Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes
著者
書誌事項
Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes
（Memoirs of the American Mathematical Society, no. 1170）
American Mathematical Society, c2016
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注記
"Volume 247, number 1170 (third of 7 numbers), May 2017"
Includes bibliographical references (p. 111-113)
内容説明・目次
内容説明
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.
目次
Introduction
Kerr-Newman-de-Sitter black holes
The massless charged Dirac equation
The direct scattering problem
Uniqueness results in the inverse scattering problem at fixed energy
The angular equation and partial inverse result
The radial equation: complexification of the angular momentum
Large $z$ asymptotics of the scattering data
The inverse scattering problem
Appendix A. Growth estimate of the eigenvalues $\mu_{kl}(\lambda)$
Appendix B. Limiting Absorption principles and scattering theory for $H_0$ and $H$
Bibliography
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