Essential spectrum and Weyl asymptotics for discrete Laplacians

Michel Bonnefont (1) , Sylvain Golenia (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: discrete Laplacian; locally finite graphs; asympotic of eigenvalues; spectrum; essential spectrum; markov chains; functional inequalities

Domaines: Théorie spectrale [math.SP]; Probabilités [math.PR]

Michel Bonnefont, Sylvain Golenia. Essential spectrum and Weyl asymptotics for discrete Laplacians. 2014. ⟨hal-01010730⟩