Capoferri, Matteo;
Vassiliev, Dmitri;
(2022)
Global Propagator for the Massless Dirac Operator and Spectral Asymptotics.
Integral Equations and Operator Theory
, 94
, Article 30. 10.1007/s00020-022-02708-1.
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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.
| Type: | Article |
|---|---|
| Title: | Global Propagator for the Massless Dirac Operator and Spectral Asymptotics |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00020-022-02708-1 |
| Publisher version: | https://doi.org/10.1007/s00020-022-02708-1 |
| Language: | English |
| Additional information: | ©The Author(s) 2022. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
| Keywords: | math.AP, math.AP, math-ph, math.DG, math.MP, math.SP, primary 35L45, secondary 35Q41, 58J40, 58J45, 35P20 |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10153872 |
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