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.

After giving a general introduction to the main known results on the anisotropic Calderón problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5,6] for the anisotropic Calderón problem at fixed frequency, in dimension n ≥ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n ≥ 3, we first show that there exist in the conformal class of g an infinite number of Riemannian metricsg such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency coincide when the Dirichlet data ΓD and Neumann data ΓN are measured on disjoint sets and satisfy ΓD ∪ ΓN = ∂M . The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe type on (M, g) and arise from a natural but subtle gauge invariance of the Calderón when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n ≥ 3 to the anisotropic Calderón problem at fixed frequency with data on disjoint sets, which do not arise from this gauge invariance. They are given by cylindrical Riemannian manifolds with boundary having two ends, equipped with a suitably chosen warped product metric. This survey concludes with some remarks on the case of manifolds with corners.

A direct and an inverse scattering method is developed for Schrödinger operators with electromagnetic fields in the case of obstacles in order to study the well-known Aharonov–Bohm effect. In dimension greater or equal to three, we show that the electric potential and the magnetic field are uniquely determined by the S-operator. In the two-dimensional case, some obstruction appears based on a quantification of the magnetic flux.