Inverse Scattering at Fixed Energy in de Sitter–Reissner–Nordström Black Holes

Abstract

In this paper, we consider massless Dirac fields propagating in the outer region of de SitterReissner-Nordström black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ = 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ = 0 is the fixed energy and n ∈ N * denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M , the square of the charge Q 2 and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T (λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ∈ L where L is a subset of N * that satisfies the Müntz condition n∈L 1 n = +∞. Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the "unphysical" scattering matrix S(λ, z) in the complex variable z. We show in particular that the quantitiesbelong to the Nevanlinna class in the region {z ∈ C, Re(z) > 0} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstrution formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.

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

“…5. Contrary to [13], the case of a zero energy is not an obstruction for our inverse problem if we suppose that the electric charge q of the Dirac fields is non vanishing.…”

“…We follow the strategy of [7,8,13]. Considering the differential equations given in Proposition 3.5, we use a Liouville transformation, i.e.…”

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

“…This same approach has been used recently to study scattering inverse problems for asymptotically hyperbolic manifolds (see [11,12,10]). In the hyperbolic setting, a Liouville transformation changes the angular momentum variable in a spectral variable, and we can see the Jost solutions as suitable perturbations of the modified Bessel functions I ν (z).…”

Resonance expansions of massless Dirac fields propagating in the exterior of a de Sitter–Reissner–Nordström black hole

“…where the one-dimensional Dirac operator D l,m does not depend on index m. Now, the scattering of massless charged Dirac fields in de Sitter-Reissner-Nordström black holes is described (see [13]) by the scattering on the line for the massless Dirac system…”

“…Recall that the energy is conserved along the evolution (6) By orthogonal decomposition (13) and (14) we get also the spaces…”