Accurate estimates of asymptotic indices via fractional calculus

2022

Symmetry

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.

“…A close result is achieved by optimization according to (34) and (35), as in the fourth order, we arrive at A 4,2 (u 4 ) = A 4,1 (u 4 ) = 1.33898, (u 4 = 1.34757).…”

Section: Examplesupporting

confidence: 67%

Iterative Borel Summation with Self-Similar Iterated Roots

2022

Symmetry

“…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].…”

Section: Discussionmentioning

confidence: 99%

“…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),…”

Section: Indeterminate Problem Calculation Of Critical Exponentsmentioning

confidence: 99%

Continued Roots, Power Transform and Critical Properties

2021

Symmetry

We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to the indeterminate (divergent) problem my means of power transformation. Most challenging indeterminate for the continued roots problems of calculating critical amplitudes, can be successfully attacked by performing proper power transformation to be found from the optimization imposed on the parameters of power transform. The self-similar continued roots were derived by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the fundamental symmetry principle of functional self-similarity, realized constructively in the space of approximations. Our approach to the solution of the indeterminate problem is to replace it with the determinate problem, but with some unknown control parameter b in place of the known critical index β. From optimization conditions b is found in the way making the problem determinate and convergent. The index β is hidden under the carpet and replaced by b. The idea is applied to various, mostly quantum-mechanical problems. In particular, the method allows us to solve the problem of Bose-Einstein condensation temperature with good accuracy.

“…The form (28) hints that it could be feasible to combine the technique of Borel summation with fractional calculus [7,8]. We would like to introduce fractional derivatives in such a way that a nice asymptotic property of asymptotic scale invariance [9] given by the expression ( 1) is preserved.…”

Section: Commentsmentioning

confidence: 99%

Modified Padé–Borel Summation

2023

Axioms

We revisit the problem of calculating amplitude at infinity for the class of functions with power-law behavior at infinity by means of a resummation procedure based on the truncated series for small variables. Iterative Borel summation is applied by employing Padé approximants of the “odd” and “even” types modified to satisfy the power-law. The odd approximations are conventional and are asymptotically equivalent with an odd number of terms in the truncated series. Even approximants are new, and they are constructed based on the idea of corrected approximants. They are asymptotically equivalent to the even number of terms in truncated series. Odd- and even-modified Padé approximants could be applied with and without a Borel transformation. The four methods are applied to some basic examples from condensed matter physics. We found that modified Padé–Borel summation works well in the case of zero-dimensional field theory with fast-growing coefficients and for similar examples. Remarkably, the methodology of modified Padé–Borel summation appears to be extendible to the instances with slow decay or non-monotonous behavior. In such situations, exemplified by the problem of Bose condensation temperature shift, the results are still very good.