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The spectrum of Dirichlet-to-Neumann maps for radial conductivities

Thierry Daudé, Fabricio Macià, Cristóbal Meroño, François Nicoleau

TL;DR

This paper characterizes the spectrum of Dirichlet-to-Neumann maps for radial conductivities on the ball by decomposing the DtN spectrum into a universal boundary term and Hausdorff moments of a Born approximation γ^{\mathrm{B}}. The authors build a two-step Calderón problem: first, recover γ^{\mathrm{B}} linearly from the spectral data by solving a moment problem, then reconstruct γ from γ^{\mathrm{B}} with Hölder stability in suitable Sobolev spaces. A core contribution is the Schrödinger reduction: radial γ induce a 1D Schrödinger problem with potential Q_V, linking the DtN spectrum to the Weyl–Titchmarsh m-function and the A-amplitude; the singular structure of γ^{\mathrm{B}} at the origin is characterized via resonances and zero-energy phenomena. The results establish local uniqueness for the radial Calderón problem, provide Hölder stability for the γ←→γ^{\mathrm{B}} map, and illuminate how regularity and resonances govern the Born approximation, with explicit examples illustrating the theory.

Abstract

The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator $\nabla \cdot (γ\nabla \centerdot )$ in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity $γ$, plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of $γ$. We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calderón Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is Hölder stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.

The spectrum of Dirichlet-to-Neumann maps for radial conductivities

TL;DR

This paper characterizes the spectrum of Dirichlet-to-Neumann maps for radial conductivities on the ball by decomposing the DtN spectrum into a universal boundary term and Hausdorff moments of a Born approximation γ^{\mathrm{B}}. The authors build a two-step Calderón problem: first, recover γ^{\mathrm{B}} linearly from the spectral data by solving a moment problem, then reconstruct γ from γ^{\mathrm{B}} with Hölder stability in suitable Sobolev spaces. A core contribution is the Schrödinger reduction: radial γ induce a 1D Schrödinger problem with potential Q_V, linking the DtN spectrum to the Weyl–Titchmarsh m-function and the A-amplitude; the singular structure of γ^{\mathrm{B}} at the origin is characterized via resonances and zero-energy phenomena. The results establish local uniqueness for the radial Calderón problem, provide Hölder stability for the γ←→γ^{\mathrm{B}} map, and illuminate how regularity and resonances govern the Born approximation, with explicit examples illustrating the theory.

Abstract

The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity , plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of . We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calderón Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is Hölder stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.
Paper Structure (15 sections, 22 theorems, 136 equations, 1 figure)

This paper contains 15 sections, 22 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence of the Born approximation
  3. Linearized DtN operators in the radial case
  4. The Born approximation for Schrödinger operator DtN maps
  5. The Born approximation
  6. The Born approximation at the origin: singularity and resonances
  7. Weyl-Titchmarsh functions and DtN maps
  8. Asymptotics of A-amplitudes
  9. Singularities of the Born approximation
  10. The conductivity case
  11. Explicit examples
  12. Stability properties of the Born approximation
  13. Hölder stability
  14. Proof of Theorem 3
  15. Proof of Lemma 4.1

Key Result

Theorem 1

Let $d\ge 2$, and let $\gamma \in W^{2,p}_\mathrm{rad}(\mathbb{B}^d)$Throughout this article, the subscript $\mathrm{rad}$ added to some function space over $\mathbb{B}^d$ will mean that we are considering the subspace of its radial elements. with $d/2<p \le \infty$ be a conductivity. Then there ex and In addition, $\gamma^{\mathrm{B}}$ is uniquely determined from $\Lambda_\gamma$ since e:specdt

Figures (1)

  • Figure 1: Plots of the radial profiles of $\gamma_{d,\mu,\nu}$ (blue) and $\gamma^{\mathrm{B}}_{d,\mu,\nu}$ (orange) defined in \ref{['sec:ex']}.

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2: Uniqueness
  • Corollary 3
  • Theorem 4: Stability
  • Theorem 5: Regularity
  • Proposition 2.1
  • proof : Proof of \ref{['p:eigenvalues_dPsi']}
  • Proposition 2.2
  • proof : Proof of \ref{['p:Vb_space']}
  • Lemma 2.3
  • ...and 34 more