eBook ISBN:  9781470437015 
Product Code:  MEMO/247/1170.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470437015 
Product Code:  MEMO/247/1170.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 247; 2016; 113 ppMSC: Primary 81; 35; Secondary 58
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 KerrNewmande Sitter black hole. In the first part, they establish the existence and asymptotic completeness of timedependent wave operators associated to our Dirac fields. This leads to the definition of the timedependent scattering operator that encodes the farfield 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 timedependent 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

Chapters

1. Introduction

2. KerrNewmandeSitter 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 $H$


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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 KerrNewmande Sitter black hole. In the first part, they establish the existence and asymptotic completeness of timedependent wave operators associated to our Dirac fields. This leads to the definition of the timedependent scattering operator that encodes the farfield 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 timedependent 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.

Chapters

1. Introduction

2. KerrNewmandeSitter 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 $H$