Accurate estimates of asymptotic indices via fractional calculus

TL;DR

This paper introduces a novel fractional calculus-based method with a three-parameter search to accurately estimate asymptotic indices from divergent series, demonstrated on quantum oscillator models.

Contribution

It presents a new fractional calculus approach combined with Padé approximants for precise asymptotic index estimation from divergent series.

Findings

Effective in estimating large-coupling amplitudes and exponents

Demonstrated high accuracy on anharmonic oscillator series

Outperforms traditional methods in pilot tests

Abstract

We devise a three-parameter random search strategy to obtain accurate estimates of the large-coupling amplitude and exponent of an observable from its divergent Taylor expansion, known to some desired order. The endeavor exploits the power of fractional calculus, aided by an auxiliary series and subsequent construction of Pad\'e approximants. Pilot calculations on the ground-state energy perturbation series of the octic anharmonic oscillator reveal the spectacular performance.