A C*-algebra associated with dynamics on a graph of strings

TL;DR

This paper introduces a C*-algebra linked to wave dynamics on a metric graph, exploring its structure as a sum of elementary blocks and its potential for inverse problem applications.

Contribution

It defines a new C*-algebra generated by eikonals for wave systems on graphs and analyzes its structure and boundary data dependence.

Findings

The algebra decomposes into a direct sum of elementary blocks.

Each block is an algebra of operators with matrix-valued functions.

The algebra is determined by boundary inverse data.

Abstract

A C*-algebra $\mathfrak E$ associated with a dynamical system on a metric graph is introduced. The system is governed by the wave equation and controlled from boundary vertices. Algebra $\mathfrak E$ is generated by the so-called {\it eikonals}, which are self-adjoint operators related with reachable sets of the system. Its structure is the main subject of the paper. We show that $\mathfrak E$ is a direct sum of "elementary blocks". Each block is an algebra of operators multiplying ${\mathbb R}^n$-valued functions by continuous matrix-valued functions of special kind. The eikonal algebra is determined by the boundary inverse data. This shows promise of its possible applications to inverse problems.