Pade' summability of the cubic oscillator

We prove the Pade (Stieltjes) summability of the perturbation series of anyenergy level En,1(β), n \in N, of the cubic anharmonic oscillator, H1(β) =p^2 +x^2 +i√βx^3, as suggested by the numerical studies of Bender andWeniger.At the same time, we give a simple proof of the positivity of every levelof the PT -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justinconjecture). The n zeros, of a state ψ_n,1(β), stable at β = 0, are confinedfor β on the cut complex plane, and are related to the level En,1(β) by theBohr–Sommerfeld quantization rule (semiclassical phase-integral condition).We also prove the absence of non-perturbative eigenvalues and the simplicityof the spectrum of our Hamiltonians.