Isospectral infinite graphs and networks and infinite eigenvalue multiplicities

Joachim von Below; José A. Lubary; Joachim von Below; José A. Lubary

doi:10.3934/nhm.2009.4.453

Networks and Heterogeneous Media

2009, Volume 4, Issue 3: 453-468. doi: 10.3934/nhm.2009.4.453

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Isospectral infinite graphs and networks and infinite eigenvalue multiplicities

Joachim von Below ^{1
,},
José A. Lubary ^{2
,}

1.
LMPA Joseph Liouville, FCNRS 2956, Université du Littoral Côte d’Opale, 50, rue F. Buisson, B.P. 699, F–62228 Calais Cedex
2.
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona, 1–3, 08034 Barcelona

Received: 01 February 2008 Revised: 01 February 2009
Primary: 34B45, 05C50; Secondary: 05C10, 35J05, 34L10, 35P10.

We consider the continuous Laplacian on infinite locally finite networks under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a $L^\infty$-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to $\l^\infty(\ZZ)$ is investigated, in particular in trees and periodic graphs.
- Locally finite graphs and networks,
- Laplacian,
- eigenvalue problems,
- adjacency and transition operators
Citation: Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities[J]. Networks and Heterogeneous Media, 2009, 4(3): 453-468. doi: 10.3934/nhm.2009.4.453

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Abstract

We consider the continuous Laplacian on infinite locally finite networks under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a $L^\infty$-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to $\l^\infty(\ZZ)$ is investigated, in particular in trees and periodic graphs.
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