The second Weyl coefficient for a first order system

TL;DR

This paper simplifies the calculation of the second Weyl coefficient for first order elliptic systems by analyzing the resolvent as a pseudodifferential operator, building on prior asymptotic results.

Contribution

It provides a simplified method to compute the second Weyl coefficient for first order elliptic systems, extending previous complex calculations.

Findings

Simplified the computation of the second Weyl coefficient.

Confirmed the existence of two-term asymptotics for first order systems.

Utilized pseudodifferential operator analysis of the resolvent.

Abstract

For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between zero and lambda when lambda tends to infinity, under an additional dynamical condition. This is a well known result of J.J.Duistermaat and V.W.Guillemin from 1975. In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in 2013. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptotics already established, it suffices to study the resolvent as a pseudodifferential operator in order to identify and compute the second coefficient.