The resolvent of a first order elliptic system and spectral asymptotics

Weyl's law for the distribution of eigenvalues of the Laplacian on a compact domain $\Omega\subset\mathbb{R}^n$ is widely considered as the birth of the subject of spectral asymptotics. While the spectral asymptotics of a single elliptic partial or pseudo differential operator (PDO or $\Psi$DO) on a compact manifold is by now well understood, and asymptotic coefficients of two-digit orders are often known explicitly, far less is known about systems of several operators beyond the original Weyl's law which is amazingly universal. The main difficulty lies in the presence of additional fiberwise degrees of freedom, if we think of the system as a single (matrix) operator acting on a vector field. There have been well known naive attempts to reduce the problem to the scalar case, which have failed dramatically.
On a compact smooth manifold without boundary we consider an elliptic first order $\Psi DO$ acting on columns of functions or half-densities. Under suitable conditions (see here [1] for a review) this operator admits spectral asymptotic expansion, and we want to find an explicit formula for the second asymptotic coefficients (the first ones being given by Weyl's law). One of the most successful and commonly used approach, based on the heat kernel expansion, is not effective in this case since our operator is not semibounded and we need to study the positive and negative parts of the spectrum separately. Working along the lines of the wave equation method instead, the desired coefficients were finally established in 2013 in a fairly long and complicated way [2].
The most natural and elegant approach to the problem uses an approximate resolvent, which was suggested by Ivrii in 1984, but has never been accomplished due to the apparently unmanageable calculations arising on the way. Namely, a direct substitution into Ivrii's formula yields more than a hundred terms, which has left this problem open for more than 30 years. Thankfully, we succeeded in finding an algebraic identity that drastically simplifies the expression and quickly yields the desired formula. Moreover, in the course of deriving our results we came up with a novel approach for finding spectral asymptotics of pseudo-differential systems explicitly, which can potentially give further asymptotic coefficients. This work is in collaboration with Prof. D. Vassiliev and Prof. J. Sjöstrand [3].