The spectral function of a first order elliptic system

Abstract

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 $\lambda \to +\infty$.