Padé summability of the cubic oscillator

We prove the Pad?e (Stieltjes) summability of the perturbation series of any energy level En,1(?), n ? N, of the cubic anharmonic oscillator, H1(?) = p2 +x2 +i??x3, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the PT -symmetric Hamiltonian H1(?) for positive ? (Bessis–Zinn Justin conjecture). The n zeros, of a state ?n,1(?), stable at ? = 0, are confined for ? on the cut complex plane, and are related to the level En,1(?) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.