The second Weyl coefficient for a first order system

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.

Edited in accordance with referee's recommendations + updated bibliography

References in corpus (2)

• Spectral asymptotics for first order systems
• Pseudodifferential operators on manifolds: a coordinate-free approach

Cited by in corpus (5)

• Invariant subspaces of elliptic systems II: spectral theory
• Diagonalization of elliptic systems via pseudodifferential projections
• Global propagator for the massless Dirac operator and spectral asymptotics
• Geometric wave propagator on Riemannian manifolds
• Spectral asymptotics for linear elasticity: the case of mixed boundary conditions