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Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph

E. E. Chitorkin, N. P. Bondarenko

TL;DR

This work studies the inverse Sturm–Liouville problem on a star-shaped graph with $m$ edges, focusing on recovering edge potentials $q_j$ from spectral data. It develops a two-pronged approach: establishing uniform stability for reconstruction from $m$ spectra (plus auxiliary spectra) and for reconstruction from eigenvalues with weight numbers, using spectral mappings and uniform control of analytic data derived from zeros. The authors derive explicit Lipschitz-type bounds in terms of spectral deviations, employing kernel representations, Paley–Wiener bounds, and Borg-type inversion on graphs. These results advance the stability theory for inverse problems on quantum graphs and provide a rigorous foundation for numerical methods in this setting.

Abstract

In this paper, we study the inverse spectral problem for the Sturm-Liouville operators on a star-shaped graph, which consists in the recovery of the potentials from specral data or several spectra. The uniform stability of these inverse problems on the whole graph is proved.

Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph

TL;DR

This work studies the inverse Sturm–Liouville problem on a star-shaped graph with edges, focusing on recovering edge potentials from spectral data. It develops a two-pronged approach: establishing uniform stability for reconstruction from spectra (plus auxiliary spectra) and for reconstruction from eigenvalues with weight numbers, using spectral mappings and uniform control of analytic data derived from zeros. The authors derive explicit Lipschitz-type bounds in terms of spectral deviations, employing kernel representations, Paley–Wiener bounds, and Borg-type inversion on graphs. These results advance the stability theory for inverse problems on quantum graphs and provide a rigorous foundation for numerical methods in this setting.

Abstract

In this paper, we study the inverse spectral problem for the Sturm-Liouville operators on a star-shaped graph, which consists in the recovery of the potentials from specral data or several spectra. The uniform stability of these inverse problems on the whole graph is proved.
Paper Structure (4 sections, 9 theorems, 41 equations)

This paper contains 4 sections, 9 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Uniform stability of the inverse problem by the set of spectra
  4. Uniform stability of the inverse problem by spectral data

Key Result

Theorem 2.2

Let $Q > 0$ and $\textbf{q}^{(1)}, \textbf{q}^{(2)} \in \textbf{P}_Q$. Then where $\delta = ( \sum\limits_{n=1}^{\infty} (\sum\limits_{k=1}^{m}|n(\rho^{(1)}_{nk} - \rho^{(2)}_{nk})|^2 + \sum\limits_{k=1}^{m}\sum\limits_{j=1}^{m-1}|n(\theta_{nkj}^{(1)}-\theta_{nkj}^{(2)})|^2 ) )^{\frac{1}{2}}$.

Theorems & Definitions (12)

  • Theorem 2.2
  • Theorem 2.4
  • Proposition 3.1: FY01
  • Lemma 3.2: Corollary 3.3 from Bond25cycle
  • Lemma 3.3
  • proof
  • Lemma 3.4: Theorem 2.3 from Bond25cycle
  • Lemma 3.5: Corollary 3.3 from Bond25cycle
  • Lemma 3.6: Corollary 3.5 from Bond25cycle
  • Lemma 3.7
  • ...and 2 more