Eigenvalue asymptotics for Schrödinger operators on sparse graphs
[Asymptotique des valeurs propres pour les opérateurs de Schrödinger agissant sur des graphes éparses]
Bonnefont, Michel1 ; Golénia, Sylvain1 ; Keller, Matthias2
1 Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F-33405 Talence cedex (France)
2 Friedrich Schiller Universität Jena Mathematisches Institut 07745 Jena (Germany)
Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1969-1998.

Nous considérons des opérateurs de Schrödinger agissant sur des graphes éparses. Le fait d’être éparse est équivalent à une inégalité fonctionnelle pour le Laplacien. En particulier il y a des conséquences spectrales fortes pour le Laplacien quand le graphe est éparse : caractérisation de son domaine de forme et de l’absence du spectre essentiel. Dans ce dernier cas, nous calculons l’asymptotique des valeurs propres.

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

DOI : 10.5802/aif.2979
Classification : 47A10, 34L20, 05C63, 47B25, 47A63
Keywords: discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality
Mot clés : Laplacien discret, graphe locallement fini, valeurs propres, asymptotique, planarité, éparse, inégalité fonctionelle
Bonnefont, Michel 1 ; Golénia, Sylvain 1 ; Keller, Matthias 2

1 Institut de Mathématiques de Bordeaux Université Bordeaux 1 351, cours de la Libération F-33405 Talence cedex (France)
2 Friedrich Schiller Universität Jena Mathematisches Institut 07745 Jena (Germany) 
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Bonnefont, Michel; Golénia, Sylvain; Keller, Matthias. Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 1969-1998. doi : 10.5802/aif.2979. http://www.numdam.org/articles/10.5802/aif.2979/

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