4704-3701-5
Product Code: MEMO/247/1170.E**

List Price: $75.00

AMS Member Price: $45.00

MAA Member Price: $67.50

# Direct and Inverse Scattering at Fixed Energy for Massless Charged Dirac Fields by Kerr-Newman-de Sitter Black Holes

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*Thierry Daudé; François Nicoleau*

In this paper, the authors 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, they 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. The authors 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, the authors 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, the authors are finally able to determine uniquely
the metric of the black hole from the knowledge of the scattering
matrix at a fixed energy.

#### Table of Contents

# Table of Contents

## Direct and Inverse Scattering at Fixed Energy for Massless Charged Dirac Fields by Kerr-Newman-de Sitter Black Holes

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Kerr-Newman-de-Sitter black holes 1320
- Chapter 3. The massless charged Dirac equation 1724
- Chapter 4. The direct scattering problem 2734
- Chapter 5. Uniqueness results in the inverse scattering problem at fixed energy 5562
- Chapter 6. The angular equation and partial inverse result 6168
- Chapter 7. The radial equation: complexification of the angular momentum 6774
- Chapter 8. Large 𝑧 asymptotics of the scattering data 7986
- Chapter 9. The inverse scattering problem 93100
- Appendix A. Growth estimate of the eigenvalues \muk 97104
- Appendix B. Limiting Absorption Principles and scattering theory for 𝐻₀ and 𝐻 101108
- Bibliography 111118
- Back Cover Back Cover1126