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 at a fixed energy λ. More precisely, we consider the partial wave scattering operators (here is the energy and 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 , for all , where is a subset of that satisfies the Müntz condition , makes it possible to recover the mass, the charge, and the cosmological constant of a de Sitter–Reissner–Nordström 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.