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

In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dinger equation to prove \emph{local} uniqueness results in the setting of inverse \emph{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 $T(\lambda,n)$ and reflection coefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixed energy $\lambda$ and angular momentum $n$. For instance, the reflection coefficients $L(\lambda,n)$ correspond to the scattering experiment in which a wave is sent from the \emph{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 $\lambda \not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp. $R(\lambda,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with B\textgreater{}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.Comment: 24 page

Inverse scattering at fixed energy on three-dimensional asymptotically hyperbolic Stäckel manifolds

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

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

International audienceIn this paper, we consider massive charged Dirac fields propagating in the exterior region of de Sitter-Reissner-Nordstr\"{o}m black holes.We show that the parameters of such black holes are uniquely determined by the partial knowledge of the corresponding scattering operator $S(\lambda)$ at a fixed energy $\lambda$.More precisely, we consider the partial wave scattering operators $S(\lambda,n)$ (here $\lambda \in \mathbb{R}$ is the energy and $n \in \mathbb{N}^{\star}$ 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 $S(\lambda,n)$, for all $n \in \mathcal{L}$, where $\mathcal{L}$ is a subset of $\mathbb{N}^{\star}$ that satisfies the M\"{u}ntz condition $\sum_{n \in \mathcal{L}} \frac{1}{n} = + \infty$, 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 $z$

Étude de problèmes de diffusion inverse à énergie fixée pour des variétés asymptotiquement hyperboliques

We study inverse scattering problems at fixed energy for different geometries with more or less symmetries. First, we obtain a local inverse scattering result at fixed energy for the massless and chargeless Dirac equation on asymptotically hyperbolic manifolds with spherical symmetry. In a second chapter, we are interested in Reissner-Nordström-de Sitter black holes which are spherically symmetric and electrically charged solutions of the Einstein equation. We then obtain an inverse scattering result at fixed energy for the massive and charged Dirac equation. Finally, we are interested in Stäckel manifolds of dimension three with the topology of a toric cylinder, satisfying the Robertson condition and endowed with an asymptotically hyperbolic structure. On these manifolds we use the variable separation theory for the Helmholtz equation and a multivariable version of the method of Complexification of the Angular Momentum in order to obtain an inverse scattering result at fixed energy.On étudie des problèmes de diffusion inverse à énergie fixée pour différents types de géométries ayant plus ou moins de symétries. On commence par obtenir un résultat de diffusion inverse local à énergie fixée pour l'équation de Dirac sans masse et sans charge sur des variétés asymptotiquement hyperboliques et à symétrie sphérique. Dans un second chapitre on s'intéresse aux trous noirs de type Reissner-Nordström-de Sitter qui sont des solutions à symétrie sphérique et électriquement chargées de l'équation d'Einstein. On obtient alors un résultat de diffusion inverse à énergie fixée pour l'équation de Dirac massive et chargée. Enfin, on s'intéresse à des variétés de Stäckel de dimension trois ayant la topologie d'un cylindre torique, satisfaisant la condition de Robertson et munies d'une structure asymptotiquement hyperbolique. Sur ces variétés on utilise la théorie de séparation des variables pour l'équation de Helmholtz et une version multivariable de la méthode de Complexification du Moment Angulaire afin d'obtenir un résultat de diffusion inverse à énergie fixée

Inverse scattering at fixed energy for radial magnetic Schrödinger operators with obstacle in dimension two

We study an inverse scattering problem at fixed energy for radial magnetic Schrödinger operators on R^2 \ B(0, r_0), where r_0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and (\tilde{V} , \tilde{B}) are two couples belonging to C, we then show that if the corresponding phase shifts δ_l and \tilde{δ}_l (i.e. the scattering data at fixed energy) coincide for all l ∈ L, where L ⊂ N^⋆ satisfies the Müntz condition \sum_{l∈L} \frac{1}{l} = +∞, then V (x) = \tilde{V}(x) and B(x) = \tilde{B}(x) outside the obstacle B(0, r_0). The proof use the Complex Angular Momentum method and is close in spirit to the celebrated Börg-Marchenko uniqueness Theorem

Comparison of 3 diffractive IOLs: one monofocal (achromatic), one bifocal and one trifocal lens implant

Purpose: To describe the optical performance of 3 types (monofocal, bifocal and trifocal) of diffractive hydrophobic intraocular lenses (IOLs). Setting: Fondation Rothschild, Paris, France Methods: An achromatic monofocal, a bifocal and trifocal diffractive IOLs were measured with an optical bench, designed to measure Modulation transfer function (MTF) and point spread function (PSF) of diffractive intraocular lenses. The measurements were performed at 3 wavelengths (480 nm, 546 nm and 650 nm), using an aberration free cornea and an aspherical artificial cornea generating a +0.28 µm of positive spherical aberration (ISO 11979-2 guideline). The through-focus MTF was recorded with the 3 IOLs for various pupil apertures. The evaluation of the magnitude of the halos was performed from PSF measurement obtained at each focal spot location. Results: The monofocal and the two bifocal IOLs were presenting 2 peaks on the through-focus MTF in the green: for intermediate vision (at 1.75D) for the monofocal achromatic lens, and for far and for near vision( at +4D) for the bifocal IOL of the same material. Three peaks were recorded with the trifocal IOL at any wavelength. The achromatic monofocal was monofocal for far in the red and monofocal for near in the blue. The amount of energy allocated to each focal point was not very different for the three IOLs except for smaller apertures where the distance vision with the trifocal IOL had a significant lower peak than the two other IOLs. The amount of halos were comparable between lenses. Conclusions: The diffractive and refractive optics generated opposite chromatic aberration. A achromatic diffractive IOL behaves like a bifocal IOL with an intermediate addition The bifocal and trifocal IOL induce similar chromatic effects for the near (bifocal, trifocal) and intermediate (trifocal) foci. The tested diffractive patterns wee not efficient to correct the chromatic aberration at the distance foci for all tested IOLs

Comparison of 3 Diffractive IOLs in 3 Wavelengths Bifocal/ EDOF/Trifocal.

Purpose To describe the optical performances of 3 types of diffractive hydrophobic intraocular lenses (IOLs): monofocal achromatic, bifocal and trifocal, using various visible wavelengths. Methods 3 IOLs were measured with an optical bench (PMTF,LambdaX,Nivelles, Belgium) to determine modulation transfer function (MTF) and point spread function (PSF). Measurements were performed at 480 nm,546 nm,650 nm using an aberration free cornea and aspheric artificial cornea generating +0.28µm positive spherical aberration by ISO11979-2 guidelines. Through-focus MTF was recorded for various pupil apertures. Evaluation of halos was performed from PSF measurements obtained at each focal spot location. Images of USAF targets were recorded with simulated distances from 1m to 25cm. Amplitude of off-axis peaks of radial profile of PSF enabled to quantify percentage of energy within the halos. Results The monofocal and the bifocal IOLs presented 2 peaks on the through-focus MTF in green light: intermediate vision (at +1.75D) for the monofocal achromatic lens, and far and near vision (at +4D) for the bifocal IOL of the same material. Three peaks (Distance, Intermediate at +1.75D, Near at +3.5D) were recorded with the trifocal IOL at any wavelength. The achromatic monofocal IOL was monofocal for far vision with red light and monofocal for near vision with blue light. The influence of the cornea models was limited. The amount of halos were comparable between lenses (3 to 5 % of the enclosed central light energy). Conclusion The achromatic diffractive IOL behaved like a bifocal IOL with an intermediate addition foci in green light. The bifocal and trifocal IOL induced similar chromatic effects for the near (bifocal, trifocal) and intermediate (trifocal) foci