An index theorem on asymptotically static spacetimes with compact Cauchy surface

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then is Fredholm. This generalizes a theorem due to B\"ar-Strohmaier in case of finite times, and we also show corresponding index formula extends setting. Furthermore, demonstrate existence Fredholm inverse which same time Feynman parametrix sense Duistermaat-H\"ormander. The proof combines methods from time-dependent scattering theory variant Egorov's for pseudo-differential hyperbolic systems.