Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of $2 \times 2$ 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 for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we show that the knowledge of the reflection operators at a fixed non zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration.Comment: 39 p

Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes

40 pagesIn this paper, we consider massless Dirac fields propagating in the outer region of de Sitter-Reissner-Nordström black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix $S(\lambda)$ at a fixed energy $\lambda \ne 0$. More precisely, we consider the partial wave scattering matrices $S(\lambda,n)$ (here $\lambda \ne 0$ is the fixed energy and $n \in \N^*$ 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 $\Lambda$ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients $T(\lambda, n)$, or the reflexion coefficients $R(\lambda, n)$ (resp. $L(\lambda, n)$), for all $n \in {\mathcal{L}}$ where $\mathcal{L}$ is a subset of $\N^*$ that satisfies the Müntz condition $\sum_{n \in {\mathcal{L}}} \frac{1}{n} = +\infty$. Our main tool consists in complexifying the angular momentum $n$ and in studying the analytic properties of the "unphysical" scattering matrix $S(\lambda,z)$ in the complex variable $z$. We show in particular that the quantities $\frac{1}{T(\lambda,z)}$, $\frac{R(\lambda,z)}{T(\lambda,z)}$ and $\frac{L(\lambda,z)}{T(\lambda,z)}$ belong to the Nevanlinna class in the region \{z \in \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 in de Sitter-Reissner-Nordstr\"om black hole spacetimes

In this paper, we study the inverse scattering of massive charged Dirac fields in the exterior region of (de Sitter)-Reissner-Nordstr\"om black holes. First we obtain a precise high-energy asymptotic expansion of the diagonal elements of the scattering matrix (i.e. of the transmission coefficients) and we show that the leading terms of this expansion allows to recover uniquely the mass, the charge and the cosmological constant of the black hole. Second, in the case of nonzero cosmological constant, we show that the knowledge of the reflection coefficients of the scattering matrix on any interval of energy also permits to recover uniquely these parameters.Comment: 44 page

Inverse scattering in de Sitter-Reissner-Nordström black hole spacetimes

44 pagesIn this paper, we study the inverse scattering of massive charged Dirac fields in the exterior region of (de Sitter)-Reissner-Nordström black holes. First we obtain a precise high-energy asymptotic expansion of the diagonal elements of the scattering matrix (i.e. of the transmission coefficients) and we show that the leading terms of this expansion allows to recover uniquely the mass, the charge and the cosmological constant of the black hole. Second, in the case of nonzero cosmological constant, we show that the knowledge of the reflection coefficients of the scattering matrix on any interval of energy also permits to recover uniquely these parameters

Local H\"older Stability in the Inverse Steklov and Calder\'on Problems for Radial Schr\"odinger operators and Quantified Resonances

We obtain H\"older stability estimates for the inverse Steklov and Calder\'on problems for Schr\"odinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in \cite{DKN5} in the case of the the Schr\"odinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: i) A formula relating the difference of the Steklov spectra of the Schr\"odinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon \cite{Si1} in his representation formula for the Weyl-Titchmarsh function, and ii) A key moment stability estimate due to Still \cite{St}. It is noteworthy that with respect to the original Schr\"odinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.Comment: To appear in Annales Henri Poincar\'