H\"older stability of an inverse spectral problem for the magnetic Schr\"odinger operator on a simple manifold

TL;DR

This paper establishes H"older stability in recovering electric and magnetic potentials on a simple manifold from boundary spectral data, linking spectral information to geometric inverse problems.

Contribution

It proves H"older stability for inverse spectral problems for magnetic Schr"odinger operators on simple manifolds, connecting spectral data to potential recovery.

Findings

Boundary spectral data can be stably derived from the Dirichlet-to-Neumann map.

Stable inversion of the geodesic ray transform is achieved for lower order terms.

The method links spectral data to geometric inverse problems.

Abstract

We show that on a simple Riemannian manifold, the electric potential and the solenoidal part of the magnetic potential appearing in the magnetic Schr\"odinger operator can be recovered H\"older stably from the boundary spectral data. This data contains the eigenvalues and the Neumann traces of the corresponding sequence of Dirichlet eigenfunctions of the operator. Our proof contains two parts, which we present in the reverse order. (1) We show that the boundary spectral data can be stably obtained from the Dirichlet-to-Neumann map associated with the respective initial boundary value problem for a hyperbolic equation, whose leading order terms are a priori known. (2) We construct geometric optics solutions to the hyperbolic equation, which reduce the stable recovery of the lower order terms to the stable inversion of the geodesic ray transform of one-forms and functions.