We work on a compact manifold without boundary and consider the spectral problem for an elliptic self-adjoint first order system of (pseudo)differential equations. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. The objective is to derive a two-term asymptotic formula for the counting function (number of eigenvalues between zero and a positive lambda) as lambda tends to plus infinity.

The author has recently discovered [1] that all previous publications on first order systems give formulae for the second asymptotic coefficient that are either incorrect or incomplete (i.e. an algorithm for the calculation of the second asymptotic coefficient rather than an explicit formula). The aim of the talk is to present the correct formula for the second asymptotic coefficient and to discuss its geometric meaning.

[1] Preprint arXiv:1208.6015. To appear in Journal of Spectral Theory.