Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian

A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian $H=p^2+1/4x^2+i λ x^3$, is performed using high-order Rayleigh-Schrödinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. The Rayleigh-Schrödinger perturbation series is Borel summable, and Padé summation provides excellent agreement with the real energy spectrum. Padé analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of $λ^2$ is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian $H=p^2+1/4x^2-ε x^3$.