The Spectrum of the Cubic Oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, $$begin{array}{ll}H(beta)=-frac{d^2}{dx^2}+x^2+isqrt{beta}x^3end{array}$$ , for β in the cut plane ${mathcal{C}_c:=mathcal{C}backslash mathcal{R}_-}$ . Moreover, we prove that the spectrum consists of the perturbative eigenvalues {E _n (β)} _{n ≥ 0} labeled by the constant number n of nodes of the corresponding eigenfunctions. In addition, for all ${beta in mathcal{C}_c, E_n(beta)}$ can be computed as the Stieltjes-Padé sum of its perturbation series at β = 0. This also gives an alternative proof of the fact that the spectrum of H(β) is real when β 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.