Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition

TL;DR

This paper investigates inverse spectral problems for a class of nonlocal differential operators on star-shaped graphs, focusing on recovering integral condition coefficients from eigenvalues, with spectrum characterization and reconstruction methods.

Contribution

It introduces two approaches for inverse spectral problems on nonlocal operators on metric graphs, including spectrum characterization, algorithms, and proof of uniqueness.

Findings

Spectrum characterization for the nonlocal operator

Reconstruction algorithms for integral condition coefficients

Proof of uniqueness of the inverse problem solution

Abstract

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral matching condition. This operator is adjoint to the functional-differential operator with frozen argument at the central vertex of the graph. We study the inverse problem that consists in the recovery of the integral condition coefficients from the eigenvalues. We obtain the spectrum characterization, reconstruction algorithms, and prove the uniqueness of the inverse problem solution.