Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

TL;DR

This paper investigates the spectral properties of the magnetic discrete Laplacian, including its form-domain, essential spectrum absence, and asymptotic behavior of eigenvalues, contributing to the understanding of discrete spectral theory.

Contribution

It provides a detailed analysis of the magnetic discrete Laplacian's spectral properties, including form-domain characterization and eigenvalue asymptotics, which are novel insights in discrete spectral analysis.

Findings

Identified the form-domain of the magnetic discrete Laplacian

Proved the absence of essential spectrum for this operator

Derived the asymptotic distribution of eigenvalues

Abstract

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.