Canonical form of $C^*$-algebra of eikonals related to the metric graph

TL;DR

This paper characterizes the canonical block form of the eikonal $C^*$-algebra associated with a metric graph, facilitating inverse problems of reconstructing the graph from spectral data.

Contribution

It provides a canonical block form of the eikonal algebra for any compact connected metric graph, enabling a functional model for inverse spectral problems.

Findings

Canonical block form of the eikonal algebra derived

Functional model as algebra of matrix-valued functions constructed

Application to inverse graph reconstruction outlined

Abstract

The eikonal algebra $\mathfrak E$ of the metric graph $\Omega$ is an operator $C^*$--algebra defined by the dynamical system which describes the propagation of waves generated by sources supported in the boundary vertices of $\Omega$. This paper describes the canonical block form of the algebra $\mathfrak E$ of an arbitrary compact connected metric graph. Passing to this form is equivalent to constructing a functional model which realizes $\mathfrak E$ as an algebra of continuous matrix-valued functions on its spectrum $\widehat{\mathfrak{E}}$. The results are intended to be used in the inverse problem of reconstruction of the graph by spectral and dynamical boundary data. Bibliography: 28 items.