Anharmonic oscillator, negative dimensions and inverse factorial convergence of large orders to the asymptotic form

The spectral problem for O(D) symmetric polynomial potentials allows for a partial algebraic solution after analytical continuation to negative even dimensions D. This fact is closely related to the disappearance of the factorial growth of large orders of the perturbation theory at negative even D. As a consequence, certain quantities constructed from the perturbative coefficients exhibit fast inverse factorial convergence to the asymptotic values in the limit of large orders. This quantum mechanical construction can be generalized to the case of quantum field theory.