Global propagator for the massless Dirac operator and spectral asymptotics

TL;DR

This paper constructs a global propagator for the massless Dirac operator on a 3-manifold, providing explicit formulas and geometric invariants, and computes spectral asymptotics including Weyl coefficients.

Contribution

It introduces a global invariant construction of the Dirac propagator, with explicit formulas and algorithms for symbol calculation, and derives spectral asymptotics.

Findings

Explicit formulas for propagator symbols

Algorithm for symbol component calculation

Computed third Weyl coefficient

Abstract

We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- correspond to positive and negative eigenvalues of $W$, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of $W$.