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Magnetic spectral inverse problems on compact Anosov manifolds

David dos Santos Ferreira, Benjamin Florentin

TL;DR

This work addresses how magnetic effects modify classical spectral inverse problems on compact manifolds. By exploiting principal wave trace invariants and an adapted transport equation analysis, it proves that on closed Anosov manifolds with simple length spectrum, the spectrum of the magnetic Schrödinger operator $P_{a,q}$ uniquely determines the electric potential $q$ and the magnetic potential $a$ up to gauge, while on manifolds with boundary, the magnetic Steklov spectrum determines boundary jets of $a$ and $q$, and, under analyticity, fixes them entirely at the boundary. The results rely on a refined wave-trace framework (Duistermaat–Guillemin/Flo) together with a boundary-jet induction using trace formulas for magnetic Dirichlet-to-Neumann maps, and they reveal how the Aharonov–Bohm-type gauge freedom interacts with spectral data. Collectively, they extend positive inverse results to magnetic settings and build a gauge-aware boundary-detection theory that connects geometric optics, X-ray transforms, and boundary control in the presence of magnetic fields.

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 show 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 Schrödinger operator. This extends a particular instance of a recent positive result on the spectral inverse problem for the Bochner Laplacian in negative curvature, obtained by M.Cekić and T.Lefeuvre (2023). Similarly, we prove that the spectrum of the magnetic Dirichlet-to-Neumann map (or Steklov operator) determines at the boundary both a magnetic potential, up to gauge, and an electric potential, provided the boundary 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.

Magnetic spectral inverse problems on compact Anosov manifolds

TL;DR

This work addresses how magnetic effects modify classical spectral inverse problems on compact manifolds. By exploiting principal wave trace invariants and an adapted transport equation analysis, it proves that on closed Anosov manifolds with simple length spectrum, the spectrum of the magnetic Schrödinger operator uniquely determines the electric potential and the magnetic potential up to gauge, while on manifolds with boundary, the magnetic Steklov spectrum determines boundary jets of and , and, under analyticity, fixes them entirely at the boundary. The results rely on a refined wave-trace framework (Duistermaat–Guillemin/Flo) together with a boundary-jet induction using trace formulas for magnetic Dirichlet-to-Neumann maps, and they reveal how the Aharonov–Bohm-type gauge freedom interacts with spectral data. Collectively, they extend positive inverse results to magnetic settings and build a gauge-aware boundary-detection theory that connects geometric optics, X-ray transforms, and boundary control in the presence of magnetic fields.

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 show 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 Schrödinger operator. This extends a particular instance of a recent positive result on the spectral inverse problem for the Bochner Laplacian in negative curvature, obtained by M.Cekić and T.Lefeuvre (2023). Similarly, we prove that the spectrum of the magnetic Dirichlet-to-Neumann map (or Steklov operator) determines at the boundary both a magnetic potential, up to gauge, and an electric potential, provided the boundary 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.
Paper Structure (21 sections, 23 theorems, 230 equations)

This paper contains 21 sections, 23 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Spectral inverse problems on compact manifolds
  3. Notations
  4. The magnetic Schrödinger operator
  5. The magnetic DN map
  6. Main results
  7. Wave trace formulas
  8. A transport equation with real-valued potential
  9. The two-dimensional case
  10. The higher-dimensional case
  11. The Hamiltonian vector field
  12. Algebra on the cosphere bundle
  13. The equations
  14. The inverse problem for magnetic Schrödinger operators
  15. Gauge invariance
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian manifold of dimension $n \geq 2$. Assume that the cogeodesic flow $\phi_t$ is Anosov with simple length spectrum. If the spectra of the magnetic Schrödinger operators $P_{a,q}$ and $P_{\tilde{a},\tilde{q}}$ coincide then the electric potentials are equal, and the magnetic potentials are gauge-equivalent, namely there is a smooth function $\vartheta \in C^{\inft

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: DG
  • Corollary 2.2
  • Theorem 2.3: Flo
  • Corollary 2.4
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 35 more