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We give simple conditions implying the well-posedness of the Cauchy problemfor the propagation of classical scalar fields in general (n+2)-dimensionalstatic and spherically symmetric spacetimes. They are related to properties ofthe underlying spatial part of the wave operator, one of which being thestandard essentially self-adjointness. However, in many examples the spatialpart of the wave operator turns out to be not essentially selfadjoint, but itdoes satisfy a weaker property that we call here quasi essentiallyself-adjointness, which is enough to ensure the desired well-posedness. This iswhy we also characterize this second property.  We state abstract results, then general results for a class of operatorsencompassing many examples in the literature, and we finish with the explicitanalysis of some of them.

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes