Spectral theory of non-symmetric operators in time-dependent external fields and in globally hyperbolic spacetimes

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For the proposed research project, we plan to analyze non-symmetric operators defined on the space of solutions of the Dirac equation containing a time-dependent gravitational field. In more detail, we consider the Dirac equation defined on a globally hyperbolic spacetime background. The goal of this project is to develop a method for introducing a specific class of spectral projectors non-perturbatively, which have only been defined through a formal perturbation-expansion so far. In the case of a static spacetime those spectral projectors are supposed to be the well-known projectors on the negative (or positive) spectral subspaces of the relativistic hamilton operator. In order to develop this method step by step, we analyze the case of a closed Friedmann-Robertson-Walker (FRW) spacetime first.