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

TL;DR

This paper demonstrates that the scattering operator at a fixed energy uniquely determines the metric of three-dimensional asymptotically hyperbolic Stäckel manifolds, using a novel multivariable Complex Angular Momentum method.

Contribution

It introduces a new multivariable Complex Angular Momentum method to solve the inverse scattering problem on these manifolds, establishing uniqueness results.

Findings

Scattering operator at fixed energy determines the manifold's metric.

Reflection coefficients are generalized Weyl-Titchmarsh functions.

The method applies to manifolds with toric cylinder topology.

Abstract

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