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Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential

Mourad Choulli, Abdelmalek Metidji, Éric Soccorsi

Abstract

This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schrödinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either full or partial Dirichlet measurement of the eigenfunctions on the boundary of the domain.

Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential

Abstract

This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schrödinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either full or partial Dirichlet measurement of the eigenfunctions on the boundary of the domain.
Paper Structure (15 sections, 12 theorems, 175 equations)

This paper contains 15 sections, 12 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. The Robin Laplacian
  3. Boundary spectral data
  4. Statement of the results
  5. A short bibliography of the existing literature
  6. Outline
  7. Preliminaries
  8. Resolvent estimates
  9. Asymptotic spectral analysis
  10. $H^2$-regularity of the eigenfunctions
  11. Proof of Theorems \ref{['theorem1']}, \ref{['theorem2']} and \ref{['theorem3']}
  12. Proof of Theorem \ref{['theorem1']}
  13. Proof of Theorem \ref{['theorem2']}
  14. Proof of Theorem \ref{['theorem3']}
  15. Proof of the continuity of $\mathfrak{a}_0$

Key Result

Theorem 1.1

Let $q$ and $\tilde{q}$ be in $L^{r}(\Omega,\mathbb{R})$, where $r=n/2$ when $n \ge 4$ and $r >n/2$ when $n=3$, and let $\ell \in \mathbb{N}$. Then, the conditions yield that $q=\tilde{q}$ in $\Omega$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • ...and 13 more