Topology in One Point Interaction Problem on Extended Non-Local Star Graphs and its Eigenvalues

Technical Summary: Topology in One Point Interaction Problem on Extended Non-Local Star Graphs and its Eigenvalues

Problem Statement
This paper investigates the inverse spectral problem for Sturm-Liouville operators defined on star-shaped metric graphs. The specific model involves a central vertex $v_0$ connected to $m$ edges ($e_j$) of known lengths $l_j$. The system is governed by a differential equation with complex-valued non-local potentials $q_j(x)$ and a "frozen argument" condition at the central vertex, representing a one-point interaction. The boundary conditions include Dirichlet conditions at the outer vertices, continuity at the central vertex, and a generalized non-local Kirchhoff law involving an integral term of the potential.

A key objective of the study is to recover topological information regarding the network structure using non-local spectral data. To facilitate this analysis, the author constructs an "extended closed graph" $\bar{T}$ by connecting the outer endpoints of the star graph edges, forming a cyclic metric graph with new edges $\bar{e}_j$ of lengths $\bar{l}_j$. The study aims to determine the uniqueness of the lengths of these extended edges and the potential functions based on the spectrum of this extended system.

Methodology
The analysis proceeds through the construction of special solutions and the derivation of a characteristic function $\Phi(z)$ whose zeros correspond to the eigenvalues of the system.

1. Construction of Special Solutions: The author modifies Nizhnik's construction to define special solutions $\phi_j(x; z)$ on the original edges and $\bar{\phi}_j(x; z)$ on the extended edges. These solutions are designed to satisfy the continuity conditions at the central vertex and the Dirichlet conditions at the outer vertices.
2. Fourier Analysis: The non-local potentials $q_j(x)$ are expanded into Fourier-sine series. The coefficients of these series are utilized to express the solutions and their derivatives.
3. Characteristic Function Derivation: By summing the fluxes (derivatives) at the vertices and incorporating the non-local boundary condition (the generalized Kirchhoff law), the author derives a characteristic function $\Phi(z)$. This function is an entire function of finite type, and its zero set defines the spectrum of the problem.
4. Complex Analysis and Density Arguments: The paper employs tools from the theory of entire functions, specifically:
   • Titchmarsh's Lemma: To relate the counting function of zeros to the convex hull of the support.
   • Cartwright's Theorem: To establish that if a Fourier transform vanishes on the integers, the underlying function is trivial.
   • Zero Density: The density of zeros ($\delta$) is calculated for products and sums of entire functions to compare the spectral data of two different graphs.

Key Contributions and Results

• Existence and Well-Posedness: The paper establishes that the differential system is well-posed. It proves that under the assumption of rational independence of the edge lengths $\{l_j, \bar{l}_j\}$, the set of eigenvalues is non-empty and not entirely contained within the zeros of the trivial sine terms (i.e., non-trivial non-local eigenvalues exist).
• Uniqueness of Extended Edge Lengths: Proposition 3.11 demonstrates that if two extended star graphs share the same potential functions and their characteristic functions are identical (up to a constant), then the lengths of the extended edges $\bar{l}_j$ must be equal. The author explicitly notes that while this proves the uniqueness of the extended edge lengths, it does not automatically imply that the original graphs are concurrent (geometrically identical in all aspects) without further constraints.
• Uniqueness of Potentials: Theorem 3.12 proves that if the characteristic functions of two systems (with known, rationally independent edge lengths) are identical, then the potential functions $q_j(x)$ on each edge are identical almost everywhere. This is achieved by showing that the difference in potentials leads to a vanishing Fourier transform, which implies the potentials are zero.

Significance and Claims
The author claims that the system is solvable under "very necessary conditions," specifically the rational independence of edge lengths and the non-triviality of the potentials. The primary significance of the work lies in:

1. Recovering Network Geometry: It provides a method to recover the lengths of the extended edges connecting the endpoints of the star graph from spectral data. While the paper aims to recover angular information, the rigorous result establishes the uniqueness of the extended edge lengths, which relates to the angles under specific geometric configurations.
2. Non-Local Modeling: It addresses the inverse problem for operators with non-local potentials and non-local boundary conditions (frozen argument), which model physical phenomena such as vibrations or fluxes monitored at a central control center.
3. Mathematical Rigor: The paper establishes the uniqueness of the inverse spectral problem for this specific class of extended non-local star graphs, showing that the spectral data uniquely determines the potential functions and the lengths of the extended edges.

The paper concludes that the inverse spectral problem admits a unique and solvable formulation, allowing for the reconstruction of the potential functions and the lengths of the extended edges from the non-local spectral data.