Magnetic spectral inverse problems on compact Anosov manifolds

TL;DR

This paper proves that on certain negatively curved manifolds, the spectrum of magnetic Schrödinger operators and Steklov maps uniquely determine magnetic and electric potentials, extending previous inverse spectral results to magnetic settings.

Contribution

It establishes new inverse spectral results for magnetic potentials on Anosov manifolds with boundary, including boundary determination and uniqueness of potentials from spectral data.

Findings

Magnetic and electric potentials are recoverable from the spectrum on closed Anosov manifolds.

The magnetic Steklov spectrum determines the full Taylor series of potentials at the boundary.

Unique determination of analytic magnetic and electric potentials from spectral data.

Abstract

In this paper, we establish positive results for two spectral inverse problems in the presence of a magnetic potential. Exploiting the principal wave trace invariants, we first observe that on closed Anosov manifolds with simple length spectrum, one can recover an electric and a magnetic (up to a natural gauge) potential from the spectrum of the associated magnetic Schr\"odinger operator. This simple observation extends a particular instance of a recent positive result on the spectral inverse problem for the Bochner Laplacian in negative curvature, obtained by M.Ceki\'c and T$.$Lefeuvre (2023)$.$ Similarly, we prove that the spectrum of the magnetic Dirichlet-to-Neumann map (or magnetic Steklov operator) on a compact Riemannian manifold with boundary determines both a magnetic potential (up to gauge) and an electric potential at the boundary, provided the latter is Anosov with simple length spectrum. Under this assumption, one can actually show that the magnetic Steklov spectrum determines the full Taylor series at the boundary of any smooth magnetic field and electric potential. As a simple consequence, in this case, both an analytic magnetic field and an analytic electric potential are uniquely determined by their Steklov spectrum.