  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.
