Recovering the Topology in One Point Interaction Problem on Extended Non-Local Star Graphs

TL;DR

This paper addresses the inverse spectral problem on star-like graphs with non-local potentials, demonstrating conditions for reconstructing the network topology from spectral data and constructing solutions edge by edge.

Contribution

It introduces a method to recover the topology of a star graph with non-local potentials using inverse spectral analysis, including constructing solutions edge by edge.

Findings

System is solvable under necessary conditions

Topology of the network can be recovered from spectral data

Constructs solutions systematically edge by edge

Abstract

The author studies the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this star-like graph centered at the origin as its vertex, there are attached $m$ edges that imposed the Sturm-Liouville operator with certain non-local potential functions with some suitable local boundary value conditions. At the vertex, we consider one point interaction condition at vertex to model a network that fixed on the end of the edges on the graph. The vibration and flow changes are monitored at that vertex which serves as certain control/regulation center. The author shows that the system is solvable under very necessary conditions. It is crucial to recover the topology of the network. In this paper, author constructs the special solution edge by edge and point to point.