09-82 Vincenzo Grecchi, Marco Maioli, Andre' Martinez
Pade' summability of the cubic oscillator (306K, pdf) May 26, 09
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Abstract. We prove the Pad\'e (Stieltjes) summability of the perturbation series of the energy levels of the cubic anharmonic oscillator, $H_1(\beta)=p^2+x^2+i\sqrt{\beta} x^3$, 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 $\mathcal{PT}$-symmetric operator $H_1(\beta)$ for real $\beta$ (Bessis-Zinn Justin conjecture). All the $n\in\N$ zeros of an eigenfunction, real at $\beta=0$, become complex with negative imaginary part, for complex, non-negative $\beta\neq 0$.