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On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta

Damien Gobin, Benoît Grébert, Bernard Helffer, François Nicoleau

TL;DR

The paper addresses the inverse spectral problem for radial Schrödinger operators with a centrifugal singularity, asking whether the potential q is determined by Dirichlet spectra for multiple angular momenta without norming constants. It develops a spectral map and proves its real-analyticity, computes its Fréchet derivative at the zero potential, and uses a decomposition into a main isomorphism plus a compact part to apply the inverse function theorem, yielding local uniqueness for (\ell_1,\ell_2) = (0,1) and (0,2). A global uniqueness result is also established: if Dirichlet spectra are known for infinitely many angular momenta (with Müntz-type divergence condition \sum 1/\ell_k = \infty), the potential is uniquely determined. The analysis hinges on the corrected Kneser–Sommerfeld formula, transformation operators mapping Bessel kernels to trigonometric ones, and completeness arguments via Müntz–Szász, providing partial confirmation of Rundell and Sacks’ conjecture in the linearized setting. The work highlights an explicit pathway to uniqueness from angular-momentum-dependent spectral data and suggests the conjectured general case remains a challenging but natural direction for future study.

Abstract

We consider an inverse spectral problem for radial Schr\" odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~$\ell$ satisfying a Müntz-type condition uniquely determines the potential. Then, in a neighborhood of the zero potential, we prove that the potential is uniquely determined by two Dirichlet spectra associated with distinct angular momenta in the cases \((\ell_1,\ell_2) = (0,1)\) and \((0,2)\). Our approach relies on an explicit analysis of the corresponding singular differential equation, combined with the classical Kneser--Sommerfeld formula. These results confirm, in the linearized setting and in these configurations, a conjecture originally formulated by Rundell and Sacks (2001).

On uniqueness of radial potentials for given Dirichlet spectra with distinct angular momenta

TL;DR

The paper addresses the inverse spectral problem for radial Schrödinger operators with a centrifugal singularity, asking whether the potential q is determined by Dirichlet spectra for multiple angular momenta without norming constants. It develops a spectral map and proves its real-analyticity, computes its Fréchet derivative at the zero potential, and uses a decomposition into a main isomorphism plus a compact part to apply the inverse function theorem, yielding local uniqueness for (\ell_1,\ell_2) = (0,1) and (0,2). A global uniqueness result is also established: if Dirichlet spectra are known for infinitely many angular momenta (with Müntz-type divergence condition \sum 1/\ell_k = \infty), the potential is uniquely determined. The analysis hinges on the corrected Kneser–Sommerfeld formula, transformation operators mapping Bessel kernels to trigonometric ones, and completeness arguments via Müntz–Szász, providing partial confirmation of Rundell and Sacks’ conjecture in the linearized setting. The work highlights an explicit pathway to uniqueness from angular-momentum-dependent spectral data and suggests the conjectured general case remains a challenging but natural direction for future study.

Abstract

We consider an inverse spectral problem for radial Schr\" odinger operators with singular potentials. First, we show that the knowledge of the Dirichlet spectra for infinitely many angular momenta~ satisfying a Müntz-type condition uniquely determines the potential. Then, in a neighborhood of the zero potential, we prove that the potential is uniquely determined by two Dirichlet spectra associated with distinct angular momenta in the cases \((\ell_1,\ell_2) = (0,1)\) and \((0,2)\). Our approach relies on an explicit analysis of the corresponding singular differential equation, combined with the classical Kneser--Sommerfeld formula. These results confirm, in the linearized setting and in these configurations, a conjecture originally formulated by Rundell and Sacks (2001).

Paper Structure

This paper contains 18 sections, 23 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. The spectral map
  4. Proof of Theorem \ref{['thm:main-uniqueness']}
  5. The Kneser--Sommerfeld formula
  6. An integral Green identity
  7. A first completeness result
  8. Transformation Operators
  9. Green's identity via transformation operators
  10. From Green's integral identities to differential relations
  11. The case $\ell =0$
  12. The case $\ell=1$
  13. The case $\ell \geq 2$
  14. Uniqueness result in the case l = 0 and l = 1
  15. Uniqueness result in the case l = 0 and l = 2
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(\ell_k)_{k \geq 1}$ be a strictly increasing sequence of positive integers such that Then the potential $q \in L^2(0,1)$ is uniquely determined by the Dirichlet spectra $\{\lambda_{\ell_k,n}(q)\}_{k,n\geq 1}$.

Theorems & Definitions (36)

  • Theorem 1.1: Global uniqueness
  • Theorem 1.2: Completeness
  • Theorem 1.3: Local uniqueness near the zero potential
  • Proposition 2.1
  • Theorem 2.2: Injectivity of the differential of the spectral map
  • Theorem 2.3: Finite-dimensional kernel
  • proof : Sketch of proof
  • Theorem 3.1: A completeness result
  • proof
  • Lemma 4.1
  • ...and 26 more