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

• URL: http://arxiv.org/abs/2302.03891v1
• Date: Wed, 8 Feb 2023 05:54:33 GMT
• Title: Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration
• Authors: Christian D. Tica and Eric A. Galapon
• Abstract summary: We devise a prescription to utilize a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral. The novel expansion makes use of divergent negative-power moments which we treated as Hadamard's finite part integrals.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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