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.
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Zhirayr Avetisyan, Yan-Long Fang, Dmitri Vassiliev, Spectral asymptotics for first order systems. J. Spectr. Theory 6 (2016), no. 4, pp. 695–715DOI 10.4171/JST/137