Symmetry (Oct 2024)

Borel Summation Can Be Controlled by Critical Indices

  • Simon Gluzman

DOI
https://doi.org/10.3390/sym16111438
Journal volume & issue
Vol. 16, no. 11
p. 1438

Abstract

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We consider application of the self-similarity principle in approximation theory under the conditions of asymptotic scale-invariance. For the effective summation of the asymptotic series methods, an iterative Borel summation with self-similar iterated roots is applied. The approximants follow from the self-similarity considerations and behave asymptotically as a power-law satisfying the asymptotic scale invariance. Optimal conditions on convergence of the sequence of approximants are imposed through the critical indices defined from the approximants. The indices are understood as control parameters for the optimal convergence of the asymptotic series. Such interpretation of the indices leads to an overall improvement of accuracy in calculations of the indices. The statement is supported by fifteen examples from condensed matter physics, quantum mechanics and field theory.

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