The magnetic Laplacian acting on discrete cusps

TL;DR

This paper introduces discrete cusps in weighted graphs, analyzes the magnetic Laplacian's spectral properties, and compares it with the non-magnetic case, revealing differences in their form-domains and spectral asymptotics.

Contribution

It defines discrete cusps for weighted graphs and studies the spectral properties of the magnetic Laplacian in this setting, including essential spectrum and eigenvalue asymptotics.

Findings

The form-domain of magnetic and non-magnetic Laplacians can differ.

The essential spectrum of the magnetic Laplacian is empty.

Eigenvalues have specific asymptotic behavior.

Abstract

We introduce the notion of discrete cusp for a weighted graph. In this context, we provethat the form-domain of the magnetic Laplacian and that of thenon-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute theasymptotic of eigenvalues for the magnetic Laplacian.