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Michel Bonnefont; Sylvain Golénia; Matthias Keller
Eigenvalue asymptotics for Schrödinger operators on sparse graphs
(Asymptotique des valeurs propres pour les opérateurs de Schrödinger agissant sur des graphes éparses)
Annales de l'institut Fourier, 65 no. 5 (2015), p. 1969-1998, doi: 10.5802/aif.2979
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Class. Math.: 47A10, 34L20, 05C63, 47B25, 47A63
Keywords: discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, planarity, sparse, functional inequality

Résumé - Abstract

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.

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