Essential spectrum and Weyl asymptotics for discrete Laplacians

TL;DR

This paper analyzes the spectral properties of discrete Laplacians, improving bounds and deriving Weyl asymptotics, using Hardy inequalities, super-harmonic functions, and probabilistic methods, with detailed study of symmetric graphs.

Contribution

It introduces new bounds and Weyl asymptotics for discrete Laplacians, employing probabilistic and comparison techniques, and explores spectral properties on symmetric graphs.

Findings

Improved lower bounds for the spectrum and essential spectrum

Derived Weyl asymptotics for eigenvalues in certain cases

Established comparison results for spectral bottom and stochastic completeness

Abstract

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.