An index theorem on asymptotically static spacetimes with compact Cauchy surface
| Autor: | Shen, Dawei, Wrochna, Michał |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Pure Appl. Analysis 4 (2022) 727-766 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/paa.2022.4.727 |
| Popis: | We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat-H\"ormander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems. Comment: 41 pages; v3: minor fixes, references added, accepted in Pure Appl. Anal |
| Databáze: | arXiv |
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