Continued Roots, Power Transform and Critical Properties
TLDR
In this paper, the authors considered the problem of computing the critical amplitudes at infinity by means of the self-similar continued root approximants, which can be found from the optimization imposed on the parameters of power transform.Abstract:
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.read more
Citations
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Iterative Borel Summation with Self-Similar Iterated Roots
TL;DR: In this paper , the iterative Borel summation is applied iteratively in conjunction with self-similar iterated roots to find critical indices and amplitudes directly and explicitly, and the number of steps employed in the course of iterations is used as a continuous control parameter.
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Symmetry and Approximation Methods
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On the Optimal Conductivity of Packed Two-Dimensional Dispersed Composites
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On the quartic anharmonic oscillator and the Padé-approximant averaging method
Victor Babenko,N. M. Petrov +1 more
TL;DR: In this paper , the convergence of the Padé-type approximations is studied and the ground state energy of the anharmonic oscillator with the Hamiltonian is calculated for a wide range of variation of the coupling constant.
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