  We study overlapping Schwarz methods for the Helmholtz equation posed in any
dimension with large, real wavenumber and smooth variable wave speed. The
radiation condition is approximated by a Cartesian perfectly-matched layer
(PML). The domain-decomposition subdomains are overlapping hyperrectangles with
Cartesian PMLs at their boundaries. The overlaps of the subdomains and the
widths of the PMLs are all taken to be independent of the wavenumber.
  For both parallel (i.e., additive) and sequential (i.e., multiplicative)
methods, we show that after a specified number of iterations -- depending on
the behaviour of the geometric-optic rays -- the error is smooth and smaller
than any negative power of the wavenumber. For the parallel method, the
specified number of iterations is less than the maximum number of subdomains,
counted with their multiplicity, that a geometric-optic ray can intersect.
  These results, which are illustrated by numerical experiments, are the first
wavenumber-explicit results about convergence of overlapping Schwarz methods
for the Helmholtz equation, and the first wavenumber-explicit results about
convergence of any domain-decomposition method for the Helmholtz equation with
a non-trivial scatterer (here a variable wave speed).
