Numerical evidence that the perturbation expansion for a non-Hermitian PT-symmetric Hamiltonian is Stieltjes

Abstract

Recently, several studies of non-Hermitian Hamiltonians having 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 PT-symmetric Hamiltonian H=p2+14x2+iλx3 recently studied by Bender and Dunne. For this purpose the first 193 (nonzero) coefficients of the Rayleigh–Schrödinger perturbation series in powers of λ2 for the ground-state energy were calculated. Padé-summation and Padé-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.