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 and Strohmaier (Amer. J. Math. 141:5 (2019), 1421–1455) 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 and Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov’s theorem for pseudodifferential hyperbolic systems.
Citation
Dawei Shen. Michał Wrochna. "AN INDEX THEOREM ON ASYMPTOTICALLY STATIC SPACETIMES WITH COMPACT CAUCHY SURFACE." Pure Appl. Anal. 4 (4) 727 - 766, 2022. https://doi.org/10.2140/paa.2022.4.727
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