  A long time ago, it has been conjectured that a Hamiltonian with a potential
of the form x^2+i v x^3, v real, has a real spectrum. This conjecture has been
generalized to a class of so-called PT symmetric Hamiltonians and some proofs
have been given. Here, we show by numerical investigation that the divergent
perturbation series can be summed efficiently by an order-dependent mapping
(ODM) in the whole complex plane of the coupling parameter v^2, and that some
information about the location of level crossing singularities can be obtained
in this way. Furthermore, we discuss to which accuracy the strong-coupling
limit can be obtained from the initially weak-coupling perturbative expansion,
by the ODM summation method. The basic idea of the ODM summation method is the
notion of order-dependent "local" disk of convergence and analytic continuation
by an order-dependent mapping of the domain of analyticity augmented by the
local disk of convergence onto a circle. In the limit of vanishing local radius
of convergence, which is the limit of high transformation order, convergence is
demonstrated both by numerical evidence as well as by analytic estimates.
