Essential spectrum and Weyl asymptotics for discrete Laplacians

Bonnefont, Michel; Golénia, Sylvain

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 563-624.

Dans cet article, nous étudions le spectre de Laplaciens discrets. Notre travail est basé sur l’inégalité de Hardy et l’étude des fonctions super-harmoniques. Nous retrouvons et améliorons des bornes inférieures pour le bas du spectre et le bas du spectre essentiel. Dans certains cas, nous obtenons des asymptotiques de Weyl pour les valeurs propres. Nous donnons aussi une représentation probabiliste des fonctions super-harmoniques, puis avec des arguments de type couplage, nous établissons des résultats de comparaison pour le bas du spectre, le bas du spectre essentiel et la complétude stochastique de différents Laplaciens discrets. Une classe de graphes faiblement symétriques est aussi étudiée en grand détail.

In this paper, we investigate spectral properties of discrete Laplacians. Our study is based on the Hardy inequality and the use of super-harmonic functions. We recover and improve lower bounds for the bottom of the spectrum and of the essential spectrum. In some situation, we obtain Weyl asymptotics for the eigenvalues. We also provide a probabilistic representation of super-harmonic functions. Using coupling arguments, we set comparison results for the bottom of the spectrum, the bottom of the essential spectrum and the stochastic completeness of different discrete Laplacians. The class of weakly spherically symmetric graphs is also studied in full detail.

Classification : 34L20, 05C63, 47B25, 47A63, 47A10
Mots clés : discrete Laplacian, locally finite graphs, asympotic of eigenvalues, spectrum, essential spectrum, markov chains, functional inequalities

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