On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes
arXiv:1304.7041 · doi:10.1088/0264-9381/30/23/235014
Abstract
We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n+2)-dimensional static and spherically symmetric spacetimes. They are related to properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially selfadjoint, but it does satisfy a weaker property that we call here quasi essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.
36 pages. Final version to appear in Classical and Quantum Gravity