In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schrödinger equation to prove local uniqueness results in the setting of inverse metric problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients T (λ, n) and reflection coefficients L(λ, n) and R(λ, n) of a Dirac wave having a fixed energy λ and angular momentum n. For instance, the reflection coefficients L(λ, n) correspond to the scattering experiment in which a wave is sent from the left end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed λ = 0, the knowledge of the reflection coefficients L(λ, n) (resp. R(λ, n)) -up to a precise error term of the form O(e −2nB ) with B > 0 -determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude B of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic Stäckel manifolds having the topology of toric cylinders and satisfying the Robertson condition. On these manifolds the Helmholtz equation can be separated into a system of a radial ODE and two angular ODEs. We can thus decompose the full scattering operator onto generalized harmonics and the resulting partial scattering matrices consist in a countable set of 2 × 2 matrices whose coefficients are the so-called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions associated with the radial ODE. Using a novel multivariable version of the Complex Angular Momentum method, we show that the knowledge of the scattering operator at a fixed non-zero energy is enough to determine uniquely the metric of the three-dimensional Stäckel manifold up to natural obstructions.R ij = 0, ∀i = j.Remark 1.4. We note that the Robertson condition is satisfied for Einstein manifolds. Indeed, an Einstein manifold is a riemannian manifold whose Ricci tensor is proportional to the metric which is diagonal in the orthogonal case we study.As shown by Eisenhart in [29,30] and by Kalnins and Miller in [44] the separation of the Hamilton-Jacobi equation for the geodesic flow is related to the existence of Killing tensors of order two (whose presence highlights the presence of hidden symmetries). We thus follow [3,44] in order to study this relation. We use the natural symplectic structure on the cotangent bundle T M of the manifold (M, g). Let {x i } be local coordinates on M and {x i , p i } the associated coordinates on T M. Let

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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(λ) at a fixed energy λ. More precisely, we consider the partial wave scattering operators S(λ, n) (here λ ∈ R is the energy and n ∈ N ⋆ 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(λ, n), for all n ∈ L, where L is a subset of N ⋆ that satisfies the Müntz condition n∈L 1 n = +∞, 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.

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