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

TL;DR

The paper explores how negative dimensions affect the convergence of perturbation series in quantum mechanics and field theory, revealing inverse factorial convergence and potential for algebraic solutions.

Contribution

It demonstrates the algebraic solution of spectral problems in negative dimensions and links this to improved convergence properties of perturbation series.

Findings

Factorial growth of perturbation series diminishes at negative even dimensions

Inverse factorial convergence of quantities constructed from perturbative coefficients

Potential generalization to quantum field theory

Abstract

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.