Global Propagator for the Massless Dirac Operator and Spectral Asymptotics

Matteo Capoferri*, Dmitri Vassiliev

*Corresponding author for this work

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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.

Original languageEnglish
Article number30
JournalIntegral Equations and Operator Theory
Volume94
Issue number3
Early online date9 Aug 2022
DOIs
Publication statusPublished - Sep 2022

Keywords

  • Dirac operator
  • Global Fourier integral operators
  • Hyperbolic propagators
  • Weyl coefficients

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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