Stability of the inverse Sturm-Liouville problem on a quantum tree

TL;DR

This work establishes a constructive and stable framework for solving the inverse Sturm-Liouville problem on a metric tree with distributional potentials in $W_2^{-1}$. It combines contour-based boundary-edge reconstructions with a vertex-transition sampling approach, and proves both uniform stability on an $L_2$-ball and local stability under spectral-data perturbations. The approach is grounded in the method of spectral mappings and leverages Lipschitz continuity of transformation operators, explicit Weyl-function relations, and a carefully designed algorithm that recovers edge potentials sequentially from the boundary to the interior. The results provide robust, scalable procedures for recovering singular potentials on quantum graphs, with potential applications in physics and engineering.

Abstract

This paper deals with the Sturm-Liouville operators with distribution potentials of the space $W_2^{-1}$ on a metric tree. We study an inverse spectral problem that consists in the recovery of the potentials from the characteristic functions related to various boundary conditions. We prove the uniform stability of this inverse problem for potentials in a ball of any fixed radius, as well as the local stability under small perturbations of the spectral data. Our approach is based on a stable algorithm for the unique reconstruction of the potentials relying on the ideas of the method of spectral mappings.

Figures (6)

• Figure 1: Example of tree $G$
• Figure 2: Splitting of the tree $G$ by the vertex $u$
• Figure 3: Star-shaped graph in Example \ref{['ex:star']}
• Figure 4: Graphs $G_p$ and $g_p$
• Figure 5: Contour $\Gamma$

Theorems & Definitions (26)

• Definition 3.1: Bond19
• Proposition 3.2
• Proposition 3.3: HM04; Bond21, Theorem 2.1
• Lemma 3.4
• proof
• Corollary 3.5
• Example 3.6
• Lemma 3.7
• Corollary 3.8
• Corollary 3.9