Recovering Variable Order Differential Operators with Regular Singularities on Graphs

TL;DR

This paper investigates inverse spectral problems for differential operators with regular singularities on graphs, introducing Weyl-type matrices to reconstruct operators uniquely, extending classical Sturm-Liouville theory to complex graph structures.

Contribution

It introduces Weyl-type matrices for variable order differential operators on graphs and provides a unique reconstruction method for inverse problems.

Findings

Weyl-type matrices generalize classical Weyl functions.

Unique solution procedure for inverse spectral problems.

Extension of inverse spectral theory to graph-based differential operators.

Abstract

We study inverse spectral problems for ordinary differential equations with regular singularities on compact star-type graphs when differential equations have different orders on diferent edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.