Inverse Scattering at Fixed Energy on Three-Dimensional Asymptotically Hyperbolic Stäckel Manifolds

Abstract

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

The anisotropic Calder{ó}n problem on 3-dimensional conformally St{ä}ckel manifolds

“…Theorem 1.1 solves positively the uniqueness issue in the Calderón problem on conformally Stäckel manifolds. It is an extension of the results in [33] where the inverse scattering problem at a fixed energy on Stäckel asymptotically hyperbolic manifolds was considered.…”

“…Though a complete answer of the inverse scattering problem at a fixed energy is not known in that setting, some interesting and important results in that direction have been proved in [49,69,47,48,40,41] for some general asymptotically hyperbolic manifolds. However, a complete positive answer was found in [22,24,14,19,33] for some Riemannian asymptotically hyperbolic manifolds or de-Sitter like black holes spacetimes having a sufficient amount of (hidden) symmetries.…”

“…In this paper, we wish to continue the series of works [22,24,14,19,33] by studying the anisotropic Calderón problem on conformally Stäckel manifolds, a class of n-dimensional completely integrable Riemannian manifolds with the property that the Laplace-Beltrami operator possess n − 1 commuting second order conformal symmetry operators that allows to solve the corresponding Laplace PDE by separation of variables. Important to say in this introduction are the facts that :…”

Inverse Scattering at Fixed Energy for Radial Magnetic Schrödinger Operators with Obstacle in Dimension Two

“…We mention that this tool was already used in the field of inverse problems for one angular momentum in [11,21,32] for Schrödinger operators, in [9,10] in the context of general relativity, in [6,7] on asymptotically hyperbolic manifolds and in [8] to study counterexamples for the Calderón problem which is closely related to inverse scattering problems at fixed energy on asymptotically hyperbolic manifolds. Moreover, this method was also used for two angular momenta in [16] and we note that it is also a useful tool in high energy physics (see [5]). This work is a continuation and is really close to the spirit of the paper [11] of Daudé and Nicoleau in which the authors treat the same question for the Schrödinger operators with no magnetic fields in all dimensions and for particular classes of radial potentials.…”

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

“…• If r = 3, then 3D Painlevé metrics are in fact Stäckel metrics. According to Examples 2.2 and 2.3, we have the following possible expressions for Stäckel metrics g satisfying the Robertson conditions (see also [21]): • If r = 2 and l 1 + l 2 = 4, then according to Example 2.1, Painlevé metrics that satisfy the generalized Robertson conditions are warped products of the type…”