Eigenvalue asymptotics for Schrödinger operators on sparse graphs
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, p. 1969-1998

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.

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.

DOI : https://doi.org/10.5802/aif.2979
Classification:  47A10,  34L20,  05C63,  47B25,  47A63
Keywords: discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality
@article{AIF_2015__65_5_1969_0,
     author = {Bonnefont, Michel and Gol\'enia, Sylvain and Keller, Matthias},
     title = {Eigenvalue asymptotics for Schr\"odinger operators on sparse graphs},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     pages = {1969-1998},
     doi = {10.5802/aif.2979},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_5_1969_0}
}
Bonnefont, Michel; Golénia, Sylvain; Keller, Matthias. Eigenvalue asymptotics for Schrödinger operators on sparse graphs. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 1969-1998. doi : 10.5802/aif.2979. http://www.numdam.org/item/AIF_2015__65_5_1969_0/

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