A Deal with the Devil: From Divergent Perturbation Theory to an Exponentially-Convergent Self-Consistent Expansion

TL;DR

This paper demonstrates that the self-consistent expansion (SCE) method converges exponentially fast for nonlinear systems, outperforming traditional perturbation theory and other resummation techniques, especially for large nonlinearity parameters.

Contribution

The paper rigorously proves exponential convergence of the SCE for the classical anharmonic oscillator's partition function, extending its applicability and comparing it favorably with other methods.

Findings

SCE converges exponentially fast in order N for the anharmonic oscillator.

SCE outperforms Borel resummation, hyperasymptotics, Padé, and Lanczos methods for large nonlinearity.

SCE successfully captures the partition function for double-well potentials and generalizes to complex nonlinearities.

Abstract

For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the non-linear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "an invention of the devil." An alternative method, the self-consistent expansion (SCE), has been introduced by Schwartz and Edwards. Its basic idea is a rescaling of the zeroth-order system around which the solution is expanded, to achieve optimal results. While low-order SCEs have been remarkably successful in describing the dynamics of non-equilibrium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this issue we apply this technique to the canonical partition function of the classical harmonic oscillator with a quartic $gx^{4}$ anharmonicity, for which perturbation theory's divergence is well-known. We obtain the $N$th order SCE for the partition function, which is rigorously found to converge exponentially fast in $N$, and uniformly in $g\ge0$. We use our results to elucidate the relation between the SCE and the class of approaches based on the so-called "order-dependent mapping." Moreover, we put the SCE to test against other methods that improve upon perturbation theory (Borel resummation, hyperasymptotics, Pad\'e approximants, and the Lanczos $\tau$-method), and find that it compares favorably with all of them for small $g$ and dominates over them for large $g$. The SCE is shown to successfully capture the correct partition function for the double-well potential case, where no perturbative expansion exists. Our treatment is generalized to the case of many oscillators, as well as to any nonlinearity of the form $g|x|^{q}$ with $q\ge0$ and complex $g$. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the SCE.