From divergent perturbation theory to an exponentially convergent self-consistent expansion

For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the nonlinear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "an invention of the devil." Although a lot of progress has been made on understanding the mathematics and physics behind this, new approaches are still called for. A related 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 nonequilibrium 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 gx4 anharmonicity, for which perturbation theory's divergence is well-known. We obtain the Nth order SCE for the partition function, which is rigorously found to converge exponentially fast in N, and uniformly in g≥0. 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, Padé approximants, and the Lanczos τ-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 converge to the correct partition function for the double-well potential case, even when expanding around the local maximum. Our treatment is generalized to the case of many oscillators, as well as to any nonlinearity of the form g|x|q with q≥0 and complex g. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the SCE.