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Can one hear the shape of a network? (English) Zbl 0973.35143

Ali Mehmeti, Felix (ed.) et al., Partial differential equations on multistructures. Proceedings of the conference, Luminy, France. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 219, 19-36 (2001).
From the introduction: We show that, in contrast to the one-dimensional domain case, one cannot recover the shape of a network from the spectrum of its Laplacian under the continuity condition at ramification nodes and the Kirchhoff condition at all vertices. Thus, in that regard, networks behave like higher-dimensional objects. Moreover, we shall discuss the eigenvalue asymptotics as well as the distinction of network immanent eigenvalues from those stemming from single branches.
For the entire collection see [Zbl 0960.00045].

MSC:

35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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