Abstract
We consider the continuous Laplacian on an infinite locally finite network with equal edge lengths under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices. It is shown that eigenvalues of the Laplacian in a L∞-setting are closely related to those of the adjacency and transition operator of the network. In this way the point spectrum is determined completely in terms of combinatorial quantities and properties of the underlying graph as in the finite case [2]. Moreover, the occurrence of infinite geometric multiplicity on trees and some periodic graphs is investigated.
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Partially supported by DGI-MCYT (BFM2002-04613-C03-01), Spain
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von Below, J., Lubary, J.A. The eigenvalues of the Laplacian on locally finite networks. Results. Math. 47, 199–225 (2005). https://doi.org/10.1007/BF03323026
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DOI: https://doi.org/10.1007/BF03323026