Vincenzo Grecchi, Marco Maioli, Andre' Martinez Pade' summability of the cubic oscillator (306K, pdf) ABSTRACT. We prove the Pad\'e (Stieltjes) summability of the perturbation series of the energy levels of the cubic anharmonic oscillator, $H_1(\beta)=p^2+x^2+i\sqrt{\beta} x^3$, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple and independent proof of the positivity of the eigenvalues of the $\mathcal{PT}$-symmetric operator $H_1(\beta)$ for real $\beta$ (Bessis-Zinn Justin conjecture). All the $n\in\N$ zeros of an eigenfunction, real at $\beta=0$, become complex with negative imaginary part, for complex, non-negative $\beta\neq 0$.