Weighted eigenvalues of Dirac operators: complete continuity and comparison

TL;DR

This paper provides a min-max characterization of weighted Dirac eigenvalues, proves their continuity under weak $L^p$ convergence, and establishes comparison results in the absence of harmonic spinors.

Contribution

It introduces a new min-max framework for weighted Dirac eigenvalues and demonstrates their stability and comparison properties under specific convergence conditions.

Findings

Weighted eigenvalues are continuous with respect to weak $L^p$ convergence.

A min-max characterization of weighted Dirac eigenvalues is established.

Comparison results are proved when no harmonic spinors are present.

Abstract

We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$. Moreover, we establish a comparison result for such weighted eigenvalue problems when there are no harmonic spinors.