Uniqueness of Inverse Spectral Problem of Non-Local Sturm-Liouville Operators on Star Graph

TL;DR

This paper investigates the inverse spectral problem for non-local Sturm-Liouville operators on star graphs, establishing uniqueness and solvability, and providing spectral asymptotics for the system.

Contribution

It introduces a novel inverse spectral problem for non-local Sturm-Liouville operators on star graphs with unique boundary conditions, proving the system's solvability.

Findings

Proved the inverse spectral problem has a unique solution.

Established spectral asymptotics similar to Weyl's law.

Demonstrated the system's solvability under certain boundary conditions.

Abstract

In this paper, we explore the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this fixed star-like graph centered at the origin as its vertex, we attach $m$ edges. On each edge, we impose the Sturm-Liouville operator with certain non-local potential functions with some suitable non-local boundary value conditions. At the vertex, we consider a frozen argument type of condition at zero to model a network that fixed on the end of each edge on the graph. The vibration and flow changes are monitored at that vertex which serves as certain control center. There is an inverse uniqueness subject to the suitable non-local boundary condition. We show that the system is solvable. Additionally, we give a Weyl's type of spectral asymptotics.