Nonperturbative Quantum Physics from Low-Order Perturbation Theory

TL;DR

This paper demonstrates how low-order divergent perturbation series, combined with analytic continuation functions like hypergeometric functions, can accurately approximate nonperturbative quantum effects such as ionization and energy broadening.

Contribution

It introduces a novel method using hypergeometric functions and Bender-Wu dispersion relations to improve approximations of nonperturbative quantum phenomena from low-order perturbation data.

Findings

Hypergeometric functions outperform Padé approximants in approximating complex eigenvalues.

Excellent agreement with exact results using only fourth-order perturbation theory.

Method effectively captures nonperturbative effects like tunneling and energy broadening.

Abstract

The Stark effect in hydrogen and the cubic anharmonic oscillator furnish examples of quantum systems where the perturbation results in a certain ionization probability by tunneling processes. Accordingly, the perturbed ground-state energy is shifted and broadened, thus acquiring an imaginary part which is considered to be a paradigm of nonperturbative behavior. Here we demonstrate how the low order coefficients of a divergent perturbation series can be used to obtain excellent approximations to both real and imaginary parts of the perturbed ground state eigenenergy. The key is to use analytic continuation functions with a built in analytic structure within the complex plane of the coupling constant, which is tailored by means of Bender-Wu dispersion relations. In the examples discussed the analytic continuation functions are Gauss hypergeometric functions, which take as input fourth order perturbation theory and return excellent approximations to the complex perturbed eigenvalue. These functions are Borel-consistent and dramatically outperform widely used Pad\'e and Borel-Pad\'e approaches, even for rather large values of the coupling constant.