Padé summability of the cubic oscillator

Abstract

We prove the Padé (Stieltjes) summability of the perturbation series of any energy level En,1(β), , of the cubic anharmonic oscillator, , as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the -symmetric Hamiltonian H1(β) for positive β (Bessis–Zinn Justin conjecture). The n zeros, of a state ψn,1(β), stable at β = 0, are confined for β on the cut complex plane, and are related to the level En,1(β) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.