Asymptotic response of observables from divergent weak-coupling expansions: A fractional-calculus-assisted Padé technique

Abstract

Appropriate constructions of Padé approximants are believed to provide reasonable estimates of the asymptotic (large-coupling) amplitude and exponent of an observable, given its weak-coupling expansion to some desired order. In many instances, however, sequences of such approximants are seen to converge very poorly. We outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior. Pilot calculations on the ground-state perturbative energy series of quartic, sextic, and octic anharmonic oscillators reveal clearly the worth of our endeavor.

Accurate estimates of asymptotic indices via fractional calculus

“…All displayed data are rounded off at the 3 rd decimal place. Here, method I refers to the parent scheme and method II the bare FC-assisted one [3]. Notably, the latter involves a 1parameter variation, the order of fractional differentiation.…”

“…In the context of phase transitions involving power series with a finite radius of convergence, FC can be implemented [2] to obtain improved estimates of critical indices. We have recently showed [3] how it can be employed fruitfully in assessing asymptotic indices too from power series with zero radius of convergence. Here, we put forward a remarkably powerful strategy that exploits the idea of embedding an auxiliary series in conjunction.…”

“…The power-law form (2) is, in cases, intuitively obvious [3], e.g., when F stands for an observable and x is a tunable physical variable. In a few situations, however, form (2) represents the leading behavior of an asymptotic expansion whose structure is derivable from (1).…”

Continued Roots, Power Transform and Critical Properties

“…It appears to be quite good and better in prediction than others considered for some other trial values of b * . The convergence of the method is much faster than for the method of corrected approximants [14,33], or for the fractional-calculus applied together with Padé approximants [37,38].…”

“…as shown in [4,37,38]. In terms of the small-variable truncation φ k (x), we can express the truncated DLog function ψ k (x),…”