Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 057, 27 pages      arXiv:2303.02993
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

Modified Green-Hyperbolic Operators

Christopher J. Fewster
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK

Received April 20, 2023, in final form July 29, 2023; Published online August 08, 2023

Green-hyperbolic operators – partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators – play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.

Key words: Green-hyperbolic operators; nonlocal equations; Fredholm theory.

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