Inverse scattering at fixed energy on three-dimensional asymptotically hyperbolic St{ä}ckel manifolds

Abstract

In this paper, we study an inverse scattering problem at fixed energy on three-dimensional asymptotically hyperbolic St{ä}ckel 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 $2\times 2$ 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 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 St{ä}ckel manifold up to natural obstructions.

arXiv commentarXiv admin note: text overlap with arXiv:1409.6229, arXiv:1510.06559 by other authors

References in corpus (5)

• The Calderon problem with partial data on manifolds and applications
• Calderon inverse Problem with partial data on Riemann Surfaces
• Weyl-Titchmarsh Theory for Schroedinger Operators with Strongly Singular Potentials
• Separability in Riemannian Manifolds
• Multivariable approximate Carleman-type theorems for complex measures

Cited by in corpus (1)

• Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations