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

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Abstract

The eikonal algebra $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-algebra defined by the dynamical system which describes the propagationof waves generated by sources supported at the boundary vertices of $\Omega$. This paper describes the canonical block form of the algebra $\mathfrak E$ for an arbitrary compact connected metric graph. Passing tothis form is equivalent to constructing a functional model which realizes$\mathfrak E$ as an algebra of continuous matrix-valued functions on itsspectrum $\widehat{\mathfrak{E}}$. The results are intended to be used inthe inverse problem of recovering the graph from spectral and dynamical boundary data.

About the authors

Mikhail Igorevich Belishev

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Email: belishev@pdmi.ras.ru
Doctor of physico-mathematical sciences, no status

Aleksandr Vladimirovich Kaplun

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Email: alex.v.kaplun@gmail.com

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