Inverse Regge poles problem on a warped ball

TL;DR

This paper investigates the inverse problem of determining the warping function of warped ball manifolds using Regge poles, which are linked to the spectral properties of a related Schrödinger equation, establishing uniqueness results.

Contribution

It introduces the concept of Regge poles for warped balls and proves their ability to uniquely identify the warping function, advancing inverse spectral geometry methods.

Findings

Regge poles are characterized as eigenvalues and resonances of a Schrödinger operator.

Asymptotic localization of Regge poles in the complex plane is established.

Regge poles uniquely determine the warping function of the manifold.

Abstract

In this paper, we study a new type of inverse problem on warped product Riemannian manifolds with connected boundary that we name warped balls. Using the symmetry of the geometry, we first define the set of Regge poles as the poles of the meromorphic continuation of the Dirichlet-to-Neumann map with respect to the complex angular momentum appearing in the separation of variables procedure. These Regge poles can also be viewed as the set of eigenvalues and resonances of a one-dimensional Schr\"odinger equation on the half-line, obtained after separation of variables. Secondly, we find a precise asymptotic localisation of the Regge poles in the complex plane and prove that they uniquely determine the warping function of the warped balls.