Recovering Higher Order Differential Systems with Regular Singularities on Star-type Graphs

TL;DR

This paper addresses the inverse spectral problem for high-order differential equations with regular singularities on star-shaped graphs, introducing Weyl-type matrices as key spectral data and establishing a unique reconstruction method.

Contribution

It introduces Weyl-type matrices for arbitrary order differential equations with singularities on star graphs and proves the uniqueness of the inverse spectral problem solution.

Findings

Weyl-type matrices generalize classical Weyl functions for complex differential systems.

A constructive procedure for solving the inverse problem is developed.

Uniqueness of the solution to the inverse spectral problem is proven.

Abstract

We study an inverse spectral problem for arbitrary order ordinary differential equations on compact star-type graphs when differential equations have regular singularities at boundary vertices. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.