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Quantitative Borg-Levinson theorem for the magnetic Schödinger operator with unbounded electrical potential

Mourad Choulli, Hiroshi Takase

TL;DR

This work extends the quantitative Borg-Levinson framework to magnetic Schrödinger operators with optimal unbounded potentials in $L^{\frac{n}{2}}$, addressing both isotropic and anisotropic (geometric) settings. It develops a robust spectral-resolvent-DtN pipeline: spectral analysis of the magnetic Schrödinger operator, resolvent estimates, and a boundary data–to–DtN map reduction, culminating in integral identities that connect BSD to interior quantities. The main contributions are Hölder stability results for recovering the electric potential and the magnetic field from boundary spectral data, with precise exponents depending on dimension and geometry, and a detailed treatment on simple manifolds via geodesic ray transforms. These results advance inverse spectral theory by handling unbounded potentials and extending to anisotropic manifolds through CGO constructions and transport equations, offering potential impact in quantum mechanics and PDE-based imaging where boundary measurements are the primary data source.

Abstract

The first author established in [8] a quantitative Borg-Levinson theorem for the Schrödinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schrödinger operator. We discuss both the isotropic and anisotropic cases. We establish Hölder stability inequalities of determining the electrical potential or magnetic field from the corresponding boundary spectral data.

Quantitative Borg-Levinson theorem for the magnetic Schödinger operator with unbounded electrical potential

TL;DR

This work extends the quantitative Borg-Levinson framework to magnetic Schrödinger operators with optimal unbounded potentials in , addressing both isotropic and anisotropic (geometric) settings. It develops a robust spectral-resolvent-DtN pipeline: spectral analysis of the magnetic Schrödinger operator, resolvent estimates, and a boundary data–to–DtN map reduction, culminating in integral identities that connect BSD to interior quantities. The main contributions are Hölder stability results for recovering the electric potential and the magnetic field from boundary spectral data, with precise exponents depending on dimension and geometry, and a detailed treatment on simple manifolds via geodesic ray transforms. These results advance inverse spectral theory by handling unbounded potentials and extending to anisotropic manifolds through CGO constructions and transport equations, offering potential impact in quantum mechanics and PDE-based imaging where boundary measurements are the primary data source.

Abstract

The first author established in [8] a quantitative Borg-Levinson theorem for the Schrödinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schrödinger operator. We discuss both the isotropic and anisotropic cases. We establish Hölder stability inequalities of determining the electrical potential or magnetic field from the corresponding boundary spectral data.
Paper Structure (24 sections, 22 theorems, 366 equations)

This paper contains 24 sections, 22 theorems, 366 equations.

Table of Contents

  1. Introduction
  2. Main notations
  3. Statement of the main results
  4. Spectral analysis and resolvent estimates
  5. Spectral analysis of the magnetic Schrödinger operator
  6. Regularity of eigenfunctions
  7. Resolvent estimates
  8. From BSD to a family of DtN maps
  9. DtN maps
  10. BSD determines uniquely a family of DtN maps
  11. The inverse spectral problem
  12. Integral formula
  13. Proof of Theorem \ref{['thm1']}
  14. Proof of Theorem \ref{['thm2']}
  15. Introduction
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $a\in \mathcal{A}_0$, $(V_1,V_2)\in \mathcal{V}_0$ and $b_j=(a,V_j)$, $j=1,2$, and assume that $\delta_+(b_1,b_2)<\infty$. Then we have where and $\mathbf{c}_\ast=\mathbf{c}_\ast(n,\Omega,\Omega_0,a,V_0,W_0,\mathfrak{c},\tilde{\mathfrak{c}},t)>0$ is a constant.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • proof
  • Corollary 1.2
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 23 more