  We prove the simplicity and analyticity of the eigenvalues of the cubic
oscillator Hamiltonian,$H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3$,for $\beta$ in
the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the
spectrum consists of the perturbative eigenvalues $\{E_n(\beta)\}_{n\geq 0}$
labeled by the constant number $n$ of nodes of the corresponding
eigenfunctions. In addition, for all $\beta\in\C_c$, $E_n(\beta)$ can be
computed as the Stieltjes-Pad\'e sum of its perturbation series at $\beta=0$.
This also gives an alternative proof of the fact that the spectrum of
$H(\beta)$ is real when $\beta $ is a positive number. This way, the main
results on the repulsive PT-symmetric and on the attractive quartic oscillators
are extended to the cubic case.
