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A quantitative Borg-Levinson theorem for a large class of unbounded potentials

Mourad Choulli

TL;DR

The article delivers a quantitative Borg-Levinson stability result for unbounded potentials in $L^{n/2}(\Omega)$ on bounded $C^{1,1}$ domains with $n\ge 5$, tying spectral data (eigenvalues and boundary traces) to the potential via explicit distance functionals and a Hölder-type bound in $H^{-1}(\Omega)$. The approach hinges on Weyl asymptotics, resolvent and Dirichlet-to-Neumann map analysis, and Isozaki-type representations to connect the Fourier transform of the potential to boundary measurements, yielding a concrete stability exponent. The framework accommodates lower dimensions with modifications and extends to anisotropic media through geometric-optics constructions and the geodesic-ray transform. This advances inverse spectral theory by providing a quantitative stability mechanism for unbounded potentials and supports anisotropic generalizations relevant to practical PDE inverse problems.

Abstract

We prove a quantitative Borg-Levinson theorem for a large class of unbounded potentials. We give a detailed proof when the dimension of the space is greater than or equal to five. We also indicate the modifications necessary to cover lower dimensions. In the last section, we briefly show how to extend our result to the anisotropic case.

A quantitative Borg-Levinson theorem for a large class of unbounded potentials

TL;DR

The article delivers a quantitative Borg-Levinson stability result for unbounded potentials in on bounded domains with , tying spectral data (eigenvalues and boundary traces) to the potential via explicit distance functionals and a Hölder-type bound in . The approach hinges on Weyl asymptotics, resolvent and Dirichlet-to-Neumann map analysis, and Isozaki-type representations to connect the Fourier transform of the potential to boundary measurements, yielding a concrete stability exponent. The framework accommodates lower dimensions with modifications and extends to anisotropic media through geometric-optics constructions and the geodesic-ray transform. This advances inverse spectral theory by providing a quantitative stability mechanism for unbounded potentials and supports anisotropic generalizations relevant to practical PDE inverse problems.

Abstract

We prove a quantitative Borg-Levinson theorem for a large class of unbounded potentials. We give a detailed proof when the dimension of the space is greater than or equal to five. We also indicate the modifications necessary to cover lower dimensions. In the last section, we briefly show how to extend our result to the anisotropic case.
Paper Structure (11 sections, 20 theorems, 233 equations)

This paper contains 11 sections, 20 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weyl's asymptotic formula
  4. Dirichlet to Neumann map
  5. Analyticity of solutions
  6. Resolvent estimates
  7. The difference between two Dirichlet to Neumann maps
  8. Proof of Theorem \ref{['mainthm1']}
  9. Extension to the anisotropic case
  10. Coercivity of the sesquilinear form $\mathfrak{a}_V$
  11. BVPs with unbounded lower order term

Key Result

Theorem 1.1

Let $\ell \ge 1$ be an integer and $V_1,V_2\in L^m(\Omega,\mathbb{R})$, where $m=n/2$ when $n\ge 4$ and $m>n/2$ when $n=3$, satisfy Then $V_1=V_2$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 24 more