Abstract. We study the perturbation theory for H=pZ+x2+flx 2~+1, n= 1,2 ..... It is proved that when Imfl~:0, H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schr6dinger perturbation expansion, and admits an analytic continuation to Im/~=0 where it can be interpreted as a resonance of the problem.

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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.

We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the "resonances" of the anharmonic and double well oscillators.