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

Christian D. Tica, Eric A. Galapon

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.

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

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 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.
Paper Structure (11 sections, 84 equations, 1 figure, 6 tables)

This paper contains 11 sections, 84 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. The method
  3. Finite-Part Integration
  4. Application to Summation Problems
  5. Singular Perturbation Theory in Quantum Mechanics
  6. $\mathcal{PT}$ Symmetric Cubic Oscillator
  7. Quartic Anharmonic Oscillator
  8. Funnel Potential
  9. Sextic Anharmonic Oscillator
  10. Summary, Recommendations, and Conclusion
  11. Hadamard's Finite Part Integrals

Figures (1)

  • Figure 1: The contour of integration. The upper limit $a$ can be infinite. The points represent poles which for the case of the Stieltjes integral is at $z=-1/\beta$. $\epsilon$ is a small positive parameter.