Spectral asymptotics for first order systems

TL;DR

This review paper discusses recent advances in the spectral analysis of elliptic self-adjoint first order systems on closed manifolds, focusing on asymptotic formulas for positive and negative eigenvalue counts.

Contribution

It provides a comprehensive overview of spectral asymptotics for first order systems, emphasizing the asymmetry in positive and negative eigenvalues.

Findings

Derived asymptotic formulas for eigenvalue counting functions

Analyzed spectral asymmetry in first order systems

Reviewed recent progress in spectral analysis techniques

Abstract

This is a review paper outlining recent progress in the spectral analysis of first order systems. We work on a closed manifold and study an elliptic self-adjoint first order system of linear partial differential equations. The aim is to examine the spectrum and derive asymptotic formulae for the two counting functions. Here the two counting functions are those for the positive and the negative eigenvalues. One has to deal with positive and negative eigenvalues separately because the spectrum is, generically, asymmetric.