Odd anharmonic oscillators and shape resonances

By adapting a stability argument of Vock and Hunziker it is proved that the Borel sum of the Rayleigh-Schroedinger perturbation expansion of any odd anharmonic oscillator p2 + x2 + gxsup(2k+1), k = 1, 2,..., is the limit of a sequence of resonances in the standard sense of dilation analyticity. The same method yields, for a suitable class of potentials, an existence proof of shape resonances