Recovering Differential Operators on Spatial Networks

TL;DR

This paper reviews inverse spectral problems for differential operators on spatial networks, focusing on recovering coefficients from spectral data with known graph structures, covering Sturm-Liouville, higher-order, and noncompact graphs.

Contribution

It provides a comprehensive overview of recent results on inverse spectral problems for differential operators on various types of spatial networks.

Findings

Inverse Sturm-Liouville problems on compact graphs solved.

Results extended to higher-order differential operators.

Main results on inverse problems for noncompact graphs presented.

Abstract

We give a short review of results on inverse spectral problems for ordinary differential operators on a spatial networks (geometrical graphs). We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential equations from spectral characteristics provided that the structure of the graph is known a priori. In the first half of the review we provide results related to inverse Sturm-Liouville problems on arbitrary compact graphs. Further, results on inverse problems for arbitrary order differential operators on compact graphs are presented. At the end we provide the main results on inverse problems on noncompact graphs.