Benjamin Florentin | Searcharxiv

arXiv2026

Magnetic spectral inverse problems on compact Anosov manifolds

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.

math.SP↗

arXiv2025

Steklov isospectrality of conformal metrics

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n\geq 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calder\'on problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide. Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schr\"odinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.

math.SP↗