Iterative Borel Summation with Self-Similar Iterated Roots

Abstract

Borel summation is applied iteratively in conjunction with self-similar iterated roots. In general form, the iterative Borel summation is presented in the form of a multi-dimensional integral. It can be developed only numerically and is rarely used. Such a technique is developed in the current paper analytically and is shown to be more powerful than the original Borel summation. The self-similar nature of roots and their asymptotic scale invariance allow us to find critical indices and amplitudes directly and explicitly. The locations of poles remain the same with the uncontrolled self-similar Borel summation. The number of steps employed in the course of iterations is used as a continuous control parameter. To introduce control into the discrete version of the iterative Borel summation, instead of the exponential function, we use a stretched (compacted) exponential function. For the poles, considering inverse quantities is prescribed. The simplest scheme of the iterative Borel method, based on averaging over the one-step and two-step Borel iterations, works well when lower and upper bounds are established by making those steps. In the situations when only a one-sided bound is found, the iterative Borel summation with the number of iterations employed as the control works best by extrapolating beyond the bound. Several key examples from condensed matter physics are considered. Iterative application of Borel summation leads to an improvement compared with a conventional, single-step application of the Borel summation.

Modified Padé–Borel Summation

“…Our choice throughout the current paper of the modified Padé approximants allows for analytical calculation of the amplitudes while keeping the calculations rather simple and straightforward. Again, just like in the paper [9], we can extend the technique from the amplitude A calculations to the indices β.…”

“…Borel summation is applied for the effective summation of the functions with known truncation at small x [9,[11][12][13][14][15][16][17][18][19][20]. More references on Borel summation can be found in our recent paper [11].…”

“…in the same form for odd and even cases [9]. Let us consider the very popular in field theory and statistical mechanics, zerodimensional anharmonic model represented by the integral…”

“…While for the very long truncations, the difference between odd and even cases may be insignificant and ignored, for short truncations, the difference can very well be detectable. Of course, to avoid the problem of odd-even approximants altogether, one can resort to the self-similar iterated roots, which assimilate the coefficients a n one-by-one [9]. However, in contrast, the Padé approximants can be easily and routinely extended to very high orders.…”

“…The modified Padé-Borel summation takes into account an arbitrary power-law behavior at infinity, making it superior to the standard Padé-Borel approach which considers only integer powers. In addition, the approach is much simpler compared with optimal Borel-Leroy, Mittag-Leffler and iterative Borel techniques [9,11], allowing us to go easily to very high orders of perturbation theory.…”