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arXiv · 2210.13246

Canonical forms of metric graph eikonal algebra and graph geometry

M. I. Belishev·A. V. Kaplun
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Abstract

The algebra of eikonals $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-algebra determined by dynamical system with boundary control that describes wave propagation on the graph. In this paper, two canonical block forms (algebraic and geometric) of the algebra $\mathfrak E$ are provided for an arbitrary connected locally compact graph. These forms determine some metric graphs (frames) $\mathfrak F^{\,\rm a}$ and $\mathfrak F^{\,\rm g}$. Frame $\mathfrak F^{\,\rm a}$ is determined by the boundary inverse data. Frame $\mathfrak F^{\,\rm g}$ is related to graph geometry. A class of ordinary graphs is introduced, whose frames are identical: $\mathfrak F^{\,\rm a}\equiv\mathfrak F^{\,\rm g}$. The results are supposed to be used in the inverse problem that consists in determination of the graph from its boundary inverse data.

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M. I. Belishev, A. V. Kaplun. 2022-10-24. Canonical forms of metric graph eikonal algebra and graph geometry. https://arxiv.org/abs/2210.13246

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