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• W2792119550 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 $gge0$. 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 $qge0$ and complex $g$. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the SCE." @default.
• W2792119550 created "2018-03-29" @default.
• W2792119550 creator A5062694019 @default.
• W2792119550 creator A5064570911 @default.
• W2792119550 date "2018-09-19" @default.
• W2792119550 modified "2023-10-17" @default.
• W2792119550 title "From divergent perturbation theory to an exponentially convergent self-consistent expansion" @default.
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• W2792119550 doi "https://doi.org/10.1103/physrevd.98.056017" @default.
• W2792119550 hasPublicationYear "2018" @default.
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