  We consider an elliptic self-adjoint first order pseudodifferential operator
acting on columns of complex-valued half-densities over a connected compact
manifold without boundary. 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. We study the following objects: the propagator
(time-dependent operator which solves the Cauchy problem for the dynamic
equation), the spectral function (sum of squares of Euclidean norms of
eigenfunctions evaluated at a given point of the manifold, with summation
carried out over all eigenvalues between zero and a positive lambda) and the
counting function (number of eigenvalues between zero and a positive lambda).
We derive explicit two-term asymptotic formulae for all three. For the
propagator "asymptotic" is understood as asymptotic in terms of smoothness,
whereas for the spectral and counting functions "asymptotic" is understood as
asymptotic with respect to lambda tending to plus infinity.
