Global propagator for the massless Dirac operator and spectral asymptotics

Capoferri, Matteo ORCID: https://orcid.org/0000-0001-6226-1407 and Vassiliev, Dmitri 2022. Global propagator for the massless Dirac operator and spectral asymptotics. Integral Equations and Operator Theory 94 (3) , 30. 10.1007/s00020-022-02708-1

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Abstract

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.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Additional Information:	License information from Publisher: LICENSE 1: URL: http://creativecommons.org/licenses/by/4.0/, Type: open-access
Publisher:	Springer
ISSN:	0378-620X
Date of First Compliant Deposit:	10 August 2022
Date of Acceptance:	11 July 2022
Last Modified:	07 May 2023 01:31
URI:	https://orca.cardiff.ac.uk/id/eprint/151866

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