The spectrum of the cubic oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, $$ H(beta)=-frac{d^2}{dx^2}+x^2+isqrt{beta}x^3, $$ for $beta$ in the cut plane $C_c:=Cbackslash R_-$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues ${E_n(beta)}_{ngeq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $betainC_c$, $E_n(beta)$ 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.