Spectral stability of metric-measure Laplacians

TL;DR

This paper investigates the spectral stability of a convolution-based Laplacian operator on metric-measure spaces, demonstrating that its spectrum remains stable under perturbations and providing estimates on eigenvalue counts.

Contribution

It introduces a convolution mm-Laplacian on metric-measure spaces and proves spectral stability and Weyl-type eigenvalue estimates for this operator.

Findings

Spectral stability under metric-measure perturbations

Weyl-type estimates on eigenvalue counts

Applicability to reasonably nice metric-measure spaces

Abstract

We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of convolution Laplacian's spectrum with respect to metric-measure perturbations and obtain Weyl-type estimates on the number of eigenvalues.