Spectral comparison results for Laplacians on discrete graphs

TL;DR

This paper develops a general framework for spectral comparison of Laplacians on discrete graphs, including a novel discrete local Weyl law that avoids traditional continuous analysis techniques.

Contribution

It introduces a broad spectral comparison framework for possibly infinite discrete graphs and proves a discrete local Weyl law without relying on Tauberian theorems.

Findings

Established spectral comparison results for discrete Laplacians.

Proved a discrete local Weyl law independent of continuous methods.

Extended spectral analysis techniques to infinite graph settings.

Abstract

In the recent literature, various authors have studied spectral comparison results for Schr\"odinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local Weyl law whose proof does neither involve any Tauberian theorem nor the Weyl law as used in the continuous case.