Numerical Evidence that the Perturbation Expansion for a Non-Hermitian $\mathcal{PT}$-Symmetric Hamiltonian is Stieltjes

TL;DR

This paper provides numerical evidence that the divergent perturbation series for a specific non-Hermitian $ ext{PT}$-symmetric Hamiltonian behaves as a Stieltjes series, indicating a deep mathematical similarity to Hermitian systems.

Contribution

The study computes extensive perturbation series coefficients and applies Padé techniques to demonstrate the Stieltjes nature of the series for a $ ext{PT}$-symmetric Hamiltonian, extending understanding of its mathematical structure.

Findings

Perturbation series coefficients calculated up to 193 terms.

Padé summation techniques applied to the series.

Series exhibits properties of a Stieltjes series, similar to Hermitian cases.

Abstract

Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry have been conducted. Most striking about these complex Hamiltonians is how closely their properties resemble those of conventional Hermitian Hamiltonians. This paper presents further evidence of the similarity of these Hamiltonians to Hermitian Hamiltonians by examining the summation of the divergent weak-coupling perturbation series for the ground-state energy of the $\mathcal{PT}$-symmetric Hamiltonian $H=p^2+{1/4}x^2+i\lambda x^3$ recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of $\lambda^2$ for the ground-state energy were calculated. Pad\'e-summation and Pad\'e-prediction techniques recently described by Weniger are applied to this perturbation series. The qualitative features of the results obtained in this way are indistinguishable from those obtained in the case of the perturbation series for the quartic anharmonic oscillator, which is known to be a Stieltjes series.