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Inverse spectral problems for higher-order coefficients

Mourad Choulli

TL;DR

This work addresses inverse spectral problems for higher-order coefficients of elliptic operators using boundary spectral data (BSD). By relating BSD to both elliptic and hyperbolic Dirichlet-to-Neumann maps, the authors derive rigorous uniqueness and stability results for reconstructing conformal factors, Riemannian metrics, conductivities, and potentials, including explicit modulus-type bounds. The approach develops a comprehensive framework: Weyl asymptotics and resolvent representations provide technical control, BSD is shown to determine the full family of elliptic DtN maps and a hyperbolic DtN map, and extensions to conductivity and potential problems yield quantitative stability (often with log-type or Hölder-type rates) under natural geometric and regularity assumptions. The results unify inverse spectral data with boundary maps, offering stable reconstruction formulas and shedding light on limitations and open questions, such as stability in the partial-data regime and the sharpness of the stability exponents. The methods have potential implications for practical inverse problems in anisotropic media and Schrödinger-type systems, linking spectral information to boundary measurements through a robust, map-based calculus.

Abstract

We establish uniqueness and stability inequalities for the problem of determining the higher-order coefficients of an elliptic operator from the corresponding boundary spectral data (BSD). Our analysis relies on the relationship between boundary spectral data and elliptic and hyperbolic Dirichlet to Neumann (DtN) maps. We also show how to adapt our analysis to obtain uniqueness and stability inequalities for determining the conductivity or the potential in an elliptic operator from the corresponding BSD.

Inverse spectral problems for higher-order coefficients

TL;DR

This work addresses inverse spectral problems for higher-order coefficients of elliptic operators using boundary spectral data (BSD). By relating BSD to both elliptic and hyperbolic Dirichlet-to-Neumann maps, the authors derive rigorous uniqueness and stability results for reconstructing conformal factors, Riemannian metrics, conductivities, and potentials, including explicit modulus-type bounds. The approach develops a comprehensive framework: Weyl asymptotics and resolvent representations provide technical control, BSD is shown to determine the full family of elliptic DtN maps and a hyperbolic DtN map, and extensions to conductivity and potential problems yield quantitative stability (often with log-type or Hölder-type rates) under natural geometric and regularity assumptions. The results unify inverse spectral data with boundary maps, offering stable reconstruction formulas and shedding light on limitations and open questions, such as stability in the partial-data regime and the sharpness of the stability exponents. The methods have potential implications for practical inverse problems in anisotropic media and Schrödinger-type systems, linking spectral information to boundary measurements through a robust, map-based calculus.

Abstract

We establish uniqueness and stability inequalities for the problem of determining the higher-order coefficients of an elliptic operator from the corresponding boundary spectral data (BSD). Our analysis relies on the relationship between boundary spectral data and elliptic and hyperbolic Dirichlet to Neumann (DtN) maps. We also show how to adapt our analysis to obtain uniqueness and stability inequalities for determining the conductivity or the potential in an elliptic operator from the corresponding BSD.

Paper Structure

This paper contains 14 sections, 22 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. The conformal factor
  3. The metric
  4. Comments
  5. Preliminaries
  6. Weyl's asymptotic formula
  7. Useful formula
  8. Elliptic a priori estimates
  9. Analyticity properties
  10. BSD determines a family of elliptic DtN maps
  11. BSD determines a hyperbolic DtN map
  12. Extensions
  13. The conductivity
  14. The potential

Key Result

Theorem 1.1

There exist three constants $\kappa>0$, $0<\varsigma<e^{-1}$ and $0<\theta <1$, depending only on $n$, $\Omega$, $\tilde{g}$, $\alpha$, $\tilde{\beta}$ such that for all $g_1=c_1\tilde{g}\in \mathbf{G}_0^{\rm{c}}$ and $g_2=c_2\tilde{g}\in \mathbf{G}_0^{\rm{c}}$ satisfying $\delta_+(g_1,g_2)<\infty$ where $\|\cdot\|_\infty$ represents the $L^\infty$ norm.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • proof
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 32 more