Global propagator for the massless Dirac operator and spectral asymptotics

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 $W$.

Final version, to appear in Integral Equations and Operator Theory

References in corpus (7)

• Invariant subspaces of elliptic systems II: spectral theory
• Global wave parametrices on globally hyperbolic spacetimes
• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Geometric wave propagator on Riemannian manifolds
• Diagonalization of elliptic systems via pseudodifferential projections
• Classification of first order sesquilinear forms
• Global and microlocal aspects of Dirac operators: propagators and Hadamard states

Cited by in corpus (7)

• Invariant subspaces of elliptic systems II: spectral theory
• Global wave parametrices on globally hyperbolic spacetimes
• Invariant subspaces of elliptic systems I: pseudodifferential projections
• Diagonalization of elliptic systems via pseudodifferential projections
• On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary
• Topological obstructions to the diagonalisation of pseudodifferential systems
• Propagators in curved spacetimes from operator theory