Functional-differential operators on geometrical graphs with global delay and inverse spectral problems

TL;DR

This paper introduces a new class of functional-differential operators with global delay on geometrical graphs, extending the modeling of nonlocal processes and exploring inverse spectral problems with theoretical analysis.

Contribution

It proposes globally nonlocal operators on graphs, extending previous locally nonlocal models, and studies inverse spectral problems including uniqueness and stability for these operators.

Findings

Introduction of globally nonlocal operators on graphs.

Extension of the concept to arbitrary trees.

Analysis of inverse spectral problems with results on uniqueness and stability.

Abstract

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves {\it global} delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a {\it locally} nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce {\it globally} nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area of further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.