The spectral function of a first order elliptic system

TL;DR

This paper derives explicit two-term asymptotic formulas for the propagator, spectral function, and counting function of a first order elliptic pseudodifferential operator on a compact manifold, regardless of eigenvalue sign assumptions.

Contribution

It provides the first detailed asymptotic analysis of these spectral objects for general first order elliptic systems without semi-boundedness assumptions.

Findings

Explicit two-term asymptotics for the propagator

Asymptotic formulas for spectral and counting functions as lambda approaches infinity

Results applicable to operators with eigenvalues of arbitrary sign

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 tending to plus infinity.