Entire hypergeometric approximants for the ground state energy perturbation series of the quartic, sextic and octic anharmonic oscillators - I.S. Elkamash (16810689) (2023)
In resumming a divergent series, one is frequently confronted with situations where predictions from traditional resummation techniques might not accurate when considering strong-coupling values. Besides, in literature, we barely can find a resummation algorithm for which the convergence is faster for stronglydivergent ( Gevrey − k , k > 1) series than a divergent one ( Gevrey − 1 ). In this work, we use appropriate hypergeometric functions to approximate a divergent series ( n! large order growth factor) and strongly-divergent series ( (2n)! and (3n)! large order growth factors). We found that the convergence of the predicted non-perturbative parameters to their exact values is faster for the challenging Gevrey−3 than the Gevrey−2 series which in turn is faster than the divergent Gevrey − 1 series. To explain such interesting feature, we suggest a type of analytic continuation where the parametrized divergent hypergeometric series is represented as a finite sum over entire hypergeometric functions. The algorithm is applied to sum the weak-coupling series of the ground state energy of the quartic (divergent), sextic (Gevrey−2) and Octic (Gevrey−3) anharmonic oscillators. Accurate results are obtained for the ground state energy as well as the predicted non-perturbative parameters. While traditional resummation techniques fail to give reliable results for large coupling values, our algorithm gives the expected limit as the coupling goes to infinity. Other Information Published in: Annals of Physics License: http://creativecommons.org/licenses/by/4.0/ See article on publisher's website: https://dx.doi.org/10.1016/j.aop.2023.169427