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Abstract. We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,for β in the cut plane Cc := C\R−. Moreover, we prove that the spectrum consists of the perturbative eigenvalues {En(β)} n≥0 labeled by the constant number n of nodes of the corresponding eigenfunctions. In addition, for all β ∈ Cc, En(β) 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.

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Abstract. It is proved that the action of a weak electric field shifts the eigenvalues of the Hydrogen atom into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel method. This is obtained by combining the Balslev-Combes technique of analytic dilatations with Simon's results on anharmonic oscillators.

Abstract.We prove the Padé (Stieltjes) summability of the perturbation series of the energy levels of the cubic anharmonic oscillator, H 1 (β) = p 2 + x 2 + i p βx 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 PT -symmetric operator H 1 (β) for real β (Bessis-Zinn Justin conjecture). All the n 2 N zeros of an eigenfunction, real at β = 0, become complex with negative imaginary part, for complex, non-negative β 6 = 0.