Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration

TL;DR

This paper introduces a new method using finite-part integrals to sum divergent Stieltjes series, enabling accurate calculations of quantum oscillator energies across all regimes from a single expansion.

Contribution

It develops a novel convergent expansion based on Hadamard's finite part integrals to sum divergent series in the strong-asymptotic regime for Stieltjes integrals.

Findings

Successfully computed energies of anharmonic oscillators in all regimes.

Unified approach from weak to strong coupling regimes.

Improved accuracy over traditional perturbation methods.

Abstract

We devise a prescription to utilize a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments which we treated as Hadamard's finite part integrals. The result allowed us to compute the ground-state energy of the quartic, sextic anharmonic oscillators as well as the $\mathcal{PT}$ symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behavior of the spectra.