Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

Daudé, Thierry; Gobin, Damien; Nicoleau, François

doi:10.48550/arxiv.1310.0733

Cited by 8 publications

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“…Actually, it is enough to prove our uniqueness result to assume that the reflection coefficients are known up to an error O(e −2nB ) for some B ∈]0, min(A, Ã)[. Indeed, we can use the idea of [10] which proves a local inverse scattering result at a fixed energy in spherically symmetric asymptotically hyperbolic manifolds.…”

Section: The Scattering Matrix and Statement Of The Resultsmentioning

confidence: 99%

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

Gobin¹

2015

Inverse Problems

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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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Section: The Scattering Matrix and Statement Of The Resultsmentioning

confidence: 99%

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

Gobin¹

2015

Inverse Problems

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“…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).…”

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confidence: 96%

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

Daudé

Nicoleau

2015

Ann. Henri Poincaré

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We study inverse scattering problems at a fixed energy for radial Schrödinger operators on R n , n ≥ 2. First, we consider the class A of potentials q(r) which can be extended analytically in. If q andq are two such potentials and if the corresponding phase shifts δ l andδ l are super-exponentially close, then q =q. Secondly, we study the class of potentials q(r) which can be split into q(r) = q1(r) + q2(r) such that q1(r) has compact support and q2(r) ∈ A. If q andq are two such potentials, we show that for any fixed a > 0,when l → +∞ if and only if q(r) =q(r) for almost all r ≥ a. The proofs are close in spirit with the celebrated Borg-Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in ℜz ≥ 0 with | q(z) |≤ C (1+ | z |) −ρ , ρ > 1 , we show that the Regge poles are confined in a vertical strip in the complex plane.

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“…Hence, the sectional curvature of σ tends to the constant negative values −κ 2 ± on the corresponding ends {x → ±∞}. Such spherically symmetric manifolds are very particular cases of the much broader class of asymptotically hyperbolic manifolds (see references in [11]). We mention also [40] for a very general analysis of meromorphic continuation for de Sitter black holes and perturbations.…”

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confidence: 99%

“…in the exterior region of de Sitter-Reissner-Nordström black hole takes the same form as a representation of Dirac operator D σ on the so called Spherically Symmetric Asymptotically Hyperbolic (SSAH) Manifolds Σ = R x ×S 2 θ,ϕ (see [11]) equipped with the Riemannian metric…”

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

Iantchenko

2017

Journal of Mathematical Analysis and Applications

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We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-Reissner-Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semiclassical Schrödinger operators from [4]. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.

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Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

Cited by 8 publications

References 0 publications

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

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

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

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

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