Inverse scattering at fixed energy for massive charged Dirac fields in de Sitter-Reissner-Nordström black holes

Abstract

In this paper, we consider massive charged Dirac fields propagating in the exterior region of de Sitter-Reissner-Nordström black holes.We show that the parameters of such black holes are uniquely determined by the partial knowledge of the corresponding scattering operator $S(\lambda )$ at a fixed energy $\lambda$.More precisely, we consider the partial wave scattering operators $S(\lambda ,n)$ (here $\lambda \in \mathbb{R}$ is the energy and $n\in {\mathbb{N}}^{\star }$ denotes the angular momentum) defined as the restrictions of the full scattering operator on a well chosen basis of spin-weighted spherical harmonics.We prove that the knowledge of the scattering operators $S(\lambda ,n)$, for all $n\in \mathcal{L}$, where $\mathcal{L}$ is a subset of ${\mathbb{N}}^{\star }$ that satisfies the Müntz condition $\sum \_n\in \mathcal{L}\frac{1}{n}=+\infty$, allows to recover the mass, the charge and the cosmological constant of a dS-RN black hole.The main tool consists in the complexification of the angular momentum $n$ and in studying the analytic properties of the "unphysical" corresponding data in the complex variable $z$.

arXiv admin note: text overlap with arXiv:1004.1466, arXiv:0904.2064 by other authors