Solving the Stieltjes summation problem by employing various mappings of finite-part integration prescription

Abstract

We devise and compare various prescriptions to sum a divergent Stieltjes series into its associated Stieltjes transform. Central to the prescriptions are the various mappings of the perturbation coefficients to the positive-power moments of some unknown positive function. The knowledge of the large-order growth of the perturbation coefficients is required to successfully sum and continue the divergent Stieltjes. We applied the prescription by implementing various mappings of the coefficients of the divergent expansion for the ground state energy of the sextic anharmonic oscillator. The results reveal that all mappings theoretically provide accurate approximations in the perturbative regime. However, at constant value at which we truncate the close form of the approximation, there will be an apparent difference between the mappings that overestimate and underestimate the actual growth of coefficients. The patterns observed in these two types of mappings can be used to pinpoint the proper mapping for systems with unknown correction growth.