  A precise calculation of the ground-state energy of the complex PT-symmetric
Hamiltonian $H=p^2+{1/4}x^2+i \lambda x^3$, is performed using high-order
Rayleigh-Schr\"odinger perturbation theory. The energy spectrum of this
Hamiltonian has recently been shown to be real using numerical methods. The
Rayleigh-Schr\"odinger perturbation series is Borel summable, and Pad\'e
summation provides excellent agreement with the real energy spectrum. Pad\'e
analysis provides strong numerical evidence that the once-subtracted
ground-state energy considered as a function of $\lambda^2$ is a Stieltjes
function. The analyticity properties of this Stieltjes function lead to a
dispersion relation that can be used to compute the imaginary part of the
energy for the related real but unstable Hamiltonian $H=p^2+{1/4}x^2-\epsilon
x^3$.
