Inverse spectral theory for perturbed torus

TL;DR

This paper develops an inverse spectral theory for Laplacians on rotationally symmetric tori, establishing a bijective relationship between geometric profiles and spectral data, along with stability estimates.

Contribution

It introduces an infinite dimensional analytic isomorphism linking torus profiles to spectral data, advancing inverse spectral analysis for these manifolds.

Findings

Constructed an analytic isomorphism between profiles and spectral data.

Established stability estimates for the inverse problem.

Extended inverse spectral theory to rotationally symmetric manifolds.

Abstract

We consider an inverse problem for Laplacians on rotationally symmetric manifolds, which are finite for the transversal direction and periodic with respect to the axis of the manifold, i.e., Laplacians on tori. We construct an infinite dimensional analytic isomorphism between the space of profiles (the radius of the rotation) of the torus and the spectral data as well as the stability estimates: those for the spectral data in terms of the profile and conversely, for the profile in term of the spectral data.