Eikonal Algebra on a Graph of Simple Structure

Abstract

An eikonal algebra E(Ω) is a C*-algebra related to a metric graph Ω. It is determined by trajectories and reachable sets of a dynamical system associated with the graph. The system describes the waves, which are initiated by boundary sources (controls) and propagate into the graph with finite velocity. Motivation and interest to eikonal algebras comes from the inverse problem of reconstruction of the graph via its dynamical and/or spectral boundary data. Algebra E(Ω) is determined by these data. In the mean time, its structure and algebraic invariants (irreducible representations) are connected with topology of Ω. We demonstrate such connections and study E(Ω) by the example of Ω of a simple structure. Hopefully, in future, these connections will provide an approach to reconstruction. * Saint-Petersburg Department of the Steklov Mathematical Institute, RAS, beli-shev@pdmi.ras.ru; Saint-Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia, m.belishev@spbu.ru.

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

“…The introductory part of the present paper is rather long and essentially repeats the corresponding sections of [7] and [8]. This is unavoidable since the presentation of facts and results related to E T Σ requires a solid preparation.…”

“…A similar cluster may exist at the right end t = b. At the same time, all the points π t with t ∈ (a, b) are separable from each other and from the clusters (see [7], [8]). The spectrum of…”

“…Variants of the BC-method for inverse problems on graphs are proposed in [4]- [6]. The version using the eikonal algebra was suggested in [7] and extended in [8]. Our work develops this approach.…”

“…For the simplest graphs (three-beam stars). the nature of these relations and their evolution with respect to T were considered in [8].…”

“…The technical part is rather complicated since we deal with an extremely general object: an arbitrary compact connected metric graph having a boundary. The study of simple examples in [7], [8] with illustrations may be instructive. We also recommend [22], where the reduction of E T Σ to a canonical form is described in detail for a simple graph.…”