An index theorem on asymptotically static spacetimes with compact Cauchy surface
arXiv:2104.02816 · doi:10.2140/paa.2022.4.727
Abstract
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 Bär-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-Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov's theorem for pseudo-differential hyperbolic systems.
arXiv comment41 pages; v3: minor fixes, references added, accepted in Pure Appl. Anal
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