Inverse eigenvalue problem for a simple star graph

Abstract

A Schrödinger operator and associated spectra may be defined for a graph by identifying edges with intervals of $\mathbb{R}$, on which coefficient functions are defined, imposing appropriate matching conditions at the internal vertices and boundary conditions at the external vertices. Following earlier work of Pivovarchik [14], we consider an inverse eigenvalue problem for a graph consisting of three equal length edges meeting at a single point, where the spectral data is the Dirichlet eigenvalues of the graph together with the Dirichlet spectra of the three individual edges. We derive, discuss and demonstrate a constructive solution method, obtain an alternative uniqueness proof, and discuss several kinds of generalizations.